charge-transfer excitations and photophysical properties of molecular building...

Charge-transfer Excitations and

Photophysical Properties of Molecular

Building Blocks

Oscar Rubio Pons

Theoretical Chemistry

Department of Biotechnology

Royal Institute of Technology

Stockholm 2005

c© Oscar Rubio Pons, 2005

ISBN 91-7178-020-3

Printed by Universitetsservice US AB,

Stockholm, Sweden, 2005

3

Abstract

This thesis reports a state-of-the-art theoretical study of photophysical properties of organic

charge-transfer aromatic molecules. These molecules are building blocks of molecular func-

tional materials used in modern photonics technology and play essential roles in chemistry

and biology in general. A good understanding of these systems is thus important.

The theoretical results for permanent dipole moments of some substituted benzenes have

been obtained using the coupled cluster singles and doubles (CCSD) method. The perfor-

mance of density functional theory (DFT) for the geometry and electronic properties has

been compared with that of traditional ab initio methods, such as Hartree-Fock, second-

order Møller Plesset perturbation theory (MP2), CCSD and CCSD(T). Limitations of

the DFT methods for charge transfer molecules have been demonstrated. The multi-

configuration self-consistent field (MCSCF) method has been applied to understand proper-

ties of the triplet states of benzene derivatives by studying their phosphorescence with the

inclusion of contributions from vibronic coupling. It has also been employed to calculate the

photophysics of the thioxanthone molecule containing three benzene rings in combination

with the CASPT2 method, resolving a long-standing problem concerning the possible stable

conformations of the molecule.

With knowledge of the building blocks a series of porphyrin derivatives with exceptionally

large two-photon absorption cross sections were designed, and proposed for use in bioimaging

applications. The static and dynamic properties of a few zinc and platinum organometallic

compounds, being possible candidates for optical limiting devices, have also investigated.

4

Preface

The work presented in this thesis has been carried out at Laboratory of Theoretical Chem-

istry, Royal Institute of technology, Stockholm, Sweden.

Publications included in the thesis

Paper I The Permanent Dipole moment of gas phase p-aminobenzoic acid revisited, O.

Rubio-Pons, and Y. Luo, J. Chem. Phys. 157, 121, (2004).

Paper II Electronic Properties of Ground and Charge Transfer States of p-Substituted-

Anilines O. Rubio-Pons, Y. Luo, P. Sa lek and H. Agren J. Chem. Phys, Submitted. (2005)

Paper III CASSCF calculations of triplet-state properties. Applications to benzene deriva-

tives, O. Rubio-Pons, O. Loboda, B. Minaev, B.Schimmelpfennig, O. Vahtras and H. Agren,

Mol. Phys. 101, 2103 (2003).

Paper IV Ab initio calculations of vibronic activity in phosphorescence microwave double

resonance spectra of para-dichlorobenzene, O. Rubio-Pons, B. Minaev, O. Loboda, and H

Agren, Theor. Chem. Accounts, 113, 15 (2005)

Paper V A butterfly effect as a clue to the unique photophysics of thioxanthone, O.

Rubio-Pons, L. Serrano-Andres, D. Burget and P. Jacques. Phys. Chem. Chem. Phys.,

Submitted. (2005)

Paper VI Charge-transfer Zn-porphyrin derivatives with very large two-photon absorp-

tion cross sections at the fundamental wavelengths of 1.3-1.5 µm, Y. Luo, O. Rubio-Pons,

J.-D. Guo, and H. Agren, J. Chem. Phys. 122, 096101, (2005).

Paper VII Effects of Conjugation Length, Electron Donor and Acceptor Strengths on

the Two-Photon Absorption Cross Sections of Asymmetric Zinc-Porphyrin Derivatives O.

Rubio-Pons, and Y. Luo, and H. Agren in Preparation.

Paper VIII Optical limiting properties of Zinc and Platinum based organometallic com-

pounds, A. Baev, O. Rubio-Pons, F. Gel’mukhanov, and H. Agren, J. Phys. Chem. A, 108,

7406, (2004).

5

List of Papers not included in the thesis

Paper IX Characterization of aza-fullerene C58N2 by X-ray spectroscopy, S. Kashtanov,

O. Rubio-Pons, Y. Luo, H. Agren, S. Stafstrom, and S. Csillag, Chem. Phys. Lett., 371, 98

(2003).

Paper X Upconverted lasing based on many-photon absorption: An all dynamic descrip-

tion, A. Baev, F. Gel’mukhanov, O. Rubio-Pons, P. Cronstrand, and H. Agren, J. Opt. Soc.

Am. B, 21, 384 (2004).

6

Comments on my contribution to the Papers included

• I was responsible for all calculations and writing of the manuscripts for Papers I and

II, III, IV, VII.

• I was responsible for part of the calculations and collaborated in writing the manuscript

of Papers V, VI, VIII.

7

Acknowledgments

This PhD Thesis would have been very different or almost impossible to produce without

the help of many people, whom I would like to thank:

I would like to express sincere gratitude my supervisors Prof. Hans Agren and Dr Yi Luo,

for constant help in my studies and for giving me the opportunity to come to Stockholm to

realize my PhD studies.

Thanks to Prof. Faris Gel’mukhanov and Prof. Boris Minaev, for explaining to me very

clearly all physics behind the spectroscopies.

Thanks to Dr Bernd Schimmelpfening, for his supervision in the first year, and his invaluable

help related to quantum chemical calculations.

Thanks to Dr Pawel, and Dr Olav for the constant help with the computers problems and

quantum chemical calculations.

Thanks to Dr Matteo, Dr Javier and Dr Nicole, and Dr Nathalie, and the friend for the

closer house, Dr Marcus, Mattias, Dr Arianna, Dr Tomek, Valentin for all the good moments

in Stockholm.

To Mia, I will never let you to give up your dreams. All love that has not friendship for its

base, is like a mansion built upon sand

My Indian family, Dr Rajeev Prabhakar, Rajeev Natarajan and Parvathi, Ritu, Dr Vinod,

Vipin and Nilima, Shankha, and Vickas, all of them “Pappajis”, for all the good moments,

indian dinners, and pure friendship that we have shared a long these years, I will never

forget it.

Tusen Tack, to my Swedish classmates, and friends in Stockholm, Niina, Caterina, Andreaa,

Karina, Martin, Karen, Gabriella, and Fayyaz, and our best teacher Pelle, we learnt some

Swedish and enjoyed our time.

Muito Obrigado to Viviane, it was very nice to spend a year working in Stockholm, I will

never forget you.

Thanks, to my previous officemates and friends for the all good moments Dr Sacha, Dr

Brano and Dr Jing-Dong. Now it is my time!

Thanks to my officemates and friends, Dr Luca, Kathrin and Polina, and the colleagues at

the Theoretical Chemistry Lab, Dr Fahmi, Viktor, Barbara, Freddy, Mathias, Ivo, Linnea,

Stepan, Yanhua, Liu, Zilvinas, Lyudmila, Elias, Cornel, Jun Jiang, Wenyong Su, Sergey,

Tu, Emanuel, Emil, Quande, Hakan, Peter, and Laban. It is a good environment to work

together.

8

To my professors in Valencia, Dr Luis Serrano Andres, and Dr Manuela Merchan. ”Teachers

open the door. You enter by yourself”, I did and this is the result. Thanks.

All friends in Valencia, Jose Miguel, Begona, Boutaina, Ximo, Ernest, Ernesto, Victoria,

David, Pepa, Jesus, Victor, Julio, Alejandro, Montse, Vicente, Rafa, Rosendo, Nacho, En-

rique, Pepe.

No words I can find to express the constant support from my relatives, siblings, and family

in Spain. This thesis is dedicated to my parents: Julio and Pilar

Contents

1 Introduction 11

2 Quantum Chemical Methods 13

2.1 Hartree-Fock Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Multi-Configuration Self-Consistent Field method . . . . . . . . . . . . . . . 14

2.2.1 Perturbation Theory with a CAS Reference State . . . . . . . . . . . 15

2.3 Coupled Cluster Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . . . 20

2.4.2 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.3 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Response theory for an Exact State . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.1 Linear Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.2 Quadratic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Excited State Properties 27

3.1 Phosphorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.2 Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.3 Herzberg-Teller Approach . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Two-photon absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9

10 CONTENTS

3.2.1 Two-state model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Chemical Dynamics as Transitions between States . . . . . . . . . . . . . . . 33

3.3.1 Theory of Dynamics Simulations . . . . . . . . . . . . . . . . . . . . 35

4 Summary of articles 37

Chapter 1

Introduction

Happiness is when what you think, what you say,

and what you do are in harmony.

Mahatma Gandhi

In the beginning it was dark and cold. There was no Sun, no light, no Earth, no solar

system. There was nothing, just the empty void of space. Then slowly, about 4.5 billion

years ago, a swirling nebula, - a huge cloud of gas and dust was formed. Eventually this

cloud contracted and grew into a central mass that became our Sun.

At first the Sun was a molten glow. As the core pressure increased, and the temperature rose

to millions of degrees - a star was born. Through the process of thermonuclear hydrogen

fusion, the Sun began to shine. Somehow the Sun, and the Earth have been created, the

Sun light reached the Earth and life just started. From that period the human being has

always been fascinated by the environment, by the objects around him. Fire, was discovered

and the light, electromagnetic radiation, has been the most powerful tool to learn about

properties of matter, from the beginning of history to now.

By analysis of an object’s light, astronomers can know the physical properties of that object

(such as temperature, mass, luminosity and composition). The interaction of light with

matter was some of the most difficult conceptual challenges in the long history of physics.

Only in last century, with the creation of quantum mechanics, we have gained a quantitative

understanding of how light and atoms interact. A new way of looking at our Universe

emerged.

Ions, atoms, and molecules have electronically excited states that can be populated by the

absorption of light of appropriate energy. The electronic structure of a molecule is deter-

mined by the quantum behavior of electrons inside the system. Molecular dipole moments

11

12 CHAPTER 1. INTRODUCTION

are one of the basic electric properties that can be used to describe the behavior of the

molecule in media and the static interaction between a molecule and an electromagnetic

wave. More advanced spectroscopic methods, as well as the theoretical methods, are needed

to study the dynamic molecular properties such as the transition probabilities and the ex-

ited state properties. The excited state can be reached by absorbing photons. One-photon

excitations can only reach states that follow the dipole selection rule. One way to improve

the situation is to use two-photon absorption for which the quadrupole selection rules op-

erate. According to the energy conservation law, the excited electron will eventually loose

its energy and go back to the ground state. Sometimes, the lost enegy can be coverted to

light emission, resulting in fluorescence. Transitions between states of different multiplicities

such as triplet-singlet transitions are controlled by spin selection rules. The emission from

the triplet state to the singlet state is allowed only when spin-orbit coupling is taken into

account, which can result in so-called phosphorescence.

State-of-the-art theoretical methods have been applied to describe the interaction between

molecules and light in this thesis, focusing on the photophysical properties of the ground

and excited states of charge-transfer (CT) molecules. The CT molecules under investigation

often serve as molecular building blocks for constructing larger systems with exceptional

nonlinear optical properties. Accurate theoretical results for some of the basic photophysical

properties of these building blocks have been provided. The performance of the popular

and powerful density functional theory for charge-transfer molecules has been under close

scrutiny. New Zn-porphyrin derivatives with very large two-photon absorption cross sections

at the fundamental wavelengths suitable for bioimaging and optical communications have

been predicted. The dynamic of two-photon absorption in metallorganic nonlinear optical

materials have been explored.

The applications described in this thesis are relevant to a variety of emerging technologies,

ranging from optical limiting, non-destructive bioimaging, 3D optical data storage, lasering

and photodynamic therapy.

Chapter 2

Quantum Chemical Methods

2.1 Hartree-Fock Theory

Hartree-Fock theory is a molecular orbital based approximation which has served as a corner

stone in quantum chemistry. The idea of relating the electronic structure with molecular

orbitals is particularly intriguing for chemists because of its instructive and illustrative

nature. The Hartree-Fock (HF) method is based on the assumption that every electron

moves in the potential created by the nucleus plus the average potential of all the other

electrons. It is an independent-particle model in which the electron correlation is neglected.

In restricted Hartree-Fock (RHF) theory, the electronic state is represented by a single

configuration state function (CSF)1–3

CSF =∑

i

Ci|i〉 (2.1)

where |i〉 are Slater determinants with coefficients Ci fixed by the spin symmetry of the

wave function. Each Slater determinant is constructed as a product of the canonical spin

orbitals (cs).

|cs〉 =∏

i

a†iαa†

iβ|vac〉 (2.2)

where |vac〉 is the vacuum space and a†iαa†

iβ are the creation operators.

13

14 CHAPTER 2. QUANTUM CHEMICAL METHODS

The Hartree-Fock equations are described as pseudo-eigenvalue equations, to be solved in

an iterative process where the Fock matrix is repeatedly constructed and diagonalized until

a set of orbitals that satisfy the canonical conditions is obtained.

The Fock operator is

fpq =1

2

∑

σ

〈cs|[a†qσ, [apσH]]+|cs〉 (2.3)

The Hamiltonian operator may be written in the form

H = h + g + hnun =∑

pq

hpqEpq +1

2

∑

pqrs

gpqrsepqrs + hnuc (2.4)

The Hamiltonian contains the full set of electronic interactions and is independent of the

electronic state studied. The Fock operator is by contrast an effective one-electron operator,

designated with a particular state in mind.1,3

2.2 Multi-Configuration Self-Consistent Field method

The single-configuration HF approximation is incapable of representing systems dominated

by several configurations. The Multi-Configurational Self Consistent Field (MCSCF), method

represents a flexible solution to the configuration problem in quantum chemistry.3,4

The energy in MCSCF can be described using the second quantization formalism:

Eij =∑

σ=α,β

= a†iσajσ = a†

iαajα + a†iβajβ (2.5)

The Hamiltonian in the operator basis is constructed from the excitation energy operator,

Eij

H =∑

ij

hijEij +1

2

∑

i,j,k,l

gijkl(EijEkl − δjkEil) (2.6)

where, hij, are the one-electron integrals including the electron kinetic energy and the

electron-nuclear attraction terms, gijkl, are the two-electron repulsion integrals.

For a normalized Configuration Interaction wave function,

|Ψ〉 =∑

m

Cm|m〉 (2.7)

2.2. MULTI-CONFIGURATION SELF-CONSISTENT FIELD METHOD 15

expanded in the determinant basis, we obtain the energy as the expectation value of the

Hamiltonian.

E = 〈Ψ|H|Ψ〉 =∑

ij

hijDij +∑

i,j,k,l

gijklPijkl (2.8)

Here, Dij, is the first-order reduced density matrix, and Pijkl, the second order reduced

density matrix. The density matrices D and P contain information about the CI coefficients.

In the MCSCF wave function the variables are the CI coefficients and the molecular orbitals

in an orthonormalized vectorial space.4,5 One of the often used MCSCF methods is the so-

called complete active space self-consistent field (CASSCF) approach. In a CASSCF wave

function the occupied orbital space is divided into a set of inactive or closed-shell orbitals and

a set of active orbitals. All inactive orbitals are doubly occupied in each Slater determinant.

On the other hand, the active orbitals have varying occupations, and all possible Slater

determinants (or CSFs) are taken into account which can be generated by distributing the

electrons in all possible ways among the active orbitals. The CASSCF method mainly treats

the so-called static correlation.6

2.2.1 Perturbation Theory with a CAS Reference State

The MCSCF method should be considered as an extension of the HF method to deal with

the near-degeneracy of different electronic configurations. Thus, like in HF, we still have to

solve the problem of the dynamic electron correlation, which is necessary in order to obtain

quantitative results.

The early treatments of the electron correlation often employed the multi-reference CI

method. This approach approximates the wave function as an expansion in all singly and

doubly excited configurations with respect to a set of the chosen reference configurations.

Another approach, CASPT2, is based on perturbation theory, which has turned out to be

accurate in a wide variety of applications, in particular in electronic spectroscopies. Starting

from a CASSCF reference function the remaining correlation effects are estimated using

second order perturbation theory. As the theory is formulated, it is valid for any reference

state constructed as a full CI wave function in some orbital space. however, optimization of

the orbitals often leads to simplifications in the computational procedure. The configuration

space, in which the wave function is expanded can be divided into four subspaces: V0, VK ,

VSD and VTQ..., where V0 is one-dimensional space spanned by the CAS reference function,

|0〉 for the state under consideration, VK is the space spanned by the orthogonal complement


to |0〉 in the restricted full CI subspace used to generate CAS wave function, VSD is the space

spanned by all single and double replacement states generated from V0, and finally VTQ... is

the space which contains all higher order excitations not included in the previous terms.7,8

It is only the functions in subspace VK , that interact with the reference function via the

total Hamiltonian, and the zeroth-order Hamiltonian is formulated in such a way that only

VSD, contributes to the expansion of the first order wave function. All functions in VSD, can

be generated from the functions EpqErs, where Epq are the normal spin-averaged excitation

operators. Not all these functions are needed, functions of VK space, are generated when all

four indexes p, q, r, and s refer to active orbitals. The functions needed in the expansion

of the first-order wave function can be divided into three big groups, internal, semiinternal,

and external:

Internal =

{

EtiEuv|0〉

EtiEuj|0〉(2.9)

Semiinternal =

EatEuv|0〉

EaiEtu|0〉, EtiEau|0〉

EtiEaj|0〉

(2.10)

External =

EatEbu|0〉

EaiEbt|0〉

EaiEbj|0〉

(2.11)

where (i, j) are inactive, (t, u, v) are active, and (a, b) are secondary orbital indices. All the

functions 2.9-2.11 are referred to as internal, semiinternal, and external when none, one, or

two orbitals belong to the secondary subspace, respectively.

The first-order wave function is now expended into a set of functions |j〉 from VSD

|Ψ1〉 =M

∑

j=1

Cj|j〉 (2.12)

where M ≥ dim VSD and {Cj, j = 1, ...,M} is a solution of the system of linear equations,

M∑

j=1

Cj〈i|H0 − E0|j〉 = −〈i|H0|0〉 (2.13)

2.2. MULTI-CONFIGURATION SELF-CONSISTENT FIELD METHOD 17

i = 1, ...,M and where E0 = 〈0|H0|0〉 is the zeroth-order energy. The expansion functions

|j〉 are not necessary orthogonal and may also be linearly dependent.

The first order wave function appears in the expression for the second order energy. This is

equivalent to Møller-Plesset,9 if we assume that Ψ0 is a HF determinant. The zeroth order

Hamiltonian H0 is:

H0 = P0F P0 + PXF PX (2.14)

where P0 = |Ψ0〉〈Ψ0| is a projection operator onto the reference function, PX is a corre-

sponding projection operator for the rest of the configurations space, and F is the Fock

operator, which usually is assumed to be diagonal in the chosen orbital space. Now, if

we extend the above treatment to the the multiconfigurational case, where Ψ0 is a CAS

function, the zeroth order Hamiltonian will be chosen in analogy as equation 2.14:

H0 = P0F P0 + PKF PK + PSDF PSD + PTQ..F PTQ.. (2.15)

where now P0 = |Ψ0〉〈Ψ0| is a projection operator onto V0, PK is the projector onto VK ,

PSD is the projector onto VSD, and PTQ.. is the projector onto VTQ.., and F is a sum of

one-particle operators.

With the definition of the zeroth order Hamiltonian given in Equation 2.15, only those

configurations which interact directly with the CAS reference function have to be included

into the first order wave function. They all belong to the SD space, thus we can write the

first order wave function as

|Ψ1〉 =M

∑

p.q,r,s

Cp,q,r,s|pqrs〉 (2.16)

where |pqrs〉 = EpqErs|Ψ0〉, and all four indexes, p,q,r,s can not be active since the gener-

ated function then belongs to the CAS CI K space. All singly and doubly excited states

are included in the wave function, except those which have all four indices in the active

space. The different functions that contribute to the SD space are, however, in general not

orthogonal to each other or even always linearly independent.

The remaining operator to be defined is the one-particle operator F , since we want to

reproduce the results from closed-shell Møller Plesset second-order perturbation theory.

F =∑

pq

fpqEpq (2.17)

The first order equation for the coefficients Cp,q,r,s, in the equation 2.16, therefore takes the

more general form:

(F − E0S)C = −V (2.18)


where F is the Fock matrix, in the space VSD, S is the overlap matrix, C is the vector of

the coefficient in equation 2.16, and V is the representing the interaction between VSD and

the reference function. It has the elements Vpqrs = 〈Ψ0|H|pqrs〉.

We note that MP2, where F is diagonal and the inter acting space is orthogonal, cwandnote

that the equation will be identical to the Moller Plesset equation when there are no active

orbitals. The elements of the matrix F are rather complicated, they have the general

structure:

〈pqrs|F |p′q′r′s′〉 =∑

α,β

fαβ〈Ψ0|EsrEqpEαβEp′q′Er′s′ |Ψ0〉 (2.19)

The calculation of these matrices will occupy a major part of the time used for a CASPT2

calculation.

2.3 Coupled Cluster Theory

In the coupled cluster theory, the wave function is written as an exponential of a cluster

operator, acting on a single-determinant reference state. In constructing the wave function,

the excitations included in the cluster operator are not selected individually. A hierarchy

of approximations is established by dividing the cluster into classes, single, double, triple,

excitations and so on.1,4, 10

T = T1 + T2 + ... + TN (2.20)

where the one- and two-electron parts are given by

T1 =∑

AI

tAI a†AaI =

∑

AI

tAI τAI (2.21)

T2 =∑

A>B

tABIJ a†

AaIa†BaJ =

1

4

∑

AIBJ

tABIJ τAB

IJ (2.22)

One can compare the excitation based coupled-cluster model with the configuration based

CI model by expanding the exponential operator and collecting terms in the same order of

the excitation level:1

2.4. DENSITY FUNCTIONAL THEORY 19

exp(T )|HF 〉 =N

∑

i=0

Ci|HF 〉 (2.23)

These equations show which excitation processes contribute at each excitation level. The

lowest order operators Ci are given by:

C0 = 1 (2.24)

C1 = T1 (2.25)

C2 = T2 +1

2T 2

1 (2.26)

The most common approximation in coupled cluster theory is to truncate the cluster opera-

tor at second order, yielding the coupled-cluster singles and doubles(CCSD) model. In this

model the T2 operator describes the important electron-pair interactions and T1 carries out

the orbital relaxations induced by the field set up by the pair interactions.1

The single-reference CCSD wave function is given by the exponential ansatz

|CC〉 = exp(T1 + T2)|HF 〉 (2.27)

where the cluster operator contains contributions from single and double excitations only.

The CC energy is obtained by projecting the CC Schrodinger equation against the Hartree-

Fock operator reference state.

ECC = 〈HF |Hexp(T1 + T2)|HF 〉 (2.28)

and the cluster amplitudes are determined by projecting the CC Schrodinger equation

against the excitation manifold of Hartree-Fock operator reference state. A detailed deriva-

tion of the CCSD model is well documented in many places, see1

For higher accuracy, we must take into account triple excitations, truncating the expansion

at the T3 level, but this model is computationally very demanding and can be applied only

to small molecules.

2.4 Density Functional Theory

The methods described in previous sections deal with the wave function of the system

studied. Density functional Theory (DFT) works instead with the electronic charge density


as the fundamental quantity. It incorporates both exchange and correlation effects, and

is more computationally efficient than wave function based methods. DFT has become

prominent over the last decade as a method potentially capable of accurate results at low

cost. In practice, approximations are required to implement the theory, and a significantly

variable accuracy of the results are often seen. Calibration studies are therefore required to

establish the accuracy for a given class of systems.

2.4.1 The Hohenberg-Kohn theorems

DFT is based on the theorems of Hohenberg and Kohn, which state11

I. Every observable of a stationary quantum mechanical system (including energy), can

be calculated, in principle exactly, from the ground-state density alone, i.e., every

observable can be written as a functional of the ground-state density.

II. The ground state density can be calculated, in principle exactly, using the variational

method involving only the density.

The original theorems refer to the time independent (stationary) ground state, but are being

extended to excited states and time-dependent potentials.

2.4.2 Kohn-Sham equations

It is noticed that the kinetic energy is easily calculated from the wave function, provided

that it is known. For that reason, Kohn & Sham (1965)12 proposed a clever method of

marrying the wave function and density approaches. They repartitioned the total energy

functional into the following parts:

E[ρ] = T0[ρ] +

∫

[

Vext(r) + Ucl(r)]

ρ(r)dr + Exc[ρ] (2.29)

where T0[ρ] is the kinetic energy of electrons in a system which has the same density ρ as

the real system, but in which there are no electron-electron interactions.

Ucl(r) =

∫

ρ(r′)

|r′ − r|dr′ (2.30)

2.4. DENSITY FUNCTIONAL THEORY 21

is a pure Coulomb interaction between electrons. It includes electron self-interaction explic-

itly, since the corresponding energy is

Ecl[ρ] =

∫ ∫

ρ(r′)ρ(r)

|r′ − r|drdr′ (2.31)

Vext(r) is the external potential, i.e., the potential coming from the nuclei:

Vext =∑

α

−Zα

|Rα − r|(2.32)

The last functional, Exc[ρ], called the exchange-correlation functional, is in fact including

all the energy contributions which were not accounted for by previous terms, and several

approximations for this functional have been proposed.

2.4.3 Functionals

There is no straightforward way in which the exchange correlation functional, Exc[ρ], can

be systematically improved. It is customary to separate it in two parts, a pure exchange Ex

and a correlation part Ec. Over the years, many functionals have been proposed. We will

briefly discuss some of the important ones.

Local Density and Gradient Corrected Methods

Local Density Approximation (LDA): It is assumed that the density is local and can be

treated as a uniform electron gas. The exchange energy for a uniform electron gas has been

formulated by Dirac. In general, LDA is replaced by the Local Spin Density Approximation

(LSDA) where the densities are divided for the α and β spins. For a closed shell system

LDA is equal to LSDA.

The LSDA approximation in general underestimates the exchange energy by ≈ 10%. Despite

the simplicity of the assumptions involved, the LSDA method often provides results with

an accuracy similar to that obtained by Hartree-Fock methods.

Improvements over LSDA can be made by considering a non-uniform gas. These methods are

called Generalized Gradient Approximation, (GGA), assuming that the functional depends

not only of the density and but also on its derivatives. Several gradient corrected functionals

have been proposed for the correlation energy. One popular functional was given by Lee,

Yang, and Parr (LYP)13


Hybrid Methods

It is known that GGA methods give a substantial improvement over LDA. A more gen-

eralized method can be defined by writing the exchange energy as a suitable combination

of LSDA, exact exchange and the gradient correction term. The correlation energy may

similarly be taken as the LSDA formula plus a gradient term.

EB3xc = (1 − a)ELSDA

x + aEexactx + b∆EB88

x + ELSDAc + c∆EGCA

c (2.33)

The a, b, and c parameters are determined by fitting to experimental data and depend on the

form chosen for EGCAc , typical values are a= 0.2, b=0.7, and c= 0.8.14 This is the so-called

Becke-3 (B3) parameter exchange functional.14 The combination of B3 exchange and LYP

correlation functionals (B3LYP) has become one of the most accurate DFT method.

The hybrid functional B3LYP could, however, be problematic for a number of important

applications. One example is charge-transfer excitation. The reason for its failure is well-

understood15 since at large distance, the exchange potential of the B3LYP functional behaves

like −0.2r−1, differing considerably from the exact −r−1 behaviour. Yanai et al. proposed a

so-called Coulomb attenuating method by introducing two extra parameters α and β to the

B3LYP functionals. The parameter α allows to incorporate the HF exchange contribution

over the whole range, while β makes the DFT counterpart over the whole range attenuated

by a factor 1 − (α + β). With that, the CAM-B3LYP hybrid functional was established.15

2.5 Response Theory

Figure 2.1: p-AminoBenzoic Acid molecule and Dipole moment representation scheme.

2.5. RESPONSE THEORY 23

In classical theory, the electrical properties of a system of particles are described by its

multipole moments of various orders, expressed in terms of the charges and coordinates of

the particles. In quantum theory, the definitions of these quantities are of the same form,

but must now be regarded as operators. The first multipole moment is the dipole moment,

defined as a vector.

µ =∑

er (2.34)

where the summation runs over all particles.

In the presence of uniform electrostatic field or light, the energy E of a molecular system be-

comes field-dependent. When the field is weak, the field-dependent energy can be expanded

in terms of the field strength F .

E(F ) = E0 − F∂E

∂F 0−

1

2F 2∂2E

∂2F 0+ −

1

3!F 3 ∂3E

∂3F 0+ .... (2.35)

where ∂E∂F 0

= µ0, is the molecular dipole moment, − ∂2E∂2F 0

= α, is the dipole-polarizability

tensor, and −∂3E∂3F 0

= β, is the hyperpolarizability tensor.

The permanent molecular dipole moment and the polarizability tensor are the basic param-

eters that characterize the interaction of a molecular system with an external electric field

generated by neighboring molecules or by an experimental apparatus. By expanding the

energy to even higher orders in F , we obtain the hyperpolarizability tensors of higher orders

which are needed for an accurate description of the molecular system in strong fields. In this

thesis the studied molecular properties are not going beyond the third order perturbation

of the energy, and the main focus is concentrated on the transition dipole moments.

By inspecting the Eq. 2.35, one can realize that it is possible to obtain the molecular

properties by calculating the field-dependent energy of the system, for instance by means of

the finite field method. It should, however, be noted that this approach can only provides

the static properties. The dynamic properties as well as the optical excitations of molecules

are beyond its reach.

One of the best ways of computing the dynamic properties of molecules is the use of response

theory, which is based on time-dependent perturbation theory. Over the years, response the-

ory has been successfully applied to many molecular systems for a wide range of properties.

In this thesis, we will only brifely introduce the concept of the response theory and discuss

its applications to few properties related to the work presented.


2.6 Response theory for an Exact State

The time development of the exact wave function |0〉, is governed by the time-dependent

Schodinger equation16–18

H|0(t)〉 = ı∂

∂t|0(t)〉 (2.36)

where H is the total Hamiltonian operator.

H = H0 + V t (2.37)

with H0 being the time-independent Hamiltonian of the unperturbed system, and V t is the

time-dependent perturbation.

We could assume that the perturbation V t is switched on adiabatically at t = −∞. In the

frequency domain, the perturbation operator can be written as:

V t =

∫ ∞

−∞

dωV ω exp((−ıω + ε)t) (2.38)

where ε is a small positive infinitesimal that ensures that the field is switched on adiabatically

and that V t vanishes at t=−∞. The perturbation can include a time-independent term.

The perturbation V t is required to be Hermitian which also is the imposed condition for the

frequency component of V t.

(V ω)† = V (−ω) (2.39)

We will assume that |0(t)〉 is an eigenfunction of |0〉 of H0 at t = −∞

H0|0〉 = E0|0〉 (2.40)

At the finite time t, we can write the perturbed wave function as a perturbation expansion.

2.6. RESPONSE THEORY FOR AN EXACT STATE 25

0(t)〉 = |0〉 +

∫ ∞

−∞

dω|0(ω)1 〉 exp[(−ıω + ε)t]+

+

∫ ∞

−∞

∫ ∞

−∞

dω1dω2|0(ω1,ω2)1 〉 exp[(−ı(ω1 + ω2) + 2ε)t] + ..., (2.41)

where |0(ω)1 〉 contains terms linear in the perturbation and |0

(ω1,ω2)1 〉, terms that are quadratic

in the perturbation.

The time-dependent expectation value of an operator A can be expanded in the series:

〈0(t)|A|0(t)〉 = 〈0|A|0〉 +

∫ ∞

−∞

dω1 exp((−ıω1 + ε)t)〈〈A; V ω1〉〉ω1+

1

2

∫ ∞

−∞

dω1

∫ ∞

−∞

dω2 exp((−ı(ω1 + ω2) + 2ε)t)〈〈A; V ω1 , V ω2〉〉ω1,ω2+ (2.42)

1

6

∫ ∞

−∞

dω1

∫ ∞

−∞

dω2

∫ ∞

−∞

dω3 exp((−ı(ω1 + ω2 + ω3) + 3ε)t)〈〈A; V ω1 , V ω2 , V ω3〉〉ω1,ω2,ω3+

which in principle determines all time-dependent properties of the molecule. For instance,

the time-dependent expectation value of a dipole operator µ in an electromagnetic field gives

information about the linear and nonlinear optical properties of the molecule.

2.6.1 Linear Response

The function 〈〈A; V ω1〉〉ω contains all terms that are linear in V ω, and is denoted as a linear

response function. The functions 〈〈A; V ω1 , V ω2〉〉ω1,ω2and 〈〈A; V ω1 , V ω2 , V ω3〉〉ω1,ω2,ω3

contain

all contributions to the expectation values that are linear in V ω1 , V ω2 and V ω1 , V ω2 , V ω3 ,

respectively, and are know as quadratic and cubic response functions.

The response functions determine the changes of observables when the system is subject

to one or several fields. By suitable choices of fields a wide range of different effects can

be described within a common framework. The response functions also contain informa-

tion about transition properties. A linear response function can be written as a sum over

eigenstates, in the basis of eigenfunctions (|0〉, |n〉)

〈〈A; V ω〉〉ω =∑

n6=0

〈0|A|n〉〈n|V ω|0〉

ω − (En − E0)−

∑

n6=0

〈0|V ω|n〉〈n|A|0〉

ω + (En − E0)(2.43)


where En is the energy corresponding to state |n〉. The linear response function 〈〈A; V ω〉〉ωhas poles at frequencies equal to plus or minus the excitation energies of the unperturbed

system. The corresponding residues are:

limω→ωk

(ω − ωk)〈〈A; V ω〉〉ω = 〈0|A|k〉〈k|V ω|0〉 (2.44)

limω→ωk

(ω + ωk)〈〈A; V ω〉〉ω = −〈0|V ω|k〉〈k|A|0〉 (2.45)

where ωk = Ek − E0. The linear response functions thus contains information about the

excitation energies from the reference state |0〉 and the corresponding transition matrix

elements. Such information is sufficient to describe all one-photon processes.

2.6.2 Quadratic Response

Assuming that |n〉 diagonalizes H0, we obtain the spectral representation of the quadratic

response function.

〈〈V ω0 ; V ω1 ; V ω2〉〉ω1,ω2= −P (0, 1, 2)

∑

nm

〈0|V ω0 |n〉〈n|(V ω1 − 〈0|V ω1 |0〉)|m〉〈m|V ω2 |0〉

(ω0 + ωn)(ω2 − ωm)(2.46)

where P (0, 1, 2) is a function to permute indices, and ω0 = −(ω1 + ω2).

If now we consider the case where the homogeneous periodic electric field of frequency ω1

and one of frequency ω2, are applied to the molecular system and V ω, it could be the electric

dipole operator or the spin-orbit operator, then we would obtain the two-photon transition

elements and singlet–triplet matrix elements. From the the residue of the quadratic response

function, the transition elements between non-references states, i.e. the transitions moments

between excited states can be obtained.

All together, linear and quadratic linear response functions describe the most important

photophysical properties of any system.16–18

Chapter 3

Excited State Properties

When a molecule interacts with external electromagnetic fields with proper frequencies, it

absorbs the energy and transfers itself into an excited state. This absorption process follows

various selection rules. Emission is the reverse process of absorption which obeys similar

rules as absorption. However, it should be noted that emission is not the usual mechanism

for energy loss from the excited state; a competing one is vibrational relaxation where the

excess energy is lost through vibrations of the molecule.19 In this thesis, we have studied

one-, and two-photon absorption and emission from the triplet to the singlet ground state

of different molecular systems. Only related theories will therefore be discussed here.

S0

T

T

1

2S1

S2

Figure 3.1: Molecular electronic energy level diagram of ground and low-lying excited states.

27

28 CHAPTER 3. EXCITED STATE PROPERTIES

3.1 Phosphorescence

Phosphorescence is the luminescence produced by certain substances after absorbing radiant

energy or other types of energy. It is distinguished from fluorescence in that it continues

even after the radiation causing it has ceased. It is a radiative transition involving a change

in the spin multiplicity of the molecule, for instance from a triplet state to a singlet state.

Such a transition is normally forbidden according to the spin selection rule. It can become

allowed only when spin-orbit coupling is taken into account.

3.1.1 Spin-Orbit Coupling

The spin-orbit coupling (SOC) is a magnetic interaction between the spin and the orbital

angular momenta. It results in a coupling between the triplet state to the singlet state of a

molecule20–23 The electronic spin-orbit coupling is described as

Hso =α2

2

[

∑

i,A

ZA

~liA · ~si

r3iA

−∑

ij

~lij · (~si + 2~sj)

r3ij

]

, (3.1)

where ZA is the atomic number of nucleus A, riA and rij are distances between electrons i

and nucleus A and electrons i and j, respectively, ~si is the spin operator for electron i, and~liA and ~lij are the orbital angular momentum operators for electron i with respect to the

position of the nucleus A and electron j, respectively.

Note that the relative orbital angular momentum between two particles, lij= (ri − rj) ×pi

is asymmetric, i.e. lij 6= lji. This means that the two-electron integrals over this operator

do not have permutation symmetry as do the electron repulsion integrals.

The results from response theory are most conveniently cast into a formalism based on

second quantization. The second quantization representation of the spin-operators will

transfer to triplet excitation operators which are weighted by the integrals over orbital pars.

The z−component will have the form:

Hzso =

∑

ij(ij|Z)E−ij +

∑

ijkl

(ij|kl)(e−+ijkl + 2e+−

ijkl) (3.2)

where the one-particle excitations are given E+ij = a†

iαajα+a†iβajβ and two-particle excitations

are given by:

e−+ijkl = E+

ijE−kl − E−

il δkj (3.3)

3.1. PHOSPHORESCENCE 29

and the integrals, written in a Mulliken-like notation, by

(ij|Z) =α2

2

∑

A

ZA

∫

drφi(r)lzAr3A

φj(r) (3.4)

(ij|kl) = −α2

2

∫

dr1dr2φi(r1)φk(r2)lz12r312

φj(r1)φl(r2) (3.5)

A large number of spin-orbit properties can now be derived from the response functions,

from linear response we can deduce the second-order energy correction due to SOC

E(2) =1

2〈〈HSO; HSO〉〉0 =

1

2H

[1]†SO

1

E(2)H

[1]SO (3.6)

where H[1]SO is the spin-orbit gradient, and the SOC constants are derived form the residue

limω→ωf

(ω → ωf )〈〈HSO; HSO〉〉ω → 〈0|HSO|f〉 (3.7)

and which is formed by solving the eigenvalue equation (E(2) − ωS(2))Xf = 0 and forming

〈0|HSO|f〉 = H[1]†SO Xf

One of the most important applications of quadratic response theory, pertaining to spin-orbit

properties, is the calculation of the spin-orbit induced dipole moment, i.e. phosphorescence,

which can be derived from the residue.

limω→ωf

(ω → ωf )〈〈r; HSO; C〉〉0,ω → 〈10|HSO|3f〉 (3.8)

where C is an arbitrary triplet operator, and |3f〉 is a triplet state.

3.1.2 Lifetime

The phosphorescence phenomenon was early identified as a radiative process between the

lowest triplet state and the singlet ground state. The nature of triplet phosphorescence

radiation can be established by magnetic resonance measurements.20,22


The triplet state of a molecule at room temperature is a rapidly equilibrating mixture of

zero field split (ZFS) three states - the spin sublevels, T x, T y, and T z. Absorption by an

individual molecule produces only one of the three sublevels initially. At low temperatures,

this process could in favorable cases be observed by several experimental techniques.

The mechanism of phosphorescence is provided by spin orbit coupling between the triplet

state T1 and the singlet ground state S0. The strongest mixing occurs between T1 and

singlet states which possess the correct electronic symmetry. The electronic symmetry of

the magnetic spin-sublevels could be different. The mechanism of populating the spin-

sublevels from the first singlet excited state through the inter-system crossing (ISC) leads

to different rates of population of the T x, T y, and T z substates. The shape of the emission

emission spectrum of each sublevel is different since the vibrational perturbations which

induce spin-orbit coupling are different for each sublevel.

Under these conditions the phosphorescence spectrum, the lifetime and polarization depend

on the emission properties of each spin sublevel.21,24,25

The radiative lifetimes for excited states, i.e. the average time in which a molecule is in an

electronically excited state before the energy is lost, can be calculated with the following

formula.

1

τk

=1

to

4

3α3E3µ2

k (3.9)

where to is a combination of factors to express the τ in seconds,

to =(4πε0)

2h3

mee4= 2.4189 10−17s (3.10)

α is the fine-structure constant, α ≈ 1137

, µk is the electric dipole transition moment between

the ground state and the spin-sublevel k (T k) of the triplet state, and E is the transition

energy.

The l-component of the S0 → T1 transition dipole moment connected with a particular spin

sublevel k is determined by the well-known equation

µl(Tk) = < S0 | rl | T k

1 >=∑

n

< S00 | rl | S0

n >< S0n | Hk

so | T k,01 >

E(T 01 ) − E(S0

s )

+∑

m

< S00 | Hk

so | T k,0m >< T k,0

m | rl | T k,01 >

E(S00) − E(T 0

m)(3.11)

3.2. TWO-PHOTON ABSORPTION 31

where Hkso is the k:th component (k, l∈x, y, z) of the SOC operator (T 0

1 and T1 denote the

zero and first order wave functions of perturbation theory). Each k spin sublevel of the

triplet state T k is associated with a specific S → T k transition probability.

Finally, for the triplet state, an average is calculated to predict the radiative lifetime of

the triplet state. The total radiative lifetime τn determined by the Tn → S0 spontaneous

emission is thus obtained by the averaging of all spin components.

3

τn

=1

τx

+1

τy

+1

τz

(3.12)

3.1.3 Herzberg-Teller Approach

The vibronic analysis of phosphorescence spectra have in this thesis been treated within the

framework of the Herzberg-Teller approximation26

µv,v′

l (T k) =< S0v′ | rl | T k

1 v > (3.13)

where v, v′ are vibrational quantum numbers for a particular vibrational mode α in the

lower and upper states, respectively. The complete equation for the transition moment is

equal to:

µv′,vl (T k) = µl(T

k)(Q0)

∫

XS,v′XT,vdQ +∂[µS,T (Q)]QO=0

∂Qα

∫

XSv′QαXTvdQ (3.14)

where Q0 is the equilibrium point geometry.26,27

3.2 Two-photon absorption

Quantum theory indicates that a molecule can simultaneously absorb two or more photons.

However, such multi-photon absorption events can not be observed under ordinary radiation

because of very low probability. The invention of lasers turned these processes into powerful

spectroscopic tools. In recent years two-photon absorption (TPA) has also become relevant

to a variety of technical applications, such as optical limiting, data storage and photo-

dynamic therapy, owing to the discovery of organic molecules with very large TPA cross

sections.28–32 Theoretical calculations have played, and will continue to play, very important

roles in the design of two-photon active molecules.33,34


The strength of two-photon absorption is proportional to the square of the incident intensity.

The two-photon absorption (TPA) cross section is related to the imaginary part on the

second hyperpolarizability. The two-photon transition matrix element Sαβ between the

initial state 0 and the final state f reads

Sαβ =∑

k

(〈0|µα|k〉〈k|µα|f〉

ωk −ωf

2

+〈0|µβ|k〉〈k|µβ|f〉

ωk −ωf

2

)

(3.15)

where ωf denotes the excitation energies of the excites states |k〉, and the summation includes

the ground state, |k〉= |0〉.

The TPA cross section (δTPA) is obtained as:

δTPA =∑

αβ

(2SααS∗ββ + 4SαβS∗

αβ) (3.16)

where the summation is performed over the molecular axis α, β =(x, y, z).

The TPA cross section directly comparable with experiment is defined as

σTPA =4π2a5

0α

15c0

ω2g(w)

Γf

δTPA (3.17)

where a0 is the Bohr radius, c0 is the speed of light and α is the fine structure constant, ω is

the photon energy, g(w) is the spectral line profile which is assumed to be a delta function

(δij) for many systems and Γf is the lifetime broadening for the final state, commonly

assumed to be 0.1 eV.

In the response theory framework, the TPA transition matrix can be determined by the

single residue of the quadratic response function.

limω1→−ωf

(ω2 − ωf )[ limω2→ωi

〈〈µa; µb; µc〉〉ωf

2,ω2

] = Sα,β〈f |µc|0〉 (3.18)

where the perturbation, V ω0 , refers to the electric dipole operator. Furthermore, the dou-

ble residue of the quadratic response function gives the transition moments between non-

references states. Hence the transition dipole moments between excited states can be de-

termined

limω1→−ωf

(ω1 + ωf )[ limω2→ωi

〈〈µa; µb; µc〉〉ω1,ω2] = −〈0|µb|f〉〈f |(µa − 〈0|µa|0〉)|i〉〈i|µc|0〉 (3.19)

3.3. CHEMICAL DYNAMICS AS TRANSITIONS BETWEEN STATES 33

Excited state absorption is important processe when the pulse of radiation is longer than

the lifetime of the excited state, some of the excited molecules are pumped up to even higher

states through excited state absorption, rather than go back to the ground state by emission.

This point will be reviewed in next section concerning the dynamics of the nonlinear optical

absoprtion.

3.2.1 Two-state model

The size of two-photon active molecules is normally quite large and the usefulness of highly

correlated methods to study TPA is therefore limited. We have addressed this situation by

applying DFT methods to calculate the TPA cross sections.

It is known that for large charge-transfer conjugated systems few state models can often be

adequate for describing the TPA cross section.33,35 This allows us to deal with systems with

large size, as porphyrin derivatives. The TPA cross section of one-dimensional molecules is

completely dominated by the component along the molecular axis Sxx. For asymmetrical

charge-transfer molecules it is sufficient to include only two states, the ground and the final

state33

STPAxx =

2µofx (µff

x − µ00x )

∆E(3.20)

∆E = hωf − ω =ωf

2(3.21)

where µofx is the transition dipole moment between the ground state and the final state, and

µffx −µ00

x is the difference between the ground and excited state permanent dipole moment.

The microscopic TPA cross section δTPA can be expressed as

δTPA = 24(µff

x − µ00x )2(µof

x )2

(ωf

2)2

(3.22)

3.3 Chemical Dynamics as Transitions between States

The experimental observables are often related to the macroscopic properties of the sys-

tems under investigation. They are determined not only by the microscopic properties of


the individual molecules, but also the features of the external electromagnetic fields. A

complete description of the experimental measurements thus needs to combine the quan-

tum chemical modeling and Maxwell’s equations. One way to do that is to use the density

matrix formalism as developed in our laboratory.36 It allows to account for any type of

coherent and stepwise molecular excitations, to describe the saturation and propagation

effects that are caused by the laser pulses, and to include the desaturation effects induced

by inter-system-crossing (ISC) and emission.

Figure 3.3 shows a typical energy level diagram involved in the dynamics processes. During

the first absorption step, the molecule is excited from the ground to any (singlet) excited

state. It then tends to release the excess of energy by fast relaxation to lower excited

states. This could be done in a non-radiative way through the so-called internal conversion

(IC) process, or in a radiative way by emitting light, i.e. a fluorescence process (F). The

non-radiative process, IC, is estimated to be in the timescale of picoseconds, while the

radiative lifetime of the lowest excited singlet state (S1) is often much larger, in the range

of nanoseconds.

When a molecule possesses a heavy atom, large spin-orbit coupling could occur which can

open up more channels for absorption and emission. This can be excitations between triplet

states, as well as between triplet and singlet states. In the latter case, the triplet state

is initially populated by the inter-system-crossing process, which is in the time scale of

nanoseconds.

Figure 3.2: Low-lying electronic energy level diagram of the main states to describe the

dynamic processes

3.3. CHEMICAL DYNAMICS AS TRANSITIONS BETWEEN STATES 35

3.3.1 Theory of Dynamics Simulations

We consider to solve the problem strictly making use of an expansion of the density matrix

into Fourier series over the harmonics of the light beams.36

E =E

2e−ıωt+ıkz + c.c. (3.23)

with a many-level molecule. The polarization of the medium has the same structure:

P = Tr(dρ) =∑

βα

dβα(t)ραβ = Pe−ıωt+ıkz + c.c. (3.24)

To know the polarization we need the transition dipole moments, dβα(t) = dβα exp(ıωαβt),

and the density matrix, ραβ, of the medium. Here, ωαβ = (Eα − Eβ)/h, is the frequency of

the transition α → β.

We make the reasonable approximation of having slowly-varying phases and amplitudes:

ω � |E/E|; k � |∂ ln E/∂z|. The substitution of E (3.23) and P (3.24) in Maxwell’s

equations and a selection of contributions with the frequencies ω results in the following

paraxial wave equations for the amplitude of the field, using atomic units:

( ∂

∂z+

1

c

∂

∂t−

ı

2k∆⊥

)

E =ık

ε0

P . (3.25)

The density matrix equation for nonlinear media reads

( ∂

∂t+ Γ

)

ρ =ı

h[ρ, V ], Trρ = N, V = E · d, (3.26)

where the concentration of the absorbing molecules is denoted by N , and Γ is the relaxation

matrix. The kinetic equations for populations, ραα, and off-diagonal elements, ραβ, of the

density matrix read:

( ∂

∂t+ Γαβ

)

ραβ = δα,β

∑

γ>α

Γαγργγ +

ı

h

∑

γ

(ραγVγβ − Vαγργβ). (3.27)

Here we neglect the space derivatives at the left-hand side of the kinetic equations which are

responsible for the Doppler effect (exp(±ıkz) → 1). The Doppler broadening in the studied

condensed absorber is indeed negligible compared to the dephasing rate, Γαβ. Due to non-

radiative conversion the rate Γαβ of decay transitions β → α is large in organic molecules

and almost completely coincides with the total decay rate Γαα.


The decay rate of the α th level is the sum of the partial decay rates of this level to all lower

levels

Γαα =∑

β(<α)

Γβα (3.28)

We seek the solution of the density matrix equations making use of the Fourier expansion:

ραβ = eıωαβt[

rαβ +∞

∑

n=1

(

r(n)αβ e−ınωt + r

(n) ∗βα eınωt

)]

(3.29)

The substitution of the Fourier expansion, 3.29, in the density matrix equation, 3.27, results

in the equations to solve the dynamical problem through a nonlinear many-level system. The

solution of the coupled eqs. 3.23, 3.24, and 3.25 gives the dynamics of the pulse propagation

through the non-linear absorbing medium and allows to determine the transmission though

the absorber.

Chapter 4

Summary of articles

In Paper I, we studied the permanent dipole moment of para−amino benzoic acid, a basic

charge transfer molecule, commonly used in photonics applications. It has been calculated

by different quantum methods, including Hartree-Fock, second-order Møller Plesset per-

turbation theory, coupled cluster theory with singles, doubles, and triples corrections, and

hybrid density functional theory at the B3LYP level. The B3LYP method gives a planar

geometry and larger dipole moment. The use of highly correlated methods are found to

be necessary for providing accurate values for geometry and dipole moment of para−amino

benzoic acid.

An extension of the work presented in Paper I was carried out for a series of aromatics amines

in Paper II, focusing on their electronic properties of the ground and the charge-transfer

states. Dipole moments have been studied by several theoretical methods as used in Paper I.

Moreover, the excited states properties have been studied using the hybrid density methods

B3LYP and CAM-B3LYP and compared with the CCSD method. The calculations seem to

suggest that the use of highly correlated methods is necessary for describing accurately the

electronic properties in both in the ground and the excited states.

In Paper III we carried out calculations on the phosphorescence effect in the benzene deriva-

tives: p−, o−, and m−dichlorobenzene, and p−, o−, and m−dibromobenzene, indole,

styren, aniline, and its halogenated derivates and nitrobenzene. The information gained

from such calculations concerns polarization directions, oscillator strengths, radiative life-

times, and excitation energies for the specific triplet spin sublevels.

One of molecules studied in Paper III, p-dichlorobenzene, was selected in Paper IV as a

model molecule to study vibrational coupling effects on a phosphorescence spectrum. The

spectrum has been calculated using multi-configuration self-consistent field wave functions

and the quadratic response technique. Attention was paid to the intensity distribution

37

38 CHAPTER 4. SUMMARY OF ARTICLES

of the singlet-triplet (3B1u→1Ag) transition through a number of vibronic subbands. The

electric dipole activity of the spin sub-levels in the triplet-singlet transition to the ground

state vibrational levels is estimated by calculations of derivatives using distorted geometries

being shifted from the equilibrium position along different vibrational modes. In selected

cases we have also performed calculations of zero-field splittings and nuclear quadrupole

constants.

Paper V is an experimental and theoreticals collaboration work that studies the photophysics

of the thioxanthone molecule, describing the one-photon absorption, its dipole moments, and

these properties in different solvents. Two conformers have been found close in energy, a

planar and a non-planar differing in the dihedral angle representing the bend of the side

benzene rings. The thioxanthone moleculeis found to have ”butterfly effect” in the molecular

conformation that explains its unique photophysical properties.

In Paper VI we have theoretically designed a series of charge-transfer Zn-porphyrin deriva-

tives that possess very large two-photon absorption cross sections in the wavelength interval

of 1.3-1.5 µ m. This wavelength region is known to be useful for fiber communications

and other technological applications. The designed chromophores have a Donor-π-Acceptor

structure. All Zn-porphyrin derivatives show a strong one-photon transition moment. The

feature which mostly separates one from another in terms of the TPA activity, is the differ-

ence of permanent dipole moment between the ground and excited states. The result shown

in this article provides criteria to design TPA materials for technological applications.

More studies on the Zn-porphyrin derivatives have been carried out in Paper VII. The

formation of charge-transfer state with respect to the conjugation length has there been

evaluated. The charge distributions inside the molecule have been analyzed in order to

understand the origin of the exceptional large two-photon absorption cross sections. Possible

applications of these newly designed compounds were proposed.

In Paper VIII optical power limiting was theoretically studied using an approach that com-

bines quantum electronic structure calculations of multi-photon excitations and classical

calculations of the dynamical wave propagation. A number of organometallic compounds,

such as zinc-porphyrin, magnesium-porphyrin, lead-porphyrin, free-base porphyrin, vinyl-

phenylamine zinc-porphyrin and a conjugated platinum compound were addressed in the

paper. Their electronic properties related to nonlinear absorption, two-photon absorption,

excited state absorption, and triplet-triplet absorption, have been calculated using the DFT

quadratic response technique. It was shown that the platinum compound exhibits better

optical limiting properties than porphyrins for short-pulse excitations. It was found that

different organic groups can bring in different functionalities. For instance, the use of the

triple C-C bonds can shift the energy of the first excited singlet state, and the use of the

39

phenyl and the thiophene groups provides enhanced nonlinear optical response. All this

leads to larger TPA cross sections.

40 CHAPTER 4. SUMMARY OF ARTICLES

Bibliography

[1] T Helgaker, P Jørgensen, and J Olsen. Molecular electronic-structure theory. John

Wiley & Sons Ltd, 2000.

[2] A. Szabo and N.S Ostlund. Modern quantum chemistry. Dover Publications, Inc New

York, 1989.

[3] B. O. Roos and Eds. Per-Olof Widmark. European summerschool in quantum chem-

istry, book i. Lund University, 2000.

[4] B. O. Roos and Eds. Per-Olof Widmark. European summerschool in quantum chem-

istry, book ii. Lund University, 2000.

[5] B. O. Roos, K. Andersson, M. P. Fulscher, L. Serrano-Andres P.-A. Malmqvist, K. Pier-

loot, and M. Merchan. Multiconfigurational perturbation theory: Applications in elec-

tronic spectroscopy. In I. Prigogine and S. A. Rice, editors, Advances in Chemical

Physics: New Methods in Computational Quantum Mechanics, Vol. XCIII:219–331,

John Wiley & Sons, New York, 1996.

[6] M. Merchan, L. Serrano-Andres, M. P. Fulscher, , and B. O. Roos. Multiconfigurational

perturbation theory applied to excited states of organic compounds. In K. Hirao (Univ.

of Tokyo), editor, Recent Advances in Multireference Theory. Vol. IV.

[7] K. Andersson, P A Malmqvist, B. O. Roos, A. J. Sadlej, and K. Wolinski. J. Chem.

Phys., 94:5483, 1990.

[8] K. Andersson, P A Malmqvist, and B. O. Roos. J. Chem. Phys., 96:1218, 1992.

[9] C Møller and M. S. Plesset. Phys. Rev., 46:618, 1934.

[10] J. Cizek. J. Chem. Phys., 45:4256, 1966.

[11] P. Hohenberg and W. Kohn. Phys. Rev. B., 76:6062, 1964.

41

42 BIBLIOGRAPHY

[12] W. Kohn and L. J. Sham. Phys. Rev. A., 140:1133, 1965.

[13] C. Lee, W. Yang, and R.G. Parr. Phys. Rev. B., 37:785, 1988.

[14] A.D. Becke. J. Chem. Phys., 98:5648, 1993.

[15] D. P. Tew T. Yanai and N. C. Handy. Chem. Phys. Lett., 393:51, 2004.

[16] J Olsen and P Jørgensen. J. Chem. Phys., 82:3235, 1985.

[17] P Jørgensen, H. J.Aa Jensen, and J Olsen. J. Chem. Phys., 89:3654, 1988.

[18] P Jørgensen and Jack Simons. Second quantization-based methods in quantum chem-

istry. Academic Press, Inc. 1981, 1981.

[19] Nicholas J Turro. Modern molecular photochemistry. University Science Books.

[20] H. Agren, Olav Vahtras, and B.F. Minaev. Adv. Quant. Chem., 27:71, 1996.

[21] B.F. Minaev, S. Knuts, H. Agren, and O. Vahtras. Chem. Phys., 175:245, 1993.

[22] S. P. McGlynn, T. Azumi, and M. Kinoshita. Molecular spectroscopy of the triplet

state. University Science Books, 1991gelwood Cliffs, New Jersey: Prentice Hall.

[23] B. Schimmelpfenning. AMFI, An Atomic Mean-Field Spin-Orbit Integral Program.

Stockholm University, Stockholm University, Sweden, 1996.

[24] S. Knuts, H. Agren, and B.F. Minaev. J. Mol. Struc., 311:185, 1994.

[25] S. Knuts, B.F. Minaev, H. Agren, and Olav Vahtras. Theor. Chim. Acta, 87:343, 1994.

[26] G. Herzberg and E. Teller. Z. Physik Chem, page 410, 1933.

[27] Roche M. and H. Jaffe. J. Chem. Phys., 60:1193, 1974.

[28] M Albota et altri. Science, 281:1653, 1998.

[29] Dimitri A. P. and Peter. M. R. Science, New Series, 245.

[30] Wenhui Zhou et altri. Science, 296:1106, 2002.

[31] Daniel. R. Larson et altri. Science, 300:1434, 2000.

[32] Satshi Kawata et altri. Nature, 412:697, 2001.

[33] J.D. Guo and H. Agren C.K. Wang, Y. Luo. Phys Chem. Chem. Phys, page 3869, 2003.

BIBLIOGRAPHY 43

[34] Y. Luo, O. Rubio-Pons, J-D Gou, and H. Agren. J. Chem. Phys., 122:096101, 2005.

[35] P. Constrand, Y Luo, and H Agren. Chem. Phys. Lett., 352:262, 2003.

[36] A. Baev, F. Gelmukhanov, O. Rubio-Pons, P. Cronstrand, and H. Agren. J. Opt. Soc.

Am. B, 21:384, 2004.

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