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Home , Graphene, Triangulene, Zethrene

A Theoretical Study of -Based Functionalized Materials

by

Reed Nieman, B.S.

A Dissertation

In

Chemistry

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Approved

Clemens Krempner, Ph.D. Chair of Committee

Hans Lischka, Ph.D.

Adelia J. A. Aquino, Ph.D.

Mark Sheridan Dean of the Graduate School

May, 2019

Copyright 2019, Reed Nieman

Texas Tech University, Reed Nieman, May 2019

ACKNOWLEDGMENTS

I would first like to thank my advisor, Dr. Hans Lischka, whose passion for science and dedication to his work are truly inspiring. Working with him on my various projects has been a tremendous learning experience these past several years. One of his foremost characteristics that I have made an attempt to emulate is his tireless attention to detail, no matter how seemingly small, as all contributing parts are significant to a finished product. Finally, I have always been able to rely on his patience and encouragement to see through any difficult moment in a project. I would also like to thank Dr. Adelia Aquino who first saw potential in me when I did my undergraduate research project with her. She provided my first lessons in performing computational chemistry calculations and I still fondly remember sitting at her side hurriedly taking notes and trying to keep up as she quickly went through commands. Her constant kindness and support over the years is very much appreciated. I am very grateful too for Dr. Clemens Krempner for not only serving as the chairperson of my committee, but for going above and beyond in, among other things, helping me to secure several of my fellowships.

I also thank Dr. Anita Das, and Nadeesha Jayani Silva for being good group members and great friends. I am thankful as well to Megan Gonzalez and Dr. Veronica Lyons for their friendship and support, without which graduate school would not have been nearly as colorful let alone bearable. I am also very thankful for my family and their unending encouragement even through our lowest moments.

I acknowledge and thank our collaborators, specifically Dr. Francisco Machado with whom I worked closely while he was visiting. Thanks is also given to the groups of Dr. Toma Susi at the University of Vienna and Dr. Sergei Tretiak at Los Alamos National Laboratory for their excellent work and contributions.

Finally, I gratefully acknowledge the following for helping to fund my education: the National Science Foundation, Texas Tech University for the Provost Scholarship and Doctoral Dissertation Completion Fellowship, the AT&T Foundation for the AT&T President’s Fellowship, and the Community Foundation of West Texas for the Lubbock Achievement Rewards for College Scientists Chapter (ARCS) Memorial Scholarship. ii Texas Tech University, Reed Nieman, May 2019

TABLE OF CONTENTS

ACKNOWLEDGMENTS ii ABSTRACT vi LIST OF TABLES viii LIST OF FIGURES xiii LIST OF ABBREVIATIONS xxi

I. BACKGROUND 1 1.1 2 1.2 Organic Photovoltaic Bulk Heterojunctions and Electronic Excited 4 States 1.3 Radical Character in Polycyclic Aromatic 6 1.4 Bibliography 10

II. THEORETICAL METHODS 16 2.1 Density Functional Theory 18 2.2 Second-Order Møller-Plesset Perturbation Theory 22 2.3 Second-Order Algebraic Diagrammatic Construction 25 2.4 Basis Set Superposition Error and the Counterpoise Correction 27 Method 2.5 Environmental Simulation Approaches 29 2.6 Multiconfigurational Self-Consistent Field 31 2.7 Multireference Configuration Interaction 32 2.8 Multireference Averaged Quadratic Coupled-Cluster 34 2.9 Bibliography 36

III. SINGLE AND DOUBLE CARBON VACANCIES IN AS FIRST MODELS FOR GRAPHENE DEFECTS: A SURVEY OF THE 38 CHEMICAL REACTIVITY TOWARD 3.1 Abstract 39 3.2 Introduction 40 3.3 Computational Details 44 3.4 Results and Discussion 45 3.4.1 Potential Energy Scan 45 3.4.2 Geometry Optimizations 49 3.4.3 Energetic Stabilities 52 3.5 Conclusions 59 3.5 Acknowledgments 61 3.7 Bibliography 61

IV. STRUCTURE AND ELECTRONIC STATES OF A GRAPHENE 63 DOUBLE VACANCY WITH EMBEDDED SI 4.1 Abstract 64 4.2 Introduction 65 4.3 Computational Details 66 iii Texas Tech University, Reed Nieman, May 2019

4.4 Results and Discussion 68 4.4.1 The Pyrene-2C+Si Defect Structure 68 4.4.2 The Circumpyrene-2C+Si Defect Structure 71 4.4.3 The Graphene-2C+Si Defect Structure 73 4.4.4 Natural Bond Orbital Analysis 73 4.4.5 Molecular Electrostatic Potential 79 4.4.6 Energy Loss 80 4.5 Conclusions 82 4.5 Acknowledgments 83 4.7 Bibliography 83

V. THE CRUCIAL ROLE OF A SPACER MATERIAL ON THE EFFICIENCY OF CHARGE TRANSFER PROCESSES IN ORGANIC 87 DONOR-ACCEPTOR JUNCTION SOLAR CELLS 5.1 Abstract 88 5.2 Introduction 89 5.3 Methods 92 5.3.1 Experimental 92 5.3.1.1 Bilayer Solar Cells Fabrication 92 5.3.1.2 Device Characterization 93 5.3.2 Computational Details 93 5.4 Results and Discussion 95 5.4.1 Experimental Observations 95 5.4.2 Analysis of Energy Levels 98 5.4.3 Characterization of Electronic States 102 5.5 Conclusions 104 5.6 Acknowledgments 106 5.7 Bibliography 106

VI. BENCHMARK AB INITIO CALCULATIONS OF POLY(P- PHENYLENEVINYLENE) DIMERS: STRUCTURE AND 111 ANALYSIS 6.1 Abstract 112 6.2 Introduction 112 6.3 Computational Details 116 6.4 Results and Discussion 119 6.4.1 Structural Description 119 6.4.2 Excited States 124 6.5 Conclusions 133 6.6 Bibliography 134

VII. THE BIRADICALOID CHARACTER IN RELATION TO SINGLET/TRIPLET SPLITTINGS FOR CIS-/TRANS- DIINDENOACENES AND TRANS- 139 INDENOINDENODI(ACENOTHIOPHENES) BASED ON EXTENDED MULTIREFERENCE CALCULATIONS iv Texas Tech University, Reed Nieman, May 2019

7.1 Abstract 140 7.2 Introduction 141 7.3 Computational Details 143 7.4 Results and Discussion 146 7.4.1 Trans-diindenoacenes 146 7.4.2 Cis-diindenoacenes 152 7.4.3 Trans-indenoindenodi(acenothiophenes) 156 7.5 Conclusions 160 7.6 Bibliography 161 169 VIII. CONCLUDING REMARKS

APPENDICES A. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN CHAPTER III 174 B. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN CHAPTER IV 181 C. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN CHAPTER V 189 D. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN CHAPTER VI 198 E. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN CHAPTER VII 216

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ABSTRACT This dissertation will examine the electronic properties of several materials used in state-of-the art technology and ongoing scientific investigation by means of high-level ab initio quantum chemical calculations. In this way, the application of such calculations to graphene reactivity and defects, organic photovoltaic processes, and excited states and biradical character of polycyclic aromatic hydrocarbons (PAH) is presented.

Graphene reactivity and defect characterization is currently the source of intense scrutiny. The reactivity of pristine graphene and single and double carbon vacancies toward a hydrogen radical was assessed using complete active space self-consistent field theory (CASSCF) and multireference configuration interaction singles and doubles (MR-CISD) calculations using a pyrene model. While all carbon centers formed stable, novel C-H bonds, the single carbon defect was found to be most stabilizing thanks to the dangling bond at the six-membered ring. Bonding at this carbon center shows interaction with several closely-spaced electronic excited states. A rigid scan was also used to show the dynamic landscape of the potential energy surfaces of these structures.

Double carbon vacancy pyrene and circumpyrene were used to investigate the structure and electronic states of an intrinsic impurity in graphene by density functional theory (DFT) methods. The ground state was found to be low spin, non-planar and D2 symmetric with a nearby low spin, planar state with D2h symmetry. Characterization with natural bond orbital (NBO) analyses showed that the hybridization in the non-planar structure is sp3 and sp2d in the planar structure. Charge transfer from the silicon dopant to the surrounding carbon is demonstrated via analysis with natural charges and molecular electrostatic potential (MEP) plots.

Experiment shows drastically enhanced power conversion efficiency (PCE) in a bulk heterojunction (BHJ) device consisting of poly(3-hexylthiophene-2, 5-diyl) (P3HT), terthiophene (T3), and (C60). This BHJ scheme was investigated using DFT and several environmental models. The state-specific (SS) environment was found to best reproduce the experimental findings showing that initial photoexcitation of a local, bright exciton at P3HT was able to decay through several charge transfer (CT) bands of

P3HT→C60 and T3→C60 character explaining the increased PCE. The spectra calculated vi Texas Tech University, Reed Nieman, May 2019 for the isolated system and linear response (LR) environment were very similar to each other and showed the initial photoexcitation at P3HT decayed instead into bands of local

C60 excitonic states quenching the CT states.

Crucial benchmark calculations of the structure and electronic excited states missing from the current literature were performed for oligomers of poly(p- phenylenevinylene) (PPV) dimers using high-level scaled opposite-spin (SOS) second- order Møller-Plesset (SOS-MP2) and second-order algebraic diagrammatic construction (SOS-ADC(2)) calculations. Comparisons were also made to DFT methods. The dimer structures were studied in two conformations: sandwich-stacked and displaced; the latter exhibited greater stacking interactions. Good agreement was found between the cc-pVQZ and aug-cc-pVQZ basis sets and the cc-pVTZ basis. Comparisons of the interaction energies were made to the complete basis set limit. Basis set superposition error was also accounted for. The excited state spectra presented a varied and dynamic picture that was analyzed using transition density matrices and natural transition orbitals (NTOs).

The assessment of the biradical character of several trans-diindenoacenes, cis- diindenoacenes, and trans-indenoindenodi(acenothiophenes) was accomplished utilizing the multireference averaged quadratic coupled-cluster (MR-AQCC) method. These systems have been previously identified as highly reactive in experiments, often subject to unplanned side-reactions. The biradical character was characterized using the singlet- triplet splitting energy (∆ES-T), effective number of unpaired (NU), and unpaired electron density. For all structures, low spin ground states were found. In addition, increasing the number of core rings decreased the ∆ES-T, increased the NU, and increased the unpaired electron density. Good agreement to experimental values for the

∆ES-T is also demonstrated.

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LIST OF TABLES 3.1 Vertical energy differences (eV) of the lowest two A’ and A” states relative to the C15 ground state computed for the C1, C8 and C15 pyrene-1C+Hr structures using the SA4-CASSCF(5,5), MR-CISD and MR-CISD+Q approaches and the 6-31G* basis set. 54

3.2 Dissociation energies Ediss (eV) for the C15 global minimum pyrene- 1C+Hr structure and estimated energy barriers Ebarr(est.) (eV) for the approach of Hr on the ground state surface toward the different carbon atoms of pyrene-1C using different theoretical methods and the 6-31G* basis set. 57

3.3 Dissociation energies Ediss (eV) for the double vacancy pyrene-2C+Hr and pristine pyrene + Hr cases using different methods and the 6-31G* basis se 58

4.1 Open-shell character, structural symmetry, spin multiplicity, bond distances (Å), and relative energies (eV) of selected optimized structures of pyrene- 2C+Si. 69

4.2 Structural symmetry, open-shell character, spin multiplicity, bond distances (Å), and relative energies (eV) of selected optimized structures of circumyrene-2C+Si. 72

4.3 NBO bonding character analysis of pyrene-2C+Si using B3LYP/6- 311G(2d,1p). The discussed carbon is one of the four bonded to silicon. 74

4.4 NBO bonding character analysis of circumpyrene-2C+Si. The discussed carbon atom is the one of the four bonded to silicon. 76

5.1 Stabilization energy of the lowest P3HT→C60 CT state of P3HT-T3-C60 and P3HT-C60 system calculated in the isolated system and the SS environmen 102

6.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer computed at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets using C2h symmetry. 120

6.2 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV3 dimer computed at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets using C2h symmetry. 121

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6.3 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV2 dimer computed at SOS-MP2 level using C1 symmetry. 122

6.4 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV3 dimer computed at SOS-MP2 level using C1 symmetry 122

6.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVQZ and SOS-MP2/cc-pVQZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) 125

6.6 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) 127

6.7 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) 131

6.8 Interchain distances (Å) in the stacked PPV dimers optimized for π-π* S1 1 (1 Bg) excited state using the cc-pVTZ basis and C2h symmetry 132

3 1 7.1 Singlet/triplet splitting energy (∆ES-T, E(1 Bu) - E(1 Ag)) calculated using the MR-AQCC/6-31G* method for trans-diindenoacenes, 1-4. 149

3 1 7.2 Singlet/triplet splitting energy (∆ES-T, E(1 B1) - E(1 A1)) calculated using the MR-AQCC/6-31G* method for cis-diindenoacenes, 5-8. 153

3 1 7.3 Singlet/triplet splitting energy (∆ES-T, E(1 Bu) - E(1 Ag)) calculated using the MR-AQCC/6-31G*(6-311G*)S method for trans- indenoindenodi(acenothiophene) (9-12). 157

A.1 Relative energy differences ΔErel (eV) of the local energy minima in the electronic ground state computed for the different structures of pyrene- 1C+Hr using different theoretical methods and the 6-31G* approach. 174

A.2 Relative energy differences ΔErel (eV) of the local energy minima in the electronic ground state computed for the different structures of pyrene- 2C+Hr using different theoretical methods and the 6-31G* approach. 174

A.3 Relative energy differences ΔErel (eV) of the local energy minima in the electronic ground state computed for the different structures of pyrene+Hr using different theoretical methods and the 6-31G* basis set. 175 ix Texas Tech University, Reed Nieman, May 2019

B.1 NBO bonding character analysis of pyrene-2C+Si using the CAM- B3LYP/6-311G(2d,1p) method. The carbon atom discussed is the carbon bonded to silicon. 181

C.1 Excitation energies Ω of the isolated standard C60-T3-P3HT system using the CAM-B3LYP/6-31G approach 189

C.2 Excitation energies Ω of the isolated standard C60-T3-P3HT system using the CAM-B3LYP/6-31G* approach 190

C.3 Lowest excitation energies Ω of the isolated components P3HT, T3 and C60 using the CAM-B3LYP/6-31G* approach. 191

C.4 Excitation energies Ω of the standard C60-T3-P3HT system the using the CAM-B3LYP/6-31G* approach in the LR environment 192

C.5 Excitation energies Ω of the standard C60-T3-P3HT system the using the CAM-B3LYP/6-31G approach in the LR environment 193

C.6 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM- B3LYP/6-31G* approach in the SS environment 193

C.7 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM- B3LYP/6-31G approach in the SS environment 194

C.8 Excitation energies Ω of the isolated standard C60-T3-P3HT system using the CAM-B3LYP/6-31G* approach (only core orbitals were frozen) 194

C.9 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system using the CAM-B3LYP/6-31G* approach 195

C.10 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system using the CAM-B3LYP/6-31G approach 196

C.11 Excitation energies Ω of the extended C60-T3-P3HT system using the CAM-B3LYP/6-31G approach in the SS environment 197

D.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer computed at MP2 and MP2/BSSE levels with several basis sets using C2h symmetry. 198

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D.2 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV4 dimer computed at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets using C2h symmetry. 199

D.3 Interchain distances (Å) and ΔE (kcal/mol) in the diplaced PPV4 dimer computed at SOS-MP2 level with several basis sets using C2h symmetry. 200

D.4 Single chain PPV2 vertical excitations using SOS-ADC(2) and SOS-MP2 optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength) 200

D.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2) and SOS-MP2 optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) 201

D.6 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) 202

D.7 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) 205

D.8 Stacked PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) 208

D.9 Stacked PPV4 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) 208

D.10 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) 209

D.11 Stacked PPV3 dimer vertical excitations using B3LYP-D3/cc-pVTZ and the B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) 212

D.12 Stacked PPV3 dimer vertical excitations using CAM-B3LYP-D3/cc-pVTZ and the CAM-B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) –

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vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) 213

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

1 E.2 NU value (e) of trans-diindenoacene (1-4) singlet 1 Ag state and vertically 3 excited 1 Bu state using the MR-AQCC(CAS(4,4))/6-31G* method. 217

1 E.3 NU value (e) of cis-diindenoacene (5-8) singlet 1 A1 state and vertically 3 excited 1 B1 state using the MR-AQCC(CAS(4,4))/6-31G* method. 218

1 E.4 NU value (e) of trans-indenoindenodi(acenothiophene) (9-12) singlet 1 Ag 3 state and vertically excited 1 Bu state using the MR-AQCC(CAS(4,4))/6- 31G*(6-311G*)S method. 219

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LIST OF FIGURES

1.1 Structures: a) n-, b) , c) 1,2:9,10-dibenzoheptazethrene, d) (ma,nz) periacene ( model), e) indeno[1,2-b] 7

3.1 PES scans in terms of the location of Hr for the four lowest states of pyrene- 1C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energies: a -574.222889 Hartree, b -574.198174 Hartree, c -574.149530 Hartree, d -574.117820 Hartree. The energy (color scale) is given in eV, relative to the ground state lowest minimum. 46

3.2 PES scan in terms of the location of Hr for the lowest state of pyrene-2C+ Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energy = -536.385046 Hartree. The energy (color scale) is given in eV, relative to the ground state lowest minimum. 47

3.3 PES scan in terms of the location of Hr for the 22A state of pyrene-2C+ Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energy = -536.347629 Hartree. The energy (color scale) is given in eV, relative to the ground state lowest minimum at -536.385046 Hartree. 48

3.4 PES scan of the lowest state of pyrene+Hr using the CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energy = -612.270253 Hartree. The energy (color scale) is given in eV, relative to the ground state lowest minimum. 48

3.5 Plots of the different minimum structures (labeled according to the bonding carbon atom) on the ground state energy surface of pyrene-1C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. Bond distances are given in Å. 50

3.6 Plots of the different minimum structures (labeled according to the bonding carbon atom) of the ground state energy surface of pyrene-2C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. Bond distances are given in Å. 51

3.7 Plots of the different minimum structures (labeled according to the bonding carbon atom) on the ground state energy surface of pyrene+Hr using the CASSCF(5,5)/6-31G* approach. Bond distances are given in Å. 53

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3.8 Location and relative stability (eV) of the minima on the ground state energy surface of pyrene-1C+Hr using the MR-CISD+Q/6-31G* method. Blue: 2A ground state, red: lowest vertical excitation. Note: the rigid pyrene- 1C structure shown does not reflect the actual structure of the optimized minima. 54

2 3.9 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C15 using the MR-CISD/6-31G* approach. Minimum energy = -574.277274 Hartree 56

2 3.10 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C1 using the MR-CISD/6-31G* approach. Minimum energy = - 574.277274 Hartree 56

3.11 Location and relative stability (eV) of the minima on the ground state energy surface of pyrene-2C+Hr defect using the MR-CISD+Q/6-31G* method. Blue: 2A ground state, red: lowest vertical excitation. Note: the rigid pyrene-2C structure shown does not reflect the actual structure of the optimized minima. 58

3.12 Location and relative stability (eV) of the minima on the ground state energy surface of pyrene+Hr using the MR-CISD+Q/6-31G* method. Note: the rigid pyrene structure shown does not reflect the actual structure of the optimized minima. 59

4.1 Optimized pyrene-2C+Si structure P1 (D2 symmetry) using the B3LYP/6- 311G(2d,1p) method with frozen carbon atoms marked by gray circles 68

4.2 Optimized circumpyrene-2C+Si structure CP1 (D2 symmetry) using the B3LYP/6-311G(2d,1p) method with frozen carbon atoms marked by gray circles. 68

4.3 NBO plots for pyrene-2C+Si using the B3LYP/6-311G(2d,1p) approach for structures P1 and P2. Isovalue = ±0.02 e/Bohr3 77

4.4 Natural charges (e) of carbon and silicon in circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p) method. 78

4.5 Molecular electrostatic potential for circumpyrene-2C+Si mapped onto the electron density isosurface with an isovalue of 0.0004 e/Bohr3 (B3LYP/6- 311G(2d,1p)). 80

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4.6 Comparison of simulated and experimental EELS spectra of the Si-C4 defect in graphene. Despite the out-of-plane distorted structure being the ground state, the originally proposed flat spectrum provides an overall better match to the experiment, apart from the small peak at ~96 eV that is not predicted for the flat structure. The inset shows a colored medium angle annular dark field STEM image of the Si-C4 defect (the Si atom shows brighter due to its greater ability to scatter the imaging electrons). 81

5.1 (a) Standard model of the molecular arrangement in the trimer. (b) Front and (c) side views of the extended model. The picture shows molecular notations, geometrical arrangement and selected nonbonding distances (Å). Atoms kept fixed during the geometry optimization are circled in black. 92

5.2 (a) Scheme of bilayer device structure used in this study, the interface layer with vertical alignment was produced by Langmuir-Blodgett (LB) method illustrated on right. (b) photocurrent measured at short circuit condition as function of illumination wavelength for bilayer device without spacer and with 1-bilayer of O3 between P3HT and C60. (c) GIWAXS map for 1 bilayer of O3 deposited on P3HT by LB method. (d) peak photocurrent at 550 nm illumination as a function of number of O3 bilayer at interface. 97

5.3 Excited state energies Ω of the standard P3HT-T3-C60 trimer using the CAM-B3LYP method and (a) the 6-31G* and (b) the 6-31G basis sets both as isolated systems and in the LR and SS environments (for numerical data see Table C.1, Table C.2, and Table C.4-Table C.8). For the isolated system using the 6-31G* basis set (farthest left of (a)), the fifteen lowest-energy locally excited C60 states have been computed also. Excited states of (c) the extended P3HT-T3-C60 trimer using the 6-31G* and (d) 6-31G basis sets both as isolated systems and in the SS environment (Table C.9-Table C.11). 100

5.4 Natural Transition Orbitals (NTO), particle/hole populations and MEP of (a, b, c) the lowest-energy P3HT→(C60)3 CT state, (d, e, f) the lowest- energy T3→(C60)3 CT state, (j, k, l) the lowest-energy P3HT bright excitonic state of the extended thiophene trimer calculated using the CAM- B3LYP/6-31G method in the SS environment, and (g, h, i) the lowest- energy C60 dark excitonic state of the isolated standard thiophene trimer calculated using the CAM-B3LYP/6-31G* method. Here we use NTO isovalue ±0.015 e/Bohr3 and MEP isovalue ±0.0003 e/Bohr3. 103

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

6.2 Omega matrices for vertical excitations of stacked PPV2 dimer using the SOS-ADC(2)/cc-pVQZ method with the SOS-MP2/cc-pVQZ optimized xv Texas Tech University, Reed Nieman, May 2019

geometry. Each vinylene and phenylene group is a separate fragment: fragments 1-3 belong to one chain, fragments 4-6 to the other. The y-axis represents the electron and the x-axis represents the hole. Omega matrix plots calculated using the TheoDORE program package. 126

6.3 Omega matrices for vertical excitations of stacked PPV3 dimer using the SOS-ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each vinylene and phenylene group is a separate fragment: fragments 1-5 belong to one chain, fragments 6-10 to the other. The y-axis represents the electron and the x-axis represents the hole. Omega matrix plots calculated using the TheoDORE program package. 128

6.4 Calculated line spectra (right y-axis) and density of states (left y-axis) for the stacked PPV oligomers optimized using the SOS-ADC(2), B3LYP-D3, and CAM-B3LYP-D3 methods. Stacked a) PPV2 (Table 6.5), b-d) PPV3 (Table D.8, S11-S12), e) PPV4 (Table D.9). 130

7.1 Quinoid Kekulé and biradical resonance forms of trans-diindenoanthracene, cis-diindenoanthracene, and trans-indenoindenodi(anthracenothiophene). 142

3 1 7.2 Comparison of the singlet/triplet energy (∆ES-T, E(1 Bu) - E(1 Ag)) and 1 number of unpaired electrons (NU and NU(H-L), Table E.2) of the 1 Ag state with the number of interior six-membered rings of trans-diindenoacenes (1- 4) using the MR-AQCC(CAS(4,4))/6-31G* method. 148

7.3 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of unpaired electrons (NU) and b) the number of unpaired electrons in the 1 HONO-LUNO pair (NU(H-L)) of the 1 Ag state (Table E.2) of trans- diindenoacenes (1-4) using the MR-AQCC(CAS(4,4))/6-31G* method. 150

7.4 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the 1 1 Ag state of trans-diindenobenzene (1-4) using the MR- AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring. 151

3 1 7.5 Comparison of the singlet/triplet energy (∆ES-T, E(1 B1) - E(1 A1)) and 1 number of unpaired electrons (NU and NU(H-L), Table E.3) of the 1 A1 state with the number of interior six-membered rings of cis-diindenoacenes (5-8) using the MR-AQCC(CAS(4,4))/6-31G* method. 153

7.6 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of unpaired electrons (NU) and b) the number of unpaired electrons in the

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1 HONO-LUNO pair (NU(H-L)) of the 1 A1 state (Table E.3) of cis- diindenoacenes (5-8) using the MR-AQCC(CAS(4,4))/6-31G* method. 154

7.7 Total unpaired electron density (sum over all NOs, Table E.3), in red, of the 1 1 A1 state of cis-diindenobenzene (5-8) using the MR-AQCC(CAS(4,4))/6- 31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring. 155

3 1 7.8 Comparison of the singlet/triplet energy (∆ES-T, E(1 Bu) - E(1 Ag)) and 1 number of unpaired electrons (NU and NU(H-L), Table E.4) of the 1 Ag state with the number of interior six-membered rings of trans- indenoindenodi(acenothiophene) (9-12) using the MR- AQCC(CAS(4,4))/6-31G*(6-311G*)S method. 157

7.9 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of unpaired electrons (NU) and b) the number of unpaired electrons in the 1 HONO-LUNO pair (NU(H-L)) of the 1 Ag state (Table E.4) of trans- indenoindenodi(acenothiophene) (9-12) using the MR- AQCC(CAS(4,4))/6-31G*(6-311G*)S method. 159

7.10 Total unpaired electron density (sum over all NOs, Table E.4), in red, of the 1 1 Ag state of trans-indenoindenodi(acenothiophene) (9-12) using the MR- AQCC(CAS(4,4))/6-31G*/6-311G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring. 160

A.1 Plots of the active orbitals of the optimized ground states of pyrene+Hr using the CASSCF(5,5)/6-31G* approach. 175

A.2 Plots of the active orbitals of the optimized ground states of pyrene-1C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. 176

A.3 Plots of the active orbitals of the optimized ground states of pyrene-2C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. 176

2 2 A.4 Potential energy scans in terms of the location of Hr for the 3 A and 4 A states of pyrene-2C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energies: a -574.149530 Hartree, b - 574.117820 Hartree. The energy (color scale) is given in eV, relative to the ground state lowest minimum at -536.385046 Hartree. 177

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2 A.5 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C2 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -574.277274 Hartree 177

2 A.6 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C3 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -574.277274 Hartree 178

2 A.7 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C4 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -574.277274 Hartree 178

2 A.8 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C5 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -574.277274 Hartree 179

2 A.9 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C6 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -574.277274 Hartree 179

2 A.10 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C7 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -574.277274 Hartree 180

2 A.11 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C8 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -574.277274 Hartree 180

B.1 NBO plots for circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p) 3 approach (CP2 D2h). Isovalue = ±0.02 e/Bohr 182

B.2 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using the B3LYP/6-311G(2d,1p) method. 183

B.3 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using the CAM-B3LYP/6-311G(2d,1p) method. 184

B.4 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the B3LYP/6-311G(2d,1p) method. 185

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B.5 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the CAM-B3LYP/6-311G(2d,1p) method. 187

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

B.7 First two unoccupied states immediately above the Fermi level spanning about 1 eV for the flat and corrugated Si/graphene structures from the periodic DFT calculations. Electron density isovalue is 0.277 e/Bohr3. 188

D.1 Omega matrices for vertical excitations of displaced PPV2 dimer using the SOS-ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each vinylene and phenylene group is a separate fragment: fragments 1-3 belong to one chain, fragments 4-6 to the other. The y-axis represents the electron and the x-axis represents the hole. Omega matrix plots calculated using the TheoDORE program package. 214

D.2 Omega matrices for vertical excitations of stacked PPV4 dimer using the SOS-ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each vinylene and phenylene group is a separate fragment: fragments 1-7 belong to one chain, fragments 8-14 to the other. The y-axis represents the electron and the x-axis represents the hole. Omega matrix plots calculated using the TheoDORE program package. 214

D.3 Omega matrices for vertical excitations of displaced PPV3 dimer using the SOS-ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each vinylene and phenylene group is a separate fragment: fragments 1-5 belong to one chain, fragments 6-10 to the other. The y-axis represents the electron and the x-axis represents the hole. Omega matrix plots calculated using the TheoDORE program package. 215

E.1 Quinoid Kekulé and biradical resonance forms of trans-diindenoacenes (1- 4), cis-diindenoacenes (5-8), and trans-indenoindenodi(acenothiophenes) (9-12). 220

E.2 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the 3 vertically excited 1 Bu state of trans-diindenobenzene (1-4) using the MR- AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring. 221

E.3 Total unpaired electron density (sum over all NOs, Table E.3), in red, of the 3 vertically excited 1 B1 state of cis-diindenobenzene (5-8) using the MR-

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AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring. 221

E.4 Total unpaired electron density (sum over all NOs, Table E.4), in red, of the 3 vertically excited 1 Bu state of trans-indenoindenodi(acenothiophene) (9- 12) using the MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring. 222

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LIST OF ABBREVIATIONS

ADC(2) Second-Order Algebraic Diagrammatic Construction AUX Auxiliary Orbitals BHJ Bulk Hetero-Junction BSSE Basis Set Superposition Error

C60 Fullerene CASSCF Complete Active Space Self-Consistent Field CI Configuration Interaction CP Counterpoise Correction CSF Configuration State Function CT Charge Transfer DFT Density Functional Theory HF Hartree-Fock HOMA Harmonic Oscillator Measure of Aromaticity K-S Kohn-Sham LR Linear Response MCSCF Multiconfigurational Self-Consistent Field MEP Molecular Electrostatic Potential MP2 Second-Order Møller-Plesset MR-AQCC Multireference-Averaged Quadratic Coupled-Cluster MR-CISD Multireference-Configuration Interaction Singles and Doubles NBO Natural Bond Orbital NTO Natural Transition Orbital

NU Number of Unpaired Electrons OPV Organic Photovoltaic P3HT Poly(3-hexylthiophene-2, 5-diyl) PAH Polycyclic Aromatic PCE Power Conversion Efficiency PES Potential Energy Surface

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PP Polarization Propagator PPV Poly(p-phenylenevinylene) SA State-Averaged SOS Scaled Opposite-Spin SS State-Specific T3 Terthiophene TD-DFT Time-Dependent Density Functional Theory

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CHAPTER I

BACKGROUND

This chapter will present a general overview of each research project of this dissertation. Research into graphene, an allotrope of carbon in which the atoms are arranged in a hexagonal network, has grown into a major area of investigation since its discovery and characterization as a semi-metal with promise of semi-conductor capability. Experimental and theoretical investigations of graphene have concentrated on, among a wide variety other topics, engineering by way of defects. Though many classes of graphene defects have been characterized, the first two projects in this dissertation will discuss two defects in detail: i) the ground state and three lowest excited states involved in the chemisorption of a hydrogen radical to pristine graphene and graphene with a one and two carbon atom vacancy utilizing pyrene as a graphene model system, and ii) the structure and electronic states of silicon doped in two-carbon vacancy graphene using pyrene and circumpyrene as graphene model systems. In the former case, CASSCF and MR-CISD calculations were utilized to show that the bonding of hydrogen to graphene, specifically at the dangling bond (a radical site) found in the single-carbon , involves several closely spaced electronic excited states, and in ii) the non-planar ground state and nearby planar excited state both exhibit hybridization at the silicon dopant that would interrupt graphene’s sp2 hybridized network creating a band gap.

Another area of intense research is organic (OPV), which show great promise in providing an alternative to traditional inorganic photovoltaic devices as they consist of more abundant and potentially cheaper materials. Unfortunately, they do not yet operate at the same PCE in converting photons into electricity, necessitating both experimental and theoretical attention in order to both optimize the operational device makeup and understand the contributing physical processes. The third research project in this dissertation presents theoretical calculations of the electronic excited states an OPV BHJ device that was found to have dramatically increased PCE upon addition of a terthiophene spacer material between the electron donor/acceptor layers. The utilization of the SS solvation environment was instrumental in the correct ordering of excited states. 1 Texas Tech University, Reed Nieman, May 2019

The fourth research project presents high-level benchmark calculations of the structure and electronic excited states of several PPV oligomer dimers using the SOS-MP2 and SOS- ADC(2) methods. PPV is an important material for the elucidation of excitation energy transfer (EET) mechanisms in organic polyconjugated systems.

Finally, PAHs are a class of materials consisting of fused aromatic rings primarily found to have close-shell, low spin ground states. Some cases have been found, however, to have unpaired electron spins in their low spin ground state. These have proven to be highly reactive compared to close-shell PAHs and are subject to unforeseen side reactions and with better characterization of their radical properties, their reactivity can be better controlled in the future. In the final research project of this dissertation, MR-AQCC calculations were utilized to determine the singlet-triplet splitting energy, effective number of unpaired electrons, and unpaired electron density in order to characterize a series of singlet biradical indenofluorenes and thiophene-fused indenofluorenes.

1.1 Graphene

The initial discovery of graphene by Geim and Novoselov1 using the aptly-named “Scotch-Tape” method revolutionized material science by characterizing an allotrope of carbon composed of a one-atom thick layer of hexagonally-arranged carbon atoms forming a diffuse, delocalized sp2 hybridized network. Graphene was found to be a semi-metal with a plethora of mechanical, thermal, and electronic properties making it very useful in next- generation technologies like spin-tronics devices, single , photovoltaic devices and more. Graphene has two main difficulties that prevent its wide-spread adoption to industrial applications. The first is that there is no effective method for obtaining the large amounts of pristine, one-atom thick graphene sheets that would be necessary, and the second is that, as a semi-metal, graphene has a zero-energy band gap.

One of the primary reasons single-layer graphene has proven difficult to obtain, is that once graphene layers are separated from , the individual layers will bond together and reform graphite. One of the first methods used to separate graphene layers

2 Texas Tech University, Reed Nieman, May 2019 was using with similar surface energies as graphene to exfoliate graphene from graphite.2-3 Sonication of graphite dispersed in liquid mediums has also been shown to transform flakes of graphite into single- or multi-layer graphene.4-5 The use of dispersion stabilizing agents with high energy (higher than that of graphene) promotes exfoliation by adsorbing to the separate layers and preventing reaggregation.6-7

The problem of a zero-band gap material like graphene presents an interesting avenue towards exploiting its many electronic properties as band gap engineering allows for the semi-metal to be transformed into a semi-conductor via chemical functionalization.8-9 , the chemisorption of hydrogen onto graphene surfaces, has shown to be an effective method towards opening and tuning its band gap.10- 11 The chemisorption of a species such as hydrogen opens the band gap of graphene by disrupting the local sp2 hybridized bond network in the diffuse π system creating a sp3 hybridized bond with the incoming chemisorbate. This new sp3 hybridized bond disrupts the once-continuous flow of conducting electrons opening the band gap, and, as an additional benefit, increasing the reactivity of nearby carbon centers toward atomic hydrogen chemisorption. Regions of the graphene surface made of sp3 hybridized centers are referred to as islands and have been confirmed using scanning tunneling microscopy (STM) and angle-resolved transmission electron microscopy (TEM).12 Graphane island arrangement and their proximity to one another contribute to the overall electronic character of the graphene sheet and have been shown to open band gaps.12-13

Defects in the carbon network of graphene sheets such as vacancies (removed carbon atoms in the sheet) and dopant species (carbon atoms replaced with heteroatoms) have also displayed the capability to induce a band gap compared to pristine, unaltered graphene.8-9 The removal of carbon atoms from the hexagonal structure either leaves dangling bonds, radical centers that are much more reactive, or results in bond restructuring using Jahn-Teller rearrangements. Such defects have demonstrated several properties such as magnetism,14-15 and increased reactivity by way of greater binding energy was found for vacancy defects in graphene toward species such as metals, hydrogen, fluorine, aryl groups and thiol groups.16-18 Using pyrene as a model for a larger graphene sheet, it was shown 3 Texas Tech University, Reed Nieman, May 2019 using multireferernce (MR) calculations that in single (SV)19 and double (DV)20 vacancy defects there are several closely-spaced singlet and triplet electronic states. These closely spaced states were later shown to be instrumental in bond formation with atomic hydrogen at these sites.21

Graphene with heteroatoms is another interesting method to tuning and regulating the electronic properties of graphene sheets. such as , , and have been substituted into the carbon network using a variety of techniques (e.g. , arc discharge, plasma exposure, chemical vapor deposition (CVD), etc.) and have shown to change the band character.22-24 Silicon is another common graphene dopant species that is especially interesting as it is isovalent to carbon and can be found as intrinsic impurities when graphene is grown using CVD25-26 and by thermal decomposition of .27 Two identified forms of silicon impurity have been identified and studied: the nonplanar threefold- coordinated and planar fourfold-coordinated cases, Si-C3 and Si-C4 respectively. Experimental techniques such as scanning tunneling electron microscopy (STEM) and annular dark field (ADF) imaging together with DFT calculations confirmed that the hybridization at Si-C3 is sp3 and at Si-C4 is sp2d.28 The d-band hybridization of the Si-C4 case is thought to be responsible for the disruption of electron 29 density at the site of the defect. Further theoretical work suggested that the planar Si-C4 structure is an equilibrium between a planar and more stable non-planar structure.30

1.2 Organic Photovoltaic Bulk Heterojunctions and Electronic Excited States

OPV BHJs consist of interfaces between electron donor layers and electron acceptor layers both of which are composed of π-conjugated organic polymers. BHJ devices are fundamental to the understanding in processes (dissociation of generating CT states) necessary to generate electricity and to improving their conversion efficiency.31-32 Bulk heterojunction devices convert light into electricity in a four-step process: i) light is absorbed by bands of bright π→π* transitions of the conjugated organic polymer generating excitonic states, ii) and migration of the 4 Texas Tech University, Reed Nieman, May 2019 exciton through the bulk polymer to the interface and eventual segregation, iii) excitons dissociate at the heterojunction interfaces creating CT states, leading to iv) charge separation collected at the contacts.33 Heterogeneous distributions of donor and acceptor materials in real BHJ devices causes the electricity generating processes to be more complicated than what has been described here due to numerous structural defects and conformations, large numbers of internal degrees of freedom for the various polymer chains, and complicated manifolds of electronic states.34-35 In addition, the rate of exciton diffusion at the interface competes with the rate of charge recombination, and controlling these kinetic factors is key to the utilization of BHJ devices.

BHJ devices have been previously studied in a wide variety of compositions. A PCE of 8.2% was obtained with a BHJ setup consisting of the electron donor, poly[4,8- bis(5-(2-ethylhexyl)thiophen-2-yl)benzo[1,2-b; 4,5-b']dithiophene-2,6-diyl-alt-(4-(2- ethylhexyl)-3-fluorothieno[3,4-b]thiophene-)-2-carboxylate-2-6-diyl)] (PTB7-Th), and 36 electron acceptor, [6,6]-phenyl-C70-butyric acid methyl ester (CN-PC70BM). Switching the common electron acceptor, fullerene (C60) for one based on a fluroene-core design saw an increase in the PCE to 10.7%.37 A BHJ PV device with modifications showed photoresponse beyond 800 nm together with a high open-circuit voltage and PCE,38-39 while hybrid organic/inorganic BHJ PV devices utilizing both porphyrin parts and perovskite parts achieved a PCE of 19%.40 In another BHJ OPV device consisting of electron donor, P3HT, and electron acceptor, fullerene, a dramatic increase in photocurrent, open-circuit voltage, and overall PCE (2-5 times greater) was observed when utilizing a spacer layer consisting of either fluoride, terthiophene-derivative (for which the greatest increase in PCE was found), or one of two metal-organic complexes involving iridium.41

PPV and is a paradigmatic electron donor material for the understanding of EET processes in organic π-conjugated polymers at the molecular level. Such understanding will aid in the fine-tuning of their attractive absorption and emission properties useful in, among other things, electroluminescent and OPV devices.42-44 It has been concluded using experimental evidence that coherent PPV dynamics are a result of structural effects coupling to vibrational modes.43, 45 The evolution of excited states resulting in EET 5 Texas Tech University, Reed Nieman, May 2019 dynamics is a product of structural relaxation and these processes are coupled to electronic states requiring non-adiabatic considerations as state-crossing occurs with structural evolution.33

BHJ devices have been investigated utilizing PPV as the electron donor. A study of the compositional dependence of performance of a OPV BJH composed of (poly(2- methoxy-5-(3’,7’-dimethyloctyloxy)-p-phenylene vinyl-ene):methanofullerene [6,6]- phenyl C61-butyric acid methyl ester) (OC1C10-PPV:PCBM) showed that the hole mobility in the PPV phase increases with increasing composition of the PCBM phase until the point of hole saturation, allowing for the optimal design tuning.46 Another device showed that polymer/polymer BHJ devices where electron-withdrawing cyano-group substituted PPV could act as the electron acceptor with another PPV derivative, poly[2-methoxy-5-(2- ethylhexyloxy)-1,4-phenylenevinylene] (MEH-PPV), acting as the electron donor.47

Quantum chemical calculations on electron donor/acceptor complexes are important in describing the electronic processes in BHJ devices. The electronic excited state spectrum of a system composed of poly(thieno[3,4-b]-thiophene benzodithiophene) (PTB1) and [6,6]-phenyl-C61-butyric acid methyl ester (PCBM) was calculated showing how internal conversion processes can convert the bright, light-absorbing π→π* transition to CT states which were stabilized by inter-chain charge delocalization.48 Local, low energy, dark excitonic transitions on fullerene units were shown to be problematic as they can possibly quench CT states once formed, emphasizing the correct ordering and characterizing of electronic excited states.49 The CT states were investigated in a system of n-acene/tetracyanoethylene (TCNE) showing the necessity of the SOS approach with the second-order algebraic diagrammatic construction (ADC(2)) to the accurate description of the electronic excited states of donor/acceptor complexes, the strongly stabilizing effects of interactions on the lowest calculated CT states, and the change of decay mechanisms from radiative to nonradiative going from benzene to with acene- type electron donors.50

1.3 Radical Character in Polycyclic Aromatic Hydrocarbons

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PAHs are composed of fused aromatic rings, whose ground state in most cases is expected to be closed-shell and low spin. Several types of PAH have been found to exist however where the ground state is low spin, but open-shell in character suggesting the presence of radicals.51 When found in nature, PAHs are often environmental pollutants52 and are good indicators of chemical reactivity of black carbon interfaces in soil.53 They can even be found in surprising abundance in the interstellar medium54 and are hypothesized to be the origin of unidentified mid- range bands in the of the spectrum.55-56 Many classes of radical PAHs have been studied such as zethrenes,57 anthenes,58 phenalenyl,59 besphenalenyls,60 triangulene,61 quinodimethanes,62 indofluorenes,63-65 high-order- ,66-67 and graphene nanoribbons.68-70 Several of these classes will be discussed below.

Figure 1.1 Structures: a) n-acene, b) triangulene, c) 1,2:9,10-dibenzoheptazethrene, d) (ma,nz) periacene (graphene nanoribbon model), e) indeno[1,2-b]fluorene Acenes (Figure 1.1a) are linear arrangements of n number of fused benzene rings (referred to n-acenes) that are studied not only for their own properties, but their value as models for larger PAH systems. Acenes are interesting cases for the investigation of radical character as their number of unpaired electrons in their low spin or singlet ground state has been found to increase with increasing chain length.71 Fluorine substituted (five rings or 5-acene), 2,3,9,10-tetrafluoropentacene, was found to have a larger energy gap

7 Texas Tech University, Reed Nieman, May 2019 between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) when compared to unsubstituted pentacene and persistent radical cationic and dicationic states when exposed to oxidizing solvents.72 Synthesizing acenes with extended chain lengths larger than pentacene is difficult due to developing radical character, but it is possible by functionalizing with additional groups or heteroatom doping.73 Theoretical studies found that doping the outer more aromatic rings of octacene increased the radical character by breaking the π-electron bonding, while doping in the center acted to quench radical character and decrease the effective number of unpaired electrons.74

Triangulene (Figure 1.1b) is a non-Kekulé extended phenalenyl derivative with an two traditional radical centers. Its radical nature makes it difficult to synthesize as polymerization side reactions occur. The ground state of triangulene was determined to be a high spin triplet so the introduction of bulky groups such as tert-butyl is necessary to protect reactive carbon sites.61 Tert-butyl substituted triangulene was also thought to have ferromagnetic potential due to its triplet ground state.75 This magnetic behavior has been exploited in interesting ways such as with spin-photovoltaic devices where pure spin current can be generated without charge current upon irradiation with light.76 Theoretical studies using high-level ab initio calculations support the high spin ground state, showing degenerate first excited states, relatively small triplet-singlet splitting (0.8 eV), and unpaired electron density that is delocalized over the entire molecule but is greatest at the edges.77 The σ- and π-dimerization schemes of trioxo-triangulene monomers have also been investigated using unrestricted density functional theory (UDFT).78

Zethrenes, so named for their z-shaped configurations of fused rings (Figure 1.1c), are molecules possessing a singlet biradical ground state that is stabilized by enhanced aromaticity as described by Clar’s aromatic sextet rule.79 Functionalized molecules have shown singlet-fission processes by forming a bound, spin-entangled pair of triplet states that can thermally dissociate and is enhanced with greater diradical character.80 The large, laterally extended zethrene, superoctazethrene, was found to be relatively stable, have significant radical character, have a smaller single-triplet gap than previous octazethrenes (showing that its first triplet excited state is more thermally 8 Texas Tech University, Reed Nieman, May 2019 accessible), and have aromatic character covering the whole surface.81 High-level multireference calculations found singlet-triplet splitting energies for heptazethrene in good agreement with experiment and decreased with increasing number of Clar’s sextet rings.77 Doping zethrenes with heteroatoms has been found to efficiently tune the electronic properties of PAHs.82 This was shown utilizing multireference calculations in nitrogen and boron doped heptazene where the effective number of unpaired electrons and singlet-triplet splitting energy provided an excellent measure of radical character.83

Graphene nanoribbons (Figure 1.1d) are created by cutting graphene sheets into slices possessing distinct open edges that have been shown to influence the overall electronic properties of the nanoribbon with respect to bulk graphene. The edges come in two varieties: zigzag edges that have a localized non-bonding π-state, and armchair edges that do not. Because of the difference in character between these two edges, edge effects and reactivity are intensely studied. Quarteranthene was synthesized and shown to have magnetic susceptibility above 50 K, a small singlet triplet splitting of 30 meV, and a large biradical index, y, of 0.91 (the singlet ground state possessed 91% biradical character).84 Multireference calculations revealed a multiradical ground state with a near-zero singlet- triplet splitting.85 Additionally the unpaired electron density is concentrated at the zigzag edges. The effects of ionization were also investigated in periacene cases showing decreased unpaired electron density at the edges resulting from removing an electron.86

Indenofluroenes (Figure 1.1e) consist of fused indene and fluorene moieties and have unique electronic characteristics that are targeted for intense investigation and exploitation. Thanks to the two five-membered rings incorporated into the structures, indenofluorenes have demonstrated reversible electron donation useful for OPV devices and organic field effect (OPET).87 Ambipolar charge transport properties were found for diindenobenzene derivatives in singlet- and thin-film OFETs.88 Fluorene- based dispiro[fluorene-indenofluorene] was synthesized and functioned as a hole transport material in perovskite solar cells with PCE of about 18.5 %.89 Magnetic susceptibility studies of thiophene-fused structures found the high-spin structure of indenoindenodi(naphthalenothiophene) exists only above 450 K while maintaining moderate biradical character in the low-spin ground state and a relatively small singlet- 9 Texas Tech University, Reed Nieman, May 2019 triplet splitting.90 In this same study, indenofluorenes were found to have increasing singlet biradical character and smaller singlet-triplet splitting with increasing core acene length.

1.4 Bibliography

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(78) Mou, Z.; Kertesz, M., Sigma- versus Pi-Dimerization Modes of Triangulene. Chemistry – A European Journal 2018, 24 (23), 6140-6147. (79) Clar, E., The Aromatic Sextet. Wiley: London, 1972. (80) Lukman, S.; Richter, J. M.; Yang, L.; Hu, P.; Wu, J.; Greenham, N. C.; Musser, A. J., Efficient Singlet Fission and Triplet-Pair Emission in a Family of Zethrene Diradicaloids. Journal of the American Chemical Society 2017, 139 (50), 18376- 18385. (81) Zeng, W.; Gopalakrishna, T. Y.; Phan, H.; Tanaka, T.; Herng, T. S.; Ding, J.; Osuka, A.; Wu, J., Superoctazethrene: An Open-Shell Graphene-like Molecule Possessing Large Diradical Character but Still with Reasonable Stability. Journal of the American Chemical Society 2018, 140 (43), 14054-14058. (82) Yu, S.-S.; Zheng, W.-T., Effect of N/B doping on the electronic and field emission properties for carbon nanotubes, carbon nanocones, and graphene nanoribbons. Nanoscale 2010, 2 (7), 1069-1082. (83) Das, A.; Pinheiro Jr, M.; Machado, F. B. C.; Aquino, A. J. A.; Lischka, H., Tuning the Biradicaloid Nature of Polycyclic Aromatic Hydrocarbons: The Effect of Graphitic Nitrogen Doping in Zethrenes. ChemPhysChem 2018, 19 (19), 2492- 2499. (84) Konishi, A.; Hirao, Y.; Matsumoto, K.; Kurata, H.; Kishi, R.; Shigeta, Y.; Nakano, M.; Tokunaga, K.; Kamada, K.; Kubo, T., Synthesis and characterization of quarteranthene: elucidating the characteristics of the edge state of graphene nanoribbons at the molecular level. Journal of the American Chemical Society 2013, 135 (4), 1430-1437. (85) Horn, S.; Plasser, F.; Müller, T.; Libisch, F.; Burgdörfer, J.; Lischka, H., A comparison of singlet and triplet states for one- and two-dimensional graphene nanoribbons using multireference theory. Theoretical Chemistry Accounts 2014, 133 (8), 1511. (86) Horn, S.; Lischka, H., A comparison of neutral and charged species of one- and two-dimensional models of graphene nanoribbons using multireference theory. The Journal of Chemical Physics 2015, 142 (5), 054302. (87) Rose, B. D.; Sumner, N. J.; Filatov, A. S.; Peters, S. J.; Zakharov, L. N.; Petrukhina, M. A.; Haley, M. M., Experimental and Computational Studies of the Neutral and Reduced States of Indeno[1,2-b]fluorene. Journal of the American Chemical Society 2014, 136 (25), 9181-9189. (88) Chase, D. T.; Fix, A. G.; Kang, S. J.; Rose, B. D.; Weber, C. D.; Zhong, Y.; Zakharov, L. N.; Lonergan, M. C.; Nuckolls, C.; Haley, M. M., 6,12- Diarylindeno[1,2-b]: Syntheses, Photophysics, and Ambipolar OFETs. Journal of the American Chemical Society 2012, 134 (25), 10349-10352. (89) Yu, W.; Zhang, J.; Wang, X.; Liu, X.; Tu, D.; Zhang, J.; Guo, X.; Li, C., A Dispiro- Type Fluorene-Indenofluorene-Centered Hole Transporting Material for Efficient Planar Perovskite Solar Cells. Solar RRL 2018, 2 (7), 1800048. (90) Dressler, J. J.; Teraoka, M.; Espejo, G. L.; Kishi, R.; Takamuku, S.; Gómez-García, C. J.; Zakharov, L. N.; Nakano, M.; Casado, J.; Haley, M. M., Thiophene and its sulfur inhibit indenoindenodibenzothiophene diradicals from low-energy lying thermal triplets. Nature Chemistry 2018, 10 (11), 1134-1140.

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CHAPTER II

THEORETICAL METHODS

The Schrödinger equation (shown in Equation 2.1 in its one-dimension, time- independent form) allows the total energy of a system, E, to be calculated from the system of mass, m, with , 휓(푥). The wave function is important as it describes the state of the system. The first term is the kinetic energy, and the second term, 푉(푥), is the potential energy function. As the Schrödinger equation can only be solved exactly for very specific cases, several approximations have been made to make it accessible to a larger range of systems. The Born-Oppenheimer (BO) approximation for instance states that since the mass of electrons is much less than that of atomic nuclei, their velocities can be decoupled. The consequences of this is that the nuclei are stationary from the perspective of electrons, so the wave function depends only on electrons moving in the field of the stationary nuclei (that is, it depends only parametrically on the nuclear coordinates). The Hartree-Fock (HF) treatment for a many-electron system, itself utilizing the BO approximation, and upon which several more advanced method (called post-HF methods) are built, calculates the total energy of a system of electrons in only the average potential created by the other particles.

ħ2 푑2휓(푥) − + 푉(푥)휓(푥) = 퐸휓(푥) (2.1) 2푚 푑푥2

The choice of which method is key when one wants to calculate specific molecular properties. One priority is simply computational rigor, i.e. how much time will the calculation require? The HF method for instance scales exponentially with the number of basis functions, N, as N4, second-order Møller Plesset perturbation theory (MP2) is more accurate but scales as N5, and multiconfigurational and multireference are more complex as they depend not only on the number of basis functions, but also the number of reference configurations considered (scaling linearly).

Besides the computational cost, one must consider the properties one wishes to calculate. For example, HF theory provides nice molecular orbital shapes, but as it is only

17 Texas Tech University, Reed Nieman, May 2019 a single reference method, electron correlation energy is lost and it will not reliably describe correct dissociation behavior. To account for this, multiconfigurational and multireference methods should be used. DFT is quite efficient with respect to computational cost as it foregoes reliance on the wave function for the total electron density, but the correct choice of DFT functional is non-trivial and not easily determined.

This remainder of chapter will provide a brief discussion and derivation of the various quantum chemical methods implemented in the chapters that follow. Unless otherwise specified, the derivations and discussion presented in this chapter are adapted from Levine’s Quantum Chemisty.1

2.1 Density Functional Theory

Hohenberg and Kohn provided the foundation for DFT by showing that the total energy of a non-degenerate ground state, 퐸0, is determined by the ground state electron probability density, 휌0(푟) in a given external field, υ(r), where r represents the electron coordinates.2 In addition, as the electron probability density can be used to find the ground- state total energy, the ground state wave function, ψ0, can be determined which would allow one to find excited state wave functions and energies as well.

The general expression for the ground state total energy is given in Equation 2.2.

The term 푇[휌0] is the kinetic energy, 푉푁푒[휌0] is the nuclear-electron attraction energy, and

푉푒푒[휌0] is the electron-electron repulsion energy. Of these, only 푉푁푒[휌0] is known

(Equation 2.3), while 푇[휌0] and 푉푒푒[휌0] are not explicitly known.

퐸0 = 퐸휐[휌0] = 푇[휌0] + 푉푁푒[휌0] + 푉푒푒[휌0] (2.2)

푛 푉푁푒[휌0] = ⟨휓0| ∑푖=1 υ(r푖) |휓0⟩ = ∫ 휌0υ(r)dr (2.3)

With this definition of 푉푁푒[휌0], Equation 2.2 can be simplified into Equation 2.2, and by gathering the kinetic energy and electron-electron repulsion energy into a new term,

퐹[휌0], showing a functional dependent on the electron probability density, Equation 2.5. 18 Texas Tech University, Reed Nieman, May 2019

This new functional is independent of the external potential, but cannot be used to calculate the total ground state energy from 휌0 since 퐹[휌0] is not explicitly defined.

퐸0 = 퐸휐[휌0] = ∫ 휌0υ(r)dr + 푇[휌0] + 푉푒푒[휌0] (2.4)

퐸0 = 퐸휐[휌0] = ∫ 휌0υ(r)dr + 퐹[휌0] (2.5)

The Hohenberg-Kohn theorem that was just defined shows how the density, 휌0, can be used in lieu of the wave function to calculate the ground state properties of a molecular system, but since 퐹[휌0] and the method of obtaining 휌0 without a wave function are unknown, it does not explain how this is accomplished. Kohn and Sham provided a method by which 휌0 could be determined and used to calculate 퐸0.

The Kohn-Sham (K-S) approach3 to DFT begins with a reference system composed of non-interacting electrons all experiencing the same external potential energy function,

υref(r). The function υref(r) gives the same ground state electron probability density,

휌푟푒푓(푟), for the reference as for the exact ground state electron probability density, 휌0(푟).

Finding 휌푟푒푓(푟) in this way allows for the determination of the external potential for the reference system.

The K-S Hamiltonian, 퐻̂푟푒푓, for this non-interacting reference system is a sum of ̂ one-electron Hamiltonians, ℎ푖, according to Equation 2.6, and the K-S reference system is related back to the real, fully interacting system by Equation 2.7. The factor λ describes the amount of electron-electron interaction and varies between λ=0 for the non-interacting system and λ=1 for the real system (fully interacting). The external potential is such that it makes the ground state electron probability density with the Hamiltonian, 퐻̂휆, equal to the ground state electron probability density of the real system. Eigenfunctions of the one- electron Hamiltonian are the spatial part of the K-S spin orbitals and the eigenvalues are the K-S orbital energies as shown by Equation 2.8.

19 Texas Tech University, Reed Nieman, May 2019

푛 푛 1 퐻̂ = ∑ [− ∇2 + 휐 (푟 )] ≡ ∑ ℎ̂ (2.6) 푟푒푓 2 푖 푟푒푓 푖 푖 푖 = 1 푖=1

̂ ̂ ̂ 퐻휆 ≡ 푇 + ∑ 휐휆(푟푖) + 휆푉푒푒 (2.7) 푖

̂ 퐾푆 퐾푆 퐾푆 ℎ푖 휓푖 = 휀푖 휓푖 (2.8)

With these, another equation can be derived from Equation 2.5 to find the total energy of the ground state, Equation 2.9. In Equation 2.9, Δ푇̅[휌] is the difference between the ground state electronic kinetic energy and that of the non-interacting system, and

Δ푉̅푒푒[휌] is the difference between the average ground state electron-electron repulsion energy and the electrostatic repulsion for a system of electrons that has been smeared into a continuous distribution of charge with a density, ρ. The sum of Δ푇̅[휌] and Δ푉̅푒푒[휌], each of which is unknown, gives the exchange-correlation energy functional, 퐸푋퐶[휌], rewriting Equation 2.9 as Equation 2.10.

1 휌(푟1)휌(푟2) 퐸휐[휌0] = ∫ 휌0υ(r)dr + 푇̅푟푒푓[휌0] + ∬ 푑푟1푑푟2 + Δ푇̅[휌] 2 푟12 (2.9)

+ Δ푉̅푒푒[휌]

1 휌(푟1)휌(푟2) 퐸휐[휌0] = ∫ 휌0υ(r)dr + 푇̅푟푒푓[휌0] + ∬ 푑푟1푑푟2 + 퐸푋퐶[휌] (2.10) 2 푟12

The first three terms in Equation 2.10 are found from the ground state electron density and provide the largest contribution. The last term, the exchange-correlation energy, contributes less, but is more difficult to obtain accurately. All four terms in Equation 2.10 require the ground state electron density for the reference system of non- interacting electrons such that it equals the electron density of the real system. The electron density of the n-electron system can then be found using a wave function expressed as a Slater determinant as in Equation 2.11, where the K-S spatial orbitals are restricted to remain orthonormal.

20 Texas Tech University, Reed Nieman, May 2019

푛 퐾푆 2 휌 = 휌푟푒푓 = ∑|휓푖 | (2.11) 푖=1

With this definition of the electron density, Slater-Condon rules4-5 allow one to rewrite nuclear-electron (nuclear coordinates given by α) attraction energy in Equation 2.10 as Equation 2.12. One can see that finding K-S orbitals that minimize the total ground state energy will follow Equation 2.13 (following Equation 2.8), where 휐푋퐶(1) is the exchange- correlation potential and is the functional derivative of the exchange-correlation energy with respect to the ground state electron density. Finally, Equation 2.13 can be written succinctly as Equation 2.14.

푛 ( ) 휌 푟1 1 퐾푆 2 퐾푆 퐸휐[휌0] = − ∑ 푍훼 ∫ 푑푟1 − ∑⟨휓푖 (1)|∇1|휓푖 (1)⟩ 푟1훼 2 훼 푖=1 (2.12) 1 휌(푟1)휌(푟2) + ∬ 푑푟1푑푟2 + 퐸푋퐶[휌] 2 푟12

( ) 1 2 푍훼 휌 푟1 퐾푆 퐾푆 퐾푆 [− ∇1 − ∑ + ∫ 푑푟1 + 휐푋퐶(1)] 휓푖 (1) = 휀푖 (1)휓푖 (1) (2.13) 2 푟1훼 푟1훼 훼

̂ 퐾푆 퐾푆 퐾푆 ℎ푖 휓푖 (1) = 휀푖 (1)휓푖 (1) (2.14)

The most difficult part in evaluating DFT energies is the finding the correct functional form of the exchange-correlation energy, 퐸푋퐶, as it makes the most significant contribution to the total energy. One common procedure for overcoming this problem called the Local Density Approximation (LDA) is to assume a uniform electron gas and split 퐸푋퐶 into exchange and correlation energies, 퐸푋 and 퐸퐶 respectively, which are then summed together. A specific example of a LDA functional is from Vosko, Wilk, and Nuair (VWN).6 Gradient-corrected functionals build on the LDA formalism and take into account non-uniform gases utilizing gradients. This approach improve accuracy by having the exchange-correlation energy depend on the electron density as well as the first derivative of the electron density. Common examples of gradient-corrected functionals are Perdew, Burke and Ernzerhof (PBE)7-8, and Becke, Lee, Yang, Parr (BLYP).9-11 Hybrid density functionals were developed to combine Hartree-Fock exchange energy with the exchange- 21 Texas Tech University, Reed Nieman, May 2019 correlation energy of gradient-corrected functionals. An example of a hybrid functional is the popular Becke 3-parameter Lee, Yang, Parr (B3LYP).12 Lastly, range-separated functionals were developed where the Coulomb potential is treated separately at short and long range in the form of a hybrid functional. Examples of such range-separated functional are the Coulomb attenuated method B3LYP (CAM-B3LYP)13 and ωB97XD14.

Non-covalent interactions are especially important in weakly bonded complexes, aggregates, and condensed phases made up of separable units, but these dispersion/Van der Waals interactions require special consideration outside of computationally laborious ab initio methods. One approach is DFT-D (D stands for dispersion),15 where the dispersion energy correction, 퐸푑푖푠푝, to the DFT energy, 퐸퐷퐹푇, is applied via Equations 2.15 and 2.16. 푡푓 In Equation 2.16, 푁푎푡 is the number of atoms, 퐶6 is the dispersion coefficient for the atom pairs f and t, 푆6 is a global scaling factor which is dependent on the density functional,

푅푡푓is the distance between atoms t and f, and 푓푑푎푚푝 is a dampening function applied to correct for small R and electron double-counting.

퐸퐷퐹푇−퐷 = 퐸퐾푆−퐷퐹푇 + 퐸푑푖푠푝 (2.15)

푁푎푡−1 푁푎푡 퐶푡푓 퐸 = −푆 ∑ ∑ 6 푓 (푅 ) (2.16) 푑푖푠푝 6 푅6 푑푎푚푝 푡푓 푡=1 푓=푡+1 푡푓

2.2 Second-Order Møller-Plesset Perturbation Theory

This section will cover perturbation theory (PT) as described by Møller and Plesset (MP)16 for second-order corrections (MP2)17-21 for closed-shell, ground state systems and utilizing spin orbitals, 푢푖. They detailed a method for the treatment of molecules in which the HF wave function is the unperturbed wave function. The Hartree-Fock equation for an electron, m, in a many-electron system is given by Equation 2.17 where the Fock operator expression is Equation 2.18.

̂ 푓(푚)푢푖(푚) = 휀푖푢푖(푚) (2.17)

22 Texas Tech University, Reed Nieman, May 2019

푛 1 푍 ̂ 2 훼 ̂ 푓(푚) = − ∇푚 − ∑ + ∑[푗푗̂ (푚) − 푘푗(푚)] (2.18) 2 푟푚훼 훼 푗=1

The MP unperturbed Hamiltonian is a sum of one-electron Fock operators,

Equation 2.19, with the HF wave function, Φ0, also referred to in this context as the zeroth- order wave function, is shown in Equation 2.20 and is a Slater determinant composed of any of the n possible different spin orbitals. Each term in its expansion is a permutation of the electrons in the spin orbitals, and is an eigenfunction of 퐻̂0, the MP zeroth-order 0 Hamiltonian. The total wave function, Φ0, is like-wise an eigenvalue of 퐻̂ with eigenvalues given by Equation 2.21.

푛 퐻̂0 ≡ ∑ 푓̂(푚) (2.19) 푚=1

|푢1푢2푢3 … 푢푛| = Φ0 (2.20)

푛 0 (0) 퐻̂ Φ0 = ( ∑ 휀푚) Φ0 = 퐸 Φ0 (2.21) 푚=1

The Hamiltonian describing the perturbation, 퐻̂′ is the difference between the real molecular Hamiltonian, 퐻̂, and the zeroth-order Hamiltonian, 퐻̂0, as in Equation 2.22. In this way, 퐻̂′ is the difference between the real interelectronic repulsion and the HF interelectronic repulsion. The first-order correction to the ground state energy is given in

Equation 2.23, which is also the variational integral for the HF wave function, Φ0, giving the HF energy, E퐻퐹, Equation 2.24.

푛 푛 1 ̂ 0 ̂ 퐻′ = 퐻̂ − 퐻̂ = ∑ ∑ − ∑ ∑[푗푗̂ (푚) − 푘푗(푚)] (2.22) 푟푙푚 푙 푚>푙 푚=1 푗=1

(1) ̂ 퐸0 = ⟨Φ0|퐻′|Φ0⟩ (2.23)

(0) (1) 퐸0 + 퐸0 = E퐻퐹 (2.24)

23 Texas Tech University, Reed Nieman, May 2019

With Equation 2.24 in mind, corrections up to the second-order are shown to be (2) necessary in order to improve upon the HF energy, 퐸0 , as in Equation 2.25, where the (0) unperturbed functions, 휓푠 , are all possible Slater determinants formed from n possible different spin orbitals. One must also apply excitations in the ground state electron (0) configuration such that 휓푠 differs from Φ0 by the number of unoccupied orbitals, also referred to virtual orbitals, (represented by indices a, b, c, d, …) replacing occupied orbitals 푎 (represented by indices i, j, k, l, …). An excitation level of one is shown as Φ푖 , where occupied spin orbital, 푢푖 is replaced by unoccupied spin orbital, 푢푎, an excitation level of 푎푏 two is shown as Φ푖푗 , where occupied spin orbitals, 푢푖 and 푢푗, are replaced by unoccupied spin orbitals, 푢푎 and 푢푏, and so on for excitation levels of three and higher. Single 푎 ̂ excitations can be neglected as the integral, ⟨휓푖 |퐻′|Φ0⟩, is zero for all occupied orbitals, i, and all unoccupied orbitals, a. Similarly, triple excitations and higher, 푎푏푐 ̂ 4-5 ⟨휓푖푗푘 |퐻′|Φ0⟩, 푒푡푐 …, are zero following Slater-Condon rules . Therefore, only an 푎푏 ̂ excitation level of two is needed to be evaluated, ⟨휓푖푗 |퐻′|Φ0⟩. This same discussion is the (1) reason why only double excitations are present in the first order correction, 휓푠 , to the (0) unperturbed wave function, 휓푠 .

2 (0) ̂ ⟨휓 |퐻′|Φ0⟩ (2) 푠 (2.25) 퐸0 = ∑ (0) (0) 푠≠0 퐸0 − 퐸푠

푎푏 The doubly excited wave functions, Φ푖푗 , are eigenfunctions of Equation 2.19 whose eigenvalues differ from those of the HF wave function, Φ0, by replacing occupied orbital energies, 휀푖 and 휀푗, by unoccupied orbital energies, 휀푎 and 휀푏, as in Equation 2.26,

(0) 푎푏 (which is the denominator of Equation 2.25) when 휓푠 is equal to Φ푖푗 . With the expression for 퐻′ in Equation 2.22 and Slater-Condon rules, the evaluation of integrals 푎푏 with Φ푖푗 proceeds as in Equation 2.27 with the integral in Equation 2.28, where n is the number of electrons.

24 Texas Tech University, Reed Nieman, May 2019

(0) (0) 퐸0 − 퐸푠 = 휀푖 + 휀푗 + 휀푎 + 휀푏 (2.26)

푛 푛−1 ∞ ∞ −1 −1 2 (2) |⟨푎푏|푟 |푖푗⟩ − ⟨푎푏|푟 |푗푖⟩| 퐸0 = ∑ ∑ ∑ ∑ (2.27) 휀푖 + 휀푗 + 휀푎 + 휀푏 푏=푎+1 푎=푛+1 푖=푗+1 푗=1

−1 ∗ ∗ −1 ⟨푎푏|푟 |푖푗⟩ = ∫ ∫ 푢푎(1)푢푏(2)푟12 푢푖(1)푢푗(2)푑휏 (2.28)

Integrals involving spin orbitals (including the actual spin) are evaluated with respect to the electron-repulsion integrals, and sums over all a, b, i, and j in Equation 2.27 (0) allow for all possible doubly excited terms of 휓푠 . With this, the molecular MP2 energy,

퐸푀푃2, is finally given in Equation 2.29. Consequently, MP2 energies are size-extensive, but since they are not variational, they can produce energies below the true energy.

(0) (1) (2) (2) 퐸 + 퐸 + 퐸 = 퐸퐻퐹 + 퐸 = 퐸푀푃2 (2.29)

2.3 Second-Order Algebraic Diagrammatic Construction

The algebraic diagrammatic construction (ADC)22 for the polarization propagator (PP) allows for the calculation of electronic excited state information for a molecular system and the following derivation is adapted from Ref 23. The derivation of the ADC approach starts with many-body Green’s function theory that is used to solve inhomogeneous differential equations.24 The electronic Hamiltonian for a many-bodied system does not have a uniquely defined Green’s function, however, but propagators can be created from the foundations of a Green’s function in order to solve the problem.25

Several propagators can be constructed such as the one-electron propagator which gives the probability of an electron travelling a given length, l, in a given time, t; the two- electron propagator gives the same property but for two correlated electrons. The expectation values for the calculated properties can be found by knowing these two propagators.26 The one-electron propagator can also be used for other one-electron properties such as ionization energies and electron affinities.27-29 The PP describes the time-

25 Texas Tech University, Reed Nieman, May 2019 evolution of polarization in a many-body system by acting on a time-dependent (TD) ground state wave function and propagates the TD change in density.25

The electronic Hamiltonian needs the wave function of the electronic excited states and the PP contains the implicit information of the electronic excited states for the molecular system. The spectral representation of the PP as a matric function, also called the Lehmann representation, is given in Equation 2.30, where 휓0 is the ground state wave 푁 † function, 퐸0 is the grounds state energy, 푐푞 is the creation operator for the one-electron state, q, and 푐푝 is the annihilation operator for the one-electron state, p (both the creation and annihilation operators are associated with HF orbitals). In this form, the PP gives the eigenstates of the molecular system to diagonalize the Hamiltonian matrix. The summation 푁 in the PP is done for all electronic excited states, 휓푛, with total energy, 퐸푛 . This will give 푁 푁 the exciation spectrum since the poles of the PP (when 휔 = 퐸푛 − 퐸0 ) are at the vertical excitation energies and the residues are the transition probabilities. The spectral representation of the PP can therefore be written in a compact, diagonalized form as in

Equation 2.31, where 훀 is the matrix of vertical excitation energies, 휔푛, and 풙 is the matrix of transition or spectroscopic amplitudes.25

⟨휓 |푐†푐 |휓 ⟩⟨휓 |푐†푐 |휓 ⟩ ( ) 0 푞 푝 푛 푛 푟 푠 0 횷푝푞,푟푠 휔 = ∑ 푁 푁 휔 + 퐸 − 퐸푛 푛≠0 0 (2.30) † † ⟨휓0|푐푟 푐푠|휓푛⟩⟨휓푛|푐푞 푐푝|휓0⟩ + ∑ 푁 푁 −휔 + 퐸 − 퐸푛 푛≠0 0

횷(휔) = 풙†(휔 − 훀)−1풙 (2.31)

A non-diagonal representation of the PP is needed for the Feynmann-Goldstone diagrammatic perturbation scheme, and is shown in Equation 2.32, where M and f are respectively the non-diagonal matrix representation of the effective Hamiltonian and the matrix of effective transition moments.22 The effective Hamiltonian and transition amplitudes are expanded to the PT order in the fluctuation potential, and, following that, the correlation energy in the MP Hamiltonian partitioning is shown in Equations 2.33 and

26 Texas Tech University, Reed Nieman, May 2019

(푛) (푛) 2.34. Using the PT up to order n expands 횷(휔) to give the expression of 푀휇휈 and 푓휇 (μ and ν give the excitation level).

횷(휔) = 퐟†(휔 − 퐌)−1퐟 (2.32)

푴 = 푴(0) + 푴(1) + 푴(2) + ⋯ (2.33)

퐟 = 퐟(0) + 퐟(1) + 퐟(2) + ⋯ (2.34)

The ADC approximation up to order n, ADC(n), has all the terms necessary for a PT application of 횷(휔) to that same order. The ADC scheme is thusly a reformulation of the diagrammatic perturbation series of the PP. Since the excitation energies are the poles of PP, they can be found by diagonalizing the M matrix to the specified order using PT.

The Hamiltonian eigenvalue problem gives the vertical excitation energies, 휔푛, whose eigenvectors, y, give the spectroscopic amplitudes using Equation 2.36.

풙 = 풚†퐟 (2.35)

In summary, the ADC method allows for the use of PT to reformulate the approximations of the PP, which formally contains the information on the exact excited states of a given molecular system, so that it only has approximate information on the excited states. The diagrammatic method together with the PT series up to a specified order gives the exact algebraic formulas for the computation of the excited states, with higher orders being more accurate.

2.4 Basis Set Superposition Error and the Counterpoise Correction Method

Basis set superposition error (BSSE) manifests as artificially shorter intermolecular distances and greater (more stabilizing) interaction energy in weakly bonded dimers or clusters; the error is more prominent for smaller basis set sizes (fewer basis functions) and is vanishing at the complete basis set limit.30-31 The following is adapted from Ref. 32. In dimer systems composed of two monomers, A and B, the dimer is stabilized artificially as monomer A, as a result of proximity, begins to take advantage of the basis functions

27 Texas Tech University, Reed Nieman, May 2019 localized to the atoms on monomer B (and vice versa). The increase in basis set size as a result of the extra basis functions is not necessarily the source of the error, but rather it is the unequal treatment found for each monomer: smaller intermolecular distances are able to access more basis function from the other monomer than larger intermolecular distances where the overlap integral becomes too small.33

The counterpoise correction (CP) method by Boys and Bernardi34 is a way to removing BSSE from interacting systems. The usual method for calculating the interaction energy, Δ퐸푖푛푡(퐴퐵), of a dimer system, AB, composed of monomer A interacting with monomer B without correcting for BSSE is shown in Equation 2.36, where the parenthetical symbol is the system of interest, the superscripts are the basis set, and the 퐴퐵 subscripts are the geometries. In this way, 퐸퐴퐵 (퐴퐵), is the energy of the dimer with the dimer geometry in the basis of the dimer. The same procedure is carried out for the energies of both monomers A and B (respectively the second and third terms in Equation 2.36).

퐴퐵 퐴 퐵 Δ퐸푖푛푡(퐴퐵) = 퐸퐴퐵 (퐴퐵) − 퐸퐴 (퐴) − 퐸퐵 (퐵) (2.36)

The CP method corrects for BSSE by finding the energy by which monomer A (or B) is stabilized artificially by the accessible basis functions of monomer B (or A). Equations 2.37 and 2.38 represent the differences between monomer A in the basis of the dimer and monomer A in the basis of monomer A (same for monomer B). The CP energy is then negative as the energy of a monomer in the basis of the dimer must be lower than that of the monomer in its own basis. Inserting Equations 2.37 and 2.38 into Equation 2.36 gives the expression for the CP correction energy, Equation 2.39.

퐴퐵 퐴 퐸퐵푆푆퐸(퐴) = 퐸퐴 (퐴) − 퐸퐴 (퐴) (2.37)

퐴퐵 퐵 퐸퐵푆푆퐸(퐵) = 퐸퐵 (퐵) − 퐸퐵 (퐵) (2.38)

퐶푃 퐴퐵 퐴퐵 퐴퐵 Δ퐸푖푛푡(퐴퐵) = 퐸퐴퐵 (퐴퐵) − 퐸퐴 (퐴) − 퐸퐵 (퐵) (2.39)

In practice, evaluating monomer A in the basis of the dimer requires that the basis functions of monomer B are placed on the atomic centers of monomer B while simultaneously neglecting both the atomic charges and electrons found on monomer B. In this case, the atoms and basis functions of monomer B are thusly referred to as ghost atoms 28 Texas Tech University, Reed Nieman, May 2019 and ghost functions, respectively. The procedure is performed also for monomer B in the basis of the dimer with the ghost atoms and ghost functions of monomer A.

2.5 Environmental Simulation Approaches

Since most chemistry occurs in solution, not the gas phase with isolated molecules, solvent effects are critical. For example, for a given solute in solution, the solute can induce a dipole moment in the surrounding solvent molecules and the bulk of the solvent is then polarized in the region of each solute molecule generating an electric field, or reaction field, at every solute molecule. The reaction field distorts the solute’s electronic wave function from the gas phase giving an induced dipole moment in addition to the solute gas phase dipole moment.

There are two methods for simulating solvent or environmental effects. Explicit solvation is where the solvent molecules are directly modelled and a potential is applied such as the Lennard-Jones 6-12 or TIP4P approaches. More common however is implicit solvation utilizing a continuum model. In continuum models, the molecular structure of the solvent is ignored and the solvent environment is constructed as a continuous (something that cannot conduct electricity) surrounding a cavity centered on the solute molecule. The continuous dielectric is described using the dielectric constant of the chosen solvent.

In quantum mechanical calculations, a potential term, 푉̂푖푛푡 , is included in the electronic Hamiltonian, 퐻̂(0) , of the isolated molecular system to account for the interactions of the solute and dielectric continuum of the solvent environment. For a given calculation using the continuum solvation method, the electronic wave function and electronic probability density of the solute change iteratively going from vacuum to solution until self-consistency between the solute’s charge distribution and the solvent reaction field. This method is referred to the self-consistent reaction field (SCRF) method.

The polarizable continuum model (PCM) is an application of the SCRF approach where a sphere is centered on each nucleus of the solute with radius equal to 1.2 times the 29 Texas Tech University, Reed Nieman, May 2019 van der Waals radius of the atom. A smoothing effect is generally applied to take care of sharp crevices created by regions of the overlapping spheres. Apparent surface charges (ASC) are distributed on the surface to create an electric potential equal to the electric potential, 휙휎, created by the polarized dielectric constant. The ASC are approximated by using point charges on the cavity surface and the cavity is divided into many small regions, k, each with an apparent charge, 푄푘 (Equation 2.40), where 푟푘 is the point 푄푘 is located,

휙휎(푟) is the electric potential (Equation 2.41) due to the polarization of the dielectric, 퐴푘 is the area of the region, ∇휙푖푛(푟푘) is the gradient of the electric potential in the cavity, and

푛푘 is a unit vector perpendicular to the cavity surface at 푟푘 pointing out of the cavity. The electric potential in the cavity, 휙푖푛, is the sum of the charge distribution of the solute, 휙휇,푖푛, and the polarized dielectric, 휙휎,푖푛: 휙푖푛 = 휙휇,푖푛 + 휙휎,푖푛. The electric potential, 휙푖푛, and 푄푘 are not known and are found iteratively until convergence of the charges, which are then used to find the interacting potential term, 푉̂푖푛푡, as in Equation 2.43. The interacting (0) potential, 푉̂푖푛푡, is then added to the electronic Hamiltonian, 퐻̂ , and the entire process is repeated iteratively until convergence.

휀푟 − 1 푄푘 = [ ] 퐴푘∇휙푖푛(푟푘) ∙ 푛푘 (2.40) 푟휋휀푟

푄푘 휙휎(푟) = ∑ (2.41) |푟 − 푟푘| 푘

̂ 푉푖푛푡 = − ∑ 휙휎(푟푖) + ∑ 푍훼휙휎(푟훼) (2.42) 푖 훼

A similar treatment to the PCM method is the conductor-like screening model (COSMO).35 In this approach realistic solute-solvent shapes are still constructed and surface charges are still used on the solute cavity. The difference comes when the charges are calculated initially with a condition appropriate for the solvent that is not a dielectric, but a conductor.

30 Texas Tech University, Reed Nieman, May 2019

2.6 Multiconfigurational Self-Consistent Field

Multiconfigurational self-consistent field (MCSCF)36 is a method that corrects the dissociation behavior of molecules and incorrect description of systems at increasing intermolecular distances in comparison to the HF method. The HF method does not account for static electron correlation effects as only a single electron configuration is considered in a HF wave function and MCSCF is an improvement over HF since it utilizes several electron configurations in the MCSCF wave function. The MCSCF wave function,

휓푀퐶푆퐶퐹, is a linear combination of different electron configurations represented by configuration state functions (CSF), Φ푛, as in Equation 2.43, where n is the total number of CSFs and 푏푛 are the expansion coefficients that are minimized in the variational integral. The MCSCF method, in addition to optimizing the expansion coefficients, also optimizes the form of the molecular orbitals (MOs) by varying the expansion coefficients, c, that connect the MOs, 휙, to the basis functions, 휒, shown in Equation 2.44 for the ground state of He in the minimal basis set. Uncovered terms inside of bars represent α spin orbitals and the covered terms are β spin orbitals. The coefficients b and c are varied to minimize the variational integral in an iterative process similar to a HF SCF procedure and while keeping

휙 orthonormal and 휓푀퐶푆퐶퐹 normalized. Because both the MOs and the CSF expansion coefficients are optimized in the process, only relatively few CSFs are necessary for normally acceptable results.

|휓푀퐶푆퐶퐹⟩ = ∑ 푏푛|Φ푛⟩ (2.43) 푛

|휓푀퐶푆퐶퐹⟩ = 푏1| 휙1 ̅휙̅̅1̅| + 푏2| 휙2 ̅휙̅̅2̅|

= 푏1|(푐11휒1 + 푐21휒1)(̅푐̅11̅̅̅휒̅1̅̅+̅̅̅푐̅̅21̅̅̅휒̅2̅)| (2.44)

+ 푏2|(푐12휒1 + 푐22휒2)(̅푐̅12̅̅̅휒̅1̅̅+̅̅̅푐̅̅22̅̅̅휒̅2̅)|

The CASSCF method is the most common approach to MCSCF calculations.37 The orbitals are still expanded as a linear combination of CSFs, though in CASSCF, the CSFs are divided into active and inactive orbitals. The inactive orbitals are kept either doubly occupied (if in the occupied space) or unoccupied (if in the virtual space) in all CSFs. The remaining electrons and orbitals constitute the active space. The CASSCF wave function 31 Texas Tech University, Reed Nieman, May 2019

is then a linear combination of all CSFs, Φ푛, formed by distributing all the active electrons in the active orbitals in all possible permutations while maintaining the desired state to be calculated. An active space consisting of x number of electrons in y number of orbitals has the notation, CAS(x,y). A MCSCF procedure is performed optimizing the coefficients, b and c.

The selection of the active space for a CASSCF calculation is a non-trivial process and of great consequence. One reliable method is found in Pulay’s unrestricted natural orbital-complete active space (UNOCAS) approach.38 Here an unrestricted HF calculation is performed in the chosen basis and the natural orbitals are analyzed. All natural orbitals (NOs) whose occupation is between 1.98 e and 0.02 e are considered fractionally occupied and will constitute the largest extent of the active space.

2.7 Multireference Configuration Interaction

Configuration interaction (CI) is a natural improvement of the single configuration HF method and corrects for dissociation behavior and increasing intermolecular distances The full CI wave function is a linear combination of CSFs, where the CSFs are first the reference wave function (the HF wave function, 휓0) and subsequent CSFs represent excitations of an electron in an occupied orbital (indices i, j, k, etc…) into a virtual orbital (indices a, b, c, etc…). For full CI, the wave function, shown in Equation 2.45, contains all possible excitations (single, double, triple excitations, etc…) of the reference wave function while respecting the symmetry of the state of interest. The variational principle is then used to minimize the CI energy with respect to the CSF expansion coefficients, 푏푛. Full CI, by taking into account all excitations of the reference, produces a very large number of CSFs and is computationally infeasible for all but the smallest systems (number of atoms) and basis sets (number of basis functions). In light of this, truncations can be made on the CI wave functions restricting the excitation level most commonly to either single and/or double, CIS and CISD respectively, though truncated CI methods, including MCSCF, are not size-extensive.

32 Texas Tech University, Reed Nieman, May 2019

푎 푎 푎푏 푎푏 |휓퐶퐼⟩ = 푏0|휓0⟩ + ∑ 푏푖 |Φ푖 ⟩ + ∑ 푏푖푗 |Φ푖푗 ⟩ 푖,푎 푖>푗,푎>푏 (2.45) 푎푏푐 푎푏푐 + ∑ 푏푖푗푘 |Φ푖푗푘 ⟩ + ⋯ 푖>푗>푘,푎>푏>푐

Multireference configuration interaction (MR-CI)36 is built from the traditional CI and MCSCF methods. A MCSCF calculation is first performed to optimize the MOs and CSFs of the MCSCF wave function. Reference CSFs are created for the MR-CI wave function by exciting electrons from occupied orbitals into virtual ones. In practice, the excitation level for MR-CI calculations is truncated to only singles and doubles (MR- CISD). The variational principle is again used to minimize the energy by varying the CSF expansion coefficients. When considering computational cost, the savings are great when compared to full CI (in that the calculations are at all feasible at modest system sizes), but the cost still scales linearly with the number of CSFs.

Since MR-CISD is a truncated CI method, it is not size-extensive. This is not a large issue as the amount of correlation energy lost is not great, but it becomes increasingly important with increasing system size. The Davidson correction36, 39-40 (designated “+Q”) is an a posteriori treatment that recovers some of the correlation energy lost because the wave function neglects higher order terms. The expression for the energy correction, 퐸푄, 2 is shown in Equation 2.46 where 푏0 is the sum of the square of the coefficients of the reference CSFs in the MR-CISD expansion. It can be seen that this is only a correction to the energy and does not improve any other properties calculated at the MR-CISD level, but the improvement to the energy gives favorable agreement with full CI and comes at substantially reduced computational cost.

2 (1 − 푏0 )(퐸푀푅−퐶퐼푆퐷 − 퐸푀퐶푆퐶퐹) 퐸푄 = 2 (2.46) 푏0

MR-CI calculations are ideal for calculations of systems where there is a high amount of resonance and aromaticity or in the case of radical systems. The multireference character of the wave function provided by multiple configurations provides the flexibility

33 Texas Tech University, Reed Nieman, May 2019 to characterize and predict reactivity in these species so long as the computational cost, and therefore the choice of reference space and basis set, is closely considered.

2.8 Multireference Averaged Quadratic Coupled-Cluster

The MCSCF and MR-CI methods account for static and dynamic electron correlation, but the problem with size-extensivity in larger molecular systems remains even with the Davidson correction to MR-CI. In an effort to remedy this, several other multireference methods were developed such as the multireference linearized coupled- cluster method (MR-LCCM), coupled electron pair approximation (MR-CEPA), and averaged couple pair functional (MR-ACPF) that were shown to be approximately size- extensive alternatives to MR-CI.41 The MR-AQCC method was developed to approach nearly size-extensive results while simultaneously utilizing smaller reference spaces. The following derivation follows what is presented in Ref.41

For a given reference space consisting of 푛푟 reference functions, Φ푟, the solution to the Schrödinger equation is the zeroth-order reference, Ψ0 = ∑푟 푐푟Φ푟, and the correlated wave function is found to be Equation 2.47. In this expression, 푇푟 is the cluster operator, 푟 exp(푇 ) and 푃푟 = |Φ푟⟩⟨Φ푟| are found relative to each Φ푟 to define the cluster operator, and Ψ푃 is the part of the wave function made up of the remaining orthogonal combinations 푎 of reference functions in the set of all Φ푟 in the space of P. The singly (Ψ푖 (푟)) and doubly 푎푏 (Ψ푖푗 ) excited portion of the wave function is presented in Equation 2.48. With this, the full correction, Ψ푐, to the zeroth-order wave function is Ψ푐 = Ψ푃 = Ψ푄.

푟 Ψ = ∑ exp(푇 ) 푃푟Ψ0 + Ψ푃 (2.47) 푟

푎 푎 푎푏 푎푏 Ψ푄 = ∑ (∑ 푐푖 Ψ푖 (푟) + ∑ 푐푖푗 Ψ푖푗 (푟)) (2.48) 푟 푖푎 푖푗푎푏

The functional form of the expectation value for the energy of the single reference case from the cluster operator, T, and the reference wave function, Φ0, is shown in Equation 34 Texas Tech University, Reed Nieman, May 2019

2.49. In this equation, 퐸0 is the reference energy and the c indicates that only connected terms are included. Exclusion principle violating (EPV) terms (terms do not come from pure interactions, but from groupings of interactions) are included as well, so the expression is considered infinite. Equation 2.49 is for true coupled-cluster which is computationally expensive even without considering multireference, so approximations are made to neglect the virtual EPV terms in the disconnected diagrams of the cluster expansion. This is accomplished by writing the cluster expansion in its disconnected form and substituting the MR-CISD correlation energy, ∆E푀푅−퐶퐼푆퐷, into the coupled-cluster energy functional, Equation 2.50. Doing this illustrates that the MR-AQCC method, despite its name, is not a true coupled-cluster method, but is in actuality an advanced form of MR-CISD.

퐹(Δ퐸) = ⟨푒푇Φ |퐻̂ − 퐸 |푒푇Φ ⟩ 0 0 0 푐 (2.49)

푎푏2 푎푏 푎푏 ̂ 푐푑 푐푑 퐹(Δ퐸) = ∆E푀푅−퐶퐼푆퐷 + ∑ 푐푖푗 ⟨Ψ푖푗 |Ψ푖푗 ⟩ ∑⟨Ψ0|퐻 − 퐸0|Ψ푘푙 ⟩ c푘푙 (2.50) 푖푗푎푏 푘푙푐푑

By also neglecting the virtual EPV terms and considering the rest in only an average manner, the second term of Equation 2.50 becomes that shown in Equation 2.51, where 푛푒 is the number of electrons. Finally, the functional form of the MR-AQCC correlation energy can be found in Equation 2.52 by combining Equations 2.50 and 2.51 and by writing

∆E푀푅−퐶퐼푆퐷 in a more explicit fashion.

(푛푒 − 2)(푛푒 − 3) ⟨Ψ푐|Ψ푐⟩ ∆E푀푅−퐶퐼푆퐷 (2.51) 푛푒(푛푒 − 1)

⟨Ψ0 + Ψ푐|퐻̂ − 퐸0|Ψ0 + Ψ푐⟩ 퐹푀푅−퐴푄퐶퐶(∆E) = (푛푒 − 2)(푛푒 − 3) (2.52) 1 + [1 − ] ⟨Ψ푐|Ψ푐⟩ 푛푒(푛푒 − 1)

This approach is still only nearly size-extensive since only the quadratic unlinked EPV terms are included. The primary advantage of the MR-AQCC method is that it allows for high-level calculations that do not have the computational cost of full MR-CI. In addition, the Davidson correction to the MR-CI method only adds size-extensivity to the total energy, whereas the size-extensivity correction with the MR-AQCC method is applied 35 Texas Tech University, Reed Nieman, May 2019 a priori, so meaningful electronic properties of the investigated system can be found such as orbital occupations and spin-couplings among others.

2.9 Bibliography

(1) Levine, I. N., Quantum Chemistry. 7th ed.; Pearson Prentice Hall: New Jersey, 2014. (2) Hohenberg, P.; Kohn, W., Inhomogeneous Electron Gas. Physical Review 1964, 136 (3B), B864-B871. (3) Kohn, W.; Sham, L. J., Self-Consistent Equations Including Exchange and Correlation Effects. Physical Review 1965, 140 (4A), A1133-A1138. (4) Slater, J. C., The Theory of Complex Spectra. Physical Review 1929, 34 (10), 1293- 1322. (5) Condon, E. U., The Theory of Complex Spectra. Physical Review 1930, 36 (7), 1121-1133. (6) Vosko, S. H.; Wilk, L.; Nusair, M., Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Canadian Journal of Physics 1980, 58 (8), 1200-1211. (7) Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple. Physical Review Letters 1996, 77 (18), 3865-3868. (8) Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple [Phys. Rev. Lett. 77, 3865 (1996)]. Physical Review Letters 1997, 78 (7), 1396-1396. (9) Becke, A. D., Density-functional exchange-energy approximation with correct asymptotic behavior. Physical Review A 1988, 38 (6), 3098-3100. (10) Lee, C.; Yang, W.; Parr, R. G., Development of the Colle-Salvetti correlation- energy formula into a functional of the electron density. Physical Review B 1988, 37 (2), 785-789. (11) Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H., Results obtained with the correlation energy density functionals of becke and Lee, Yang and Parr. Chemical Physics Letters 1989, 157 (3), 200-206. (12) Becke, A. D., Density‐functional thermochemistry. III. The role of exact exchange. The Journal of Chemical Physics 1993, 98 (7), 5648-5652. (13) Yanai, T.; Tew, D. P.; Handy, N. C., A new hybrid exchange–correlation functional using the Coulomb-attenuating method (CAM-B3LYP). Chemical Physics Letters 2004, 393 (1), 51-57. (14) Chai, J.-D.; Head-Gordon, M., Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections. Physical Chemistry Chemical Physics 2008, 10 (44), 6615-6620. (15) Grimme, S.; Antony, J.; Schwabe, T.; Mück-Lichtenfeld, C., Density functional theory with dispersion corrections for supramolecular structures, aggregates, and complexes of (bio)organic molecules. Organic & Biomolecular Chemistry 2007, 5 (5), 741-758. 36 Texas Tech University, Reed Nieman, May 2019

(16) Møller, C.; Plesset, M. S., Note on an Approximation Treatment for Many-Electron Systems. Physical Review 1934, 46 (7), 618-622. (17) Frisch, M. J.; Head-Gordon, M.; Pople, J. A., A direct MP2 gradient method. Chemical Physics Letters 1990, 166 (3), 275-280. (18) Frisch, M. J.; Head-Gordon, M.; Pople, J. A., Semi-direct algorithms for the MP2 energy and gradient. Chemical Physics Letters 1990, 166 (3), 281-289. (19) Head-Gordon, M.; Pople, J. A.; Frisch, M. J., MP2 energy evaluation by direct methods. Chemical Physics Letters 1988, 153 (6), 503-506. (20) Sæbø, S.; Almlöf, J., Avoiding the integral storage bottleneck in LCAO calculations of electron correlation. Chemical Physics Letters 1989, 154 (1), 83-89. (21) Head-Gordon, M.; Head-Gordon, T., Analytic MP2 frequencies without fifth-order storage. Theory and application to bifurcated hydrogen bonds in the water hexamer. Chemical Physics Letters 1994, 220 (1), 122-128. (22) Schirmer, J., Beyond the random-phase approximation: A new approximation scheme for the polarization propagator. Physical Review A 1982, 26 (5), 2395-2416. (23) Dreuw, A.; Wormit, M., The algebraic diagrammatic construction scheme for the polarization propagator for the calculation of excited states. Wiley Interdisciplinary Reviews: Computational Molecular Science 2015, 5 (1), 82-95. (24) Bayin, S. S., Mathematical Methods in Science and Engineering. 2006. (25) Fetter, A. L.; Walecka, J. D., Quantum theory of many-particle systems. McGraw- Hill: New York, 1971. (26) Linderberg, J.; Öhrn, Y., Propagators in quantum chemistry. Academic Press: New York, 1973. (27) Danovich, D., Green's function methods for calculating ionization potentials, electron affinities, and excitation energies. Wiley Interdisciplinary Reviews: Computational Molecular Science 2011, 1 (3), 377-387. (28) Von Niessen, W.; Schirmer, J.; Cederbaum, L. S., Computational methods for the one-particle green's function. Computer Physics Reports 1984, 1 (2), 57-125. (29) Cederbaum, L. S., Green's Functions and Propagators for Chemistry. John Wiley & Sons: Chichester, 1998. (30) Jansen, H. B.; Ros, P., Non-empirical molecular orbital calculations on the of . Chemical Physics Letters 1969, 3 (3), 140-143. (31) Liu, B.; McLean, A. D., Accurate calculation of the attractive interaction of two ground state helium atoms. The Journal of Chemical Physics 1973, 59 (8), 4557- 4558. (32) Sherrill, C. D., Lecture Notes on Counterpoise Correction and Basis Set Superposition Error (http://vergil.chemistry.gatech.edu/notes/cp.pdf). 2010. (33) van Duijneveldt, F. B.; van Duijneveldt-van de Rijdt, J. G. C. M.; van Lenthe, J. H., State of the Art in Counterpoise Theory. Chemical Reviews 1994, 94 (7), 1873- 1885. (34) Boys, S. F.; Bernardi, F., The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Molecular Physics 1970, 19 (4), 553-566.

37 Texas Tech University, Reed Nieman, May 2019

(35) Klamt, A.; Jonas, V.; Bürger, T.; Lohrenz, J. C. W., Refinement and Parametrization of COSMO-RS. The Journal of Physical Chemistry A 1998, 102 (26), 5074-5085. (36) Szalay, P. G.; Müller, T.; Gidofalvi, G.; Lischka, H.; Shepard, R., Multiconfiguration Self-Consistent Field and Multireference Configuration Interaction Methods and Applications. Chemical Reviews 2012, 112 (1), 108-181. (37) Roos, B. O., The Complete Active Space Self-Consistent Field Method and its Applications in Electronic Structure Calculations. Advances in Chemical Physics 1987, 69, 399. (38) Pulay, P.; Hamilton, T. P., UHF natural orbitals for defining and starting MC‐SCF calculations. The Journal of Chemical Physics 1988, 88 (8), 4926-4933. (39) Langhoff, S. R.; Davidson, E. R., Configuration interaction calculations on the nitrogen molecule. International Journal of Quantum Chemistry 1974, 8 (1), 61- 72. (40) Luken, W. L., Unlinked cluster corrections for configuration interaction calculations. Chemical Physics Letters 1978, 58 (3), 421-424. (41) Szalay, P. G.; Bartlett, R. J., Multi-reference averaged quadratic coupled-cluster method: a size-extensive modification of multi-reference CI. Chemical Physics Letters 1993, 214 (5), 481-488.

38 Texas Tech University, Reed Nieman, May 2019

CHAPTER III

SINGLE AND DOUBLE CARBON VACANCIES IN PYRENE AS FIRST MODELS FOR GRAPHENE DEFECTS: A SURVEY OF THE CHEMICAL REACTIVITY TOWARD HYDROGEN

Authors: Reed Nieman,a Anita Das,a Adelia J. A. Aquino,a,b Rodrigo G. Amorim,c,d Francisco B. C. Machado,d and Hans Lischkaa,b,e

Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University, Lubbock, TX 79409-1061, USA, bSchool of Pharmaceutical Sciences and Technology, Tianjin University, Tianjin 300072, PR China, cDepartamento de Física, Universidade Federal Fluminense, Volta Redonda, 27213-145, Rio de Janeiro, Brazil, dDepartamento de Química, Instituto Tecnológico de Aeronáutica, São José dos Campos 12228-900, São Paulo, Brazil, eInstitute for Theoretical Chemistry, University of Vienna, Währingerstrasse 17, A-1090 Vienna, Austria

Published: January 2017 in Chemical Physics, DOI:10.1016/j.chemphys.2016.08.007

3.1 Abstract

Graphene is regarded as one of the most promising materials for applications. Defects play an important role in modulating its electronic properties and also enhance its chemical reactivity. In this work the reactivity of single vacancies (SV) and double vacancies (DV) in reaction with a hydrogen atom Hr is studied. Because of the complicated open shell electronic structures of these defects due to dangling bonds, multireference configuration interaction (MR-CI) methods are being used in combination with a previously developed defect model based on pyrene. Comparison of the stability of products derived from C-Hr bond formation with different carbon atoms of the different polyaromatic hydrocarbons is made. In the single vacancy case the most stable structure is the one where the incoming hydrogen is bound to the carbon atom carrying the dangling bond. However, stable C-Hr bonded structures are also observed in the five-membered ring of the single vacancy. In the double vacancy, most stable bonding of the reactant Hr atom is found in the five-membered rings. In total, C-Hr bonds, corresponding to local energy minimum structures, are formed with all carbon atoms in the different defect systems and the pyrene itself. Reaction profiles for the four lowest electronic states show in the case of

39 Texas Tech University, Reed Nieman, May 2019 a single vacancy a complex picture of curve crossings and avoided crossings which will give rise to a complex nonadiabatic reaction dynamics involving several electronic states.

3.2 Introduction

Graphene has many extremely appealing electrical, physical, and thermal properties which have made it an exciting material that is the subject of intensive research ever since it was first isolated.1 Graphene is a particular allotrope of carbon in which a two- dimensional hexagonal lattice is composed of benzene rings. Though it has been suggested that graphene’s ability to conduct electricity could allow it to replace silicon in next- generation organic semi-conductors, two problems remain, however, which detract from this potential. The first issue is that there are no effective methods for obtaining amounts of pristine graphene sheets of one atom thickness necessary for industrial applications. The second problem is that graphene’s electronic properties come with a down side, though it conducts electricity well, it is a semi-metal with a zero-energy band gap.

The chemical functionalization of graphene sheets for the purpose of tuning its band gap has shown great promise.2-3 The hydrogen atom is a natural starting point when beginning the investigation of the ability of graphene sheets to form sp3 bonds between an incoming atom or molecule and the bonding carbon atom. Hydrogen chemisorption on a graphene surface has been studied in detail by Casolo et al.4 using density functional theory (DFT) and periodic boundary conditions. In fact, hydrogenation has been shown to be an efficient tool for band gap tuning of graphene.5-8 Formation of the sp3 bond disrupts the existing sp2 bond network by breaking the local π system. This has the effect of not only increasing the reactivity of the carbon atoms neighboring the new C-H bond, but it also creates a potential barrier to the conducting electrons opening a band gap, which is tunable depending on the concentration of the chemisorbed species and properties of the sp3 network that has been formed. By reacting graphene sheets with atomic hydrogen, hydrogenated regions of graphene are formed which are referred to as graphane islands. The presence and extent of hydrogenation has been previously confirmed through the use of scanning tunneling microscopy (STM) and angle-resolved transmission electron 40 Texas Tech University, Reed Nieman, May 2019 microscopy (TEM).6 These images also show details pertaining to the arrangement of graphane islands with respect to each other. PAHs and their reaction with hydrogen are not only of interest as molecular models for graphene but also as possible catalysts for H2 formation in interstellar space.9 Hydrogenation studies of pyrene have been performed10 to investigate energetic stabilities of hydrogen attached to different carbon atoms and the existence of energy barriers.

The arrangement of bonded hydrogen atoms in graphane islands and the proximity of islands to one another are also contributing factors to the overall electronic character of the hydrogenated graphene sheet due the dependence of the created band gap on such. A small group of low hydrogen-concentration graphane islands have shown clearly the beginning of the formation of a band gap.6 On a larger scale, graphane islands created in strips parallel to each other, called graphene roads, have been shown to create environments similar to graphene nanoribbons.11 Hydrogen atoms adsorbed on graphene have also been reported to induce magnetic moments.12 The spin-polarized state is shown to be essentially located on the carbon sublattice opposite to the one belonging to the adsorbed hydrogen atom. This finding goes very well along with previous success in experimentally engineering band gaps and has been shown by the Haddon group using substituted benzene radicals, nitrophenylene, interacting with graphene sheets.3 As a consequence of this functionalization, residual spins are available to couple and allow the possibility of at room temperatures.

Vacancies defects, absences of carbon atoms in the hexagonal structure, have shown to be an interesting area with regards to the investigation of the chemical reactivity of graphene. When one or multiple carbons are removed, the created dangling bonds represent areas of greater chemical reactivity. The presence of these dangling bonds then implies high polyradical character with many closely spaced locally excited electronic states which are difficult to describe computationally. This situation then strongly advises the use of flexible multireference theory13 in which quasi-degenerate orbitals are treated at equal footing and the correct construction of wavefunctions of well-defined symmetry and spin properties is possible.

41 Texas Tech University, Reed Nieman, May 2019

The electronic structures of both SV and DV pyrene (Scheme 3.1) which has been used as a prototype for a graphene defect structure, have recently been described using mulireference configuration interaction MR-CI14-15 and DFT.16 The single vacancy investigation of pyrene using MR-CI calculations14 showed several closely spaced electronic states of triplet and singlet multiplicity (Scheme 3.1, Pyrene-1C). The electronic ground state of the double vacancy defect15 (Scheme 3.1, pyrene-2C) possesses 50% closed shell character which indicates a significantly different character as compared to the SV case. The pristine pyrene is depicted in Scheme 3.1, top, and as is, of course, to be expected the chemically most stable structure of the three candidates discussed.

The purpose of this work is to explore the chemical reactivity of pyrene and the two defect structures by studying their interaction with hydrogen radical (Hr). The role of Hr is two-fold. It is used in the spirit of a test particle by moving it in a grid search across the different pyrene structures as the arrows in Scheme 3.1 indicate. In this way a stability surface with respect to C-Hr bond formation is constructed. In the second step optimizations of C-Hr structures are performed and C-Hr dissociation curves are constructed to the lowest dissociation products. The previous MR-CI calculations on the SV and DV pyrene defect structures had shown the existence of several low-lying excited states. We wanted to reproduce this asymptotic limit at complete C-H dissociation and, thus, computed in all respective cases potential energy scans and C-H dissociation curves for the four lowest electronic states. In addition to the determination of local energy minima for different C-H bonding situations, these data should provide an overview of the most important aspects for the approach and bonding of a hydrogen atom to a pristine or defect graphene sheet surface.

42 Texas Tech University, Reed Nieman, May 2019

Pyrene

Pyrene-1C

Pyrene-2C

Scheme 3.1 Atomic numbering scheme for pristine pyrene, pyrene-1C (single vacancy) and pyrene-2C (double vacancy), respectively. Hr is the hydrogen atom moving across the different pyrene structures as shown by arrows.

43 Texas Tech University, Reed Nieman, May 2019

3.3 Computational Details

The geometries of each pyrene structure (Scheme 1) were optimized using complete active space (CAS) self-consistent field (CASSCF) theory.17-18 Based on the experience with previous pyrene defect calculations14-15 a CAS(5,5) was used comprised of five electrons in five orbitals where, in Cs symmetry, three orbitals belonged to the A’ irreducible representation and two orbitals belonged to the A” irreducible representation. The CAS(4,4) is chosen for the description of the isolated pyrene defect and one active orbital and electron is added to account for the hydrogen atom. The active orbitals used in the CAS for pyrene+Hr were 31a’, 32a’, 33a’, 22a”, 23a” (Figure A.1 of Appendix A). For pyrene-

1C+Hr active orbitals were 29a’, 30a’, 31a’, 21a”, 22a” (Figure A.2) and for pyrene-2C+Hr active orbitals were 27a’, 28a’, 29a’, 20a”, 21a” (Figure A.3). The 6-31G* basis set19 was used. The state-averaging included four states (SA4). In case of Cs symmetry the two lowest 2A’ and two lowest 2A” states were chosen. Single point calculations were carried out using SA4-CASSCF(5,5)/6-31G* and MR-CI with all singles and doubles (MR-CISD) again using the 6-31G* basis set. As reference space, the same CAS(5,5) as used in the CASSCF(5,5) calculations was selected. Single and double excitations were constructed from the doubly-occupied and active orbitals applying the interactive space restriction.20 The 1s orbitals of all carbon atoms were kept frozen. Size-extensivity corrections were included by means of the extended Davidson correction13, 21-22 denoted as +Q (MR-

2 CISD+Q). C0 is the sum of square coefficients of the reference configurations in the MR-

CISD expansion and EREF denotes the reference energy.

2 1 C0  ECI  EREF  EQ  2 (3.1) 2C0 1

The potential energy surface (PES) scan was carried out using the SA4-

CASSCF(5,5)/6-31G* approach for pyrene-1C+Hr and pyrene-2C+Hr and the

CASSCF(5,5)/6-31G* approach for pyrene+Hr. For that purpose, the previously relaxed pyrene defect (pyrene-1C and -2C) structures computed earlier14-15 and an optimized pyrene structure using the DFT and the hybrid Becke, 3-parameter, Lee-Yang-Parr 23 19 (B3LYP) functional together with the 6-31G** basis set were used to move Hr around 44 Texas Tech University, Reed Nieman, May 2019 in a grid with an increment of 0.3 Å and at a distance of 1.0 Å from the plane structures. Single-point calculations were performed for each point on the grid.

Using the local minima in the PES´s scans as starting structures, geometry optimizations were carried out. To simulate the constraints present in a larger graphene system, the coordinates of the carbons connected to hydrogen atoms at the periphery of the pyrene and pyrene defect systems were always frozen in all geometry optimizations since they would be connected to neighboring carbon atoms in a graphene sheet. All other atoms were allowed to relax. Geometry optimizations using SA4-CASSCF(5,5)/6-31G* for the defect cases plus Hr and CASSCF(5,5) calculations for the pristine pyrene+Hr were performed by placing Hr at an initial distance of 1.1 Å over each unique carbon in each of the pyrene system.

The CASSCF, MR-CISD and MR-CISD+Q calculations were carried out using COLUMBUS;24-28 the geometry optimizations used the program system DL-FIND29 interfaced to COLUMBUS. The DFT calculations were performed with the Turbomole suite of programs.30

3.4 Results and Discussion 3.4.1 Potential Energy Scans

The two-dimensional PES scans of the single vacancy pyrene-1C+Hr for the ground state and the first three excited states are shown in Figure 3.1. For that purpose, the Hr atom has been moved systematically in a grid 1.0 Å above the fixed, planar structure of pyrene-

1C. These scans detail the ability of the test particle, Hr, to bond in a particular region of pyrene-1C. For the ground state, deep local minima can be seen at the carbons in the five- membered ring, C1, C2, C3, C13 and C14 (see Scheme 1 for numbering of atoms). The lowest minimum at C15 persists through all four states; for the excited states the local minimum at the five-membered ring is considerably higher in energy. Shallower minima are present for C5, C7, C9, and C11. Generally, the minima for carbons in the six-

45 Texas Tech University, Reed Nieman, May 2019 membered ring, aside from C15, give minima much shallower than for the remaining carbon atoms.

1 2Aa

2 2Ab

3 2Ac

4 2Ad

Figure 3.1 PES scans in terms of the location of Hr for the four lowest states of pyrene- 1C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane 46 Texas Tech University, Reed Nieman, May 2019 parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energies: a - 574.222889 Hartree, b -574.198174 Hartree, c -574.149530 Hartree, d -574.117820 Hartree. The energy (color scale) is given in eV, relative to the ground state lowest minimum.

3-D pictures of the PES scan of the double vacancy defect pyrene-2C+Hr for the electronic ground- and first excited states are displayed in Figure 3.2 and Figure 3.3, respectively. The plots for the second and third excited state are shown in Figure A.4. For the ground state deep potential wells are seen at C1, C2, C14 and C7, C8, C9, these being the carbons at the outer sections of the five-membered rings, with the strongest minima present at C1 and C8. Additional shallower minima are seen at each of the remaining carbons. The ground and first excitation are seen to be on average energetically closer as compared to the single vacancy defect just discussed above. On the other hand, the scans for the second and third vertical excitations (Figure A.4) are closely spaced and quite higher, by ~ 2-2.5 eV, than the lowest minimum.

Figure 3.2 PES scan in terms of the location of Hr for the lowest state of pyrene-2C+ Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energy = -536.385046 Hartree. The energy (color scale) is given in eV, relative to the ground state lowest minimum. The ground state PES for pyrene+Hr is shown in Figure 3.4. Pronounced minima can be seen at C4, C5, C11, and C12. While for each of the remaining carbon atoms in the pyrene system local minima can be found, C2, C7, C9, and C14 show to be especially stabilized among them.

47 Texas Tech University, Reed Nieman, May 2019

Figure 3.3 PES scan in terms of the location of Hr for the 22A state of pyrene-2C+ Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energy = -536.347629 Hartree. The energy (color scale) is given in eV, relative to the ground state lowest minimum at -536.385046 Hartree.

Figure 3.4 PES scan of the lowest state of pyrene+Hr using the CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by

48 Texas Tech University, Reed Nieman, May 2019 the frozen carbon atoms. Lowest energy = -612.270253 Hartree. The energy (color scale) is given in eV, relative to the ground state lowest minimum.

3.4.2 Geometry Optimizations

The PES scans discussed in the previous section give a good global overview of the chemical reactivity of the different pyrene defects (including the pristine case) concerning the strength of different C-Hr bonds as well as to the location of the centers of most stable bond formation. In the following sections more detailed information on the properties of individual structures will be discussed.

In the search for local minima structures containing C-Hr bonds, geometry optimizations were performed as described in the Computational Details by placing Hr on top of each carbon atom and performing a geometry optimization in each case only freezing selected carbon atoms as discussed above to simulate geometrical restrictions due to neighboring carbon atoms in a graphene sheet. Each carbon tested was found to give rise to a local minimum for all structures investigated.

Starting with the single vacancy pyrene-1C+Hr case, the nine ground state structures found are described in Figure 3.5. The global minimum is given by bonding Hr to C15 with a C-Hr bond distance of 1.060 Å. This is the smallest C-H bond distance determined for the nine structures and is much smaller than the C-Hr bond distance of the other minima whose bond distance is around 1.095 Å. When Hr is bonded to C1, C2, C4, and C5, planarity of the ring system is mostly maintained. However, the existing hydrogen 3 is moved out-of-plane away from the incoming Hr due to the formation of a tetrahedral sp hybrid at this carbon atom. This behavior repeated in all analogous cases. When Hr is bonded to C3 and C6, the carbon atoms reacted by rising out of plane. When Hr was bonded to C7 and C8, the bonding carbon could not move since it was frozen, but C6, C10, and

C15 moved out of plane. Finally, when Hr bonded to C15 in the global minimum structure, again C6, C10, and C15 moved out of plane.

49 Texas Tech University, Reed Nieman, May 2019

C1 C2

C3 C4

C5 C6

C7 C8

C15

Figure 3.5 Plots of the different minimum structures (labeled according to the bonding carbon atom) on the ground state energy surface of pyrene-1C+Hr using the SA4- CASSCF(5,5)/6-31G* approach. Bond distances are given in Å.

In the double vacancy pyrene-2C+Hr case, the four ground state structures are characterized in Figure 3.6. The global minimum was found when Hr is bonded to C2 in the five-membered ring with a C-Hr distance of 1.093 Å and gave the smallest C-Hr distance of all the minimum structures. The largest C-Hr bond distance of 1.121 Å to C1 is 50 Texas Tech University, Reed Nieman, May 2019 notable here as it is the largest one of all three pyrene structures. For C3, as before, since it was not frozen and only bonded to other carbons, the carbon atom rose out of the plane of the frozen carbon atoms when rehybridizing from sp2 to sp3.

C1

C2

C3

C4

Figure 3.6 Plots of the different minimum structures (labeled according to the bonding carbon atom) of the ground state energy surface of pyrene-2C+Hr using the SA4- CASSCF(5,5)/6-31G* approach. Bond distances are given in Å.

Finally, for pyrene+Hr, the characterization of the five ground states is presented in

Figure 3.7. The global minimum was found when Hr was bonded to C4 with a C-Hr bond distance of 1.097 Å. This C-Hr distance was the smallest one of the four minima found, though it can be noted that the C-Hr bond distances for this structure do not vary greatly as the largest C-Hr bond distance is only slightly larger with 1.103 Å given for C1-Hr. In cases where the bonding carbon was already bonded to hydrogen (carbons 1, 2, 11), Hr formed a 51 Texas Tech University, Reed Nieman, May 2019

C-H bond by pushing the other hydrogen away, as reported also in previous cases. When

Hr bonded to a hydrogen-free carbon (carbons 3 and 16), the bonding carbon responded by rising out of the plane of the frozen carbon atoms by about 0.5 Å. A similar observation was made by Rasmussen et al.10 by means of DFT calculations where binding of a hydrogen atom on the interior positions (C3 and C16) of the pyrene led to a significant puckering of these carbon atoms (~0.4 Å) out of the molecular plane. This behavior on hydrogenation of graphene is generally feasible in the free graphene sheet4 but is also true for the pyrene defect plus Hr cases as explained above

3.4.3 Energetic Stabilities

Table 3.1 shows the vertical excitation energies for pyrene-1C+Hr the lowest two

A’ and A” states for structures having Cs symmetry, i.e. Hr bonded to C15 (global minimum), C1 and C8 using the SA4-CASSCF, MR-CISD and MR-CISD+Q approaches. The 12A’ state is the ground state. At MR-CISD+Q level, in the C15 structure the first excited state is only ~0.8 eV above the ground state, followed by states about 2.4 eV and 3.6 eV above the ground state. The ground state C1 structure is about 1.4 eV higher at MR- CISD+Q level than the respective one for C15; the C8 structure is located significantly higher (4.2 eV) above the C15 ground state. The order of the states computed at MR-CISD and MR-CISD+Q levels is reproduced well by the SA4-CASSCF(5,5) calculations even though there are non-negligible numerical differences up to 0.8 eV.

52 Texas Tech University, Reed Nieman, May 2019

C1

C2

C3

C4

C16

Figure 3.7 Plots of the different minimum structures (labeled according to the bonding carbon atom) on the ground state energy surface of pyrene+Hr using the CASSCF(5,5)/6- 31G* approach. Bond distances are given in Å.

53 Texas Tech University, Reed Nieman, May 2019

Table 3.1 Vertical energy differences (eV) of the lowest two A’ and A” states relative to the C15 ground state computed for the C1, C8 and C15 pyrene-1C+Hr structures using the SA4-CASSCF(5,5), MR-CISD and MR-CISD+Q approaches and the 6-31G* basis set.

CASSCF MR-CISD MR-CISD+Q State ΔErel ΔErel ΔErel C15 1 2A’ 0.000a,d 0.000b,d 0.000c,d 1 2A” 1.154 1.023 0.834 2 2A’ 2.571 2.504 2.375 2 2A” 4.100 3.904 3.594 C1 1 2A’ 0.636d 0.960d 1.393d 2 2A’ 3.775 4.073 4.444 1 2A” 4.828 4.973 5.074 2 2A” 5.854 6.069 6.186 C8 1 2A’ 3.542d 3.930d 4.246d 2 2A’ 3.900 4.214 4.456 1 2A” 4.216 4.604 4.947 2 2A” 4.643 4.994 5.187 a Total energy= -574.277274 Hartree, b Total energy= -575.610208 Hartree, cTotal Energy= -576.244081 Hartree, d The SA4-CASSCF(5,5) optimized geometry was used.

Figure 3.8 Location and relative stability (eV) of the minima on the ground state energy 2 surface of pyrene-1C+Hr using the MR-CISD+Q/6-31G* method. Blue: A ground state, red: lowest vertical excitation. Note: the rigid pyrene-1C structure shown does not reflect the actual structure of the optimized minima.

54 Texas Tech University, Reed Nieman, May 2019

In Figure 3.8 (see also Table A.1) are the relative energies of each carbon minimum in pyrene-1C+Hr given with respect to the global minimum structure at C15 computed at MR-CISD+Q level. The first vertical excitations are given in the figure as well. C15 marks the global minimum followed by C2, C3, and C1, respectively, in the five-membered ring. Except for the structure C15, all first excited states are located in energy higher by ~3.5 eV or more. This shows that in case of the reaction of Hr with a single vacancy graphene defect at thermalized kinetic energies nonadiabatic interactions between different electronic states will be relevant mostly for the C15 approach, which, is however, energetically the most relevant one.

The energy profiles for the reaction of Hr with the pyrene-1C defect are shown in Figure 3.9 for the case of bond formation with C15. The reaction curves have been created by fixing the C15-Hr distance and optimizing the remaining part of the pyrene defect structure for the ground state at SA4-CASSCF(5,5)/6-31G* level. Selected carbon atoms as discussed above were frozen to simulate also in this case geometrical restrictions due to neighboring carbon atoms in a graphene sheet. Vertical excitations have been used to compute the higher electronic states. These calculations have been performed in C1 symmetry. Nevertheless, the Cs symmetry analysis of Table 3.1 for the ground state minimum, as indicated in Figure 3.9, can be used for the analysis of states. For the 12A’ ground state the figure shows a smooth dissociation without change of the character of the wave function. The second state is 12A” at the energy minimum. A crossing between this 2 state and the 2 A’ state is observed at a C15-Hr distance of about 2 Å. For large distances the two 2A’states become practically degenerate. The 22A’ and 22A” states show a spike at 1.5 Å indicating the crossing with higher states of different character. These states are repulsive and are responsible for the correct asymptotic behavior of the dissociation. The energy minima of the first three states are located below the lowest dissociation limit so that on reaction with an incoming hydrogen atom several electronic states could be involved in the relaxation process to the ground state.

55 Texas Tech University, Reed Nieman, May 2019

2 Figure 3.9 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C15 using the MR-CISD/6-31G* approach. Minimum energy = -574.277274 Hartree

Figure 3.10 shows the dissociation/bonding process of Hr from/to C1. Evident is the lack of a potential barrier in the ground state curve, though potential energy barriers and avoided crossings exist in the excited state curves. However, differently to the case of the C15-Hr bond dissociation, the excited states are not expected to play a major role for the relaxation process to the energy minimum on formation of the C1-Hr bond due to their high energies around the ground state minimum.

2 Figure 3.10 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C1 using the MR-CISD/6-31G* approach. Minimum energy = -574.277274 Hartree

56 Texas Tech University, Reed Nieman, May 2019

The Hr dissociation curves computed at SA4-CASSCF(5,5) level for bonding to C2 to C8, respectively, are collected in the SI. For C2 and C3, Figure A.5 and Figure A.6, there is no potential barrier for the incoming Hr ground state. In the remaining cases (Figure A.7 – Figure A.11), however, there are potential barriers present for the bond formation to C4, C5, C6, C7 and C8. Estimated barrier heights with respect to the dissociation limit are collected in Table 3.2. They show a variation between 0.4 eV to 1.4 eV indicating that the approach from that side of pyrene-1C is not so favorable.

The dissociation energy of the C15 global minimum pyrene-1C + Hr structure into the lowest state of pyrene-1C + Hr is given in Table 3.2. At MR-CISD+Q level a value of 4.61 eV is found. Comparison with the MR-CISD (4.13 eV) and SA4-CASSCF(5,5) (3.66 eV) values shows the importance of including size-extensivity and dynamic electron correlation contributions to the dissociation energy. Dissociation of Hr for the other structures can be obtained from Figure 3.8 by subtracting the relative stabilities form the just-mentioned dissociation limit of the C15 structure. The strongest C-Hr bonds following the one to C15 can be found for C-H bonding to carbons in the five-membered ring. Some of the other structures, though being local minima, are close in energy to the C-Hr bond dissociation limit.

Table 3.2 Dissociation energies Ediss (eV) for the C15 global minimum pyrene-1C+Hr structure and estimated energy barriers Ebarr(est.) (eV) for the approach of Hr on the ground state surface toward the different carbon atoms of pyrene-1C using different theoretical methods and the 6-31G* basis set.

Ediss SA4-CASSCF(5,5) MR-CISD MR-CISD+Q 3.660 4.128 4.605 a Ebarr(est.) (SA4-CASSCF(5,5)) C1, C2, C3, C15: no barrier C4 C5 C6 0.914 0.426 1.403 C7 C8 1.138 1.051

The relative stabilities in the electronic ground state and the lowest vertical excitations of pyrene-2C+Hr are presented in Figure 3.11 for the MR-CISD+Q/6-31G* approach. SA4-CASSCF(5,5) and MR-CISD results are collected in Table A.2. Numerical 57 Texas Tech University, Reed Nieman, May 2019 agreement between the different methods is within 0.1 – 0. 2 eV. The global minimum was found to be for Hr bonded to C2 in the five-membered ring. The minima are ordered from most stable to least stable as C2

Figure 3.11 Location and relative stability (eV) of the minima on the ground state energy 2 surface of pyrene-2C+Hr defect using the MR-CISD+Q/6-31G* method. Blue: A ground state, red: lowest vertical excitation. Note: the rigid pyrene-2C structure shown does not reflect the actual structure of the optimized minima.

The dissociation energy for the process pyrene-2C+Hr is given in Table 3.3 for the global minimum structure with Hr bound to C2 for the CASSCF, MR-CISD and MR-

CISD+Q methods. Compared to the pyrene-1C+Hr case (Table 3.2), the C-Hr bond strength is significantly reduced. However, it should be noted that all local ground state minima (Figure 3.11) are below the dissociation limit of 2.269 eV.

Table 3.3 Dissociation energies Ediss (eV) for the double vacancy pyrene-2C+Hr and pristine pyrene + Hr cases using different methods and the 6-31G* basis set. SA4-CASSCF(5,5) MR-CISD MR-CISD+Q

pyrene-2C+Hr Ediss(C2) 2.224 2.257 2.269 a pyrene+Hr Ediss(C2) 0.833 0.902 0.943

For the reaction of pyrene with Hr, pyrene+Hr, only the electronic ground state has been investigated. Relative stabilities of the different C-Hr bonded local minima are shown in Figure 3.12. CASSCF(5,5), MR-CISD and MR-CISD+Q results are collected in Table

58 Texas Tech University, Reed Nieman, May 2019

A.3. A similar good agreement between different methods is obtained as for the pyrene-

2C+Hr case discussed above in Table A.2. The minima are ordered energetically as

C2

Figure 3.12 Location and relative stability (eV) of the minima on the ground state energy surface of pyrene+Hr using the MR-CISD+Q/6-31G* method. Note: the rigid pyrene structure shown does not reflect the actual structure of the optimized minima.

3.5 Conclusions

The chemical reactivity of a single and double vacancy in graphene and of a pristine surface toward C-H bond formation with a reactive hydrogen atom Hr has been investigated using a pyrene model with peripheral carbon atoms connected to H atoms frozen. Multireference methods (MR-CISD and MR-CISD+Q) have been used to describe (i) the dissociation processes properly and (ii) the manifold of quasi-degenerate states of the 14-15 isolated defect. In a first step energy grids have been computed by moving an atom Hr across the rigid surface of the pyrene defects structures in order to obtain an overview of the stabilization energies in the different regions in space. The results show for the single vacancy a preference for bonding to C15 which possessing a dangling bond. The relative 59 Texas Tech University, Reed Nieman, May 2019 stabilization in this position is also observed in the lowest excited states. However, the carbon atoms in the five-membered ring show also strongly attractive positions in the ground state. The double vacancy displays similar stability preferences in the five membered rings; also the pristine pyrene model indicates attractive interactions primarily around the peripheral carbon atoms.

Geometry optimizations performed for the ground state at CASSCF level followed by single-point MR-CISD and MR-CISD+Q calculations refine this picture. Importantly, separate local energy minima for C-Hr bond formation have been determined for each of the carbon atoms in all defect structures and in pyrene itself. In the single vacancy case, C15 remains the position with greatest stability followed by C2 towards the top of the five- membered ring. In case of the double vacancy position C2 in the five-membered ring gives rise to the strongest C-Hr bond and in pyrene itself these are positions C2 and C4.

C-Hr dissociation curves have been computed for the four lowest electronic states for the single vacancy model starting from all minima found. Barrierless ground state processes have been found for C15 and the C-Hr bonds involving the five-membered ring. An incoming hydrogen atom though approaching the pyrene-C defect toward the other carbon atoms would encounter barriers of about 0.5 – 1 eV, depending on the incoming approach angle. Curve crossings and avoided crossings are found especially for Hr approaches to the most stable carbon positions which lead to the expectation of a complex nonadiabatic reaction dynamics. In summary, a remarkably complex picture of the bonding options of a hydrogen atom reacting with a graphene vacancy structures is observed both in terms of different bonding positions within the defect but also in terms of the reaction dynamics involving several excited states. It is expected that these properties found for the pyrene model are fairly representative for a localized graphene defect. Preliminary density functional studies confirm this view. More detailed investigations are in preparation.

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3.6 Acknowledgments

This material is based upon work supported by the National Science Foundation under Project No. CHE-1213263 and by the Austrian Science Fund (SFB F41, ViCoM). We are grateful for computer time at the Vienna Scientific Cluster (VSC), project 70376. FBCM wishes to thank Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under Projects Process Nos. 2014/24155-6 and 2015/500118-9, to Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) under Project CAPES/ITA 005/2014 and to Conselho Nacional de Desenvolvimento Científico and Tecnológico (CNPq) for the research fellowship under Process No. 304914/2013-4. We also want to thank the FAPESP/Texas Tech University SPRINT program (project no. 2015/50018-9) for travel support.

3.7 Bibliography

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(7) Guisinger, N. P.; Rutter, G. M.; Crain, J. N.; First, P. N.; Stroscio, J. A., Exposure of Epitaxial Graphene on SiC(0001) to Atomic Hydrogen. Nano Letters 2009, 9 (4), 1462-1466. (8) Elias, D. C.; Nair, R. R.; Mohiuddin, T. M. G.; Morozov, S. V.; Blake, P.; Halsall, M. P.; Ferrari, A. C.; Boukhvalov, D. W.; Katsnelson, M. I.; Geim, A. K.; Novoselov, K. S., Control of Graphene's Properties by Reversible Hydrogenation: Evidence for Graphane. Science 2009, 323 (5914), 610. (9) Bauschlicher, C. W., The reaction of polycyclic aromatic hydrocarbon cations with hydrogen atoms: The astrophysical implications. Astrophys J 1998, 509 (2), L125- L127. (10) Rasmussen, J. A.; Henkelman, G.; Hammer, B., Pyrene: Hydrogenation, hydrogen evolution, and pi-band model. Journal of Chemical Physics 2011, 134 (16), 164703. (11) Chernozatonskii, L. A.; Kvashnin, D. G.; Sorokin, P. B.; Kvashnin, A. G.; Brüning, J. W., Strong Influence of Graphane Island Configurations on the Electronic Properties of a Mixed Graphene/Graphane Superlattice. The Journal of Physical Chemistry C 2012, 116 (37), 20035-20039. (12) Gonzalez-Herrero, H.; Gomez-Rodriguez, J. M.; Mallet, P.; Moaied, M.; Palacios, J. J.; Salgado, C.; Ugeda, M. M.; Veuillen, J.-Y.; Yndurain, F.; Brihuega, I., Atomic-scale control of graphene magnetism by using hydrogen atoms. Science 2016, 352, 437-441. (13) Szalay, P. G.; Müller, T.; Gidofalvi, G.; Lischka, H.; Shepard, R., Multiconfiguration Self-Consistent Field and Multireference Configuration Interaction Methods and Applications. Chemical Reviews 2012, 112 (1), 108-181. (14) Machado, F. B. C.; Aquino, A. J. A.; Lischka, H., The Diverse Manifold of Electronic States Generated by a Single Carbon Defect in a Graphene Sheet: Multireference Calculations Using a Pyrene Defect Model. ChemPhysChem 2014, 15 (15), 3334-3341. (15) Machado, F. B. C.; Aquino, A. J. A.; Lischka, H., The electronic states of a double carbon vacancy defect in pyrene: a model study for graphene. Physical Chemistry Chemical Physics 2015, 17 (19), 12778-12785. (16) Haldar, S.; Amorim, R. G.; Sanyal, B.; Scheicher, R. H.; Rocha, A. R., Energetic stability, STM fingerprints and electronic transport properties of defects in graphene and . Rsc Adv 2016, 6 (8), 6702-6708. (17) Ruedenberg, K.; Cheung, L. M.; Elbert, S. T., MCSCF optimization through combined use of natural orbitals and the brillouin-levy-berthier theorem. International Journal of Quantum Chemistry 1979, 16, 1069-1101. (18) Roos, B.; Taylor, P. R.; Siegbahn, P. E. M., A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chemical Physics 1980, 48 (2), 157-173. (19) Hehre, W. J.; Ditchfield, R.; Pople, J. A., Self—Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian—Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules. The Journal of Chemical Physics 1972, 56 (5), 2257-2261.

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(20) Bunge, A., Electronic Wavefunctions for Atoms. III. Partition of Degenerate Spaces and Ground State of C. The Journal of Chemical Physics 1970, 53 (1), 20- 28. (21) Langhoff, S. R.; Davidson, E. R., Configuration interaction calculations on the nitrogen molecule. International Journal of Quantum Chemistry 1974, 8 (1), 61- 72. (22) Luken, W. L., Unlinked cluster corrections for configuration interaction calculations. Chemical Physics Letters 1978, 58 (3), 421-424. (23) Becke, A. D., Density‐functional thermochemistry. III. The role of exact exchange. The Journal of Chemical Physics 1993, 98 (7), 5648-5652. (24) Lischka, H.; Shepard, R.; Pitzer, R. M.; Shavitt, I.; Dallos, M.; Müller, T.; Szalay, P. t. G.; Seth, M.; Kedziora, G. S.; Yabushita, S.; Zhang, Z., High-level multireference methods in the quantum-chemistry program system COLUMBUS: Analytic MR-CISD and MR-AQCC gradients and MR-AQCC-LRT for excited states, GUGA spin–orbit CI and parallel CI density. Physical Chemistry Chemical Physics 2001, 3, 664-673. (25) Lischka, H.; Müller, T.; Szalay, P. G.; Shavitt, I.; Pitzer, R. M.; Shepard, R., Columbus-a program system for advanced multireference theory calculations. Wiley Interdisciplinary Reviews-Computational Molecular Science 2011, 1, 191- 199. (26) Dachsel, H.; Lischka, H.; Shepard, R.; Nieplocha, J.; Harrison, R. J., A massively parallel multireference configuration interaction program: The parallel COLUMBUS program. Journal of Computational Chemistry 1997, 18 (3), 430- 448. (27) Müller, T., Large-scale parallel uncontracted multireference-averaged quadratic coupled cluster: the ground state of the chromium dimer revisited. The Journal of Physical Chemistry A 2009, 113, 12729-40. (28) H. Lischka, R. S., I. Shavitt, R. M. Pitzer, M. Dallos, Th. Müller, P. G. Szalay, F. B. Brown, R. Ahlrichs, H. J. Böhm, A. Chang, D. C. Comeau, R. Gdanitz, H. Dachsel, C. Ehrhardt, M. Ernzerhof, P. Höchtl, S. Irle, G. Kedziora, T. Kovar, V. Parasuk, M. J. M. Pepper, P. Scharf, H. Schiffer, M. Schindler, M. Schüler, M. Seth, E. A. Stahlberg, J.-G. Zhao, S. Yabushita, Z. Zhang, M. Barbatti, S. Matsika, M. Schuurmann, D. R. Yarkony, S. R. Brozell, E. V. Beck, and J.-P. Blaudeau, M. Ruckenbauer, B. Sellner, F. Plasser, and J. J. Szymczak COLUMBUS 7.0. (29) Kästner, J.; Carr, J. M.; Keal, T. W.; Thiel, W.; Wander, A.; Sherwood, P., DL- FIND: an open-source geometry optimizer for atomistic simulations. The journal of physical chemistry. A 2009, 113, 11856-65. (30) Ahlrichs, R.; Bär, M.; Häser, M.; Horn, H.; Kölmel, C., Electronic structure calculations on workstation computers: The program system turbomole. Chemical Physics Letters 1989, 162 (3), 165-169.

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CHAPTER VI

STRUCTURE AND ELECTRONIC STATES OF A GRAPHENE DOUBLE VACANCY WITH EMBEDDED SI DOPANT Authors: Reed Nieman,a Adelia J. A. Aquino,a,b Tevor P. Hardcastle,c,d Jani Kotakoski,e Toma Susi,e and Hans Lischkaa,b Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University, Lubbock, TX 79409-1061, USA, bSchool of Pharmaceutical Sciences and Technology, Tianjin University, Tianjin 300072, PR China, cSuperSTEM Laboratory, STFC Daresbury Campus, Daresbury WA4 4AD, United Kingdom, dFaculty of Engineering, School of Chemical and Process Engineering, University of Leeds, 211 Clarendon Rd., Leeds LS2 9JT, United Kingdom, eFaculty of Physics, University of Vienna, Boltzmanngasse 5, A- 1090 Vienna, Austria Published: October 2017 in The Journal of Chemical Physics, DOI:10.1063/1.4999779

4.1 Abstract

Silicon represents a common intrinsic impurity in graphene, commonly bonding to either three or four carbon neighbors respectively in a single or double carbon vacancy. We investigate the effect of the latter defect (Si-C4) on the structural and electronic properties of graphene using density functional theory (DFT). Calculations based both on molecular models and with periodic boundary conditions have been performed. The two-carbon vacancy was constructed from pyrene (pyrene-2C) which was then expanded to circumpyrene-2C. The structural characterization of these cases revealed that the ground state is slightly non-planar, with the bonding carbons displaced from the plane by up to ±0.2 Å. This non-planar structure was confirmed by embedding the defect into a 10×8 supercell of graphene, resulting in 0.22 eV lower energy than the previously considered planar structure. Natural bond orbital (NBO) analysis showed sp3 hybridization at the silicon atom for the non-planar structure and sp2d hybridization for the planar structure. Atomically resolved electron energy loss spectroscopy (EELS) and corresponding spectrum simulations provide a mixed picture: a flat structure provides a slightly better overall spectrum match, but a small observed pre-peak is only present in the corrugated simulation. Considering the small energy barrier between the two equivalent corrugated

64 Texas Tech University, Reed Nieman, May 2019 conformations, both structures could plausibly exist as a superposition over the experimental timescale of seconds.

4.2 Introduction

The modification of the physical properties of graphene sheets and nanoribbons, in particular the introduction of a band gap via chemical adsorption, carbon vacancies, and the incorporation of dopant species has gained a great deal of attention.1-2 Silicon is an interesting element as a dopant as it is isovalent to carbon. It is also a common intrinsic impurity in graphene sheets grown by chemical vapor deposition3-4 (CVD) and in graphene epitaxially grown by the thermal decomposition of silicon carbide.5-6 Further, dopant impurities have been shown to have significant effects on the transport properties of graphene.7-9

Silicon dopants in graphene exist predominantly in two previously identified forms: a non-planar, threefold-coordinated silicon atom in a single carbon vacancy, referred to as

Si-C3, and a fourfold-coordinated silicon atom in a double carbon vacancy, called Si-C4, which is also thought to be planar. An experimental work from 2012 by Zhou and coauthors10 showed via simultaneous scanning transmission electron microscopy (STEM) annular dark field imaging (ADF) and electron energy loss spectroscopy (EELS) and, additionally, by DFT calculations that the two forms of single silicon doped graphene differ 3 energetically, and that the silicon atom in Si-C3 has sp orbital hybridization and that the 2 silicon atom in Si-C4 is sp d hybridized. Similarly, EELS experiments combined with DFT spectrum simulations confirmed the puckering of the Si in the Si-C3 structure and supported 11 the planarity of the Si-C4 structure. However, a less satisfactory agreement of the simulated spectrum with the experimental one was noted in the latter case. The results suggested that d-band hybridization of the Si was responsible for electronic density disruption found at these sites. Experiments and dynamical simulations performed on these 12 defects have demonstrated how the Si-C4 structure is formed when an adjacent carbon atom to the silicon is removed. However, it was concluded that the Si-C3 structure is more stable and can be readily reconstructed by a diffusing carbon atom. 65 Texas Tech University, Reed Nieman, May 2019

Past investigations of the electronic structure of graphene defects have shown the usefulness of the molecule pyrene as a compact representation of the essential features of single (SV) and double vacancy (DV) defects. Using this model, chemical bonding and the manifold of electronic states were investigated by multireference configuration interaction (MR-CI) calculations.13-16 In case of the SV defect,13 the MR-CI calculations showed four electronic states (two singlet and two triplet) within a narrow margin of 0.1 eV, whereas for the DV structure,14 a gap of ~1 eV to the lowest excited state was found. For the DV defect, comparison with density functional theory using the hybrid density functional Becke three-parameter Lee, Yang, and Parr (B3-LYP) functional17 showed good agreement with the MR-CI results. Whereas for the covalently bonded Si-C3 structure no further low- lying states are to be expected, the situation is different for the Si-C4 defect since in this case an open-shell Si atom is inserted into a defect structure containing low-lying electronic states.

Based on these experiences, a pyrene model will be adopted also in this work to investigate the geometric arrangement and orbital hybridization around the silicon atom in a double carbon vacancy (Figure 4.1). In a second step, a larger circumpyrene structure (Figure 4.2) will be used to study the effect of increasing number of surrounding benzene rings. Two types of functionals will be employed, the afore-mentioned B3LYP and, for comparison, the long-range corrected Coulomb-attenuating B3LYP method (CAM- B3LYP).18 Special emphasis will be devoted to verifying the electronic stability of the closed-shell wavefunction with respect to triplet instability19-20 and of the optimized structures with respect to structural relaxation. Finally, periodic DFT simulations of graphene with the Perdew-Burke-Ernzerhof (PBE) functional21-22 will be compared to the molecular calculations and an EELS spectrum simulated based on the computed Si-C4 structure compared to an atomic resolution STEM/EELS measurement.

4.3 Computational Details

Pyrene doped with a single silicon atom was investigated by first optimizing the structure of pristine pyrene using the density functional B3LYP17 with the 6-31G*23 basis 66 Texas Tech University, Reed Nieman, May 2019 set. The two interior carbon atoms were then removed and replaced with one silicon atom, referred to in this investigation as pyrene-2C+Si. This unrelaxed structure was subjected to a variety of closed-shell [restricted DFT (RDFT)] and open-shell [unrestricted DFT (UDFT)] geometry optimizations using the hybrid density functionals B3LYP and CAM- B3LYP in combination with the 6-311G(2d,1p)24-28 basis set. To extend to a larger system, pyrene-2C+Si was surrounded by benzene rings (circumpyrene-2C) and this structure was optimized using the B3LYP/6-31G** approach. Finally, the geometry optimization of cirumpyrene-2C+Si was performed using the B3LYP functional and the 6-311G(2d,1d) basis.

To simulate the geometric restrictions present in a large graphene sheet, the Cartesian coordinates of all carbon atoms on the outer rims of pyrene-2C+Si (Figure 4.1) and cirumpyrene-2C+Si (Figure 4.2) that are bonded to hydrogen atoms were frozen during the course of geometry optimization to their values in the pristine structures. The coordinates of both pyrene-2C+Si and circumpyrene-2C+Si were oriented along the y-axis (the long axis) in the xy-plane.

The orbital occupations of the open-shell structures were determined using the natural orbital population analysis (NPA)29 calculated from the total density matrix, while the bonding character at silicon for each structure was investigated using the natural bond orbital analysis.30-34

The discovered corrugated ground state structure of the Si-C4 defect was then introduced into a periodic 10×8 supercell of graphene, and the structure relaxed to confirm that this corrugation is reproduced at this level of theory. Finally, the Si L2,3 EELS response of the defect was simulated11, 35 by evaluating the perturbation matrix elements of transitions from the Si 2p core states to the unoccupied states calculated up to 3042 bands, with no explicit core hole.36 The resulting densities of state were broadened using the OptaDOS package37 with a 0.4 eV Gaussian instrumental broadening and semi-empirical 0.015 eV Lorentzian lifetime broadening.

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The Gaussian 0938 and TURBOMOLE39 program packages were used for the molecular calculations, and the GPAW40 and CASTEP41 packages for the graphene simulations.

Figure 4.1 Optimized pyrene-2C+Si structure P1 (D2 symmetry) using the B3LYP/6- 311G(2d,1p) method with frozen carbon atoms marked by gray circles

Figure 4.2 Optimized circumpyrene-2C+Si structure CP1 (D2 symmetry) using the B3LYP/6-311G(2d,1p) method with frozen carbon atoms marked by gray circles.

4.4 Results and Discussion 4.4.1 The Pyrene-2C+Si Defect Structure

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The geometry of the unrelaxed, planar pyrene-2C+Si was first optimized using the

B3LYP/6-311G(2d,1p) method at closed-shell level resulting in the D2h structure denoted P2. There were two imaginary frequencies present in the Hessian matrix. A displacement of the geometry of structure P2 along either of these imaginary frequencies led to a distortion of the planarity, lowering the point group to D2 and producing the minimum energy structure P1, which was 0.736 eV lower in energy than structure P2 (Table 4.1). The carbon atoms bonded to the silicon atom (atoms 3, 6, 10, and 13) were displaced from planarity by about ±0.2 Å while the silicon atom remained co-planar with the frozen carbon atoms on the periphery of pyrene-2C+Si. A similar tetrahedral geometry was found for 42 doping Al and Ga into DV graphene. No triplet instability was found for either of the D2h and D2 structures. Optimizing the planar pyrene-2C+Si structure at UDFT/B3LYP/6- 311G(2d,1p) level for the high spin case resulted in structure P3, which was planar and belonging to the D2h point group. No imaginary frequencies were found in the Hessian matrix of structure P3; it is located 0.922 eV above the minimum energy structure P1.

Table 4.1 Open-shell character, structural symmetry, spin multiplicity, bond distances (Å), and relative energies (eV) of selected optimized structures of pyrene-2C+Si. Closed/ High/ Open Triplet Low C3- Structure Shell instab. #imaga Spin Sib-C C13 C3-C6 ΔE B3LYP/6-311G(2d,1p) c P1 (D2) Closed no 0 Low 1.940 2.652 2.860 0.000 P2 (D2h) Closed no 2 Low 1.932 2.634 2.826 0.736 P3 (D2h) Open - 0 High 1.933 2.610 2.852 0.922 CAM-B3LYP/6-311G(2d,1p) c d d d P4 (D2) Open no 0 Low 1.933 2.645 2.847 0.000 c P5 (D2) Weakly no 0 Low 1.936 2.648 2.852 0.040 open c P6 (D2) Closed yes 0 Low 1.938 2.648 2.857 0.056 P7 (D2h) Open no 1 Low 1.923 2.616 2.820 0.602 P8 (D2h) Closed yes 2 Low 1.931 2.631 2.828 0.734 e P9 (D2) Open - 0 High 1.900 2.599 2.777 0.770 P10 (D2h) Open - 1 High 1.900 2.598 2.772 0.826 aNumber of imaginary frequencies. bSilicon always located in-plane. cThe out-of-plane d distance of carbon atoms that were not frozen in D2 structures is ±0.2 Å Averaged value of the four bonding carbons. eThe out-of-plane distance of carbon atoms that were not frozen is ±0.07 Å

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Structures P1-P3 were investigated also using the CAM-B3LYP/6-311G(2d,1p) method (Table 4.1). For the planar geometry (D2h symmetry) two low spin structures were found. The first one was the closed-shell structure P8 that, however, turned out to be triplet instable. Re-optimization at UDFT/CAM-B3LYP level under D2h restrictions led to the open-shell structure P7 that was 0.13 eV more stable than the corresponding closed-shell structure. Structure P8 also had two imaginary vibrational frequencies, both of which were out-of-plane, leading to D2 symmetry. Displacement along these modes and optimization at closed-shell level gave the non-planar structure P6 of D2 symmetry. Similarly, following the out-of-plane imaginary frequency found in the Hessian of structure P7 led to a new structure P5 (D2 symmetry) that was 0.016 eV lower in energy than structure P6. It had only a small open-shell character as can be seen from a natural orbital occupation of 0.089 and 1.911 e respectively for the lowest unoccupied natural orbital (LUNO) and highest occupied natural orbital (HONO). This is much smaller than the NO occupation of the open-shell low spin structure P7 whose LUNO-HONO occupations were 0.302 and 1.698 e. Searching along the imaginary frequencies found for the previous structures and considering the triplet instability present in the wave functions led to structure P4 (D2 symmetry), the new low spin, open-shell minimum-energy CAM-B3LYP structure which was 0.04 eV more stable than structure P5. The HONO-LUNO occupation for structure P4 was 1.899 e and 0.101 e respectively. Comparing the occupations from CAM-B3LYP to B3LYP shows that the range-corrected functional CAM-B3LYP presents a more varied picture than found with the B3LYP functional. In particular, open shell structures with non- negligible deviations of NO occupations from the closed shell reference values of two and zero, respectively, were found. However, the energy differences between these different states are quite small, only a few hundredths of an eV.

Structure P10 of Table 4.1 was obtained using the unrestricted high spin approach in the geometry optimization. The Hessian contained one imaginary frequency, an out-of- plane bending mode leading to D2 symmetry. Following this imaginary frequency led to structure P9 that was about 0.06 eV more stable than structure P10. The bonding carbons of this new high spin structure were slightly out-of-plane by 0.07 Å.

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Selected bond distances of the silicon-2C+Si complex are given in Table 4.1 for the B3-LYP and CAM-B3LYP results. Looking first at the distances from the silicon atom to its bonding carbon atoms computed at B3LYP/6-311G(2d,1p) level, the Si-C bond distances are found to vary by less than 0.01 Å in the different structures. The D2 structure has the longest Si-C bond distance as would be expected from moving the bonding carbon atoms out-of-plane from the D2h to D2 symmetry. The internuclear distance between adjacent, non-bonded carbon atoms C3 and C13 along the direction perpendicular to the long axis is 2.65 Å for structure P1, which is the largest among the three structures. This is again a result of the out-of-plane character of structure P1. The distance C3-C6 along the long axis of pyrene is ~0.2 Å longer than the perpendicular distance, reflecting the restrictions imposed by freezing the carbon atoms which would be connected to the surrounding graphene sheet (Figure 4.1).

Next, we characterize the structures computed with the CAM-B3LYP/6- 311G(2d,1p) method. The largest C3-C13 bond distance (Table 4.1) is 2.65 Å for the low spin structures P5 and P6 (D2), while the smallest, 2.60 Å, is found for the high spin, open- shell structure P10. The dependence of the Si-C distances on the different structures and spin states is not very pronounced, though the high spin planar structures, structures P9 and P10, were found to have the smallest Si-C and C3-C13 distances; the same situation that was seen using B3LYP/6-311G(2d,1p).

Comparing the two above methods for pyrene-2C+Si, the most noticeable effect is that there is greater open-shell character for the low spin structures when using the CAM- B3LYP functional. The Si-C bond distance and C3-C13 intranuclear distances are also slightly smaller when using the CAM-B3LYP functional. But overall, agreement between the two methods is quite good.

4.4.2 The Circumpyrene-2C+Si Defect Structure

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Pyrene-2C+Si was then surrounded by benzene rings to create the larger circumpyrene-2C+Si structure (Figure 4.2) as a better model for embedding the defect into a graphene sheet. The edge carbon atoms linked to hydrogen atoms were again frozen. Description of the geometry of circumpyrene-2C+Si is presented in Table 4.2 for the B3LYP/6-311G(2d,1p) method. The geometry of the planar, unrelaxed circumpyrene-

2C+Si was optimized using a closed-shell approach that led to the D2h structure CP2, which had one out-of-plane imaginary frequency. Following this mode, the non-planar D2 symmetric structure CP1 was obtained that contained no imaginary frequencies. Optimizing the geometry of the planar, unrelaxed circumpyrene-2C+Si using an open- shell, high spin approach resulted in a geometrically stable (no imaginary frequencies) planar structure CP3 with D2h symmetry. The wave functions of circumpyrene-2C+Si structures CP1 and CP2 were triplet stable.

Table 4.2 Structural symmetry, open-shell character, spin multiplicity, bond distances (Å), and relative energies (eV) of selected optimized structures of circumyrene-2C+Si.

Closed/ High/ Open Triplet Low C29- C29- Structure Shell instab. #imaga Spin Sib-C C39 C32 ΔE B3LYP/6-311G(2d,1p) c CP1 (D2) Closed no 0 Low 1.909 2.690 2.741 0.000 CP2 (D2h) Closed no 1 Low 1.902 2.668 2.713 0.118 CP3 (D2h) Open - 0 High 1.912 2.662 2.745 1.254 aNumber of imaginary frequencies. bSilicon always stays in-plane cThe out-of-plane distance of non-frozen carbon atoms is ±0.2 Å in structures that are D2 symmetric The Si-C bond distances for the three investigated structures are very similar (Table 4.2). The C29-C39 distance is largest for the non-planar, closed-shell structure P1 and smallest for the planar structures P2 and P3, but the differences are less than 0.03 Å.

The difference in energy between the non-planar and planar low spin structures decreases markedly when the 2C+Si defect is embedded in a larger hexagonal sheet (compare ΔE values in Table 4.1 and Table 4.2). The energy difference decreases by ~0.6 eV from 0.74 eV in pyrene-2C+Si to 0.12 eV in circumpyrene-2C+Si. Also noteworthy, when comparing pyrene-2C+Si and circumpyrene-2C+Si is that the Si-C bond distances in circumpyrene-2C+Si are shorter by about 0.03 Å on average for the low spin cases. The differences in the internuclear distance between adjacent, non-bonded carbon atoms C29 72 Texas Tech University, Reed Nieman, May 2019 and C39 (perpendicular to the long axis) and C29-C32 (parallel) is much less pronounced than in the pyrene case, which is a consequence of the more flexible embedding into the carbon network where none of the four C atoms surrounding Si is bonded to another atom which is frozen.

4.4.3 The Graphene-2C+Si Defect Structure

To confirm the obtained geometry and to validate that periodic DFT simulations reproduce the observed ground state structure, the Si-C4 defect with an alternating ±0.2 Å corrugation of the four C atoms was placed in a 10×8 supercell of graphene and the structure and cell relaxed using the PBE functional. The structural optimization preserved the out-of-plane corrugation, and resulted in a total energy 0.22 eV lower than for the flat structure. Thus, standard DFT is able to find the correct, non-planar structure when initializing away from the flat geometry. Further, a nudged elastic band calculation43 shows that the two equivalent conformations (alternating which C atoms are up and which are down) are separated only by this energy barrier, leading to a Boltzmann factor of 1.7×10-4 at room temperature and thus a rapid oscillation between the two equivalent structures for any reasonable vibration frequency.

Bader analysis44 of the charge density shows that the Si-C bonds are rather polarized, with the Si donating 2.61 electrons shared by the four neighboring carbons. This is even larger than the bond polarization in 2D-SiC, where a Si donates 1.2 electrons (although each C has three Si neighbors, bringing the total to 1.2 per C instead of 0.65 here).45

4.4.4 Natural Bond Orbital Analysis

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NBO analysis was used to further characterize the bonding of pyrene-2C+Si and circumpyrene-2C+Si and to describe in detail the charge transfer from silicon to carbon in the various structures.

The NBO analysis of pyrene-2C+Si is presented in Table 4.3. The linear combination factor of the Si is less than half of that of the bonding C, a sign of the greater electronegativity of carbon compared to silicon. The non-planar, low spin, closed-shell 3 3 structure P1 (D2 symmetry) shows sp orbital hybridization at the silicon and sp orbital hybridization at the bonding carbon, consistent with the slight tetrahedral arrangement about the silicon center. The planar closed-shell structure P2 shows sp2d hybridization at the silicon center, consistent with previous experimental and computational interpretations,10 and approximately sp3 hybridization at the bonding carbons. The high 2 spin structure P3, which also has D2h symmetry, shows also sp d hybridization at the silicon. Compared to B3LYP (structures P1-P3), bonding from CAM-B3LYP (Table B.1 of Appendix B) is essentially the same.

Table 4.3 NBO bonding character analysis of pyrene-2C+Si using B3LYP/6-311G(2d,1p). The discussed carbon atom is one of the four bonded to silicon. Structure Bonding Character B3LYP/6-311G(2d,1p) P1 (D ) 2 0.54(sp2.98d0.02) +0.84(sp3.24d0.01) closed sh. Si C P2 (D ) 2h 0.46(sp2.00d1.00) +0.89(sp2.97d0.01) closed sh. Si C P3 (D ) 2h α: 0.46(sp2.00d1.00) +0.89(sp2.81d0.01) open sh., Si C β: 0.55(sp2.00d1.00) +0.83(sp3.61d0.01) high spin Si C

The NBO analysis of circumpyrene-2C+Si is presented in Table 4.4. The bonding character of structure CP1 is essentially sp3 for both the silicon and the bonding carbon as in structure P1 of pyrene-2C+Si, although the p orbital character on the carbon atom of this structure is reduced by about 0.60 e. Structure CP2 presents an interesting case, as no bonding NBOs were found. Instead, a series of four unoccupied valence lone pairs (LP*), one of s orbital character and three of p character, were located on the silicon and a corresponding set of occupied valence lone pairs (LP), one for each bonding carbon, was

74 Texas Tech University, Reed Nieman, May 2019 found with sp3.58d0.01 character. It is noted that the Si-C bonding character of the high spin structure CP3 changes markedly between the alpha shell and beta shell. In the alpha shell, the silicon and carbon atoms both exhibit sp1 orbital hybridization with no d orbital contribution on the silicon, but in the beta shell, the silicon atom shows sp2d hybridization and the carbon atoms show sp3 hybridization as in the other planar cases.

The shapes of the NBOs describing the bond between Si and C is depicted in Figure

4.3 for structures P1 (D2 symmetry) and P2 (D2h symmetry) using the B3LYP/6- 311G(2d,1p) method. They appear very similar in spite of the different hybridization on Si (sp3 vs. sp2d, Table 4.3). The plots of the NBOs for all other structures, even those for circumpyrene-2C+Si, are almost identical. The exception is structure CP2 of circumpyrene-2C+Si in which no Si-C bonds were found during the NBO analysis. The NBOs shown in Figure B.1 for this structure consist of the valence lone pairs found on each carbon atom (Table 4.4) possessing sp3.58 orbital hybridization with a lobe of electron density found facing the silicon center. The NBOs of silicon are low-occupancy valence lone pairs, representing one s and three p NBOs. Even though this classification of bonding by means of lone pairs according to the NBO analysis looks quite different, the combination of these lone pairs is not expected to look substantially different from the localized bonds shown in the other structures, especially considering the strong polarity of the silicon-carbon bonds as discussed below.

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Table 4.4 NBO bonding character analysis of circumpyrene-2C+Si. The discussed carbon atom is the one of the four bonded to silicon. Structure Bonding Character B3LYP/6-311G(2d,1p) CP1 (D ) 2 0.51(sp2.96d0.04) +0.86(sp2.62d0.01) closed sh. Si C

a CP2 (D2h) Four LP*: one [1.0(s)Si] + three [1.00(p)Si] 3.58 0.01 b closed sh. four[1.00(sp d )C] CP3 (D ) 2h α: 0.68(sp1.00d0.00) +0.74(sp1.00d0.00) open sh., Si C β: 0.55(sp2.00d1.00) +0.84(sp3.41 d0.01) high spin Si C aFour unfilled valence lone pairs (LP*) on silicon (occupation less than 1 e), bOne occupied valence lone pair on each bonding carbon Natural charges are instructive for illustrating the polarity of the silicon-carbon bonds and the charge transfer between silicon and the surrounding graphitic lattice. These are shown in Figure 4.4 for circumpyrene-2C+Si (Structures CP2 and CP3) using the B3LYP/6-311G(2d,1p) method. The silicon center is strongly positive in all cases and more positive in circumpyrene-2C+Si for the low spin cases (1.7–1.8 e, structures CP1 and CP2) than the high spin case (1.4 e, structure CP3). It thus seems that the Bader analysis presented above overestimates the charge transfer by over 40 %, despite being qualitatively correct. The bonding carbon in each case shows a larger negative charge compared to the remaining carbon atoms of circumpyrene. The charge on the bonding carbon in the low spin structures CP1 and CP2 is more negative than that of the high spin structure CP3.

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Figure 4.3 NBO plots for pyrene-2C+Si using the B3LYP/6-311G(2d,1p) approach for structures P1 and P2. Isovalue = ±0.02 e/Bohr3

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Figure 4.4 Natural charges (e) of carbon and silicon in circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p) method.

The natural charges of the remaining carbon atoms in circumpyrene-2C vary depending on whether they are bonded to other carbon atoms or to hydrogen. For the low spin structures CP1 and CP2, the natural charges on C atoms on the periphery of circumpyrene bonded to other C atoms is about one-third as negative as the natural charges of C atoms bonded to H atoms, while the natural charges of carbon atoms in the interior of circumpyrene and not bonded to Si are much smaller. Taken together, the magnitude of the Si-graphene charge transfer can be seen to be largest for the planar, closed-shell, low spin structure CP2, and smallest for the planar, high spin structure CP3. 78 Texas Tech University, Reed Nieman, May 2019

The natural charges for pyrene-2C+Si using B3LYP/6-311G(2d,1p) are plotted in Figure B.2. The low spin structures P1 and P2 have respective positive charges on the silicon atom of 1.596 and 1.953 e, a larger difference than in the low spin structures CP1 and CP2 of circumpyrene-2C+Si where the silicon atom has positive charges of 1.726 and 1.821 e. Correspondingly, the difference in the negative charges of the bonding carbon atoms is also larger between these two structures than in circumpyrene. For the other carbons in pyrene, the magnitude of their charges varies less than in circumpyrene. For pyrene-2C+Si using the CAM-B3LYP/6-311G(2d,1p) method, the natural charges are plotted in Figure B.3. The same trend is seen for this method as in our other cases, namely that the closed-shell, low spin, planar structure P8 has the largest positive charge on silicon and the largest negative charge on the bonding carbon compared to the other structures.

4.4.5 Molecular Electrostatic Potential

The molecular electrostatic potential (MEP) was computed for each structure and is presented in Figure 4.5 for circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p) method. The plots illustrate the natural charge transfer discussed previously: for structures CP1-CP3, the positive potential at the silicon corresponds to the positive charge buildup as discussed above. This positive potential at the silicon is more positive for structures CP1 and CP2 than for CP3 following the trend of the natural charges. There is also a negative potential diffusely distributed on the carbon atoms in circumpyrene.

The MEP plots for pyrene-2C+Si using the B3LYP/6-311G(2d,1p) method is shown in Figure B.4. Similar results are obtained when using the CAM-B3LYP functional (shown in Figure B.5). Finally, the low spin planar structures P2, P7, and P8 have a much more positive potential than either the non-planar structures or the high spin planar structures, represented by the deeper blue coloring in the figures.

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Figure 4.5 Molecular electrostatic potential for circumpyrene-2C+Si mapped onto the electron density isosurface with an isovalue of 0.0004 e/Bohr3 (B3LYP/6-311G(2d,1p)).

4.4.6 Electron Energy Loss Spectroscopy

The Si-C4 defect in graphene was originally identified through a combination of atomic resolution STEM and atomically resolved EELS. In that study11 it was concluded that a Si-C3 defect is non-planar due to a significantly better match between the measured and simulated EELS spectrum; however, the simulated spectrum of a flat Si-C4 defect matched the experiment less well. Because of this discrepancy and the findings described above, we also simulated the EELS spectrum of the corrugated Si-C4 graphene defect. Figure 4.6 displays the simulated spectra of both the flat and the corrugated defect (normalized to the π* peak at 100 eV), overlaid on a new background-subtracted experimental signal that we have recorded with a higher dispersion than the original 11 spectrum (average of 20 spot spectra, with the Si-C4 structure verified after acquisition by imaging; see Figure B.6 for the unprocessed spectrum).

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1.5

Signal Flat Corrugated

1.0

s

t

n

u

o

c

4

0

1 ´ 0.5

0.0

100 110 120 130 Energy loss (eV)

Figure 4.6 Comparison of simulated and experimental EELS spectra of the Si-C4 defect in graphene. Despite the out-of-plane distorted structure being the ground state, the originally proposed flat spectrum provides an overall better match to the experiment, apart from the small peak at ~96 eV that is not predicted for the flat structure. The inset shows a colored medium angle annular dark field STEM image of the Si-C4 defect (the Si atom shows brighter due to its greater ability to scatter the imaging electrons). The new spectrum closely resembles the originally reported one apart from a small additional peak around 96 eV. Comparing the recorded signal to the simulated spectra, the overall relative intensities of the π* and σ* contributions match the flat structure better, but both simulations overestimate the π* intensity at 100 eV. Intriguingly, despite the better overall match with the flat structure, the small pre-peak present in the new signal is only present in the spectrum simulated for the lowest energy corrugated structure, providing spectroscopic evidence for its existence. The lowest unoccupied states responsible for this peak are π* in character and form a „cross-like“ pattern across the Si site, as shown in Figure B.7, markedly different from the lowest unoccupied states of the flat structure. However, considering the small energy barrier between the equivalent corrugated conformations discussed above, even at room temperature the spectrum may be integrated over a superposition of corrugated and nearly flat morphologies. 81 Texas Tech University, Reed Nieman, May 2019

4.5 Conclusions

Density functional theory calculations (both restricted and unrestricted) were performed on pyrene-2C+Si and circumpyrene-2C+Si producing a variety of structures that were characterized using natural orbital occupations, natural bond orbital analyses, and molecular electrostatic potential plots. A non-planar low spin structure of D2 symmetry was found to be the minimum for both cases, followed energetically by a low spin planar structure, 0.6 eV higher in energy for pyrene-2C+Si, but with a much smaller (only ~0.1 eV) energy difference for circumpyrene-2C+Si. A periodic graphene model with the non- planar defect had a 0.22 eV lower energy than a flat one. In the molecular calculations, the out-of-plane structure was shown to be a minimum by means of harmonic frequency calculations. The structures of the high spin state are much higher in energy than the minimum: about 0.85 eV higher for pyrene-2C+Si and about 1.3 eV higher for circumpyrene-2C+Si.

For pyrene-2C+Si, B3LYP and CAM-B3LYP were compared showing that there was not much difference either energetically, in the NBO characterization, or in the MEP plots. One feature of interest is that CAM-B3LYP shows greater open-shell character in the low spin case than was seen for B3LYP. This open-shell character demonstrates the variety of electronic structures which can be found in seemingly ordinary closed-shell cases.

NBO analyses showed the expected bonding configurations for the two structural symmetries present. The bonding of the silicon atom in the planar structure was found to have sp2d orbital character while in the non-planar structures, the bonding orbitals are sp3 hybridized. The natural charges illustrated the charge transfer that occurs, demonstrating the buildup of positive charge at the silicon and the diffusion of negative charge into the pyrene-2C/cirumpyrene-2C structure, which was further evident in the MEP plots.

Despite carefully establishing the non-planar ground state nature of the Si-C4 defect in graphene, electron energy loss spectra still seem to match the flat structure better, but imperfectly especially concerning a small peak at ~96 eV. Considering the large amounts 82 Texas Tech University, Reed Nieman, May 2019 of kinetic energy that the energetic electrons used in transmission electron microscopy can impart, and the available thermal energy in the system, perhaps the defect is not able to stay in its ground state, but rather exists in a superposition of slightly different configurations that result in an average spectrum resembling the flat state.

4.6 Acknowledgments

We are grateful to Michael Zehetbauer for continuous support and interest in this work and also thank Quentin Ramasse for many useful discussions. T.S. and J.K. acknowledge the Austrian Science Fund (FWF) for funding via projects P 28322-N36 and I 3181-N36. H.L. and T.S. are grateful to the Vienna Scientific Cluster (Austria) for computational resources (H.L.: project no. P70376). H.L. also acknowledges computer time at the computer cluster of the School for Pharmaceutical Science and Technology of the Tianjin University, China, T.P.H. acknowledges the Engineering and Physical Sciences Research Council (EPSRC) Doctoral Prize Fellowship that funded this research, and the ARC1 and ARC2 high-performance computing facilities at the University of Leeds.

4.7 Bibliography

(1) Bekyarova, E.; Itkis, M. E.; Ramesh, P.; Berger, C.; Sprinkle, M.; de Heer, W. A.; Haddon, R. C., Chemical Modification of Epitaxial Graphene: Spontaneous Grafting of Aryl Groups. Journal of the American Chemical Society 2009, 131 (4), 1336-1337. (2) Bekyarova, E.; Sarkar, S.; Wang, F.; Itkis, M. E.; Kalinina, I.; Tian, X.; Haddon, R. C., Effect of Covalent Chemistry on the Electronic Structure and Properties of Carbon Nanotubes and Graphene. Accounts of Chemical Research 2013, 46 (1), 65- 76. (3) Kim, K. S.; Zhao, Y.; Jang, H.; Lee, S. Y.; Kim, J. M.; Kim, K. S.; Ahn, J.-H.; Kim, P.; Choi, J.-Y.; Hong, B. H., Large-scale pattern growth of graphene films for stretchable transparent electrodes. Nature 2009, 457, 706. (4) Lee, Y.; Bae, S.; Jang, H.; Jang, S.; Zhu, S.-E.; Sim, S. H.; Song, Y. I.; Hong, B. H.; Ahn, J.-H., Wafer-Scale Synthesis and Transfer of Graphene Films. Nano Letters 2010, 10 (2), 490-493. (5) Emtsev, K. V.; Bostwick, A.; Horn, K.; Jobst, J.; Kellogg, G. L.; Ley, L.; McChesney, J. L.; Ohta, T.; Reshanov, S. A.; Röhrl, J.; Rotenberg, E.; Schmid, A. K.; Waldmann, D.; Weber, H. B.; Seyller, T., Towards wafer-size graphene layers 83 Texas Tech University, Reed Nieman, May 2019

by atmospheric pressure graphitization of silicon carbide. Nature Materials 2009, 8, 203. (6) Berger, C.; Song, Z.; Li, T.; Li, X.; Ogbazghi, A. Y.; Feng, R.; Dai, Z.; Marchenkov, A. N.; Conrad, E. H.; First, P. N.; de Heer, W. A., Ultrathin Epitaxial Graphite: 2D Electron Gas Properties and a Route toward Graphene-based Nanoelectronics. The Journal of Physical Chemistry B 2004, 108 (52), 19912- 19916. (7) Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K., The electronic properties of graphene. Reviews of Modern Physics 2009, 81 (1), 109-162. (8) Rutter, G. M.; Crain, J. N.; Guisinger, N. P.; Li, T.; First, P. N.; Stroscio, J. A., Scattering and interference in epitaxial graphene. Science 2007, 317 (5835), 219- 22. (9) Zhao, L.; He, R.; Rim, K. T.; Schiros, T.; Kim, K. S.; Zhou, H.; Gutierrez, C.; Chockalingam, S. P.; Arguello, C. J.; Palova, L.; Nordlund, D.; Hybertsen, M. S.; Reichman, D. R.; Heinz, T. F.; Kim, P.; Pinczuk, A.; Flynn, G. W.; Pasupathy, A. N., Visualizing individual nitrogen dopants in graphene. Science 2011, 333 (6045), 999-1003. (10) Zhou, W.; Kapetanakis, M. D.; Prange, M. P.; Pantelides, S. T.; Pennycook, S. J.; Idrobo, J.-C., Direct Determination of the Chemical Bonding of Individual Impurities in Graphene. Physical Review Letters 2012, 109 (20), 206803. (11) Ramasse, Q. M.; Seabourne, C. R.; Kepaptsoglou, D.-M.; Zan, R.; Bangert, U.; Scott, A. J., Probing the Bonding and Electronic Structure of Single Atom Dopants in Graphene with Electron Energy Loss Spectroscopy. Nano Letters 2013, 13 (10), 4989-4995. (12) Susi, T.; Kotakoski, J.; Kepaptsoglou, D.; Mangler, C.; Lovejoy, T. C.; Krivanek, O. L.; Zan, R.; Bangert, U.; Ayala, P.; Meyer, J. C.; Ramasse, Q., Silicon-carbon bond inversions driven by 60-keV electrons in graphene. Physical Review Letters 2014, 113 (11), 115501. (13) Machado, F. B. C.; Aquino, A. J. A.; Lischka, H., The Diverse Manifold of Electronic States Generated by a Single Carbon Defect in a Graphene Sheet: Multireference Calculations Using a Pyrene Defect Model. ChemPhysChem 2014, 15 (15), 3334-3341. (14) Machado, F. B. C.; Aquino, A. J. A.; Lischka, H., The electronic states of a double carbon vacancy defect in pyrene: a model study for graphene. Physical Chemistry Chemical Physics 2015, 17 (19), 12778-12785. (15) Haldar, S.; Amorim, R. G.; Sanyal, B.; Scheicher, R. H.; Rocha, A. R., Energetic stability, STM fingerprints and electronic transport properties of defects in graphene and silicene. Rsc Adv 2016, 6 (8), 6702-6708. (16) Nieman, R.; Das, A.; Aquino, A. J. A.; Amorim, R. G.; Machado, F. B. C.; Lischka, H., Single and double carbon vacancies in pyrene as first models for graphene defects: A survey of the chemical reactivity toward hydrogen. Chemical Physics 2017, 482, 346-354. (17) Becke, A. D., Density‐functional thermochemistry. III. The role of exact exchange. The Journal of Chemical Physics 1993, 98 (7), 5648-5652.

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(18) Yanai, T.; Tew, D. P.; Handy, N. C., A new hybrid exchange–correlation functional using the Coulomb-attenuating method (CAM-B3LYP). Chemical Physics Letters 2004, 393 (1), 51-57. (19) Seeger, R.; Pople, J. A., Self‐consistent molecular orbital methods. XVIII. Constraints and stability in Hartree–Fock theory. The Journal of Chemical Physics 1977, 66 (7), 3045-3050. (20) Bauernschmitt, R.; Ahlrichs, R., Stability analysis for solutions of the closed shell Kohn–Sham equation. The Journal of Chemical Physics 1996, 104 (22), 9047- 9052. (21) Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple. Physical Review Letters 1996, 77 (18), 3865-3868. (22) Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple [Phys. Rev. Lett. 77, 3865 (1996)]. Physical Review Letters 1997, 78 (7), 1396-1396. (23) Hehre, W. J.; Ditchfield, R.; Pople, J. A., Self—Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian—Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules. The Journal of Chemical Physics 1972, 56 (5), 2257-2261. (24) Binning, R. C.; Curtiss, L. A., Compact contracted basis sets for third-row atoms: Ga–Kr. Journal of Computational Chemistry 1990, 11 (10), 1206-1216. (25) Curtiss, L. A.; McGrath, M. P.; Blaudeau, J. P.; Davis, N. E.; Binning, R. C.; Radom, L., Extension of Gaussian‐2 theory to molecules containing third‐row atoms Ga–Kr. The Journal of Chemical Physics 1995, 103 (14), 6104-6113. (26) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A., Self‐consistent molecular orbital methods. XX. A basis set for correlated wave functions. The Journal of Chemical Physics 1980, 72 (1), 650-654. (27) McGrath, M. P.; Radom, L., Extension of Gaussian‐1 (G1) theory to bromine‐ containing molecules. The Journal of Chemical Physics 1991, 94 (1), 511-516. (28) McLean, A. D.; Chandler, G. S., Contracted Gaussian basis sets for molecular calculations. I. Second row atoms, Z=11–18. The Journal of Chemical Physics 1980, 72 (10), 5639-5648. (29) Reed, A. E.; Weinstock, R. B.; Weinhold, F., Natural population analysis. The Journal of Chemical Physics 1985, 83 (2), 735-746. (30) Carpenter, J. E.; Weinhold, F., Analysis of the geometry of the hydroxymethyl radical by the “different hybrids for different spins” natural bond orbital procedure. Journal of Molecular Structure: THEOCHEM 1988, 169, 41-62. (31) Foster, J. P.; Weinhold, F., Natural hybrid orbitals. Journal of the American Chemical Society 1980, 102 (24), 7211-7218. (32) Reed, A. E.; Curtiss, L. A.; Weinhold, F., Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint. Chemical Reviews 1988, 88 (6), 899-926. (33) Reed, A. E.; Weinhold, F., Natural bond orbital analysis of near‐Hartree–Fock water dimer. The Journal of Chemical Physics 1983, 78 (6), 4066-4073. (34) Reed, A. E.; Weinhold, F., Natural localized molecular orbitals. The Journal of Chemical Physics 1985, 83 (4), 1736-1740.

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(35) Susi, T.; Hardcastle, T. P.; Hofsäss, H.; Mittelberger, A.; Pennycook, T. J.; Mangler, C.; Drummond-Brydson, R.; Scott, A. J.; Meyer, J. C.; Kotakoski, J. Single-atom spectroscopy of phosphorus dopants implanted into graphene, in: arXiv:1610.03419, 2016 2016. (36) Hardcastle, T. P.; Seabourne, C. R.; Kepaptsoglou, D. M.; Susi, T.; Nicholls, R. J.; Brydson, R. M. D.; Scott, A. J.; Ramasse, Q. M., Robust theoretical modelling of core ionisation edges for quantitative electron energy loss spectroscopy of B- and N-doped graphene. Journal of Physics: Condensed Matter 2017, 29 (22), 225303. (37) Nicholls, R. J.; Morris, A. J.; Pickard, C. J.; Yates, J. R., OptaDOS - a new tool for EELS calculations. Journal of Physics: Conference Series 2012, 371 (1), 012062. (38) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery Jr., J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J.; Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, Gaussian, Inc.: Wallingford, CT, USA, 2009. (39) Ahlrichs, R.; Bär, M.; Häser, M.; Horn, H.; Kölmel, C., Electronic structure calculations on workstation computers: The program system turbomole. Chemical Physics Letters 1989, 162 (3), 165-169. (40) Enkovaara, J.; Rostgaard, C.; Mortensen, J. J.; Chen, J.; Dulak, M.; Ferrighi, L.; Gavnholt, J.; Glinsvad, C.; Haikola, V.; Hansen, H. A.; Kristoffersen, H. H.; Kuisma, M.; Larsen, A. H.; Lehtovaara, L.; Ljungberg, M.; Lopez-Acevedo, O.; Moses, P. G.; Ojanen, J.; Olsen, T.; Petzold, V.; Romero, N. A.; Stausholm-Møller, J.; Strange, M.; Tritsaris, G. A.; Vanin, M.; Walter, M.; Hammer, B.; Häkkinen, H.; Madsen, G. K. H.; Nieminen, R. M.; Nørskov, J. K.; Puska, M.; Rantala, T. T.; Schiøtz, J.; Thygesen, K. S.; Jacobsen, K. W., Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method. Journal of Physics: Condensed Matter 2010, 22 (25), 253202. (41) Clark, S. J.; Segall, M. D.; Pickard, C. J.; Hasnip, P. J.; Probert, M. J.; Refson, K.; Payne, M. C., First principles methods using CASTEP. Zeitschrift für Kristallographie - Crystalline Materials 2005, 220 (5-6), 567-570. (42) Tsetseris, L.; Wang, B.; Pantelides, S. T., Substitutional doping of graphene: The role of carbon divacancies. Physical Review B 2014, 89 (3), 035411. (43) Henkelman, G.; Jónsson, H., Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. The Journal of Chemical Physics 2000, 113 (22), 9978-9985.

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(44) Tang, W.; Sanville, E.; Henkelman, G., A grid-based Bader analysis algorithm without lattice bias. Journal of Physics: Condensed Matter 2009, 21 (8), 084204. (45) Susi, T.; Skakalova, V.; Mittelberger, A.; Kotrusz, P.; Hulman, M.; Pennycook, T. J.; Mangler, C.; Kotakoski, J.; Meyer, J. C., Computational insights and the observation of SiC nanograin assembly: towards 2D silicon carbide. Scientific Reports 2017, 7 (1), 4399.

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CHAPTER V

THE CRUCIAL ROLE OF A SPACER MATERIAL ON THE EFFICIENCY OF CHARGE TRANSFER PROCESSES IN ORGANIC DONOR-ACCEPTOR JUNCTION SOLAR CELLS Authors: Reed Nieman,a Hsinhan Tsai,b Wanyi Nie,b Adelia J. A. Aquino,a,c Aditya D. Mohite,b Sergei Tretiak,b Hao Li,d Hans Lischkaa,c Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University Lubbock, TX 79409-1061, USA, bLos Alamos National Laboratory, Los Alamos, New Mexico 87545, USA, cSchool of Pharmaceutical Sciences and Technology, Tianjin University, Tianjin, 300072 P.R.China, dDepartment of Chemistry, University of Houston, Houston, Texas 77204. USA Published: December 2017 in Nanoscale, DOI:10.1039/C7NR07125F

5.1 Abstract

Organic photovoltaic donor-acceptor junction devices composed of π-conjugated polymer electron donors (D) and fullerene electron acceptors (A) show greatly increased performance when a spacer material is inserted between the two layers.1 For instance, experimental results reveal significant improvement of photocurrent when a terthiophene oligomer derivative is inserted in between π-conjugated poly(3-hexylthiophene-2, 5-diyl

(P3HT) donor and C60 acceptor. These results indicate favorable charge separation dynamics, which is addressed by our present joint theoretical/experimental study establishing the beneficial alignment of electronic levels due to the specific morphology of the material. Namely, based on the experimental data we have constructed extended structural interface models containing C60 and P3HT separated by aligned oligomer chains. Our time-dependent density functional theory (TD-DFT) calculations based on a long-range corrected functional, allowed us to address the energetics of essential electronic states and analyze them in terms of charge transfer (CT) character. Specifically, the simulations reveal the electronic spectra composed of a ladder of excited states evolving excitation toward spatial charge separation: an initial excitonic excitation at P3HT decomposes into charges by sequentially relaxing through bands of C60-centric, oligomer→C60 and P3HT→C60 CT states. Our modeling exposes a critical role of dielectric environment effects and electronic couplings in the self-assembled spacer oligomer layer 88 Texas Tech University, Reed Nieman, May 2019 on the energetics of critical CT states leading to a reduced back-electron transfer, preventing recombination losses, and thus rationalizes physical processes underpinning experimental observations.

5.2 Introduction

The interfaces of organic photovoltaic bulk heterojunction (BHJ) donor (D)/acceptor (A) devices between π-conjugated electron donor polymers and electron acceptors, such as fullerenes, are of fundamental importance for understanding organic photovoltaic processes and for improving their efficiency.2-4 These interfaces control the dissociation of excitons and generation of CT states. The conversion of light into electricity in a BHJ photovoltaic device is accomplished by a four-step process: (i) light absorption by bright π→π* transition of the organic conjugated polymer and generation of excitons, (ii) diffusion of the excitons through the bulk polymer and segregation to the interface, (iii) dissociation of the excitons at afore-mentioned heterojunctions and creation of CT states, and (iv) charge separation and collection at contacts.2, 5 The actual processes occurring in the BHJ material are significantly more complex because of the heterogeneous distribution of donor and acceptor in the BHJ material, a broad variety of structural defects and conformations, the large number of internal degrees of freedom of the polymer chains, and the complicated manifold of electronic states.6-8 Therefore, finessing the mechanism of ultrafast and loss-less free charge generation in BHJ devices via intercede design is still a very active research area.9-13

While traditional organic photovoltaic (OPV) BHJ devices involve only a pair of materials, i.e., electron donor and acceptor complexes, recent experimental work1 has employed a new strategy of using a third material acting as a spacer between the donor and acceptor regions,14-15 which mitigates the electron-hole recombination rate (i.e. the backward charge transfer processes) but without affecting the efficiency of charge separation (i.e., forward electron transfer). In particular, these studies in a simple bi-layer device have shown a large increase in both the photocurrent (up to 800%) and open circuit voltage (VOC) when a spacer material such as a terthiophene-derivative (O3) is added 89 Texas Tech University, Reed Nieman, May 2019

between the donor, a P3HT, and a fullerene (C60) acceptor. Subsequently, in the BHJ device a significant increase in the power conversion efficiency (PCE) from 4% to greater than 7% was observed.1 These experimental findings followed the hypothesis of a “fast energy transfer of the exciton from the donor across O3 to the acceptor and the back cascading of the hole to the donor” due to a favorable alignment of the corresponding energy levels. Such arrangement ensures that the spacer material is also physically separating the donor and acceptor interface thus working to mitigate the CT recombination rate.

Quantum-chemical modeling of these materials can provide important atomistic insights into the nature of underlining processes and ultimately help to formulate design strategies toward optimizing the photophysical dynamics. The first important step is the determination of the energy level alignment occurring at the complex donor-acceptor interfaces. For such large systems density functional theory (DFT) and time-dependent (TD)-DFT are the most widely used techniques providing access to the ground and excited state properties, respectively, being a reasonable compromise between accuracy and numerical cost. However, in spite of the overwhelming success of DFT in ground-state computation, the correct description of charge-separated excited states is still problematic for many developed exchange-correlation functionals due to the significant delocalized nature of the density, especially for intermolecular CT states at BHJ interfaces.16-19 A variety of strategies have been developed to correctly account for delocalized excitations. Long-range corrected functionals20-21 in combination with optimization of the parameters determining the separation range22-25 have been used successfully to correct several TD- DFT artifacts. In previous work,26 we have shown that long-range corrected functionals of CAM-B3LYP27-28 and ωB97XD29 gave very good agreement for excitonic and CT states for thiolated oligothiophene dimers in comparison to the second-order algebraic diagrammatic construction (ADC(2)) approach. Thus, long-range corrected functionals appear to be well applicable for the calculation of electronic spectra of π-conjugated semiconductor polymers including capacity of describing both excitonic and CT states.30 They are also computationally efficient so that the capability of treating large molecular aggregate systems can be well managed.

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Regarding the discussed bilayer interfaces,31 previous calculations have shown that the bright, light absorbing π→π* transition can be located above CT states and can convert to the CT state via internal conversion processes. However, it has been concluded from TD-DFT calculations32 that dark fullerene states could be located even lower in energy and would have the possibility to quench the CT states after they have been formed. Other calculations based on the already mentioned ADC(2) method have shown a slightly different picture indicating CT states as the lowest singlet states.31, 33

The energetic ordering of states and the stabilization of CT states obviously represents a crucial issue affecting the reduction of unwanted leaking processes. This joint experimental and computational study wants to demonstrate the specific factors affecting especially the stability of the CT states. For that purpose, the experimental P3HT/O3/C60 system discussed above is being used together with our previously benchmarked TD-DFT methodology.26 Even though the calculations are specific for this system, as we will show, the conclusions are quite general and can certainly be transferred to many other polar interface systems. Two structural models were employed in order to represent the donor- spacer-acceptor interface: (i) the donor P3HT polymer is connected via one spacer T3 chain to the acceptor C60 (Figure 5.1a, denoted as standard model) and (ii) an extended model

(Figure 5.1b,c) where P3HT is connected via three T3 spacer molecules to three C60 electron acceptors. A spacer material, thiolated terthiophene, T3, was used which can be considered as a good approximation to O3 used in the experiments. This specific mutual arrangement of molecules underscores our experimental fabrication technique utilizing the Langmuir-Blodgett (LB) method (Figure 5.2a) ensuring orthogonal orientation of the oligomer with respect to the polymer P3HT layer, as demonstrated by our X-ray measurements. Furthermore, the extended model aims to account for π-stacking effects in the oligomer, which were shown to be important for conventional, simple donor-acceptor interfaces.34-39

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Figure 5.1 (a) Standard model of the molecular arrangement in the trimer. (b) Front and (c) side views of the extended model. The picture shows molecular notations, geometrical arrangement and selected nonbonding distances (Å). Atoms kept fixed during the geometry optimization are circled in black. The article is organized as follows: Section 5.3 describes experimental and computational methodology, Section 5.4 presents our results, and finally Section 5.5 summarizes our findings and conclusions.

5.3 Methods 5.3.1 Experimental 5.3.1.1 Bilayer Solar Cells Fabrication

Solar cell fabrication started from the cleaning of (ITO) transparent substrate with sonication bath in water, and isopropyl alcohol for 15 min respectively. The hole transporting polymer poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS) solution was then coated on the cleaned ITO substrates by spin coating method. After drying, the coated substrates were transferred to an filled glovebox for P3HT coating. The in-stock P3HT solution (2 mg/ml in chlorobenzene) was spin coated on the substrates at 2000 rpm for 45 Sec, forming a uniform, 40 nm without further annealing. The coated substrates were then taken outside the glovebox for interfacial layer deposition by LB technique between air-water 92 Texas Tech University, Reed Nieman, May 2019 interfaces on a spin coated P3HT film. After LB deposition, the devices were completed by thermally evaporating C60 (30nm) for acceptor layer and electrode deposition through shadow mask.

5.3.1.2 Device Characterization

The completed solar cells were mounted in cryostat with BNC connections from the electrodes to the instruments. The photocurrent was collected in AC mode by illuminating the device with monochromatic light while measuring the short circuit current using lock-in amplifier.

5.3.2 Computational Details

The computational methods are based on DFT using the Perdew–Burke–Ernzerhof (PBE)40 functional with the SV(P)41 basis for geometry optimizations, and the long-range separated density functional, CAM-B3LYP, for the calculation of excited states using two basis sets, 6-31G42 and 6-31G*.43-44 Solvent effects were taken into account using the conductor-like polarizable continuum model (C-PCM)45. Environmental effects on the electronically excited states were investigated on the basis of the linear response (LR)46-47 and state-specific (SS)48 approaches. Dichloromethane (dielectric constant, ε=8.93, refractive index, n=1.42) was chosen to act as a modestly polar environment. Geometry optimizations were performed by freezing selected atoms as shown in Figure 5.1 in order to maintain the desired structure of the aggregate as deduced from experiment.

The two types of trimer structures are shown in Figure 5.1a (standard trimer) and Figure 5.1b,c (extended trimer). All structures were optimized as isolated complexes using DFT together with the PBE40 functional, the SV(P)41 basis, and the Multipole-Accelerated Resolution of the Identity for the Coulomb energy (MARI-J).49 Empirical dispersion corrections were added by means of the D350 approach. Excited states were calculated using the long-range separated density functional, CAM-B3LYP, and two basis sets, 6-

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31G42 and 6-31G*43-44. D3 dispersion corrections were used for the CAM-B3LYP TD-DFT calculations as well.

Full geometry optimizations of the standard and extended trimer structures would lead to distorted geometries which do not resemble the intended structural arrangements of a donor-spacer-C60 interface since geometrical restrictions of the surrounding environment are missing. To achieve the desired model arrangement, geometry optimization was performed in steps and by freezing a few selected atoms (see Figure 5.1) to restrict the structural manifold. For the standard trimer, the C60 and T3 monomers were combined and optimized freezing selected atoms. To this C60-T3 structure, P3HT was added as shown in

Figure 5.1a. The whole complex (C60-T3-P3HT) was then optimized. The frozen atoms were located at the end of the chains and at the opposite side of C60, respectively, so that the subunits had sufficient flexibility to come in contact allowing the intersystem distances to be optimized. For the extended trimer, an analogous restricted geometry optimization technique was used as shown in Figure 5.1b, c.

Since the main objective of this investigation was to determine the relative stability of the CT states with excitations from the π-conjugated polymers to the C60s and the local excitations in the polymers, excitations within the fullerenes were suppressed in most of the calculations by freezing in the TD-DFT calculations the occupied orbitals located on the C60 units. Otherwise, about a dozen of local excitations per C60 unit would have impeded the already expensive calculations and would not have allowed the focus on the actual states of interest. In practice, all the core orbitals along with twenty percent of the highest occupied orbitals, which were localized at the C60 and (C60)3 units, were kept frozen. Comparison between full and frozen orbital calculations have been performed as well, showing good agreement in energetic ordering of the remaining states in the two calculations.

Solvent effects were taken into account using the conductor-like polarizable continuum model (C-PCM).45 Environmental effects on the electronically excited states were investigated on the basis of the LR46-47 and SS48 approaches. In the LR approach, Hartree and exchange-correlation (XC) potentials are linearly expanded with respect to the variation of the time-dependent density of the ground state. In the SS method, the solvation 94 Texas Tech University, Reed Nieman, May 2019 field adapts to changes in the solute wave function for electronic excitations where the energy correction from fast electronic relaxation of the solvent is state-dependent. SS is considered more accurate for describing the excited electronic states51-53 with charge distributions significantly different from the ground state such as with polar solvents or CT states in the solute. The SS method is computationally significantly more demanding though and requires that each state be calculated independently. As such, only five states were calculated using the CAM-B3LYP/6-31G* method for the extended trimer which contained, however, representatives of the most important excitation types. For the extended trimer using the CAM-B3LYP/6-31G method, twenty states were calculated, and for the standard trimer using both basis sets, ten states were considered. Dichloromethane (dielectric constant, ε=8.93, refractive index, n=1.42) was chosen to act as a modestly polar environment.

Geometry optimizations were carried out using the TURBOMOLE54 program suite, while all TD-DFT calculations were performed with the Gaussian G09 program package.55 The character of the electronic states has been analyzed by means of natural transition orbitals (NTOs)56-58 and molecular electrostatic potential plots (MEP). NTOs represent the weighted contribution to each excited state transition from a hole to a particle (electron) derived from the one-electron transition density matrix. NTOs and particle/hole populations were performed with the TheoDORE program package.59-61

5.4 Results and Discussion 5.4.1 Experimental Observations

We fabricate the bilayer device as illustrated in Figure 5.2a comprised with 20 nm

P3HT as donor and 40 nm of fullerene (C60) as acceptor for photocurrent measurements as summarized in Figure 5.2. To insert oriented O3 between the (D/A) interface, we employed the O3 with Langmuir-Blodgett (LB) method after spin coating P3HT on hole transporting material, followed by C60 deposition by thermal evaporation (Figure 5.2a). The obtained bilayer device was taken for photocurrent spectrum measurement in Figure 5.2b. The photocurrent follows the absorption spectrum of P3HT (band gap 1.9 eV), while the 95 Texas Tech University, Reed Nieman, May 2019 magnitude of the photocurrent increase by 1.8 times from the device without spacer layer (black) to the device with 1-bilayer of O3 inserted (red). The optical band gap remains the same in both cases indicating the oligomer absorption does not contribute to the photocurrent increase. To characterize the molecular orientation of the O3 molecules at the interface, we took the LB deposited O3 molecules on P3HT thin-film for gracing incidence wide angle X-ray scattering (GIWAXS) measurement. The result in Figure 5.2c clearly -1 shows a diffraction spot along the qz axis. The spot is located near qz = 0.322 Å corresponding to a d-spacing of 19.4 Å which matches with the molecule length. In addition, the Bragg spots are parallel to the qy axis (substrate) indicating that the O3 molecules are aligning in the out-of-plane62 orientation by LB method (Figure 5.2a) and pointing from P3HT towards C60. The distance between the donor-acceptor is thus determined by the length of molecule and number of bilayers deposited. This is consistent with previous measurements of three self-assembled oligomers.1, 63 We further performed device photocurrent measurement as a function of the number of O3 bilayers as shown in Figure 5.2d, we found the maximum gain in photocurrent can be obtained when inserting two bilayers of O3 molecules between D/A interface. This is suggesting that such separation between donor and acceptor materials is optimal to suppress the electron-hole recombination through CT state at the interface after a suitable thickness of spacer layer insertion. The subsequent decrease of the photocurrent is associated with the decrease of efficiency for direct CT process.

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b a Bilayer device 1-bilayer O3

) 150

A

Langmuir-Blodgett 9

-

0

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(

t

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r Al S r Al N u

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o O O

t

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S No spacer 0 400 450 500 550 600 650 700 c d Wavelength (nm) 400

) 300

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n

(

t

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0 0 2 4 6 8 layer number

Figure 5.2 (a) Scheme of bilayer device structure used in this study, the interface layer with vertical alignment was produced by Langmuir-Blodgett (LB) method illustrated on right. (b) photocurrent measured at short circuit condition as function of illumination wavelength for bilayer device without spacer and with 1-bilayer of O3 between P3HT and C60. (c) GIWAXS map for 1 bilayer of O3 deposited on P3HT by LB method. (d) peak photocurrent at 550 nm illumination as a function of number of O3 bilayer at interface. At open circuit condition, the photo-voltage value is determined by the carrier generation and recombination, following the equation:

푛퐾푏푇 퐽푅푒푐 푉푂퐶 = 퐸푔 − 푙푛( ) (5.1) 푞 퐽푆퐶 where Eg is the effective band gap, Kb is the Boltzmann constant, T is the temperature, q is the elementary charge, JRec is the recombination current and JSC is the final short circuit photocurrent density.64 Considering a dielectric constant of 3~4 in a typical organic system, the Coulomb radius is estimated to be 16 nm,65 and the energetic disorder further reduces this value to 4 nm, probed experimentally.11 This means that if the electron-hole pair bonded at the charge transfer states can be separated to a distance greater than the Coulomb radius, significant amount of carrier recombination can be suppressed, that reduces JREC

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and increases JSC, which ultimately reduces the voltage loss through non-radiative recombination. In fact, our experimental data in Figure 5.2 show an increase in JSC (photocurrent at short circuit condition) which is an indication of reduced recombination that will lead to increase in VOC.

The purpose of the following electronic structure calculations is to show that such a large electron-hole separation is energetically feasible using the detailed interface models shown in Figure 5.1. At this point it should be mentioned that in the single spacer level model used because of computational economy, the electron-hole pair will have a separation of only ~3 nm.

5.4.2 Analysis of Energy Levels

The geometrical arrangement of the three components of the standard model is depicted in Figure 5.1. In this model only a single T3 layer is used because of computational efficiency. Nevertheless, as the results below will show, the major stabilization effects for charge separated states can be seen already in this case. The spacer T3 chain faces the C60 unit via the thiol group with interatomic distances of 2.6 – 2.8 Å and P3HT via the alkane chain at distances of 2.7 – 2.85 Å. The distance between the center of C60 and P3HT is

~28.6 Å. In the extended model (Figure 5.1b, c) three C60 units are arranged horizontally with distances of around 3.3 Å. The three T3 chains are stacked (side view, Figure 5.1c). Typical intermolecular distances are displayed in Figure 5.1.

The calculations on the standard trimer have been used to establish the general structure of the electronic spectrum and to test the validity of the smaller 6-31G basis set for use with the significantly larger extended trimer. The scheme of the lowest excited singlet states of the isolated complex is given in Figure 5.3a, b using the 6-31G* and 6- 31G basis sets, respectively. The full list of the 20 excited singlet state energies Ω calculated can be found in Table C.1 (6-31G basis) and Table C.2 (6-31G* basis) of

Appendix C. For the 6-31G* basis (Figure 5.3a, isolated system), the local C60 states have been calculated as well. This figure shows that in case of the calculation for the isolated system they form the lowest band of excitonic states. If this would be the true energy 98 Texas Tech University, Reed Nieman, May 2019 ordering, these states would act as a sink for conversion of CT states, which would lead to serious losses for the efficiency of the photovoltaic process. The C60 states are followed by a local π-π* state on P3HT which has the largest oscillator strength, f, of all states calculated. Next appears another band of C60 states followed by two closely spaced triples of CT states from P3HTC60 and T3C60 at 3.3 and 3.4 eV, respectively. There are two more locally excited states, one on T3 and the other one on P3HT. Triples of CT states alternating between excitations from P3HT and T3 follow. This sequence is interrupted only by one local state on P3HT. Inclusion of the environment via the LR method Figure 5.3a, b and Table C.4 and Table C.5) does not change much in the relative ordering of states compared to the isolated system. However, the more sophisticated, but also numerically more expensive SS solvent calculations change the situation dramatically (Figure 5.3a, Table C.6). With respect to the P3HT excitonic bright state, the SS approach leads to a quite dramatic stabilization of nine CT states to be located below the P3HT state.

The energetic spacing between the P3HT→C60 CT states and the T3→C60 CT states is rather large being about 0.6 eV. There is also a large energy gap of about 0.75 eV between the P3HT state and the following T3→C60 CT state. Thus, from this model one can already see the crucial environmental effect which significantly stabilizes the CT state relative to the others. On the other side, the environmental effect on the energetic location of the locally excited P3HT and C60 states is small. As a result, the CT states are the lowest ones and the leaking of the CT states into locally excited C60 states is avoided.

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Figure 5.3 Excited state energies Ω of the standard P3HT-T3-C60 trimer using the CAM- B3LYP method and (a) the 6-31G* and (b) the 6-31G basis sets both as isolated systems and in the LR and SS environments (for numerical data see Table C.1, Table C.2, and Table C.4-Table C.8). For the isolated system using the 6-31G* basis set (farthest left of (a)), the fifteen lowest-energy locally excited C60 states have been computed also. Excited states of (c) the extended P3HT-T3-C60 trimer using the 6-31G* and (d) 6-31G basis sets both as isolated systems and in the SS environment (Table C.9-Table C.11). The strongly different behavior of the two solvation models is to be expected since in the LR method dynamic solvent polarization is computed from the transition density which is, however, very small for the present case of charge transfer states. Consequently, the solvent effects are correspondingly small and not adequately represented by this approach in the case of charge transfer states. On the other hand, the SS approach has been developed having charge-transfer transitions in mind by representing the dynamic solvent 100 Texas Tech University, Reed Nieman, May 2019 polarization by means of the difference of the electronic densities of the initial and final states. For a more detailed discussion of this point see e.g. Ref. 66. Therefore, it is concluded that the SS scheme allows for a better characterization of environmental effects on CT states and the LR approach will not be considered further here.

Comparison between the 6-31G* and 6-31G basis sets shows that the smaller 6- 31G basis represents the energy spectrum of the standard complex very well and also that omission of the C60 states by freezing respective orbitals does not affect the remaining spectrum. These findings are important for the calculations on the extended trimer where the SS solvent calculations are time consuming. Therefore, they have been performed only with the 6-31G basis (see below).

The extended trimer was investigated next as an extension of our standard trimer structure providing a significantly enhanced interface model. (Figure 5.3c, d, Table C.9- Table C.11). For the isolated system, the lowest bright state is located on P3HT at ~2.65 eV, very similar to the excitation energy in the standard model. The difference to the standard model is that now several (T3)3→(C60)3 CT states are located below even for the isolated complex. A rich variety of states is also found above the P3HT state (Figure 5.3c).

Including the SS environment using the 6-31G basis, Figure 5.3d, results in significant changes with respect to the CT states, the most important one being that the P3HT CT states are lowest in energy. Below the P3HT bright excitonic state, we computed nineteen CT states: fifteen (T3)3→(C60)3 CT states and four P3HT→(C60)3 CT states. The

(T3)3→(C60)3 CT states occupy a CT band of 0.44 eV, while the P3HT→(C60)3 CT states are found in a much narrower range of 0.02 eV. As has already been found for the standard model, the mentioned CT states are located significantly below the C60 states (which are around 2.5 eV) and that, therefore, the former states cannot leak into C60 states as soon as they are formed. In contrast to the results presented here, in TD-DFT thiophene/fullerene 32 complexes locally excited C60 states comprised the lowest-energy band of states. These differences are probably due to the use of the LR solvation model used in this reference. It is important to point out that the presence of any -conjugated chromophores adds an effective dielectric medium and facilitates further stabilization of CT states even though

101 Texas Tech University, Reed Nieman, May 2019 the wavefunction is not directly delocalized into these molecules. To illustrate this trait, we removed the spacer and calculated only the C60/P3HT system. Table 5.1 records the calculated energies of the lowest P3HT→C60 CT states. In the P3HT-T3-C60 system, the solvent stabilization is 1.62 eV, but in the P3HT-C60 system only 0.42 eV. This comparison shows the strong sensitivity of the electrostatic interaction in the charge separated systems which are better stabilized by the environment in the case where the charge separation is more pronounced due to the spacer in between.

Table 5.1 Stabilization energy of the lowest P3HT→C60 CT state of P3HT-T3-C60 and P3HT-C60 system calculated in the isolated system and the SS environment.

P3HT-T3-C60 P3HT-C60 Isolated system 2.91 2.20 SS environment 1.29 1.78 ΔE (eV) 1.62 0.42

5.4.3 Characterization of Electronic States

The properties of the different types of electronic states are characterized in Figure 5.4 by means of two descriptors. The first one is represented by the most important NTO’s of typical electronic transitions of the systems calculated in the SS environment (Figure 5.4). All transitions are well represented with one hole/particle pair as the corresponding weights are above 0.9 e. As discussed above, the lowest excited singlet state band has

P3HT→(C60)3 CT character. This can be nicely seen from the NTO picture of Figure 5.4a, b, which shows, respectively, the hole density in P3HT and the particle density in the central C60 unit. These two densities are well separated with essentially no overlap between the two systems. As second descriptor, the MEP plot displayed in Figure 5.4c shows that the P3HT unit has a strongly positive potential which extends significantly into the (T3)3 moiety. The electrostatic potential of the middle C60 molecule of the (C60)3 unit is strongly negative in agreement with the location of the particle NTO in Figure 5.4b. The other two

C60 molecules are slightly negative as well. Overall, the MEP plots give a stronger delocalized picture of the CT process in comparison to the NTOs, since the localized CT density affects much larger regions via Coulomb interactions.

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Figure 5.4 Natural Transition Orbitals (NTO), particle/hole populations and MEP of (a, b, c) the lowest-energy P3HT→(C60)3 CT state, (d, e, f) the lowest-energy T3→(C60)3 CT state, (j, k, l) the lowest-energy P3HT bright excitonic state of the extended thiophene trimer calculated using the CAM-B3LYP/6-31G method in the SS environment, and (g, h, i) the lowest-energy C60 dark excitonic state of the isolated standard thiophene trimer calculated using the CAM-B3LYP/6-31G* method. Here we use NTO isovalue ±0.015 e/Bohr3 and MEP isovalue ±0.0003 e/Bohr3.

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The next band consists of (T3)3(C60)3 CT states. Here the lowest-energy state of this type (Figure 5.4d,e) shows a transition from an occupied orbital diffusely distributed across the (T3)3 unit to an orbital located on a single C60 molecule in the (C60)3 unit. The

MEP plot given in Figure 5.4f depicts similar character to the P3HT→(C60)3 delocalized CT character, however as expected the positive region is spread over the T3 unit and P3HT is essentially neutral.

The C60 exciton band is represented in Figure 5.4g,h by the lowest-energy C60 state calculated for the isolated standard trimer. This band is optically dark consisting of an excitation localized to the C60 molecule, while the MEP plot (Figure 5.4i) shows very little polarity, which is typical for neutral excitonic states. For the P3HT bright excitonic state, the primary contribution to the electronic excitation is localized on the P3HT unit (Figure 5.4j,k). The MEP plot of this state displayed in Figure 5.4l shows only little polarity.

5.5 Conclusions

Efficient operation of organic solar cells based on the BHJ architecture is ensured by tuning sophisticated organic-organic interfaces to align excitonic and charge transfer excited states to provide robust photo-generated exciton dissociation, while minimizing the recombination of the dissociated electron and hole across the donor-acceptor interface. The present study focuses on a detailed molecular model of a polar interface in a bulk heterojunction, which is suggested by our experiment. Here a self-assembled spacer unit

(oligomer T3) is included in between a P3HT donor and C60 acceptor materials. Following a non-trivial geometry optimization routine, extensive TD-DFT calculations based on long- range corrected functional mode and including environmental effects have been performed to characterize the series of lowest excited singlet states in terms of charge transfer and excitonic character. The calculations show unequivocally the strong stabilization of the CT states by environmental effects. The amount of stabilization increases with the distance of the charges (P3HT→C60 vs. T3→C60 CT) due to improved environmental stabilization. By comparison with a system of an ion pair without spacer layer (P3HT→C60), a stabilization of more than 1 eV is computed for the single layer model. Inclusion of more layers are 104 Texas Tech University, Reed Nieman, May 2019 expected to increase the stabilization somewhat further but are currently too expensive in the framework of an explicit molecular model and state-specific TDDFT. Both spacer models show that with consideration of environmental interaction the lowest excited states are formed by bands of CT states, the lowest ones being of P3HTC60 character followed by T3C60 transitions. The bright excitonic P3HT state is located 0.5 eV above the first

CT below and 1.15 eV above the lowest CT state. Local C60 excitations were found below the P3HT state, but significantly (~0.9 eV) above the lowest CT state.

Such calculated and analyzed electronic structure fully support our hypothesis which emerged from experimental study1: photo-generated exciton in the P3HT component at the BHJ interface either goes to C60 via energy transfer and subsequently dissociates into a T3C60 CT state, or directly dissociates into a P3HTC60 state. This state is followed by a transition to the next P3HTC60 state with lower energy, which has even stronger CT character. Our calculation of the stability of the CT states clearly indicate via the use of Equation 5.1 that the spacer facilitates separation of the electron and hole, thus diminishing their Coulomb binding energy and reducing the efficiency of recombination. Moreover, our simulations demonstrate how the interface ordering and structure affects the energetics of relevant states via electronic coupling in self-assembled oligomer layer, which further translates to macroscopic device performance. Such physical processes are general and underpin manipulation in any donor-acceptor based electronic devices, such as organic light emitting diodes, , sensors, and comprise the design principles for tailoring the interface properties of organic electronic devices. All our simulations of electronically excited states are based on the equilibrium molecular geometries. To capture the effects of vibrational broadening and conformational disorder, it is possible to use multiple snapshots of structures obtained from classical simulations. However, modeling excited states in such a case is currently a numerically intractable task for the present high level of theory. Likewise, direct modeling of excited state relaxation with, for instance, surface hopping approaches,6762 are numerically costly and require reduced Hamiltonian models. This work goes far beyond the scope of this paper and will be addressed in future investigations.

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5.6 Acknowledgments

The work at Los Alamos National Laboratory (LANL) was supported by the LANL LDRD program (W.N., H.T., A.D.M., S.T.). This work was done in part at Center for Nonlinear Studies (CNLS) and the Center for Integrated CINT, project no. 2017AC0027), a U.S. Department of Energy and Office of Basic Energy Sciences user facility, at LANL. This research used resources provided by the LANL Institutional Computing Program. We also thank for computer time at the Linux cluster arran of the School for Pharmaceutical Science and Technology of the University of Tianjin, China.

5.7 Bibliography

(1) Nie, W.; Gupta, G.; Crone, B. K.; Liu, F.; Smith, D. L.; Ruden, P. P.; Kuo, C.-Y.; Tsai, H.; Wang, H.-L.; Li, H.; Tretiak, S.; Mohite, A. D., Interface Design Principles for High-Performance Organic Semiconductor Devices. Advanced Science 2015, 2 (6), 1500024. (2) Thompson, B. C.; Fréchet, J. M. J., Polymer–Fullerene Composite Solar Cells. Angewandte Chemie International Edition 2008, 47 (1), 58-77. (3) Liang, Y.; Feng, D.; Wu, Y.; Tsai, S.-T.; Li, G.; Ray, C.; Yu, L., Highly Efficient Solar Cell Polymers Developed via Fine-Tuning of Structural and Electronic Properties. Journal of the American Chemical Society 2009, 131 (22), 7792-7799. (4) Yu, G.; Gao, J.; Hummelen, J. C.; Wudl, F.; Heeger, A. J., Polymer Photovoltaic Cells: Enhanced Efficiencies via a Network of Internal Donor-Acceptor Heterojunctions. Science 1995, 270 (5243), 1789. (5) Brédas, J.-L.; Norton, J. E.; Cornil, J.; Coropceanu, V., Molecular Understanding of Organic Solar Cells: The Challenges. Accounts of Chemical Research 2009, 42 (11), 1691-1699. (6) Baumeier, B.; May, F.; Lennartz, C.; Andrienko, D., Challenges for in silico design of organic semiconductors. Journal of Materials Chemistry 2012, 22 (22), 10971- 10976. (7) Kanai, Y.; Grossman, J. C., Insights on Interfacial Charge Transfer Across P3HT/Fullerene Photovoltaic Heterojunction from Ab Initio Calculations. Nano Letters 2007, 7 (7), 1967-1972. (8) Yi, Y.; Coropceanu, V.; Brédas, J.-L., Exciton-Dissociation and Charge- Recombination Processes in Pentacene/C60 Solar Cells: Theoretical Insight into the Impact of Interface Geometry. Journal of the American Chemical Society 2009, 131 (43), 15777-15783. (9) Sariciftci, N. S.; Heeger, A. J., Reversible, Metastable, Ultrafast Photoinduced Electron-Transfer from Semiconducting Polymers to and in

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the Corresponding Donor-Acceptor Heterojunctions. International Journal of Modern Physics B 1994, 8 (3), 237-274. (10) Grancini, G.; Maiuri, M.; Fazzi, D.; Petrozza, A.; Egelhaaf, H. J.; Brida, D.; Cerullo, G.; Lanzani, G., Hot exciton dissociation in polymer solar cells. Nature Materials 2013, 12 (1), 29-33. (11) Gelinas, S.; Rao, A.; Kumar, A.; Smith, S. L.; Chin, A. W.; Clark, J.; van der Poll, T. S.; Bazan, G. C.; Friend, R. H., Ultrafast Long-Range Charge Separation in Organic Semiconductor Photovoltaic Diodes. Science 2014, 343 (6170), 512-516. (12) Bittner, E. R.; Silva, C., Noise-induced quantum coherence drives photo-carrier generation dynamics at polymeric semiconductor heterojunctions. Nature Communications 2014, 5, 3119. (13) Vella, E.; Li, H.; Gregoire, P.; Tuladhar, S. M.; Vezie, M. S.; Few, S.; Bazan, C. M.; Nelson, J.; Silva-Acuna, C.; Bittner, E. R., Ultrafast decoherence dynamics govern photocarrier generation efficiencies in polymer solar cells. Scientific Reports 2016, 6, 29437. (14) Campbell, I. H.; Crone, B. K., Improving an organic photodiode by incorporating a tunnel barrier between the donor and acceptor layers. Applied Physics Letters 2012, 101 (2), 023301. (15) Sampat, S.; Mohite, A. D.; Crone, B.; Tretiak, S.; Malko, A. V.; Taylor, A. J.; Yarotski, D. A., Tunable Charge Transfer Dynamics at Tetracene/LiF/C60 Interfaces. Journal of Physical Chemistry C 2015, 119 (3), 1286-1290. (16) Adamo, C.; Jacquemin, D., The calculations of excited-state properties with Time- Dependent Density Functional Theory. Chemical Society Reviews 2013, 42 (3), 845-856. (17) Dreuw, A.; Weisman, J. L.; Head-Gordon, M., Long-range charge-transfer excited states in time-dependent density functional theory require non-local exchange. Journal of Chemical Physics 2003, 119 (6), 2943-2946. (18) Dreuw, A.; Head-Gordon, M., Single-reference ab initio methods for the calculation of excited states of large molecules. Chemical Reviews 2005, 105 (11), 4009-4037. (19) Kaduk, B.; Kowalczyk, T.; Van Voorhis, T., Constrained Density Functional Theory. Chemical Reviews 2012, 112 (1), 321-370. (20) Iikura, H.; Tsuneda, T.; Yanai, T.; Hirao, K., A long-range correction scheme for generalized-gradient-approximation exchange functionals. The Journal of Chemical Physics 2001, 115, 3540. (21) Vydrov, O. A.; Scuseria, G. E., Assessment of a long-range corrected hybrid functional. Journal of Chemical Physics 2006, 125, 234109. (22) Stein, T.; Kronik, L.; Baer, R., Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory. Journal of the American Chemical Society 2009, 131 (8), 2818-2820. (23) Lange, A. W.; Herbert, J. M., Both Intra- and Interstrand Charge-Transfer Excited States in Aqueous B-DNA Are Present at Energies Comparable To, or Just Above, the 1ππ* Excitonic Bright States. Journal of the American Chemical Society 2009, 131 (11), 3913-3922.

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(24) Kronik, L.; Stein, T.; Refaely-Abramson, S.; Baer, R., Excitation Gaps of Finite- Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals. Journal of Chemical Theory and Computation 2012, 8 (5), 1515-1531. (25) Körzdörfer, T.; Sears, J. S.; Sutton, C.; Brédas, J.-L., Long-range corrected hybrid functionals for π-conjugated systems: Dependence of the range-separation parameter on conjugation length. The Journal of Chemical Physics 2011, 135 (20), 204107. (26) Li, H.; Nieman, R.; Aquino, A. J. A.; Lischka, H.; Tretiak, S., Comparison of LC- TDDFT and ADC(2) Methods in Computations of Bright and Charge Transfer States in Stacked Oligothiophenes. Journal of Chemical Theory and Computation 2014, 10 (8), 3280-3289. (27) Yanai, T.; Tew, D. P.; Handy, N. C., A new hybrid exchange–correlation functional using the Coulomb-attenuating method (CAM-B3LYP). Chemical Physics Letters 2004, 393 (1), 51-57. (28) Limacher, P. A.; Mikkelsen, K. V.; Lüthi, H. P., On the accurate calculation of polarizabilities and second hyperpolarizabilities of polyacetylene oligomer chains using the CAM-B3LYP density functional. Journal of Chemical Physics 2009, 130, 194114. (29) Chai, J.-D.; Head-Gordon, M., Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections. Physical Chemistry Chemical Physics 2008, 10 (44), 6615-6620. (30) Manna, A. K.; Lee, M. H.; McMahon, K. L.; Dunietz, B. D., Calculating High Energy Charge Transfer States Using Optimally Tuned Range-Separated Hybrid Functionals. Journal of Chemical Theory and Computation 2015, 11 (3), 1110- 1117. (31) Borges, I.; Aquino, A. J. A.; Köhn, A.; Nieman, R.; Hase, W. L.; Chen, L. X.; Lischka, H., Ab Initio Modeling of Excitonic and Charge-Transfer States in Organic Semiconductors: The PTB1/PCBM Low Band Gap System. Journal of the American Chemical Society 2013, 135 (49), 18252-18255. (32) Sen, K.; Crespo-Otero, R.; Weingart, O.; Thiel, W.; Barbatti, M., Interfacial States in Donor–Acceptor Organic Heterojunctions: Computational Insights into Thiophene-Oligomer/Fullerene Junctions. Journal of Chemical Theory and Computation 2013, 9 (1), 533-542. (33) Balamurugan, D.; Aquino, A. J. A.; de Dios, F.; Flores, L.; Lischka, H.; Cheung, M. S., Multiscale Simulation of the Ground and Photo-Induced Charge-Separated States of a Molecular Triad in Polar Organic Solvent: Exploring the Conformations, Fluctuations, and Free Energy Landscapes. Journal of Physical Chemistry B 2013, 117 (40), 12065-12075. (34) Perrine, T. M.; Berto, T.; Dunietz, B. D., Enhanced Conductance via Induced II- Stacking Interactions in Cobalt(II) Terpyridine Bridged Complexes. Journal of Physical Chemistry B 2008, 112 (50), 16070-16075. (35) Katz, H. E., Organic molecular solids as thin film semiconductors. Journal of Materials Chemistry 1997, 7 (3), 369-376.

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(36) Cornil, J.; Beljonne, D.; Calbert, J. P.; Brédas, J. L., Interchain Interactions in Organic π-Conjugated Materials: Impact on Electronic Structure, Optical Response, and Charge Transport. Advanced Materials 2001, 13 (14), 1053-1067. (37) Bredas, J. L.; Calbert, J. P.; da Silva, D. A.; Cornil, J., Organic semiconductors: A theoretical characterization of the basic parameters governing charge transport. Proceedings of the National Academy of Sciences of the United States of America 2002, 99 (9), 5804-5809. (38) Lee, M. H.; Geva, E.; Dunietz, B. D., The Effect of Interfacial Geometry on Charge- Transfer States in the /Fullerene Organic Photovoltaic System. Journal of Physical Chemistry A 2016, 120 (19), 2970-2975. (39) Lee, M. H.; Dunietz, B. D.; Geva, E., Donor-to-Donor vs Donor-to-Acceptor Interfacial Charge Transfer States in the Phthalocyanine-Fullerene Organic Photovoltaic System. Journal of Physical Chemistry Letters 2014, 5 (21), 3810- 3816. (40) Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple. Physical Review Letters 1996, 77 (18), 3865-3868. (41) Schafer, A.; Horn, H.; Ahlrichs, R., Fully Optimized Contracted Gaussian-Basis Sets for Atoms Li to Kr. Journal of Chemical Physics 1992, 97, 2571-2577. (42) Ditchfield, R.; Hehre, W. J.; Pople, J. A., Self‐Consistent Molecular‐Orbital Methods. IX. An Extended Gaussian‐Type Basis for Molecular‐Orbital Studies of Organic Molecules. The Journal of Chemical Physics 1971, 54 (2), 724-728. (43) Petersson, G. A.; Bennett, A.; Tensfeldt, T. G.; Al‐Laham, M. A.; Shirley, W. A.; Mantzaris, J., A complete basis set model chemistry. I. The total energies of closed‐ shell atoms and hydrides of the first‐row elements. The Journal of Chemical Physics 1988, 89 (4), 2193-2218. (44) Petersson, G. A.; Al‐Laham, M. A., A complete basis set model chemistry. II. Open‐shell systems and the total energies of the first‐row atoms. The Journal of Chemical Physics 1991, 94 (9), 6081-6090. (45) Cossi, M.; Rega, N.; Scalmani, G.; Barone, V., Energies, structures, and electronic properties of molecules in solution with the C-PCM solvation model. Journal of Computational Chemistry 2003, 24 (6), 669-681. (46) Runge, E.; Gross, E. K. U., Density-Functional Theory for Time-Dependent Systems. Physical Review Letters 1984, 52 (12), 997-1000. (47) Petersilka, M.; Gossmann, U. J.; Gross, E. K. U., Excitation Energies from Time- Dependent Density-Functional Theory. Physical Review Letters 1996, 76 (8), 1212- 1215. (48) Improta, R.; Barone, V.; Scalmani, G.; Frisch, M. J., A state-specific polarizable continuum model time dependent density functional theory method for excited state calculations in solution. Journal of Chemical Physics 2006, 125 (5), 054103. (49) Hättig, C.; Weigend, F., CC2 excitation energy calculations on large molecules using the resolution of the identity approximation. Journal of Chemical Physics 2000, 113, 5154-5161. (50) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H., A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. The Journal of Chemical Physics 2010, 132 (15), 154104.

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(51) Corni, S.; Cammi, R.; Mennucci, B.; Tomasi, J., Electronic excitation energies of molecules in solution within continuum solvation models: investigating the discrepancy between state-specific and linear-response methods. Journal of Chemical Physics 2005, 123 (13), 134512. (52) Cammi, R.; Corni, S.; Mennucci, B.; Tomasi, J., Electronic excitation energies of molecules in solution: state specific and linear response methods for nonequilibrium continuum solvation models. Journal of Chemical Physics 2005, 122 (10), 104513. (53) Caricato, M., A comparison between state-specific and linear-response formalisms for the calculation of vertical electronic transition energy in solution with the CCSD-PCM method. Journal of Chemical Physics 2013, 139 (4), 044116. (54) Ahlrichs, R.; Bär, M.; Häser, M.; Horn, H.; Kölmel, C., Electronic structure calculations on workstation computers: The program system turbomole. Chemical Physics Letters 1989, 162 (3), 165-169. (55) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery Jr., J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J.; Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, Gaussian, Inc.: Wallingford, CT, USA, 2009. (56) Luzanov, A. V.; Sukhorukov, A. A.; Umanskii, V. É., Application of transition density matrix for analysis of excited states. Theoretical and Experimental Chemistry 1976, 10 (4), 354-361. (57) Mayer, I., Using singular value decomposition for a compact presentation and improved interpretation of the CIS wave functions. Chemical Physics Letters 2007, 437 (4–6), 284-286. (58) Martin, R. L., Natural transition orbitals. The Journal of Chemical Physics 2003, 118 (11), 4775-4777. (59) Plasser, F.; Lischka, H., Analysis of Excitonic and Charge Transfer Interactions from Quantum Chemical Calculations. Journal of Chemical Theory and Computation 2012, 8 (8), 2777-2789. (60) Plasser, F.; Wormit, M.; Dreuw, A., New tools for the systematic analysis and visualization of electronic excitations. I. Formalism. The Journal of Chemical Physics 2014, 141 (2), 024106. (61) Plasser, F. TheoDORE: a package for theoretical density, orbital relaxation, and exciton analysis, available from http://theodore-qc.sourceforge.net.

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(62) Smith, K. A.; Lin, Y. H.; Mok, J. W.; Yager, K. G.; Strzalka, J.; Nie, W. Y.; Mohite, A. D.; Verduzco, R., Molecular Origin of Photovoltaic Performance in Donor- block-Acceptor All-Conjugated Block Copolymers. 2015, 48 (22), 8346-8353. (63) Kuo, C. Y.; Liu, Y. H.; Yarotski, D.; Li, H.; Xu, P.; Yen, H. J.; Tretiak, S.; Wang, H. L., Synthesis, , STM investigation of oligothiophene self- assemblies with superior structural order and electronic properties. Chemical Physics 2016, 481, 191-197. (64) Credgington, D.; Durrant, J. R., Insights from Transient Optoelectronic Analyses on the Open-Circuit Voltage of Organic Solar Cells. Journal of Physical Chemistry Letters 2012, 3 (11), 1465-1478. (65) Lakhwani, G.; Rao, A.; Friend, R. H., Bimolecular Recombination in Organic Photovoltaics. Annual Review of Physical Chemistry, Vol 65 2014, 65, 557-581. (66) Guido, C. A.; Jacquemin, D.; Adamo, C.; Mennucci, B., Electronic Excitations in Solution: The Interplay between State Specific Approaches and a Time-Dependent Density Functional Theory Description. Journal of Chemical Theory and Computation 2015, 11 (12), 5782-5790. (67) Tully, J. C., Molecular dynamics with electronic transitions. The Journal of Chemical Physics 1990, 93 (2), 1061-1071.

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CHAPTER VI BENCHMARK AB INITIO CALCULATIONS OF POLY(P- PHENYLENEVINYLENE) DIMERS: STRUCTURE AND EXCITON ANALYSIS Authors: Reed Nieman,a Adelia J. A. Aquino,a,b and Hans Lischkaa,b

Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University, Lubbock, TX 79409-1061, USA, bSchool of Pharmaceutical Sciences and Technology, Tianjin University, Tianjin 300072, PR China 6.1 Abstract

Crucial benchmark calculations of the structure and electronic excited states missing from the current literature were performed for oligomers of poly(p- phenylenevinylene) (PPV) dimers using high-level scaled opposite-spin (SOS) second- order Møller-Plesset (SOS-MP2) and second-order algebraic diagrammatic construction (SOS-ADC(2)) calculations. Comparisons were also made to DFT methods. The dimer structures were studied in two conformations: sandwich-stacked and displaced; the latter exhibited greater stacking interactions. Good agreement was found between the cc-pVQZ and aug-cc-pVQZ basis sets and the cc-pVTZ basis. Comparisons of the interaction energies were made to the complete basis set limit. Basis set superposition error was also accounted for. The excited state spectra presented a varied and dynamic picture that was analyzed using transition density matrices and natural transition orbitals (NTOs).

6.2 Introduction

Poly(p-phenylenevinylene) (PPV) is paradigmatic for the study of the mechanisms affecting the photophysical excitation energy transfer (EET) behavior of organic π- conjugated polymers on the molecular level that allows the fine-tuning of their absorption and emission properties which make them so attractive as materials for use in electroluminescent and photovoltaic devices.1-6 Experiments conclude that features of coherent PPV dynamics can be attributed to structural effects from coupling to a vibrational

112 Texas Tech University, Reed Nieman, May 2019 mode,5, 7 and structural relaxation is connected to the evolution of excited states necessary for EET dynamics.

Accurate electronic structure calculations are a prerequisite for the characterization of these properties, but are hindered by the necessity of large molecular systems and reliable descriptions of excited states. The Pariser-Parr-Pople (PPP) π electron Hamiltonian combined with a mechanical described the effect of the σ electrons has been used previously to model surface-hopping dynamics of PPV,8 while later the Pariser-Parr-Pople-Peierls (PPPP) Hamiltonian was used together with the Configuration Interaction Singles (CIS) method to compute and classify PPV excited states.9 Calculations of PPV chains with the collective electronic oscillator (CEO) method together with the Austin Model 1 (AM1) approach showed exciton self-trapping over six PV units and demonstrated close agreement with experimental spectra.10 The vibronic structure of the lowest optical transition of PPV was found using the CIS method with an empirical description of the electron- coupling.11 A time-dependent density functional theory (TDDFT) method was used to calculate the singlet-triplet splitting of PPV among other π- conjugated systems.12 The exchange-correlation (XC) functional dependence of the localization of the electronic excitation was investigated for polymeric PPV chains.13 Charged polaron localization was found to be sensitive to the polarization of the environment whereas neutral states are not affected to the same degree.14 A diabatic Hamiltonian has been developed for use studying wavepacket dynamics in the torsional modes of PPV vinylene single bonds.15 The kinetic Monte-Carlo method was used to model fluorescent depolarization to study the interactions between linked chromophoric units in poly-[2-methoxy-5-((2-ethylhexyl)oxy)-phenylenevinylene (MEH-PPV).16

Though more computationally expensive than semi-empirical or TDDFT calculations, ab initio methods have also been used to model PPV and PPV-like systems, though they focus on smaller molecular sizes. The electronic spectrum of stilbene (PPV2) has been investigated using complete active space perturbation theory to second-order,17 and the electronic spectrum of PPV oligomers up to four repeat PV units (PPV4) has been studied with symmetry-adapted cluster-configuration interaction (SAC-CI).18 Multi- configurational time-dependent Hartree (MCTDH) simulations found that dynamic 113 Texas Tech University, Reed Nieman, May 2019 electron-hole transients involve rapid expansion and contraction of the exciton coherence size in PPV-type systems.19

There is a need to calculate larger oligomer lengths and the approximate coupled- cluster method to second-order (CC2) has shown to be adequate for the calculations of -bridged oligofluorenes and oligo-p-phenylenes.20-21 The resolution of the identity method (RI) together with CC2 has allowed for efficient, less expensive calculations of excited states as well as having analytical gradients available.22 The second- order algebraic diagrammatic construction (ADC(2)) is related to the CC2 method and provides similar results, though it has an advantage over the CC2 method in that excited states are obtained as eigenvalues of a Hermitian matrix instead of a non-Hermitian Jacobi matrix as with the coupled-cluster response in the CC2 method.23 Calculations based on the ADC(2) method investigated the exciton and charge transfer (CT) states of nucleobase dimers.24 Benchmark calculations using the equation of motion excitation energy coupled- cluster (EOMEE-CC) method on DNA nucleobases showed that CC2 was able to reproduce π-π* excitations well compared to EOMEE-CC with singles, doubles, and non- iterative triples (CCSD(T)).25 CC2 has shown deficiencies however in describing n-π* states, though these are not types of excitations normally relevant to the UV spectrum of PPV.25

Excited state properties of PPV oligomers has been calculated in the past by the RI- ADC(2) method when investigating vertical excitation energies, torsional potentials, and defect localization in the S1 state in varying sized oligomers.26 The ADC(2) method has also been used to calculate the absorbance and fluorescence band spectra of PPV oligomers of increasing chain length,27 while high-level ADC(3) benchmark calculations on PPV oligomers have shown that the ADC(2) method is adequate for excited state energies, oscillator strengths, and wave function behavior.28 The scaled opposite spin (SOS) approach applied to ADC(2) (SOS-ADC(2)) described well the pi-pi* electronic excited states of several electron donor-acceptor complexes with excitation energies that compared favorably to experimental results.29

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Most calculations of PPV oligomers focus only on single chain systems, with few calculations of dimer systems. Semi-empirical calculations were carried out using the CEO method together with the AM1 approach to show the dependence on dimer conformation and intermolecular distance of the formation of interchain electronic excitations in PPV dimers.30 The AM1 method was used in another investigation finding that the luminescence capabilities of cofacial PPV dimers is strongly quenched by transition from ground state to the first excited state.31 The Intermediate Neglect of the Differential Overlap (INDO) Hamiltonian together with the Single Configuration Interaction (SCI) approach was used to show that the most stable photogenerated species in interacting PPV chains are intrachain excitons.32 Mixed quantum/classical calculations of non-adiabatic excitation processes of long-chain, stacked PPV oligomers showed that the photoexcitation process leads to thermally-driven polaron hopping between adjacent chains.33 A study on linearly coupled PPV chains showed the effects of intrinsic conjugation defects on the mobility of excitons using a tight-binding method with the Su–Schrieffer–Heeger (SSH) model for which an interchain coupling term is included in the Hamiltonian.34 TDDFT comparisons of the method to experimental two-photon absorption measurements of PPV derivatives showed that the hybrid exchange-correlation density functional, Becke three-parameter Lee, Yang, and Parr (B3LYP) compared well to experimental observations.35 Experimental results on interchain interactions suggest that interchain electronic phenomena such as excimers may be responsible for the promotion of charge transport in PPV chains.36-38

The goal of this work is to provide crucial reference data lacking in the literature by characterizing the geometry and excited states of PPV oligomer dimer systems with high-level ab initio calculations and wave function analysis tools. To that end, PPV oligomers chains of from two (stilbene) to four repeating phenylenevinylene (PV) units

(referred to here as PPV2, PPV3, and PPV4) were arranged in either in stacked-sandwich or displaced dimer conformations. The basis set dependence of the geometric characteristics was investigated using the SOS approach39-40 applied to second-order Møller-Plesset Perturbation theory (MP2)41 with several correlation-consistent basis sets with emphasis on extrapolation of the stabilization energy to the complete basis set (CBS) limit. Excited states and S1 excited state geometries were studied using the SOS-ADC(2) method.42-43

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The SOS-ADC(2) method was also compared to DFT utilizing B3LYP and the long-range corrected Coulomb Attenuating Method with B3LYP (CAM-B3LYP).

6.3 Computational Details

The geometry of PPV dimers was investigated using PPV chains of from two to four repeating PV units arranged in either stacked-sandwich, belonging to the C2h point group, or displaced, belonging to the C1 point group, structural conformations (Figure 6.1). Geometry optimizations for both structural conformations were carried out using the SOS- MP2 method using the RI approximation.44-45 A variety of Dunning-type correlation consistent basis sets were used in order to show the dependence of the basis on the intermolecular stacking distances and the stabilization energies. For sandwich-stacked

PPV2 dimer, the geometry was studied with the largest number of basis sets, cc-pVXZ, and aug-cc-pVXZ where X=D, T, and Q,46-47 as its relatively small size allowed for more extensive investigations. Smaller basis sets were considered for sandwich-stacked PPV3 and PPV4 dimers, cc-pVDZ and cc-pVTZ. Though geometry optimization calculations for the sandwich-stacked PPV3 and PPV4 dimers with the cc-pVQZ basis set would be too computationally taxing, single-point calculations were made for both systems with the cc- pVQZ basis set at the cc-pVTZ optimized geometry. The displaced dimers lack the symmetry advantages of the sandwich-stacked dimers and so were treated with a fewer basis sets, cc-pVXZ (X=D, T, and Q) and aug-cc-pVDZ for PPV2 dimer, cc-pVDZ and cc- pVTZ for PPV3 and PPV4. For the sandwich-stacked PPV2 dimer, the effect of using the SOS approach is compared to geometries optimized with the MP2 method. Sandwich- stacked PPV3 dimer geometries were compared to density functional theory (DFT) methods using the B3LYP48 and CAM-B3LYP49-50 methods together with the D351 dispersion correction approach and the cc-pVTZ basis set. Basis set superposition error (BSSE) was included using the counterpoise (CP) correction method52 for MP2 and SOS- MP2 calculations.

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

Extrapolation to the CBS limit of the stabilization energies was carried out using the two-point fit approach given by Halkier et al in Equation 6.1 for the TQ extrapolation,53

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퐸 푄3 − 퐸 푇3 퐸∞ = 푄 푇 (6.1) 푇푄 푄3 − 푇3 ∞ where 퐸푇푄 is the CBS limit stabilization energy, 푄 and 푇 are the cardinal numbers four (for the cc-pVQZ or aug-cc-pVQZ basis sets) and three (for the cc-pVTZ or aug-cc-pVTZ basis sets), respectively, and 퐸푄 and 퐸푇 are the stabilization energies found for those individual basis sets.

Vertical excitations were calculated using the SOS-ADC(2) method with the RI approximation and the basis sets specified earlier for stacked PPV2, but only cc-pVTZ was considered for the stacked PPV3 and PPV4 dimers. Two states per irreducible representation were calculated for each case. The optimized geometry of the S1 excited state was found for each stacked dimer using the SOS-ADC(2) method with the cc-pVTZ basis set. For stacked PPV3 dimers, comparative TDDFT calculations were carried out using the B3LYP-D3, CAM-B3LYP-D3 methods and the cc-pVTZ basis set. Due to the computational efficiency afforded by TDDFT methods, thirty states were calculated with each method.

All excited states were analyzed via natural transition orbitals (NTO)54 and the

NTO participation ratio (PRNTO), while charge transfer between fragments for a given electronic transition was described using the q(CT) value55 computed from the transition density, 퐷0훼. The value, q(CT), is calculated first by using the transition density to define 훼 the omega matrix, Ω퐴퐵,

1 Ω훼 = ∑(퐷0훼.[퐴푂]푆[퐴푂]) (푆[퐴푂]퐷0훼.[퐴푂]) 퐴퐵 2 푎푏 푎푏 (6.2) 푎∈퐴 푏∈퐵

훼 such that α is the electronic state and AO are the atomic orbitals. The omega matrix, Ω퐴퐵, describes the charge transfer contribution from fragment A to fragment B (A ≠ B), and contributions from excitations occurring on the same fragment (A = B). The CT character for a given system composed of multiple fragments is then given by q(CT):

1 푞(CT) = ∑ ∑ Ω훼 Ω훼 퐴퐵 (6.3) 퐴 퐴≠퐵 118 Texas Tech University, Reed Nieman, May 2019 where Ω훼 is the sum of the charge transfer values for all fragment pairs, A and B. When q(CT) = 1 e, one electron has been transferred, and when q(CT) = 0 e, the transition is local a excitation or Frenkel exciton state.

The value of PRNTO indicates the number of essential participating NTOs and, therefore, how many configurations are important to the description of the excited state.

PRNTO is calculated as:

2 (∑푖 휆푖) 푃푅푁푇푂 = 2 (4) ∑푖 휆푖 where 휆푖 is the weight for each hole-to-electron orbital transition.

All MP2, SOS-MP2, and SOS-ADC(2) calculations were performed using the TURBOMOLE program package,56 and all DFT and TDDFT calculations were done with the Gaussian G09 program package.57 Excited states were analyzed via natural transition orbitals (NTO), q(CT), and omega matrices utilizing the wave function analysis program TheoDORE.55, 58-59

6.4 Results and Discussion 6.4.1 Structural Description

The geometry of stacked PPV2 has been optimized utilizing the SOS-MP2 method at several basis sets and described via intermolecular stacking distances in Table 6.1 with labeling provided in Figure 6.1. In generally, these distances for the phenylene units were larger than the vinylene unit. The aug-cc-pVDZ and aug-cc-pVTZ basis sets show considerably smaller intermolecular distances compared to the cc-pVDZ and cc-pVTZ basis sets, respectively, especially going toward the interior. Going from cc-pVDZ to cc- pVTZ, the distances decrease only slightly and the structure found for cc-pVTZ agreed well with the cc-pVQZ structure. The distances found for the cc-pVQZ and the aug-cc- pVQZ basis sets are nearly identical. BSSE counterpoise corrections showed increased intermolecular stacking distances of an average of about 0.37 Å for the cc-pVDZ basis set.

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This increase in distance is an average of about 0.14 Å for the cc-pVTZ basis, and is relatively unchanged for the aug-cc-pVDZ and cc-pVQZ basis sets. In spite of the relatively large changes in the distances, the differences in distances between the basis sets with and without the diffuse functions and the BSSE corrections do converge, but only at the QZ basis. Finally, when using the MP2 method without the SOS approach (Table D.1), the stacking distances were smaller in general with each basis set utilized than those found with the SOS-MP2 method, an effect that extended to the BSSE corrections as well.

Table 6.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer computed at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets using C2h symmetry. 1-1’ 2-2’ 3-3’ 4-4’ 5-5’ ΔE SOS-MP2 cc-pVDZ 3.83 3.82 3.80 3.85 3.86 -3.99 aug-cc-pVDZ 3.67 3.61 3.61 3.66 3.72 -12.24 cc-pVTZ 3.80 3.79 3.77 3.82 3.83 -5.41 aug-cc-pVTZ 3.80 3.63 3.64 3.68 3.86 -8.26 cc-pVQZ 3.81 3.80 3.77 3.83 3.84 -5.07 aug-cc-pVQZ 3.81 3.80 3.77 3.83 3.84 -5.96 CBS limit extrapolation: TQa -4.79 aug-TQb -4.27 SOS-MP2/BSSE cc-pVDZ 4.21 4.18 4.17 4.21 4.23 -1.26 aug-cc-pVDZ 3.67 3.62 3.61 3.67 3.72 -2.58 cc-pVTZ 3.95 3.93 3.93 3.94 3.96 -3.40 cc-pVQZ 3.81 3.80 3.77 3.83 3.83 -4.04 CBS limit extrapolation: TQa -4.51

Stacked PPV3 (Table 6.2) showed only a slight decrease in the intermolecular stacking distances when going from cc-pVDZ to cc-pVTZ. BSSE counterpoise corrections again calculate much larger distances between monomers. The average intermolecular distances found using the cc-pVDZ and cc-pVTZ basis sets for stacked PPV3 are only marginally smaller than those found for stacked PPV2 using the SOS-MP2 method. Similarly, there were only small differences between the two stacked systems for the two basis sets after BSSE counterpoise correction. Comparing the SOS-MP2 method to DFT methods at the cc-pVTZ basis set for stacked PPV3, the stacking distances are marginally 120 Texas Tech University, Reed Nieman, May 2019 smaller when the B3LYP-D3 method was used, but are larger by about 0.05 Å with the long range-corrected CAM-B3LYP-D3 method. The intermolecular stacking distances of stacked PPV4 are shown in Table D.2 for the SOS-MP2 and SOS-MP2/BSSE methods. An increase in basis set size from cc-pVDZ to cc-pVTZ showed a similar in intermolecular distances as with stacked PPV3, and the average stacking distance decreased in general for stacked PPV4 compared to stacked PPV2.

Table 6.2 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV3 dimer computed at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets using C2h symmetry. 1-1’ 2-2’ 3-3’ 4-4’ 5-5’ 6-6’ 7-7’ 8-8’ ΔE SOS-MP2 cc-pVDZ 3.82 3.81 3.77 3.77 3.77 3.80 3.82 3.83 -7.12 cc-pVTZ 3.81 3.81 3.76 3.76 3.76 3.79 3.82 3.83 -9.54 cc-pVQZa ------8.97 CBS limit extrapolation: TQb -8.55 SOS-MP2/BSSE cc-pVDZ 4.16 4.15 4.07 4.08 4.07 4.14 4.18 4.18 -2.64 cc-pVTZ 3.94 3.93 3.88 3.87 3.87 3.91 3.95 3.96 -6.26 cc-pVQZa ------7.37 CBS limit extrapolation: TQb -8.18 B3LYP-D3 cc-pVTZ 3.77 3.83 3.75 3.75 3.73 3.79 3.83 3.77 - CAM-B3LYP-D3 10.62 cc-pVTZ 3.89 3.84 3.81 3.80 3.80 3.85 3.85 3.89 -8.99 aSingle-point energy at SOS-MP2/cc-pVTZ optimized geometry, bExtrapolation made using cc-pVTZ and cc-pVQC

The sandwich-stacked PPV dimers were compared to a displaced dimer geometry configuration using the SOS-MP2 method, Table 6.3. Displaced PPV2 dimer displayed smaller intermolecular distances than the sandwich stacked PPV2 dimer by an average of about 0.3 Å, but showed similar trends. The intermolecular distances increased from cc- pVDZ to cc-pVQZ, with increasing basis set size. Displaced PPV3 dimer, Table 6.4, using the cc-pVTZ basis, had smaller intermolecular distances compared to the sandwich-stacked

PPV3 dimer and displaced PPV2 dimer using the same basis set. The intermolecular

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distances of displaced PPV4, Table D.3, were smaller compared to the sandwich-stacked

PPV4 dimer by about 0.3 Å, but larger than either displaced PPV2 or PPV3 dimers.

Table 6.3 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV2 dimer computed at SOS-MP2 level using C1 symmetry. 1-1’ 2-2’ 3-3’ 4-4’ ΔE cc-pVDZ 3.52 3.44 3.44 3.52 -7.69 cc-pVTZ 3.54 3.44 3.45 3.54 -8.73 cc-pVQZ 3.54 3.45 3.45 3.54 -8.12 CBS limit extrapolation: TQa -7.68 aExtrapolation made using cc-pVTZ and cc-pVQC

Table 6.4 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV3 dimer computed at SOS-MP2 level using C1 symmetry 1-1’ 2-2’ 3-3’ 4-4’ 5-5’ 6-6’ ΔE cc-pVTZ 3.35 3.43 3.39 3.39 3.34 3.53 -15.10 cc-pVQZa ------14.04 CBS limit extrapolation: TQb -13.28 aSingle-point energy at SOS-MP2/cc-pVTZ optimized geometry, bExtrapolation made using cc-pVTZ and cc-pVQC

The stability of these PPV dimer systems was described by calculating the interaction energy with comparison to the extrapolated CBS limit. For the stacked PPV2 dimer (Table 6.1), the extrapolation of the CBS limit interaction energy is found to be less stabilizing than that found with the cc-pVDZ basis. The interaction energy given for the aug-cc-pVDZ is greatly overstabilized compared to the CBS limit energy, a larger difference than is found for any other case. The cc-pVQZ interaction energy is found very close to the CBS limit. The interaction energy at the CBS limit using the SOS-MP2/BSSE method was less stabilizing than that found for the SOS-MP2 method, but was more stabilizing than the interaction energies found for the individual basis sets using counterpoise corrections; using the cc-pVDZ basis was much less stabilizing than the CBS limit by about 3.3 kcal/mol, though the interaction energy at the cc-pVQZ basis was only

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slightly less stable. The interaction energy of the displaced PPV2 dimer is more stabilizing compared to the sandwich stacked system, reflective of the smaller intermolecular distances, and the energy at the CBS limit compares most closely to that found using the cc-pVQZ basis. This stabilization of π-conjugated aromatic systems coming from displacement has been rigorously shown in previous investigations of benzene and dimers.60-65

Without the SOS approach, the MP2 method, Table D.1, the interaction energy was found to be more stabilizing for all basis sets by about 6 kcal/mol without BSSE. This showed the necessity of the SOS approach to correct the over-estimation of the stabilization energy of dimer complexes when using the MP2 method. A similar effect was demonstrated for stacked complexes of benzene and naphthalene with tetracyanoethylene (TCNE) (benzene/TCNE and naphthalene/TCNE, respectively)29 where the MP2 method alone considerably overestimated the interaction energy compared to the SOS-MP2 approach. Calculations with the highly correlated CCSD(F12*)(T) method were also performed for the benzene/TCNE system (which was found to agree quite well with experimental findings). The SOS-ADC(2) and CCSD(F12*)(T) methods differed only slightly from each other.

The interaction energies of the sandwich-stacked PPV3 dimer (Table 6.2) using the SOS-MP2 method at the CBS limit was less stable than when using the cc-pVQZ and cc- pVTZ basis sets but was more stable than for the cc-pVDZ basis. CP corrections for the interaction energy at the CBS limit became slightly less stable compared to the extrapolated value with the SOS-MP2 method. Using the CAM-B3LYP-D3/cc-pVTZ method compares well to the SOS-MP2 interaction energy at the CBS limit, being only a little more stable, though the B3LYP-D3/cc-pVTZ method showed an over-stabilization of about 2 kcal/mol.

The interaction energy at the CBS limit of the displaced PPV3 dimer (Table 6.4) was about

4.7 kcal/mol more stable than for the sandwich-stacked PPV3 dimer and is less stable compared to the cc-pVQZ basis set.

The CBS limit of the interaction energy found for the stacked PPV4 dimer (Table

D.2) was more stable than the cc-pVQZ energy, and the displaced PPV4 dimer (Table D.3)

123 Texas Tech University, Reed Nieman, May 2019 is stabilized by about 6 kcal/mol at the CBS limit compared to the stacked dimer. The CBS limit for the displaced dimer is less stable than the cc-pVQZ basis.

Several general trends are apparent across all investigated structures. With increasing basis set size (not including augmented basis sets or BSSE corrections), the interchain stacking distances increase slightly and interaction energies become more stabilizing. Decreases in stacking distances and increasingly stable interaction energies are found with larger chain lengths. The displaced geometries have much smaller stacking distances compared to the sandwich stacked dimers, and are greatly stabilized as a result. The BSSE counterpoise correction largely overestimates the interchain stacking distances and greatly decreases the stabilization energies. The augmented basis sets in general displayed much smaller interchain stacking distances and greatly overestimated interaction energies. In light of this, augmented basis sets and counterpoise BSSE corrections are not recommended for intermolecular geometries.

6.4.2 Excited States

The excited states of the stacked PPV dimers were investigated using the SOS-

ADC(2) method and compared to TDDFT for stacked PPV3. The vertical excitations for stacked PPV2 using the SOS-ADC(2)/cc-pVQZ method are presented in Table 6.5. Two 1 states per irreducible representation were calculated. The S1 state (1 Bg) is dark, possesses partial interchain CT character, and is found at 4.07 eV. Two intensive transitions, or bright 1 1 states, were found, the S4 and S6 states (1 Bu and 2 Bu, respectively) where the latter has greater oscillator strength. Both were found to be local exciton states with negligible interchain charge transfer. In the spectrum of the monomer (Table D.4), the bright states,

S1 and S3, occur at approximately the same energy as the dimer, though the oscillator strength of the S2 state is reduced by about 0.8.

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Table 6.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVQZ and SOS- MP2/cc-pVQZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer)

State Excitation ΔE (eV) f CT (e) 1 S1 1 Bg 4.07 0.00 0.17 1 S2 1 Au 4.49 0.00 0.09 1 S3 2 Bg 4.50 0.00 0.08 1 S4 1 Bu 4.52 0.83 0.01 1 S5 1 Ag 4.64 0.00 0.02 1 S6 2 Bu 4.75 1.18 0.01 1 S7 2 Au 5.59 0.00 0.21 1 S8 2 Ag 5.96 0.00 0.02

Omega matrices revealed more detailed information about the nature of each excitation and was examined for stacked PPV2 dimer using the SOS-MP2/cc-pVQZ method (Figure 6.2). The small amount of off-diagonal character seen in the dark S1 and

S7 states correlates with calculated CT values (Table 6.5). Distinct nodes between the individual chains (fragments 1-3 belong to one chain and fragments 4-6 belong to the other) 1 1 are visible for remaining six excitations. The bright 1 Bu and 2 Bu excitations, the S4 and

S6 states, respectively, are characterized by local excitations in the phenylene groups.

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S1 S2 S3 S4

S5 S6 S7 S8

Figure 6.2 Omega matrices for vertical excitations of stacked PPV2 dimer using the SOS- ADC(2)/cc-pVQZ method with the SOS-MP2/cc-pVQZ optimized geometry. Each vinylene and phenylene group is a separate fragment: fragments 1-3 belong to one chain, fragments 4-6 to the other. The y-axis represents the electron and the x-axis represents the hole. Omega matrix plots calculated using the TheoDORE program package.

As with the structural investigation, the basis set dependence of stacked PPV2 was studied comparing the cc-pVQZ basis (Table 6.5) to the cc-pVDZ and cc-pVTZ basis sets

(Table D.5). The relative ordering of the states remains constant. The S1 state is stabilized with increasing basis set size. For each basis, the oscillator strengths of the bright states (S4 and S6) were found to be sensitive to the basis set. Compared to the monomers, the first bright state was found to be well preserved, though the oscillator strength of the second bright state is greatly reduced as was seen with the cc-pVQZ case. An exception to this trend was the cc-pVDZ basis where the second bright state is stabilized by about 0.07 eV and the oscillator strengths are increased for the two bright states.

The NTOs for the vertical excitations were plotted for stacked PPV2 (Table D.6).

Each excitation had contributions from at least two primary transitions. For the S1 state, the primary contributing hole NTO is located on the separated phenylene and vinylene groups while the electron NTO is somewhat mixed. This same intrachain fragment interaction is present in several other states and explains some of the off-diagonal character seen in the omega matrices (Figure 6.2). The NTOs of the partial CT state, S7, displays,

126 Texas Tech University, Reed Nieman, May 2019 while not much interchain interaction, transfer of electron density between intrachain fragments.

The spectrum of displaced PPV2 dimer was calculated (Table 6.6) to compare to the stacked dimer configuration showing non-negligible destabilization of the first several excitations and the introduction of interchain CT states. Displacement from the stacked conformation raises the energy of the S1 state compared to the stacked, though it remains a dark, local exciton. The first bright state is now the S2 state and was destabilized in the displaced dimer by about 0.08 eV. The S2 and S3 states in the stacked dimers were both destabilized to above the bright state. Two pronounced intermolecular CT states, S7 and

S8, were found for the displaced dimer at approximately 5.4 eV. In the associated omega matrices (Figure D.1), the CT states can be seen to have primarily involved the phenylene units. This can be seen in the NTOs (Table D.7) for the S7 state, though the NTOs for the

S8 state are more complex. The S4 and S5 states showed unique intrachain CT character compared to the sandwich-stacked dimer involving only electron/hole density on one phenylene group per chain. These two excitations both have NTO participation ratios

(PRNTO) of about four in which the intrachain exciton character was represented well.

Table 6.6 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) State Excitation ΔE (eV) f CT (e) 1 S1 1 A 4.40 0.00 0.03 1 S2 2 A 4.58 0.53 0.05 1 S3 3 A 4.64 0.01 0.05 1 S4 4 A 4.64 0.00 0.09 1 S5 5 A 4.66 0.00 0.09 1 S6 6 A 4.77 1.43 0.04 1 S7 7 A 5.39 0.00 0.95 1 S8 8 A 5.40 0.08 0.94

The spectra of stacked PPV3 (Table D.8) and PPV4 (Table D.9) show that the S1 states occur at lower energies compared to stacked PPV2 (Table 6.5), and are also lower in energy compared to the experimental values by 0.22 eV 0.27 eV, respectivley. The bright

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states, S2 for both PPV3 and PPV4, are likewise stabilized considerably going to longer chain lengths. In the omega matrices (Figure 6.3) of the vertical excitations of stacked

PPV3, five of the eight calculated states are similar in character to the excitations found in stacked PPV2: transitions beginning and ending on the capping phenylene groups. The remaining three states, S1, S2, and S4, are transitions primarily involving the interior phenlylene groups. The excitation with the largest interchain CT character (Table D.8, theS1 state) is seen to mostly involve the interior phenylene groups on each chain. Other partial interchain CT states, S5 and S6, appear to have very similar transitions: mainly phenylene-phenelyene with some phenylene-vinylene involvement. The S1, S2, and S8 states are shown to have the most phenylene-vinylene CT character. The omega matrices of stacked PPV4 (Figure D.2) were similar in character to those of stacked PPV3.

S1 S2 S3 S4

S5 S6 S7 S8

Figure 6.3 Omega matrices for vertical excitations of stacked PPV3 dimer using the SOS- ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each vinylene and phenylene group is a separate fragment: fragments 1-5 belong to one chain, fragments 6-10 to the other. The y-axis represents the electron and the x-axis represents the hole. Omega matrix plots calculated using the TheoDORE program package.

The density of states is shown in Figure 6.4a,b,e for stacked PPV2, stacked PPV3, and PPV4 using the SOS-ADC(2) method. The energy range containing the highest density of states was stabilized from the 4.5-5.0 eV range in stacked PPV2 (four states) to the 4.0-

4.5 eV range in stacked PPV3 and PPV4 (three and four states respectively) coinciding with the stabilization of the bright states. While there are two states in the 5.5-6.0 eV range, this too shifted to lower energies, as no calculated state was found higher in energy than 5.5 eV for stacked PPV3 and 5.0 eV in stacked PPV4.

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Examining the spectrum of the displaced PPV3 dimer (Table 6.7), the same destabilization compared to the sandwich stacked dimer is found that was seen for the PPV2 dimers. The S1 state was destabilized more than most of the calculated states. There was a third state (S5) with significant oscillator strength appearing about 0.07 eV higher than the second bright state that displayed partial interchain CT character (~0.5 e). The omega matrices for this excitation (Figure D.3) showed off-diagonal elements corresponding to transitions primarily from/to the interior phenylene groups of both chains, though also involving the vinylene and capping phenylene groups. The NTOs (Table D.10) for the two primary contributing transitions showed symmetric distribution of hole density shifting to one side of both dimers. There were four other excitations in addition to the S5 state with

CT character greater than 0.1 e. Of these the S6 and S7 states had significant off-diagonal elements in the omega matrices with transitions from a variety of fragments.

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a) b)

c) d)

e)

Figure 6.4 Calculated line spectra (right y-axis) and density of states (left y-axis) for the stacked PPV oligomers optimized using the SOS-ADC(2), B3LYP-D3, and CAM-B3LYP- D3 methods. Stacked a) PPV2 (Table 6.5), b-d) PPV3 (Table D.8, S11-S12), e) PPV4 (Table D.9).

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Table 6.7 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) State Excitation ΔE (eV) f CT (e) 1 S1 1 A 3.77 0.00 0.04 1 S2 2 A 4.02 3.47 0.06 1 S3 3 A 4.34 0.00 0.17 1 S4 4 A 4.38 0.32 0.07 1 S5 5 A 4.55 0.11 0.52 1 S6 6 A 4.58 0.00 0.39 1 S7 7 A 4.59 0.02 0.30 1 S8 8 A 4.61 0.00 0.13

TDDFT methods were compared to the SOS-ADC(2) using the cc-pVTZ basis for stacked PPV3. The relative ordering of the first eight excited states is mostly preserved with the B3LYP-D3 method (Table D.8). The S1 state is stabilized using both TDDFT methods compared to the previous method, though much more so with B3LYP-D3. The first bright state (Table D.8, S2) was found to be stabilized using the B3LYP-D3 (Table D.11, S3) and

CAM-B3LYP-D3 (Table D.12, S2) methods (about 0.7 and 0.3 eV respectively). The oscillator strength of this transition is also reduced for B3LYP-D3 and CAM-B3LYP-D3. Though only this one bright state was found when utilizing the B3LYP-D3 method, a third bright state was found with the CAM-B3LYP-D3 method, located about 0.2 eV higher than the second and is a partial CT state. The density of states (Figure 6.4c,d) using both TDDFT methods is much larger than with SOS-ADC(2) approach. The majority of transitions occur in the energy range of 4.0-4.5 eV (twelve states), similar to what was discussed before. This peak in states is destabilized to the range of 5.0-5.5 eV (eight states) using the CAM- B3LYP-D3 method.

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(1 (1

1

π* π* S

-

symmetry

2h

pVTZ basis and C pVTZand basis

-

Interchain distances (Å) Interchain in the stacked PPV dimers optimized for π

8

) excited state using the cc using state excited )

g

B

Table Table 6. 1

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The B3LYP-D3 and CAM-B3LYP-D3 methods presented more varied CT character compared to the SOS-ADC(2) method. Though only one bright state was found using the B3LYP-D3 method in the thirty states calculated (S3), it is located energetically higher than one prominent interchain CT state (S2) and one partial interchain CT state (S1).

The first prominent CT state using the CAM-B3LYP-D3 method (S3) was located about 0.3 eV higher in energy than the bright state and had oscillator strength greater than zero.

1 The geometry of the S1 state (1 Bg) was optimized (Table 6.8) and compared to the ground state. In all cases a decrease in intermolecular stacking distances was observed and was greatest for stacked PPV2 and least for stacked PPV4. The largest decrease in stacking distances occurred for the vinylene units (about 0.9 Å for stacked PPV2) and generally becomes greater toward the interior of the stacked dimers. Comparing the SOS-ADC(2) method to TDDFT in stacked PPV3, the CAM-B3LYP-D3 method deviated slightly less on average than the B3LYP-D3 method which deviated quite strongly (over 0.1 Å) in the interior phenylene and vinylene groups.

6.5 Conclusions

Ab initio calculations were performed on stacked PPV oligomers of from two to four repeating PV units with the purpose of providing necessary benchmark calculations. Geometry optimizations showed that the intermolecular stacking distances became generally shorter with increasing basis set size. In stacked PPV2, the structure obtained with the cc-pVTZ basis closely matched those found with the cc-pVQZ and aug-cc-pVQZ basis sets and the CP method to correct reproduced the cc-pVQZ structure well, though the cc-pVDZ and aug-cc-pVDZ basis sets were found to, respectively, overestimate and underestimate intermolecular distances. Increasing system size increased the intermolecular stacking distances going to PPV3 and PPV4. Comparisons with DFT methods showed that the B3LYP-D3 method more closely reproduced the structure found with the SOS-MP2 method. Displacement from the exact sandwich structure was found to play a large role as the displaced PPV dimers had much smaller stacking distances than the sandwich-stacked dimers. 133 Texas Tech University, Reed Nieman, May 2019

The calculated interaction energies were compared to the extrapolated CBS limit.

The interactions energy found for stacked PPV2 using the aug-cc-pVDZ basis set strongly overestimated the stabilization, though the interaction energy found with the cc-pVQZ basis was close to the CBS limit. CP corrections decreased the magnitude of the interaction energy, though the calculated CBS limit for SOS-MP2/BSSE method was still relatively close to stabilization energy found for the SOS-MP2 method. The CAM-B3LYP-D3 method gave stabilization energies slightly more stabilizing than the SOS-MP2 mothod, but the B3LYP-D3 method differed more strongly.

Vertical excitation calculation using the SOS-ADC(2) method found two intensive transitions in the eight calculated excited states for each case. The oscillator strength of the first bright state increased with going from PPV2 to PPV3 and PPV4, but the oscillator strength of the second decreased. Though interchain CT character is more or less constant with increasing chain size, intrachain CT increases with a larger number of contributing phenylene and vinylene fragments as analyzed using omega matrices and NTO transitions. Comparative DFT methods displayed a much more diverse range of intermolecular CT states than was found for the SOS-ADC(2) method.

6.6 Bibliography

(1) Brédas, J.-L.; Cornil, J.; Beljonne, D.; dos Santos, D. A.; Shuai, Z., Excited-State Electronic Structure of Conjugated Oligomers and Polymers: A Quantum- Chemical Approach to Optical Phenomena. Accounts of Chemical Research 1999, 32 (3), 267-276. (2) Meier, H.; Stalmach, U.; Kolshorn, H., Effective conjugation length and UV/vis spectra of oligomers. Acta Polymerica 1997, 48 (9), 379-384. (3) Natasha, K., Understanding excitons in optically active polymers. Polymer International 2008, 57 (5), 678-688. (4) Peeters, E.; Ramos, A. M.; Meskers, S. C. J.; Janssen, R. A. J., Singlet and triplet excitations of chiral dialkoxy-p-phenylene vinylene oligomers. The Journal of Chemical Physics 2000, 112 (21), 9445-9454. (5) Collini, E.; Scholes, G. D., Coherent Intrachain Energy Migration in a Conjugated Polymer at Room Temperature. Science 2009, 323 (5912), 369. (6) Facchetti, A., π-Conjugated Polymers for Organic Electronics and Photovoltaic Cell Applications. Chemistry of Materials 2011, 23 (3), 733-758.

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(7) Brédas, J.-L.; Silbey, R., Excitons Surf Along Conjugated Polymer Chains. Science 2009, 323 (5912), 348. (8) Sterpone, F.; Rossky, P. J., Molecular Modeling and Simulation of Conjugated Polymer Oligomers: Ground and Excited State Chain Dynamics of PPV in the Gas Phase. The Journal of Physical Chemistry B 2008, 112 (16), 4983-4993. (9) Sun, Z.; Li, S.; Xie, S.; An, Z., A study on excited-state properties of conjugated polymers using the Pariser-Parr-Pople-Peierls model combined with configuration- interaction-singles. Organic Electronics 2018, 57, 277-284. (10) Tretiak, S.; Saxena, A.; Martin, R. L.; Bishop, A. R., Conformational Dynamics of Photoexcited Conjugated Molecules. Physical Review Letters 2002, 89 (9), 097402. (11) Karabunarliev, S.; Bittner, E. R., Polaron–excitons and electron–vibrational band shapes in conjugated polymers. The Journal of Chemical Physics 2003, 118 (9), 4291-4296. (12) Pogantsch, A.; Heimel, G.; Zojer, E., Quantitative prediction of optical excitations in conjugated organic oligomers: A density functional theory study. The Journal of Chemical Physics 2002, 117 (12), 5921-5928. (13) Nayyar, I. H.; Batista, E. R.; Tretiak, S.; Saxena, A.; Smith, D. L.; Martin, R. L., Localization of Electronic Excitations in Conjugated Polymers Studied by DFT. The Journal of Physical Chemistry Letters 2011, 2 (6), 566-571. (14) Nayyar, I. H.; Batista, E. R.; Tretiak, S.; Saxena, A.; Smith, D. L.; Martin, R. L., Role of Geometric Distortion and Polarization in Localizing Electronic Excitations in Conjugated Polymers. Journal of Chemical Theory and Computation 2013, 9 (2), 1144-1154. (15) Sterpone, F.; Martinazzo, R.; Panda Aditya, N.; Burghardt, I., Coherent Excitation Transfer Driven by Torsional Dynamics: a Model Hamiltonian for PPV Type Systems. In Zeitschrift für Physikalische Chemie, 2011; Vol. 225, p 541. (16) Singh, J.; Bittner, E. R.; Beljonne, D.; Scholes, G. D., Fluorescence depolarization in poly[2-methoxy-5-((2-ethylhexyl)oxy)-1,4-phenylenevinylene]: Sites versus eigenstates hopping. The Journal of Chemical Physics 2009, 131 (19), 194905. (17) Molina, V.; Merchán, M.; Roos, B. O., Theoretical Study of the Electronic Spectrum of trans-Stilbene. The Journal of Physical Chemistry A 1997, 101 (19), 3478-3487. (18) Saha, B.; Ehara, M.; Nakatsuji, H., Investigation of the Electronic Spectra and Excited-State Geometries of Poly(para-phenylene vinylene) (PPV) and Poly(para- phenylene) (PP) by the Symmetry-Adapted Cluster Configuration Interaction (SAC-CI) Method. The Journal of Physical Chemistry A 2007, 111 (25), 5473- 5481. (19) Binder, R.; Wahl, J.; Romer, S.; Burghardt, I., Coherent exciton transport driven by torsional dynamics: a quantum dynamical study of phenylene-vinylene type conjugated systems. Faraday Discussions 2013, 163 (0), 205-222. (20) Lukeš, V.; Aquino, A.; Lischka, H., Theoretical Study of Vibrational and Optical Spectra of Methylene-Bridged Oligofluorenes. The Journal of Physical Chemistry A 2005, 109 (45), 10232-10238.

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(21) Lukeš, V.; Aquino, A. J. A.; Lischka, H.; Kauffmann, H.-F., Dependence of Optical Properties of Oligo-para-phenylenes on Torsional Modes and Chain Length. The Journal of Physical Chemistry B 2007, 111 (28), 7954-7962. (22) Köhn, A.; Hättig, C., Analytic gradients for excited states in the coupled-cluster model CC2 employing the resolution-of-the-identity approximation. The Journal of Chemical Physics 2003, 119 (10), 5021-5036. (23) Hättig, C., Structure Optimizations for Excited States with Correlated Second- Order Methods: CC2 and ADC(2). In Advances in Quantum Chemistry, Jensen, H. J. Å., Ed. Academic Press: 2005; Vol. 50, pp 37-60. (24) Aquino, A., J. A.; Nachtigallova, D.; Hobza, P.; Truhlar Donald, G.; Hättig, C.; Lischka, H., The charge‐transfer states in a stacked nucleobase dimer complex: A benchmark study. Journal of Computational Chemistry 2010, 32 (7), 1217-1227. (25) Szalay, P. G.; Watson, T.; Perera, A.; Lotrich, V. F.; Bartlett, R. J., Benchmark Studies on the Building Blocks of DNA. 1. Superiority of Coupled Cluster Methods in Describing the Excited States of Nucleobases in the Franck–Condon Region. The Journal of Physical Chemistry A 2012, 116 (25), 6702-6710. (26) Panda, A. N.; Plasser, F.; Aquino, A. J. A.; Burghardt, I.; Lischka, H., Electronically Excited States in Poly(p-phenylenevinylene): Vertical Excitations and Torsional Potentials from High-Level Ab Initio Calculations. The Journal of Physical Chemistry A 2013, 117 (10), 2181-2189. (27) Cardozo, T. M.; Aquino, A. J. A.; Barbatti, M.; Borges, I.; Lischka, H., Absorption and Fluorescence Spectra of Poly(p-phenylenevinylene) (PPV) Oligomers: An ab Initio Simulation. The Journal of Physical Chemistry A 2015, 119 (9), 1787-1795. (28) Mewes, S. A.; Mewes, J.-M.; Dreuw, A.; Plasser, F., Excitons in poly(para phenylene vinylene): a quantum-chemical perspective based on high-level ab initio calculations. Physical Chemistry Chemical Physics 2016, 18 (4), 2548-2563. (29) Aquino, A. A. J.; Borges, I.; Nieman, R.; Kohn, A.; Lischka, H., Intermolecular interactions and charge transfer transitions in aromatic hydrocarbon- tetracyanoethylene complexes. Physical Chemistry Chemical Physics 2014, 16 (38), 20586-20597. (30) Tretiak, S.; Saxena, A.; Martin, R. L.; Bishop, A. R., Interchain Electronic Excitations in Poly(phenylenevinylene) (PPV) Aggregates. The Journal of Physical Chemistry B 2000, 104 (30), 7029-7037. (31) Cornil, J.; dos Santos, D. A.; Crispin, X.; Silbey, R.; Brédas, J. L., Influence of Interchain Interactions on the Absorption and Luminescence of Conjugated Oligomers and Polymers: A Quantum-Chemical Characterization. Journal of the American Chemical Society 1998, 120 (6), 1289-1299. (32) Cornil, J.; Beljonne, D.; Calbert, J. P.; Brédas, J. L., Interchain Interactions in Organic π-Conjugated Materials: Impact on Electronic Structure, Optical Response, and Charge Transport. Advanced Materials 2001, 13 (14), 1053-1067. (33) Bedard-Hearn, M. J.; Sterpone, F.; Rossky, P. J., Nonadiabatic Simulations of Exciton Dissociation in Poly-p-phenylenevinylene Oligomers. The Journal of Physical Chemistry A 2010, 114 (29), 7661-7670.

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(34) Meng, R.; Li, Y.; Li, C.; Gao, K.; Yin, S.; Wang, L., Exciton transport in [small pi]-conjugated polymers with conjugation defects. Physical Chemistry Chemical Physics 2017, 19 (36), 24971-24978. (35) Nayyar, I. H.; Masunov, A. E.; Tretiak, S., Comparison of TD-DFT Methods for the Calculation of Two-Photon Absorption Spectra of Oligophenylvinylenes. The Journal of Physical Chemistry C 2013, 117 (35), 18170-18189. (36) Yan, M.; Rothberg, L. J.; Kwock, E. W.; Miller, T. M., Interchain Excitations in Conjugated Polymers. Physical Review Letters 1995, 75 (10), 1992-1995. (37) Nguyen, T.-Q.; Kwong, R. C.; Thompson, M. E.; Schwartz, B. J., Improving the performance of conjugated polymer-based devices by control of interchain interactions and polymer film morphology. Applied Physics Letters 2000, 76 (17), 2454-2456. (38) Nguyen, T.-Q.; Martini, I. B.; Liu, J.; Schwartz, B. J., Controlling Interchain Interactions in Conjugated Polymers: The Effects of Chain Morphology on Exciton−Exciton Annihilation and Aggregation in MEH−PPV Films. The Journal of Physical Chemistry B 2000, 104 (2), 237-255. (39) Grimme, S., Improved second-order Møller–Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies. The Journal of Chemical Physics 2003, 118 (20), 9095-9102. (40) Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M., Scaled opposite-spin second order Møller–Plesset correlation energy: An economical electronic structure method. The Journal of Chemical Physics 2004, 121 (20), 9793-9802. (41) Møller, C.; Plesset, M. S., Note on an Approximation Treatment for Many-Electron Systems. Physical Review 1934, 46 (7), 618-622. (42) Winter, N. O. C.; Hättig, C., Scaled opposite-spin CC2 for ground and excited states with fourth order scaling computational costs. The Journal of Chemical Physics 2011, 134 (18), 184101. (43) Hellweg, A.; Grun, S. A.; Hattig, C., Benchmarking the performance of spin- component scaled CC2 in ground and electronically excited states. Physical Chemistry Chemical Physics 2008, 10 (28), 4119-4127. (44) Hättig, C., Geometry optimizations with the coupled-cluster model CC2 using the resolution-of-the-identity approximation. The Journal of Chemical Physics 2003, 118 (17), 7751-7761. (45) Hättig, C.; Weigend, F., CC2 excitation energy calculations on large molecules using the resolution of the identity approximation. Journal of Chemical Physics 2000, 113, 5154-5161. (46) Dunning, T. H., Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. The Journal of Chemical Physics 1989, 90 (2), 1007-1023. (47) Weigend, F.; Köhn, A.; Hättig, C., Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations. The Journal of Chemical Physics 2002, 116 (8), 3175-3183. (48) Becke, A. D., Density‐functional thermochemistry. III. The role of exact exchange. The Journal of Chemical Physics 1993, 98 (7), 5648-5652.

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(49) Yanai, T.; Tew, D. P.; Handy, N. C., A new hybrid exchange–correlation functional using the Coulomb-attenuating method (CAM-B3LYP). Chemical Physics Letters 2004, 393 (1), 51-57. (50) Limacher, P. A.; Mikkelsen, K. V.; Lüthi, H. P., On the accurate calculation of polarizabilities and second hyperpolarizabilities of polyacetylene oligomer chains using the CAM-B3LYP density functional. Journal of Chemical Physics 2009, 130, 194114. (51) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H., A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. The Journal of Chemical Physics 2010, 132 (15), 154104. (52) Boys, S. F.; Bernardi, F., The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Molecular Physics 1970, 19 (4), 553-566. (53) Halkier, A.; Helgaker, T.; Jørgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K., Basis-set convergence in correlated calculations on Ne, N2, and H2O. Chemical Physics Letters 1998, 286 (3), 243-252. (54) Martin, R. L., Natural transition orbitals. The Journal of Chemical Physics 2003, 118 (11), 4775-4777. (55) Plasser, F.; Lischka, H., Analysis of Excitonic and Charge Transfer Interactions from Quantum Chemical Calculations. Journal of Chemical Theory and Computation 2012, 8 (8), 2777-2789. (56) Ahlrichs, R.; Bär, M.; Häser, M.; Horn, H.; Kölmel, C., Electronic structure calculations on workstation computers: The program system turbomole. Chemical Physics Letters 1989, 162 (3), 165-169. (57) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery Jr., J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J.; Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, Gaussian, Inc.: Wallingford, CT, USA, 2009. (58) Plasser, F.; Wormit, M.; Dreuw, A., New tools for the systematic analysis and visualization of electronic excitations. I. Formalism. The Journal of Chemical Physics 2014, 141 (2), 024106. (59) Plasser, F. TheoDORE: a package for theoretical density, orbital relaxation, and exciton analysis, available from http://theodore-qc.sourceforge.net.

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(60) Lee, N. K.; Park, S.; Kim, S. K., Ab initio studies on the van der Waals complexes of polycyclic aromatic hydrocarbons. II. Naphthalene dimer and naphthalene– anthracene complex. The Journal of Chemical Physics 2002, 116 (18), 7910-7917. (61) Tsuzuki, S.; Honda, K.; Uchimaru, T.; Mikami, M., High-level ab initio computations of structures and interaction energies of naphthalene dimers: Origin of attraction and its directionality. The Journal of Chemical Physics 2003, 120 (2), 647-659. (62) Sinnokrot, M. O.; Sherrill, C. D., Highly Accurate Coupled Cluster Potential Energy Curves for the Benzene Dimer: Sandwich, T-Shaped, and Parallel- Displaced Configurations. The Journal of Physical Chemistry A 2004, 108 (46), 10200-10207. (63) Sato, T.; Tsuneda, T.; Hirao, K., A density-functional study on π-aromatic interaction: Benzene dimer and naphthalene dimer. The Journal of Chemical Physics 2005, 123 (10), 104307. (64) Grimme, S., Do Special Noncovalent π–π Stacking Interactions Really Exist? Angewandte Chemie International Edition 2008, 47 (18), 3430-3434. (65) Pitoňák, M.; Neogrády, P.; R̆ ezáč, J.; Jurečka, P.; Urban, M.; Hobza, P., Benzene Dimer: High-Level Wave Function and Density Functional Theory Calculations. Journal of Chemical Theory and Computation 2008, 4 (11), 1829-1834.

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CHAPTER VII

THE BIRADICALOID CHARACTER IN RELATION TO SINGLET/TRIPLET SPLITTINGS FOR CIS-/TRANS- DIINDENOACENES AND TRANS- INDENOINDENODI(ACENOTHIOPHENES) BASED ON EXTENDED MULTIREFERENCE CALCULATIONS Authors: Reed Nieman,a Nadeesha J. Silva,a Adelia J. A. Aquino,a,b and Hans Lischkaa,b

Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University, Lubbock, TX 79409-1061, USA, bSchool of Pharmaceutical Sciences and Technology, Tianjin University, Tianjin 300072, PR China 7.1 Abstract

The assessment of the biradical character of select trans-diindenoacenes, cis- diindenoacenes, and trans-indenoindenodi(acenothiophenes) was accomplished utilizing the multireference averaged quadratic coupled-cluster (MR-AQCC) method. These systems have been previously identified as highly reactive in experiments, often subject to unplanned side-reactions. The biradical character of these species was characterized using the singlet/triplet splitting energy (∆ES-T), effective number of unpaired electrons (NU), and unpaired electron density. Additionally, the aromaticity has been described by the harmonic oscillator measure of aromaticity (HOMA). For all structures, low spin singlet ground states were found, in agreement with previous investigations. In addition, increasing the number of core six-membered rings decreased the ∆ES-T, and increased the

NU and total unpaired electron density of the singlet state. The apical carbon of the five- membered ring has the largest amount of unpaired electron density in each investigated structure. The cis-diindenoacenes (5-8) displayed a greatly increased value of NU and unpaired electron density corresponding to much smaller ∆ES-T values compared to the other structures. The thiophene rings in the trans-indenoindenodi(acenothiophenes) structures (9-12) were found to simultaneously limit the unpaired electron density in the terminal six-membered rings and stabilize the density found in the core rings. Finally excellent agreement to experimental values for the ∆ES-T has been demonstrated for structures 5 and 10.

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7.2 Introduction

The study of the properties and reactivity of biradical polycyclic aromatic hydrocarbons (PAHs) is important to a wide variety of scientific fields due to their unique electronic nature. Since the discovery of open-shell PAHs systems,1 they have provided invaluable insight into how the radicals influence the π-bonding network to generate properties such as magnetism. Diradical PAH systems often display a range of properties such as small energy gaps between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO),2 electron spin resonance and peak broadening in proton NMR spectra,2 low-lying doubly-excited electron configurations allowing low-energy electronic absorptions,3 and amphoteric redox character and multicenter bonds.4 These expansive properties have led biradical PAHs to be prime candidates for use in molecular electronic and spintronic devices,5 battery technology,6 nonlinear optical devices,7-9 and singlet fission materials.10-12 Several classes of radical and biradical PAHs have been studied including zethrenes,13-17 anthenes,18-19 phenalenyl,5 bisphenalenyls,20-22 triangulene23-25 and extended triangulene,5 quinodimethanes,26-28 indofluorenes,29-35 high-order acenes,36-40 and graphene nanoribbons.40-44 The biradical character in these cases is derived from open-shell and closed-shell resonance structures,45- 47 the interconversion between which cleaves the exocyclic π-bonds increasing the number of Clar sextets and energetically driving the stability of the open-shell state.15, 48 Double spin-polarization effects give biradical PAHs with singlet ground states low-lying triplet excited states that are more easily populated.49

The unique characteristics of idenofluorenes (representative structures shown in Figure 7.1) have been targeted for investigation and utilization. The five-member carbon rings in indenofluorenes have shown to make the structures reversible electron donors.50 Diindenobenzene dervatives show ambipolar charge transport properties in single-crystal and thin-film organic field effect transitors (OFETs).30, 51 Extensive work on trans- diindenoanthracene found that it was chemically robust due to only modest biradical character and moderate singlet/triplet splitting energy, ∆ES-T, of 4.2 kcal/mol,4 and this

141 Texas Tech University, Reed Nieman, May 2019 small energy gap showed that the triplet state was thermally accessible and could be populated by an entropy-driven singlet-triplet fission process.52 Only the outer six- membered rings of trans-diindenonaphthalene (fluoreno[3,2-b]fluorene) were shown to be aromatic, while the core rings displayed anti-aromatic character.35 Experimental results for cis-diindenobenzene (fluoreno[2,1-b]fluorene) show it has a singlet ground state, modest 31 ∆ES-T of 4.2 kcal/mol, and a small HOMO-LUMO band gap (1.26 eV). The fusion of thiophene rings into the indenofluorene frame yield interesting experimental results. Indacenedithiophenes have smaller energy level gaps than pure indenofluorenes as they have increased paratropicity in their indacene core.32 It has been shown that the inclusion of fused benzothia groups in indenofluorenes leads to pro-aromatic properties that show larger diradical character and longer lifetimes of singlet excited states.53 Indenoindenodibenzothiophene molecules were studied experimentally and theoretically and were found to have relatively large ∆ES-T (8.0 kcal/mol) together with large diradial character.54

Figure 7.1 Quinoid Kekulé and biradical resonance forms of trans-diindenoanthracene, cis- diindenoanthracene, and trans-indenoindenodi(anthracenothiophene). Quantum chemical calculations are of great importance to the determination of molecular properties. While the computationally efficient density functional theory (DFT) is a natural first choice, the unrestricted DFT (UDFT) method suffers heavily from spin- contamination effects especially for broken-symmetry singlet state biradical cases.37, 55 To remedy this, PAH systems with singlet biradical cases have been investigated with a wide 142 Texas Tech University, Reed Nieman, May 2019 variety of methods such as spin-flip (SF) approaches,56-57 projected Hartree-Fock,58 active space variational two-electron reduced density matrix (2RDM),59-60 density matrix renormalization group (DMRG),61-62 and CCSD(T).63-64 The multireference averaged quadratic coupled-cluster (MR-AQCC) method65 is able to measure reliably the biradicaloid character of molecules as it includes quasi-degenerate configurations in the reference wave function and dynamic electron correlation including size-extensivity contributions by explicitly considering single and double excitations.65-66 Previous reports have had success using the MR-AQCC method to describe n-acenes,40, 67-68 periacenes, zethrenes,69-70 among other challenging biradical cases.71-72

To this end, the MR-AQCC method has been employed in this work to investigate the biradical character of several extended indenofluorene and indenoindenodi(acenothiophene) structures (Figure 7.1 and Figure E.1) by describing their singlet ground state and first vertical and adiabatic triplet excited state. From these calculations, relatively simple descriptors such as the singlet/triplet splitting energy, ∆ES-

T, effective number of unpaired electrons (NU), and density of unpaired electrons will be used to characterize singlet biradical nature of twelve PAH molecules. The harmonic oscillator measure of aromaticity (HOMA) index will also be calculated at the DFT level in order to assess the aromaticity of each structure.

7.3 Computational Details

The biradicaloid character of the structures shown in Figure 7.1 and Figure E.1 of the Supporting Information were investigated using the MR-AQCC65 method by relating the ∆ES-T values to NU data. To begin, the low and high spin ground states of all structures were optimized using the range-corrected ωB97XD73 density functional and the def2- TZVP74-75 basis set. Systems 1-4 (trans-diindenoacenes) and 9-12 (trans- indenoindenodi(acenothiophenes)) belong to the C2h point group while 5-8 (cis- diindenoacenes) belong to the C2v point group. The correct low spin ground state electron configuration was determined by way of wave function instability analysis76 of the Kohn- Sham determinants.77 All structures, except 1, were found to have triplet instabilities 143 Texas Tech University, Reed Nieman, May 2019 present in their wave functions at restricted DFT (RDFT) level, which were followed to reoptimize the geometry using an unrestricted DFT (UDFT) approach. The harmonic vibrational frequencies for all structures were calculated and found to be non-imaginary in all cases showing that each structure is a minimum in the potential curve.

Based on the optimized geometry using the ωB97XD/def2-TZVP method, the aromaticity of each ring in each structure was determined using the harmonic oscillator measure of aromaticity (HOMA):78-79

훼 2 퐻푂푀퐴 = 1 − ∑(푅 − 푅 ) 푁 표푝푡 푖 (6.1) 푖 This method uses Kekulé benzene for the reference, and the index of aromaticity is determined based on how much the bond distances, 푅푖, differ from the optimal C-C bond length, 푅표푝푡. A normalization constant, 훼, keeps the HOMA index of zero for Kekulé benzene. The total number of bonds in a given ring is given by N. Here, 푅표푝푡 for C-C bonds is 1.388 Å, and for C-S bonds is 1.677 Å. The constant, 훼, for C-C bonds is 257.7 Å2, and for C-S bonds is 94.09 Å2.

The starting orbitals for the MR-AQCC calculations were found using the complete active space self-consistent field theory (CASSCF) method80 and the ωB97XD/def2-TZVP optimized geometries. The 6-31G*81-83 basis set was used for all CASSCF and MR-AQCC calculations. For 9-12, a split basis set was used, designated here as 6-31G*(6-311G*)S, where 6-31G* was applied for carbon and hydrogen, and the triple-zeta basis set, 6- 311G*,84-85 was utilized for sulfur. The σ orbitals (both occupied and virtual) were frozen in all CASSCF and MR-AQCC calculations at SCF level while the π orbitals were kept active. For 1-4 and 9-12, orbitals belonging to the Ag and Bu irreducible representations were σ in character, and orbitals belonging to the Au and Bg irreducible representations were π in character. For 5-8, orbitals belonging to the A1 and B1 irreducible representations were σ in character, and orbitals belonging to the A2 and B2 irreducible representations were π in character. The molecular orbitals constituting the active space were determined with the method of Pulay et al86 using the natural orbital (NO) occupation numbers from the unrestricted Hartree-Fock (UHF) wave function. For all structures, the optimized

144 Texas Tech University, Reed Nieman, May 2019 orbitals from a two-state state-averaged (SA2) complete active space consisting of four electrons in four orbitals (CAS(4,4)) where the state-averaging was performed over the lowest singlet and triplet states. These orbitals were taken for the subsequent MR-AQCC calculations with a CAS(4,4) reference space (MR-AQCC(CAS(4,4)). The states taken in 1 3 1 3 SA2 were the 1 Ag and 1 Bu states for 1-4 and 9-12, and 1 A1 and 1 B1 for 5-8. Expanded reference spaces were also considered for 1, 2, and 9 to show the effect on the ∆ES-T by utilizing in addition to the CAS a restricted active space (RAS) (doubly occupied orbitals from which one electron may be excited to the CAS or auxiliary (AUX) orbitals) and AUX space (unoccupied or virtual orbitals into which one electron may be excited). For 1 and 9, CASSCF calculations were done with RAS(3)/CAS(4,4)/AUX(3) to optimize orbitals for MR-AQCC(CAS(10,7)/AUX(3)) calculations, and for 9, CASSCF calculations were performed with RAS(4)/CAS(4,4)/AUX(4) to optimize orbitals for MR- AQCC(CAS(12,8)/AUX(4)) calculations. A further expanded reference space was considered for 1, where CASSCF calculations were done with RAS(5)/CAS(4,4)/AUX(3) to optimize orbitals for MR-AQCC(CAS(14,9)/AUX(3)) calculations. In addition, to show the effect on the ∆ES-T when keeping the σ orbitals active, calculations were carried out for 1 using the MR-AQCC(CAS(4,4)) reference space whereby only the core orbitals were frozen. For each reference space investigated, the active orbitals are given in Table E.1 of the Supporting Information. In a few cases, intruder states (configurations not found in the reference space and with ≥1% weight) were found to occur at the MR-AQCC level. This was accounted for by utilizing more active orbitals to include the intruder state as a reference configuration in the wave function.

The singlet/triplet splitting energy (∆ES-T) is calculated according to ∆ES-T = ET -

ES, such that positive values mean that the low spin or singlet state is lower in energy than the high spin or triplet state. Three different cases of the ∆ES-T are considered: the vertical splitting, ∆ES-T, vert., where the energy of the triplet state is found at the singlet state geometry, the adiabatic splitting, ∆ES-T, adiab., where the energy of the triplet state is found at the triplet state geometry, and the adiabatic splitting that includes the zero-point energy correction (ZPE), ∆ES-T, adiab.+ZPE, for which the ZPE was calculated at the ωB97XD/def2- TZVP level.

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The effective number of unpaired electrons (NU) and unpaired electron density are calculated87-88 in order to assess the radical nature of the discussed structures by means of the non-linear formula developed by Head-Gordon:89-90

푚 2 2 푁푈 = ∑ 푛푖 (2 − 푛푖) (6.2) 푖=1 In this approach, orbitals with occupations close to one are emphasized over those orbitals that are either nearly double-occupied or are unoccupied. m is the total number of NOs and ni is the occupation of the ith NO. Two schemes for NU have been considered: as a sum over all NO occupations (NU) to assess the total radical character, and as a sum only over the HONO-LUNO pair (H-L, NU(H-L)) to account only for the biradical character. For each triplet state calculated, the NU values and unpaired electron density are practically the same for the vertically excited triplet state and the adiabatic triplet state, so only the values for the vertical triplet states are reported here.

The geometry optimizations, stability analysis, and frequency calculations were carried out using the Gaussian program package (Gaussian 09 Rev.E.01).91 The UHF and natural orbitals calculations were done using the TURBOMOLE program.92 All CASSCF and MR-AQCC calculations were performed using the COLUMBUS program suite.93-95 The unpaired electron population analysis was accomplished using the TheoDORE program.96-97

7.4 Results and Discussion 7.4.1 Trans-diindenoacenes

The ∆ES-T values for 1-4 (trans-diindenoacenes) and the effective number of unpaired electrons, respectively, have been plotted in Figure 7.2 against the number of interior six-membered rings using the MR-AQCC(CAS(4,4))/6-31G* method. One finds that with increasing number of rings, that the ∆ES-T decreases. This decrease is greatest when going from 1 to 2 (the vertical ∆ES-T decreases by about 22.3 kcal/mol and the ∆ES-T decrease by 10.6 and 9.0 kcal/mol for adiabatic and adiabatic+ZPE, respectively). The

146 Texas Tech University, Reed Nieman, May 2019

decrease in ∆ES-T going from 2-4 shows to be practically linear. The effect of using the adiabatic and adiabatic+ZPE ∆ES-T is especially clear in 1, where the ∆ES-T is smaller than the vertically excited case by about 14.1 and 16.2 kcal/mol, respectively. This decrease is less pronounced in 2 (-2.5 and -2.9 kcal/mol, respectively), 3 (-0.8 and -0.9 kcal/mol, respectively), and 4 (+0.05 and +0.07 kcal/mol, respectively). For 3 and 4, the ZPE correction to the adiabatic ∆ES-T makes nearly no difference. For 4, the adiabatic and adiabatic+ZPE ∆ES-T is increased from the vertical ∆ES-T, though by less than 0.1 kcal/mol.

In addition, 4 has the smallest ∆ES-T at only 6.5 kcal/mol, meaning that it has the most energetically accessible triplet state of the investigated trans-diindenoacenes. Overall, there is good agreement to recent theoretical ∆ES-T values calculated using the spin-flip non- colinear time-dependent DFT (SF-NC-TDDFT) method for 1-3,54 though the MR-AQCC values here are slightly larger in each case. The experimental value obtained for a derivative of 3 (with substituted mesityl (2,4,6-trimethylphenyl) groups at the apical carbon of the five-membered rings and (triisopropylsilyl)ethynyl groups at the center-most six- 4 membered ring) shows a ∆ES-T of 4.18 kcal/mol, smaller than the calculated adiabatic+ZPE value of 10.5 kcal/mol reported here. The calculated ∆ES-T value for 3 reported in ref. 54 via the SF-NC-TDDFT method is about 7.1 kcal/mol, intermediate of our MR-AQCC result and the experimental value, though the ∆ES-T values computed with the complete active space configuration interaction (CASCI) method were approximately twice as large as those presented here for 1-3 using the MR-AQCC(CAS(4,4)) reference space.

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3 1 Figure 7.2 Comparison of the singlet/triplet energy (∆ES-T, E(1 Bu) - E(1 Ag)) and number 1 of unpaired electrons (NU and NU(H-L), Table E.2) of the 1 Ag state with the number of interior six-membered rings of trans-diindenoacenes (1-4) using the MR- AQCC(CAS(4,4))/6-31G* method. The standard MR-AQCC(CAS(4,4)) method is compared to expanded reference spaces for 1 and 2 in Table 7.1. First looking at 1, the vertical ∆ES-T calculated using the MR-AQCC(CAS(4,4)) method (frozen σ orbitals) compares well to those using the MR- AQCC(CAS(10,7)/AUX(3)) and MR-AQCC(CAS(14,9)/AUX(3)) methods, decreasing only by about 0.06 and 0.6 kcal/mol, respectively. Using the MR-

AQCC(CAS(10,7)/AUX(3)) method for 1, the adiabatic and adiabatic+ZPE ∆ES-T are increased by about 0.9 and 0.7 kcal/mol, respectively. This is likely a result of intruder states (configurations from outside the reference space) in the triplet state, the necessary inclusion of which as reference configurations increased the total energy and thus the ∆ES-

T. The inclusion of sigma orbitals (excluding the core orbitals that remained frozen) in the

MR-AQCC(CAS(4,4)) method for 1 decreases the adiabatic and adiabatic+ZPE ∆ES-T by about 2.0 and 2.1 kcal/mol, respectively, compared to the MR-AQCC(CAS(4,4)) method where the σ orbitals remained frozen. The vertically excited ∆ES-T could not be obtained with the MR-AQCC(CAS(4,4)) method and including the σ orbitals due to persistent intruder states in the triplet state keeping the calculation from converging. When

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considering 2, the vertical ∆ES-T decreases with the MR-AQCC(CAS(12,8)/AUX(4)) method by about 1.9 kcal/mol compared to the MR-AQCC(CAS(4,4)) method, but the adiabatic and adiabatic+ZPE ∆ES-T increase by about 0.6 kcal/mol.

3 1 Table 7.1 Singlet/triplet splitting energy (∆ES-T, E(1 Bu) - E(1 Ag)) calculated using the MR-AQCC/6-31G* method for trans-diindenoacenes, 1-4. System Reference Space ∆ES-T, vert. ∆ES-T, adiab. ∆ES-T, adiab.+ZPE (kcal/mol) (kcal/mol) (kcal/mol) 1 MR-AQCC(CAS(4,4))a -b 23.7 21.6 MR-AQCC(CAS(4,4)) 39.8 25.7 23.6 MR-AQCC 38.8 26.5 24.4 (CAS(10,7)/AUX(3)) MR-AQCC 38.2 23.0 20.9 (CAS(14,9)/AUX(3))

2 MR-AQCC(CAS(4,4)) 17.5 15.1 14.7 MR-AQCC 15.6 15.6 15.2 (CAS(12,8)/AUX(4))

3 MR-AQCC(CAS(4,4)) 11.5 10.7 10.5 Expt.4 4.18

4 MR-AQCC(CAS(4,4)) 6.4 6.5 6.5 aonly core orbitals are frozen, bvertically excited triplet state had persistent intruder states.

1 The effective number of unpaired electrons of the 1 Ag state increases linearly, whether one considers the sum over all NOs or just the H-L pair, with the number of interior rings going from 1-4. 1 has the smallest NU values of any of the trans-diindenoacenes (NU is 0.55 e and NU(H-L) is 0.23 e), correlating with its large ∆ES-T, Figure 7.3 and Table E.2. The total effective number of unpaired electrons for 2 is about 1.1 e, about 1.6 e for 3, and 2.2 e for 4. Using only the H-L pair reduces the total number of unpaired electrons by about 0.3 e for 1 to about 0.7 e for 4. The total number of unpaired electrons for 4 is 2.2 e, showing a true singlet biradical system, which correlates to its small ∆ES-T, Figure 7.3. When considering only the H-L pair for each vertically excited triplet state, Table E.2, the number of unpaired electrons is almost constant at 2.0 e, consistent with triplet spin multiplicity and singly occupied HONOs and LUNOs. The largest difference in NU values between the singlet and vertical triplet state comes for 1 at about 1.8 e. 149 Texas Tech University, Reed Nieman, May 2019

a)

b)

Figure 7.3 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of unpaired electrons (NU) and b) the number of unpaired electrons in the HONO-LUNO pair 1 (NU(H-L)) of the 1 Ag state (Table E.2) of trans-diindenoacenes (1-4) using the MR- AQCC(CAS(4,4))/6-31G* method.

1 The unpaired electron density of 1-4 for the 1 Ag ground state is shown in Figure 7.4. The density is greatest on the carbon atom of the five-membered rings connected to hydrogen, referred to as the apical carbon, correlating to the location of the radical center shown in the valence bond (VB) structures shown in Figure 7.1 and Figure E.1, and the density then alternates going toward the center of the molecules. The unpaired electron density on carbon atoms not bonded to hydrogen is nearly negligible. The unpaired electron 150 Texas Tech University, Reed Nieman, May 2019 density of the vertically excited triplet state of 1-4 (Figure E.2) is greater than for the singlet state. There is greater density found on the indene groups in the triplet state. The greatest density is still found on the apical carbon atom of the five-membered ring in the triplet state as in the singlet state, but the density decreases at this location with increasing system size. Despite the decrease in unpaired electron density at the location of greatest density, the slight increase of the NU (Table E.2) with increasing system size (about 0.1 e per additional core ring) can be seen to come from the increase in number of centers of unpaired electron density.

Figure 7.4 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the 1 1 Ag state of trans-diindenobenzene (1-4) using the MR-AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring. The HOMA indices for the singlet ground state geometry are also shown in Figure 7.4 for each trans-diindenoacene structured investigated (1-4). Structure 1 is noteworthy as the five-membered ring and interior six-membered ring are almost anti-aromatic in character as the HOMA indices are nearly zero, consistent with the previous findings that concluded the local anti-aromatic nature arises from the large odd-electron density distribution.98 The six-membered ring of the indene is aromatic however. This is substantially different from the triplet ground state geometry (Figure E.2) of 1 where the five-membered ring has a HOMA index of 0.37 and the interior six-membered ring has an index of 0.93. This great difference in HOMA indices between the singlet and triplet states,

151 Texas Tech University, Reed Nieman, May 2019 implying also a large change in the geometry, is the origin of the substantial difference in the NU discussed above for 1. In the remaining structures, 2-4, the HOMA index of the five-membered is about 0.3, with indices increasing going toward the center of each molecule, reaching a maximum value of about 0.75 for 3 and 4. As shown, the aromaticity of each ring (correlating to the HOMA index) increases as the unpaired electron density at each carbon of that ring decreases. For the triplet ground states structures, the interior most six-membered ring has a larger HOMA index than those found in the singlet ground state structures, though they are nearly the same in 4.

7.4.2 Cis-diindenoacenes

The singlet/triplet splitting of the cis-diindenoacene structures, 5-8, Figure 7.5 and

Table 7.2, is much smaller compared to the trans-isomers, 1-4. The ∆ES-T decreases as the number of rings in each system increases. The largest vertical ∆ES-T is found for 5 and is 3 only 3.9 kcal/mol while the vertical case for 8 is negative indicating that the 1 B1 state is 1 more stable than the 1 A1 state. For 7 though, the ∆ES-T, while negative, is close to zero so the singlet and triplet states are almost isoelectronic. The adiabatic and adiabatic+ZPE ∆ES-

T for 5 and 6 are smaller than the vertical ∆ES-T by about 0.4 kcal/mol, but 7 and 8 become positive when considering the adiabatic+ZPE ∆ES-T, increasing to 0.5 and 0.2 kcal/mol.

The ZPE correction does not contribute greatly to the adiabatic ∆ES-T, adding or subtracting by only about 0.1 kcal/mol. The ∆ES-T calculated for 5, 3.6 kcal/mol, compares well with the experimental value of 4.2 kcal/mol.31

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3 1 Figure 7.5 Comparison of the singlet/triplet energy (∆ES-T, E(1 B1) - E(1 A1)) and number 1 of unpaired electrons (NU and NU(H-L), Table E.3) of the 1 A1 state with the number of interior six-membered rings of cis-diindenoacenes (5-8) using the MR- AQCC(CAS(4,4))/6-31G* method.

3 1 Table 7.2 Singlet/triplet splitting energy (∆ES-T, E(1 B1) - E(1 A1)) calculated using the MR-AQCC/6-31G* method for cis-diindenoacenes, 5-8. System Reference Space ∆ES-T, vert. ∆ES-T, adiab. ∆ES-T, adiab.+ZPE (kcal/mol) (kcal/mol) (kcal/mol) 5 MR-AQCC(CAS(4,4)) 3.9 3.5 3.6 Expt.31 4.2

6 MR-AQCC(CAS(4,4)) 1.2 0.8 0.7

7 MR-AQCC(CAS(4,4)) -0.3 0.5 0.5

8 MR-AQCC(CAS(4,4)) -1.2 0.3 0.2

The decrease in ∆ES-T with increasing number of interior six-membered rings 1 correlates to an increase in the effective number of unpaired electrons of the 1 A1 state

(Figure 7.5), and as was shown with the trans-isomers (Figure 7.3), the decrease in ∆ES-T correlates to increasing effective unpaired electrons for the cis-diindenoacenes, shown in

Figure 7.6. The NU for 5 and 6 (Table E.3) are much larger than their counterparts, 1 and 2 153 Texas Tech University, Reed Nieman, May 2019

(Table E.2) by about 1.0 e, showing much more radical character, and the NU values for 7 and 8 are larger than those for 3 and 4 by about 0.6 e and 0.4 e, respectively. Taking only the HONO-LUNO pair, the NU is least 1.0 e for 5 and increases to almost 2.0 e for 8. The

NU and NU(H-L) for the triplet state is very similar to those found in the trans-isomers.

a)

b)

Figure 7.6 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of unpaired electrons (NU) and b) the number of unpaired electrons in the HONO-LUNO pair 1 (NU(H-L)) of the 1 A1 state (Table E.3) of cis-diindenoacenes (5-8) using the MR- AQCC(CAS(4,4))/6-31G* method.

1 The unpaired electron density of the 1 A1 state for 5-8 is presented in Figure 7.7. The The greatest amount of unpaired electron density is found located on the apical carbon in the five-membered ring of the indene, like for 1-4 (Figure 7.4), correlating to the VB 154 Texas Tech University, Reed Nieman, May 2019 structures. The apical carbon atoms were also found by Shimizu et al. to have the largest spin-density amplitudes in 1.31 For 7 and 8, nearly 0.5 e are present on these carbon atoms.

As was discussed for the NU values above, the unpaired electron density is greater for the cis-diindenoacene structures than for trans-indenoacene. The same trend that was seen for 1-4 (Figure 7.4) is present for the singlet states of 5-8 (Figure 7.7) whereby the unpaired density increases, especially at the apical carbon of the five-membered ring, with increasing number of interior rings. There is greater unpaired electron density present on the six-membered rings of the capping indene groups of the cis-isomers compared to 3 transisomers. The unpaired density of the vertically excited 1 B1 states is similar to the triplet states calculated for the trans-isomers (Figure E.3), where the value of the total NU increases with increasing system size due to the larger number of centers of unpaired electron density, but the density on the carbon in the five-membered ring decreases. While the difference in density for 5 between the singlet and triplet states is rather large (with the triplet density being larger), going from 6-7 the difference decreases, and for 8 the density is only slightly larger in the triplet state than in the singlet state.

Figure 7.7 Total unpaired electron density (sum over all NOs, Table E.3), in red, of the 1 1 A1 state of cis-diindenobenzene (5-8) using the MR-AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring. HOMA indices were calculated for the singlet and triplet states of the cis-diindenoacenes (5-8), Figure 7.7. The HOMA indices of the five-membered rings decrease from 5 (0.51)

155 Texas Tech University, Reed Nieman, May 2019 to 8 (0.31) while the indices of six-membered ring of the indene group remains practically constant. The HOMA index also shows to increase from the five-membered ring toward the center. The HOMA indices reported here for 5 and 6 are slightly larger for all rings than those presented previously by Miyoshi and coworkers.34 For structures with odd numbers of rings, 5 and 7, the HOMA index of the center-most ring decreases by 0.07, while for even numbers of rings, 6 and 8, the HOMA indices for the center-most rings increases slightly by 0.04. Comparing to the trans-isomers, the five-membered rings in the cis-isomers have higher HOMA indices, while the six-membered rings of the indene groups are decreases slightly. The interior rings are shown to be more aromatic in the cis-isomers, 3 as the HOMA is larger for 5-8 than 1-4. The HOMA indices of the 1 B1 state (Figure E.3) 1 are smaller for all structures than the 1 A1 state except for the six-membered ring of the indene groups, which are only slightly larger.

7.4.3 Trans-indenoindenodi(acenothiophenes)

The ∆ES-T of the investigated trans-indenoindenodi(acenothiophenes) (9-12) is compared to the effective NU and number of interior rings in Figure 7.8. The trend of decreasing ∆ES-T with increasing ring number is found also for 9-12 as for the previous structures. The adiabatic ∆ES-T value decreases going from 9 to 10 by about 16 kcal/mol (primarily due to the large value found for 9), though the values for 11 and 12 are practically the same. The large decrease in all considered ∆ES-T values is observed also going from 1 to 2 due to the much larger ∆ES-T of 1. The ZPE correction to the adiabatic

∆ES-T is similarly small; while the adiabatic+ZPE ∆ES-T decreases going from 9-11 with increasing number of rings, going from 11 to 12 is basically constant. As with 1 and 2, an expanded reference space was calculated showing that the vertical ∆ES-T increased by about

0.3 kcal/mol, but the adiabatic and adiabatic+ZPE ∆ES-T decreased by about 2.3 kcal/mol compared to the MR-AQCC(CAS(4,4)) reference space. There is excellent agreement with 54 previous SF-NC-TDDFT calculations for 9-11 (the CASCI values for the ∆ES-T of these structures54 are in general larger by about 10 kcal/mol for each structure compared to our MR-AQCC values), and the experimental value (8.0 kcal/mol)54 was found to agree quite

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well and was only 0.4 kcal/mol smaller than the calculated value of the adiabatic+ZPE ∆ES-

T for 10, given in Table 7.3.

3 1 Figure 7.8 Comparison of the singlet/triplet energy (∆ES-T, E(1 Bu) - E(1 Ag)) and number 1 of unpaired electrons (NU and NU(H-L), Table E.4) of the 1 Ag state with the number of interior six-membered rings of trans-indenoindenodi(acenothiophene) (9-12) using the MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method.

3 1 Table 7.3 Singlet/triplet splitting energy (∆ES-T, E(1 Bu) - E(1 Ag)) calculated using the MR-AQCC/6-31G*(6-311G*)S method for trans-indenoindenodi(acenothiophene) (9-12). System Reference Space ∆ES-T, vert. ∆ES-T, adiab. ∆ES-T, adiab.+ZPE (kcal/mol) (kcal/mol) (kcal/mol) 9 MR-AQCC(CAS(4,4)) 24.5 19.7 19.2 MR-AQCC 24.8 17.4 17.0 (CAS(10,7)/AUX(3))

10 MR-AQCC(CAS(4,4)) 8.8 8.6 8.4 Expt.54 8.0

11 MR-AQCC(CAS(4,4)) 4.4 4.4 4.2

12 MR-AQCC(CAS(4,4)) 4.5 4.5 4.4

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1 As shown in Figure 7.9, the energy decreases with increasing NU in the 1 Ag state.

The total NU values (Table E.4) for the trans-indenoindenodi(acenothiophene) (9-12) are larger than those of the trans-diindenoacenes, 1-4 (Table E.2), by about 0.2 e to 0.5 e, and

0.1 e to 0.4 e when considering only the H-L pair. The total NU for the cis-diindenoacenes (5-8, Table E.3) are larger than for 9-12 by 0.7 e comparing 5 and 9, but the difference becomes smaller, about 0.1 e, with increasing molecular size for 8 and 12. The vertically excited triplet state has larger values of NU than for the singlet state, though the difference while large for 9 (by about 1.7 e) decreases with increasing system size and is only 0.45 e for 12. As with 1-8, when taking only H-L, the NU values become essentially 2.0 e. The total NU for the vertical triplet state is larger than the triplet states of 1-8 by about 0.1 e.

1 The unpaired electron density of the 1 Ag state of trans- indenoindenodi(acenothiophenes) (9-12) is presented in Figure 7.10. As for structures 1-8, the greatest amount of unpaired density is located on the apical carbon atom of the five- membered ring (again correlating to the VB structures as with 1-8) and increases with increasing system size, 9 to 12. The unpaired electron density is much reduced in the thiophene rings and terminal six-membered rings compared to that in the core six- 3 membered rings. This same feature is found for the 1 Bu state, Figure E.4. The triplet state features greater unpaired electron density compared to the singlet state, but the magnitude of the density localized on each center decreases as the system size grows from 9 to 12. As suggested by the NU values, Table E.4, the unpaired density of the singlet state is larger than for the trans-diindenoacenes, though smaller than for the cis-diindenoacenes.

158 Texas Tech University, Reed Nieman, May 2019

a)

b)

Figure 7.9 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of unpaired electrons (NU) and b) the number of unpaired electrons in the HONO-LUNO pair 1 (NU(H-L)) of the 1 Ag state (Table E.4) of trans-indenoindenodi(acenothiophene) (9-12) using the MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method.

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Figure 7.10 Total unpaired electron density (sum over all NOs, Table E.4), in red, of the 1 1 Ag state of trans-indenoindenodi(acenothiophene) (9-12) using the MR- AQCC(CAS(4,4))/6-31G*/6-311G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring.

1 The HOMA indices for the 1 Ag state structures is presented in Figure 7.10 as well. The index of the thiophene group decreases by about 0.08 by increasing the system size after 9. The capping six-membered rings have HOMA indices of approximately 1 for 9-12. The HOMA index of the five-membered rings increases from 9 to 10, but then decreases from 10 to 12. The innermost six-membered rings increase in HOMA index from 9 to 11, but then decreases with 12. The HOMA indices of the singlet state structures indicate that 9-12 are more aromatic than 1-4, but less aromatic than 5-8. The triplet state structures’ HOMA indices are shown in Figure E.4, and are found to be more aromatic than the singlet state structures. As was the case with 1 of the trans-diindenoacenes, the large difference in

NU of 9 between the singlet and triplet states originates in the large difference in HOMA indices, indicating a large difference in geometry between the two states.

7.5 Conclusions

MR-AQCC calculations were carried out to investigate the biradicaloid character of trans- diindenoacenes (1-4), cis-diindenoacenes (5-8), and trans- indenoindenodi(acenothiophenes) (9-12) by calculating the ∆ES-T and NU and comparing to the number of interior or core rings. For each structure, the ground state was found to be

160 Texas Tech University, Reed Nieman, May 2019 low spin, with the high spin state becoming increasing thermally accessible with increasing number of core rings. The singlet ground state 1 and 9 have only slight radical or biradical character judged by their small NU and NU(H-L) values, respectively, but these values increase greatly when increasing the number of core rings by only one to obtain 2 and 10, respectively. The cis-diindenoacene isomers are very different from the other eight structures, as even the smallest considered structure, 5, has a small ∆ES-T (3.57 kcal/mol) and large NU (1.41 e) and NU(H-L) (1.08 e). The greatest unpaired electron density for each considered structure is found on the five-membered ring, and in the case of the thiophene- fused structures, does not extend to the capping six-membered rings. The triplet structures were stabilized with increasing core rings while the effective number of unpaired electrons remained quite unchanged in magnitude, but became more diffuse and delocalized. In summary, the MR-AQCC approach has been proven to be a single method able to successfully assess the biradicaloid character of PAH molecules and compares exceptionally well to experimental values for the ∆ES-T in the cases of 5 and 10. This method provides necessary insight and analysis of these highly reactive species.

7.6 Bibliography

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CHAPTER VIII CONCLUDING REMARKS

This dissertation presents the application of several quantum chemical methods to the description of carbon-functionalized material properties used in cutting edge technologies. The necessary use of these high-level ab initio methods would inform future scientists on how best to utilize the discussed materials in future experiments and real- world applications.

The reactivity of pristine graphene as well as single and double carbon vacancy defects was assessed toward chemisorption of a hydrogen radical test particle using the CASSCF and MR-CISD methods with pyrene as a graphene model. While all carbon centers for each structure formed stable C-H bonds, single carbon vacancy pyrene possessed the strongest reaction (was most stabilizing) toward the incoming hydrogen radical thanks to the dangling bond left in the six-membered ring as a result of the defect. Bond formation with the hydrogen radical at this site was found to involve at least three closely spaced electronic excited states near the ground state and complicated diabatic electronic behavior at extended C-H bond distances. The carbon centers comprising the five-membered ring were found to quite stabilizing as well, though without the same degree of involvement from the complex manifold of excited states. A similar situation was found for the five-membered rings of the double vacancy structure. A rigid potential energy scan was performed to provide a survey of reactivity across the face of each structure confirming the stability at the carbon centers and finding more repulsive regions at the existing and in the center of rings.

The structure and electronic states of double vacancy graphene with the intrinsic dopant, silicon, were investigated by way of DFT calculations using pyrene and circumpyrene as model systems. The silicon defect was characterized by means of NO occupations, NBO analyses, and MEP surfaces. The dopant produced a non-planar low- spin ground state with D2 symmetry, with a small energy gap between it and the planar low spin D2h symmetric structure. The high spin structures were planar and found higher in energy. The hybridization of the non-planar structures was sp3 while the planar structures 170 Texas Tech University, Reed Nieman, May 2019 were sp2d showing d-orbital involvement which is thought to be responsible for opening the band gap. Comparisons between the B3LYP and CAM-B3LYP density functionals found similar energies and structures, though the CAM-B3LYP low spin structures displayed more open-shell character. Natural charge and MEP analyses showed charge transfer from the silicon center to the bonding carbon atoms. Supporting experimental results suggest a more complex picture, however, as features of the EEL spectrum indicate an equilibrium existing between the non-planar and planar structures.

The electronic excited states of a BHJ scheme composed of P3HT, T3, and fullerene

(C60) (electron donor, spacer material, and electron acceptor respectively), were studied using DFT with consideration to environmental effects. Experimental findings using this BHJ scheme showed dramatically increased PCE with the terthiophene spacer material. The excited state spectra calculated for isolated system and using the LR environmental model did not explain the marked increase in PCE as the local excitons generated in the bright π→π* of P3HT would decay into local excitons located on the fullerene units, thereby quenching any generated CT states. The electronic excited state spectra calculate using the SS environment however, showed a drastically different result as the local excitons generated in the bright π→π* of P3HT were shown to decay into significantly stabilized P3HT→C60 and T3→C60 CT states helping to explain the increased PCE. The increase in PCE was thusly explained as resulting from increased physical separation caused by the spacer decreasing the Coulomb binding energy and reducing the rate of charge recombination, and stabilizing the CT states below local C60 excitons increasing the rate of charge separation.

High-level SOS-MP2 and SOS-ADC(2) calculations were performed on several oligomer dimers of PPV for the purpose of providing much needed benchmark calculations of the structure and electronic excited states of the material that is paradigmatic in the description of EET processes in PAHs. Dimers composed of two, three, and four repeating phenylenevinylene (PPV2, PPV3, and PPV4 respectively) units were investigated using a variety of correlation consistent basis sets. As PPV2 is the smallest structure, it was subject to study with basis sets up cc-pVQZ and aug-cc-pVQZ, showing good agreement with the smaller cc-pVTZ basis. The conformation was studied comparing the sandwich-stacked 171 Texas Tech University, Reed Nieman, May 2019 system with a displaced dimer, showing smaller stacking distances and increased π-π* interactions in the latter. DFT methods exhibited larger stabilization energies. Two intensive electronic transitions were found for each dimer using the SOS-ADC(2) method, where the oscillatory strength increased with increasing chain length. A similar trend was found for the magnitude of interchain charge transfer as analyzed using omega matrices and NTO transitions. DFT results calculated a more diverse interchain CT spectra than with the SOS-ADC(2) method.

The biradical character of trans-diindenoacenes, cis-diindenoacenes, and trans- indenoindenodi(acenothiophenes) with increasing number of core fused-benzene rings were investigated using MR-AQCC calculations where the biradical character was described by the ∆ES-T and NU. For each structure studied, a low spin singlet ground state was found, and aside from trans-diindenobenzene and trans- indenoindenodi(acenobenzene), each had significant singlet diradical character. The ∆ES-T was shown to decrease with increasing number of core rings for each set of structures correlating to an increase in the NU. There is good agreement to experiment for the calculated values for ∆ES-T. The cis-diindenoacene structures were remarkable by comparison as the ∆ES-T was lower and NU larger than the other structures. Increasing NU correlated with the greater unpaired electron density in the singlet ground state and was greatest for the carbon bonded to hydrogen in the five-membered ring. While the NU in the first triplet excited state was relatively unchanged with increasing number of core rings, the larger structures stabilized the unpaired electron density, thus lowering the energy of the triplet state and the ∆ES-T gap. In the thiophene-fused structures, the thiophene groups disrupted the aromaticity and concentrated the unpaired density in the core of the structures. Consequently, relatively little unpaired electron density extended to the capping six-membered rings.

The study of graphene functionalization and defect formation represents an exciting field of computational study. To this end, as one hydrogenation defect has been discussed already, the addition of a second hydrogen radical is being studied using molecular dynamics simulations and circumcoronene model where the potential energy is determined using DFT. The calculations are being performed in order to determine the energy transfer 172 Texas Tech University, Reed Nieman, May 2019

process upon bond formation, the significance of H2 abstraction pathways, and the probability and location of the second C-H formation at room temperature. Molecular dynamics simulations are also being carried out in a similar fashion to determine the dynamic dissociation energy required to remove a single carbon atom from pristine graphene using the circumcoronene model generating the single carbon vacancy that has been investigated previously.

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APPENDIX A

ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN CHAPTER III

Table A.1 Relative energy differences ΔErel (eV) of the local energy minima in the electronic ground state computed for the different structures of pyrene-1C+Hr using different theoretical methods and the 6-31G* approach.

ΔErel CASSCF MR-CISD MR-CISD+Q C1 0.637 0.960 1.393 C2 0.230 0.681 1.244 C3 0.855 1.163 1.474 C4 2.752 3.179 3.729 C5 1.627 1.809 2.032 C6 3.107 3.638 4.049 C7 2.517 2.700 2.816 C8 3.542 3.930 4.246 C15 0.000a 0.000b 0.000c aTotal energy = -574.277274 Hartree, bTotal energy = -575.610208 Hartree, cTotal energy = -576.244082 Hartree, d The SA4-CASSCF(5,5) optimized geometry was used.

Table A.2 Relative energy differences ΔErel (eV) of the local energy minima in the electronic ground state computed for the different structures of pyrene-2C+Hr using different theoretical methods and the 6-31G* approach.

ΔErel CASSCF MR-CISD MR-CISD+Q C1 0.881 0.829 0.767 C2 0.000a 0.000b 0.000c C3 0.612 0.735 0.806 C4 1.059 1.054 1.047 aTotal energy = -536.452180 Hartree, bTotal energy = -537.716100 Hartree, cTotal energy = -538.283379 Hartree

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Table A.3 Relative energy differences ΔErel (eV) of the local energy minima in the electronic ground state computed for the different structures of pyrene+Hr using different theoretical methods and the 6-31G* basis set.

ΔErel CASSCF MR-CISD MR-CISD+Q C1 0.647 0.634 0.578 C2 0.137 0.080 0.000c C3 0.630 0.649 0.559 C4 0.000a 0.000b 0.018 C16 0.820 0.814 0.691 aTotal energy = -612.336622 Hartree, bTotal energy = -613.718399 Hartree, cTotal energy = -614.381588 Hartree

Hr bonded to Carbon 1

31a’ 32a’ 33a’ 22a” 23a”

Figure A.1 Plots of the active orbitals of the optimized ground states of pyrene+Hr using the CASSCF(5,5)/6-31G* approach.

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Hr optimized on Carbon 15

29a’ 30a’ 31a’ 21a” 22a”

Figure A.2 Plots of the active orbitals of the optimized ground states of pyrene-1C+Hr using the SA4-CASSCF(5,5)/6-31G* approach.

Hr optimized on Carbon 1

27a’ 28a’ 29a’ 20a” 21a”

Figure A.3 Plots of the active orbitals of the optimized ground states of pyrene-2C+Hr using the SA4-CASSCF(5,5)/6-31G* approach.

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3 2Aa

4 2Ab

2 2 Figure A.4 Potential energy scans in terms of the location of Hr for the 3 A and 4 A states of pyrene-2C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energies: a -574.149530 Hartree, b -574.117820 Hartree. The energy (color scale) is given in eV, relative to the ground state lowest minimum at -536.385046 Hartree.

2 Figure A.5 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C2 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = - 574.277274 Hartree

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2 Figure A.6 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C3 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = - 574.277274 Hartree

2 Figure A.7 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C4 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = - 574.277274 Hartree

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2 Figure A.8 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C5 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = - 574.277274 Hartree

2 Figure A.9 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C6 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = - 574.277274 Hartree

179 Texas Tech University, Reed Nieman, May 2019

2 Figure A.10 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C7 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = - 574.277274 Hartree

2 Figure A.11 Dissociation curve of the lowest four A states of pyrene-1C+Hr when Hr is bonded to C8 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = - 574.277274 Hartree

180 Texas Tech University, Reed Nieman, May 2019

APPENDIX B

ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN CHAPTER IV

Table B.1 NBO bonding character analysis of pyrene-2C+Si using the CAM-B3LYP/6- 311G(2d,1p) method. The carbon atom discussed is the carbon bonded to silicon. Structure Bonding Character CAM-B3LYP/6-311G(2d,1p) a 2.98 0.03 2.97 0.01 α : 0.53(sp d )Si+0.85(sp d )C a 2.98 1.03 2.98 0.01 P4 (D2) β : 0.53(sp d )Si+0.85(sp d )C a 2.98 0.03 3.00 0.01 α : 0.53(sp d )Si+0.85(sp d )C a 2.98 0.03 3.00 0.01 P5 (D2) β : 0.53(sp d )Si+0.85(sp d )C

2.98 0.02 3.20 0.01 P6 (D2) 0.54(sp d )Si+0.84(sp d )C 2.01 1.01 2.80 0.01 P7 (D2h) α: 0.45(sp d )Si+0.89(sp d )C 1.99 0.99 2.78 0.01 β: 0.46(sp d )Si+0.89(sp d )C 2.00 1.00 2.92 0.01 P8 (D2h) 0.45(sp d )Si+0.89(sp d )C 2.38 0.62 2.39 0.01 P9 (D2) α: 0.48(sp d )Si+0.88(sp d )C 2.92 0.09 2.45 0.01 β: 0.48(sp d )Si+0.87(sp d )C 2.00 1.00 2.41 0.01 P10 (D2h) α: 0.46(sp d )Si+0.89(sp d )C 2.00 1.00 2.69 0.01 β: 0.45(sp d )Si+0.89(sp d )C

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s sp3.58

p sp3.58

p sp3.58

p sp3.58

Figure B.1 NBO plots for circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p) approach 3 (CP2 D2h). Isovalue = ±0.02 e/Bohr

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P1 (D2) P2 (D2h) P3 (D2h)

Figure B.2 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using the B3LYP/6-311G(2d,1p) method.

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P4 (D2) P5 (D2) P6 (D2)

P7 (D2h) P8 (D2h) P9 (D2)

P10 (D2h)

Figure B.3 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using the CAM-B3LYP/6-311G(2d,1p) method.

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P1 (D2) P2 (D2h)

P3 (D2h)

Figure B.4 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the B3LYP/6- 311G(2d,1p) method.

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P4 (D2) P5 (D2)

P6 (D2) P7 (D2h)

P8 (D2h) P9 (D2)

P10 (D2h)

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Figure B.5 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the CAM-B3LYP/6- 311G(2d,1p) method.

8.5

8.0

s

t

n

u

o

c

7.5

5

0

1 7.0

6.5 90 100 110 120 130 Energy loss (eV)

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

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Flat Corrugated

Figure B.7 First two unoccupied states immediately above the Fermi level spanning about 1 eV for the flat and corrugated Si/graphene structures from the periodic DFT calculations. Electron density isovalue is 0.277 e/Bohr3.

188 Texas Tech University, Reed Nieman, May 2019

APPENDIX C

ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN CHAPTER V

Table C.1 Excitation energies Ω of the isolated standard C60-T3-P3HT system using the CAM-B3LYP/6-31G approach Exc. Ω(eV) f CT Direction of Location of CT exciton S1 2.69 2.08 0.00 P3HT S2 2.83 0.00 0.97 P3HTC60 S3 2.84 0.00 1.00 P3HTC60 S4 2.85 0.00 0.99 P3HTC60 S5 2.87 0.00 0.99 T3C60 S6 2.87 0.00 0.99 T3C60 S7 2.93 0.00 0.99 T3C60 S8 3.41 1.18 0.01 T3 S9 3.47 0.10 0.00 P3HT S10 3.63 0.00 0.97 P3HTC60 S11 3.63 0.00 1.00 P3HTC60 S12 3.65 0.00 0.99 P3HTC60 S13 3.92 0.02 0.00 P3HT S14 3.92 0.02 0.94 T3C60 S15 3.93 0.01 0.94 T3C60 S16 4.10 0.00 0.98 P3HTC60 S17 4.11 0.00 0.98 P3HTC60 S18 4.12 0.00 0.99 P3HTC60 S19 4.14 0.02 0.95 T3C60 S20 4.16 0.00 0.99 T3C60

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Table C.2 Excitation energies Ω of the isolated standard C60-T3-P3HT system using the CAM-B3LYP/6-31G* approach Exc. Ω (eV) f CT Direction of Location of CT exciton S1 2.62 2.04 0.00 P3HT S2 2.91 0.00 0.97 P3HTC60 S3 2.91 0.00 1.00 P3HTC60 S4 2.92 0.00 0.99 P3HTC60 S5 2.93 0.00 0.99 T3C60 S6 2.94 0.00 0.99 T3C60 S7 2.99 0.00 0.99 T3C60 S8 3.33 1.16 0.01 T3 S9 3.39 0.10 0.00 P3HT S10 3.70 0.00 0.97 P3HTC60 S11 3.70 0.00 1.00 P3HTC60 S12 3.71 0.00 0.99 P3HTC60 S13 3.87 0.02 0.00 P3HT S14 3.99 0.02 0.94 T3C60 S15 3.99 0.03 0.94 T3C60 S16 4.20 0.03 0.95 T3C60 S17 4.23 0.00 0.98 P3HTC60 S18 4.24 0.00 0.98 P3HTC60 S19 4.25 0.00 0.99 P3HTC60 S20 4.27 0.00 0.99 T3C60

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Table C.3 Lowest excitation energies Ω of the isolated components P3HT, T3 and C60 using the CAM-B3LYP/6-31G* approach. Exc. Ω (eV) f P3HT S1 2.62 2.10 S2 3.39 0.00 S3 3.87 0.02 S4 4.06 0.14 S5 4.49 0.01 S6 4.63 0.00 S7 4.84 0.04 S8 4.86 0.00 S9 4.89 0.00 S10 4.93 0.01 T3 S1 3.30 1.09 S2 4.54 0.00 S3 4.91 0.00 S4 5.07 0.00 S5 5.13 0.00 S6 5.15 0.01 S7 5.24 0.01 S8 5.30 0.06 S9 5.41 0.00 S10 5.43 0.02 C60 S1 2.47 0.00 S2 2.47 0.00 S3 2.47 0.00 S4 2.53 0.00 S5 2.53 0.00 S6 2.53 0.00 S7 2.56 0.00 S8 2.56 0.00 S9 2.56 0.00 S10 2.57 0.00

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Table C.4 Excitation energies Ω of the standard C60-T3-P3HT system the using the CAM- B3LYP/6-31G* approach in the LR environment Exc. Ω (eV) f CT Direction of Location of CT exciton S1 2.53 2.22 0.00 P3HT S2 3.12 0.00 0.97 P3HTC60 S3 3.12 0.00 1.00 P3HTC60 S4 3.13 0.00 0.99 P3HTC60 S5 3.22 0.60 0.57 T3C60 S6 3.23 0.14 0.90 T3C60 S7 3.24 0.48 0.56 T3C60 S8 3.27 0.10 0.00 P3HT S9 3.29 0.02 0.99 T3C60 S10 3.87 0.01 0.00 P3HT S11 3.91 0.00 0.97 P3HTC60 S12 3.91 0.00 1.00 P3HTC60 S13 3.92 0.00 0.99 P3HTC60 S14 3.98 0.19 0.00 P3HT S15 4.28 0.01 0.85 T3C60 S16 4.29 0.13 0.86 T3C60 S17 4.37 0.00 1.00 P3HTT3 S18 4.45 0.00 0.97 P3HTC60 S19 4.45 0.00 0.99 P3HTC60 S20 4.46 0.00 0.99 P3HTC60

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Table C.5 Excitation energies Ω of the standard C60-T3-P3HT system the using the CAM- B3LYP/6-31G approach in the LR environment Exc. Ω (eV) f CT Direction of Location of CT exciton S1 2.60 2.26 0.0 P3HT S2 3.08 0.00 0.97 P3HTC60 S3 3.08 0.00 1.00 P3HTC60 S4 3.09 0.00 0.98 P3HTC60 S5 3.20 0.01 0.99 T3C60 S6 3.21 0.01 0.99 T3C60 S7 3.27 0.02 0.99 T3C60 S8 3.32 1.21 0.01 T3 S9 3.35 0.09 0.00 P3HT S10 3.87 0.00 0.97 P3HTC60 S11 3.88 0.00 1.00 P3HTC60 S12 3.89 0.00 0.98 P3HTC60 S13 3.92 0.01 0.00 P3HT S14 4.08 0.20 0.00 P3HT S15 4.28 0.05 0.91 T3C60 S16 4.28 0.04 0.91 T3C60 S17 4.35 0.00 1.00 P3HTC60 S18 4.35 0.00 0.98 P3HTC60 S19 4.36 0.00 0.99 P3HTC60 S20 4.40 0.00 1.00 P3HTT3

Table C.6 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM- B3LYP/6-31G* approach in the SS environment

Exc. Ω (eV) f Direction of Location of CT Exciton

S1 1.29 0.00 P3HTC60 S2 1.29 0.00 P3HTC60 S3 1.29 0.00 P3HTC60 S4 1.29 0.00 P3HTC60 S5 1.88 0.00 T3C60 S6 1.88 0.00 T3C60 S7 1.88 0.00 T3C60 S8 1.88 0.00 T3C60 S9 1.90 0.00 T3C60 S10 2.63 2.04 P3HT

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Table C.7 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM- B3LYP/6-31G approach in the SS environment

Exc. Ω (eV) f CT Direction of Location of CT Exciton

S1 1.23 0.00 1.00 P3HTC60 S2 1.23 0.00 1.00 P3HTC60 S3 1.24 0.00 1.00 P3HTC60 S4 1.84 0.00 1.00 T3C60 S5 1.84 0.00 1.00 T3C60 S6 1.86 0.00 1.00 T3C60 S7 2.06 0.00 1.00 P3HTC60 S8 2.06 0.00 1.00 P3HTC60 S9 2.07 0.00 1.00 P3HTC60 S10 2.70 2.07 0.00 P3HT

Table C.8 Excitation energies Ω of the isolated standard C60-T3-P3HT system using the CAM-B3LYP/6-31G* approach (only core orbitals were frozen) Exc. Ω (eV) f CT Direction of Location of CT exciton S1 2.47 0.00 0.01 C60 S2 2.47 0.00 0.00 C60 S3 2.48 0.00 0.01 C60 S4 2.52 0.00 0.01 C60 S5 2.53 0.00 0.00 C60 S6 2.53 0.00 0.01 C60 S7 2.56 0.00 0.01 C60 S8 2.56 0.00 0.01 C60 S9 2.57 0.00 0.00 C60 S10 2.57 0.00 0.00 C60 S11 2.62 2.03 0.00 P3HT S12 2.84 0.00 0.01 C60 S13 2.85 0.00 0.01 C60 S14 2.85 0.00 0.01 C60 S15 2.86 0.00 0.01 C60 S16 2.86 0.00 0.01 C60 S17 2.91 0.00 1.00 P3HTC60 S18 2.91 0.00 1.00 P3HTC60 S19 2.92 0.00 1.00 P3HTC60 S20 2.93 0.00 1.00 T3C60

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Table C.9 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system using the CAM-B3LYP/6-31G* approach

Exc. Ω (eV) f CT Direction of Location of CT exciton

1 1 A 2.33 0.00 1.00 (T3)3(C60)3 1 2 A 2.37 0.00 1.00 (T3)3(C60)3 1 3 A 2.38 0.00 1.00 (T3)3(C60)3 1 4 A 2.56 0.01 0.99 (T3)3(C60)3 1 5 A 2.59 0.00 1.00 (T3)3(C60)3 1 6 A 2.62 0.00 0.99 (T3)3(C60)3 7 1A 2.67 1.92 0.00 P3HT 1 8 A 2.74 0.00 1.00 (T3)3(C60)3 1 9 A 2.75 0.00 1.00 (T3)3(C60)3 1 10 A 2.76 0.00 1.00 (T3)3(C60)3 1 11 A 2.78 0.00 1.00 (T3)3(C60)3 1 12 A 2.79 0.00 0.99 (T3)3(C60)3 1 13 A 2.80 0.00 1.00 (T3)3(C60)3 1 14 A 2.81 0.00 0.98 (T3)3(C60)3 1 15 A 2.82 0.00 1.00 (T3)3(C60)3 1 16 A 2.82 0.00 1.00 P3HT(C60)3 1 17 A 2.83 0.00 1.00 (T3)3(C60)3 1 18 A 2.85 0.00 1.00 P3HT(C60)3 1 19 A 2.89 0.00 1.00 P3HT(C60)3 1 20 A 2.96 0.00 0.97 (T3)3(C60)3 1 21 A 2.96 0.00 0.99 (T3)3(C60)3 1 22 A 2.97 0.00 0.98 (T3)3(C60)3 1 23 A 3.02 0.00 1.00 (T3)3(C60)3 1 24 A 3.04 0.00 1.00 (T3)3(C60)3 1 25 A 3.05 0.00 1.00 (T3)3(C60)3 1 26 A 3.06 0.00 1.00 P3HT(C60)3 1 27 A 3.07 0.00 1.00 P3HT(C60)3 1 28 A 3.08 0.00 1.00 P3HT(C60)3 1 29 A 3.10 0.00 0.10 (T3)3 1 30 A 3.14 0.00 1.00 (T3)3(C60)3 1 31 A 3.15 0.00 1.00 (T3)3(C60)3 1 32 A 3.15 0.00 1.00 P3HT(C60)3 1 33 A 3.16 0.00 1.00 P3HT(C60)3 1 34 A 3.16 0.00 1.00 (T3)3(C60)3 1 35 A 3.17 0.00 1.00 P3HT(C60)3 195 Texas Tech University, Reed Nieman, May 2019

Table C.9, Continued

1 36 A 3.26 0.07 0.03 (T3)3 1 37 A 3.28 0.00 0.99 (T3)3(C60)3 1 38 A 3.29 0.00 0.99 (T3)3(C60)3 1 39 A 3.29 0.00 0.97 (T3)3(C60)3 1 40 A 3.43 1.38 0.03 P3HT/(T3)3

Table C.10 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system using the CAM-B3LYP/6-31G approach

Exc. Ω (eV) f CT Direction of Location of CT exciton

S1 2.24 0.00 1.00 (T3)3(C60)3 S2 2.28 0.00 1.00 (T3)3(C60)3 S3 2.30 0.00 1.00 (T3)3(C60)3 S4 2.47 0.01 0.99 (T3)3(C60)3 S5 2.51 0.00 1.00 (T3)3(C60)3 S6 2.54 0.00 1.00 (T3)3(C60)3 S7 2.68 0.00 1.00 (T3)3(C60)3 S8 2.69 0.00 1.00 (T3)3(C60)3 S9 2.70 0.00 1.00 (T3)3(C60)3 S10 2.71 0.00 1.00 (T3)3(C60)3 S11 2.72 0.00 1.00 (T3)3(C60)3 S12 2.73 0.00 1.00 (T3)3(C60)3 S13 2.73 0.00 1.00 P3HT(C60)3 S14 2.74 1.96 0.00 P3HT S15 2.75 0.00 0.98 (T3)3(C60)3 S16 2.75 0.00 1.00 (T3)3(C60)3 S17 2.76 0.00 1.00 P3HT(C60)3 S18 2.77 0.00 0.97 (T3)3(C60)3 S19 2.79 0.00 1.00 P3HT(C60)3 S20 2.92 0.00 0.98 (T3)3(C60)3 S21 2.92 0.00 1.00 (T3)3(C60)3 S22 2.93 0.00 0.97 (T3)3(C60)3 S23 2.96 0.00 1.00 (T3)3(C60)3 S24 2.97 0.00 1.00 (T3)3(C60)3 S25 2.99 0.00 1.00 (T3)3(C60)3 S26 3.00 0.00 1.00 P3HT(C60)3 S27 3.01 0.00 1.00 P3HT(C60)3 S28 3.02 0.00 1.00 P3HT(C60)3 196 Texas Tech University, Reed Nieman, May 2019

Table C.10, Continued

S29 3.07 0.00 1.00 (T3)3(C60)3 S30 3.08 0.00 1.00 (T3)3(C60)3 S31 3.09 0.00 1.00 (T3)3(C60)3 S32 3.10 0.00 1.00 P3HT(C60)3 S33 3.11 0.00 1.00 P3HT(C60)3 S34 3.11 0.00 1.00 P3HT(C60)3 S35 3.15 0.00 0.11 (T3)3 S36 3.23 0.00 0.99 (T3)3(C60)3 S37 3.23 0.00 0.99 (T3)3(C60)3 S38 3.24 0.00 0.97 (T3)3(C60)3 S39 3.32 0.10 0.03 (T3)3 S40 3.49 1.97 0.08 (T3)3

Table C.11 Excitation energies Ω of the extended C60-T3-P3HT system using the CAM- B3LYP/6-31G approach in the SS environment

Exc. Ω (eV) f CT Direction of Location of CT exciton

S1 1.59 0.00 1.00 P3HT(C60)3 S2 1.59 0.00 1.00 P3HT(C60)3 S3 1.60 0.00 1.00 P3HT(C60)3 S4 1.61 0.00 1.00 P3HT(C60)3 S5 1.77 0.00 1.00 (T3)3(C60)3 a S6 1.77 0.00 1.00 (T3)3(C60)3 S7 1.88 0.00 1.00 (T3)3(C60)3 S8 1.91 0.00 1.00 (T3)3(C60)3 S9 1.91 0.00 1.00 (T3)3(C60)3 S10 1.91 0.00 1.00 (T3)3(C60)3 S11 1.91 0.00 1.00 (T3)3(C60)3 S12 2.10 0.00 1.00 (T3)3(C60)3 S13 2.10 0.00 1.00 (T3)3(C60)3 S14 2.10 0.00 1.00 (T3)3(C60)3 S15 2.12 0.00 1.00 (T3)3(C60)3 S16 2.12 0.00 1.00 (T3)3(C60)3 S17 2.13 0.00 1.00 (T3)3(C60)3 S18 2.13 0.00 1.00 (T3)3(C60)3 b S19 2.21 0.00 1.00 (T3)3(C60)3 S20 2.75 1.92 0.00 P3HT aExcitation energy averaged over two oscillating energies: 1.77 eV and 1.76 eV, bExcitation energy averaged over three oscillating energies: 2.12 eV, 2.21 eV, 2.29 eV 197 Texas Tech University, Reed Nieman, May 2019

APPENDIX D

ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN CHAPTER VI

Table D.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer computed at MP2 and MP2/BSSE levels with several basis sets using C2h symmetry.

1-1’ 2-2’ 3-3’ 4-4’ 5-5’ ΔE MP2 cc-pVDZ 3.58 3.56 3.54 3.61 3.62 -9.13 aug-cc-pVDZ 3.48 3.45 3.44 3.50 3.52 -20.23 cc-pVTZ 3.57 3.55 3.53 3.59 3.60 -11.80 aug-cc-pVTZ 3.57 3.45 3.46 3.53 3.64 -15.10 MP2/BSSE cc-pVDZ 3.78 3.77 3.77 3.78 3.79 -4.73 cc-pVTZ 3.66 3.64 3.63 3.67 3.68 -8.95

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computed

symmetry.

h

dimer dimer

2

4

C

in the stacked PPV stacked the in

several basis sets using sets basis several

(kcal/mol)

ΔE

and and

MP2/BSSE levels with MP2/BSSE levels

-

distances (Å) distances

Interchain Interchain

2

MP2 and SOS MP2and

-

Table D. Table SOS at

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Table D.3 Interchain distances (Å) and ΔE (kcal/mol) in the diplaced PPV4 dimer computed at SOS-MP2 level with several basis sets using C2h symmetry.

1-1’ 2-2’ 3-3’ 4-4’ 5-5’ 6-6’ 7-7’ 8-8’ ΔE cc-pVTZ 3.57 3.50 3.46 3.45 3.45 3.46 3.50 3.57 -21.17 cc-pVQZa ------19.56 CBS limit extrapolation: TQb -18.39 aSingle-point energy at SOS-MP2/cc-pVTZ optimized geometry, bExtrapolation made using cc-pVTZ and cc-pVQC

Table D.4 Single chain PPV2 vertical excitations using SOS-ADC(2) and SOS-MP2 optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength)

Basis State Excitation ΔE (eV) f 1 cc-pVDZ S1 1 Bu 4.57 0.51 1 S2 1 Ag 4.65 0.00 1 S3 2 Bu 4.74 0.72 1 S4 2 Ag 6.10 0.00 1 S5 1 Bg 7.63 0.00 1 S6 1 Au 7.76 0.00 1 S7 2 Bg 7.80 0.00 1 S8 2 Au 8.22 0.00

1 cc-pVTZ S1 1 Bu 4.52 0.75 1 S2 1 Ag 4.66 0.00 1 S3 2 Bu 4.72 0.43 1 S4 2 Ag 6.00 0.00 1 S5 1 Bg 7.24 0.00 1 S6 1 Au 7.26 0.00 1 S7 2 Bg 7.56 0.00 1 S8 2 Au 8.04 0.01

1 cc-pVQZ S1 1 Bu 4.49 0.77 1 S2 1 Ag 4.66 0.00 1 S3 2 Bu 4.71 0.40 1 S4 2 Ag 5.95 0.00 1 S5 1 Au 6.86 0.00 1 S6 1 Bg 6.95 0.00 200 Texas Tech University, Reed Nieman, May 2019

Table D.4, Continued

1 S7 2 Bg 7.26 0.00 1 S8 2 Au 7.66 0.02

Table D.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2) and SOS-MP2 optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) Basis State Excitation ΔE (eV) f CT (e) 1 cc-pVDZ S1 1 Bg 4.21 0.00 0.17 1 S2 1 Au 4.51 0.00 0.05 1 S3 2 Bg 4.53 0.00 0.08 1 S4 1 Bu 4.58 0.36 0.01 1 S5 1 Ag 4.63 0.00 0.01 1 S6 2 Bu 4.81 1.82 0.01 1 S7 2 Au 5.78 0.00 0.21 1 S8 2 Ag 6.11 0.00 0.02

1 cc-pVTZ S1 1 Bg 4.10 0.00 0.18 1 S2 1 Au 4.50 0.00 0.08 1 S3 2 Bg 4.51 0.00 0.08 1 S4 1 Bu 4.55 0.74 0.03 1 S5 1 Ag 4.65 0.00 0.02 1 S6 2 Bu 4.76 1.32 0.02 1 S7 2 Au 5.64 0.00 0.22 1 S8 2 Ag 6.01 0.00 0.03

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Table D.6 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS- MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) Excitation % Hole Electron

1 1 Bg 82

12

1 1 Au 28

26

1 2 Bg 40

16

11

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Table D.6, Continued

1 1 Bu 41

30

1 1 Ag 19

18

16

1 2 Bu 46

36

1 2 Au 55

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Table D.6, Continued

25

1 2 Ag 33

25

14

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Table D.7 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) Excitation % Hole Electron

1 1A 46

45

2 1A 40

29

1 3 A 23

19

17

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Table D.7, Continued

13

4 1A 28

21

19

13

5 1A 35

20

16

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Table D.7, Continued

14

6 1A 53

32

7 1A 55

40

8 1A 62

33

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Table D.8 Stacked PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS- MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) State Excitation ΔE (eV) f CT (e) 1 S1 1 Bg 3.47 0.00 0.18 1 S2 1 Bu 3.98 3.60 0.02 1 S3 2 Bg 4.21 0.00 0.10 1 S4 2 Bu 4.38 0.34 0.02 1 S5 1 Au 4.39 0.00 0.16 1 S6 2 Au 4.57 0.00 0.15 1 S7 1 Ag 4.60 0.00 0.04 1 S8 2 Ag 5.11 0.00 0.02

Table D.9 Stacked PPV4 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS- MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer)

State Excitation ΔE (eV) f CT (e) 1 S1 1 Bg 3.20 0.00 0.19 1 S2 1 Bu 3.69 5.58 0.02 1 S3 1 Au 3.86 0.00 0.21 1 S4 2 Bg 4.17 0.00 0.13 1 S5 2 Au 4.20 0.00 0.14 1 S6 1 Ag 4.31 0.00 0.04 1 S7 2 Bu 4.34 0.31 0.23 1 S8 2 Ag 4.53 0.00 0.04

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Table D.10 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) Excitation % Hole Electron

1 1A 50

40

2 1A 55

35

1 3 A 38

24

13

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Table D.10, Continued

11

4 1A 31

24

15

14

5 1A 50

30

6 1A 28

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Table D.10, Continued

24

7 1A 26

20

8 1A 16

16

14

12

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Table D.11 Stacked PPV3 dimer vertical excitations using B3LYP-D3/cc-pVTZ and the B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer) State Excitation ΔE (eV) f CT (e) 1 S1 1 Bg 2.40 0.00 0.50 1 S2 1 Bu 2.83 0.04 0.99 1 S3 2 Bu 3.28 3.10 0.01 1 S4 2 Bg 3.38 0.00 0.50 1 S5 1 Au 3.38 0.00 0.50 1 S6 2 Au 3.45 0.00 0.50 1 S7 1 Ag 3.77 0.00 0.99 1 S8 3 Bg 3.79 0.00 0.50 1 S9 2 Ag 3.84 0.00 0.08 1 S10 4 Bg 3.95 0.00 0.50 1 S11 3 Au 3.97 0.00 0.50 1 S12 3 Ag 3.99 0.00 0.92 1 S13 5 Bg 4.07 0.00 0.50 1 S14 6 Bg 4.13 0.00 0.50 1 S15 4 Au 4.15 0.00 0.50 1 S16 3 Bu 4.18 0.06 0.08 1 S17 4 Ag 4.33 0.00 0.05 1 S18 4 Bu 4.35 0.01 0.07 1 S19 5 Au 4.41 0.00 0.50 1 S20 7 Bg 4.44 0.00 0.50 1 S21 8 Bg 4.47 0.00 0.50 1 S22 6 Au 4.47 0.00 0.50 1 S23 5 Bu 4.47 0.02 0.90 1 S24 5 Ag 4.48 0.00 0.03 1 S25 6 Bu 4.52 0.00 1.00 1 S26 7 Bu 4.58 0.01 0.94 1 S27 6 Ag 4.58 0.00 0.99 1 S28 8 Bu 4.60 0.01 0.83 1 S29 7 Ag 4.63 0.00 0.96 1 S30 9 Bg 4.71 0.00 0.54

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Table D.12 Stacked PPV3 dimer vertical excitations using CAM-B3LYP-D3/cc-pVTZ and the CAM-B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT – interchain charge transfer)

State Excitation ΔE (eV) f CT (e) 1 S1 1 Bg 3.17 0.00 0.25 1 S2 1 Bu 3.73 3.16 0.05 1 S3 2 Bu 4.09 0.13 0.96 1 S4 1 Au 4.17 0.00 0.28 1 S5 2 Bg 4.35 0.00 0.77 1 S6 3 Bg 4.47 0.00 0.16 1 S7 3 Bu 4.67 0.07 0.02 1 S8 4 Bg 4.77 0.00 0.19 1 S9 2 Au 4.77 0.00 0.28 1 S10 1 Ag 4.78 0.00 0.09 1 S11 3 Au 4.88 0.00 0.50 1 S12 4 Bu 4.98 0.01 0.01 1 S13 2 Ag 5.02 0.00 0.12 1 S14 3 Ag 5.03 0.00 0.88 1 S15 5 Bg 5.07 0.00 0.50 1 S16 4 Ag 5.17 0.00 0.03 1 S17 6 Bg 5.27 0.00 0.50 1 S18 4 Au 5.33 0.00 0.73 1 S19 5 Au 5.39 0.00 0.50 1 S20 7 Bg 5.47 0.00 0.50 1 S21 5 Bu 5.59 0.34 0.25 1 S22 5 Ag 5.67 0.00 0.99 1 S23 6 Bu 5.73 0.15 0.40 1 S24 8 Bg 5.79 0.00 0.27 1 S25 7 Bu 5.84 0.28 0.48

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S1 S2 S3 S4

S5 S6 S7 S8

Figure D.1 Omega matrices for vertical excitations of displaced PPV2 dimer using the SOS- ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each vinylene and phenylene group is a separate fragment: fragments 1-3 belong to one chain, fragments 4-6 to the other. The y-axis represents the electron and the x-axis represents the hole. Omega matrix plots calculated using the TheoDORE program package.

S1 S2 S3 S4

S5 S6 S7 S8

Figure D.2 Omega matrices for vertical excitations of stacked PPV4 dimer using the SOS- ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each vinylene and phenylene group is a separate fragment: fragments 1-7 belong to one chain, fragments 8-14 to the other. The y-axis represents the electron and the x-axis represents the hole. Omega matrix plots calculated using the TheoDORE program package.

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S1 S2 S3 S4

S5 S6 S7 S8

Figure D.3 Omega matrices for vertical excitations of displaced PPV3 dimer using the SOS- ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each vinylene and phenylene group is a separate fragment: fragments 1-5 belong to one chain, fragments 6-10 to the other. The y-axis represents the electron and the x-axis represents the hole. Omega matrix plots calculated using the TheoDORE program package.

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APPENDIX E

ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN CHAPTER VII

Table E.1 Active orbitals for each MR-AQCC reference space Reference Space CAS AUX 1 MR-AQCC(CAS(4,4))a 5au, 6au, 5bg, 6bg MR-AQCC(CAS(4,4)) 5au, 6au, 5bg, 6bg MR-AQCC 3au, 4au, 5au, 7au, 7bg, 8bg (CAS(10,7)/AUX(3)) 6au, 4bg, 5bg, 6bg 3au, 4au, 5au, MR-AQCC 6au, 2bg, 3bg, 7au, 7bg, 8bg (CAS(14,9)/AUX(3)) 4bg, 5bg, 6bg 2 MR-AQCC(CAS(4,4)) 6au, 7au, 6bg, 7bg 4au, 5au, 6au, MR-AQCC 7au, 4bg, 5bg, 8au, 9au, 8bg, 9bg (CAS(12,8)/AUX(4)) 6bg, 7bg 3 MR-AQCC(CAS(4,4)) 7au, 8au, 7bg, 8bg 4 MR-AQCC(CAS(4,4)) 8au, 9au, 8bg, 9bg 5 MR-AQCC(CAS(4,4)) 5b2, 6b2, 5a2, 6a2 6 MR-AQCC(CAS(4,4)) 7b2, 8b2, 5a2, 6a2 7 MR-AQCC(CAS(4,4)) 7b2, 8b2, 7a2, 8a2 8 9b2, 10b2, MR-AQCC(CAS(4,4)) 7a2, 8a2 9 MR-AQCC(CAS(4,4)) 8au, 9au, 8bg, 9bg MR-AQCC 6au, 7au, 8au, 10au, 10bg, 11bg (CAS(10,7)/AUX(3)) 9au, 7bg, 8bg, 9bg 10 9au, 10au, MR-AQCC(CAS(4,4)) 9bg, 10bg 11

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Table E.1, Continued 10au, 11au, MR-AQCC(CAS(4,4)) 10bg, 11bg 12 11au, 12au, MR-AQCC(CAS(4,4)) 11bg, 12bg aonly core orbitals are frozen

1 Table E.2 NU value (e) of trans-diindenoacene (1-4) singlet 1 Ag state and vertically 3 excited 1 Bu state using the MR-AQCC(CAS(4,4))/6-31G* method.

System State NU (e) 1 a 1 1 Ag NU 0.547 b NU(H-L) 0.230 3 a 1 Bu NU 2.376 b NU(H-L) 1.991

1 a 2 1 Ag NU 1.095 b NU(H-L) 0.672 3 a 1 Bu NU 2.455 b NU(H-L) 1.997

1 a 3 1 Ag NU 1.602 b NU(H-L) 1.048 3 a 1 Bu NU 2.563 b NU(H-L) 1.998

1 a 4 1 Ag NU 2.178 b NU(H-L) 1.456 3 a 1 Bu NU 2.722 b NU(H-L) 1.998 aSum over all NO occupations, bSum over only HONO-LUNO pair

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1 Table E.3 NU value (e) of cis-diindenoacene (5-8) singlet 1 A1 state and vertically excited 3 1 B1 state using the MR-AQCC(CAS(4,4))/6-31G* method.

System State NU (e) 1 a 5 1 A1 NU 1.414 b NU(H-L) 1.083 3 a 1 B1 NU 2.345 b NU(H-L) 1.999

1 a 6 1 A1 NU 1.923 b NU(H-L) 1.506 3 a 1 B1 NU 2.429 b NU(H-L) 2.000

1 a 7 1 A1 NU 2.280 b NU(H-L) 1.764 3 a 1 B1 NU 2.541 b NU(H-L) 2.000

1 a 8 1 A1 NU 2.550 b NU(H-L) 1.902 3 a 1 B1 NU 2.703 b NU(H-L) 2.000 aSum over all NO occupations, bSum over only HONO-LUNO pair

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1 Table E.4 NU value (e) of trans-indenoindenodi(acenothiophene) (9-12) singlet 1 Ag state 3 and vertically excited 1 Bu state using the MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method.

System State NU (e) 1 a 9 1 Ag NU 0.784 b NU(H-L) 0.362 3 a 1 Bu NU 2.461 b NU(H-L) 1.997

1 a 10 1 Ag NU 1.505 b NU(H-L) 0.979 3 a 1 Bu NU 2.552 b NU(H-L) 1.999

1 a 11 1 Ag NU 2.088 b NU(H-L) 1.451 3 a 1 Bu NU 2.670 b NU(H-L) 1.999

1 a 12 1 Ag NU 2.482 b NU(H-L) 1.623 3 a 1 Bu NU 2.834 b NU(H-L) 1.999 aSum over all NO occupations, bSum over only HONO-LUNO pair

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Figure E.1 Quinoid Kekulé and biradical resonance forms of trans-diindenoacenes (1-4), cis-diindenoacenes (5-8), and trans-indenoindenodi(acenothiophenes) (9-12).

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Figure E.2 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the 3 vertically excited 1 Bu state of trans-diindenobenzene (1-4) using the MR- AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring.

Figure E.3 Total unpaired electron density (sum over all NOs, Table E.3), in red, of the 3 vertically excited 1 B1 state of cis-diindenobenzene (5-8) using the MR- AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring.

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Figure E.4 Total unpaired electron density (sum over all NOs, Table E.4), in red, of the 3 vertically excited 1 Bu state of trans-indenoindenodi(acenothiophene) (9-12) using the 3 MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method. (isovalue=0.004 e/Bohr ). HOMA index is shown in the center of each ring.

222