a computational study of free radicals in chemistry …

A COMPUTATIONAL STUDY OF FREE RADICALS IN

CHEMISTRY AND BIOLOGY

ADRIAN MATTHEW MAK WENG KIN (B.Sc.(Hons.)), NUS

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

NUS GRADUATE SCHOOL OF INTEGRATIVE SCIENCES AND

ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2007

ii

Abstract

Free radicals are involved in many aspects of our daily lives, but due to their transient

nature and high reactivity, many free radicals and free radical precursors prove

difficult to study experimentally. The use of computational quantum chemical

methods to study these intriguing molecules has become popular today, given the

increasing availability and lowering cost of computing power. In this work,

computational studies on two families of reactive species are presented. First, a study

on intermediate species formed from the action of nitric oxide, NO•, with particular

emphasis on the direct effects of the free radical precursor peroxynitrite, ONOO–, and

the second, a study of bis(thiocarbonyl)disulfides (RCS2)2 in industrial sulfur

vulcanization of rubber, in particular, tetramethylthiuram disulfide (TMTD).

Knowledge of the intricate chemistry behind these reactive species will be useful in

designing experiments in this area, and better understanding of the way they work in

nature can thus be achieved.

iii

Acknowledgements The greatest thanks go to God, without whom all this would not have been possible.

Next, I am grateful to my family and Juliana, for being by my side as sound pillars of

support; for being there for me when I needed them most.

To Dr. Richard Wong, my thesis supervisor, thanks for the knowledge imparted to

this student of yours, and for being so much more than just a mere supervisor to me,

and also to my co-supervisor Dr. Matt Whiteman, for his valuable advice and help

along the way. Many thanks also go to Prof. Ralf Steudel and Dr. Yana Steudel of

Technical University of Berlin for opportunities in collaborative work.

Thanks goes to the Agency for Science, Technology and Research (A*STAR) for

providing the funding for graduate study, and the opportunity for a brighter future. To

those at NGS, here’s a big ‘thank you’ for your efficiency in administering to us

graduate students.

I am thankful for the many friends I made throughout the course of my study in the

Molecular Modeling Laboratory. To my friends, Henry, Jason, Ted, St. Francis Music

Ministry, Nicholas, Sabrina, Paul, Celene, Mr. & Mrs. Kodama, I am grateful to you,

for cheering me on.

Thank you, Mother Mary, for praying for me.

iv

Dedicated to my late grandfather.

v

List of Publications Included in this Thesis

The following papers were published as a direct consequence of the work undertaken

in this thesis:

1. Thermochemistry of Reactive Nitrogen Oxide Species and Reaction

Enthalpies for Decomposition of ONOO– and ONOOH. Mak, A. M.; Wong, M. W. Chem. Phys. Lett. 2005, 403, 192.

2. Homolytic Dissociation of the Vulcanization Accelerator Tetramethylthiuram Disulfide (TMTD) and Structures and Stabilities of the Related Radicals Me2NCSn

• (n = 1–4). Steudel, R.; Steudel, Y.; Mak, A. M.; Wong, M. W. J. Org. Chem. 2006, 71, 9302.

3. Reaction of the Radical Pair NO2• and CO3

•– with 2-[6-(4′-Amino)phenoxy-3H-xanthen-3-on-9-yl]benzoic Acid (APF). Mak, A. M.; Whiteman, M.; Wong, M. W. J. Phys. Chem. A, 2007, 111, 8202.

vi

Table of Contents Abstract ii

Acknowledgements iii

Table of Contents vi

Summary Xi

List of Tables 1

List of Figures 4

List of Reaction Schemes 7

1. Introduction 8

2. Theory and Methodology 16

2.1 Introduction 16

2.2 The Schrödinger Equation 16

2.3 Time-independence 17

2.4 The Hamiltonian Operator 18

2.5 Atomic Units 19

2.6 Born-Oppenheimer Approximation 19

2.7 Hartree-Fock Method 20

2.7.1 The Wave Function Φ as a Slater Determinant 20

2.7.2 Basis Sets 21

2.7.3 The Variation Principle 24

vii

2.7.4 The Fock Matrix 25

2.7.5 The Roothaan-Hall Equations 25

2.8 Restricted and Unrestricted Hartree-Fock Methods 27

2.9 Correlation Effects 29

2.9.1 Configuration Interaction (CI) 30

2.9.2 Limited Configuration Interaction 31

2.9.3 Coupled Cluster Methods 31

2.9.4 Møller-Plesset Perturbation Theory 32

2.10 Multi-Configuration SCF 34

2.11 Density Functional Theory 35

2.11.1 Local Density Approximation (LDA) and Local Spin Density

Approximation (LSDA) 36

2.11.2 Generalized Gradient Approximation (GGA) 37

2.11.3 Hybrid Functionals 39

2.12 Composite Methods for Accurate Thermochemistry 40

2.12.1 Gaussian-n Methods 40

2.12.2 CBS-n Methods 43

2.13 Methods of Calculation of ESR G-shifts 45

2.14 Solvation Models 46

2.14.1 Onsager Reaction Field Theory 47

2.14.2 Integral Equation Formalism for Polarizable Continuum

Solvent Model (IEFPCM) 48

3. Thermochemistry of Reactive Nitrogen Oxide Species 55

3.1 Introduction 55

viii

3.2 Computational Methods 57

3.3 Results and Discussion 59

3.3.1 Enthalpies of Formation of RNOS 59

3.3.2 Enthalpies of Reactions Involving ONOO– and ONOOH

Decomposition 66

3.4 Conclusion 68

4. Decomposition of Nitrosoperoxycarbonate Anion: Gas Phase and

Solvent Modeled Studies 73

4.1 Introduction 73



4.3.1 Gas-phase Reaction Profile of ONOO– Reaction with CO2 77

4.3.2 Implicit Solvation Model (ISM) Results 82

4.3.3 Explicit Solvation Model (ESM) Results 84

4.3.4 Hybrid Solvation Model (HSM) Results 89

4.3.5 UV-Visible Spectra of ONOO–, ONOOCO2– and CO3

•– 91

4.4 Concluding Remarks 95

5. Reaction of the Radical Pair NO2• and CO3

•– with 2-[6-(4′-

Amino)phenoxy-3H-xanthen-3-on-9-yl]benzoic Acid (APF) 100

5.1 Introduction 100

5.2 Methods 103

5.2.1 Computational Methods 103

5.2.2 Experimental Methods 104

ix


5.3.1 Influence of HCO3− concentration on ONOO− mediated

oxidation of APF 106

5.3.2 Gas Phase Calculations 107

5.3.3 Solvent Effects 118


6. Homolytic Dissociation of the Vulcanization Accelerator

Tetramethylthiuram Disulfide (TMTD) and Structures and Stabilities of

the Related Radicals Me2NCSn• (n = 1−4) 125




6.3.1 Structures and Stabilities of Tetramethylthiuram Sulfides and

Their Related Radicals 129

6.3.2 Impact of Metal Cations on the Dissociation of

Tetramethylthiuram Di- and Trisulfides 143


7. Dissociation Enthalpies and Activation Barriers of Various

Bis(thiocarbonyl)disulfides R–C(=S)–S–S–C(=S)–R 159




7.3.1 Substituted bis(thiocarbonyl)disulfides, R = H, Me, F, Cl, OMe, 164

x

SMe, NMe2, PMe2

7.3.2 Novel bis(thiocarbonyl)disulfides, R = OSF5, Gu1, Gu2 174


Appendix 1 183

xi

Summary

Free radicals are isolated atoms or molecules that possess one or more unpaired

electrons, and are capable of independent existence. Electrons within an atomic or

molecular system seek to be paired with another electron with opposite spin, resulting

in stable electronic configurations. Free radicals can react readily with many

molecules, in a myriad of mechanisms.

The study of free radicals is an interest of many research groups worldwide. This

comes as no surprise, since radicals play important roles in many chemical and

biological processes. For instance, nitric oxide (NO•) and superoxide (O2•–) play

important roles in biochemical pathways, triplet oxygen (O2) is a biradical that is

essential in combustion processes, benzoyloxyl radicals (PhCOO•) are used as

initiators in radical polymerization processes. Since most free radicals and free radical

precursors are transient by nature, the experimental study of them is challenging. The

enormous computing power available in the present day allows the use of

computational methods to study the chemistry of these intriguing compounds.

The work described in this thesis is based on computational studies of the molecular

properties, reactions and hence chemistry of free radicals and is centered on two

themes. The first is the study of reactive nitrogen species (RNS) and reactive oxygen

species (ROS), particularly those involving intriguing free radical precursor

peroxynitrite, ONOO–. The thermochemistry of reactions involving peroxynitrite was

evaluated and reviewed using high level ab initio and composite methods. As the

xii

most likely compound to react directly with ONOO– is CO2, a solvation study to

provide insight into solvent effects on the energetics and mechanisms of this reaction

was attempted, and its results presented. The detection and quantitation of RNS such

as ONOO– is carried out in a number of ways, of which, one method involves the use

of a pre-fluorescent probe in vivo in vitro. The mechanism of reaction of ONOO– with

such a fluorescent probe, APF, was studied, to provide a better picture of the reactive

species directly responsible for oxidation of the probe.

The second theme in this thesis is the study of bis(thiocarbonyl)disulfides (RCS2)2 and

thiocarbamate radicals RCS2• in the sulfur vulcanization of rubber. A theoretical study

on the energetics, molecular structures and properties of the intermediates involved in

tetramethylthiuram disulfide (TMTD) mediated sulfur vulcanization is presented.

Also presented is an exploration of novel bis(thiocarbonyl)disulfides to examine their

feasibility as alternatives to TMTD.

1

List of Tables PageTable 2.1. Components of CBS-4, CBS-q, CBS-Q, CBS-QCI/APNO, G3(MP2) and G3 models.

45

Table 3.1. Calculated standard enthalpies of formation (ΔH°f,298, kJ

mol–1) of sixteen RNOS using various Gaussian-n methods.

60

Table 3.2. Calculated standard enthalpies of formation (ΔH°f,298, kJ

mol–1) of sixteen RNOS using various CBS methods.

61

Table 3.3. Selected calculated bond lengths of RNOS in Ångstroms.

62

Table 3.4. Calculated structural parameters of planar cis-peroxynitrite, ONOO–, at various levels of theory.

64

Table 3.5. Calculated reaction enthalpies, in kJ mol–1, for reactions (1) to (3) at 298 K.

67

Table 4.1. Zero point corrected relative energies of 1-7, TS1 and TS2, in kJ mol–1.

82

Table 4.2. B3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d) + ZPE and spin correction relative energies of 1-7, TS1 and TS2, in kJ mol–1, using implicit solvation models (ISM).

82

Table 4.3. B3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d) + ZPE + Spin correction relative energies of 1a-7a, TS1a and TS2a, in kJ mol–1, using the explicit solvation model (ESM).

85

Table 4.4. B3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d) + ZPE + Spin correction relative energies of 1a-7a, TS1a and TS2a, in kJ mol–1, using the hybrid solvation model (HSM).

89

Table 4.5. Predicted UV-vis absorption bands of cis-ONOO– (1), trans-ONOO– (2), cis-ONOOCO2

– (4), trans-ONOOCO2– (5), and

CO3•– (7) using various methods, based geometry obtained at

B3LYP/6-31+G(d) level.

91

Table 4.6. Calculated UV-Visible excitation wavelengths λexcitation, and corresponding oscillator strength values, of CO3

•– with 0, 1 and 2 explicit H2O molecules, and incorporating IEFPCM for modeling bulk effects of an aqueous solvent, using the TD-B3LYP method and the aug-cc-pVTZ basis set.

94

2

Table 5.1. Calculated activation barriers (kJ mol–1) of the two reaction pathways in the gas phase and in a polar medium at the MPW1K/6-311+G(d,p)//MPW1K/6-31G(d) level.

119

Table 6.1. Calculated Reaction Enthalpies (∆H°298, kJ mol–1) Using the G3X(MP2) and G3X(MP2)-RAD Methods.

135

Table 6.2. NBO Atomic Charges of the Donor Atoms and Dipole Moments μ in Debye of Species 1–4, 7, 7a and 7b.

139

Table 6.3. Binding energies, in kJ mol–1, and dipole moments μ, in Debye, of the lithium complexes 8–12, 14 and 15. Energies correspond to G3X(MP2) values, and dipole moments are calculated at the B3LYP/6-31G(2df,p) level.

146

Table 7.1. Selected calculated bond lengths (in Ångstroms) of (RCS2)2. Molecular geometry was optimized at the B3LYP/6-31G(2df,p) level of theory. Average experimental values for C—R bond lengths are included for comparison.

165

Table 7.2. Selected bond lengths (in Ångstroms) of RCS2•. Molecular

geometry was optimized at the B3LYP/6-31G(2df,p) level of theory.

167

Table 7.3. ∆H°298 and ∆G°298 for the homolytic S—S bond dissociation reaction (RCS2)2 → 2 RCS2

• calculated at the G3X(MP2) level of theory.

168

Table 7.4. NBO charges of (RCS2)2 and RCS2•, evaluated at the

B3LYP/6-31G(2df,p) level of theory, for substituents R = H, Me, F, Cl, OMe, SMe, NMe2 and PMe2.

169

Table 7.5. Selected calculated bond lengths (in Ångstroms) of [Li(RCS2)2]+. Molecular geometry was optimized at the B3LYP/6-31G(2df,p) level of theory.

170

Table 7.6. Selected calculated bond lengths (in Ångstroms) of [Li(RCS2)]•+. Molecular geometry was optimized at the B3LYP/6-31G(2df,p) level of theory.

170

Table 7.7. Selected calculated bond lengths (in Ångstroms) of triplet [(RCS2)Li(S2CR)]+. Molecular geometry was optimized at the B3LYP/6-31G(2df,p) level of theory.

171

Table 7.8. ∆H°298 and ∆G°298 for the coordination of (RCS2)2 and RCS2

• to Li+ in kJ mol–1, calculated at the G3X(MP2) level of theory.

172

Table 7.9. ∆H°298 and ∆G°298 for the reactions [Li(RCS2)2]+ → [Li(RCS2)]•+ + (RCS2)•, [Li(RCS2)2]+ → [(RCS2)Li(S2CR)]+, and [Li(RCS2)2]+ + Li+ → 2 [Li(RCS2)]•+ in kJ mol–1, calculated at the G3X(MP2) level of theory. 173

3

Table 7.10. Zero-point corrected reaction energies (∆E0) for the reaction (RCS2)2 → 2 RCS2

• , in kJ mol–1, for R = NMe2, OSF5, Gu1 and Gu2.

176

4

List of Figures PageFigure 2.1. A graphical representation of the variation of the energy of the 1s electron with its distance from the nucleus using a Slater type orbital (STO) basis function and a Gaussian type basis function (STO-3G basis set).

23

Figure 2.2. Flowchart summary of the procedure for solving the Roothaan-Hall equations.

27

Figure 2.3. Graphical representation of electron placement in orbitals following the RHF, UHF and ROHF formalisms.

29

Figure 2.4. Schematic diagram of electron correlation versus size of basis set.

34

Figure 2.5. A graph of eij(N) versus (N + δ)–1 for the helium PNOs.

44

Figure 3.1. Structures, element numbering, state and symmetry symbols of the sixteen RNOS: NO•, NO+, NO–, HNO, NO2

•, NO2+,

NO2–, HNO2, N2O, NO3

•, ONOO–, ONOOH, NOCl, NO2Cl, N2O3, N2O4.

57

Figure 4.1. Structures of local minima 1-7, TS1 and TS2, calculated at the B3LYP/6-31+G(d) level. Distances are given in Ångstroms.

79

Figure 4.2. Highest occupied molecular orbital (HOMO) of cis-ONOO– calculated at the B3LYP/6-31+G(d) level, showing extent of π*–π* interaction.

80

Figure 4.3. Structures of local minima 1a, 2a, 3a, 6a and 7a, calculated at the B3LYP/6-31+G(d) level. Distances are given in Ångstroms.

86

Figure 4.4. Structures of local minima 4a and 5a, and transition states TS1a and TS2a, calculated at the B3LYP/6-31+G(d) level. Distances are given in Ångstroms.

87

Figure 5.1. Plot of percentage yield of fluorescein, relative to ONOO–, versus the concentration of HCO3

– in the reaction mixture.

107

Figure 5.2. Calculated structural parameters of dianionic fluorescein (2E), neutral fluorescein, at the MPW1K/6-31G(d) level in the gas phase. X-ray crystal structural parameters are included for comparison.

110

5

Figure 5.3. Schematic gas-phase reaction energy profile for step 1 of pathway I, calculated at the MPW1K/6-311+G(d,p)//MPW1K/6-31G(d) +ZPE level with spin correction.

111


113

Figure 5.5. Schematic gas-phase reaction energy profile for pathway II, calculated at the MPW1K/6-311+G(d,p)//MPW1K/6-31G(d) +ZPE level with spin correction.

114

Figure 5.6. MPW1K/6-31G(d) optimized geometries of 1A, 1B, 1D and 1E. Distances shown are in Ångstroms. Non essential hydrogens are omitted for clarity.

115

Figure 5.7. MPW1K/6-31G(d) optimized geometries of 2B, 2D, 3B and 3D. Distances shown are in Ångstroms. Non essential hydrogens are omitted for clarity.

116

Figure 5.8. MPW1K/6-31G(d) optimized geometries of transition structures TS1C, TS2A, TS2C, TS3A and TS3C. Distances shown are in Ångstroms. Non essential hydrogens are omitted for clarity.

117

Figure 5.9. Atom numbering scheme for 1A, 1E, TS2C and TS3C.

118

Figure 6.1. Chemical structure of tetramethylthiuram disulfide (TMTD).

125

Figure 6.2. B3LYP/6-31G(2df,p) optimized geometries of the two conformers of TMTD. Distances are in Ångstroms.

130

Figure 6.3. B3LYP/6-31G(2df,p) optimized geometries of the radicals Me2NCSn

• (n = 1–3). Distances are in Ångstroms.

131

Figure 6.4. UB3LYP/6-31G(2df,p) spin density plots of the Me2NCSn•

radicals (n = 1–3)

132

Figure 6.5. Chemical drawing of 2, showing resonance structures.

132

Figure 6.6. B3LYP/6-31G(2df,p) optimized geometries of the conformers of the Me2NCS4

• radical, 7, 7a and 7b, together with the numbering convention of sulfur atoms. Distances are given in Ångstroms.

139

Figure 6.7. UB3LYP/6-31G(2df,p) spin density plots of 7, 7a and 7b.

140

Figure 6.8. B3LYP/6-31G(2df,p) optimized geometries of TMTU, TMTM and TMTT. Distances are in Ångstroms.

142

6

Figure 6.9. B3LYP/6-31G(2df,p) optimized geometries of the cationic lithium complexes [Li(TMTD)]+ (8), [Li(SCNMe2)]•+ (9), [Li(S2CNMe2)]•+ (10) and [Li(S3CNMe2)]•+ (11). Distances are in Ångstroms.

144

Figure 6.10. B3LYP/6-31G(2df,p) optimized geometry of the triplet lithium complex [Li(S2CNMe2)2]+. Distances are given in Ångstroms. 148Figure 6.11. Schematic energy level diagram for the homolytic dissociation and complex formation reactions of TMTD with and without Li+ cations, based on G3X(MP2) computed reaction enthalpies.

149

Figure 6.12. B3LYP/6-31G(2df,p) optimized geometries of the triplet species 13 and 15 and singlet species 14. Distances are given in Ångstroms.

152

Figure 7.1. Examples of different sulfur vulcanization reagents commercially available.

160

Figure 7.2. The homolytic dissociation of (RCS2)2 with novel substituents R.

162

Figure 7.3. Typical structures of (RCS2)2, RCS2•, [Li(RCS2)2]+,

[Li(RCS2)]•+ and triplet [(RCS2)Li(S2CR)]+.

164

Figure 7.4. The B3LYP/6-31G(2df,p) optimized structure of (Me2PCS2)2, showing the lack of planarity at the C2PCS2 framework.

166

Figure 7.5. UB3LYP/6-31G(2df,p) spin density plots of the radicals RCS2

•.

167

Figure 7.6. B3LYP/6-31G(d) optimized structures of (F5SOCS2)2, ((Gu1)CS2)2 and ((Gu2)CS2)2. Distances are in Ångstroms.

175

Figure 7.7. B3LYP/6-31G(d) optimized structures of F5SOCS2•,

(Gu1)CS2• and (Gu2)CS2

•. Distances are in Ångstroms.

176

Figure 7.8. Schematic potential energy profile of the homolytic S—S bond dissociation reaction (RCS2)2 → 2 RCS2

•, for R = NMe2, OSF5, Gu1 and Gu2, calculated at the MP2/6-311+G(d,p)//B3LYP/6-31G(d) + ZPE level of theory. Relative energies are given in parentheses.

178

7

List of Reaction Schemes PageScheme 4.1. Proposed reaction scheme and labeling of various species involved in the reaction ONOO– + CO2 → ONO−OCO2

– → NO2• +

CO3•–.

77

Scheme 5.1. Action of RNS on APF.

102

Scheme 5.2. Pathway I of APF oxidation.

109

Scheme 5.3. Pathway II of APF oxidation.

109

8

Chapter One

Introduction

Free radicals are commonly defined as atomic or molecular species that possess one

or more unpaired electrons (i.e. nonzero spin), and are capable of independent

existence. These free radicals and other highly reactive closed-shell (i.e. all electrons

paired) molecules form a collective group known as reactive species (RS). The

considerable importance of RS in chemistry and biology has come to light only within

the last few decades, upon the realization that these RS are part of important chemical

reactions in the atmosphere, in industrial processes and in the human body. It is

certainly remarkable to see how free radical chemistry is involved in much of our

daily lives, for instance, the destruction of the ozone layer surrounding our planet, the

industrial preparation of plastics, and in cellular respiration and signaling [1].

Free radicals can be studied in a number of ways. Experimental methods employed to

study them include infra-red (IR) spectroscopy, mass spectrometry (MS), electron

paramagnetic resonance (EPR, also known as electron spin resonance, ESR), pulse

radiolysis and high speed electrochemistry [2]. Aside from these experimental

methods, ab initio or density functional theory calculations can be employed to study

the geometries, energies and hence molecular properties of these radicals in question.

The emergence of the field of computational chemistry followed the formulation and

development of quantum theory in the early twentieth century. Calculations based on

the fundamental principles that govern the behavior of chemical systems can be

9

carried out to determine their physical and chemical properties, thus revolutionizing

our understanding of physics and chemistry. Given the transient nature of most RS,

the use of computational tools to study their electronic structure has become a popular

method of scientific inquiry [3, 4]. We can now predict the structural and hence

molecular properties of these RS to a reasonably good degree of accuracy.

Chapter two of this thesis describes the theoretical framework behind the methods

employed in each topic, namely the ab initio methods, density functional methods,

and basis sets. The calculation of molecular properties such as thermochemistry and

electron spin resonance is briefly touched on in this chapter as well.

Following the description of the theoretical framework and methodology, the rest of

the work described in this thesis centers around two themes. First is the study of

reactive nitrogen and oxygen species that result from reactions of nitric oxide (NO•),

and the second, the study of bis(thiocarbonyl) disulfides ((RCS2)2) and related radicals

involved in the vulcanization of rubber by sulfur and sulfur-rich compounds.

Despite their reputation for causing damage, many biological processes are found to

be mediated by free radicals. The discovery of NO• as a biological mediator is

credited to the 1998 winners of the Nobel Prize for Physiology or Medicine:

Furchgott, Ignarro and Murad. Murad first observed in 1977 that NO• activates an

enzyme guanylyl cyclase and activates smooth muscle relaxation [5, 6]. Ignarro made

two key observations – that NO• has an effect of relaxing an artery [7] and this same

gas inhibits platelet aggregation and activates the cellular messenger cyclic guanosine

monophosphate (cGMP) [8]. Furchgott observed that the endothelium in smooth

10

muscles releases a factor, tentatively named endothelium-derived relaxing factor

(EDRF) that causes vasodilation [9]. It was established in 1986 that EDRF and NO•

are one and the same [10, 11]. This was a surprising discovery, given that during that

time, NO• was believed to be an atmospheric pollutant and little else.

The free radical NO• is formed by the action of nitric oxide synthases (NOS) on the

amino acid L-arginine [12], and can react with a host of other molecules to form

reactive nitrogen species (RNS). For instance, NO• can combine with superoxide

(O2•−), a free radical produced by phagocytic cells, to form peroxynitrite (ONOO–), an

essential precursor of other RNS such as NO2•. The direct and indirect effects of NO•

are diverse and can be cytoprotective, regulatory or cytotoxic [13-15]. Such diverse

effects of RNS sparked off tremendous research interest by many research groups

worldwide to better understand their biological behavior.

In chapter three, the thermochemistry of a series of biologically relevant reactive

nitrogen oxide species (RNOS) is discussed. The standard enthalpies of formation at

298K of NO•, NO+, NO–, HNO, NO2•, NO2

+, NO2–, HNO2, N2O, NO3

•, ONOO–,

ONOOH, NOCl, NO2Cl, N2O3, and N2O4 were computed using various composite

methods, namely the Gaussian-n (Gn) and complete basis set extrapolation (CBS)

methods, and the standard enthalpies of decomposition reactions involving ONOO–

and ONOOH were investigated at various levels of theory, namely the composite

methods mentioned above, and coupled-cluster theory. The goal of this work is to find

a suitable method that can well describe the thermochemistry of such nitrogen-

centered RS, and predict the standard heats of formation of ONOO– and ONOOH

which have not been experimentally ascertained.

11

A study on the effects of solvation on the reaction of cis- and trans-ONOO– with CO2

is described in chapter four. Formation of the nitrosoperoxycarbonate anion,

ONOOCO2–, and its subsequent decomposition into NO2

• and CO3•− free radicals have

been attributed as a source of free radicals in biological systems [16]. A

computational study of the above reaction using density functional theory was carried

out, to gain some chemical insight on the influence of an aqueous solvent on the

energetics involved. In addition, the transition energies of the RS ONOO–,

ONOOCO2–, and CO3

•− were investigated using a variety of theoretical methods, to

substantiate the assignments of their experimental UV absorption bands.

The detection and quantitation of RNS and reactive oxygen species (ROS) remains a

constant challenge to many scientists today. The use of fluorescent probes in various

assays is the most common method used today for such purposes. 2-[6-(4′-

Amino)phenoxy-3H-xanthen-3-on-9-yl]benzoic acid or 4-aminophenylfluorescein

(APF), developed by Setsukinai et al in 2003 [17], displayed excellent selectivity for

the RNS ONOO– over other RS such as superoxide (O2•−) and hydrogen peroxide

(H2O2). The mechanism of APF reaction with species related to ONOO– reaction with

CO2 was investigated, using a combination of experimental and computational

methods, to determine the identities of the species directly involved in the two-

electron oxidation of the probe, and the energetics involved. This work is described in

chapter five of this thesis.

Other than the involvement of free radicals in biological systems, free radicals have

also been employed in organic synthesis [18, 19] and other industrial applications.

12

Aside from being used for free radical chain substitution reactions in hydrocarbons

and free radical polymerization – common topics in most high school chemistry

textbooks – free radicals are also involved in functional group transformation (e.g.

decarboxylation, deamination, halogen reduction) [20], cyclization reactions and

aromatic substitutions. Reactive precursors such as bromine gas, tributyltin hydride

(Bu3SnH), and 2,2′-azobis(2-methylpropionitrile) (AIBN) are used to generate free

radicals in situ for such applications in organic synthesis. Free radicals can also be

used in cross-linking polymerization, of which one major industrial application would

be the sulfur vulcanization of rubber.

Sulfur vulcanization of rubber involves sulfide and disulfide cross-link generation

across neighboring poly(cis-isoprene) chains using elemental sulfur and accelerators

such as tetramethylthiuram disulfide (TMTD), thus providing strength to the overall

molecular structure. TMTD, having a formula (CH3)2NC(=S)SSC(=S)N(CH3)2 can be

thought of as a ‘reactive precursor’ as mentioned above, where the free radical

(CH3)2NCS2• is generated in situ. The cross-linking process is believed to take place

via various radical pathways. A detailed examination of the homolytic dissociation of

TMTD and the radicals involved in this process is described in chapter six. In addition,

the chemistry surrounding coordination of TMTD and associated radicals with a

lithium cation is explored. The thermochemistry surrounding their reactions are

derived using high-level composite methods, and calculation of molecular properties

such as spin density and electron spin resonance spectra are also covered in this study.

The work done in this chapter is a collaborative effort with Professor Ralf Steudel and

Dr. Yana Steudel from the Institute of Chemistry in the Technical University of Berlin.

13

While TMTD is already an adequate choice for an accelerator in rubber vulcanization,

there is still reliance on zinc oxide to facilitate the formation of thiocarbamoyl

radicals RCS2• as the homolytic S—S bond dissociation enthalpy in room temperature

conditions is about 150 kJ mol–1. In chapter seven, an exploration of other substituted

bis(thiocarbonyl) disulfides (RCS2)2 is attempted, to find suitable alternative rubber

vulcanization accelerators. The homolytic S—S bond dissociation enthalpy of

(RCS2)2, and coordination enthalpy of Li+ ions to (RCS2)2 and RCS2• for R = H, Me, F,

Cl, OMe, SMe, NMe2 and PMe2 were studied and compared. Additionally, the S—S

bond dissociation enthalpy and activation barriers for a number of novel substituents

R were examined, in an effort to find suitable substituents with can stabilize the RCS2•

radical and result in a more energetically favorable generation of these radicals.

References

1. Parsons, A. F., An Introduction to Free Radical Chemistry. Blackwell Science:

Oxford, 2000.

2. Fossey, J.; Lefort, D.; Sorba, J., Free Radicals in Organic Chemistry. Masson:

Paris, 1995.

3. Anders, M. W.; Yin, H.; Jones, J. P., Application of Computational Chemistry

in the Study of Biologically Reactive Intermediates In Advances in

Experimental Medicine and Biology: Biological Reactive Intermediates V:

Basic Mechanistic Research in Toxicology and Human Risk Assessment,

Snyder, R.; Kocsis, J. J.; Jollow, D. J.; Witmer, C. M.; Greim, H.; Sipes, I. G.;

14

Monks, T. J.; Kalf, G. F., Eds. Plenum Press: New York, 1996; Vol. 387, pp

347.

4. Ventura, O. N.; Kieninger, M. Pure Appl. Chem. 1998, 70, 2301.

5. Arnold, W. P.; Mittal, C. K.; Katsuki, S.; Murad, F. Proc. Natl. Acad. Sci. U. S.

A. 1977, 74, 3203.

6. Katsuki, S.; Arnold, W.; Mittal, C.; Murad, F. J. Cyclic Nucleotide Res. 1977,

3, 23.

7. Gruetter, C. A.; Barry, B. K.; McNamara, D. B.; Gruetter, D. Y.; Kadowitz, P.

J.; Ignarro, L. J. J. Cyclic Nucleotide Res. 1979, 5, 211.

8. Ignarro, L. J.; Lippton, H.; Edwards, J. C.; Baricos, W. H.; Hyman, A. L.;

Kadowitz, P. J.; Gruetter, C. A. J. Pharmacol. Exp. Ther. 1981, 218, 739.

9. Furchgott, R. F.; Zawadzki, J. V. Nature (London, U. K.) 1980, 288, 373.

10. Ignarro, L. J.; Buga, G. M.; Wood, K. S.; Byrns, R. E.; Chaudhuri, G. Proc.

Natl. Acad. Sci. U. S. A. 1987, 84, 9265.

11. Palmer, R. M. J.; Ferrige, A. G.; Moncada, S. Nature (London, U. K.) 1987,

327, 524.

12. Palmer, R. M. J.; Ashton, D. S.; Moncada, S. Nature (London, U. K.) 1988,

333, 664.

13. Espey, M. G.; Miranda, K. M.; Thomas, D. D.; Xavier, S.; Citrin, D.; Vitek, M.

P.; Wink, D. A. Ann. N. Y. Acad. Sci. 2002, 962, 195.

14. Miranda, K. M.; Espey, M. G.; Jourd'heuil, D.; Grisham, M. B.; Fukuto, J. M.;

Feelisch, M.; Wink, D. A., The Chemical Biology of Nitric Oxide. In Nitric

Oxide : Biology and Pathobiology, Ignarro, L. J., Ed. Academic Press: San

Diego, 2000; pp 41.

15. Wink, D. A.; Mitchell, J. B. Free Radical Biol. Med. 1998, 25, 434.

15

16. Veselá, A.; Wilhelm, J. Physiol. Res. 2002, 51, 335.

17. Setsukinai, K.-i.; Urano, Y.; Kakinuma, K.; Majima, H. J.; Nagano, T. J. Biol.

Chem. 2003, 278, 3170.

18. Barton, D. H. R. Tetrahedron 1992, 48, 2529.

19. Jasperse, C. P.; Curran, D. P.; Fevig, T. L. Chem. Rev. (Washington, DC, U. S.)

1991, 91, 1237.

20. Ramaiah, M. Tetrahedron 1987, 43, 3541.

16

Chapter 2

Theoretical Background and Methodology

2.1 Introduction

This chapter outlines the theoretical framework used in the quantum chemical

calculations that have been carried out. Sections 2.2 to 2.10 describe the theoretical

framework of ab initio methods and basis sets. Section 2.11 describes density

functional theory (DFT) and framework. Sections 2.12 to 2.13 describe methods used

to calculate and hence predict molecular properties such as thermochemistry, and

electron spin resonance (ESR) g-shifts. Section 2.14 briefly describes the methods

used to model solvation. Details of the intermediate steps in deriving equations can be

found in most quantum chemistry textbooks [1–3], and are included in Appendix 1.

2.2 The Schrödinger Equation

The time dependence of any quantum chemical system is described by the time-

dependent Schrödinger equation [4, 5]:

),(),(ˆ

00 txt

itxH Ψ∂∂

=Ψ (2-1)

17

Where ħ is the Planck’s constant divided by 2π, m is the mass of the system, V is a

potential energy operator and Ψ0( x ,t) is the wave function of the system as a function

of position and spin, x and time t.

2.3 Time-independence

In a real world problem, we are interested in the variation of the energy of the states

of a molecular system with x .

Assuming that the wave function Ψ( x ,t) can be factorized into two functions, one

being a function of x and one being a function of t, the factorized equations can be

substituted into equation (2-1) to yield the time independent Schrödinger equation.

The detailed steps can be found in section A1.1 of Appendix 1.

)()(ˆ xExH Ψ⋅=Ψ (2-2)

Ĥ is the time-independent Hamiltonian operator.

Solving the time-independent Schrödinger equation yields E, the energy term of the

state of interest, and the state wave function as a function of x . A knowledge of the

ground state wave function, Ψ0( x ), provides information on the physical properties of

the system.

18

An exact analytical solution of the Schrödinger equation for any system larger than

that of the hydrogen atom is too complicated, and currently not possible. However, a

number of assumptions and approximations can be employed to yield an approximate

solution to the Schrödinger equation.

2.4 The Hamiltonian Operator

The molecular Hamiltonian operator is made up of a kinetic energy and a potential

energy term, T and V respectively:

VTH ˆˆˆ += (2-3)

where

∇∑−= 2

2

2ˆ

kk km

T (2-4)

jk

kj

j jk ree

V ⋅= ∑∑< 04

1ˆπε

(2-5)

An atom or molecule is considered as a system of interacting nuclei and electrons.

The Hamiltonian operator for such a system is constructed as follows:

AB

BAM

A

M

ABij

N

i

N

ijiA

AN

i

M

AA

M

A Ai

N

i e ReZZ

re

reZ

MmH

i

2

1 0

21

1 1 0

2

1 1 0

2

1

22

1

2

41

41

41

22ˆ ∑∑∑∑∑∑∇∑∇∑

= >

−

= +== ===

++−−−=πεπεπε

Hkin

elecˆ= H

kin

nucˆ+ H

pot

nucelecˆ

−+ H

pot

elecelecˆ

−+ H

pot

nucnucˆ

−+

(2-6)

19

The above terms in the Hamiltonian correspond to the operators for kinetic energy

contribution of the electrons, kinetic energy contribution of the nuclei, potential

energy contribution from electron-nuclei interaction, potential energy contribution

from electron-electron interaction, and potential energy contribution from nuclei-

nuclei interaction respectively.

2.5 Atomic Units

While dealing with atomic and molecular systems, it is often useful to use atomic

units to simplify the results of our calculations. By setting , me, |e| and 4πε0 to equal

to 1 [6, 7], we obtain a less complicated expression for Ĥ:

∑∑∑∑∑∑∑∇∑∇= >

−

= +== ===

++−−−=M

A

M

AB AB

BAN

i

N

ij ij

N

i

M

A iA

AM

AA

N

ii R

ZZrr

ZH1

1

1 11 11

221

1

221 1ˆ

(2-7)

Energies are then measured in Hartrees, where 1 Hartree is defined to be the

Coulombic repulsion between two electrons separated by a distance of 1 Bohr radius,

or 0.52911725 Å. One Hartree is equivalent to 27.2107 eV or 4.35974394 × 10–18 J.

2.6 Born-Oppenheimer Approximation

The physical idea behind the Born-Oppenheimer approximation is that electrons move

much faster than nuclei, since the mass of a nucleus is typically thousands of times

that of an electron. For a system of nuclei with initial fixed positions, the electrons

20

adjust their positions with respect to these nuclei before the nuclei actually move [8].

By separation of nuclear and electronic motions, the problem can be simplified to the

electronic Schrödinger equation.

eleceffelecelec EH Ψ=Ψˆ (2-8)

The detailed steps can be found in section A1.2 of Appendix 1.

2.7 Hartree-Fock Method

Despite the approximations mentioned so far, the electronic Schrödinger equation (2-

8) for a molecule is a complex differential equation, and still cannot be solved

analytically except for the smallest systems.

The Hartree-Fock (HF) method [9, 10], or the self consistent field (SCF) method is a

numerical iterative technique employed to calculate the Slater determinant and an

expectation value of the electronic Hamiltonian. It uses a number of approximations

and assumptions to provide an approximate solution to the electronic Schrödinger

equation. The following subsections outline the steps taken in the method.

2.7.1 The wave function Φ as a Slater determinant

The electronic wave function Ψ should have two important properties. First, it should

21

be normalized, and second, because electrons are fermions, it should also be

antisymmetric, that is it should change sign when two electrons are interchanged.

An approximate electronic closed shell wave function, the Slater determinant, is

worked out in section A1.3 of Appendix 1. The Slater determinant yielded fulfills the

above requirements of being normalized and antisymmetric.

≈Ψ )(xe )(xeΦ

= )()()()()()()()()()()()(

)2()()2()()2()()2()()2()()2()()1()()1()()1()()1()()1()()1()(

!1

222211

222222222121

121212121111

NxNxNxNxNxNx

xxxxxxxxxxxx

NNNNNNNNN

NN

NN

βϕαϕβϕαϕβϕαϕ

βϕαϕβϕαϕβϕαϕβϕαϕβϕαϕβϕαϕ

(2-10)

The essential assumption made here in the Hartree-Fock method is that the wave

function of an electron, which is not a physical observable, can be mathematically

represented as an antisymmetric product of molecular orbitals. Each electron here is

assumed to experience an average charge distribution due to other electrons in the

system, and not interact explicitly with each and every one of the other electrons.

2.7.2 Basis Sets

Individual molecular orbitals φ in the Slater determinant (eq. 2-10) are expressed as

linear sums of normalized basis functions, χ. Basis functions are simply atomic

orbitals described by a mathematical function based on spherical or Cartesian

coordinates.

22

∑=

=Nbasis

ii c1μ

μμ χϕ (2-11)

Slater-type orbitals (STOs) were historically used as basis functions of molecular

orbitals because of their similarity to the atomic orbitals of the hydrogen atom.

),(),,;,,,( 1 φθφθζχ ζlm

rni YeNrrmln −−= (2-12)

N is a normalization constant; r, θ , φ are spherical coordinates; n, l, m are principal,

angular momentum and magnetic quantum numbers respectively; ζ is called an

“exponent”, and Ylm is a function describing the shape (the angular momentum part).

STOs have the advantage of describing wavefunctions well. The exponential term, e-ζr,

well describes the energy of electrons near the nucleus. However they also have the

disadvantage of being unsuitable for calculating two-electron integrals quickly. A

solution to this is the implementation of Gaussian-type functions (GTFs).

nmlzyx zyxNezyxnmlg )( 222

),,;,,,( ++−= αα

nmlr zyxNe2α−= (2-13)

N is a normalization constant; x, y, and z are spherical coordinates; n, l, m are integral

exponents at Cartesian coordinates, and α is called the “exponent” here.

GTFs allow fast calculation of two electron integrals, but representing an atomic

orbital with a GTF leads to error as GTFs do not have a cusp at the nucleus like STOs

do. However, addition of a number of GTFs yields a good approximation to an STO,

and is still computationally cheaper than the integration of the STO itself. GTFs are

23

grouped, or contracted, in linear combinations to form basis functions. A comparison

of an STO with an STO-3G (STO approximated by 3 GTFs) basis set for the 1s

orbital of the hydrogen atom is graphically presented in Figure 2.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2

Distance, Angstroms

Figure 2.1. A graphical representation of the variation of the energy of the 1s electron with its distance from the nucleus using a Slater type orbital (STO) basis function and a Gaussian type basis function (STO-3G basis set). The filled line represents the trend for the STO, while the dotted line represents the trend for the STO-3G basis.

Basis sets can be grouped into a number of categories – minimal, split valence,

polarized basis sets, and basis sets with diffuse functions.

Minimal basis sets such as STO-3G use fixed-size atomic-type orbitals. They contain

the minimum number of basis functions needed for each atom.

Split valence basis sets have a larger number of basis functions per atom. Two or

more sizes of basis functions are assigned to the valence shells of each atom, allowing

the orbitals to change size. A double-split valence basis set, such as 3-21G, has the

24

valence shells of each atom split into two basis functions, the first basis function is a

contraction of two Gaussian primitives, and the second basis function consists of one

Gaussian primitive. Within the split valence basis set category, there are also n-ζ type,

where molecular orbitals are formed from linear combinations of n sizes of functions

for each atomic orbital, for example the double-ζ Dunning-Huzinaga (D95), and the

correlation consistent, polarized valence triple-ζ (cc-pVTZ) basis sets.

Polarized basis sets involve adding orbitals of a higher angular momentum than the

ground state electronic configuration to the description of each atom. This allows

orbitals to effectively change shape. Examples of polarized basis sets are 6-31G(d)

and 6-31G(d,p).

Diffuse functions can be added to basis sets to allow orbitals to occupy larger regions

in space. This allows better description of species where electrons are further away

from the nucleus, such as anions, and molecules with lone pairs of electrons.

Examples of such basis sets are 6-31+G(d) and 6-311++G(d,p).

2.7.3 The Variation Principle

The variation principle is a starting point for almost all methods that seek to find an

approximate solution to the Schrödinger equation. It states that for any acceptable

normalized function Φ, the expectation value of Ĥ will be an upper bound to the

lowest eigenvalue of Ĥ.

25

A consequence of the variation principle is that as the trial wave function, Φtrial,

approaches the true wave function Ψ0, the expectation value of Ĥelec, Etrial, approaches

the true energy of the system, E0.

2.7.4 The Fock Matrix

The Fock matrix, F, is an approximation of the true Hamiltonian operator of the

system. It is made up of Fock operators, )(ˆ iF , with the following general formula

(see section A1.4 of Appendix 1 for derivation)

∑=

−+=N

jijj

corei KJHiF

1)( )]}ˆˆ2{ˆ)(ˆ

(2-14)

coreiH )(

ˆ is the core Hamiltonian operator, jJ and ijK are Coulomb and exchange

operators respectively.

The equations that connect the electronic Hamiltonian, eH , and the orbital wave

functions φi can now be written in terms of the Fock operator F , φi, and a set of

constants, εi. in a form that resembles the Schrödinger equation. These equations are

also commonly referred to as the Hartree-Fock equations.

iiiF ϕεϕ =ˆ (2-15)

2.7.5 The Roothaan-Hall equations

Roothaan and Hall independently devised a method to represent the Hartree-Fock

26

equations using non-orthonormal Gaussian type or Slater type basis sets [11, 12]. The

Roothaan-Hall equations are:

∑=

=−Nbasis

ii CSF1

0)(ν

νμνμν ε ;

for μ = 1,2,…,Nbasis.

(2-16)

The electronic energy can be calculated from the elements of the core Hamiltonian

matrix coreH μν , the Fock matrix Fμν and the one electron density matrix Pμν. The nuclear

repulsion energy is then added to obtain the HF energy.

The Hartree-Fock energy is obtained by adding Eelec to the nuclear repulsion energy.

∑∑∑ ∑= <= =

++=M

A

M

BA AB

BANbasis Nbasis

coreHF R

ZZHFPE

11 1)(

21

μ νμνμνμν

(2-17)

The program structure for solving the Roothaan-Hall equations [1] is summarized in

Figure 2.2. The detailed derivation of the Roothaan-Hall equations is provided in

Section A1.5 of Appendix 1.

27

Form Fock Matrix

Solve Roothaan-Hall equations

∑=

=−Nbasis

ii CSF1

0)(ν

νμνμν ε

Form density matrix

∑=

=occ

iii CCP

1

*2 σλλσ

Calculate energy

)(21

1 1∑ ∑= =

+=Nbasis Nbasis

coreelec HFPE

μ νμνμνμν

Initial guess Pμν

Eelec and/or Pμνconverged?

Pμν

No

)]|(21)|[(

1 1

νσμλλσμνλ σ

λσμνμν ∑ ∑= =

−+=Nbasis Nbasis

core PHF

Calculate 2-electronIntegrals


Sμν

coreH μν

(μν|λσ)

Done

Form Fock Matrix

Solve Roothaan-Hall equations

∑=

=−Nbasis

ii CSF1

0)(ν

νμνμν ε

Form density matrix

∑=

=occ

iii CCP

1

*2 σλλσ

Calculate energy

)(21

1 1∑ ∑= =

+=Nbasis Nbasis

coreelec HFPE

μ νμνμνμν

Initial guess Pμν

Eelec and/or Pμνconverged?

Pμν

No

)]|(21)|[(

1 1

νσμλλσμνλ σ

λσμνμν ∑ ∑= =

−+=Nbasis Nbasis

core PHF



Sμν

coreH μν

(μν|λσ)

Done

Figure 2.2. Flowchart summary of the procedure for solving the Roothaan-Hall equations.

2.8 Restricted and Unrestricted Hartree-Fock methods

The Restricted Hartree Fock (RHF) method described in the previous section assumes

a closed-shell system where each spatial orbital is doubly occupied. For open-shell

systems where there are singly occupied orbitals, the RHF method is not suitable.

28

The common method used to treat open-shell systems is to separate α and β electrons

into singly occupied spatial orbitals, and solve for two sets of Roothaan-Hall

equations, known as the Pople-Nesbet equations [13]:

FαCα = SCαεα (2-18)

FβCβ = SCβεβ (2-19)

This approach is termed as the spin-unrestricted Hartree-Fock (UHF) method.

The major drawback of the UHF method is that one-electron Slater determinants for α

and β electrons are not satisfactory eigenfunctions of the total spin operator, Ŝ2. Thus

the total wave function may be contaminated by wave functions of higher states; for

example, a doublet state contaminated with quartet or higher spin states. The spin

contamination – the deviation of the expectation value of the total spin operator, ⟩⟨ 2S ,

from S(S+1) where S is the 0.5 times the number of unpaired electrons – gives an

indication of the contamination from higher spin states.

Another modification to the RHF method to accommodate open-shell systems is the

Restricted Open-Shell Hartree-Fock (ROHF) method. The ROHF method follows the

RHF method as much as possible, and does not suffer from spin contamination from

higher spin states. However, the Fock matrix for the ROHF method is not unique, and

different Fock matrices can yield the same wave function.

A graphical representation of the orbital treatment for RHF, UHF and ROHF methods

is given in Figure 2.3.

29

RHF UHF ROHF

Figure 2.3. Graphical representation of electron placement in orbitals following the RHF, UHF and ROHF formalisms.

2.9 Correlation Effects

Throughout the formulation of the HF method, one assumption held was that each

electron moves in a time-averaged potential field caused by the other electrons in the

system. In reality, the motions of electrons are correlated, since they actually repel

each other individually, rather than each one of them reacting to an average field

generated by the other electrons. The energy calculated using the HF method is higher

than that when electron correlation is explicitly considered in the wave function. This

energy difference is known as the correlation energy.

ncorrelatioHFexact EEE += (2-20)

There are many techniques that are used to calculate this correlation energy, and are

collectively termed post SCF methods. The following subsections describe their

formulation.

30

2.9.1 Configuration Interaction (CI)

In the HF method, the wave function was a single determinant for a closed shell

system. To implement configuration interaction, one has to take into account other

electronic configurations that can be adopted by the same number of electrons in the

system. Taking the HF single determinant wave function Φ0 as reference, the

eigenvector corresponding to the exact solution of the Schrödinger equation, ΨCI, for a

given one-electron basis set, is written as a linear combination of determinant wave

functions (or configuration state functions, CSFs) from all electronic states, Φ:

…+Φ+Φ+Φ=Ψ ∑∑∑∑

< <

occ

ji

virt

qp

pqij

pqij

occ

i

virt

p

pi

piCI aaa 00

(2-21)

The above expression is the eigenvector for the full CI (FCI) method, wherein all

possible excited configurations have been included. The single-indexed term

∑∑ Φocc

i

virt

p

pi

pia represents the contributions from single excitations, and the double-

indexed term ∑∑< <

Φocc

ji

virt

qp

pqij

pqija represents the contributions from double excitations. The

use of the infinite expansion ΨCI in ⟩ΨΨ⟨ CIeCI H |ˆ| would yield the sum of HF

energy and all energy due to explicit electron correlation within the given basis set.

The disadvantage of carrying out a full CI calculation is that it will involve a great

number of CSFs to be computed, and is not practical except for very small systems.

31

2.9.2 Limited Configuration Interaction

Limited CI is carried out by adding only a limited set of substitutions to the HF

determinantal wave function. The CIS, CID and CISD methods (see section A1.6 of

Appendix 1) involve adding single, double, and both single and double substitutions

to the CI wave function respectively. However, these methods suffer the disadvantage

of not being size-consistent.

The quadratic configuration interaction method, using singles and doubles substitution

(QCISD) was developed by Pople et al [14] to ensure size consistency in the resulting

total energy. The QCISD equations are described in section A1.7 in Appendix 1.

The full triples substitution (QCISDT method) is also possible, but is computationally

impractical for large systems. To address this, a partial triples correction based on

perturbation at the singles-doubles level, ΔET is included in the QCISD formulation to

yield improved QCISD(T) energies [14].

2.9.3 Coupled Cluster Methods

Coupled cluster methods, derived by Purvis III et al [15], work on the ansatz that the

wave function can be expressed as exp( T )Ψ0, where T is the cluster operator. The

formulation of the coupled cluster method using singles and doubles substitution

(CCSD) equations is included in section A1.8 of Appendix 1.

32

There are a number of different formulations to address the triples correction for the

CCSD method [16-18]. One commonly used is that proposed by Raghavachari et al,

which is similar to that adopted in the QCISD(T) method, to yield CCSD(T) energies

[18].

2.9.4 Møller-Plesset Perturbation Theory

An alternative way to treat electron correlation is using Møller-Plesset perturbation

theory (MPPT) [19], whose principles are drawn from techniques in many body

perturbation theory (MBPT).

In MPPT, the Hamiltonian operator is first divided into two parts:

VHH ˆˆˆ0 λ+= (2-22)

Where Ĥ0 is the sum of one-electron Fock operators, which is soluble exactly, and

Vλ is a small perturbation. According to Rayleigh-Schrödinger perturbation theory

(RSPT), the ground state wave function and energy of the system can be expanded as

a power series:

…)3(3)2(2)1()0( Ψ+Ψ+Ψ+Ψ=Ψ λλλ (2-23)

…)3(3)2(2)1()0( EEEEE λλλ +++= (2-24)

Substituting these equations back into the time-independent Schrödinger equation

gives:

))(())(ˆˆ( )()()(

0 ∑∑∑ Ψ=Ψ+n

i

iin

i

iin

i

ii EVH λλλλ (2-25)

33

For n=0, the sum of the electronic orbital energies εi is obtained and for n=1, the HF

energy EHF = E(0) + E(1) and the HF wave function Ψ(0) are obtained.

Solving these equations to the nth order will yield the corresponding MPn energies.

The detailed derivation of the expressions of the MPn energies is provided in section

A1.9 of Appendix 1. MP2 energies include contributions where doubly excited

configurations interact with the ground state configuration. MP3 energies include

contributions of doubly excited configurations interacting with each other. MP4

energies include energy contributions of interactions that involve single, double and

quadruple excitations.

It is noted that MPPT is size consistent, but not variationally bound. It cannot be

assumed that adopting higher orders of perturbation will yield energies that are

progressively closer to the true energy.

Figure 2.4 summarizes the effects of correlation versus the effects of basis sets in

attempts to solve the Schrödinger equation using ab initio techniques. The horizontal

axis depicts the extent of electron correlation considered, and the vertical axis depicts

the size of the basis set. The true solution of the Schrödinger equation is reached when

a full configuration interaction using a complete basis set is performed.

34

Increasing electron correlation HF MP2 MP3 MP4 QCISD QCISD(T) … Full CI

Minimal e.g. STO-3G

Split-valence e.g. 3-21G

Polarized e.g. 6-31G(d)

Diffuse e.g. 6-311+G(d)

High angular momentum e.g. 6-311+G(2df,p)

…

Increasing size of basis set Complete basis set HF

limit Exact

Solution Ψ=Ψ EH

Figure 2.4. Schematic diagram of electron correlation versus size of basis set.

2.10 Multi-Configuration SCF

Instead of considering just a single electronic configuration in the construction of the

HF wave function from the Slater determinant, the multi-configuration SCF (MCSCF)

method considers multiple electronic configurations in the construction of the

electronic wave function [20]. Essentially it is an extension of the single configuration

HF method. The most common approach is the complete active space SCF (CASSCF)

method [21-23], in which a set of chemically important molecular orbitals from the

HF wave function – the active space – are chosen, and a full CI carried out within this

active space.

35

2.11 Density Functional Theory

Density Functional Theory (DFT) methods are in principle an exact theory of

electronic structure, with its base on the electron density ρ(r) instead of the

approximate many electron wave function Ψ[24]. DFT methods are extremely popular

today due to its versatility and reduced computational costs compared to conventional

ab initio methods. A great deal of information on DFT is available in most textbooks

[25, 26]; only a brief overview of DFT will be provided in this section.

Modern DFT methods stem from the Hohenberg-Kohn theorem [27], which states that

all properties of a system defined by an external potential, Vext, are uniquely

determined by the ground state electron density )(rρ . Hence, as a result of the

variational principle, the )(rρ that gives the minimum total energy is the ground state

electron density.

The Hamiltonian of a given system, and its electronic energy, is partitioned using the

Kohn-Sham approach [28].

H eeNe UTV ˆˆˆ ++= (2-26)

)]([ rE ρ

)]([)(

)]([)]([)]([

rFrdVr

rErTrE

Ne

eeNe

ρρ

ρρρ

+=

++=

∫

(2-27)

)]([ rF ρ is a functional of the electron density )(rρ , as Ψ is. The components of

)]([ rF ρ are: )]([ rTs ρ , the kinetic energy of the non-interacting system with density

36

)(rρ , )]([ rJ ρ , the classical electrostatic electron-electron repulsion energy, and the

remainder, )]([ rEXC ρ , the exchange-correlation energy.

The entire expression for the energy calculated using the DFT approach is written in

equation (2-28). A detailed explanation is provided in section A1.10 of Appendix 1.

=)]([ rE ρ ∑ ⟩∇⟨−

N

iii ϕϕ ||

21 2

)]([

|)(|

|)(|1|)(|21

12

11

212

212

21

rE

rdrrZ

rdrdrr

r

XC

N

i

M

Ai

A

A

j

N

i

N

ji

ρ

ϕ

ϕϕ

+

−

+

∑∫∑

∑∑∫∫

(2-28)

The exact form of )]([ rEXC ρ is unknown. Finding accurate forms of )]([ rEXC ρ is

still an agendum of modern research in quantum chemical methods.

2.11.1 Local Density Approximation (LDA) and Local Spin Density Approximation

(LSDA)

The center of the LDA and LSDA is the idea of a hypothetical uniform electron gas,

with number of electrons N and volume of the gas V. As N and V approach infinity,

the electron density ρ attains a constant value everywhere. EXC can then be assumed to

take the following form:

rdrrE XCLDAXC ))(()(][ ρερρ ∫= (2-29)

37

))(( rXC ρε is the exchange-correlation energy per particle of a uniform electron gas

with density )(rρ . The arrival of the overall expression for EXC involves the

multiplication of this term with the probability of finding an electron at that position

in space – the electron density )(rρ – as the weighting function.

))(( rXC ρε can be split into the exchange and correlation contributions, ))(( rX ρε and

))(( rC ρε respectively. The Slater exchange functional ))(( rX ρε is equal to:

31

)(343))(( ⎟

⎠⎞

⎜⎝⎛−=

πρρε rrX

(2-30)

Substituting equation (2-30) back to (2-29) yields the 34

ρ dependence of the exchange

energy.

Extension of the LDA to the unrestricted case leads to the LSDA. The exchange

energy for the LSDA is written as:

rdrrrrrE XCLSDAXC ))(),(()()](),([ βαβα ρρερρρ ∫= (2-31)

The LDA and LSDA are drastic approximations, since they assume that the density in

a molecular system is constant and can be approximated by a homogeneous uniform

electron gas.

2.11.2 Generalized Gradient Approximation (GGA)

The LDA and LSDA fail when the density undergoes rapid changes. An improvement

38

to the LDA and LSDA is the GGA, where the gradient of the electron density is taken

into account when computing the exchange energy, which can translate to an

improvement in accuracy of results. The exchange energy as a result of the GGA can

be written generically as:

∫ ∇∇= rdrrrrfrrE GGAXC ))(),(),(),(()](),([ βαβαβα ρρρρρρ (2-32)

Several parameterizations of the GGA exist today, such as the B88 [29] and the PW91

[30, 31] exchange functionals, some include the incorporation of empirical parameters,

and some are completely free of such. For instance, the B88 exchange functional

includes an empirical parameter – the exactly known exchange energies of the rare

gas atoms He through Rn, whereas the PBE functional [32, 33] is free of such

empirical parameters.

A similar approach has also been taken to obtain analytical forms of gradient

corrected correlation functionals, but their explicit forms are complicated and not

easily understood by simple physically motivated reasoning. A list of the explicit

forms of some exchange and correlation functionals is provided in Section A1.11 of

Appendix 1.

In principle, exchange functionals can be combined with correlation functionals to

calculate the exchange energy of a system. An example is the combination of

Becke’s functional with the correlation functional by Lee, Yang and Parr [34] – often

abbreviated as the BLYP functional.

39

2.11.3 Hybrid Functionals

Density based functionals often overestimate a quantity which HF theory

underestimates, for instance, bond lengths [35]. To address this, Becke suggested that

the exact exchange-correlation functional must include a fraction of HF exchange [36].

The exchange-correlation of the Kohn-Sham scheme is obtained through the

coupling-strength integrated exchange-correlation hole.

λλ dEE nclXC ∫=

1

0

(2-33)

The case when λ = 0 refers to an interaction free system, where the contribution to EXC

is the exchange energy only, and the correlation term is neglected. The case when λ =

1 refers to a fully interacting system, with exchange and correlation parts taken into

consideration in the computation of EXC.

The simplest case to consider is to assume that λnclE varies linearly with respect to λ.

This results in a half-and-half combination of exact exchange and density functional

exchange-correlation [37].

The most popular hybrid functional is Becke’s three parameter exchange functional

combined with the correlation functional of Lee, Yang and Parr (B3LYP). The

B3LYP exchange-correlation expression for the energy is as follows:

VWNC

LYPC

BXXC

DiracX

LYPBXC EccEbEaEEaE )1()1( 8803 −++++−= =λ (2-34)

40

The three parameters are optimized to be a = 0.20, b = 0.72 and c = 0.81.

Hybrid functionals such as B3LYP have achieved great success in the prediction of

molecular geometries and harmonic vibrational frequencies in recent years [38],

hence their widespread use in computational chemistry packages.

2.12 Composite Methods for Accurate Thermochemistry

From Figure 2-4, the exact solution of the Schrödinger equation is reached by using

full configuration interaction (FCI) and the complete basis set (CBS). However, this

task is not currently achievable, as the computing resources required (CPU, memory

and disk) is vast.

However, several composite methods have been proposed in recent years to calculate

accurate thermochemistry data. They usually incorporate a method for obtaining

accurate geometries compared to experiment, and an algorithm for extrapolating the

FCI/CBS energy. Two widely used composite methods are the Gaussian-n and CBS-n

methods developed by Pople et al and Petersson et al respectively.

2.12.1 Gaussian-n methods

The first Gaussian-n method, G1, was developed in 1989 by Pople et al [39]. The G1

method performed fairly well, yielding an accuracy of above 2 kcal mol–1 accuracy in

41

comparison with experimentally known atomization energies, ionization potentials,

and electron and proton affinities of a group of 31 molecules. In 1991, the second

generation of Gaussian-n methods was developed, namely the G2 method [40]. The

G2 method eliminated several deficiencies of G1 theory, and yielded an improvement

of 0.50 kcal mol–1 in mean absolute deviation from experimental heat of atomization

for 39 first row compounds. The use of reduced Møller-Plesset orders in calculation

of Gaussian-n energies was also introduced to reduce computation time and yet

minimize the compromise in accuracy [41].

The most current Gaussian-n method in use is the G3 family of methods, first

developed in 1998 [42]. Some key improvements of the G3 method over its

predecessors are the use of the G3Large basis set, addition of an empirical higher

level correction (HLC) term, and the consideration of core correlation effects. The

G3Large basis set is a modification of the 6-311+G(3df,2p) basis set used in G2, and

includes additional polarization functions for second row atoms (3d2f) and less for the

first row atoms (2df). Neutral basis sets were also used for the atoms P, S and Cl. The

HLC term is parameterized to obtain the least mean absolute deviation to that of

known experimental energies of the G2/97 test set [43, 44]. Core correlation effects

are included in a MP2(full)/G3Large energy calculation, which improves the accuracy

of calculated energies for systems with unsaturated ring systems.

The procedure for the G3 method is as follows: geometry optimization at the

MP2(full)/6-31G(d) level; zero point energies (ZPE) calculated at the HF/6-31G(d)

level, scaled by a factor of 0.8929; single point calculations carried out at the

42

QCISD(T)(fc)/6-31G(d), MP4(fc)/6-31+G(d), MP4(fc)/6-31G(2df,p), and

MP2(fu)/G3Large levels based on the MP2(fu)/6-31G(d) geometry.

The G3 energy for a molecule at 0 K is given as:

E0(G3) = E[MP4/6-31G(d)] + ΔE(+) + ΔE(2df) + ΔE(QCI)

+ ΔE(G3Large) + E(SO) + E(HLC) + E(ZPE) (2-35)

Where

ΔE(+) = E[MP4/6-31+G(d)] – E[MP4/6-31G(d)] (2-36)

ΔE(2df) = E[MP4/6-31G(2df,p)] – E[MP4/6-31G(d)] (2-37)

ΔE(QCI) = E[QCISD(T)(fc)/6-31G(d)] – E[MP4/6-

31G(d)] (2-38)

ΔE(G3Large)= E[MP2(fu)/G3Large]

– E[MP2(fc)/6-31G(2df,p)]

– E[MP2(fc)/6-31+G(d)]

+ E[MP2(fc)/6-31G(d)] (2-39)

E(HLC) = Anβ – B(nα – nβ) for molecules;

Cnβ – D(nα – nβ) for atoms (2-40)

E(SO) is the experimental or theoretically calculated spin-orbit correction. A, B, C

and D in E(HLC) are parameters having the values 6.386, 2.977, 6.219 and 1.185

millihartrees respectively.

Several variants of the G3 method are available as well. They include G3S, using

scaled energies for the HLC [45], G3X, using improved B3LYP/6-31G(2df,p)

43

geometry and ZPE [46], and G3-RAD, optimized for calculations of thermochemistry

of open-shell species [47]. G3 theory, together with its variants, can be carried out

with lower orders of Møller-Plesset perturbation [48], for example G3(MP2),

G3(MP3), G3X(MP2) and G3(MP2)-RAD. These are used for larger systems where

MP4 calculations are prohibitive, while minimizing loss of accuracy. Details of the

other Gaussian-n calculations used in this thesis are provided in section A2.12 of

Appendix 2.

2.12.2 CBS-n methods

The complete basis set (CBS) extrapolation methods were proposed by Petersson and

co-workers, who observed that basis set truncation was one of the main sources of

error in accurate calculation of molecular correlation energies. CBS extrapolations are

based on the asymptotic convergence of pair natural orbital (PNO) expansions of the

first order wave function [49].

∑=

=ΨN

CN1

)1( )(μ

μμϕ (2-41)

A graph of eij(N) versus N–1 for the helium atom (Figure 2.5) summarizes the general

idea of CBS extrapolation. The term eij(N) represents the correlation energy between

an electron pair, using N PNO expansions. The useful points for the extrapolation are

those corresponding to completely filled shells [50].

44

Figure 2.5. A graph of eij(N) versus (N + δ)–1 for the helium PNOs, adapted from reference 50.

The complete basis set–atomic pair natural orbital (CBS–APNO) method was

developed in 1993, and provided excellent accuracy (mean absolute error of 0.53 kcal

mol–1) for bond dissociation energies, ionization potentials and electron affinities for

the G2 test set [51]. The components of various CBS-n methods, with the components

of the G3 and G3(MP2) methods alongside are summarized in Table 2.1.

45

CBS-4 CBS-q CBS-Q CBS-QCI/APNO G3(MP2) G3 Geometry Method UHF/3-21G(*) UHF/3-21G(*) MP2(fc)/6-31G† QCISD/6-311G** MP2(fu)/6-31G* MP2(fu)/6-31G* ZPE Method UHF/3-21G(*) UHF/3-21G(*) UHF/6-31G† UHF/6-311G** UHF/6-31G* UHF/6-31G* Scale Factor 0.91671 0.91671 0.91844 0.9251 0.8929 0.8929 Core Correlation Method None None 3.92q(Na) +

2.83q2(Na) CBS2/KK,KL,LL,LL′ None None

SCF Method UHF UHF UHF UHF UHF UHF Basis sets H, He 31+G(p) 31+G(p) 311+G(2p) 4s2p1d 311+G(2p) 311+G(2p) Li – Ne 6-311+G(2df) 6-311+G(2df) 6-311+G(2df) 6s6p3d2f G3MP2Large G3Large Na – Ar 6-311+G(3d2f) 6-311+G(3d2f) 6-311+G(3d2f) N/A G3MP2Large G3Large MP2 Method CBS2 CBS2 CBS2 CBS2 MP2 MP2 Nmin 5 5 10 10 Npts 40 000 40 000 100 000 100 000 Basis sets H, He 31G† 31G† 311+G(2p) 4s2p1d 311+G(2p) 311+G(2p) Li – Ne 6-31+G† 6-31+G† 6-311+G(2df) 6s6p3d2f G3MP2Large G3Large Na – Ar 6-31+G† 6-31+G† 6-311+G(3d2f) N/A G3MP2Large G3Large MP4 Method MP4(SDQ) MP4(SDQ) MP4(SDQ) MP4(SDTQ) Basis sets 6-31G 6-31+G† 6-31+G(d(f),d,p) 6-31G* 6-31+G* 6-31G(2df,p) QCISD(T) Method QCISD(T) QCISD(T) QCISD(T) QCISD(T) QCISD(T) Basis sets 6-31G 6-31+G† 6-311G(d,p) 6-31G* 6-31G* Empirical corrections (mEh)

2 electron iiii

IS 252.5− iiii

IS 294.5− iiii

IS 233.5− iiii

IS 274.1− – 4.808npairs – 3.409npairs

1 electron – 4.55 (nα + nβ) – 4.03 (nα + nβ) 0 0 – 4.471 (nα + nβ) – 2.977 (nα + nβ) Spin – 38.43 Δ‹S2› 0 – 9.2 Δ‹S2› 0 0 0 Nmin = minimum number of natural orbital configurations used for CBS extrapolations Npts = number of mesh points per electron for the Monte Carlo numerical integration of |S|ij |S|ii = ∫|αφi

βφi|dτ , the absolute overlap integral Iii = (ΣCμ)2, the intraorbital interference factor

Table 2.1. Components of CBS-4, CBS-q, CBS-Q, CBS-QCI/APNO, G3(MP2) and G3 models. Adapted from Ref. 50.

2.13 Methods of Calculation of ESR G-shifts

G-tensors are an important part of any ESR spectrum. Calculations of EPR g-tensors

used in this thesis are based on DFT and the use of “gauge-including atomic orbitals”

(GIAO) schemes [52]. The g-tensor of EPR spectroscopy is considered as a second-

order property. The perturbation parameters are one Cartesian component of the

46

constant external magnetic field BS, and the net electronic spin component along a

given coordinate axis, St. The st tensor component of g is given by:

0

21

==∂∂

∂=

SBtSBst SB

Egμ

(2-42)

E is the total energy of the many-electron system. μB is the Bohr magneton.

The g-shift, Δg, is defined as the deviation of the molecular g-value from the free

electron value ge = 2.002 319 277 8.

ggg e Δ+= 1 (2-43)

1 is the unit tensor, g and gΔ are second-rank tensors. The isotropic g-shift, Δg is

the trace of gΔ . Contributions to Δg come from the operators hZ−KE, hSO−N, hSO−e,

hSOO, hSO−Ndia, hSO−e

dia, and hSOOdia, details of which are found in section A1.12 of

Appendix 1.

2.14 Solvation Models

Ab initio and DFT calculations are typically carried out for molecules in vacuo, to

determine their intrinsic properties. A great deal of experimental work is carried out

with reactants dissolved in a solvent, which typically changes the characteristics of

the molecules in question as compared to them in the gas phase. Solvent effects can

be modeled in the framework of self-consistent reaction field (SCRF) theory. A

47

macroscopic property of the solvent, such as the dielectric constant, is often used to

describe a continuous medium.

The solute is first placed in a cavity which is immersed in a continuous medium with

reaction field R . The solute is then treated quantum mechanically, and the

interactions with the solvent are taken into account by a perturbation on the solute

Hamiltonian.

Two continuum solvation models are used in this thesis. They are the Onsager

reaction field method [53, 54], and the integral equation formalism for the polarizable

continuum model (IEF-PCM) [55].

2.14.1 Onsager Reaction Field Theory

The solute is placed in a spherical cavity surrounded by a continuous medium of

dielectric constant, ε. A dipole in the solute induces a dipole in the solvent, and the

electric field experienced by the solute interacts with the molecular dipole.

The electrostatic solvent effect may be taken as a perturbation in the Hamiltonian of

the isolated molecule. This perturbation term describes the coupling between the

molecular dipole operator and the reaction field, R .

RHH fieldrxn ⋅−=− μ0 (2-44)

48

The reaction field R is proportional to the molecular dipole moment, μ , hence

μgR = , where the proportionality constant, g is related to ε and the radius of the

spherical cavity, a0.

3

0)12()1(2

ag

+−

=εε

(2-45)

For an SCF wave function, the effects of R are incorporated into the Fock matrix as

an additional term.

⟩⟨−= σλλσλσ ϕμϕμ |ˆ|0 gFF (2-46)

The energy of the system is hence given by

RHE ⋅−⟩ΨΨ⟨= μ21

0 || (2-47)

In the case of a molecular ion with a total charge Q, a Born charge term is added to

the energy expression, to account for the ion-dipole interaction. The quantities E, Q

and a0 are in atomic units, namely Hartree, units of electron charge, and Bohr radii,

respectively.

0

2

)(11

21

aQEE ion ⎟

⎠⎞

⎜⎝⎛ −−=

ε

(2-48)

2.14.2 Integral equation formalism for polarizable continuum solvent model

(IEFPCM)

The IEFPCM procedure considers an infinitely dilute solution at equilibrium [56].

The reference state of a solvent at a given temperature T, supplemented by a number

49

of non interacting electrons and nuclei sufficient to build the solute M, is used to

consider the Gibbs free energy of solvation. A cavity is constructed around M, formed

by interlocking spheres, centred on solute atoms or atomic groups. Several sets of

atomic radii, such as the united atom topological model (UATM) [57] can be used to

create such a cavity. The work spent in constructing the cavity in the solvent

corresponds to Gcav, the cavitation free energy. Free electrons and nuclei are then

collected in the cavity to form M. The work spent in this second process gives a

second contribution to the free energy of the system. Phenomenological partitioning

of these free energies gives:

cavMmdisrepel

S GGGGGG ++++= (2-49)

Where GS is the difference of free energy with respect to the reference state, Gel, Grep,

Gdis and GMm are the free energy contributions in solution due to electrostatic

interactions, repulsion, dispersion and molecular motion respectively. The Gibbs free

energy change of solvation would then correspond to the difference between GS and

the Gibbs free energy of M in the gaseous state, i.e.

0GGG Ssol −=Δ (2-50)

And hence:

Mmcavdisrepelsol GGGGGG Δ++++Δ=Δ (2-51)

The interaction potential can also be decomposed as:

repdisel VVVV ++=int (2-52)

50

The quantum mechanical problem can be expressed in terms of the following

Schrödinger equation:

)()())(( int0 xExVH M Ψ=Ψ+ ρ (2-53)

Vint depends on the charge distribution of M. In standard PCM, the Vdis and Vrep terms

are eliminated, and corresponding GS is obtained with the aid of empirical expressions

which do not exhibit explicit dependence on ρM. The framework of the Born-

Oppenheimer approximation is still used, and thus only the electronic component of Ψ

in equation (2-60) is considered. The linear dependence of Vel is reduced to the ρeM

term, and Vint is partitioned into two components:

intintint )( VVV e

M ′′+′= ρ (2-54)

where *)(int ΨΨ=′′ AV (2-55)

A is an appropriate integral operator. The free energy functional JS subject to

minimization is written as:

⟩Ψ′+′′+Ψ⟨= |)(ˆ

21ˆˆ| intint

0 eM

S VVHJ ρ(2-56)

In the Hartree-Fock case, JS is reduced to the matrix equation

F′C=SCε (2-57)

The Fock matrix here includes terms accounting for the presence of the solvent field,

i.e.

F′= h + ½(j+y) + G(P) + X(P) (2-58)

51

The matrices h and G(P) contain the one and two-electron integrals collected in the

SCF calculations in vacuo, P is the one-electron density matrix, j, y, and X(P) collect

the one- and two-electron integrals added when the interactions with the polarized

dielectric medium are explicitly taken into account within the PCM framework [55].

The final output is an electronic wave function from which properties can be derived,

together with a free energy of solvation.

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51. Ochterski, J. W.; Petersson, G. A.; Montgomery, Jr. J. A. J. Chem. Phys. 1994,

101, 5900.

52. Schreckenbach, G.; Ziegler, T. J. Phys. Chem. A 1997, 101, 3388.

53. Wong, M. W.; Frisch, M. J.; Wiberg, K. B. J. Am. Chem. Soc. 1991, 113, 4776.

54. Wong, M. W.; Wiberg, K. B.; Frisch, M. J. J. Chem. Phys. 1991, 95, 8991.

55. Cancès, E.; Mennucci, B.; Tomasi, J. J. Chem. Phys. 1997, 107, 3032.

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55

Chapter Three

Thermochemistry of Reactive Nitrogen Oxide Species

3.1 Introduction

Reactive nitrogen oxide species (RNOS) are of great importance in many areas of

chemistry and biology. For instance, nitric oxide (NO•) is known to have contrastive

roles. Its important effects range from cytoprotective, to regulatory, and even to

cytotoxic [1– 4]. There is also strong interest in the spectrum of reactions of RNOS.

For instance, the RNOS peroxynitrite (ONOO–) and peroxynitrous acid (ONOOH) are

involved in atmospheric chemistry [5] and biochemical processes [6, 7]. They are

capable of undergoing a vast number of biological reactions, resulting in the

formation of other RNOS, which in turn have direct or indirect effects in many

important biochemical processes.

RNOS, being highly reactive, are transient in nature. Hence it is not easy to ascertain

their chemical properties, such as their standard enthalpies of formation, and

molecular structures, experimentally. To better understand the thermochemistry of

these RNOS, we have performed high-level ab initio molecular orbital calculations to

benchmark the suitability of various theoretical methods, and also to provide more

definitive estimates of the heats of formation of ONOO– and ONOOH, for which

experimental thermochemical data are lacking.

56

To this end, a series of sixteen biologically relevant RNOS, namely NO•, NO+, NO–,

HNO, NO2•, NO2

+, NO2–, HNO2, N2O, NO3

•, ONOO–, ONOOH, NOCl, NO2Cl, N2O3

and N2O4 (see Figure 3-1) were studied using various popular composite methods.

These are the Gaussian-3 methods and their variants (G3(MP2), G3X(MP2), G3 and

G3X) and the complete basis set (CBS) extrapolation methods (CBS-4, CBS-q, CBS-

Q, CBS-QB3 and CBS-APNO), which are well established for giving reliable

thermochemistry. The standard enthalpies of formation (ΔH°f) at 298 K were

calculated. In addition, the 298 K reaction enthalpies of three reactions related to the

decomposition of ONOO– and ONOOH (eqs. (1)–(3)) were also examined using the

CBS-RAD and G3-RAD methods of Radom and co-workers. These RAD methods

were designed to provide better theoretical treatment for free radicals by using

restricted open-shell (ROHF) wave functions in place of unrestricted (UHF) wave

functions, thus minimizing the effects of unwanted spin contamination from higher

electronic states in UHF formalisms.

ONOO– → NO• + O2•– (1)

ONOO– → NO2• + O2

•– (2)

ONOOH → NO2• + OH• (3)

57

N1 O 1

NO•2ΠC∞v

N1 O 1

NO+1Σ+

C∞v

N1 O1

NO–3Σ–

C∞v

N1 O1

H

HNO1A′Cs

N1 O1

O2

NO2•

2A1C2v

N1 O1O2

NO2+

1Σg+

D∞h

N1 O1

O2

NO2–

1A1C2v

HO1

O2 N1

HNO21A′Cs

N1 N2O1

N2O1Σg

+

C∞v

N1 O1

O2

O3

NO3•

2A2′D3h

N1 O1

O2

O3

N1 O1

O2

O3

H

N1 O1

Cl 1

N1 O1

O2

Cl 1

N1 O1

O2

N2

O3

O4

ONOO–1A′Cs

ONOOH1A′Cs

N1 O1

N2O2

O3

NOCl1A′Cs

NO2Cl1A1C2v

N2O31A′Cs

N2O41AgD2h

N1 O 1

NO•2ΠC∞v

N1 O 1

NO+1Σ+

C∞v

N1 O1

NO–3Σ–

C∞v

N1 O1

H

HNO1A′Cs

N1 O1

O2

NO2•

2A1C2v

N1 O1O2

NO2+

1Σg+

D∞h

N1 O1

O2

NO2–

1A1C2v

HO1

O2 N1

HNO21A′Cs

N1 N2O1

N2O1Σg

+

C∞v

N1 O1

O2

O3

NO3•

2A2′D3h

N1 O1

O2

O3

N1 O1

O2

O3

H

N1 O1

Cl 1

N1 O1

O2

Cl 1

N1 O1

O2

N2

O3

O4

ONOO–1A′Cs

ONOOH1A′Cs

N1 O1

N2O2

O3

NOCl1A′Cs

NO2Cl1A1C2v

N2O31A′Cs

N2O41AgD2h

Figure 3.1. Structures, element numbering, state and symmetry symbols of the sixteen RNOS in this study.

3.2 Computational Methods

The standard enthalpies of formation at 298 K (ΔH°f,298) of the sixteen RNOS

mentioned earlier were studied using the following Gaussian-3 and CBS methods:

58

G3(MP2) [8], G3X(MP2) [9], G3 [10], G3X [9], CBS-4 [11, 12], CBS-q [13], CBS-Q

[11], CBS-QB3 [14] and CBS-APNO [15]. These composite methods are designed to

predict reliable thermochemistry, and are benchmarked with known experimental

energies in the G2/97 test set. The overall mean absolute deviations (MAD) of

G3(MP2), G3X(MP2), G3 and G3X theories from experiment for 299 energies in the

G2/97 test set are 1.30 [8], 1.25 [9], 1.02 [9] and 0.96 kcal mol–1 [10] respectively.

For the CBS methods, MAD for the CBS-4 method is 3.26 kcal mol–1 on the G2/97

test set [12]. The benchmarks for the CBS-q, CBS-Q, CBS-QB3 and CBS-APNO

methods were 125 energies of the G2 test set for CBS-q and CBS-Q, the entire G2 test

set for CBS-QB3, and 64 energies involving only first row atoms from the G2 test set

for CBS-APNO. The reported MAD for these methods are 1.71 [11], 1.01 [11], 0.87

[14] and 0.53 kcal mol–1 [15] for the CBS-q, CBS-Q, CBS-QB3 and CBS-APNO

methods respectively. The low values of MAD with experimental values for these

composite methods gave confidence that this was the method of choice to predict

ΔH°f,298 for those species without known experimental values, namely NO2

+, ONOO–,

and ONOOH.

Reaction enthalpies at 298 K of reactions (1)–(3) were also examined by CBS-RAD

[16], G3(MP2)-RAD [17], G3X(MP2)-RAD [17], G3(MP2), G3X(MP2) and

CCSD(T) theories. The standard enthalpies of formation were computed using the

atomization approach of Radom and co-workers [18]. This method involves the use of

experimental heats of formation of gaseous atoms and ions at 0 K, obtained from

NIST-JANAF thermochemical tables [19]. For instance, the ΔH°f,298 of nitric oxide

(NO•) is computed as follows:

59

ΔH°f,0 (NO•) = H°0 (NO•) – H°0 (N(g)) – H°0 (O(g))

+ ΔH°f,0 (N(g)) + ΔH°

f,0 (O(g))

ΔH°f,298 (NO•) = ΔH°

f,0 (NO•) + [H°f,298 (NO•) – H°0 (NO•)]

– [H°f,298 (N(g)) – H°0 (N(g))]

– [H°f,298 (O(g)) – H°0 (O(g))]

The cis conformation of ONOO–, and the (cis,cis)-planar form of ONOOH were

employed in this study, as they were shown to be the preferred conformations based

on theoretical calculations and NMR studies [20]. For ONOO–, the cis conformation

is in accord with the observed geometry from the crystal structure of tetraammonium

peroxynitrite [21].

Full valence complete-active-space SCF (CASSCF) geometry optimizations were also

carried out for ONOO–. URCCSD(T) calculations employed in the G3(MP2)-RAD

and G3X(MP2)-RAD methods, as well as the CASSCF calculations were performed

with the MOLPRO 2002 [22] program. All other calculations were carried out using

the GAUSSIAN 98 [23] suite of programs.

3.3 Results and Discussion

3.3.1 Enthalpies of Formation of RNOS

The calculated enthalpies of formation (ΔH°f,298) of the sixteen RNOS are presented in

Tables 3.1 and 3.2. There is fair to excellent agreement of the calculated values

60

compared with the known experimental values [19, 24], with mean average deviations

(MAD) ranging from 5.1 to 12.9 kJ mol–1. However, we note that the MAD values are

somewhat larger than the corresponding values reported originally for a larger set of

molecules, the G2/97 test set. Among the various composite methods considered here,

the G3X(MP2) method has the smallest MAD of 4.9 kJ mol–1, and smallest maximum

deviation (MD) of 12.0 kJ mol–1. The G3 method performs marginally better, with a

MAD and MD of 5.1 and 12.9 kJ mol–1 respectively. The Gaussian-n methods

generally perform somewhat better than the CBS methods.

Species G3(MP2) G3 G3X(MP2) G3X Expt. NO• 88.0 87.5 85.9 86.8 90.29 ± 0.17 NO+ 999.1 997.6 987.9 985.8 990.19 ± 0.8 NO– 76.7 75.9 76.7 76.4 88.70 ± 1.3 HNO 105.0 104.8 100.4 100.2 99.58 NO2

• 32.6 28.5 29.3 25.3 33.10 ± 0.8 NO2

+ 951.9 958.6 956.9 946.9 NO2

– –200.8 –203.4 –203.1 –205.9 –202.72 ± 5.8 HNO2 –77.7 –80.1 –82.6 –83.9 –76.73 ± 1.34 N2O 86.8 83.8 77.2 73.6 82.05 ± 0.4 NO3

• 94.9 84.0 76.2 65.6 71.13 ± 20.9 ONOO– –103.8 –108.2 –108.0 –112.3 ONOOH –12.6 –17.6 –17.7 –22.7 NOCl 48.5 56.9 47.5 49.0 51.71 ± 0.42 NO2Cl 8.4 9.4 6.6 1.3 12.13 ± 1.7 N2O3 88.7 84.8 75.5 70.8 82.84 ± 0.8 N2O4 14.0 3.6 2.4 –8.9 9.08 ± 1.7 Mean abs. dev.

6.0 5.1 4.9 7.4

Max. dev. 23.8 12.9 12.0 18.0 Table 3.1. Calculated standard enthalpies of formation (ΔH°

f,298, kJ mol–1) of sixteen RNOS using various Gaussian-n methods.

61

Species CBS-4 CBS-q CBS-Q CBS-QB3 CBS-APNO

Expt.

NO• 108.3 79.9 84.6 83.1 88.5 90.29 ± 0.17 NO+ 998.9 987.7 988.4 985.3 984.2 990.19 ± 0.8 NO– 83.7 84.3 73.5 73.7 80.0 88.70 ± 1.3 HNO 107.4 103.0 100.5 97.2 103.6 99.58 NO2

• 24.1 6.6 18.0 19.6 21.2 33.10 ± 0.8 NO2

+ 968.7 939.9 942.5 932.3 938.8 NO2

– –191.5 –205.1 –205.6 –208.4 –196.0 –202.72 ± 5.8 HNO2 –73.3 –87.8 –89.0 –90.3 –83.9 –76.73 ± 1.34 N2O 99.2 82.9 74.3 68.7 71.6 82.05 ± 0.4 NO3

• 78.4 35.5 63.7 51.1 69.6 71.13 ± 20.9 ONOO– –50.4 –87.3 –114.9 –115.0 –37.0 ONOOH 9.1 –17.5 –31.1 –31.8 –25.1 NOCl 52.0 46.6 40.1 43.6 – 51.71 ± 0.42 NO2Cl 3.0 –12.3 –10.3 –8.5 – 12.13 ± 1.7 N2O3 83.9 58.7 68.2 66.0 75.1 82.84 ± 0.8 N2O4 –0.8 –32.0 –13.9 –17.2 –7.3 9.08 ± 1.7 Mean abs. dev.

8.3 14.8 10.8 12.9 7.5

Max. dev.

18.0 41.1 23.0 26.3 16.4

Table 3.2. Calculated standard enthalpies of formation (ΔH°

f,298, kJ mol–1) of sixteen RNOS using various CBS methods. For the three RNOS systems without known experimental heats of formation, namely

NO2+, ONOO–, and ONOOH, the agreement between G3 and CBS-4, CBS-q and

CBS-APNO values is rather poor. For instance, the predicted heats of formation of

ONOO– by the G3 and CBS-APNO methods differ greatly, by 71 kJ mol–1! Other

methods predict values (–104 to –115 kJ mol–1) close to that of the G3 result (–108 kJ

mol–1). A careful inspection of the geometries employed by the G3 and CBS-APNO

methods reveals significant differences in the central N—O bond length in ONOO–,

which may be a possible cause in the error in the calculated heat of formation of

ONOO–.

Selected structural parameters of the sixteen RNOS obtained by the various composite

methods are tabulated in Table 3.3. Included are CCSD(T)/cc-pVTZ and experimental

geometrical parameters [19] for comparison. It is observed that the experimental N—

62

O and N—N bond lengths of RNOS are readily produced by B3LYP and MP2

methods, with B3LYP giving better agreement than MP2. This is in accord with

previous findings based on a comparative study of the geometries of nitrogen oxides

by Bohr and co-workers [5].

Species Parameter HF/ a CBS-4 CBS-q

B3LYP/ b CBS-QB3

B3LYP/ c G3X

MP2(fu)/ d G3

MP2(fc)/ e CBS-Q

QCISD/ f CBS-APNO

CCSD(T)/ g

Expt.

NO• r(N1—O1) 1.201 1.148 1.151 1.143 1.137 1.159 1.153 1.151 NO+ r(N1—O1) 1.047 1.060 1.064 1.103 1.090 1.067 1.069 NO– r(N1—O1) 1.312 1.273 1.272 1.280 1.270 1.275 1.275 HNO r(N1—O1) 1.216 1.200 1.201 1.236 1.223 1.207 1.214 1.211 NO2

• r(N1—O1) = r(N1—O2)

1.204 1.195 1.196 1.165 1.205 1.194 1.199 1.193

NO2+ r(N1—O1)

= r(N1—O2) 1.137 1.137 1.121 1.091 1.137 1.137 1.126

NO2– r(N1—O1)

= r(N1—O2) 1.287 1.265 1.263 1.281 1.271 1.261 1.266

HNO2 r(N1—O1) r(N1—O2)

1.190 1.420

1.181 1.388

1.184 1.382

1.209 1.385

1.198 1.376

1.184 1.379

1.187 1.392

1.169 1.442

N2O r(N1—N2) r(N1—O1)

1.095 1.260

1.126 1.184

1.128 1.186

1.092 1.179

1.169 1.182

1.129 1.189

1.133 1.190

1.128 1.184

NO3• r(N1—O1)

=r(N1—O2) =r(N1—O3)

1.245 1.233 1.233 1.255 1.246 1.230 1.237

ONOO– r(N1—O1) r(N1—O2) r(O2—O3)

1.227 1.343 1.533

1.218 1.370 1.382

1.221 1.365 1.370

1.188 1.270 1.428

1.239 1.340 1.358

1.214 1.721 1.292

1.222 1.361 1.389

1.16 1.35 1.41

ONOOH r(N1—O1) r(N1—O2) r(O2—O3)

1.184 1.447 1.463

1.184 1.390 1.429

1.189 1.379 1.420

1.211 1.385 1.446

1.197 1.384 1.433

1.184 1.390 1.419

1.199 1.402 1.438

NOCl r(N1—O1) r(N1—Cl1)

1.148 1.907

1.126 2.044

1.134 1.990

1.156 2.031

1.141 2.057

– –

1.141 1.982

1.139 1.975

NO2Cl r(N1—O1) =r(N1—O2) r(N1—Cl1)

1.221

1.762

1.184

1.939

1.188

1.903

1.206

1.906

1.192

1.943

–

–

1.205

1.825

1.202

1.840 N2O3 r(N1—N2)

r(N1—O1) r(N2—O2) r(N2—O3)

1.671 1.153 1.217 1.220

1.879 1.135 1.200 1.201

1.856 1.140 1.201 1.202

1.935 1.173 1.220 1.222

1.929 1.161 1.209 1.211

1.751 1.148 1.200 1.201

1.836 1.147 1.205 1.205

1.864 1.142 1.217 1.202

N2O4 r(N1—N2) r(N1—O1) =r(N1—O2) =r(N2—O3) =r(N2—O4)

1.637 1.204

1.798 1.188

1.780 1.190

1.820 1.212

1.828 1.202

1.688 1.190

1.744 1.194

1.782 1.190

Table 3.3. Selected calculated bond lengths of RNOS in Ångstroms. Basis sets: (a) 3-21G(d), (b) 6-311G(2d,d,p), (c) 6-31G(2df,p), (d) 6-31G(d), (e) 6-31G†, (f) 6-311G(d,p), (g) cc-pVTZ.

The QCISD bond lengths are in good agreement with the CCSD(T)/cc-pVTZ results

and experimental values, with a few notable exceptions. The N—O bond length in

ONOO– and the N—N bond length in N2O3 and N2O4. For ONOO–, QCISD theory

63

predicts a rather unexpectedly long bond length of 1.72–1.73 Å. MP2, B3LYP and

CCSD(T) methods yield a typical N—O bond length of 1.340–1.375 Å. For the N—N

bonds of N2O3 and N2O4, QCISD bond distances are too short by 0.1 Å compared to

the experimental values.

To further shed light on this intriguing discrepancy in molecular geometries, the

structure of ONOO– was optimized using various single-reference and multi-reference

based methods with the 6-31G(d) basis set. The default convergence criteria for

geometry optimization in the GAUSSIAN98 and MOLPRO2002 software packages

were employed. For the GAUSSIAN98 program, the values of maximum force, root-

mean square (RMS) force, maximum step, and RMS step have to fall within threshold

values of 0.000450, 0.000300, 0.001800 and 0.001200 a.u., respectively, to achieve

convergence. For the MOLPRO2002 program, the values for maximum force and

maximum step have to be less than 0.000300 a.u. to achieve convergence. The results

are summarized in Table 3.4. Most methods predict a ‘normal’ N—O bond distance

of 1.30–1.40 Å except for QCISD. The closely related CCSD method yields an

unexpectedly different geometry when compared to QCISD. Somewhat longer N—O

bond lengths are also obtained by pure functionals SVWN and BLYP in density

functional theory. It is interesting to note that inclusion of triple excitations to QCISD

(i.e. QCISD(T)) gives good geometry, compared to the higher-level results. A similar

trend is obtained for the larger cc-pVTZ basis set (see Table 3.4), indicating that the

QCISD problem for ONOO– is independent of the basis set used. The best calculated

N—O bond lengths at the CCSD(T)/cc-pVTZ and full-valence-CASSCF/cc-pVTZ

levels are 1.361 and 1.358 Å respectively, in excellent accord with the experimental

value of 1.35 Å, derived from the X-ray structure of tetraammonium peroxynitrite

64

[21]. To the best of our knowledge, we are unaware of any example where QCISD

and CCSD theories yield significantly different molecular geometries. It is worth

noting that there are problematic closed-shell systems such as O22+ [25] which exhibit

slow convergence of the Møller-Plesset perturbation theory. However, the MPn series

appears to converge normally for ONOO–.

Level r(N1—O1)

r(N1—O2) r(O2—O3) α(O1N1O2) α(N1O2O3)

6-31G(d) basis set

HF 1.188 1.269 1.428 118.5 118.3 SVWN 1.222 1.428 1.330 112.6 115.9 BLYP 1.234 1.541 1.367 112.3 115.6 B3LYP 1.227 1.374 1.377 115.3 117.9 MP2 1.251 1.348 1.370 114.9 117.6 MP3 1.220 1.314 1.404 116.5 117.3 MP4 1.261 1.391 1.361 113.9 117.5 BD 1.226 1.330 1.414 116.5 117.2 CISD 1.211 1.301 1.407 117.3 117.6 QCISD 1.232 1.729 1.306 104.1 108.4 QCISD(T) 1.236 1.397 1.402 114.8 116.8 CCSD 1.227 1.340 1.411 116.2 117.1 CCSD(T) 1.238 1.383 1.397 114.9 116.9 CAS(24,16) 1.229 1.367 1.437 116.3 115.6 cc-pVTZ basis set HF 1.178 1.264 1.415 118.8 118.5 B3LYP 1.215 1.361 1.385 116.2 118.3 MP2 1.234 1.336 1.361 115.4 117.9 QCISD 1.210 1.709 1.296 105.2 109.5 QCISD(T) 1.209 1.383 1.371 115.2 116.5 CCSD 1.210 1.322 1.400 116.9 117.5 CCSD(T) 1.222 1.361 1.389 115.8 117.4 CAS(24,16) 1.219 1.358 1.432 116.8 117.2 Table 3.4. Calculated structural parameters of planar cis-peroxynitrite, ONOO–, at various levels of theory.

From the above discussion, it is clear that the significant difference in CBS-APNO

heat of formation of ONOO– compared with experiment and with other composite

methods, is due to inadequate QCISD geometry. Hence, the best predicted heats of

formation of ONOO–, NO2+, and ONOOH are the G3 values (Table 3.2).

65

The exothermicity of formation for ONOO– (–108 kJ mol–1) and ONOOH (–18 kJ

mol–1) at 298 K, indicates the stabilities of these compounds with respect to their

constituent elements in their standard states. It must be noted, however, that these

species are considerably less stable than the commonly found nitrate ion (NO3–) and

nitric acid (HNO3). The G3 relative energies of NO3– and HNO3 are –219.3 and –

122.3 kJ mol–1 respectively. Both species are predicted to have a sizable activation

barrier at the G3 level for the exothermic unimolecular rearrangement, 242.6 and

208.9 kJ mol–1 for ONOO– and ONOOH respectively. Although these unimolecular

rearrangements are unlikely due to their prohibitively high barriers, these processes

are catalyzed by proteins such as human hemoglobin, within a biological system [26].

The predicted proton affinity for ONOO– at 298 K is 1446.5 kJ mol–1 at the G3 level,

which is consistent with the value reported by Klasinc and co-workers [27].

Although the G3 method shows promising results in predicting heats of formation, it

should be noted that significant deviations occur for open shell species NO– (3Σ–) and

NO3• (2A2′) (–12.8 and +12.9 kJ mol–1 respectively). The use of UHF wave functions

in G3 theory on these species may be the possible source of error in the calculated

enthalpies. For NO3•, the discrepancy between calculated and experimental energies

can also be explained in terms of the pseudo Jahn-Teller effect. There are strong

interactions between its ground state (2A2′) and the second excited state (2E′), as

explored by Janoschek and co-workers [28].

Interestingly, the G3X heats of formation are found to be less accurate than the

corresponding G3 values, with G3X having a MAD of 7.4 kJ mol–1 compared to 5.1

kJ mol–1 for G3. This finding is somewhat surprising, given the fact that B3LYP

66

geometries are more reliable than MP2 geometries for this set of RNOS. Perhaps the

high level correction (HLC) term for G3X theory may not be optimum for the RNOS

studied here.

It is important to note that there are two ways of computing atomization energy for

the ionic species. For instance, the two possible reaction channels of NO – correspond

to N + O– and N– + O. For all the anions, the two approaches give rise to two sets of

significantly different heats of formation. By comparing with well established heats of

formation of nitrogen-containing compounds, it is found that the heat of formation of

the nitrogen anion (N–) cited in the JANAF tables is unreliable. The reported value of

473.5 kJ mol–1 is likely to be underestimated by about 15 kJ mol–1.

3.3.2 Enthalpies of reactions involving ONOO– and ONOOH decomposition

The calculated reaction enthalpies for the decomposition reactions of ONOO– and

ONOOH (reactions (1) – (3)) are presented in Table 3.5. The results, using various

levels of theory, are generally in close agreement with each other, typically within

~10 kJ mol–1.

Notable exceptions are the enthalpies of reactions (1) and (2) calculated using the

CBS-RAD and CBS-APNO methods. The computed enthalpies using these two

methods are considerably different from other calculated values, differing by up to 75

kJ mol–1. Both CBS-RAD and CBS-APNO methods are based on QCISD geometry,

in conjunction with the 6-31G(d) and 6-311G(d,p) basis sets respectively. As shown

in the previous section, the geometry of ONOO– is poorly described by QCISD theory.

67

Hence, the large enthalpy deviation using these methods can be attributed to the use

of inadequate QCISD geometries. This is readily confirmed by performing modified

CBS-RAD and CBS-APNO calculations, where CCSD geometries are employed

instead of QCISD geometries. The revised reaction enthalpies (Table 3.5) derived

from the CCSD geometry are now in good agreement with other computed values. A

comparison of these results for reaction enthalpies with those reported by Olson et al

[29] and Pfeiffer et al [30] shows consistency with calculated values for CBS-QB3

and G3(MP2)//B3 methods. Interestingly, the geometry of the protonated form of

ONOO− (i.e. ONOOH) is well described by QCISD theory (Table 3.3). As a

consequence, the computed reaction enthalpies by various methods for reaction (3)

agree well with each other. The DFT method B3LYP/aug-cc-pVTZ//B3LYP/cc-

pVDZ predicts a fairly large deviation of 20% from the G3 enthalpy in all cases

(Table 3.5).

Method Reaction (1) enthalpy Reaction (2) enthalpy Reaction (3) enthalpy G3(MP2) 147.0 242.2 76.2 G3X(MP2) 148.7 243.2 76.8 CBS-Q 141.9 238.8 82.8 G3(MP2)-RAD 135.6 244.4 79.1 G3X(MP2)-RAD 136.1 243.4 77.9 CBS-RAD 96.8 (136.4) 199.1 (239.8) 84.7 CBS-APNO 75.9 (142.7) 164.0 (231.7) 80.5 G3 146.2 236.4 77.5 G3X 149.1 243.4 78.3 B3LYP/aug-cc-pVTZ/ /B3LYP/cc-pVDZ

124.1 209.1 62.6

CCSD(T)/aug-cc-pVTZ// CCSD/cc-pVTZ

129.8 225.9 82.7

CCSD(T)/cc-pVQZ// CCSD/cc-pVTZ

138.3 258.8 83.9

Table 3.5. Calculated reaction enthalpies, in kJ mol–1, for reactions (1) to (3) at 298 K. Values in parentheses indicate reaction enthalpies calculated based on CCSD instead of QCISD geometries for CBS-RAD and CBS-APNO methods.

Reactions involving the decomposition of ONOO− and ONOOH via various pathways

have attracted strong interest due to their biological importance, and their

68

thermochemistry is a subject of intense debate among various research groups. For

instance, the harmful effects of ONOOH has been attributed to the apparent release of

NO2• and OH• radicals upon unimolecular decomposition and the direct reaction of

these radicals on other substrates [31]. However, this release of OH• radicals as a

result of O—O bond homolysis has recently been questioned [32]. It is worth noting

that molecular orbital calculations by Rauk and co-workers [2] have shown that

ONOOH can abstract hydrogen atoms from aliphatic compounds directly. Calculated

results here clearly indicate that decomposition of ONOO− into NO2• and O•− is

unlikely to happen at physiological temperatures, given the high enthalpy of reaction.

The activation barrier for such a process is expected to exceed 240 kJ mol–1. The

decomposition of ONOO− into NO− and singlet O2 has been ruled out [33]. CASSCF

calculations carried out on this reaction confirm that this fragmentation has a rather

high enthalpy, thus this decomposition pathway was not considered in detail. Whether

these decomposition reactions of ONOOH and ONOO− actually occur in a

straightforward manner is yet to be proven. Although potential energy surfaces for

ONOOH has been calculated and extensively mapped out [34–36], information

related to the transition states corresponding to O—O bond breaking in ONOO− and

ONOOH is lacking. Calculations of these enthalpies at high levels of theory can

provide at least some insights into the feasibility of these fragmentation processes.

3.4 Conclusions

The heats of formation, ΔH°f,298, for a series of sixteen biologically relevant RNOS

were calculated using various composite methods. The G3 and G3X(MP2) methods

give the best performance with a mean absolute deviation from experiment of 5.1 and

69

4.9 kJ mol–1 respectively. However, the less impressive performance of the G3X

method compared with the G3X(MP2) method suggests that the apparent good

performance of G3X(MP2) could be fortuitous. The best estimates for the heats of

formation of ONOO– and ONOOH are –108 and –18 kJ mol–1, respectively. The

reaction enthalpies for these reactions involving ONOO– and ONOOH decomposition

were also examined at various levels of theory. The geometry of ONOO– is found to

be poorly described by the QCISD method. As a result, CBS-APNO and CBS-RAD

give rather unsatisfactory thermochemistry related to peroxynitrite.

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21. Woerle, M.; Latel, P.; Kissner, R.; Nesper, R.; Koppenol, W. H. Chem. Res.

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Montgomery Jr., J. A.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J.

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73

Chapter Four

Decomposition of Nitrosoperoxycarbonate Anion: Gas

Phase and Solvent Modeled Studies.

4.1 Introduction

Peroxynitrite (ONOO–), formed from the interaction of nitric oxide (NO•) and

superoxide (O2•–) [1], is a potent two electron oxidant responsible for direct and

indirect effects of reactive nitrogen oxide species. ONOO– reacts with several

substrates such as dimethylsulfoxide and deoxyribose with zero order kinetics [2], and

with substrates such as methionine [3] and ascorbate [4] with small second-order rate

constants.

Biologically, ubiquitous CO2 can accelerate oxidation of the above substrates by

ONOO– [5, 6]. The direct reaction of ONOO– with CO2 produces an intermediate –

nitrosoperoxycarbonate, ONOOCO2– [7] – which undergoes homolytic scission of the

O—O bond to yield NO2• and CO3

•– radicals [8-11].

ONOO– + CO2 → ONO−OCO2– → NO2

• + CO3•–

Previous theoretical calculations have shown that the O—O bond in ONOOCO2– is

exceptionally weak [12] and this is substantiated by the derivation of the very high

overall rate constant that governs this process (k = 1.9 × 109 s–1) [11]. These radicals

74

can diffuse out and initiate other free radical reactions, playing a role in various

pathophysiological processes [13]. The CO3•– radical is relatively less reactive than

•OH, but being more stable, it can diffuse further from the site origin and propagate

oxidative damage. NO2• is able to initiate free radical reactions, for instance, hydrogen

abstraction from fatty acids, tyrosine residues or tocopherol [6, 14].

Currently, due to the transient nature of the reactive species discussed, it is difficult to

study the factors that govern the formation of the free radicals NO2• and CO3

•– that are

reported to cause so much harm. To this end, a computational study of the reaction

shown above, in the gas phase and in solvent, is attempted and presented.

Modeling of the solvent environment can be carried out in using implicit solvent

models (ISM), explicit solvent models (ESM) and hybrid solvent models (HSM). ISM,

or continuum models of solvent, focuses on dielectric models of electrostatic effects.

The solvent is modeled as a continuous, homogeneous and isotropic medium and is

characterized by a scalar value – the dielectric constant ε [15, 16]. ESM involves the

addition of discrete solvent molecules around the solute to constitute the first

solvation shell, and subsequent calculations for this system carried out in the gas

phase. HSM used in this work involves the addition of solvent molecules to constitute

the first solvation shell of the solute as in ESM, followed by the treatment of this

solvent-bound complex using ISM, similar to the approach adopted by Kiruba et al in

the study of tautomeric equilibria of 3-hydroxypyridine derivatives [17]. Calculations

in the gas phase and in solvent using various approaches would provide valuable

information regarding solvent effect on the reaction profile.

75

The detection and monitoring of reactive species related to ONOO– reaction with CO2

is commonly carried out using UV-visible spectroscopy. For instance, it was proposed

by Koppenol et al that the adduct formed by the reaction of ONOO– with CO2 absorbs

at 640 nm [9]. In an attempt to provide a better understanding of this observation, the

absorption spectra of the reactive species ONOO–, ONOOCO2– and CO3

•– were also

studied using various computational methods.


The hybrid B3LYP functional [18], a three-parameter combination of Becke’s 1988

exchange functional [19], the Lee-Yang-Parr functional for non-local correlation [20]

and the Vosko-Wilk-Nusair functional III for local correlation [21] was employed for

DFT calculations. Full geometry optimizations of all species studied were carried out

at the B3LYP/6-31+G(d) level. Frequency calculations were carried out for the

stationary points obtained, to determine whether they were indeed local minima with

all real harmonic frequencies, or first order saddle points, with one imaginary

frequency. Additional higher level energy calculations were carried out at the

B3LYP/6-311+G(2df,p) level, based on B3LYP/6-31+G(d) geometries. Selected gas-

phase benchmark calculations were performed at the G3 [22], CBS-QB3 [23] and

CCSD(T)/aug-cc-pVTZ//CCSD/aug-cc-pVDZ levels of theory.

The continuum solvation models employed were the Onsager self consistent reaction

field model (SCRF=Dipole) [24-26] and the integral equation formalism for the

polarizable continuum model or IEFPCM [27-29]. The dielectric constant ε, used in

76

these models is 78.39, corresponding to that for water. For the IEFPCM methods, the

united atom topological model (UA0) method was used to generate atomic radii for

construction of the molecular cavity. Calculations using explicit and hybrid solvation

models used four discrete water molecules to form the first solvation shell.

For calculation of the gas-phase transition energies (λexcitation) of ONOO–, ONOOCO2–

and CO3•–, Zerner’s intermediate neglect of differential overlap (ZINDO/S) [30],

single-excitation configuration interaction (CIS) [31] and time-dependent density

functional theory (TD-DFT) [32, 33] methods were employed, based on gas-phase

geometries obtained at the B3LYP/6-31+G(d) level.

Closed shell species were treated using the restricted Hartree-Fock (RHF) formalism

and open shell species using the unrestricted Hartree-Fock (UHF) formalism, except

for CCSD calculations, in which the ROHF reference wave functions are used for

open-shell systems instead. Singlet diradical species were treated using UHF, with the

HOMO and LUMO mixed so as to destroy α-β and spatial symmetries. Spin

correction for open shell species were treated using the approximate spin correction

procedure proposed by Yamaguchi et al [34, 35]. Unless otherwise noted, energies

reported are calculated at the B3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d) + ZPE +

spin correction level.

All calculations were carried out using the GAUSSIAN 03 [36] and MOLPRO 2002

[37] suite of programs.

77


4.3.1 Gas-phase reaction profile of ONOO– reaction with CO2

The following detailed reaction scheme and labeling of the various species involved is

proposed. Cis- and trans-peroxynitrite (1 and 2, respectively) undergo reaction with

CO2 (3) to form cis- and trans-nitrosoperoxycarbonate (4 and 5). These adducts

undergo O—O bond homolysis, via transition states TS1 and TS2, respectively, to

form nitrogen dioxide (6) and carbonate radical anion (7) (See Scheme 4.1).

Scheme 4.1. Proposed reaction scheme and labeling of various species involved in the reaction ONOO– + CO2 → ONO−OCO2

– → NO2• + CO3

•–.

Structures of the local minima 1 to 7 and transition states TS1 and TS2, calculated at

the B3LYP/6-31+G(d) level, are presented in Figure 4.1. Relative gas-phase energies

78

of these species (including ZPE and spin correction) are presented in Table 4.1, with

G3, CBS-QB3 and CCSD(T)/aug-cc-pVTZ//CCSD/aug-cc-pVDZ energies included

for comparison. ONOO– and ONOOCO2– in the cis conformation is preferred, and

they are 12.3 and 5.1 kJ mol–1 lower in energy than their trans counterparts,

respectively. Preference of the ONOO– molecule to adopt the cis conformation can be

explained by a favorable π*–π* interaction formed from the overlap of O1=N1 and

O2—O3 π* orbitals, as seen in the highest occupied molecular orbital (HOMO) of

cis-ONOO– (See Figure 4.2). The CO2 molecule forms an adduct with ONOO– above

the O–N–O–O plane. The gas-phase dihedral angles N1–O2–O3–C1 for 4 and 5 are –

89.7° and –84.4°, respectively.

79

N1 O2

O1O3

O3

O1N1

O2

C1O1 O2

O2

O3

N1

O4

C1

O1

O5

O3O2

O4C1

N1

O5

O1

O1

N1

O2

O1

O3

C1

O2

O1

N1

O3O4

O2C1

O5

O1

N1

O2

O3

C1

O5

O4

1.222

(1.214)

[1.201]

1.367

(1.364)

[1.391]1.390

(1.402)

[1.427] 1.237

(1.221)

[1.196]

1.339

(1.350)

[1.412]1.410

(1.421)

[1.446]

1.169

(1.169)

[1.169]

1.192

(1.181)

[1.185]

1.381

(1.412)

[1.397]

1.440

(1.442)

[1.449]

1.571

(1.488)

[1.503]

1.223

(1.231)

[1.230]

1.227

(1.239)

[1.242]

1.439

(1.434)

[1.439]

1.383

(1.428)

[1.423]

1.200

(1.183)

[1.185]

1.559

(1.485)

[1.495]

1.225

(1.232)

[1.230]1.228

(1.239)

[1.245]

1.202

(1.201)

[1.202]

1.278

(1.275)

[1.278]

1.217

(1.203)

[1.188]

1.251

(1.240)

[1.257]

1.903

(1.858)

[1.789]

1.363

(1.371)

[1.378]

1.230

(1.239)

[1.247]

1.274

(1.266)

[1.278]

1.235

(1.227)

[1.218]

1.279

(1.278)

[1.295]

2.012

(2.000)

[1.964]

1.319

(1.323)

[1.336]

1.230

(1.233)

[1.236]

1.294

(1.286)

[1.284]

1.169

(1.169)

[1.169]

1.202

(1.201)

[1.202]

1.278

(1.275)

[1.278]

1.278

(1.275)

[1.278]

1 23

4 5

6 7

TS1 TS2

Figure 4.1. Structures of local minima 1-7, TS1 and TS2, calculated at the B3LYP/6-31+G(d) level. Distances are given in Ångstroms. Bond lengths in regular notation, parentheses, and square brackets correspond to gas-phase, IEFPCM solvation modeled, and Onsager’s SCRF solvation modeled values, respectively.

80

Figure 4.2. Highest occupied molecular orbital (HOMO) of cis-ONOO– calculated at the B3LYP/6-31+G(d) level, showing extent of π*–π* interaction.

The reactions 1 + 3 → 4 and 2 + 3 → 5 are exothermic by 61.7 and 56.6 kJ mol–1,

respectively, at the B3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d) level, an

underestimation of the binding energy of CO2 to ONOO–, when compared to

benchmark results. Other DFT methods reproduce similar trends in the

underestimation of the binding energy, for instance, the density functionals TPSS and

VSXC with the same basis sets employed yielded binding energies of 62.5 and 59.0

kJ mol–1, respectively. The calculated overall enthalpy at 0 K for the reactions 1 + 3

→ 6 + 7 and 2 + 3 → 6 + 7 generally do not agree well, even among benchmark

calculations. Spin contamination in the CO3•− radical could be the cause of such

discrepancy in the G3 method, since the energies presented are not spin-corrected.

81

The overall enthalpy is expected to be close to the CBS-QB3 and CCSD(T) value, i.e.

−47.3 and −35.4 kJ mol–1, respectively, of which B3LYP does not reproduce too well.

Experimentally, ONOOCO2– is not accumulated during the reaction of ONOO– with

CO2, and cannot be detected spectrophotochemically [1, 7, 38, 39]. Also, attempts to

chemically trap or scavenge ONOOCO2– have not been successful [40]. Calculated

gas-phase O—O homolytic dissociation barriers for cis- and trans-

nitrosoperoxycarbonate are low, at 17.7 and 35.1 kJ mol–1, respectively. The low

barrier for 4 → TS1 shows the relative ease of O—O homolytic bond scission. As the

O—O homolytic dissociation of ONOOCO2– is expected to involve a curve crossing

from the singlet potential energy hypersurface (PES) to the triplet PES of the two

radicals NO2• and CO3

•–, the transition states TS1 and TS2 are expected to be singlet

and diradicaloid in nature, a result of contamination from the triplet state wave

function. The diradical wave function is also expected to be more stable than the

singlet and triplet wave functions. Expectation values of the spin-squared operator

⟩⟨2S for such singlet diradicals are between that of a singlet (i.e. ⟩⟨

2S = 0) and a

triplet (i.e. ⟩⟨2S = 2). A ⟩⟨

2S value of unity indicates 50:50 singlet and triplet, and

the singlet and triplet states are nearly degenerate. The ⟩⟨2S values for TS1 and TS2

are 0.27 and 0.54, and their corresponding spin-corrections to the energy are 11.5 and

16.3 kJ mol–1, respectively.

In our various attempts to find the O2NOCO2– species described by many authors as a

rearrangement product of ONOOCO2–, no local minima were found that corresponded

to the nitrogen in NO2• forming a proper covalent bond with oxygen in CO3

•−. This is

substantiated by previous theoretical work that concluded that O2NOCO2– is not a

82

local energy minimum in the PES [41]. The overall process is expected to proceed in

the following scheme: 1 + 3 → 4 → TS1 → 6 + 7.

Species B3LYPa G3 CBS-QB3 CCSD(T)b

1 + 3 0.0 0.0 0.0 0.0 2 + 3 12.3 13.3 14.0 16.7 4 −61.7 −73.7 −78.2 −77.7 5 −56.6 −66.9 −73.0 −74.9

6 + 7 −68.8 −19.5 −47.3 −35.4

TS1 −44.0 (17.1) TS2 −21.5 (35.1)

Table 4.1. Zero point corrected relative energies of 1-7, TS1 and TS2, in kJ mol–1. Values in parentheses indicate intrinsic barriers for 4 → TS1 and 5 → TS2. aB3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d) + ZPE + spin correction. bCCSD(T)/aug-cc-pVTZ//CCSD/aug-cc-pVDZ + ZPE.

4.3.2 Implicit Solvation Model (ISM) Results

The gas-phase structures 1-7, TS1 and TS2 were re-optimized using the Onsager

SCRF model and Tomasi’s IEFPCM solvation models for implicit solvation, to

examine the bulk solvent effects. Calculated relative energies of the species involved

are shown in Table 4.2 and geometric parameters of the local minima and transition

states using ISM are shown in Figure 4.1.

Species ISM (Onsager) ISM (IEFPCM)

1 + 3 0.0 0.0 2 + 3 6.6 16.4 4 −52.9 −17.2 5 −49.6 −9.5

6 + 7 −12.5 −54.4

TS1 −51.5 (1.4) 0.7 (17.9) TS2 2.2 (51.8) 42.3 (51.8)

Table 4.2. B3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d) + ZPE and spin correction relative energies of 1-7, TS1 and TS2, in kJ mol–1, using implicit solvation models. Values in parentheses indicate intrinsic barriers for 4 → TS1 and 5 → TS2.

83

Compared to the gas phase and ISM (IEFPCM) values, the cis- and trans-ONOO–

energy difference for ISM (Onsager) appears to be understated, at 6.6 kJ mol–1. The

binding energy of cis- and trans-ONOO– to CO2 differs greatly across both methods

as well, from −52.9 (1 + 3 → 4) and −56.2 kJ mol–1 (2 + 3 → 5) in ISM (Onsager) to

−17.2 (1 + 3 → 4) and −25.9 kJ mol–1 (2 + 3 → 5) in ISM (IEFPCM). The overall

energy change from 1 + 3 → 6 + 7 and 2 + 3 → 6 + 7 seems to differ greatly across

both methods as well. However, both ISMs employed predict low reaction barriers for

4 → TS1, at 1.4 kJ mol–1 for ISM (Onsager) and 17.9 kJ mol–1 for ISM (IEFPCM),

and a higher barrier for 5 → TS2, at 51.8 kJ mol–1 for both methods.

As seen from Figure 4.1, the incorporation of ISM did not drastically change the

structures of 1-7, TS1 and TS2. Particularly, in the small molecules 3, 6 and 7,

geometric parameters remain somewhat unchanged. The C=O bond lengths in CO2

show no change at 1.169 Å in the gas phase, ISM (IEFPCM) and ISM (Onsager)

models. For NO2• and CO3

•–, N—O and C—O bond lengths are only slightly shorter

at 1.201 and 1.275 Å, respectively, compared to 1.202 and 1.278 Å for gas-phase and

ISM (Onsager) modeled geometries. In cis-ONOO–, the N1=O1 bond is shortened

slightly, from 1.222 Å in the gas phase, to 1.214 Å for ISM (IEFPCM) and 1.201 Å

for ISM (Onsager). The N1—O2 bond length is lengthened, from 1.367 Å in the gas

phase to 1.364 Å for ISM (IEFPCM) and 1.391 Å for ISM (Onsager). The O2—O3

bond is also lengthened, from 1.390 Å in the gas phase to 1.402 Å for ISM (IEFPCM)

and 1.427 Å for ISM (Onsager). Similar trends are seen for trans-ONOO–; the

terminal N=O bond is shortened, the central N—O and the terminal O—O bonds are

lengthened in solution. Again, there are no significant changes in bond lengths for cis-

84

and trans-ONOOCO2–. However it should be noted that C1—O3 bond lengths show

noticeable shortening from gas phase to solution. In cis-ONOOCO2–, the C1—O3

bond lengths shortened from 1.571 Å to 1.488 and 1.503 Å for ISM (IEFPCM) and

ISM (Onsager), respectively, and in trans-ONOOCO2–, they shortened from 1.559 Å

to 1.485 and 1.495 Å.

The O2—O3 bond lengths in cis- and trans-ONOOCO2– are similar in length, at about

1.44 Å in both gas phase and in solvent. It is also observed that both ISMs predict a

lowering of the O2—O3 distance in TS1 and TS2, more so for TS1, where the O2—

O3 distance is shortened from 1.903 Å in the gas phase to 1.789 Å for ISM (Onsager).

However, the O—O bond dissociation energies (BDE) in ONO–OCO2– differ greatly

in gas-phase and solvent modeled calculations. O2—O3 bond lengths are seen here as

not being reflective of the actual BDE.

The lack of consistency in the energy results yielded from both ISMs employed

suggests that the use of continuum methods alone is insufficient to adequately

describe the solvent effects on the energetics of this system. There is a need, therefore,

to use methods that can account for more than just the nonspecific solute-solvent

interactions as considered in ISMs. A closer look at solvent effects of the first

solvation shell is described in the following sections.

4.3.3 Explicit Solvation Model (ESM) Results

Discrete solvation of the solutes (ESM) was carried out as described in the following

reaction scheme:

85

(H2O)3·ONOO– + (H2O)·CO2 → (H2O)4·ONOOCO2– → (H2O)·NO2

• + (H2O)3·CO3•–

Four of the solvent molecules are treated as discrete entities, used to describe the

microscopic properties of the solvent at the first solvation shell. The structures of

these clusters (local minima and transition states) 1a-7a, TS1a and TS2a, calculated

using ESM are shown in Figures 4.3 and 4.4. Relative energies of these solutes

B3LYP/6-311+G(2df,p)//B3LYP/6-31G(d) + ZPE + Spin correction level are shown

in Table 4.3.

Species ESM (4H2O) 1a + 3a 0.0 2a + 3a 14.9 4a −42.2 5a −41.4

6a + 7a −32.7

TS1a −22.9 (19.3) TS2a 9.3 (50.7)

Table 4.3. B3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d) + ZPE + Spin correction relative energies of 1a-7a, TS1a and TS2a, in kJ mol–1, using the explicit solvation model (ESM). Values in parentheses indicate intrinsic barriers for 4a → TS1a and 5a → TS2a.

86

O6

O4

O3

O2

O1

N1

O5

O2

O3

C1O1

N1

O2O1

O3

O4

O3O2

C1

O5O1

O6

O3

N1

O2

O4

O1

O5

O6

2.771

(2.771)

2.850

(2.649)

1a 2a

3a 6a7a

1.212

(1.203)

1.360

(1.376)

1.405

(1.413)

1.214

(1.207)

1.359

(1.374)

1.426

(1.429)

1.168

(1.170)

1.171

(1.170)

1.201

(1.203)

1.201

(1.203)

1.290

(1.293)

1.280

(1.275)

1.262

(1.265)

Figure 4.3. Structures of local minima 1a, 2a, 3a, 6a and 7a, calculated at the B3LYP/6-31+G(d) level. Distances are given in Ångstroms. Bond lengths in regular notation and parentheses correspond to gas-phase (ESM), and Onsager’s SCRF solvation modeled (HSM) values, respectively. Dotted lines show hydrogen bond interactions (1a, 2a, 7a) and dipole-dipole interactions (3a, 6a) between solute and solvent, which have distances less than the sum of van der Waals’ radii for the two atoms concerned.

87

O8

O9

O2O3

O4

O7

C1

N1

O5

O1

O6

O9

O8

O2O3

O7

O1 N1

C1

O5

O4

O6

O2

N1

O7

O3

O9

O1

C1

O5

O4

O8

O6

O9

O7

O1 N1

O2

O5

O6

O3

C1

O4

O8

1.869

(1.849)

1.964

(1.953)

4a 5a

TS1a TS2a

1.181

(1.174)

1.416

(1.441)1.437

(1.436)

1.514

(1.479)

1.230

(1.234)

1.231

(1.240)

1.188

(1.178)

1.414

(1.448)

1.434

(1.431)

1.473

(1.453)

1.237

(1.236)

1.238

(1.248)

1.204

(1.199)

1.248

(1.239)1.370

(1.371)

1.238

(1.242)

1.261

(1.265)

1.228

(1.226)

1.268

(1.270)

1.328

(1.330)

1.239

(1.241)

1.277

(1.275)

Figure 4.4. Structures of local minima 4a and 5a, and transition states TS1a and TS2a, calculated at the B3LYP/6-31+G(d) level. Distances are given in Ångstroms. Bond lengths in regular notation and parentheses correspond to gas-phase (ESM), and Onsager’s SCRF solvation modeled (HSM) values, respectively. Dotted lines show hydrogen bonding interactions between solute and solvent, which have distances less than the sum of van der Waals’ radii for the two atoms concerned.

The preferred sites for hydrogen bonding in 1a and 2a are on O3 (See Figure 4.3). It

is interesting to note, that for 1a, all three water molecules exclusively bind to O3,

compared to two to O3 and one to O2 in 2a. Solvation studies of ONOO– using the

PBE functional predicted a similar outcome with two discrete solvent molecules [42].

NBO charge analysis of 1 and 2 at the B3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d)

level shows that the charge on O3 is the most negative in –0.592 for 1, and –0.603 for

88

2, hence O3 would most likely be the preferred site for hydrogen bond formation

between water molecules and the solutes 1 and 2. Attempts to optimize 6a and 7a

with the water molecule forming a hydrogen bond with oxygen atoms in CO2 and

NO2• as starting structures resulted in the local minimum structure of the dipole-dipole

complex shown in Figure 4.3. The C1–O3 and N1–O3 distances are still less than the

sum of the van der Waals’ radii of the atoms involved, which are 3.22 and 3.07 Å,

respectively. In 7a, water molecules are found to bind to O2 and O3 via one hydrogen

bond per water molecule. The approximate molecular symmetry of 7a is D3h, while

the actual symmetry is C1. Attempts to optimize the structure at D3h symmetry

resulted in a number of small imaginary frequencies corresponding to rotations of

water molecules along the plane defined by CO3•–. The tendency for water to form

one hydrogen bond interaction instead of two is also shown in the results of

Armstrong et al [43], where the optimization of (H2O)·CO3•– resulted in a CS structure

instead of a C2v structure. The four water molecules in 4a, and TS1a show preference

of bonding to the CO3 moiety, whereas for 5a and TS2a, one water molecule is shown

to bind to the NO2 moiety and the other three to the CO3 moiety. Calculated bond

lengths of the solutes 1a-7a, TS1a and TS2a using ESM appear to be consistent with

those obtained for ISM (IEFPCM) and ISM (Onsager) (See Figure 4.1) in the

previous section.

The energy difference between the solvated forms of the two cis- and trans-ONOO–

conformers (1a and 2a) in the ESM is 14.9 kJ mol–1, similar to that calculated in the

gas phase. The binding energy of cis- and trans-ONOO– to CO2 is less exothermic

than gas-phase binding energies, at –42.2 and –56.3 kJ mol–1, for 1a + 3a → 4a and

2a + 3a → 5a, respectively. The overall reaction energy is less exothermic, at –32.7

89

and –47.6 kJ mol–1 for the reactions 1a + 3a → 6a + 7a and 2a + 3a → 6a + 7a,

respectively, compared to the B3LYP gas-phase results of –68.8 and –81.1 kJ mol–1.

4.3.4 Hybrid Solvation Model (HSM) Results

The structures obtained in ESM were re-optimized using the Onsager SCRF, using a

dielectric constant ε = 78.39. This method provides both microscopic (discrete water

molecules) and macroscopic representations of the solvent in theoretical treatments to

the solutes. Relative energies of the species 1a-7a, TS1a and TS2a are shown in

Table 4.4, and selected geometric parameters are shown in Figures 4.3 and 4.4.

Species HSM (Onsager, 4H2O) 1a + 3a 0.0 2a + 3a 19.8 4a −32.4 5a −32.0

6a + 7a −25.4

TS1a −18.0 (14.4) TS2a 32.7 (64.7)

Table 4.4. B3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d) + ZPE + Spin correction relative energies of 1a-7a, TS1a and TS2a, in kJ mol–1, using the hybrid solvation model (HSM). Values in parentheses indicate intrinsic barriers for 4a → TS1a and 5a → TS2a.

Molecular geometries of 1a-7a, TS1a and TS2a for HSM do not differ much from

those obtained in ESM, which are in turn consistent with those obtained using ISM

(IEFPCM) and ISM (Onsager), as shown in Figures 4.3 and 4.4, thus it is noted, that

these geometric parameters are not changed significantly with increasing

sophistication of solvation models.

90

The energy gap between cis- and trans-ONOO– in HSM is calculated to be 19.8 kJ

mol–1, which is slightly greater than those predicted in gas-phase, ISM and ESM

calculations. Binding energies of ONOO– to CO2 are –32.4 and –51.8 kJ mol–1 for 1a

+ 3a → 4a and 2a + 3a → 4a, respectively, which are slightly less exothermic than

those for ESM (–42.2 and –56.3 kJ mol–1). The overall reaction energy for 1a + 3a →

6a + 7a and 2a + 3a → 6a + 7a is –25.4 and –45.2 kJ mol–1, respectively, again

slightly less exothermic than those for ESM (–32.7 and –47.6 kJ mol–1). Incorporation

of a solvent reaction field to the solute-solvent clusters in ESM is observed to lower

reaction energies. Another interesting observation is that the energy difference

between the cis and trans conformers 4a and 5a is very small, at 0.4 kJ mol–1 in HSM.

The intrinsic activation barrier 4a → TS1a has decreased from 19.3 kJ mol–1 (ESM)

to 14.4 kJ mol–1 (HSM) upon the incorporation of a solvent reaction field, while the

barrier for 5a → TS1a has increased from 50.7 kJ mol–1 to 64.7 kJ mol–1 (HSM).

So far, continuum approaches and first solvation shell approaches have been

employed to model the reaction profile in an aqueous solvent. It is noted, however,

that there are still problems with such approaches. Applicability of continuum

approaches is also limited by the specific interactions of the solvent molecules with

the solute, effects of which dominate in the first solvation shell. In first solvation shell

approaches (e.g. ESM and HSM), it is not clear how the number and orientation of

non-bulk water molecules should be determined. There are also a very large number

of states which correspond to local minima, and the averaging of these states should

be considered as representative of the solution described [15]. QM/MM approaches

were adopted by González Lebrero et al in the study of solvated cis-ONOO– [42], and

these showed similar results to structures obtained here using ESM and HSM. The use

91

of QM/MM is attractive, but will still be computationally more expensive than ISM

approaches. Nevertheless, there remains a need for quantitatively accurate solvation

models without incurring significant computational cost that can provide a definitive

description of the system studied. The HSM approach used is perhaps the closest that

has been achieved here.

4.3.5 UV-visible spectra of ONOO–, ONOOCO2

– and CO3

•–

UV-visible absorption bands of cis-ONOO– (1), trans-ONOO– (2), cis-ONOOCO2–

(4), trans-ONOOCO2– (5), and CO3

•– (7) were predicted using ZINDO/S, CIS/6-

311+G(2df,p) and TD-B3LYP/6-311+G(2df,p) levels using geometries obtained at the

B3LYP/6-31+G(d) level. Table 4.5 shows the bands obtained and their corresponding

oscillator strengths.

ZINDO/S CISa TD-B3LYPb IEFPCM-TD-B3LYPc

Species Symd λexc/nm f λexc/nm f λexc/nm f λexc/nm f

1 A′ 445.0 0.098 209.3 0.128 312.6 0.092 304.1 0.107 2 A′ 420.7 0.117 213.1 0.168 366.1 0.095 349.9 0.110 4 A 326.9 0.000 152.0 0.001 373.3 0.001 351.6 0.001 5 A 325.6 0.004 152.1 0.001 393.5 0.002 374.7 0.002 7 E′ 165.4 0.157 218.8 0.039 603.1 0.025 620.3 0.030

Table 4.5. Predicted UV-vis absorption bands of cis-ONOO– (1), trans-ONOO– (2), cis-ONOOCO2

– (4), trans-ONOOCO2

– (5), and CO3•– (7) using various methods, based geometry obtained at B3LYP/6-

31+G(d) level. aCIS/6-311+G(2df,p). bTD-B3LYP/6-311+G(2df,p). cTD-B3LYP/6-311+G(2df,p) in IEFPCM SCRF, solvent dielectric=78.39. dTransition symmetry label.

Absorption wavelengths of cis- and trans-ONOO– were previously calculated by

Krauss to be 324.3 and 368.5 nm, respectively, using MCSCF methods [44]. This

λexcitation is calculated to be blue-shifted in a solvent reaction field, to 310.4 and 338.8

92

nm. The experimental absorption spectra of cis- and trans-ONOOK isolated in solid

Ar and were obtained by Lo et al, with absorption wavelengths of 325 and 375 nm

[45]. Gas-phase TD-DFT calculated values (312.6 and 366.1 nm) are in good accord

with the experimental and MCSCF-calculated values, while the ZINDO/S and CIS

methods yield significant deviations (≥ 100 nm). This relatively good agreement of

TD-DFT predicted values provides some confidence in the applicability of the TD-

DFT method to the other species investigated.

Electronic transitions in cis- and trans-ONOOCO2– (4 and 5, respectively) occur at

373.3 and 393.5 nm, respectively, in the gas phase. Higher-level calculations were

also performed using Nakatsuji’s symmetry-adapted-cluster configuration-interaction

(SAC-CI) methods [46-48] incorporated into the GAUSSIAN 03 code. Absorption

bands of cis- and trans-ONOOCO2– at the SAC-CI/6-31+G(d)//B3LYP/6-31G(d)

level are 372.9 (f = 0.0025) and 392.4 (f = 0.0021) nm, respectively. The almost

perfect agreement of the TD-DFT results with SAC-CI results lends further

confidence that the TD-DFT method was a good choice for elucidating the absorption

bands of the cis- and trans-ONOOCO2– system. As with cis- and trans-ONOO–, a

blue-shift is expected in a polar solvent reaction field, to 351.6 and 374.7 nm.

However, the oscillator strength f for these transitions is low, between 0.001 and

0.002. Additional TD-DFT calculations were carried out for cis-ONOOCO2 – with

two explicit water molecules, incorporating bulk solvent effects using IEFPCM, at the

B3LYP/aug-cc-pVTZ level. At this level of theory, no electronic transitions above

450 nm were obtained. This could partially explain why ONOOCO2– is not detected

spectrophotometrically, as electronic transitions for both cis- and trans-ONOOCO2–

do not occur within the UV and visible regions in the electromagnetic spectrum.

93

Although some authors claim that ONOOCO2– was indeed spectrophotometrically

observed as an intermediate in the reaction of ONOO– with CO2 [9], it has also been

reported that the observed peak is due to the effect from stray light [49].

The observed band at 640 nm in the reaction of ONOO– with CO2, which decreases in

intensity as the reaction proceeds, is probably not due to the adduct ONOOCO2–, but

possibly CO3•–. The calculated absorption band for CO3

•– using TD-DFT is at 603.1

nm. However, upon incorporating solvent effects using IEFPCM, this band is

observed to be red-shifted to 620.3 nm. Upon calculation of the absorption spectrum

for (H2O)·CO3•– using TD-B3LYP/6-311+G(2df,p)//B3LYP/6-31+G(d), it was found

that the band was split to the two A′ states, 624.3 (f = 0.0152) and 593.3 (f = 0.0303)

nm. When IEFPCM SCRF was incorporated (i.e. IEFPCM-TD-B3LYP/6-

311+G(2df,p)//B3LYP/6-31+G(d)), these bands were further red-shifted to 650.6 (f =

0.0177) and 602.9 (f = 0.0368) nm, respectively. The good correspondence of the

calculated 650.6 nm band with the broad 640 nm band in the spectrum obtained by

Meli et al lends weight to the claim that what was spectrophotometrically observed

100 ms after mixing ONOO– with CO2 was CO3•– instead of ONOOCO2

– as

previously concluded [9]. However, the observed red-shift of the 603.1 nm absorption

band to 650.6 nm upon solvation is unusual, as a previous study by Armstrong et al

reported this band to be blue-shifted and split by explicit solvation from 608.2 nm (E′)

to 596.0 (A′) and 569.3 nm (A′) [43] using IEFPCM-TD-B3LYP/6-311+G(3df,2p).

In an attempt to clarify if the CO3•– absorption band at ~600 nm is indeed red-shifted

or blue-shifted upon solvation by an aqueous solvent, the spectra of CO3•–,

(H2O)·CO3•–, and (H2O)2·CO3

•– were calculated using IEFPCM-TD-B3LYP/aug-cc-

94

pVTZ based on IEFPCM-B3LYP/aug-cc-pVTZ geometries. The results are shown in

Table 4.6. It is observed that the addition of coordinated water molecules to CO3•–

actually blue-shifts the absorption band from 605.5 nm (no explicit H2O) to 573.6 nm

(two explicit H2O molecules). These results are now in accord with the results found

by Armstrong et al.

IEFPCM-TD-B3LYP/aug-cc-pVTZ CO3

•– Explicit H2O molecules (n)

0 1 2

λexcitation (1) / nm 605.5 (E′) 605.2 (A′) 573.6 (A′) f (osc. strength) 0.0292 0.0228 0.0182 λexcitation (2) / nm 605.5 (E′) 572.9 (A′) 551.6 (A′) f (osc. strength) 0.0292 0.0359 0.0374 Table 4.6. Calculated UV-Visible excitation wavelengths λexcitation, and corresponding oscillator strength values, of CO3

•– with 0, 1 and 2 explicit H2O molecules, and incorporating IEFPCM for modeling bulk effects of an aqueous solvent, using the TD-B3LYP method and the aug-cc-pVTZ basis set.

A probable explanation can now be provided for the observed phenomena by Meli et

al, in the light of these calculated TD-DFT results. The observed shift of the 640 nm

absorption band at 100 ms after mixing ONOO– and CO2 to a broad band centered at

620 nm at 170 ms is probably due to the combined effects of increasing explicit

solvation of CO3•– and the depletion of CO3

•– due to its reaction with other species in

the buffered solution. This significant solvent effect leading to the blue-shift in the

CO3•– absorption spectrum provides theoretical evidence to warrant some caution in

the interpretation of spectrophotochemical data obtained from the study of this

intriguing reaction.

95

4.4 Concluding Remarks

The potential energy surface of the reaction between ONOO– and CO2 was explored

using DFT calculations, both in the gas-phase and using various solvent models. The

overall reaction is expected to proceed via the lower energy path, that is, cis-ONOO–

+ CO2 → cis-ONOOCO2– → TS1 → NO2

• + CO3•–. The calculated gas-phase overall

reaction energy is exothermic by 68.8 kJ mol–1. Upon the incorporation of an aqueous

solvent, the reaction energy becomes less exothermic. The barrier for O—O

homolytic bond scission is expected to be about 17 kJ mol–1 both in the gas phase and

in an aqueous solvent. Electronic transition energies were calculated for cis-ONOO–,

trans-ONOO–, cis-ONOOCO2–, trans-ONOOCO2

–, and CO3•–, using ZINDO/S, CIS

and TD-DFT methods, of which TD-DFT results agreed best with experimental

absorption wavelengths and benchmark SAC-CI calculations. The TD-DFT results

suggest that the observation of a 640 nm band in the reaction of ONOO– with CO2 is

not due to ONOOCO2– but CO3

•–, and that the solvation effect of an aqueous solvent

is responsible for the blue shift in the absorption spectrum of CO3•–.

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38. Goldstein, S.; Czapski, G.; Lind, J.; Merényi, G. Chem. Res. Toxicol. 2001, 14,

1273.

39. Lymar, S. V.; Hurst, J. K. Inorg. Chem. 1998, 37, 294.

40. Radi, R.; Peluffo, G.; Alvarez, M. N.; Naviliat, M.; Cayota, A. Free Radical

Biol. Med. 2001, 30, 463.

41. Bartberger, M. D.; Olson, L. P.; Houk, K. N. Chem. Res. Toxicol. 1998, 11,

710.

42. González Lebrero, M. C.; Perissinotti, L. L.; Estrin, D. A. J. Phys. Chem. A

2005, 109, 9598.

43. Armstrong, D. A.; Waltz, W. L.; Rauk, A. Can. J. Chem. 2006, 84, 1614.

44. Krauss, M. Chem. Phys. Lett. 1994, 222, 513.

99

45. Lo, W.-J.; Lee, Y.-P.; Tsai, J.-H. M.; Beckman, J. S. Chem. Phys. Lett. 1995,

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46. Nakatsuji, H. Chem. Phys. Lett. 1978, 59, 362.



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100

Chapter Five

Reaction of the Radical Pair NO2• and CO3•– with 2-[6-(4′-

Amino)phenoxy-3H-xanthen-3-on-9-yl]benzoic Acid (APF)

5.1 Introduction

Reactive nitrogen species (RNS) and reactive oxygen species (ROS) play numerous

and diverse roles in physiological pathways. They are capable of mediating nitrosative

and oxidative stress and exhibit both cytotoxic and cytoprotective effects [1]. The

RNS peroxynitrite (ONOO–) is a free radical precursor that is formed in cells by

diffusion-controlled reaction of nitric oxide (NO•) and superoxide (O2•–). The gas-

phase reaction enthalpy for the reaction of NO• with O2•– to form ONOO– at 298 K

was earlier calculated to be −138.3 kJ mol–1 at the CCSD(T)/cc-pVQZ//CCSD/cc-

pVTZ level [2]. It has been established recently that ONOO– reacts rapidly with

ubiquitous CO2 (k = 3 × 104 M–1 s–1) [3] to form nitrosoperoxycarbonate anions

(ONOOCO2–) which then undergo O–O bond homolysis to give NO2

• and CO3•–

radicals at a yield of about 30% [4-7]. These radicals can in turn react with nucleotide

bases in DNA and RNA, or amino acid residues in proteins, resulting in significant

structural changes and irreversible damage [8], although such reactions have not been

well characterized experimentally. It is widely believed that the action of these RNS

and ROS contributes to Alzheimer’s disease [9], Parksinson’s disease [10],

atherosclerosis [11] and cancer [12, 13].

101

The detection of RNS and ROS is a difficult task due to two main reasons: their short

lifespan, and their high reactivity with numerous other molecules. As a consequence,

detection, measurement and monitoring of RNS and ROS levels remain a constant

challenge to scientists. Fluorescent probes are highly sensitive and useful in

monitoring RNS and ROS. An ideal fluorescent probe should display high

fluorescence quantum yield for the species to be detected, and should be both

selective and accurate. At present, there are several fluorescent probes commercially

available for detecting and measuring ONOO–, for instance, 2′,7′-dicholorofluorescin

(DCFH), dihydrofluorescein (HFLUOR), dihydrorhodamine 123 (DHR), Rhodamine

B hydrazide, 2-[6-(4′-amino)phenoxy-3H-xanthen-3-on-9-yl]benzoic acid (APF) [14,

15] and the more recently developed HKGreen-1 [16]. In particular, Setsukinai et al

have shown that APF exhibits excellent selectivity for ONOO– over other radicals and

radical precursors such as NO•, O2•–, hydrogen peroxide (H2O2) and alkyl peroxide

radicals (ROO•), when compared with DCFH [17]. The APF molecule undergoes O-

dearylation in the presence of RNS generated by ONOO– derivatives to yield

fluorescein, with excitation and emission wavelengths of 490 and 520 nm,

respectively (Scheme 5.1). Although these probes are known to be oxidized to their

fluorescent forms in the presence of ONOO–, information on the exact identities of the

species that react with these probes is lacking.

102

Scheme 5.1. Action of RNS on APF, adapted from Ref.17

Experiments carried out by various research groups have shed some light on the

mechanism of fluorescent probe oxidation by ONOO– and its derivatives. The

exclusion of a direct reaction of ONOO– with a fluorescent probe was established by

Jourd’heuil et al [18]. The authors demonstrated that the oxidation of DHR by

ONOO– occurs via free radical intermediates NO2• and OH• formed from the

spontaneous decomposition of ONOOH. Kinetic studies by Koppenol et al reveal that

reaction of DCFH with ONOO– is essentially zero order with respect to the indicator,

eliminating the possibility of a direct reaction between ONOO– and the probe

molecule [19]. They proposed that the reactive species that oxidized DCFH is the

radical pair NO2• and CO3

•–.

Despite the fact that APF is used by many research groups to detect and measure

ONOO– in various experiments, both in vivo and in vitro, the reasons for which APF

exhibits selectivity for ONOO– over other RNS and ROS are still unclear. In this

chapter, a computational study of the reaction of APF with NO2• and CO3

•– is

presented, to shed some light into its reaction mechanism, and the factors that

influence the selectivity of the probe. An experimental examination of the influence

103

of HCO3– concentration on the ONOO– mediated two-electron oxidation of APF was

also carried out to better understand the chemistry pertinent to this intriguing reaction.

5.2 Methods

5.2.1 Computational Methods

A hybrid DFT method, the modified Perdew-Wang 1-parameter method for kinetics

(MPW1K), was employed in this study. Hybrid DFT methods have been shown to be

successful in obtaining accurate molecular geometries, vibrational frequencies and

bond energies [20]. The MPW1K functional was parameterized to yield improved

performance for kinetics calculations and bond energies as compared to other hybrid

DFT functionals such as B3LYP, BH&HLYP and mPWPW91 [21]. In particular, the

MPW1K functional yields improved accuracy in transition state geometries and

barrier heights compared to the abovementioned functionals [22], making the

MPW1K method an adequate choice for investigating the system in question.

Full geometry optimizations of all local minima and saddle points were carried out at

the MPW1K/6-31G(d) level of theory. Frequency calculations for all species were

carried out to verify if they were indeed local minima with all real frequencies, or

first-order saddle points (i.e. transition states) with one imaginary frequency. Closed-

and open-shell species were treated using the RHF and UHF formalisms, respectively.

Additional energy calculations were carried out at the MPW1K/6-311+G(d,p) level,

based on the MPW1K/6-31G(d) optimized geometry, to obtain a more accurate

104

estimate of the energetics of the reaction studied. The energies of open-shell species

were also spin corrected using the method developed by Yamaguchi et al [23, 24].

The polarizable continuum model of Tomasi et al [25-27] (PCM) was initially used to

investigate the effect of a solvent reaction field on the energetics of the reaction

studied. However, convergence problems were encountered in the optimization of

structures. Hence, geometry optimizations and frequency calculations were carried

out for selected local minima and transition states using the more robust Onsager

reaction field (SCRF=Dipole) method [28-30] at the MPW1K/6-31G(d) level. A

dielectric constant (ε) of 40.0 is used to model a polar medium. For a molecular ion

with a total charge Q, a Born charge term [31], –½(1 – 1/ε)Q2/a0 (where a0 is the

cavity radius), was added to the expression of solvation energy to account for the ion-

dipole interaction. It is important to note that the SCRF is not a suitable method to

model aqueous solvation where specific solvent-solute interaction is expected to be

important. Therefore the scope of this present solvation study is limited to an aprotic

polar solvent such as acetonitrile. Higher-level relative energies were obtained using

this solvation model at the MPW1K/6-311+G(d,p) level, together with spin correction

for open-shell species. All calculations were performed using the GAUSSIAN 03 [32]

program suite.

5.2.2 Experimental methods

APF {2-[6-(4’-amino)phenoxy-3H-xanthen-3-on-9-yl]benzoic acid} was purchased

from Alexis Corporation (Lausen, Switzerland). All other chemicals were obtained

from Sigma Ltd (St Louis, MO, USA), and freshly prepared on the day of the

105

experiment. Distilled water passed through a Millipore water purification system was

used for all purposes.

Sodium phosphate buffer (0.1 M) and sodium hydrogen carbonate solution (0.25 M)

were added along with 2 µl of stock APF solution into a 1 ml sample tube and

protected from light. This mixture is allowed to stand for about fifteen minutes to

allow equilibriation of hydrogen carbonate, carbon dioxide and carbonate in solution,

before other reagents were added. Hydrogen peroxide-free peroxynitrite was prepared

using the method described by Beckman et al [33], and added to the mixture in the

sample tube. Removal of residual H2O2 was confirmed using a commercial kit

(Amplex Red; Molecular Probes) [34]. Concentrations of ONOO– were redetermined

before each experiment using UV-visible spectroscopy, with ONOO– having a molar

extinction coefficient of 1670 cm–1, at 302 nm. ONOO– concentrations of 250 – 300

mM were usually obtained. This concentrated peroxynitrite (about 2 – 5 µl) was then

added to the reaction mixture in the sample tube. The sample tubes were then

vortexed and left to stand for five minutes before the fluorescence intensity was

measured. The pH of the well buffered reaction mixture upon adding of a small

amount of concentrated peroxynitrite is not expected to significantly change, as

already reported in previous work [35]. The final concentrations of the components in

the reaction mixture after mixing were: 10 µM APF, 1.0 µM ONOO–, 0.1 M

phosphate buffer, and varying concentrations of HCO3–, from 0 to 25 mM. The

fluorescence intensity that resulted from peroxynitrite-mediated oxidation of APF was

monitored with λexcitation = 490 nm and λemission = 520 nm at room temperature, using a

Gemini EM fluorescence microplate reader from Molecular Devices.

106


5.3.1 Influence of HCO3− concentration on ONOO

− mediated oxidation of APF

In an attempt to obtain some insight as to whether HCO3− substantially influence the

ONOO− mediated oxidation of APF, we studied the percentage yield of fluorescein as

a function of [HCO3–]. It was found that with increasing concentrations of HCO3

− in

solution, oxidation of APF was inhibited, though not completely (Figure 5.1). This

finding is consistent with the results obtained by Koppenol et al for the fluorogenic

indicators DCFH and DHR [19]. Increasing [HCO3–] shifts the equilibrium of the

reaction HCO3– = CO2 + OH

– further to the right, and the increased CO2 concentration

leads to the formation of larger amounts of ONOOCO2– from the reaction ONOO– +

CO2 = ONOOCO2–. The observation that increasing [HCO3

–] progressively lowers the

yield of fluorescent products instead of increasing it, appears to exclude the pathway

where APF reacts directly with ONOOCO2–. A similar conclusion can be reached, i.e.

the direct reaction between APF and ONOOCO2– is not likely.

107

HCO3– concentration / µM

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16

%age yield of fluorescein, relative to ONOO–

HCO3– concentration / µM

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16

%age yield of fluorescein, relative to ONOO–

Figure 5.1. Plot of percentage yield of fluorescein, relative to ONOO–, versus the concentration of HCO3

– in the reaction mixture. Other experimental conditions: 10 µM APF, 1.0 µM ONOO–, 0.1 M phosphate buffer. Total volume of the mixture is 1 ml.

The O—O bond dissociation energy (BDE) of ONOOCO2– was previously estimated

at the CBS-QB3 level to be 31.4 kJ mol–1 [36]. This magnitude of O—O BDE in

ONOOCO2– is unexpectedly low compared to O—O BDEs in other compounds such

as hydrogen peroxide HO–OH (213.4 kJ mol–1), dimethyl peroxide CH3O–OCH3

(158.6 kJ mol–1) and pernitric acid HO–ONO2 (163.2 kJ mol–1) [37]. The relative ease

of decomposition of ONOOCO2– supports the explanation proposed by Koppenol et al

for the DCFH and DHR case, that these fluorogenic indicators react with that the

decomposition products of ONOOCO2–, namely the radical pair NO2

• and CO3•– [19].

This explanation can be extended for the fluorogenic indicator APF.

5.3.2 Gas-Phase Calculations

The final product, fluorescein, can exist in a number of structural (quinoid,

zwitterionic and lactoid) and prototropic (neutral, monoanionic and dianionic) forms.

108

It exists purely as the dianionic form at pH 9.1 and the monoanionic form at pH 5.4.

At pH 7.4, fluorescein exists as a mixture of both forms [38]. Quantum mechanical

studies on the charge transfer transition for both forms show that the transition for the

dianion has markedly higher oscillator strength than that for the monoanion [39].

Thus, the monoanionic carboxylate form of APF, which yields dianionic fluorescein

after oxidation, was employed in this study.

Based on the two-electron oxidation products of APF reaction with RNS, namely

fluorescein and p-benzoquinone imine, and the reactive species NO2• and CO3

•–

involved as reactants, the reaction of APF can be interpreted as sequences of reactions

illustrated in Schemes 5.2 and 5.3. The reaction can occur via two plausible pathways.

The first, pathway I (Scheme 5.2), involves two rapid one-electron oxidations. The

first one-electron oxidation step involves a proton coupled electron transfer (PCET)

from the amino group in APF (1A) by NO2•. The second step involves O– addition by

CO3•– to form the resonance stabilized anilinyl fluorescein radical 1E, followed by

decomposition of the adduct 2B to yield fluorescein, 2E, p-benzoquinone imine, 2F,

and CO2. The second pathway, pathway II (Scheme 5.3), involves a direct addition of

CO3•– to 1A to give a radical adduct 3B. Decomposition of the adduct yields 2E, p-

aminophenoxyl radical, 3E, and CO2.

109

OR

NH

O2CO

OR

NH

O2COO

R

NH

OO2C

TS2A 2B TS2C

OR

NH

1E

OR

O

HN

CO2 NH

O

RO

2E

2F

+ + CO2

2D

OR

NH2

OR

HNH ONO

OR

HN H ONO

OR

HN HONO

OR

NH

OR

NH

1A 1B TS1C 1D 1E

R= OO

COO

CO3

NO2+ HONO

Scheme 5.2. Pathway I of APF oxidation.

OR

NH2

O2CO

OR

NH2

O2COO

R

NH2

O

TS3A 3B TS3C

OR

O

H2N

CO2

3D

OR

NH2

1A

NH2

O

RO

2E

3E

+ + CO2

R= OO

COO

O2C

CO3

Scheme 5.3. Pathway II of APF oxidation.

The calculated structural parameters of neutral fluorescein (Figure 5.2), are in good

accord with known experimental values [40], with maximum relative errors for bond

lengths at 5%. Calculated structural parameters of dianionic fluorescein 2E, are

essentially identical to those of the neutral form. A comparison of 11 known

vibrational modes of fluorescein [38] with the calculated MPW1K/6-31G(d)

frequencies yielded errors between 4−18%. However, upon applying a frequency

110

scaling factor of 0.905, the relative errors were reduced to 1.1−5.7%. The relatively

good agreement of calculated geometries and frequencies with experimental values

provided the confidence that the choice of the MPW1K/6-31G(d) method is adequate

for the study of the reaction in question. With the lack of reliable zero-point energy

data for fluorescein, the scaling factor for fundamentals was applied to calculated the

zero-point vibrational energies.

1.219

[1.219]

(1.222)

1.469

[1.469]

(1.398)

1.451

[1.451]

(1.415)

1.344

[1.345]

(1.402)

1.355

[1.355]

(1.364)1.446

[1.446]

(1.398)

1.339

[1.339]

(1.412)

1.442

[1.442]

(1.410)

1.360

[1.360]

(1.414)

1.437

[1.437]

(1.398)

1.342

[1.342]

(1.356)

1.399

[1.399]

(1.407)

1.381

[1.381]

(1.407)

1.384

[1.384]

(1.392)

1.342

[1.343]

(1.365)1.397

[1.397]

(1.409)

1.397

[1.397]

(1.411)

1.375

[1.375]

(1.410)

1.486

[1.486]

(1.495)1.391

[1.391]

(1.410)

1.384

[1.384]

(1.406)

1.385

[1.385]

(1.400)

1.380

[1.380]

(1.400)

1.392

[1.392]

(1.415)

1.401

[1.401]

(1.401)

1.485

[1.486]

(1.511)

1.329

[1.330]

(1.339)

1.201

[1.202]

(1.235)

1.219

[1.219]

(1.222)

1.469

[1.469]

(1.398)

1.451

[1.451]

(1.415)

1.344

[1.345]

(1.402)

1.355

[1.355]

(1.364)1.446

[1.446]

(1.398)

1.339

[1.339]

(1.412)

1.442

[1.442]

(1.410)

1.360

[1.360]

(1.414)

1.437

[1.437]

(1.398)

1.342

[1.342]

(1.356)

1.399

[1.399]

(1.407)

1.381

[1.381]

(1.407)

1.384

[1.384]

(1.392)

1.342

[1.343]

(1.365)1.397

[1.397]

(1.409)

1.397

[1.397]

(1.411)

1.375

[1.375]

(1.410)

1.486

[1.486]

(1.495)1.391

[1.391]

(1.410)

1.384

[1.384]

(1.406)

1.385

[1.385]

(1.400)

1.380

[1.380]

(1.400)

1.392

[1.392]

(1.415)

1.401

[1.401]

(1.401)

1.485

[1.486]

(1.511)

1.329

[1.330]

(1.339)

1.201

[1.202]

(1.235)

Figure 5.2. Calculated structural parameters of dianionic fluorescein, 2E, at the MPW1K/6-31G(d) level in the gas phase. Calculated structural parameters for neutral fluorescein at the MPW1K/6-31G(d) level are given in square brackets. X-ray structural parameters of neutral fluorescein (from Ref. 38) are given in parentheses. Non essential hydrogens are omitted for clarity. Distances are in Ångstroms. Oxygen atoms are in dark color.

Gas-phase transition energy (λexcitation) for dianionic fluorescein was calculated using

ZINDO/S [41] and TD-DFT [42, 43] methods. The predicted λexcitation using the

ZINDO/S method is 475 nm, in close agreement with the experimental value of 490

nm [44]. TD-B3LYP/6-311+G(d,p)//B3LYP/6-31G(d) and TD-MPW1K/6-

111

311+G(d,p)//MPW1K/6-31G(d) predict λexcitation values of 430 nm and 392 nm,

respectively. It is important to note that time-dependent density-functional theory

yields unsatisfactory results for excitation energies of long-range charge-transfer

states [45].

Schematic energy profiles of the APF oxidation reactions in the gas phase, calculated

at MPW1K/6-311+G(d,p)//MPW1K/6-31G(d) + ZPE level, are presented in Figures

5.3 to 5.5. Gas-phase MPW1K/6-31G(d) optimized structures of selected local

minima (1A, 1B, 1D, 1E, 2B, 2D, 3B, 3D and 3E) and all transition states (TS1C,

TS2A, TS2C, TS3A and TS3C) are presented in Figures 5.6 to 5.8. The atom

numbering scheme used for 1A, 1E, TS2C and TS3C is given in Figure 5.9.

-10

0

10

20

30

40

50

60

70

80

90

1A + NO2•0.0

1B

-65.4

TS1C

-18.9

1D

-35.7

1E + HONO18.2

Energy (kJ mol−1 )


In pathway I, NO2• first interacts with 1A to form a hydrogen-bonded reactant

complex 1B, with a relatively large binding energy of 65.4 kJ mol–1. The hydrogen

112

transfer reaction involves the transfer of the amino hydrogen of 1B to NO2•. It occurs

via TS1C, with an activation barrier of 46.5 kJ mol–1. The N…H and O…H bond

distances in TS1C are 1.143 and 1.367 Å, respectively. The N…H…O moiety is close

to linearity (Figure 5.8). Because of the larger interaction of the pre-transition state

complex 1B, the hydrogen transfer reaction has a negative barrier with respect to 1A

+ NO2•. The hydrogen bonded product complex 1D is slightly less stable than the

reactant complex 1B by 29.7 kJ mol–1.

It is important to note that the carbonate radical anion (CO3•–) is a strong one-electron

oxidant that oxidizes appropriate electron donors via electron transfer mechanisms

[46-48]. The redox potential of CO3•– (equation 5-1) is well established [49, 50] and is

strongly positive, 1.6 (pH 12.5) to 1.78 V (pH 7.0) vs. NHE. In fact, this value is

greater than the redox potential of NO2• (equation 5-2) [51].

CO3•– + e– + H+ → HCO3

– E° = 1.78 V vs. NHE (5-1)

NO2• + e– → NO2

– E° ≈ 1.04 V vs. NHE (5-2)

Thus, the first one-electron oxidation step of pathway I may involve CO3•– (equation

7-3).

1A + CO3•– → 1E + HCO3

– (5-3)

The proton coupled electron transfer (PCET) from the amino group in APF (1A) by

CO3•– will also lead to the formation of 1E together with HCO3

–, a fairly stable

bicarbonate anion. The calculated reaction energy of this one-electron oxidation

reaction is exothermic, by 45.0 kJ mol–1. However, this electron transfer reaction is

expected to be unfavorable kinetically as the transition state involves two anions

113

approaching each other. Indeed, this proton coupled electron transfer reaction is

calculated to have a large barrier of 106.9 kJ mol–1. The dianionic transition state is

significantly higher in energy (by 125.8 kJ mol–1) compared to the transition state

involving NO2• (TS1C). This result suggests that the alternate one-electron oxidation

via carbon radical anion is less viable.

-100

-50

0

50

100

150

200

1E + CO3•−

0.0

TS2A

162.2

2B

28.7

TS2C

36.4

2D

-40.2

2E + 2F + CO2

61.8



The CO3•− radical can add to 1E in three plausible positions: C(23), C(25) and C(27)

(See Figure 5.9). The only position involving CO3•− radical addition to yield the final

products is at C(23) and, hence, is the only one considered here. Addition of the CO3•−

radical to 1E to form adduct 2B is endothermic by 28.7 kJ mol–1, and proceeds via a

transition state TS2A with strong biradical character. The forming C…O bond length

in TS2A is 2.035 Å. The expectation value of the spin-squared operator ⟩⟨2S of

TS2A is 0.742, indicating that there is a significant degree of spin contamination from

the higher energy spin states. The activation barrier for CO3•− addition via TS2A

before spin correction is 193.0 kJ mol–1. After the addition of the spin correction term,

this barrier is reduced significantly to 162.2 kJ mol–1.

114

Decomposition of 2B to form 2E, 2F and CO2 is endothermic by 33.1 kJ mol–1 and

takes place via TS2C. Two C—O bonds break simultaneously, between C(23) and

O(22) and between C(31) and O(30) in TS2C (See Figure 5.8). C(23)—O(22) and

C(31)—O(30) distances are 1.853 Å and 2.521 Å, respectively in TS2C. Remarkably

this fragmentation process is inhibited by a small barrier of just 7.7 kJ mol–1. As

expected, the final products form a stable complex 2D (2E…2F

…CO2) with a fairly

large binding energy of 102.0 kJ mol–1.

-100

-50

0

50

100

150

200

250

1A + CO3•−

0.0

TS3A

233.5

3B

132.4

TS3C

150.1

3D

-44.6

2E+3E+CO2

80.8


Figure 5.5. Schematic gas-phase reaction energy profile for pathway II, calculated at the MPW1K/6-311+G(d,p)//MPW1K/6-31G(d) +ZPE level with spin correction.

The radical addition process via pathway II is similar to the second step of pathway I.

The CO3•− radical is expected to undergo radical addition with the initial reactant 1A

instead of the intermediate formed after hydrogen transfer. As with pathway I, the site

considered for CO3•− addition is at C(23). The direct addition of CO3

•− radical anion to

1A to form 3B is highly endothermic, by 132.4 kJ mol–1. The activation barrier for

this process is notably high at 233.5 kJ mol–1, via TS3A. The transition state of this

115

reaction, TS3C, has similar structural characteristics as TS2C of pathway I. The

C(23)—O(22) and C(31)—O(30) distances in TS3C are comparatively shorter, at

1.685 and 2.239 Å respectively. The products 2E, 3E and CO2 form a complex with

substantial binding energy of 125.4 kJ mol–1. Overall, addition of CO3•− to 1A to form

3B is the rate determining step of pathway II.

1A 1B

1D 1E

1.769

1.710

N

NN

N

N

N

Figure 5.6. MPW1K/6-31G(d) optimized geometries of 1A, 1B, 1D and 1E. Distances shown are in Ångstroms. Non essential hydrogens are omitted for clarity. Oxygen atoms are in dark color.

116

2B 2D

3B 3D

1.3621.788

2.673

1.369

1.558

2.559

N

N

N

N

Figure 5.7. MPW1K/6-31G(d) optimized geometries of 2B, 2D, 3B and 3D. Distances shown are in Ångstroms. Non essential hydrogens are omitted for clarity. Oxygen atoms are in dark color.

117

TS1C

TS2A TS2C

TS3CTS3A

1.143

1.367

2.035

1.853

2.521

1.875

1.685

2.239

N

N

N N

NN

Figure 5.8. MPW1K/6-31G(d) optimized geometries of transition structures TS1C, TS2A, TS2C, TS3A and TS3C. Distances shown are in Ångstroms. Non essential hydrogens are omitted for clarity. Oxygen atoms are in dark color.

118

Figure 5.9. Atom numbering scheme for 1A, 1E, TS2C and TS3C used in this study.

Based on the calculated gas-phase potential energy surface, oxidation of APF to form

fluorescein can take place via two pathways. Pathway I involves initial proton

coupled electron transfer (PCET) from APF to NO2•, followed by O•– addition of

CO3•− to the radical 1E, which, with a barrier of 162.2 kJ mol–1, is the rate limiting

step. Pathway II is independent of NO2•, but is not likely to proceed due to a

prohibitively high barrier of 233.5 kJ mol–1 for the rate determining step.

5.3.3 Solvent Effects

To investigate the influence of a solvent reaction field on the reaction mechanism,

119

species 1A, 1B, 1E, 2B, 3B, TS1C, TS2A, TS2C, TS3A and TS3C were re-

optimized at the MPW1K/6-31G(d) level incorporated with the Onsager reaction field

model with a dielectric medium of ε = 40.0. The MPW1K/6-311+G(d,p)//MPW1K/6-

31G(d) activation barriers corresponding to 1B → TS1C, 1E + CO3•− → TS2A, 2B

→ TS2C, 1A + CO3•− → TS3A, and 3B → TS3C in the solvent medium are

presented in Table 5.1, with the gas-phase MPW1K/6-311+G(d,p)//MPW1K/6-31G(d)

barriers shown alongside for comparison.

For pathway I, activation barriers 1B → TS1C and 1E + CO3•− → TS2A decreased by

19.2 and 101.3 kJ mol–1, respectively, in a polar dielectric medium. The vast decrease

in barrier height for 1E + CO3•− → TS2A is due mainly to the additional contribution

of an ion-dipole (Born charge) term in the dianionic transition state TS2A. The barrier

for 2B → TS2C in the polar dielectric medium, on the other hand, increased by 36.4

kJ mol–1. The rate limiting step is still the addition of CO3•− radical to 1E. However,

APF oxidation via pathway I is generally more favorable in solvent than in the gas

phase, with barriers ranging from 27.2 to 60.9 kJ mol–1.

Reaction Gas Phase (ε =1) Polar medium (ε = 40)a

Pathway I

1B → TS1C 46.4 27.2

1E + CO3•− → TS2A 162.2 60.9

2B → TS2C 7.7 44.1

Pathway II

1A + CO3•− → TS3A 233.5 218.2

3B → TS3C 17.7 19.5

Table 5.1. Calculated activation barriers (kJ mol–1) of the two reaction pathways in the gas phase and in a polar medium at the MPW1K/6-311+G(d,p)//MPW1K/6-31G(d) level. a SCRF=dipole calculations.

120

For pathway II, the activation barrier 1A + CO3•− → TS3A is lowered by 15.3 kJ mol–

1, and the barrier 3B → TS3C is slightly increased by 1.8 kJ mol–1. Comparing with

pathway I, pathway II is less likely to proceed because of the prohibitively high

barrier of CO3•− radical addition to 1A.


The reaction of APF with the radicals generated by the reaction of ONOO– and CO2,

namely NO2• and CO3

•−, was explored. Experimentally, HCO3– was found to inhibit

the two-electron oxidation of APF, suggesting that the direct reaction of APF with

ONOOCO2– is unlikely. Two possible pathways of APF oxidation were explored

using DFT calculations. Pathway I involves initial proton coupled electron transfer

from APF to NO2 to obtain a resonance stabilized anilinyl fluorescein radical. CO3•−

addition to this radical follows, and subsequently decomposition of the adduct yields

fluorescein, p-benzoquinone imine and CO2. Pathway II involves the direct addition

of CO3•− to APF, and the subsequent decomposition of the APF-CO3

•− adduct to form

fluorescein, p-aminophenoxyl radical and CO2. Addition of CO3•− is the rate

determining step in both pathways, and the activation barrier for this step in pathway

II is prohibitively high even in a polar aprotic solvent. Hence, the two-electron

oxidation of APF by is likely to involve two one-electron oxidations by both NO2• and

CO3•− radicals and proceeds via pathway I.

121

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125

Chapter Six

Homolytic Dissociation of the Vulcanization Accelerator

Tetramethylthiuram Disulfide (TMTD) and Structures and

Stabilities of the Related Radicals Me2NCSn• (n = 1−4)

6.1 Introduction

The global annual production of rubber is approximately 17 million tons. In order to

turn natural or synthetic polyolefins or “rubbers” into useful elastomers, sulfur

vulcanization is usually performed to generate sulfidic and disulfidic cross-links

between neighboring macromolecules. Tetramethylthiuram disulfide (TMTD) is one

of the most important accelerators of this type of vulcanization (see Figure 6.1).

Figure 6.1. Chemical structure of tetramethylthiuram disulfide (TMTD)

TMTD is usually applied together with zinc oxide, and the function of the latter in this

context has been studied extensively [1-3]. The formation of zinc thiolate complexes

has been postulated, which are believed to add more sulfur atoms either from each

other by disproportionation or from added elemental sulfur. These polysulfido

126

complexes are assumed to attack the polyolefin of natural or synthetic rubbers,

eventually resulting in cross-linking between polymeric chain molecules [4, 5].

The thermal behavior of pure TMTD has repeatedly been studied by several groups.

Free radicals have been detected by electron spin resonance (ESR) spectroscopy when

TMTD was heated to 130–150°C [6, 7]. The intriguing question here is which of the

covalent bonds of TMTD was broken homolytically. A triplet signal at g = 2.03 with a

peak separation of ca. 15 G was observed after heating TMTD followed by cooling to

room temperature. This signal was assigned to the thiuram radical Me2NCS2•. A

singlet signal at g = 2.02 was observed when TMTD was heated and subsequently

quenched to –56°C. This signal was assigned to the persulfenyl radical Me2NCS3•. In

addition, a weak signal at g = 2.006 was recorded after heating TMTD for 2 hours to

145°C; this signal was assigned to an unknown carbon radical [6]. All these

assignments have not been substantiated by theoretical calculations as yet, and the

structures of various Me2NCSn• species except for the thiuram radical Me2NCS2

• are

as yet unknown. The structure of Me2NCS2• was investigated using MNDO RHF-CI

calculations by Plyusnin et al [8]. A planar C2NCS2 skeleton was predicted with two

identical C—S bonds 1.61 Å in length.

Nichols et al generated the radical iPr2NCS2• by thermolysis of the related

tetraisopropylthiuram disulfide [9]. Detection of this radical was achieved by ESR

spectroscopy. The precursor molecule was first dissolved in decalin and heated to

temperatures above 45°C. A single line at g = 2.015 was observed in the ESR

spectrum. From the temperature dependence of the ESR signal intensity, the

dissociation enthalpy of (iPr2NCS2)2 was calculated at 104 ± 2 kJ mol–1. The

127

dissociation enthalpy of the corresponding methyl derivative (TMTD) was roughly

estimated to lie in the range of 100–130 kJ mol–1 [9]. The iPr2NCS2• radical generated

using laser flash photolysis from the corresponding disulfide with acetonitrile as a

solvent showed an absorption band at 580 nm [8].

Depending on the nature and concentration of the Me2NCSn• radicals, the

vulcanization of rubber accelerated by TMTD can take place by either an anionic or a

radical mechanism or by both. Under equilibrium conditions, the concentration of

radicals depends on the various dissociation enthalpies of TMTD, which are not

known accurately. To address this question, the structures of the radicals Me2NCSn• (n

= 1–4) have been calculated and the various homolytic dissociation enthalpies of

TMTD have been studied using high-level ab initio calculations according to the

G3X(MP2) and G3X(MP2)-RAD theories. In addition, the influence of a metal cation

on the dissociation was investigated using lithium ions as a probe and as a model ion

for the impact of zinc cations during rubber vulcanization accelerated by the

TMTD/ZnO system. The zinc ions on the surface of solid ZnO particles have an

atomic charge much smaller than +2. Their interaction with TMTD and other

substrates can therefore be modeled by a singly charged ion such as Li+.


The structures and energies of TMTD and its various derivatives were examined using

G3X(MP2) theory [10]. This composite method corresponds effectively to

QCISD(T)/G3XLarge//B3LYP/6-31G(2df,p) energy calculations together with zero-

128

point vibrational and isogyric corrections. The G3X(MP2) theory represents a

modification of G3(MP2) [11] theory with three important changes: (1) B3LYP/6-

31G(2df,p) instead of MP2(full)/6-31G(d) geometry, (2) B3LYP/6-31G(2df,p) instead

of HF/6-31G(d) zero-point vibrational energy, and (3) the addition of a g polarization

function to the G3Large basis set for second-row atoms at the Hartree-Fock level. All

three modifications are particularly important for the proper description of sulfur-

containing species. Harmonic fundamental vibrations were calculated at the

B3LYP/6-31G(2df,p) level to characterize stationary points as equilibrium structures

with all real frequencies and transition states with one and only one imaginary

frequency. Selected reactions were also examined using G3X(MP2)-RAD theory [12],

which was designed to provide better theoretical treatment for free radicals. The RHF

and UHF formalisms were employed for closed-shell and open-shell systems,

respectively, except for certain G3X(MP2)-RAD calculations.

For all investigated species, a charge density analysis was performed using the natural

bond orbital (NBO) [13] approach based on the B3LYP/6-31G(2df,p) wave function.

NBO atomic charges of small molecules have recently been demonstrated to agree

well with experimental values obtained from X-ray diffraction data [14]. Unless

otherwise noted, relative energies reported in this chapter correspond to G3X(MP2)

∆H°298 values. The same temperature of 298 K holds for the reported reaction

enthalpies and Gibbs free energies. All reported structural parameters correspond to

those obtained at the B3LYP/6-31G(2df,p) level. The electronic transition energies of

the Me2NCS2• radical and its isopropyl analogue were performed using the time-

dependent DFT (TD-DFT) method [15] at the B3LYP/6-31G(2df,p) level. The DFT

calculations of ESR g-tensors and g-values were based on the gauche-including

129

atomic orbitals (GIAO) approach [16-18] and were carried out using the Amsterdam

density functional package ADF [19-21]. All other calculations were performed using

the GAUSSIAN 03 [22] and MOLPRO 2002 [23] programs.


6.3.1 Structures and Stabilities of Tetramethylthiuram Sulfides and Their Related

Radicals

The calculated structure of the TMTD molecule is shown in Figure 6.2, together with

the structural data obtained from X-ray diffraction on single crystals at 138 K. Solid

TMTD crystallizes as two allotropes: a low-temperature phase of unknown structure

below 133 K and a monoclinic high-temperature phase in which the two independent

molecules of the asymmetric unit are of C2 symmetry [24]. The most stable calculated

structure 1 is also of C2 symmetry. Calculated and experimental bond lengths and

generally agree well to within 0.03 Å. There exists a rotational isomer of 1 (1a) that is

less stable by 13.9 kJ mol–1. The main difference between 1 and 1a are the N–C–S1–

S2 and the N–C–S3–S4 improper torsional angles. These angles are both –177.3° in 1,

compared to 9.5° and 155.0° in 1a. The C–S2–S3–C torsional angle is 88.7° in 1 and

78.3° in 1a. It is worth noting that the C—N bonds in 1 and 1a are relatively short

(1.332 – 1.347 Å) compared to a typical C—N bond length of 1.46 Å. This indicates

some extent of delocalization of the π electrons in the NCS frameworks.

130

1.347

(1.340)

1.840

(1.809)

2.023

(2.019)

1.650

(1.653)

1.347

(1.340)

1.840

(1.809)

2.023

(2.019)

1.650

(1.653) 1.3491.658

1.821

2.105

1.791

1.666

1.332

1.3491.658

1.821

2.105

1.791

1.666

1.332

1 (C2) 1a (C1)

S1

S2

S3

S4

Figure 6.2. B3LYP/6-31G(2df,p) optimized geometries of the two conformers of TMTD. Distances are in Ångstroms. X-ray structural parameters are given in parentheses.

Previous authors have discussed the possibilities that on heating TMTD dissociates

symmetrically (S—S bond cleavage) giving two Me2NCS2• radicals (2) or

asymmetrically (C—S bond cleavage) yielding the radicals Me2NCS• (3) and

Me2NCS3• (4) [6]. The calculated structures of these radicals are shown in Figure 6.3.

It is interesting to note that the structural units C2NCSn (n = 1–3) are either exactly

planar (2 and 3) or nearly planar (4). Expectedly, the most dramatic structural changes

on dissociation of TMTD take place at the CS2 unit, particularly with the bond lengths.

In TMTD there are two regular C=S double bonds (1.650 Å) and two C—S single

bonds (1.840 Å). Radical 3 has just one very short C=S bond of length 1.610 Å. Also,

the C—N bond length has decreased from 1.347 Å in 1 to 1.310 Å in 3. Evidently, the

π electron system of 3 extends from sulfur to nitrogen. In contrast, radical 2 has two

identical C—S bonds (1.710 Å) and a relatively short C—N bond (1.335 Å).

Structurally, the persulfenyl radical 4 resembles the parent compound 1.

131

1.456 1.335 1.700 1.467

1.461

1.3101.610 1.459

1.460

1.342

1.698

1.738

2.075

2 (C2v, 2B2) 3 (Cs,

2A′) 4 (C1, 2A)

S1 S1

S1

S2

S3

S2

1.456 1.335 1.700 1.467

1.461

1.3101.610 1.459

1.460

1.342

1.698

1.738

2.075

2 (C2v, 2B2) 3 (Cs,

2A′) 4 (C1, 2A)

S1 S1

S1

S2

S3

S2

Figure 6.3. B3LYP/6-31G(2df,p) optimized geometries of the radicals Me2NCSn• (n = 1–3). Distances

are in Ångstroms. Molecular symmetry and electronic states are given in parentheses.

It can be seen, from spin density plots in Figure 6.4, that all three radicals 2, 3 and 4

correspond to a σ-type radical. For radical 2 (2B2 ground electronic state), the

unpaired spin density is located almost exclusively in the two sulfur atoms.

Essentially, the SOMO is a linear combination of the two sulfur lone pair orbitals. The

resonance hybrids of 2 clearly show the delocalization of the π orbitals over the four-

center NCS2 framework as well as the delocalization of the spin density (see Figure

6.5).

132

Figure 6.4. UB3LYP/6-31G(2df,p) spin density plots of the Me2NCSn• radicals (n = 1–3)

Figure 6.5. Chemical drawing of 2, showing resonance structures.

The radical 2 has a large dipole moment of 4.92 D as expected from contributing

dipolar resonance forms. For comparison, the parent radical HCS2• has a smaller

dipole moment of 1.49 D at the same level of theory. It is noted that the π-type radical

(2A2) of 2 corresponds to an excited state with a relative enthalpy of 71.3 kJ mol–1.

The stability of this radical is readily reflected in the S—H bond dissociation enthalpy

133

(BDE) of the corresponding acid Me2NC(=S)SH. The calculated value of 301.7 kJ

mol–1 is the lowest among various sulfur-containing compounds as the following

examples demonstrate: 381.6 (H2S), 365.7 (CH3SH), 349.4 (PhSH), 330.5 (HOSH),

and 318.0 (HSSH) kJ mol–1 [25]. The stabilizing effect of the dimethylamino group in

radical 2 can be estimated by the following isodesmic reaction:

Me3N + HCS2• → Me2NCS2

• + CH4

∆H°298 = −159.0 kJ mol–1

(1)

The computed radical stabilization energy is substantially larger than those calculated

for carbon radicals XCH2• [26]. As in the case of 2, the spin density of the radical 4

resides mainly in the two terminal sulfur atoms. In the case of 3, however, the

unpaired electron spin is located in both the carbon and sulfur atoms (see Figure 6.4).

The ESR g-values of radicals 2−4 were investigated by the DFT-GIAO method at the

BLYP using Slater-type orbitals of the TZ2P basis set. The calculated isotropic g-

values are 2.017, 2.001 and 2.017 for 2, 3, and 4, respectively. Both radicals 2 and 4

are predicted to have similar g-values and would yield triplet signals on the ESR

spectrum, as the lone electron resides on the sulfur atoms which are β from the

nitrogen atom having nuclear spin I = 1 which causes peak splitting in the ratio 1:1:1.

The question of whether radical 2 or 4 is responsible for the observed ESR signal

arises. Homolytic cleavage of the S—S bond in TMTD yields two molecules of 2.

Similarly, the cleavage of the C—S bond in TMTD would yield 3 and 4. As predicted

from the calculated isotropic g-values, a mixture of 3 and 4 would show up as two

134

distinct triplets with g-values of 2.001 and 2.017, respectively. It is thus safe to assign

the observed ESR signal to the radical 2. The homolytic dissociation energies of the

S—S bond and of one of the C—S bonds in TMTD further provides evidence for this

claim. S—S and C—S bond dissociation energies differ considerably by 111.3 kJ

mol–1 shown in equations (2) and (3). The more favorable process yields radical 2 as

the product.

(Me2NCS2)2 (1) → 2Me2NCS2• (2)

∆H°298 = 155.1 kJ mol–1 ∆G°298 = 100.3 kJ mol–1

(2)

(Me2NCS2)2 (1) → Me2NCS• (3) + Me2NCS3

• (4)

∆H°298 = 266.4 kJ mol–1 ∆G°298 = 206.3 kJ mol–1

(3)

The calculated g-value of 3 appears to match well with the ESR signal previously

assigned to an “unknown carbon radical” [6]. The isotropic g-value of the radical

iPr2NCS2•, which has a definitive ESR spectrum, was computed to be 2.015, in perfect

agreement with the observed g-value. This computed g-value for iPr2NCS2• is very

close to that calculated for the methyl analogue (g = 2.017). This is not unexpected as

both radicals have similar electronic structures. It thus appears that the experimental

g-value of Me2NCS2• (g = 2.03) is somewhat high and may require re-examination.

The low S—S bond dissociation enthalpy is attributed to the remarkable stability of

the Me2NCS2• radical. To provide more definitive estimates of the various bond

135

dissociation enthalpies of 1, the structures and energies of species 1 to 5 were also

examined by the G3X(MP2)-RAD method, which yields more reliable predictions of

thermochemical data of free radicals [27]. From the data in Table 6.1, the computed

dissociation enthalpies obtained by both methods agree within 5 kJ mol–1. This result

is perhaps not surprising as the radicals 2, 3 and 4 do not have a significant spin

contamination in their UHF-based wave functions ( ⟩⟨2S = 0.77 in all cases). Hence,

the best theoretical estimate of the lowest dissociation enthalpy of TMTD to form two

Me2NCS2• radicals is 150.0 kJ mol–1, in perfect agreement with the value of 151 kJ

mol–1 reported by Blokh [28] who investigated the isotopic exchange between TMTD

and S8 using radioactive 35S. In this context, it is interesting to note that TMTD

triggers radical polymerization reactions of styrene and of methyl-2-methaylacrylate

already at 70°C [29].

Reaction in equation G3X(MP2) G3X(MP2)-RAD

2 155.1 150.0

3 266.4 268.0

4 111.3 117.8

6 267.0 265.2

7 111.9 115.3

Table 6.1. Calculated Reaction Enthalpies (∆H°298, kJ mol–1) Using the G3X(MP2) and G3X(MP2)-RAD Methods.

From the above data, it follows that the disproportionation or “self-saturation” of

radical 2 is thermodynamically unfavorable:

2 Me2NCS2• (2) → Me2NCS

• (3) + Me2NCS3• (4)

136

∆H°298 = 111.3 kJ mol–1 ∆G°298 = 120.5 kJ mol–1

(4)

The dissociation of one of the C—N bonds of 1 with formation of another radical (6)

has also been discussed in the literature:

(Me2NCS2)2 (1) → Me2N• (5) + Me2NC2S4

• (6)

(5)

However, radical 6 is not a stable equilibrium structure. Attempts to optimize its

geometry at the B3LYP/6-31G(2df,p) level results in a CS2 fragment dissociating

from Me2NCS2•. The overall reaction is therefore as follows and no carbon radical is

formed:

(Me2NCS2)2 (1) → Me2N• (5) + CS2 + Me2NCS2

• (2)

∆H°298 = 267.0 kJ mol–1 ∆G°298 = 173.4 kJ mol–1

(6)

Based on the calculated energies, the reported formation of a carbon radical by

thermal dissociation of TMTD at 145°C is unlikely.

The homolytic dissociation of radical 2 is less endothermic than the homolysis of the

S—S bond of TMTD:

Me2NCS2• (2) → Me2N

• (5) + CS2

137

∆H°298 = 111.9 kJ mol–1 ∆G°298 = 73.2 kJ mol–1

(7)

Therefore, at temperatures high enough to initiate the reaction in equation (2), the

reaction in equation (7) will also take place. However, since the concentration of

radical 2 will always be very small, the concentration of Me2N• will be even smaller

unless CS2 is allowed to escape, in which case Me2N• will be continuously produced.

The thermal decomposition of TMTD has been reported to yield eventually CS2,

tetramethylthiurea (TMTU) and elemental sulfur [6]. This reaction is predicted to be

slightly endothermic, but exergonic if all species are gaseous.

(Me2NCS2)2 (1) → CS2 + (Me2N)2CS + ⅛S8

∆H°298 = 2.7 kJ mol–1 ∆G°298 = −35.2 kJ mol–1

(8)

However, other studies show that the product of thermal TMTD decomposition was

not TMTU but tetramethylthiuram monosulfide (TMTM) [30]. The radicals 2, 4 and 5,

as well as tetramethylthiuram polysulfides (TMTP) are believed to be intermediates in

these reactions. In this context, the hitherto unknown structure of Me2NCS4• was

calculated to find out if it was a cyclic or chain-like species. Three structural

conformers were obtained (7, 7a and 7b) and their structures are given in Figure 6.6.

The two conformers 7 and 7a differ mainly by their torsional angles. The global

minimum structure 7 has all non-hydrogen atoms practically in one plane. Dipole

moment and NBO atomic charges are given in Table 6.2. The relative enthalpy of 7a

138

with respect to 7 is 11.3 kJ mol–1. In 7a, the sulfur atoms S3 and S4 are out of the

plane defined by the NCS2 group. Torsional angles for 7a are: N−C−S−S, −32.3°;

S−C−S−S, 149.9°; C−S−S−S, −59.1°. The third conformer 7b was obtained when a

cyclic structure with a five-membered CS4 heterocycle was constructed and geometry

optimization attempted. This species, with a relative enthalpy of 62.6 kJ mol–1 with

respect to 7, is of C2 symmetry with one rather long S—S bond of 2.700 Å connecting

the two disulfide groups. Therefore, radical 7b may be denoted as Me2NC(S2)2•. The

spin density plots of the three structures are presented in Figure 6.7. Essentially, 7 and

7a are π-type radicals with unpaired spin densities located in the terminal S—S

moiety. 7b is better described as a σ-type radical. The predicted isotropic g-values of

7, 7a and 7b are 2.024, 2.029 and 2.011 respectively. These ESR g-values may be

used to differentiate these species from the related radicals 2, 3 and 4.

139

1.348

1.662

1.796

2.148 1.957

S1

S2

S3

S4

1.342

1.660

1.789

2.165

1.960

S1

S2

S3

S4

1.342

1.740

2.040

2.700

S1

S2 S3

S4

7 (C1)

7a (C1) 7b (C2)

Figure 6.6. B3LYP/6-31G(2df,p) optimized geometries of the conformers of the Me2NCS4• radical, 7,

7a and 7b, together with the numbering convention of sulfur atoms. Distances are given in Ångstroms.

Species N S1 S2 S3 S4 µ

TMTD (1) −0.44 −0.12 +0.17 +0.17 −0.12 0.95

Me2NCS• (3) −0.45 −0.02 4.44

Me2NCS2• (2) −0.42 −0.04 −0.04 4.92

Me2NCS3• (4) −0.43 +0.06 +0.24 −0.21 6.03

Me2NCS4• (7) −0.43 −0.12 +0.15 +0.20 −0.16 5.28

Me2NCS4• (7a) −0.43 +0.06 +0.16 +0.09 −0.13 4.51

Me2NC(S2)2• (7b) −0.41 +0.26 −0.23 +0.26 −0.23 7.73

Table 6.2. NBO Atomic Charges of the Donor Atoms and Dipole Moments µ in Debye of Species 1–4, 7, 7a and 7b.

140

Figure 6.7. UB3LYP/6-31G(2df,p) spin density plots of 7, 7a and 7b.

The formation of 4 and 7 by sulfuration of 2 with elemental sulfur is slightly

endothermic and endergonic, and the corresponding equilibria lie on the left hand side

at moderate temperatures:

Me2NCS2• (2) + ⅛ S8 → Me2NCS3

• (4)

∆H°298 = 41.6 kJ mol–1 ∆G°298 = 40.1 kJ mol–1

(9)

141

Me2NCS2• (2) + ¼ S8 → Me2NCS4

• (7)

∆H°298 = 9.3 kJ mol–1 ∆G°298 = 16.8 kJ mol–1

(10)

By recombination reactions between radicals 2, 4 and 7, various tetramethylthiuram

polysides (TMTP) can be formed. Thus, one possible reaction sequence for the

formation of TMTP on heating of TMTD is 1 → 2 → 4 → 7 → TMTP. The

disproportionation of the disulfide TMTD into the monosulfide (TMTM) and the

trisulfide (TMTT) has also been discussed in the literature [31].

2 (Me2NCS)2S2 (TMTD, 1) → (Me2NCS)2S (TMTM) + (Me2NCS)2S3 (TMTT)

∆H°298 = 6.5 kJ mol–1 ∆G°298 = 9.5 kJ mol–1

(11)

(Me2NCS)2S2 (TMTD, 1) + Me2NCS2• (2) → (Me2NCS)2S (TMTM) + Me2NCS3

• (4)

∆H°298 = 47.3 kJ mol–1 ∆G°298 = 45.1 kJ mol–1

(12)

The structures of TMTU, TMTM and TMTT are shown in Figure 6.8. The trisulfide

molecule is of C2 symmetry with normal geometrical parameters. S—S bond lengths

are 2.053 Å in length, and the C−S−S−S torsional angle is 85.4°. The substituents

Me2NC(=S) are trans to each other. It is not known whether the reaction in equation

(11) proceeds via the intermediate formation of radicals, for example, as in equations

(2), (12) and (13):

142

Me2NCS2• (2) + Me2NCS3

• (4) → (Me2NCS)2S3 (TMTT)

∆H°298 = 195.9 kJ mol–1 ∆G°298 = 135.9 kJ mol–1

(13)

1.3741.677

1.341

1.6561.817

1.803

1.6621.352

1.3491.652

1.8362.053

TMTU (C2) TMTM (C1)

TMTT (C2)

Figure 6.8. B3LYP/6-31G(2df,p) optimized geometries of TMTU, TMTM and TMTT. Distances are in Ångstroms. Molecular symmetry is given in parentheses.

It is remarkable that the S—S bond dissociation enthalpy of TMTT (195.9 kJ mol–1) is

larger than that of TMTD (150.0 kJ mol –1), whereas normally disulfide bonds are

thermally more stable than the tri- and tetrasulfide chains of organic polysulfides [32,

33].

143

6.3.2 Impact of Metal Cations on the Dissociation of Tetramethylthiuram Di- and

Trisulfides.

The various radicals as well as TMTD itself contain several donor atoms with lone

electron pairs, and these are expected to interact with the positively charged metal

atoms of zinc oxide or zinc complexes added to a rubber vulcanization mixture as

essential ingredients or “activators”. The NBO atomic charges of 1 to 4 (Table 6.2)

together with the well known polarizability values indicate that the terminal sulfur

atoms are likely to be the preferred sites for any cation coordination, while the planar

coordinated nitrogen atoms are of lower Lewis basicity despite their higher negative

charge. For initial structures with a Li+ ion directly linked to the nitrogen atoms of

radicals 2 and 4, the Li+ ion moved to the sulfur atoms on geometry optimization. This

finding is in line with the structures of several metal complexes containing TMTD as

a bidentate chelate ligand linked to a metal atom by the thione sulfur atoms [34], e.g.

in [ZnI2(TMTD)] [35], and [HgI2(TMTD)] [36]. Certain organometallic complexes

react with TMTD or the corresponding ethyl derivative at 20°C with cleavage of the

S—S bond and oxidative addition of the thiocarbamato units to the metal atoms. This

holds true for [MeAuPPh3] yielding [MeAu(S2CNMe2)2] [37] and for [(CpCr(CO)3)2]

yielding [CpCr(CO)2(S2CNEt2)] [38].

To study the influence of a metal cation on the S—S bond dissociation energy of

TMTD, the smaller lithium atom was used as a probe to save computational time and

to allow for a high level of theory for theoretical investigation. All species 1 to 4 form

stable chelate complexes with a Li+ ion, which becomes two- or three-coordinated.

The optimized structures of these lithium complexes are shown in Figure 6.9.

144

2.3492.371

2.5171.681

1.683

1.323

1.8202.070

1.824

1.332

Li

1.291

2.273

1.633

2.331

Li

2.420

1.7251.310

Li 1.328

1.845

1.679

2.310

1.968

2.413

Li

8 (C1) 9 (Cs)

10 (Cs) 11 (Cs)

S1

S2

S3

S4

Figure 6.9. B3LYP/6-31G(2df,p) optimized geometries of the cationic lithium complexes [Li(TMTD)]+ (8), [Li(SCNMe2)]

•+ (9), [Li(S2CNMe2)]•+ (10) and [Li(S3CNMe2)]

•+ (11). Distances are in Ångstroms. Molecular symmetry is given in parentheses.

The structure of [Li(TMTD)]+ (8) is of C1 symmetry and characterized by a seven-

membered ring since the metal ion forms a bridge between the two thione sulfur

atoms of TMTD. The Li—S bond lengths of 2.349 and 2.371 Å are in line with the

data of other lithiated sulfur-rich compounds [39-41]. There is one additional Li…S

contact of 2.517 Å to one side of the disulfide sulfur atoms, and the metal atom may

thus be considered as three-coordinated. The structural parameters of the TMTD

ligand changed only slightly from those of the free molecule, except for the two

torsional angles S3−S2−C=S1 (134.3°) and S3−S2−C−N (−50.5°) on the left side of

the molecule 8 in Figure 6.9. A structure of C2 symmetry with two-coordinated Li+ is

145

also a minimum on the potential energy hypersurface of [Li(TMTD)]+, but this isomer

is by 10.7 kJ mol–1 less stable than 8.

The small Me2NCS• radical coordinates to a lithium cation via both the sulfur and the

carbon atom resulting in the formation of a three-membered ring. All non-hydrogen

atoms of [Li(SCNMe2)]•+ (9) are coplanar, and the molecular symmetry is Cs. The

shortest S…H contact is 2.840 Å. The structure of [Li(S2CNMe2)]•+ (10) is also of Cs

symmetry and characterized by a non-planar four-membered CS2Li heterocycle. The

metal ion is not in the plane of the C2NCS2 skeleton; the mirror plane bisects the

angles C2N and CS2. This structure is obviously a compromise between the ion-dipole

attraction, which would be maximum if Li+ was in the C2NCS2 plane and located on

the dipole vector that is in the direction of the N—C (sp2) bond, and a covalent bond

between Li+ and the sulfur 3p lone pairs, which are perpendicular to the C2NCS2 plane

and are part of the four-center π electron system of the radical. The torsional angles

S−C−S−Li are ±37.9°. The sulfur-rich radical cation [Li(S3CNMe2)]•+ (11) is also of

Cs symmetry with all non-hydrogen atoms in one plane. Both the C=S and S—S

bonds contract on Li+ coordination while the C—S bond considerably lengthens by

0.107 Å. Interestingly, the C—N bonds of 1 to 4 become shorter upon complexation

with Li+ (See Figures 6.2, 6.3 and 6.9). In general, the geometrical features of the

S…Li+ coordination in 8 to 11 are similar to those calculated for the lithium

complexes of sulfur clusters [39-41].

146

Complex ∆H°298 ∆G°298 Μ

[Li(TMTD)]+ (8) −284.9 −243.8 3.01

[Li(SCNMe2)]•+ (9) −166.1 −137.2 6.16

[Li(S2CNMe2)]•+ (10) −181.1 −149.0 4.69

[Li(S3CNMe2)]•+ (11) −253.6 −203.9 2.46

[Li(S2CNMe2)2]•+ (12) −314.3 −235.2 4.74

[Li(TMTT)]+ (14) −291.5 −254.9 3.17

[Li(S3CNMe2)2]•+ (15) −409.2 −326.0 1.42

Table 6.3. Binding energies, in kJ mol–1, and dipole moments µ, in Debye, of the lithium complexes 8–12, 14 and 15. Energies correspond to G3X(MP2) values, and dipole moments are calculated at the B3LYP/6-31G(2df,p) level.

The binding energies of the four complexes 8 to 11, that is, the energy differences

between Li+ and the free ligand on one side and the complex on the other side, are

listed in Table 6.3. These data demonstrate that the binding energies of the two-

coordinated species [Li(SnCNMe2)]•+, are smaller than that for [Li(TMTD)]+, but

these energies increase with the sulfur content of the ligand and reach a maximum for

n = 3. In the latter case, the radical cation complex contains a five-membered

metallacycle. As a consequence of the increase in binding energy of with sulfur

content, the sulfurization of 10 by elemental sulfur to produce the trithiolane

derivative 11 is exothermic and exergonic:

[Li(S2CNMe2)]•+ (10) + ⅛S8 → [Li(S3CNMe2)]

•+ (11)

∆H°298 = −19.9 kJ mol–1 ∆G°298 = −13.7 kJ mol–1

(14)

147

This finding is important, since only the polysulfidic complexes are assumed to

abstract allylic hydrogen atoms from rubber molecules with the formation of H2S.

Evidently, the polysulfido complexes are formed spontaneously if elemental sulfur is

added, in contrastto the endothermic sulfuration reactions of the free radical 2 (see

equation (9)). Using the more reactive (metastable) polymeric sulfur “Crystex” [42],

the equilibrium of the reaction in equation (14) will lie even more toward the right.

Due to the high stability of the complex [Li(TMTD)]+, the homolytic dissociation

energies of the TMTD ligand with formation of one free radical and one ligand-

stabilized radical are drastically different from those of free TMTD:

[Li(TMTD)]+ (8) → [Li(S2CNMe2)]•+ (10) + Me2NCS2

• (2)

∆H°298 = 257.6 kJ mol–1 ∆G°298 = 193.7 kJ mol–1

(15)

[Li(TMTD)]+ (8) → [Li(S3CNMe2)]•+ (11) + Me2NCS

• (3)

∆H°298 = 307.4 kJ mol–1 ∆G°298 = 245.9 kJ mol–1

(16)

[Li(TMTD)]+ (8) → [Li(SCNMe2)]•+ (9) + Me2NCS3

• (4)

∆H°298 = 384.9 kJ mol–1 ∆G°298 = 312.7 kJ mol–1

(17)

148

A more likely thermal reaction of 8 is the isomerization to the bis-chelate complex 12

with a four coordinated lithium atom and the TMTD ligand dissociated into two equal

fragments:

[Li(TMTD)]+ (8) → [Li(S2CNMe2)2]+ (12)

∆H°298 = 127.1 kJ mol–1 ∆G°298 = 153.7 kJ mol–1

(18)

The electronic ground state of species 12 is a triplet, which is by 12.0 kJ mol–1 more

stable than the singlet state. The geometry of 12 is shown in Figure 6.10. It is

interesting to note that the Li is tetrahedrally coordinated to four S atoms in this

species. The mode of binding for the Li(S2CNMe2) moieties is similar to that in 10,

where the Li+ ion binds slightly above the CS2 plane. The fact that the dissociation of

the TMTD ligand in 8 to give the dissociated ligands present in 12 requires only 127.1

kJ mol–1 compared to 155.1 kJ mol–1 for the dissociation of free TMTD at the

G3X(MP2) level, shows that the lithium cation facilitates S—S bond homolysis in

TMTD.

Li

2.506

1.717

1.314

Figure 6.10. B3LYP/6-31G(2df,p) optimized geometry of the triplet lithium complex [Li(S2CNMe2)2]+.

Distances are given in Ångstroms.

149

If additional lithium cations are available, complex 8 may be cleaved into two

identical complexes 10:

[Li(TMTD)]+ (8) + Li+ → 2 [Li(S2CNMe2)]•+ (10)

∆H°298 = 75.4 kJ mol–1 ∆G°298 = 43.6 kJ mol–1

(19)

The overall reaction of TMTD with an excess of Li+ is thus strongly exothermic:

TMTD (1) + 2 Li+ → 2 [Li(S2CNMe2)]•+ (10)

∆H°298 = −209.2 kJ mol–1 ∆G°298 = −200.0 kJ mol–1

(20)

The various dissociations of TMTD are shown schematically in Figure 6.11.

-300

-150

150

0

H°298 (kJ mol–1)

TMTD

2 Me2NCS2•

+155.1

–284.6+Li+

[Li(TMTD)]+

+127.1

–157.5+Li+

[Li(S2CNMe2)2]+

+75.4

–209.2+2 Li+

+Li+

2 [Li(S2CNMe2)]•+

Figure 6.11. Schematic energy level diagram for the homolytic dissociation and complex formation reactions of TMTD with and without Li+ cations, based on G3X(MP2) computed reaction enthalpies.

150

The reaction of 12 with S8 to the mixed ligand complex [Li(S2CNMe2)(S3CNMe2)]+

(13) in its triplet state is slightly endothermic:

[Li(S2CNMe2)2]+ (12) + ⅛ S8 → [Li(S2CNMe2)(S3CNMe2)]

+ (13)

∆H°298 = 2.2 kJ mol–1 ∆G°298 = –11.3 kJ mol–1

(21)

Attempts to optimize the structure of [Li(S2CNMe2)(S3CNMe2)]+ as a singlet resulted

in the novel species [Li(TMTT)]+ (14) in which the trisulfide ligand TMTT forms

three Li—S bonds to the metal cation. This complex is by 133.1 kJ mol–1 more stable

than 13.

TMTT + Li+ → [Li(TMTT)]+ (14)

∆H°298 = −291.5 kJ mol–1 ∆G°298 = −254.9 kJ mol–1

(22)

As a result of the high stability of 14, the following sulfurization reaction with spin-

crossover is exothermic:

[Li(S2CNMe2)2]+ (12) + ⅛ S8 → [Li(TMTT)]+ (14)

∆H°298 = −130.3 mol–1 ∆G°298 = −156.5 kJ mol–1

(23)

Dissociation of the trisulfide TMTT in the presence of 2 equivalents of Li+ is also an

exothermic process:

151

TMTT + 2 Li+ → [Li(S2CNMe2)]•+ (10) + [Li(S3CNMe2)]

•+ (11)

∆H°298 = −229.9 kJ mol–1 ∆G°298 = −218.2 kJ mol–1

(24)

The formation of the paramagnetic lithium complex [Li(S3CNMe2)2]+ (15) as shown

in equation (25), is a thermodynamically favorable process. Interestingly, the Li+ ions

of 13 and 15 are coordinated to four sulfur atoms in a tetrahedral manner.

TMTD + Li+ + ¼ S8→ [Li(S3CNMe2)2]+ (14)

∆H°298 = −187.1 kJ mol–1 ∆G°298 = −158.5 kJ mol–1

(25)

The structures of 13, 14 and 15 are shown in Figure 6.12.

152

2.377

2.490 2.528

2.530

1.717

1.717

1.315

1.963

1.671

1.849

1.331

2.707

2.508

2.318

1.818

1.681

1.332

2.387

2.507

1.963

1.669

1.8501.332

13 (C1)

14 (Cs)

15 (C2)

Li

Li

Li

Figure 6.12. B3LYP/6-31G(2df,p) optimized geometries of the triplet species 13 and 15 and singlet species 14. Distances are given in Ångstroms. Molecular symmetry is given in parentheses.

The presence of free metal cations and elemental sulfur, TMTD and the

corresponding polysulfides TMTP are predicted to spontaneously dissociate with the

formation of chelate complexes provided the activation energy is available. This

activation energy is expected to be low, since the interaction of TMTD and TMTT

with a cation starts with the exothermic coordination to thione sulfur atoms.

Zinc chloride and tetramethylthiourea form a complex [ZnCl2(TMTU)2] in which the

metal atom is coordinated by two chloride anions and the thione sulfur atoms of two

TMTU molecules in an approximately tetrahedral manner [43]. Tetracoordinated zinc

153

complexes with the related anions RCS2– [44] and RCS3

– [45, 46] have been prepared

and tetra- and hexacoordinated zinc complexes with thiocarbamato anions Me2NCS2−

[47] and Me2NCS3− [4, 5] are well known. It can reasonably be assumed that the

radicals 2–4 can function as ligands in such complexes too. Preliminary DFT

investigations using lithium ions as a probe support this view.

Three-coordinated zinc ions on the surface of bulk ZnO particles are expected to

facilitate the formation of radicals from TMTD during the vulcanization of rubber

with this accelerator system in a similar fashion as the reaction in equation (20). Since

ca. 95% of a typical vulcanization mixture consists of rather non-polar materials such

as polyolefins and carbon black, there should be no strong solvent effects. Radical

reactions may then play a major role during the vulcanization of rubber with this

particular accelerator system. The predicted paramagnetic thiocarbamato zinc

complex [Zn(S2CNMe2)]2+ is expected to be sulfurized by elemental sulfur with

formation of the analogous polysulfidic trithiolane species, which can then react when

the polyolefins with crosslinking. At higher temperatures, additional redox reactions

take place between ZnO and TMTD yielding zinc dithiocarbamate, TMTU, elemental

sulfur, CS2, and zinc sulfate [48]. Possible intermediates of this reaction have recently

been characterized [49].


Tetramethylthiuram disulfide (TMTD) and the corresponding trisulfide (TMTT) are

predicted to dissociate on heating preferentially with homolytic dissociation of the

154

S—S bonds to give radicals. The reported g-values of some of these radicals match

those obtained experimentally from ESR spectra. The formed radicals Me2NCSn• (n =

1–4) are stabilized by electron delocalization. Interestingly, the S—S bond

dissociation enthalpy of TMTD is considerably smaller than that of TMTT, which is

rather uncommon in organosulfur chemistry. The low S—S bond dissociation

enthalpy of TMTD (150.0 kJ mol–1) is attributed to the remarkable stability of

Me2NCS2•, a σ-type radical with the unpaired spin density shared equally between

two sulfur atoms and a four-centered delocalized π system. Both the experimental

dissociation enthalpy and the ESR g-value of tetraisopropylthiuram disulfide are

reproduced well by these calculations. TMTD, TMTT and the radicals Me2NCSn• (n =

1–3) are excellent ligands toward metal cations. The high binding energies of the

complexes [Li(SnCNMe2)2]+ (n = 2, 3) are responsible for the unexpected finding that

the homolytic dissociation of the S—S bonds of both TMTD and TMTT are

facilitated by lithium cations, especially in the presence of elemental sulfur. Therefore,

spontaneous dissociation of these vulcanization accelerators is also expected in the

presence of zinc ions and consequently, also in a rubber vulcanization mixture

containing TMTD as an accelerator and zinc oxide as an activator.

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159

Chapter Seven

Dissociation Enthalpies and Activation Barriers of Various

Bis(thiocarbonyl)disulfides R−C(=S)−S−S−C(=S)−R

7.1 Introduction

The curing or vulcanization of natural rubber remains the essential crux of rubber

processing. Rubber vulcanization essentially involves the cross-linking of individual

rubber polymer chains (cis-polyisoprene) via sulfide bridges, making the overall

material tougher and resistant to chemical attack. The discovery of vulcanization

using elemental sulfur has been credited to Charles Goodyear, but it has recently been

discovered that Mesoamericans have achieved this method of rubber vulcanization as

early as 1600BC [1]. In this present day, a vast number of reagents associated with

sulfur vulcanization of natural rubber have been developed. These include

dithiocarbamates, thiurams, thiourea, thiazoles, and sulfenamides [2]. Examples of

these reagents are shown in Figure 7.1.

160

Dithiocarbamates

e.g. Zinc Dimethyldithiocarbamate

(ZDMC)

Thiurams

e.g. Tetramethylthiuram Disulfide

(TMTD)

Thioureas Thiazoles

e.g. Ethylene Thiourea

(ETU)

e.g. Mercaptobenzathiazole Disulfide

(MBTS)

Sulfenamides

e.g. N-cyclohexyl-2-benzothiazole sulfenamide

(CBS)

Dithiocarbamates


(ZDMC)

Dithiocarbamates


(ZDMC)

Thiurams


(TMTD)

Thiurams


(TMTD)

Thioureas Thiazoles

e.g. Ethylene Thiourea

(ETU)

e.g. Mercaptobenzathiazole Disulfide

(MBTS)

Sulfenamides

e.g. N-cyclohexyl-2-benzothiazole sulfenamide

(CBS)

Figure 7.1. Examples of different sulfur vulcanization reagents commercially available.

Previous work on the accelerator tetramethylthiuram disulfide (TMTD) showed that

its S—S bond dissociation enthalpy (150 kJ mol–1) was lower than the corresponding

trisulfide (TMTT) (190 kJ mol–1) based on high level calculations using the

G3X(MP2) composite method [3]. The dependence of the dissociation enthalpies of

the bis(thiocarbonyl)disulfides R−C(=S)−S−S−C(=S)−R, with R = H, Me, F, Cl, OMe,

SMe, NMe2 and PMe2 is investigated in this work. The better known

bis(thiocarbonyl)disulfides are those with R = Alkyl/Aryl [4], OMe [5], SMe [6, 7]

and NMe2 [8]. Less is known about the chlorine substituted case (R = Cl) [9], and no

references have been found for R = F and PMe2. Experimental structure

determinations of these compounds have not been published to date. However, limited

theoretical structures have been reported for R = H (at the HF/STO-3G level) [4] and

OMe (at the CNDO level) [10]. An important goal of this work is to obtain structural

and thermochemical data for the eight compounds mentioned above at a high level of

theory for the first time.

161

Many coordination compounds are known which contain the above compounds as

ligands or anionic ligands derived from these species by reductive cleavage of the S—

S bond [11]. To systematically investigate the ligand properties of the neutral species

(RCS2)2 and the radicals RCS2• that form as a result of S—S bond homolysis, with R

= H, Me, F, Cl, OMe, SMe, NMe2 and PMe2, the lithium cation is used as a model

metal cation for ligand coordination. The small Li+ cation allows for a high level of

theory to be used in quantum-chemical calculations. Since the binding energies of Li+

complexes towards a large number of Lewis bases are linearly related to those of

other cations such as Na+ and Mg2+ [12], the use of Li+ as a model cation seems

justified.

As mentioned in previous work, the stability of the Me2NCS2• radical is remarkable,

as a result of effective delocalization of the π orbitals and the spin density over the

four-center NCS2 framework [3]. However, the S—S bond dissociation enthalpy

(BDE) for TMTD is still endothermic by 155.1 kJ mol–1 at the G3X(MP2) level [3].

In industrial rubber vulcanization, activators such as zinc oxide are added to facilitate

S—S bond breaking by the coordination of TMTD to zinc. The binding of small

sulfur containing compounds to zinc oxide clusters was investigated by Steudel et al,

and it was found that the binding energy Zn4O4 clusters to S6, Me2S2, MeSH and

Me2S was exothermic by –60.7 to –106.7 kJ mol–1 [13]. Zinc oxide, however, is an

environmental pollutant that displays eco-toxicity [14] and the inhalation of zinc

oxide can result in severe pulmonary conditions [15, 16]. In an effort to find ways that

can further stabilize the RCS2• radical and hence reduce the reliance on zinc oxide

activators, suitable substituents R were considered based on their electron donating

162

ability as well as their ability to form resonance stabilized structures. The electron

donating ability of the –OSF5 group is demonstrated by its strong conjugate acid

HOSF5 [17], and the other two guanidine substituents (hereby denoted Gu1 and Gu2)

(See Figure 7.2) are noted for their strong basicity [18] as well as ability to form

numerous resonance-stabilized structures that result in the stabilization of the RCS2•

radical. Thus, bis(thiocarbonyl)disulfides with these chosen novel substituents were

also examined theoretically.

(Gu1) (Gu2)

Figure 7.2. The homolytic dissociation of (RCS2)2 with novel substituents R.


The structures and energies of the disulfides (RCS2)2, radicals RCS2•, and their

lithiated complexes for R = H, Me, F, Cl, OMe, SMe, NMe2 and PMe2 were examined

at the G3X(MP2) level of theory [19]. The G3X(MP2) method corresponds

effectively to energies calculated at the QCISD(T)/G3XL//B3LYP/6-31G(2df,p) level,

with certain assumptions made about additivity [20]. Correction terms considered

include the spin-orbit correction term, and a high-level empirical correction term to

address remaining deficiencies. Three important changes in G3X(MP2) theory from

G3(MP2) theory are important for the proper description of the molecules studied.

163

First, the use of B3LYP/6-31G(2df,p) geometry, second, the use of B3LYP/6-

31G(2df,p) zero-point vibrational energy scaled by an empirical factor of 0.9854, and

third, an addition of a g polarization function to the G3Large basis set for second-row

atoms at the Hartree-Fock level.

Structures of the novel bis(thiocarbonyl)disulfides (F5SOCS2)2, ((Gu1)CS2)2 and

(Gu2)CS2)2 and their associated radicals F5SOCS2•, (Gu1)CS2

• and (Gu2)CS2• were

optimized at the B3LYP/6-31G(d) level. Singlet biradical transition states

corresponding to homolytic S—S bond cleavage were optimized at this level of theory

as well, using the UHF formalism and the mixing of the HOMO and LUMO to

destroy α and β spatial symmetries. Energy calculations were then carried out at the

MP2(fc)/6-311+G(d,p) level of theory based on the B3LYP/6-31G(d) optimized

structures.

The RHF formalism was employed for all closed shell species, and the UHF

formalism was employed for all open-shell and singlet diradical species. Frequency

calculations were carried out to determine the nature of minima calculated, those with

all real frequencies correspond to local minima, and those with one and only one

imaginary frequency correspond to transition state structures. All calculations were

carried out using the GAUSSIAN 03 [21] and MOLPRO 2002 [22] suite of programs.


The typical structures of (RCS2)2, RCS2•, [Li(RCS2)2]

+, [Li(RCS2)]•+ and triplet

164

[(RCS2)Li(S2CR)]+ together with their atom numbering conventions, are shown in

Figure 7.3.

(RCS2)2

(C2)RCS

2•

(C2v or Cs)

[Li(RCS2)2]+

(C1)

[Li(RCS2)]•+

(Cs or C1)[(RCS2)Li(S2CR)]

+

(C2)

Figure 7.3. Typical structures of (RCS2)2, RCS2

•, [Li(RCS2)2]+, [Li(RCS2)]

•+ and triplet [(RCS2)Li(S2CR)]+. Molecular symmetry is indicated in parentheses.

7.3.1 Substituted bis(thiocarbonyl)disulfides, R = H, Me, F, Cl, OMe, SMe, NMe2,

PMe2.

Selected calculated structural parameters of (RCS2)2 for R = H, Me, F, Cl, OMe, SMe,

NMe2, PMe2 are presented in Table 7.1. Consistent with previous work done for

tetramethylthiuram disulfide (TMTD) [3], the most stable conformers of substituted

(RCS2)2 are of C2 symmetry. C=S bond lengths range from 1.603 Å for the fluorine

substituent, to 1.650 Å for the dimethylamino substituent. S—S bond lengths range

from 2.023 Å for the dimethylamino substituent, to 2.065 Å for the parent compound

(R = H). The C—R bond length interestingly shows the most variation for the various

165

substituents. For R = H, Me, F and Cl, the calculated C—R distances are slightly

shorter than the average experimental value. This trend is reversed for R = OMe, SMe,

NMe2 and PMe2. The biggest changes occur for R = OMe and NMe2 (difference in

bond lengths of 0.076 and 0.081 Å, respectively), which is indicative of the

delocalization of π electrons in the OCS2 and NCS2 frameworks. This extent of

delocalization becomes less pronounced for the third row analogues R = SMe and

PMe2, with the difference in bond lengths at 0.048 and 0.014 Å, respectively. It is also

interesting to note that in the case of the structure of (Me2PCS2)2, the C2PCS2 skeleton

is not planar but pyramidal at the phosphorus atom, unlike TMTD which has an

almost planar C2NCS2 framework (See Figure 7.4). This is not unusual, as phosphorus

prefers non-hybridization and a high s-orbital character of the lone-pair electrons, and

hence shows a reluctance to undergo s,p-hybridization to form planar coordinated

compounds [23, 24].

Substituent C1=S2

(= C5=S6) S3—S4

C1—R7 (= C5—R8)

Average Expt’l C—R

H 1.614 2.065 1.095 1.092 (C—H) Me 1.622 2.049 1.512 1.501 (C—C) F 1.603 2.045 1.331 1.327 (C—F) Cl 1.604 2.058 1.767 1.752 (C—Cl) OMe 1.628 2.034 1.327 1.405 (C—O) SMe 1.626 2.037 1.755 1.803 (C—S) NMe2 1.650 2.023 1.347 1.428 (C—N) PMe2 1.631 2.059 1.844 1.858 (C—P) Table 7.1. Selected calculated bond lengths (in Ångstroms) of (RCS2)2. Molecular geometry was optimized at the B3LYP/6-31G(2df,p) level of theory. Average experimental values for C—R bond lengths are included for comparison.

166

C5

S3S2

C2

C6

C4 P2

C3

S4

P1

S1

C1

Figure 7.4. The B3LYP/6-31G(2df,p) optimized structure of (Me2PCS2)2, showing the lack of planarity at the C2PCS2 framework.

The radicals RCS2• are typically of C2v (R = H, F, Cl, NMe2) or Cs symmetry (R = Me,

OMe, SMe, PMe2). Selected calculated structural parameters are shown in Table 7.2.

The RCS2• radical is resonance stabilized, as evidenced from the C—S bond lengths

of 1.659 to 1.700 Å, intermediate between the average length of a C—S single bond

(1.803 Å) and a C=S double bond (1.584 Å). The C—R bonds are also slightly shorter

than those in the corresponding dimers (RCS2)2, suggesting some increased extent of

delocalization of π electrons into the attachment atom of the R group (i.e. C for Me, O

for OMe, N for NMe2) (See Figure 6.5 in Chapter 6). The spin density plots of the

RCS2• radicals (see Figure 7.5) show that they are essentially σ-type, with the

unpaired spin density located in the two sulfur atoms.

167

Substituent C1—R4 C1—S H 1.089 1.665 Me 1.495 1.674 F 1.315 1.659 Cl 1.740 1.659 OMe 1.315 1.683a

1.679b SMe 1.728 1.685c

1.679d NMe2 1.335 1.700 PMe2 1.816 1.683

Table 7.2. Selected bond lengths (in Ångstroms) of RCS2

•. Molecular geometry was optimized at the B3LYP/6-31G(2df,p) level of theory. atrans- with respect to O–CH3.

bcis- with respect to O–CH3.

ctrans- with respect to S–CH3.

dcis- with respect to S–CH3.

Figure 7.5. UB3LYP/6-31G(2df,p) spin density plots of the radicals RCS2•.

168

∆H°298 and ∆G°298 for the homolytic S—S bond dissociation reaction (RCS2)2 → 2

RCS2• calculated at the G3X(MP2) level of theory are tabulated in Table 7.3. From

these results, TMTD has the lowest homolytic S—S bond dissociation enthalpy (BDE)

by far, attributed to the significant stability of Me2NCS2•, while the other substituted

(RCS2)2 have homolytic S—S BDEs in excess of 190 kJ mol–1.

(RCS2)2 → 2 RCS2

• Substituent ∆H°298 (kJ mol–1) ∆G°298 (kJ mol–1) H 210.8 158.4 Me 193.0 139.1 F 190.0 137.1 Cl 215.2 161.9 OMe 206.5 153.1 SMe 195.7 146.5 NMe2 155.1 100.2 PMe2 198.9 141.7

Table 7.3. ∆H°298 and ∆G°298 for the homolytic S—S bond dissociation reaction (RCS2)2 → 2 RCS2

• calculated at the G3X(MP2) level of theory.

An NBO charge analysis of (RCS2)2 and RCS2• was carried out at the B3LYP/6-

31G(2df,p) level, results of which are presented in Table 7.4. The NBO charge on S3

and S4 for (RCS2)2 is generally above 0.200, except for TMTD (R = NMe2) showing

a less positive charge of 0.172. For the RCS2• radical, the charges on S2, S3, S4 and

S6 even out, as compared to a more polarized situation in (RCS2)2. Along with the

two C1—S bonds effectively reaching the same bond length, this spreading of

electron density as seen in the NBO charges in C1, S2 and S3 is indicative of

resonance stabilization in the RCS2• radical. For the RCS2

• radical, charges on the R

groups for R = F, OMe and NMe2 are negative, possibly due to the electron

withdrawing effect of the electronegative attachment atoms F, O and N respectively.

169

NBO Charges R7 (=R8)a C1 (=C5) S2 (=S6) S3 (=S4) (RCS2)2 R = H 0.263 –0.608 0.090 0.255 Me 0.068 –0.373 0.069 0.236 F –0.292 –0.012 0.078 0.226 Cl 0.063 –0.503 0.164 0.277 OMe –0.135 –0.038 –0.040 0.212 SMe 0.352 –0.626 0.036 0.237 NMe2 0.068 –0.126 –0.115 0.172 PMe2 0.455 –0.742 0.044 0.242 RCS2

• R = H 0.255 –0.606 0.176 0.176 Me 0.071 –0.392 0.160 0.160 F –0.283 –0.044 0.164 0.164 Cl 0.087 –0.547 0.230 0.230 OMe –0.120 –0.075 0.057 0.137 SMe 0.385 –0.671 0.123 0.164 NMe2 –0.175 –0.165 0.040 0.040 PMe2 0.480 –0.770 0.145 0.145 Table 7.4. NBO charges of (RCS2)2 and RCS2

•, evaluated at the B3LYP/6-31G(2df,p) level of theory, for substituents R = H, Me, F, Cl, OMe, SMe, NMe2 and PMe2.

aRepresents the sum of NBO charges on the atomic centers in substituent R.

The effect of metal ion coordination on the S—S BDEs was investigated, using the

Li+ ion as a model metal cation. As with TMTD, the preferred Li+ coordination sites

for (RCS2)2 were actually at the terminal sulfur atoms instead of the attachment atom

of the substituent. Coordination of (RCS2)2 to Li+ ions generally leads to (RCS2)2

forming three favorable Li+…S interactions with a reduction in molecular symmetry

from C2 to C1 (see Figure 7.3). Selected calculated geometrical parameters of the

[Li(RCS2)2]+ complexes are presented in Table 7.5. All three Li+…S interactions for

all eight complexes studied are below 2.8 Å in length, and are, thus, expected to be

bonding interactions with a electron density maximum along their path [25]. For the

radicals [Li(RCS2)]•+, the Li+ ion is slightly out of the plane defined by C1–S2–S3

170

(See Figure 7.3). As already mentioned in previous work with the Me2NCS2• radical,

Li+ is at a compromised position between two forces: the ion-dipole attraction which

is maximum on the CS2 plane, and a covalent bond interaction with the sulfur 3p lone

pairs perpendicular to the CS2 plane. This positioning of the Li+ ion is also seen for

the triplet [(RCS2)Li(S2CR)]+ complexes. The Li+ is coordinated to two RCS2•

radicals via their terminal sulfur atoms, and in a tetrahedral manner. Overall C2

molecular symmetry is observed for these complexes. Selected geometrical

parameters for the [Li(RCS2)]•+ and triplet [(RCS2)Li(S2CR)]+ complexes are shown

in Table 7.6 and 7.7, respectively.

Substituent Li—S2 Li—S3 Li—S5 S3—S4 C1—R7 C6—R8 H 2.510 2.541 2.456 2.087 1.091 1.091 Me 2.474 2.483 2.433 2.080 1.489 1.508 F 2.506 2.561 2.451 2.073 1.296 1.308 Cl 2.494 2.515 2.456 2.076 1.701 1.722 OMe 2.452 2.496 2.410 2.062 1.291 1.301 SMe 2.441 2.482 2.409 2.064 1.704 1.725 NMe2 2.371 2.517 2.349 2.069 1.323 1.332 PMe2 2.423 2.476 2.399 2.086 1.791 1.824 Table 7.5. Selected calculated bond lengths (in Ångstroms) of [Li(RCS2)2]

+. Molecular geometry was optimized at the B3LYP/6-31G(2df,p) level of theory.

Substituent Li—S2 Li—S3 C1—S2 C1—S3 C1—R7 H 2.521 2.521 1.675 1.675 1.089 Me 2.479 2.479 1.690 1.690 1.484 F 2.513 2.513 1.679 1.679 1.287 Cl 2.504 2.504 1.684 1.684 1.689 OMe 2.463 2.450 1.704 1.708 1.282 SMe 2.456 2.439 1.705 1.713 1.691 NMe2 2.419 2.419 1.725 1.725 1.310 PMe2 2.431 2.431 1.712 1.712 1.771 Table 7.6. Selected calculated bond lengths (in Ångstroms) of [Li(RCS2)]

•+. Molecular geometry was optimized at the B3LYP/6-31G(2df,p) level of theory.

171

Substituent Li—S2

(=Li—S5) Li—S3

(=Li—S4) C1—S2

(=C6—S5) C1—S3

(=C6—S4) C1—R7

(=C6—R8) H 2.585 2.584 1.671 1.672 1.089 Me 2.551 2.550 1.685 1.685 1.486 F 2.567 2.568 1.675 1.675 1.290 Cl 2.565 2.564 1.678 1.679 1.695 OMe 2.516 2.539 1.701 1.698 1.288 SMe 2.513 2.541 1.706 1.698 1.697 NMe2 2.505 2.506 1.717 1.718 1.314 PMe2 2.520 2.523 1.702 1.702 1.782 Table 7.7. Selected calculated bond lengths (in Ångstroms) of triplet [(RCS2)Li(S2CR)]+. Molecular geometry was optimized at the B3LYP/6-31G(2df,p) level of theory.

Coordination of (RCS2)2 and RCS2• to Li+ is usually exothermic and exergonic, as

shown in Table 7.8. The case for R = NMe2 (TMTD) has the highest binding energy

of the bis(thiocarbonyl)disulfide and the disulfide radical to Li+, while the case for R

= F has the lowest binding energy. The low binding energy for (FCS2)2 and FCS2• can

be explained by the highly electronegative fluorine atom drawing electron density

away from the CS2 moiety, resulting in less pronounced ion-dipole attraction. This is

evidenced by NBO charges on fluorine of –0.292 for (FCS2)2 and –0.283 for FCS2•;

the NBO charge on the sulfur atoms of (FCS2)2 are 0.078 and 0.226, and those on

FCS2• are 0.164. Although the attachment atoms for for R = OMe and NMe2 are also

highly electronegative, the contribution of electron density from the methyl groups to

the oxygen and nitrogen atoms leads to effective back-donation of electron density

from oxygen and nitrogen lone pairs to the CS2 moiety, resulting in less positive NBO

charges on the sulfur atoms and more favorable Li+…S interaction. The NBO charges

on sulfur atoms for the cases R = OMe and NMe2 are: for (MeOCS2)2, –0.040 and

0.212, for MeOCS2•, 0.057 and 0.137, for (Me2NCS2)2, –0.115 and 0.172, and for

Me2NCS2•, 0.040 (See Table 7.4).

172

Li+ + (RCS2)2 → [Li(RCS2)2]

+ Li+ + (RCS2)• → [Li(RCS2)]

•+ Substituent ∆H°298 (kJ mol–

1) ∆G°298 (kJ mol–

1) ∆H°298 (kJ mol–

1) ∆G°298 (kJ mol–

1) H –166.3 –131.4 –101.6 –72.1 Me –199.3 –163.7 –122.7 –93.7 F –137.6 –103.2 –80.6 –50.6 Cl –152.6 –117.5 –101.5 –71.3 OMe –219.1 –184.4 –138.9 –108.7 SMe –227.0 –188.9 –142.6 –111.8 NMe2 –284.9 –243.8 –181.1 –149.0 PMe2 –223.6 –192.0 –139.9 –109.9 Table 7.8. ∆H°298 and ∆G°298 for the coordination of (RCS2)2 and RCS2

• to Li+ in kJ mol–1, calculated at the G3X(MP2) level of theory.

Subsequent Li+ mediated S—S homolytic bond scission in (RCS2)2 can occur via a

number of modes. These can also be facilitated by additional Li+ ions or the

incorporation of elemental sulfur to form bis(thiocarbonyl) oligosulfides, followed by

formation of disulfidic, trisulfidic, and polysulfidic radicals. The following three

modes are considered in this study:

[Li(RCS2)2]+ → [Li(RCS2)]

•+ + (RCS2)• (7-1)

[Li(RCS2)2]+ → [(RCS2)Li(S2CR)] + (7-2)

[Li(RCS2)2]+ + Li+ → 2 [Li(RCS2)]

•+ (7-3)

Gas-phase enthalpies and free energies of the above reactions are generally

endothermic, as shown in Table 7.9. S—S BDEs for reaction (7-1) are highly

endothermic by more than 200 kJ mol–1 for all substituents. The fluorine substituent

shows the lowest enthalpy at 247.0 kJ mol–1 and the highest is that for the methoxy

substituent, at 286.7 kJ mol–1. The enthalpies for reaction (7-1) show little variation

across different R, and are higher than those for reaction (7-2) and (7-3). The lower

enthalpies for reaction (7-2) compared to reaction (7-1) show that it is more favorable

173

for S—S bond dissociation with associated curve crossing to occur while the (RCS2)2

ligand is still chelated to the Li+ ion. Another explanation for the lower enthalpy is the

formation of an additional Li+…S contact which provides some energy for dissociation

of the S—S bond. Trends for the S—S BDE for reaction (7-2) are not the same for

those for reaction (7-1). The highest S—S BDE for reaction (7-3) is that for the parent

compound (R = H) at 198.3 kJ mol–1, and the lowest is that of TMTD (R = NMe2) at

127.0 kJ mol–1. The addition of Li+ ions aid the process of S—S bond dissociation

further, shown in the enthalpies for reaction (7-3). The enthalpy trend for reaction (7-

3) is similar to that for reaction (7-2), again with the parent compound having the

highest reaction enthalpy of 173.9 kJ mol–1 and TMTD having the lowest reaction

enthalpy of 78.2 kJ mol–1.

[Li(RCS2)2]+

→ [Li(RCS2)]•+ +

(RCS2)•

[Li(RCS2)2]+

→ [(RCS2)Li(S2CR)]+ [Li(RCS2)2]

+ + Li+ → 2 [Li(RCS2)]

•+

Substituent ∆H°298 (kJ mol–1)

∆G°298 (kJ mol–1)

∆H°298 (kJ mol–1)

∆G°298 (kJ mol–1)

∆H°298 (kJ mol–1)

∆G°298 (kJ mol–1)

H 275.5 217.7 198.3 174.2 173.9 145.6 Me 269.6 209.2 176.7 151.1 146.9 115.5 F 247.0 189.7 183.7 161.7 166.4 139.1 Cl 266.2 208.0 187.3 164.1 164.6 136.6 OMe 286.7 228.9 180.2 157.3 147.9 120.2 SMe 280.2 223.7 175.2 154.6 137.6 111.9 NMe2 259.3 195.4 127.0 100.3 78.2 46.4 PMe2 282.5 223.8 173.9 151.2 142.6 113.9 Table 7.9. ∆H°298 and ∆G°298 for the reactions [Li(RCS2)2]

+ → [Li(RCS2)]•+ + (RCS2)

•, [Li(RCS2)2]+ →

[(RCS2)Li(S2CR)]+, and [Li(RCS2)2]+ + Li+ → 2 [Li(RCS2)]

•+ in kJ mol–1, calculated at the G3X(MP2) level of theory.

Breaking of the S—S bonds in bis(thiocarbonyl)disulfides seem to be unfavorable

processes, with S—S BDEs being generally endothermic by more than 100 kJ mol–1.

The addition of Li+ ions help facilitate this process by forming energetically favorable

Li+…S interactions. Taking the enthalpy values from Tables 7.8 and 7.9, Li+-mediated

S—S bond dissociation via reaction (7-3) is most favorable for TMTD (R = NMe2)

174

with ∆H°298 and ∆G°298 values being highly exothermic and exergonic, by –206.7 and

–197.5 kJ mol–1, respectively.

7.3.2. Novel bis(thiocarbonyl)disulfides, R = OSF5, Gu1, Gu2

As discussed in the previous section, the substituted bis(thiocarbonyl)disulfides

(RCS2)2 (R = H, CH3, F, Cl, OMe, SMe, and PMe2) display more endothermic S—S

BDE than TMTD, and thus, there remains a reliance on a metal cation to facilitate

S—S homolysis. In an attempt to find suitable substituents R than can lower this S—S

BDE, three novel bis(thiocarbonyl)disulfides with R = OSF5, Gu1 and Gu2 are

explored and evaluated. B3LYP/6-31G(d) optimized structures of these novel

bis(thiocarbonyl)disulfides are shown, together with selected geometrical parameters,

in Figure 7.6. Optimized structures of their associated radicals RCS2• are shown in

Figure 7.7.

The three novel bis(thiocarbonyl)disulfides studied generally display C2 molecular

symmetry. S—S bond lengths for all three cases are generally greater than 2.0 Å, and

it is observed that the S—S bond length for ((Gu1)CS2)2 is markedly longer than the

others, at 2.111 Å. From the molecular structure in Figure 7.6, it can be seen that the 2

Gu1 groups are relatively far apart, thereby ruling out steric effect as a reason for the

relatively long S—S bond. The C—O bond length in (F5SOCS2)2, and the C—N bond

lengths in ((Gu1)CS2)2 and (Gu2)CS2)2 are calculated to be 1.367, 1.379 and 1.380 Å,

respectively. Similar to the case for R = OMe and NMe2, these bond lengths are

shorter than the average experimental C—O and C—N single bond lengths (see Table

7.1), which suggests that there is some extent of π back-donation from O and N back

175

to the CS2 moiety, leading to increased C—O and C—N bond order and hence shorter

bond lengths. The C—O and C—N bond lengths of the RCS2• radicals show little

change with respect to their corresponding dilsulfides. It is interesting to note, that the

structure of the (Gu1)CS2• radical shows a noticeable difference in the C1—S1 and

C1—S2 bonds (See Figure 7.7), which suggests a lesser degree of delocalization of

electron density in the CS2 moiety.

S2S3

F2

O1

F6

C1

C2O2

F5

S6

F1

F9

S4S5

F3

S1

F7

F4

F8

F10

C11

C12

C7

N6S1S4N3

C13

C8

C10

C2

C3

S3

N5

C1

N1

N4

C9

N2

C4

S2

C14

C5

C16

C6

C15

C4

C5

S2S3

N1

C7

N3

C1

C8 C2

C11

C3

N2

S4

S1

N5

N4

C6

C10

C9

C13N6

C14

C12

1.6131.367

1.8302.047

(F5SOCS2)2

1.6641.379

1.827 2.111

((Gu1)CS2)2

1.380

1.627

1.867 2.033

((Gu2)CS2)2

Figure 7.6. B3LYP/6-31G(d) optimized structures of (F5SOCS2)2, ((Gu1)CS2)2 and ((Gu2)CS2)2. Distances are in Ångstroms.

176

S1

C1

F2

O1

S2

F5

S3

F1

F3

F4

C4

S1C3

C1N1

S2

C5

C2

N3

N2

C6

C8

C7

S1

C3

C4

C1N1

N2

S2

C2

C6

N3

C5

C7

F5SOCS2• (Gu1)CS2

•

(Gu2)CS2•

1.356 1.682

1.671

1.390 1.684

1.717

1.3721.685

1.702

Figure 7.7. B3LYP/6-31G(d) optimized structures of F5SOCS2•, (Gu1)CS2

• and (Gu2)CS2•. Distances

are in Ångstroms.

MP2/6-311+G(d,p)//B3LYP/6-31G(d) zero-point corrected energies (∆E0) for the

homolytic S—S bond dissociation in these novel bis(thiocarbonyl)disulfides are

presented in Table 7.10. Calculated ∆E0 using this method for R = NMe2 and

G3X(MP2) ∆E0 are presented alongside for purposes of comparison.

(RCS2)2 → 2 RCS2

• Substituent Method ∆E0 (kJ mol–1) NMe2 G3X(MP2) 155.6 NMe2 MP2/B3LYPa 149.2 OSF5 MP2/B3LYPa 188.9 Gu1 MP2/B3LYPa 115.9 Gu2 MP2/B3LYPa 151.4

Table 7.10. Zero-point corrected reaction energies (∆E0) for the reaction (RCS2)2 → 2 RCS2

• , in kJ mol–1, for R = NMe2, OSF5, Gu1 and Gu2. aMP2/6-311+G(d,p)//B3LYP/6-31G(d) + ZPE.

177

For the TMTD case (R = NMe2), the relatively good agreement of ∆E0 obtained at the

MP2/6-311+G(d,p)//B3LYP/6-31G(d) level of theory with G3X(MP2) results

(difference of 6.4 kJ mol–1 or about 1.5 kcal mol–1) gives the confidence that this is an

adequate method of choice for studying the energetics of these

bis(thiocarbonyl)disulfide systems. As seen from Table 7.10, the cases for R = OSF5

and Gu2 are more endothermic than in the TMTD case, by 39.7 and 2.2 kJ mol–1,

respectively. However, the homolytic S—S dissociation reaction ((Gu1)CS2)2 → 2

(Gu1)CS2• is less endothermic than that for TMTD, by 33.3 kJ mol–1! This finding

suggests that ((Gu1)CS2)2 may be a viable alternative to TMTD as a vulcanization

reagent, as (Gu1)CS2• radicals can be formed more easily. Since S—S BDE for R =

NMe2 is lower than those for the other substituents studied, that is R = H, Me, F, Cl,

OMe, SMe, and PMe2, it can be concluded that the S—S BDE for the case R = Gu1 is

expected to be less endothermic than those for the other substituents.

A glance at the S—S dissociation barriers would shed some more light on the

energetics of this reaction for various substituents R. The reaction (RCS2)2 → 2 RCS2•

is expected to proceed via a singlet diradical transition state near the point where the

singlet potential energy hypersurface (PES) of (RCS2)2 crosses with the triplet PES of

2 RCS2•. Schematic potential energy surfaces of homolytic S—S dissociation reaction

in (RCS2)2 for R = NMe2, OSF5, Gu1 and Gu2 are shown in Figure 7.8.

178

-50

0

50

100

150

200

250

300

Reaction Coordinate

Energy (kJ mol–1)

(RCS2)2

[RCS2•…•S2CR]‡

2 RCS2•

(0.0)

(193.0)

(211.2)

(213.3)

(268.5)

(115.9)

(149.2)

(151.4)

(188.9)

Figure 7.8. Schematic potential energy profile of the homolytic S—S bond dissociation reaction (RCS2)2 → 2 RCS2

•, for R = NMe2, OSF5, Gu1 and Gu2, calculated at the MP2/6-311+G(d,p)//B3LYP/6-31G(d) + ZPE level of theory. Relative energies are given in parentheses.

The activation barrier for homolytic S—S bond dissociation reactions as shown in

Figure 7.8 is about 200 kJ mol–1. In the case for R = OSF5, the barrier is significantly

higher than that for R = NMe2, Gu1 and Gu2, at 268.5 kJ mol–1. The barrier for R =

Gu2 is slightly higher than that for TMTD, by 2.1 kJ mol–1. These two novel

bis(thiocarbonyl)disulfides are still less effective than TMTD in providing RCS2•

radicals without the aid of metal ions or elemental sulfur. However, the barrier for R =

Gu1 is calculated to be 193.0 kJ mol–1, which is 18.2 kJ mol–1 lower than that for S—

S bond dissociation in TMTD. On the basis of these results, ((Gu1)CS2)2 is predicted

to be more effective than TMTD in providing RCS2• radicals in the process of rubber

vulcanization. As the reaction energy for ((Gu1)CS2)2 → 2 (Gu1)CS2• is still

179

endothermic by about 115.9 kJ mol–1, there still would be some reliance on activators

like ZnO to drive the reaction forward. In summary, ((Gu1)CS2)2 is predicted to be a

potential reagent for sulfur vulcanization, displaying energetics which are both

kinetically and thermodynamically more favorable than TMTD. This finding also

provides an avenue to explore the use of highly electron donating substituents R, for

instance, polyguanidine substituted bis(thiocarbonyl)disulfides as viable reagents in

rubber vulcanization. Such efforts would be limited by the size of the substituents

themselves, as a bulky group could hinder the cross-linking process instead of

promoting it.


The structures and energetics of substituted bis(thiocarbonyl)disulfides (RCS2)2, their

associated thiocarbamate radicals RCS2•, and their coordinated compounds with

lithium cation, for R = H, Me, F, Cl, OMe, SMe, NMe2 and PMe2 have been studied

theoretically, to determine their usefulness as reagents in the sulfur vulcanization of

rubber. It was found that S—S BDEs for these (RCS2)2 are about 200 kJ mol–1, at the

G3X(MP2) level, with the exception of TMTD (R = NMe2) having the least

endothermic S—S BDE at 155.1 kJ mol–1. The addition of Li+ ions to (RCS2)2

promotes the homolytic S—S bond dissociation process by providing energy from

the exothermic association of Li+ to (RCS2)2. This process is likely to proceed via the

formation of a triplet complex [(RCS2)Li(S2CR)]+; the two RCS2• radicals are bound

to the Li+ ion via their sulfur atoms and in a tetrahedral manner. Additional Li+ ions

would facilitate the process by forming [Li(S2CR)]•+ radicals, and for the TMTD case,

this process is by far the most exothermic (∆H°298 = –206.7 kJ mol–1).

180

Three novel bis(thiocarbonyl)disulfides with R = OSF5, Gu1 and Gu2 were explored,

and their S—S BDEs and activation barriers for homolytic S—S dissociation

calculated at the MP2/6-311+G(d,p)//B3LYP/6-31G(d) level. (F5SOCS2)2 and

((Gu2)CS2)2 were found to have S—S BDEs higher than that for TMTD, although the

reaction energy for the ((Gu2)CS2)2 case was just more endothermic than TMTD by

2.2 kJ mol–1. Similarly, activation barriers for homolytic S—S dissociation for

(F5SOCS2)2 and ((Gu2)CS2)2 were higher than TMTD, by 57.3 and 2.1 kJ mol–1,

respectively. The bis(thiocarbonyl)disulfide ((Gu1)CS2)2 showed the most promising

results, with S—S BDE at 115.9 kJ mol–1 and the corresponding barrier at 193.0 kJ

mol–1, clearly lower than that for TMTD. These calculations provide strong

theoretical evidence of ((Gu1)CS2)2 being an effective reagent in sulfur vulcanization

of rubber, which may reduce the reliance of ZnO in reactant mixtures.

References

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1999, 284, 1988.

2. Akiba, M.; Hashim, A. S. Prog. Polym. Sci. 1997, 22, 475.

3. Steudel, R.; Steudel, Y.; Mak, A. M.; Wong, M. W. J. Org. Chem. 2006, 71,

9302.

4. Kato, S.; Nishiwaki, M.; Inagaki, S.; Ohshima, S.; Ohno, Y.; Mizuta, M.;

Murai, T. Chem. Ber. 1985, 118, 1684.

5. Schroll, A. L.; Barany, G. J. Org. Chem. 1986, 51, 1866.

6. Chiefari, J.; Mayadunne, R. T. A.; Moad, C. L.; Moad, G.; Rizzardo, E.;

Postma, A.; Skidmore, M. A.; Thang, S. H. Macromolecules 2003, 36, 2273.

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7. Thang, S. H.; Chong, Y. K.; Mayadunne, R. T. A.; Moad, G.; Rizzardo, E.

Tetrahedron Lett. 1999, 40, 2435.

8. Ghosh, P. J. Polym. Sci: Macromol. Rev. 1971, 5, 195.

9. Klabunde, K. J.; Kramer, M. P.; Senning, A.; Moltzen, E. K. J. Am. Chem. Soc.

1984, 106, 263.

10. Suoninen, E. J.; Thomas, T. D.; Anderson, S. E.; Runyan, M. T.; Ungier, L. J.

Electron Spectrosc. Relat. Phenom. 1985, 35, 259.

11. Victoriano, L. I. Coord. Chem. Rev. 2000, 196, 383.

12. Remko, M.; Rode, B. M. J. Mol. Struct. (Theochem) 2000, 505, 269.

13. Steudel, R.; Steudel, Y. Chem.--Eur. J. 2006, 12, 8589.

14. Adams, L. K.; Lyon, D. Y.; Alvarez, P. J. J. Water Res. 2006, 40, 3527.

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Toxicol. 1989, 63, 336.

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Frampton, M. W.; Utell, M. J.; Huang, L.-S.; Cox, C.; Zareba, W.;

Oberdörster, G. Am. J. Respir. Crit. Care Med. 2005, 171, 1129.

17. Seppelt, K. Angew. Chem. Int. Ed. 1982, 21, 877.

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183

Appendix 1

A1.1 Time independence Approximation

Assuming that the wave function Ψ( x�,t) can be factorized into two functions, one

being a function of x� and one being a function of time t:

)()(),(0 tAxtx��

Ψ=Ψ (A1-1)

),(ˆ0 txH�

Ψ⇒

t

tAxi

tAxH

∂Ψ∂

=

Ψ=

)()(

)]()([ˆ�

ℏ

�

(A1-2)

Again, assuming that H is a linear operator that does not depend on t, dividing the

entire equation by )()( tAx�

Ψ gives:

t

tAxixHtA

∂∂

Ψ=Ψ)(

)()(ˆ)(�

ℏ�

)()(

)(ˆ )(

tAi

x

xH t

tA

∂∂

=ΨΨ

ℏ�

�

(A1-3)

(A1-4)

184

The expression on the left hand side of equation (2-5) is entirely dependent on x�

while that on the right hand side is entirely dependent on t. This can only be possible

when both are equal to a constant. Taking this constant to be E, the two equations that

are obtained are:

)(

)(tAE

t

tAi ⋅=

∂∂ℏ

(A1-5)

and

)()(ˆ xExH��

Ψ=Ψ (A1-6)

which is the time-independent Schrödinger equation.

A1.2 Application of the Born-Oppenheimer Approximation

The wave function of a molecular system can be approximately factorized into two

functions, one being a function of nuclear coordinates X�

, and the other being a

function of electronic coordinates x�

, with parametric dependence on the nuclear

coordinates.

);()(),( xXXxX elecnuc

��Ψ⋅Ψ=Ψ (A1-7)

185

Using the same technique employed for equations (2-2) to (2-5), the electronic

Schrödinger equation is obtained:

)(

)()ˆˆ(

);(

);()ˆˆˆ(

X

XHHE

xX

xXHHH

nuc

nuc

pot

nucnuc

kin

nuc

elec

elec

pot

elecnuc

pot

elecelec

kin

elec �

�

��

��

Ψ

Ψ+−=

Ψ

Ψ++ −−−

)(XEelec

�=

⇒ );()();()ˆˆˆ( xXXExXHHH elecelecelec

pot

elecnuc

pot

elecelec

kin

elec

��Ψ=Ψ++ −− (A1-

8)

As a consequence of the Born-Oppenheimer approximation, the expectation values

for the nuclear Hamiltonians, kin

nucH and pot

nucnucH −ˆ , can be added as constants to the

energy of the system to yield the electronic energy )(XEelec

�.

A1.3 Construction of the approximate wave function, Φ

The simplest wave function that can be constructed is the Hartree product – a product

of normalized, orthogonal orbital wave functions, φi,

≈Ψ )(x�

)(x�

Φ

= )()()( 2211 nn xxx�

…��

ϕϕϕ ⋅⋅⋅ (A1-9)

with the following condition for orthogonality:

186

ijji d δτϕϕ =∫ (A1-10)

The symbol δij denotes the Kronecker delta function, and equals to unity if i and j are

equal and zero otherwise. The differential dτ denotes an integration over space and

spin coordinates of the electrons.

The Hartree product is not an acceptable wave function, since it does not fulfill the

antisymmetric condition, that is, it must change sign when identical electrons are

interchanged:

),,,,,,(),,,,,,( 2121 xxxxxxxx ijji

�…

�…

�…

��…

�…

�…

�Ψ−=Ψ (A1-11)

To construct an antisymmetric wave function, a simplified function is first considered:

)()()(),,( 332211321 xfxfxfxxxF = (A1-12)

To make the above function antisymetric with respect to x1 and x2,

=),,( 321 xxxF

)()()(

)()()(

331221

332211

xfxfxf

xfxfxf

−

(A1-13)

Continuing the same process to make it antisymmetric with respect to changing x1 and

x2, x2 and x3, and x1 and x3, a determinantal function is obtained.

187

=),,( 321 xxxF

)()()()()()()()()(

)()()()()()()()()(

132231233211331221

231231133221332211

xfxfxfxfxfxfxfxfxf

xfxfxfxfxfxfxfxfxf

−−−

++

=

)()()(

)()()(

)()()(

333231

232221

131211

xfxfxf

xfxfxf

xfxfxf

(A1-14)

The above strategy can be applied to )(x�

Φ , to obtain the determinantal wave function.

Since electrons possess spin, spin functions α and β are also introduced. α(i) is defined

to be 1 if the spin of electron i is +½ (or spin up) and 0 otherwise; β(i) is defined to be

1 if the spin of electron i is –½ (or spin down) and 0 otherwise. The product of a

molecular orbital function φ with its spin function α or β is defined to be a spin orbital

function, with the electrons’ position and spin as its arguments. The wave function is

normalized by multiplying it by !/1 N .

=Φ )(xe

�

)()()()()()()()()()()()(

)2()()2()()2()()2()()2()()2()(

)1()()1()()1()()1()()1()()1()(

!

1

222211

222222222121

121212121111

NxNxNxNxNxNx

xxxxxx

xxxxxx

N

NNNNNNNN

NN

NN

βϕαϕβϕαϕβϕαϕ

βϕαϕβϕαϕβϕαϕβϕαϕβϕαϕβϕαϕ

��⋯

��⋮⋮⋱⋮⋮⋮⋮

��⋯

��

��⋯

��

(A1-15)

A1.4 The Fock Matrix

The electronic Hamiltonian in equation (2-11) and (2-12) can be decomposed into a

sum of one- and two-electron operators.

188

=eH kin

elecH pot

elecnucH −+ ˆ pot

elecelecH −+ ˆ

= ∑=

∇−N

i

i

1

2

2

1 ∑∑

= =

−N

i

M

A iA

A

r

Z

1 1

∑ ∑−

= +=

+1

1 1

1N

i

N

ij ijr

= ∑=

N

i

core

iH1

)(ˆ ∑ ∑

−

= +=

+1

1 1

1N

i

N

ij ijr

(A1-16)

The one-electron operator core Hamiltonian core

iH )(ˆ deals with the kinetic energy and

the potential energy of nuclear-electron attraction of electron i. The two-electron

operator deals with interelectronic repulsion energy.

The two-electron operator in equation (2-21) can be further decomposed into two

operators, the Coulomb and exchange operators, iJ and iK respectively. They are

defined as:

2

12

)2(1

)2(*)1(ˆ τϕϕ dr

J iii ∫= (A1-17)

)1()2(

1)2(*)1()1(ˆ

2

12

ijiji dr

K ϕτϕϕϕ

= ∫

(A1-18)

The one-electron and two-electron operators can be combined to form the Fock

operator:

189

∑=

−+=N

j

ijj

core

i KJHiF1

)( )]}ˆˆ2{ˆ)(ˆ (A1-19)

A1.5 Derivation of the Roothaan-Hall equations

The orbital wave functions φi in the Hartree-Fock equations are first expanded in a set

of fixed basis functions, χ as in equation (2-11). Putting these expressions for φi into

equation (2-20):

=∑

=

Nbasis

iFC1

ˆ

µµµ χ ( )∑ ∑

= =

−+

Nbasis N

j

jj

core

ii KJHC1 1

)( )1(ˆ)1(ˆ2ˆ

µµµ χ

= ∑

=

Nbasis

ii C1µ

µµ χε (A1-20)

Where the operators iJ and iK in this case are the following:

2

121 1

)2(1

)2(*)1(ˆ τχχ dr

CCJ nm

Nbasis

m

Nbasis

n

njmjj ∫∑ ∑= =

= (A1-21)

)1()2(

1)2(*)1(ˆ

1 1

2

12

n

Nbasis

m

Nbasis

n

mnjmjj dr

CCK χτχχχ µµ ∑ ∑ ∫= =

⋅= (A1-22)

Multiplying equation (2-21) by χν and integrating over all space,

∑ ∫∑ ∫

==

=Nbasis

ii

Nbasis

i dCdFC11

ˆ

µµνµ

µµνµ τχχετχχ

(A1-23)

190

Using Fµν to represent the integral matrix on the left hand side (Fock matrix) and Sµν

the integral matrix on the right hand side (overlap matrix) of equation (2-24), the

more familiar form of the Roothaan-Hall equations is arrived at after some

rearrangement:

∑=

=−Nbasis

ii CSF1

0)(ν

νµνµν ε ;

for µ = 1,2,…,Nbasis.

(A1-24)

With normalization conditions:

∑ ∑

= =

=Nbasis Nbasis

ii CSC1 1

1*µ ν

νµνµ (A1-25)

Using the following notation for two-electron repulsion integrals,

21

12

)2()2(*1

)1()1(*)|( ττϕϕϕϕλσµν σλνµ ddr∫∫

=

(A1-26)

The elements of the Fock matrix are:

)]|(

2

1)|[(

1 1

νσµλλσµνλ σ

λσµνµν ∑ ∑= =

−+=Nbasis Nbasis

core PHF (A1-27)

Where Pλσ is the one electron density matrix,

191

∑=

=occ

i

ii CCP1

*2 σλλσ (A1-28)

The electronic energy is given by:

)(

2

1

1 1

∑ ∑= =

+=Nbasis Nbasis

core

elec HFPEµ ν

µνµνµν (A1-29)

A1.6 Configuration Interaction, Singles (CIS), Configuration Interaction,

Doubles (CID) and Configuration Interaction (CISD) methods

Starting from the expression for full configuration interaction (FCI),

…+Φ+Φ+Φ=Ψ ∑∑∑∑

< <

occ

ji

virt

qp

pq

ij

pq

ij

occ

i

virt

p

p

i

p

iCI aaa 00 (A1-30)

The wave function corresponding to inclusion of single substitutions only would be:

∑∑ Φ+Φ=Ψocc

i

virt

p

p

i

p

iCIS aa 00 (A1-31)

Similarly, the wave function corresponding to the inclusion of double substitutions

would be:

192

∑∑< <

Φ+Φ=Ψocc

ji

virt

qp

pq

ij

pq

ijCID aa 00 (A1-32)

The CISD wave function, which includes single and double substitutions, is thus:

∑∑∑∑< <

Φ+Φ+Φ=Ψocc

ji

virt

qp

pq

ij

pq

ij

occ

i

virt

p

p

i

p

iCISD aaa 00 (A1-33)

A1.7 Quadratic Configuration Interaction

In QCI, the first three substitution operators, 1T , 2T and 3T are defined as:

p

i

pi

p

i taT ˆˆ

,

1 ∑ ⋅=

pq

ij

pqij

pq

ij taT ˆ4

1ˆ

,

2 ∑ ⋅=

pqr

ijk

pqrijk

pqr

ijk taT ˆ36

1ˆ

,

3 ∑ ⋅= (A1-34)

Where …,ˆ,ˆ pq

ij

p

i tt are the elementary substitution operators and the arrays

…,, pq

ij

p

i aa are coefficients to be determined. The QCISD equations are:

QCISDETH =⟩Φ⋅Φ⟨ 020

ˆ||

QCISD

p

i

p

i EaTTTTH =⟩Φ⋅++Φ⟨ 02121 )ˆˆˆˆ(|| pi,∀

193

QCISD

pq

ij

pq

ij EaTTTH =⟩Φ+++Φ⟨ 0

2

221

21 )ˆˆˆ1(|| qpji <<∀ ,

(A1-35)

A partial triples correction based on perturbation theory is applied to QCISD energies:

=∆ )(QCISDET utusts

T

t

D

u t

S

s

D

s

aVVaEE

∑∑∑ ∑ −

+

)(

12

0

= pqr

ijk

pqr

ijk

pqr

ijk

ijk pqr

pqr

ijkuuu )2()(

36

1 1 +− −∑∑ ∆ (A1-36)

The quantity (E0 – Et) is the triples excitation energy using the Fock Hamiltonian, sa

and ua are QCISD coefficients, and the u-array elements are:

=upqr

ijk

)||()||()||(

)||()||()||(

)||()||()||(

pqijarpijaqrija

pqkiarpkiaqrkia

pqjkarpjkaqrjka

r

i

q

i

p

i

r

i

q

i

p

i

r

i

q

i

p

i

+++

+++

++

(A1-37)

=upqr

ijk ∑

+++

+++

++

e re

jk

qe

jk

pe

jk

re

ki

qe

ki

pe

ki

re

ij

qe

ij

pe

ij

eipqaeirpaeiqra

ejpqaejrpaejqra

ekpqaekrpaekqra

)||()||()||(

)||()||()||(

)||()||()||(

∑

+++

+++

++

+m

re

km

qe

km

pe

km

re

jm

qe

jm

pq

jm

re

ij

qe

ij

pq

im

ijqmaijpmaijrma

kiqmakipmakirma

jkqmajkpmajkrma

)||()||()||(

)||()||()||(

)||()||()||(

(A1-38)

Where (ij||pq) denotes:

194

=)||( pqij 21

12

)]1()2()2()1([1

)2(*)1(* ττχχχχχχ ddr

qpqpji −

∫∫

(A1-39)

A1.8 Coupled Cluster Method With Singles and Doubles Substitution (CCSD)

equations

The CCSD method works on the following ansatz:

…++++= 32 ˆ

!3

1ˆ!2

1ˆ1)ˆexp( TTTT (A1-40)

Where T is the cluster operator:

…++= 21ˆˆˆ TTT (A1-41)

The CCSD equations are:

0)ˆˆexp(|)(| 0210 =⟩Φ⋅+−Φ⟨ TTEH CCSD


p

i pi,∀


pq

ij qpji <<∀ ,

(A1-42)

The triples correction to CCSD to obtain CCSD(T) energies is given as:

195

utusts

T

t

D

u t

S

s

D

s

T aVVaEE

CCSDE ∑∑∑ ∑ −

+=∆

)(

1)(

0

(A1-43)

A1.9 Møller-Plesset Perturbation Energies

The initial premise of MPPT is the division of the true Hamiltonian into two parts,

VHH ˆˆˆ0 λ+= (A1-44)

Where 0H is first defined as the sum of one-electron Fock operators, V is a small

perturbation, and λ an arbitrary parameter.

The ground state wave function and the corresponding true energy of the system can

be expanded as a power series as a result of Rayleigh-Schrödinger perturbation theory.

…)3(3)2(2)1()0( Ψ+Ψ+Ψ+Ψ=Ψ λλλ (A1-45)

…)3(3)2(2)1()0( EEEEE λλλ +++= (A1-46)

Substituting these equations back into the Schrödinger equation, and equating the

coefficients of λ, a series of equations is obtained. The first three equations are shown

after some rearrangement:

0)ˆ( )0()0(

0 =Ψ− EH (A1-47)

196

)0()1()1()0(

0 )ˆ()ˆ( Ψ−=Ψ− VEEH (A1-48)

)0()2()1()1()2()0(

0 )ˆ()ˆ( Ψ+Ψ−=Ψ− EVEEH (A1-49)

Ψ(0)

is the HF wave function.

From equation (A1-47),

=⟩Ψ− )0()0(

0 |ˆ EH 0

=⟩Ψ−Ψ⟨ )0()0(

0

)0( |ˆ| EH 0

=⟩ΨΨ⟨ )0(

0

)0( |ˆ| H ⟩ΨΨ⟨ )0()0()0( |E

= E(0)

= ∑

i

iε (A1-50)

Since 0H is just the sum of one-electron Fock operators, E(0)

is simply the sum of

orbital energies.

From equation (A1-48),

=⟩Ψ− )1()0(

0 |ˆ EH ⟩Ψ− )0()1( |VE

=⟩Ψ−Ψ⟨ )1()0(

0

)0( |ˆ| EH ⟩Ψ−Ψ⟨ )0()1()0( |ˆ| VE

=⟩ΨΨ⟨−⟩ΨΨ⟨ )1()0()0()1(

0

)0( ||ˆ| EH ⟩ΨΨ⟨−⟩ΨΨ⟨ )0()0()0()0()1( |ˆ|| VE

(A1-51)

197

0H is also Hermitian, which leads to

=⟩ΨΨ⟨ )0()0()1( |E ⟩ΨΨ⟨ )0()0( |ˆ|V

=)1(E ⟩ΨΨ⟨ )0()0( |ˆ|V (A1-52)

The addition of E(0)

and E(1)

yields the HF energy, EHF.

=+ )1()0( EE ⟩ΨΨ⟨+⟩ΨΨ⟨ )0()0()0(

0

)0( |ˆ||ˆ| VH

= ⟩Ψ+Ψ⟨ )0(

0

)0( |ˆˆ| VH

= HFE (A1-53)

The first-order contribution to the wave function, Ψ(1)

, is written as a linear

combination of substituted wave functions with coefficients sa .

∑ Ψ=Ψs

ssa)1(

(A1-54)

Substituting equation (A1-54) into (A1-48),

=Ψ− ∑s

ssaEH )ˆ( )0(

0 )0()1( )ˆ( Ψ−VE

=⟩Ψ−Ψ⟨ ∑s

sst aEH |)ˆ(| )0(

0 ⟩Ψ−Ψ⟨ )0()1( |)ˆ(| VEt

=⟩Ψ−Ψ⟨∑ st

s

s EHa |)ˆ(| )0(

0 ⟩ΨΨ⟨−⟩ΨΨ⟨ )0()0()1( |ˆ|| VE tt

198

( )=⟩ΨΨ⟨−⟩ΨΨ⟨∑ stst

s

s EHa |||ˆ| )0(

0 ⟩ΨΨ⟨−⟩ΨΨ⟨ )0()0()1( |ˆ|| VE tt

(A1-55)

The left hand side of equation (A1-55) is only nonzero in the case where s=t, therefore:

=− )( )0(EEa ss ⟩ΨΨ⟨− )0(|ˆ|Vs

=∴ sa

s

s

EE

V

−

⟩ΨΨ⟨)0(

)0(|ˆ|

(A1-56)

And therefore Ψ(1)

is:

=Ψ )1( ∑ Ψ

−

⟩ΨΨ⟨

s

s

s

s

EE

V)0(

)0(|ˆ|

= ∑ −

Ψ

s s

ss

EE

V

0

0 (A1-57)

The subscript t indicates a double substitution ij→pq. Vt0 are the matrix elements

involving the perturbation operator V .

Vs0

=

∫ ∫ ΨΨ ns dddV τττ …⋯ 210ˆ

For

ij→pq,

Vs0

=

(ij||pq)

= 21

12

)]1()2()2()1([1

)2(*)1(* ττχχχχχχ ddr

qpqpji −

∫∫

(A1-58)

199

The second-order contribution to the MP contribution to the MP energy, E(2)

, can be

obtained:

E(2)

= ∑ −

−D

s s

s

EE

V

0

2

0 ||

= ∑∑∑∑< < −−+

occ

ji

virt

qp jiqp

pqij

)(

|)||(| 2

εεεε

(A1-59)

The above processes can be repeated for E(3)

, E(4)

, etc. to yield higher order MP

energies (MP3, MP4, etc.).

∑ −−

−=

D

st ts

tststs

EEEE

VVVVE

))((

)(

00

0000)3( δ

(A1-60)

=)4(E

∑ ∑

∑

−−−

−−+

−−−

D

su

SDTQ

t uts

ttutuststs

D

st ts

ttss

EEEEEE

VVVVVV

EEEE

VVVV

))()((

))((

))((

000

000000

2

00

0000

δδ

(A1-61)

A1.10 Density Functional Theory

For a system of N non relativistic interacting electrons in a nonmagnetic state, in a

field of nuclei with nuclear charges ZA and nuclear coordinates RA, the electronic

Hamiltonian can be written:

200

elecH ∑∑ ∑∑∑∇= = = >=

+−−=N

i

M

A

N

i

N

j ijiA

AN

ii rr

Z

1 1 1 11

2 1

2

1

eeNe UVT ˆˆˆ ++= (A1-62)

The expectation value of NeV – the external potential, hereby written VNe – is the

attractive potential exerted by the nuclei on the electrons.

From the Hohenberg-Kohn theorem, VNe is a unique functional of )(r�

ρ and since the

Hamiltonian H is determined by VNe, the full many particle ground state is a unique

functional of the ground state electron density )(0 r�

ρ . In other words, once )(0 r�

ρ is

known, the Hamiltonian H , the ground state wave function of the system Ψ0, and the

ground state energy E0, together with all other properties, can be determined.

Accordingly the ground state energy of the system is a functional of the ground state

electron density:

)]([ 00 rE�

ρ ∫ ++= )]([)]([)( 000 rErTrdVr eeNe

��ρρρ (A1-63)

The first term depends on the actual system, and the other two terms are independent

of N, ZA and RA. The system independent terms are grouped into a functional known

as the Hohenberg-Kohn functional, )]([ rF�

ρ .

)]([ rF�

ρ )]([)]([ rErT ee

��ρρ +=

201

⟩Ψ+Ψ⟨= |ˆˆ| eeUT (A1-64)

From Eee[ρ] the classical electron-electron coulombic repulsion term J[ρ] can be

extracted. The remainder is termed the non-classical contribution to electron-electron

interaction, Encl[ρ]. The non-classical contribution includes self-interaction correction,

exchange, and Coulomb correlation.

][ρeeE ][

)()(

2

121

12

21 ρρρ

nclErrdr

rr+= ∫∫��

��

][][ ρρ nclEJ += (A1-65)

Similarly, from T[ρ], the exact kinetic energy of a non-interacting reference system

Ts[ρ] can be extracted, and the remainder will fall into the residual part, Tc[ρ].

][][][ ρρρ cs TTT += (A1-66)

Where

⟩∇⟨−= ∑ i

N

i

isT ϕϕ ||2

1 2 (A1-67)

The expression for the energy of the interacting, real system can be written now as

][][][][][)]([ ρρρρρρ nclcsNe ETTJErE ++++=�

(A1-68)

202

Writing ENe[ρ], J[ρ] and Ts[ρ] in their explicit forms, and grouping the terms for

which there is no known explicit expression for, Tc[ρ] and Encl[ρ] into an exchange

term )]([ rEXC

�ρ , the following expression for )]([ rE�ρ is obtained.

=)]([ rE

�ρ ][][)()(

2

1)(

12

21 ρρρρ

ρ XCsNe ETr

rrrdrV +−+ ∫∫∫

��

=

)]([2

1

)(1

)(2

1

)(

2

21

2

1

12

2

1

1

2

1

1

rE

rdrdrr

r

rdrr

Z

XCii

N

i

j

N

i

N

j

i

i

N

i

M

A A

A

�

��

��

ρϕϕ

ϕϕ

ϕ

+⟩∇⟨−

+

−

∑

∑∑∫∫

∑∫∑

(A1-69)

A1.11 Common Exchange and Correlation Functionals

Listed below are some explicit forms of common exchange and correlation

functionals used in most DFT calculations.

Dirac exchange functional based on the uniform electron gas (LDA/LSDA):

rdrE Dirac

X

��3

43

1

)(4

3

2

3][ ∫

−= ρπ

ρ (A1-70)

Becke 88 (B88) exchange functional:

203

rd

xx

xrEE Dirac

X

B

X

��∫ −+

−=1

2

3

4

88

sinh61)(][

βρβρ

(A1-71)

Where

3

4

)(

|)(|)(

r

rrx

�

��

ρ

ρ∇=

Lee-Yang-Parr correlation functional:

=],[ βα ρρLYP

CE

( ) ( ) rdtt

tCCb

d

a

c

WW

WFF

�

∇+∇+++

−++

+

−

−−

−

−∫

3

1

exp18

1

9

1

222

1

22

3

83

2

3

83

2

3

5

3

1

ρββαα

αα

αα

βα

ρρρρρρ

ρρρρρ

ρ

γ

(A1-72)

Where

∇−

∇=

∇−

∇=

=

+

+−=

)()(

)(|

8

1

)()(

)(|

8

1

)3(10

3

)()(

)()(12

2

2

2

2

2

2

3

2

2

22

rr

rt

rr

rt

C

rr

rr

W

W

F

��

�

��

�

��

��

σσ

σσ

βα

βα

ρρρ

ρρρ

π

ρρ

ρργ

(A1-73)

The constants a = 0.04918, b = 0.132, c = 0.2533 and d = 0.349 are derived from the

Colle-Salvetti fit to the correlation energy of the helium atom.

Perdew-Wang 1991 GGA correlation functional

204

)),,(),(( βαβα ρρρρερ dHK += (A1-74)

Where

3/23/2

6/1

6/5

)),(1(2

1)),(1(

2

1),(

),(

3

12

1

βαζβαζβα

πρ

ρρρ

σ

βα

−++=

=

u

u

d

1

23

1

3/15

24

23

1

422

4223

1)),((

),(2exp2),(

)3(

)),((400exp)),()(

7

3)),(((),,(

))),((),(1(

))),((21ln)),((

2

1),,(

),,(),,(),,(

−

−

−

−

−=

−−−=

++

++=

+=

λρρβαιε

ιλβα

ρπ

ρρρρκβαφνβα

ιβαβαλ

βαιλρρβα

βαβαβα

βα

βαβα

βα

uA

duduZrdJ

dAdA

dAdudL

dJdLdH

001667.0

004235.0

316

09.0

3/1

−=

=

=

=

=

Z

κπ

ν

νκλι

32

2

07389.0472.0723.81

007389.0266.23568.2

1000

1)(

)()(

rrr

rrr

Zrr

+++

++⋅=

−=

θ

θφ

=),( βαε

4

1111111

2222222

4

3333333

1111111

)),())(,((),,,,,,),,((

),,,,,,),,((

))),((1))(,((),,,,,,),,((

),,,,,,),,((

βαζβαζωβα

βα

βαζβαζωβα

βα

PYXWVUTre

PYXWVUTre

c

PYXWVUTre

PYXWVUTre

−

+

−−

205

709921.1

22

2)1()1()(

),(

)(4

3),(

3/4

3/43/4

3/1

=−

−−++=

+

−=

+=

c

zzz

r

ω

βαβα

βαζ

βαπβα

+++++−=

+ )(2

11ln)1(2),,,,,,,(

12/3 pyrxrwrrvturtpyxwvutre

),,(),,(),,( βαβαβα QMQKdC +=

ssQ

N

dNdN

dNddK

ddZrdM

σπρ

ρ

λβαιε

ιλβα

ιβαβαλ

βαιλβα

ρπκβαφνβα

6/16/53/1

1

2

1

1

422

422

3/15

23/22

12

32

1),(4

exp2),(

))),((),(1(

))),((21ln25.0),,(

)(

39789467.335exp)

7

3)),(((5.0),,(

−

−

−

−

=

−

−=

++

++=

−−−=

T=[0.031091, 0.015545, 0.016887]

U=[0.21370, 0.20548, 0.11125]

V=[7.5957, 14.1189, 10.357]

W=[3.5876, 6.1977, 3.6231]

X=[1.6382, 3.3662, 0.88026]

Y=[0.49294, 0.62517, 0.49671]

P=[1,1,1]

To avoid singularities in the limit 0→S

ρ

))0,,()0,(( SS QCG ρρερ +=

206

A1.12 Perturbation Operators in the Calculation of ESR G-Tensors

The operators relevant for the calculation of ESR g-tensors are:

BSgh eZ

��⋅=

2

α

(A1-75)

BSpgh eKEZ

��⋅−=−

23

4

α

(A1-76)

∑ ∑

×−

−⋅

′=−

NUCN

A

n

j

jAj

Aj

AjNSO pRr

Rr

ZS

gh

��)(

||4 3

2α

(A1-77)

∑∑≠

−

×

−

−⋅

′−=

n

kj

n

k

j

kj

kj

jeSO prr

rrS

gh

��

��

3

2

||4

α

(A1-78)

∑∑≠

×

−

−⋅−=

n

kj

n

k

k

jk

jk

jSOO prr

rrSh

��

��

3

2

||α

(A1-79)

]}))[(()](){[(||8 3

3

BRrrSBSrRrRr

Zgh AjjjjjAj

N

A

n

j Aj

Adia

NSO

NUC ��⋅−⋅−⋅⋅−

−

′= ∑ ∑−

α

(A1-80)

]}))[(()](){[(||

1

8 3

3

BrrrSBSrrrrr

gh jkjjjjjk

n

kj

n

k kj

dia

eSO

�� ⋅−⋅−⋅⋅−

−

′−= ∑∑

≠−

α

(A1-81)

]}))[(()](){[(||

1

2 3

3

BrrrSBSrrrrr

h kjkjjkkj

n

kj

n

k kj

dia

SOO

�� ⋅−⋅−⋅⋅−

−−= ∑∑

≠

α

(A1-82)

Where: hZ is the electron spin Zeeman operator; hZ−KE

is the kinetic energy correction

to the electron spin Zeeman operator; hSO−N is the (electron-nuclear) spin-orbit

operator; hSO−e is the electron-electron spin-orbit operator; hSOO is the spin-other-orbit

operator; hSO−Ndia

, hSO−edia

, hSOOdia

are the diamagnetic correction terms to hSO−N, hSO−e,

and hSOO respectively, also known as ‘gauge correction terms’. ZA is the charge of

nucleus A, the total number of nuclei is NNUC and the total number of electrons is n.

207

The terms jp�

, jS�

, jr�

correspond to the momentum, spin and position operators for

electron j.

∑=n

j

jSS��

(A1-83)

The value ge is the electronic Zeeman g-factor, ge=2.0023192778, and g′ is the

electronic spin-orbit g-factor g′= 2.0046385556.

a computational study of free radicals in chemistry …

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