copyright 2019, reed nieman

A Theoretical Study of Carbon-Based Functionalized Materials

by

Reed Nieman, B.S.

A Dissertation

In

Chemistry

Submitted to the Graduate Faculty

of Texas Tech University in

Partial Fulfillment of

the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Clemens Krempner, Ph.D.

Chair of Committee

Hans Lischka, Ph.D.

Adelia J. A. Aquino, Ph.D.

Mark Sheridan

Dean of the Graduate School

May, 2019

Texas Tech University, Reed Nieman, May 2019

ii

ACKNOWLEDGMENTS

I would first like to thank my advisor, Dr. Hans Lischka, whose passion for science

and dedication to his work are truly inspiring. Working with him on my various projects

has been a tremendous learning experience these past several years. One of his foremost

characteristics that I have made an attempt to emulate is his tireless attention to detail, no

matter how seemingly small, as all contributing parts are significant to a finished product.

Finally, I have always been able to rely on his patience and encouragement to see through

any difficult moment in a project. I would also like to thank Dr. Adelia Aquino who first

saw potential in me when I did my undergraduate research project with her. She provided

my first lessons in performing computational chemistry calculations and I still fondly

remember sitting at her side hurriedly taking notes and trying to keep up as she quickly

went through commands. Her constant kindness and support over the years is very much

appreciated. I am very grateful too for Dr. Clemens Krempner for not only serving as the

chairperson of my committee, but for going above and beyond in, among other things,

helping me to secure several of my fellowships.

I also thank Dr. Anita Das, and Nadeesha Jayani Silva for being good group

members and great friends. I am thankful as well to Megan Gonzalez and Dr. Veronica

Lyons for their friendship and support, without which graduate school would not have been

nearly as colorful let alone bearable. I am also very thankful for my family and their

unending encouragement even through our lowest moments.

I acknowledge and thank our collaborators, specifically Dr. Francisco Machado

with whom I worked closely while he was visiting. Thanks is also given to the groups of

Dr. Toma Susi at the University of Vienna and Dr. Sergei Tretiak at Los Alamos National

Laboratory for their excellent work and contributions.

Finally, I gratefully acknowledge the following for helping to fund my education:

the National Science Foundation, Texas Tech University for the Provost Scholarship and

Doctoral Dissertation Completion Fellowship, the AT&T Foundation for the AT&T

President’s Fellowship, and the Community Foundation of West Texas for the Lubbock

Achievement Rewards for College Scientists Chapter (ARCS) Memorial Scholarship.


iii

TABLE OF CONTENTS

ACKNOWLEDGMENTS ii

ABSTRACT vi

LIST OF TABLES viii

LIST OF FIGURES xiii

LIST OF ABBREVIATIONS xxi

I. BACKGROUND 1

1.1 Graphene 2

1.2 Organic Photovoltaic Bulk Heterojunctions and Electronic Excited

States 4

1.3 Radical Character in Polycyclic Aromatic Hydrocarbons 6

1.4 Bibliography 10

II. THEORETICAL METHODS 16

2.1 Density Functional Theory 18

2.2 Second-Order Møller-Plesset Perturbation Theory 22

2.3 Second-Order Algebraic Diagrammatic Construction 25

2.4 Basis Set Superposition Error and the Counterpoise Correction

Method 27

2.5 Environmental Simulation Approaches 29

2.6 Multiconfigurational Self-Consistent Field 31

2.7 Multireference Configuration Interaction 32

2.8 Multireference Averaged Quadratic Coupled-Cluster 34

2.9 Bibliography 36

III. SINGLE AND DOUBLE CARBON VACANCIES IN PYRENE AS

FIRST MODELS FOR GRAPHENE DEFECTS: A SURVEY OF THE

CHEMICAL REACTIVITY TOWARD HYDROGEN

38

3.1 Abstract 39

3.2 Introduction 40

3.3 Computational Details 44

3.4 Results and Discussion 45

3.4.1 Potential Energy Scan 45

3.4.2 Geometry Optimizations 49

3.4.3 Energetic Stabilities 52

3.5 Conclusions 59

3.5 Acknowledgments 61

3.7 Bibliography 61

IV. STRUCTURE AND ELECTRONIC STATES OF A GRAPHENE

DOUBLE VACANCY WITH EMBEDDED SI DOPANT 63

4.1 Abstract 64

4.2 Introduction 65



iv


4.4.1 The Pyrene-2C+Si Defect Structure 68

4.4.2 The Circumpyrene-2C+Si Defect Structure 71

4.4.3 The Graphene-2C+Si Defect Structure 73

4.4.4 Natural Bond Orbital Analysis 73

4.4.5 Molecular Electrostatic Potential 79

4.4.6 Electron Energy Loss Spectroscopy 80

4.5 Conclusions 82


4.7 Bibliography 83

V. THE CRUCIAL ROLE OF A SPACER MATERIAL ON THE

EFFICIENCY OF CHARGE TRANSFER PROCESSES IN ORGANIC

DONOR-ACCEPTOR JUNCTION SOLAR CELLS

87

5.1 Abstract 88

5.2 Introduction 89

5.3 Methods 92

5.3.1 Experimental 92

5.3.1.1 Bilayer Solar Cells Fabrication 92

5.3.1.2 Device Characterization 93

5.3.2 Computational Details 93


5.4.1 Experimental Observations 95

5.4.2 Analysis of Energy Levels 98

5.4.3 Characterization of Electronic States 102

5.5 Conclusions 104


5.7 Bibliography 106

VI. BENCHMARK AB INITIO CALCULATIONS OF POLY(P-

PHENYLENEVINYLENE) DIMERS: STRUCTURE AND EXCITON

ANALYSIS

111

6.1 Abstract 112

6.2 Introduction 112



6.4.1 Structural Description 119

6.4.2 Excited States 124

6.5 Conclusions 133


VII. THE BIRADICALOID CHARACTER IN RELATION TO

SINGLET/TRIPLET SPLITTINGS FOR CIS-/TRANS-

DIINDENOACENES AND TRANS-

INDENOINDENODI(ACENOTHIOPHENES) BASED ON EXTENDED

MULTIREFERENCE CALCULATIONS

139


v

7.1 Abstract 140

7.2 Introduction 141



7.4.1 Trans-diindenoacenes 146

7.4.2 Cis-diindenoacenes 152

7.4.3 Trans-indenoindenodi(acenothiophenes) 156

7.5 Conclusions 160


VIII. CONCLUDING REMARKS 169

APPENDICES

A. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN

CHAPTER III 174

B. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN

CHAPTER IV 181

C. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN

CHAPTER V 189

D. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN

CHAPTER VI 198

E. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN

CHAPTER VII 216


vi

ABSTRACT

This dissertation will examine the electronic properties of several materials used in

state-of-the art technology and ongoing scientific investigation by means of high-level ab

initio quantum chemical calculations. In this way, the application of such calculations to

graphene reactivity and defects, organic photovoltaic processes, and excited states and

biradical character of polycyclic aromatic hydrocarbons (PAH) is presented.

Graphene reactivity and defect characterization is currently the source of intense

scrutiny. The reactivity of pristine graphene and single and double carbon vacancies toward

a hydrogen radical was assessed using complete active space self-consistent field theory

(CASSCF) and multireference configuration interaction singles and doubles (MR-CISD)

calculations using a pyrene model. While all carbon centers formed stable, novel C-H

bonds, the single carbon defect was found to be most stabilizing thanks to the dangling

bond at the six-membered ring. Bonding at this carbon center shows interaction with

several closely-spaced electronic excited states. A rigid scan was also used to show the

dynamic landscape of the potential energy surfaces of these structures.

Double carbon vacancy pyrene and circumpyrene were used to investigate the

structure and electronic states of an intrinsic silicon impurity in graphene by density

functional theory (DFT) methods. The ground state was found to be low spin, non-planar

and D2 symmetric with a nearby low spin, planar state with D2h symmetry. Characterization

with natural bond orbital (NBO) analyses showed that the hybridization in the non-planar

structure is sp3 and sp2d in the planar structure. Charge transfer from the silicon dopant to

the surrounding carbon atoms is demonstrated via analysis with natural charges and

molecular electrostatic potential (MEP) plots.

Experiment shows drastically enhanced power conversion efficiency (PCE) in a

bulk heterojunction (BHJ) device consisting of poly(3-hexylthiophene-2, 5-diyl) (P3HT),

terthiophene (T3), and fullerene (C60). This BHJ scheme was investigated using DFT and

several environmental models. The state-specific (SS) environment was found to best

reproduce the experimental findings showing that initial photoexcitation of a local, bright

exciton at P3HT was able to decay through several charge transfer (CT) bands of

P3HT→C60 and T3→C60 character explaining the increased PCE. The spectra calculated


vii

for the isolated system and linear response (LR) environment were very similar to each

other and showed the initial photoexcitation at P3HT decayed instead into bands of local

C60 excitonic states quenching the CT states.

Crucial benchmark calculations of the structure and electronic excited states

missing from the current literature were performed for oligomers of poly(p-

phenylenevinylene) (PPV) dimers using high-level scaled opposite-spin (SOS) second-

order Møller-Plesset (SOS-MP2) and second-order algebraic diagrammatic construction

(SOS-ADC(2)) calculations. Comparisons were also made to DFT methods. The dimer

structures were studied in two conformations: sandwich-stacked and displaced; the latter

exhibited greater stacking interactions. Good agreement was found between the cc-pVQZ

and aug-cc-pVQZ basis sets and the cc-pVTZ basis. Comparisons of the interaction

energies were made to the complete basis set limit. Basis set superposition error was also

accounted for. The excited state spectra presented a varied and dynamic picture that was

analyzed using transition density matrices and natural transition orbitals (NTOs).

The assessment of the biradical character of several trans-diindenoacenes, cis-

diindenoacenes, and trans-indenoindenodi(acenothiophenes) was accomplished utilizing

the multireference averaged quadratic coupled-cluster (MR-AQCC) method. These

systems have been previously identified as highly reactive in experiments, often subject to

unplanned side-reactions. The biradical character was characterized using the singlet-

triplet splitting energy (∆ES-T), effective number of unpaired electrons (NU), and unpaired

electron density. For all structures, low spin ground states were found. In addition,

increasing the number of core benzene rings decreased the ∆ES-T, increased the NU, and

increased the unpaired electron density. Good agreement to experimental values for the

∆ES-T is also demonstrated.


viii

LIST OF TABLES

3.1 Vertical energy differences (eV) of the lowest two A’ and A” states relative

to the C15 ground state computed for the C1, C8 and C15 pyrene-1C+Hr

structures using the SA4-CASSCF(5,5), MR-CISD and MR-CISD+Q

approaches and the 6-31G* basis set. 54

3.2 Dissociation energies Ediss (eV) for the C15 global minimum pyrene-

1C+Hr structure and estimated energy barriers Ebarr(est.) (eV) for the

approach of Hr on the ground state surface toward the different carbon

atoms of pyrene-1C using different theoretical methods and the 6-31G*

basis set. 57

3.3 Dissociation energies Ediss (eV) for the double vacancy pyrene-2C+Hr and

pristine pyrene + Hr cases using different methods and the 6-31G* basis se 58

4.1 Open-shell character, structural symmetry, spin multiplicity, bond distances

(Å), and relative energies (eV) of selected optimized structures of pyrene-

2C+Si. 69

4.2 Structural symmetry, open-shell character, spin multiplicity, bond distances

(Å), and relative energies (eV) of selected optimized structures of

circumyrene-2C+Si. 72

4.3 NBO bonding character analysis of pyrene-2C+Si using B3LYP/6-

311G(2d,1p). The discussed carbon atom is one of the four bonded to

silicon. 74

4.4 NBO bonding character analysis of circumpyrene-2C+Si. The discussed

carbon atom is the one of the four bonded to silicon. 76

5.1 Stabilization energy of the lowest P3HT→C60 CT state of P3HT-T3-C60 and

P3HT-C60 system calculated in the isolated system and the SS environmen 102

6.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer

computed at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets

using C2h symmetry. 120

6.2 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV3 dimer




ix

6.3 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV2 dimer

computed at SOS-MP2 level using C1 symmetry. 122

6.4 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV3 dimer

computed at SOS-MP2 level using C1 symmetry 122

6.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVQZ and

SOS-MP2/cc-pVQZ optimized geometry (ΔE (eV) – vertical excitation

energy, f – oscillator strength, CT – interchain charge transfer) 125

6.6 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ

and SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation


6.7 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ

and SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation


6.8 Interchain distances (Å) in the stacked PPV dimers optimized for π-π* S1

(1 1Bg) excited state using the cc-pVTZ basis and C2h symmetry 132

7.1 Singlet/triplet splitting energy (∆ES-T, E(13Bu) - E(11Ag)) calculated using

the MR-AQCC/6-31G* method for trans-diindenoacenes, 1-4. 149

7.2 Singlet/triplet splitting energy (∆ES-T, E(13B1) - E(11A1)) calculated using

the MR-AQCC/6-31G* method for cis-diindenoacenes, 5-8. 153

7.3 Singlet/triplet splitting energy (∆ES-T, E(13Bu) - E(11Ag)) calculated using

the MR-AQCC/6-31G*(6-311G*)S method for trans-

indenoindenodi(acenothiophene) (9-12). 157

A.1 Relative energy differences ΔErel (eV) of the local energy minima in the

electronic ground state computed for the different structures of pyrene-

1C+Hr using different theoretical methods and the 6-31G* approach. 174


electronic ground state computed for the different structures of pyrene-

2C+Hr using different theoretical methods and the 6-31G* approach. 174


electronic ground state computed for the different structures of pyrene+Hr

using different theoretical methods and the 6-31G* basis set. 175


x

B.1 NBO bonding character analysis of pyrene-2C+Si using the CAM-

B3LYP/6-311G(2d,1p) method. The carbon atom discussed is the carbon

bonded to silicon. 181

C.1 Excitation energies Ω of the isolated standard C60-T3-P3HT system using

the CAM-B3LYP/6-31G approach 189


the CAM-B3LYP/6-31G* approach 190

C.3 Lowest excitation energies Ω of the isolated components P3HT, T3 and C60

using the CAM-B3LYP/6-31G* approach. 191

C.4 Excitation energies Ω of the standard C60-T3-P3HT system the using the

CAM-B3LYP/6-31G* approach in the LR environment 192

C.5 Excitation energies Ω of the standard C60-T3-P3HT system the using the

CAM-B3LYP/6-31G approach in the LR environment 193

C.6 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM-

B3LYP/6-31G* approach in the SS environment 193

C.7 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM-

B3LYP/6-31G approach in the SS environment 194


the CAM-B3LYP/6-31G* approach (only core orbitals were frozen) 194

C.9 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system

using the CAM-B3LYP/6-31G* approach 195

C.10 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system

using the CAM-B3LYP/6-31G approach 196

C.11 Excitation energies Ω of the extended C60-T3-P3HT system using the

CAM-B3LYP/6-31G approach in the SS environment 197

D.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer

computed at MP2 and MP2/BSSE levels with several basis sets using C2h

symmetry. 198


xi

D.2 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV4 dimer



D.3 Interchain distances (Å) and ΔE (kcal/mol) in the diplaced PPV4 dimer

computed at SOS-MP2 level with several basis sets using C2h symmetry. 200

D.4 Single chain PPV2 vertical excitations using SOS-ADC(2) and SOS-MP2

optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator

strength) 200

D.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2) and SOS-MP2

optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator

strength, CT – interchain charge transfer) 201

D.6 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and

SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing

hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) 202

D.7 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ

and SOS-MP2/cc-pVTZ optimized geometry showing the largest

contributing hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) 205


SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation



SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation


D.10 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ

and SOS-MP2/cc-pVTZ optimized geometry showing the largest

contributing hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) 209

D.11 Stacked PPV3 dimer vertical excitations using B3LYP-D3/cc-pVTZ and the

B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation


D.12 Stacked PPV3 dimer vertical excitations using CAM-B3LYP-D3/cc-pVTZ

and the CAM-B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) –


xii

vertical excitation energy, f – oscillator strength, CT – interchain charge

transfer) 213

E.1 Active orbitals for each MR-AQCC reference space 216

E.2 NU value (e) of trans-diindenoacene (1-4) singlet 1 1Ag state and vertically

excited 1 3Bu state using the MR-AQCC(CAS(4,4))/6-31G* method. 217

E.3 NU value (e) of cis-diindenoacene (5-8) singlet 1 1A1 state and vertically

excited 1 3B1 state using the MR-AQCC(CAS(4,4))/6-31G* method. 218

E.4 NU value (e) of trans-indenoindenodi(acenothiophene) (9-12) singlet 1 1Ag

state and vertically excited 1 3Bu state using the MR-AQCC(CAS(4,4))/6-

31G*(6-311G*)S method. 219


xiii

LIST OF FIGURES

1.1 Structures: a) n-acene, b) triangulene, c) 1,2:9,10-dibenzoheptazethrene, d)

(ma,nz) periacene (graphene nanoribbon model), e) indeno[1,2-b]fluorene 7

3.1 PES scans in terms of the location of Hr for the four lowest states of pyrene-

1C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is

located in a plane parallel and 1 Å above the plane defined by the frozen

carbon atoms. Lowest energies: a -574.222889 Hartree, b -574.198174

Hartree, c -574.149530 Hartree, d -574.117820 Hartree. The energy (color

scale) is given in eV, relative to the ground state lowest minimum. 46

3.2 PES scan in terms of the location of Hr for the lowest state of pyrene-2C+

Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located

in a plane parallel and 1 Å above the plane defined by the frozen carbon

atoms. Lowest energy = -536.385046 Hartree. The energy (color scale) is

given in eV, relative to the ground state lowest minimum. 47

3.3 PES scan in terms of the location of Hr for the 22A state of pyrene-2C+ Hr

using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a

plane parallel and 1 Å above the plane defined by the frozen carbon atoms.

Lowest energy = -536.347629 Hartree. The energy (color scale) is given in

eV, relative to the ground state lowest minimum at -536.385046 Hartree. 48

3.4 PES scan of the lowest state of pyrene+Hr using the CASSCF(5,5)/6-31G*

approach. The atom Hr is located in a plane parallel and 1 Å above the plane

defined by the frozen carbon atoms. Lowest energy = -612.270253 Hartree.

The energy (color scale) is given in eV, relative to the ground state lowest

minimum. 48

3.5 Plots of the different minimum structures (labeled according to the bonding

carbon atom) on the ground state energy surface of pyrene-1C+Hr using the

SA4-CASSCF(5,5)/6-31G* approach. Bond distances are given in Å. 50


carbon atom) of the ground state energy surface of pyrene-2C+Hr using the

SA4-CASSCF(5,5)/6-31G* approach. Bond distances are given in Å. 51


carbon atom) on the ground state energy surface of pyrene+Hr using the

CASSCF(5,5)/6-31G* approach. Bond distances are given in Å. 53


xiv

3.8 Location and relative stability (eV) of the minima on the ground state

energy surface of pyrene-1C+Hr using the MR-CISD+Q/6-31G* method.

Blue: 2A ground state, red: lowest vertical excitation. Note: the rigid pyrene-

1C structure shown does not reflect the actual structure of the optimized

minima. 54

3.9 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is

bonded to C15 using the MR-CISD/6-31G* approach. Minimum energy =

-574.277274 Hartree 56

3.10 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is

bonded to C1 using the MR-CISD/6-31G* approach. Minimum energy = -

574.277274 Hartree 56


energy surface of pyrene-2C+Hr defect using the MR-CISD+Q/6-31G*

method. Blue: 2A ground state, red: lowest vertical excitation. Note: the

rigid pyrene-2C structure shown does not reflect the actual structure of the

optimized minima. 58


energy surface of pyrene+Hr using the MR-CISD+Q/6-31G* method. Note:

the rigid pyrene structure shown does not reflect the actual structure of the

optimized minima. 59

4.1 Optimized pyrene-2C+Si structure P1 (D2 symmetry) using the B3LYP/6-

311G(2d,1p) method with frozen carbon atoms marked by gray circles 68

4.2 Optimized circumpyrene-2C+Si structure CP1 (D2 symmetry) using the

B3LYP/6-311G(2d,1p) method with frozen carbon atoms marked by gray

circles. 68

4.3 NBO plots for pyrene-2C+Si using the B3LYP/6-311G(2d,1p) approach for

structures P1 and P2. Isovalue = ±0.02 e/Bohr3 77

4.4 Natural charges (e) of carbon and silicon in circumpyrene-2C+Si using the

B3LYP/6-311G(2d,1p) method. 78

4.5 Molecular electrostatic potential for circumpyrene-2C+Si mapped onto the

electron density isosurface with an isovalue of 0.0004 e/Bohr3 (B3LYP/6-

311G(2d,1p)). 80


xv

4.6 Comparison of simulated and experimental EELS spectra of the Si-C4

defect in graphene. Despite the out-of-plane distorted structure being the

ground state, the originally proposed flat spectrum provides an overall

better match to the experiment, apart from the small peak at ~96 eV that is

not predicted for the flat structure. The inset shows a colored medium angle

annular dark field STEM image of the Si-C4 defect (the Si atom shows

brighter due to its greater ability to scatter the imaging electrons). 81

5.1 (a) Standard model of the molecular arrangement in the trimer. (b) Front

and (c) side views of the extended model. The picture shows molecular

notations, geometrical arrangement and selected nonbonding distances (Å).

Atoms kept fixed during the geometry optimization are circled in black. 92

5.2 (a) Scheme of bilayer device structure used in this study, the interface layer

with vertical alignment was produced by Langmuir-Blodgett (LB) method

illustrated on right. (b) photocurrent measured at short circuit condition as

function of illumination wavelength for bilayer device without spacer and

with 1-bilayer of O3 between P3HT and C60. (c) GIWAXS map for 1 bilayer

of O3 deposited on P3HT by LB method. (d) peak photocurrent at 550 nm

illumination as a function of number of O3 bilayer at interface. 97

5.3 Excited state energies Ω of the standard P3HT-T3-C60 trimer using the

CAM-B3LYP method and (a) the 6-31G* and (b) the 6-31G basis sets both

as isolated systems and in the LR and SS environments (for numerical data

see Table C.1, Table C.2, and Table C.4-Table C.8). For the isolated system

using the 6-31G* basis set (farthest left of (a)), the fifteen lowest-energy

locally excited C60 states have been computed also. Excited states of (c) the

extended P3HT-T3-C60 trimer using the 6-31G* and (d) 6-31G basis sets

both as isolated systems and in the SS environment (Table C.9-Table C.11).

100

5.4 Natural Transition Orbitals (NTO), particle/hole populations and MEP of

(a, b, c) the lowest-energy P3HT→(C60)3 CT state, (d, e, f) the lowest-

energy T3→(C60)3 CT state, (j, k, l) the lowest-energy P3HT bright

excitonic state of the extended thiophene trimer calculated using the CAM-

B3LYP/6-31G method in the SS environment, and (g, h, i) the lowest-

energy C60 dark excitonic state of the isolated standard thiophene trimer

calculated using the CAM-B3LYP/6-31G* method. Here we use NTO

isovalue ±0.015 e/Bohr3 and MEP isovalue ±0.0003 e/Bohr3. 103

6.1 Top/side views of the stacked PPV oligomers and labeling scheme. 117

6.2 Omega matrices for vertical excitations of stacked PPV2 dimer using the

SOS-ADC(2)/cc-pVQZ method with the SOS-MP2/cc-pVQZ optimized


xvi

geometry. Each vinylene and phenylene group is a separate fragment:

fragments 1-3 belong to one chain, fragments 4-6 to the other. The y-axis

represents the electron and the x-axis represents the hole. Omega matrix

plots calculated using the TheoDORE program package. 126

6.3 Omega matrices for vertical excitations of stacked PPV3 dimer using the

SOS-ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized





6.4 Calculated line spectra (right y-axis) and density of states (left y-axis) for

the stacked PPV oligomers optimized using the SOS-ADC(2), B3LYP-D3,

and CAM-B3LYP-D3 methods. Stacked a) PPV2 (Table 6.5), b-d) PPV3

(Table D.8, S11-S12), e) PPV4 (Table D.9). 130

7.1 Quinoid Kekule and biradical resonance forms of trans-diindenoanthracene,

cis-diindenoanthracene, and trans-indenoindenodi(anthracenothiophene). 142

7.2 Comparison of the singlet/triplet energy (∆ES-T, E(13Bu) - E(11Ag)) and

number of unpaired electrons (NU and NU(H-L), Table E.2) of the 11Ag state

with the number of interior six-membered rings of trans-diindenoacenes (1-

4) using the MR-AQCC(CAS(4,4))/6-31G* method. 148

7.3 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of

unpaired electrons (NU) and b) the number of unpaired electrons in the

HONO-LUNO pair (NU(H-L)) of the 1 1Ag state (Table E.2) of trans-

diindenoacenes (1-4) using the MR-AQCC(CAS(4,4))/6-31G* method. 150

7.4 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the

1 1Ag state of trans-diindenobenzene (1-4) using the MR-

AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA

index is shown in the center of each ring. 151

7.5 Comparison of the singlet/triplet energy (∆ES-T, E(13B1) - E(11A1)) and

number of unpaired electrons (NU and NU(H-L), Table E.3) of the 11A1 state

with the number of interior six-membered rings of cis-diindenoacenes (5-8)

using the MR-AQCC(CAS(4,4))/6-31G* method. 153




xvii

HONO-LUNO pair (NU(H-L)) of the 1 1A1 state (Table E.3) of cis-

diindenoacenes (5-8) using the MR-AQCC(CAS(4,4))/6-31G* method. 154


1 1A1 state of cis-diindenobenzene (5-8) using the MR-AQCC(CAS(4,4))/6-

31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the

center of each ring. 155

7.8 Comparison of the singlet/triplet energy (∆ES-T, E(13Bu) - E(11Ag)) and

number of unpaired electrons (NU and NU(H-L), Table E.4) of the 11Ag state

with the number of interior six-membered rings of trans-

indenoindenodi(acenothiophene) (9-12) using the MR-

AQCC(CAS(4,4))/6-31G*(6-311G*)S method. 157



HONO-LUNO pair (NU(H-L)) of the 1 1Ag state (Table E.4) of trans-

indenoindenodi(acenothiophene) (9-12) using the MR-

AQCC(CAS(4,4))/6-31G*(6-311G*)S method. 159


1 1Ag state of trans-indenoindenodi(acenothiophene) (9-12) using the MR-

AQCC(CAS(4,4))/6-31G*/6-311G* method. (isovalue=0.004 e/Bohr3).

HOMA index is shown in the center of each ring. 160

A.1 Plots of the active orbitals of the optimized ground states of pyrene+Hr

using the CASSCF(5,5)/6-31G* approach. 175

A.2 Plots of the active orbitals of the optimized ground states of pyrene-1C+Hr

using the SA4-CASSCF(5,5)/6-31G* approach. 176

A.3 Plots of the active orbitals of the optimized ground states of pyrene-2C+Hr

using the SA4-CASSCF(5,5)/6-31G* approach. 176

A.4 Potential energy scans in terms of the location of Hr for the 3 2A and 4 2A

states of pyrene-2C+Hr using the SA4-CASSCF(5,5)/6-31G* approach.

The atom Hr is located in a plane parallel and 1 Å above the plane defined

by the frozen carbon atoms. Lowest energies: a -574.149530 Hartree, b -

574.117820 Hartree. The energy (color scale) is given in eV, relative to the

ground state lowest minimum at -536.385046 Hartree. 177


xviii

A.5 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is

bonded to C2 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum

energy = -574.277274 Hartree 177



















B.1 NBO plots for circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p)

approach (CP2 D2h). Isovalue = ±0.02 e/Bohr3 182

B.2 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using

the B3LYP/6-311G(2d,1p) method. 183

B.3 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using

the CAM-B3LYP/6-311G(2d,1p) method. 184

B.4 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the

electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the

B3LYP/6-311G(2d,1p) method. 185


xix

B.5 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the

electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the

CAM-B3LYP/6-311G(2d,1p) method. 187

B.6 Unprocessed EEL spectrum of the Si-C4 site. 187

B.7 First two unoccupied states immediately above the Fermi level spanning

about 1 eV for the flat and corrugated Si/graphene structures from the

periodic DFT calculations. Electron density isovalue is 0.277 e/Bohr3. 188

D.1 Omega matrices for vertical excitations of displaced PPV2 dimer using the






D.2 Omega matrices for vertical excitations of stacked PPV4 dimer using the






D.3 Omega matrices for vertical excitations of displaced PPV3 dimer using the






E.1 Quinoid Kekule and biradical resonance forms of trans-diindenoacenes (1-

4), cis-diindenoacenes (5-8), and trans-indenoindenodi(acenothiophenes)

(9-12). 220

E.2 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the

vertically excited 1 3Bu state of trans-diindenobenzene (1-4) using the MR-




vertically excited 1 3B1 state of cis-diindenobenzene (5-8) using the MR-


xx




vertically excited 1 3Bu state of trans-indenoindenodi(acenothiophene) (9-

12) using the MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method.

(isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring.

222


xxi

LIST OF ABBREVIATIONS

ADC(2) Second-Order Algebraic Diagrammatic Construction

AUX Auxiliary Orbitals

BHJ Bulk Hetero-Junction

BSSE Basis Set Superposition Error

C60 Fullerene

CASSCF Complete Active Space Self-Consistent Field

CI Configuration Interaction

CP Counterpoise Correction

CSF Configuration State Function

CT Charge Transfer

DFT Density Functional Theory

HF Hartree-Fock

HOMA Harmonic Oscillator Measure of Aromaticity

K-S Kohn-Sham

LR Linear Response

MCSCF Multiconfigurational Self-Consistent Field

MEP Molecular Electrostatic Potential

MP2 Second-Order Møller-Plesset

MR-AQCC Multireference-Averaged Quadratic Coupled-Cluster

MR-CISD Multireference-Configuration Interaction Singles and Doubles

NBO Natural Bond Orbital

NTO Natural Transition Orbital

NU Number of Unpaired Electrons

OPV Organic Photovoltaic

P3HT Poly(3-hexylthiophene-2, 5-diyl)

PAH Polycyclic Aromatic Hydrocarbon

PCE Power Conversion Efficiency

PES Potential Energy Surface


xxii

PP Polarization Propagator

PPV Poly(p-phenylenevinylene)

SA State-Averaged

SOS Scaled Opposite-Spin

SS State-Specific

T3 Terthiophene

TD-DFT Time-Dependent Density Functional Theory


1

CHAPTER I

BACKGROUND

This chapter will present a general overview of each research project of this

dissertation. Research into graphene, an allotrope of carbon in which the atoms are

arranged in a hexagonal network, has grown into a major area of investigation since its

discovery and characterization as a semi-metal with promise of semi-conductor capability.

Experimental and theoretical investigations of graphene have concentrated on, among a

wide variety other topics, band gap engineering by way of defects. Though many classes

of graphene defects have been characterized, the first two projects in this dissertation will

discuss two defects in detail: i) the ground state and three lowest excited states involved in

the chemisorption of a hydrogen radical to pristine graphene and graphene with a one and

two carbon atom vacancy utilizing pyrene as a graphene model system, and ii) the structure

and electronic states of silicon doped in two-carbon vacancy graphene using pyrene and

circumpyrene as graphene model systems. In the former case, CASSCF and MR-CISD

calculations were utilized to show that the bonding of hydrogen to graphene, specifically

at the dangling bond (a radical site) found in the single-carbon vacancy defect, involves

several closely spaced electronic excited states, and in ii) the non-planar ground state and

nearby planar excited state both exhibit hybridization at the silicon dopant that would

interrupt graphene’s sp2 hybridized network creating a band gap.

Another area of intense research is organic photovoltaics (OPV), which show great

promise in providing an alternative to traditional inorganic photovoltaic devices as they

consist of more abundant and potentially cheaper materials. Unfortunately, they do not yet

operate at the same PCE in converting photons into electricity, necessitating both

experimental and theoretical attention in order to both optimize the operational device

makeup and understand the contributing physical processes. The third research project in

this dissertation presents theoretical calculations of the electronic excited states an OPV

BHJ device that was found to have dramatically increased PCE upon addition of a

terthiophene spacer material between the electron donor/acceptor layers. The utilization of

the SS solvation environment was instrumental in the correct ordering of excited states.


2

The fourth research project presents high-level benchmark calculations of the structure and

electronic excited states of several PPV oligomer dimers using the SOS-MP2 and SOS-

ADC(2) methods. PPV is an important material for the elucidation of excitation energy

transfer (EET) mechanisms in organic polyconjugated systems.

Finally, PAHs are a class of materials consisting of fused aromatic rings primarily

found to have close-shell, low spin ground states. Some cases have been found, however,

to have unpaired electron spins in their low spin ground state. These molecules have proven

to be highly reactive compared to close-shell PAHs and are subject to unforeseen side

reactions and with better characterization of their radical properties, their reactivity can be

better controlled in the future. In the final research project of this dissertation, MR-AQCC

calculations were utilized to determine the singlet-triplet splitting energy, effective number

of unpaired electrons, and unpaired electron density in order to characterize a series of

singlet biradical indenofluorenes and thiophene-fused indenofluorenes.

1.1 Graphene

The initial discovery of graphene by Geim and Novoselov1 using the aptly-named

“Scotch-Tape” method revolutionized material science by characterizing an allotrope of

carbon composed of a one-atom thick layer of hexagonally-arranged carbon atoms forming

a diffuse, delocalized sp2 hybridized network. Graphene was found to be a semi-metal with

a plethora of mechanical, thermal, and electronic properties making it very useful in next-

generation technologies like spin-tronics devices, single molecule sensors, photovoltaic

devices and more. Graphene has two main difficulties that prevent its wide-spread adoption

to industrial applications. The first is that there is no effective method for obtaining the

large amounts of pristine, one-atom thick graphene sheets that would be necessary, and the

second is that, as a semi-metal, graphene has a zero-energy band gap.

One of the primary reasons single-layer graphene has proven difficult to obtain, is

that once graphene layers are separated from graphite, the individual layers will bond

together and reform graphite. One of the first methods used to separate graphene layers


3

was using solvents with similar surface energies as graphene to exfoliate graphene from

graphite.2-3 Sonication of graphite dispersed in liquid mediums has also been shown to

transform flakes of graphite into single- or multi-layer graphene.4-5 The use of dispersion

stabilizing agents with high adsorption energy (higher than that of graphene) promotes

exfoliation by adsorbing to the separate layers and preventing reaggregation.6-7

The problem of a zero-band gap material like graphene presents an interesting

avenue towards exploiting its many electronic properties as band gap engineering allows

for the semi-metal to be transformed into a semi-conductor via chemical

functionalization.8-9 Hydrogenation, the chemisorption of hydrogen onto graphene

surfaces, has shown to be an effective method towards opening and tuning its band gap.10-

11 The chemisorption of a species such as hydrogen opens the band gap of graphene by

disrupting the local sp2 hybridized bond network in the diffuse π system creating a sp3

hybridized bond with the incoming chemisorbate. This new sp3 hybridized bond disrupts

the once-continuous flow of conducting electrons opening the band gap, and, as an

additional benefit, increasing the reactivity of nearby carbon centers toward atomic

hydrogen chemisorption. Regions of the graphene surface made of sp3 hybridized carbons

centers are referred to as graphane islands and have been confirmed using scanning

tunneling microscopy (STM) and angle-resolved transmission electron microscopy

(TEM).12 Graphane island arrangement and their proximity to one another contribute to the

overall electronic character of the graphene sheet and have been shown to open band

gaps.12-13

Defects in the carbon network of graphene sheets such as vacancies (removed

carbon atoms in the sheet) and dopant species (carbon atoms replaced with heteroatoms)

have also displayed the capability to induce a band gap compared to pristine, unaltered

graphene.8-9 The removal of carbon atoms from the hexagonal structure either leaves

dangling bonds, radical centers that are much more reactive, or results in bond restructuring

using Jahn-Teller rearrangements. Such defects have demonstrated several properties such

as magnetism,14-15 and increased reactivity by way of greater binding energy was found for

vacancy defects in graphene toward species such as metals, hydrogen, fluorine, aryl groups

and thiol groups.16-18 Using pyrene as a model for a larger graphene sheet, it was shown


4

using multireferernce (MR) calculations that in single (SV)19 and double (DV)20 vacancy

defects there are several closely-spaced singlet and triplet electronic states. These closely

spaced states were later shown to be instrumental in bond formation with atomic hydrogen

at these sites.21

Graphene doping with heteroatoms is another interesting method to tuning and

regulating the electronic properties of graphene sheets. Dopants such as nitrogen, boron,

and oxygen have been substituted into the carbon network using a variety of techniques

(e.g. annealing, arc discharge, plasma exposure, chemical vapor deposition (CVD), etc.)

and have shown to change the band character.22-24 Silicon is another common graphene

dopant species that is especially interesting as it is isovalent to carbon and can be found as

intrinsic impurities when graphene is grown using CVD25-26 and by thermal decomposition

of silicon carbide.27 Two identified forms of silicon impurity have been identified and

studied: the nonplanar threefold- coordinated and planar fourfold-coordinated cases, Si-C3

and Si-C4 respectively. Experimental techniques such as scanning tunneling electron

microscopy (STEM) and annular dark field (ADF) imaging together with DFT calculations

confirmed that the hybridization at Si-C3 is sp3 and at Si-C4 is sp2d.28 The d-band

hybridization of the Si-C4 case is thought to be responsible for the disruption of electron

density at the site of the defect.29 Further theoretical work suggested that the planar Si-C4

structure is an equilibrium between a planar and more stable non-planar structure.30

1.2 Organic Photovoltaic Bulk Heterojunctions and Electronic Excited

States

OPV BHJs consist of interfaces between electron donor layers and electron

acceptor layers both of which are composed of π-conjugated organic polymers. BHJ

devices are fundamental to the understanding in electron mobility processes (dissociation

of excitons generating CT states) necessary to generate electricity and to improving their

light conversion efficiency.31-32 Bulk heterojunction devices convert light into electricity

in a four-step process: i) light is absorbed by bands of bright π→π* transitions of the

conjugated organic polymer generating excitonic states, ii) diffusion and migration of the


5

exciton through the bulk polymer to the interface and eventual segregation, iii) excitons

dissociate at the heterojunction interfaces creating CT states, leading to iv) charge

separation collected at the contacts.33 Heterogeneous distributions of donor and acceptor

materials in real BHJ devices causes the electricity generating processes to be more

complicated than what has been described here due to numerous structural defects and

conformations, large numbers of internal degrees of freedom for the various polymer

chains, and complicated manifolds of electronic states.34-35 In addition, the rate of exciton

diffusion at the interface competes with the rate of charge recombination, and controlling

these kinetic factors is key to the utilization of BHJ devices.

BHJ devices have been previously studied in a wide variety of compositions. A

PCE of 8.2% was obtained with a BHJ setup consisting of the electron donor, poly[4,8-

bis(5-(2-ethylhexyl)thiophen-2-yl)benzo[1,2-b; 4,5-b']dithiophene-2,6-diyl-alt-(4-(2-

ethylhexyl)-3-fluorothieno[3,4-b]thiophene-)-2-carboxylate-2-6-diyl)] (PTB7-Th), and

electron acceptor, [6,6]-phenyl-C70-butyric acid methyl ester (CN-PC70BM).36 Switching

the common electron acceptor, fullerene (C60) for one based on a fluroene-core design saw

an increase in the PCE to 10.7%.37 A BHJ PV device with porphyrin modifications showed

photoresponse beyond 800 nm together with a high open-circuit voltage and PCE,38-39

while hybrid organic/inorganic BHJ PV devices utilizing both porphyrin parts and

perovskite parts achieved a PCE of 19%.40 In another BHJ OPV device consisting of

electron donor, P3HT, and electron acceptor, fullerene, a dramatic increase in photocurrent,

open-circuit voltage, and overall PCE (2-5 times greater) was observed when utilizing a

spacer layer consisting of either lithium fluoride, terthiophene-derivative (for which the

greatest increase in PCE was found), or one of two metal-organic complexes involving

iridium.41

PPV and is a paradigmatic electron donor material for the understanding of EET

processes in organic π-conjugated polymers at the molecular level. Such understanding

will aid in the fine-tuning of their attractive absorption and emission properties useful in,

among other things, electroluminescent and OPV devices.42-44 It has been concluded using

experimental evidence that coherent PPV dynamics are a result of structural effects

coupling to vibrational modes.43, 45 The evolution of excited states resulting in EET

https://www.sciencedirect.com/topics/chemistry/methyl-ester


6

dynamics is a product of structural relaxation and these processes are coupled to electronic

states requiring non-adiabatic considerations as state-crossing occurs with structural

evolution.33

BHJ devices have been investigated utilizing PPV as the electron donor. A study

of the compositional dependence of performance of a OPV BJH composed of (poly(2-

methoxy-5-(3’,7’-dimethyloctyloxy)-p-phenylene vinyl-ene):methanofullerene [6,6]-

phenyl C61-butyric acid methyl ester) (OC1C10-PPV:PCBM) showed that the hole mobility

in the PPV phase increases with increasing composition of the PCBM phase until the point

of hole saturation, allowing for the optimal design tuning.46 Another device showed that

polymer/polymer BHJ devices where electron-withdrawing cyano-group substituted PPV

could act as the electron acceptor with another PPV derivative, poly[2-methoxy-5-(2-

ethylhexyloxy)-1,4-phenylenevinylene] (MEH-PPV), acting as the electron donor.47

Quantum chemical calculations on electron donor/acceptor complexes are

important in describing the electronic processes in BHJ devices. The electronic excited

state spectrum of a system composed of poly(thieno[3,4-b]-thiophene benzodithiophene)

(PTB1) and [6,6]-phenyl-C61-butyric acid methyl ester (PCBM) was calculated showing

how internal conversion processes can convert the bright, light-absorbing π→π* transition

to CT states which were stabilized by inter-chain charge delocalization.48 Local, low

energy, dark excitonic transitions on fullerene units were shown to be problematic as they

can possibly quench CT states once formed, emphasizing the correct ordering and

characterizing of electronic excited states.49 The CT states were investigated in a system of

n-acene/tetracyanoethylene (TCNE) showing the necessity of the SOS approach with the

second-order algebraic diagrammatic construction (ADC(2)) to the accurate description of

the electronic excited states of donor/acceptor complexes, the strongly stabilizing effects

of solvent interactions on the lowest calculated CT states, and the change of decay

mechanisms from radiative to nonradiative going from benzene to anthracene with acene-

type electron donors.50

1.3 Radical Character in Polycyclic Aromatic Hydrocarbons


7

PAHs are composed of fused aromatic rings, whose ground state in most cases is

expected to be closed-shell and low spin. Several types of PAH have been found to exist

however where the ground state is low spin, but open-shell in character suggesting the

presence of radicals.51 When found in nature, PAHs are often environmental pollutants52

and are good indicators of chemical reactivity of black carbon interfaces in soil.53 They can

even be found in surprising abundance in the interstellar medium54 and are hypothesized

to be the origin of unidentified mid-infrared range bands in the of the spectrum.55-56 Many

classes of radical PAHs have been studied such as zethrenes,57 anthenes,58 phenalenyl,59

besphenalenyls,60 triangulene,61 quinodimethanes,62 indofluorenes,63-65 high-order-

acenes,66-67 and graphene nanoribbons.68-70 Several of these classes will be discussed

below.

Figure 1.1 Structures: a) n-acene, b) triangulene, c) 1,2:9,10-dibenzoheptazethrene, d)

(ma,nz) periacene (graphene nanoribbon model), e) indeno[1,2-b]fluorene

Acenes (Figure 1.1a) are linear arrangements of n number of fused benzene rings

(referred to n-acenes) that are studied not only for their own properties, but their value as

models for larger PAH systems. Acenes are interesting cases for the investigation of radical

character as their number of unpaired electrons in their low spin or singlet ground state has

been found to increase with increasing chain length.71 Fluorine substituted pentacene (five

rings or 5-acene), 2,3,9,10-tetrafluoropentacene, was found to have a larger energy gap


8

between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular

orbital (LUMO) when compared to unsubstituted pentacene and persistent radical cationic

and dicationic states when exposed to oxidizing solvents.72 Synthesizing acenes with

extended chain lengths larger than pentacene is difficult due to developing radical

character, but it is possible by functionalizing with additional groups or heteroatom

doping.73 Theoretical studies found that doping the outer more aromatic rings of octacene

increased the radical character by breaking the π-electron bonding, while doping in the

center acted to quench radical character and decrease the effective number of unpaired

electrons.74

Triangulene (Figure 1.1b) is a non-Kekulé extended phenalenyl derivative with an

two traditional radical centers. Its radical nature makes it difficult to synthesize as

polymerization side reactions occur. The ground state of triangulene was determined to be

a high spin triplet so the introduction of bulky groups such as tert-butyl is necessary to

protect reactive carbon sites.61 Tert-butyl substituted triangulene was also thought to have

ferromagnetic potential due to its triplet ground state.75 This magnetic behavior has been

exploited in interesting ways such as with spin-photovoltaic devices where pure spin

current can be generated without charge current upon irradiation with light.76 Theoretical

studies using high-level ab initio calculations support the high spin ground state, showing

degenerate first excited states, relatively small triplet-singlet splitting (0.8 eV), and

unpaired electron density that is delocalized over the entire molecule but is greatest at the

edges.77 The σ- and π-dimerization schemes of trioxo-triangulene monomers have also

been investigated using unrestricted density functional theory (UDFT).78

Zethrenes, so named for their z-shaped configurations of fused rings (Figure 1.1c),

are molecules possessing a singlet biradical ground state that is stabilized by enhanced

aromaticity as described by Clar’s aromatic sextet rule.79 Functionalized zethrene

molecules have shown singlet-fission processes by forming a bound, spin-entangled pair

of triplet states that can thermally dissociate and is enhanced with greater diradical

character.80 The large, laterally extended zethrene, superoctazethrene, was found to be

relatively stable, have significant radical character, have a smaller single-triplet gap than

previous octazethrenes (showing that its first triplet excited state is more thermally


9

accessible), and have aromatic character covering the whole surface.81 High-level

multireference calculations found singlet-triplet splitting energies for heptazethrene in

good agreement with experiment and decreased with increasing number of Clar’s sextet

rings.77 Doping zethrenes with heteroatoms has been found to efficiently tune the electronic

properties of PAHs.82 This was shown utilizing multireference calculations in nitrogen and

boron doped heptazene where the effective number of unpaired electrons and singlet-triplet

splitting energy provided an excellent measure of radical character.83

Graphene nanoribbons (Figure 1.1d) are created by cutting graphene sheets into

slices possessing distinct open edges that have been shown to influence the overall

electronic properties of the nanoribbon with respect to bulk graphene. The edges come in

two varieties: zigzag edges that have a localized non-bonding π-state, and armchair edges

that do not. Because of the difference in character between these two edges, edge effects

and reactivity are intensely studied. Quarteranthene was synthesized and shown to have

magnetic susceptibility above 50 K, a small singlet triplet splitting of 30 meV, and a large

biradical index, y, of 0.91 (the singlet ground state possessed 91% biradical character).84

Multireference calculations revealed a multiradical ground state with a near-zero singlet-

triplet splitting.85 Additionally the unpaired electron density is concentrated at the zigzag

edges. The effects of ionization were also investigated in periacene cases showing

decreased unpaired electron density at the edges resulting from removing an electron.86

Indenofluroenes (Figure 1.1e) consist of fused indene and fluorene moieties and

have unique electronic characteristics that are targeted for intense investigation and

exploitation. Thanks to the two five-membered rings incorporated into the structures,

indenofluorenes have demonstrated reversible electron donation useful for OPV devices

and organic field effect transistors (OPET).87 Ambipolar charge transport properties were

found for diindenobenzene derivatives in singlet-crystal and thin-film OFETs.88 Fluorene-

based dispiro[fluorene-indenofluorene] was synthesized and functioned as a hole transport

material in perovskite solar cells with PCE of about 18.5 %.89 Magnetic susceptibility

studies of thiophene-fused structures found the high-spin structure of

indenoindenodi(naphthalenothiophene) exists only above 450 K while maintaining

moderate biradical character in the low-spin ground state and a relatively small singlet-


10

triplet splitting.90 In this same study, indenofluorenes were found to have increasing singlet

biradical character and smaller singlet-triplet splitting with increasing core acene length.

1.4 Bibliography

(1) Geim, A. K.; Novoselov, K. S., The rise of graphene. Nature Materials 2007, 6,

183.

(2) Hernandez, Y.; Nicolosi, V.; Lotya, M.; Blighe, F. M.; Sun, Z.; De, S.; McGovern,

I. T.; Holland, B.; Byrne, M.; Gun'Ko, Y. K.; Boland, J. J.; Niraj, P.; Duesberg, G.;

Krishnamurthy, S.; Goodhue, R.; Hutchison, J.; Scardaci, V.; Ferrari, A. C.;

Coleman, J. N., High-yield production of graphene by liquid-phase exfoliation of

graphite. Nature Nanotechnology 2008, 3, 563.

(3) Lotya, M.; Hernandez, Y.; King, P. J.; Smith, R. J.; Nicolosi, V.; Karlsson, L. S.;

Blighe, F. M.; De, S.; Wang, Z.; McGovern, I. T.; Duesberg, G. S.; Coleman, J. N.,

Liquid Phase Production of Graphene by Exfoliation of Graphite in

Surfactant/Water Solutions. Journal of the American Chemical Society 2009, 131

(10), 3611-3620.

(4) Feng, R.; Zhao, Y.; Zhu, C.; Mason, T. J., Enhancement of ultrasonic cavitation

yield by multi-frequency sonication. Ultrasonics Sonochemistry 2002, 9 (5), 231-

236.

(5) Coleman, J. N., Liquid Exfoliation of Defect-Free Graphene. Accounts of Chemical

Research 2013, 46 (1), 14-22.

(6) Ciesielski, A.; Haar, S.; El Gemayel, M.; Yang, H.; Clough, J.; Melinte, G.; Gobbi,

M.; Orgiu, E.; Nardi, M. V.; Ligorio, G.; Palermo, V.; Koch, N.; Ersen, O.;

Casiraghi, C.; Samorì, P., Harnessing the Liquid-Phase Exfoliation of Graphene

Using Aliphatic Compounds: A Supramolecular Approach. Angewandte Chemie

International Edition 2014, 53 (39), 10355-10361.

(7) Haar, S.; Ciesielski, A.; Clough, J.; Yang, H.; Mazzaro, R.; Richard, F.; Conti, S.;

Merstorf, N.; Cecchini, M.; Morandi, V.; Casiraghi, C.; Samorì, P., A

Supramolecular Strategy to Leverage the Liquid-Phase Exfoliation of Graphene in

the Presence of Surfactants: Unraveling the Role of the Length of Fatty Acids.

Small 2015, 11 (14), 1691-1702.

(8) Bekyarova, E.; Itkis, M. E.; Ramesh, P.; Berger, C.; Sprinkle, M.; de Heer, W. A.;

Haddon, R. C., Chemical Modification of Epitaxial Graphene: Spontaneous

Grafting of Aryl Groups. Journal of the American Chemical Society 2009, 131 (4),

1336-1337.

(9) Bekyarova, E.; Sarkar, S.; Wang, F.; Itkis, M. E.; Kalinina, I.; Tian, X.; Haddon,

R. C., Effect of Covalent Chemistry on the Electronic Structure and Properties of

Carbon Nanotubes and Graphene. Accounts of Chemical Research 2013, 46 (1), 65-

76.

(10) Balog, R.; Jørgensen, B.; Wells, J.; Lægsgaard, E.; Hofmann, P.; Besenbacher, F.;

Hornekær, L., Atomic Hydrogen Adsorbate Structures on Graphene. Journal of the

American Chemical Society 2009, 131 (25), 8744-8745.


11

(11) Elias, D. C.; Nair, R. R.; Mohiuddin, T. M. G.; Morozov, S. V.; Blake, P.; Halsall,

M. P.; Ferrari, A. C.; Boukhvalov, D. W.; Katsnelson, M. I.; Geim, A. K.;

Novoselov, K. S., Control of Graphene's Properties by Reversible Hydrogenation:

Evidence for Graphane. Science 2009, 323 (5914), 610.

(12) Balog, R.; Jørgensen, B.; Nilsson, L.; Andersen, M.; Rienks, E.; Bianchi, M.;

Fanetti, M.; Lægsgaard, E.; Baraldi, A.; Lizzit, S.; Sljivancanin, Z.; Besenbacher,

F.; Hammer, B.; Pedersen, T. G.; Hofmann, P.; Hornekær, L., Bandgap opening in

graphene induced by patterned hydrogen adsorption. Nature Materials 2010, 9,

315.

(13) Chernozatonskii, L. A.; Kvashnin, D. G.; Sorokin, P. B.; Kvashnin, A. G.; Brüning,

J. W., Strong Influence of Graphane Island Configurations on the Electronic

Properties of a Mixed Graphene/Graphane Superlattice. The Journal of Physical

Chemistry C 2012, 116 (37), 20035-20039.

(14) Valencia, A. M.; Caldas, M. J., Single vacancy defect in graphene: Insights into its

magnetic properties from theoretical modeling. Physical Review B 2017, 96 (12),

125431.

(15) Zhang, Y.; Li, S.-Y.; Huang, H.; Li, W.-T.; Qiao, J.-B.; Wang, W.-X.; Yin, L.-J.;

Bai, K.-K.; Duan, W.; He, L., Scanning Tunneling Microscopy of the

$\ensuremath{\pi}$ Magnetism of a Single Carbon Vacancy in Graphene. Physical

Review Letters 2016, 117 (16), 166801.

(16) Kim, G.; Jhi, S.-H.; Lim, S.; Park, N., Effect of vacancy defects in graphene on

metal anchoring and hydrogen adsorption. Applied Physics Letters 2009, 94 (17),

173102.

(17) Denis, P. A.; Iribarne, F., Comparative Study of Defect Reactivity in Graphene.

The Journal of Physical Chemistry C 2013, 117 (37), 19048-19055.

(18) Denis, P. A., Density Functional Investigation of Thioepoxidated and Thiolated

Graphene. The Journal of Physical Chemistry C 2009, 113 (14), 5612-5619.

(19) Machado, F. B. C.; Aquino, A. J. A.; Lischka, H., The Diverse Manifold of

Electronic States Generated by a Single Carbon Defect in a Graphene Sheet:

Multireference Calculations Using a Pyrene Defect Model. ChemPhysChem 2014,

15 (15), 3334-3341.

(20) Machado, F. B. C.; Aquino, A. J. A.; Lischka, H., The electronic states of a double

carbon vacancy defect in pyrene: a model study for graphene. Physical Chemistry

Chemical Physics 2015, 17 (19), 12778-12785.

(21) Nieman, R.; Das, A.; Aquino, A. J. A.; Amorim, R. G.; Machado, F. B. C.; Lischka,

H., Single and double carbon vacancies in pyrene as first models for graphene

defects: A survey of the chemical reactivity toward hydrogen. Chemical Physics

2017, 482, 346-354.

(22) Guo, B.; Liu, Q.; Chen, E.; Zhu, H.; Fang, L.; Gong, J. R., Controllable N-Doping

of Graphene. Nano Letters 2010, 10 (12), 4975-4980.

(23) Wang, H.; Zhou, Y.; Wu, D.; Liao, L.; Zhao, S.; Peng, H.; Liu, Z., Synthesis of

Boron-Doped Graphene Monolayers Using the Sole Solid Feedstock by Chemical

Vapor Deposition. Small 2013, 9 (8), 1316-1320.

(24) Liu, L.; Ryu, S.; Tomasik, M. R.; Stolyarova, E.; Jung, N.; Hybertsen, M. S.;

Steigerwald, M. L.; Brus, L. E.; Flynn, G. W., Graphene Oxidation: Thickness-


12

Dependent Etching and Strong Chemical Doping. Nano Letters 2008, 8 (7), 1965-

1970.

(25) Kim, K. S.; Zhao, Y.; Jang, H.; Lee, S. Y.; Kim, J. M.; Kim, K. S.; Ahn, J.-H.; Kim,

P.; Choi, J.-Y.; Hong, B. H., Large-scale pattern growth of graphene films for

stretchable transparent electrodes. Nature 2009, 457, 706.

(26) Lee, Y.; Bae, S.; Jang, H.; Jang, S.; Zhu, S.-E.; Sim, S. H.; Song, Y. I.; Hong, B.

H.; Ahn, J.-H., Wafer-Scale Synthesis and Transfer of Graphene Films. Nano

Letters 2010, 10 (2), 490-493.

(27) Emtsev, K. V.; Bostwick, A.; Horn, K.; Jobst, J.; Kellogg, G. L.; Ley, L.;

McChesney, J. L.; Ohta, T.; Reshanov, S. A.; Röhrl, J.; Rotenberg, E.; Schmid, A.

K.; Waldmann, D.; Weber, H. B.; Seyller, T., Towards wafer-size graphene layers

by atmospheric pressure graphitization of silicon carbide. Nature Materials 2009,

8, 203.

(28) Zhou, W.; Kapetanakis, M. D.; Prange, M. P.; Pantelides, S. T.; Pennycook, S. J.;

Idrobo, J.-C., Direct Determination of the Chemical Bonding of Individual

Impurities in Graphene. Physical Review Letters 2012, 109 (20), 206803.

(29) Ramasse, Q. M.; Seabourne, C. R.; Kepaptsoglou, D.-M.; Zan, R.; Bangert, U.;

Scott, A. J., Probing the Bonding and Electronic Structure of Single Atom Dopants

in Graphene with Electron Energy Loss Spectroscopy. Nano Letters 2013, 13 (10),

4989-4995.

(30) Nieman, R.; Aquino, A. J. A.; Hardcastle, T. P.; Kotakoski, J.; Susi, T.; Lischka,

H., Structure and electronic states of a graphene double vacancy with an embedded

Si dopant. The Journal of Chemical Physics 2017, 147 (19), 194702.

(31) Liang, Y.; Feng, D.; Wu, Y.; Tsai, S.-T.; Li, G.; Ray, C.; Yu, L., Highly Efficient

Solar Cell Polymers Developed via Fine-Tuning of Structural and Electronic

Properties. Journal of the American Chemical Society 2009, 131 (22), 7792-7799.

(32) Yu, G.; Gao, J.; Hummelen, J. C.; Wudl, F.; Heeger, A. J., Polymer Photovoltaic

Cells: Enhanced Efficiencies via a Network of Internal Donor-Acceptor

Heterojunctions. Science 1995, 270 (5243), 1789.

(33) Brédas, J.-L.; Norton, J. E.; Cornil, J.; Coropceanu, V., Molecular Understanding

of Organic Solar Cells: The Challenges. Accounts of Chemical Research 2009, 42

(11), 1691-1699.

(34) Baumeier, B.; May, F.; Lennartz, C.; Andrienko, D., Challenges for in silico design

of organic semiconductors. Journal of Materials Chemistry 2012, 22 (22), 10971-

10976.

(35) Kanai, Y.; Grossman, J. C., Insights on Interfacial Charge Transfer Across

P3HT/Fullerene Photovoltaic Heterojunction from Ab Initio Calculations. Nano

Letters 2007, 7 (7), 1967-1972.

(36) Nagarjuna, P.; Bagui, A.; Gupta, V.; Singh, S. P., A highly efficient PTB7-Th

polymer donor bulk hetero-junction solar cell with increased open circuit voltage

using fullerene acceptor CN-PC70BM. Organic Electronics 2017, 43, 262-267.

(37) Suman; Bagui, A.; Garg, A.; Tyagi, B.; Gupta, V.; Singh, S. P., A fluorene-core-

based electron acceptor for fullerene-free BHJ organic solar cells—towards power

conversion efficiencies over 10%. Chemical Communications 2018, 54 (32), 4001-

4004.


13

(38) Gao, K.; Li, L.; Lai, T.; Xiao, L.; Huang, Y.; Huang, F.; Peng, J.; Cao, Y.; Liu, F.;

Russell, T. P.; Janssen, R. A. J.; Peng, X., Deep Absorbing Porphyrin Small

Molecule for High-Performance Organic Solar Cells with Very Low Energy

Losses. Journal of the American Chemical Society 2015, 137 (23), 7282-7285.

(39) Li, M.; Gao, K.; Wan, X.; Zhang, Q.; Kan, B.; Xia, R.; Liu, F.; Yang, X.; Feng, H.;

Ni, W.; Wang, Y.; Peng, J.; Zhang, H.; Liang, Z.; Yip, H.-L.; Peng, X.; Cao, Y.;

Chen, Y., Solution-processed organic tandem solar cells with power conversion

efficiencies >12%. Nature Photonics 2016, 11, 85.

(40) Gao, K.; Zhu, Z.; Xu, B.; Jo, S. B.; Kan, Y.; Peng, X.; Jen, A. K. Y., Highly

Efficient Porphyrin-Based OPV/Perovskite Hybrid Solar Cells with Extended

Photoresponse and High Fill Factor. Advanced Materials 2017, 29 (47), 1703980.

(41) Nie, W.; Gupta, G.; Crone, B. K.; Liu, F.; Smith, D. L.; Ruden, P. P.; Kuo, C.-Y.;

Tsai, H.; Wang, H.-L.; Li, H.; Tretiak, S.; Mohite, A. D., Interface Design

Principles for High-Performance Organic Semiconductor Devices. Advanced

Science 2015, 2 (6), 1500024.

(42) Natasha, K., Understanding excitons in optically active polymers. Polymer

International 2008, 57 (5), 678-688.

(43) Collini, E.; Scholes, G. D., Coherent Intrachain Energy Migration in a Conjugated

Polymer at Room Temperature. Science 2009, 323 (5912), 369.

(44) Facchetti, A., π-Conjugated Polymers for Organic Electronics and Photovoltaic

Cell Applications. Chemistry of Materials 2011, 23 (3), 733-758.

(45) Brédas, J.-L.; Silbey, R., Excitons Surf Along Conjugated Polymer Chains. Science

2009, 323 (5912), 348.

(46) Mihailetchi, V. D.; Koster, L. J. A.; Blom, P. W. M.; Melzer, C.; de Boer, B.;

van Duren, J. K. J.; Janssen, R. A. J., Compositional Dependence of the

Performance of Poly(p-phenylene vinylene):Methanofullerene Bulk-

Heterojunction Solar Cells. Advanced Functional Materials 2005, 15 (5), 795-801.

(47) Wan, M.; Zhu, H.; Guo, J.; Peng, Y.; Deng, H.; Jin, L.; Liu, M., High open circuit

voltage polymer solar cells with blend of MEH-PPV as donor and fumaronitrile

derivate as acceptor. Synthetic Metals 2013, 178, 22-26.

(48) Borges, I.; Aquino, A. J. A.; Köhn, A.; Nieman, R.; Hase, W. L.; Chen, L. X.;

Lischka, H., Ab Initio Modeling of Excitonic and Charge-Transfer States in

Organic Semiconductors: The PTB1/PCBM Low Band Gap System. Journal of the


(49) Sen, K.; Crespo-Otero, R.; Weingart, O.; Thiel, W.; Barbatti, M., Interfacial States

in Donor–Acceptor Organic Heterojunctions: Computational Insights into

Thiophene-Oligomer/Fullerene Junctions. Journal of Chemical Theory and

Computation 2013, 9 (1), 533-542.

(50) Aquino, A. A. J.; Borges, I.; Nieman, R.; Köhn, A.; Lischka, H., Intermolecular

interactions and charge transfer transitions in aromatic hydrocarbon–

tetracyanoethylene complexes. Physical Chemistry Chemical Physics 2014, 16

(38), 20586-20597.

(51) Haddon, R. C., Quantum chemical studies in the design of organic metals. III. Odd-

alternant hydrocarbons&;The phenalenyl (ply) system. Australian Journal of

Chemistry 1975, 28 (11), 2343-2351.


14

(52) Jacob, J., The significance of polycyclic aromatic hydrocarbons as environmental

carcinogens. Pure and applied chemistry 1996, 68 (2), 301-308.

(53) Heymann, K.; Lehmann, J.; Solomon, D.; Schmidt, M. W. I.; Regier, T., C 1s K-

edge near edge X-ray absorption fine structure (NEXAFS) spectroscopy for

characterizing functional group chemistry of black carbon. Organic Geochemistry

2011, 42 (9), 1055-1064.

(54) Weisman, J. L.; Lee, T. J.; Salama, F.; Head‐Gordon, M., Time‐dependent Density

Functional Theory Calculations of Large Compact Polycyclic Aromatic

Hydrocarbon Cations: Implications for the Diffuse Interstellar Bands. The

Astrophysical Journal 2003, 587 (1), 256-261.

(55) Salama, F., Molecular spectroscopy in astrophysics: the case of polycyclic aromatic

hydrocarbons. Journal of Molecular Structure 2001, 563-564, 19-26.

(56) Bréchignac, P.; Pino, T., Electronic spectra of cold gas phase PAH cations:

Towards the identification of the Diffuse Interstellar Bands carriers. Astronomy and

Astrophysics 1999, 343, L49-L52.

(57) Hu, P.; Wu, J., Modern zethrene chemistry. Canadian Journal of Chemistry 2016,

95 (3), 223-233.

(58) Konishi, A.; Hirao, Y.; Kurata, H.; Kubo, T.; Nakano, M.; Kamada, K., Anthenes:

Model systems for understanding the edge state of graphene nanoribbons. Pure and

Applied Chemistry 2014, 86 (4), 497-505.

(59) Morita, Y.; Suzuki, S.; Sato, K.; Takui, T., Synthetic organic spin chemistry for

structurally well-defined open-shell graphene fragments. Nature Chemistry 2011,

3, 197.

(60) Shimizu, A.; Hirao, Y.; Matsumoto, K.; Kurata, H.; Kubo, T.; Uruichi, M.;

Yakushi, K., Aromaticity and π-bond covalency: prominent intermolecular

covalent bonding interaction of a Kekulé hydrocarbon with very significant singlet

biradical character. Chemical Communications 2012, 48 (45), 5629-5631.

(61) Inoue, J.; Fukui, K.; Kubo, T.; Nakazawa, S.; Sato, K.; Shiomi, D.; Morita, Y.;

Yamamoto, K.; Takui, T.; Nakasuji, K., The First Detection of a Clar's

Hydrocarbon, 2,6,10-Tri-tert-Butyltriangulene: A Ground-State Triplet of Non-

Kekulé Polynuclear Benzenoid Hydrocarbon. Journal of the American Chemical

Society 2001, 123 (50), 12702-12703.

(62) Casado, J., Para-Quinodimethanes: A Unified Review of the Quinoidal-Versus-

Aromatic Competition and its Implications. Topics in Current Chemistry 2017, 375

(4), 73.

(63) Frederickson, C. K.; Rose, B. D.; Haley, M. M., Explorations of the

Indenofluorenes and Expanded Quinoidal Analogues. Accounts of Chemical

Research 2017, 50 (4), 977-987.

(64) Miyoshi, H.; Miki, M.; Hirano, S.; Shimizu, A.; Kishi, R.; Fukuda, K.; Shiomi, D.;

Sato, K.; Takui, T.; Hisaki, I.; Nakano, M.; Tobe, Y., Fluoreno[2,3-b]fluorene vs

Indeno[2,1-b]fluorene: Unusual Relationship between the Number of π Electrons

and Excitation Energy in m-Quinodimethane-Type Singlet Diradicaloids. The

Journal of Organic Chemistry 2017, 82 (3), 1380-1388.


15

(65) Barker, J. E.; Frederickson, C. K.; Jones, M. H.; Zakharov, L. N.; Haley, M. M.,

Synthesis and Properties of Quinoidal Fluorenofluorenes. Organic Letters 2017, 19

(19), 5312-5315.

(66) Purushothaman, B.; Bruzek, M.; Parkin, S. R.; Miller, A.-F.; Anthony, J. E.,

Synthesis and Structural Characterization of Crystalline Nonacenes. Angewandte

Chemie 2011, 123 (31), 7151-7155.

(67) Zimmerman, P. M.; Bell, F.; Casanova, D.; Head-Gordon, M., Mechanism for

Singlet Fission in Pentacene and Tetracene: From Single Exciton to Two Triplets.

Journal of the American Chemical Society 2011, 133 (49), 19944-19952.

(68) Plasser, F.; Pašalić, H.; Gerzabek, M. H.; Libisch, F.; Reiter, R.; Burgdörfer, J.;

Müller, T.; Shepard, R.; Lischka, H., The Multiradical Character of One- and Two-

Dimensional Graphene Nanoribbons. Angewandte Chemie International Edition

2013, 52 (9), 2581-2584.

(69) Barone, V.; Hod, O.; Peralta, J. E.; Scuseria, G. E., Accurate Prediction of the

Electronic Properties of Low-Dimensional Graphene Derivatives Using a Screened

Hybrid Density Functional. Accounts of Chemical Research 2011, 44 (4), 269-279.

(70) Lambert, C., Towards Polycyclic Aromatic Hydrocarbons with a Singlet Open-

Shell Ground State. Angewandte Chemie International Edition 2011, 50 (8), 1756-

1758.

(71) Hachmann, J.; Dorando, J. J.; Avilés, M.; Chan, G. K.-L., The radical character of

the acenes: A density matrix renormalization group study. The Journal of Chemical

Physics 2007, 127 (13), 134309.

(72) Shen, B.; Geiger, T.; Einholz, R.; Reicherter, F.; Schundelmeier, S.; Maichle-

Mössmer, C.; Speiser, B.; Bettinger, H. F., Bridging the Gap between Pentacene

and Perfluoropentacene: Synthesis and Characterization of 2,3,9,10-

Tetrafluoropentacene in the Neutral, Cationic, and Dicationic States. The Journal

of Organic Chemistry 2018, 83 (6), 3149-3158.

(73) Anthony, J. E., The Larger Acenes: Versatile Organic Semiconductors.

Angewandte Chemie International Edition 2008, 47 (3), 452-483.

(74) Pinheiro, M.; Ferrão, L. F. A.; Bettanin, F.; Aquino, A. J. A.; Machado, F. B. C.;

Lischka, H., How to efficiently tune the biradicaloid nature of acenes by chemical

doping with boron and nitrogen. Physical Chemistry Chemical Physics 2017, 19

(29), 19225-19233.

(75) Fukui, K.; Inoue, J.; Kubo, T.; Nakazawa, S.; Aoki, T.; Morita, Y.; Yamamoto, K.;

Sato, K.; Shiomi, D.; Nakasuji, K.; Takui, T., The first non-Kekulé polynuclear

aromatic high-spin hydrocarbon: Generation of a triangulene derivative and band

structure calculation of triangulene-based high-spin hydrocarbons. Synthetic Metals

2001, 121 (1), 1824-1825.

(76) Jin, H.; Li, J.; Wang, T.; Yu, Y., Photoinduced pure spin-current in triangulene-

based nano-devices. Carbon 2018, 137, 1-5.

(77) Das, A.; Müller, T.; Plasser, F.; Lischka, H., Polyradical Character of Triangular

Non-Kekulé Structures, Zethrenes, p-Quinodimethane-Linked Bisphenalenyl, and

the Clar Goblet in Comparison: An Extended Multireference Study. The Journal of

Physical Chemistry A 2016, 120 (9), 1625-1636.


16

(78) Mou, Z.; Kertesz, M., Sigma- versus Pi-Dimerization Modes of Triangulene.

Chemistry – A European Journal 2018, 24 (23), 6140-6147.

(79) Clar, E., The Aromatic Sextet. Wiley: London, 1972.

(80) Lukman, S.; Richter, J. M.; Yang, L.; Hu, P.; Wu, J.; Greenham, N. C.; Musser, A.

J., Efficient Singlet Fission and Triplet-Pair Emission in a Family of Zethrene

Diradicaloids. Journal of the American Chemical Society 2017, 139 (50), 18376-

18385.

(81) Zeng, W.; Gopalakrishna, T. Y.; Phan, H.; Tanaka, T.; Herng, T. S.; Ding, J.;

Osuka, A.; Wu, J., Superoctazethrene: An Open-Shell Graphene-like Molecule

Possessing Large Diradical Character but Still with Reasonable Stability. Journal

of the American Chemical Society 2018, 140 (43), 14054-14058.

(82) Yu, S.-S.; Zheng, W.-T., Effect of N/B doping on the electronic and field emission

properties for carbon nanotubes, carbon nanocones, and graphene nanoribbons.

Nanoscale 2010, 2 (7), 1069-1082.

(83) Das, A.; Pinheiro Jr, M.; Machado, F. B. C.; Aquino, A. J. A.; Lischka, H., Tuning

the Biradicaloid Nature of Polycyclic Aromatic Hydrocarbons: The Effect of

Graphitic Nitrogen Doping in Zethrenes. ChemPhysChem 2018, 19 (19), 2492-

2499.

(84) Konishi, A.; Hirao, Y.; Matsumoto, K.; Kurata, H.; Kishi, R.; Shigeta, Y.; Nakano,

M.; Tokunaga, K.; Kamada, K.; Kubo, T., Synthesis and characterization of

quarteranthene: elucidating the characteristics of the edge state of graphene

nanoribbons at the molecular level. Journal of the American Chemical Society

2013, 135 (4), 1430-1437.

(85) Horn, S.; Plasser, F.; Müller, T.; Libisch, F.; Burgdörfer, J.; Lischka, H., A

comparison of singlet and triplet states for one- and two-dimensional graphene

nanoribbons using multireference theory. Theoretical Chemistry Accounts 2014,

133 (8), 1511.

(86) Horn, S.; Lischka, H., A comparison of neutral and charged species of one- and

two-dimensional models of graphene nanoribbons using multireference theory. The

Journal of Chemical Physics 2015, 142 (5), 054302.

(87) Rose, B. D.; Sumner, N. J.; Filatov, A. S.; Peters, S. J.; Zakharov, L. N.; Petrukhina,

M. A.; Haley, M. M., Experimental and Computational Studies of the Neutral and

Reduced States of Indeno[1,2-b]fluorene. Journal of the American Chemical

Society 2014, 136 (25), 9181-9189.

(88) Chase, D. T.; Fix, A. G.; Kang, S. J.; Rose, B. D.; Weber, C. D.; Zhong, Y.;

Zakharov, L. N.; Lonergan, M. C.; Nuckolls, C.; Haley, M. M., 6,12-

Diarylindeno[1,2-b]fluorenes: Syntheses, Photophysics, and Ambipolar OFETs.


(89) Yu, W.; Zhang, J.; Wang, X.; Liu, X.; Tu, D.; Zhang, J.; Guo, X.; Li, C., A Dispiro-

Type Fluorene-Indenofluorene-Centered Hole Transporting Material for Efficient

Planar Perovskite Solar Cells. Solar RRL 2018, 2 (7), 1800048.

(90) Dressler, J. J.; Teraoka, M.; Espejo, G. L.; Kishi, R.; Takamuku, S.; Gómez-García,

C. J.; Zakharov, L. N.; Nakano, M.; Casado, J.; Haley, M. M., Thiophene and its

sulfur inhibit indenoindenodibenzothiophene diradicals from low-energy lying

thermal triplets. Nature Chemistry 2018, 10 (11), 1134-1140.


17

CHAPTER II

THEORETICAL METHODS

The Schrödinger equation (shown in Equation 2.1 in its one-dimension, time-

independent form) allows the total energy of a system, E, to be calculated from the system

of mass, m, with wave function, 𝜓(𝑥). The wave function is important as it describes the

state of the system. The first term is the kinetic energy, and the second term, 𝑉(𝑥), is the

potential energy function. As the Schrödinger equation can only be solved exactly for very

specific cases, several approximations have been made to make it accessible to a larger

range of systems. The Born-Oppenheimer (BO) approximation for instance states that since

the mass of electrons is much less than that of atomic nuclei, their velocities can be

decoupled. The consequences of this is that the nuclei are stationary from the perspective

of electrons, so the wave function depends only on electrons moving in the field of the

stationary nuclei (that is, it depends only parametrically on the nuclear coordinates). The

Hartree-Fock (HF) treatment for a many-electron system, itself utilizing the BO

approximation, and upon which several more advanced method (called post-HF methods)

are built, calculates the total energy of a system of electrons in only the average potential

created by the other particles.

−ħ2

2𝑚

𝑑2𝜓(𝑥)

𝑑𝑥2+ 𝑉(𝑥)𝜓(𝑥) = 𝐸𝜓(𝑥) (2.1)

The choice of which method is key when one wants to calculate specific molecular

properties. One priority is simply computational rigor, i.e. how much time will the

calculation require? The HF method for instance scales exponentially with the number of

basis functions, N, as N4, second-order Møller Plesset perturbation theory (MP2) is more

accurate but scales as N5, and multiconfigurational and multireference are more complex

as they depend not only on the number of basis functions, but also the number of reference

configurations considered (scaling linearly).

Besides the computational cost, one must consider the properties one wishes to

calculate. For example, HF theory provides nice molecular orbital shapes, but as it is only


18

a single reference method, electron correlation energy is lost and it will not reliably

describe correct dissociation behavior. To account for this, multiconfigurational and

multireference methods should be used. DFT is quite efficient with respect to

computational cost as it foregoes reliance on the wave function for the total electron

density, but the correct choice of DFT functional is non-trivial and not easily determined.

This remainder of chapter will provide a brief discussion and derivation of the

various quantum chemical methods implemented in the chapters that follow. Unless

otherwise specified, the derivations and discussion presented in this chapter are adapted

from Levine’s Quantum Chemisty.1

2.1 Density Functional Theory

Hohenberg and Kohn provided the foundation for DFT by showing that the total

energy of a non-degenerate ground state, 𝐸0, is determined by the ground state electron

probability density, 𝜌0(𝑟) in a given external field, υ(r), where r represents the electron

coordinates.2 In addition, as the electron probability density can be used to find the ground-

state total energy, the ground state wave function, ψ0, can be determined which would allow

one to find excited state wave functions and energies as well.

The general expression for the ground state total energy is given in Equation 2.2.

The term 𝑇[𝜌0] is the kinetic energy, 𝑉𝑁𝑒[𝜌0] is the nuclear-electron attraction energy, and

𝑉𝑒𝑒[𝜌0] is the electron-electron repulsion energy. Of these, only 𝑉𝑁𝑒[𝜌0] is known

(Equation 2.3), while 𝑇[𝜌0] and 𝑉𝑒𝑒[𝜌0] are not explicitly known.

𝐸0 = 𝐸𝜐[𝜌0] = 𝑇[𝜌0] + 𝑉𝑁𝑒[𝜌0] + 𝑉𝑒𝑒[𝜌0] (2.2)

𝑉𝑁𝑒[𝜌0] = ⟨𝜓0| ∑ υ(r𝑖)𝑛𝑖=1 |𝜓0⟩ = ∫ 𝜌0υ(r)dr (2.3)

With this definition of 𝑉𝑁𝑒[𝜌0], Equation 2.2 can be simplified into Equation 2.2,

and by gathering the kinetic energy and electron-electron repulsion energy into a new term,

𝐹[𝜌0], showing a functional dependent on the electron probability density, Equation 2.5.


19

This new functional is independent of the external potential, but cannot be used to calculate

the total ground state energy from 𝜌0 since 𝐹[𝜌0] is not explicitly defined.

𝐸0 = 𝐸𝜐[𝜌0] = ∫ 𝜌0υ(r)dr + 𝑇[𝜌0] + 𝑉𝑒𝑒[𝜌0] (2.4)

𝐸0 = 𝐸𝜐[𝜌0] = ∫ 𝜌0υ(r)dr + 𝐹[𝜌0] (2.5)

The Hohenberg-Kohn theorem that was just defined shows how the density, 𝜌0, can

be used in lieu of the wave function to calculate the ground state properties of a molecular

system, but since 𝐹[𝜌0] and the method of obtaining 𝜌0 without a wave function are

unknown, it does not explain how this is accomplished. Kohn and Sham provided a method

by which 𝜌0 could be determined and used to calculate 𝐸0.

The Kohn-Sham (K-S) approach3 to DFT begins with a reference system composed

of non-interacting electrons all experiencing the same external potential energy function,

υref(r). The function υref(r) gives the same ground state electron probability density,

𝜌𝑟𝑒𝑓(𝑟), for the reference as for the exact ground state electron probability density, 𝜌0(𝑟).

Finding 𝜌𝑟𝑒𝑓(𝑟) in this way allows for the determination of the external potential for the

reference system.

The K-S Hamiltonian, ��𝑟𝑒𝑓, for this non-interacting reference system is a sum of

one-electron Hamiltonians, ℎ𝑖, according to Equation 2.6, and the K-S reference system is

related back to the real, fully interacting system by Equation 2.7. The factor λ describes the

amount of electron-electron interaction and varies between λ=0 for the non-interacting

system and λ=1 for the real system (fully interacting). The external potential is such that it

makes the ground state electron probability density with the Hamiltonian, ��𝜆, equal to the

ground state electron probability density of the real system. Eigenfunctions of the one-

electron Hamiltonian are the spatial part of the K-S spin orbitals and the eigenvalues are

the K-S orbital energies as shown by Equation 2.8.


20

��𝑟𝑒𝑓 = ∑ [−1

2∇𝑖

2 + 𝜐𝑟𝑒𝑓(𝑟𝑖)] ≡ ∑ ℎ𝑖

𝑛

𝑖=1

𝑛

𝑖 = 1

(2.6)

��𝜆 ≡ �� + ∑ 𝜐𝜆(𝑟𝑖)

𝑖

+ 𝜆��𝑒𝑒 (2.7)

ℎ𝑖 𝜓𝑖𝐾𝑆 = 𝜀𝑖

𝐾𝑆 𝜓𝑖𝐾𝑆 (2.8)

With these, another equation can be derived from Equation 2.5 to find the total

energy of the ground state, Equation 2.9. In Equation 2.9, Δ��[𝜌] is the difference between

the ground state electronic kinetic energy and that of the non-interacting system, and

Δ��𝑒𝑒[𝜌] is the difference between the average ground state electron-electron repulsion

energy and the electrostatic repulsion for a system of electrons that has been smeared into

a continuous distribution of charge with a density, ρ. The sum of Δ��[𝜌] and Δ��𝑒𝑒[𝜌], each

of which is unknown, gives the exchange-correlation energy functional, 𝐸𝑋𝐶[𝜌], rewriting

Equation 2.9 as Equation 2.10.

𝐸𝜐[𝜌0] = ∫ 𝜌0υ(r)dr + ��𝑟𝑒𝑓[𝜌0] +1

2∬

𝜌(𝑟1)𝜌(𝑟2)

𝑟12𝑑𝑟1𝑑𝑟2 + Δ��[𝜌]

+ Δ��𝑒𝑒[𝜌]

(2.9)

𝐸𝜐[𝜌0] = ∫ 𝜌0υ(r)dr + ��𝑟𝑒𝑓[𝜌0] +1

2∬


𝑟12𝑑𝑟1𝑑𝑟2 + 𝐸𝑋𝐶[𝜌] (2.10)

The first three terms in Equation 2.10 are found from the ground state electron

density and provide the largest contribution. The last term, the exchange-correlation

energy, contributes less, but is more difficult to obtain accurately. All four terms in

Equation 2.10 require the ground state electron density for the reference system of non-

interacting electrons such that it equals the electron density of the real system. The electron

density of the n-electron system can then be found using a wave function expressed as a

Slater determinant as in Equation 2.11, where the K-S spatial orbitals are restricted to

remain orthonormal.


21

𝜌 = 𝜌𝑟𝑒𝑓 = ∑|𝜓𝑖𝐾𝑆|

2𝑛

𝑖=1

(2.11)

With this definition of the electron density, Slater-Condon rules4-5 allow one to

rewrite nuclear-electron (nuclear coordinates given by α) attraction energy in Equation 2.10

as Equation 2.12. One can see that finding K-S orbitals that minimize the total ground state

energy will follow Equation 2.13 (following Equation 2.8), where 𝜐𝑋𝐶(1) is the exchange-

correlation potential and is the functional derivative of the exchange-correlation energy

with respect to the ground state electron density. Finally, Equation 2.13 can be written

succinctly as Equation 2.14.

𝐸𝜐[𝜌0] = − ∑ 𝑍𝛼 ∫𝜌(𝑟1)

𝑟1𝛼𝑑𝑟1

𝛼

−1

2∑⟨𝜓𝑖

𝐾𝑆(1)|∇12|𝜓𝑖

𝐾𝑆(1)⟩

𝑛

𝑖=1

+1

2∬


𝑟12𝑑𝑟1𝑑𝑟2 + 𝐸𝑋𝐶[𝜌]

(2.12)

[−1

2∇1

2 − ∑𝑍𝛼

𝑟1𝛼+ ∫

𝜌(𝑟1)

𝑟1𝛼𝑑𝑟1 + 𝜐𝑋𝐶(1)

𝛼

] 𝜓𝑖𝐾𝑆(1) = 𝜀𝑖

𝐾𝑆(1)𝜓𝑖𝐾𝑆(1) (2.13)

ℎ𝑖 𝜓𝑖𝐾𝑆(1) = 𝜀𝑖

𝐾𝑆(1)𝜓𝑖𝐾𝑆(1) (2.14)

The most difficult part in evaluating DFT energies is the finding the correct

functional form of the exchange-correlation energy, 𝐸𝑋𝐶, as it makes the most significant

contribution to the total energy. One common procedure for overcoming this problem

called the Local Density Approximation (LDA) is to assume a uniform electron gas and

split 𝐸𝑋𝐶 into exchange and correlation energies, 𝐸𝑋 and 𝐸𝐶 respectively, which are then

summed together. A specific example of a LDA functional is from Vosko, Wilk, and Nuair

(VWN).6 Gradient-corrected functionals build on the LDA formalism and take into account

non-uniform gases utilizing gradients. This approach improve accuracy by having the

exchange-correlation energy depend on the electron density as well as the first derivative

of the electron density. Common examples of gradient-corrected functionals are Perdew,

Burke and Ernzerhof (PBE)7-8, and Becke, Lee, Yang, Parr (BLYP).9-11 Hybrid density

functionals were developed to combine Hartree-Fock exchange energy with the exchange-


22

correlation energy of gradient-corrected functionals. An example of a hybrid functional is

the popular Becke 3-parameter Lee, Yang, Parr (B3LYP).12 Lastly, range-separated

functionals were developed where the Coulomb potential is treated separately at short and

long range in the form of a hybrid functional. Examples of such range-separated functional

are the Coulomb attenuated method B3LYP (CAM-B3LYP)13 and ωB97XD14.

Non-covalent interactions are especially important in weakly bonded complexes,

aggregates, and condensed phases made up of separable units, but these dispersion/Van der

Waals interactions require special consideration outside of computationally laborious ab

initio methods. One approach is DFT-D (D stands for dispersion),15 where the dispersion

energy correction, 𝐸𝑑𝑖𝑠𝑝, to the DFT energy, 𝐸𝐷𝐹𝑇, is applied via Equations 2.15 and 2.16.

In Equation 2.16, 𝑁𝑎𝑡 is the number of atoms, 𝐶6𝑡𝑓

is the dispersion coefficient for the atom

pairs f and t, 𝑆6 is a global scaling factor which is dependent on the density functional,

𝑅𝑡𝑓is the distance between atoms t and f, and 𝑓𝑑𝑎𝑚𝑝 is a dampening function applied to

correct for small R and electron double-counting.

𝐸𝐷𝐹𝑇−𝐷 = 𝐸𝐾𝑆−𝐷𝐹𝑇 + 𝐸𝑑𝑖𝑠𝑝 (2.15)

𝐸𝑑𝑖𝑠𝑝 = −𝑆6 ∑ ∑𝐶6

𝑡𝑓

𝑅𝑡𝑓6

𝑁𝑎𝑡

𝑓=𝑡+1

𝑁𝑎𝑡−1

𝑡=1

𝑓𝑑𝑎𝑚𝑝(𝑅𝑡𝑓) (2.16)

2.2 Second-Order Møller-Plesset Perturbation Theory

This section will cover perturbation theory (PT) as described by Møller and Plesset

(MP)16 for second-order corrections (MP2)17-21 for closed-shell, ground state systems and

utilizing spin orbitals, 𝑢𝑖. They detailed a method for the treatment of molecules in which

the HF wave function is the unperturbed wave function. The Hartree-Fock equation for an

electron, m, in a many-electron system is given by Equation 2.17 where the Fock operator

expression is Equation 2.18.

𝑓(𝑚)𝑢𝑖(𝑚) = 𝜀𝑖𝑢𝑖(𝑚) (2.17)


23

𝑓(𝑚) = −1

2∇𝑚

2 − ∑𝑍𝛼

𝑟𝑚𝛼𝛼

+ ∑[𝑗��(𝑚) − ��𝑗(𝑚)]

𝑛

𝑗=1

(2.18)

The MP unperturbed Hamiltonian is a sum of one-electron Fock operators,

Equation 2.19, with the HF wave function, Φ0, also referred to in this context as the zeroth-

order wave function, is shown in Equation 2.20 and is a Slater determinant composed of

any of the n possible different spin orbitals. Each term in its expansion is a permutation of

the electrons in the spin orbitals, and is an eigenfunction of ��0, the MP zeroth-order

Hamiltonian. The total wave function, Φ0, is like-wise an eigenvalue of ��0 with

eigenvalues given by Equation 2.21.

��0 ≡ ∑ 𝑓(𝑚)

𝑛

𝑚=1

(2.19)

|𝑢1𝑢2𝑢3 … 𝑢𝑛| = Φ0 (2.20)

��0Φ0 = ( ∑ 𝜀𝑚

𝑛

𝑚=1

) Φ0 = 𝐸(0)Φ0 (2.21)

The Hamiltonian describing the perturbation, 𝐻′ is the difference between the real

molecular Hamiltonian, ��, and the zeroth-order Hamiltonian, ��0, as in Equation 2.22. In

this way, 𝐻′ is the difference between the real interelectronic repulsion and the HF

interelectronic repulsion. The first-order correction to the ground state energy is given in

Equation 2.23, which is also the variational integral for the HF wave function, Φ0, giving

the HF energy, E𝐻𝐹, Equation 2.24.

𝐻′ = �� − ��0 = ∑ ∑1

𝑟𝑙𝑚− ∑ ∑[𝑗��(𝑚) − ��𝑗(𝑚)]

𝑛

𝑗=1

𝑛

𝑚=1𝑚>𝑙𝑙

(2.22)

𝐸0(1)

= ⟨Φ0|𝐻′|Φ0⟩ (2.23)

𝐸0(0)

+ 𝐸0(1)

= E𝐻𝐹 (2.24)


24

With Equation 2.24 in mind, corrections up to the second-order are shown to be

necessary in order to improve upon the HF energy, 𝐸0(2)

, as in Equation 2.25, where the

unperturbed functions, 𝜓𝑠(0)

, are all possible Slater determinants formed from n possible

different spin orbitals. One must also apply excitations in the ground state electron

configuration such that 𝜓𝑠(0)

differs from Φ0 by the number of unoccupied orbitals, also

referred to virtual orbitals, (represented by indices a, b, c, d, …) replacing occupied orbitals

(represented by indices i, j, k, l, …). An excitation level of one is shown as Φ𝑖𝑎, where

occupied spin orbital, 𝑢𝑖 is replaced by unoccupied spin orbital, 𝑢𝑎, an excitation level of

two is shown as Φ𝑖𝑗𝑎𝑏, where occupied spin orbitals, 𝑢𝑖 and 𝑢𝑗 , are replaced by unoccupied

spin orbitals, 𝑢𝑎 and 𝑢𝑏, and so on for excitation levels of three and higher. Single

excitations can be neglected as the integral, ⟨𝜓𝑖𝑎|𝐻′|Φ0⟩, is zero for all occupied orbitals,

i, and all unoccupied orbitals, a. Similarly, triple excitations and higher,

⟨𝜓𝑖𝑗𝑘𝑎𝑏𝑐|𝐻′|Φ0⟩, 𝑒𝑡𝑐 …, are zero following Slater-Condon rules4-5. Therefore, only an

excitation level of two is needed to be evaluated, ⟨𝜓𝑖𝑗𝑎𝑏|𝐻′|Φ0⟩. This same discussion is the

reason why only double excitations are present in the first order correction, 𝜓𝑠(1)

, to the

unperturbed wave function, 𝜓𝑠(0)

.

𝐸0(2)

= ∑⟨𝜓𝑠

(0)|𝐻′|Φ0⟩

2

𝐸0(0)

− 𝐸𝑠(0)

𝑠≠0

(2.25)

The doubly excited wave functions, Φ𝑖𝑗𝑎𝑏, are eigenfunctions of Equation 2.19

whose eigenvalues differ from those of the HF wave function, Φ0, by replacing occupied

orbital energies, 𝜀𝑖 and 𝜀𝑗, by unoccupied orbital energies, 𝜀𝑎 and 𝜀𝑏, as in Equation 2.26,

(which is the denominator of Equation 2.25) when 𝜓𝑠(0)

is equal to Φ𝑖𝑗𝑎𝑏. With the

expression for 𝐻′ in Equation 2.22 and Slater-Condon rules, the evaluation of integrals

with Φ𝑖𝑗𝑎𝑏 proceeds as in Equation 2.27 with the integral in Equation 2.28, where n is the

number of electrons.


25

𝐸0(0)

− 𝐸𝑠(0)

= 𝜀𝑖 + 𝜀𝑗 + 𝜀𝑎 + 𝜀𝑏 (2.26)

𝐸0(2)

= ∑ ∑ ∑ ∑|⟨𝑎𝑏|𝑟−1|𝑖𝑗⟩ − ⟨𝑎𝑏|𝑟−1|𝑗𝑖⟩|2

𝜀𝑖 + 𝜀𝑗 + 𝜀𝑎 + 𝜀𝑏

𝑛−1

𝑗=1

𝑛

𝑖=𝑗+1

∞

𝑎=𝑛+1

∞

𝑏=𝑎+1

(2.27)

⟨𝑎𝑏|𝑟−1|𝑖𝑗⟩ = ∫ ∫ 𝑢𝑎∗ (1)𝑢𝑏

∗ (2)𝑟12−1𝑢𝑖(1)𝑢𝑗(2)𝑑𝜏 (2.28)

Integrals involving spin orbitals (including the actual spin) are evaluated with

respect to the electron-repulsion integrals, and sums over all a, b, i, and j in Equation 2.27

allow for all possible doubly excited terms of 𝜓𝑠(0)

. With this, the molecular MP2 energy,

𝐸𝑀𝑃2, is finally given in Equation 2.29. Consequently, MP2 energies are size-extensive,

but since they are not variational, they can produce energies below the true energy.

𝐸(0) + 𝐸(1) + 𝐸(2) = 𝐸𝐻𝐹 + 𝐸(2) = 𝐸𝑀𝑃2 (2.29)

2.3 Second-Order Algebraic Diagrammatic Construction

The algebraic diagrammatic construction (ADC)22 for the polarization propagator

(PP) allows for the calculation of electronic excited state information for a molecular

system and the following derivation is adapted from Ref 23. The derivation of the ADC

approach starts with many-body Green’s function theory that is used to solve

inhomogeneous differential equations.24 The electronic Hamiltonian for a many-bodied

system does not have a uniquely defined Green’s function, however, but propagators can

be created from the foundations of a Green’s function in order to solve the problem.25

Several propagators can be constructed such as the one-electron propagator which

gives the probability of an electron travelling a given length, l, in a given time, t; the two-

electron propagator gives the same property but for two correlated electrons. The

expectation values for the calculated properties can be found by knowing these two

propagators.26 The one-electron propagator can also be used for other one-electron

properties such as ionization energies and electron affinities.27-29 The PP describes the time-


26

evolution of polarization in a many-body system by acting on a time-dependent (TD)

ground state wave function and propagates the TD change in density.25

The electronic Hamiltonian needs the wave function of the electronic excited states

and the PP contains the implicit information of the electronic excited states for the

molecular system. The spectral representation of the PP as a matric function, also called

the Lehmann representation, is given in Equation 2.30, where 𝜓0 is the ground state wave

function, 𝐸0𝑁 is the grounds state energy, 𝑐𝑞

† is the creation operator for the one-electron

state, q, and 𝑐𝑝 is the annihilation operator for the one-electron state, p (both the creation

and annihilation operators are associated with HF orbitals). In this form, the PP gives the

eigenstates of the molecular system to diagonalize the Hamiltonian matrix. The summation

in the PP is done for all electronic excited states, 𝜓𝑛, with total energy, 𝐸𝑛𝑁. This will give

the exciation spectrum since the poles of the PP (when 𝜔 = 𝐸𝑛𝑁 − 𝐸0

𝑁) are at the vertical

excitation energies and the residues are the transition probabilities. The spectral

representation of the PP can therefore be written in a compact, diagonalized form as in

Equation 2.31, where 𝛀 is the matrix of vertical excitation energies, 𝜔𝑛, and 𝒙 is the matrix

of transition or spectroscopic amplitudes.25

𝚷𝑝𝑞,𝑟𝑠(𝜔) = ∑⟨𝜓0|𝑐𝑞

†𝑐𝑝|𝜓𝑛⟩⟨𝜓𝑛|𝑐𝑟†𝑐𝑠|𝜓0⟩

𝜔 + 𝐸0𝑁 − 𝐸𝑛

𝑁𝑛≠0

+ ∑⟨𝜓0|𝑐𝑟

†𝑐𝑠|𝜓𝑛⟩⟨𝜓𝑛|𝑐𝑞†𝑐𝑝|𝜓0⟩

−𝜔 + 𝐸0𝑁 − 𝐸𝑛

𝑁𝑛≠0

(2.30)

𝚷(𝜔) = 𝒙†(𝜔 − 𝛀)−1𝒙 (2.31)

A non-diagonal representation of the PP is needed for the Feynmann-Goldstone

diagrammatic perturbation scheme, and is shown in Equation 2.32, where M and f are

respectively the non-diagonal matrix representation of the effective Hamiltonian and the

matrix of effective transition moments.22 The effective Hamiltonian and transition

amplitudes are expanded to the PT order in the fluctuation potential, and, following that,

the correlation energy in the MP Hamiltonian partitioning is shown in Equations 2.33 and


27

2.34. Using the PT up to order n expands 𝚷(𝜔) to give the expression of 𝑀𝜇𝜈(𝑛)

and 𝑓𝜇(𝑛)

(μ

and ν give the excitation level).

𝚷(𝜔) = 𝐟†(𝜔 − 𝐌)−1𝐟 (2.32)

𝑴 = 𝑴(0) + 𝑴(1) + 𝑴(2) + ⋯ (2.33)

𝐟 = 𝐟(0) + 𝐟(1) + 𝐟(2) + ⋯ (2.34)

The ADC approximation up to order n, ADC(n), has all the terms necessary for a

PT application of 𝚷(𝜔) to that same order. The ADC scheme is thusly a reformulation of

the diagrammatic perturbation series of the PP. Since the excitation energies are the poles

of PP, they can be found by diagonalizing the M matrix to the specified order using PT.

The Hamiltonian eigenvalue problem gives the vertical excitation energies, 𝜔𝑛, whose

eigenvectors, y, give the spectroscopic amplitudes using Equation 2.36.

𝒙 = 𝒚†𝐟 (2.35)

In summary, the ADC method allows for the use of PT to reformulate the

approximations of the PP, which formally contains the information on the exact excited

states of a given molecular system, so that it only has approximate information on the

excited states. The diagrammatic method together with the PT series up to a specified order

gives the exact algebraic formulas for the computation of the excited states, with higher

orders being more accurate.

2.4 Basis Set Superposition Error and the Counterpoise Correction Method

Basis set superposition error (BSSE) manifests as artificially shorter intermolecular

distances and greater (more stabilizing) interaction energy in weakly bonded dimers or

clusters; the error is more prominent for smaller basis set sizes (fewer basis functions) and

is vanishing at the complete basis set limit.30-31 The following is adapted from Ref. 32. In

dimer systems composed of two monomers, A and B, the dimer is stabilized artificially as

monomer A, as a result of proximity, begins to take advantage of the basis functions


28

localized to the atoms on monomer B (and vice versa). The increase in basis set size as a

result of the extra basis functions is not necessarily the source of the error, but rather it is

the unequal treatment found for each monomer: smaller intermolecular distances are able

to access more basis function from the other monomer than larger intermolecular distances

where the overlap integral becomes too small.33

The counterpoise correction (CP) method by Boys and Bernardi34 is a way to

removing BSSE from interacting systems. The usual method for calculating the interaction

energy, Δ𝐸𝑖𝑛𝑡(𝐴𝐵), of a dimer system, AB, composed of monomer A interacting with

monomer B without correcting for BSSE is shown in Equation 2.36, where the

parenthetical symbol is the system of interest, the superscripts are the basis set, and the

subscripts are the geometries. In this way, 𝐸𝐴𝐵𝐴𝐵(𝐴𝐵), is the energy of the dimer with the

dimer geometry in the basis of the dimer. The same procedure is carried out for the energies

of both monomers A and B (respectively the second and third terms in Equation 2.36).

Δ𝐸𝑖𝑛𝑡(𝐴𝐵) = 𝐸𝐴𝐵𝐴𝐵(𝐴𝐵) − 𝐸𝐴

𝐴(𝐴) − 𝐸𝐵𝐵(𝐵) (2.36)

The CP method corrects for BSSE by finding the energy by which monomer A (or

B) is stabilized artificially by the accessible basis functions of monomer B (or A).

Equations 2.37 and 2.38 represent the differences between monomer A in the basis of the

dimer and monomer A in the basis of monomer A (same for monomer B). The CP energy

is then negative as the energy of a monomer in the basis of the dimer must be lower than

that of the monomer in its own basis. Inserting Equations 2.37 and 2.38 into Equation 2.36

gives the expression for the CP correction energy, Equation 2.39.

𝐸𝐵𝑆𝑆𝐸(𝐴) = 𝐸𝐴𝐴𝐵(𝐴) − 𝐸𝐴

𝐴(𝐴) (2.37)

𝐸𝐵𝑆𝑆𝐸(𝐵) = 𝐸𝐵𝐴𝐵(𝐵) − 𝐸𝐵

𝐵(𝐵) (2.38)

Δ𝐸𝑖𝑛𝑡𝐶𝑃 (𝐴𝐵) = 𝐸𝐴𝐵

𝐴𝐵(𝐴𝐵) − 𝐸𝐴𝐴𝐵(𝐴) − 𝐸𝐵

𝐴𝐵(𝐵) (2.39)

In practice, evaluating monomer A in the basis of the dimer requires that the basis

functions of monomer B are placed on the atomic centers of monomer B while

simultaneously neglecting both the atomic charges and electrons found on monomer B. In

this case, the atoms and basis functions of monomer B are thusly referred to as ghost atoms


29

and ghost functions, respectively. The procedure is performed also for monomer B in the

basis of the dimer with the ghost atoms and ghost functions of monomer A.

2.5 Environmental Simulation Approaches

Since most chemistry occurs in solution, not the gas phase with isolated molecules,

solvent effects are critical. For example, for a given solute in solution, the solute can induce

a dipole moment in the surrounding solvent molecules and the bulk of the solvent is then

polarized in the region of each solute molecule generating an electric field, or reaction

field, at every solute molecule. The reaction field distorts the solute’s electronic wave

function from the gas phase giving an induced dipole moment in addition to the solute gas

phase dipole moment.

There are two methods for simulating solvent or environmental effects. Explicit

solvation is where the solvent molecules are directly modelled and a potential is applied

such as the Lennard-Jones 6-12 or TIP4P approaches. More common however is implicit

solvation utilizing a continuum model. In continuum models, the molecular structure of the

solvent is ignored and the solvent environment is constructed as a continuous dielectric

(something that cannot conduct electricity) surrounding a cavity centered on the solute

molecule. The continuous dielectric is described using the dielectric constant of the chosen

solvent.

In quantum mechanical calculations, a potential term, ��𝑖𝑛𝑡 , is included in the

electronic Hamiltonian, ��(0) , of the isolated molecular system to account for the

interactions of the solute and dielectric continuum of the solvent environment. For a given

calculation using the continuum solvation method, the electronic wave function and

electronic probability density of the solute change iteratively going from vacuum to

solution until self-consistency between the solute’s charge distribution and the solvent

reaction field. This method is referred to the self-consistent reaction field (SCRF) method.

The polarizable continuum model (PCM) is an application of the SCRF approach

where a sphere is centered on each nucleus of the solute with radius equal to 1.2 times the


30

van der Waals radius of the atom. A smoothing effect is generally applied to take care of

sharp crevices created by regions of the overlapping spheres. Apparent surface charges

(ASC) are distributed on the surface to create an electric potential equal to the electric

potential, 𝜙𝜎, created by the polarized dielectric constant. The ASC are approximated by

using point charges on the cavity surface and the cavity is divided into many small regions,

k, each with an apparent charge, 𝑄𝑘 (Equation 2.40), where 𝑟𝑘 is the point 𝑄𝑘 is located,

𝜙𝜎(𝑟) is the electric potential (Equation 2.41) due to the polarization of the dielectric, 𝐴𝑘

is the area of the region, ∇𝜙𝑖𝑛(𝑟𝑘) is the gradient of the electric potential in the cavity, and

𝑛𝑘 is a unit vector perpendicular to the cavity surface at 𝑟𝑘 pointing out of the cavity. The

electric potential in the cavity, 𝜙𝑖𝑛, is the sum of the charge distribution of the solute, 𝜙𝜇,𝑖𝑛,

and the polarized dielectric, 𝜙𝜎,𝑖𝑛: 𝜙𝑖𝑛 = 𝜙𝜇,𝑖𝑛 + 𝜙𝜎,𝑖𝑛. The electric potential, 𝜙𝑖𝑛, and 𝑄𝑘

are not known and are found iteratively until convergence of the charges, which are then

used to find the interacting potential term, ��𝑖𝑛𝑡, as in Equation 2.43. The interacting

potential, ��𝑖𝑛𝑡, is then added to the electronic Hamiltonian, ��(0), and the entire process is

repeated iteratively until convergence.

𝑄𝑘 = [𝜀𝑟 − 1

𝑟𝜋𝜀𝑟] 𝐴𝑘∇𝜙𝑖𝑛(𝑟𝑘) ∙ 𝑛𝑘 (2.40)

𝜙𝜎(𝑟) = ∑𝑄𝑘

|𝑟 − 𝑟𝑘|𝑘

(2.41)

��𝑖𝑛𝑡 = − ∑ 𝜙𝜎(𝑟𝑖)

𝑖

+ ∑ 𝑍𝛼𝜙𝜎(𝑟𝛼)

𝛼

(2.42)

A similar treatment to the PCM method is the conductor-like screening model

(COSMO).35 In this approach realistic solute-solvent shapes are still constructed and

surface charges are still used on the solute cavity. The difference comes when the charges

are calculated initially with a condition appropriate for the solvent that is not a dielectric,

but a conductor.


31

2.6 Multiconfigurational Self-Consistent Field

Multiconfigurational self-consistent field (MCSCF)36 is a method that corrects the

dissociation behavior of molecules and incorrect description of systems at increasing

intermolecular distances in comparison to the HF method. The HF method does not account

for static electron correlation effects as only a single electron configuration is considered

in a HF wave function and MCSCF is an improvement over HF since it utilizes several

electron configurations in the MCSCF wave function. The MCSCF wave function,

𝜓𝑀𝐶𝑆𝐶𝐹 , is a linear combination of different electron configurations represented by

configuration state functions (CSF), Φ𝑛, as in Equation 2.43, where n is the total number

of CSFs and 𝑏𝑛 are the expansion coefficients that are minimized in the variational integral.

The MCSCF method, in addition to optimizing the expansion coefficients, also optimizes

the form of the molecular orbitals (MOs) by varying the expansion coefficients, c, that

connect the MOs, 𝜙, to the basis functions, 𝜒, shown in Equation 2.44 for the ground state

of He in the minimal basis set. Uncovered terms inside of bars represent α spin orbitals and

the covered terms are β spin orbitals. The coefficients b and c are varied to minimize the

variational integral in an iterative process similar to a HF SCF procedure and while keeping

𝜙 orthonormal and 𝜓𝑀𝐶𝑆𝐶𝐹 normalized. Because both the MOs and the CSF expansion

coefficients are optimized in the process, only relatively few CSFs are necessary for

normally acceptable results.

|𝜓𝑀𝐶𝑆𝐶𝐹⟩ = ∑ 𝑏𝑛|Φ𝑛⟩

𝑛

(2.43)

|𝜓𝑀𝐶𝑆𝐶𝐹⟩ = 𝑏1| 𝜙1 𝜙1 | + 𝑏2| 𝜙2 𝜙2

|

= 𝑏1|(𝑐11𝜒1 + 𝑐21𝜒1)(𝑐11𝜒1 + 𝑐21𝜒2 )|

+ 𝑏2|(𝑐12𝜒1 + 𝑐22𝜒2)(𝑐12𝜒1 + 𝑐22𝜒2 )|

(2.44)

The CASSCF method is the most common approach to MCSCF calculations.37 The

orbitals are still expanded as a linear combination of CSFs, though in CASSCF, the CSFs

are divided into active and inactive orbitals. The inactive orbitals are kept either doubly

occupied (if in the occupied space) or unoccupied (if in the virtual space) in all CSFs. The

remaining electrons and orbitals constitute the active space. The CASSCF wave function


32

is then a linear combination of all CSFs, Φ𝑛, formed by distributing all the active electrons

in the active orbitals in all possible permutations while maintaining the desired state to be

calculated. An active space consisting of x number of electrons in y number of orbitals has

the notation, CAS(x,y). A MCSCF procedure is performed optimizing the coefficients, b

and c.

The selection of the active space for a CASSCF calculation is a non-trivial process

and of great consequence. One reliable method is found in Pulay’s unrestricted natural

orbital-complete active space (UNOCAS) approach.38 Here an unrestricted HF calculation

is performed in the chosen basis and the natural orbitals are analyzed. All natural orbitals

(NOs) whose occupation is between 1.98 e and 0.02 e are considered fractionally occupied

and will constitute the largest extent of the active space.

2.7 Multireference Configuration Interaction

Configuration interaction (CI) is a natural improvement of the single configuration

HF method and corrects for dissociation behavior and increasing intermolecular distances

The full CI wave function is a linear combination of CSFs, where the CSFs are first the

reference wave function (the HF wave function, 𝜓0) and subsequent CSFs represent

excitations of an electron in an occupied orbital (indices i, j, k, etc…) into a virtual orbital

(indices a, b, c, etc…). For full CI, the wave function, shown in Equation 2.45, contains all

possible excitations (single, double, triple excitations, etc…) of the reference wave

function while respecting the symmetry of the state of interest. The variational principle is

then used to minimize the CI energy with respect to the CSF expansion coefficients, 𝑏𝑛.

Full CI, by taking into account all excitations of the reference, produces a very large

number of CSFs and is computationally infeasible for all but the smallest systems (number

of atoms) and basis sets (number of basis functions). In light of this, truncations can be

made on the CI wave functions restricting the excitation level most commonly to either

single and/or double, CIS and CISD respectively, though truncated CI methods, including

MCSCF, are not size-extensive.


33

|𝜓𝐶𝐼⟩ = 𝑏0|𝜓0⟩ + ∑ 𝑏𝑖𝑎|Φ𝑖

𝑎⟩

𝑖,𝑎

+ ∑ 𝑏𝑖𝑗𝑎𝑏|Φ𝑖𝑗

𝑎𝑏⟩

𝑖>𝑗,𝑎>𝑏

+ ∑ 𝑏𝑖𝑗𝑘𝑎𝑏𝑐|Φ𝑖𝑗𝑘

𝑎𝑏𝑐⟩

𝑖>𝑗>𝑘,𝑎>𝑏>𝑐

+ ⋯

(2.45)

Multireference configuration interaction (MR-CI)36 is built from the traditional CI

and MCSCF methods. A MCSCF calculation is first performed to optimize the MOs and

CSFs of the MCSCF wave function. Reference CSFs are created for the MR-CI wave

function by exciting electrons from occupied orbitals into virtual ones. In practice, the

excitation level for MR-CI calculations is truncated to only singles and doubles (MR-

CISD). The variational principle is again used to minimize the energy by varying the CSF

expansion coefficients. When considering computational cost, the savings are great when

compared to full CI (in that the calculations are at all feasible at modest system sizes), but

the cost still scales linearly with the number of CSFs.

Since MR-CISD is a truncated CI method, it is not size-extensive. This is not a

large issue as the amount of correlation energy lost is not great, but it becomes increasingly

important with increasing system size. The Davidson correction36, 39-40 (designated “+Q”)

is an a posteriori treatment that recovers some of the correlation energy lost because the

wave function neglects higher order terms. The expression for the energy correction, 𝐸𝑄,

is shown in Equation 2.46 where 𝑏02 is the sum of the square of the coefficients of the

reference CSFs in the MR-CISD expansion. It can be seen that this is only a correction to

the energy and does not improve any other properties calculated at the MR-CISD level, but

the improvement to the energy gives favorable agreement with full CI and comes at

substantially reduced computational cost.

𝐸𝑄 =(1 − 𝑏0

2)(𝐸𝑀𝑅−𝐶𝐼𝑆𝐷 − 𝐸𝑀𝐶𝑆𝐶𝐹)

𝑏02 (2.46)

MR-CI calculations are ideal for calculations of systems where there is a high

amount of resonance and aromaticity or in the case of radical systems. The multireference

character of the wave function provided by multiple configurations provides the flexibility


34

to characterize and predict reactivity in these species so long as the computational cost, and

therefore the choice of reference space and basis set, is closely considered.

2.8 Multireference Averaged Quadratic Coupled-Cluster

The MCSCF and MR-CI methods account for static and dynamic electron

correlation, but the problem with size-extensivity in larger molecular systems remains even

with the Davidson correction to MR-CI. In an effort to remedy this, several other

multireference methods were developed such as the multireference linearized coupled-

cluster method (MR-LCCM), coupled electron pair approximation (MR-CEPA), and

averaged couple pair functional (MR-ACPF) that were shown to be approximately size-

extensive alternatives to MR-CI.41 The MR-AQCC method was developed to approach

nearly size-extensive results while simultaneously utilizing smaller reference spaces. The

following derivation follows what is presented in Ref.41

For a given reference space consisting of 𝑛𝑟 reference functions, Φ𝑟, the solution

to the Schrödinger equation is the zeroth-order reference, Ψ0 = ∑ 𝑐𝑟Φ𝑟𝑟 , and the correlated

wave function is found to be Equation 2.47. In this expression, 𝑇𝑟 is the cluster operator,

exp(𝑇𝑟) and 𝑃𝑟 = |Φ𝑟⟩⟨Φ𝑟| are found relative to each Φ𝑟 to define the cluster operator,

and Ψ𝑃 is the part of the wave function made up of the remaining orthogonal combinations

of reference functions in the set of all Φ𝑟 in the space of P. The singly (Ψ𝑖𝑎(𝑟)) and doubly

(Ψ𝑖𝑗𝑎𝑏) excited portion of the wave function is presented in Equation 2.48. With this, the

full correction, Ψ𝑐, to the zeroth-order wave function is Ψ𝑐 = Ψ𝑃 = Ψ𝑄.

Ψ = ∑ exp(𝑇𝑟) 𝑃𝑟Ψ0 + Ψ𝑃

𝑟

(2.47)

Ψ𝑄 = ∑ (∑ 𝑐𝑖𝑎Ψ𝑖

𝑎(𝑟)

𝑖𝑎

+ ∑ 𝑐𝑖𝑗𝑎𝑏Ψ𝑖𝑗

𝑎𝑏(𝑟)

𝑖𝑗𝑎𝑏

)

𝑟

(2.48)

The functional form of the expectation value for the energy of the single reference

case from the cluster operator, T, and the reference wave function, Φ0, is shown in Equation


35

2.49. In this equation, 𝐸0 is the reference energy and the c indicates that only connected

terms are included. Exclusion principle violating (EPV) terms (terms do not come from

pure interactions, but from groupings of interactions) are included as well, so the

expression is considered infinite. Equation 2.49 is for true coupled-cluster which is

computationally expensive even without considering multireference, so approximations

are made to neglect the virtual EPV terms in the disconnected diagrams of the cluster

expansion. This is accomplished by writing the cluster expansion in its disconnected form

and substituting the MR-CISD correlation energy, ∆E𝑀𝑅−𝐶𝐼𝑆𝐷, into the coupled-cluster

energy functional, Equation 2.50. Doing this illustrates that the MR-AQCC method,

despite its name, is not a true coupled-cluster method, but is in actuality an advanced form

of MR-CISD.

𝐹(Δ𝐸) = ⟨𝑒𝑇Φ0|�� − 𝐸0|𝑒𝑇Φ0⟩𝑐 (2.49)

𝐹(Δ𝐸) = ∆E𝑀𝑅−𝐶𝐼𝑆𝐷 + ∑ 𝑐𝑖𝑗𝑎𝑏2

𝑖𝑗𝑎𝑏

⟨Ψ𝑖𝑗𝑎𝑏|Ψ𝑖𝑗

𝑎𝑏⟩ ∑⟨Ψ0|�� − 𝐸0|Ψ𝑘𝑙𝑐𝑑⟩

𝑘𝑙𝑐𝑑

c𝑘𝑙𝑐𝑑 (2.50)

By also neglecting the virtual EPV terms and considering the rest in only an average

manner, the second term of Equation 2.50 becomes that shown in Equation 2.51, where 𝑛𝑒

is the number of electrons. Finally, the functional form of the MR-AQCC correlation

energy can be found in Equation 2.52 by combining Equations 2.50 and 2.51 and by writing

∆E𝑀𝑅−𝐶𝐼𝑆𝐷 in a more explicit fashion.

⟨Ψ𝑐|Ψ𝑐⟩(𝑛𝑒 − 2)(𝑛𝑒 − 3)

𝑛𝑒(𝑛𝑒 − 1)∆E𝑀𝑅−𝐶𝐼𝑆𝐷 (2.51)

𝐹𝑀𝑅−𝐴𝑄𝐶𝐶(∆E) =⟨Ψ0 + Ψ𝑐|�� − 𝐸0|Ψ0 + Ψ𝑐⟩

1 + [1 −(𝑛𝑒 − 2)(𝑛𝑒 − 3)

𝑛𝑒(𝑛𝑒 − 1)] ⟨Ψ𝑐|Ψ𝑐⟩

(2.52)

This approach is still only nearly size-extensive since only the quadratic unlinked

EPV terms are included. The primary advantage of the MR-AQCC method is that it allows

for high-level calculations that do not have the computational cost of full MR-CI. In

addition, the Davidson correction to the MR-CI method only adds size-extensivity to the

total energy, whereas the size-extensivity correction with the MR-AQCC method is applied


36

a priori, so meaningful electronic properties of the investigated system can be found such

as orbital occupations and spin-couplings among others.

2.9 Bibliography

(1) Levine, I. N., Quantum Chemistry. 7th ed.; Pearson Prentice Hall: New Jersey,

2014.

(2) Hohenberg, P.; Kohn, W., Inhomogeneous Electron Gas. Physical Review 1964,

136 (3B), B864-B871.

(3) Kohn, W.; Sham, L. J., Self-Consistent Equations Including Exchange and

Correlation Effects. Physical Review 1965, 140 (4A), A1133-A1138.

(4) Slater, J. C., The Theory of Complex Spectra. Physical Review 1929, 34 (10), 1293-

1322.

(5) Condon, E. U., The Theory of Complex Spectra. Physical Review 1930, 36 (7),

1121-1133.

(6) Vosko, S. H.; Wilk, L.; Nusair, M., Accurate spin-dependent electron liquid

correlation energies for local spin density calculations: a critical analysis. Canadian

Journal of Physics 1980, 58 (8), 1200-1211.

(7) Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation

Made Simple. Physical Review Letters 1996, 77 (18), 3865-3868.


Made Simple [Phys. Rev. Lett. 77, 3865 (1996)]. Physical Review Letters 1997, 78

(7), 1396-1396.

(9) Becke, A. D., Density-functional exchange-energy approximation with correct

asymptotic behavior. Physical Review A 1988, 38 (6), 3098-3100.

(10) Lee, C.; Yang, W.; Parr, R. G., Development of the Colle-Salvetti correlation-

energy formula into a functional of the electron density. Physical Review B 1988,

37 (2), 785-789.

(11) Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H., Results obtained with the correlation

energy density functionals of becke and Lee, Yang and Parr. Chemical Physics

Letters 1989, 157 (3), 200-206.

(12) Becke, A. D., Density‐functional thermochemistry. III. The role of exact exchange.

The Journal of Chemical Physics 1993, 98 (7), 5648-5652.

(13) Yanai, T.; Tew, D. P.; Handy, N. C., A new hybrid exchange–correlation functional

using the Coulomb-attenuating method (CAM-B3LYP). Chemical Physics Letters

2004, 393 (1), 51-57.

(14) Chai, J.-D.; Head-Gordon, M., Long-range corrected hybrid density functionals

with damped atom–atom dispersion corrections. Physical Chemistry Chemical

Physics 2008, 10 (44), 6615-6620.

(15) Grimme, S.; Antony, J.; Schwabe, T.; Mück-Lichtenfeld, C., Density functional

theory with dispersion corrections for supramolecular structures, aggregates, and

complexes of (bio)organic molecules. Organic & Biomolecular Chemistry 2007, 5

(5), 741-758.


37

(16) Møller, C.; Plesset, M. S., Note on an Approximation Treatment for Many-Electron

Systems. Physical Review 1934, 46 (7), 618-622.

(17) Frisch, M. J.; Head-Gordon, M.; Pople, J. A., A direct MP2 gradient method.

Chemical Physics Letters 1990, 166 (3), 275-280.

(18) Frisch, M. J.; Head-Gordon, M.; Pople, J. A., Semi-direct algorithms for the MP2

energy and gradient. Chemical Physics Letters 1990, 166 (3), 281-289.

(19) Head-Gordon, M.; Pople, J. A.; Frisch, M. J., MP2 energy evaluation by direct

methods. Chemical Physics Letters 1988, 153 (6), 503-506.

(20) Sæbø, S.; Almlöf, J., Avoiding the integral storage bottleneck in LCAO

calculations of electron correlation. Chemical Physics Letters 1989, 154 (1), 83-89.

(21) Head-Gordon, M.; Head-Gordon, T., Analytic MP2 frequencies without fifth-order

storage. Theory and application to bifurcated hydrogen bonds in the water hexamer.


(22) Schirmer, J., Beyond the random-phase approximation: A new approximation

scheme for the polarization propagator. Physical Review A 1982, 26 (5), 2395-2416.

(23) Dreuw, A.; Wormit, M., The algebraic diagrammatic construction scheme for the

polarization propagator for the calculation of excited states. Wiley Interdisciplinary

Reviews: Computational Molecular Science 2015, 5 (1), 82-95.

(24) Bayin, S. S., Mathematical Methods in Science and Engineering. 2006.

(25) Fetter, A. L.; Walecka, J. D., Quantum theory of many-particle systems. McGraw-

Hill: New York, 1971.

(26) Linderberg, J.; Öhrn, Y., Propagators in quantum chemistry. Academic Press: New

York, 1973.

(27) Danovich, D., Green's function methods for calculating ionization potentials,

electron affinities, and excitation energies. Wiley Interdisciplinary Reviews:

Computational Molecular Science 2011, 1 (3), 377-387.

(28) Von Niessen, W.; Schirmer, J.; Cederbaum, L. S., Computational methods for the

one-particle green's function. Computer Physics Reports 1984, 1 (2), 57-125.

(29) Cederbaum, L. S., Green's Functions and Propagators for Chemistry. John Wiley

& Sons: Chichester, 1998.

(30) Jansen, H. B.; Ros, P., Non-empirical molecular orbital calculations on the

protonation of carbon monoxide. Chemical Physics Letters 1969, 3 (3), 140-143.

(31) Liu, B.; McLean, A. D., Accurate calculation of the attractive interaction of two

ground state helium atoms. The Journal of Chemical Physics 1973, 59 (8), 4557-

4558.

(32) Sherrill, C. D., Lecture Notes on Counterpoise Correction and Basis Set

Superposition Error (http://vergil.chemistry.gatech.edu/notes/cp.pdf). 2010.

(33) van Duijneveldt, F. B.; van Duijneveldt-van de Rijdt, J. G. C. M.; van Lenthe, J.

H., State of the Art in Counterpoise Theory. Chemical Reviews 1994, 94 (7), 1873-

1885.

(34) Boys, S. F.; Bernardi, F., The calculation of small molecular interactions by the

differences of separate total energies. Some procedures with reduced errors.

Molecular Physics 1970, 19 (4), 553-566.

http://vergil.chemistry.gatech.edu/notes/cp.pdf


38

(35) Klamt, A.; Jonas, V.; Bürger, T.; Lohrenz, J. C. W., Refinement and

Parametrization of COSMO-RS. The Journal of Physical Chemistry A 1998, 102

(26), 5074-5085.

(36) Szalay, P. G.; Müller, T.; Gidofalvi, G.; Lischka, H.; Shepard, R.,

Multiconfiguration Self-Consistent Field and Multireference Configuration

Interaction Methods and Applications. Chemical Reviews 2012, 112 (1), 108-181.

(37) Roos, B. O., The Complete Active Space Self-Consistent Field Method and its

Applications in Electronic Structure Calculations. Advances in Chemical Physics

1987, 69, 399.

(38) Pulay, P.; Hamilton, T. P., UHF natural orbitals for defining and starting MC‐SCF

calculations. The Journal of Chemical Physics 1988, 88 (8), 4926-4933.

(39) Langhoff, S. R.; Davidson, E. R., Configuration interaction calculations on the

nitrogen molecule. International Journal of Quantum Chemistry 1974, 8 (1), 61-

72.

(40) Luken, W. L., Unlinked cluster corrections for configuration interaction

calculations. Chemical Physics Letters 1978, 58 (3), 421-424.

(41) Szalay, P. G.; Bartlett, R. J., Multi-reference averaged quadratic coupled-cluster

method: a size-extensive modification of multi-reference CI. Chemical Physics

Letters 1993, 214 (5), 481-488.


39

CHAPTER III

SINGLE AND DOUBLE CARBON VACANCIES IN PYRENE AS

FIRST MODELS FOR GRAPHENE DEFECTS: A SURVEY OF THE

CHEMICAL REACTIVITY TOWARD HYDROGEN

Authors: Reed Nieman,a Anita Das,a Adelia J. A. Aquino,a,b Rodrigo G. Amorim,c,d

Francisco B. C. Machado,d and Hans Lischkaa,b,e

Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University,

Lubbock, TX 79409-1061, USA, bSchool of Pharmaceutical Sciences and Technology,

Tianjin University, Tianjin 300072, PR China, cDepartamento de Física, Universidade

Federal Fluminense, Volta Redonda, 27213-145, Rio de Janeiro, Brazil, dDepartamento de

Química, Instituto Tecnológico de Aeronáutica, São José dos Campos 12228-900, São

Paulo, Brazil, eInstitute for Theoretical Chemistry, University of Vienna, Währingerstrasse

17, A-1090 Vienna, Austria

Published: January 2017 in Chemical Physics, DOI:10.1016/j.chemphys.2016.08.007

3.1 Abstract

Graphene is regarded as one of the most promising materials for nanoelectronics

applications. Defects play an important role in modulating its electronic properties and also

enhance its chemical reactivity. In this work the reactivity of single vacancies (SV) and

double vacancies (DV) in reaction with a hydrogen atom Hr is studied. Because of the

complicated open shell electronic structures of these defects due to dangling bonds,

multireference configuration interaction (MR-CI) methods are being used in combination

with a previously developed defect model based on pyrene. Comparison of the stability of

products derived from C-Hr bond formation with different carbon atoms of the different

polyaromatic hydrocarbons is made. In the single vacancy case the most stable structure is

the one where the incoming hydrogen is bound to the carbon atom carrying the dangling

bond. However, stable C-Hr bonded structures are also observed in the five-membered ring

of the single vacancy. In the double vacancy, most stable bonding of the reactant Hr atom

is found in the five-membered rings. In total, C-Hr bonds, corresponding to local energy

minimum structures, are formed with all carbon atoms in the different defect systems and

the pyrene itself. Reaction profiles for the four lowest electronic states show in the case of


40

a single vacancy a complex picture of curve crossings and avoided crossings which will

give rise to a complex nonadiabatic reaction dynamics involving several electronic states.

3.2 Introduction

Graphene has many extremely appealing electrical, physical, and thermal

properties which have made it an exciting material that is the subject of intensive research

ever since it was first isolated.1 Graphene is a particular allotrope of carbon in which a two-

dimensional hexagonal lattice is composed of benzene rings. Though it has been suggested

that graphene’s ability to conduct electricity could allow it to replace silicon in next-

generation organic semi-conductors, two problems remain, however, which detract from

this potential. The first issue is that there are no effective methods for obtaining amounts

of pristine graphene sheets of one atom thickness necessary for industrial applications. The

second problem is that graphene’s electronic properties come with a down side, though it

conducts electricity well, it is a semi-metal with a zero-energy band gap.

The chemical functionalization of graphene sheets for the purpose of tuning its band

gap has shown great promise.2-3 The hydrogen atom is a natural starting point when

beginning the investigation of the ability of graphene sheets to form sp3 bonds between an

incoming atom or molecule and the bonding carbon atom. Hydrogen chemisorption on a

graphene surface has been studied in detail by Casolo et al.4 using density functional theory

(DFT) and periodic boundary conditions. In fact, hydrogenation has been shown to be an

efficient tool for band gap tuning of graphene.5-8 Formation of the sp3 bond disrupts the

existing sp2 bond network by breaking the local π system. This has the effect of not only

increasing the reactivity of the carbon atoms neighboring the new C-H bond, but it also

creates a potential barrier to the conducting electrons opening a band gap, which is tunable

depending on the concentration of the chemisorbed species and properties of the sp3

network that has been formed. By reacting graphene sheets with atomic hydrogen,

hydrogenated regions of graphene are formed which are referred to as graphane islands.

The presence and extent of hydrogenation has been previously confirmed through the use

of scanning tunneling microscopy (STM) and angle-resolved transmission electron


41

microscopy (TEM).6 These images also show details pertaining to the arrangement of

graphane islands with respect to each other. PAHs and their reaction with hydrogen are not

only of interest as molecular models for graphene but also as possible catalysts for H2

formation in interstellar space.9 Hydrogenation studies of pyrene have been performed10 to

investigate energetic stabilities of hydrogen attached to different carbon atoms and the

existence of energy barriers.

The arrangement of bonded hydrogen atoms in graphane islands and the proximity

of islands to one another are also contributing factors to the overall electronic character of

the hydrogenated graphene sheet due the dependence of the created band gap on such. A

small group of low hydrogen-concentration graphane islands have shown clearly the

beginning of the formation of a band gap.6 On a larger scale, graphane islands created in

strips parallel to each other, called graphene roads, have been shown to create environments

similar to graphene nanoribbons.11 Hydrogen atoms adsorbed on graphene have also been

reported to induce magnetic moments.12 The spin-polarized state is shown to be essentially

located on the carbon sublattice opposite to the one belonging to the adsorbed hydrogen

atom. This finding goes very well along with previous success in experimentally

engineering band gaps and has been shown by the Haddon group using substituted benzene

radicals, nitrophenylene, interacting with graphene sheets.3 As a consequence of this

functionalization, residual spins are available to couple and allow the possibility of

ferromagnetism at room temperatures.

Vacancies defects, absences of carbon atoms in the hexagonal structure, have

shown to be an interesting area with regards to the investigation of the chemical reactivity

of graphene. When one or multiple carbons are removed, the created dangling bonds

represent areas of greater chemical reactivity. The presence of these dangling bonds then

implies high polyradical character with many closely spaced locally excited electronic

states which are difficult to describe computationally. This situation then strongly advises

the use of flexible multireference theory13 in which quasi-degenerate orbitals are treated at

equal footing and the correct construction of wavefunctions of well-defined symmetry and

spin properties is possible.


42

The electronic structures of both SV and DV pyrene (Scheme 3.1) which has been

used as a prototype for a graphene defect structure, have recently been described using

mulireference configuration interaction MR-CI14-15 and DFT.16 The single vacancy

investigation of pyrene using MR-CI calculations14 showed several closely spaced

electronic states of triplet and singlet multiplicity (Scheme 3.1, Pyrene-1C). The electronic

ground state of the double vacancy defect15 (Scheme 3.1, pyrene-2C) possesses 50% closed

shell character which indicates a significantly different character as compared to the SV

case. The pristine pyrene is depicted in Scheme 3.1, top, and as is, of course, to be expected

the chemically most stable structure of the three candidates discussed.

The purpose of this work is to explore the chemical reactivity of pyrene and the two

defect structures by studying their interaction with hydrogen radical (Hr). The role of Hr is

two-fold. It is used in the spirit of a test particle by moving it in a grid search across the

different pyrene structures as the arrows in Scheme 3.1 indicate. In this way a stability

surface with respect to C-Hr bond formation is constructed. In the second step

optimizations of C-Hr structures are performed and C-Hr dissociation curves are

constructed to the lowest dissociation products. The previous MR-CI calculations on the

SV and DV pyrene defect structures had shown the existence of several low-lying excited

states. We wanted to reproduce this asymptotic limit at complete C-H dissociation and,

thus, computed in all respective cases potential energy scans and C-H dissociation curves

for the four lowest electronic states. In addition to the determination of local energy minima

for different C-H bonding situations, these data should provide an overview of the most

important aspects for the approach and bonding of a hydrogen atom to a pristine or defect

graphene sheet surface.


43

Scheme 3.1 Atomic numbering scheme for pristine pyrene, pyrene-1C (single vacancy)

and pyrene-2C (double vacancy), respectively. Hr is the hydrogen atom moving across the

different pyrene structures as shown by arrows.

Pyrene

Pyrene-1C

Pyrene-2C


44

3.3 Computational Details

The geometries of each pyrene structure (Scheme 1) were optimized using complete active

space (CAS) self-consistent field (CASSCF) theory.17-18 Based on the experience with

previous pyrene defect calculations14-15 a CAS(5,5) was used comprised of five electrons

in five orbitals where, in Cs symmetry, three orbitals belonged to the A’ irreducible

representation and two orbitals belonged to the A” irreducible representation. The

CAS(4,4) is chosen for the description of the isolated pyrene defect and one active orbital

and electron is added to account for the hydrogen atom. The active orbitals used in the CAS

for pyrene+Hr were 31a’, 32a’, 33a’, 22a”, 23a” (Figure A.1 of Appendix A). For pyrene-

1C+Hr active orbitals were 29a’, 30a’, 31a’, 21a”, 22a” (Figure A.2) and for pyrene-2C+Hr

active orbitals were 27a’, 28a’, 29a’, 20a”, 21a” (Figure A.3). The 6-31G* basis set19 was

used. The state-averaging included four states (SA4). In case of Cs symmetry the two

lowest 2A’ and two lowest 2A” states were chosen. Single point calculations were carried

out using SA4-CASSCF(5,5)/6-31G* and MR-CI with all singles and doubles (MR-CISD)

again using the 6-31G* basis set. As reference space, the same CAS(5,5) as used in the

CASSCF(5,5) calculations was selected. Single and double excitations were constructed

from the doubly-occupied and active orbitals applying the interactive space restriction.20

The 1s orbitals of all carbon atoms were kept frozen. Size-extensivity corrections were

included by means of the extended Davidson correction13, 21-22 denoted as +Q (MR-

CISD+Q). 2

0C is the sum of square coefficients of the reference configurations in the MR-

CISD expansion and EREF denotes the reference energy.

12

12

0

2

0

C

EECE REFCI

Q (3.1)

The potential energy surface (PES) scan was carried out using the SA4-

CASSCF(5,5)/6-31G* approach for pyrene-1C+Hr and pyrene-2C+Hr and the

CASSCF(5,5)/6-31G* approach for pyrene+Hr. For that purpose, the previously relaxed

pyrene defect (pyrene-1C and -2C) structures computed earlier14-15 and an optimized

pyrene structure using the DFT and the hybrid Becke, 3-parameter, Lee-Yang-Parr

(B3LYP) functional23 together with the 6-31G**19 basis set were used to move Hr around


45

in a grid with an increment of 0.3 Å and at a distance of 1.0 Å from the plane structures.

Single-point calculations were performed for each point on the grid.

Using the local minima in the PES´s scans as starting structures, geometry

optimizations were carried out. To simulate the constraints present in a larger graphene

system, the coordinates of the carbons connected to hydrogen atoms at the periphery of the

pyrene and pyrene defect systems were always frozen in all geometry optimizations since

they would be connected to neighboring carbon atoms in a graphene sheet. All other atoms

were allowed to relax. Geometry optimizations using SA4-CASSCF(5,5)/6-31G* for the

defect cases plus Hr and CASSCF(5,5) calculations for the pristine pyrene+Hr were

performed by placing Hr at an initial distance of 1.1 Å over each unique carbon in each of

the pyrene system.

The CASSCF, MR-CISD and MR-CISD+Q calculations were carried out using

COLUMBUS;24-28 the geometry optimizations used the program system DL-FIND29

interfaced to COLUMBUS. The DFT calculations were performed with the Turbomole

suite of programs.30

3.4 Results and Discussion

3.4.1 Potential Energy Scans

The two-dimensional PES scans of the single vacancy pyrene-1C+Hr for the ground

state and the first three excited states are shown in Figure 3.1. For that purpose, the Hr atom

has been moved systematically in a grid 1.0 Å above the fixed, planar structure of pyrene-

1C. These scans detail the ability of the test particle, Hr, to bond in a particular region of

pyrene-1C. For the ground state, deep local minima can be seen at the carbons in the five-

membered ring, C1, C2, C3, C13 and C14 (see Scheme 1 for numbering of atoms). The

lowest minimum at C15 persists through all four states; for the excited states the local

minimum at the five-membered ring is considerably higher in energy. Shallower minima

are present for C5, C7, C9, and C11. Generally, the minima for carbons in the six-


46

membered ring, aside from C15, give minima much shallower than for the remaining

carbon atoms.

Figure 3.1 PES scans in terms of the location of Hr for the four lowest states of pyrene-

1C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane

1 2Aa

2 2Ab

3 2Ac

4 2Ad


47

parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energies: a -

574.222889 Hartree, b -574.198174 Hartree, c -574.149530 Hartree, d -574.117820 Hartree.

The energy (color scale) is given in eV, relative to the ground state lowest minimum.

3-D pictures of the PES scan of the double vacancy defect pyrene-2C+Hr for the

electronic ground- and first excited states are displayed in Figure 3.2 and Figure 3.3,

respectively. The plots for the second and third excited state are shown in Figure A.4. For

the ground state deep potential wells are seen at C1, C2, C14 and C7, C8, C9, these being

the carbons at the outer sections of the five-membered rings, with the strongest minima

present at C1 and C8. Additional shallower minima are seen at each of the remaining

carbons. The ground and first excitation are seen to be on average energetically closer as

compared to the single vacancy defect just discussed above. On the other hand, the scans

for the second and third vertical excitations (Figure A.4) are closely spaced and quite

higher, by ~ 2-2.5 eV, than the lowest minimum.

Figure 3.2 PES scan in terms of the location of Hr for the lowest state of pyrene-2C+ Hr

using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel

and 1 Å above the plane defined by the frozen carbon atoms. Lowest energy = -536.385046

Hartree. The energy (color scale) is given in eV, relative to the ground state lowest

minimum.

The ground state PES for pyrene+Hr is shown in Figure 3.4. Pronounced minima

can be seen at C4, C5, C11, and C12. While for each of the remaining carbon atoms in the

pyrene system local minima can be found, C2, C7, C9, and C14 show to be especially

stabilized among them.


48

Figure 3.3 PES scan in terms of the location of Hr for the 22A state of pyrene-2C+ Hr using

the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1

Å above the plane defined by the frozen carbon atoms. Lowest energy = -536.347629

Hartree. The energy (color scale) is given in eV, relative to the ground state lowest

minimum at -536.385046 Hartree.

Figure 3.4 PES scan of the lowest state of pyrene+Hr using the CASSCF(5,5)/6-31G*

approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by


49

the frozen carbon atoms. Lowest energy = -612.270253 Hartree. The energy (color scale)

is given in eV, relative to the ground state lowest minimum.

3.4.2 Geometry Optimizations

The PES scans discussed in the previous section give a good global overview of the

chemical reactivity of the different pyrene defects (including the pristine case) concerning

the strength of different C-Hr bonds as well as to the location of the centers of most stable

bond formation. In the following sections more detailed information on the properties of

individual structures will be discussed.

In the search for local minima structures containing C-Hr bonds, geometry

optimizations were performed as described in the Computational Details by placing Hr on

top of each carbon atom and performing a geometry optimization in each case only freezing

selected carbon atoms as discussed above to simulate geometrical restrictions due to

neighboring carbon atoms in a graphene sheet. Each carbon tested was found to give rise

to a local minimum for all structures investigated.

Starting with the single vacancy pyrene-1C+Hr case, the nine ground state

structures found are described in Figure 3.5. The global minimum is given by bonding Hr

to C15 with a C-Hr bond distance of 1.060 Å. This is the smallest C-H bond distance

determined for the nine structures and is much smaller than the C-Hr bond distance of the

other minima whose bond distance is around 1.095 Å. When Hr is bonded to C1, C2, C4,

and C5, planarity of the ring system is mostly maintained. However, the existing hydrogen

is moved out-of-plane away from the incoming Hr due to the formation of a tetrahedral sp3

hybrid at this carbon atom. This behavior repeated in all analogous cases. When Hr is

bonded to C3 and C6, the carbon atoms reacted by rising out of plane. When Hr was bonded

to C7 and C8, the bonding carbon could not move since it was frozen, but C6, C10, and

C15 moved out of plane. Finally, when Hr bonded to C15 in the global minimum structure,

again C6, C10, and C15 moved out of plane.


50

Figure 3.5 Plots of the different minimum structures (labeled according to the bonding

carbon atom) on the ground state energy surface of pyrene-1C+Hr using the SA4-

CASSCF(5,5)/6-31G* approach. Bond distances are given in Å.

In the double vacancy pyrene-2C+Hr case, the four ground state structures are

characterized in Figure 3.6. The global minimum was found when Hr is bonded to C2 in

the five-membered ring with a C-Hr distance of 1.093 Å and gave the smallest C-Hr

distance of all the minimum structures. The largest C-Hr bond distance of 1.121 Å to C1 is

C1 C2

C3 C4

C5 C6

C7 C8

C15


51

notable here as it is the largest one of all three pyrene structures. For C3, as before, since

it was not frozen and only bonded to other carbons, the carbon atom rose out of the plane

of the frozen carbon atoms when rehybridizing from sp2 to sp3.


carbon atom) of the ground state energy surface of pyrene-2C+Hr using the SA4-

CASSCF(5,5)/6-31G* approach. Bond distances are given in Å.

Finally, for pyrene+Hr, the characterization of the five ground states is presented in

Figure 3.7. The global minimum was found when Hr was bonded to C4 with a C-Hr bond

distance of 1.097 Å. This C-Hr distance was the smallest one of the four minima found,

though it can be noted that the C-Hr bond distances for this structure do not vary greatly as

the largest C-Hr bond distance is only slightly larger with 1.103 Å given for C1-Hr. In cases

where the bonding carbon was already bonded to hydrogen (carbons 1, 2, 11), Hr formed a

C1

C2

C3

C4


52

C-H bond by pushing the other hydrogen away, as reported also in previous cases. When

Hr bonded to a hydrogen-free carbon (carbons 3 and 16), the bonding carbon responded by

rising out of the plane of the frozen carbon atoms by about 0.5 Å. A similar observation

was made by Rasmussen et al.10 by means of DFT calculations where binding of a hydrogen

atom on the interior positions (C3 and C16) of the pyrene led to a significant puckering of

these carbon atoms (~0.4 Å) out of the molecular plane. This behavior on hydrogenation

of graphene is generally feasible in the free graphene sheet4 but is also true for the pyrene

defect plus Hr cases as explained above

3.4.3 Energetic Stabilities

Table 3.1 shows the vertical excitation energies for pyrene-1C+Hr the lowest two

A’ and A” states for structures having Cs symmetry, i.e. Hr bonded to C15 (global

minimum), C1 and C8 using the SA4-CASSCF, MR-CISD and MR-CISD+Q approaches.

The 12A’ state is the ground state. At MR-CISD+Q level, in the C15 structure the first

excited state is only ~0.8 eV above the ground state, followed by states about 2.4 eV and

3.6 eV above the ground state. The ground state C1 structure is about 1.4 eV higher at MR-

CISD+Q level than the respective one for C15; the C8 structure is located significantly

higher (4.2 eV) above the C15 ground state. The order of the states computed at MR-CISD

and MR-CISD+Q levels is reproduced well by the SA4-CASSCF(5,5) calculations even

though there are non-negligible numerical differences up to 0.8 eV.


53


carbon atom) on the ground state energy surface of pyrene+Hr using the CASSCF(5,5)/6-

31G* approach. Bond distances are given in Å.

C1

C2

C3

C4

C16


54

Table 3.1 Vertical energy differences (eV) of the lowest two A’ and A” states relative to

the C15 ground state computed for the C1, C8 and C15 pyrene-1C+Hr structures using the

SA4-CASSCF(5,5), MR-CISD and MR-CISD+Q approaches and the 6-31G* basis set.

CASSCF MR-CISD MR-CISD+Q

State ΔErel ΔErel ΔErel

C15

1 2A’ 0.000a,d 0.000b,d 0.000c,d

1 2A” 1.154 1.023 0.834

2 2A’ 2.571 2.504 2.375

2 2A” 4.100 3.904 3.594

C1

1 2A’ 0.636d 0.960d 1.393d

2 2A’ 3.775 4.073 4.444

1 2A” 4.828 4.973 5.074

2 2A” 5.854 6.069 6.186

C8

1 2A’ 3.542d 3.930d 4.246d

2 2A’ 3.900 4.214 4.456

1 2A” 4.216 4.604 4.947

2 2A” 4.643 4.994 5.187

a Total energy= -574.277274 Hartree, b Total energy= -575.610208 Hartree, cTotal

Energy= -576.244081 Hartree, d The SA4-CASSCF(5,5) optimized geometry was used.

Figure 3.8 Location and relative stability (eV) of the minima on the ground state energy

surface of pyrene-1C+Hr using the MR-CISD+Q/6-31G* method. Blue: 2A ground state,

red: lowest vertical excitation. Note: the rigid pyrene-1C structure shown does not reflect

the actual structure of the optimized minima.


55

In Figure 3.8 (see also Table A.1) are the relative energies of each carbon minimum

in pyrene-1C+Hr given with respect to the global minimum structure at C15 computed at

MR-CISD+Q level. The first vertical excitations are given in the figure as well. C15 marks

the global minimum followed by C2, C3, and C1, respectively, in the five-membered ring.

Except for the structure C15, all first excited states are located in energy higher by ~3.5 eV

or more. This shows that in case of the reaction of Hr with a single vacancy graphene defect

at thermalized kinetic energies nonadiabatic interactions between different electronic states

will be relevant mostly for the C15 approach, which, is however, energetically the most

relevant one.

The energy profiles for the reaction of Hr with the pyrene-1C defect are shown in

Figure 3.9 for the case of bond formation with C15. The reaction curves have been created

by fixing the C15-Hr distance and optimizing the remaining part of the pyrene defect

structure for the ground state at SA4-CASSCF(5,5)/6-31G* level. Selected carbon atoms

as discussed above were frozen to simulate also in this case geometrical restrictions due to

neighboring carbon atoms in a graphene sheet. Vertical excitations have been used to

compute the higher electronic states. These calculations have been performed in C1

symmetry. Nevertheless, the Cs symmetry analysis of Table 3.1 for the ground state

minimum, as indicated in Figure 3.9, can be used for the analysis of states. For the 12A’

ground state the figure shows a smooth dissociation without change of the character of the

wave function. The second state is 12A” at the energy minimum. A crossing between this

state and the 22A’ state is observed at a C15-Hr distance of about 2 Å. For large distances

the two 2A’states become practically degenerate. The 22A’ and 22A” states show a spike at

1.5 Å indicating the crossing with higher states of different character. These states are

repulsive and are responsible for the correct asymptotic behavior of the dissociation. The

energy minima of the first three states are located below the lowest dissociation limit so

that on reaction with an incoming hydrogen atom several electronic states could be

involved in the relaxation process to the ground state.


56

Figure 3.9 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is

bonded to C15 using the MR-CISD/6-31G* approach. Minimum energy = -574.277274

Hartree

Figure 3.10 shows the dissociation/bonding process of Hr from/to C1. Evident is

the lack of a potential barrier in the ground state curve, though potential energy barriers

and avoided crossings exist in the excited state curves. However, differently to the case of

the C15-Hr bond dissociation, the excited states are not expected to play a major role for

the relaxation process to the energy minimum on formation of the C1-Hr bond due to their

high energies around the ground state minimum.

Figure 3.10 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is

bonded to C1 using the MR-CISD/6-31G* approach. Minimum energy = -574.277274

Hartree


57

The Hr dissociation curves computed at SA4-CASSCF(5,5) level for bonding to C2

to C8, respectively, are collected in the SI. For C2 and C3, Figure A.5 and Figure A.6, there

is no potential barrier for the incoming Hr ground state. In the remaining cases (Figure A.7

– Figure A.11), however, there are potential barriers present for the bond formation to C4,

C5, C6, C7 and C8. Estimated barrier heights with respect to the dissociation limit are

collected in Table 3.2. They show a variation between 0.4 eV to 1.4 eV indicating that the

approach from that side of pyrene-1C is not so favorable.

The dissociation energy of the C15 global minimum pyrene-1C + Hr structure into

the lowest state of pyrene-1C + Hr is given in Table 3.2. At MR-CISD+Q level a value of

4.61 eV is found. Comparison with the MR-CISD (4.13 eV) and SA4-CASSCF(5,5) (3.66

eV) values shows the importance of including size-extensivity and dynamic electron

correlation contributions to the dissociation energy. Dissociation of Hr for the other

structures can be obtained from Figure 3.8 by subtracting the relative stabilities form the

just-mentioned dissociation limit of the C15 structure. The strongest C-Hr bonds following

the one to C15 can be found for C-H bonding to carbons in the five-membered ring. Some

of the other structures, though being local minima, are close in energy to the C-Hr bond

dissociation limit.

Table 3.2 Dissociation energies Ediss (eV) for the C15 global minimum pyrene-1C+Hr

structure and estimated energy barriers Ebarr(est.) (eV) for the approach of Hr on the

ground state surface toward the different carbon atoms of pyrene-1C using different

theoretical methods and the 6-31G* basis set.

Ediss

SA4-CASSCF(5,5) MR-CISD MR-CISD+Q

3.660 4.128 4.605

Ebarr(est.)a (SA4-CASSCF(5,5))

C1, C2, C3, C15: no barrier

C4 C5 C6

0.914 0.426 1.403

C7 C8

1.138 1.051

The relative stabilities in the electronic ground state and the lowest vertical

excitations of pyrene-2C+Hr are presented in Figure 3.11 for the MR-CISD+Q/6-31G*

approach. SA4-CASSCF(5,5) and MR-CISD results are collected in Table A.2. Numerical


58

agreement between the different methods is within 0.1 – 0. 2 eV. The global minimum was

found to be for Hr bonded to C2 in the five-membered ring. The minima are ordered from

most stable to least stable as C2<C1<C3<C4. The variation in the relative energies is much

smaller than observed for the pyrene-1C+Hr case (Figure 3.9).


surface of pyrene-2C+Hr defect using the MR-CISD+Q/6-31G* method. Blue: 2A ground

state, red: lowest vertical excitation. Note: the rigid pyrene-2C structure shown does not

reflect the actual structure of the optimized minima.

The dissociation energy for the process pyrene-2C+Hr is given in Table 3.3 for the

global minimum structure with Hr bound to C2 for the CASSCF, MR-CISD and MR-

CISD+Q methods. Compared to the pyrene-1C+Hr case (Table 3.2), the C-Hr bond strength

is significantly reduced. However, it should be noted that all local ground state minima

(Figure 3.11) are below the dissociation limit of 2.269 eV.

Table 3.3 Dissociation energies Ediss (eV) for the double vacancy pyrene-2C+Hr and

pristine pyrene + Hr cases using different methods and the 6-31G* basis set.

SA4-CASSCF(5,5) MR-CISD MR-CISD+Q

pyrene-2C+Hr

Ediss(C2) 2.224 2.257 2.269

pyrene+Hra

Ediss(C2) 0.833 0.902 0.943

For the reaction of pyrene with Hr, pyrene+Hr, only the electronic ground state has

been investigated. Relative stabilities of the different C-Hr bonded local minima are shown

in Figure 3.12. CASSCF(5,5), MR-CISD and MR-CISD+Q results are collected in Table


59

A.3. A similar good agreement between different methods is obtained as for the pyrene-

2C+Hr case discussed above in Table A.2. The minima are ordered energetically as

C2<C4<C3<C1<C16. Though the structure where Hr bound to C2 is most stable than C4,

the difference is negligibly small, which is in good agreement with literature.10 The

stabilization energy of the various minima was found to be quite close to each other, with

all five minima being found within 0.7 eV of the global minimum. With the dissociation

energy of 0.943 eV computed at MR-CISD+Q/6-31G* level (Table 3.3) all of the minima

are located below the dissociation limit, even though several cases are very close to it. A

similar value has been found by Casolo et al.4 in their hydrogenation studies of graphene

performed at DFT level.


surface of pyrene+Hr using the MR-CISD+Q/6-31G* method. Note: the rigid pyrene

structure shown does not reflect the actual structure of the optimized minima.

3.5 Conclusions

The chemical reactivity of a single and double vacancy in graphene and of a pristine

surface toward C-H bond formation with a reactive hydrogen atom Hr has been investigated

using a pyrene model with peripheral carbon atoms connected to H atoms frozen.

Multireference methods (MR-CISD and MR-CISD+Q) have been used to describe (i) the

dissociation processes properly and (ii) the manifold of quasi-degenerate states of the

isolated defect.14-15 In a first step energy grids have been computed by moving an atom Hr

across the rigid surface of the pyrene defects structures in order to obtain an overview of

the stabilization energies in the different regions in space. The results show for the single

vacancy a preference for bonding to C15 which possessing a dangling bond. The relative


60

stabilization in this position is also observed in the lowest excited states. However, the

carbon atoms in the five-membered ring show also strongly attractive positions in the

ground state. The double vacancy displays similar stability preferences in the five

membered rings; also the pristine pyrene model indicates attractive interactions primarily

around the peripheral carbon atoms.

Geometry optimizations performed for the ground state at CASSCF level followed

by single-point MR-CISD and MR-CISD+Q calculations refine this picture. Importantly,

separate local energy minima for C-Hr bond formation have been determined for each of

the carbon atoms in all defect structures and in pyrene itself. In the single vacancy case,

C15 remains the position with greatest stability followed by C2 towards the top of the five-

membered ring. In case of the double vacancy position C2 in the five-membered ring gives

rise to the strongest C-Hr bond and in pyrene itself these are positions C2 and C4.

C-Hr dissociation curves have been computed for the four lowest electronic states

for the single vacancy model starting from all minima found. Barrierless ground state

processes have been found for C15 and the C-Hr bonds involving the five-membered ring.

An incoming hydrogen atom though approaching the pyrene-C defect toward the other

carbon atoms would encounter barriers of about 0.5 – 1 eV, depending on the incoming

approach angle. Curve crossings and avoided crossings are found especially for Hr

approaches to the most stable carbon positions which lead to the expectation of a complex

nonadiabatic reaction dynamics. In summary, a remarkably complex picture of the bonding

options of a hydrogen atom reacting with a graphene vacancy structures is observed both

in terms of different bonding positions within the defect but also in terms of the reaction

dynamics involving several excited states. It is expected that these properties found for the

pyrene model are fairly representative for a localized graphene defect. Preliminary density

functional studies confirm this view. More detailed investigations are in preparation.


61

3.6 Acknowledgments

This material is based upon work supported by the National Science Foundation under

Project No. CHE-1213263 and by the Austrian Science Fund (SFB F41, ViCoM). We are

grateful for computer time at the Vienna Scientific Cluster (VSC), project 70376. FBCM

wishes to thank Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under

Projects Process Nos. 2014/24155-6 and 2015/500118-9, to Coordenação de

Aperfeiçoamento de Pessoal de Nível Superior (CAPES) under Project CAPES/ITA

005/2014 and to Conselho Nacional de Desenvolvimento Científico and Tecnológico

(CNPq) for the research fellowship under Process No. 304914/2013-4. We also want to

thank the FAPESP/Texas Tech University SPRINT program (project no. 2015/50018-9)

for travel support.

3.7 Bibliography

(1) Geim, A. K.; Novoselov, K. S., The rise of graphene. Nature Materials 2007, 6,

183.




1336-1337.




76.

(4) Casolo, S.; Lovvik, O. M.; Martinazzo, R.; Tantardini, G. F., Understanding

adsorption of hydrogen atoms on graphene. Journal of Chemical Physics 2009, 130

(5), 054704.

(5) Balog, R.; Jørgensen, B.; Wells, J.; Lægsgaard, E.; Hofmann, P.; Besenbacher, F.;

Hornekær, L., Atomic Hydrogen Adsorbate Structures on Graphene. Journal of the


(6) Balog, R.; Jørgensen, B.; Nilsson, L.; Andersen, M.; Rienks, E.; Bianchi, M.;

Fanetti, M.; Laegsgaard, E.; Baraldi, A.; Lizzit, S.; Sljivancanin, Z.; Besenbacher,

F.; Hammer, B.; Pedersen, T. G.; Hofmann, P.; Hornekaer, L., Bandgap opening in

graphene induced by patterned hydrogen adsorption. Nature materials 2010, 9 (4),

315-319.


62

(7) Guisinger, N. P.; Rutter, G. M.; Crain, J. N.; First, P. N.; Stroscio, J. A., Exposure

of Epitaxial Graphene on SiC(0001) to Atomic Hydrogen. Nano Letters 2009, 9

(4), 1462-1466.

(8) Elias, D. C.; Nair, R. R.; Mohiuddin, T. M. G.; Morozov, S. V.; Blake, P.; Halsall,

M. P.; Ferrari, A. C.; Boukhvalov, D. W.; Katsnelson, M. I.; Geim, A. K.;

Novoselov, K. S., Control of Graphene's Properties by Reversible Hydrogenation:

Evidence for Graphane. Science 2009, 323 (5914), 610.

(9) Bauschlicher, C. W., The reaction of polycyclic aromatic hydrocarbon cations with

hydrogen atoms: The astrophysical implications. Astrophys J 1998, 509 (2), L125-

L127.

(10) Rasmussen, J. A.; Henkelman, G.; Hammer, B., Pyrene: Hydrogenation, hydrogen

evolution, and pi-band model. Journal of Chemical Physics 2011, 134 (16),

164703.

(11) Chernozatonskii, L. A.; Kvashnin, D. G.; Sorokin, P. B.; Kvashnin, A. G.; Brüning,

J. W., Strong Influence of Graphane Island Configurations on the Electronic

Properties of a Mixed Graphene/Graphane Superlattice. The Journal of Physical

Chemistry C 2012, 116 (37), 20035-20039.

(12) Gonzalez-Herrero, H.; Gomez-Rodriguez, J. M.; Mallet, P.; Moaied, M.; Palacios,

J. J.; Salgado, C.; Ugeda, M. M.; Veuillen, J.-Y.; Yndurain, F.; Brihuega, I.,

Atomic-scale control of graphene magnetism by using hydrogen atoms. Science

2016, 352, 437-441.







15 (15), 3334-3341.




(16) Haldar, S.; Amorim, R. G.; Sanyal, B.; Scheicher, R. H.; Rocha, A. R., Energetic

stability, STM fingerprints and electronic transport properties of defects in

graphene and silicene. Rsc Adv 2016, 6 (8), 6702-6708.

(17) Ruedenberg, K.; Cheung, L. M.; Elbert, S. T., MCSCF optimization through

combined use of natural orbitals and the brillouin-levy-berthier theorem.

International Journal of Quantum Chemistry 1979, 16, 1069-1101.

(18) Roos, B.; Taylor, P. R.; Siegbahn, P. E. M., A complete active space SCF method

(CASSCF) using a density matrix formulated super-CI approach. Chemical Physics

1980, 48 (2), 157-173.

(19) Hehre, W. J.; Ditchfield, R.; Pople, J. A., Self—Consistent Molecular Orbital

Methods. XII. Further Extensions of Gaussian—Type Basis Sets for Use in

Molecular Orbital Studies of Organic Molecules. The Journal of Chemical Physics

1972, 56 (5), 2257-2261.


63

(20) Bunge, A., Electronic Wavefunctions for Atoms. III. Partition of Degenerate

Spaces and Ground State of C. The Journal of Chemical Physics 1970, 53 (1), 20-

28.

(21) Langhoff, S. R.; Davidson, E. R., Configuration interaction calculations on the

nitrogen molecule. International Journal of Quantum Chemistry 1974, 8 (1), 61-

72.

(22) Luken, W. L., Unlinked cluster corrections for configuration interaction

calculations. Chemical Physics Letters 1978, 58 (3), 421-424.



(24) Lischka, H.; Shepard, R.; Pitzer, R. M.; Shavitt, I.; Dallos, M.; Müller, T.; Szalay,

P. t. G.; Seth, M.; Kedziora, G. S.; Yabushita, S.; Zhang, Z., High-level

multireference methods in the quantum-chemistry program system COLUMBUS:

Analytic MR-CISD and MR-AQCC gradients and MR-AQCC-LRT for excited

states, GUGA spin–orbit CI and parallel CI density. Physical Chemistry Chemical

Physics 2001, 3, 664-673.

(25) Lischka, H.; Müller, T.; Szalay, P. G.; Shavitt, I.; Pitzer, R. M.; Shepard, R.,

Columbus-a program system for advanced multireference theory calculations.

Wiley Interdisciplinary Reviews-Computational Molecular Science 2011, 1, 191-

199.

(26) Dachsel, H.; Lischka, H.; Shepard, R.; Nieplocha, J.; Harrison, R. J., A massively

parallel multireference configuration interaction program: The parallel

COLUMBUS program. Journal of Computational Chemistry 1997, 18 (3), 430-

448.

(27) Müller, T., Large-scale parallel uncontracted multireference-averaged quadratic

coupled cluster: the ground state of the chromium dimer revisited. The Journal of

Physical Chemistry A 2009, 113, 12729-40.

(28) H. Lischka, R. S., I. Shavitt, R. M. Pitzer, M. Dallos, Th. Müller, P. G. Szalay, F.

B. Brown, R. Ahlrichs, H. J. Böhm, A. Chang, D. C. Comeau, R. Gdanitz, H.

Dachsel, C. Ehrhardt, M. Ernzerhof, P. Höchtl, S. Irle, G. Kedziora, T. Kovar, V.

Parasuk, M. J. M. Pepper, P. Scharf, H. Schiffer, M. Schindler, M. Schüler, M.

Seth, E. A. Stahlberg, J.-G. Zhao, S. Yabushita, Z. Zhang, M. Barbatti, S. Matsika,

M. Schuurmann, D. R. Yarkony, S. R. Brozell, E. V. Beck, and J.-P. Blaudeau, M.

Ruckenbauer, B. Sellner, F. Plasser, and J. J. Szymczak COLUMBUS 7.0.

(29) Kästner, J.; Carr, J. M.; Keal, T. W.; Thiel, W.; Wander, A.; Sherwood, P., DL-

FIND: an open-source geometry optimizer for atomistic simulations. The journal

of physical chemistry. A 2009, 113, 11856-65.

(30) Ahlrichs, R.; Bär, M.; Häser, M.; Horn, H.; Kölmel, C., Electronic structure

calculations on workstation computers: The program system turbomole. Chemical

Physics Letters 1989, 162 (3), 165-169.


64

CHAPTER VI

STRUCTURE AND ELECTRONIC STATES OF A GRAPHENE

DOUBLE VACANCY WITH EMBEDDED SI DOPANT

Authors: Reed Nieman,a Adelia J. A. Aquino,a,b Tevor P. Hardcastle,c,d Jani Kotakoski,e

Toma Susi,e and Hans Lischkaa,b



Tianjin University, Tianjin 300072, PR China, cSuperSTEM Laboratory, STFC Daresbury

Campus, Daresbury WA4 4AD, United Kingdom, dFaculty of Engineering, School of

Chemical and Process Engineering, University of Leeds, 211 Clarendon Rd., Leeds LS2

9JT, United Kingdom, eFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-

1090 Vienna, Austria

Published: October 2017 in The Journal of Chemical Physics, DOI:10.1063/1.4999779

4.1 Abstract

Silicon represents a common intrinsic impurity in graphene, commonly bonding to

either three or four carbon neighbors respectively in a single or double carbon vacancy. We

investigate the effect of the latter defect (Si-C4) on the structural and electronic properties

of graphene using density functional theory (DFT). Calculations based both on molecular

models and with periodic boundary conditions have been performed. The two-carbon

vacancy was constructed from pyrene (pyrene-2C) which was then expanded to

circumpyrene-2C. The structural characterization of these cases revealed that the ground

state is slightly non-planar, with the bonding carbons displaced from the plane by up to

±0.2 Å. This non-planar structure was confirmed by embedding the defect into a 10×8

supercell of graphene, resulting in 0.22 eV lower energy than the previously considered

planar structure. Natural bond orbital (NBO) analysis showed sp3 hybridization at the

silicon atom for the non-planar structure and sp2d hybridization for the planar structure.

Atomically resolved electron energy loss spectroscopy (EELS) and corresponding

spectrum simulations provide a mixed picture: a flat structure provides a slightly better

overall spectrum match, but a small observed pre-peak is only present in the corrugated

simulation. Considering the small energy barrier between the two equivalent corrugated


65

conformations, both structures could plausibly exist as a superposition over the

experimental timescale of seconds.

4.2 Introduction

The modification of the physical properties of graphene sheets and nanoribbons, in

particular the introduction of a band gap via chemical adsorption, carbon vacancies, and

the incorporation of dopant species has gained a great deal of attention.1-2 Silicon is an

interesting element as a dopant as it is isovalent to carbon. It is also a common intrinsic

impurity in graphene sheets grown by chemical vapor deposition3-4 (CVD) and in graphene

epitaxially grown by the thermal decomposition of silicon carbide.5-6 Further, dopant

impurities have been shown to have significant effects on the transport properties of

graphene.7-9

Silicon dopants in graphene exist predominantly in two previously identified forms:

a non-planar, threefold-coordinated silicon atom in a single carbon vacancy, referred to as

Si-C3, and a fourfold-coordinated silicon atom in a double carbon vacancy, called Si-C4,

which is also thought to be planar. An experimental work from 2012 by Zhou and

coauthors10 showed via simultaneous scanning transmission electron microscopy (STEM)

annular dark field imaging (ADF) and electron energy loss spectroscopy (EELS) and,

additionally, by DFT calculations that the two forms of single silicon doped graphene differ

energetically, and that the silicon atom in Si-C3 has sp3 orbital hybridization and that the

silicon atom in Si-C4 is sp2d hybridized. Similarly, EELS experiments combined with DFT

spectrum simulations confirmed the puckering of the Si in the Si-C3 structure and supported

the planarity of the Si-C4 structure.11 However, a less satisfactory agreement of the

simulated spectrum with the experimental one was noted in the latter case. The results

suggested that d-band hybridization of the Si was responsible for electronic density

disruption found at these sites. Experiments and dynamical simulations performed on these

defects have demonstrated12 how the Si-C4 structure is formed when an adjacent carbon

atom to the silicon is removed. However, it was concluded that the Si-C3 structure is more

stable and can be readily reconstructed by a diffusing carbon atom.


66

Past investigations of the electronic structure of graphene defects have shown the

usefulness of the molecule pyrene as a compact representation of the essential features of

single (SV) and double vacancy (DV) defects. Using this model, chemical bonding and the

manifold of electronic states were investigated by multireference configuration interaction

(MR-CI) calculations.13-16 In case of the SV defect,13 the MR-CI calculations showed four

electronic states (two singlet and two triplet) within a narrow margin of 0.1 eV, whereas

for the DV structure,14 a gap of ~1 eV to the lowest excited state was found. For the DV

defect, comparison with density functional theory using the hybrid density functional

Becke three-parameter Lee, Yang, and Parr (B3-LYP) functional17 showed good agreement

with the MR-CI results. Whereas for the covalently bonded Si-C3 structure no further low-

lying states are to be expected, the situation is different for the Si-C4 defect since in this

case an open-shell Si atom is inserted into a defect structure containing low-lying electronic

states.

Based on these experiences, a pyrene model will be adopted also in this work to

investigate the geometric arrangement and orbital hybridization around the silicon atom in

a double carbon vacancy (Figure 4.1). In a second step, a larger circumpyrene structure

(Figure 4.2) will be used to study the effect of increasing number of surrounding benzene

rings. Two types of functionals will be employed, the afore-mentioned B3LYP and, for

comparison, the long-range corrected Coulomb-attenuating B3LYP method (CAM-

B3LYP).18 Special emphasis will be devoted to verifying the electronic stability of the

closed-shell wavefunction with respect to triplet instability19-20 and of the optimized

structures with respect to structural relaxation. Finally, periodic DFT simulations of

graphene with the Perdew-Burke-Ernzerhof (PBE) functional21-22 will be compared to the

molecular calculations and an EELS spectrum simulated based on the computed Si-C4

structure compared to an atomic resolution STEM/EELS measurement.


Pyrene doped with a single silicon atom was investigated by first optimizing the

structure of pristine pyrene using the density functional B3LYP17 with the 6-31G*23 basis


67

set. The two interior carbon atoms were then removed and replaced with one silicon atom,

referred to in this investigation as pyrene-2C+Si. This unrelaxed structure was subjected to

a variety of closed-shell [restricted DFT (RDFT)] and open-shell [unrestricted DFT

(UDFT)] geometry optimizations using the hybrid density functionals B3LYP and CAM-

B3LYP in combination with the 6-311G(2d,1p)24-28 basis set. To extend to a larger system,

pyrene-2C+Si was surrounded by benzene rings (circumpyrene-2C) and this structure was

optimized using the B3LYP/6-31G** approach. Finally, the geometry optimization of

cirumpyrene-2C+Si was performed using the B3LYP functional and the 6-311G(2d,1d)

basis.

To simulate the geometric restrictions present in a large graphene sheet, the

Cartesian coordinates of all carbon atoms on the outer rims of pyrene-2C+Si (Figure 4.1)

and cirumpyrene-2C+Si (Figure 4.2) that are bonded to hydrogen atoms were frozen during

the course of geometry optimization to their values in the pristine structures. The

coordinates of both pyrene-2C+Si and circumpyrene-2C+Si were oriented along the y-axis

(the long axis) in the xy-plane.

The orbital occupations of the open-shell structures were determined using the

natural orbital population analysis (NPA)29 calculated from the total density matrix, while

the bonding character at silicon for each structure was investigated using the natural bond

orbital analysis.30-34

The discovered corrugated ground state structure of the Si-C4 defect was then

introduced into a periodic 10×8 supercell of graphene, and the structure relaxed to confirm

that this corrugation is reproduced at this level of theory. Finally, the Si L2,3 EELS response

of the defect was simulated11, 35 by evaluating the perturbation matrix elements of

transitions from the Si 2p core states to the unoccupied states calculated up to 3042 bands,

with no explicit core hole.36 The resulting densities of state were broadened using the

OptaDOS package37 with a 0.4 eV Gaussian instrumental broadening and semi-empirical

0.015 eV Lorentzian lifetime broadening.


68

The Gaussian 0938 and TURBOMOLE39 program packages were used for the

molecular calculations, and the GPAW40 and CASTEP41 packages for the graphene

simulations.

Figure 4.1 Optimized pyrene-2C+Si structure P1 (D2 symmetry) using the B3LYP/6-

311G(2d,1p) method with frozen carbon atoms marked by gray circles

Figure 4.2 Optimized circumpyrene-2C+Si structure CP1 (D2 symmetry) using the

B3LYP/6-311G(2d,1p) method with frozen carbon atoms marked by gray circles.


4.4.1 The Pyrene-2C+Si Defect Structure


69

The geometry of the unrelaxed, planar pyrene-2C+Si was first optimized using the

B3LYP/6-311G(2d,1p) method at closed-shell level resulting in the D2h structure denoted

P2. There were two imaginary frequencies present in the Hessian matrix. A displacement

of the geometry of structure P2 along either of these imaginary frequencies led to a

distortion of the planarity, lowering the point group to D2 and producing the minimum

energy structure P1, which was 0.736 eV lower in energy than structure P2 (Table 4.1).

The carbon atoms bonded to the silicon atom (atoms 3, 6, 10, and 13) were displaced from

planarity by about ±0.2 Å while the silicon atom remained co-planar with the frozen carbon

atoms on the periphery of pyrene-2C+Si. A similar tetrahedral geometry was found for

doping Al and Ga into DV graphene.42 No triplet instability was found for either of the D2h

and D2 structures. Optimizing the planar pyrene-2C+Si structure at UDFT/B3LYP/6-

311G(2d,1p) level for the high spin case resulted in structure P3, which was planar and

belonging to the D2h point group. No imaginary frequencies were found in the Hessian

matrix of structure P3; it is located 0.922 eV above the minimum energy structure P1.

Table 4.1 Open-shell character, structural symmetry, spin multiplicity, bond distances (Å),

and relative energies (eV) of selected optimized structures of pyrene-2C+Si.

Structure

Closed/

Open

Shell

Triplet

instab. #imaga

High/

Low

Spin Sib-C

C3-

C13 C3-C6 ΔE

B3LYP/6-311G(2d,1p)

P1 (D2)c Closed no 0 Low 1.940 2.652 2.860 0.000

P2 (D2h) Closed no 2 Low 1.932 2.634 2.826 0.736

P3 (D2h) Open - 0 High 1.933 2.610 2.852 0.922

CAM-B3LYP/6-311G(2d,1p)

P4 (D2)c Open no 0 Low 1.933d 2.645d 2.847d 0.000

P5 (D2)c Weakly

open

no 0 Low 1.936 2.648 2.852 0.040

P6 (D2)c Closed yes 0 Low 1.938 2.648 2.857 0.056

P7 (D2h) Open no 1 Low 1.923 2.616 2.820 0.602

P8 (D2h) Closed yes 2 Low 1.931 2.631 2.828 0.734

P9 (D2)e Open - 0 High 1.900 2.599 2.777 0.770

P10 (D2h) Open - 1 High 1.900 2.598 2.772 0.826

aNumber of imaginary frequencies. bSilicon always located in-plane. cThe out-of-plane

distance of carbon atoms that were not frozen in D2 structures is ±0.2 Å dAveraged value

of the four bonding carbons. eThe out-of-plane distance of carbon atoms that were not

frozen is ±0.07 Å


70

Structures P1-P3 were investigated also using the CAM-B3LYP/6-311G(2d,1p)

method (Table 4.1). For the planar geometry (D2h symmetry) two low spin structures were

found. The first one was the closed-shell structure P8 that, however, turned out to be triplet

instable. Re-optimization at UDFT/CAM-B3LYP level under D2h restrictions led to the

open-shell structure P7 that was 0.13 eV more stable than the corresponding closed-shell

structure. Structure P8 also had two imaginary vibrational frequencies, both of which were

out-of-plane, leading to D2 symmetry. Displacement along these modes and optimization

at closed-shell level gave the non-planar structure P6 of D2 symmetry. Similarly, following

the out-of-plane imaginary frequency found in the Hessian of structure P7 led to a new

structure P5 (D2 symmetry) that was 0.016 eV lower in energy than structure P6. It had

only a small open-shell character as can be seen from a natural orbital occupation of 0.089

and 1.911 e respectively for the lowest unoccupied natural orbital (LUNO) and highest

occupied natural orbital (HONO). This is much smaller than the NO occupation of the

open-shell low spin structure P7 whose LUNO-HONO occupations were 0.302 and 1.698

e. Searching along the imaginary frequencies found for the previous structures and

considering the triplet instability present in the wave functions led to structure P4 (D2

symmetry), the new low spin, open-shell minimum-energy CAM-B3LYP structure which

was 0.04 eV more stable than structure P5. The HONO-LUNO occupation for structure P4

was 1.899 e and 0.101 e respectively. Comparing the occupations from CAM-B3LYP to

B3LYP shows that the range-corrected functional CAM-B3LYP presents a more varied

picture than found with the B3LYP functional. In particular, open shell structures with non-

negligible deviations of NO occupations from the closed shell reference values of two and

zero, respectively, were found. However, the energy differences between these different

states are quite small, only a few hundredths of an eV.

Structure P10 of Table 4.1 was obtained using the unrestricted high spin approach

in the geometry optimization. The Hessian contained one imaginary frequency, an out-of-

plane bending mode leading to D2 symmetry. Following this imaginary frequency led to

structure P9 that was about 0.06 eV more stable than structure P10. The bonding carbons

of this new high spin structure were slightly out-of-plane by 0.07 Å.


71

Selected bond distances of the silicon-2C+Si complex are given in Table 4.1 for the

B3-LYP and CAM-B3LYP results. Looking first at the distances from the silicon atom to

its bonding carbon atoms computed at B3LYP/6-311G(2d,1p) level, the Si-C bond

distances are found to vary by less than 0.01 Å in the different structures. The D2 structure

has the longest Si-C bond distance as would be expected from moving the bonding carbon

atoms out-of-plane from the D2h to D2 symmetry. The internuclear distance between

adjacent, non-bonded carbon atoms C3 and C13 along the direction perpendicular to the

long axis is 2.65 Å for structure P1, which is the largest among the three structures. This is

again a result of the out-of-plane character of structure P1. The distance C3-C6 along the

long axis of pyrene is ~0.2 Å longer than the perpendicular distance, reflecting the

restrictions imposed by freezing the carbon atoms which would be connected to the

surrounding graphene sheet (Figure 4.1).

Next, we characterize the structures computed with the CAM-B3LYP/6-

311G(2d,1p) method. The largest C3-C13 bond distance (Table 4.1) is 2.65 Å for the low

spin structures P5 and P6 (D2), while the smallest, 2.60 Å, is found for the high spin, open-

shell structure P10. The dependence of the Si-C distances on the different structures and

spin states is not very pronounced, though the high spin planar structures, structures P9 and

P10, were found to have the smallest Si-C and C3-C13 distances; the same situation that

was seen using B3LYP/6-311G(2d,1p).

Comparing the two above methods for pyrene-2C+Si, the most noticeable effect is

that there is greater open-shell character for the low spin structures when using the CAM-

B3LYP functional. The Si-C bond distance and C3-C13 intranuclear distances are also

slightly smaller when using the CAM-B3LYP functional. But overall, agreement between

the two methods is quite good.

4.4.2 The Circumpyrene-2C+Si Defect Structure


72

Pyrene-2C+Si was then surrounded by benzene rings to create the larger

circumpyrene-2C+Si structure (Figure 4.2) as a better model for embedding the defect into

a graphene sheet. The edge carbon atoms linked to hydrogen atoms were again frozen.

Description of the geometry of circumpyrene-2C+Si is presented in Table 4.2 for the

B3LYP/6-311G(2d,1p) method. The geometry of the planar, unrelaxed circumpyrene-

2C+Si was optimized using a closed-shell approach that led to the D2h structure CP2, which

had one out-of-plane imaginary frequency. Following this mode, the non-planar D2

symmetric structure CP1 was obtained that contained no imaginary frequencies.

Optimizing the geometry of the planar, unrelaxed circumpyrene-2C+Si using an open-

shell, high spin approach resulted in a geometrically stable (no imaginary frequencies)

planar structure CP3 with D2h symmetry. The wave functions of circumpyrene-2C+Si

structures CP1 and CP2 were triplet stable.

Table 4.2 Structural symmetry, open-shell character, spin multiplicity, bond distances (Å),

and relative energies (eV) of selected optimized structures of circumyrene-2C+Si.

Structure

Closed/

Open

Shell

Triplet

instab. #imaga

High/

Low

Spin Sib-C

C29-

C39

C29-

C32 ΔE

B3LYP/6-311G(2d,1p)

CP1 (D2)c Closed no 0 Low 1.909 2.690 2.741 0.000

CP2 (D2h) Closed no 1 Low 1.902 2.668 2.713 0.118

CP3 (D2h) Open - 0 High 1.912 2.662 2.745 1.254 aNumber of imaginary frequencies. bSilicon always stays in-plane cThe out-of-plane

distance of non-frozen carbon atoms is ±0.2 Å in structures that are D2 symmetric

The Si-C bond distances for the three investigated structures are very similar (Table

4.2). The C29-C39 distance is largest for the non-planar, closed-shell structure P1 and

smallest for the planar structures P2 and P3, but the differences are less than 0.03 Å.

The difference in energy between the non-planar and planar low spin structures

decreases markedly when the 2C+Si defect is embedded in a larger hexagonal sheet

(compare ΔE values in Table 4.1 and Table 4.2). The energy difference decreases by ~0.6

eV from 0.74 eV in pyrene-2C+Si to 0.12 eV in circumpyrene-2C+Si. Also noteworthy,

when comparing pyrene-2C+Si and circumpyrene-2C+Si is that the Si-C bond distances in

circumpyrene-2C+Si are shorter by about 0.03 Å on average for the low spin cases. The

differences in the internuclear distance between adjacent, non-bonded carbon atoms C29


73

and C39 (perpendicular to the long axis) and C29-C32 (parallel) is much less pronounced

than in the pyrene case, which is a consequence of the more flexible embedding into the

carbon network where none of the four C atoms surrounding Si is bonded to another atom

which is frozen.

4.4.3 The Graphene-2C+Si Defect Structure

To confirm the obtained geometry and to validate that periodic DFT simulations

reproduce the observed ground state structure, the Si-C4 defect with an alternating ±0.2 Å

corrugation of the four C atoms was placed in a 10×8 supercell of graphene and the

structure and cell relaxed using the PBE functional. The structural optimization preserved

the out-of-plane corrugation, and resulted in a total energy 0.22 eV lower than for the flat

structure. Thus, standard DFT is able to find the correct, non-planar structure when

initializing away from the flat geometry. Further, a nudged elastic band calculation43 shows

that the two equivalent conformations (alternating which C atoms are up and which are

down) are separated only by this energy barrier, leading to a Boltzmann factor of 1.7×10-4

at room temperature and thus a rapid oscillation between the two equivalent structures for

any reasonable vibration frequency.

Bader analysis44 of the charge density shows that the Si-C bonds are rather

polarized, with the Si donating 2.61 electrons shared by the four neighboring carbons. This

is even larger than the bond polarization in 2D-SiC, where a Si donates 1.2 electrons

(although each C has three Si neighbors, bringing the total to 1.2 per C instead of 0.65

here).45

4.4.4 Natural Bond Orbital Analysis


74

NBO analysis was used to further characterize the bonding of pyrene-2C+Si and

circumpyrene-2C+Si and to describe in detail the charge transfer from silicon to carbon in

the various structures.

The NBO analysis of pyrene-2C+Si is presented in Table 4.3. The linear

combination factor of the Si is less than half of that of the bonding C, a sign of the greater

electronegativity of carbon compared to silicon. The non-planar, low spin, closed-shell

structure P1 (D2 symmetry) shows sp3 orbital hybridization at the silicon and sp3 orbital

hybridization at the bonding carbon, consistent with the slight tetrahedral arrangement

about the silicon center. The planar closed-shell structure P2 shows sp2d hybridization at

the silicon center, consistent with previous experimental and computational

interpretations,10 and approximately sp3 hybridization at the bonding carbons. The high

spin structure P3, which also has D2h symmetry, shows also sp2d hybridization at the

silicon. Compared to B3LYP (structures P1-P3), bonding from CAM-B3LYP (Table B.1

of Appendix B) is essentially the same.

Table 4.3 NBO bonding character analysis of pyrene-2C+Si using B3LYP/6-311G(2d,1p).

The discussed carbon atom is one of the four bonded to silicon.

Structure Bonding Character

B3LYP/6-311G(2d,1p)

P1 (D2)

closed sh. 0.54(sp2.98d0.02)Si+0.84(sp3.24d0.01)C

P2 (D2h)


P3 (D2h)

open sh.,

high spin

α: 0.46(sp2.00d1.00)Si+0.89(sp2.81d0.01)C

β: 0.55(sp2.00d1.00)Si+0.83(sp3.61d0.01)C

The NBO analysis of circumpyrene-2C+Si is presented in Table 4.4. The bonding

character of structure CP1 is essentially sp3 for both the silicon and the bonding carbon as

in structure P1 of pyrene-2C+Si, although the p orbital character on the carbon atom of this

structure is reduced by about 0.60 e. Structure CP2 presents an interesting case, as no

bonding NBOs were found. Instead, a series of four unoccupied valence lone pairs (LP*),

one of s orbital character and three of p character, were located on the silicon and a

corresponding set of occupied valence lone pairs (LP), one for each bonding carbon, was


75

found with sp3.58d0.01 character. It is noted that the Si-C bonding character of the high spin

structure CP3 changes markedly between the alpha shell and beta shell. In the alpha shell,

the silicon and carbon atoms both exhibit sp1 orbital hybridization with no d orbital

contribution on the silicon, but in the beta shell, the silicon atom shows sp2d hybridization

and the carbon atoms show sp3 hybridization as in the other planar cases.

The shapes of the NBOs describing the bond between Si and C is depicted in Figure

4.3 for structures P1 (D2 symmetry) and P2 (D2h symmetry) using the B3LYP/6-

311G(2d,1p) method. They appear very similar in spite of the different hybridization on Si

(sp3 vs. sp2d, Table 4.3). The plots of the NBOs for all other structures, even those for

circumpyrene-2C+Si, are almost identical. The exception is structure CP2 of

circumpyrene-2C+Si in which no Si-C bonds were found during the NBO analysis. The

NBOs shown in Figure B.1 for this structure consist of the valence lone pairs found on

each carbon atom (Table 4.4) possessing sp3.58 orbital hybridization with a lobe of electron

density found facing the silicon center. The NBOs of silicon are low-occupancy valence

lone pairs, representing one s and three p NBOs. Even though this classification of bonding

by means of lone pairs according to the NBO analysis looks quite different, the

combination of these lone pairs is not expected to look substantially different from the

localized bonds shown in the other structures, especially considering the strong polarity of

the silicon-carbon bonds as discussed below.


76

Table 4.4 NBO bonding character analysis of circumpyrene-2C+Si. The discussed carbon

atom is the one of the four bonded to silicon.


B3LYP/6-311G(2d,1p)

CP1 (D2)


CP2 (D2h)

closed sh. Four LP*: one [1.0(s)Si] + three [1.00(p)Si]

a

four[1.00(sp3.58d0.01)C]b

CP3 (D2h)

open sh.,

high spin

α: 0.68(sp1.00d0.00)Si+0.74(sp1.00d0.00)C

β: 0.55(sp2.00d1.00)Si+0.84(sp3.41 d0.01)C

aFour unfilled valence lone pairs (LP*) on silicon (occupation less than 1 e), bOne occupied

valence lone pair on each bonding carbon

Natural charges are instructive for illustrating the polarity of the silicon-carbon

bonds and the charge transfer between silicon and the surrounding graphitic lattice. These

are shown in Figure 4.4 for circumpyrene-2C+Si (Structures CP2 and CP3) using the

B3LYP/6-311G(2d,1p) method. The silicon center is strongly positive in all cases and more

positive in circumpyrene-2C+Si for the low spin cases (1.7–1.8 e, structures CP1 and CP2)

than the high spin case (1.4 e, structure CP3). It thus seems that the Bader analysis

presented above overestimates the charge transfer by over 40 %, despite being qualitatively

correct. The bonding carbon in each case shows a larger negative charge compared to the

remaining carbon atoms of circumpyrene. The charge on the bonding carbon in the low

spin structures CP1 and CP2 is more negative than that of the high spin structure CP3.


77

Figure 4.3 NBO plots for pyrene-2C+Si using the B3LYP/6-311G(2d,1p) approach for

structures P1 and P2. Isovalue = ±0.02 e/Bohr3


78

Figure 4.4 Natural charges (e) of carbon and silicon in circumpyrene-2C+Si using the

B3LYP/6-311G(2d,1p) method.

The natural charges of the remaining carbon atoms in circumpyrene-2C vary

depending on whether they are bonded to other carbon atoms or to hydrogen. For the low

spin structures CP1 and CP2, the natural charges on C atoms on the periphery of

circumpyrene bonded to other C atoms is about one-third as negative as the natural charges

of C atoms bonded to H atoms, while the natural charges of carbon atoms in the interior of

circumpyrene and not bonded to Si are much smaller. Taken together, the magnitude of the

Si-graphene charge transfer can be seen to be largest for the planar, closed-shell, low spin

structure CP2, and smallest for the planar, high spin structure CP3.


79

The natural charges for pyrene-2C+Si using B3LYP/6-311G(2d,1p) are plotted in

Figure B.2. The low spin structures P1 and P2 have respective positive charges on the

silicon atom of 1.596 and 1.953 e, a larger difference than in the low spin structures CP1

and CP2 of circumpyrene-2C+Si where the silicon atom has positive charges of 1.726 and

1.821 e. Correspondingly, the difference in the negative charges of the bonding carbon

atoms is also larger between these two structures than in circumpyrene. For the other

carbons in pyrene, the magnitude of their charges varies less than in circumpyrene. For

pyrene-2C+Si using the CAM-B3LYP/6-311G(2d,1p) method, the natural charges are

plotted in Figure B.3. The same trend is seen for this method as in our other cases, namely

that the closed-shell, low spin, planar structure P8 has the largest positive charge on silicon

and the largest negative charge on the bonding carbon compared to the other structures.

4.4.5 Molecular Electrostatic Potential

The molecular electrostatic potential (MEP) was computed for each structure and

is presented in Figure 4.5 for circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p)

method. The plots illustrate the natural charge transfer discussed previously: for structures

CP1-CP3, the positive potential at the silicon corresponds to the positive charge buildup as

discussed above. This positive potential at the silicon is more positive for structures CP1

and CP2 than for CP3 following the trend of the natural charges. There is also a negative

potential diffusely distributed on the carbon atoms in circumpyrene.

The MEP plots for pyrene-2C+Si using the B3LYP/6-311G(2d,1p) method is

shown in Figure B.4. Similar results are obtained when using the CAM-B3LYP functional

(shown in Figure B.5). Finally, the low spin planar structures P2, P7, and P8 have a much

more positive potential than either the non-planar structures or the high spin planar

structures, represented by the deeper blue coloring in the figures.


80

Figure 4.5 Molecular electrostatic potential for circumpyrene-2C+Si mapped onto the

electron density isosurface with an isovalue of 0.0004 e/Bohr3 (B3LYP/6-311G(2d,1p)).

4.4.6 Electron Energy Loss Spectroscopy

The Si-C4 defect in graphene was originally identified through a combination of

atomic resolution STEM and atomically resolved EELS. In that study11 it was concluded

that a Si-C3 defect is non-planar due to a significantly better match between the measured

and simulated EELS spectrum; however, the simulated spectrum of a flat Si-C4 defect

matched the experiment less well. Because of this discrepancy and the findings described

above, we also simulated the EELS spectrum of the corrugated Si-C4 graphene defect.

Figure 4.6 displays the simulated spectra of both the flat and the corrugated defect

(normalized to the π* peak at 100 eV), overlaid on a new background-subtracted

experimental signal that we have recorded with a higher dispersion than the original

spectrum11 (average of 20 spot spectra, with the Si-C4 structure verified after acquisition

by imaging; see Figure B.6 for the unprocessed spectrum).


81

Figure 4.6 Comparison of simulated and experimental EELS spectra of the Si-C4 defect in

graphene. Despite the out-of-plane distorted structure being the ground state, the originally

proposed flat spectrum provides an overall better match to the experiment, apart from the

small peak at ~96 eV that is not predicted for the flat structure. The inset shows a colored

medium angle annular dark field STEM image of the Si-C4 defect (the Si atom shows

brighter due to its greater ability to scatter the imaging electrons).

The new spectrum closely resembles the originally reported one apart from a small

additional peak around 96 eV. Comparing the recorded signal to the simulated spectra, the

overall relative intensities of the π* and σ* contributions match the flat structure better, but

both simulations overestimate the π* intensity at 100 eV. Intriguingly, despite the better

overall match with the flat structure, the small pre-peak present in the new signal is only

present in the spectrum simulated for the lowest energy corrugated structure, providing

spectroscopic evidence for its existence. The lowest unoccupied states responsible for this

peak are π* in character and form a „cross-like“ pattern across the Si site, as shown in Figure

B.7, markedly different from the lowest unoccupied states of the flat structure. However,

considering the small energy barrier between the equivalent corrugated conformations

discussed above, even at room temperature the spectrum may be integrated over a

superposition of corrugated and nearly flat morphologies.

100 110 120 130

0.0

0.5

1.0

1.5

´1

04

co

un

ts

Energy loss (eV)

Signal

Flat

Corrugated


82

4.5 Conclusions

Density functional theory calculations (both restricted and unrestricted) were

performed on pyrene-2C+Si and circumpyrene-2C+Si producing a variety of structures that

were characterized using natural orbital occupations, natural bond orbital analyses, and

molecular electrostatic potential plots. A non-planar low spin structure of D2 symmetry

was found to be the minimum for both cases, followed energetically by a low spin planar

structure, 0.6 eV higher in energy for pyrene-2C+Si, but with a much smaller (only ~0.1

eV) energy difference for circumpyrene-2C+Si. A periodic graphene model with the non-

planar defect had a 0.22 eV lower energy than a flat one. In the molecular calculations, the

out-of-plane structure was shown to be a minimum by means of harmonic frequency

calculations. The structures of the high spin state are much higher in energy than the

minimum: about 0.85 eV higher for pyrene-2C+Si and about 1.3 eV higher for

circumpyrene-2C+Si.

For pyrene-2C+Si, B3LYP and CAM-B3LYP were compared showing that there

was not much difference either energetically, in the NBO characterization, or in the MEP

plots. One feature of interest is that CAM-B3LYP shows greater open-shell character in

the low spin case than was seen for B3LYP. This open-shell character demonstrates the

variety of electronic structures which can be found in seemingly ordinary closed-shell

cases.

NBO analyses showed the expected bonding configurations for the two structural

symmetries present. The bonding of the silicon atom in the planar structure was found to

have sp2d orbital character while in the non-planar structures, the bonding orbitals are sp3

hybridized. The natural charges illustrated the charge transfer that occurs, demonstrating

the buildup of positive charge at the silicon and the diffusion of negative charge into the

pyrene-2C/cirumpyrene-2C structure, which was further evident in the MEP plots.

Despite carefully establishing the non-planar ground state nature of the Si-C4 defect

in graphene, electron energy loss spectra still seem to match the flat structure better, but

imperfectly especially concerning a small peak at ~96 eV. Considering the large amounts


83

of kinetic energy that the energetic electrons used in transmission electron microscopy can

impart, and the available thermal energy in the system, perhaps the defect is not able to

stay in its ground state, but rather exists in a superposition of slightly different

configurations that result in an average spectrum resembling the flat state.

4.6 Acknowledgments

We are grateful to Michael Zehetbauer for continuous support and interest in this

work and also thank Quentin Ramasse for many useful discussions. T.S. and J.K.

acknowledge the Austrian Science Fund (FWF) for funding via projects P 28322-N36 and

I 3181-N36. H.L. and T.S. are grateful to the Vienna Scientific Cluster (Austria) for

computational resources (H.L.: project no. P70376). H.L. also acknowledges computer

time at the computer cluster of the School for Pharmaceutical Science and Technology of

the Tianjin University, China, T.P.H. acknowledges the Engineering and Physical Sciences

Research Council (EPSRC) Doctoral Prize Fellowship that funded this research, and the

ARC1 and ARC2 high-performance computing facilities at the University of Leeds.

4.7 Bibliography




1336-1337.




76.

(3) Kim, K. S.; Zhao, Y.; Jang, H.; Lee, S. Y.; Kim, J. M.; Kim, K. S.; Ahn, J.-H.; Kim,

P.; Choi, J.-Y.; Hong, B. H., Large-scale pattern growth of graphene films for

stretchable transparent electrodes. Nature 2009, 457, 706.

(4) Lee, Y.; Bae, S.; Jang, H.; Jang, S.; Zhu, S.-E.; Sim, S. H.; Song, Y. I.; Hong, B.

H.; Ahn, J.-H., Wafer-Scale Synthesis and Transfer of Graphene Films. Nano

Letters 2010, 10 (2), 490-493.

(5) Emtsev, K. V.; Bostwick, A.; Horn, K.; Jobst, J.; Kellogg, G. L.; Ley, L.;

McChesney, J. L.; Ohta, T.; Reshanov, S. A.; Röhrl, J.; Rotenberg, E.; Schmid, A.

K.; Waldmann, D.; Weber, H. B.; Seyller, T., Towards wafer-size graphene layers


84

by atmospheric pressure graphitization of silicon carbide. Nature Materials 2009,

8, 203.

(6) Berger, C.; Song, Z.; Li, T.; Li, X.; Ogbazghi, A. Y.; Feng, R.; Dai, Z.;

Marchenkov, A. N.; Conrad, E. H.; First, P. N.; de Heer, W. A., Ultrathin Epitaxial

Graphite: 2D Electron Gas Properties and a Route toward Graphene-based

Nanoelectronics. The Journal of Physical Chemistry B 2004, 108 (52), 19912-

19916.

(7) Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K.,

The electronic properties of graphene. Reviews of Modern Physics 2009, 81 (1),

109-162.

(8) Rutter, G. M.; Crain, J. N.; Guisinger, N. P.; Li, T.; First, P. N.; Stroscio, J. A.,

Scattering and interference in epitaxial graphene. Science 2007, 317 (5835), 219-

22.

(9) Zhao, L.; He, R.; Rim, K. T.; Schiros, T.; Kim, K. S.; Zhou, H.; Gutierrez, C.;

Chockalingam, S. P.; Arguello, C. J.; Palova, L.; Nordlund, D.; Hybertsen, M. S.;

Reichman, D. R.; Heinz, T. F.; Kim, P.; Pinczuk, A.; Flynn, G. W.; Pasupathy, A.

N., Visualizing individual nitrogen dopants in monolayer graphene. Science 2011,

333 (6045), 999-1003.

(10) Zhou, W.; Kapetanakis, M. D.; Prange, M. P.; Pantelides, S. T.; Pennycook, S. J.;

Idrobo, J.-C., Direct Determination of the Chemical Bonding of Individual

Impurities in Graphene. Physical Review Letters 2012, 109 (20), 206803.

(11) Ramasse, Q. M.; Seabourne, C. R.; Kepaptsoglou, D.-M.; Zan, R.; Bangert, U.;

Scott, A. J., Probing the Bonding and Electronic Structure of Single Atom Dopants

in Graphene with Electron Energy Loss Spectroscopy. Nano Letters 2013, 13 (10),

4989-4995.

(12) Susi, T.; Kotakoski, J.; Kepaptsoglou, D.; Mangler, C.; Lovejoy, T. C.; Krivanek,

O. L.; Zan, R.; Bangert, U.; Ayala, P.; Meyer, J. C.; Ramasse, Q., Silicon-carbon

bond inversions driven by 60-keV electrons in graphene. Physical Review Letters

2014, 113 (11), 115501.




15 (15), 3334-3341.




(15) Haldar, S.; Amorim, R. G.; Sanyal, B.; Scheicher, R. H.; Rocha, A. R., Energetic

stability, STM fingerprints and electronic transport properties of defects in

graphene and silicene. Rsc Adv 2016, 6 (8), 6702-6708.

(16) Nieman, R.; Das, A.; Aquino, A. J. A.; Amorim, R. G.; Machado, F. B. C.; Lischka,

H., Single and double carbon vacancies in pyrene as first models for graphene

defects: A survey of the chemical reactivity toward hydrogen. Chemical Physics

2017, 482, 346-354.




85



2004, 393 (1), 51-57.

(19) Seeger, R.; Pople, J. A., Self‐consistent molecular orbital methods. XVIII.

Constraints and stability in Hartree–Fock theory. The Journal of Chemical Physics

1977, 66 (7), 3045-3050.

(20) Bauernschmitt, R.; Ahlrichs, R., Stability analysis for solutions of the closed shell

Kohn–Sham equation. The Journal of Chemical Physics 1996, 104 (22), 9047-

9052.




Made Simple [Phys. Rev. Lett. 77, 3865 (1996)]. Physical Review Letters 1997, 78

(7), 1396-1396.

(23) Hehre, W. J.; Ditchfield, R.; Pople, J. A., Self—Consistent Molecular Orbital

Methods. XII. Further Extensions of Gaussian—Type Basis Sets for Use in

Molecular Orbital Studies of Organic Molecules. The Journal of Chemical Physics

1972, 56 (5), 2257-2261.

(24) Binning, R. C.; Curtiss, L. A., Compact contracted basis sets for third-row atoms:

Ga–Kr. Journal of Computational Chemistry 1990, 11 (10), 1206-1216.

(25) Curtiss, L. A.; McGrath, M. P.; Blaudeau, J. P.; Davis, N. E.; Binning, R. C.;

Radom, L., Extension of Gaussian‐2 theory to molecules containing third‐row

atoms Ga–Kr. The Journal of Chemical Physics 1995, 103 (14), 6104-6113.

(26) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A., Self‐consistent molecular

orbital methods. XX. A basis set for correlated wave functions. The Journal of


(27) McGrath, M. P.; Radom, L., Extension of Gaussian‐1 (G1) theory to bromine‐

containing molecules. The Journal of Chemical Physics 1991, 94 (1), 511-516.

(28) McLean, A. D.; Chandler, G. S., Contracted Gaussian basis sets for molecular

calculations. I. Second row atoms, Z=11–18. The Journal of Chemical Physics

1980, 72 (10), 5639-5648.

(29) Reed, A. E.; Weinstock, R. B.; Weinhold, F., Natural population analysis. The

Journal of Chemical Physics 1985, 83 (2), 735-746.

(30) Carpenter, J. E.; Weinhold, F., Analysis of the geometry of the hydroxymethyl

radical by the “different hybrids for different spins” natural bond orbital procedure.

Journal of Molecular Structure: THEOCHEM 1988, 169, 41-62.

(31) Foster, J. P.; Weinhold, F., Natural hybrid orbitals. Journal of the American

Chemical Society 1980, 102 (24), 7211-7218.

(32) Reed, A. E.; Curtiss, L. A.; Weinhold, F., Intermolecular interactions from a natural

bond orbital, donor-acceptor viewpoint. Chemical Reviews 1988, 88 (6), 899-926.

(33) Reed, A. E.; Weinhold, F., Natural bond orbital analysis of near‐Hartree–Fock

water dimer. The Journal of Chemical Physics 1983, 78 (6), 4066-4073.

(34) Reed, A. E.; Weinhold, F., Natural localized molecular orbitals. The Journal of



86

(35) Susi, T.; Hardcastle, T. P.; Hofsäss, H.; Mittelberger, A.; Pennycook, T. J.;

Mangler, C.; Drummond-Brydson, R.; Scott, A. J.; Meyer, J. C.; Kotakoski, J.

Single-atom spectroscopy of phosphorus dopants implanted into graphene, in:

arXiv:1610.03419, 2016 2016.

(36) Hardcastle, T. P.; Seabourne, C. R.; Kepaptsoglou, D. M.; Susi, T.; Nicholls, R. J.;

Brydson, R. M. D.; Scott, A. J.; Ramasse, Q. M., Robust theoretical modelling of

core ionisation edges for quantitative electron energy loss spectroscopy of B- and

N-doped graphene. Journal of Physics: Condensed Matter 2017, 29 (22), 225303.

(37) Nicholls, R. J.; Morris, A. J.; Pickard, C. J.; Yates, J. R., OptaDOS - a new tool for

EELS calculations. Journal of Physics: Conference Series 2012, 371 (1), 012062.

(38) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.;

Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.;

Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.;

Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.;

Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven,

T.; Montgomery Jr., J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J.;

Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.;

Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi,

M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.;

Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A.

J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.;

Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.;

Daniels, A. D.; Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J.

Gaussian 09, Gaussian, Inc.: Wallingford, CT, USA, 2009.



Physics Letters 1989, 162 (3), 165-169.

(40) Enkovaara, J.; Rostgaard, C.; Mortensen, J. J.; Chen, J.; Dulak, M.; Ferrighi, L.;

Gavnholt, J.; Glinsvad, C.; Haikola, V.; Hansen, H. A.; Kristoffersen, H. H.;

Kuisma, M.; Larsen, A. H.; Lehtovaara, L.; Ljungberg, M.; Lopez-Acevedo, O.;

Moses, P. G.; Ojanen, J.; Olsen, T.; Petzold, V.; Romero, N. A.; Stausholm-Møller,

J.; Strange, M.; Tritsaris, G. A.; Vanin, M.; Walter, M.; Hammer, B.; Häkkinen, H.;

Madsen, G. K. H.; Nieminen, R. M.; Nørskov, J. K.; Puska, M.; Rantala, T. T.;

Schiøtz, J.; Thygesen, K. S.; Jacobsen, K. W., Electronic structure calculations with

GPAW: a real-space implementation of the projector augmented-wave method.

Journal of Physics: Condensed Matter 2010, 22 (25), 253202.

(41) Clark, S. J.; Segall, M. D.; Pickard, C. J.; Hasnip, P. J.; Probert, M. J.; Refson, K.;

Payne, M. C., First principles methods using CASTEP. Zeitschrift für

Kristallographie - Crystalline Materials 2005, 220 (5-6), 567-570.

(42) Tsetseris, L.; Wang, B.; Pantelides, S. T., Substitutional doping of graphene: The

role of carbon divacancies. Physical Review B 2014, 89 (3), 035411.

(43) Henkelman, G.; Jónsson, H., Improved tangent estimate in the nudged elastic band

method for finding minimum energy paths and saddle points. The Journal of



87

(44) Tang, W.; Sanville, E.; Henkelman, G., A grid-based Bader analysis algorithm

without lattice bias. Journal of Physics: Condensed Matter 2009, 21 (8), 084204.

(45) Susi, T.; Skakalova, V.; Mittelberger, A.; Kotrusz, P.; Hulman, M.; Pennycook, T.

J.; Mangler, C.; Kotakoski, J.; Meyer, J. C., Computational insights and the

observation of SiC nanograin assembly: towards 2D silicon carbide. Scientific

Reports 2017, 7 (1), 4399.


88

CHAPTER V

THE CRUCIAL ROLE OF A SPACER MATERIAL ON THE

EFFICIENCY OF CHARGE TRANSFER PROCESSES IN ORGANIC

DONOR-ACCEPTOR JUNCTION SOLAR CELLS

Authors: Reed Nieman,a Hsinhan Tsai,b Wanyi Nie,b Adelia J. A. Aquino,a,c Aditya D.

Mohite,b Sergei Tretiak,b Hao Li,d Hans Lischkaa,c

Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University

Lubbock, TX 79409-1061, USA, bLos Alamos National Laboratory, Los Alamos, New

Mexico 87545, USA, cSchool of Pharmaceutical Sciences and Technology, Tianjin

University, Tianjin, 300072 P.R.China, dDepartment of Chemistry, University of Houston,

Houston, Texas 77204. USA

Published: December 2017 in Nanoscale, DOI:10.1039/C7NR07125F

5.1 Abstract

Organic photovoltaic donor-acceptor junction devices composed of π-conjugated polymer

electron donors (D) and fullerene electron acceptors (A) show greatly increased

performance when a spacer material is inserted between the two layers.1 For instance,

experimental results reveal significant improvement of photocurrent when a terthiophene

oligomer derivative is inserted in between π-conjugated poly(3-hexylthiophene-2, 5-diyl

(P3HT) donor and C60 acceptor. These results indicate favorable charge separation

dynamics, which is addressed by our present joint theoretical/experimental study

establishing the beneficial alignment of electronic levels due to the specific morphology of

the material. Namely, based on the experimental data we have constructed extended

structural interface models containing C60 fullerenes and P3HT separated by aligned

oligomer chains. Our time-dependent density functional theory (TD-DFT) calculations

based on a long-range corrected functional, allowed us to address the energetics of essential

electronic states and analyze them in terms of charge transfer (CT) character. Specifically,

the simulations reveal the electronic spectra composed of a ladder of excited states evolving

excitation toward spatial charge separation: an initial excitonic excitation at P3HT

decomposes into charges by sequentially relaxing through bands of C60-centric,

oligomer→C60 and P3HT→C60 CT states. Our modeling exposes a critical role of dielectric

environment effects and electronic couplings in the self-assembled spacer oligomer layer


89

on the energetics of critical CT states leading to a reduced back-electron transfer,

preventing recombination losses, and thus rationalizes physical processes underpinning

experimental observations.

5.2 Introduction

The interfaces of organic photovoltaic bulk heterojunction (BHJ) donor

(D)/acceptor (A) devices between π-conjugated electron donor polymers and electron

acceptors, such as fullerenes, are of fundamental importance for understanding organic

photovoltaic processes and for improving their efficiency.2-4 These interfaces control the

dissociation of excitons and generation of CT states. The conversion of light into electricity

in a BHJ photovoltaic device is accomplished by a four-step process: (i) light absorption

by bright π→π* transition of the organic conjugated polymer and generation of excitons,

(ii) diffusion of the excitons through the bulk polymer and segregation to the interface, (iii)

dissociation of the excitons at afore-mentioned heterojunctions and creation of CT states,

and (iv) charge separation and collection at contacts.2, 5 The actual processes occurring in

the BHJ material are significantly more complex because of the heterogeneous distribution

of donor and acceptor in the BHJ material, a broad variety of structural defects and

conformations, the large number of internal degrees of freedom of the polymer chains, and

the complicated manifold of electronic states.6-8 Therefore, finessing the mechanism of

ultrafast and loss-less free charge generation in BHJ devices via intercede design is still a

very active research area.9-13

While traditional organic photovoltaic (OPV) BHJ devices involve only a pair of

materials, i.e., electron donor and acceptor complexes, recent experimental work1 has

employed a new strategy of using a third material acting as a spacer between the donor and

acceptor regions,14-15 which mitigates the electron-hole recombination rate (i.e. the

backward charge transfer processes) but without affecting the efficiency of charge

separation (i.e., forward electron transfer). In particular, these studies in a simple bi-layer

device have shown a large increase in both the photocurrent (up to 800%) and open circuit

voltage (VOC) when a spacer material such as a terthiophene-derivative (O3) is added


90

between the donor, a P3HT, and a fullerene (C60) acceptor. Subsequently, in the BHJ device

a significant increase in the power conversion efficiency (PCE) from 4% to greater than

7% was observed.1 These experimental findings followed the hypothesis of a “fast energy

transfer of the exciton from the donor across O3 to the acceptor and the back cascading of

the hole to the donor” due to a favorable alignment of the corresponding energy levels.

Such arrangement ensures that the spacer material is also physically separating the donor

and acceptor interface thus working to mitigate the CT recombination rate.

Quantum-chemical modeling of these materials can provide important atomistic

insights into the nature of underlining processes and ultimately help to formulate design

strategies toward optimizing the photophysical dynamics. The first important step is the

determination of the energy level alignment occurring at the complex donor-acceptor

interfaces. For such large systems density functional theory (DFT) and time-dependent

(TD)-DFT are the most widely used techniques providing access to the ground and excited

state properties, respectively, being a reasonable compromise between accuracy and

numerical cost. However, in spite of the overwhelming success of DFT in ground-state

computation, the correct description of charge-separated excited states is still problematic

for many developed exchange-correlation functionals due to the significant delocalized

nature of the density, especially for intermolecular CT states at BHJ interfaces.16-19 A

variety of strategies have been developed to correctly account for delocalized excitations.

Long-range corrected functionals20-21 in combination with optimization of the parameters

determining the separation range22-25 have been used successfully to correct several TD-

DFT artifacts. In previous work,26 we have shown that long-range corrected functionals of

CAM-B3LYP27-28 and ωB97XD29 gave very good agreement for excitonic and CT states

for thiolated oligothiophene dimers in comparison to the second-order algebraic

diagrammatic construction (ADC(2)) approach. Thus, long-range corrected functionals

appear to be well applicable for the calculation of electronic spectra of π-conjugated

semiconductor polymers including capacity of describing both excitonic and CT states.30

They are also computationally efficient so that the capability of treating large molecular

aggregate systems can be well managed.


91

Regarding the discussed bilayer interfaces,31 previous calculations have shown that

the bright, light absorbing π→π* transition can be located above CT states and can convert

to the CT state via internal conversion processes. However, it has been concluded from

TD-DFT calculations32 that dark fullerene states could be located even lower in energy and

would have the possibility to quench the CT states after they have been formed. Other

calculations based on the already mentioned ADC(2) method have shown a slightly

different picture indicating CT states as the lowest singlet states.31, 33

The energetic ordering of states and the stabilization of CT states obviously

represents a crucial issue affecting the reduction of unwanted leaking processes. This joint

experimental and computational study wants to demonstrate the specific factors affecting

especially the stability of the CT states. For that purpose, the experimental P3HT/O3/C60

system discussed above is being used together with our previously benchmarked TD-DFT

methodology.26 Even though the calculations are specific for this system, as we will show,

the conclusions are quite general and can certainly be transferred to many other polar

interface systems. Two structural models were employed in order to represent the donor-

spacer-acceptor interface: (i) the donor P3HT polymer is connected via one spacer T3 chain

to the acceptor C60 (Figure 5.1a, denoted as standard model) and (ii) an extended model

(Figure 5.1b,c) where P3HT is connected via three T3 spacer molecules to three C60

electron acceptors. A spacer material, thiolated terthiophene, T3, was used which can be

considered as a good approximation to O3 used in the experiments. This specific mutual

arrangement of molecules underscores our experimental fabrication technique utilizing the

Langmuir-Blodgett (LB) method (Figure 5.2a) ensuring orthogonal orientation of the

oligomer with respect to the polymer P3HT layer, as demonstrated by our X-ray

measurements. Furthermore, the extended model aims to account for π-stacking effects in

the oligomer, which were shown to be important for conventional, simple donor-acceptor

interfaces.34-39


92

Figure 5.1 (a) Standard model of the molecular arrangement in the trimer. (b) Front and (c)

side views of the extended model. The picture shows molecular notations, geometrical

arrangement and selected nonbonding distances (Å). Atoms kept fixed during the geometry

optimization are circled in black.

The article is organized as follows: Section 5.3 describes experimental and

computational methodology, Section 5.4 presents our results, and finally Section 5.5

summarizes our findings and conclusions.

5.3 Methods

5.3.1 Experimental

5.3.1.1 Bilayer Solar Cells Fabrication

Solar cell fabrication started from the cleaning of indium tin oxide (ITO)

transparent substrate with sonication bath in water, acetone and isopropyl alcohol for 15

min respectively. The hole transporting polymer poly(3,4-ethylenedioxythiophene)

polystyrene sulfonate (PEDOT:PSS) solution was then coated on the cleaned ITO

substrates by spin coating method. After drying, the coated substrates were transferred to

an argon filled glovebox for P3HT coating. The in-stock P3HT solution (2 mg/ml in

chlorobenzene) was spin coated on the substrates at 2000 rpm for 45 Sec, forming a

uniform, 40 nm thin film without further annealing. The coated substrates were then taken

outside the glovebox for interfacial layer deposition by LB technique between air-water


93

interfaces on a spin coated P3HT film. After LB deposition, the devices were completed

by thermally evaporating C60 (30nm) for acceptor layer and electrode deposition through

shadow mask.

5.3.1.2 Device Characterization

The completed solar cells were mounted in cryostat with BNC connections from

the electrodes to the instruments. The photocurrent was collected in AC mode by

illuminating the device with monochromatic light while measuring the short circuit current

using lock-in amplifier.

5.3.2 Computational Details

The computational methods are based on DFT using the Perdew–Burke–Ernzerhof

(PBE)40 functional with the SV(P)41 basis for geometry optimizations, and the long-range

separated density functional, CAM-B3LYP, for the calculation of excited states using two

basis sets, 6-31G42 and 6-31G*.43-44 Solvent effects were taken into account using the

conductor-like polarizable continuum model (C-PCM)45. Environmental effects on the

electronically excited states were investigated on the basis of the linear response (LR)46-47

and state-specific (SS)48 approaches. Dichloromethane (dielectric constant, ε=8.93,

refractive index, n=1.42) was chosen to act as a modestly polar environment. Geometry

optimizations were performed by freezing selected atoms as shown in Figure 5.1 in order

to maintain the desired structure of the aggregate as deduced from experiment.

The two types of trimer structures are shown in Figure 5.1a (standard trimer) and

Figure 5.1b,c (extended trimer). All structures were optimized as isolated complexes using

DFT together with the PBE40 functional, the SV(P)41 basis, and the Multipole-Accelerated

Resolution of the Identity for the Coulomb energy (MARI-J).49 Empirical dispersion

corrections were added by means of the D350 approach. Excited states were calculated

using the long-range separated density functional, CAM-B3LYP, and two basis sets, 6-


94

31G42 and 6-31G*43-44. D3 dispersion corrections were used for the CAM-B3LYP TD-DFT

calculations as well.

Full geometry optimizations of the standard and extended trimer structures would

lead to distorted geometries which do not resemble the intended structural arrangements of

a donor-spacer-C60 interface since geometrical restrictions of the surrounding environment

are missing. To achieve the desired model arrangement, geometry optimization was

performed in steps and by freezing a few selected atoms (see Figure 5.1) to restrict the

structural manifold. For the standard trimer, the C60 and T3 monomers were combined and

optimized freezing selected atoms. To this C60-T3 structure, P3HT was added as shown in

Figure 5.1a. The whole complex (C60-T3-P3HT) was then optimized. The frozen atoms

were located at the end of the chains and at the opposite side of C60, respectively, so that

the subunits had sufficient flexibility to come in contact allowing the intersystem distances

to be optimized. For the extended trimer, an analogous restricted geometry optimization

technique was used as shown in Figure 5.1b, c.

Since the main objective of this investigation was to determine the relative stability

of the CT states with excitations from the π-conjugated polymers to the C60s and the local

excitations in the polymers, excitations within the fullerenes were suppressed in most of

the calculations by freezing in the TD-DFT calculations the occupied orbitals located on

the C60 units. Otherwise, about a dozen of local excitations per C60 unit would have

impeded the already expensive calculations and would not have allowed the focus on the

actual states of interest. In practice, all the core orbitals along with twenty percent of the

highest occupied orbitals, which were localized at the C60 and (C60)3 units, were kept

frozen. Comparison between full and frozen orbital calculations have been performed as

well, showing good agreement in energetic ordering of the remaining states in the two

calculations.

Solvent effects were taken into account using the conductor-like polarizable

continuum model (C-PCM).45 Environmental effects on the electronically excited states

were investigated on the basis of the LR46-47 and SS48 approaches. In the LR approach,

Hartree and exchange-correlation (XC) potentials are linearly expanded with respect to the

variation of the time-dependent density of the ground state. In the SS method, the solvation


95

field adapts to changes in the solute wave function for electronic excitations where the

energy correction from fast electronic relaxation of the solvent is state-dependent. SS is

considered more accurate for describing the excited electronic states51-53 with charge

distributions significantly different from the ground state such as with polar solvents or CT

states in the solute. The SS method is computationally significantly more demanding

though and requires that each state be calculated independently. As such, only five states

were calculated using the CAM-B3LYP/6-31G* method for the extended trimer which

contained, however, representatives of the most important excitation types. For the

extended trimer using the CAM-B3LYP/6-31G method, twenty states were calculated, and

for the standard trimer using both basis sets, ten states were considered. Dichloromethane

(dielectric constant, ε=8.93, refractive index, n=1.42) was chosen to act as a modestly polar

environment.

Geometry optimizations were carried out using the TURBOMOLE54 program suite,

while all TD-DFT calculations were performed with the Gaussian G09 program package.55

The character of the electronic states has been analyzed by means of natural transition

orbitals (NTOs)56-58 and molecular electrostatic potential plots (MEP). NTOs represent the

weighted contribution to each excited state transition from a hole to a particle (electron)

derived from the one-electron transition density matrix. NTOs and particle/hole

populations were performed with the TheoDORE program package.59-61


5.4.1 Experimental Observations

We fabricate the bilayer device as illustrated in Figure 5.2a comprised with 20 nm

P3HT as donor and 40 nm of fullerene (C60) as acceptor for photocurrent measurements as

summarized in Figure 5.2. To insert oriented O3 between the (D/A) interface, we employed

the O3 with Langmuir-Blodgett (LB) method after spin coating P3HT on hole transporting

material, followed by C60 deposition by thermal evaporation (Figure 5.2a). The obtained

bilayer device was taken for photocurrent spectrum measurement in Figure 5.2b. The

photocurrent follows the absorption spectrum of P3HT (band gap 1.9 eV), while the


96

magnitude of the photocurrent increase by 1.8 times from the device without spacer layer

(black) to the device with 1-bilayer of O3 inserted (red). The optical band gap remains the

same in both cases indicating the oligomer absorption does not contribute to the

photocurrent increase. To characterize the molecular orientation of the O3 molecules at the

interface, we took the LB deposited O3 molecules on P3HT thin-film for gracing incidence

wide angle X-ray scattering (GIWAXS) measurement. The result in Figure 5.2c clearly

shows a diffraction spot along the qz axis. The spot is located near qz = 0.322 Å-1

corresponding to a d-spacing of 19.4 Å which matches with the molecule length. In

addition, the Bragg spots are parallel to the qy axis (substrate) indicating that the O3

molecules are aligning in the out-of-plane62 orientation by LB method (Figure 5.2a) and

pointing from P3HT towards C60. The distance between the donor-acceptor is thus

determined by the length of molecule and number of bilayers deposited. This is consistent

with previous measurements of three self-assembled oligomers.1, 63 We further performed

device photocurrent measurement as a function of the number of O3 bilayers as shown in

Figure 5.2d, we found the maximum gain in photocurrent can be obtained when inserting

two bilayers of O3 molecules between D/A interface. This is suggesting that such

separation between donor and acceptor materials is optimal to suppress the electron-hole

recombination through CT state at the interface after a suitable thickness of spacer layer

insertion. The subsequent decrease of the photocurrent is associated with the decrease of

efficiency for direct CT process.


97

Figure 5.2 (a) Scheme of bilayer device structure used in this study, the interface layer with

vertical alignment was produced by Langmuir-Blodgett (LB) method illustrated on right.

(b) photocurrent measured at short circuit condition as function of illumination wavelength

for bilayer device without spacer and with 1-bilayer of O3 between P3HT and C60. (c)

GIWAXS map for 1 bilayer of O3 deposited on P3HT by LB method. (d) peak photocurrent

at 550 nm illumination as a function of number of O3 bilayer at interface.

At open circuit condition, the photo-voltage value is determined by the carrier

generation and recombination, following the equation:

𝑉𝑂𝐶 = 𝐸𝑔 −𝑛𝐾𝑏𝑇

𝑞𝑙𝑛(

𝐽𝑅𝑒𝑐

𝐽𝑆𝐶) (5.1)

where Eg is the effective band gap, Kb is the Boltzmann constant, T is the temperature, q is

the elementary charge, JRec is the recombination current and JSC is the final short circuit

photocurrent density.64 Considering a dielectric constant of 3~4 in a typical organic system,

the Coulomb radius is estimated to be 16 nm,65 and the energetic disorder further reduces

this value to 4 nm, probed experimentally.11 This means that if the electron-hole pair

bonded at the charge transfer states can be separated to a distance greater than the Coulomb

radius, significant amount of carrier recombination can be suppressed, that reduces JREC

S

S

S

N

Si

O

O

O

S

S

S

N

Si

O

O

O

AlAl

b

c

Langmuir-Blodgett

a

d

Bilayer device

400

300

200

100

0

Ph

oto

cu

rre

nt

(nA

)86420

layer number

150

100

50

0

Ph

oto

cu

rre

nt

(10

-9 A

)

700650600550500450400Wavelength (nm)

No spacer

1-bilayer O3


98

and increases JSC, which ultimately reduces the voltage loss through non-radiative

recombination. In fact, our experimental data in Figure 5.2 show an increase in JSC

(photocurrent at short circuit condition) which is an indication of reduced recombination

that will lead to increase in VOC.

The purpose of the following electronic structure calculations is to show that such

a large electron-hole separation is energetically feasible using the detailed interface models

shown in Figure 5.1. At this point it should be mentioned that in the single spacer level

model used because of computational economy, the electron-hole pair will have a

separation of only ~3 nm.

5.4.2 Analysis of Energy Levels

The geometrical arrangement of the three components of the standard model is

depicted in Figure 5.1. In this model only a single T3 layer is used because of computational

efficiency. Nevertheless, as the results below will show, the major stabilization effects for

charge separated states can be seen already in this case. The spacer T3 chain faces the C60

unit via the thiol group with interatomic distances of 2.6 – 2.8 Å and P3HT via the alkane

chain at distances of 2.7 – 2.85 Å. The distance between the center of C60 and P3HT is

~28.6 Å. In the extended model (Figure 5.1b, c) three C60 units are arranged horizontally

with distances of around 3.3 Å. The three T3 chains are stacked (side view, Figure 5.1c).

Typical intermolecular distances are displayed in Figure 5.1.

The calculations on the standard trimer have been used to establish the general

structure of the electronic spectrum and to test the validity of the smaller 6-31G basis set

for use with the significantly larger extended trimer. The scheme of the lowest excited

singlet states of the isolated complex is given in Figure 5.3a, b using the 6-31G* and 6-

31G basis sets, respectively. The full list of the 20 excited singlet state energies Ω

calculated can be found in Table C.1 (6-31G basis) and Table C.2 (6-31G* basis) of

Appendix C. For the 6-31G* basis (Figure 5.3a, isolated system), the local C60 states have

been calculated as well. This figure shows that in case of the calculation for the isolated

system they form the lowest band of excitonic states. If this would be the true energy


99

ordering, these states would act as a sink for conversion of CT states, which would lead to

serious losses for the efficiency of the photovoltaic process. The C60 states are followed by

a local π-π* state on P3HT which has the largest oscillator strength, f, of all states

calculated. Next appears another band of C60 states followed by two closely spaced triples

of CT states from P3HTC60 and T3C60 at 3.3 and 3.4 eV, respectively. There are two

more locally excited states, one on T3 and the other one on P3HT. Triples of CT states

alternating between excitations from P3HT and T3 follow. This sequence is interrupted

only by one local state on P3HT. Inclusion of the environment via the LR method Figure

5.3a, b and Table C.4 and Table C.5) does not change much in the relative ordering of

states compared to the isolated system. However, the more sophisticated, but also

numerically more expensive SS solvent calculations change the situation dramatically

(Figure 5.3a, Table C.6). With respect to the P3HT excitonic bright state, the SS approach

leads to a quite dramatic stabilization of nine CT states to be located below the P3HT state.

The energetic spacing between the P3HT→C60 CT states and the T3→C60 CT states is

rather large being about 0.6 eV. There is also a large energy gap of about 0.75 eV between

the P3HT state and the following T3→C60 CT state. Thus, from this model one can already

see the crucial environmental effect which significantly stabilizes the CT state relative to

the others. On the other side, the environmental effect on the energetic location of the

locally excited P3HT and C60 states is small. As a result, the CT states are the lowest ones

and the leaking of the CT states into locally excited C60 states is avoided.


100

Figure 5.3 Excited state energies Ω of the standard P3HT-T3-C60 trimer using the CAM-

B3LYP method and (a) the 6-31G* and (b) the 6-31G basis sets both as isolated systems

and in the LR and SS environments (for numerical data see Table C.1, Table C.2, and Table

C.4-Table C.8). For the isolated system using the 6-31G* basis set (farthest left of (a)), the

fifteen lowest-energy locally excited C60 states have been computed also. Excited states of

(c) the extended P3HT-T3-C60 trimer using the 6-31G* and (d) 6-31G basis sets both as

isolated systems and in the SS environment (Table C.9-Table C.11).

The strongly different behavior of the two solvation models is to be expected since

in the LR method dynamic solvent polarization is computed from the transition density

which is, however, very small for the present case of charge transfer states. Consequently,

the solvent effects are correspondingly small and not adequately represented by this

approach in the case of charge transfer states. On the other hand, the SS approach has been

developed having charge-transfer transitions in mind by representing the dynamic solvent


101

polarization by means of the difference of the electronic densities of the initial and final

states. For a more detailed discussion of this point see e.g. Ref. 66. Therefore, it is

concluded that the SS scheme allows for a better characterization of environmental effects

on CT states and the LR approach will not be considered further here.

Comparison between the 6-31G* and 6-31G basis sets shows that the smaller 6-

31G basis represents the energy spectrum of the standard complex very well and also that

omission of the C60 states by freezing respective orbitals does not affect the remaining

spectrum. These findings are important for the calculations on the extended trimer where

the SS solvent calculations are time consuming. Therefore, they have been performed only

with the 6-31G basis (see below).

The extended trimer was investigated next as an extension of our standard trimer

structure providing a significantly enhanced interface model. (Figure 5.3c, d, Table C.9-

Table C.11). For the isolated system, the lowest bright state is located on P3HT at ~2.65

eV, very similar to the excitation energy in the standard model. The difference to the

standard model is that now several (T3)3→(C60)3 CT states are located below even for the

isolated complex. A rich variety of states is also found above the P3HT state (Figure 5.3c).

Including the SS environment using the 6-31G basis, Figure 5.3d, results in

significant changes with respect to the CT states, the most important one being that the

P3HT CT states are lowest in energy. Below the P3HT bright excitonic state, we computed

nineteen CT states: fifteen (T3)3→(C60)3 CT states and four P3HT→(C60)3 CT states. The

(T3)3→(C60)3 CT states occupy a CT band of 0.44 eV, while the P3HT→(C60)3 CT states

are found in a much narrower range of 0.02 eV. As has already been found for the standard

model, the mentioned CT states are located significantly below the C60 states (which are

around 2.5 eV) and that, therefore, the former states cannot leak into C60 states as soon as

they are formed. In contrast to the results presented here, in TD-DFT thiophene/fullerene

complexes32 locally excited C60 states comprised the lowest-energy band of states. These

differences are probably due to the use of the LR solvation model used in this reference. It

is important to point out that the presence of any -conjugated chromophores adds an

effective dielectric medium and facilitates further stabilization of CT states even though


102

the wavefunction is not directly delocalized into these molecules. To illustrate this trait, we

removed the spacer and calculated only the C60/P3HT system. Table 5.1 records the

calculated energies of the lowest P3HT→C60 CT states. In the P3HT-T3-C60 system, the

solvent stabilization is 1.62 eV, but in the P3HT-C60 system only 0.42 eV. This comparison

shows the strong sensitivity of the electrostatic interaction in the charge separated systems

which are better stabilized by the environment in the case where the charge separation is

more pronounced due to the spacer in between.

Table 5.1 Stabilization energy of the lowest P3HT→C60 CT state of P3HT-T3-C60 and

P3HT-C60 system calculated in the isolated system and the SS environment.

P3HT-T3-C60 P3HT-C60

Isolated system 2.91 2.20

SS environment 1.29 1.78

ΔE (eV) 1.62 0.42

5.4.3 Characterization of Electronic States

The properties of the different types of electronic states are characterized in Figure

5.4 by means of two descriptors. The first one is represented by the most important NTO’s

of typical electronic transitions of the systems calculated in the SS environment (Figure

5.4). All transitions are well represented with one hole/particle pair as the corresponding

weights are above 0.9 e. As discussed above, the lowest excited singlet state band has

P3HT→(C60)3 CT character. This can be nicely seen from the NTO picture of Figure 5.4a,

b, which shows, respectively, the hole density in P3HT and the particle density in the

central C60 unit. These two densities are well separated with essentially no overlap between

the two systems. As second descriptor, the MEP plot displayed in Figure 5.4c shows that

the P3HT unit has a strongly positive potential which extends significantly into the (T3)3

moiety. The electrostatic potential of the middle C60 molecule of the (C60)3 unit is strongly

negative in agreement with the location of the particle NTO in Figure 5.4b. The other two

C60 molecules are slightly negative as well. Overall, the MEP plots give a stronger

delocalized picture of the CT process in comparison to the NTOs, since the localized CT

density affects much larger regions via Coulomb interactions.


103

Figure 5.4 Natural Transition Orbitals (NTO), particle/hole populations and MEP of (a, b,

c) the lowest-energy P3HT→(C60)3 CT state, (d, e, f) the lowest-energy T3→(C60)3 CT

state, (j, k, l) the lowest-energy P3HT bright excitonic state of the extended thiophene

trimer calculated using the CAM-B3LYP/6-31G method in the SS environment, and (g, h,

i) the lowest-energy C60 dark excitonic state of the isolated standard thiophene trimer

calculated using the CAM-B3LYP/6-31G* method. Here we use NTO isovalue ±0.015

e/Bohr3 and MEP isovalue ±0.0003 e/Bohr3.


104

The next band consists of (T3)3(C60)3 CT states. Here the lowest-energy state of

this type (Figure 5.4d,e) shows a transition from an occupied orbital diffusely distributed

across the (T3)3 unit to an orbital located on a single C60 molecule in the (C60)3 unit. The

MEP plot given in Figure 5.4f depicts similar character to the P3HT→(C60)3 delocalized

CT character, however as expected the positive region is spread over the T3 unit and P3HT

is essentially neutral.

The C60 exciton band is represented in Figure 5.4g,h by the lowest-energy C60 state

calculated for the isolated standard trimer. This band is optically dark consisting of an

excitation localized to the C60 molecule, while the MEP plot (Figure 5.4i) shows very little

polarity, which is typical for neutral excitonic states. For the P3HT bright excitonic state,

the primary contribution to the electronic excitation is localized on the P3HT unit (Figure

5.4j,k). The MEP plot of this state displayed in Figure 5.4l shows only little polarity.

5.5 Conclusions

Efficient operation of organic solar cells based on the BHJ architecture is ensured

by tuning sophisticated organic-organic interfaces to align excitonic and charge transfer

excited states to provide robust photo-generated exciton dissociation, while minimizing the

recombination of the dissociated electron and hole across the donor-acceptor interface. The

present study focuses on a detailed molecular model of a polar interface in a bulk

heterojunction, which is suggested by our experiment. Here a self-assembled spacer unit

(oligomer T3) is included in between a P3HT donor and C60 acceptor materials. Following

a non-trivial geometry optimization routine, extensive TD-DFT calculations based on long-

range corrected functional mode and including environmental effects have been performed

to characterize the series of lowest excited singlet states in terms of charge transfer and

excitonic character. The calculations show unequivocally the strong stabilization of the CT

states by environmental effects. The amount of stabilization increases with the distance of

the charges (P3HT→C60 vs. T3→C60 CT) due to improved environmental stabilization. By

comparison with a system of an ion pair without spacer layer (P3HT→C60), a stabilization

of more than 1 eV is computed for the single layer model. Inclusion of more layers are


105

expected to increase the stabilization somewhat further but are currently too expensive in

the framework of an explicit molecular model and state-specific TDDFT. Both spacer

models show that with consideration of environmental interaction the lowest excited states

are formed by bands of CT states, the lowest ones being of P3HTC60 character followed

by T3C60 transitions. The bright excitonic P3HT state is located 0.5 eV above the first

CT below and 1.15 eV above the lowest CT state. Local C60 excitations were found below

the P3HT state, but significantly (~0.9 eV) above the lowest CT state.

Such calculated and analyzed electronic structure fully support our hypothesis

which emerged from experimental study1: photo-generated exciton in the P3HT component

at the BHJ interface either goes to C60 via energy transfer and subsequently dissociates into

a T3C60 CT state, or directly dissociates into a P3HTC60 state. This state is followed

by a transition to the next P3HTC60 state with lower energy, which has even stronger CT

character. Our calculation of the stability of the CT states clearly indicate via the use of

Equation 5.1 that the spacer facilitates separation of the electron and hole, thus diminishing

their Coulomb binding energy and reducing the efficiency of recombination. Moreover,

our simulations demonstrate how the interface ordering and structure affects the energetics

of relevant states via electronic coupling in self-assembled oligomer layer, which further

translates to macroscopic device performance. Such physical processes are general and

underpin manipulation in any donor-acceptor based electronic devices, such as organic

light emitting diodes, photodetectors, sensors, and comprise the design principles for

tailoring the interface properties of organic electronic devices. All our simulations of

electronically excited states are based on the equilibrium molecular geometries. To capture

the effects of vibrational broadening and conformational disorder, it is possible to use

multiple snapshots of structures obtained from classical molecular dynamics simulations.

However, modeling excited states in such a case is currently a numerically intractable task

for the present high level of theory. Likewise, direct modeling of excited state relaxation

with, for instance, surface hopping approaches,6762 are numerically costly and require

reduced Hamiltonian models. This work goes far beyond the scope of this paper and will

be addressed in future investigations.


106

5.6 Acknowledgments

The work at Los Alamos National Laboratory (LANL) was supported by the LANL LDRD

program (W.N., H.T., A.D.M., S.T.). This work was done in part at Center for Nonlinear

Studies (CNLS) and the Center for Integrated Nanotechnologies CINT, project no.

2017AC0027), a U.S. Department of Energy and Office of Basic Energy Sciences user

facility, at LANL. This research used resources provided by the LANL Institutional

Computing Program. We also thank for computer time at the Linux cluster arran of the

School for Pharmaceutical Science and Technology of the University of Tianjin, China.

5.7 Bibliography

(1) Nie, W.; Gupta, G.; Crone, B. K.; Liu, F.; Smith, D. L.; Ruden, P. P.; Kuo, C.-Y.;

Tsai, H.; Wang, H.-L.; Li, H.; Tretiak, S.; Mohite, A. D., Interface Design

Principles for High-Performance Organic Semiconductor Devices. Advanced

Science 2015, 2 (6), 1500024.

(2) Thompson, B. C.; Fréchet, J. M. J., Polymer–Fullerene Composite Solar Cells.


(3) Liang, Y.; Feng, D.; Wu, Y.; Tsai, S.-T.; Li, G.; Ray, C.; Yu, L., Highly Efficient

Solar Cell Polymers Developed via Fine-Tuning of Structural and Electronic

Properties. Journal of the American Chemical Society 2009, 131 (22), 7792-7799.

(4) Yu, G.; Gao, J.; Hummelen, J. C.; Wudl, F.; Heeger, A. J., Polymer Photovoltaic

Cells: Enhanced Efficiencies via a Network of Internal Donor-Acceptor

Heterojunctions. Science 1995, 270 (5243), 1789.

(5) Brédas, J.-L.; Norton, J. E.; Cornil, J.; Coropceanu, V., Molecular Understanding

of Organic Solar Cells: The Challenges. Accounts of Chemical Research 2009, 42

(11), 1691-1699.

(6) Baumeier, B.; May, F.; Lennartz, C.; Andrienko, D., Challenges for in silico design

of organic semiconductors. Journal of Materials Chemistry 2012, 22 (22), 10971-

10976.

(7) Kanai, Y.; Grossman, J. C., Insights on Interfacial Charge Transfer Across

P3HT/Fullerene Photovoltaic Heterojunction from Ab Initio Calculations. Nano

Letters 2007, 7 (7), 1967-1972.

(8) Yi, Y.; Coropceanu, V.; Brédas, J.-L., Exciton-Dissociation and Charge-

Recombination Processes in Pentacene/C60 Solar Cells: Theoretical Insight into

the Impact of Interface Geometry. Journal of the American Chemical Society 2009,

131 (43), 15777-15783.

(9) Sariciftci, N. S.; Heeger, A. J., Reversible, Metastable, Ultrafast Photoinduced

Electron-Transfer from Semiconducting Polymers to Buckminsterfullerene and in


107

the Corresponding Donor-Acceptor Heterojunctions. International Journal of

Modern Physics B 1994, 8 (3), 237-274.

(10) Grancini, G.; Maiuri, M.; Fazzi, D.; Petrozza, A.; Egelhaaf, H. J.; Brida, D.;

Cerullo, G.; Lanzani, G., Hot exciton dissociation in polymer solar cells. Nature

Materials 2013, 12 (1), 29-33.

(11) Gelinas, S.; Rao, A.; Kumar, A.; Smith, S. L.; Chin, A. W.; Clark, J.; van der Poll,

T. S.; Bazan, G. C.; Friend, R. H., Ultrafast Long-Range Charge Separation in

Organic Semiconductor Photovoltaic Diodes. Science 2014, 343 (6170), 512-516.

(12) Bittner, E. R.; Silva, C., Noise-induced quantum coherence drives photo-carrier

generation dynamics at polymeric semiconductor heterojunctions. Nature

Communications 2014, 5, 3119.

(13) Vella, E.; Li, H.; Gregoire, P.; Tuladhar, S. M.; Vezie, M. S.; Few, S.; Bazan, C.

M.; Nelson, J.; Silva-Acuna, C.; Bittner, E. R., Ultrafast decoherence dynamics

govern photocarrier generation efficiencies in polymer solar cells. Scientific

Reports 2016, 6, 29437.

(14) Campbell, I. H.; Crone, B. K., Improving an organic photodiode by incorporating

a tunnel barrier between the donor and acceptor layers. Applied Physics Letters

2012, 101 (2), 023301.

(15) Sampat, S.; Mohite, A. D.; Crone, B.; Tretiak, S.; Malko, A. V.; Taylor, A. J.;

Yarotski, D. A., Tunable Charge Transfer Dynamics at Tetracene/LiF/C60

Interfaces. Journal of Physical Chemistry C 2015, 119 (3), 1286-1290.

(16) Adamo, C.; Jacquemin, D., The calculations of excited-state properties with Time-

Dependent Density Functional Theory. Chemical Society Reviews 2013, 42 (3),

845-856.

(17) Dreuw, A.; Weisman, J. L.; Head-Gordon, M., Long-range charge-transfer excited

states in time-dependent density functional theory require non-local exchange.

Journal of Chemical Physics 2003, 119 (6), 2943-2946.

(18) Dreuw, A.; Head-Gordon, M., Single-reference ab initio methods for the

calculation of excited states of large molecules. Chemical Reviews 2005, 105 (11),

4009-4037.

(19) Kaduk, B.; Kowalczyk, T.; Van Voorhis, T., Constrained Density Functional

Theory. Chemical Reviews 2012, 112 (1), 321-370.

(20) Iikura, H.; Tsuneda, T.; Yanai, T.; Hirao, K., A long-range correction scheme for

generalized-gradient-approximation exchange functionals. The Journal of

Chemical Physics 2001, 115, 3540.

(21) Vydrov, O. A.; Scuseria, G. E., Assessment of a long-range corrected hybrid

functional. Journal of Chemical Physics 2006, 125, 234109.

(22) Stein, T.; Kronik, L.; Baer, R., Reliable Prediction of Charge Transfer Excitations

in Molecular Complexes Using Time-Dependent Density Functional Theory.


(23) Lange, A. W.; Herbert, J. M., Both Intra- and Interstrand Charge-Transfer Excited

States in Aqueous B-DNA Are Present at Energies Comparable To, or Just Above,

the 1ππ* Excitonic Bright States. Journal of the American Chemical Society 2009,

131 (11), 3913-3922.


108

(24) Kronik, L.; Stein, T.; Refaely-Abramson, S.; Baer, R., Excitation Gaps of Finite-

Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals.

Journal of Chemical Theory and Computation 2012, 8 (5), 1515-1531.

(25) Körzdörfer, T.; Sears, J. S.; Sutton, C.; Brédas, J.-L., Long-range corrected hybrid

functionals for π-conjugated systems: Dependence of the range-separation

parameter on conjugation length. The Journal of Chemical Physics 2011, 135 (20),

204107.

(26) Li, H.; Nieman, R.; Aquino, A. J. A.; Lischka, H.; Tretiak, S., Comparison of LC-

TDDFT and ADC(2) Methods in Computations of Bright and Charge Transfer

States in Stacked Oligothiophenes. Journal of Chemical Theory and Computation

2014, 10 (8), 3280-3289.



2004, 393 (1), 51-57.

(28) Limacher, P. A.; Mikkelsen, K. V.; Lüthi, H. P., On the accurate calculation of

polarizabilities and second hyperpolarizabilities of polyacetylene oligomer chains

using the CAM-B3LYP density functional. Journal of Chemical Physics 2009, 130,

194114.



Physics 2008, 10 (44), 6615-6620.

(30) Manna, A. K.; Lee, M. H.; McMahon, K. L.; Dunietz, B. D., Calculating High

Energy Charge Transfer States Using Optimally Tuned Range-Separated Hybrid

Functionals. Journal of Chemical Theory and Computation 2015, 11 (3), 1110-

1117.

(31) Borges, I.; Aquino, A. J. A.; Köhn, A.; Nieman, R.; Hase, W. L.; Chen, L. X.;

Lischka, H., Ab Initio Modeling of Excitonic and Charge-Transfer States in

Organic Semiconductors: The PTB1/PCBM Low Band Gap System. Journal of the


(32) Sen, K.; Crespo-Otero, R.; Weingart, O.; Thiel, W.; Barbatti, M., Interfacial States

in Donor–Acceptor Organic Heterojunctions: Computational Insights into

Thiophene-Oligomer/Fullerene Junctions. Journal of Chemical Theory and

Computation 2013, 9 (1), 533-542.

(33) Balamurugan, D.; Aquino, A. J. A.; de Dios, F.; Flores, L.; Lischka, H.; Cheung,

M. S., Multiscale Simulation of the Ground and Photo-Induced Charge-Separated

States of a Molecular Triad in Polar Organic Solvent: Exploring the Conformations,

Fluctuations, and Free Energy Landscapes. Journal of Physical Chemistry B 2013,

117 (40), 12065-12075.

(34) Perrine, T. M.; Berto, T.; Dunietz, B. D., Enhanced Conductance via Induced II-

Stacking Interactions in Cobalt(II) Terpyridine Bridged Complexes. Journal of

Physical Chemistry B 2008, 112 (50), 16070-16075.

(35) Katz, H. E., Organic molecular solids as thin film transistor semiconductors.

Journal of Materials Chemistry 1997, 7 (3), 369-376.


109

(36) Cornil, J.; Beljonne, D.; Calbert, J. P.; Brédas, J. L., Interchain Interactions in

Organic π-Conjugated Materials: Impact on Electronic Structure, Optical

Response, and Charge Transport. Advanced Materials 2001, 13 (14), 1053-1067.

(37) Bredas, J. L.; Calbert, J. P.; da Silva, D. A.; Cornil, J., Organic semiconductors: A

theoretical characterization of the basic parameters governing charge transport.

Proceedings of the National Academy of Sciences of the United States of America

2002, 99 (9), 5804-5809.

(38) Lee, M. H.; Geva, E.; Dunietz, B. D., The Effect of Interfacial Geometry on Charge-

Transfer States in the Phthalocyanine/Fullerene Organic Photovoltaic System.

Journal of Physical Chemistry A 2016, 120 (19), 2970-2975.

(39) Lee, M. H.; Dunietz, B. D.; Geva, E., Donor-to-Donor vs Donor-to-Acceptor

Interfacial Charge Transfer States in the Phthalocyanine-Fullerene Organic

Photovoltaic System. Journal of Physical Chemistry Letters 2014, 5 (21), 3810-

3816.



(41) Schafer, A.; Horn, H.; Ahlrichs, R., Fully Optimized Contracted Gaussian-Basis

Sets for Atoms Li to Kr. Journal of Chemical Physics 1992, 97, 2571-2577.

(42) Ditchfield, R.; Hehre, W. J.; Pople, J. A., Self‐Consistent Molecular‐Orbital

Methods. IX. An Extended Gaussian‐Type Basis for Molecular‐Orbital Studies of

Organic Molecules. The Journal of Chemical Physics 1971, 54 (2), 724-728.

(43) Petersson, G. A.; Bennett, A.; Tensfeldt, T. G.; Al‐Laham, M. A.; Shirley, W. A.;

Mantzaris, J., A complete basis set model chemistry. I. The total energies of closed‐

shell atoms and hydrides of the first‐row elements. The Journal of Chemical

Physics 1988, 89 (4), 2193-2218.

(44) Petersson, G. A.; Al‐Laham, M. A., A complete basis set model chemistry. II.

Open‐shell systems and the total energies of the first‐row atoms. The Journal of


(45) Cossi, M.; Rega, N.; Scalmani, G.; Barone, V., Energies, structures, and electronic

properties of molecules in solution with the C-PCM solvation model. Journal of

Computational Chemistry 2003, 24 (6), 669-681.

(46) Runge, E.; Gross, E. K. U., Density-Functional Theory for Time-Dependent

Systems. Physical Review Letters 1984, 52 (12), 997-1000.

(47) Petersilka, M.; Gossmann, U. J.; Gross, E. K. U., Excitation Energies from Time-

Dependent Density-Functional Theory. Physical Review Letters 1996, 76 (8), 1212-

1215.

(48) Improta, R.; Barone, V.; Scalmani, G.; Frisch, M. J., A state-specific polarizable

continuum model time dependent density functional theory method for excited state

calculations in solution. Journal of Chemical Physics 2006, 125 (5), 054103.

(49) Hättig, C.; Weigend, F., CC2 excitation energy calculations on large molecules

using the resolution of the identity approximation. Journal of Chemical Physics

2000, 113, 5154-5161.

(50) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H., A consistent and accurate ab initio

parametrization of density functional dispersion correction (DFT-D) for the 94

elements H-Pu. The Journal of Chemical Physics 2010, 132 (15), 154104.


110

(51) Corni, S.; Cammi, R.; Mennucci, B.; Tomasi, J., Electronic excitation energies of

molecules in solution within continuum solvation models: investigating the

discrepancy between state-specific and linear-response methods. Journal of

Chemical Physics 2005, 123 (13), 134512.

(52) Cammi, R.; Corni, S.; Mennucci, B.; Tomasi, J., Electronic excitation energies of

molecules in solution: state specific and linear response methods for

nonequilibrium continuum solvation models. Journal of Chemical Physics 2005,

122 (10), 104513.

(53) Caricato, M., A comparison between state-specific and linear-response formalisms

for the calculation of vertical electronic transition energy in solution with the

CCSD-PCM method. Journal of Chemical Physics 2013, 139 (4), 044116.



Physics Letters 1989, 162 (3), 165-169.















(56) Luzanov, A. V.; Sukhorukov, A. A.; Umanskii, V. É., Application of transition

density matrix for analysis of excited states. Theoretical and Experimental

Chemistry 1976, 10 (4), 354-361.

(57) Mayer, I., Using singular value decomposition for a compact presentation and

improved interpretation of the CIS wave functions. Chemical Physics Letters 2007,

437 (4–6), 284-286.

(58) Martin, R. L., Natural transition orbitals. The Journal of Chemical Physics 2003,

118 (11), 4775-4777.

(59) Plasser, F.; Lischka, H., Analysis of Excitonic and Charge Transfer Interactions

from Quantum Chemical Calculations. Journal of Chemical Theory and

Computation 2012, 8 (8), 2777-2789.

(60) Plasser, F.; Wormit, M.; Dreuw, A., New tools for the systematic analysis and

visualization of electronic excitations. I. Formalism. The Journal of Chemical

Physics 2014, 141 (2), 024106.

(61) Plasser, F. TheoDORE: a package for theoretical density, orbital relaxation, and

exciton analysis, available from http://theodore-qc.sourceforge.net.

http://theodore-qc.sourceforge.net/


111

(62) Smith, K. A.; Lin, Y. H.; Mok, J. W.; Yager, K. G.; Strzalka, J.; Nie, W. Y.; Mohite,

A. D.; Verduzco, R., Molecular Origin of Photovoltaic Performance in Donor-

block-Acceptor All-Conjugated Block Copolymers. Macromolecules 2015, 48

(22), 8346-8353.

(63) Kuo, C. Y.; Liu, Y. H.; Yarotski, D.; Li, H.; Xu, P.; Yen, H. J.; Tretiak, S.; Wang,

H. L., Synthesis, electrochemistry, STM investigation of oligothiophene self-

assemblies with superior structural order and electronic properties. Chemical

Physics 2016, 481, 191-197.

(64) Credgington, D.; Durrant, J. R., Insights from Transient Optoelectronic Analyses

on the Open-Circuit Voltage of Organic Solar Cells. Journal of Physical Chemistry

Letters 2012, 3 (11), 1465-1478.

(65) Lakhwani, G.; Rao, A.; Friend, R. H., Bimolecular Recombination in Organic

Photovoltaics. Annual Review of Physical Chemistry, Vol 65 2014, 65, 557-581.

(66) Guido, C. A.; Jacquemin, D.; Adamo, C.; Mennucci, B., Electronic Excitations in

Solution: The Interplay between State Specific Approaches and a Time-Dependent

Density Functional Theory Description. Journal of Chemical Theory and

Computation 2015, 11 (12), 5782-5790.

(67) Tully, J. C., Molecular dynamics with electronic transitions. The Journal of



112

CHAPTER VI

BENCHMARK AB INITIO CALCULATIONS OF POLY(P-

PHENYLENEVINYLENE) DIMERS: STRUCTURE AND EXCITON

ANALYSIS

Authors: Reed Nieman,a Adelia J. A. Aquino,a,b and Hans Lischkaa,b



Tianjin University, Tianjin 300072, PR China

6.1 Abstract

Crucial benchmark calculations of the structure and electronic excited states

missing from the current literature were performed for oligomers of poly(p-

phenylenevinylene) (PPV) dimers using high-level scaled opposite-spin (SOS) second-

order Møller-Plesset (SOS-MP2) and second-order algebraic diagrammatic construction

(SOS-ADC(2)) calculations. Comparisons were also made to DFT methods. The dimer

structures were studied in two conformations: sandwich-stacked and displaced; the latter

exhibited greater stacking interactions. Good agreement was found between the cc-pVQZ

and aug-cc-pVQZ basis sets and the cc-pVTZ basis. Comparisons of the interaction

energies were made to the complete basis set limit. Basis set superposition error was also

accounted for. The excited state spectra presented a varied and dynamic picture that was

analyzed using transition density matrices and natural transition orbitals (NTOs).

6.2 Introduction

Poly(p-phenylenevinylene) (PPV) is paradigmatic for the study of the mechanisms

affecting the photophysical excitation energy transfer (EET) behavior of organic π-

conjugated polymers on the molecular level that allows the fine-tuning of their absorption

and emission properties which make them so attractive as materials for use in

electroluminescent and photovoltaic devices.1-6 Experiments conclude that features of

coherent PPV dynamics can be attributed to structural effects from coupling to a vibrational


113

mode,5, 7 and structural relaxation is connected to the evolution of excited states necessary

for EET dynamics.

Accurate electronic structure calculations are a prerequisite for the characterization

of these properties, but are hindered by the necessity of large molecular systems and

reliable descriptions of excited states. The Pariser-Parr-Pople (PPP) π electron

Hamiltonian combined with a mechanical force field described the effect of the σ electrons

has been used previously to model surface-hopping dynamics of PPV,8 while later the

Pariser-Parr-Pople-Peierls (PPPP) Hamiltonian was used together with the Configuration

Interaction Singles (CIS) method to compute and classify PPV excited states.9 Calculations

of PPV chains with the collective electronic oscillator (CEO) method together with the

Austin Model 1 (AM1) approach showed exciton self-trapping over six PV units and

demonstrated close agreement with experimental spectra.10 The vibronic structure of the

lowest optical transition of PPV was found using the CIS method with an empirical

description of the electron-phonon coupling.11 A time-dependent density functional theory

(TDDFT) method was used to calculate the singlet-triplet splitting of PPV among other π-

conjugated systems.12 The exchange-correlation (XC) functional dependence of the

localization of the electronic excitation was investigated for polymeric PPV chains.13

Charged polaron localization was found to be sensitive to the polarization of the

environment whereas neutral states are not affected to the same degree.14 A diabatic

Hamiltonian has been developed for use studying wavepacket dynamics in the torsional

modes of PPV vinylene single bonds.15 The kinetic Monte-Carlo method was used to model

fluorescent depolarization to study the interactions between linked chromophoric units in

poly-[2-methoxy-5-((2-ethylhexyl)oxy)-phenylenevinylene (MEH-PPV).16

Though more computationally expensive than semi-empirical or TDDFT

calculations, ab initio methods have also been used to model PPV and PPV-like systems,

though they focus on smaller molecular sizes. The electronic spectrum of stilbene (PPV2)

has been investigated using complete active space perturbation theory to second-order,17

and the electronic spectrum of PPV oligomers up to four repeat PV units (PPV4) has been

studied with symmetry-adapted cluster-configuration interaction (SAC-CI).18 Multi-

configurational time-dependent Hartree (MCTDH) simulations found that dynamic


114

electron-hole transients involve rapid expansion and contraction of the exciton coherence

size in PPV-type systems.19

There is a need to calculate larger oligomer lengths and the approximate coupled-

cluster method to second-order (CC2) has shown to be adequate for the calculations of

methylene-bridged oligofluorenes and oligo-p-phenylenes.20-21 The resolution of the

identity method (RI) together with CC2 has allowed for efficient, less expensive

calculations of excited states as well as having analytical gradients available.22 The second-

order algebraic diagrammatic construction (ADC(2)) is related to the CC2 method and

provides similar results, though it has an advantage over the CC2 method in that excited

states are obtained as eigenvalues of a Hermitian matrix instead of a non-Hermitian Jacobi

matrix as with the coupled-cluster response in the CC2 method.23 Calculations based on

the ADC(2) method investigated the exciton and charge transfer (CT) states of nucleobase

dimers.24 Benchmark calculations using the equation of motion excitation energy coupled-

cluster (EOMEE-CC) method on DNA nucleobases showed that CC2 was able to

reproduce π-π* excitations well compared to EOMEE-CC with singles, doubles, and non-

iterative triples (CCSD(T)).25 CC2 has shown deficiencies however in describing n-π*

states, though these are not types of excitations normally relevant to the UV spectrum of

PPV.25

Excited state properties of PPV oligomers has been calculated in the past by the RI-

ADC(2) method when investigating vertical excitation energies, torsional potentials, and

defect localization in the S1 state in varying sized oligomers.26 The ADC(2) method has

also been used to calculate the absorbance and fluorescence band spectra of PPV oligomers

of increasing chain length,27 while high-level ADC(3) benchmark calculations on PPV

oligomers have shown that the ADC(2) method is adequate for excited state energies,

oscillator strengths, and wave function behavior.28 The scaled opposite spin (SOS)

approach applied to ADC(2) (SOS-ADC(2)) described well the pi-pi* electronic excited

states of several electron donor-acceptor complexes with excitation energies that compared

favorably to experimental results.29


115

Most calculations of PPV oligomers focus only on single chain systems, with few

calculations of dimer systems. Semi-empirical calculations were carried out using the CEO

method together with the AM1 approach to show the dependence on dimer conformation

and intermolecular distance of the formation of interchain electronic excitations in PPV

dimers.30 The AM1 method was used in another investigation finding that the luminescence

capabilities of cofacial PPV dimers is strongly quenched by transition from ground state

to the first excited state.31 The Intermediate Neglect of the Differential Overlap (INDO)

Hamiltonian together with the Single Configuration Interaction (SCI) approach was used

to show that the most stable photogenerated species in interacting PPV chains are

intrachain excitons.32 Mixed quantum/classical calculations of non-adiabatic excitation

processes of long-chain, stacked PPV oligomers showed that the photoexcitation process

leads to thermally-driven polaron hopping between adjacent chains.33 A study on linearly

coupled PPV chains showed the effects of intrinsic conjugation defects on the mobility of

excitons using a tight-binding method with the Su–Schrieffer–Heeger (SSH) model for

which an interchain coupling term is included in the Hamiltonian.34 TDDFT comparisons

of the method to experimental two-photon absorption measurements of PPV derivatives

showed that the hybrid exchange-correlation density functional, Becke three-parameter

Lee, Yang, and Parr (B3LYP) compared well to experimental observations.35 Experimental

results on interchain interactions suggest that interchain electronic phenomena such as

excimers may be responsible for the promotion of charge transport in PPV chains.36-38

The goal of this work is to provide crucial reference data lacking in the literature

by characterizing the geometry and excited states of PPV oligomer dimer systems with

high-level ab initio calculations and wave function analysis tools. To that end, PPV

oligomers chains of from two (stilbene) to four repeating phenylenevinylene (PV) units

(referred to here as PPV2, PPV3, and PPV4) were arranged in either in stacked-sandwich or

displaced dimer conformations. The basis set dependence of the geometric characteristics

was investigated using the SOS approach39-40 applied to second-order Møller-Plesset

Perturbation theory (MP2)41 with several correlation-consistent basis sets with emphasis

on extrapolation of the stabilization energy to the complete basis set (CBS) limit. Excited

states and S1 excited state geometries were studied using the SOS-ADC(2) method.42-43


116

The SOS-ADC(2) method was also compared to DFT utilizing B3LYP and the long-range

corrected Coulomb Attenuating Method with B3LYP (CAM-B3LYP).


The geometry of PPV dimers was investigated using PPV chains of from two to

four repeating PV units arranged in either stacked-sandwich, belonging to the C2h point

group, or displaced, belonging to the C1 point group, structural conformations (Figure 6.1).

Geometry optimizations for both structural conformations were carried out using the SOS-

MP2 method using the RI approximation.44-45 A variety of Dunning-type correlation

consistent basis sets were used in order to show the dependence of the basis on the

intermolecular stacking distances and the stabilization energies. For sandwich-stacked

PPV2 dimer, the geometry was studied with the largest number of basis sets, cc-pVXZ, and

aug-cc-pVXZ where X=D, T, and Q,46-47 as its relatively small size allowed for more

extensive investigations. Smaller basis sets were considered for sandwich-stacked PPV3

and PPV4 dimers, cc-pVDZ and cc-pVTZ. Though geometry optimization calculations for

the sandwich-stacked PPV3 and PPV4 dimers with the cc-pVQZ basis set would be too

computationally taxing, single-point calculations were made for both systems with the cc-

pVQZ basis set at the cc-pVTZ optimized geometry. The displaced dimers lack the

symmetry advantages of the sandwich-stacked dimers and so were treated with a fewer

basis sets, cc-pVXZ (X=D, T, and Q) and aug-cc-pVDZ for PPV2 dimer, cc-pVDZ and cc-

pVTZ for PPV3 and PPV4. For the sandwich-stacked PPV2 dimer, the effect of using the

SOS approach is compared to geometries optimized with the MP2 method. Sandwich-

stacked PPV3 dimer geometries were compared to density functional theory (DFT)

methods using the B3LYP48 and CAM-B3LYP49-50 methods together with the D351

dispersion correction approach and the cc-pVTZ basis set. Basis set superposition error

(BSSE) was included using the counterpoise (CP) correction method52 for MP2 and SOS-

MP2 calculations.


117

Figure 6.1 Top/side views of the stacked PPV oligomers and labeling scheme.

Extrapolation to the CBS limit of the stabilization energies was carried out using

the two-point fit approach given by Halkier et al in Equation 6.1 for the TQ extrapolation,53


118

𝐸𝑇𝑄∞ =

𝐸𝑄𝑄3 − 𝐸𝑇𝑇3

𝑄3 − 𝑇3 (6.1)

where 𝐸𝑇𝑄∞ is the CBS limit stabilization energy, 𝑄 and 𝑇 are the cardinal numbers four

(for the cc-pVQZ or aug-cc-pVQZ basis sets) and three (for the cc-pVTZ or aug-cc-pVTZ

basis sets), respectively, and 𝐸𝑄 and 𝐸𝑇 are the stabilization energies found for those

individual basis sets.

Vertical excitations were calculated using the SOS-ADC(2) method with the RI

approximation and the basis sets specified earlier for stacked PPV2, but only cc-pVTZ was

considered for the stacked PPV3 and PPV4 dimers. Two states per irreducible

representation were calculated for each case. The optimized geometry of the S1 excited

state was found for each stacked dimer using the SOS-ADC(2) method with the cc-pVTZ

basis set. For stacked PPV3 dimers, comparative TDDFT calculations were carried out

using the B3LYP-D3, CAM-B3LYP-D3 methods and the cc-pVTZ basis set. Due to the

computational efficiency afforded by TDDFT methods, thirty states were calculated with

each method.

All excited states were analyzed via natural transition orbitals (NTO)54 and the

NTO participation ratio (PRNTO), while charge transfer between fragments for a given

electronic transition was described using the q(CT) value55 computed from the transition

density, 𝐷0𝛼. The value, q(CT), is calculated first by using the transition density to define

the omega matrix, Ω𝐴𝐵𝛼 ,

Ω𝐴𝐵𝛼 =

1

2∑(𝐷0𝛼.[𝐴𝑂]𝑆[𝐴𝑂])

𝑎𝑏(𝑆[𝐴𝑂]𝐷0𝛼.[𝐴𝑂])

𝑎𝑏𝑎∈𝐴𝑏∈𝐵

(6.2)

such that α is the electronic state and AO are the atomic orbitals. The omega matrix, Ω𝐴𝐵𝛼 ,

describes the charge transfer contribution from fragment A to fragment B (A ≠ B), and

contributions from excitations occurring on the same fragment (A = B). The CT character

for a given system composed of multiple fragments is then given by q(CT):

𝑞(CT) =1

Ω𝛼∑ ∑ Ω𝐴𝐵

𝛼

𝐴≠𝐵𝐴

(6.3)


119

where Ω𝛼 is the sum of the charge transfer values for all fragment pairs, A and B. When

q(CT) = 1 e, one electron has been transferred, and when q(CT) = 0 e, the transition is local

a excitation or Frenkel exciton state.

The value of PRNTO indicates the number of essential participating NTOs and,

therefore, how many configurations are important to the description of the excited state.

PRNTO is calculated as:

𝑃𝑅𝑁𝑇𝑂 =(∑ 𝜆𝑖𝑖 )2

∑ 𝜆𝑖2

𝑖

(4)

where 𝜆𝑖 is the weight for each hole-to-electron orbital transition.

All MP2, SOS-MP2, and SOS-ADC(2) calculations were performed using the

TURBOMOLE program package,56 and all DFT and TDDFT calculations were done with

the Gaussian G09 program package.57 Excited states were analyzed via natural transition

orbitals (NTO), q(CT), and omega matrices utilizing the wave function analysis program

TheoDORE.55, 58-59


6.4.1 Structural Description

The geometry of stacked PPV2 has been optimized utilizing the SOS-MP2 method

at several basis sets and described via intermolecular stacking distances in Table 6.1 with

labeling provided in Figure 6.1. In generally, these distances for the phenylene units were

larger than the vinylene unit. The aug-cc-pVDZ and aug-cc-pVTZ basis sets show

considerably smaller intermolecular distances compared to the cc-pVDZ and cc-pVTZ

basis sets, respectively, especially going toward the interior. Going from cc-pVDZ to cc-

pVTZ, the distances decrease only slightly and the structure found for cc-pVTZ agreed

well with the cc-pVQZ structure. The distances found for the cc-pVQZ and the aug-cc-

pVQZ basis sets are nearly identical. BSSE counterpoise corrections showed increased

intermolecular stacking distances of an average of about 0.37 Å for the cc-pVDZ basis set.


120

This increase in distance is an average of about 0.14 Å for the cc-pVTZ basis, and is

relatively unchanged for the aug-cc-pVDZ and cc-pVQZ basis sets. In spite of the

relatively large changes in the distances, the differences in distances between the basis sets

with and without the diffuse functions and the BSSE corrections do converge, but only at

the QZ basis. Finally, when using the MP2 method without the SOS approach (Table D.1),

the stacking distances were smaller in general with each basis set utilized than those found

with the SOS-MP2 method, an effect that extended to the BSSE corrections as well.

Table 6.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer computed

at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets using C2h symmetry.

1-1’ 2-2’ 3-3’ 4-4’ 5-5’ ΔE

SOS-MP2

cc-pVDZ 3.83 3.82 3.80 3.85 3.86 -3.99

aug-cc-pVDZ 3.67 3.61 3.61 3.66 3.72 -12.24

cc-pVTZ 3.80 3.79 3.77 3.82 3.83 -5.41

aug-cc-pVTZ 3.80 3.63 3.64 3.68 3.86 -8.26

cc-pVQZ 3.81 3.80 3.77 3.83 3.84 -5.07

aug-cc-pVQZ 3.81 3.80 3.77 3.83 3.84 -5.96

CBS limit extrapolation:

TQa -4.79

aug-TQb -4.27

SOS-MP2/BSSE

cc-pVDZ 4.21 4.18 4.17 4.21 4.23 -1.26

aug-cc-pVDZ 3.67 3.62 3.61 3.67 3.72 -2.58

cc-pVTZ 3.95 3.93 3.93 3.94 3.96 -3.40

cc-pVQZ 3.81 3.80 3.77 3.83 3.83 -4.04


TQa -4.51

Stacked PPV3 (Table 6.2) showed only a slight decrease in the intermolecular

stacking distances when going from cc-pVDZ to cc-pVTZ. BSSE counterpoise corrections

again calculate much larger distances between monomers. The average intermolecular

distances found using the cc-pVDZ and cc-pVTZ basis sets for stacked PPV3 are only

marginally smaller than those found for stacked PPV2 using the SOS-MP2 method.

Similarly, there were only small differences between the two stacked systems for the two

basis sets after BSSE counterpoise correction. Comparing the SOS-MP2 method to DFT

methods at the cc-pVTZ basis set for stacked PPV3, the stacking distances are marginally


121

smaller when the B3LYP-D3 method was used, but are larger by about 0.05 Å with the

long range-corrected CAM-B3LYP-D3 method. The intermolecular stacking distances of

stacked PPV4 are shown in Table D.2 for the SOS-MP2 and SOS-MP2/BSSE methods. An

increase in basis set size from cc-pVDZ to cc-pVTZ showed a similar in intermolecular

distances as with stacked PPV3, and the average stacking distance decreased in general for

stacked PPV4 compared to stacked PPV2.

Table 6.2 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV3 dimer computed

at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets using C2h symmetry.

1-1’ 2-2’ 3-3’ 4-4’ 5-5’ 6-6’ 7-7’ 8-8’ ΔE

SOS-MP2

cc-pVDZ 3.82 3.81 3.77 3.77 3.77 3.80 3.82 3.83 -7.12

cc-pVTZ 3.81 3.81 3.76 3.76 3.76 3.79 3.82 3.83 -9.54

cc-pVQZa - - - - - - - - -8.97


TQb -8.55

SOS-MP2/BSSE

cc-pVDZ 4.16 4.15 4.07 4.08 4.07 4.14 4.18 4.18 -2.64

cc-pVTZ 3.94 3.93 3.88 3.87 3.87 3.91 3.95 3.96 -6.26

cc-pVQZa - - - - - - - - -7.37


TQb -8.18

B3LYP-D3

cc-pVTZ 3.77 3.83 3.75 3.75 3.73 3.79 3.83 3.77 -

10.62 CAM-B3LYP-D3

cc-pVTZ 3.89 3.84 3.81 3.80 3.80 3.85 3.85 3.89 -8.99 aSingle-point energy at SOS-MP2/cc-pVTZ optimized geometry, bExtrapolation made

using cc-pVTZ and cc-pVQC

The sandwich-stacked PPV dimers were compared to a displaced dimer geometry

configuration using the SOS-MP2 method, Table 6.3. Displaced PPV2 dimer displayed

smaller intermolecular distances than the sandwich stacked PPV2 dimer by an average of

about 0.3 Å, but showed similar trends. The intermolecular distances increased from cc-

pVDZ to cc-pVQZ, with increasing basis set size. Displaced PPV3 dimer, Table 6.4, using

the cc-pVTZ basis, had smaller intermolecular distances compared to the sandwich-stacked

PPV3 dimer and displaced PPV2 dimer using the same basis set. The intermolecular


122

distances of displaced PPV4, Table D.3, were smaller compared to the sandwich-stacked

PPV4 dimer by about 0.3 Å, but larger than either displaced PPV2 or PPV3 dimers.

Table 6.3 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV2 dimer

computed at SOS-MP2 level using C1 symmetry.

1-1’ 2-2’ 3-3’ 4-4’ ΔE

cc-pVDZ 3.52 3.44 3.44 3.52 -7.69

cc-pVTZ 3.54 3.44 3.45 3.54 -8.73

cc-pVQZ 3.54 3.45 3.45 3.54 -8.12


TQa -7.68 aExtrapolation made using cc-pVTZ and cc-pVQC

Table 6.4 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV3 dimer

computed at SOS-MP2 level using C1 symmetry

1-1’ 2-2’ 3-3’ 4-4’ 5-5’ 6-6’ ΔE

cc-pVTZ 3.35 3.43 3.39 3.39 3.34 3.53 -15.10

cc-pVQZa - - - - - - -14.04


TQb -13.28 aSingle-point energy at SOS-MP2/cc-pVTZ optimized geometry, bExtrapolation made


The stability of these PPV dimer systems was described by calculating the

interaction energy with comparison to the extrapolated CBS limit. For the stacked PPV2

dimer (Table 6.1), the extrapolation of the CBS limit interaction energy is found to be less

stabilizing than that found with the cc-pVDZ basis. The interaction energy given for the

aug-cc-pVDZ is greatly overstabilized compared to the CBS limit energy, a larger

difference than is found for any other case. The cc-pVQZ interaction energy is found very

close to the CBS limit. The interaction energy at the CBS limit using the SOS-MP2/BSSE

method was less stabilizing than that found for the SOS-MP2 method, but was more

stabilizing than the interaction energies found for the individual basis sets using

counterpoise corrections; using the cc-pVDZ basis was much less stabilizing than the CBS

limit by about 3.3 kcal/mol, though the interaction energy at the cc-pVQZ basis was only


123

slightly less stable. The interaction energy of the displaced PPV2 dimer is more stabilizing

compared to the sandwich stacked system, reflective of the smaller intermolecular

distances, and the energy at the CBS limit compares most closely to that found using the

cc-pVQZ basis. This stabilization of π-conjugated aromatic systems coming from

displacement has been rigorously shown in previous investigations of benzene and

naphthalene dimers.60-65

Without the SOS approach, the MP2 method, Table D.1, the interaction energy was

found to be more stabilizing for all basis sets by about 6 kcal/mol without BSSE. This

showed the necessity of the SOS approach to correct the over-estimation of the stabilization

energy of dimer complexes when using the MP2 method. A similar effect was

demonstrated for stacked complexes of benzene and naphthalene with tetracyanoethylene

(TCNE) (benzene/TCNE and naphthalene/TCNE, respectively)29 where the MP2 method

alone considerably overestimated the interaction energy compared to the SOS-MP2

approach. Calculations with the highly correlated CCSD(F12*)(T) method were also

performed for the benzene/TCNE system (which was found to agree quite well with

experimental findings). The SOS-ADC(2) and CCSD(F12*)(T) methods differed only

slightly from each other.

The interaction energies of the sandwich-stacked PPV3 dimer (Table 6.2) using the

SOS-MP2 method at the CBS limit was less stable than when using the cc-pVQZ and cc-

pVTZ basis sets but was more stable than for the cc-pVDZ basis. CP corrections for the

interaction energy at the CBS limit became slightly less stable compared to the extrapolated

value with the SOS-MP2 method. Using the CAM-B3LYP-D3/cc-pVTZ method compares

well to the SOS-MP2 interaction energy at the CBS limit, being only a little more stable,

though the B3LYP-D3/cc-pVTZ method showed an over-stabilization of about 2 kcal/mol.

The interaction energy at the CBS limit of the displaced PPV3 dimer (Table 6.4) was about

4.7 kcal/mol more stable than for the sandwich-stacked PPV3 dimer and is less stable

compared to the cc-pVQZ basis set.

The CBS limit of the interaction energy found for the stacked PPV4 dimer (Table

D.2) was more stable than the cc-pVQZ energy, and the displaced PPV4 dimer (Table D.3)


124

is stabilized by about 6 kcal/mol at the CBS limit compared to the stacked dimer. The CBS

limit for the displaced dimer is less stable than the cc-pVQZ basis.

Several general trends are apparent across all investigated structures. With

increasing basis set size (not including augmented basis sets or BSSE corrections), the

interchain stacking distances increase slightly and interaction energies become more

stabilizing. Decreases in stacking distances and increasingly stable interaction energies are

found with larger chain lengths. The displaced geometries have much smaller stacking

distances compared to the sandwich stacked dimers, and are greatly stabilized as a result.

The BSSE counterpoise correction largely overestimates the interchain stacking distances

and greatly decreases the stabilization energies. The augmented basis sets in general

displayed much smaller interchain stacking distances and greatly overestimated interaction

energies. In light of this, augmented basis sets and counterpoise BSSE corrections are not

recommended for intermolecular geometries.

6.4.2 Excited States

The excited states of the stacked PPV dimers were investigated using the SOS-

ADC(2) method and compared to TDDFT for stacked PPV3. The vertical excitations for

stacked PPV2 using the SOS-ADC(2)/cc-pVQZ method are presented in Table 6.5. Two

states per irreducible representation were calculated. The S1 state (1 1Bg) is dark, possesses

partial interchain CT character, and is found at 4.07 eV. Two intensive transitions, or bright

states, were found, the S4 and S6 states (1 1Bu and 2 1Bu, respectively) where the latter has

greater oscillator strength. Both were found to be local exciton states with negligible

interchain charge transfer. In the spectrum of the monomer (Table D.4), the bright states,

S1 and S3, occur at approximately the same energy as the dimer, though the oscillator

strength of the S2 state is reduced by about 0.8.


125

Table 6.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVQZ and SOS-

MP2/cc-pVQZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator

strength, CT – interchain charge transfer)

State Excitation ΔE (eV) f CT (e)

S1 1 1Bg 4.07 0.00 0.17

S2 1 1Au 4.49 0.00 0.09

S3 2 1Bg 4.50 0.00 0.08

S4 1 1Bu 4.52 0.83 0.01

S5 1 1Ag 4.64 0.00 0.02

S6 2 1Bu 4.75 1.18 0.01

S7 2 1Au 5.59 0.00 0.21

S8 2 1Ag 5.96 0.00 0.02

Omega matrices revealed more detailed information about the nature of each

excitation and was examined for stacked PPV2 dimer using the SOS-MP2/cc-pVQZ

method (Figure 6.2). The small amount of off-diagonal character seen in the dark S1 and

S7 states correlates with calculated CT values (Table 6.5). Distinct nodes between the

individual chains (fragments 1-3 belong to one chain and fragments 4-6 belong to the other)

are visible for remaining six excitations. The bright 1 1Bu and 2 1Bu excitations, the S4 and

S6 states, respectively, are characterized by local excitations in the phenylene groups.


126

Figure 6.2 Omega matrices for vertical excitations of stacked PPV2 dimer using the SOS-

ADC(2)/cc-pVQZ method with the SOS-MP2/cc-pVQZ optimized geometry. Each

vinylene and phenylene group is a separate fragment: fragments 1-3 belong to one chain,

fragments 4-6 to the other. The y-axis represents the electron and the x-axis represents the

hole. Omega matrix plots calculated using the TheoDORE program package.

As with the structural investigation, the basis set dependence of stacked PPV2 was

studied comparing the cc-pVQZ basis (Table 6.5) to the cc-pVDZ and cc-pVTZ basis sets

(Table D.5). The relative ordering of the states remains constant. The S1 state is stabilized

with increasing basis set size. For each basis, the oscillator strengths of the bright states (S4

and S6) were found to be sensitive to the basis set. Compared to the monomers, the first

bright state was found to be well preserved, though the oscillator strength of the second

bright state is greatly reduced as was seen with the cc-pVQZ case. An exception to this

trend was the cc-pVDZ basis where the second bright state is stabilized by about 0.07 eV

and the oscillator strengths are increased for the two bright states.

The NTOs for the vertical excitations were plotted for stacked PPV2 (Table D.6).

Each excitation had contributions from at least two primary transitions. For the S1 state,

the primary contributing hole NTO is located on the separated phenylene and vinylene

groups while the electron NTO is somewhat mixed. This same intrachain fragment

interaction is present in several other states and explains some of the off-diagonal character

seen in the omega matrices (Figure 6.2). The NTOs of the partial CT state, S7, displays,

S1 S2 S3 S4

S5 S6 S7 S8


127

while not much interchain interaction, transfer of electron density between intrachain

fragments.

The spectrum of displaced PPV2 dimer was calculated (Table 6.6) to compare to

the stacked dimer configuration showing non-negligible destabilization of the first several

excitations and the introduction of interchain CT states. Displacement from the stacked

conformation raises the energy of the S1 state compared to the stacked, though it remains

a dark, local exciton. The first bright state is now the S2 state and was destabilized in the

displaced dimer by about 0.08 eV. The S2 and S3 states in the stacked dimers were both

destabilized to above the bright state. Two pronounced intermolecular CT states, S7 and

S8, were found for the displaced dimer at approximately 5.4 eV. In the associated omega

matrices (Figure D.1), the CT states can be seen to have primarily involved the phenylene

units. This can be seen in the NTOs (Table D.7) for the S7 state, though the NTOs for the

S8 state are more complex. The S4 and S5 states showed unique intrachain CT character

compared to the sandwich-stacked dimer involving only electron/hole density on one

phenylene group per chain. These two excitations both have NTO participation ratios

(PRNTO) of about four in which the intrachain exciton character was represented well.

Table 6.6 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and

SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f –

oscillator strength, CT – interchain charge transfer)


S1 1 1A 4.40 0.00 0.03

S2 2 1A 4.58 0.53 0.05

S3 3 1A 4.64 0.01 0.05

S4 4 1A 4.64 0.00 0.09

S5 5 1A 4.66 0.00 0.09

S6 6 1A 4.77 1.43 0.04

S7 7 1A 5.39 0.00 0.95

S8 8 1A 5.40 0.08 0.94

The spectra of stacked PPV3 (Table D.8) and PPV4 (Table D.9) show that the S1

states occur at lower energies compared to stacked PPV2 (Table 6.5), and are also lower in

energy compared to the experimental values by 0.22 eV 0.27 eV, respectivley. The bright


128

states, S2 for both PPV3 and PPV4, are likewise stabilized considerably going to longer

chain lengths. In the omega matrices (Figure 6.3) of the vertical excitations of stacked

PPV3, five of the eight calculated states are similar in character to the excitations found in

stacked PPV2: transitions beginning and ending on the capping phenylene groups. The

remaining three states, S1, S2, and S4, are transitions primarily involving the interior

phenlylene groups. The excitation with the largest interchain CT character (Table D.8,

theS1 state) is seen to mostly involve the interior phenylene groups on each chain. Other

partial interchain CT states, S5 and S6, appear to have very similar transitions: mainly

phenylene-phenelyene with some phenylene-vinylene involvement. The S1, S2, and S8

states are shown to have the most phenylene-vinylene CT character. The omega matrices

of stacked PPV4 (Figure D.2) were similar in character to those of stacked PPV3.

Figure 6.3 Omega matrices for vertical excitations of stacked PPV3 dimer using the SOS-

ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each




The density of states is shown in Figure 6.4a,b,e for stacked PPV2, stacked PPV3,

and PPV4 using the SOS-ADC(2) method. The energy range containing the highest density

of states was stabilized from the 4.5-5.0 eV range in stacked PPV2 (four states) to the 4.0-

4.5 eV range in stacked PPV3 and PPV4 (three and four states respectively) coinciding with

the stabilization of the bright states. While there are two states in the 5.5-6.0 eV range, this

too shifted to lower energies, as no calculated state was found higher in energy than 5.5 eV

for stacked PPV3 and 5.0 eV in stacked PPV4.

S1 S2 S3 S4

S5 S6 S7 S8


129

Examining the spectrum of the displaced PPV3 dimer (Table 6.7), the same

destabilization compared to the sandwich stacked dimer is found that was seen for the PPV2

dimers. The S1 state was destabilized more than most of the calculated states. There was a

third state (S5) with significant oscillator strength appearing about 0.07 eV higher than the

second bright state that displayed partial interchain CT character (~0.5 e). The omega

matrices for this excitation (Figure D.3) showed off-diagonal elements corresponding to

transitions primarily from/to the interior phenylene groups of both chains, though also

involving the vinylene and capping phenylene groups. The NTOs (Table D.10) for the two

primary contributing transitions showed symmetric distribution of hole density shifting to

one side of both dimers. There were four other excitations in addition to the S5 state with

CT character greater than 0.1 e. Of these the S6 and S7 states had significant off-diagonal

elements in the omega matrices with transitions from a variety of fragments.


130

Figure 6.4 Calculated line spectra (right y-axis) and density of states (left y-axis) for the

stacked PPV oligomers optimized using the SOS-ADC(2), B3LYP-D3, and CAM-B3LYP-

D3 methods. Stacked a) PPV2 (Table 6.5), b-d) PPV3 (Table D.8, S11-S12), e) PPV4 (Table

D.9).

a)

b)

c)

d)

e)


131

Table 6.7 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and

SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f –



S1 1 1A 3.77 0.00 0.04

S2 2 1A 4.02 3.47 0.06

S3 3 1A 4.34 0.00 0.17

S4 4 1A 4.38 0.32 0.07

S5 5 1A 4.55 0.11 0.52

S6 6 1A 4.58 0.00 0.39

S7 7 1A 4.59 0.02 0.30

S8 8 1A 4.61 0.00 0.13

TDDFT methods were compared to the SOS-ADC(2) using the cc-pVTZ basis for

stacked PPV3. The relative ordering of the first eight excited states is mostly preserved with

the B3LYP-D3 method (Table D.8). The S1 state is stabilized using both TDDFT methods

compared to the previous method, though much more so with B3LYP-D3. The first bright

state (Table D.8, S2) was found to be stabilized using the B3LYP-D3 (Table D.11, S3) and

CAM-B3LYP-D3 (Table D.12, S2) methods (about 0.7 and 0.3 eV respectively). The

oscillator strength of this transition is also reduced for B3LYP-D3 and CAM-B3LYP-D3.

Though only this one bright state was found when utilizing the B3LYP-D3 method, a third

bright state was found with the CAM-B3LYP-D3 method, located about 0.2 eV higher than

the second and is a partial CT state. The density of states (Figure 6.4c,d) using both TDDFT

methods is much larger than with SOS-ADC(2) approach. The majority of transitions occur

in the energy range of 4.0-4.5 eV (twelve states), similar to what was discussed before.

This peak in states is destabilized to the range of 5.0-5.5 eV (eight states) using the CAM-

B3LYP-D3 method.


132

Tab

le 6

.8 I

nte

rchai

n d

ista

nce

s (Å

) in

the

stac

ked

PP

V d

imer

s o

pti

miz

ed f

or

π-π

* S

1 (

1

1B

g)

exci

ted s

tate

usi

ng t

he

cc-p

VT

Z b

asis

and C

2h s

ym

met

ry


133

The B3LYP-D3 and CAM-B3LYP-D3 methods presented more varied CT

character compared to the SOS-ADC(2) method. Though only one bright state was found

using the B3LYP-D3 method in the thirty states calculated (S3), it is located energetically

higher than one prominent interchain CT state (S2) and one partial interchain CT state (S1).

The first prominent CT state using the CAM-B3LYP-D3 method (S3) was located about

0.3 eV higher in energy than the bright state and had oscillator strength greater than zero.

The geometry of the S1 state (1 1Bg) was optimized (Table 6.8) and compared to the

ground state. In all cases a decrease in intermolecular stacking distances was observed and

was greatest for stacked PPV2 and least for stacked PPV4. The largest decrease in stacking

distances occurred for the vinylene units (about 0.9 Å for stacked PPV2) and generally

becomes greater toward the interior of the stacked dimers. Comparing the SOS-ADC(2)

method to TDDFT in stacked PPV3, the CAM-B3LYP-D3 method deviated slightly less

on average than the B3LYP-D3 method which deviated quite strongly (over 0.1 Å) in the

interior phenylene and vinylene groups.

6.5 Conclusions

Ab initio calculations were performed on stacked PPV oligomers of from two to

four repeating PV units with the purpose of providing necessary benchmark calculations.

Geometry optimizations showed that the intermolecular stacking distances became

generally shorter with increasing basis set size. In stacked PPV2, the structure obtained

with the cc-pVTZ basis closely matched those found with the cc-pVQZ and aug-cc-pVQZ

basis sets and the CP method to correct reproduced the cc-pVQZ structure well, though the

cc-pVDZ and aug-cc-pVDZ basis sets were found to, respectively, overestimate and

underestimate intermolecular distances. Increasing system size increased the

intermolecular stacking distances going to PPV3 and PPV4. Comparisons with DFT

methods showed that the B3LYP-D3 method more closely reproduced the structure found

with the SOS-MP2 method. Displacement from the exact sandwich structure was found to

play a large role as the displaced PPV dimers had much smaller stacking distances than the

sandwich-stacked dimers.


134

The calculated interaction energies were compared to the extrapolated CBS limit.

The interactions energy found for stacked PPV2 using the aug-cc-pVDZ basis set strongly

overestimated the stabilization, though the interaction energy found with the cc-pVQZ

basis was close to the CBS limit. CP corrections decreased the magnitude of the interaction

energy, though the calculated CBS limit for SOS-MP2/BSSE method was still relatively

close to stabilization energy found for the SOS-MP2 method. The CAM-B3LYP-D3

method gave stabilization energies slightly more stabilizing than the SOS-MP2 mothod,

but the B3LYP-D3 method differed more strongly.

Vertical excitation calculation using the SOS-ADC(2) method found two intensive

transitions in the eight calculated excited states for each case. The oscillator strength of the

first bright state increased with going from PPV2 to PPV3 and PPV4, but the oscillator

strength of the second decreased. Though interchain CT character is more or less constant

with increasing chain size, intrachain CT increases with a larger number of contributing

phenylene and vinylene fragments as analyzed using omega matrices and NTO transitions.

Comparative DFT methods displayed a much more diverse range of intermolecular CT

states than was found for the SOS-ADC(2) method.

6.6 Bibliography

(1) Brédas, J.-L.; Cornil, J.; Beljonne, D.; dos Santos, D. A.; Shuai, Z., Excited-State

Electronic Structure of Conjugated Oligomers and Polymers: A Quantum-

Chemical Approach to Optical Phenomena. Accounts of Chemical Research 1999,

32 (3), 267-276.

(2) Meier, H.; Stalmach, U.; Kolshorn, H., Effective conjugation length and UV/vis

spectra of oligomers. Acta Polymerica 1997, 48 (9), 379-384.

(3) Natasha, K., Understanding excitons in optically active polymers. Polymer

International 2008, 57 (5), 678-688.

(4) Peeters, E.; Ramos, A. M.; Meskers, S. C. J.; Janssen, R. A. J., Singlet and triplet

excitations of chiral dialkoxy-p-phenylene vinylene oligomers. The Journal of


(5) Collini, E.; Scholes, G. D., Coherent Intrachain Energy Migration in a Conjugated

Polymer at Room Temperature. Science 2009, 323 (5912), 369.

(6) Facchetti, A., π-Conjugated Polymers for Organic Electronics and Photovoltaic

Cell Applications. Chemistry of Materials 2011, 23 (3), 733-758.


135

(7) Brédas, J.-L.; Silbey, R., Excitons Surf Along Conjugated Polymer Chains. Science

2009, 323 (5912), 348.

(8) Sterpone, F.; Rossky, P. J., Molecular Modeling and Simulation of Conjugated

Polymer Oligomers: Ground and Excited State Chain Dynamics of PPV in the Gas

Phase. The Journal of Physical Chemistry B 2008, 112 (16), 4983-4993.

(9) Sun, Z.; Li, S.; Xie, S.; An, Z., A study on excited-state properties of conjugated

polymers using the Pariser-Parr-Pople-Peierls model combined with configuration-

interaction-singles. Organic Electronics 2018, 57, 277-284.

(10) Tretiak, S.; Saxena, A.; Martin, R. L.; Bishop, A. R., Conformational Dynamics of

Photoexcited Conjugated Molecules. Physical Review Letters 2002, 89 (9), 097402.

(11) Karabunarliev, S.; Bittner, E. R., Polaron–excitons and electron–vibrational band

shapes in conjugated polymers. The Journal of Chemical Physics 2003, 118 (9),

4291-4296.

(12) Pogantsch, A.; Heimel, G.; Zojer, E., Quantitative prediction of optical excitations

in conjugated organic oligomers: A density functional theory study. The Journal of


(13) Nayyar, I. H.; Batista, E. R.; Tretiak, S.; Saxena, A.; Smith, D. L.; Martin, R. L.,

Localization of Electronic Excitations in Conjugated Polymers Studied by DFT.

The Journal of Physical Chemistry Letters 2011, 2 (6), 566-571.

(14) Nayyar, I. H.; Batista, E. R.; Tretiak, S.; Saxena, A.; Smith, D. L.; Martin, R. L.,

Role of Geometric Distortion and Polarization in Localizing Electronic Excitations

in Conjugated Polymers. Journal of Chemical Theory and Computation 2013, 9

(2), 1144-1154.

(15) Sterpone, F.; Martinazzo, R.; Panda Aditya, N.; Burghardt, I., Coherent Excitation

Transfer Driven by Torsional Dynamics: a Model Hamiltonian for PPV Type

Systems. In Zeitschrift für Physikalische Chemie, 2011; Vol. 225, p 541.

(16) Singh, J.; Bittner, E. R.; Beljonne, D.; Scholes, G. D., Fluorescence depolarization

in poly[2-methoxy-5-((2-ethylhexyl)oxy)-1,4-phenylenevinylene]: Sites versus

eigenstates hopping. The Journal of Chemical Physics 2009, 131 (19), 194905.

(17) Molina, V.; Merchán, M.; Roos, B. O., Theoretical Study of the Electronic

Spectrum of trans-Stilbene. The Journal of Physical Chemistry A 1997, 101 (19),

3478-3487.

(18) Saha, B.; Ehara, M.; Nakatsuji, H., Investigation of the Electronic Spectra and

Excited-State Geometries of Poly(para-phenylene vinylene) (PPV) and Poly(para-

phenylene) (PP) by the Symmetry-Adapted Cluster Configuration Interaction

(SAC-CI) Method. The Journal of Physical Chemistry A 2007, 111 (25), 5473-

5481.

(19) Binder, R.; Wahl, J.; Romer, S.; Burghardt, I., Coherent exciton transport driven by

torsional dynamics: a quantum dynamical study of phenylene-vinylene type

conjugated systems. Faraday Discussions 2013, 163 (0), 205-222.

(20) Lukeš, V.; Aquino, A.; Lischka, H., Theoretical Study of Vibrational and Optical

Spectra of Methylene-Bridged Oligofluorenes. The Journal of Physical Chemistry

A 2005, 109 (45), 10232-10238.


136

(21) Lukeš, V.; Aquino, A. J. A.; Lischka, H.; Kauffmann, H.-F., Dependence of Optical

Properties of Oligo-para-phenylenes on Torsional Modes and Chain Length. The

Journal of Physical Chemistry B 2007, 111 (28), 7954-7962.

(22) Köhn, A.; Hättig, C., Analytic gradients for excited states in the coupled-cluster

model CC2 employing the resolution-of-the-identity approximation. The Journal

of Chemical Physics 2003, 119 (10), 5021-5036.

(23) Hättig, C., Structure Optimizations for Excited States with Correlated Second-

Order Methods: CC2 and ADC(2). In Advances in Quantum Chemistry, Jensen, H.

J. Å., Ed. Academic Press: 2005; Vol. 50, pp 37-60.

(24) Aquino, A., J. A.; Nachtigallova, D.; Hobza, P.; Truhlar Donald, G.; Hättig, C.;

Lischka, H., The charge‐transfer states in a stacked nucleobase dimer complex: A

benchmark study. Journal of Computational Chemistry 2010, 32 (7), 1217-1227.

(25) Szalay, P. G.; Watson, T.; Perera, A.; Lotrich, V. F.; Bartlett, R. J., Benchmark

Studies on the Building Blocks of DNA. 1. Superiority of Coupled Cluster Methods

in Describing the Excited States of Nucleobases in the Franck–Condon Region. The

Journal of Physical Chemistry A 2012, 116 (25), 6702-6710.

(26) Panda, A. N.; Plasser, F.; Aquino, A. J. A.; Burghardt, I.; Lischka, H.,

Electronically Excited States in Poly(p-phenylenevinylene): Vertical Excitations

and Torsional Potentials from High-Level Ab Initio Calculations. The Journal of


(27) Cardozo, T. M.; Aquino, A. J. A.; Barbatti, M.; Borges, I.; Lischka, H., Absorption

and Fluorescence Spectra of Poly(p-phenylenevinylene) (PPV) Oligomers: An ab

Initio Simulation. The Journal of Physical Chemistry A 2015, 119 (9), 1787-1795.

(28) Mewes, S. A.; Mewes, J.-M.; Dreuw, A.; Plasser, F., Excitons in poly(para

phenylene vinylene): a quantum-chemical perspective based on high-level ab initio

calculations. Physical Chemistry Chemical Physics 2016, 18 (4), 2548-2563.

(29) Aquino, A. A. J.; Borges, I.; Nieman, R.; Kohn, A.; Lischka, H., Intermolecular

interactions and charge transfer transitions in aromatic hydrocarbon-

tetracyanoethylene complexes. Physical Chemistry Chemical Physics 2014, 16

(38), 20586-20597.

(30) Tretiak, S.; Saxena, A.; Martin, R. L.; Bishop, A. R., Interchain Electronic

Excitations in Poly(phenylenevinylene) (PPV) Aggregates. The Journal of Physical

Chemistry B 2000, 104 (30), 7029-7037.

(31) Cornil, J.; dos Santos, D. A.; Crispin, X.; Silbey, R.; Brédas, J. L., Influence of

Interchain Interactions on the Absorption and Luminescence of Conjugated

Oligomers and Polymers: A Quantum-Chemical Characterization. Journal of the


(32) Cornil, J.; Beljonne, D.; Calbert, J. P.; Brédas, J. L., Interchain Interactions in

Organic π-Conjugated Materials: Impact on Electronic Structure, Optical

Response, and Charge Transport. Advanced Materials 2001, 13 (14), 1053-1067.

(33) Bedard-Hearn, M. J.; Sterpone, F.; Rossky, P. J., Nonadiabatic Simulations of

Exciton Dissociation in Poly-p-phenylenevinylene Oligomers. The Journal of



137

(34) Meng, R.; Li, Y.; Li, C.; Gao, K.; Yin, S.; Wang, L., Exciton transport in [small

pi]-conjugated polymers with conjugation defects. Physical Chemistry Chemical

Physics 2017, 19 (36), 24971-24978.

(35) Nayyar, I. H.; Masunov, A. E.; Tretiak, S., Comparison of TD-DFT Methods for

the Calculation of Two-Photon Absorption Spectra of Oligophenylvinylenes. The

Journal of Physical Chemistry C 2013, 117 (35), 18170-18189.

(36) Yan, M.; Rothberg, L. J.; Kwock, E. W.; Miller, T. M., Interchain Excitations in

Conjugated Polymers. Physical Review Letters 1995, 75 (10), 1992-1995.

(37) Nguyen, T.-Q.; Kwong, R. C.; Thompson, M. E.; Schwartz, B. J., Improving the

performance of conjugated polymer-based devices by control of interchain

interactions and polymer film morphology. Applied Physics Letters 2000, 76 (17),

2454-2456.

(38) Nguyen, T.-Q.; Martini, I. B.; Liu, J.; Schwartz, B. J., Controlling Interchain

Interactions in Conjugated Polymers: The Effects of Chain Morphology on

Exciton−Exciton Annihilation and Aggregation in MEH−PPV Films. The Journal

of Physical Chemistry B 2000, 104 (2), 237-255.

(39) Grimme, S., Improved second-order Møller–Plesset perturbation theory by separate

scaling of parallel- and antiparallel-spin pair correlation energies. The Journal of


(40) Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M., Scaled opposite-spin

second order Møller–Plesset correlation energy: An economical electronic structure

method. The Journal of Chemical Physics 2004, 121 (20), 9793-9802.

(41) Møller, C.; Plesset, M. S., Note on an Approximation Treatment for Many-Electron

Systems. Physical Review 1934, 46 (7), 618-622.

(42) Winter, N. O. C.; Hättig, C., Scaled opposite-spin CC2 for ground and excited states

with fourth order scaling computational costs. The Journal of Chemical Physics

2011, 134 (18), 184101.

(43) Hellweg, A.; Grun, S. A.; Hattig, C., Benchmarking the performance of spin-

component scaled CC2 in ground and electronically excited states. Physical

Chemistry Chemical Physics 2008, 10 (28), 4119-4127.

(44) Hättig, C., Geometry optimizations with the coupled-cluster model CC2 using the

resolution-of-the-identity approximation. The Journal of Chemical Physics 2003,

118 (17), 7751-7761.

(45) Hättig, C.; Weigend, F., CC2 excitation energy calculations on large molecules

using the resolution of the identity approximation. Journal of Chemical Physics

2000, 113, 5154-5161.

(46) Dunning, T. H., Gaussian basis sets for use in correlated molecular calculations. I.

The atoms boron through neon and hydrogen. The Journal of Chemical Physics

1989, 90 (2), 1007-1023.

(47) Weigend, F.; Köhn, A.; Hättig, C., Efficient use of the correlation consistent basis

sets in resolution of the identity MP2 calculations. The Journal of Chemical Physics

2002, 116 (8), 3175-3183.




138



2004, 393 (1), 51-57.

(50) Limacher, P. A.; Mikkelsen, K. V.; Lüthi, H. P., On the accurate calculation of

polarizabilities and second hyperpolarizabilities of polyacetylene oligomer chains

using the CAM-B3LYP density functional. Journal of Chemical Physics 2009, 130,

194114.

(51) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H., A consistent and accurate ab initio

parametrization of density functional dispersion correction (DFT-D) for the 94

elements H-Pu. The Journal of Chemical Physics 2010, 132 (15), 154104.

(52) Boys, S. F.; Bernardi, F., The calculation of small molecular interactions by the

differences of separate total energies. Some procedures with reduced errors.

Molecular Physics 1970, 19 (4), 553-566.

(53) Halkier, A.; Helgaker, T.; Jørgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson,

A. K., Basis-set convergence in correlated calculations on Ne, N2, and H2O.


(54) Martin, R. L., Natural transition orbitals. The Journal of Chemical Physics 2003,

118 (11), 4775-4777.

(55) Plasser, F.; Lischka, H., Analysis of Excitonic and Charge Transfer Interactions

from Quantum Chemical Calculations. Journal of Chemical Theory and

Computation 2012, 8 (8), 2777-2789.



Physics Letters 1989, 162 (3), 165-169.

















Physics 2014, 141 (2), 024106.





139

(60) Lee, N. K.; Park, S.; Kim, S. K., Ab initio studies on the van der Waals complexes

of polycyclic aromatic hydrocarbons. II. Naphthalene dimer and naphthalene–

anthracene complex. The Journal of Chemical Physics 2002, 116 (18), 7910-7917.

(61) Tsuzuki, S.; Honda, K.; Uchimaru, T.; Mikami, M., High-level ab initio

computations of structures and interaction energies of naphthalene dimers: Origin

of attraction and its directionality. The Journal of Chemical Physics 2003, 120 (2),

647-659.

(62) Sinnokrot, M. O.; Sherrill, C. D., Highly Accurate Coupled Cluster Potential

Energy Curves for the Benzene Dimer: Sandwich, T-Shaped, and Parallel-

Displaced Configurations. The Journal of Physical Chemistry A 2004, 108 (46),

10200-10207.

(63) Sato, T.; Tsuneda, T.; Hirao, K., A density-functional study on π-aromatic

interaction: Benzene dimer and naphthalene dimer. The Journal of Chemical

Physics 2005, 123 (10), 104307.

(64) Grimme, S., Do Special Noncovalent π–π Stacking Interactions Really Exist?


(65) Pitoňák, M.; Neogrády, P.; Rezáč, J.; Jurečka, P.; Urban, M.; Hobza, P., Benzene

Dimer: High-Level Wave Function and Density Functional Theory Calculations.

Journal of Chemical Theory and Computation 2008, 4 (11), 1829-1834.


140

CHAPTER VII

THE BIRADICALOID CHARACTER IN RELATION TO

SINGLET/TRIPLET SPLITTINGS FOR CIS-/TRANS-

DIINDENOACENES AND TRANS-

INDENOINDENODI(ACENOTHIOPHENES) BASED ON

EXTENDED MULTIREFERENCE CALCULATIONS

Authors: Reed Nieman,a Nadeesha J. Silva,a Adelia J. A. Aquino,a,b and Hans Lischkaa,b



Tianjin University, Tianjin 300072, PR China

7.1 Abstract

The assessment of the biradical character of select trans-diindenoacenes, cis-

diindenoacenes, and trans-indenoindenodi(acenothiophenes) was accomplished utilizing

the multireference averaged quadratic coupled-cluster (MR-AQCC) method. These

systems have been previously identified as highly reactive in experiments, often subject to

unplanned side-reactions. The biradical character of these species was characterized using

the singlet/triplet splitting energy (∆ES-T), effective number of unpaired electrons (NU), and

unpaired electron density. Additionally, the aromaticity has been described by the

harmonic oscillator measure of aromaticity (HOMA). For all structures, low spin singlet

ground states were found, in agreement with previous investigations. In addition,

increasing the number of core six-membered rings decreased the ∆ES-T, and increased the

NU and total unpaired electron density of the singlet state. The apical carbon of the five-

membered ring has the largest amount of unpaired electron density in each investigated

structure. The cis-diindenoacenes (5-8) displayed a greatly increased value of NU and

unpaired electron density corresponding to much smaller ∆ES-T values compared to the

other structures. The thiophene rings in the trans-indenoindenodi(acenothiophenes)

structures (9-12) were found to simultaneously limit the unpaired electron density in the

terminal six-membered rings and stabilize the density found in the core rings. Finally

excellent agreement to experimental values for the ∆ES-T has been demonstrated for

structures 5 and 10.


141

7.2 Introduction

The study of the properties and reactivity of biradical polycyclic aromatic

hydrocarbons (PAHs) is important to a wide variety of scientific fields due to their unique

electronic nature. Since the discovery of open-shell PAHs systems,1 they have provided

invaluable insight into how the radicals influence the π-bonding network to generate

properties such as magnetism. Diradical PAH systems often display a range of properties

such as small energy gaps between the highest occupied molecular orbital (HOMO) and

the lowest unoccupied molecular orbital (LUMO),2 electron spin resonance and peak

broadening in proton NMR spectra,2 low-lying doubly-excited electron configurations

allowing low-energy electronic absorptions,3 and amphoteric redox character and

multicenter bonds.4 These expansive properties have led biradical PAHs to be prime

candidates for use in molecular electronic and spintronic devices,5 battery technology,6

nonlinear optical devices,7-9 and singlet fission materials.10-12 Several classes of radical and

biradical PAHs have been studied including zethrenes,13-17 anthenes,18-19 phenalenyl,5

bisphenalenyls,20-22 triangulene23-25 and extended triangulene,5 quinodimethanes,26-28

indofluorenes,29-35 high-order acenes,36-40 and graphene nanoribbons.40-44 The biradical

character in these cases is derived from open-shell and closed-shell resonance structures,45-

47 the interconversion between which cleaves the exocyclic π-bonds increasing the number

of Clar sextets and energetically driving the stability of the open-shell state.15, 48 Double

spin-polarization effects give biradical PAHs with singlet ground states low-lying triplet

excited states that are more easily populated.49

The unique characteristics of idenofluorenes (representative structures shown in

Figure 7.1) have been targeted for investigation and utilization. The five-member carbon

rings in indenofluorenes have shown to make the structures reversible electron donors.50

Diindenobenzene dervatives show ambipolar charge transport properties in single-crystal

and thin-film organic field effect transitors (OFETs).30, 51 Extensive work on trans-

diindenoanthracene found that it was chemically robust due to only modest biradical

character and moderate singlet/triplet splitting energy, ∆ES-T, of 4.2 kcal/mol,4 and this


142

small energy gap showed that the triplet state was thermally accessible and could be

populated by an entropy-driven singlet-triplet fission process.52 Only the outer six-

membered rings of trans-diindenonaphthalene (fluoreno[3,2-b]fluorene) were shown to be

aromatic, while the core rings displayed anti-aromatic character.35 Experimental results for

cis-diindenobenzene (fluoreno[2,1-b]fluorene) show it has a singlet ground state, modest

∆ES-T of 4.2 kcal/mol, and a small HOMO-LUMO band gap (1.26 eV).31 The fusion of

thiophene rings into the indenofluorene frame yield interesting experimental results.

Indacenedithiophenes have smaller energy level gaps than pure indenofluorenes as they

have increased paratropicity in their indacene core.32 It has been shown that the inclusion

of fused benzothia groups in indenofluorenes leads to pro-aromatic properties that show

larger diradical character and longer lifetimes of singlet excited states.53

Indenoindenodibenzothiophene molecules were studied experimentally and theoretically

and were found to have relatively large ∆ES-T (8.0 kcal/mol) together with large diradial

character.54

Figure 7.1 Quinoid Kekule and biradical resonance forms of trans-diindenoanthracene, cis-

diindenoanthracene, and trans-indenoindenodi(anthracenothiophene).

Quantum chemical calculations are of great importance to the determination of

molecular properties. While the computationally efficient density functional theory (DFT)

is a natural first choice, the unrestricted DFT (UDFT) method suffers heavily from spin-

contamination effects especially for broken-symmetry singlet state biradical cases.37, 55 To

remedy this, PAH systems with singlet biradical cases have been investigated with a wide


143

variety of methods such as spin-flip (SF) approaches,56-57 projected Hartree-Fock,58 active

space variational two-electron reduced density matrix (2RDM),59-60 density matrix

renormalization group (DMRG),61-62 and CCSD(T).63-64 The multireference averaged

quadratic coupled-cluster (MR-AQCC) method65 is able to measure reliably the

biradicaloid character of molecules as it includes quasi-degenerate configurations in the

reference wave function and dynamic electron correlation including size-extensivity

contributions by explicitly considering single and double excitations.65-66 Previous reports

have had success using the MR-AQCC method to describe n-acenes,40, 67-68 periacenes,

zethrenes,69-70 among other challenging biradical cases.71-72

To this end, the MR-AQCC method has been employed in this work to investigate

the biradical character of several extended indenofluorene and

indenoindenodi(acenothiophene) structures (Figure 7.1 and Figure E.1) by describing their

singlet ground state and first vertical and adiabatic triplet excited state. From these

calculations, relatively simple descriptors such as the singlet/triplet splitting energy, ∆ES-

T, effective number of unpaired electrons (NU), and density of unpaired electrons will be

used to characterize singlet biradical nature of twelve PAH molecules. The harmonic

oscillator measure of aromaticity (HOMA) index will also be calculated at the DFT level

in order to assess the aromaticity of each structure.


The biradicaloid character of the structures shown in Figure 7.1 and Figure E.1 of

the Supporting Information were investigated using the MR-AQCC65 method by relating

the ∆ES-T values to NU data. To begin, the low and high spin ground states of all structures

were optimized using the range-corrected ωB97XD73 density functional and the def2-

TZVP74-75 basis set. Systems 1-4 (trans-diindenoacenes) and 9-12 (trans-

indenoindenodi(acenothiophenes)) belong to the C2h point group while 5-8 (cis-

diindenoacenes) belong to the C2v point group. The correct low spin ground state electron

configuration was determined by way of wave function instability analysis76 of the Kohn-

Sham determinants.77 All structures, except 1, were found to have triplet instabilities


144

present in their wave functions at restricted DFT (RDFT) level, which were followed to

reoptimize the geometry using an unrestricted DFT (UDFT) approach. The harmonic

vibrational frequencies for all structures were calculated and found to be non-imaginary in

all cases showing that each structure is a minimum in the potential curve.

Based on the optimized geometry using the ωB97XD/def2-TZVP method, the

aromaticity of each ring in each structure was determined using the harmonic oscillator

measure of aromaticity (HOMA):78-79

𝐻𝑂𝑀𝐴 = 1 −𝛼

𝑁∑(𝑅𝑜𝑝𝑡 − 𝑅𝑖)

2

𝑖

(6.1)

This method uses Kekulé benzene for the reference, and the index of aromaticity is

determined based on how much the bond distances, 𝑅𝑖, differ from the optimal C-C bond

length, 𝑅𝑜𝑝𝑡. A normalization constant, 𝛼, keeps the HOMA index of zero for Kekulé

benzene. The total number of bonds in a given ring is given by N. Here, 𝑅𝑜𝑝𝑡 for C-C bonds

is 1.388 Å, and for C-S bonds is 1.677 Å. The constant, 𝛼, for C-C bonds is 257.7 Å2, and

for C-S bonds is 94.09 Å2.

The starting orbitals for the MR-AQCC calculations were found using the complete active

space self-consistent field theory (CASSCF) method80 and the ωB97XD/def2-TZVP

optimized geometries. The 6-31G*81-83 basis set was used for all CASSCF and MR-AQCC

calculations. For 9-12, a split basis set was used, designated here as 6-31G*(6-311G*)S,

where 6-31G* was applied for carbon and hydrogen, and the triple-zeta basis set, 6-

311G*,84-85 was utilized for sulfur. The σ orbitals (both occupied and virtual) were frozen

in all CASSCF and MR-AQCC calculations at SCF level while the π orbitals were kept

active. For 1-4 and 9-12, orbitals belonging to the Ag and Bu irreducible representations

were σ in character, and orbitals belonging to the Au and Bg irreducible representations

were π in character. For 5-8, orbitals belonging to the A1 and B1 irreducible representations

were σ in character, and orbitals belonging to the A2 and B2 irreducible representations

were π in character. The molecular orbitals constituting the active space were determined

with the method of Pulay et al86 using the natural orbital (NO) occupation numbers from

the unrestricted Hartree-Fock (UHF) wave function. For all structures, the optimized


145

orbitals from a two-state state-averaged (SA2) complete active space consisting of four

electrons in four orbitals (CAS(4,4)) where the state-averaging was performed over the

lowest singlet and triplet states. These orbitals were taken for the subsequent MR-AQCC

calculations with a CAS(4,4) reference space (MR-AQCC(CAS(4,4)). The states taken in

SA2 were the 1 1Ag and 1 3Bu states for 1-4 and 9-12, and 1 1A1 and 1 3B1 for 5-8. Expanded

reference spaces were also considered for 1, 2, and 9 to show the effect on the ∆ES-T by

utilizing in addition to the CAS a restricted active space (RAS) (doubly occupied orbitals

from which one electron may be excited to the CAS or auxiliary (AUX) orbitals) and AUX

space (unoccupied or virtual orbitals into which one electron may be excited). For 1 and 9,

CASSCF calculations were done with RAS(3)/CAS(4,4)/AUX(3) to optimize orbitals for

MR-AQCC(CAS(10,7)/AUX(3)) calculations, and for 9, CASSCF calculations were

performed with RAS(4)/CAS(4,4)/AUX(4) to optimize orbitals for MR-

AQCC(CAS(12,8)/AUX(4)) calculations. A further expanded reference space was

considered for 1, where CASSCF calculations were done with RAS(5)/CAS(4,4)/AUX(3)

to optimize orbitals for MR-AQCC(CAS(14,9)/AUX(3)) calculations. In addition, to show

the effect on the ∆ES-T when keeping the σ orbitals active, calculations were carried out for

1 using the MR-AQCC(CAS(4,4)) reference space whereby only the core orbitals were

frozen. For each reference space investigated, the active orbitals are given in Table E.1 of

the Supporting Information. In a few cases, intruder states (configurations not found in the

reference space and with ≥1% weight) were found to occur at the MR-AQCC level. This

was accounted for by utilizing more active orbitals to include the intruder state as a

reference configuration in the wave function.

The singlet/triplet splitting energy (∆ES-T) is calculated according to ∆ES-T = ET -

ES, such that positive values mean that the low spin or singlet state is lower in energy than

the high spin or triplet state. Three different cases of the ∆ES-T are considered: the vertical

splitting, ∆ES-T, vert., where the energy of the triplet state is found at the singlet state

geometry, the adiabatic splitting, ∆ES-T, adiab., where the energy of the triplet state is found

at the triplet state geometry, and the adiabatic splitting that includes the zero-point energy

correction (ZPE), ∆ES-T, adiab.+ZPE, for which the ZPE was calculated at the ωB97XD/def2-

TZVP level.


146

The effective number of unpaired electrons (NU) and unpaired electron density are

calculated87-88 in order to assess the radical nature of the discussed structures by means of

the non-linear formula developed by Head-Gordon:89-90

𝑁𝑈 = ∑ 𝑛𝑖2

𝑚

𝑖=1

(2 − 𝑛𝑖)2 (6.2)

In this approach, orbitals with occupations close to one are emphasized over those orbitals

that are either nearly double-occupied or are unoccupied. m is the total number of NOs and

ni is the occupation of the ith NO. Two schemes for NU have been considered: as a sum

over all NO occupations (NU) to assess the total radical character, and as a sum only over

the HONO-LUNO pair (H-L, NU(H-L)) to account only for the biradical character. For each

triplet state calculated, the NU values and unpaired electron density are practically the same

for the vertically excited triplet state and the adiabatic triplet state, so only the values for

the vertical triplet states are reported here.

The geometry optimizations, stability analysis, and frequency calculations were

carried out using the Gaussian program package (Gaussian 09 Rev.E.01).91 The UHF and

natural orbitals calculations were done using the TURBOMOLE program.92 All CASSCF

and MR-AQCC calculations were performed using the COLUMBUS program suite.93-95

The unpaired electron population analysis was accomplished using the TheoDORE

program.96-97


7.4.1 Trans-diindenoacenes

The ∆ES-T values for 1-4 (trans-diindenoacenes) and the effective number of

unpaired electrons, respectively, have been plotted in Figure 7.2 against the number of

interior six-membered rings using the MR-AQCC(CAS(4,4))/6-31G* method. One finds

that with increasing number of rings, that the ∆ES-T decreases. This decrease is greatest

when going from 1 to 2 (the vertical ∆ES-T decreases by about 22.3 kcal/mol and the ∆ES-T

decrease by 10.6 and 9.0 kcal/mol for adiabatic and adiabatic+ZPE, respectively). The


147

decrease in ∆ES-T going from 2-4 shows to be practically linear. The effect of using the

adiabatic and adiabatic+ZPE ∆ES-T is especially clear in 1, where the ∆ES-T is smaller than

the vertically excited case by about 14.1 and 16.2 kcal/mol, respectively. This decrease is

less pronounced in 2 (-2.5 and -2.9 kcal/mol, respectively), 3 (-0.8 and -0.9 kcal/mol,

respectively), and 4 (+0.05 and +0.07 kcal/mol, respectively). For 3 and 4, the ZPE

correction to the adiabatic ∆ES-T makes nearly no difference. For 4, the adiabatic and

adiabatic+ZPE ∆ES-T is increased from the vertical ∆ES-T, though by less than 0.1 kcal/mol.

In addition, 4 has the smallest ∆ES-T at only 6.5 kcal/mol, meaning that it has the most

energetically accessible triplet state of the investigated trans-diindenoacenes. Overall, there

is good agreement to recent theoretical ∆ES-T values calculated using the spin-flip non-

colinear time-dependent DFT (SF-NC-TDDFT) method for 1-3,54 though the MR-AQCC

values here are slightly larger in each case. The experimental value obtained for a

derivative of 3 (with substituted mesityl (2,4,6-trimethylphenyl) groups at the apical carbon

of the five-membered rings and (triisopropylsilyl)ethynyl groups at the center-most six-

membered ring) shows a ∆ES-T of 4.18 kcal/mol,4 smaller than the calculated

adiabatic+ZPE value of 10.5 kcal/mol reported here. The calculated ∆ES-T value for 3

reported in ref. 54 via the SF-NC-TDDFT method is about 7.1 kcal/mol, intermediate of

our MR-AQCC result and the experimental value, though the ∆ES-T values computed with

the complete active space configuration interaction (CASCI) method were approximately

twice as large as those presented here for 1-3 using the MR-AQCC(CAS(4,4)) reference

space.


148

Figure 7.2 Comparison of the singlet/triplet energy (∆ES-T, E(13Bu) - E(11Ag)) and number

of unpaired electrons (NU and NU(H-L), Table E.2) of the 11Ag state with the number of

interior six-membered rings of trans-diindenoacenes (1-4) using the MR-

AQCC(CAS(4,4))/6-31G* method.

The standard MR-AQCC(CAS(4,4)) method is compared to expanded reference

spaces for 1 and 2 in Table 7.1. First looking at 1, the vertical ∆ES-T calculated using the

MR-AQCC(CAS(4,4)) method (frozen σ orbitals) compares well to those using the MR-

AQCC(CAS(10,7)/AUX(3)) and MR-AQCC(CAS(14,9)/AUX(3)) methods, decreasing

only by about 0.06 and 0.6 kcal/mol, respectively. Using the MR-

AQCC(CAS(10,7)/AUX(3)) method for 1, the adiabatic and adiabatic+ZPE ∆ES-T are

increased by about 0.9 and 0.7 kcal/mol, respectively. This is likely a result of intruder

states (configurations from outside the reference space) in the triplet state, the necessary

inclusion of which as reference configurations increased the total energy and thus the ∆ES-

T. The inclusion of sigma orbitals (excluding the core orbitals that remained frozen) in the

MR-AQCC(CAS(4,4)) method for 1 decreases the adiabatic and adiabatic+ZPE ∆ES-T by

about 2.0 and 2.1 kcal/mol, respectively, compared to the MR-AQCC(CAS(4,4)) method

where the σ orbitals remained frozen. The vertically excited ∆ES-T could not be obtained

with the MR-AQCC(CAS(4,4)) method and including the σ orbitals due to persistent

intruder states in the triplet state keeping the calculation from converging. When


149

considering 2, the vertical ∆ES-T decreases with the MR-AQCC(CAS(12,8)/AUX(4))

method by about 1.9 kcal/mol compared to the MR-AQCC(CAS(4,4)) method, but the

adiabatic and adiabatic+ZPE ∆ES-T increase by about 0.6 kcal/mol.

Table 7.1 Singlet/triplet splitting energy (∆ES-T, E(13Bu) - E(11Ag)) calculated using the

MR-AQCC/6-31G* method for trans-diindenoacenes, 1-4.

System Reference Space ∆ES-T, vert.

(kcal/mol)

∆ES-T, adiab.

(kcal/mol)

∆ES-T, adiab.+ZPE

(kcal/mol)

1 MR-AQCC(CAS(4,4))a -b 23.7 21.6

MR-AQCC(CAS(4,4)) 39.8 25.7 23.6

MR-AQCC

(CAS(10,7)/AUX(3))

38.8 26.5 24.4

MR-AQCC

(CAS(14,9)/AUX(3))

38.2 23.0 20.9

2 MR-AQCC(CAS(4,4)) 17.5 15.1 14.7

MR-AQCC

(CAS(12,8)/AUX(4))

15.6 15.6 15.2

3 MR-AQCC(CAS(4,4)) 11.5 10.7 10.5

Expt.4 4.18

4 MR-AQCC(CAS(4,4)) 6.4 6.5 6.5 aonly core orbitals are frozen, bvertically excited triplet state had persistent intruder states.

The effective number of unpaired electrons of the 1 1Ag state increases linearly,

whether one considers the sum over all NOs or just the H-L pair, with the number of interior

rings going from 1-4. 1 has the smallest NU values of any of the trans-diindenoacenes (NU

is 0.55 e and NU(H-L) is 0.23 e), correlating with its large ∆ES-T, Figure 7.3 and Table E.2.

The total effective number of unpaired electrons for 2 is about 1.1 e, about 1.6 e for 3, and

2.2 e for 4. Using only the H-L pair reduces the total number of unpaired electrons by about

0.3 e for 1 to about 0.7 e for 4. The total number of unpaired electrons for 4 is 2.2 e, showing

a true singlet biradical system, which correlates to its small ∆ES-T, Figure 7.3. When

considering only the H-L pair for each vertically excited triplet state, Table E.2, the number

of unpaired electrons is almost constant at 2.0 e, consistent with triplet spin multiplicity

and singly occupied HONOs and LUNOs. The largest difference in NU values between the

singlet and vertical triplet state comes for 1 at about 1.8 e.


150

a)

b)

Figure 7.3 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of

unpaired electrons (NU) and b) the number of unpaired electrons in the HONO-LUNO pair

(NU(H-L)) of the 1 1Ag state (Table E.2) of trans-diindenoacenes (1-4) using the MR-


The unpaired electron density of 1-4 for the 1 1Ag ground state is shown in Figure

7.4. The density is greatest on the carbon atom of the five-membered rings connected to

hydrogen, referred to as the apical carbon, correlating to the location of the radical center

shown in the valence bond (VB) structures shown in Figure 7.1 and Figure E.1, and the

density then alternates going toward the center of the molecules. The unpaired electron

density on carbon atoms not bonded to hydrogen is nearly negligible. The unpaired electron


151

density of the vertically excited triplet state of 1-4 (Figure E.2) is greater than for the singlet

state. There is greater density found on the indene groups in the triplet state. The greatest

density is still found on the apical carbon atom of the five-membered ring in the triplet state

as in the singlet state, but the density decreases at this location with increasing system size.

Despite the decrease in unpaired electron density at the location of greatest density, the

slight increase of the NU (Table E.2) with increasing system size (about 0.1 e per additional

core ring) can be seen to come from the increase in number of centers of unpaired electron

density.

Figure 7.4 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the 1 1Ag state of trans-diindenobenzene (1-4) using the MR-AQCC(CAS(4,4))/6-31G* method.


The HOMA indices for the singlet ground state geometry are also shown in Figure

7.4 for each trans-diindenoacene structured investigated (1-4). Structure 1 is noteworthy as

the five-membered ring and interior six-membered ring are almost anti-aromatic in

character as the HOMA indices are nearly zero, consistent with the previous findings that

concluded the local anti-aromatic nature arises from the large odd-electron density

distribution.98 The six-membered ring of the indene is aromatic however. This is

substantially different from the triplet ground state geometry (Figure E.2) of 1 where the

five-membered ring has a HOMA index of 0.37 and the interior six-membered ring has an

index of 0.93. This great difference in HOMA indices between the singlet and triplet states,


152

implying also a large change in the geometry, is the origin of the substantial difference in

the NU discussed above for 1. In the remaining structures, 2-4, the HOMA index of the

five-membered is about 0.3, with indices increasing going toward the center of each

molecule, reaching a maximum value of about 0.75 for 3 and 4. As shown, the aromaticity

of each ring (correlating to the HOMA index) increases as the unpaired electron density at

each carbon of that ring decreases. For the triplet ground states structures, the interior most

six-membered ring has a larger HOMA index than those found in the singlet ground state

structures, though they are nearly the same in 4.

7.4.2 Cis-diindenoacenes

The singlet/triplet splitting of the cis-diindenoacene structures, 5-8, Figure 7.5 and

Table 7.2, is much smaller compared to the trans-isomers, 1-4. The ∆ES-T decreases as the

number of rings in each system increases. The largest vertical ∆ES-T is found for 5 and is

only 3.9 kcal/mol while the vertical case for 8 is negative indicating that the 1 3B1 state is

more stable than the 1 1A1 state. For 7 though, the ∆ES-T, while negative, is close to zero so

the singlet and triplet states are almost isoelectronic. The adiabatic and adiabatic+ZPE ∆ES-

T for 5 and 6 are smaller than the vertical ∆ES-T by about 0.4 kcal/mol, but 7 and 8 become

positive when considering the adiabatic+ZPE ∆ES-T, increasing to 0.5 and 0.2 kcal/mol.

The ZPE correction does not contribute greatly to the adiabatic ∆ES-T, adding or subtracting

by only about 0.1 kcal/mol. The ∆ES-T calculated for 5, 3.6 kcal/mol, compares well with

the experimental value of 4.2 kcal/mol.31


153

Figure 7.5 Comparison of the singlet/triplet energy (∆ES-T, E(13B1) - E(11A1)) and number

of unpaired electrons (NU and NU(H-L), Table E.3) of the 11A1 state with the number of

interior six-membered rings of cis-diindenoacenes (5-8) using the MR-


Table 7.2 Singlet/triplet splitting energy (∆ES-T, E(13B1) - E(11A1)) calculated using the

MR-AQCC/6-31G* method for cis-diindenoacenes, 5-8.


(kcal/mol)

∆ES-T, adiab.

(kcal/mol)

∆ES-T, adiab.+ZPE

(kcal/mol)

5 MR-AQCC(CAS(4,4)) 3.9 3.5 3.6

Expt.31 4.2

6 MR-AQCC(CAS(4,4)) 1.2 0.8 0.7

7 MR-AQCC(CAS(4,4)) -0.3 0.5 0.5

8 MR-AQCC(CAS(4,4)) -1.2 0.3 0.2

The decrease in ∆ES-T with increasing number of interior six-membered rings

correlates to an increase in the effective number of unpaired electrons of the 1 1A1 state

(Figure 7.5), and as was shown with the trans-isomers (Figure 7.3), the decrease in ∆ES-T

correlates to increasing effective unpaired electrons for the cis-diindenoacenes, shown in

Figure 7.6. The NU for 5 and 6 (Table E.3) are much larger than their counterparts, 1 and 2


154

(Table E.2) by about 1.0 e, showing much more radical character, and the NU values for 7

and 8 are larger than those for 3 and 4 by about 0.6 e and 0.4 e, respectively. Taking only

the HONO-LUNO pair, the NU is least 1.0 e for 5 and increases to almost 2.0 e for 8. The

NU and NU(H-L) for the triplet state is very similar to those found in the trans-isomers.



(NU(H-L)) of the 1 1A1 state (Table E.3) of cis-diindenoacenes (5-8) using the MR-


The unpaired electron density of the 1 1A1 state for 5-8 is presented in Figure 7.7.

The The greatest amount of unpaired electron density is found located on the apical carbon

in the five-membered ring of the indene, like for 1-4 (Figure 7.4), correlating to the VB

a)

b)


155

structures. The apical carbon atoms were also found by Shimizu et al. to have the largest

spin-density amplitudes in 1.31 For 7 and 8, nearly 0.5 e are present on these carbon atoms.

As was discussed for the NU values above, the unpaired electron density is greater for the

cis-diindenoacene structures than for trans-indenoacene. The same trend that was seen for

1-4 (Figure 7.4) is present for the singlet states of 5-8 (Figure 7.7) whereby the unpaired

density increases, especially at the apical carbon of the five-membered ring, with

increasing number of interior rings. There is greater unpaired electron density present on

the six-membered rings of the capping indene groups of the cis-isomers compared to

transisomers. The unpaired density of the vertically excited 1 3B1 states is similar to the

triplet states calculated for the trans-isomers (Figure E.3), where the value of the total NU

increases with increasing system size due to the larger number of centers of unpaired

electron density, but the density on the carbon in the five-membered ring decreases. While

the difference in density for 5 between the singlet and triplet states is rather large (with the

triplet density being larger), going from 6-7 the difference decreases, and for 8 the density

is only slightly larger in the triplet state than in the singlet state.

Figure 7.7 Total unpaired electron density (sum over all NOs, Table E.3), in red, of the 1 1A1 state of cis-diindenobenzene (5-8) using the MR-AQCC(CAS(4,4))/6-31G* method.


HOMA indices were calculated for the singlet and triplet states of the cis-diindenoacenes

(5-8), Figure 7.7. The HOMA indices of the five-membered rings decrease from 5 (0.51)


156

to 8 (0.31) while the indices of six-membered ring of the indene group remains practically

constant. The HOMA index also shows to increase from the five-membered ring toward

the center. The HOMA indices reported here for 5 and 6 are slightly larger for all rings

than those presented previously by Miyoshi and coworkers.34 For structures with odd

numbers of rings, 5 and 7, the HOMA index of the center-most ring decreases by 0.07,

while for even numbers of rings, 6 and 8, the HOMA indices for the center-most rings

increases slightly by 0.04. Comparing to the trans-isomers, the five-membered rings in the

cis-isomers have higher HOMA indices, while the six-membered rings of the indene groups

are decreases slightly. The interior rings are shown to be more aromatic in the cis-isomers,

as the HOMA is larger for 5-8 than 1-4. The HOMA indices of the 1 3B1 state (Figure E.3)

are smaller for all structures than the 1 1A1 state except for the six-membered ring of the

indene groups, which are only slightly larger.

7.4.3 Trans-indenoindenodi(acenothiophenes)

The ∆ES-T of the investigated trans-indenoindenodi(acenothiophenes) (9-12) is

compared to the effective NU and number of interior rings in Figure 7.8. The trend of

decreasing ∆ES-T with increasing ring number is found also for 9-12 as for the previous

structures. The adiabatic ∆ES-T value decreases going from 9 to 10 by about 16 kcal/mol

(primarily due to the large value found for 9), though the values for 11 and 12 are

practically the same. The large decrease in all considered ∆ES-T values is observed also

going from 1 to 2 due to the much larger ∆ES-T of 1. The ZPE correction to the adiabatic

∆ES-T is similarly small; while the adiabatic+ZPE ∆ES-T decreases going from 9-11 with

increasing number of rings, going from 11 to 12 is basically constant. As with 1 and 2, an

expanded reference space was calculated showing that the vertical ∆ES-T increased by about

0.3 kcal/mol, but the adiabatic and adiabatic+ZPE ∆ES-T decreased by about 2.3 kcal/mol

compared to the MR-AQCC(CAS(4,4)) reference space. There is excellent agreement with

previous SF-NC-TDDFT calculations for 9-1154 (the CASCI values for the ∆ES-T of these

structures54 are in general larger by about 10 kcal/mol for each structure compared to our

MR-AQCC values), and the experimental value (8.0 kcal/mol)54 was found to agree quite


157

well and was only 0.4 kcal/mol smaller than the calculated value of the adiabatic+ZPE ∆ES-

T for 10, given in Table 7.3.

Figure 7.8 Comparison of the singlet/triplet energy (∆ES-T, E(13Bu) - E(11Ag)) and number

of unpaired electrons (NU and NU(H-L), Table E.4) of the 11Ag state with the number of

interior six-membered rings of trans-indenoindenodi(acenothiophene) (9-12) using the

MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method.

Table 7.3 Singlet/triplet splitting energy (∆ES-T, E(13Bu) - E(11Ag)) calculated using the

MR-AQCC/6-31G*(6-311G*)S method for trans-indenoindenodi(acenothiophene) (9-12).


(kcal/mol)

∆ES-T, adiab.

(kcal/mol)

∆ES-T, adiab.+ZPE

(kcal/mol)

9 MR-AQCC(CAS(4,4)) 24.5 19.7 19.2

MR-AQCC

(CAS(10,7)/AUX(3))

24.8 17.4 17.0

10 MR-AQCC(CAS(4,4)) 8.8 8.6 8.4

Expt.54 8.0

11 MR-AQCC(CAS(4,4)) 4.4 4.4 4.2

12 MR-AQCC(CAS(4,4)) 4.5 4.5 4.4


158

As shown in Figure 7.9, the energy decreases with increasing NU in the 1 1Ag state.

The total NU values (Table E.4) for the trans-indenoindenodi(acenothiophene) (9-12) are

larger than those of the trans-diindenoacenes, 1-4 (Table E.2), by about 0.2 e to 0.5 e, and

0.1 e to 0.4 e when considering only the H-L pair. The total NU for the cis-diindenoacenes

(5-8, Table E.3) are larger than for 9-12 by 0.7 e comparing 5 and 9, but the difference

becomes smaller, about 0.1 e, with increasing molecular size for 8 and 12. The vertically

excited triplet state has larger values of NU than for the singlet state, though the difference

while large for 9 (by about 1.7 e) decreases with increasing system size and is only 0.45 e

for 12. As with 1-8, when taking only H-L, the NU values become essentially 2.0 e. The

total NU for the vertical triplet state is larger than the triplet states of 1-8 by about 0.1 e.

The unpaired electron density of the 1 1Ag state of trans-

indenoindenodi(acenothiophenes) (9-12) is presented in Figure 7.10. As for structures 1-8,

the greatest amount of unpaired density is located on the apical carbon atom of the five-

membered ring (again correlating to the VB structures as with 1-8) and increases with

increasing system size, 9 to 12. The unpaired electron density is much reduced in the

thiophene rings and terminal six-membered rings compared to that in the core six-

membered rings. This same feature is found for the 1 3Bu state, Figure E.4. The triplet state

features greater unpaired electron density compared to the singlet state, but the magnitude

of the density localized on each center decreases as the system size grows from 9 to 12. As

suggested by the NU values, Table E.4, the unpaired density of the singlet state is larger

than for the trans-diindenoacenes, though smaller than for the cis-diindenoacenes.


159



(NU(H-L)) of the 1 1Ag state (Table E.4) of trans-indenoindenodi(acenothiophene) (9-12)

using the MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method.

a)

b)


160

Figure 7.10 Total unpaired electron density (sum over all NOs, Table E.4), in red, of the 1 1Ag state of trans-indenoindenodi(acenothiophene) (9-12) using the MR-

AQCC(CAS(4,4))/6-31G*/6-311G* method. (isovalue=0.004 e/Bohr3). HOMA index is

shown in the center of each ring.

The HOMA indices for the 1 1Ag state structures is presented in Figure 7.10 as well.

The index of the thiophene group decreases by about 0.08 by increasing the system size

after 9. The capping six-membered rings have HOMA indices of approximately 1 for 9-12.

The HOMA index of the five-membered rings increases from 9 to 10, but then decreases

from 10 to 12. The innermost six-membered rings increase in HOMA index from 9 to 11,

but then decreases with 12. The HOMA indices of the singlet state structures indicate that

9-12 are more aromatic than 1-4, but less aromatic than 5-8. The triplet state structures’

HOMA indices are shown in Figure E.4, and are found to be more aromatic than the singlet

state structures. As was the case with 1 of the trans-diindenoacenes, the large difference in

NU of 9 between the singlet and triplet states originates in the large difference in HOMA

indices, indicating a large difference in geometry between the two states.

7.5 Conclusions

MR-AQCC calculations were carried out to investigate the biradicaloid character

of trans- diindenoacenes (1-4), cis-diindenoacenes (5-8), and trans-

indenoindenodi(acenothiophenes) (9-12) by calculating the ∆ES-T and NU and comparing

to the number of interior or core rings. For each structure, the ground state was found to be


161

low spin, with the high spin state becoming increasing thermally accessible with increasing

number of core rings. The singlet ground state 1 and 9 have only slight radical or biradical

character judged by their small NU and NU(H-L) values, respectively, but these values

increase greatly when increasing the number of core rings by only one to obtain 2 and 10,

respectively. The cis-diindenoacene isomers are very different from the other eight

structures, as even the smallest considered structure, 5, has a small ∆ES-T (3.57 kcal/mol)

and large NU (1.41 e) and NU(H-L) (1.08 e). The greatest unpaired electron density for each

considered structure is found on the five-membered ring, and in the case of the thiophene-

fused structures, does not extend to the capping six-membered rings. The triplet structures

were stabilized with increasing core rings while the effective number of unpaired electrons

remained quite unchanged in magnitude, but became more diffuse and delocalized. In

summary, the MR-AQCC approach has been proven to be a single method able to

successfully assess the biradicaloid character of PAH molecules and compares

exceptionally well to experimental values for the ∆ES-T in the cases of 5 and 10. This

method provides necessary insight and analysis of these highly reactive species.

7.6 Bibliography

(1) Tschitschibabin, A. E., Über einige phenylierte Derivate des p, p-Ditolyls. Berichte

der deutschen chemischen Gesellschaft 1907, 40 (2), 1810-1819.

(2) Y. Gopalakrishna, T.; Zeng, W.; Lu, X.; Wu, J., From open-shell singlet

diradicaloids to polyradicaloids. Chemical Communications 2018, 54 (18), 2186-

2199.

(3) Di Motta, S.; Negri, F.; Fazzi, D.; Castiglioni, C.; Canesi, E. V., Biradicaloid and

Polyenic Character of Quinoidal Oligothiophenes Revealed by the Presence of a

Low-Lying Double-Exciton State. The Journal of Physical Chemistry Letters 2010,

1 (23), 3334-3339.

(4) Rudebusch, G. E.; Zafra, J. L.; Jorner, K.; Fukuda, K.; Marshall, J. L.; Arrechea-

Marcos, I.; Espejo, G. L.; Ponce Ortiz, R.; Gómez-García, C. J.; Zakharov, L. N.;

Nakano, M.; Ottosson, H.; Casado, J.; Haley, M. M., Diindeno-fusion of an

anthracene as a design strategy for stable organic biradicals. Nature Chemistry

2016, 8, 753.

(5) Morita, Y.; Suzuki, S.; Sato, K.; Takui, T., Synthetic organic spin chemistry for

structurally well-defined open-shell graphene fragments. Nature Chemistry 2011,

3, 197.


162

(6) Morita, Y.; Nishida, S.; Murata, T.; Moriguchi, M.; Ueda, A.; Satoh, M.; Arifuku,

K.; Sato, K.; Takui, T., Organic tailored batteries materials using stable open-shell

molecules with degenerate frontier orbitals. Nature Materials 2011, 10, 947.

(7) Nakano, M.; Kishi, R.; Nitta, T.; Kubo, T.; Nakasuji, K.; Kamada, K.; Ohta, K.;

Champagne, B.; Botek, E.; Yamaguchi, K., Second Hyperpolarizability (γ) of

Singlet Diradical System: Dependence of γ on the Diradical Character. The Journal

of Physical Chemistry A 2005, 109 (5), 885-891.

(8) Nakano, M.; Kishi, R.; Ohta, S.; Takahashi, H.; Kubo, T.; Kamada, K.; Ohta, K.;

Botek, E.; Champagne, B., Relationship between Third-Order Nonlinear Optical

Properties and Magnetic Interactions in Open-Shell Systems: A New Paradigm for

Nonlinear Optics. Physical Review Letters 2007, 99 (3), 033001.

(9) Nakano, M.; Champagne, B., Theoretical Design of Open-Shell Singlet Molecular

Systems for Nonlinear Optics. The Journal of Physical Chemistry Letters 2015, 6

(16), 3236-3256.

(10) Minami, T.; Nakano, M., Diradical Character View of Singlet Fission. The Journal

of Physical Chemistry Letters 2012, 3 (2), 145-150.

(11) Smith, M. B.; Michl, J., Recent Advances in Singlet Fission. Annual Review of

Physical Chemistry 2013, 64 (1), 361-386.

(12) Varnavski, O.; Abeyasinghe, N.; Aragó, J.; Serrano-Pérez, J. J.; Ortí, E.; López

Navarrete, J. T.; Takimiya, K.; Casanova, D.; Casado, J.; Goodson, T., High Yield

Ultrafast Intramolecular Singlet Exciton Fission in a Quinoidal Bithiophene. The

Journal of Physical Chemistry Letters 2015, 6 (8), 1375-1384.

(13) Hu, P.; Wu, J., Modern zethrene chemistry. Canadian Journal of Chemistry 2016,

95 (3), 223-233.

(14) Zeng, W.; Sun, Z.; Herng, T. S.; Gonçalves, T. P.; Gopalakrishna, T. Y.; Huang, K.

W.; Ding, J.; Wu, J., Super‐heptazethrene. Angewandte Chemie International

Edition 2016, 55 (30), 8615-8619.

(15) Sun, Z.; Lee, S.; Park, K. H.; Zhu, X.; Zhang, W.; Zheng, B.; Hu, P.; Zeng, Z.; Das,

S.; Li, Y.; Chi, C.; Li, R.-W.; Huang, K.-W.; Ding, J.; Kim, D.; Wu, J.,

Dibenzoheptazethrene Isomers with Different Biradical Characters: An Exercise of

Clar’s Aromatic Sextet Rule in Singlet Biradicaloids. Journal of the American


(16) Li, Y.; Heng, W.-K.; Lee, B. S.; Aratani, N.; Zafra, J. L.; Bao, N.; Lee, R.; Sung,

Y. M.; Sun, Z.; Huang, K.-W., Kinetically blocked stable heptazethrene and

octazethrene: Closed-shell or open-shell in the ground state? Journal of the


(17) Wu, T. C.; Chen, C. H.; Hibi, D.; Shimizu, A.; Tobe, Y.; Wu, Y. T., Synthesis,

structure, and photophysical properties of dibenzo [de, mn] naphthacenes.


(18) Konishi, A.; Hirao, Y.; Kurata, H.; Kubo, T.; Nakano, M.; Kamada, K., Anthenes:

Model systems for understanding the edge state of graphene nanoribbons. Pure and

Applied Chemistry 2014, 86 (4), 497-505.

(19) Konishi, A.; Hirao, Y.; Matsumoto, K.; Kurata, H.; Kishi, R.; Shigeta, Y.; Nakano,

M.; Tokunaga, K.; Kamada, K.; Kubo, T., Synthesis and characterization of

quarteranthene: elucidating the characteristics of the edge state of graphene


163

nanoribbons at the molecular level. Journal of the American Chemical Society

2013, 135 (4), 1430-1437.

(20) Kubo, T.; Sakamoto, M.; Akabane, M.; Fujiwara, Y.; Yamamoto, K.; Akita, M.;

Inoue, K.; Takui, T.; Nakasuji, K., Four-Stage Amphoteric Redox Properties and

Biradicaloid Character of Tetra-tert-butyldicyclopenta[b;d]thieno[1,2,3-cd;5,6,7-

c′d′]diphenalene. Angewandte Chemie International Edition 2004, 43 (47), 6474-

6479.

(21) Shimizu, A.; Kubo, T.; Uruichi, M.; Yakushi, K.; Nakano, M.; Shiomi, D.; Sato,

K.; Takui, T.; Hirao, Y.; Matsumoto, K.; Kurata, H.; Morita, Y.; Nakasuji, K.,

Alternating Covalent Bonding Interactions in a One-Dimensional Chain of a

Phenalenyl-Based Singlet Biradical Molecule Having Kekulé Structures. Journal

of the American Chemical Society 2010, 132 (41), 14421-14428.

(22) Shimizu, A.; Hirao, Y.; Matsumoto, K.; Kurata, H.; Kubo, T.; Uruichi, M.;

Yakushi, K., Aromaticity and π-bond covalency: prominent intermolecular

covalent bonding interaction of a Kekulé hydrocarbon with very significant singlet

biradical character. Chemical Communications 2012, 48 (45), 5629-5631.

(23) Inoue, J.; Fukui, K.; Kubo, T.; Nakazawa, S.; Sato, K.; Shiomi, D.; Morita, Y.;

Yamamoto, K.; Takui, T.; Nakasuji, K., The First Detection of a Clar's

Hydrocarbon, 2,6,10-Tri-tert-Butyltriangulene: A Ground-State Triplet of Non-

Kekulé Polynuclear Benzenoid Hydrocarbon. Journal of the American Chemical

Society 2001, 123 (50), 12702-12703.

(24) Fukui, K.; Inoue, J.; Kubo, T.; Nakazawa, S.; Aoki, T.; Morita, Y.; Yamamoto, K.;

Sato, K.; Shiomi, D.; Nakasuji, K.; Takui, T., The first non-Kekulé polynuclear

aromatic high-spin hydrocarbon: Generation of a triangulene derivative and band

structure calculation of triangulene-based high-spin hydrocarbons. Synthetic Metals

2001, 121 (1), 1824-1825.

(25) Allinson, G.; Bushby, R. J.; Paillaud, J. L.; Oduwole, D.; Sales, K., ESR spectrum

of a stable triplet .pi. biradical: trioxytriangulene. Journal of the American


(26) Casado, J., Para-Quinodimethanes: A Unified Review of the Quinoidal-Versus-

Aromatic Competition and its Implications. Topics in Current Chemistry 2017, 375

(4), 73.

(27) Zeng, Z.; Wu, J., Stable π-Extended p-Quinodimethanes: Synthesis and Tunable

Ground States. The Chemical Record 2015, 15 (1), 322-328.

(28) Zeng, Z.; Ishida, M.; Zafra, J. L.; Zhu, X.; Sung, Y. M.; Bao, N.; Webster, R. D.;

Lee, B. S.; Li, R.-W.; Zeng, W.; Li, Y.; Chi, C.; Navarrete, J. T. L.; Ding, J.;

Casado, J.; Kim, D.; Wu, J., Pushing Extended p-Quinodimethanes to the Limit:

Stable Tetracyano-oligo(N-annulated perylene)quinodimethanes with Tunable

Ground States. Journal of the American Chemical Society 2013, 135 (16), 6363-

6371.

(29) Frederickson, C. K.; Rose, B. D.; Haley, M. M., Explorations of the

Indenofluorenes and Expanded Quinoidal Analogues. Accounts of Chemical

Research 2017, 50 (4), 977-987.

(30) Chase, D. T.; Fix, A. G.; Kang, S. J.; Rose, B. D.; Weber, C. D.; Zhong, Y.;

Zakharov, L. N.; Lonergan, M. C.; Nuckolls, C.; Haley, M. M., 6,12-


164

Diarylindeno[1,2-b]fluorenes: Syntheses, Photophysics, and Ambipolar OFETs.


(31) Shimizu, A.; Kishi, R.; Nakano, M.; Shiomi, D.; Sato, K.; Takui, T.; Hisaki, I.;

Miyata, M.; Tobe, Y., Indeno[2,1-b]fluorene: A 20-π-Electron Hydrocarbon with

Very Low-Energy Light Absorption. Angewandte Chemie 2013, 125 (23), 6192-

6195.

(32) Young, B. S.; Chase, D. T.; Marshall, J. L.; Vonnegut, C. L.; Zakharov, L. N.;

Haley, M. M., Synthesis and properties of fully-conjugated indacenedithiophenes.

Chemical Science 2014, 5 (3), 1008-1014.

(33) Marshall, J. L.; Uchida, K.; Frederickson, C. K.; Schütt, C.; Zeidell, A. M.; Goetz,

K. P.; Finn, T. W.; Jarolimek, K.; Zakharov, L. N.; Risko, C.; Herges, R.; Jurchescu,

O. D.; Haley, M. M., Indacenodibenzothiophenes: synthesis, optoelectronic

properties and materials applications of molecules with strong antiaromatic

character. Chemical Science 2016, 7 (8), 5547-5558.

(34) Miyoshi, H.; Miki, M.; Hirano, S.; Shimizu, A.; Kishi, R.; Fukuda, K.; Shiomi, D.;

Sato, K.; Takui, T.; Hisaki, I.; Nakano, M.; Tobe, Y., Fluoreno[2,3-b]fluorene vs

Indeno[2,1-b]fluorene: Unusual Relationship between the Number of π Electrons

and Excitation Energy in m-Quinodimethane-Type Singlet Diradicaloids. The


(35) Barker, J. E.; Frederickson, C. K.; Jones, M. H.; Zakharov, L. N.; Haley, M. M.,

Synthesis and Properties of Quinoidal Fluorenofluorenes. Organic Letters 2017, 19

(19), 5312-5315.

(36) Bendikov, M.; Duong, H. M.; Starkey, K.; Houk, K. N.; Carter, E. A.; Wudl, F.,

Oligoacenes: Theoretical Prediction of Open-Shell Singlet Diradical Ground

States. Journal of the American Chemical Society 2004, 126 (24), 7416-7417.

(37) Jiang, D.-e.; Dai, S., Electronic Ground State of Higher Acenes. The Journal of


(38) Purushothaman, B.; Bruzek, M.; Parkin, S. R.; Miller, A.-F.; Anthony, J. E.,

Synthesis and Structural Characterization of Crystalline Nonacenes. Angewandte

Chemie 2011, 123 (31), 7151-7155.

(39) Zimmerman, P. M.; Bell, F.; Casanova, D.; Head-Gordon, M., Mechanism for

Singlet Fission in Pentacene and Tetracene: From Single Exciton to Two Triplets.


(40) Plasser, F.; Pašalić, H.; Gerzabek, M. H.; Libisch, F.; Reiter, R.; Burgdörfer, J.;

Müller, T.; Shepard, R.; Lischka, H., The Multiradical Character of One- and Two-

Dimensional Graphene Nanoribbons. Angewandte Chemie International Edition

2013, 52 (9), 2581-2584.

(41) Jiang, D.-e.; Sumpter, B. G.; Dai, S., Unique chemical reactivity of a graphene

nanoribbon’s zigzag edge. The Journal of Chemical Physics 2007, 126 (13),

134701.

(42) Nagai, H.; Nakano, M.; Yoneda, K.; Kishi, R.; Takahashi, H.; Shimizu, A.; Kubo,

T.; Kamada, K.; Ohta, K.; Botek, E.; Champagne, B., Signature of multiradical

character in second hyperpolarizabilities of rectangular graphene nanoflakes.



165

(43) Barone, V.; Hod, O.; Peralta, J. E.; Scuseria, G. E., Accurate Prediction of the

Electronic Properties of Low-Dimensional Graphene Derivatives Using a Screened

Hybrid Density Functional. Accounts of Chemical Research 2011, 44 (4), 269-279.

(44) Lambert, C., Towards Polycyclic Aromatic Hydrocarbons with a Singlet Open-

Shell Ground State. Angewandte Chemie International Edition 2011, 50 (8), 1756-

1758.

(45) Kubo, T., Recent Progress in Quinoidal Singlet Biradical Molecules. Chemistry

Letters 2014, 44 (2), 111-122.

(46) Sun, Z.; Zeng, Z.; Wu, J., Zethrenes, Extended p-Quinodimethanes, and Periacenes

with a Singlet Biradical Ground State. Accounts of Chemical Research 2014, 47

(8), 2582-2591.

(47) Abe, M., Diradicals. Chemical Reviews 2013, 113 (9), 7011-7088.

(48) Clar, E., The Aromatic Sextet. Wiley: London, 1972.

(49) Karafiloglou, P., The double (or dynamic) spin polarization in π diradicals. Journal

of Chemical Education 1989, 66 (10), 816.

(50) Rose, B. D.; Sumner, N. J.; Filatov, A. S.; Peters, S. J.; Zakharov, L. N.; Petrukhina,

M. A.; Haley, M. M., Experimental and Computational Studies of the Neutral and

Reduced States of Indeno[1,2-b]fluorene. Journal of the American Chemical

Society 2014, 136 (25), 9181-9189.

(51) Nishida, J.-i.; Deno, H.; Ichimura, S.; Nakagawa, T.; Yamashita, Y., Preparation,

physical properties and n-type FET characteristics of substituted

diindenopyrazinediones and bis(dicyanomethylene) derivatives. Journal of

Materials Chemistry 2012, 22 (10), 4483-4490.

(52) Burdett, J. J.; Bardeen, C. J., Quantum Beats in Crystalline Tetracene Delayed

Fluorescence Due to Triplet Pair Coherences Produced by Direct Singlet Fission.


(53) Shi, X.; Quintero, E.; Lee, S.; Jing, L.; Herng, T. S.; Zheng, B.; Huang, K.-W.;

López Navarrete, J. T.; Ding, J.; Kim, D.; Casado, J.; Chi, C., Benzo-thia-fused

[n]thienoacenequinodimethanes with small to moderate diradical characters: the

role of pro-aromaticity versus anti-aromaticity. Chemical Science 2016, 7 (5),

3036-3046.

(54) Dressler, J. J.; Teraoka, M.; Espejo, G. L.; Kishi, R.; Takamuku, S.; Gómez-García,

C. J.; Zakharov, L. N.; Nakano, M.; Casado, J.; Haley, M. M., Thiophene and its

sulfur inhibit indenoindenodibenzothiophene diradicals from low-energy lying

thermal triplets. Nature Chemistry 2018, 10 (11), 1134-1140.

(55) Yamaguchi, K.; Jensen, F.; Dorigo, A.; Houk, K. N., A spin correction procedure

for unrestricted Hartree-Fock and Møller-Plesset wavefunctions for singlet

diradicals and polyradicals. Chemical Physics Letters 1988, 149 (5), 537-542.

(56) Krylov, A. I., Spin-flip configuration interaction: an electronic structure model that

is both variational and size-consistent. Chemical Physics Letters 2001, 350 (5),

522-530.

(57) Casanova, D.; Head-Gordon, M., Restricted active space spin-flip configuration

interaction approach: theory, implementation and examples. Physical Chemistry



166

(58) Jiménez-Hoyos, C. A.; Henderson, T. M.; Tsuchimochi, T.; Scuseria, G. E.,

Projected Hartree–Fock theory. The Journal of Chemical Physics 2012, 136 (16),

164109.

(59) Gidofalvi, G.; Mazziotti, D. A., Active-space two-electron reduced-density-matrix

method: Complete active-space calculations without diagonalization of the N-

electron Hamiltonian. The Journal of Chemical Physics 2008, 129 (13), 134108.

(60) Pelzer, K.; Greenman, L.; Gidofalvi, G.; Mazziotti, D. A., Strong Correlation in

Acene Sheets from the Active-Space Variational Two-Electron Reduced Density

Matrix Method: Effects of Symmetry and Size. The Journal of Physical Chemistry

A 2011, 115 (22), 5632-5640.

(61) Hachmann, J.; Dorando, J. J.; Avilés, M.; Chan, G. K.-L., The radical character of

the acenes: A density matrix renormalization group study. The Journal of Chemical

Physics 2007, 127 (13), 134309.

(62) Mizukami, W.; Kurashige, Y.; Yanai, T., More π Electrons Make a Difference:

Emergence of Many Radicals on Graphene Nanoribbons Studied by Ab Initio

DMRG Theory. Journal of Chemical Theory and Computation 2013, 9 (1), 401-

407.

(63) Hajgató, B.; Huzak, M.; Deleuze, M. S., Focal Point Analysis of the Singlet–Triplet

Energy Gap of Octacene and Larger Acenes. The Journal of Physical Chemistry A

2011, 115 (33), 9282-9293.

(64) Hajgató, B.; Szieberth, D.; Geerlings, P.; De Proft, F.; Deleuze, M. S., A benchmark

theoretical study of the electronic ground state and of the singlet-triplet split of

benzene and linear acenes. The Journal of Chemical Physics 2009, 131 (22),

224321.

(65) Szalay, P. G.; Bartlett, R. J., Multi-reference averaged quadratic coupled-cluster

method: a size-extensive modification of multi-reference CI. Chemical Physics

Letters 1993, 214 (5), 481-488.




(67) Horn, S.; Plasser, F.; Müller, T.; Libisch, F.; Burgdörfer, J.; Lischka, H., A

comparison of singlet and triplet states for one- and two-dimensional graphene

nanoribbons using multireference theory. Theoretical Chemistry Accounts 2014,

133 (8), 1511.

(68) Horn, S.; Lischka, H., A comparison of neutral and charged species of one- and

two-dimensional models of graphene nanoribbons using multireference theory. The

Journal of Chemical Physics 2015, 142 (5), 054302.

(69) Das, A.; Müller, T.; Plasser, F.; Lischka, H., Polyradical Character of Triangular

Non-Kekulé Structures, Zethrenes, p-Quinodimethane-Linked Bisphenalenyl, and

the Clar Goblet in Comparison: An Extended Multireference Study. The Journal of


(70) Das, A.; Pinheiro Jr, M.; Machado, F. B. C.; Aquino, A. J. A.; Lischka, H., Tuning

the Biradicaloid Nature of Polycyclic Aromatic Hydrocarbons: The Effect of

Graphitic Nitrogen Doping in Zethrenes. ChemPhysChem 2018, 19 (19), 2492-

2499.


167

(71) Antol, I.; Eckert-Maksić, M.; Lischka, H.; Maksić, Z. B., [2.2.2]Propellane

Isomerization by Grob Rearrangement: An Ab Initio MR-AQCC Study. European


(72) Wang, E. B.; Parish, C. A.; Lischka, H., An extended multireference study of the

electronic states of para-benzyne. The Journal of Chemical Physics 2008, 129 (4),

044306.



Physics 2008, 10 (44), 6615-6620.

(74) Weigend, F.; Ahlrichs, R., Balanced basis sets of split valence, triple zeta valence

and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy.

Physical Chemistry Chemical Physics 2005, 7 (18), 3297-3305.

(75) Weigend, F., Accurate Coulomb-fitting basis sets for H to Rn. Physical Chemistry


(76) Čížek, J.; Paldus, J., Stability Conditions for the Solutions of the Hartree—Fock

Equations for Atomic and Molecular Systems. Application to the Pi‐Electron

Model of Cyclic Polyenes. The Journal of Chemical Physics 1967, 47 (10), 3976-

3985.

(77) Bauernschmitt, R.; Ahlrichs, R., Stability analysis for solutions of the closed shell

Kohn–Sham equation. The Journal of Chemical Physics 1996, 104 (22), 9047-

9052.

(78) Kruszewski, J.; Krygowski, T. M., Definition of aromaticity basing on the

harmonic oscillator model. Tetrahedron Letters 1972, 13 (36), 3839-3842.

(79) Krygowski, T. M., Crystallographic studies of inter- and intramolecular

interactions reflected in aromatic character of .pi.-electron systems. Journal of

Chemical Information and Computer Sciences 1993, 33 (1), 70-78.

(80) Roos, B.; Taylor, P. R.; Siegbahn, P. E. M., A complete active space SCF method

(CASSCF) using a density matrix formulated super-CI approach. Chemical Physics

1980, 48 (2), 157-173.

(81) Ditchfield, R.; Hehre, W. J.; Pople, J. A., Self‐Consistent Molecular‐Orbital

Methods. IX. An Extended Gaussian‐Type Basis for Molecular‐Orbital Studies of

Organic Molecules. The Journal of Chemical Physics 1971, 54 (2), 724-728.

(82) Petersson, G. A.; Bennett, A.; Tensfeldt, T. G.; Al‐Laham, M. A.; Shirley, W. A.;

Mantzaris, J., A complete basis set model chemistry. I. The total energies of closed‐

shell atoms and hydrides of the first‐row elements. The Journal of Chemical

Physics 1988, 89 (4), 2193-2218.

(83) Petersson, G. A.; Al‐Laham, M. A., A complete basis set model chemistry. II.

Open‐shell systems and the total energies of the first‐row atoms. The Journal of


(84) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A., Self‐consistent molecular

orbital methods. XX. A basis set for correlated wave functions. The Journal of


(85) McLean, A. D.; Chandler, G. S., Contracted Gaussian basis sets for molecular

calculations. I. Second row atoms, Z=11–18. The Journal of Chemical Physics

1980, 72 (10), 5639-5648.


168

(86) Pulay, P.; Hamilton, T. P., UHF natural orbitals for defining and starting MC‐SCF

calculations. The Journal of Chemical Physics 1988, 88 (8), 4926-4933.

(87) Takatsuka, K.; Fueno, T.; Yamaguchi, K., Distribution of odd electrons in ground-

state molecules. Theoretica chimica acta 1978, 48 (3), 175-183.

(88) Staroverov, V. N.; Davidson, E. R., Distribution of effectively unpaired electrons.


(89) Head-Gordon, M., Characterizing unpaired electrons from the one-particle density

matrix. Chemical Physics Letters 2003, 372 (3), 508-511.

(90) Head-Gordon, M., Reply to comment on ‘characterizing unpaired electrons from

the one-particle density matrix’. Chemical Physics Letters 2003, 380 (3), 488-489.

















Physics Letters 1989, 162 (3), 165-169.

(93) Lischka, H.; Müller, T.; Szalay, P. G.; Shavitt, I.; Pitzer, R. M.; Shepard, R.,

Columbus-a program system for advanced multireference theory calculations.

Wiley Interdisciplinary Reviews-Computational Molecular Science 2011, 1, 191-

199.

(94) Lischka, H.; Shepard, R.; Pitzer, R. M.; Shavitt, I.; Dallos, M.; Müller, T.; Szalay,

P. t. G.; Seth, M.; Kedziora, G. S.; Yabushita, S.; Zhang, Z., High-level

multireference methods in the quantum-chemistry program system COLUMBUS:

Analytic MR-CISD and MR-AQCC gradients and MR-AQCC-LRT for excited

states, GUGA spin–orbit CI and parallel CI density. Physical Chemistry Chemical

Physics 2001, 3, 664-673.

(95) H. Lischka; R. Shepard; I. Shavitt; R. M. Pitzer; M. Dallos; Th. Müller; P. G.

Szalay; F. B. Brown; R. Ahlrichs; H. J. Böhm; A. Chang; D. C. Comeau; R.

Gdanitz; H. Dachsel; C. Ehrhardt; M. Ernzerhof; P. Höchtl; S. Irle; G. Kedziora; T.

Kovar; V. Parasuk; M. J. M. Pepper; P. Scharf; H. Schiffer; M. Schindler; M.

Schüler; M. Seth; E. A. Stahlberg; J.-G. Zhao; S. Yabushita; Z. Zhang; M. Barbatti;

S. Matsika; M. Schuurmann; D. R. Yarkony; S. R. Brozell; E. V. Beck; J.-P.

Blaudeau; M. Ruckenbauer; B. Sellner; F. Plasser; J. J. Szymczak; R. F. K. Spada;

Das, A., COLUMBUS, an ab initio electronic structure program, release 7.0. 2017.


169



Physics 2014, 141 (2), 024106.



(98) Fukuda, K.; Nagami, T.; Fujiyoshi, J.-y.; Nakano, M., Interplay between Open-

Shell Character, Aromaticity, and Second Hyperpolarizabilities in Indenofluorenes.

The Journal of Physical Chemistry A 2015, 119 (42), 10620-10627.



170

CHAPTER VIII

CONCLUDING REMARKS

This dissertation presents the application of several quantum chemical methods to

the description of carbon-functionalized material properties used in cutting edge

technologies. The necessary use of these high-level ab initio methods would inform future

scientists on how best to utilize the discussed materials in future experiments and real-

world applications.

The reactivity of pristine graphene as well as single and double carbon vacancy

defects was assessed toward chemisorption of a hydrogen radical test particle using the

CASSCF and MR-CISD methods with pyrene as a graphene model. While all carbon

centers for each structure formed stable C-H bonds, single carbon vacancy pyrene

possessed the strongest reaction (was most stabilizing) toward the incoming hydrogen

radical thanks to the dangling bond left in the six-membered ring as a result of the defect.

Bond formation with the hydrogen radical at this site was found to involve at least three

closely spaced electronic excited states near the ground state and complicated diabatic

electronic behavior at extended C-H bond distances. The carbon centers comprising the

five-membered ring were found to quite stabilizing as well, though without the same degree

of involvement from the complex manifold of excited states. A similar situation was found

for the five-membered rings of the double vacancy structure. A rigid potential energy scan

was performed to provide a survey of reactivity across the face of each structure confirming

the stability at the carbon centers and finding more repulsive regions at the existing

hydrogens and in the center of rings.

The structure and electronic states of double vacancy graphene with the intrinsic

dopant, silicon, were investigated by way of DFT calculations using pyrene and

circumpyrene as model systems. The silicon defect was characterized by means of NO

occupations, NBO analyses, and MEP surfaces. The dopant produced a non-planar low-

spin ground state with D2 symmetry, with a small energy gap between it and the planar low

spin D2h symmetric structure. The high spin structures were planar and found higher in

energy. The hybridization of the non-planar structures was sp3 while the planar structures


171

were sp2d showing d-orbital involvement which is thought to be responsible for opening

the band gap. Comparisons between the B3LYP and CAM-B3LYP density functionals

found similar energies and structures, though the CAM-B3LYP low spin structures

displayed more open-shell character. Natural charge and MEP analyses showed charge

transfer from the silicon center to the bonding carbon atoms. Supporting experimental

results suggest a more complex picture, however, as features of the EEL spectrum indicate

an equilibrium existing between the non-planar and planar structures.

The electronic excited states of a BHJ scheme composed of P3HT, T3, and fullerene

(C60) (electron donor, spacer material, and electron acceptor respectively), were studied

using DFT with consideration to environmental effects. Experimental findings using this

BHJ scheme showed dramatically increased PCE with the terthiophene spacer material.

The excited state spectra calculated for isolated system and using the LR environmental

model did not explain the marked increase in PCE as the local excitons generated in the

bright π→π* of P3HT would decay into local excitons located on the fullerene units,

thereby quenching any generated CT states. The electronic excited state spectra calculate

using the SS environment however, showed a drastically different result as the local

excitons generated in the bright π→π* of P3HT were shown to decay into significantly

stabilized P3HT→C60 and T3→C60 CT states helping to explain the increased PCE. The

increase in PCE was thusly explained as resulting from increased physical separation

caused by the spacer decreasing the Coulomb binding energy and reducing the rate of

charge recombination, and stabilizing the CT states below local C60 excitons increasing the

rate of charge separation.

High-level SOS-MP2 and SOS-ADC(2) calculations were performed on several

oligomer dimers of PPV for the purpose of providing much needed benchmark calculations

of the structure and electronic excited states of the material that is paradigmatic in the

description of EET processes in PAHs. Dimers composed of two, three, and four repeating

phenylenevinylene (PPV2, PPV3, and PPV4 respectively) units were investigated using a

variety of correlation consistent basis sets. As PPV2 is the smallest structure, it was subject

to study with basis sets up cc-pVQZ and aug-cc-pVQZ, showing good agreement with the

smaller cc-pVTZ basis. The conformation was studied comparing the sandwich-stacked


172

system with a displaced dimer, showing smaller stacking distances and increased π-π*

interactions in the latter. DFT methods exhibited larger stabilization energies. Two

intensive electronic transitions were found for each dimer using the SOS-ADC(2) method,

where the oscillatory strength increased with increasing chain length. A similar trend was

found for the magnitude of interchain charge transfer as analyzed using omega matrices

and NTO transitions. DFT results calculated a more diverse interchain CT spectra than with

the SOS-ADC(2) method.

The biradical character of trans-diindenoacenes, cis-diindenoacenes, and trans-

indenoindenodi(acenothiophenes) with increasing number of core fused-benzene rings

were investigated using MR-AQCC calculations where the biradical character was

described by the ∆ES-T and NU. For each structure studied, a low spin singlet ground state

was found, and aside from trans-diindenobenzene and trans-

indenoindenodi(acenobenzene), each had significant singlet diradical character. The ∆ES-T

was shown to decrease with increasing number of core rings for each set of structures

correlating to an increase in the NU. There is good agreement to experiment for the

calculated values for ∆ES-T. The cis-diindenoacene structures were remarkable by

comparison as the ∆ES-T was lower and NU larger than the other structures. Increasing NU

correlated with the greater unpaired electron density in the singlet ground state and was

greatest for the carbon bonded to hydrogen in the five-membered ring. While the NU in the

first triplet excited state was relatively unchanged with increasing number of core rings,

the larger structures stabilized the unpaired electron density, thus lowering the energy of

the triplet state and the ∆ES-T gap. In the thiophene-fused structures, the thiophene groups

disrupted the aromaticity and concentrated the unpaired density in the core of the

structures. Consequently, relatively little unpaired electron density extended to the capping

six-membered rings.

The study of graphene functionalization and defect formation represents an exciting

field of computational study. To this end, as one hydrogenation defect has been discussed

already, the addition of a second hydrogen radical is being studied using molecular

dynamics simulations and circumcoronene model where the potential energy is determined

using DFT. The calculations are being performed in order to determine the energy transfer


173

process upon bond formation, the significance of H2 abstraction pathways, and the

probability and location of the second C-H formation at room temperature. Molecular

dynamics simulations are also being carried out in a similar fashion to determine the

dynamic dissociation energy required to remove a single carbon atom from pristine

graphene using the circumcoronene model generating the single carbon vacancy that has

been investigated previously.


174

APPENDIX A

ADDITIONAL TABLES AND FIGURES

NOT INCLUDED IN CHAPTER III

Table A.1 Relative energy differences ΔErel (eV) of the local energy minima in the

electronic ground state computed for the different structures of pyrene-1C+Hr using

different theoretical methods and the 6-31G* approach.

ΔErel


C1 0.637 0.960 1.393

C2 0.230 0.681 1.244

C3 0.855 1.163 1.474

C4 2.752 3.179 3.729

C5 1.627 1.809 2.032

C6 3.107 3.638 4.049

C7 2.517 2.700 2.816

C8 3.542 3.930 4.246

C15 0.000a 0.000b 0.000c aTotal energy = -574.277274 Hartree, bTotal energy = -575.610208 Hartree, cTotal energy

= -576.244082 Hartree, d The SA4-CASSCF(5,5) optimized geometry was used.


electronic ground state computed for the different structures of pyrene-2C+Hr using

different theoretical methods and the 6-31G* approach.

ΔErel


C1 0.881 0.829 0.767

C2 0.000a 0.000b 0.000c

C3 0.612 0.735 0.806

C4 1.059 1.054 1.047 aTotal energy = -536.452180 Hartree, bTotal energy = -537.716100 Hartree, cTotal energy

= -538.283379 Hartree


175


electronic ground state computed for the different structures of pyrene+Hr using different

theoretical methods and the 6-31G* basis set.

ΔErel


C1 0.647 0.634 0.578

C2 0.137 0.080 0.000c

C3 0.630 0.649 0.559

C4 0.000a 0.000b 0.018

C16 0.820 0.814 0.691 aTotal energy = -612.336622 Hartree, bTotal energy = -613.718399 Hartree, cTotal energy

= -614.381588 Hartree

Hr bonded to Carbon 1

31a’ 32a’ 33a’ 22a” 23a”

Figure A.1 Plots of the active orbitals of the optimized ground states of pyrene+Hr using

the CASSCF(5,5)/6-31G* approach.


176

Hr optimized on Carbon 15

29a’ 30a’ 31a’ 21a” 22a”

Figure A.2 Plots of the active orbitals of the optimized ground states of pyrene-1C+Hr

using the SA4-CASSCF(5,5)/6-31G* approach.

Hr optimized on Carbon 1

27a’ 28a’ 29a’ 20a” 21a”

Figure A.3 Plots of the active orbitals of the optimized ground states of pyrene-2C+Hr

using the SA4-CASSCF(5,5)/6-31G* approach.


177

3 2Aa

4 2Ab

Figure A.4 Potential energy scans in terms of the location of Hr for the 3 2A and 4 2A states

of pyrene-2C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located

in a plane parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest

energies: a -574.149530 Hartree, b -574.117820 Hartree. The energy (color scale) is given

in eV, relative to the ground state lowest minimum at -536.385046 Hartree.

Figure A.5 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is

bonded to C2 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -

574.277274 Hartree


178



574.277274 Hartree



574.277274 Hartree


179



574.277274 Hartree



574.277274 Hartree


180



574.277274 Hartree



574.277274 Hartree


181

APPENDIX B


NOT INCLUDED IN CHAPTER IV

Table B.1 NBO bonding character analysis of pyrene-2C+Si using the CAM-B3LYP/6-

311G(2d,1p) method. The carbon atom discussed is the carbon bonded to silicon.


CAM-B3LYP/6-311G(2d,1p)

P4 (D2) αa: 0.53(sp2.98d0.03)Si+0.85(sp2.97d0.01)C

βa: 0.53(sp2.98d1.03)Si+0.85(sp2.98 d0.01)C

P5 (D2) αa: 0.53(sp2.98d0.03)Si+0.85(sp3.00d0.01)C

βa: 0.53(sp2.98d0.03)Si+0.85(sp3.00 d0.01)C

P6 (D2) 0.54(sp2.98d0.02)Si+0.84(sp3.20d0.01)C

P7 (D2h) α: 0.45(sp2.01d1.01)Si+0.89(sp2.80d0.01)C

β: 0.46(sp1.99d0.99)Si+0.89(sp2.78d0.01)C

P8 (D2h) 0.45(sp2.00d1.00)Si+0.89(sp2.92d0.01)C

P9 (D2) α: 0.48(sp2.38d0.62)Si+0.88(sp2.39d0.01)C

β: 0.48(sp2.92d0.09)Si+0.87(sp2.45d0.01)C

P10 (D2h) α: 0.46(sp2.00d1.00)Si+0.89(sp2.41d0.01)C

β: 0.45(sp2.00d1.00)Si+0.89(sp2.69d0.01)C


182

s

sp3.58

p

sp3.58

p

sp3.58

p

sp3.58

Figure B.1 NBO plots for circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p) approach

(CP2 D2h). Isovalue = ±0.02 e/Bohr3


183

P1 (D2) P2 (D2h) P3 (D2h)

Figure B.2 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using the

B3LYP/6-311G(2d,1p) method.


184

P4 (D2) P5 (D2) P6 (D2)

P7 (D2h) P8 (D2h) P9 (D2)

P10 (D2h)

Figure B.3 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using the

CAM-B3LYP/6-311G(2d,1p) method.


185

P1 (D2) P2 (D2h)

P3 (D2h)

Figure B.4 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the

electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the B3LYP/6-

311G(2d,1p) method.


186

P4 (D2) P5 (D2)

P6 (D2) P7 (D2h)

P8 (D2h) P9 (D2)

P10 (D2h)


187

Figure B.5 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the

electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the CAM-B3LYP/6-

311G(2d,1p) method.

Figure B.6 Unprocessed EEL spectrum of the Si-C4 site.

90 100 110 120 130

6.5

7.0

7.5

8.0

8.5

10

5 c

ou

nts

Energy loss (eV)


188

Flat Corrugated

Figure B.7 First two unoccupied states immediately above the Fermi level spanning about

1 eV for the flat and corrugated Si/graphene structures from the periodic DFT calculations.

Electron density isovalue is 0.277 e/Bohr3.


189

APPENDIX C


NOT INCLUDED IN CHAPTER V

Table C.1 Excitation energies Ω of the isolated standard C60-T3-P3HT system using the

CAM-B3LYP/6-31G approach

Exc. Ω(eV) f CT Direction of

CT

Location of

exciton

S1 2.69 2.08 0.00 P3HT

S2 2.83 0.00 0.97 P3HTC60

S3 2.84 0.00 1.00 P3HTC60

S4 2.85 0.00 0.99 P3HTC60

S5 2.87 0.00 0.99 T3C60

S6 2.87 0.00 0.99 T3C60

S7 2.93 0.00 0.99 T3C60

S8 3.41 1.18 0.01 T3

S9 3.47 0.10 0.00 P3HT

S10 3.63 0.00 0.97 P3HTC60

S11 3.63 0.00 1.00 P3HTC60

S12 3.65 0.00 0.99 P3HTC60

S13 3.92 0.02 0.00 P3HT

S14 3.92 0.02 0.94 T3C60

S15 3.93 0.01 0.94 T3C60

S16 4.10 0.00 0.98 P3HTC60

S17 4.11 0.00 0.98 P3HTC60

S18 4.12 0.00 0.99 P3HTC60

S19 4.14 0.02 0.95 T3C60

S20 4.16 0.00 0.99 T3C60


190


CAM-B3LYP/6-31G* approach

Exc. Ω (eV) f CT Direction of

CT

Location of

exciton

S1 2.62 2.04 0.00 P3HT

S2 2.91 0.00 0.97 P3HTC60

S3 2.91 0.00 1.00 P3HTC60

S4 2.92 0.00 0.99 P3HTC60

S5 2.93 0.00 0.99 T3C60

S6 2.94 0.00 0.99 T3C60

S7 2.99 0.00 0.99 T3C60

S8 3.33 1.16 0.01 T3

S9 3.39 0.10 0.00 P3HT

S10 3.70 0.00 0.97 P3HTC60

S11 3.70 0.00 1.00 P3HTC60

S12 3.71 0.00 0.99 P3HTC60

S13 3.87 0.02 0.00 P3HT

S14 3.99 0.02 0.94 T3C60

S15 3.99 0.03 0.94 T3C60

S16 4.20 0.03 0.95 T3C60

S17 4.23 0.00 0.98 P3HTC60

S18 4.24 0.00 0.98 P3HTC60

S19 4.25 0.00 0.99 P3HTC60

S20 4.27 0.00 0.99 T3C60


191

Table C.3 Lowest excitation energies Ω of the isolated components P3HT, T3 and C60 using

the CAM-B3LYP/6-31G* approach.

Exc. Ω (eV) f

P3HT

S1 2.62 2.10

S2 3.39 0.00

S3 3.87 0.02

S4 4.06 0.14

S5 4.49 0.01

S6 4.63 0.00

S7 4.84 0.04

S8 4.86 0.00

S9 4.89 0.00

S10 4.93 0.01

T3

S1 3.30 1.09

S2 4.54 0.00

S3 4.91 0.00

S4 5.07 0.00

S5 5.13 0.00

S6 5.15 0.01

S7 5.24 0.01

S8 5.30 0.06

S9 5.41 0.00

S10 5.43 0.02

C60

S1 2.47 0.00

S2 2.47 0.00

S3 2.47 0.00

S4 2.53 0.00

S5 2.53 0.00

S6 2.53 0.00

S7 2.56 0.00

S8 2.56 0.00

S9 2.56 0.00

S10 2.57 0.00


192

Table C.4 Excitation energies Ω of the standard C60-T3-P3HT system the using the CAM-

B3LYP/6-31G* approach in the LR environment


CT

Location of

exciton

S1 2.53 2.22 0.00 P3HT

S2 3.12 0.00 0.97 P3HTC60

S3 3.12 0.00 1.00 P3HTC60

S4 3.13 0.00 0.99 P3HTC60

S5 3.22 0.60 0.57 T3C60

S6 3.23 0.14 0.90 T3C60

S7 3.24 0.48 0.56 T3C60

S8 3.27 0.10 0.00 P3HT

S9 3.29 0.02 0.99 T3C60

S10 3.87 0.01 0.00 P3HT

S11 3.91 0.00 0.97 P3HTC60

S12 3.91 0.00 1.00 P3HTC60

S13 3.92 0.00 0.99 P3HTC60

S14 3.98 0.19 0.00 P3HT

S15 4.28 0.01 0.85 T3C60

S16 4.29 0.13 0.86 T3C60

S17 4.37 0.00 1.00 P3HTT3

S18 4.45 0.00 0.97 P3HTC60

S19 4.45 0.00 0.99 P3HTC60

S20 4.46 0.00 0.99 P3HTC60


193

Table C.5 Excitation energies Ω of the standard C60-T3-P3HT system the using the CAM-

B3LYP/6-31G approach in the LR environment


CT

Location of

exciton

S1 2.60 2.26 0.0 P3HT

S2 3.08 0.00 0.97 P3HTC60

S3 3.08 0.00 1.00 P3HTC60

S4 3.09 0.00 0.98 P3HTC60

S5 3.20 0.01 0.99 T3C60

S6 3.21 0.01 0.99 T3C60

S7 3.27 0.02 0.99 T3C60

S8 3.32 1.21 0.01 T3

S9 3.35 0.09 0.00 P3HT

S10 3.87 0.00 0.97 P3HTC60

S11 3.88 0.00 1.00 P3HTC60

S12 3.89 0.00 0.98 P3HTC60

S13 3.92 0.01 0.00 P3HT

S14 4.08 0.20 0.00 P3HT

S15 4.28 0.05 0.91 T3C60

S16 4.28 0.04 0.91 T3C60

S17 4.35 0.00 1.00 P3HTC60

S18 4.35 0.00 0.98 P3HTC60

S19 4.36 0.00 0.99 P3HTC60

S20 4.40 0.00 1.00 P3HTT3

Table C.6 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM-

B3LYP/6-31G* approach in the SS environment

Exc. Ω (eV) f Direction of

CT

Location of

Exciton

S1 1.29 0.00 P3HTC60

S2 1.29 0.00 P3HTC60

S3 1.29 0.00 P3HTC60

S4 1.29 0.00 P3HTC60

S5 1.88 0.00 T3C60

S6 1.88 0.00 T3C60

S7 1.88 0.00 T3C60

S8 1.88 0.00 T3C60

S9 1.90 0.00 T3C60

S10 2.63 2.04 P3HT


194

Table C.7 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM-

B3LYP/6-31G approach in the SS environment


CT

Location of

Exciton

S1 1.23 0.00 1.00 P3HTC60

S2 1.23 0.00 1.00 P3HTC60

S3 1.24 0.00 1.00 P3HTC60

S4 1.84 0.00 1.00 T3C60

S5 1.84 0.00 1.00 T3C60

S6 1.86 0.00 1.00 T3C60

S7 2.06 0.00 1.00 P3HTC60

S8 2.06 0.00 1.00 P3HTC60

S9 2.07 0.00 1.00 P3HTC60

S10 2.70 2.07 0.00 P3HT


CAM-B3LYP/6-31G* approach (only core orbitals were frozen)


CT

Location of

exciton

S1 2.47 0.00 0.01 C60

S2 2.47 0.00 0.00 C60

S3 2.48 0.00 0.01 C60

S4 2.52 0.00 0.01 C60

S5 2.53 0.00 0.00 C60

S6 2.53 0.00 0.01 C60

S7 2.56 0.00 0.01 C60

S8 2.56 0.00 0.01 C60

S9 2.57 0.00 0.00 C60

S10 2.57 0.00 0.00 C60

S11 2.62 2.03 0.00 P3HT

S12 2.84 0.00 0.01 C60

S13 2.85 0.00 0.01 C60

S14 2.85 0.00 0.01 C60

S15 2.86 0.00 0.01 C60

S16 2.86 0.00 0.01 C60

S17 2.91 0.00 1.00 P3HTC60

S18 2.91 0.00 1.00 P3HTC60

S19 2.92 0.00 1.00 P3HTC60

S20 2.93 0.00 1.00 T3C60


195

Table C.9 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system using

the CAM-B3LYP/6-31G* approach


CT

Location of

exciton

1 1A 2.33 0.00 1.00 (T3)3(C60)3

2 1A 2.37 0.00 1.00 (T3)3(C60)3

3 1A 2.38 0.00 1.00 (T3)3(C60)3

4 1A 2.56 0.01 0.99 (T3)3(C60)3

5 1A 2.59 0.00 1.00 (T3)3(C60)3

6 1A 2.62 0.00 0.99 (T3)3(C60)3

7 1A 2.67 1.92 0.00 P3HT

8 1A 2.74 0.00 1.00 (T3)3(C60)3

9 1A 2.75 0.00 1.00 (T3)3(C60)3

10 1A 2.76 0.00 1.00 (T3)3(C60)3

11 1A 2.78 0.00 1.00 (T3)3(C60)3

12 1A 2.79 0.00 0.99 (T3)3(C60)3

13 1A 2.80 0.00 1.00 (T3)3(C60)3

14 1A 2.81 0.00 0.98 (T3)3(C60)3

15 1A 2.82 0.00 1.00 (T3)3(C60)3

16 1A 2.82 0.00 1.00 P3HT(C60)3

17 1A 2.83 0.00 1.00 (T3)3(C60)3

18 1A 2.85 0.00 1.00 P3HT(C60)3

19 1A 2.89 0.00 1.00 P3HT(C60)3

20 1A 2.96 0.00 0.97 (T3)3(C60)3

21 1A 2.96 0.00 0.99 (T3)3(C60)3

22 1A 2.97 0.00 0.98 (T3)3(C60)3

23 1A 3.02 0.00 1.00 (T3)3(C60)3

24 1A 3.04 0.00 1.00 (T3)3(C60)3

25 1A 3.05 0.00 1.00 (T3)3(C60)3

26 1A 3.06 0.00 1.00 P3HT(C60)3

27 1A 3.07 0.00 1.00 P3HT(C60)3

28 1A 3.08 0.00 1.00 P3HT(C60)3

29 1A 3.10 0.00 0.10 (T3)3

30 1A 3.14 0.00 1.00 (T3)3(C60)3

31 1A 3.15 0.00 1.00 (T3)3(C60)3

32 1A 3.15 0.00 1.00 P3HT(C60)3

33 1A 3.16 0.00 1.00 P3HT(C60)3

34 1A 3.16 0.00 1.00 (T3)3(C60)3

35 1A 3.17 0.00 1.00 P3HT(C60)3


196

Table C.9, Continued

36 1A 3.26 0.07 0.03 (T3)3

37 1A 3.28 0.00 0.99 (T3)3(C60)3

38 1A 3.29 0.00 0.99 (T3)3(C60)3

39 1A 3.29 0.00 0.97 (T3)3(C60)3

40 1A 3.43 1.38 0.03 P3HT/(T3)3

Table C.10 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system using

the CAM-B3LYP/6-31G approach


CT

Location of

exciton

S1 2.24 0.00 1.00 (T3)3(C60)3

S2 2.28 0.00 1.00 (T3)3(C60)3

S3 2.30 0.00 1.00 (T3)3(C60)3

S4 2.47 0.01 0.99 (T3)3(C60)3

S5 2.51 0.00 1.00 (T3)3(C60)3

S6 2.54 0.00 1.00 (T3)3(C60)3

S7 2.68 0.00 1.00 (T3)3(C60)3

S8 2.69 0.00 1.00 (T3)3(C60)3

S9 2.70 0.00 1.00 (T3)3(C60)3

S10 2.71 0.00 1.00 (T3)3(C60)3

S11 2.72 0.00 1.00 (T3)3(C60)3

S12 2.73 0.00 1.00 (T3)3(C60)3

S13 2.73 0.00 1.00 P3HT(C60)3

S14 2.74 1.96 0.00 P3HT

S15 2.75 0.00 0.98 (T3)3(C60)3

S16 2.75 0.00 1.00 (T3)3(C60)3

S17 2.76 0.00 1.00 P3HT(C60)3

S18 2.77 0.00 0.97 (T3)3(C60)3

S19 2.79 0.00 1.00 P3HT(C60)3

S20 2.92 0.00 0.98 (T3)3(C60)3

S21 2.92 0.00 1.00 (T3)3(C60)3

S22 2.93 0.00 0.97 (T3)3(C60)3

S23 2.96 0.00 1.00 (T3)3(C60)3

S24 2.97 0.00 1.00 (T3)3(C60)3

S25 2.99 0.00 1.00 (T3)3(C60)3

S26 3.00 0.00 1.00 P3HT(C60)3

S27 3.01 0.00 1.00 P3HT(C60)3

S28 3.02 0.00 1.00 P3HT(C60)3


197

Table C.10, Continued

S29 3.07 0.00 1.00 (T3)3(C60)3

S30 3.08 0.00 1.00 (T3)3(C60)3

S31 3.09 0.00 1.00 (T3)3(C60)3

S32 3.10 0.00 1.00 P3HT(C60)3

S33 3.11 0.00 1.00 P3HT(C60)3

S34 3.11 0.00 1.00 P3HT(C60)3

S35 3.15 0.00 0.11 (T3)3

S36 3.23 0.00 0.99 (T3)3(C60)3

S37 3.23 0.00 0.99 (T3)3(C60)3

S38 3.24 0.00 0.97 (T3)3(C60)3

S39 3.32 0.10 0.03 (T3)3

S40 3.49 1.97 0.08 (T3)3

Table C.11 Excitation energies Ω of the extended C60-T3-P3HT system using the CAM-

B3LYP/6-31G approach in the SS environment


CT

Location of

exciton

S1 1.59 0.00 1.00 P3HT(C60)3

S2 1.59 0.00 1.00 P3HT(C60)3

S3 1.60 0.00 1.00 P3HT(C60)3

S4 1.61 0.00 1.00 P3HT(C60)3

S5 1.77 0.00 1.00 (T3)3(C60)3

S6 1.77a 0.00 1.00 (T3)3(C60)3

S7 1.88 0.00 1.00 (T3)3(C60)3

S8 1.91 0.00 1.00 (T3)3(C60)3

S9 1.91 0.00 1.00 (T3)3(C60)3

S10 1.91 0.00 1.00 (T3)3(C60)3

S11 1.91 0.00 1.00 (T3)3(C60)3

S12 2.10 0.00 1.00 (T3)3(C60)3

S13 2.10 0.00 1.00 (T3)3(C60)3

S14 2.10 0.00 1.00 (T3)3(C60)3

S15 2.12 0.00 1.00 (T3)3(C60)3

S16 2.12 0.00 1.00 (T3)3(C60)3

S17 2.13 0.00 1.00 (T3)3(C60)3

S18 2.13 0.00 1.00 (T3)3(C60)3

S19 2.21b 0.00 1.00 (T3)3(C60)3

S20 2.75 1.92 0.00 P3HT aExcitation energy averaged over two oscillating energies: 1.77 eV and 1.76 eV, bExcitation energy averaged over three oscillating energies: 2.12 eV, 2.21 eV, 2.29 eV


198

APPENDIX D


NOT INCLUDED IN CHAPTER VI

Table D.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer computed

at MP2 and MP2/BSSE levels with several basis sets using C2h symmetry.

1-1’ 2-2’ 3-3’ 4-4’ 5-5’ ΔE

MP2

cc-pVDZ 3.58 3.56 3.54 3.61 3.62 -9.13

aug-cc-pVDZ 3.48 3.45 3.44 3.50 3.52 -20.23

cc-pVTZ 3.57 3.55 3.53 3.59 3.60 -11.80

aug-cc-pVTZ 3.57 3.45 3.46 3.53 3.64 -15.10

MP2/BSSE

cc-pVDZ 3.78 3.77 3.77 3.78 3.79 -4.73

cc-pVTZ 3.66 3.64 3.63 3.67 3.68 -8.95


199

Tab

le D

.2 I

nte

rchai

n d

ista

nce

s (Å

) an

d Δ

E (

kca

l/m

ol)

in the

stac

ked

PP

V4 d

imer

com

pute

d

at S

OS

-MP

2 a

nd S

OS

-MP

2/B

SS

E l

evel

s w

ith s

ever

al b

asis

set

s usi

ng C

2h s

ym

met

ry.


200

Table D.3 Interchain distances (Å) and ΔE (kcal/mol) in the diplaced PPV4 dimer computed

at SOS-MP2 level with several basis sets using C2h symmetry.

1-1’ 2-2’ 3-3’ 4-4’ 5-5’ 6-6’ 7-7’ 8-8’ ΔE

cc-pVTZ 3.57 3.50 3.46 3.45 3.45 3.46 3.50 3.57 -21.17

cc-pVQZa - - - - - - - - -19.56


TQb -18.39 aSingle-point energy at SOS-MP2/cc-pVTZ optimized geometry, bExtrapolation made


Table D.4 Single chain PPV2 vertical excitations using SOS-ADC(2) and SOS-MP2

optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength)

Basis State Excitation ΔE (eV) f

cc-pVDZ S1 1 1Bu 4.57 0.51 S2 1 1Ag 4.65 0.00 S3 2 1Bu 4.74 0.72 S4 2 1Ag 6.10 0.00 S5 1 1Bg 7.63 0.00 S6 1 1Au 7.76 0.00 S7 2 1Bg 7.80 0.00 S8 2 1Au 8.22 0.00

cc-pVTZ S1 1 1Bu 4.52 0.75 S2 1 1Ag 4.66 0.00 S3 2 1Bu 4.72 0.43 S4 2 1Ag 6.00 0.00 S5 1 1Bg 7.24 0.00 S6 1 1Au 7.26 0.00 S7 2 1Bg 7.56 0.00 S8 2 1Au 8.04 0.01

cc-pVQZ S1 1 1Bu 4.49 0.77 S2 1 1Ag 4.66 0.00 S3 2 1Bu 4.71 0.40 S4 2 1Ag 5.95 0.00 S5 1 1Au 6.86 0.00 S6 1 1Bg 6.95 0.00


201

Table D.4, Continued

S7 2 1Bg 7.26 0.00 S8 2 1Au 7.66 0.02

Table D.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2) and SOS-MP2

optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT –

interchain charge transfer)

Basis State Excitation ΔE (eV) f CT (e)

cc-pVDZ S1 1 1Bg 4.21 0.00 0.17

S2 1 1Au 4.51 0.00 0.05

S3 2 1Bg 4.53 0.00 0.08

S4 1 1Bu 4.58 0.36 0.01

S5 1 1Ag 4.63 0.00 0.01

S6 2 1Bu 4.81 1.82 0.01

S7 2 1Au 5.78 0.00 0.21

S8 2 1Ag 6.11 0.00 0.02

cc-pVTZ S1 1 1Bg 4.10 0.00 0.18

S2 1 1Au 4.50 0.00 0.08

S3 2 1Bg 4.51 0.00 0.08

S4 1 1Bu 4.55 0.74 0.03

S5 1 1Ag 4.65 0.00 0.02

S6 2 1Bu 4.76 1.32 0.02

S7 2 1Au 5.64 0.00 0.22

S8 2 1Ag 6.01 0.00 0.03


202

Table D.6 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-

MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron NTO

pairs (Isovalue = ±0.03 Bohr3)

Excitation % Hole Electron

1 1Bg 82

12

1 1Au 28

26

2 1Bg 40

16

11


203


1 1Bu 41

30

1 1Ag 19

18

16

2 1Bu 46

36

2 1Au 55


204


25

2 1Ag 33

25

14


205

Table D.7 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and

SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron

NTO pairs (Isovalue = ±0.03 Bohr3)


1 1A 46

45

2 1A 40

29

3 1A 23

19

17


206


13

4 1A 28

21

19

13

5 1A 35

20

16


207


14

6 1A 53

32

7 1A 55

40

8 1A 62

33


208


MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator



S1 1 1Bg 3.47 0.00 0.18

S2 1 1Bu 3.98 3.60 0.02

S3 2 1Bg 4.21 0.00 0.10

S4 2 1Bu 4.38 0.34 0.02

S5 1 1Au 4.39 0.00 0.16

S6 2 1Au 4.57 0.00 0.15

S7 1 1Ag 4.60 0.00 0.04

S8 2 1Ag 5.11 0.00 0.02


MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator



S1 1 1Bg 3.20 0.00 0.19

S2 1 1Bu 3.69 5.58 0.02

S3 1 1Au 3.86 0.00 0.21

S4 2 1Bg 4.17 0.00 0.13

S5 2 1Au 4.20 0.00 0.14

S6 1 1Ag 4.31 0.00 0.04

S7 2 1Bu 4.34 0.31 0.23

S8 2 1Ag 4.53 0.00 0.04


209

Table D.10 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and

SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron

NTO pairs (Isovalue = ±0.03 Bohr3)


1 1A 50

40

2 1A 55

35

3 1A 38

24

13


210


11

4 1A 31

24

15

14

5 1A 50

30

6 1A 28


211


24

7 1A 26

20

8 1A 16

16

14

12


212

Table D.11 Stacked PPV3 dimer vertical excitations using B3LYP-D3/cc-pVTZ and the

B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f –



S1 1 1Bg 2.40 0.00 0.50

S2 1 1Bu 2.83 0.04 0.99

S3 2 1Bu 3.28 3.10 0.01

S4 2 1Bg 3.38 0.00 0.50

S5 1 1Au 3.38 0.00 0.50

S6 2 1Au 3.45 0.00 0.50

S7 1 1Ag 3.77 0.00 0.99

S8 3 1Bg 3.79 0.00 0.50

S9 2 1Ag 3.84 0.00 0.08

S10 4 1Bg 3.95 0.00 0.50

S11 3 1Au 3.97 0.00 0.50

S12 3 1Ag 3.99 0.00 0.92

S13 5 1Bg 4.07 0.00 0.50

S14 6 1Bg 4.13 0.00 0.50

S15 4 1Au 4.15 0.00 0.50

S16 3 1Bu 4.18 0.06 0.08

S17 4 1Ag 4.33 0.00 0.05

S18 4 1Bu 4.35 0.01 0.07

S19 5 1Au 4.41 0.00 0.50

S20 7 1Bg 4.44 0.00 0.50

S21 8 1Bg 4.47 0.00 0.50

S22 6 1Au 4.47 0.00 0.50

S23 5 1Bu 4.47 0.02 0.90

S24 5 1Ag 4.48 0.00 0.03

S25 6 1Bu 4.52 0.00 1.00

S26 7 1Bu 4.58 0.01 0.94

S27 6 1Ag 4.58 0.00 0.99

S28 8 1Bu 4.60 0.01 0.83

S29 7 1Ag 4.63 0.00 0.96

S30 9 1Bg 4.71 0.00 0.54


213

Table D.12 Stacked PPV3 dimer vertical excitations using CAM-B3LYP-D3/cc-pVTZ and

the CAM-B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy,

f – oscillator strength, CT – interchain charge transfer)


S1 1 1Bg 3.17 0.00 0.25

S2 1 1Bu 3.73 3.16 0.05

S3 2 1Bu 4.09 0.13 0.96

S4 1 1Au 4.17 0.00 0.28

S5 2 1Bg 4.35 0.00 0.77

S6 3 1Bg 4.47 0.00 0.16

S7 3 1Bu 4.67 0.07 0.02

S8 4 1Bg 4.77 0.00 0.19

S9 2 1Au 4.77 0.00 0.28

S10 1 1Ag 4.78 0.00 0.09

S11 3 1Au 4.88 0.00 0.50

S12 4 1Bu 4.98 0.01 0.01

S13 2 1Ag 5.02 0.00 0.12

S14 3 1Ag 5.03 0.00 0.88

S15 5 1Bg 5.07 0.00 0.50

S16 4 1Ag 5.17 0.00 0.03

S17 6 1Bg 5.27 0.00 0.50

S18 4 1Au 5.33 0.00 0.73

S19 5 1Au 5.39 0.00 0.50

S20 7 1Bg 5.47 0.00 0.50

S21 5 1Bu 5.59 0.34 0.25

S22 5 1Ag 5.67 0.00 0.99

S23 6 1Bu 5.73 0.15 0.40

S24 8 1Bg 5.79 0.00 0.27

S25 7 1Bu 5.84 0.28 0.48


214

S1 S2 S3 S4

S5 S6 S7 S8

Figure D.1 Omega matrices for vertical excitations of displaced PPV2 dimer using the SOS-





S1 S2 S3 S4

S5 S6 S7 S8

Figure D.2 Omega matrices for vertical excitations of stacked PPV4 dimer using the SOS-






215

S1 S2 S3 S4

S5 S6 S7 S8

Figure D.3 Omega matrices for vertical excitations of displaced PPV3 dimer using the SOS-






216

APPENDIX E


NOT INCLUDED IN CHAPTER VII

Table E.1 Active orbitals for each MR-AQCC reference space

Reference Space CAS AUX

1

MR-AQCC(CAS(4,4))a 5au, 6au, 5bg, 6bg

MR-AQCC(CAS(4,4)) 5au, 6au, 5bg, 6bg

MR-AQCC

(CAS(10,7)/AUX(3))

3au, 4au, 5au,

6au, 4bg, 5bg, 6bg 7au, 7bg, 8bg

MR-AQCC

(CAS(14,9)/AUX(3))

3au, 4au, 5au,

6au, 2bg, 3bg,

4bg, 5bg, 6bg

7au, 7bg, 8bg

2


MR-AQCC

(CAS(12,8)/AUX(4))

4au, 5au, 6au,

7au, 4bg, 5bg,

6bg, 7bg

8au, 9au, 8bg, 9bg

3


4


5

MR-AQCC(CAS(4,4)) 5b2, 6b2, 5a2, 6a2

6

MR-AQCC(CAS(4,4)) 7b2, 8b2, 5a2, 6a2

7

MR-AQCC(CAS(4,4)) 7b2, 8b2, 7a2, 8a2

8

MR-AQCC(CAS(4,4)) 9b2, 10b2,

7a2, 8a2

9


MR-AQCC

(CAS(10,7)/AUX(3))

6au, 7au, 8au,

9au, 7bg, 8bg, 9bg 10au, 10bg, 11bg

10

MR-AQCC(CAS(4,4)) 9au, 10au,

9bg, 10bg

11


217

Table E.1, Continued


10bg, 11bg

12


11bg, 12bg

aonly core orbitals are frozen

Table E.2 NU value (e) of trans-diindenoacene (1-4) singlet 1 1Ag state and vertically

excited 1 3Bu state using the MR-AQCC(CAS(4,4))/6-31G* method.

System State NU (e)

1 1 1Ag aNU 0.547

bNU(H-L) 0.230

1 3Bu aNU 2.376

bNU(H-L) 1.991

2 1 1Ag aNU

1.095

bNU(H-L) 0.672

1 3Bu aNU

2.455

bNU(H-L) 1.997

3 1 1Ag aNU

1.602

bNU(H-L) 1.048

1 3Bu aNU

2.563

bNU(H-L) 1.998

4 1 1Ag aNU

2.178

bNU(H-L) 1.456

1 3Bu aNU

2.722

bNU(H-L) 1.998

aSum over all NO occupations, bSum over only HONO-LUNO pair


218

Table E.3 NU value (e) of cis-diindenoacene (5-8) singlet 1 1A1 state and vertically excited

1 3B1 state using the MR-AQCC(CAS(4,4))/6-31G* method.

System State NU (e)

5 1 1A1 aNU 1.414

bNU(H-L) 1.083

1 3B1 aNU 2.345

bNU(H-L) 1.999

6 1 1A1 aNU

1.923

bNU(H-L) 1.506

1 3B1 aNU

2.429

bNU(H-L) 2.000

7 1 1A1 aNU

2.280

bNU(H-L) 1.764

1 3B1 aNU

2.541

bNU(H-L) 2.000

8 1 1A1 aNU

2.550

bNU(H-L) 1.902

1 3B1 aNU

2.703

bNU(H-L) 2.000



219

Table E.4 NU value (e) of trans-indenoindenodi(acenothiophene) (9-12) singlet 1 1Ag state

and vertically excited 1 3Bu state using the MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S

method.

System State NU (e)

9 1 1Ag aNU 0.784

bNU(H-L) 0.362

1 3Bu aNU 2.461

bNU(H-L) 1.997

10 1 1Ag aNU

1.505

bNU(H-L) 0.979

1 3Bu aNU

2.552

bNU(H-L) 1.999

11 1 1Ag aNU

2.088

bNU(H-L) 1.451

1 3Bu aNU

2.670

bNU(H-L) 1.999

12 1 1Ag aNU

2.482

bNU(H-L) 1.623

1 3Bu aNU

2.834

bNU(H-L) 1.999



220

Figure E.1 Quinoid Kekule and biradical resonance forms of trans-diindenoacenes (1-4),

cis-diindenoacenes (5-8), and trans-indenoindenodi(acenothiophenes) (9-12).


221

Figure E.2 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the

vertically excited 1 3Bu state of trans-diindenobenzene (1-4) using the MR-

AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in

the center of each ring.


vertically excited 1 3B1 state of cis-diindenobenzene (5-8) using the MR-

AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in

the center of each ring.


222


vertically excited 1 3Bu state of trans-indenoindenodi(acenothiophene) (9-12) using the

MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method. (isovalue=0.004 e/Bohr3). HOMA

index is shown in the center of each ring.

copyright 2019, reed nieman

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