Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Electronic structure and spectroscopy of radicals in the gas and condensed phases
(USC Thesis Other)
Electronic structure and spectroscopy of radicals in the gas and condensed phases
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
ELECTRONIC STRUCTURE AND SPECTROSCOPY OF RADICALS IN THE GAS AND CONDENSED PHASES by Piotr Adam Pieniazek A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMISTRY) August 2008 Copyright 2008 Piotr Adam Pieniazek Acknowledgements The past five years have been a truly wonderful experience. I have learned a lot, ran a couple of marathons, and realized that there is still much more to be learned. All this was made possible by the people around me. First of all, I would like thank my teachers and friends, Anna Krylov and Stephen Bradforth. Steve was always very patient with listening to my half-baked ideas, and telling me how to make them better. Anna taught me, that behind every rule of thumb, there are always basic physical principles. She never let me finish anything, before I was able to explain the connection. From Curt Wittig I have learned that most physics can be couched in terms of a two- level system. I tried to apply this principle throughout my work. If it was not for the recruiting effort of Hanna Reisler, I would have probably never come to USC. I thank her for that. This work has benefited from collaborations with other groups. David Sherrill and Steve Arnstein from GeorgiaTech helped with the benchmarking of the EOM-IP-CCSD method. The work on water ionization was done in collaboration with Pavel Jungwirth from the Academy of Sciences of Czech Republic. Eric Sundstrom, our summer under- graduate students, spearheaded large water cluster calculations. When I first came to the Krylov lab, I really had no idea what I was doing, and the QChem input looked quite intimidating. Ana-Maria Cristian, Lyuda Slipchenko, and Sergey Levchenko were helpful in these initial stages. Since that time, people have come and gone, and each of them has contributed in their own way to my work. Vitalii Vanovschi and Evgeny Epifanovsky always knew how to solve my numerous computer issues. Łukasz Kozioł and Vadim Mozhayskiy always made the life in SSC409 very ii exciting. Kadir Diri and Melania Oana, became close friends of mine. It is Melania that made my trip to Rome so wonderful. Kadir redefined the term working lunch for me. Our most recent group member, Prashant Uday Manohar was the one who explained the funky coupled-cluster diagrams to me. I have also enjoyed interacting with the members of the Bradforth group. Chris Elles, Chris Rivera, and Tom Zhang were full of insightful comments on the nature of the excited and ionized states in condensed phases. Diana Warren always made sure our group meetings were on time. Christi Chester answered all my DNA questions. I still do believe that it is just like H + 2 ! I would like to thank my friends, who have made my life much more pleasant. Roman always made sure things were never boring. Łukasz Rajchel is my L A T E X guru. Thanks to him I was able to present my work at Oakland and Warsaw universities. I have run my miles with Zachary Baker. Zvi Topol showed me many interesting places in Los Angeles. Iwona Hiszpanski and her kids, Krzysztof and Ania, were my family away from home. Lastly, I thank my family, my parents Lech and Bo˙ zena, my grandparents Jan and Czesława, my grandmother Zofia, and my brother Paweł. Without their unyielding sup- port none of this would be possible. iii Table of Contents Acknowledgements ii List of Tables vii List of Figures x Abstract xvii Chapter 1: Introduction 1 1.1 1+16= 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Methodological challenges . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Molecular systems studied . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Chapter 1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter 2: Methodology 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Equation-of-motion for ionized states formalism . . . . . . . . . . . . . 14 2.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Energy computation . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.3 Gradient calculation . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Molecular orbital theory of dimers . . . . . . . . . . . . . . . . . . . . 22 2.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Equivalent fragments . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.3 Nonequivalent fragments . . . . . . . . . . . . . . . . . . . . . 28 2.3.4 Localized states . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 Chapter 2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Chapter 3: Benchmark full configuration interaction and EOM-IP-CCSD results for prototypical charge transfer systems: noncovalent ionized dimers 35 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.1 EOM-CC approach to open-shell species . . . . . . . . . . . . 39 3.3.2 Generalized Mulliken-Hush model . . . . . . . . . . . . . . . . 42 3.3.3 The CT reaction coordinates . . . . . . . . . . . . . . . . . . . 44 iv 3.3.4 Computational details . . . . . . . . . . . . . . . . . . . . . . 45 3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.1 (He 2 ) + dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.2 (H 2 ) + 2 dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4.3 (Be-BH) + dimer . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4.4 (BH-H 2 ) + dimer . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.5 (LiH) + 2 dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4.6 (C 2 H 4 ) + 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.6 Chapter 3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter 4: Electronic structure of the benzene dimer cation 77 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3 Prerequisites: Bonding in the neutral benzene dimer and benzene cation 81 4.4 Electronic structure of ionized non-covalent dimers . . . . . . . . . . . 83 4.4.1 Orbital and state nomenclature for the benzene dimer cation . . 83 4.5 Theoretical methods and computational details . . . . . . . . . . . . . 85 4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.6.1 Electronic structure of the benzene cation . . . . . . . . . . . . 89 4.6.2 Potential energy scans, structural relaxation and binding ener- gies of the dimer cation isomers in the ground electronic state . 92 4.6.3 Electronic states of the t-shaped isomer . . . . . . . . . . . . . 94 4.6.4 Electronic states of sandwich isomer . . . . . . . . . . . . . . . 99 4.6.5 Electronic states of displaced sandwich isomers . . . . . . . . . 106 4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.9 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.9.2 Computational details . . . . . . . . . . . . . . . . . . . . . . 113 4.9.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.9.4 The t-shaped isomer . . . . . . . . . . . . . . . . . . . . . . . 116 4.9.5 Displaced sandwich . . . . . . . . . . . . . . . . . . . . . . . 121 4.9.6 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.9.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.10 Chapter 4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Chapter 5: Electronic structure of the water dimer cation 131 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.3 Theoretical methods and computational details . . . . . . . . . . . . . 134 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 v 5.4.1 Monomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.4.2 Dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.7 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.7.2 Dimer rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.7.3 Pentamer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.8 Chapter 5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Chapter 6: Spectroscopy of the cyano radical in an aqueous environment 177 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.3.2 The cyano radical - water dimer calculations . . . . . . . . . . 182 6.3.3 Bulk phase calculations . . . . . . . . . . . . . . . . . . . . . . 185 6.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.4.1 The cyano radical - water dimer . . . . . . . . . . . . . . . . . 187 6.4.2 Bulk phase calculations . . . . . . . . . . . . . . . . . . . . . . 195 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.6 Chapter 6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Chapter 7: Future work 215 7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.2 New ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.3 Chapter 7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Bibliography 221 Appendices Appendix A: EOM-IP-CC jobs in Q-CHEM . . . . . . . . . . . . . . . . . . 247 Appendix B: TPA calculation using Dalton . . . . . . . . . . . . . . . . . . . 251 Appendix C: Programmable EOM-IP-CC expressions . . . . . . . . . . . . . 256 Appendix D: Electron transfer rate calculation . . . . . . . . . . . . . . . . . 264 vi List of Tables 3.1 Total energy (hartree), energy splitting (cm − 1 ) between Σ + u and Σ + g states, and the transition dipole moment (au) for He + 2 calculated by EOM-IP-CCSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Total energy (hartree), energy splitting (cm − 1 ) between Σ + u and Σ + g states, and the transition dipole moment (au) for He + 2 calculated by FCI 49 3.3 Total energy (hartree), energy splitting (cm − 1 ) between Σ + u and Σ + g states, and the transition dipole moment (au) for He + 2 using open-shell reference in aug-cc-pVTZ basis set. . . . . . . . . . . . . . . . . . . . 52 3.4 Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for (H 2 ) + 2 at 3.0 ˚ A separation in aug-cc-pVTZ basis set. The R=0.4 data set is omitted from the Table. . . . . . . . . . . . . . . . . . . . . . . 55 3.5 Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for (H 2 ) + 2 at 5.0 ˚ A separation in aug-cc-pVTZ basis set. The R=0.4 data set is omitted from the Table. . . . . . . . . . . . . . . . . . . . . . . 56 3.6 Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for linear (Be-BH) + at 5.0 ˚ A separation in aug-cc-pVDZ basis set. The charge pertains to the Be fragment. . . . . . . . . . . . . . . . . . . . 60 3.7 Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for t-shaped (BH-H 2 ) + at 3.0 ˚ A separation in aug-cc-pVDZ basis set. The charge pertains to the BH fragment. The R=0.25 and R=0.75 data sets are omitted from the Table. . . . . . . . . . . . . . . . . . . . . . 63 3.8 Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for (LiH) + 2 at 4.0 ˚ A separation in 6-31+G basis set. . . . . . . . . . . . 66 3.9 Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for (C 2 H 4 ) + 2 at 4.0 ˚ A separation in 6-31+G basis set. . . . . . . . . . . 67 vii 4.1 Vertical ionization energies (eV) of benzene calculates using EOM-IP- CCSD/6-31+G* with frozen core. . . . . . . . . . . . . . . . . . . . . 90 4.2 Energies (eV), transition dipole moments (a.u.), and oscillator strengths for the transitions from the ground state benzene cation at various geome- tries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3 Estimated binding energies of (C 6 H 6 ) + 2 at various configurations. . . . . 93 4.4 IEs (eV), transition dipole moments (a.u.), and oscillator strengths of t-shaped (C 6 H 6 ) + 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 IEs (eV), transition dipole moments (a.u.), and oscillator strengths of the sandwich and the displaced sandwich (C 6 H 6 ) + 2 isomers. . . . . . . . 100 4.6 IEs (eV), transition dipole moments (a.u.), and oscillator strengths of the relaxed sandwich and the relaxed displaced sandwich (C 6 H 6 ) + 2 isomers. 101 4.7 Binding energies (kcal/mol) of (C 6 H 6 ) + 2 at various configurations. Dis- sociation limit is described as a neutral (X 1 A 1 ) and an acute (X 2 B 2g ) geometry ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.8 IEs (eV), transition dipole moments (a.u.), and oscillator strengths of the optimized t-shaped (C 6 H 6 ) + 2 obtained using EOM-IP-CCSD/6-31+G*. IE(2,3) values were obtained using EOM-IP-CC(2,3). . . . . . . . . . 117 4.9 IEs (eV), transition dipole moments (a.u.), and oscillator strengths of the sandwich and the displaced sandwich (C 6 H 6 ) + 2 isomers obtained using EOM-IP-CCSD/6-31+G*. IE(2,3) values were obtained using EOM-IP- CC(2,3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.1 Ionization energies (eV) of H 2 O, OH − , and H 3 O + . . . . . . . . . . . . 139 5.2 Excitation energies (eV) and transition properties (a.u.) of H 2 O + , OH, and H 3 O 2+ . A 6-311++G** basis set was used throughout. . . . . . . . 140 5.3 Energies (kcal/mol) of (H 2 O) + 2 relative to the proton-transferred geome- try calculated using wave function based methods. A 6-311++G** basis set was used throughout. . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.4 Energies (kcal/mol) of (H 2 O) + 2 relative to the proton-transferred geom- etry calculated using DFT methods. . . . . . . . . . . . . . . . . . . . 145 5.5 Excitation energies (eV) and transition properties (a.u.) of the (H 2 O) + 2 cation at the geometry of the neutral. All transitions are to the 2a 00 (b 1 /0) orbital. A 6-311++G** basis set was used throughout. . . . . . . . . . 150 viii 5.6 Excitation energies (eV) and transition properties (a.u.) of the (H 2 O) + 2 cation at the C 1 proton-transferred geometry. All transitions are to the 10a 1 (π oop /0) orbital. A 6-311++G** basis set was used throughout. . . 154 5.7 Excitation energies (eV) and transition properties (a.u.) of the (H 2 O) + 2 cation at the C s proton-transferred geometry. All transitions are to the 10a 1 (π oop /0) orbital. A 6-311++G** basis set was used throughout. . . 155 6.1 The ground state equilibrium properties of the cyano radical calculated at different levels of theory using cc-pVTZ basis set. . . . . . . . . . . 183 6.2 Basic SPC/E and TIP5P-E water model parameters. . . . . . . . . . . . 186 6.3 The Lennard - Jones and electrostatic interaction parameters for CN··· H 2 O. Set A has been taken from the Amber force field and Set B has been developed in this work. . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.4 Coordination number of cyano radical in water using different defini- tions of the first solvation shell. . . . . . . . . . . . . . . . . . . . . . . 198 6.5 A (A 2 Π ← X 2 Σ + ) and B (B 2 Σ + ← X 2 Σ + ) band parameters. . . . . . . . 202 C.1 Programmable expressions for the right σ -vectors in EOM-IP-CC(2,3) model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 C.2 R-independenet intermediates used in the energy EOM-IP-CCSD and EOM-IP-CC(2,3) expressions. . . . . . . . . . . . . . . . . . . . . . . 259 C.3 R-dependent intermediates used in the energy EOM-IP-CCSD and EOM- IP-CC(2,3) expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . 260 C.4 Programmable expressions for unrelaxed EOM-IP-CCSD density matri- ces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 C.5 Intermediate used in the unrealxed EOM-IP-CCSD density matrices cal- culation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 C.6 EOM-IP-CCSD amplitude responseξ -vector programmable expressions. 262 C.7 Intermediates used in theξ -vector calculation. . . . . . . . . . . . . . . 263 ix List of Figures 1.1 Ionization induces a proton transfer reaction in the water dimer, a pro- cess that does not occur in an isolated molecule. The potential energy surface along the selected intermolecular coordinates is shown. The process begins at point A. The molecules approach each other moving to point B. Between points B and C the proton transfer occurs. . . . . . 4 1.2 A snapshot from a molecular dynamics simulation of the cyano radical in water. The simulation allowed sampling of different configurations of water around the radical and and a computation its electronic spectrum in the condensed phase. . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 States with N-1/N+1 electrons can be described based on the N-electron reference using EOM-IP/EA-CC wave functions. EOM-IP-CC is par- ticularly suitable for states derived by ionization from an orbital occu- pied in the reference, while EOM-EA-CCSD describes states derived by attaching an electron to a virtual orbital in the wave function. . . . . . . 19 2.2 FMOs and DMOs of an AB dimer. . . . . . . . . . . . . . . . . . . . 24 3.1 Open-shell doublet wave functions can be described by several EOM approaches using different references/excitation operators. The EOM- IP method employs a well-behaved closed-shell reference. . . . . . . . 41 3.2 Diabatic (dashed line) and adiabatic potential energy surfaces for elec- tron transfer reactions. Diabatic states correspond to reactant and prod- uct electronic wave functions, i.e. the charge fully localized on one of the species, while adiabatic states are eigenfunctions of the electronic Hamiltonian. Marcus theory relates the coupling between diabatic states to the rate of electron/hole transfer process. . . . . . . . . . . . . . . . 43 3.3 Electronic coupling in the helium dimer as a function of distance using FCI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Electronic coupling in He + 2 at 2.5 ˚ A (a) and 5.0 ˚ A (b) calculated by FCI (squares) and EOM-IP-CCSD (diamonds). . . . . . . . . . . . . . . . . 50 3.5 The transition dipole moment in He + 2 at 2.5 ˚ A (a) and 5.0 ˚ A (b) calcu- lated by FCI (squares) and EOM-IP-CCSD (diamonds). . . . . . . . . . 51 x 3.6 Error in (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge, (e) transition dipole moment, and (f) diabatic coupling in (H 2 ) + dimer at 3.0 ˚ A separation. EOM-IP-CCSD/aug- cc-pVTZ (solid line) and EOM-EE-CCSD/aug-cc-pVTZ (dotted line) results are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.7 Error in (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge, (e) transition dipole moment, and (f) diabatic coupling in (H 2 ) + at 5.0 ˚ A separation. EOM-IP-CCSD/aug- cc-pVTZ (solid line) and EOM-EE-CCSD/aug-cc-pVTZ (dotted line) results are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.8 Error in (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge, (e) transition dipole moment, and (f) diabatic coupling in (Be-BH) + . EOM-IP-CCSD/aug-cc-pVDZ (solid line) and EOM-EE-CCSD/aug-cc-pVDZ (dotted line) results are shown. The charge pertains to the Be fragment. . . . . . . . . . . . . . . . . . 59 3.9 Error in (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge, (e) transition dipole moment, and (f) diabatic coupling in (BH-H 2 ) + . EOM-IP-CCSD/aug-cc-pVDZ (solid line) and EOM-EE-CCSD/aug-cc-pVDZ (dotted line) results are shown. The charge pertains to the BH fragment. . . . . . . . . . . . . . . . . . 62 3.10 Error in (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge, (e) transition dipole moment, and (f) diabatic coupling in (LiH) + 2 . EOM-IP-CCSD/6-31+G results are shown. 65 3.11 Changes in charge distribution and PES scans along the CT coordinate in the ground state of (C 2 H 4 ) + 2 at 4 ˚ A (panels (a) and (c)) and 6 ˚ A sepa- ration (panels (b) and (d)). . . . . . . . . . . . . . . . . . . . . . . . . 68 3.12 Differences against EOM-IP-CCSD(3h2p) for (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge in (C 2 H 4 ) + 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1 The five highest occupied orbitals of neutral benzene. Both D 6h and D 2h symmetry labels are given. . . . . . . . . . . . . . . . . . . . . . 83 4.2 Highest occupied molecular orbitals of (C 6 H 6 ) 2 at (a) D 6h ,(b) x- and (c) y-displaced sandwich configurations. Orbital energy increases from bottom to top. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 xi 4.3 Highest occupied molecular orbitals of t-shaped (C 6 H 6 ) 2 . Orbital energy increases from bottom to top. . . . . . . . . . . . . . . . . . . . . . . 87 4.4 Estimated binding energies of different (C 6 H 6 ) + 2 isomers. . . . . . . . . 92 4.5 PES scan along the x and y sliding coordinates at 3.0 (squares), 3.1 (romboids), and 3.2 ˚ A (squares) interplanar separations. . . . . . . . . . 95 4.6 Potential energy profiles along interfragment separation in t-shaped (C 6 H 6 ) + 2 . 96 4.7 Electronic states ordering and oscillator strengths of (C 6 H 6 ) + 2 at neutral (a) and relaxed (b) t-shaped configuration. Empty bars denote symmetry forbidden transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.8 Potential energy profiles along interfragment separation in sandwich (C 6 H 6 ) + 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.9 Electronic states ordering and oscillator strengths of (C 6 H 6 ) + 2 at sand- wich (a), and x- and y-displaced sandwich, (b) and (c), configurations. Empty bars denote symmetry forbidden transitions. . . . . . . . . . . . 103 4.10 Electronic states ordering and oscillator strengths of relaxed (C 6 H 6 ) + 2 at sandwich (a), and x- and y-displaced sandwich, (b) and (c), configura- tions. Empty bars denote symmetry forbidden transitions. . . . . . . . . 104 4.11 Potential energy profiles along interfragment sliding in (a) x- and (b) y-displaced sandwich (C 6 H 6 ) + 2 . Interplanar separation was held fixed at 3.1 ˚ A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.12 The geometry of the benzene ring in the (a) x-displaced and (b) y- displaced sandwich (C 6 H 6 ) + 2 (underlined numbers). The geometry of an isolated ring in a (a)π o g and (b)π a g state is also given. Angles are in degree and distances are in ˚ Angstrom. . . . . . . . . . . . . . . . . . . 115 4.13 The effect of the intermolecular relaxation and JT distortion on the four lowest electronic states of the t-shaped (C 6 H 6 ) + 2 . The energies of states relative to the vertical configuration are given in wavenumbers. The partial charge on the stem fragment is given in italics. . . . . . . . . . 118 4.14 The geometry of the (a) stem and (b) top fragments in the t-shaped (C 6 H 6 ) + 2 (underlined numbers). The geometry of an isolated ring in aπ o g state is also given. Angles are in degree and distances are in ˚ Angstrom. 119 xii 4.15 The effect of the intermolecular relaxation and JT distortion on the four lowest electronic states of (a) x-displaced and (b) y-displaced sandwich isomers of (C 6 H 6 ) + 2 . The energies of states relative to the vertical con- figuration are given in wavenumbers. . . . . . . . . . . . . . . . . . . 122 5.1 Three highest occupied MOs of (a) H 2 O, (b) OH − , and (c) H 3 O + . The geometric frame is rotated for each orbital to best show the orbital char- acter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.2 Geometries of (a) vertical neutral (H 2 O) 2 , (b) proton-transferred (H 2 O) + 2 , and (c) hemibonded(H 2 O) + 2 . Oxygen-oxygen distance has been marked on the plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.3 Six highest occupied MOs of the neutral water dimer. Ionization ener- gies were calculated using EOM-IP-CCSD/6-311++G**. The num- bers on the left are the NBO charge on the H-bond donor fragment calculated using EOM-IP-CCSD (upper number) and CCSD/EOM-EE- CCSD (lower number) wave functions. The geometric frame is rotated for each orbital to best show orbital character. The NBO analysis of the reference CCSD/6-311++G** wave function showed -0.012 charge on the H-bond donor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.4 Six highest occupied MOs of the neutral water dimer at the C s and C 1 proton-transferred configurations. Ionization energies were calculated using EOM-IP-CCSD/6-311++G**. The NBO charge on the hydroxyl radical calculated using EOM-IP-CCSD wave functions is given below. The geometric frame is rotated for each orbital to best show orbital char- acter. The NBO analysis of the reference CCSD/6-311++G** wave function showed -0.771 and -0.770 charge on the hydroxyl radical in theC s andC 1 structures, respectively. . . . . . . . . . . . . . . . . . . 149 5.5 Electronic states ordering and transition dipole moments of water dimer cation at the neutral configuration calculated using (a) EOM-IP-CCSD/6- 311++G** and (b) EOM-EE-CCSD/6-311++G** . All transitions are to theb 1 /0 orbital. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.6 Electronic states ordering and transition dipole moments of water dimer cation at theC 1 (a) andC s symmetry proton-transferred configurations calculated using EOM-IP-CCSD/6-311++G**. All transitions are to the π oop /0 orbital. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 xiii 5.7 (a) The ground state PES scan for the proton transfer reaction. The x- axis is the oxygen-oxygen distance and the y-axis is the distance between the transferring proton and the accepting oxygen.At each point a con- strained geometry optimization was conducted Points A and C corre- spond to the neutral and proton transferred geometries, respectively.Point B marks the start of the proton transfer. The black line is the steep- est descent path. (b) -(d) Vertical excitation energies, transition dipole moments and oscillator strengths along the reaction coordinate. All cal- culations were done using EOM-IP-CCSD/6-311++G**. . . . . . . . . 159 5.8 Evolution of the electronic spectrum of the water dimer cation along the reaction path at points A (red solid line), B (green dotted line), and C (blue dashed line). Points A and C correspond to vertical neutral and proton-transferred geometries. Point B marks the start of the proton transfer. 0.2 eV full width at half-maximum was assumed. See Fig. 5.7 for details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.9 Natural charge on the H-bond donor fragment in water dimer cation in six lowest electronic. Rotation is around the oxygen — oxygen axis, and the angle0 corresponds to the neutral dimer configuration. All cal- culations were done using EOM-IP-CCSD/6-311++G**.. . . . . . . . . 168 5.10 (a) Excitation energies and (b) transition dipole moments in water dimer cation. Rotation is around the oxygen — oxygen axis, and the angle 0 corresponds to the neutral dimer configuration. All calculations were done using EOM-IP-CCSD/6-311++G**. . . . . . . . . . . . . . . . . 170 5.11 MP2/6-311++G** optimized geometry of the water pentamer. . . . . . 171 5.12 NBO charge on the central fragment in a water pentamer cation as a function of the excitation energy. All calculations were done using EOM-IP-CCSD/6-31+G*. . . . . . . . . . . . . . . . . . . . . . . . . 172 6.1 Geometries of the cyano radical - water dimer used for the PES scans. Monomers are frozen at their gas phase geometries. Bond lengths are in ˚ A and angles in degrees. (a) Planar C 2v structure, (b) C s symmetry structure, C-N··· H-O formation is collinear. The distance between the fragments was varied from 2.00 ˚ A and 2.25 ˚ A up to 6.50 ˚ A, for C 2v and C s symmetry structures, respectively. . . . . . . . . . . . . . . . . . . 184 xiv 6.2 The CP corrected dimer potential energy curves along interfragment separation coordinate. (a) C 2v symmetry structure. (b) C s symme- try structure. The energies were obtained at the CCSD/6-311++G** (blue dotted line), CCSD(T)/6-311++G** (red dotted line), CCSD/aug- cc-pVTZ (blue solid line), and CCSD(T)/aug-cc-pVTZ (red solid line line) levels of theory. For comparison, the CCSD(T)/aug-cc-pVTZ scan without BSSE correction is shown (solid black line). . . . . . . . . . . 189 6.3 Comparison of the ab initio and molecular mechanics dimer potential energy surfaces. (a) C 2v symmetry structure. (b) C s symmetry struc- ture. The energies were obtained using: CCSD(T)/aug-cc-pVTZ (black solid line), Amber’s Set A and SPC/E Lennard-Jones parameters with charges reproducing gas phase dipole moments (red solid line) and con- densed phase charges (red dotted line), newly derived Set B and SPC/E Lennard-Jones parameters with charges reproducing gas phase dipole moments (blue solid line) and condensed phase charges (blue dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.4 Vertical excitation energies as a function of monomer separation cal- culated by full EOM-CCSD (black solid line), full B3LYP (blue solid line). Results of calculations with water replaced by point charges repro- ducing the water gas phase dipole moment (EOM-CCSD – black dotted line, B3LYP – blue dotted line) and charges used in SPC/E water model (EOM-CCSD – black dashed line, B3LYP – blue dashed line) are also shown. Panel (a) the A 2 Π state along the C 2v scan (only the in-plane B 1 component is shown, as the B 2 component is nearly degenerate), (b) the B 2 Σ + state along the C 2v scan, (c) the A 2 Π state along the C s scan (both the A’ and A” components are shown), (d) the B 2 Σ + state along the C s scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.5 Natural charge of the cyano radical in the ground and excited states of the cyano radical - water dimer. The CCSD/6-311++G** and EOM- CCSD/6-311++G** wave functions are analyzed along C 2v and C s scans, panels (a) and (b) respectively. The X 2 Σ + ground state (solid line), the B 2 Σ + state (dotted line), and the A 2 Π state (dashed line). . . . . . . . 196 6.6 Radial distribution functions of oxygen (a) and hydrogen (b) around the geometric center of the cyano radical. The bin size is 0.2 ˚ A. The curves are: Set B + SPC/E (solid line), Set A + SPC/E (dotted line), Set B + TIP5P/E (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . 197 xv 6.7 Angular distribution of oxygen around the cyano radical. The angular variable is the angle between CN and the geometric center - oxygen (hydrogen) vectors. Radial range is successively increased to include between (a) one and (d) four atoms. The curves are: Set B + SPC/E (solid line), Set A + SPC/E (dotted line), Set B + TIP5P/E (dashed line). 200 6.8 Angular distribution of hydrogen around the cyano radical. The angular variable is the angle between CN and the geometric center - oxygen (hydrogen) vectors. Radial range is successively increased to include between (a) one and (d) four atoms. The curves are: Set B + SPC/E (solid line), Set A + SPC/E (dotted line), Set B + TIP5P/E (dashed line). 201 6.9 Simulated spectra of the cyano radical in water: (a) the A← X transi- tion; (b) the B← X transition. The spectra are: Set B + SPC/E (solid line), Set A + SPC/E (dotted line), Set B + TIP5P/E (dashed line), Set B + TIP5P/E +TD-DFT/cc-pVTZ. Unless otherwise indicated the energies were computed using EOM-CCSD/cc-pVTZ (dash-dot line). Wiggles are a result of a finite sample of excitation energies. . . . . . . . . . . . 203 6.10 Potential energies of the cyano radical - water dimer in different elec- tronic states of the cyano radical in (a) the C 2v symmetry and (b) the C s symmetry structures. The lines are: X 2 Σ + CN··· X 1 A 1 H 2 O (black line), A 2 Π CN··· X 1 A 1 H 2 O (red line), B 2 Σ + CN··· X 1 A 1 H 2 O (violet line). Dot- ted curves were obtained by replacing water with point charges derived to reproduce its gas phase dipole moment. Only the B 1 and A’ compo- nents of the A 2 Π state in C 2v and C s structures are shown. The ground and excited state energies were calculated using CCSD/6-311++G** and EOM-CCSD/6-311++G** respectively. The CP correction is not included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 D.1 Electron transfer reaction in the sandwich benzene dimer cation. The electron hops from fragment A to fragment B. . . . . . . . . . . . . . . 264 D.2 Marcus theory picture of an electron transfer reaction. AB + and A + B are reactant and product potential energy surfaces, respectively. Δ G 0 r andΔ G† are reaction and activation free energies, respectively. λ is the reorganization energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 265 xvi Abstract Understanding radicals is central to understanding the mechanics of chemical change. In this work, quantitative and qualitative methods for the description of the electronic spectroscopy of open-shell species in the gas and condensed phases are developed. Dimer radical cations are important model systems for studying ionization and charge transfer processes. Achieving a reliable theoretical description for these systems is difficult due to the symmetry-breaking of the open-shell Hartree-Fock reference and near-degeneracies. The equation-of-motion coupled-cluster for ionization potentials (EOM-IP-CC) method implemented and benchmarked here remedies these problems by using the corresponding closed-shell reference. It is combined with a qualitative Dimer Molecular Orbitals — Linear Combination of Fragment Molecular Orbitals (DMO- LCFMO) framework to elucidate the electronic states of benzene dimer and water dimer cations. In the former case the geometry of the ground state changes from a T-shaped neutral to a displaced sandwich cation. Charge resonance bands, whose spectral location and intensity are sensitive to the geometry of the system, are present in both isomers. A reordering of electronic states in the local excitation region in the sandwich structure is observed. For the water dimer vertical ionization occurs from the hydrogen-bond donor and induces a proton transfer reaction. The evolution of the electronic spec- troscopy along the reaction coordinate is studied. The initial spectrum consists of mixed local and charge-transfer excitations, while the spectrum of the final product resembles that of the hydroxyl radical product. An empirical correction to the density functional theory (DFT) approach used in larger condensed phase Car-Parrinello simulations is benchmarked. xvii A fully condensed phase treatment is applied for the cyano radical in water. Con- flicting assignments appeared in the literature as to the position of the absorption band of aqueous CN. Classical molecular dynamics simulations are combined with accurate ab initio calculations to unravel the effect of water on the two lowest electronic states of cyano radical. Accurate dimer calculations are used to parametrize the force field. A minor blue-shift of the intervalence transition energies is observed, contradicting previ- ous experimental assignments. xviii Chapter 1: Introduction 1.1 1+16= 2 Radicals are produced whenever chemical bonds are broken, electrons added or removed from molecules, or when species are electronically excited. They can be loosely defined as species with unpaired electrons. Even the molecular oxygen we breath is a radi- cal! Tyrosine and nitric oxide radicals are important in the cell regulatory mechanisms. Radicals from photolyzed chlorofluorocarbons contribute to the depletion of the ozone layer. Combustion chemistry is that of free radicals. Thus, understanding the nature of chemical change requires us to consider species with unpaired electrons. Even the most reactive species rarely exist in nature as isolated particles. The chem- istry of the living organisms is dominated by phenomena taking place in the condensed phase, e.g. in an aqueous or protein environment. The importance of clusters and inter- faces in the atmospheric and interstellar chemistry is increasingly being recognized 1–4 . It is not a priori evident that the properties of radicals in these environments can be fully understood in terms of their gas phase properties alone. Sometimes the environ- ment becomes a part of the system, and phenomena that are not a simple sum of those of observed for constituents can occur. The title of this Section, 1+16= 2, reflects this pos- sibility of novel chemistry. Indeed, two limiting cases can be distinguished. Firstly, the surroundings may play a passive role by changing the reaction barriers and caging the reaction products. The classic example of a passive role is in nucleophilic substitution reactions. Depending on the polarity of the solvent they may proceed via a unimolec- ular or bimolecular pathway. Another example is the proton abstraction reaction from 1 acetaldehyde by hydroxyl radical 4 . The speedup observed in the gas phase reaction in the presence of water is attributed to initial formation of acetaldehyde – water complex. The water molecule does not actively participate in the abstraction reaction, but stabi- lizes the transition state. The behavior of molecules is not qualitatively different from the isolated species, and predominantly a sum of the individual components, i.e. 1+1≈ 2. Secondly, the environment may play a direct role and enable phenomena not possible for an isolated species. It can act as an energy sink, allowing the vibrational energy to flow out of a molecule before causing a chemical reaction. An illustration of this behavior is the photodissociation of H 2 SO 4 5, 6 . When an overtone of the O–H stretching vibration is excited in an isolated molecule it falls apart into H 2 O and SO 3 . This is no longer the case in an H 2 SO 4 – H 2 O complex, which dissociates into H 2 SO 4 and H 2 O. An electronically excited nucleobase quickly relaxes to a vibrationally hot ground state. This vibrational energy is then absorbed by the surroundings. In the context of the electronic properties, excimer formation, charge-transfer excitations, exciton and charge transfer processes, are other examples of phenomena that directly incorporate other molecules and are not present in the isolated species. Their understanding is contingent on considering the sur- rounding, the simplest example of which is a dimer. The distinction between the passive and active role of the environment depends on the process being studied and determines the theoretical methods used to study it. In the former case the environment can often be modeled by a Brownian bath, an electrostatic continuum, or a set of point charges. If the solvent plays an active role, the distinction between the solute and its environment is not well-defined and it must be explicitly included in the theoretical model. Incorporation of the environment into the computational treatment of radicals has been scarce due to the high level of theory needed for their description, while their reac- tive nature has hampered direct experimental observations. The focus of this work is the theoretical description and the electronic properties of doublet dimer systems. The 2 theoretical aspect of the work involved developing methodology capable of describing open-shell dimer cation systems. It was subsequently benchmarked and applied to pro- totypical systems of practical relevance. Two of them, the benzene dimer cation and water dimer cation, are examples of species where the presence of the other molecule plays an active role. It affects the nature of the ground and excited states, leading to structural or chemical changes. An example of the latter is the proton transfer reaction in water dimer cation depicted in Fig. 1.1. In the cyano radical problem, water plays a passive role by electrostatically changing its excitation energies. The magnitude of the effect was of key importance to the spectroscopic determination of CN aq as a reaction product. There are two advantages to choosing dimers as models for processes in more com- plex situations. Firstly, they are amenable to sophisticated computational methods and high resolution spectroscopic techniques. Secondly, they allow a qualitative analysis of the relevant phenomena, providing the intellectual and mathematical framework for considering more complex systems. The adequacy of the dimer model should always be validated, but this is not always possible without a prior detailed investigation. Even if it proves to be an inadequate model, it can provide invaluable insight needed in the necessarily more approximate modeling of larger systems. Thus, dimers are the first, yet essential step in understanding the chemistry and physics of radicals interacting with the environment. 1.2 Methodological challenges An important part of any computational study is the selection of an appropriate com- putational method. By their very nature ab initio methods, which solve the electronic 3 2.4 2.6 2.8 3.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 C B r O-O, Å r O-H, Å 0 4 8 12 16 20 24 A kcal/mol (A) (B) (C) Figure 1.1: Ionization induces a proton transfer reaction in the water dimer, a process that does not occur in an isolated molecule. The potential energy surface along the selected intermolecular coordinates is shown. The process begins at point A. The molecules approach each other moving to point B. Between points B and C the proton transfer occurs. 4 Schr¨ odinger equation, are in principle able to describe the physics behind most chemi- cal phenomena. The inclusion of nonadiabatic effects and relativistic effects is possible, albeit not yet standard. A bigger problem is the approximate character of the solution to the electronic problem. The approximations may introduce artifacts, which are often difficult to distinguish from the real features of the system. The computational studies of doublet dimer cations are marred by the symmetry breaking problem. It is most readily manifested in the case of open-shell symmetric dimers, i.e., when the donor and acceptor moieties are indistinguishable. Consider the example of the ethylene dimer cation. Two identical ethylene moieties are placed with their molecular planes 4 ˚ A apart forming a D 2h symmetry structure. Two solutions of the Hartree-Fock (HF) equations for the system can be found. The first one is lower in energy by 5 kcal/mol and it is the best HF solution in a variational sense. However, examination of this wave function reveals that the positive charge is localized on one of the fragments. This is an unphysical result, as the symmetry of the system requires that the electrons be equally distributed between the fragments. The second, higher energy solution exhibits the proper symmetry, but is not optimal in a variational sense. One thus faces a dilemma of which solution to choose 7 . This problem also exists in dimers composed of different fragments. A similar effect is achieved when one of the ethylene dimers is slightly distorted, thus forming a C 2v symmetry dimer. In this situation also two solutions of the HF equations can be found, differing in the extent of charge delocal- ization and energy. The lower energy can by related to the symmetry broken one at the symmetric configuration. Both symmetry and variational arguments suggest that this is the correct solution. However, even in these configurations benchmarking reveals that the extent of charge localization is excessive. The computed vibrational frequencies, excitation energies, transition dipole moments and other properties do not correspond well with experimental observables, even if correlation is taken into account. How then, 5 one can properly describe those systems? How can one decide, whether the minimum energy structure of the system features identical fragments or not? The Koopmans theorem provides an elegant way to obtain the cation wave func- tion from the corresponding neutral wave function by removing an electron. The real- ization of this Koopmans-like idea within the coupled-cluster (CC) framework is the essence of the equation-of-motion coupled-cluster model for ionized systems (EOM- IP-CC) method, which is a central methodological development of this work. Energy calculations at the EOM-IP-CCSD and EOM-IP-CC(2,3) have been benchmarked and implemented within theQ-CHEM electronic structure program 8 . We also implemented analytic gradient calculations at the EOM-IP-CCSD level. Gradient codes are essential because computational chemistry studies usually require not only single-point energies, but also characterization of the potential energy surface stationary points (e.g., equilibrium structures and transition states), as well as a variety of the state and transition properties. Properties are calculated as contrac- tions of appropriate operators with the same density matrices that are used in the gradi- ent calculations. Numerical force calculation, which requires only the computation of single-point energies, is computationally inefficient and susceptible to errors associated with the step size. For instance, a finite differences gradient calculation for a system of N atoms requires 3N single point energy calculations, while the cost of analytic gradient evaluation is only twice the cost of a single point energy calculation. In the case of transition properties, the finite differences approach requires the computation of the derivative of the overlap of the two states with respect to an external electric field. Such procedures are rarely implemented within standard electronic structure packages. Additional benefit of analytic gradient code is that the correlated electron density can be investigated by different analysis methods, e.g., the Natural Bond Orbitals (NBO) analysis 9 . 6 The interpretative power of the electronic structure theory is just as important as its ability to produce accurate and reliable numerical results. To understand the elec- tronic structure of the ionized non-covalent dimers, we developed the Dimer Molecu- lar Orbitals — Linear Combination of Fragment Molecular Orbitals (DMO-LCFMO) framework. Similarly to Molecular Orbitals – Linear Combination of Atomic Orbitals (MO-LCAO) method, it explains the dimer results in terms of its fragments, as well as considers more complex systems. DMO-LCFMO analysis also allows for a very robust nomenclature correlating the dimer and monomer states. Symmetry labels prove rather cumbersome and mask the underlying character of the states. 1.3 Molecular systems studied An important aspect of this work was understanding the electronic spectroscopy of dou- blet radicals, which are formed whenever a single bond is broken, as well as upon ioniza- tion of or electron attachment to closed-shell molecules. The insight gained in studying model systems is invaluable in understanding real-world complex molecular ensembles. In the context of ionized states the issue of charge delocalization becomes very impor- tant, as charge delocalized and localized systems possess different spectral signatures. Establishing the extent of charge delocalization is crucial in defining the active part of the system. In a condensed environment the coupling between the molecules favors delocalization. This is counteracted by polarization of the environment and the struc- tural disorder, which favor a localized charge. In many cases these limit the charge delocalization to a dimer or even monomer 10–15 . The particular molecular systems studied in this work present a range of practical and fundamental problems. Benzene dimer cation is a model system for studying charge transfer processes in extended aromatic systems frequently encountered in biology and 7 molecular engineering. Oxidative damage on DNA leads to facile hole transfer between stacked aromatic bases 16 . Cation dimers of polyaromatic hydrocarbons are suspected to be the source of broad extended interstellar emission 17 . Solvents used in radioactive element separation are susceptible to radiation induced ionization, which in the case of neat aromatic liquids leads to the formation of aromatic cations and dimer cations like (C 6 H 6 ) + 2 and (C 5 H 5 N) + 2 18–20 . Despite its status as a model system, the geometry and the electronic structure of the benzene dimer cation remained in question. Based on the presence of strong charge-resonance (CR) bands in the photodissociation spectrum a sandwich-like structure was postulated. Our studies confirmed this assignment, but also showed that CR bands are present in the spectrum of the t-shaped isomer. Thus, their presence is not an unequivocal proof of the structure. Furthermore, the ordering of the electronic states in the dimer changes compared to the monomer, a fact not previously explored. The dimer calculations also explore ways of experimentally monitoring the formation of the sandwich structure cation core in the condensed phase. Water dimer cation provides means for analyzing the initial steps in the ionization of liquid water, the most ubiquitous solvent in biological and technical applications. In the gas phase an electron is simply removed from a molecule. However, in the condensed phase the removed electron can be shared by a number of water molecules simultane- ously. If so, the positive charge will not reside on any particular water molecule, but will be delocalized. Clearly, the very definition of the problem is different from an isolated species. The dimer calculation proved not to be an adequate model for the entire process. Nonetheless, it allowed us to map out some of the coupling pathways that delocalize the hole in the condensed phase system. Mapping out the electronic spectrum along the pro- ton transfer coordinate is instrumental in the forthcoming ultrafast experiments aimed at observing the process in the condensed phase. Furthermore, semiempirical techniques suitable for simulations of larger systems were benchmarked. 8 Dimer cations also present a fundamental scientific problem. The bonding in a neutral species is typically on the order of 1-5 kcal/mol, while the typical energy of a chemical bond is 50 – 100 kcal/mol. The binding of the respective dimer cations is intermediate at 20 kcal/mol. From an electronic structure point of view, ionization of a neutral dimer changes formal interfragment bond order from zero to half-integer and, therefore, switches interaction from non-covalent to covalent. This has a strong effect on the strength of interaction and the structure of the dimer. Helium dimer is the weak- est bound dimer, with a binding energy of 0.0211 kcal/mol. Once it becomes ionized, the interaction energy increases to 57.7 kcal/mol. Thus, by simply removing an elec- tron a van der Waals dimer changed into a covalently bound molecule. Another way to perceive this problem is purely on electrostatic grounds. Imagine two molecules, each with a dipole moment of 1 Debye placed 3 ˚ A apart. Their dipole - dipole interaction energy is 1.1 kcal/mol. If one of the dipole is replaced by a unit monopole, the interac- tion energy rises to 7.7 kcal/mol, the regime of strong hydrogen bonds. Thus, ionization may induce fairly significant spectroscopic and structural changes in the system. Their nature is determined by the particular components of a given system and directly related to its properties. Finally, the CN – H 2 O dimer helped in understanding the condensed phase spec- troscopy of CN. A snapshot from a molecular dynamics simulation is presented in Fig. 1.2. The electronic absorption spectrum of aqueous CN has never been reported, but the knowledge of the transition was essential is interpreting the results from ultrafast studies of ICN photodissociation. As one can imagine, a static spectrum of a reactive radical in water cannot be measured due to its transient nature. Two conflicting assignments appeared in the literature. One group suggested that the energy of the B 2 Σ + ← X 2 Σ + transition is slightly blue shifted from its gas phase position, while another one assigned a strongly red shifted absorption band to this transition 21, 22 . Dimer calculations allowed 9 the parametrization of the molecular mechanics forcefield and also an assessment of the validity of the quantum mechanics/molecular mechanics (QM/MM) approach. The cal- culations confirmed the assignment of the slightly blue shifted band to the cyano radical. Figure 1.2: A snapshot from a molecular dynamics simulation of the cyano radical in water. The simulation allowed sampling of different configurations of water around the radical and and a computation its electronic spectrum in the condensed phase. 1.4 Thesis outline Chapter 2 presents the EOM-IP-CC and DMO-LCFMO methodologies used throughout this work. The section describing the molecular orbital approach to dimer spectroscopy is particularly useful for understanding further developments. Numerical comparisons of EOM-IP-CCSD with full configuration-interaction (FCI) results on benchmark sys- tems are discussed in Chapter 3. This worked established the accuracy of the method 10 and its suitability for the description of charge delocalization. The electronic properties of benzene dimer and water dimer cations are discussed in Chapters 4 and 5, respec- tively. The work on cyano radical in water is presented in Chapter 6. Finally, future research directions are outlined in Chapter 7. 11 1.5 Chapter 1 Bibliography [1] Aloisio, S. ; Francisco, J.S. Acc. Chem. Res. 2000, 33, 825. [2] Vaida, V . ; Kjaergaard, H.G. ; Feierabend, K.J. Int. Rev. Phys. Chem. 2003, 22, 203. [3] Klemperer, W. ; Vaida, V . Proc. Nat. Acad. Sci. 2006, 103, 10584. [4] V ohringer-Martinez, E. ; Hansmann, B. ; Hernandez, H. ; Francisco, J.S. ; Troe, J. ; Abel, B. Science 2007, 315, 497. [5] Miller, Y . ; Gerber, R.B. J. Am. Chem. Soc. 2006, 128, 9594. [6] Miller, Y . ; Gerber, R.B. ; Vaida, V . Geophys. Res. Lett. 2007, 34, L16820. [7] L¨ owdin, P.O. Rev. Mod. Phys. 1963, 35, 496. [8] Shao, Y . ; Molnar, L.F. ; Jung, Y . ; Kussmann, J. ; Ochsenfeld, C. ; Brown, S. ; Gilbert, A.T.B. ; Slipchenko, L.V . ; Levchenko, S.V . ; O’Neil, D. P. ; Distasio Jr., R.A. ; Lochan, R.C. ; Wang, T. ; Beran, G.J.O. ; Besley, N.A. ; Herbert, J.M. ; Lin, C.Y . ; Van V oorhis, T. ; Chien, S.H. ; Sodt, A. ; Steele, R.P. ; Rassolov, V . A. ; Maslen, P. ; Korambath, P.P. ; Adamson, R.D. ; Austin, B. ; Baker, J. ; Bird, E.F.C. ; Daschel, H. ; Doerksen, R.J. ; Drew, A. ; Dunietz, B.D. ; Dutoi, A.D. ; Furlani, T.R. ; Gwaltney, S.R. ; Heyden, A. ; Hirata, S. ; Hsu, C.-P. ; Kedziora, G.S. ; Khalliulin, R.Z. ; Klunziger, P. ; Lee, A.M. ; Liang, W.Z. ; Lotan, I. ; Nair, N. ; Peters, B. ; Proynov, E.I. ; Pieniazek, P.A. ; Rhee, Y .M. ; Ritchie, J. ; Rosta, E. ; Sherrill, C.D. ; Simmonett, A.C. ; Subotnik, J.E. ; Woodcock III, H.L. ; Zhang, W. ; Bell, A.T. ; Chakraborty, A.K. ; Chipman, D.M. ; Keil, F.J. ; Warshel, A. ; Herhe, W.J. ; Schaefer III, H.F. ; Kong, J. ; Krylov, A.I. ; Gill, P.M.W. ; Head-Gordon, M. Phys. Chem. Chem. Phys. 2006, 8, 3172. [9] Weinhold, F. ; Landis, C. R. Chem. Ed.: Res.& Pract. Eur. 2001, 2, 91. [10] Garcia, M.E. ; Pastor, G.M. ; Bennemann, K.H. Phys. Rev. B 1993, 48, 8388. [11] Kurnikov, I.V . ; Tong, G.S.M. ; Madrid, M. ; Beratan, D.N. J. Phys. Chem. B 2002, 106, 7. [12] Shkrob, I.A. ; Sauer Jr., M.C. ; Jonah, C.D. ; Takahashi, K. J. Phys. Chem. A 2002, 106, 11855. [13] Shkrob, I.A. J. Phys. Chem. A 2002, 106, 11871. [14] V oityuk, A.A. J. Chem. Phys. 2005, 122, 204904. 12 [15] Santoro, F. ; Barone, V . ; Improta, R. Proc. Nat. Acad. Sci. 2007, 104, 9931. [16] Delaney, S. ; Barton, J.K. J. Org. Chem. 2003, 68, 6475. [17] Rhee, Y .M. ; Lee, T.J. ; Gudipati, M.S. ; Allamandola, L.J. ; Head-Gordon, M. Proc. Nat. Acad. Sci. 2007, 104, 5274. [18] Inokuchi, Y . ; Naitoh, Y . ; Ohashi, K. ; Saitow, K. ; Yoshihara, K. ; Nishi, N. Chem. Phys. Lett. 1997, 269, 298. [19] Okamoto, K. ; Saeki, A. ; Kozawa, T. ; Yoshida, Y . ; Tagawa, S. Chem. Lett. 2003, 32, 834. [20] Enomoto, K. ; LaVerne, J.A. ; Araos, M. S. J. Phys. Chem. A 2007, 111, 9. [21] Larsen, J. ; Madsen, D. ; Poulsen, J.-A. ; Poulsen, T.D. ; Keiding, S.R. ; Thogersen, J. J. Chem. Phys. 2002, 116, 7997. [22] Moskun, A.C. ; Bradforth, S.E. J. Chem. Phys. 2003, 119, 4500. 13 Chapter 2: Methodology 2.1 Introduction This Chapter provides an overview of the theoretical methods used throughout this work. In Section 2.2 the equation-of-motion coupled-cluster for ionized states (EOM-IP-CC) formalism is discussed. It covers both the energy and the gradient computation using the Lagrangian technique. Section 2.3 elaborates on the Dimer Molecular Orbitals - Linear Combination of Fragment Molecular Orbitals (DMO-LCFMO) framework. Application to specific systems is discussed in the following chapters. 2.2 Equation-of-motion for ionized states formalism 2.2.1 Overview It has long been recognized that the description of doublet systems pose challenges for ab initio methodology due to symmetry breaking and spin-contamination of the corre- sponding open-shell Hartree-Fock (HF) wave functions 1, 2 . However, since these (N- 1)/(N+1)-electron doublet systems can formally be derived from an N-electron closed- shell system by subtraction/addition of an electron, they can be accurately and effi- ciently described by ionization potentials (IP) and electron affinities (EA) variants of the equation-of-motion coupled-cluster (EOM-CC) 3–6 , or the closely relate symmetry- adapted-cluster configuration-interaction (SAC-CI) methods 7–9 . Both EOM-IP and EOM-EA rely on the N-electron closed shell reference and are, therefore, free from 14 the symmetry breaking and spin-contamination problems that are ubiquitous in open- shell calculations. Formal properties of these methods, as well as their numerical per- formance, particularly the appropriate order of truncation of the CC and EOM expan- sions, have been studied in great detail 3–6, 10–19 . Truncation of both expansions at double excitations offers a reasonable compromise between computational feasibility and accu- racy. The latter can be improved (up to the exact result) by including higher excitations. Recently, the performance of EOM-IP for ionized non-covalent dimers has been bench- marked against full configuration interaction (CI) and multi-reference CI 20 . As pointed out by Stanton and Gauss 11 , EOM-IP-CC is equivalent to the EOM-CC for excitation energies (EOM-EE) method, in which one electron is always excited to a very diffuse orbital (e.g., of the size of the Earth), thus producing the wave function of the ionized system. This allows a quick implementation, which, unfortunately, may suffer from poor convergence and is more computationally expensive than the properly implemented EOM-IP-CC. We implemented the energy calculation at the EOM-IP-CCSD and EOM-IP-CC(2,3) levels (i.e.,2h1p and3h2p, respectively), as well as the EOM-IP-CCSD gradient calcu- lation within the Q-CHEM electronic structure program 21 . The programmable expres- sions were derived from the EOM-EE-CCSD expressions 22–24 by assuming that one of the EOM-EE excitation occurs to a very diffuse orbital. Terms including either inte- grals over the diffuse orbital, or EOM amplitudes that do not include excitation to the diffuse orbital, vanish. The prefactor and sign of the remaining terms can be efficiently determined using diagrammatic techniques. Since the derivation and the programmable expressions for EOM-IP-CCSD have been published 5 , we only briefly outline the for- malism and give the programmable expressions in the Appendix. 15 2.2.2 Energy computation The EOM-CC 22, 25–31 methods describe multi-configurational wave functions using a single-reference formalism. First, the CC equations are solved for the single- determinantal reference state|Φ 0 i: D Φ μ ¯ H− E CC Φ 0 E =0, ¯ H =e − T He T (2.1) whereΦ μ is aμ -tuply excited determinant,E CC is the CC energy of the system, andT is a linear excitation operator: E CC = D Φ 0 ¯ H Φ 0 E (2.2) T =T 1 +T 2 +...= X ia t a i a + i+ 1 4 X ijab t ab ij a + b + ji... (2.3) The choice of|Φ 0 i defines the Hartree-Fock vacuum, i.e., separation of the orbital space into occupied and virtual subspaces, and all the excitation operators T , R, L are exci- tation operators with respect to this vacuum. We adhere to the following convention: indexesi,j,... are reserved for the orbitals occupied in the reference determinant|Φ 0 >, indexesa,b,... — to unoccupied orbitals, andp,q,... are used in the general case, i.e., when an orbital can be either occupied or virtual. Once the CC amplitudes are found, the non-hermitian similarity transformed Hamil- tonian ¯ H is diagonalized in a selected subspace producing energies and amplitudes of the target EOM states: |Ψ R i=R|Φ 0 i (2.4) hΨ L |=hΦ 0 |L, (2.5) 16 hΨ L Ψ R i=δ LR , (2.6) where R and L are linear excitation operators describing the left and right eigenfunc- tions|Ψ R i andhΨ L |, respectively. By construction,Φ 0 is a right eigenfunction of ¯ H in the subspace of μ -tuply excited determinants. Although, ¯ H has the same spectrum as the original Hamiltonian H, its eigenfunctions form a biorthogonal set due to its non- hermitian nature. When ¯ H is diagonalized in an incomplete space, the quality of the target EOM states depends on how well the reference CC state is described by a single HF determinant |Φ 0 i. Thus, different sectors of Fock space can be reached by a judicious choice of the reference and the EOM R/L operators. Fock space can be thought of as the direct product of Hilbert spaces, each possessing a fixed number of particles. In other words, we can compute the ¯ H operator of anN electron system and subsequently diagonalize it in the basis ofM-electron determinants thus obtaining the eigenstates of theM electron system 32 . Let us now focus on species with one unpaired electron. The HF description of these doublet systems is complicated by the spin-contamination and symmetry break- ing, which adversely impact the properties computed at the correlated level. EOM- IP/EA overcome these difficulties by reaching this problematic sector of Fock space by combining a well-behavedN+1/N-1-electron closed-shell reference and an appropriate R operator: R IP =R IP 1 +R IP 2 +R IP 3 ...= X r i i+ 1 2 X r a ij a + ji+ (2.7) 1 12 X r ab ijk a + b + kji... (2.8) R EA =R EA 1 +R EA 2 +R EA 3 ...= X r a a + + 1 2 X r ab i a + b + i+ (2.9) 1 12 X r abc ij a + b + c + ji... (2.10) 17 Truncation of T and R operators at 2h2p and 2h1p/1h2p, respectively, yields the EOM-IP/EA-CCSD models. Including the 3h2p/2h3p excitations yields EOM-IP/EA- CC(2,3). In the matrix form, the EOM-IP/EA-CCSD equations assume the following form: ¯ H SS − E CC ¯ H SD ¯ H DS ¯ H DD − E CC R 1 (n) R 2 (n) =ω n R 1 (n) R 2 (n) (2.11) L 1 (n) L 2 (n) ! ¯ H SS − E CC ¯ H SD ¯ H DS ¯ H DD − E CC =ω n L 1 (n) L 2 (n) ! (2.12) where the IP/EA superscript is dropped. These equations are usually solved using the Davidson iterative diagonalization procedure 33–35 , which requires the computation of the σ -vectors, the products of the Hamiltonian and trial vectors. Programmable expressions for the left and right EOM-IP-CCSD σ -vectors, as well as the right EOM-IP-CC(2,3) σ -vectors are given in the Appendix. Fig. 2.1 shows electronic configurations of the target states accessible by EOM- IP/EA. EOM-IP-CCSD performs well for the states derived by single ionization, i.e., with Koopmans-like dominant configurations. The ground and low-lying excited states of doublet radicals are often of this type. Provided that properties code is available, static and transition properties between these states can be described as well. EOM-IP/EA-CC models are also suitable for the computation of ionization energies and electron affinities and the corresponding Dyson orbitals 36 , which can be used to compute ionization cross- sections if the scattering wave function of the outgoing electron is available. 18 virtual orbitals occupied orbitals EOM-IP-CC states EOM-EA-CC states reference state Figure 2.1: States with N-1/N+1 electrons can be described based on the N-electron reference using EOM-IP/EA-CC wave functions. EOM-IP-CC is particularly suitable for states derived by ionization from an orbital occupied in the reference, while EOM- EA-CCSD describes states derived by attaching an electron to a virtual orbital in the wave function. 2.2.3 Gradient calculation The Hellman-Feynman theorem states that once the wave function of a system has been fully variationally optimized, the forces on the system in response to a perturbationξ can be determined by computing the corresponding expectation value using the unperturbed wave function|Ψ i: ∂E ∂ξ = ∂hΨ H Ψ i ∂ξ = * Ψ ∂H ∂ξ Ψ + (2.13) Using the second-quantized Hamiltonian, the energy of the system can be expressed as: E =hΨ L H Ψ R i= X pq h pq γ 0 pq + 1 4 X pqrs <pq||rs>Γ 0 pqrs (2.14) γ 0 pq = 1 2 D Ψ L p + q+q + p Ψ R E (2.15) Γ 0 pqrs = 1 2 D Ψ L p + q + sr+s + r + pq Ψ R E (2.16) 19 whereγ 0 pq andΓ 0 pqrs are the one and two-particle reduced density matrices that do not depend on the perturbation. The derivative is simply: ∂E ∂ξ = X pq h ξ pq γ 0 pq + 1 4 X pqrs <pq||rs> ξ Γ 0 pqrs (2.17) where the superscriptξ denotes the derivative of the respective integrals. However, this holds true only for wave functions optimized variationally w.r.t all their parameters. The wave functions obtained by electronic structure methods (including EOM-CC) are subject to constraints, and their response to perturbationξ must be accounted for through the additional terms: ∂E ∂ξ = * Ψ ∂H ∂ξ Ψ + + * ∂Ψ ∂ξ H Ψ + + * Ψ ∂H ∂ξ ∂Ψ ∂ξ + (2.18) The direct determination of these terms is inefficient, and the Z-vector 37 and the Lagrangian 23, 38–42 techniques have been developed to compute additional terms wave function response terms. Below we summarize the Lagrangian based derivation of the EOM-IP-CCSD gradients following the presentation in Ref. 23 . The Lagrangian is constructed to incorporate the constraints on the wave function as variational parameters through the undetermined multipliers. In the case of EOM- IP-CCSD, the constraints are: (i) the CC equations are satisfied; (ii) the orbitals are eigenfunctions of the Fock operator; (iii) the orbitals are orthonormal. Thus, for unre- stricted/restricted HF references, the EOM-CC Lagrangian assumes the following form: L(L,R,T,C,Z,Λ ,Ω)= <Φ 0 L| ¯ H|RΦ 0 > <Φ 0 L|RΦ 0 > + n X μ =1 z μ <Φ μ| ¯ H− E|Φ 0 >+ 1 2 X pq λ pq (f pq − δ pq )+ X pq ω pq (S pq − δ pq ), (2.19) 20 whereZ≡{ z μ},Λ ≡{ λ pq }, andΩ ≡{ ω pq } are the undetermined Lagrange multipli- ers, andf andS are the Fock and MO overlap matrices, respectively: f pq =<p|h|q >+ X j <pj||qj > (2.20) S pq =<φ p |φ q > (2.21) C is the MO matrix and is implicit in the integrals and the MO overlap matrix. The value of the Lagrangian is the same as the energy (if the constraints are satisfied) and its derivative w.r.t. perturbationξ is the same as the respective derivative of the Lagrangian: dE dξ = dL dξ = ∂L ∂ξ + (2.22) ∂L ∂L dL dξ + ∂L ∂R dR dξ + ∂L ∂T dT dξ + ∂L ∂C dC dξ + ∂L ∂Z dC dξ + ∂L ∂Λ dC dξ + ∂L ∂Ω dC dξ (2.23) The first term describes the explicit dependence of the Hamiltonian on the perturba- tion. The second and third terms are zero, because the EOM-CC energy functional is stationary w.r.t the L and R vectors. The last three terms also vanish by virtue of the Lagrangian construction. Finally, the multipliers are defined by requiring that ∂L ∂T and ∂L ∂C are zero. This yields the amplitude and the orbital response equations that need to be solved for the gradient evaluation. Once the Lagrange multipliers Z, Λ , and Ω are computed, the derivative of the energy w.r.tξ is: dE dξ = ∂L ∂ξ = (2.24) * Φ 0 Le − T ∂H ∂ξ e T RΦ 0 + + * Φ 0 Ze − T ∂H ∂ξ e T Φ 0 + (2.25) + 1 2 X pq λ pq ∂f pq ∂ξ + X pq ω pq ∂S pq ∂ξ = X pq h ξ pq ρ pq + 1 4 X pqrs <pq||rs> ξ Π pqrs + X pq ω pq S ξ pq (2.26) 21 where ρ and Π are the effective one- and two-particle density matrices, respectively, including wave function response terms, and theh ξ pq ,hpq||rsi ξ , andS ξ pq are the deriva- tives of the respective integrals. The programmable expressions for the Lagrange multi- pliers{Z,Λ ,Ω } and the density matrices are given in the Appendix. 2.3 Molecular orbital theory of dimers 2.3.1 Overview The Dimer Molecular Orbital – Linear Combination of Fragment Molecular Orbitals (DMO-LCFMO) is a qualitative tool for the description of the electronic states of dimers. In analogy to the Molecular Orbitals – Linear Combination of Atomic Orbitals (MO-LCAO) description of diatomics and in the spirit of exciton theory 43, 44 we describe the dimer MOs (DMOs) as linear combinations of the fragment MOs. This framework results in a convenient dimer state nomenclature, and can be applied to explain the trends in the electronic state ordering, oscillator strengths, and structural relaxation. Three cases are relevant to the work presented here. In Section 2.3.2 the case of a dimer com- posed of two equivalent fragments related by a center of inversion is discussed, like the sandwich isomer of the benzene dimer cation. Both the ground state and the excited state are completely delocalized. The symmetry of the system can be broken either by distorting one of the fragments or by changing their relative orientation. The latter case, elaborated in Section 2.3.3, is relevant to the spectroscopy of the t-shaped benzene dimer cation . Finally, the nonequivalence of the fragments can lead to complete localization of one of the electronic states, like in the water dimer cation system and some of the states of the t-shaped benzene dimer cation. The resulting spectroscopy is discussed in Section 2.3.4. 22 2.3.2 Equivalent fragments We begin our DMO-LCFMO treatment of a homodimer by introducing MOs localized on fragments A and B: ν A,B and λ A,B . For equivalent fragments, the DMOs are just symmetric and antisymmetric linear combinations of the FMOs, as shown in Fig. 2.2 and in the equations below: ψ (ν )=ψ + (ν )= 1 q 2(1+s νν ) (ν A +ν B ) (2.27) ψ ∗ (ν )=ψ − (ν )= 1 q 2(1− s νν ) (ν A − ν B ) (2.28) where the subscript denotes to which monomer the FMO belongs, andψ + andψ − refer to the bonding and antibonding (w.r.t. interfragment interaction) DMOs. s νν is the overlap between the monomer MOs: s νν =hν A ν B i (2.29) Likewise,ψ ± (λ ) will refer to the bonding and antibonding DMO pair derived from the monomer orbitalsλ A andλ B . Similar notation is used in molecular electronic structure, recall, for example, σ (2p z ) in O 2 or π (p y ) and π ∗ (p y ) in ethylene. When this nam- ing scheme is applied to the benzene dimer MOs (Section 4.4.1), σ ∗ (π u ) refers to the antisymmetric combination of the monomerπ u orbitals, and so on. Following the H + 2 textbook exercise, the energy splitting between one-electron states ψ ± (ν ) can be trivially worked out and shown to be proportional to the overlap integral hν A ν B i 45 . The stabilization energy (w.r.t. isolated monomers) is simply half of this splitting. 23 O A O B < + = O A + O B < - = O A - O B Q A Q B < Q A + Q B < - = Q A - Q B case I case II CR Figure 2.2: FMOs and DMOs of an AB dimer. Each pair of doubly occupied MOs of the monomers gives rise to two dimer orbitals and a formal zero bond order. Ionization results in the 3-electrons-in-2- orbitals wave function, which can be mapped to the 1-electron-in-2-orbitals one. For example, the bond order of the (ψ + (ν )) 2 (ψ − (ν )) 1 configuration is the same as that of (ψ + (ν )) 1 (ψ − (ν )) 0 . Furthermore, the ψ + → ψ − transition in the former is equivalent to ψ + → ψ − in the latter, in terms of the symmetry and the result- ing changes in bond order. However, the motion of the hole in the 3-electrons- in-2-orbitals picture is the opposite to the motion of the electron in 1-electron-in-2- orbitals, and the(ψ + (ν )) 2 (ψ − (ν )) 1 →(ψ + (ν )) 1 (ψ − (ν )) 2 excitation is isomorphic with (ψ + (ν )) 1 (ψ − (ν )) 0 →(ψ + (ν )) 0 (ψ − (ν )) 1 . Note that the symmetry of the initial and final states is reversed in the 1-electron-in-2-orbitals (initial state has symmetry of ψ + (ν )) system relative to the 3-electrons-in-2-orbitals one (initial state has symmetry ofψ − (ν )). The transition symmetry, defined as the direct product of the symmetries of the two 24 states, remains unchanged. The transition dipole moment operator, the transition dipole moments are identical in both cases. With the above considerations, electronic states of an ionized van der Waals dimer can be mapped to those of H + 2 , and we will frequently refer to this example to demon- strate general conclusions of the model. In the case of H + 2 , the ν and λ orbitals from Fig. 2.2, could be the1s and2p orbitals of the H atom. We will consider three different types of transitions between the dimer levels. First, there are two types of transitions betweenν andλ manifolds, case I and II from Fig. 2.2. These are called local excitation bands, as they correlate with the electronic transitions of isolated fragments. In H + 2 , these correspond toσ (1s)→ σ ∗ (2p z ),σ (1s)→ π ∗ (2p x ), σ (1s)→σ (2p z ), etc, transitions that correlate with1s→2p excitations in H. The third type is specific for the dimer/diatomic and involves transitions within the same manifold, i.e., charge resonance (CR) bands, as theσ (1s)→σ ∗ (1s) transition in H + 2 . Transition dipoles for the case I and II electronic transitions from Fig. 2.2 are: hψ + (ν ) μ ψ ± (λ )i= 1 2 q (1+s νν )(1± s λλ ) (hν A μ λ A i±h ν B μ λ B i+hν B μ λ A i±h ν A μ λ B i) (2.30) The relationship between the integrals above (i.e., intrafragment AA vs BB, and inter- fragment AB vs BA), as well as the symmetry of orbitalsψ ± , depend on how the inver- sion (in the dimer frame) operator ˆ i acts onν andλ . Consider first: ˆ iν A =ν B and ˆ iλ A =λ B or (2.31) ˆ iν A =− ν B and ˆ iλ A =− λ B 25 In diatomics, the symmetric case is the case when ν and λ are s-orbitals, and the corresponding ψ + and ψ − are then gerade and ungerade, respectively. The antisym- metric case corresponds to p x,y -orbitals, and the corresponding bonding and antibond- ing orbitals are ungerade and gerade. In both cases,hν A μ λ A i =−h ν B μ λ B i and hν A μ λ B i =−h ν B μ λ A i. Thus, the transition dipole is zero for ψ + (ν )→ ψ + (λ ), as it should be for gerade→ gerade and ungerade→ ungerade transitions. The ψ + (ν )→ ψ − (λ ), i.e., case I, transition is symmetry allowed, as ψ + (ν ) and ψ − (λ ) are of different symmetry, and the transition dipole is: hψ + (ν ) μ ψ − (λ )i= 1 q (1+s νν )(1− s λλ ) (hν A μ λ A i+hν B μ λ A i) (2.32) Similarly, when: ˆ iν A =ν B and ˆ iλ A =− λ B or (2.33) ˆ iν A =− ν B and ˆ iλ A =λ B , the signs of the integrals from Eq. (2.30) change andhψ + (ν ) μ ψ − (λ )i is zero, while: hψ + (ν ) μ ψ + (λ )i= 1 q (1+s νν )(1+s λλ ) (hν A μ λ A i+hν B μ λ A i) (2.34) Equations (2.32) and (2.34) are essentially identical equations describing gerade- ungerade transition. From the bonding perspective, however, the first case is a bonding- antibonding transition, whereas the second — a bonding-bonding excitation. In H + 2 , ifν ≡ 1s andλ ≡ 2p z , both orbitals transform identically under the inversion, as in Eq. (2.31), and case I transitions, e.g., σ (1s)→ σ ∗ (2p z ) are allowed, while case II,σ (1s)→σ (2p z ), transitions are forbidden. If, however,λ ≡ 2p x,y , orbitals transform 26 according to Eq. (2.34). σ (1s)→ π ∗ (2p x,y ) and σ (1s)→ π (2p x,y ) excitations are forbidden and allowed, respectively. Let us now consider how the intensities of case I and case II transitions, which are given by Eq. (2.32) and (2.34), respectively, depend on the interfragment separation. In the united atom limit (R AB = 0) the transition becomes ν A → λ A , and its transition dipole assumes the monomer value. At largeR AB , the interfragment term in Eq. (2.32) and (2.34) vanishes, as the overlap integral decays exponentially with the distance, and the transition dipole is, again, the same as in the monomer. At intermediate distances, both the intrafragment and interfragment terms may contribute. For the CR transitions, i.e., whenν =λ , Eq. (2.30) assumes the following form: hψ + (ν ) μ ψ − (ν )i= 1 2 q (1− s 2 νν ) (hν A μ ν A i−h ν A μ ν B i+hν B μ ν A i−h ν B μ ν B i)= 1 2 q (1− s 2 νν ) (hν A μ ν A i−h ν B μ ν B i) (2.35) At large R AB , the difference between the two terms is just the difference between the average positions of monomers A and B, i.e. the interfragment distance R AB , and we arrive to: hψ + (ν ) μ ψ − (ν )i≈ R AB 2 q 1− s 2 νν (2.36) Thus, the transition dipole of such transitions increases linearly with the fragment sepa- ration. This corresponds to the transitions between complementary bonding-antibonding orbitals 46 , i.e., so-called charge transfer or charge resonance transitions (see Fig. 2.2). As the oscillator strength depends also on the energy splitting between ψ + (ν ) and ψ − (ν ), it will, therefore, go to zero at largeR AB . These types of transitions are unique 27 for the dimers, and the charge resonance band in (C 6 H 6 ) + 2 discussed below is of this type. Finally, we would like to point out that the dimer states can also be described in terms of pseudo-diabatic charge-localized states. The wave functions for the diabatic states can be expressed in the FMO basis as follows: ψ A + B =(ν A ) 0 (ν B ) 1 ψ AB + =(ν A ) 1 (ν B ) 0 (2.37) In H + 2 , these would correspond to the electron being localized on one of the 1s orbitals. When monomers are equivalent the diabatic states are degenerate, and the coupling thus is just half the splitting between the adiabatic states. 2.3.3 Nonequivalent fragments When the monomers are nonequivalent, as in the t-shaped benzene isomer, the DMOs become partially localized and so are the resulting states. This is similar to heteronuclear diatomics, where MOs are no longer equal mixtures of AOs and some of the symmetry restrictions present in homonuclear diatomics are lifted. The diabatic coupling can be inferred using the Generalized Mulliken-Hush model 47 . When monomers become non-equivalent, the relative weights of the fragment orbitals in the dimer MOs change and the terms in Eq. (2.30) do not cancel out, thus, both the ψ + (λ )→ ψ − (λ ) and the ψ + (λ )→ ψ − (λ ) transitions become allowed. The resulting DMOs are: ψ − (λ )= 1 q (1+β 2 − 2βs λλ ) (βλ A − λ B ) (2.38) 28 ψ + (λ )= 1 q (1+β 2 +2βs λλ ) (λ A +βλ B ) (2.39) ψ − (ν )= 1 q (1+β 2 − 2βs νν ) (βν A − ν B ) (2.40) ψ + (ν )= 1 q (1+β 2 +2βs νν ) (ν A +βν B ) (2.41) where 0≤ β ≤ 1. The case of equivalent fragments is recovered whenβ =1. The transi- tion dipole moments are: hψ + (ν ) μ ψ − (λ )i= 1 q (1+β 2 +2βs νν )(1+β 2 − 2βs λλ ) 2β hν A μ λ A i+(1+β 2 )hν B μ λ A i (2.42) hψ + (ν ) μ ψ + (λ )i= 1− β 2 q (1+β 2 +2βs νν )(1+β 2 +2βs λλ ) (hν A μ λ A i) (2.43) where we assumed that the symmetry rules for case I transitions still apply. Thus, a weaker line will appear at lower energy. Likewise, for the case II transitions, a higher energy transition will become allowed. We assumed that the symmetry rules for the case I transitions still apply. Consider now the transition within theν manifold: hψ + (ν ) μ ψ − (ν )i= 1 q (1+β 2 +2βs νν )(1+β 2 − 2βs νν ) β hν A μ ν A i− β hν B μ ν B i−h ν A μ ν B i+β 2 hν B μ ν A i (2.44) = βR AB q (1+β 2 +2βs νν )(1+β 2 − 2βs νν ) (2.45) This is an analogue of the CR transition, with the intensity decreased by a factor of β compared to the symmetric case. 29 Thus, when the monomers are similar, the MOs retain some of their complementary bonding-antibonding character, giving rise to strong CR bands. The intensity of a CR band is proportional to the interfragment separation, decreasing with distance due to the exponential decays of the transition energy. Regardless of how non-equivalent the fragments are, the intensities of the LE bands will be similar to that of the monomers. 2.3.4 Localized states In some systems, like the water dimer cation, the nonequivalence of the fragments leads to the complete localization of some states. In (H 2 O) + 2 , the singly occupied molecular orbital (SOMO) is the out-of-plane p orbital of the hydrogen bond donor, and all the considered excitation are transfers of an electron to this orbital. In the 1-electron-in-2- orbitals picture this orbital is vacant, while the other one is singly occupied. Let us introduce a basis of three localized fragment MOs: ω A , ν A , λ B , where the subscript denotes the fragment. We assume that the lower energy MO of the dimer is a delocalized mixture ofν A andλ B : ψ 1 =αν A +βλ B (2.46) whereα andβ satisfy the orthonormalization condition. The higher energy DMO is the localizedω A state: ψ 2 =|ω A i, (2.47) This orbital represents the localized SOMO in the case of (H 2 O) + 2 . The transition takes place fromψ 1 toψ 2 . The transition dipole moment between states 1 and 2 is: hψ 1 ˆ μ ψ 2 i=α hν A ˆ μ ω A i+β hλ B ˆ μ ω A i (2.48) 30 The equation shows that both interfragment (hλ B ˆ μ ω A i) and intrafragment (hν A ˆ μ ω A i) terms contribute to the intensity of a dimer transition. The weight of each contribution is defined by the degree of MO mixing, i.e. theα andβ coefficients. Their relative phase determines whether individual contributions add or subtract. Thus, the total intensity of the monomer bands is not necessarily conserved in the dimer. Consider first the limit of the ground state being completely localized on fragment B (α =0): hψ 1 ˆ μ ψ 2 i=hλ B ˆ μ ω A i (2.49) The transition becomes a pure charge transfer excitation in which the electron moves from B to A. Its intensity may become strong when the fragments are closer together, but will decrease rapidly with the distance due to the exponential decay of the fragment wave functions. In the limit of the excited state localized onA (β =0), we obtain: hψ 1 ˆ μ ψ 2 i=hν A ˆ μ ω A i (2.50) Thus the excitation becomes a purely local excitation on fragment A. Within this frame- work,the electron density on fragment B is not affected. Its intensity is the same as in the monomer, i.e. a forbidden excitation remains forbidden and an allowed one remains allowed. However, in a dimer the orbitals and molecular geometries become distorted relative to isolated fragments and forbidden transitions often acquire small intensity. 31 2.4 Chapter 2 Bibliography [1] Davidson, E.R. ; Borden, W.T. J. Phys. Chem. 1983, 87, 4783. [2] Russ, N.J. ; Crawford, T.D. ; Tschumper, G.S. J. Chem. Phys. 2005, 120, 7298. [3] Sinha, D. ; Mukhopadhyay, D. ; Mukherjee, D. Chem. Phys. Lett. 1986, 129, 369. [4] Pal, S. ; Rittby, M. ; Bartlett, R.J. ; Sinha, D. ; Mukherjee, D. Chem. Phys. Lett. 1987, 137, 273. [5] Stanton, J.F. ; Gauss, J. J. Chem. Phys. 1994, 101, 8938. [6] Nooijen, M. ; Bartlett, R.J. J. Chem. Phys. 1995, 102, 3629. [7] Nakatsuji, H. ; Hirao, K. J. Chem. Phys. 1978, 68, 2053. [8] Nakatsuji, H. ; Ohta, K. ; Hirao, K. J. Chem. Phys. 1981, 75, 2952. [9] Nakatsuji, H. Chem. Phys. Lett. 1991, 177, 331. [10] Sinha, D. ; Mukhopadhya, D. ; Chaudhuri, R. ; Mukherjee, D. Chem. Phys. Lett. 1989, 154, 544. [11] Stanton, J.F. ; Gauss, J. J. Chem. Phys. 1999, 111, 8785. [12] Bomble, Y .J. ; Saeh, J.C. ; Stanton, J.F. ; Szalay, P.G. ; K´ allay, M. ; J, J. Gauss J. Chem. Phys. 2005, 122, 154107. [13] Hirata, S. ; Nooijen, M. ; Bartlett, R.J. Chem. Phys. Lett. 2000, 328, 459. [14] Kamyia, M. ; Hirata, S. J. Chem. Phys. 2006, 125, 074111. [15] Ohtsuka, Y . ; P, P. Piecuch ; Gour, J.R. ; Ehara, M. ; H, H. Nakatsuji J. Chem. Phys. 2007, 126, 164111. [16] Musial, M. ; Kucharski, S.A. ; Bartlett, R.J. J. Chem. Phys. 2003, 118, 1128. [17] Musial, M. ; Bartlett, R.J. J. Chem. Phys. 2003, 119, 1901. [18] Haque, M. ; Mukherjee, D. J. Chem. Phys. 1984, 80, 5058. [19] Meissner, L. ; Bartlett, R.J. J. Chem. Phys. 1991, 94, 6670. [20] Pieniazek, P.A. ; Arnstein, S.A. ; Bradforth, S.E. ; Krylov, A.I. ; Sherrill, C.D. J. Chem. Phys. 2007, 127, 164110. 32 [21] Shao, Y . ; Molnar, L.F. ; Jung, Y . ; Kussmann, J. ; Ochsenfeld, C. ; Brown, S. ; Gilbert, A.T.B. ; Slipchenko, L.V . ; Levchenko, S.V . ; O’Neil, D. P. ; Distasio Jr., R.A. ; Lochan, R.C. ; Wang, T. ; Beran, G.J.O. ; Besley, N.A. ; Herbert, J.M. ; Lin, C.Y . ; Van V oorhis, T. ; Chien, S.H. ; Sodt, A. ; Steele, R.P. ; Rassolov, V . A. ; Maslen, P. ; Korambath, P.P. ; Adamson, R.D. ; Austin, B. ; Baker, J. ; Bird, E.F.C. ; Daschel, H. ; Doerksen, R.J. ; Drew, A. ; Dunietz, B.D. ; Dutoi, A.D. ; Furlani, T.R. ; Gwaltney, S.R. ; Heyden, A. ; Hirata, S. ; Hsu, C.-P. ; Kedziora, G.S. ; Khalliulin, R.Z. ; Klunziger, P. ; Lee, A.M. ; Liang, W.Z. ; Lotan, I. ; Nair, N. ; Peters, B. ; Proynov, E.I. ; Pieniazek, P.A. ; Rhee, Y .M. ; Ritchie, J. ; Rosta, E. ; Sherrill, C.D. ; Simmonett, A.C. ; Subotnik, J.E. ; Woodcock III, H.L. ; Zhang, W. ; Bell, A.T. ; Chakraborty, A.K. ; Chipman, D.M. ; Keil, F.J. ; Warshel, A. ; Herhe, W.J. ; Schaefer III, H.F. ; Kong, J. ; Krylov, A.I. ; Gill, P.M.W. ; Head-Gordon, M. Phys. Chem. Chem. Phys. 2006, 8, 3172. [22] Levchenko, S.V . ; Krylov, A.I. J. Chem. Phys. 2004, 120, 175. [23] Levchenko, S.V . ; Wang, T. ; Krylov, A.I. J. Chem. Phys. 2005, 122, 224106. [24] Slipchenko, L.V . ; Krylov, A.I. J. Chem. Phys. 2005, 123, 84107. [25] Rowe, D.J. Rev. Mod. Phys. 1968, 40, 153. [26] Emrich, K. Nucl. Phys. 1981, A351, 379. [27] Sekino, H. ; Bartlett, R.J. Int. J. Quant. Chem. Symp. 1984, 18, 255. [28] Geertsen, J. ; Rittby, M. ; Bartlett, R.J. Chem. Phys. Lett. 1989, 164, 57. [29] Stanton, J.F. ; Bartlett, R.J. J. Chem. Phys. 1993, 98, 7029. [30] Bartlett, R.J. ; Stanton, J.F. Rev. Comp. Chem. 1994, 5, 65. [31] Krylov, A.I. Annu. Rev. Phys. Chem. 2008, 59, 433. [32] Lindgren, I. ; D.Mukherjee Phys. Rep. 1987, 151, 93. [33] Davidson, E.R. J. Comput. Phys. 1975, 17, 87. [34] Hirao, K. ; Nakatsuji, H. J. Comput. Phys. 1982, 45, 246. [35] Rettrup, S. J. Comput. Phys. 1982, 45, 100. [36] Oana, M. ; Krylov, A.I. J. Chem. Phys. 2007, 127, 234106. [37] Handy, N.C. ; Schaefer III, H.F. J. Chem. Phys. 1984, 81, 5031. [38] Helgaker, T. ; Jørgensen, P. ; Olsen, J. Molecular electronic structure theory; Wiley & Sons, 2000. 33 [39] Szalay, P.G. Int. J. Quant. Chem. 1995, 55, 151. [40] Gwaltney, S.R. ; Bartlett, R.J. J. Chem. Phys. 1999, 110, 62. [41] Furche, F. ; Ahlrichs, R. J. Chem. Phys. 2002, 117, 7433. [42] Celani, P. ; Werner, H.-J. J. Chem. Phys. 2003, 119, 5044. [43] Birks, J.B. Photophysics of Aromatic Molecules; Wiley: New York, 1970. [44] East, A.L.L. ; Lim, E.C. J. Chem. Phys. 2000, 113,, 8981. [45] Atkins, P.W. ; Friedman, R.S. Molecular Quantum Mechanics; New York: Oxford University Press, 2005. [46] Mulliken, R.S. ; Person, W.B. Molecular Complexes; Wiley-Interscience, 1969. [47] Cave, R.J. ; Newton, M.D. Chem. Phys. Lett. 1996, 249, 15. 34 Chapter 3: Benchmark full configuration interaction and EOM-IP-CCSD results for prototypical charge transfer systems: noncovalent ionized dimers 3.1 Overview This Chapter presents benchmark full configuration interaction (FCI) and equation- of-motion coupled-cluster model with single and double substitutions for ionized sys- tems (EOM-IP-CCSD) for prototypical charge transfer species. The studied quantities are associated with the quality of the potential energy surface (PES) along the charge transfer (CT) coordinate and distribution of the charge between fragments. It is found that EOM-IP-CCSD is capable of describing accurately both the charge-localized and charge-delocalized systems, yielding accurate charge distributions and energies. This is in stark contrast with the methods based on the open-shell reference, which overlocalize the charge and produce a PES cusp when the fragments are indistinguishable. 3.2 Introduction Ionized noncovalent dimers are relevant in electron/hole transfer (E/HT) processes ubiq- uitous in biophysics and molecular electronics 1–4 , and are described by open-shell dou- blet wave functions. Solvents used in radioactive element separation are susceptible 35 to radiation induced ionization, which in the case of neat aromatic liquids leads to the initial formation of aromatic cations and dimer cations like (C 6 H 6 ) + 2 and (C 5 H 5 N) + 2 5–8 . Knowledge of the cation PES is needed in the interpretation of photoelectron spectra of neutral dimer species 9–11 . (O 2 ) + 2 is an intermediate in the formation of protonated water clusters in the lower ionosphere 12–14 . Cation dimers of polyaromatic hydrocarbons are suspected to be the source of broad extended interstellar emission 15 . However, even in the case of simple isolated systems, like (O 2 ) + 2 , (CO 2 ) + 2 , (C 6 H 6 ) + 2 and (H 2 O) + 2 , the struc- ture and properties have long eluded both theorists and experimentalists alike 8, 16–21 . For condensed phase problems Marcus theory 22–24 provides the relationship between the kinetics of the E/HT transfer processes to the electronic coupling between localized donor and acceptor sites. Often donor or acceptor sites are themselves made up of dimer or multimeric cores, for example the special pair of bacteriochlorophylls which serves as the electron donor in the photosynthetic reaction center 25, 26 . In cytochromec of bacterium Schewanella oneidensis MR-1 several heme groups acting concertedly are implicated in the reduction process, and, consequently, make the ET process more efficient. This efficiency, referred to as ‘electron harvesting’, has been attributed to the closely packed arrangement of the heme groups 27, 28 . Oxidative damage on DNA leads to facile hole transfer between stacked aromatic bases 29 . To explain and understand the function of these important biological systems, as well as to engineer new compounds, one must obtain knowledge of the energetics and properties of the states involved. Especially vital is the value of the diabatic coupling between the donor and acceptor moieties, which should be calculated with an accuracy independent of relative orientations and distance between the two. Several problems arise in approximate electronic structure calculations of dou- blet systems. Single reference approaches based on an open-shell doublet reference are plagued by symmetry breaking 30, 31 , even when highly correlated wave functions 36 are used. Typically, the initial and final states involved in the hole/electron trans- fer process are nearly degenerate, and the wave functions acquire a considerable multi-determinantal character. To this end multireference (MR) approaches have been used 32–34 ; however, artificial symmetry breaking can occur for MR wave functions as well 35 . Moreover, it requires the choice of an active space and may lead to unbal- anced description of electronic states along the CT path. Recently, the spin-flip (SF) approach 36–38 based on the quartet reference has been tested 39 . Although SF wave func- tions include all the leading electronic configurations, the quartet reference exhibits instability, which affects the quality of the PES. The symmetry breaking problem is most readily manifested in the case of open-shell symmetric dimers, i.e., when the donor and acceptor moieties are indistinguishable. In this case, there are two Hartree-Fock (HF) solutions: the delocalized wave function, which has a correct symmetry, and a lower-energy symmetry-broken one. The energetic difference persists even at correlated levels of theory and vanishes only in the FCI limit, where the correct symmetry is restored. For example, in the case of the ethylene dimer cation studied herein the difference between the symmetric and symmetry-broken HF solutions is 0.2 eV . One thus faces a dilemma of which solution to choose 40 . The sym- metric, charge delocalized solution has the correct symmetry at the symmetric nuclear configuration, but it may not be the best solution in a variational sense. On the other hand, the lower-energy solution does not exhibit the proper symmetry of the molecule at the symmetric nuclear configuration, and can therefore exhibit unphysical properties (such as artificially nonzero dipole moments). Moreover, from a practical point of view, the presence of these two different solutions can cause severe difficulties. Straightfor- ward application of electronic structure programs will typically lead to the lower-energy, symmetry-broken solution being found at nonsymmetric geometries, and the higher- energy, symmetric solution being found at the symmetric geometry. This would lead to 37 an undesirable and artificial discontinuity in the potential energy surface. Vibrational frequencies can be adversely affected no matter which solution is chosen 41 . CT systems also pose challenges to density functional theory (DFT) due to self- interaction error (SIE), of which the H + 2 dissociation curve is the most striking exam- ple 42 . SIE, which is present in many functionals, causes artificial stabilization of delo- calized charge 43–45 , which spoils the description of Rydberg and CT excited states (see, for example, Ref. 46, 47 ), vibronic interactions 48, 49 , and charge distribution in the ground- state CT systems 45 . This work presents FCI calculations of PESs and properties and demonstrates how to alleviate the problems mentioned above using single-reference equation-of-motion coupled-cluster model for ionized systems (EOM-IP-CC) methodology. A suitable ref- erence in this case is the neutral HF wave function, which does not suffer from the instability problems as all the electrons are paired, and the target ionic wave func- tions are derived by removing an electron from the reference. Implementation of this Koopmans-like idea within coupled-cluster (CC) framework is the essence of the EOM- IP-CC method 50–55 . This method has been applied earlier by Hsu to a series of alkyl compounds and ethylene dimer, but no extensive testing was performed 56 . A similar approach, albeit based on the truncated configuration interaction (CI) method, has been developed by Simons for the calculation of ionization energies and electron affinities 57 . Similar ideas are exploited by the related symmetry-adapted cluster CI (SAC-CI) family of methods 58, 59 . Here, EOM-IP-CCSD and FCI results for open-shell dimer cation species are com- pared. To the best of our knowledge there are no previous full CI benchmarks analyzing symmetry breaking in radical cations of van der Waals dimers. The evaluated quantities pertain to charge transfer states, i.e. charge localized on either fragment. We compute the absolute energy, permanent dipole, transition dipole and electronic coupling. The 38 coupling is evaluated using the Generalized Mulliken-Hush 33, 60 (GMH) model. Instead of the permanent dipole, which is origin-dependent for a charged system, we report the charge on the more positive molecule. This is also a measure of the weight of a particular diabatic (defined as a charge-localized state) state in the wave function: q A =haΨ A + B +bΨ AB + ˆ q A aΨ A + B +bΨ AB +i=|a| 2 (3.1) whereq A is the charge on fragment A and ˆ q A is the associated operator. The structure of the paper is as follows. The next section describes theoreti- cal methods (EOM-CC approach to open-shell doublet wave functions and general- ized Mulliken-Hush diabatization scheme) and computational details. Results for the selected benchmark systems are give in Section 3.4. The studied systems are: He + 2 , (H 2 ) + 2 , (BH-H 2 ) + , (Be-BH) + , and (LiH) + 2 . They were chosen based on the feasibility of FCI calculations and the difference of IEs. Finally, the EOM-IP-CCSD methodology is applied to the ethylene dimer, an often studied model system for polymer conduction andπ interactions 32, 39, 56, 61, 62 . Numerical data are available as Supplementary Informa- tion to Ref. 63 . Our final remarks are given in Section 3.5. 3.3 Methodology 3.3.1 EOM-CC approach to open-shell species The EOM-IP-CC method has been described in detail in Section 2.2. Here we compare other approaches to obtaining doublet wave functions. Conceptually, EOM is similar to CI: target EOM states are found by diagonalizing the so-called similarity transformed Hamiltonian ¯ H≡ e − T He T : ¯ HR=ER, (3.2) 39 whereT andR are general excitation operators with respect to the reference determinant |Φ 0 i. Operator T describes the dynamical correlation, while R allows one to access a variety of multideterminantal target states. Regardless of the choice ofT , the spectrum of ¯ H is exactly the same as that of the original Hamiltonian H. Thus, in the limit of the complete many-electron basis set, EOM is identical to FCI. In a more practical case of a truncated basis, e.g., whenT andR are truncated at single and double excitations, the EOM models are numerically superior to the corresponding CI models 64 , because correlation effects are “folded in” in the transformed Hamiltonian, while the computa- tional scaling remains the same. Moreover, the truncated EOM models are rigorously size-consistent (or, more precisely, size-intensive) 65–691 provided that the amplitudes T satisfy the CC equations for the reference state|Φ 0 i and are truncated at sufficiently high level of excitation consistent with that ofR: D Φ μ ¯ H Φ 0 E , (3.3) whereΦ μ denotesμ -tuply excited determinants, e.g.,{Φ a i ,Φ ab ij } in the case of CCSD. By combining different types of excitation operators and references|Φ 0 i, open-shell doublet states can be accessed in different ways, as explained in Fig. 3.1. For example, we may use the open-shell doublet reference and operatorsR that conserve the number of electrons and a total spin 65, 71, 72 . In this case, one CT state will be described at the CC level, while the other one at the equation-of-motion coupled-cluster for excitation energies (EOM-EE-CC) level. Problems arise due to the instability of the reference and 1 See, for example, footnote 32 in Ref. 38 explaining size-consistency of EOM-EE/SF, as well as related discussion in Ref. 70 . In the context of charge-transfer systems of aA + B→AB + type, this means that the EOM-IP ionization energies, PESs, and energy gaps between the PESs will not be affected by any number of closed-shell species at infinite distance. For example, all the EOM-IP results for the ionized ethylene dimer from Section III.F (except for the total energies) will remain unchanged if we add a neutral ethylene dimer at sufficiently large distance. Therefore, the quality of EOM-IP description does not deteriorate with molecular size increase. However, the lack of full size-extensivity means that EOM-IP is not capable of describing simultaneously two non-interacting ionized ethylene dimers. 40 unbalanced description of the two states. Inclusion of higher excitations, e.g., within EOM-CCSDT or EOM-CCSDt schemes 73, 74 will of course improve the description, but at the price of increased computational costs. + spin flip excitation ionization ˆ IP R ˆ SF R ˆ EE R + …. Figure 3.1: Open-shell doublet wave functions can be described by several EOM approaches using different references/excitation operators. The EOM-IP method employs a well-behaved closed-shell reference. The ionized/electron attached EOM models 50–52, 54, 55, 75 , which employ operators R that are not electron conserving (i.e., include different number of creation and anni- hilation operators) — describe ground and excited states of doublet radicals on equal footing. In our case, we start with a neutral reference and treat both CT states as ion- ized states. The truncation of EOM-IP operators deserves additional comments. For CCSD references, i.e., when operatorT includes single and double excitation, the most common strategy is to retain only1h and2h1p operators: R IP = X r i i+ X r a ij a + ji, (3.4) which gives rise to the EOM-IP-CCSD method. However, one may consider includ- ing 3h2p operators as well, as in EOM-IP-CC(2,3) 76, 77 . As demonstrated by Piecuch 41 and Bartlett, this does not break size-consistency of the resulting EOM-IP method 78 , in contrast to EOM-EE 70 , thus justifying such truncation scheme. Finally, the EOM-SF method 36–38 in which the excitation operators include spin- flip allows one to access diradicals, triradicals, and bond-breaking without using spin- and symmetry-broken unrestricted HF (UHF) references. In our cases quartet reference would be used, as first proposed by Hsu and coworkers 39 . The obtained set of determi- nants is appropriate for the description of the CT states, but the reference still exhibits instability. To summarize, the EOM-IP method avoids the HF instability problems and describes the problematic open-shell doublet states in a single reference formalism. 3.3.2 Generalized Mulliken-Hush model Fig. 3.2 presents the PESs along the CT reaction coordinate. The solid lines represent the adiabatic energies, i.e., the eigenvalues of the electronic Hamiltonian. The dotted lines are the diabatic energies. The corresponding wave functions depend only weakly on nuclear configuration and describe the charge-localized states, i.e., A + B and AB + . The magnitude of the electronic coupling between these wave functions determines the kinetics of the process within the Marcus theory. Note that electronic structure packages yield adiabatic energies and wave functions. The transformation between the two basis sets is not straightforward because dia- batic states are not rigorously defined in a general case 79 . We employed the GMH method developed by Robert Cave and Marshall Newton to compute the diabatic- adiabatic transformation matrix and the coupling elements. The method is based on the assumption that there is no dipole moment coupling between the diabatic states, and thus the dipole moment matrix is diagonal in this representation. This corresponds to the two states with the largest charge separation, i.e. charge localized on the reactants and 42 Figure 3.2: Diabatic (dashed line) and adiabatic potential energy surfaces for electron transfer reactions. Diabatic states correspond to reactant and product electronic wave functions, i.e. the charge fully localized on one of the species, while adiabatic states are eigenfunctions of the electronic Hamiltonian. Marcus theory relates the coupling between diabatic states to the rate of electron/hole transfer process. products. The so-defined transformation matrix can hence be applied to the Hamiltonian matrix in the adiabatic representation yielding the coupling as the off-diagonal element. This leads to the following expression: h ab = μ 12 Δ E ab Δ μ ab = μ 12 Δ E 12 [(Δ μ 12 ) 2 +4(μ 12 ) 2 ] 1/2 (3.5) The letter and number subscripts refer to diabatic and adiabatic quantities, respec- tively. μ 12 is the transition dipole moment andΔ μ 12 is the difference between the per- manent dipole moments. Components of each vector in the direction defined by the permanent dipole difference vector for the initial and final adiabatic states are used. In the case of a charged system the definition of the dipole moment depends on the origin. However, the diagonalization matrix depends on the difference rather than the values itself and thus is origin independent. 43 The main forte of the GMH model is its simplicity and a wide range of applicability. It can be applied both to the ground state and excited state at any nuclear configura- tion. Furthermore, the only required quantities are adiabatic, and thus easily available using standard electronic structure software. Similar diabatization schemes exploiting differences in molecular properties of the diabatic states have been explored in other applications as well 80–85 . 3.3.3 The CT reaction coordinates The two important coordinates for CT processes are the intermolecular separation and the intramolecular CT coordinate described below. In the charge transfer processes (Fig. 3.2) the reactants correspond to an electron/hole localized on one of the moieties, e.g. A + -B. At infinite separation, the geometry of A is that of the cation, whereas the geometry of B is that of its neutral. The reaction corresponds to the positive charge moving from A to B and the nuclei rearranging such that A has the geometry of the neutral form and B has a cation-like geometry. At smaller interfragment separations, the geometries of the fragments along the reaction coordinate may differ from that simple picture. In principle, the geometries along this path can be calculated by following the energy minimum, i.e., conducting constrained optimization at each point along A + B → AB + . Alternatively, a CT reaction coordinate can be approximated by arithmetic averaging of the Cartesian monomer coordinates: Q R1 =(1− R)· Q 1 +R· Q 2 Q R2 =R· Q 1 +(1− R)· Q 2 (3.6) where Q 1 is the geometry of the neutral and Q 2 is the geometry of the cation. The averaging is done with the rotation axes and the centers of mass aligned. When R=0, 44 A has its cation geometry, whereas B has its neutral geometry. At R=0.5 the geometry of each species is a simple average of the cation and neutral forms. Finally, at R=1 the geometry of A is that of the neutral, and that of B corresponds to the cation form. This approach merely ensures a smooth interpolation between the initial and final geometries and should not be taken as the path followed in a physical situation. 3.3.4 Computational details Configuration-interaction singles (CIS), EOM-IP-CCSD (IP-CCSD/2h1p), EOM-IP- CC(2,3) (IP-CCSD/3h2p), and EOM-CCSD for excitation energies (EOM-EE-CCSD) calculations, as well as all geometry optimizations were performed using theQ-CHEM ab initio package 86 . FCI calculations employed the PSI3 package 87 . Multi-reference configuration interaction (MRCI) calculations were performed using MOLPRO 88 . All basis sets were obtained from the EMSL repository 89 . The charge localized on the monomers was computed assuming that charges on individual fragments are point charges located at the COM of indivdual fragments. Only the component of the total dipole moment in this direction is considered. Charge q A is localized on fragment A at positionr A , while charge (1-q A ) is on fragment B localized atr B . This yields the following expression for the charge: q A = − μ +r B r B − r A , (3.7) where the dipole moment vectorμ is defined to point towards the positive charge. Vector quantities are computed relative to the COM of the system. Spin-restricted references were used in EOM-IP-CC and FCI calculations. EOM- EE-CCSD calculations were based on spin-unrestricted references for H 2 dimer. Oth- erwise, spin-restricted open-shell references (ROHF) were employed. For the ethylene 45 dimer, we considered both ROHF and UHF doublet references. CCSD energies were converged to 10 − 10 hartree. Davidson iteration in EOM calculations were considered converged when the residue of the excited state vectors was below 10 − 10 . EOM-IP- CC(2,3) dipole moments for (C 2 H 4 ) + 2 were computed via finite differences using field values of± 0.00001 a.u. All electrons were active in EOM calculations. EOM dipole moments were calcu- lated using fully relaxed one-particle density matrices, that is including the amplitude and orbital response contributions 90 , while transition dipoles were computed as “expec- tation values”, that is, using unrelaxed density matrices of the right-hand and left-hand eigenvectors 72 . FCI properties were computed using unrelaxed density matrices. Orbital relaxation terms are not needed in FCI property computations because the FCI properties are invari- ant to unitary transformations of the active orbitals; the exception occurs when some orbitals are frozen in the correlated computation, as was the case here for the 1s-like orbitals for Be and B atoms. However, limited tests indicate that these core-active orbital rotations did not contribute significantly for the cases considered. MRCI calculations employ the state-averaged complete active space self-consistent field (SA-CASSCF) reference and include all single and double excitations from the reference (MR-CISD) 91, 92 . To correct for the lack of size-extensivity, the resulting MR- CISD energies are augmented by the Davidson correction 93 and are denoted as MR- CISD+Q. Unfortunately, no analogue of the Davidson correction for properties is avail- able. The active space consisted of the ethylene π and π ∗ orbitals. The SA-CASSCF computations include the ground state and the first excited state with equal weights. The four 1s carbon orbitals were restricted in CASSCF calculations and frozen in MR-CISD. Geometries of H 2 , BH, and LiH were optimized using CCSD with perturbative account of triple excitations (CCSD(T)) 94, 95 and the aug-cc-pVTZ basis set. C 2 H 4 and 46 C 2 H + 4 structures were obtained using density functional theory (DFT) with B3LYP 96 functional and the 6-311+G* basis set. All geometries are given as Supplementary Information to Ref. 63 . 3.4 Results and discussion 3.4.1 (He 2 ) + dimer Our first benchmark system is the helium dimer cation. Due to symmetry, the lower and upper charge transfer states,Σ + u andΣ + g , feature the charge equally distributed between the helium atoms, and the coupling is simply half of the energy splitting between the two states. The energies and transition dipole moments were computed at intermolec- ular separations ranging from 2.5 ˚ A to 6.0 ˚ A. Representative EOM-IP-CCSD and FCI results are presented in Tables 3.1 and 3.2, respectively. Fig. 3.3 shows the distance dependence of the FCI coupling. Only basis sets with diffuse functions reproduced the correct exponential decay of the coupling. Without diffuse functions the coupling decays too fast. The magnitude of the coupling is nearly converged at the aug-cc-pVTZ basis. The convergence of the coupling as a function of the method and the distance is shown in Fig. 3.4. The behavior of the two theoretical methods is essentially identical. Adding the first set of diffuse functions increases the coupling and the magnitude of this effect increases with distance. Further basis set expansion, e.g. adding another set of diffuse or valence functions, have smaller effects. The difference between double- ζ and triple-ζ is significant, but less important than the presence of diffuse functions. One aspect is very interesting — the error of the EOM-IP-CCSD versus FCI decreases at larger distances. We attribute this effect to larger dynamical correlation at shorter distances, i.e., when the distance between electrons is smaller on average. In He + 2 at 47 Table 3.1: Total energy (hartree), energy splitting (cm − 1 ) between Σ + u and Σ + g states, and the transition dipole moment (au) for He + 2 calculated by EOM-IP-CCSD 3.00 ˚ A 4.00 ˚ A 5.00 ˚ A 6.00 ˚ A cc-pVTZ E, hartree -4.9015533 -4.8994189 -4.8992016 -4.8991721 Δ E, cm − 1 915.9 74.78 4.30 0.12 μ tr 2.815 3.769 4.718 5.664 aug-cc-pVTZ E, hartree -4.9023528 -4.8999136 -4.8996131 -4.8995577 Δ E, cm − 1 966.7 87.4 7.01 0.70 μ tr 2.796 3.758 4.711 5.660 aug-cc-pVQZ E, hartree -4.9051660 -4.9027398 -4.9024400 -4.9023808 Δ E, cm − 1 962.8 87.45 7.41 -0.60 μ tr 2.796 3.758 4.711 5.660 large separations, only two electrons need to be correlated for a good description of the dynamical correlation. A similar trend was observed in bond breaking applications of complete active space self-consistent field (CASSCF) and valence optimized orbital coupled cluster doubles (VOO-CCD) methods 97 . Similar trends are observed for the transition dipole moment, as shown in Fig. 3.5. At least a single set of diffuse functions is needed and the values converge at the aug-cc-pVDZ basis set. For this small benchmark system, we also investigated the performance of methods based on the doublet reference using the aug-cc-pVTZ basis set. These results are given in Table 3.3. UHF-CIS gives qualitatively incorrect results, i.e. it places the excited state below the ground state, and the level splitting does not decay to zero at large distances. This behavior is not rectified by including electron correlation, even at the EOM-EE- CCSD level. The ordering is only correct at small distances; however, the asymptotic distance behavior is still lacking, which is quite unexpected for this 3-electron system. 48 Table 3.2: Total energy (hartree), energy splitting (cm − 1 ) between Σ + u and Σ + g states, and the transition dipole moment (au) for He + 2 calculated by FCI 3.00 ˚ A 4.00 ˚ A 5.00 ˚ A 6.00 ˚ A cc-pVTZ E, hartree -4.9015843 -4.8994248 -4.8992035 -4.8991729 Δ E, cm − 1 921.6 75.27 4.33 0.12 μ tr 2.814 3.768 4.717 5.664 aug-cc-pVTZ E, hartree -4.9024762 -4.8999443 -4.8996242 -4.8995631 Δ E, cm − 1 981.0 89.02 7.15 0.72 μ tr 2.790 3.755 4.709 5.658 aug-cc-pVQZ E, hartree -4.9052897 -4.9027716 -4.9024514 -4.9023889 Δ E, cm − 1 976.6 89.17 7.58 0.62 μ tr 2.790 3.755 4.709 5.658 2 3 4 5 6 1E-3 0.01 0.1 1 10 100 1000 Figure 3.3: Electronic coupling in the helium dimer as a function of distance using FCI. 49 (a) cc-pVDZ cc-pVTZ aug-cc-pVDZ aug-cc-pVTZ d-aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z 1480 1520 1560 1600 (b) cc-pVDZ cc-pVTZ aug-cc-pVDZ aug-cc-pVTZ d-aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z 0.0 0.2 0.4 2 3 4 5 Figure 3.4: Electronic coupling in He + 2 at 2.5 ˚ A (a) and 5.0 ˚ A (b) calculated by FCI (squares) and EOM-IP-CCSD (diamonds). 3.4.2 (H 2 ) + 2 dimer The hydrogen dimer cation is isoelectronic with He + 2 ; however, due to the additional nuclear degree of freedom the molecular fragments no longer have to be identical, and the charge can be localized. From several possible orientations of the two fragments, we 50 (a) cc-pVDZ cc-pVTZ aug-cc-pVDZ aug-cc-pVTZ d-aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z 2.29 2.30 2.31 2.32 2.33 2.34 2.35 (b) cc-pVDZ cc-pVTZ aug-cc-pVDZ aug-cc-pVTZ d-aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z 4.708 4.710 4.712 4.714 4.716 4.718 4.720 4.722 Figure 3.5: The transition dipole moment in He + 2 at 2.5 ˚ A (a) and 5.0 ˚ A (b) calculated by FCI (squares) and EOM-IP-CCSD (diamonds). chose aC 2v symmetry configuration in which the two molecules are parallel. EOM-IP- CCSD, CCSD/EOM-EE-CCSD, and FCI calculations in the aug-cc-pVTZ basis were performed at reaction coordinate values of 0.0, 0.2, 0.4, 0.45 and 0.5 at 3.0 and 5.0 ˚ A separations. The data are summarized in Tables 3.4 and 3.5 and the error plots are 51 Table 3.3: Total energy (hartree), energy splitting (cm − 1 ) betweenΣ + u andΣ + g states, and the transition dipole moment (au) for He + 2 using open-shell reference in aug-cc-pVTZ basis set. 3.00 ˚ A 4.00 ˚ A 5.00 ˚ A 6.00 ˚ A HF/CIS spatial symmetry restricted UHF reference E, hartree -4.8401184 -4.8373567 -4.8370249 -4.8369709 Δ E, cm − 1 -3569.53 -4623.83 -4724.96 -4732.74 μ tr 2.781 3.735 4.680 5.622 HF/CIS broken spatial symmetry UHF reference E, hartree -4.8608400 -4.8603142 -4.8601885 -4.8601448 Δ E, cm − 1 21201.56 21086.06 21051.22 21039.11 μ tr 0.150 0.019 0.002 0.0003 q gr , e 0.008 0.003 0.002 0.001 q ex , e 0.994 0.996 0.998 0.999 CCSD/EOM-EE-CCSD spatial symmetry restricted UHF reference E, hartree -4.9021320 -4.8995985 -4.8992786 -4.8992177 Δ E, cm − 1 825.27 -66.65 -148.21 -154.48 μ tr 2.790 3.755 4.709 5.659 CCSD/EOM-EE-CCSD broken spatial symmetry UHF reference E, hartree -4.9011883 -4.8997508 -4.8996079 -4.8995614 Δ E, cm − 1 1365.05 988.34 981.14 979.84 μ tr 1.995 0.343 0.035 0.004 q gr , e 0.158 0.005 0.002 0.001 q ex , e 0.876 0.995 0.998 0.999 given in Fig. 3.6 and 3.7. The EOM-IP-CCSD/aug-cc-pVTZ ionization energy (IE) is 16.398 and 14.544 eV at neutral and cation geometries, respectively. A brief explanation of the plots is in order. The errors are calculated as the difference between the approximate value and the exact FCI result. Negative values mean that the given quantity is underestimated, while positive values mean the opposite. If the curve is parallel to the x axis, it means that the error is constant throughout the reaction coordinate space, a highly desirable feature. A slope, on the other hand, denotes a 52 (a) (b) (c) 0.0 0.1 0.2 0.3 0.4 0.5 0 30 60 90 120 0.0 0.1 0.2 0.3 0.4 0.5 -200 0 200 400 600 800 0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.01 0.02 0.03 (d) (e) (f) 0.0 0.1 0.2 0.3 0.4 0.5 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.0 0.1 0.2 0.3 0.4 0.5 -0.15 -0.10 -0.05 0.00 0.0 0.1 0.2 0.3 0.4 0.5 -60 -30 0 30 Figure 3.6: Error in (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge, (e) transition dipole moment, and (f) diabatic coupling in (H 2 ) + dimer at 3.0 ˚ A separation. EOM-IP-CCSD/aug-cc-pVTZ (solid line) and EOM-EE-CCSD/aug-cc-pVTZ (dotted line) results are shown. 53 (a) (b) (c) 0.0 0.1 0.2 0.3 0.4 0.5 -4 -2 0 76 78 80 82 0.0 0.1 0.2 0.3 0.4 0.5 -150 -100 -50 0 600 650 700 750 800 0.0 0.1 0.2 0.3 0.4 0.5 0.0000 0.0003 0.0006 0.0009 (d) (e) (f) 0.0 0.1 0.2 0.3 0.4 0.5 -0.0015 -0.0010 -0.0005 0.0000 0.0005 0.0 0.1 0.2 0.3 0.4 0.5 -0.15 -0.10 -0.05 0.00 0.0 0.1 0.2 0.3 0.4 0.5 -60 -58 -56 0.0 0.5 1.0 1.5 2.0 Figure 3.7: Error in (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge, (e) transition dipole moment, and (f) diabatic coupling in (H 2 ) + at 5.0 ˚ A separation. EOM-IP-CCSD/aug-cc-pVTZ (solid line) and EOM- EE-CCSD/aug-cc-pVTZ (dotted line) results are shown. 54 Table 3.4: Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for (H 2 ) + 2 at 3.0 ˚ A sep- aration in aug-cc-pVTZ basis set. The R=0.4 data set is omitted from the Table. 0.0 0.2 0.45 0.5 FCI E, hartree -1.779004 -1.777154 -1.770939 -1.770563 Δ E, cm − 1 15193 9720 4601 4369 μ tr 0.758 1.194 2.534 2.669 q gr ,e 0.957 0.924 0.648 0.500 q ex , e 0.0540 0.0822 0.353 0.500 h ab , cm − 1 2156.2 2174.4 2184.4 2184.7 EOM-IP-CCSD E, hartree -1.778861 -1.777017 -1.770830 -1.770456 Δ E, cm − 1 15187 9723 4620 4389 μ tr 0.767 1.207 2.551 2.685 q gr ,e 0.957 0.924 0.648 0.500 q ex , e 0.0549 0.0831 0.354 0.500 h ab , cm − 1 2182.4 2196.5 2195.7 2194.6 EOM-EE-CCSD E, hartree -1.778942 -1.777015 -1.770244 -1.770200 Δ E, cm − 1 15891 10303 4695 4264 μ tr 0.730 1.131 2.375 2.670 q gr ,e 0.958 0.930 0.681 0.500 q ex , e 0.04984 0.06830 0.35356 0.500 h ab , cm − 1 2167.3 2165.3 2186.6 2132.2 change in the quality of description and a non-parallelity error (NPE). The absolute values of the total energy are not important, and only NPEs are of interest — the error for the total energy of the ground state is arbitrarily set to 0 at R=0. For the ground state charge, if the error is positive is signifies that there is excessive charge separation, i.e., the state is overpolarized. Overpolarization of the excited state is manifested by a negative error. 55 Table 3.5: Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for (H 2 ) + 2 at 5.0 ˚ A sep- aration in aug-cc-pVTZ basis set. The R=0.4 data set is omitted from the Table. 0.0 0.2 0.45 0.5 FCI E, hartree -1.775487 -1.772451 -1.761042 -1.758121 Δ E, cm − 1 14882 8883 1482 133.3 μ tr 0.0412 0.0696 0.419 4.661 q gr ,e 0.995 0.994 0.991 0.500 q ex , e 0.00784 0.00742 0.00877 0.500 h ab , cm − 1 65.76 66.33 66.63 66.64 EOM-IP-CCSD E, hartree -1.775475 -1.772442 -1.761037 -1.757504 Δ E, cm − 1 14876 8880 1482 135.9 μ tr 0.0422 0.0712 0.428 4.667 q gr ,e 0.995 0.994 0.991 0.500 q ex , e 0.00807 0.00758 0.00894 0.50000 h ab , cm − 1 67.4 67.8 68.0 67.9 EOM-EE-CCSD E, hartree -1.775487 -1.772451 -1.761038 -1.757750 Δ E, cm − 1 15630 9587 2137 20.7 μ tr 0.040 0.066 0.295 4.662 q gr ,e 0.995 0.994 0.992 0.500 q ex , e 0.00804 0.00764 0.00775 0.500 h ab , cm − 1 66.9 67.4 67.5 10.3 Panels (a) and (b) in Figures 3.6 and 3.7 show the error in the ground state total energy and excitation energy, respectively. At small values of R, i.e., when there is large difference between the geometries of the two fragments both methods perform similarly. As the bond lengths become more similar the discrepancy between EOM-EE-CCSD and FCI becomes more significant. Finally, at R=0.5 the doublet HF wave function becomes unstable yielding a cusp on the PES. It is manifested as a large jump on all the plots. 56 Meanwhile the error for EOM-IP-CCSD curves remains small. At 5.0 ˚ A separation and excluding the R=0.5 point, the EOM-EE-CCSD error in excitation energy ranges between 750 and 650 cm − 1 . The EOM-IP-CCSD error is confined to the 6 - 0 cm − 1 range. Similar behavior is observed at 3.0 ˚ A separation. In other words, the EOM-EE- CCSD NPE is high and the description is not uniform throughout the reaction coordinate space. An important property of a CT system is its charge distribution. Panels (c) and (d) of Figures 3.6 and 3.7 show the ground and excited state charge of the more positively charged fragment (in the ground state) at the two interfragment separations. In both cases the EOM-IP-CCSD and FCI results are essentially identical, and the NPE is small. The quality of CCSD/EOM-EE-CCSD description degrades towards R=0.5. Both the ground and excited state become overpolarized as a consequence of the charge-localized character of the UHF doublet reference. The incorrect charge distribution in turn affects the transition dipole moment (Fig. 3.6e, 3.7e). Again, the EOM-IP-CCSD description is uniform and accurate throughout, while EOM-EE-CCSD underestimates this property. The calculated couplings depend on the energy splitting, transition and permanent dipole moment of the two states. The values are plotted in panel (f) of Figs. 3.6 and 3.7. The most striking feature is the cusp of the EOM-EE-CCSD curve at the transition state. It originates from the CCSD PES cusp, as the coupling at this point is equal to half the energy splitting between the two states. EOM-IP-CCSD systematically overestimates the coupling, due to the error in the transition dipole. The coupling weakly depends on the reaction coordinate, in agreement with the Condon approximation, i.e. the coupling only depends on the molecular coordinates that do not affect the effective donor-acceptor distance. 57 3.4.3 (Be-BH) + dimer The (Be-BH) + dimer cation was studied in a linear configuration, with the beryllium atom located on the boron side. aug-cc-pVDZ basis set was used in the calculations. The data are presented in Table 3.6 and the error plots are shown in Fig. 3.8. The BH distance was scanned from its cation geometry (R=0) to the neutral geometry (R=1). The distance between the COM of both fragments was kept constant at 5.0 ˚ A. EOM-IP-CCSD/aug- cc-pVDZ vertical IE of beryllium is 9.234 eV , while that of BH decreases from 9.687 to 9.679 eV as the bond length increases from the cation to the neutral geometry. The fact that the IE of BH at the cation geometry is larger than at the neutral geometry is due to the fact that the equilibrium bond length were optimized at a different level of theory. It is not a problem in other systems due to larger geometric change. Due to the small changes in the relative energies of both species only subtle changes are expected along the reaction coordinate. Before delving into the details note that both CCSD/EOM- EE-CCSD and EOM-IP-CCSD (in all cases except the ground state energy) capture the trends in properties along the reaction coordinate. The NPE is smaller for EOM-IP- CCSD indicating a more uniform description throughout the reaction coordinate space. The IEs of both fragments are very close, thus we expect an appreciable extent of charge delocalization. In the ground state roughly 86% of the hole is located on Be, while only 6% in the excited state. Both EOM-IP-CCSD and CCSD predict a slightly more localized structure than FCI, in both states. This in turn affects the transition dipole moment, which decreases as follows: FCI> EOM-IP-CCSD> EOM-EE-CCSD. Clearly, the more charge localized the state is, the lower the transition dipole moment is. The inverse is true for the excitation energies: FCI values are lower than EOM-EE- CCSD, while EOM-IP-CCSD is in-between. Lastly, let us look at the diabatic coupling, which is a cumulative property. Unexpectedly, all methods are in very good accord, 58 (a) (b) (c) 0.00 0.25 0.50 0.75 1.00 -6 -4 -2 0 0.00 0.25 0.50 0.75 1.00 790 800 1710 1720 0.00 0.25 0.50 0.75 1.00 0.0088 0.0089 0.0139 0.0140 0.0141 (d) (e) (f) 0.00 0.25 0.50 0.75 1.00 -0.0370 -0.0365 -0.0055 -0.0050 -0.0045 0.00 0.25 0.50 0.75 1.00 -0.334 -0.333 -0.121 -0.120 -0.119 0.00 0.25 0.50 0.75 1.00 31.75 32.00 38.25 38.50 38.75 Figure 3.8: Error in (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge, (e) transition dipole moment, and (f) diabatic coupling in (Be-BH) + . EOM-IP-CCSD/aug-cc-pVDZ (solid line) and EOM-EE-CCSD/aug- cc-pVDZ (dotted line) results are shown. The charge pertains to the Be fragment. 59 Table 3.6: Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for linear (Be-BH) + at 5.0 ˚ A separation in aug-cc-pVDZ basis set. The charge pertains to the Be fragment. 0.0 0.5 1.0 FCI E, hartree -39.505862 -39.505843 -39.505790 Δ E, cm − 1 5295.0 5282.9 5270.6 μ tr , au 1.643 1.647 1.648 q gr ,e 0.8648 0.8639 0.8630 q ex , e 0.0629 0.0634 0.0639 h ab , cm − 1 1053.1 1053.8 1054.5 EOM-IP-CCSD E, hartree -39.506545 -39.506542 -39.506503 Δ E, cm − 1 5903.8 5889.0 5873.8 μ tr , au 1.522 1.527 1.528 q gr ,e 0.8737 0.8728 0.8719 q ex , e 0.0640 0.0645 0.0650 h ab , cm − 1 1091.6 1092.3 1093.0 EOM-EE-CCSD E, hartree -39.505518 -39.505513 -39.505472 Δ E, cm − 1 6878.9 6861.8 6844.4 μ tr , au 1.310 1.313 1.315 q gr ,e 0.8788 0.8780 0.8771 q ex , e 0.0451 0.0455 0.0459 h ab , cm − 1 1085.4 1085.9 1086.4 within 5%. The agreement for EOM-IP is only slightly inferior than for EOM-EE- CCSD. This is very interesting, as both methods give slightly different picture of the states. In case of EOM-EE-CCSD, the increased transition energy is compensated by the decreased transition dipole. In the denominator the increased difference in permanent dipole moments compensates for the underestimated transition dipole, see Eq. (3.5). 60 3.4.4 (BH-H 2 ) + dimer The BH-H 2 system is an example complementary to Be-BH. The difference in vertical ionization energies is approximately 6 eV , much larger than 0.5 eV in Be-BH. At the cation geometry, the EOM-IP-CCSD/aug-cc-pVDZ IEs of BH and H 2 are 9.687 eV and 14.460 eV , respectively; values at neutral geometries are 9.679 and 16.288 eV . We studied the system in a t-shaped configuration: the H 2 molecule constitutes the top, while BH (boron atom closer to H 2 ) is the stem. At R=0, H 2 is at its neutral geometry while BH is at its cation geometry. At R=1, BH is at its neutral geometry, while H 2 is at the cation geometry. The distance between COMs is kept fixed at 3.0 ˚ A. The data are listed in Table 3.7 and error plots are given in Fig. 3.9. The aug-cc-pVDZ basis set was used in the calculations. The performance of EOM-IP-CCSD and CCSD/EOM-EE-CCSD is very similar. The errors are smaller for the latter, but the difference is not significant compared to the quantities involved. For instance, at R=0.5 the former underestimates the excitation energy by 750 cm − 1 , while the latter overestimates it by the same amount. The value of the excitation energy is c.a. 40,000 cm − 1 using all methods. With increased R the energy spacing decreases and CCSD/EOM-EE-CCSD tends to overpolarize both states while EOM-IP-CCSD underpolarizes them. Nonetheless, all methods predict the hole to be almost entirely located on BH in the ground state, and on H 2 in the excited state. This is in agreement with the difference of IEs, and according to Eq. 3.1 signifies a very similar character of diabatic and adiabatic wave functions. Another way of understanding it is by comparing the diabatic coupling with the energy difference between the adiabatic levels. Consider a two-level coupled system. If the two levels are degenerate, they will split by twice the amount for the coupling. In the case, when they are non-degenerate, the amount of splitting induced by the coupling will be less than twice its value. Thus, in (BH-H 2 ) + dimer diabatic coupling can account for utmost 13,000 cm − 1 of adiabatic 61 (a) (b) (c) 0.00 0.25 0.50 0.75 1.00 -20 -10 0 10 0.00 0.25 0.50 0.75 1.00 -800 -750 -700 -650 700 750 800 850 0.00 0.25 0.50 0.75 1.00 -0.004 -0.002 0.000 0.002 (d) (e) (f) 0.00 0.25 0.50 0.75 1.00 -0.008 -0.006 -0.004 -0.002 0.000 0.002 0.00 0.25 0.50 0.75 1.00 -0.02 0.00 0.02 0.04 0.00 0.25 0.50 0.75 1.00 0 30 60 90 120 Figure 3.9: Error in (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge, (e) transition dipole moment, and (f) diabatic coupling in (BH-H 2 ) + . EOM-IP-CCSD/aug-cc-pVDZ (solid line) and EOM-EE-CCSD/aug- cc-pVDZ (dotted line) results are shown. The charge pertains to the BH fragment. 62 Table 3.7: Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for t-shaped (BH-H 2 ) + at 3.0 ˚ A separation in aug-cc-pVDZ basis set. The charge pertains to the BH fragment. The R=0.25 and R=0.75 data sets are omitted from the Table. 0.0 0.5 1.0 FCI E, hartree -26.030624 -26.022318 -26.000430 Δ E, cm − 1 48228 40742 35094 μ tr , au 0.6300 0.7457 0.8761 q gr ,e 0.998 0.987 0.975 q ex , e 0.222 0.221 0.226 h ab , cm − 1 6643 6612 6694 EOM-IP-CCSD E, hartree -26.028457 -26.020179 -25.998359 Δ E, cm − 1 47467 39984 34373 μ tr , au 0.6469 0.7733 0.9124 q gr ,e 0.995 0.984 0.970 q ex , e 0.214 0.216 0.223 h ab , cm − 1 6654 6692 6797 EOM-EE-CCSD E, hartree -26.030734 -26.022403 -26.000479 Δ E, cm − 1 49088 41523 35783 μ tr , au 0.6273 0.7375 0.8630 q gr ,e 0.997 0.987 0.976 q ex , e 0.224 0.219 0.222 h ab , cm − 1 6754 6663 6695 state separation. The smallest difference between them occurs at R=1, and is equal to 35,000 cm − 1 . The difference, 22,000 cm − 1 is the difference between energies of the diabatic states. Since it is significantly larger than the coupling, the adiabatic states will be very similar to the diabatic states. This is additionally confirmed by good agreement of transition dipoles between EOM-IP-CCSD and EOM-EE-CCSD. For this molecular 63 system both methods perform similarly and other than the computational cost there is no preference for either one. 3.4.5 (LiH) + 2 dimer The LiH dimer cation cation was studied in an stacked antiparallel configuration. The separation between COMs was held fixed at 4.0 ˚ A. The monomer state considered here corresponds to ionization from the σ bonding orbital of the monomer. According to EOM-IP-CCSD/6-31+G it requires 6.781 and 7.589 eV at the cation and neutral geome- tries, respectively. The data are listed in Table 3.8 and error plots are given in Fig. 3.10. The 6-31+G basis set was used in the calculations. The large difference between the IEs causes a significant change in the extent of charge delocalization along the reaction coordinate. This is in stark contrast to BH. Low coupling to Be or H 2 precluded the use of LiH in the heterodimer calculations. Panels (a) and (b) of Fig. 3.10 present the error in the ground state total energy and the excitation energy, respectively. There is a significant NPE in both. EOM-IP-CCSD predicts a larger energy change when going from R=0 to R=0.5. At R=0 the excitation energy is overestimated by 600 cm − 1 , while it is underestimated by 200 cm − 1 at R=0.5. These numbers do not exceed 10%. FCI yields a less polarized state and a lower tran- sition dipole moment. The error in the transition moment increases as the monomers become more similar. Lastly, EOM-IP-CCSD predicts weaker diabatic coupling than FCI. Note that the NPE is much smaller for the coupling than for the other quantities. 3.4.6 (C 2 H 4 ) + 2 The results in this section are obtained using 6-31+G basis set. The biggest difference between the neutral and the cation geometry is the C-C bond length: 1.418 and 1.329 ˚ A, respectively. The vertical EOM-IP-CCSD/6-31+G IE of ethylene fragment at neutral 64 (a) (b) (c) 0.0 0.1 0.2 0.3 0.4 0.5 0 50 100 150 200 250 300 0.0 0.1 0.2 0.3 0.4 0.5 -200 0 200 400 600 0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.01 0.02 0.03 0.04 0.05 (d) (e) (f) 0.0 0.1 0.2 0.3 0.4 0.5 -0.08 -0.06 -0.04 -0.02 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.0 0.1 0.2 0.3 0.4 0.5 -120 -100 -80 -60 -40 Figure 3.10: Error in (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge, (e) transition dipole moment, and (f) diabatic coupling in (LiH) + 2 . EOM-IP-CCSD/6-31+G results are shown. 65 Table 3.8: Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for (LiH) + 2 at 4.0 ˚ A separation in 6-31+G basis set. 0.0 0.2 0.45 0.5 FCI E, hartree -15.7422241 -15.7428853 -15.7418536 -15.7417941 Δ E, cm − 1 6862.5 4924.4 3333.3 3275.5 μ tr 1.519 2.131 3.158 3.214 q gr ,e 0.845 0.788 0.570 0.500 q ex , e 0.222 0.252 0.436 0.500 h ab , cm − 1 1661.4 1646.9 1638.0 1637.7 EOM-IP-CCSD E, hartree -15.7383087 -15.7383529 -15.7364888 -15.7363863 Δ E, cm − 1 7478.7 5141.0 3115.0 3036.3 μ tr 1.576 2.290 3.776 3.874 q gr ,e 0.888 0.840 0.593 0.500 q ex , e 0.141 0.178 0.410 0.500 h ab , cm − 1 1614.6 1584.3 1522.7 1518.2 geometry is 10.381 eV , while that at the cation geometry is 10.049 eV . We chose the parallel stacked geometry in which the planes of the molecules are separated by 4.0 and 6.0 ˚ A. The results are in Table 3.9 and Figs. 3.11 and 3.12. Fig. 3.11 presents the ground state PES and the charge distribution along the CT reaction coordinate in the ethylene dimer cation at 4 ˚ A and 6 ˚ A separation. EOM-IP- CCSD produces a smooth change in both quantities, whereas the CCSD curve (using doublet reference) exhibits a cusp at R=0.5. Note that the cusp is present for both UHF and ROHF based curves and is due to unbalanced description of the important electronic configurations by doublet-reference based CCSD rather than spin-contamination of the reference. 66 Table 3.9: Total energy (hartree), energy splitting (cm − 1 ), transition dipole moment (au), ground and excited state charge (au), coupling (cm − 1 ) calculated for (C 2 H 4 ) + 2 at 4.0 ˚ A separation in 6-31+G basis set. 0.0 0.2 0.45 0.5 EOM-IP-CCSD E, hartree -156.078063 -156.079790 -156.080697 -156.080723 Δ E, cm − 1 5239 4841 4609 4603 μ tr , au 3.142 3.404 3.577 3.582 q gr ,e 0.733 0.652 0.527 0.500 q ex , e 0.277 0.355 0.475 0.500 h ab , cm − 1 2303.8 2302.6 2301.3 2301.3 EOM-IP-CC(2,3) E, hartree 156.083897 -156.085381 -156.086138 -156.086159 Δ E, cm − 1 5447 4948 4654 4646 q gr ,e 0.747 0.663 0.529 0.500 q ex , e 0.256 0.339 0.471 0.500 MR-CISD+Q E, hartree -156.077664 -156.079075 -156.079781 -156.079716 Δ E, cm − 1 5352 4885 4612 4610 μ tr 3.215 3.456 3.611 3.621 q gr ,e 0.722 0.643 0.525 0.500 q ex , e 0.285 0.362 0.476 0.500 h ab , cm − 1 2393.4 2334.6 2302.9 2305.2 Since this system is beyond the reach of FCI, we compare the EOM-IP-CCSD and CCSD/EE-CCSD results against more accurate EOM-IP-CC(2,3) (EOM-IP- CCSD/3h2p) values, as well as MR-CISD+Q. As expected, EOM-IP-CC(2,3) and MR- CISD+Q are in an excellent agreement. Both methods predict a deeper potential well, the difference being ca. 100 cm − 1 . MR-CISD+Q curve has a small cusp due to the frozen core, e.g., the cusp disappears if the excitations from core orbitals are included at MR-CISD level. Unfortunately, it was not possible to simultaneously unfreeze the core and employ adequately large active space in MR-CI calculations. 67 (a) (b) 0.00 0.25 0.50 0.75 1.00 -800 -600 -400 -200 0 0.00 0.25 0.50 0.75 1.00 -800 -400 0 400 (c) (d) 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Figure 3.11: Changes in charge distribution and PES scans along the CT coordinate in the ground state of (C 2 H 4 ) + 2 at 4 ˚ A (panels (a) and (c)) and 6 ˚ A separation (panels (b) and (d)). Overall, EOM-IP-CCSD, EOM-IP-CC(2,3) and MR-CISD predict very similar, smooth changes of the fragment charge. The degree of charge localization is exag- gerated in CCSD calculation. This causes an unphysically abrupt change in the polarity of the system around the transition state. Similar discrepancy between EOM-IP-CCSD and EOM-EE-CCSD is present in the excited state (not shown). 68 (a) (b) 0.0 0.1 0.2 0.3 0.4 0.5 -100 -50 0 50 0.0 0.1 0.2 0.3 0.4 0.5 -200 -150 -100 -50 (c) (d) 0.0 0.1 0.2 0.3 0.4 0.5 -0.025 -0.020 -0.015 -0.010 -0.005 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.000 0.010 0.020 0.030 Figure 3.12: Differences against EOM-IP-CCSD(3h2p) for (a) ground state energy, (b) excitation energy, (c) ground state charge, (d) excited state charge in (C 2 H 4 ) + 2 . DFT/B3LYP calculation yields qualitatively correct shapes of the PES at 4 ˚ A, how- ever, the depth and the degree of charge delocalization are severely overestimated, due to SIE. As the distance between the fragments increases, the R=0.5 point would become a transition state separating to charge-localized minima. At the EOM-IP-CCSD level, this happens at 6 ˚ A, whereas B3LYP still predicts a potential well and significant charge delocalization. In fact, there is only minor change between the B3LYP results at 4 ˚ A and 69 6 ˚ A. Of course, SIE-corrected functionals, e.g., long-range corrected functionals 98, 99 , should be able to better describe these CT systems. Since the density matrices at the EOM-IP-CC(2,3) level are not available, we have restricted ourselves to the calculation of energies and permanent dipoles via finite differ- ences. All differences shown in Fig. 3.12 were calculated relative to EOM-IP-CC(2,3). As is evident from Fig. 3.12a, the depth of the potential well along R is overes- timated by EOM-IP-CCSD. The largest error is 80 cm − 1 and occurs at R=0.5. This is best compared to the excitation energy, which at this point is 4603 cm − 1 . The MR-CISD+Q/EOM-IP-CC(2,3) difference is approximately half of the EOM-IP-CCSD error. The error in the excitation energy is shown in Fig. 3.12b. For EOM-IP-CCSD it decreases toward the transition state from 200 cm − 1 to 50 cm − 1 . In other words, the quality of the EOM-IP-CCSD wave function improves as the monomers become more similar, however NPE is appreciable. Fig. 3.12c and d show the ground and excited state charge. In both cases, the EOM-IP-CCSD and MR-CISD methods underpolarize the CT state relative to EOM-IP-CC(2,3). The magnitude of the difference is small, thus yielding credence to the transition dipole moment. As seen previously, exagger- ated delocalization leads to transition dipole moments that are typically too high. The diabatic coupling varies smoothly with R (not shown). The lack of EOM-IP-CC(2,3) transition dipole prevents us from making a direct comparison of the diabatic coupling. In other systems studied we have witnessed a peculiar error cancellation between ingre- dients of Eq. (3.5), which gives us confidence in the presented values. A comparison between MRCI and EOM-IP-CCSD is very interesting. Quantitatively the results are very similar. Note however, that while EOM-IP-CCSD coupling changes by 2 cm − 1 between R=0 and R=0.5, MRCI predicts a 90 cm − 1 change. 70 3.5 Conclusions The presented results demonstrate that EOM-IP-CCSD is a reliable method for the study of noncovalent ionized dimers. It yields smooth variation of energies and molecu- lar properties with nuclear coordinates. Most importantly, the cusp in the PES along charge transfer coordinates that is associated with the open-shell reference is completely avoided. Also, the NPE is typically small. In other words, different spatial arrangements of the fragments are described with equal accuracy. The advantages of the EOM-IP method become even more important when the ionized states of the monomers feature electronic degeneracies, as in benzene dimer cation 8 . In cases where the difference in IEs is much larger than the coupling, EOM-IP- CCSD and EOM-EE-CCSD perform similarly. Due to the lower computational scaling, the former method is preferable. An argument can be made that just like EOM-EE- CCSD overpolarizes the states, the EOM-IP-CCSD method may appreciably underpo- larize the states. In the studied systems only a small degree of underpolarization has been observed. The diabatic coupling has proved to be a fairly insensitive probe of the quality of state description. The increased polarity of EOM-EE-CCSD states is offset by lower transition dipoles and higher excitation energies. Comparison of transition dipole moments offers a better one-number descriptor of the quality of the ground and excited state wave functions. We expect that the presented results will provide useful calibration data for calcula- tion of electronic coupling elements as well as dimer properties. 71 3.6 Chapter 3 Bibliography [1] Newton, M.D. Chem. Rev. 1991, 91, 767. [2] Amini, A. ; Harriman, A. J. Photochem. Photobiol. C 2003, 4, 155. [3] Gray, H.B. ; Winkler, J.R. Q. Rev. Biophys. 2003, 36, 341. [4] Endres, R.G. ; Cox, D.L. ; Singh, R.R.P. Rev. Mod. Phys. 2004, 76, 195. [5] Inokuchi, Y . ; Naitoh, Y . ; Ohashi, K. ; Saitow, K. ; Yoshihara, K. ; Nishi, N. Chem. Phys. Lett. 1997, 269, 298. [6] Okamoto, K. ; Saeki, A. ; Kozawa, T. ; Yoshida, Y . ; Tagawa, S. Chem. Lett. 2003, 32, 834. [7] Enomoto, K. ; LaVerne, J.A. ; Araos, M. S. J. Phys. Chem. A 2007, 111, 9. [8] Pieniazek, P.A. ; Krylov, A.I. ; Bradforth, S.E. J. Chem. Phys. 2007, 127, 044317. [9] Tomoda, S. ; Achiba, Y . ; Kimura, K. Chem. Phys. Lett. 1982, 87, 197. [10] Grover, J.R. ; Walters, E.A. ; Hui, E.T. J. Phys. Chem. 1987, 91, 3233. [11] Krause, H. ; Ernstberger, B. ; Neusser, H.J. Chem. Phys. Lett. 1991, 184, 411. [12] Reid, G.C. Adv. At. Mol. Phys. 1976, 12, 375. [13] Ferguson, E.E. ; Fehsenfield, F.C. ; Albritton, D.L. In Gas Phase Ion Chemistry; Bowers, M. T. , Ed., V ol. 1; Academic Press, 1979; pages 45–82. [14] Lindh, R. ; Barnes, L.A. J. Chem. Phys. 1994, 100, 224. [15] Rhee, Y .M. ; Lee, T.J. ; Gudipati, M.S. ; Allamandola, L.J. ; Head-Gordon, M. Proc. Nat. Acad. Sci. 2007, 104, 5274. [16] Illies, A.J. ; McKee, M.L. ; Schlegel, H.B. J. Phys. Chem. 1987, 91, 3489. [17] de Visser, S. P. ; de Koning, L.J. ; Nibbering, N.M.M. J. Phys. Chem. 1995, 99, 15444. [18] Hiberty, P.C. ; Humbel, S. ; Danovich, D. ; Shaik, S. J. Am. Chem. Soc. 1995, 117, 9003. [19] Sodupe, M. ; Bertran, J. ; Rodrguez-Santiago, L. ; Baerends, E.J. J. Phys. Chem. A 1999, 103, 166. 72 [20] Ohashi, K. ; Nakai, Y . ; Shibata, T. ; Nishi, N. Laser Chem. 1994, 14, 3. [21] Shkrob, I.A. ; Sauer Jr., M.C. ; Jonah, C.D. ; Takahashi, K. J. Phys. Chem. A 2002, 106, 11855. [22] Marcus, R.A. Discussions Faraday Soc. 1960, 29, 21. [23] Marcus, R.A. J. Chem. Phys. 1965, 43, 679. [24] Marcus, R.A. ; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. [25] Norris, J.R. ; Uphaus, R.A. ; Crespi, H.L. ; Katz, J. Proc. Nat. Acad. Sci. 1971, 68, 625. [26] Zhang, L.Y . ; Friesner, R.A. Proc. Nat. Acad. Sci. 1998, 95, 13603. [27] Leys, D. ; Meyer, T.E. ; Tsapin, A.S. ; Nealson, K.H. ; Cusanovich, M.A. ; Van Beeumen, J.J. J. Biol. Chem. 2002, 277, 35703. [28] Smith, D.M.A. ; Rosso, K.M. ; Dupuis, M. ; Valiev, M. ; Straatsma, T.P. J. Phys. Chem. B 2006, 110, 15582. [29] Delaney, S. ; Barton, J.K. J. Org. Chem. 2003, 68, 6475. [30] Davidson, E.R. ; Borden, W.T. J. Phys. Chem. 1983, 87, 4783. [31] Russ, N.J. ; Crawford, T.D. ; Tschumper, G.S. J. Chem. Phys. 2005, 120, 7298. [32] Rodriguez-Monge, L. ; Larsson, S. J. Phys. Chem. 1996, 100, 6298. [33] Cave, R.J. ; Newton, M.D. J. Chem. Phys. 1997, 106, 9213. [34] Carra, C. ; Iordanova, N. ; Hammes-Schiffer, S. J. Phys. Chem. B 2002, 106, 8415. [35] Eisfeld, W. ; Morokuma, K. J. Chem. Phys. 2000, 113, 5587. [36] Krylov, A.I. Chem. Phys. Lett. 2001, 338, 375. [37] Levchenko, S.V . ; Krylov, A.I. J. Chem. Phys. 2004, 120, 175. [38] Krylov, A.I. Acc. Chem. Res. 2006, 39, 83. [39] You, Z.Q. ; Shao, Y .H. ; Hsu, C.P. Chem. Phys. Lett. 2004, 390, 116. [40] L¨ owdin, P.O. Rev. Mod. Phys. 1963, 35, 496. [41] Cohen, R.D. ; Sherrill, C.D. J. Chem. Phys. 2001, 114, 8257. [42] Bally, T. ; Sastry, G.N. J. Phys. Chem. A 1997, 101, 7923. 73 [43] Polo, V . ; Kraka, E. ; Cremer, D. Molecular Physics 2002, 100, 1771 . [44] Zhang, Y . ; Yang, W. J. Chem. Phys. 1998, 109, 2604. [45] Lundber, M. ; Siegbahn, P.E.M. J. Chem. Phys. 2005, 122, 1. [46] Dreuw, A. ; Head-Gordon, M. Chem. Rev. 2005, 105, 4009. [47] Dreuw, A. ; Head-Gordon, M. Chem. Phys. Lett. 2006, 426, 231. [48] Slipchenko, L.V . ; Krylov, A.I. J. Phys. Chem. A 2006, 110, 291. [49] Vanovschi, V . ; Krylov, A.I. ; Wenthold, P.G. Theor. Chim. Acta 2008, 120, 45. [50] Sinha, D. ; Mukhopadhyay, D. ; Mukherjee, D. Chem. Phys. Lett. 1986, 129, 369. [51] Pal, S. ; Rittby, M. ; Bartlett, R.J. ; Sinha, D. ; Mukherjee, D. Chem. Phys. Lett. 1987, 137, 273. [52] Sinha, D. ; Mukhopadhya, D. ; Chaudhuri, R. ; Mukherjee, D. Chem. Phys. Lett. 1989, 154, 544. [53] Chaudhuri, R. ; Mukhopadhyay, D. ; Mukherjee, D. Chem. Phys. Lett. 1989, 162, 393. [54] Stanton, J.F. ; Gauss, J. J. Chem. Phys. 1994, 101, 8938. [55] Stanton, J.F. ; Gauss, J. J. Chem. Phys. 1999, 111, 8785. [56] Yang, C.H. ; Hsu, C.P. J. Chem. Phys. 2006, 124, 244507. [57] Simons, J. ; Smith, W. D. J. Chem. Phys. 1973, 58, 4899. [58] Nakatsuji, H. ; Hirao, K. J. Chem. Phys. 1978, 68, 2053. [59] Nakatsuji, H. Chem. Phys. Lett. 1991, 177, 331. [60] Cave, R.J. ; Newton, M.D. Chem. Phys. Lett. 1996, 249, 15. [61] Malar, E.J.P. ; Chandra, A.K. J. Phys. Chem. 1981, 85, 2190. [62] Jungwirth, P. ; Bally, T. J. Am. Chem. Soc. 1993, 115, 5783. [63] Pieniazek, P.A. ; Arnstein, S.A. ; Bradforth, S.E. ; Krylov, A.I. ; Sherrill, C.D. J. Chem. Phys. 2007, 127, 164110. [64] Olsen, J. J. Chem. Phys. 2000, 113, 7140. 74 [65] Koch, H. ; Jensen, H.J.Aa. ; Jørgensen, P. ; Helgaker, T. J. Chem. Phys. 1990, 93, 3345. [66] Meissner, L. ; Bartlett, R.J. J. Chem. Phys. 1991, 94, 6670. [67] Nooijen, M. ; Shamasundar, K.R. ; Mukherjee, D. Mol. Phys. 2005, 103, 2277. [68] Stanton, J.F. J. Chem. Phys. 1994, 101, 8928. [69] Koch, H. ; Kobayashi, R. ; de Mer´ as, A. S. ; Jørgensen, P. J. Chem. Phys. 1994, 100, 4393. [70] Slipchenko, L.V . ; Krylov, A.I. J. Chem. Phys. 2005, 123, 84107. [71] Sekino, H. ; Bartlett, R.J. Int. J. Quant. Chem. Symp. 1984, 18, 255. [72] Stanton, J.F. ; Bartlett, R.J. J. Chem. Phys. 1993, 98, 7029. [73] Kucharski, S.A. ; Włoch, M. ; Musiał, M. ; Bartlett, R.J. J. Chem. Phys. 2001, 115, 8263. [74] Kowalski, K. ; Piecuch, P. J. Chem. Phys. 2001, 115, 643. [75] Nooijen, M. ; Bartlett, R.J. J. Chem. Phys. 1995, 102, 3629. [76] Hirata, S. ; Nooijen, M. ; Bartlett, R.J. Chem. Phys. Lett. 2000, 328, 459. [77] Kamyia, M. ; Hirata, S. J. Chem. Phys. 2006, 125, 074111. [78] Piecuch, P. ; Bartlett, R.J. Adv. Quantum Chem. 1999, 34, 295. [79] Mead, C.A. ; Truhlar, D.G. J. Chem. Phys. 1982, 77, 6090. [80] Mac´ ıas, A. ; Riera, A. J. Phys. B 1978, 11, L489. [81] Werner, H.-J. ; Meyer, W. J. Chem. Phys. 1981, 74, 5802. [82] Petrongolo, C. ; Hirsch, G. ; Buenker, R.J. Mol. Phys. 1990, 70, 825. [83] Alexander, M.H. J. Chem. Phys. 1993, 99, 6014. [84] Dobbyn, A.J. ; Knowles, P.J. Mol. Phys. 1997, 91, 1107. [85] Yarkony, D.R. J. Phys. Chem. A 1998, 102, 8073. 75 [86] Y . Shao, L.F. Molnar, Y . Jung, J. Kussmann, C. Ochsenfeld, S. Brown, A.T.B. Gilbert, L.V . Slipchenko, S.V . Levchenko, D.P. O’Neil, R.A. Distasio Jr, R.C. Lochan, T. Wang, G.J.O. Beran, N.A. Besley, J.M. Herbert, C.Y . Lin, T. Van V oorhis, S.H. Chien, A. Sodt, R.P. Steele, V .A. Rassolov, P. Maslen, P.P. Koram- bath, R.D. Adamson, B. Austin, J. Baker, E.F.C. Bird, H. Daschel, R.J. Doerksen, A. Drew, B.D. Dunietz, A.D. Dutoi, T.R. Furlani, S.R. Gwaltney, A. Heyden, S. Hirata, C.-P. Hsu, G.S. Kedziora, R.Z. Khalliulin, P. Klunziger, A.M. Lee, W.Z. Liang, I. Lotan, N. Nair, B. Peters, E.I. Proynov, P.A. Pieniazek, Y .M. Rhee, J. Ritchie, E. Rosta, C.D. Sherrill, A.C. Simmonett, J.E. Subotnik, H.L. Woodcock III, W. Zhang, A.T. Bell, A.K. Chakraborty, D.M. Chipman, F.J. Keil, A. Warshel, W.J. Herhe, H.F. Schaefer III, J. Kong, A.I. Krylov, P.M.W. Gill, M. Head-Gordon Phys. Chem. Chem. Phys. 2006, 8, 3172. [87] Crawford, T.D. ; Sherrill, C.D. ; Valeev, E.F. ; Fermann, J.T. ; King, R.A. ; Leininger, M.L. ; Brown, S.T. ; Janssen, C.L. ; Seidl, E.T. ; Kenny, J.P. ; Allen, W.D. J. Comput. Chem. 2007, 28, 1610. [88] MOLPRO 2002.6. Werner, H.-J. ; Knowles, P.J. ; Lindh, R. ; Sch¨ utz, M. et al. 2003. [89] Schuchardt, K.L. ; Didier, B.T. ; Elsethagen, T. ; Sun, L. ; Gurumoorthi, V . ; Chase, J. ; Li, J. ; Windus, T.L. J. Chem. Inf. Model. 2007, 47, 1045. [90] Levchenko, S.V . ; Wang, T. ; Krylov, A.I. J. Chem. Phys. 2005, 122, 224106. [91] Bruna, P.J. ; Peyerimhoff, S.D. In Ab initio methods in quantum chemistry, I; John Wiley & Sons, 1987; pages 1–98. [92] Werner, H.J. ; Knowles, P.J. J. Chem. Phys. 1988, 89, 5803. [93] Langhoff, S.R. ; Davidson, E.R. Int. J. Quant. Chem. 1974, 8, 61. [94] Raghavachari, K. ; Trucks, G.W. ; Pople, J.A. ; Head-Gordon, M. Chem. Phys. Lett. 1989, 157, 479. [95] Watts, J.D. ; Gauss, J. ; Bartlett, R.J. J. Chem. Phys. 1993, 98, 8718. [96] Becke, A.D. J. Chem. Phys. 1993, 98, 5648. [97] Krylov, A.I. ; Sherrill, C.D. ; Byrd, E.F.C. ; Head-Gordon, M. J. Chem. Phys. 1998, 109, 10669. [98] Tsuneda, T. ; Kamiya, M. ; Hirao, K. J. Comput. Chem. 2003, 24, 1592. [99] Tawada, Y . ; Tsuneda, T. ; Yanagisawa, S. ; Yanai, T. ; Hirao, K. J. Chem. Phys. 2004, 120, 8425. 76 Chapter 4: Electronic structure of the benzene dimer cation 4.1 Overview In this Chapter we study the benzene and benzene dimer cations using the equation-of- motion coupled-cluster model with single and double substitutions for ionized systems (EOM-IP-CCSD). The 10 lowest electronic states of the dimer at t-shaped, sandwich and displaced sandwich configurations are described and catalogued based on the character of the constituent fragment molecular orbitals. The character of the states, bonding pat- terns, and important features of the electronic spectrum are explained using qualitative dimer molecular orbitals — linear combination of fragment molecular orbitals frame- work. Relaxed ground state geometries are obtained for all isomers. Calculations reveal that the lowest energy structure of the cation has a displaced sandwich structure and a binding energy of 20 kcal/mol, while the t-shaped isomer is 6 kcal/mol higher. The calculated electronic spectra agree well with experimental gas phase action spectra and femtosecond transient absorption in liquid benzene. Both sandwich and t-shaped struc- tures feature intense charge resonance bands, whose location is very sensitive to the interfragment distance. Change in the electronic state ordering was observed between σ and π u states, which correlate to the ˜ B and ˜ C bands of the monomer, suggesting a reassignment of the local excitation peaks in the gas phase experimental spectrum. 77 4.2 Introduction The benzene dimer has attracted considerable attention as a model system for studying π − π interactions, which influence secondary structures of biological molecules, such as proteins, RNA, and DNA 1–5 , host-guest complexes 6–8 , and molecular recognition 9 . Their understanding is vital for controlling molecular organization in solid and liquid states 10 . Photoinduced processes in these complex systems involve electronic excita- tion and subsequent charge transfer and localization in which the extended aromatic π system is the key player 11–15 . In addition, oxidative damage induced either chemically or by radiation in DNA leads to aromatic radical cations. Solvents used in radioactive element separation are susceptible to radiation induced ionization, which in the case of neat aromatic liquids leads to the formation of aromatic cations and dimer cations like (C 6 H 6 ) + 2 and (C 5 H 5 N) + 2 16–18 . The benzene dimer cation is thus a prototype for study- ing ionization-induced processes in non-covalent aromatic complexes, neat liquids and aggregates. From an electronic structure point of view, ionization of a van der Waals dimer changes formal interfragment bond order from zero to half-integer and, therefore, switches interaction from non-covalent to covalent. This has a profound effect on the strength of interaction and the structure of the dimer: for example, the helium dimer is bound by only 10.6 K (0.00091 eV) and the corresponding equilibrium distance is about 5.6 bohr 19 , while He + 2 has D e =2.5 eV and r e =2.04-2.06 bohr 20 . In the case of non- spherical fragments, the relative orientation of the monomers may also change upon ion- ization. Electronic excitation, especially to diffuse Rydberg orbitals, may have a similar effect, and rare gas or aromatic excimers are well known examples 21–25 . In the ben- zene dimer, additional complexity arises due to high symmetry of the monomer, which results in degenerate states of C 6 H + 6 that undergo Jahn-Teller (JT) distortions 26–28 . Both intramolecular and intermolecular degrees of freedom may be involved. 78 The shapes of the cation potential energy surfaces (PESs) are determined by the character of molecular orbitals (MOs) hosting a hole. For example, ionization from a bonding (from the dimer point of view) MO yields a repulsive PES, whereas ionization from an antibonding orbital results in a bound potential. In both cases, the magnitude of the effect depends on the overlap between fragment MOs. Thus, ionization from the more compact inner orbitals will have a less profound effect on interfragment bonding, and the resulting states will be just like the perturbed localized states of the monomers. Consequently, the electronic spectrum of the dimer cation will feature two types of transitions: those similar to the monomers, as well as transitions unique to the dimer, that is, transitions between the states derived from ionization of strongly overlapping outer orbitals. Since the overlap depends strongly on the interfragment distance and relative orientation, dynamics of the dimer cation can be monitored through the changes in its electronic spectrum, just as in excimer studies 29, 30 . Experimentally, the dimer cation has been characterized in the gas phase by Ohashi and Nishi 31–34 . Photodissociation action spectra revealed absorption at 2.82, 2.14, 1.35, and 1.07 eV . The former two bands have been assigned as local excitations (LEs), whereas the latter two — as charge resonance (CR) bands, that is transitions unique to the dimer. In solid glasses,γ irradiated liquid benzene exhibits bands at 2.64 eV and 1.36 eV 35 , the latter band being assigned to benzene dimer cation based on its similarity to that observed in irradiated paracyclophanes 36 . In the liquid phase, photoionization of neat benzene leads to an intense transient absorption band assigned to a dimer cation core at 1.35 eV 16 . This is consistent with the adiabatic ionization energy (IE) of the neutral dimer being 8.65 eV 37, 38 , significantly lower then 9.24 eV of the monomer 28 . Ground state binding energy (BE) in the gas phase has been measured to be in the range 15-20 kcal/mol 37–43 . 79 Several computational studies have investigated the ground state structures of the dimer cation 41, 43–47 . England 44 and coworkers suggested a displaced sandwich structure based on Hartree-Fock calculations. Møller-Plesset calculations of Hiraoka 41 put for- ward the t-shaped structure as the lowest isomer. Density functional calculation predict the ground state to have a sandwich structure 43, 46 . The most exhaustive study to date is due to Miyoshi et al. 45, 47 , who characterized both ground and excited states of the dimer cation. In the earlier work 45 , they employed the complete active space self-consistent field (CASSCF) and multi-reference configuration interaction with singles and doubles (MR-CISD) methods to investigate the electronic structure of sandwich, displaced sand- wich and t-shaped isomers. The lowest energy structure at the CASSCF level was the t-shaped isomer. Inclusion of dynamical correlation at the MR-CISD and MR-CISD with Davidson correction levels changed the relative energies in favor of the displaced sandwich isomer. Electronically excited states derived by ionization of a singleπ elec- tron have also been calculated at the first-order configuration interaction level. Unfortunately, transition dipoles and oscillator strengths have not been reported. The later study explored a larger set of isomers and accounted for intramolecular relax- ation 47 . The best estimates of the cation binding energies (calculated at the multi- reference coupled pair approximation level) were 12.3 kcal/mol and 5.4 kcal/mol for the displaced sandwich and t-shaped isomers, respectively. The authors have also explored the nature of the four lowest excited charge resonance states at tilted and displaced sand- wich configurations. Contrarily to the experimental results 31 , they predicted the more intense CR band to occur at a lower energy then the less intense one. The excited states pattern in the tilted structure did not match the experimental spectrum. The goal of this work is to characterize low-lying electronic states of (C 6 H 6 ) + 2 and to outline how the bonding and electronic spectra change upon relevant geometrical dis- tortions. We start by briefly reviewing the electronic structure of the neutral dimer and 80 C 6 H + 6 in Section 4.3. We then develop a general MO framework for describing the elec- tronic states in ionized non-covalent dimers, in the spirit of exciton theory 23, 48 . Further in Section 4.4.1, we apply our MO framework to develop classification of the electronic states of the benzene dimer cation that combines symmetry and bonding properties. After this qualitative analysis, we present vertical ionization energies and the electronic spectra of the cation at equilibrium geometries of the neutral dimer. To describe the evolution of the electronic spectra with geometry changes from neutral to cation and to estimate the relaxed structures of (C 6 H 6 ) + 2 , we present scans along several relaxation coordinates. Finally, we briefly discuss connections to the experimental studies and outline future work. All numerical data are available as Supplementary Information to Ref. 49 4.3 Prerequisites: Bonding in the neutral benzene dimer and benzene cation The lowest energy structure of (C 6 H 6 ) 2 has been the subject of controversy, as the three major isomers are very close in energy and their structures are floppy 50–55 . The most accurate estimate of D 0 (close to ab initio limit) by Sherrill and coworkers 55, 56 gives the following values for the sandwich, displaced sandwich, and t-shaped structures: 1.81, 2.78, and 2.74 kcal/mol, respectively. Szalewicz et al. 57 obtained a six-dimensional PES of (C 6 H 6 ) 2 using symmetry-adapted perturbation theory of intermolecular interac- tions based on density functional theory description of monomers. Using this PES, they found that a tilted t-shaped isomer is the lowest energy one, with a BE of 2.77 kcal/mol. The BEs of the sandwich and displaced sandwich isomers were found to be 1.87 and 2.74 kcal/mol, respectively. Zero-point energy correction changed the BE of the tilted t- shaped isomer to 2.46 kcal/mol, and that of the displaced sandwich — to 2.42 kcal/mol. 81 Experimental results are consistent with the t-shaped isomer as the lowest energy struc- ture 53 . In our calculations of vertical ionization energies, we employed equilibrium struc- tures of the three isomers from Sherrill’s work. Ionization changes interaction from non-covalent to covalent, which results in larger D e and shorter r e . Moreover, as follows from the simple MO overlap considerations, ionization also significantly affects the rel- ative energies of the three isomers. We conducted several PES scans along important relaxation coordinates, which are presented in Section 4.6.2, along with our estimates of equilibrium structures and D e for the dimer cation. Unlike the dimer cation, the ionized states of C 6 H 6 have been extensively character- ized both experimentally 58–61 and theoretically 62–65 . The highest occupied MOs of the neutral benzene are presented in Fig. 4.1. The corresponding vertical IEs are 9.45, 11.7 and 12.3 eV , from the e 1g , e 2g , and a 2u orbitals, respectively 59 . At theD 6h geometry, the cation ground state is doubly degenerate and C 6 H + 6 undergoes Jahn-Teller distortions to two lower-symmetryD 2h structures, which are best described as an acute and an obtuse structure . The minimum energy structure has been the subject of intensive research, and it has been found that the acute structure is the energy minimum, whereas the obtuse structure is, in fact, a transition state. The experimental energy difference between the two is mere 8 cm − 126 . The geometries of the cation states used in our calculations have been optimized using EOM-IP-CCSD/6-311G*. The optimized acute cation structure is 1417 cm − 1 below the D 6h equilibrium structure of the neutral, whereas the obtuse structure is 1404 cm − 1 lower. Despite a relatively low level of theory, the calculated difference of 13 cm − 1 is in excellent agreement with the experimental value. Section 4.6.1 presents our calculations of IEs and transition properties between different cation states at select geometries. 82 (b 2g , S g a ) (b 3g, S g o ) (a g, V a ) (b 1u, S u ) (b 1g, V o ) e 1g , S g e 2g , V a 2u , S u y x Figure 4.1: The five highest occupied orbitals of neutral benzene. Both D 6h and D 2h symmetry labels are given. 4.4 Electronic structure of ionized non-covalent dimers 4.4.1 Orbital and state nomenclature for the benzene dimer cation Below we apply the Dimer Molecular Orbitals – Linear Combination of Fragment Molecular Orbitals (DMO-LCFMO) framework developed in Section 2.3 to low-lying states of the benzene dimer cation. For the sandwich type isomers, each benzene unit is treated as an atom, and the line connecting the centers of the monomers plays the role of 83 a bond axis. The orbital character is assigned based on the nodal structure in the plane perpendicular to the bond. Orbitals will be named as described above, that is by spec- ifying their character w.r.t. the fragments, as well as the type of the constituent FMOs, similarly to ’σ (2s)’ or ’σ ∗ (p z )’ notation in diatomics. As will become obvious, this is not only useful for explaining properties of the states, but also allows us to navigate through the 10 electronic states of each of the 4 isomers. The overall symmetry labels, although important, are not very helpful here, as they mask the character of the states, and, most importantly, change upon geometry distortions. For example, although the states of the sandwich and displaced sandwich isomers are very similar, their symmetry labels are different, as the symmetry of the latter is lowered fromD 6h toC 2h . The 10 highest occupied MOs of the benzene dimer are formed by linear combina- tion of the 5 highest MOs of each neutral monomer (see Fig. 4.1). As mentioned in Sec. 4.3, in the monomer the ionization from highest degenerate orbital pair gives rise to two degenerate states that undergo Jahn-Teller distortion to either acute or obtuse angleD 2h structures, e.g., ionization from the b 2g component leads to an acute angle configuration, whereas ionization from b 3g — to an obtuse angle structure. Thus, we will refer to these orbitals as π a g , and π o g , respectively. The e 1g orbitals are of σ character, and the corre- sponding ion states exhibit similar JT distortions. Thus, these MOs will be denoted as σ a andσ o , respectively. Finally, the lowest a 1u orbital will be calledπ u . The superscript will be dropped when referring to orbitals derived from either one of two degenerate MOs. Below we discuss the MOs of the sandwich isomers and the t-shaped dimer. The MOs at all three sandwich structures, which are shown in Fig. 4.2, are very similar. The highest MOs in the sandwich isomer, the e 1g pair, is formed from out-of-phase combinations of the π a g and π o g monomer orbitals. The e 1u in-phase combinations are lower in energy. With respect to the dimer, these orbitals are of π character. Thus, the 84 two e 1g components are named π ∗ (π a g ) and π ∗ (π o g ), whereas the e 1u orbitals are called π (π a g ) and π (π o g ). When rings slide relative to each other, the degeneracy of both pairs is lifted and their symmetry labels are changed, however, the resulting MOs retain their π -like character. The next two pairs are the e 2g and e 2u orbitals formed by in-phase and out-of-phase combinations ofσ o g andσ a g . Their nodal structure in the benzene plane is similar to φ -type diatomic orbitals. Thus, the out-of-phase combination is referred to as φ ∗ (σ a ) and φ ∗ (σ o ), whereas the in-phase combination is called φ (σ a ) and φ (σ o ). Finally, the out-of-phase (a 2u ) and in-phase a 1g combination of the monomerπ u orbitals are calledσ ∗ (π u ) andσ (π u ). The MOs of the t-shaped isomer are shown in Fig. 4.3. As the monomers are no longer equivalent, the resulting orbitals are more localized, as in heteronuclear diatomics. We will refer to them by specifying their monomer character and whether they are localized on the stem or the top fragments. For example, the HOMO is called π o g (stem), whereas the lowest orbital is called π u (top). Several orbitals feature dis- cernible delocalization between the top and stem fragments, for example theπ o g (stem) and theπ o g (top) are actually unequal mixtures ofπ o g orbitals of the monomers. Similarly, the σ a (stem) and π u (top) are mixtures of the respective fragment orbitals. The delo- calization increases at shorter distances. An interesting feature of the electronic level pattern is that the orbitals of the top fragment are lower in energy than the orbitals of the stem molecule. This is further discussed in Section 4.6.3. 4.5 Theoretical methods and computational details Describing open-shell states is a challenging task for ab initio methodology, especially in the case of orbital (near)-degeneracies. In the case of ionized dimers, even the ground state description is problematic due to symmetry breaking and spin-contamination of 85 (a) (b) (c) e 2u I ( V ) a 2u V *( S u ) a 1g V ( S u ) I ( V ) e 2g e 1g S *( S g ) e 1u S ( S g ) a g S *( S g a ) b g S *( S g o ) b u S ( S g a ) a u S ( S g o ) b g I ( V o ) b u I ( V a ) a u I ( V o ) a g I ( V a ) b u V ( S u ) a g V ( S u ) b g S *( S g a ) a g S *( S g o ) b u S ( S g o ) a u S ( S g a ) b u V *( S u ) a g V ( S u ) a g I ( V a ) b u I ( V a ) b g I ( V o ) a u I ( V R ) Figure 4.2: Highest occupied molecular orbitals of (C 6 H 6 ) 2 at (a) D 6h ,(b) x- and (c) y-displaced sandwich configurations. Orbital energy increases from bottom to top. doublet Hartree-Fock references, and the problems are only exacerbated for the excited states. EOM-IP-CCSD 66–69 overcomes these difficulties by describing the problematic 86 b 2 S g o (stem) a 2 S g a (stem) b 1 S g a (top) b 2 S g o (top) a 1 V a (stem) b 1 V o (stem) a 1 V a (top) a 2 V o (top) b 2 S u (stem) a 1 S u (top) Figure 4.3: Highest occupied molecular orbitals of t-shaped (C 6 H 6 ) 2 . Orbital energy increases from bottom to top. open-shell doublet wave functions Ψ( cation) as ionized and excited states of a well- behaved neutral wave functionΨ( neutral): Ψ( cation)=( ˆ R 1 + ˆ R 2 )Ψ( neutral), (4.1) 87 where Koopmans-like operatorsR 1 andR 2 generate all possible ionized and ionized and excited configurations out of the closed-shell reference determinant, andΨ( neutral) is a coupled-cluster with single and double substitutions (CCSD) wave function of the neutral. Our benchmark study 70 demonstrated that this approach yields smooth potential energy surfaces and accurate energy splittings, both for symmetric and asymmetric con- figurations. Thus, the t-shaped and sandwich configurations are described with equal accuracy and their relative stability can be determined. In the case when fragments are distinguishable, the extent of charge localization is also more accurately reproduced using EOM-IP-CCSD rather than doublet reference CCSD 71 and equation-of-motion coupled-cluster with single and double substitutions for excitation energies (EOM-EE- CCSD 72, 73 ). Below, we first present relevant results for the five lowest electronic states of the benzene cation, and then proceed to the dimer. All calculations were performed at the EOM-IP-CCSD level, using the Q-CHEM ab initio package 74 . The 1s core orbitals of carbon were frozen when specified. In addition to calculation of vertical IEs, we also conducted several PES scans along the important interfragment relaxation coordinates. The equilibrium geometry of C 6 H 6 (r CC =1.3915 ˚ A and r CH =1.0800 ˚ A) was taken from Ref. 75 . The 2 B 2g and 2 B 3g states geometries of C 6 H + 6 were optimized by EOM-IP- CCSD/6-311G* with tight convergence criteria. The ionization energies were calculated using EOM-IP-CCSD with 6-31G, 6-31+G*, 6-311+G*, 6-311(2+,2+)G* basis sets. In the frozen core calculation, the six 1s orbitals were frozen. All dimer calculations were performed at the EOM-IP-CCSD/6-31+G* level of the- ory with core orbitals frozen. The geometries of neutral t-shaped, sandwich, and x- displaced sandwich structures were taken from Ref. 55 . This set was augmented by a displaced sandwich structure, in which benzene rings are slipped in the y-direction. The 88 intermolecular coordinates were adapted from the structure displaced in the x direction. In the relaxation studies the minimum energy point was located with 0.1 ˚ A precision. For the t-shaped isomer the intermolecular distance was varied from 9.0 ˚ A to 4.1 ˚ A. Similarly, the interplanar distance of the sandwich isomer was changed from 9.0 ˚ A to 2.7 ˚ A. In the case of the displaced sandwich isomer, after preliminary checks, detailed scans were performed at 3.0, 3.1, and 3.2 ˚ A interplanar distance. The displacement was varied from 0 ˚ A to 1.6 ˚ A in the x and y directions separately. In all calculations, monomer rings were held at their neutral equilibrium geometry. Also, at each geometry the IEs of the ten lowest electronic states were calculated. Finally, ionization energies, transition dipole moments, and oscillator strengths for transitions between electronic states of (C 6 H 6 ) + 2 were computed at both neutral and relaxed dimer cation geometries. 4.6 Results 4.6.1 Electronic structure of the benzene cation Table 4.1 compares experimental vertical IEs with the values calculated using various basis sets. Overall, IEs increase with the basis set size. With the largest basis set employed, 6-311(2+,2+)G**, the IE to the 2 E 2g (π g ) state is still underestimated by 0.2 eV , whereas IEs to the 2 E 1g (σ ) and 2 A 2u (π u ) states are overestimated by 0.4 and 0.3 eV , respectively. Freezing the core electrons has a small effect on the IEs. The basis set dependence of energy differences between the cation states, i.e. the difference of cor- responding IEs, is much weaker due to error cancellation, e.g. the calculated changes are less than 0.1 eV for all the basis sets. The energies of the 2 E 2g → 2 E 1g ( ˜ B band) and 2 E 2g → 2 A 2u ( ˜ C band) transitions are overestimated by 0.6 eV and 0.5 eV , respectively. The electronic state ordering of C 6 H + 6 along with the transition properties at select geometries is summarized in Table 4.2. All the energies are relative to the 2 E 1g (π g ) state 89 Table 4.1: Vertical ionization energies (eV) of benzene calculates using EOM-IP- CCSD/6-31+G* with frozen core. e 1g e 2g a 2u 6-31G 8.83 11.65 12.19 6-31+G* 9.12 11.98 12.44 6-31+G* FC 9.10 11.98 12.46 6-311+G* 9.18 12.04 12.52 6-311(2+,2+)G** 9.24 12.12 12.58 experiment 59 9.45 11.7 12.3 at the neutral geometry. As expected, the 2 B 2g (π a g ) state favors the acute geometry due to antibonding character of the orbitals w.r.t two parallel carbon-carbon bonds, while 2 B 3g (π o g ) state prefers the obtuse geometry due to antibonding character w.r.t. two opposite apices. Consequently, its energy increases when the cation is distorted to the acute geometry. The 2 A g (σ a ) and 2 B 1g (σ o ) are actually destabilized by either distortion but the variations are much smaller. 2 B 1u (π u ) is stabilized relative to the neutral geometry for both acute and obtuse angle displacements. Due to these effects, both the π -π and π -σ excitations are blue shifted upon JT relaxation, the magnitude being larger in the latter case. At the neutral geometry, the corresponding target states are separated by 0.45 eV , which agrees well with 0.5 eV difference in vertical IEs. At the cation geometries the difference decreases to 0.19/0.34 eV . Only small variation of the transition dipole moment is observed. In summary, the computational method employed underestimates the IE to the 2 E 1g state of the benzene monomer by 0.35 eV , and overestimates IEs to the 2 E 2g and 2 A 2u states by 0.3 and 0.15 eV , respectively. This leads to overestimating the correspond- ing excitation energy. JT relaxation to nearly degenerate acute and obtuse geometries increases the splitting between the ground and excited states. 90 Table 4.2: Energies (eV), transition dipole moments (a.u.), and oscillator strengths for the transitions from the ground state benzene cation at various geometries acute neutral obtuse E ex μ f E ex μ f E ex μ f B 2g /π a g -0.178 E 1g /π g 0.00 B 3g /π o g -0.176 B 3g /π o g 0.265 - - B 2g /π a g 0.258 - - A g /σ a 2.919 - - E 2g /σ 2.881 - - B 1g /σ o 2.910 - - B 1g /σ o 3.072 - - A g /σ a 3.075 - - B 1u /π u 3.262 0.923 0.072 A 2u /π u 3.329 0.947 0.0731 B 1u /π u 3.259 0.919 0.0710 91 4.6.2 Potential energy scans, structural relaxation and binding energies of the dimer cation isomers in the ground electronic state In this Section, we consider PES scans of the dimer cation along several relaxation coordinates. In these calculations we ignored JT distortions and kept the monomers frozen at their neutral geometry 75 . Table 4.3 and Fig. 4.4 summarize estimates of BE at various interfragment configurations, assuming dissociation into the C 6 H + 6 and C 6 H 6 species, both at the neutral benzene equilibrium geometry. First, we calculated BEs at the geometries of the three (C 6 H 6 ) + 2 isomers from Ref. 55 . Then we have augmented this set by a displaced sandwich structure in which the ring is moved in the y direction. For the t-shaped and sandwich isomer, the PES along the interfragment distance was scanned. In the displaced isomers, the slipping coordinate was also scanned. In all these cases, minima were located with 0.1 ˚ A precision. -22 -20 -18 -16 -14 -12 -10 y x y x relaxed displaced sandwich relaxed sandwich relaxed t-shaped displaced sandwich sandwich binding energy, kcal/mol t-shaped Figure 4.4: Estimated binding energies of different (C 6 H 6 ) + 2 isomers. 92 Table 4.3: Estimated binding energies of (C 6 H 6 ) + 2 at various configurations. configuration Δ E, kcal/mol neutral t-shaped 10.74 sandwich 13.94 x-displaced sandwich 14.77 y-displaced sandwich 14.86 relaxed t-shaped 12.40 sandwich 18.34 x-displaced sandwich 20.14 y-displaced sandwich 20.18 As expected from the MO considerations (see Introduction) ionization increases the BE of the dimer and results in tighter structures. It also changes the relative energy ordering of the isomers. The t-shaped structure, a candidate for the lowest energy iso- mer of the neutral benzene dimer, happens to be the highest in the cation. The binding energy at this configuration increases from 2.6 kcal/mol in the neutral to 10.7 kcal/mol in the cation. Allowing the intermolecular distance to relax from 5.0 ˚ A to 4.6 ˚ A further increases this value to 12.4 kcal/mol. However, the relaxed t-shaped structure is still higher in energy than the sandwich, in which BE is 13.9 kcal/mol at the neutral separa- tion of 3.9 ˚ A. In the cation, the interfragment distance relaxes to 3.3 ˚ A, and the binding energy increases to 18.3 kcal/mol. Both in the neutral and the cation benzene dimer, the symmetric sandwich structures appear to be transition states between the displaced structures, possibly due to the JT effect. In the neutral, the rings move by 1.6 ˚ A in the x direction with a separation of 3.6 ˚ A. This structure of the cation is bound by 14.77 kcal/mol. The slide in the y direction results in a very similar binding energy of 14.86 kcal/mol. Thus at the level of theory used the x- and y-displaced structures are degener- ate. Two-dimensional scans shown in Fig. 4.5 result in a minimum energy displacement 93 of 1.0 ˚ A and separation of 3.1 ˚ A, for both the x and y displaced structures, the energies being 20.14 and 20.18 kcal/mol. To summarize, our PES scans suggest that the displaced sandwich isomer is the minimum energy structure, with the BE estimated to be 20 kcal/mol, the x- and y- displaced structures being essentially degenerate. 4.6.3 Electronic states of the t-shaped isomer We begin our discussion of the electronic states of (C 6 H 6 ) + 2 with the t-shaped isomer at its neutral equilibrium configuration. The important features of this isomer are: (1) the fragments are not equivalent; (2) even when the fragments are at their neutral (D 6h ) geometry orbital degeneracies are broken; (3) different types of orbitals mix, i.e. π u and σ . Excited state PES scans along the intermolecular separation coordinate are shown in Fig. 4.6. Vertical IEs, transition dipoles, and oscillator strengths for the electronic transitions between the cations states are listed in Table 4.4. The MOs and the corre- sponding stick spectrum are shown in Fig. 4.3 and Fig. 4.7, respectively. In the ground state, the π o g (stem) orbital is highest in energy and thus it is singly occupied. All the transitions discussed below involve transfer of an electron to this orbital. The corre- sponding vertical IE is 8.78 eV , which is 0.3 eV lower than the computed monomer value. Orbital ordering. The MOs of the t-shaped isomer are mostly localized on individ- ual fragments. Orbitals of the stem fragment are higher in energy than the orbitals of the top molecule. To elucidate the nature of this effect, we conducted an additional calcula- tion in which one of the fragments is replaced by a +1 point charge at its center of mass. The structure with the stem molecule substituted by a point charge was more stable by 2.5 kcal/mol at the CCSD/6-31+G* level. Thus, it appears that the charge-multipole interactions are more favorable when the hole is localized on the stem, in agreement 94 (a) 0.0 0.5 1.0 1.5 -0.10 -0.05 0.00 0.05 0.10 0.15 (b) 0.0 0.5 1.0 1.5 -0.10 -0.05 0.00 0.05 0.10 0.15 Figure 4.5: PES scan along the x and y sliding coordinates at 3.0 (squares), 3.1 (rom- boids), and 3.2 ˚ A (squares) interplanar separations. 95 4.0 5.0 6.0 7.0 8.0 0.0 1.0 2.0 3.0 Figure 4.6: Potential energy profiles along interfragment separation in t-shaped (C 6 H 6 ) + 2 . Table 4.4: IEs (eV), transition dipole moments (a.u.), and oscillator strengths of t-shaped (C 6 H 6 ) + 2 t-shaped relaxed t-shaped IE μ f IE μ f B 2 /π o g (stem) 8.776 8.638 A 2 /π a g (stem) 8.867 4.73 10 − 3 5.0 10 − 8 8.803 0.0135 7.40 10 − 7 B 1 /π a g (top) 9.163 - - 9.193 - - B 2 /π o g (top) 9.194 2.65 0.0718 9.273 2.76 0.118 A 1 /σ a (stem) 11.565 0.0928 5.88 10 − 4 11.325 0.154 1.57 10 − 3 B 1 /σ o (stem) 11.730 - - 11.656 - - A 1 /σ a (top) 12.018 1.31 10 − 3 1.37 10 − 7 12.028 2.74 10 − 3 6.25 10 − 7 A 2 /σ o (top) 12.024 1.06 10 − 3 8.9 10 − 8 12.036 1.64 10 − 3 2.23 10 − 7 B 2 /π u (stem) 12.179 0.909 0.0688 12.110 0.829 0.0585 A 1 /π u (top) 12.490 0.278 7.03 10 − 3 12.462 0.299 8.36 10 − 3 E ref a -463.019223 -463.016801 a Total CCSD energy (hartree) of (C 6 H 6 ) 2 96 (a) 0.0 1.0 2.0 3.0 4.0 0.00 0.02 0.04 0.06 0.08 S g a (s) x 10 5 S g a (t) S g o (t) V a (s) x 10 2 V o (s) V a (t), V o (t) x 10 5 S u (t) oscillator strength energy, eV S u (s) (b) 0.0 1.0 2.0 3.0 4.0 0.00 0.04 0.08 0.12 S u (t) S u (s) V a (t), V o (t) x 10 4 V o (s) V a (s) x 10 S g o (t) S g a (t) S g a (s) x 10 4 oscillator strength energy, eV Figure 4.7: Electronic states ordering and oscillator strengths of (C 6 H 6 ) + 2 at neutral (a) and relaxed (b) t-shaped configuration. Empty bars denote symmetry forbidden transi- tions. with predictions based on the Hunter-Sanders model 76 . This model pictures an aromatic molecule as a positively charged σ -framework and a negatively charged π -framework. Consequently, ionizing the stem molecules allows for a favorable interaction between the positive charge and theπ system of the top molecule. 97 PES scans. The two lowest energy states are bound and correspond to ionizations from π g orbitals of the stem. States derived by ionizing corresponding orbitals of the top fragment are repulsive. This is in agreement with the Hunter-Sanders model, as well as the antibonding character of the corresponding DMOs. A very interesting behavior is observed for the state derived from ionization from σ a (stem), which is actually a mixture of ionizations from the pure σ a (stem) and π u (top) orbitals. As the distance decreases, the weight of the latter configuration increases. Due to increased charge delocalization, this state has a higher binding energy thanπ o g (stem). In theπ u (top) state the stabilization due to delocalization is canceled out by destabilization due to placing a positive charge on the top molecules, resulting in a flat PES. Theπ u (stem) state is also bound, but the orbital is more localized. The PESs ofσ (top) states are rather flat, as the monomer charge distribution is not significantly altered. Charge resonance bands. These transitions correspond to excitations within theπ g manifold of states. Due to the presence of the top molecule, theπ a g (stem)→π o g (stem) transition becomes weakly allowed. Transitions fromπ a g (top) are symmetry forbidden. A very intense band at 0.42 eV corresponds to the π o g (top) → π o g (stem) excitation. Upon relaxation it shifts to 0.63 eV while its oscillator strength increases by 50%. The π o g (top)→π o g (stem) excitation is akin to CR bands (see Section 4.4.1) – the initial and the final MOs are in-phase and out-of-phase combinations of the unperturbed π o g (top) andπ o g (stem) monomer MOs, albeit with unequal weights. A distinguishing feature of this transition is that its dipole moment increases with decreased distance, unlike the CR bands in the sandwich isomers (see Section 4.6.4). This is because mixing becomes stronger at shorter distances, i.e. the decreased R AB is offset by increasedβ (see Section 2.3.3). Local excitations. Above 2.5 eV , transitions from the σ orbitals of the monomers appear. Excitation energies for the transitions originating from the top molecule are 98 slightly higher. They carry little oscillator strength regardless whether the σ orbital is located on the stem or the top molecule. The σ a (stem)→ π o g (stem) is a notable exception. The acquired intensity originates from the admixtures of pure π u (top) and π o g (top) in the initial and final orbitals, respectively. The transition dipole forπ u (top) is not decreased. The decreased weight of pure π u (top) is offset by increased weight of pureπ o g (top). On the other handπ u (stem)→π o g (stem) loses intensity due to decreased weight of pureπ o g (stem). Summary. Our results for the t-shaped isomer demonstrate, quite surprisingly, that low-energy strong CR bands appear even when monomers are not indistinguishable. A moderate orbital delocalization is sufficient to produce a relatively large BE, the decrease in IE, and the formation of the CR-like bands. 4.6.4 Electronic states of sandwich isomer In the sandwich configuration, the two identical monomers form a D 6h structure. Excited state PES scans along the ring separation coordinate are shown in Fig. 4.8. Neutral and relaxed configuration energies along with transition properties are listed in Tables 4.5 and 4.6, respectively. The corresponding stick spectra are shown in Fig. 4.9a and 4.10a. The HOMO (see Fig. 4.2) is the doubly degenerateπ ∗ (π g ) pair, and the cor- responding vertical IE is lowered to 8.58 eV from 9.10 eV (computed) in the monomer. In the cation, this pair of orbitals hosts three electrons and the 9 lowest excited state of the cation are derived from promoting an electron to this pair. PES scans. As expected from DMO-LCFMO considerations, energies of the states derived from ionization from bonding orbitals result in repulsive PESs, while the states derived from ionizations from anti-bonding orbitals — in bound states. The equilibrium distances of these bound states are shorter than those of the neutral, to maximize the overlap between the FMOs. For example, the lowest π ∗ (π g ) state is bound, whereas 99 Table 4.5: IEs (eV), transition dipole moments (a.u.), and oscillator strengths of the sandwich and the displaced sandwich (C 6 H 6 ) + 2 isomers. x-displaced parallel y-displaced IE μ f IE μ f IE μ f B g /π ∗ (π o g ) 8.590 E 1g /π ∗ (π g ) 8.577 B g /π ∗ (π a g ) 8.592 A g /π ∗ (π a g ) 8.796 - - A g /π ∗ (π o g ) 8.794 - - B u /π (π a g ) 8.889 0.0197 2.83 10 − 5 E 1u /π (π g ) 9.111 3.27 0.140 B u /π (π o g ) 8.890 0.0210 3.21 10 − 6 A u /π (π o g ) 9.137 3.23 0.140 A u /π (π a g ) 9.136 3.23 0.139 B g /φ (σ o ) 11.764 - - E 2u /φ ∗ (σ ) 11.753 3.02 10 − 3 7.12 10 − 7 B u /φ ∗ (σ a ) 11.761 0.133 1.37 10 − 3 B u /φ ∗ (σ a ) 11.765 0.152 1.80 10 − 3 A g /φ (σ a ) 11.766 - - A u /φ ∗ (σ o ) 11.766 0.101 7.98 10 − 4 E 2g /φ (σ ) 11.779 - - A u /φ ∗ (σ o ) 11.773 0.0544 2.31 10 − 4 A g /φ (σ a ) 11.778 - - B g /φ (σ o ) 11.775 - - B u /σ ∗ (π u ) 11.883 0.821 0.0544 A 2u /σ ∗ (π u ) 11.881 0.831 0.0559 B u /σ ∗ (π u ) 11.883 0.824 0.0547 A g /σ (π u ) 12.496 - - A 1g /σ (π u ) 12.437 - - A g /σ (π u ) 12.497 - - E ref a -463.018821 -463.017026 -463.019035 a Total CCSD energy (hartree) of (C 6 H 6 ) 2 100 Table 4.6: IEs (eV), transition dipole moments (a.u.), and oscillator strengths of the relaxed sandwich and the relaxed displaced sandwich (C 6 H 6 ) + 2 isomers. x-displaced parallel y-displaced IE μ f IE μ f IE μ f B g /π ∗ (π o g ) 8.105 E 1g /π ∗ (π g ) 8.163 B g /π ∗ (π a g ) 8.105 A g /π ∗ (π a g ) 8.325 - - A g /π ∗ (π o g ) 8.325 - - B u /π (π a g ) 9.161 0.0456 5.37 10 − 5 E 1u /π (π g ) 9.369 2.65 0.207 B u /π (π o g ) 9.161 0.0484 6.07 10 − 5 A u /π (π o g ) 9.432 2.53 0.209 A u /π (π a g ) 9.433 2.53 0.209 B g /φ (σ o ) 11.707 - - E 2u /φ ∗ (σ ) 11.659 2.03 10 − 3 4.07 10 − 7 B u /φ ∗ (σ a ) 11.685 0.0744 4.86 10 − 4 B u /φ ∗ (σ a ) 11.668 0.0741 4.79 10 − 4 A g /φ (σ a ) 11.709 - - A u /φ ∗ (σ o ) 11.685 0.0921 7.43 10 − 4 E 2g /φ (σ ) 11.751 - - A u /φ ∗ (σ o ) 11.668 0.0596 3.10 10 − 4 A g /φ (σ a ) 11.742 - - B g /φ (σ o ) 11.739 - - B u /σ ∗ (π u ) 11.367 0.795 0.0505 A 2u /σ ∗ (π u ) 11.429 0.801 0.0514 B u /σ ∗ (π u ) 11.367 0.795 0.0505 A g /σ (π u ) 12.792 - - A 1g /σ (π u ) 12.690 - - A g /σ (π u ) 12.792 - - E ref a -463.009538 -463.008808 -463.009605 a CCSD energy of (C 6 H 6 ) 2 , in hartree 101 2.5 3.0 3.5 4.0 4.5 5.0 5.5 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 I ( V ) I *( V ) V *( S u ) V ( S u ) S ( S g ) energy, eV R 1 , Å S *( S g ) Figure 4.8: Potential energy profiles along interfragment separation in sandwich (C 6 H 6 ) + 2 . π (π g ) is repulsive. Theπ ∗ (π u ) andπ (π u ) states follow the same trend, while the states derived from φ ∗ (σ ) and φ (σ ) feature flat PESs, due to small overlap between the inner σ FMOs. Thus these are perhaps more correctly described as nonbonding states. The increased attraction in the σ ∗ (π u ) state brings it below the φ states, which reverses the ordering of the LE bands relative to the monomer. Charge resonance bands. The π g orbitals of benzenes are the dimer counterparts of λ from Fig. 2.2 (see Section 2.3.2), and they are antisymmetric w.r.t. inversion in the dimer frame. Strong CR bands corresponding to transitions between bonding and complementary antibonding combinations ofπ g are expected to dominate the spectrum. At the neutral dimer geometry, the band occurs at 0.53 eV and has a transition dipole of 3.27 a.u. This value is very close to half the distance between monomers, 3.68 a.u, as follows from Eq. (2.36). As one can see from PES scans, the excitation energy of the CR band increases upon relaxation, and reaches 1.21 eV around the minimum. Although 102 (a) 0.0 1.0 2.0 3.0 4.0 0.00 0.04 0.08 0.12 0.16 I ( V ) oscillator strength energy, eV I *( V ) x 10 4 S ( S g ) V *( S u ) V ( S u ) (b) 0.0 1.0 2.0 3.0 4.0 0.00 0.04 0.08 0.12 0.16 I ( V a ), I ( V o ) I *( V a ), I *( V o ) x 10 S ( S g a ) x 10 4 S *( S g a ) V *( S u ) V ( S u ) oscillator strength energy, eV S ( S g o ) (c) 0.0 1.0 2.0 3.0 4.0 0.00 0.04 0.08 0.12 0.16 I *( V a ), I *( V o ) x 10, x 100 I ( V a ), I ( V o ) V *( S u ) S *( S g o ) S ( S g o ) x 10 4 S ( S g a ) V ( S u ) oscillator strength energy, eV Figure 4.9: Electronic states ordering and oscillator strengths of (C 6 H 6 ) + 2 at sandwich (a), and x- and y-displaced sandwich, (b) and (c), configurations. Empty bars denote symmetry forbidden transitions. 103 (a) 0.0 1.0 2.0 3.0 4.0 5.0 0.00 0.05 0.10 0.15 0.20 I ( V ) I *( V ) x 10 4 V *( S u ) V ( S u ) S ( S g ) oscillator strength energy, eV (b) 0.0 1.0 2.0 3.0 4.0 5.0 0.00 0.05 0.10 0.15 0.20 energy, eV I *( V a ), I *( V o ) x 10 I ( V a ), I ( V o ) S ( S g a ) x 10 2 S *( S g a ) S ( S g o ) V *( S u ) V ( S u ) oscillator strength (c) 0.0 1.0 2.0 3.0 4.0 5.0 0.00 0.05 0.10 0.15 0.20 I ( V a ), I ( V o ) I *( V a ), I *( V o ) x 10, x 10 S ( S g o ) x 10 2 S *( S g o ) S ( S g a ) V ( S u ) V *( S u ) oscillator strength energy, eV Figure 4.10: Electronic states ordering and oscillator strengths of relaxed (C 6 H 6 ) + 2 at sandwich (a), and x- and y-displaced sandwich, (b) and (c), configurations. Empty bars denote symmetry forbidden transitions. 104 the transition dipole moment decreases due to shorter interfragment distance following Eq. (2.36), the band gains oscillator strength due to the increase in excitation energy. Local excitations. The σ and π u orbitals are counterparts of the ν orbitals of Sec- tion 2.3 and they are symmetric w.r.t. inversion. Thus, the dipole allowed electronic transitions are of the case II type (Fig. 2.2), that is, transitions from the antibonding combinations of σ and π u to the antibonding combination of π g . As follows from the analysis Section 2.3.2, the intensities of these bands should be similar to the correspond- ing monomer transitions, i.e., the transition from σ will be weak (dipole forbidden but vibronically allowed ˜ B-band), while the transition fromπ u — relatively strong ( ˜ C-band of the monomer). LE bands occur above 3 eV . As predicted, transitions from φ ∗ (σ ) and φ (σ ) are weak, and symmetry forbidden, respectively. The intensity presumably arises from the relatively small interfragment terms in Eqns. (2.32) and (2.34). The allowed transition fromσ ∗ (π u ) is at 3.30 eV at neutral configuration. Its dipole moment is slightly lower than in the monomer due to intermolecular terms appearing in Eqns. (2.32) and (2.34). Its position changes insignificantly upon relaxation. The symmetry-forbiddenσ (π u )→ π ∗ (π g ) excitation is at 3.86 eV . Decreased inter- fragment distance in the dimer cation causes it to shift 4.53 eV . A very important feature is the (explained above) reversed order of the allowed and the forbidden bands in the 3.0 - 3.5 eV region: the strong σ ∗ (π u ) transition is now at a lower energy than the φ band, which shifted from 3.2 eV to 3.5 eV . Summary. The sandwich isomer illustrates the application of the DMO-LCFO framework developed in Section 2.3. The spectrum is clearly separated into the charge resonance and local excitation parts. Geometric relaxation from the neutral to the cation produces pronounced changes in the spectrum: the CR band shifts to the blue and gains intensity, while LE bands reverse their order. 105 4.6.5 Electronic states of displaced sandwich isomers Displaced sandwich structures are derived from the sandwich by sliding the monomers relative to each other in either x or y direction. Fig. 4.11 presents PES scans in x and y directions starting from the D 6h sandwich structure. The behavior of the states in both cases is complementary and similar to JT distortions: states that are stabilized by the displacement in one direction are destabilized by the displacement in the other. The only exception are the states derived from the monomer π u orbitals due to their cylindrical symmetry. A similar behavior was observed in the monomer relaxation to acute and obtuse geometries. IEs, transition dipoles, and oscillator strengths for transitions between cation states are given in Table 4.5 and Fig. 4.9 b, c. Overall, these are remarkably similar to the sandwich isomer. The displacement results in only minor differences in IE between the x and y isomers, i.e., the first IE 8.59 in both cases. PES scans. Due to the lower symmetry, the degeneracy betweenπ ∗ (π o g ) andπ ∗ (π a g ) is lifted, as is the degeneracy between the π (π g ) pair. Overall, the PES is rather flat in the ground state. The energy dependence is more sharp for π ∗ (π a g ) and π (π a g ) for x sliding coordinate, and π ∗ (π o g ) and π (π o g ) — for sliding in y-direction. In the 3 eV energy region, no reordering of states due to displacement occurs. The components of φ ∗ (σ ) andφ (σ ) are no longer degenerate, however, the splitting is small. Charge resonance bands. In the x-displaced isomer the highest, singly occupied orbital is π ∗ (π o g ). The complementary π (π o g ) orbital is 0.55 eV lower. As in the sand- wich, this transition is very intense, due to the nature of initial and final orbitals. Upon relaxation the band shifts to 1.33 eV . In between these two orbitals, there are theπ ∗ (π a g ) and π (π a g ) FMOs. The transition to the latter is now symmetry allowed, but carries lit- tle oscillator strength. In the y-displaced isomer, the roles of constituent π o g and π a g are simply reversed. The energy gaps between the levels remains nearly the same. 106 (a) 0.0 0.4 0.8 1.2 1.6 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 (b) 0.0 0.4 0.8 1.2 1.6 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 Figure 4.11: Potential energy profiles along interfragment sliding in (a) x- and (b) y- displaced sandwich (C 6 H 6 ) + 2 . Interplanar separation was held fixed at 3.1 ˚ A. 107 Local excitations. Higher in energy, at c.a. 3.2 eV , we find the φ states, within 0.1 eV from each other. Although dipole allowed in the displaced structures, these transitions carry essentially no oscillator strength. The excitation to σ ∗ (π u ) at 3.3 eV is symmetry allowed and is fairly intense, just like in the monomer. The σ (π u ) state is 0.6 eV higher, but it is symmetry forbidden. Upon relaxation to the cation structure the ordering of bands changes, as already discussed in Section 4.6.4. Summary. Overall, the displacement plays a relatively minor role in changing the electronic spectrum of the sandwich-type isomers. Its major effect is breaking the orbital and state degeneracies, which leads to the two structures with different orbital characters and states ordering, but nearly identical spectral signatures. Symmetry reduction and breaking the degeneracies results in additional lines, most importantly, below the major CR band. 4.7 Discussion Ionization of the benzene dimer drastically changes its bonding character from van der Waals to covalent. The change is particularly significant in the sandwich type isomers, where the HOMO has strong antibonding character with respect to the two fragments. Even in the singly occupied HOMO of the t-shaped structure, where fragments are no longer equivalent and orbitals are more localized, the fragment orbitals interfere destruc- tively in the region between monomers, i.e. have antibonding character. In the neutral species, the t-shaped and displaced sandwich isomers are nearly degenerate — the calcu- lated energy difference is less than 0.1 kcal/mol. Ionization of the dimer stabilizes most strongly the displaced sandwich isomer, and its binding energy increases to 20 kcal/mol, whereas the binding energy of the t-shaped isomer becomes 12 kcal/mol. Thus, it seems unlikely, that the t-shaped isomer will be produced in the experiments of Ref. 33, 34 This 108 supports the conclusions of the hole-burning study, which indicated presence of a single isomer in the molecular beam 34 . A t-shaped isomer will, however, be initially formed in femtosecond ionization experiments and subsequently isomerize to a displaced sand- wich. This evolution can in principle be resolved. Using a diabatic states framework, we can also comment on the nature of binding in the dimer cations. In the sandwich, the BE is 0.51 and 0.67 eV at the neutral and relaxed configurations, respectively. The diabatic states, which correspond to the charge being localized on one of the fragments, see Eq. (2.37), are exactly degenerate at the sand- wich geometry because both monomers are the same, and thus the coupling between these states is half the splitting between π ∗ (π g ) and π (π g ). At the neutral and relaxed geometries, the coupling is 0.27 and 0.60 eV , respectively (the interfragment distance changes from 3.9 ˚ A to 3.3 ˚ A). The remaining 0.24 and 0.07 eV of interaction is the interaction between the hole and a neutral benzene, i.e. the binding energy of the dia- batic state. Diabatic BE actually decreases in the dimer cation relative to the neutral geometry and the total binding is almost exclusively due to charge transfer forces. The binding energy of the t-shaped isomer is 0.39 and 0.46 eV at the neutral and relaxed configurations, respectively. Using the Generalized Mulliken-Hush model 77 , we calcu- lated the coupling to be 0.21 and 0.23 eV , while the binding energy of the diabatic state with the hole localized on the stem molecule is 0.19 and 0.36 eV . Thus, due to the larger distance the electrostatic interactions play a more significant role in the t-shaped than in the sandwich isomer. An interesting question is about the mechanism of stabilization of the displaced sandwich structure relative to the sandwich. In the neutral species, electrostatic interac- tions, as described in the Hunter-Sanders model, are the driving force, whereas in the cation, the displacement plays a role of a JT mode lifting degeneracy between the two 109 states. Both effects are characterized by similar energies. In the neutral, the displace- ment in the sandwich is associated with a 1 kcal/mol gain in binding energy, while the cation value is not much higher, 2 kcal/mol. The monomer relaxation leads to 4 kcal/mol of additional stabilization. Thus, it is not clear whether the displacement is driven by Hunter-Sanders electrostatic interactions, JT effect, or if the two contribute equally. Full geometry optimization will address the relative importance of the two effects. Further frequency calculations would allow us to characterize the displaced structures as two minima, or, perhaps, a minimum and a transition state. The photodissociation action spectra of the dimer cation exhibit peaks at 1.07, 1.35, 2.14, and 2.82 eV 33 . The two relaxed displaced structures exhibit essentially identical spectra, with excited electronic states at 1.06, 1.33, 3.26, and 3.57 eV . The positions of the two low-energy CR bands agree extremely well with the experiment, however, their intensity pattern does not. Experimentally, the ratio is 1:10, whereas the calculated ratio is 1:1000. The discrepancy could be due to strong vibronic interactions between the two states. With the current level of theory, the excitation energies of theπ u → π g andσ → π g transitions of the monomer ( ˜ B and ˜ C band) are overestimated by 0.5 and 0.6 eV respec- tively. If this correction is applied to the dimer, it yields 2.76 and 2.97 eV for the LE bands. The lower energy transition, which corresponds toπ u →π g , agrees rather poorly with the experimental peak at 2.14 eV , however, the agreement for σ → π g (2.82 vs. 2.97 eV) is much better. Most importantly, our study demonstrated that the two bands reverse their order relative to the monomer. Again, the intensity pattern is not repro- duced, because the dipole-forbidden ˜ B-band of the monomer borrows intensity due to vibronic interactions 78 . Additionally, the φ excited states have non-bonding character, while the σ states is bound. This may affect the observed intensity pattern in the pho- todissociation action spectrum. The large errors in the predicted excitation energies for 110 LE transitions are typical of the EOM-IP-CCSD method 79, 80 . They arise due to inclu- sion of only singly excited determinants (w.r.t. to the ionized state) in the wave function. Note, that the complementary nature of states involved in the CR transitions results in error cancellation and significantly more accurate transition energies. Previous studies often discounted the t-shaped dimer cation structure as a source of the CR bands because of the overlap considerations. Our calculations demonstrate, however, that such bands are also present in the t-shaped isomer and have considerable intensity due to partial delocalization of the FMOs. The oscillator strength is approxi- mately half of that of the sandwiches. It would be interesting to investigate the effect of monomer relaxation on the intensity of the bands. Two scenarios are possible. The hole may localize on the stem molecule, which would further localize orbitals and, there- fore, reduce the intensity. Alternatively, the JT distortion may bring the stem and top orbitals closer together thus allowing for more delocalization, which will yield stronger CR bands. The knowledge of both t-shaped and sandwich isomer spectra is useful from the point of view of studying ionization chemistry of aromatic solvents using femtosecond techniques. Neutron diffraction experiments on liquid benzene point to perpendicular orientation of nearest neighbors 81 , thus the t-shaped isomer will be initially produced. Subsequent nuclear dynamics will lead to charge delocalization and formation of sand- wich dimer cations. This process could be probed by monitoring the position and inten- sity of the CR band. Initially it would appear near 0.42 eV (2950 nm), as calculated for the neutral t-shaped geometry. The monomer cation would then evolve towards the dis- placed sandwich dimer cation, with the two rings rotating and coming closer together. During this process the intensity of the CR band would double and shift to 1.33 eV (940nm). Such a pronounced change in the spectrum should be easily resolved. 111 4.8 Conclusions This paper reports a thorough study of the electronic states of the benzene dimer cation. Energies of the 10 lowest electronic states of the dimer, as well as oscillator strengths for electronic transitions, are calculated by EOM-IP-CCSD/6-31+G*. Several isomers and relaxation coordinates are considered. The hallmark of the calculated spectra are the charge resonance bands, which appear both in the t-shaped and sandwich isomers. Their position and intensity can serve as extremely sensitive probe of the dimer structure. For example, the CR bands allow us to assign the displaced sandwich isomer as the ground state of the (C 6 H 6 ) + 2 system. Moreover, out results offer support support to studies aiming to resolve the dynamics of cation formation using femtosecond spectroscopy. Lastly, inversion of LE bands has been observed in the lowest energy displaced sandwich structure. This may hint at different relaxation dynamics in the cation monomer and dimer. The trends in electronic spectrum are explained by a simple DMO-LCFMO model. The results explain the nature of the intense CR bands and less intense LE transitions of the dimer, and outline the evolution of the dimer electronic spectrum upon ionization. 4.9 Postscript 4.9.1 Introduction Since this study was completed, an efficient analytic gradient EOM-IP-CCSD code was developed. While this chapter described the most important aspects of the electronic spectroscopy and relaxation effects, it neglected the Jahn-Teller (JT) distortion of the individual rings. The latter is particularly interesting in the t-shaped isomer. Although not a lowest energy structure, it serves as a model system for heterodimers, as the 112 two benzene rings (top and stem) are not equivalent. In a complex system, such as a DNA strand, the heterogeneity arises both from different chemical nature of neigh- boring bases, as well as their different local environments. Contrarily to our initial expectations, we discovered 49 a considerable delocalization of the positive charge in the t-shaped benzene dimer cation, sufficient to produce strong charge resonance bands in the spectrum. The extent of hole delocalization is determined by two opposing factors. The first one is electronic coupling between the diabatic charge-localized states, stronger diabatic coupling causing larger delocalization. The second factor, the JT distortion, sta- bilizes the hole thus favoring the charge localization. Accurate calculations are needed to determine the overall direction of the effect. Finally, fully optimized structures will allow us to refine binding energies and vertical electronic spectra of the two isomers. We refer to the structures from the PES scans and full optimization as relaxed and optimized, respectively. Vertical geometries are equilibrium structures of the neutral dimer. 4.9.2 Computational details The geometries of the JT (C 6 H 6 ) + states and the isomers of (C 6 H 6 ) + 2 were optimized at the EOM-IP-CCSD/6-31+G* level. The geometries of the t-shaped and displaced sand- wich dimer structures were restricted to C 2v and C 2h symmetries, respectively. Sub- sequently, the ten lowest electronic states of the dimer were calculated, including the transition and state properties using EOM-IP-CCSD/6-31+G*. The wave functions for the t-shaped isomer at vertical, relaxed, and optimized geometries were analyzed using the NBO 82 analysis and the charge of the individual fragments was calculated. The core 1s orbitals of carbon were held frozen in the single-point calculations, but not in the geometry optimizations. The geometries of the neutral monomer and dimer were taken from Ref. 75 and 55 . Relaxed dimer cation geometries are from Ref. 49 . The diabatic 113 couplings and splittings were calculated using the Generalized Mulliken-Hush model 77 . Unrelaxed permanent dipole moments were used. All ab initio calculations were performed using the Q-CHEM electronic structure package 83 . 4.9.3 Preliminaries The highest occupied molecular orbital (HOMO) of benzene is the doubly degener- ate π system. Upon ionization, the D 6h benzene ring undergoes distortion to two D 2h structures, best described as the acute and the obtuse geometries. The respective molec- ular orbital (MO) components are denoted as π a g and π o g . The JT effects in the ben- zene cation were thoroughly studied both experimentally 58–61 and theoretically 62–65 . The acute geometry is found experimentally to lie below the obtuse one by 8 cm − 126 . Our calculations predict this difference to be 21 cm − 1 , and that the acute and obtuse struc- tures are 1423 and 1403 cm − 1 below the neutralD 6h geometry, respectively. Their geo- metric parameters are shown in Fig. 4.12. Distorting the neutral to the geometry of the acute and obtuse cation results in an energy penalty of 757 and 935 cm − 1 , respectively. The JT distortion involves changes in the bond length and angles. In the discussion below, we use energies to quantify the JT distortions in the dimer. While there are sev- eral geometrical parameters associated with the JT distortion, energy offers a convenient measure of the effect through the following energy differences. The first quantity,Δ E 1 , is the energy difference between the optimized and relaxed structures in which the rings were frozen at their neutral geometries. Δ E 1 is a measure of the JT relaxation of the entire dimer. Optimization does not distinguish between intermolecular and intramolec- ular (JT) distortions and this energy difference represents both effects, albeit to different extent. It also implicitly accounts for the change in the JT state in the dimer relative to 114 (a) 120.83 119.54 119.59 1.419 1.383 1.411 1.389 121.53 1.370 1.370 1.429 1.429 119.23 119.23 (b) 119.26 119.04 1.395 1.425 1.396 120.48 120.30 118.12 118.12 120.94 120.94 1.445 1.392 1.392 Figure 4.12: The geometry of the benzene ring in the (a) x-displaced and (b) y-displaced sandwich (C 6 H 6 ) + 2 (underlined numbers). The geometry of an isolated ring in a (a) π o g and (b)π a g state is also given. Angles are in degree and distances are in ˚ Angstrom. the monomer. The second quantity,Δ E 2 , is the difference between energy of an individ- ual cation ring at the neutral monomer geometry and at the geometry it assumes in the 115 Table 4.7: Binding energies (kcal/mol) of (C 6 H 6 ) + 2 at various configurations. Dissocia- tion limit is described as a neutral (X 1 A 1 ) and an acute (X 2 B 2g ) geometry ring. optimized relaxed vertical t-shaped 12.41 8.33 6.67 x-displaced 19.58 16.07 10.70 y-displaced 19.81 16.11 10.79 dimer. This allows us to separate intermolecular and JT coordinates, but does not cap- ture the effect of the fragments on each other. Δ E 3 quantifies the increase in the energy of a neutral species, as it is distorted from its monomer geometry to the ring geometry in the dimer cation. Thus,Δ E 1 andΔ E 2 quantify the cation distortion in the dimer and individual ring, respectively. Δ E 3 describes the energetic penalty of the neutral species, and is useful in the discussion of the diabatic states. Table 4.7 presents the binding energies of (C 6 H 6 ) + 2 in several configurations. In the previous work 49 the dissociation limit was taken as two rings with a neutral (X 1 A 1 ) geometry. Here it is taken as a neutral (X 1 A 1 ) and an acute (X 2 B 2g ) geometry ring. The overall binding energies obtained for the relaxed structure using the first dissociation limit are similar to the optimized structures ones using the second dissociation limit. For the sandwich isomers they are∼ 20 kcal/mol, in agreement with experimental studies putting it in the 15 - 20 kcal/mol range 37–43 . The t-shaped isomer is∼ 7 kcal/mol higher. 4.9.4 The t-shaped isomer Vertical IEs, transition dipoles, and oscillator strengths for the electronic transitions between the states of t-shaped (C 6 H 6 ) + 2 are listed in Table 4.8. Fig. 4.13 shows relative energies of the four lowest electronic states along with the partial charge on the stem fragment in the vertical, relaxed and optimized geometries. Ring geometries are shown in Fig. 4.14. The ordering and character of states in the optimized structure is preserved 116 Table 4.8: IEs (eV), transition dipole moments (a.u.), and oscillator strengths of the optimized t-shaped (C 6 H 6 ) + 2 obtained using EOM-IP-CCSD/6-31+G*. IE(2,3) values were obtained using EOM-IP-CC(2,3). IE IE(2,3) μ f B 2 /π o g (stem) 8.383 8.039 A 2 /π a g (stem) 8.913 8.574 1.30 10 − 2 2.20 10 − 6 B 1 /π a g (top) 9.175 8.807 - - B 2 /π o g (top) 9.189 8.826 2.279 0.102 A 1 /σ a (stem) 11.357 10.929 0.138 0.00139 B 1 /σ o (stem) 11.544 11.187 - - A 1 /σ a (top) 12.027 11.654 8.05 10 − 3 5.78 10 − 6 A 2 /σ o (top) 12.010 11.636 5.03 10 − 4 2.2 10 − 8 B 2 /π u (stem) 11.927 11.255 0.855 0.0635 A 1 /π u (top) 12.366 11.814 0.225 4.96 10 − 3 E ref a -463.013951 a Total CCSD energy (hartree) of (C 6 H 6 ) 2 relative to the relaxed one, except for theπ u (stem) state, which is now below theσ (top) states. The four highest occupied MOs of (C 6 H 6 ) 2 , from which the four lowest states of the cation are derived, comprise the degenerateπ system of benzene. In agreement with the Hunter-Sanders model 76 , the orbitals of the stem fragment are higher in energy than on the top fragment. The π o g component is above π a g , but this order is reversed in the top fragment. Bothπ o g orbitals of the top and stem fragments belong to the same irreducible representation of the C 2v point group, and are thus allowed to mix, resulting in charge delocalization and relatively strong CRπ o g (top)← π o g (stem) band. Upon ionization, the distance between the centers of the two rings relaxes from 4.99 ˚ A to 4.59 ˚ A, while the energy of the system is lowered by 584 cm − 1 . Optimization lowers the energy by additionalΔ E 1 =1424 cm − 1 , but results in a negligible change in the ring separation (4.59 ˚ A). Such close agreement is accidental, as the relaxed distance was determined from the PES scan employing a 0.1 ˚ A step size. In the ionized monomer, 117 -0.2 0.0 0.2 0.4 0.6 Figure 4.13: The effect of the intermolecular relaxation and JT distortion on the four lowest electronic states of the t-shaped (C 6 H 6 ) + 2 . The energies of states relative to the vertical configuration are given in wavenumbers. The partial charge on the stem frag- ment is given in italics. the relaxation results in 1403 cm − 1 stabilization. Thus, the total energetic effect of JT effect is similar in the dimer and monomer. Both fragments adapt a C 2v geometry, which is very similar to the D 2h geometry of the obtuse monomer, as shown in Fig. 4.14. The extent of this JT distortion is different in each ring. The stem ring is stabilized by Δ E 2 =1395 cm − 1 , while the top ring is stabilized by Δ E 2 =603 cm − 1 , compared to the neutral geometry. The apical angle and the adjacent C-C bond length are larger in the stem fragment, which carries more charge, while the geometry of the top ring is very close to the neutral. The extent of JT distortion in the rings is also consistent with the NBO charges shown in Fig. 4.13. As dictated by symmetry considerations, the stem and topπ a g states do not mix and these two states have the charge completely localized. On the other 118 (a) 121.17 121.63 119.19 119.41 1.429 1.377 1.425 121.53 121.53 1.370 1.429 1.429 119.23 119.23 (b) 120.15 119.92 1.404 1.398 121.53 1.370 1.429 119.23 Figure 4.14: The geometry of the (a) stem and (b) top fragments in the t-shaped (C 6 H 6 ) + 2 (underlined numbers). The geometry of an isolated ring in a π o g state is also given. Angles are in degree and distances are in ˚ Angstrom. 119 hand, theπ o g states are delocalized. In the vertical structure the stem fragment has 0.880 and 0.110 positive charge in the π o g (stem) and π o g (top) states, respectively. The ring carrying more positive charge undergoes larger distortion. As shown previously, the intensity of the CR bands depends on the degree of monomer state mixing, which also manifests itself by charge delocalization. By intro- ducing diabatic states defined as the charge-localized states 49 , the fragment charge can be interpreted as the square of the weight of the corresponding diabatic state, and the delocalization is caused by the diabatic coupling between these states. The CR band transition dipole moment increases with increased delocalization. The electronic Hamil- tonian in this diabatic basis has the following form: H = H stem H st H st H top (4.2) H stem and H top are the energies of the states with the charge localized on the step and the top fragments, respectively, andH st is the coupling. We refer toΔ H =H top − H stem as the splitting between the diabatic states. At the vertical geometry, the coupling and the splitting are: H st =1067 cm − 1 and Δ H=2617 cm − 1 . As the distance between the rings relaxes, the coupling and the split- ting increase to H st =1894 and Δ H=3450 cm − 1 , respectively. The two quantities have an opposite effect on charge localization, and the calculations show that the stronger coupling prevails resulting in more significant mixing of the diabatic states, as evi- denced by the decreased partial charge on the stem fragment. The transition dipole moment changes from 2.65 a.u. to 2.76 a.u., while the charge-resonance band shifts to the blue. Upon full optimization, the JT distortion decreases charge delocalization to levels observed in the vertical structure. Greater state localization results in smaller transition dipole moment of the CR band, 2.28 a.u.. The change in the coupling is small, 120 H st =1996 cm − 1 , however the splitting increases toΔ H=5126 cm − 1 . Thus, the changes in the coupling and splitting are comparable and both needed to be considered in quanti- tative considerations of charge delocalization and spectroscopy of the system. Note that the relaxed geometry accounts only for the coupling, while full optimization includes both effects. The increased splitting at the optimized configuration is due to ring distortions, rather than intermolecular interactions. Consider the diabatic state with the positive charge on the stem fragment. It is stabilized by JT distortion of the stem fragment by Δ E 2 =1395 cm − 1 which destabilizes the neutral top fragment. In our calculations, the neutral fragment is actually stabilized by Δ E 3 =-101 cm − 1 . This is an artifact of using the experimental geometry of the benzene rings in the relaxed scans, rather than the CCSD/6-31+G* one. In the top diabatic state, the top ring is stabilized by Δ E 2 =603 cm − 1 , while the stem is destabilized byΔ E 3 =674 cm − 1 . This amounts to a 1395+100- 603+674=1566 cm − 1 increase in the splitting, thus accounting for the difference in the relaxed and optimized geometries. 4.9.5 Displaced sandwich Vertical IEs, transition dipoles, and oscillator strengths for the electronic transitions between the states of the optimized displaced (C 6 H 6 ) + 2 are listed in Table 4.9. The effect of the relaxation and optimization on the four lowest electronic states is outlined in Fig. 4.15. The geometric parameters of the rings are shown in Fig. 4.12. The previous study showed that the displaced sandwich structure is the minimum energy structure of (C 6 H 6 ) + 2 . The displacement can be either along the C-H bond (y-displaced) or perpen- dicular to a C-C bond (x-displaced structure). The vertical y-displaced structure was obtained from the x-displaced structures by using the same displacement (1.6 ˚ A) and 121 interplanar separation (3.6 ˚ A). Overall, the spectra of the relaxed and optimized geome- tries are very similar. We have observed that in the relaxed structure the ordering of σ andπ u states changes, and this is preserved upon the full geometry optimization. (a) -0.4 0.0 0.4 0.8 1.2 (b) -0.4 0.0 0.4 0.8 1.2 Figure 4.15: The effect of the intermolecular relaxation and JT distortion on the four lowest electronic states of (a) x-displaced and (b) y-displaced sandwich isomers of (C 6 H 6 ) + 2 . The energies of states relative to the vertical configuration are given in wavenumbers. 122 Table 4.9: IEs (eV), transition dipole moments (a.u.), and oscillator strengths of the sandwich and the displaced sandwich (C 6 H 6 ) + 2 isomers obtained using EOM-IP-CCSD/6-31+G*. IE(2,3) values were obtained using EOM-IP-CC(2,3). x-displaced y-displaced IE IE(2,3) μ f IE IE(2,3) μ f B g /π ∗ (π o g ) 7.886 7.536 B g /π ∗ (π a g ) 7.866 7.515 A g /π ∗ (π a g ) 8.393 8.046 - - A g /π ∗ (π o g ) 8.417 8.069 - - B u /π (π a g ) 9.174 8.824 6.03 10 − 2 1.14 10 − 4 B u /π (π o g ) 9.150 8.799 5.46 10 − 2 9.36 10 − 5 A u /π (π o g ) 9.290 8.937 2.51 0.217 A u /π (π a g ) 9.300 8.944 2.49 0.218 B g /φ (σ o ) 11.624 11.270 - - B u /φ ∗ (σ a ) 11.625 11.268 8.35 10 − 2 6.42 10 − 4 B u /φ ∗ (σ a ) 11.686 11.330 7.59 10 − 2 5.37 10 − 4 A g /φ (σ a ) 11.619 11.263 - - A u /φ ∗ (σ o ) 11.624 11.271 6.36 10 − 2 3.70 10 − 4 A u /φ ∗ (σ o ) 11.686 11.332 5.83 10 − 2 3.18 10 − 4 A g /φ (σ a ) 11.752 11.398 - - B g /φ (σ o ) 11.748 11.394 - - B u /σ ∗ (π u ) 11.208 10.599 0.778 0.0492 B u /σ ∗ (π u ) 11.203 10.593 0.778 0.0495 A g /σ (π u ) 12.671 11.763 - - A g /σ (π u ) 12.683 11.819 - - E ref a -463.007114 -463.006732 a Total CCSD energy (hartree) of (C 6 H 6 ) 2 123 The two vertical structures are nearly degenerate, the y-displaced isomer being 33 cm − 1 lower. The difference decreases to 17 cm − 1 upon relaxation of the intermolecular coordinates. Full optimization increases this gap to 78 cm − 1 . At the relaxed geome- try, the rings are separated by 3.1 ˚ A and shifted by 1.0 ˚ A, in both isomers. The distance between centers is 3.26 ˚ A. Allowing both intermolecular and intramolecular coordinates to vary has only a weak effects on the center-center distance (3.28 ˚ A), however, it differ- entiates the two isomers in ring separation and displacement. In the y-displaced isomer, the rings are shifted by 0.72 ˚ A and separated by 3.20 ˚ A. The respective numbers in the x-displaced isomer are 1.07 and 3.11 ˚ A. Overall, optimization increases the interplanar separation and the overlap between fragments’ MOs. The two displaced cation isomers correspond to different character of the singly occupied MO (SOMO). In the y-displaced structure, the antibonding combination of π a g is the SOMO, while in the x-displaced the π o g combination. The two rings in a displaced sandwich isomer are related by symmetry and equally share the effect of the JT distortion. Optimized y-displaced structure is Δ E 1 =1293 cm − 1 lower then the relaxed one, whileΔ E 1 =1231 cm − 1 for the x isomer. In the y- and x- displaced the ring relaxes byΔ E 2 =1098 cm − 1 . The neutral ring is destabilized byΔ E 3 =215 cm − 1 , in y- and by Δ E 3 =185 cm − 1 in x-displaced structures. The geometric parameters of the rings are shown in Fig. 4.12. In both cases, the values of the angles are half way between the respective cation structures and the neutral, but bond length are similar. Although the symmetry of both rings in the dimer is formally lowered fromD 2h toC 2v , the resulting asymmetry is very small. 4.9.6 Spectroscopy While the inter-fragment relaxation has dramatic spectroscopic consequences 49 , the additional effects of full geometry optimization on the electronic states is small and 124 amounts to small shifts in the excitation energies relative to the relaxed structures. Thus, we refrain from discussing this aspect of the results and merely point out the differences with our previous work. The effect of triple excitations is not uniform on all states. The ionization energy of states derived from theπ u orbitals is lowered more than that of the other states. The corresponding absorption shifts to the red in the spectra of the cation. The results are summarize in Tables 4.8 and 4.9. 4.9.7 Conclusions We reported the implementation of EOM-IP-CCSD and EOM-IP-CC(2,3) energies, as well as EOM-IP-CCSD gradients. The new code was applied to study the role and char- acter of JT distortions in the benzene dimer cations. We have shown that the JT effect in the dimer is very similar to that of the monomer. Overall, it has only a minor effect on the electronic spectra, which are much more sensitive to the inter-fragment relax- ation. We also investigated the effects of geometrical relaxation on hole localization in the t-shaped isomer. We found that the larger charge localization due to JT relax- ation is counteracted by the increased diabatic coupling causing delocalization. Thus, the detailed balance of the two effects determines the extent of charge delocalization, and the neglect of JT distortion leads to overestimation of the delocalization. In this particular case both effect largely cancel out and the observed charge distribution is similar in the vertical and optimized structures, but not in the relaxed one. Large differ- ence between the relaxed and optimized geometries in the displaced isomers points at strong coupling between the JT and intermolecular degrees of freedom. This may be of importance in a system where the freedom of rings to move around is restricted by the environment. 125 4.10 Chapter 4 Bibliography [1] Saenger, W. Principles of Nucleic Acid Structure; Springer-Verlag: New York, 1984. [2] Burley, S.K. ; Petsko, G.A Science 1985, 229, 23. [3] Hunter, C.A. ; Singh, J. ; Thornton, J.M. J. Mol. Biol. 1991, 218, 837. [4] M¨ uller-Dethlefs, K. ; Hobza, P. Chem. Rev. 2000, 100, 143. [5] Hobza, P. ; Zahradnik, R. ; M¨ uller-Dethlefs, K. Collect. Czech. Chem. Commun. 2006, 71, 443. [6] Askew, B. ; Ballester, P. ; Buhr, C. ; Jeong, K.S. ; Jones, S. ; Parris, K. ; Williams, K. ; Rebek Jr., J. J. Am. Chem. Soc. 1989, 111, 1082. [7] Hunter, C.A. Chem. Soc. Rev. 1994, 23, 101. [8] Rebek Jr., J. Chem. Soc. Rev. 1996, 25, 255. [9] Meyer, E.A. ; Castellano, R.K. ; Diederich, F. Angew. Chem. Int. Edit. 2003, 42, 1210. [10] Claessens, C.G. ; Stoddart, J.F. J. Phys. Org. Chem. 1997, 10, 254. [11] Sancar, A. Biochemistry 1994, 33, 2. [12] Bixon, M. ; Jortner, J. J. Phys. Chem. A 2001, 105, 10322. [13] Gervasio, F.L. ; Laio, A. ; Parrinello, M. ; Boero, M. Phys. Rev. Lett. 2005, 94, 158103. [14] Chebny, V .J. ; Shukla, R. ; Rathore, R. J. Phys Chem. A 2006, 110, 13003. [15] Bittner, E.R. J. Chem. Phys. 2006, 125, 094909. [16] Inokuchi, Y . ; Naitoh, Y . ; Ohashi, K. ; Saitow, K. ; Yoshihara, K. ; Nishi, N. Chem. Phys. Lett. 1997, 269, 298. [17] Okamoto, K. ; Saeki, A. ; Kozawa, T. ; Yoshida, Y . ; Tagawa, S. Chem. Lett. 2003, 32, 834. [18] Enomoto, K. ; LaVerne, J.A. ; Araos, M. S. J. Phys. Chem. A 2007, 111, 9. [19] Giese, T.J. ; York, D.M. Int. J. Quantum Chem. 2004, 98, 388. 126 [20] Huber, K.P. ; Herzberg, G. Constants of diatomic molecules; Van Nostrand Rein- hold: New York, 1979. [21] Castex, M.C. J. Chem. Phys. 1981, 74, 759. [22] Flamigni, L. ; Camaioni, N. ; Bortolus, P. ; Minto, F. ; Gleria, M. J. Phys. Chem. 1991, 95, 971. [23] East, A.L.L. ; Lim, E.C. J. Chem. Phys. 2000, 113,, 8981. [24] Minaev, B. Phys. Chem. Chem. Phys. 2003, 5, 2314. [25] Mohanambe, L. ; Vasudevan, S. J. Phys. Chem. B 2006, 110, 14345. [26] Lindner, R. ; M¨ uller-Dethlefs, K. ; Wedum, E. ; Haber, K. ; Grant, E.R. Science 1996, 271, 1698. [27] M¨ uller-Dethlefs, K. ; Peel, J.B. . [28] Ford, M. ; Lindner, R. ; M¨ uller-Dethlefs, K. Mol. Phys. 2003, 101, 705. [29] Radloff, W. ; Stert, V . ; Freudenberg, T. ; Hertel, I.V . ; Jouvet, C. ; Dedonder- Lardeux, C. ; Solgadi, D. Chem. Phys. Lett. 1997, 281, 20. [30] Hirata, T. ; Ikeda, H. ; Saigusa, H. J. Phys. Chem. A 1999, 103, 1014. [31] Ohashi, K. ; Nishi, N. J. Chem. Phys. 1991, 95, 4002. [32] Ohashi, K. ; Nishi, N. J. Phys. Chem. 1992, 96, 2931. [33] Ohashi, K. ; Nakai, Y . ; Shibata, T. ; Nishi, N. Laser Chem. 1994, 14, 3. [34] Ohashi, K. ; Inokuchi, Y . ; Nishi, N. Chem. Phys. Lett. 1996, 263, 167. [35] Shida, T. ; Hamill, W. H. J. Chem. Phys. 1966, 44, 4372. [36] Badger, B. ; Brockleh, B. Trans. Farad. Soc. 1969, 65, 2582. [37] Grover, J.R. ; Walters, E.A. ; Hui, E.T. J. Phys. Chem. 1987, 91, 3233. [38] Krause, H. ; Ernstberger, B. ; Neusser, H.J. Chem. Phys. Lett. 1991, 184, 411. [39] Field, F.H. ; Hamlet, P. ; Libby, W.F. J. Am. Chem. Soc. 1969, 91, 2839. [40] Meot-Ner (Mautner), M. ; Hamlet, P. ; Hunter, E.P. ; Field, F.H. J. Am. Chem. Soc. 1978, 100, 5466. [41] Hiraoka, K. ; Fujimaki, S. ; Aruga, K. ; Yamabe, S. J. Chem. Phys. 1991, 95, 8413. 127 [42] Rusyniak, M. ; Ibrahim, Y . ; Alsharaeh, E. ; Meot-Ner (Mautner), M. ; El-Shall, M.S. J. Phys. Chem. A 2003, 107, 7656. [43] Ibrahim, Y . ; Alsharaeh, E. ; Rusyniak, M. ; Watson, Simon ; Meot-Ner (Mautner), Michael ; El-Shall, M.S. Chem. Phys. Lett. 2003, 380, 21. [44] Milosevich, S.A. ; Saichek, K. ; Hinchey, L. ; England, W.B. ; Kovacic, P. J. Am. Chem. Soc. 1983, 105, 1088. [45] Miyoshi, E. ; Ichikawa, T. ; Sumi, T. ; Sakai, Y . ; Shida, N. Chem. Phys. Lett. 1997, 275, 404. [46] Itagaki, Y . ; Benetis, N.P. ; Kadam, R.M. ; Lund, A. Phys. Chem. Chem. Phys. 2000, 2, 2683. [47] Miyoshi, E. ; Yamamoto, N. ; Sekiya, M. ; Tanaka, K. Mol. Phys. 2003, 101, 227. [48] Birks, J.B. Photophysics of Aromatic Molecules; Wiley: New York, 1970. [49] Pieniazek, P.A. ; Krylov, A.I. ; Bradforth, S.E. J. Chem. Phys. 2007, 127, 044317. [50] Steed, J.M. ; Dixon, T.A. ; Klemperer, W. J. Chem. Phys. 1979, 70, 4940. [51] Arunan, E. ; Gutowsky, H.S. J. Chem. Phys. 1993, 98, 4294. [52] Hobza, P. ; Selzle, H.L. ; Schlag, E.W. J. Phys. Chem. 1996, 100, 18790. [53] Sun, S. ; Bernstein, E.R. J. Phys. Chem. 1996, 100, 13348. [54] Jaffe, R.L. ; Smith, G.D. J. Chem. Phys. 1996, 105, 2780. [55] Sinnokrot, M.O. ; Sherrill, C.D. J. Phys. Chem. A 2004, 108, 10200. [56] Sinnokrot, M.O. ; Sherrill, C.D. J. Phys. Chem. A 2006, 110, 10656. [57] Podeszwa, R. ; Bukowski, R. ; Szalewicz, K. J. Phys. Chem. A 2006, 110, 10345. [58] Akopian, M.E. ; Vilesov, F.I. ; Terenin, A.N. Sov. Phys. Doklady (Engl. Transl.) 1961, 6, 490. [59] Baltzer, P. ; Karlsson, L. ; Wannberg, B. ; Ohrwall, G. ; Holland, D.M.P. ; Mac- Donald, M.A. ; Hayes, M.A. ; von Niessen, D. Chem. Phys. 1997, 224, 95. [60] Rennie, E.E. ; Johnson, C.A.F. ; Parker, J.E. ; Holland, D.M.P. ; Shaw, D.A. ; Hayes, M.A. Chem. Phys. 1998, 229, 107. [61] Yencha, A.J. ; Avaldi, R.I. Hall L. ; Dawber, G. ; McConkey, A.G. ; MacDonald, M.A. ; King, G.C. Can. J. Chem. 2004, 82, 1061. 128 [62] Weikert, H.-G. ; Cederbaum, L.S. Chem. Phys. Lett. 1995, 237, 1. [63] Takeshita, K. Theo. Chem. Acc. 1999, 103, 64. [64] Doscher, M. ; Koppel, H. ; Szalay, P.G. J. Chem. Phys. 2002, 117, 2645. [65] Koppel, H. ; Doscher, M. ; Baldea, I. ; Meyer, H.D. ; Szalay, P.G. J. Chem. Phys. 2002, 117, 2657. [66] Sinha, D. ; Mukhopadhyay, D. ; Mukherjee, D. Chem. Phys. Lett. 1986, 129, 369. [67] Sinha, D. ; Mukhopadhya, D. ; Chaudhuri, R. ; Mukherjee, D. Chem. Phys. Lett. 1989, 154, 544. [68] Chaudhuri, R. ; Mukhopadhyay, D. ; Mukherjee, D. Chem. Phys. Lett. 1989, 162, 393. [69] Stanton, J.F. ; Gauss, J. J. Chem. Phys. 1999, 111, 8785. [70] Pieniazek, P.A. ; Arnstein, S.A. ; Bradforth, S.E. ; Krylov, A.I. ; Sherrill, C.D. J. Chem. Phys. 2007, 127, 164110. [71] Purvis, G.D. ; Bartlett, R.J. J. Chem. Phys 1982, 76, 1910. [72] Sekino, H. ; Bartlett, R.J. Int. J. Quant. Chem. Symp. 1984, 18, 255. [73] Stanton, J.F. ; Bartlett, R.J. J. Chem. Phys. 1993, 98, 7029. [74] Y . Shao, L.F. Molnar, Y . Jung, J. Kussmann, C. Ochsenfeld, S. Brown, A.T.B. Gilbert, L.V . Slipchenko, S.V . Levchenko, D.P. O’Neil, R.A. Distasio Jr, R.C. Lochan, T. Wang, G.J.O. Beran, N.A. Besley, J.M. Herbert, C.Y . Lin, T. Van V oorhis, S.H. Chien, A. Sodt, R.P. Steele, V .A. Rassolov, P. Maslen, P.P. Koram- bath, R.D. Adamson, B. Austin, J. Baker, E.F.C. Bird, H. Daschel, R.J. Doerksen, A. Drew, B.D. Dunietz, A.D. Dutoi, T.R. Furlani, S.R. Gwaltney, A. Heyden, S. Hirata, C.-P. Hsu, G.S. Kedziora, R.Z. Khalliulin, P. Klunziger, A.M. Lee, W.Z. Liang, I. Lotan, N. Nair, B. Peters, E.I. Proynov, P.A. Pieniazek, Y .M. Rhee, J. Ritchie, E. Rosta, C.D. Sherrill, A.C. Simmonett, J.E. Subotnik, H.L. Woodcock III, W. Zhang, A.T. Bell, A.K. Chakraborty, D.M. Chipman, F.J. Keil, A. Warshel, W.J. Herhe, H.F. Schaefer III, J. Kong, A.I. Krylov, P.M.W. Gill, M. Head-Gordon Phys. Chem. Chem. Phys. 2006, 8, 3172. [75] Gauss, J. ; Stanton, J.F. J. Phys. Chem. A 2000, 104, 2865. [76] Hunter, C.A. ; Sanders, J.K.M. J. Am. Chem. Soc. 1990, 112, 5525. [77] Cave, R.J. ; Newton, M.D. Chem. Phys. Lett. 1996, 249, 15. 129 [78] K¨ oppel, H. ; Cederbaum, L. S. J. Chem. Phys. 1988, 89, 2023. [79] Hirata, S. ; Nooijen, M. ; Bartlett, R.J. Chem. Phys. Lett. 2000, 328, 459. [80] Kamiya, M. ; Hirata, S. J. Chem. Phys. 2006, 125, 074111. [81] Misawa, M. ; Fukunaga, T. J. Chem. Phys. 1990, 93, 3495. [82] NBO 5.0. Glendening, E.D. ; Badenhoop, J.K. ; Reed, A.E. ; Carpenter, J.E. ; Bohmann, J.A. ; Morales, C.M. ; Weinhold, F. Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, 2001. [83] Shao, Y . ; Molnar, L.F. ; Jung, Y . ; Kussmann, J. ; Ochsenfeld, C. ; Brown, S. ; Gilbert, A.T.B. ; Slipchenko, L.V . ; Levchenko, S.V . ; O’Neil, D. P. ; Distasio Jr., R.A. ; Lochan, R.C. ; Wang, T. ; Beran, G.J.O. ; Besley, N.A. ; Herbert, J.M. ; Lin, C.Y . ; Van V oorhis, T. ; Chien, S.H. ; Sodt, A. ; Steele, R.P. ; Rassolov, V . A. ; Maslen, P. ; Korambath, P.P. ; Adamson, R.D. ; Austin, B. ; Baker, J. ; Bird, E.F.C. ; Daschel, H. ; Doerksen, R.J. ; Drew, A. ; Dunietz, B.D. ; Dutoi, A.D. ; Furlani, T.R. ; Gwaltney, S.R. ; Heyden, A. ; Hirata, S. ; Hsu, C.-P. ; Kedziora, G.S. ; Khalliulin, R.Z. ; Klunziger, P. ; Lee, A.M. ; Liang, W.Z. ; Lotan, I. ; Nair, N. ; Peters, B. ; Proynov, E.I. ; Pieniazek, P.A. ; Rhee, Y .M. ; Ritchie, J. ; Rosta, E. ; Sherrill, C.D. ; Simmonett, A.C. ; Subotnik, J.E. ; Woodcock III, H.L. ; Zhang, W. ; Bell, A.T. ; Chakraborty, A.K. ; Chipman, D.M. ; Keil, F.J. ; Warshel, A. ; Herhe, W.J. ; Schaefer III, H.F. ; Kong, J. ; Krylov, A.I. ; Gill, P.M.W. ; Head-Gordon, M. Phys. Chem. Chem. Phys. 2006, 8, 3172. 130 Chapter 5: Electronic structure of the water dimer cation 5.1 Overview The spectroscopic signatures of proton transfer in the water dimer cation were inves- tigated. The six lowest electronic states were characterized along the reaction coordi- nate using the equation-of-motion coupled-cluster with single and double substitutions method for ionized systems. The nature of the dimer states was explained in terms of the monomer states using a qualitative molecular orbital framework. We found that proton transfer induces significant changes in the electronic spectrum, thus suggesting that time-resolved electronic femtosecond spectroscopy is an effective strategy to mon- itor the dynamics following ionization. The electronic spectra at vertical and proton- transferred configurations include both local excitations (features similar to those of the monomers) and charge-transfer bands. Ab initio calculations were used to test the per- formance of a self-interaction correction for density functional theory (DFT). The cor- rected DFT/BLYP method is capable of reproducing the correct ordering of the (H 2 O) + 2 isomers, and thus is a reasonable approach for calculations of larger systems. 5.2 Introduction Ionized states in condensed media, for example, those produced by radiation in biologi- cal tissue, or encountered in the storage and reprocessing of fissioned nuclear materials 131 are not yet fully understood 1–3 . Even radiolysis of water, a predominant molecular target of the incident radiation in these situations, which has also served as the prototypical sys- tem for disentangling the phenomenology of high energy processes in bulk liquids, has not been completely mapped out. For example, both the nature of the initially excited states and sub-50 fs ionization/dissociation dynamics after deposition of energy have not been directly observed 3 . The major ionization channel in water radiolysis includes the following steps: n H 2 O→ (H 2 O) ∗ n → H 2 O + (H 2 O) n− 1 + e − →...→ H 3 O + aq + OH aq + e − aq (5.1) The extent of delocalization of the initial excitation, i.e., the number n of water molecules participating in (H 2 O) ∗ n , and the character of the charged species immediately after charge separation are poorly defined. Both quantities are likely to depend on the initial energy deposited 4 . In addition to reaction (5.1), energy deposited into water can also lead to homolytic bond breaking yielding H + OH, which resembles the gas phase photodissociation 5 . Beside the practical interest in the yields and spatial distribution of reactive radical products, this system also presents several fundamental questions on the mechanism and dynamics of separation and delocalization of both charge and spin in a partially disordered and rapidly fluctuating system. Both the electronic structure and molecular dynamics of water are amenable to simu- lation. While the structure, dynamics and spectroscopy of the excess electron have been extensively investigated, modeling the other initial product of photoionization 6, 7 , the ionized hole, has not been attempted in bulk, although cluster studies were reported 8–11 . The delocalization of the initial charge, the timescale and dynamics of reaction (5.1), as well as the fate and separation of the products are all important issues which a sim- ulation could help to explain. Our ongoing efforts to simulate the excess hole in bulk 132 require the benchmarking of electronic structure methods for ab initio molecular dynam- ics (AIMD). One of the goals of the present study is to compare reliable calculations of the simplest fragment of the hole in water, the water dimer cation, against less compu- tationally demanding methods suitable for AIMD 12 . Moreover, in the Bradforth laboratories at USC, photolysis studies of pure water have been carried out with pulses as short as 25 fs. With this time resolution, femtosec- ond pump-probe spectroscopy should be able to track the proton transfer step in reaction (1) and with a dispersed probe covering continuously the near UV and visible, the tran- sients should be identifiable by their spectroscopic signatures. However, it is critical to establish the signature of the H 2 O + intermediate and distinguish it from the subsequent species such as (OH-H 3 O + ) and OH (aq) 13, 14 . Therefore, another major goal of this work is to accurately describe the excited states of each of these species, at least at the level of small clusters. This will allow an assessment of what information electronic spec- troscopy can provide in tracking the ionization chemistry of bulk water. Our theoretical approach parallels a recent study we performed on the electronic spectroscopy of the benzene dimer cation, a core species in ionization of liquid benzene 15 . The results for a gas phase water dimer establish a foundation for understanding the ionized water in the condensed phase. Moreover, the gas phase species is also of interest and has attracted experimental and theoretical attention in the past, albeit not in the the context of its electronic spectroscopy. Ng et al. 16 determined the adiabatic ionization energy (IE) to be 11.2 eV by photoionization threshold measurements. De Visser 17 and coworkers investigated the reactivity of (H 2 O) + 2 towards a series of sub- strates, placing the adiabatic IE in the 10.8 - 10.9 eV bracket. Achiba and coworkers 18 measured the photoelectron spectra of (H 2 O) + 2 providing information about the two low- est electronic states of the cation. They observed two broad peaks centered at 12.1 and 13.2 eV , and determined the onset of ionization to be at 11.1 eV . This work spurred a 133 series of theoretical studies 19–25 . While the first ionization was assigned as ionization from the out-of-plane orbital of the donor molecule, there was a controversy as to the site of second ionization. Early study pointed to ionization of the a 1 donor orbital, but subsequent correlated calculations reassigned it to the b 1 acceptor orbital 23 . Also, 2D potential energy surface (PES) cuts for the proton transfer reaction along the O-O and O-H coordinates were calculated 22 . Density functional theory (DFT) calculations deter- mined the lowest-energy isomer to be the hemibonded structure 26, 27 , which later was shown to be an artifact of the self-interaction error (SIE) 28 . M¨ uller and coworkers stud- ied ionization of small water clusters using Green’s functions 29 . Their work focused on vertical structures and a hinted at delocalized nature of the hole in the ground and excited states of water cluster cations. This work presents calculations for the water dimer cation at the geometry of the neutral water dimer and along the proton transfer reaction coordinate to the OH–H 3 O + products. The character of the ground and the low-lying excited states are described within a Dimer Molecular Orbital - Linear Combination of Fragment Molecular Orbitals (DMO-LCFMO) framework. We also chart the electronic spectroscopy of the system as it evolves along the reaction coordinate. The predictions of this study will help guide ongoing laboratory femtosecond studies and AIMD simulations of ionized bulk water. 5.3 Theoretical methods and computational details We begin by characterizing the monomer fragments and then proceed to the struc- tures and excited states of water dimer cation. Theoretical descriptions of the ground and excited electronic states of doublet systems, like (H 2 O) + 2 or OH, are problematic due to symmetry breaking and spin-contamination of the doublet Hartree-Fock (HF) 134 references 30–32 . Equation-of-motion coupled-cluster for ionization energies (EOM-IP- CC) 33–36 overcomes these difficulties by describing the problematic open-shell doublet wave functionsΨ( cation) as ionized/excited states of a well-behaved neutral wave func- tionΨ( neutral): Ψ( cation)= ˆ RΨ( neutral), (5.2) where the Koopmans-like operator R generates all possible ionized, and ionized and simultaneously excited configurations out of the closed-shell reference determinant, and Ψ( neutral) is a coupled-cluster wave function, typically including only single and dou- ble substitutions (CCSD). TruncatingR at1h2p and2h3p levels gives rise to the EOM- IP-CCSD and EOM-IP-CC(2,3) models, respectively. The latter model includes more correlation and is, therefore, more accurate. However, it is not practical for extensive calculations due the higher computational cost, and we employ it in this work for bench- mark purposes. The geometries of the closed-shell species [H 2 O, OH − , H 3 O + , (H 2 O) 2 ] were opti- mized at the second-order Møller-Plesset perturbation theory level with the 6-311++G** basis set (MP2/6-311++G**), while those of the open-shell doublet species [OH, proton-transferred and hemibonded (H 2 O) + 2 ] — at the EOM-IP-CCSD/6-311++G** level. For the proton-transferred structure, both theC s and theC 1 symmetry structures were obtained. Monomer calculations of IEs were carried out, and their accuracy was assessed using the available experimental data. Calculations were performed at the EOM-IP- CCSD and EOM-IP-CC(2,3) levels using the 6-311++G** and aug-cc-pVTZ basis sets. Based on these benchmark results, the 6-311++G** basis set was selected for produc- tion calculations. Excitation energies and transition properties for monomer fragments and the dimer were computed using EOM-IP-CCSD and EOM-CCSD for excitation energies (EOM-EE-CCSD) 37–39 . EOM-EE-CCSD allows one to investigate the possible 135 presence of states derived by excitation to virtual orbitals and, therefore, not described by EOM-IP. EOM-EE results are also useful for assessing the accuracy of the excited- states description based on the open-shell reference. Additionally, excitation energies were obtained using EOM-IP-CC(2,3). The character of the dimer excited states was characterized using the Natural Bond Orbital (NBO) analysis 40 by calculating the natural charge of the fragments. The NBO analysis was also applied to the reference CCSD/6-311++G** wave functions. When appropriate, unrestricted HF (UHF) references were used in geometry optimizations, and restricted open-shell (ROHF) references employed in evaluation of excitation ener- gies and transition properties. To characterize the proton transfer reaction from the vertical ionized dimer, we computed a PES scan using EOM-IP-CCSD/6-311++G**, under theC s symmetry con- straint. To evaluate the consequences of this approximation, we compared C 1 and C s configurations of the proton-transferred structure. Briefly, excitation energies are insen- sitive to this constraint, but intensities are. The distance between the oxygen atoms was varied between 2.30 ˚ A and 3.10 ˚ A, while the distance between the hydrogen of the H-bond donor and the oxygen of the acceptor was varied from 0.90 ˚ A to 2.20 ˚ A, with 0.05 ˚ A increments. At each point a constrained geometry optimization was conducted. Subsequently, the points corresponding to the neutral (A) and proton-transferred geome- tries (C) were identified on the grid. A steepest descent path was followed between A and C. Excitation energies and transition properties were computed along the path using EOM-IP-CCSD/6-311++G**. All ab initio calculations were performed using the Q-CHEM ab initio package 41 . The basis sets were obtained from the EMSL repository 42 . With the goal of ab initio Born-Oppenheimer molecular dynamics simulations of photoionization in bulk water 12 , we benchmarked the performance of DFT methods 136 implemented within theCP2K/Quickstep software package 43 . InCP2K/Quickstep, DFT is implemented using a mixed Gaussian and plane-waves approach. We employed the BLYP functional with the double-zeta (DZVP) and the triple-zeta (TZV2P) basis sets, along with the Goedecker-Teter-Hutter pseudo-potentials for oxygen 1s electrons 44 . The energy cutoff for plane waves was set to 280 Ry. Calculations for the radical cation were performed using unrestricted (UKS) and restricted open-shell (ROKS) Kohn-Sham schemes. To eliminate the errors due to electron self-interaction 45–47 , which are signifi- cant for open-shell and charge-transfer systems, we applied a self-interaction correction (SIC) within the ROKS method 14, 48, 49 . For SIC we used the recommended values of 0.2 for Coulomb term scaling and zero for scaling of the exchange-correlation term 14 . For comparison and a stability check we also employed the value of 0.2 for the latter scaling. 5.4 Results 5.4.1 Monomers The DMO-LCFMO description (see Section 2.3) of the dimer is based on the fragment orbitals and relevant excited states. In the water dimer cation, we are interested in the states derived by single ionization of the neutral. Thus, at the vertical geometry the DMOs can be expressed in terms of the occupied H 2 O orbitals. The proton-transferred cation orbitals originate from the OH − ··· H 3 O + dimer, and the OH − and H 3 O + occu- pied orbitals form a neutral basis. The three highest occupied molecular orbitals (MOs) of H 2 O, OH − , and H 3 O + are shown in Fig. 5.1, and the corresponding IEs are given in Table 5.1. Overall, the theory and the experiment, where available, agree well. 137 (a) (b) (c) 1b 1 3a 1 1b 2 1 S 1 S 3 V + 3a 1 1e u 1e g Figure 5.1: Three highest occupied MOs of (a) H 2 O, (b) OH − , and (c) H 3 O + . The geometric frame is rotated for each orbital to best show the orbital character. It is important to determine which of the monomer transitions are relevant to the dimer cation spectroscopy. The ground-state electronic structure at the proton- transferred geometry corresponds to the ionization of the hydroxyl part of the OH − ··· H 3 O + dimer, as dictated by the IE considerations. The EOM-IP method restricts us to the electronic states derived by single ionizations of the neutrals. Thus, we only dis- cuss transitions in the ionized system for which the singly occupied MO (SOMO) is the target orbital. This restricts the types of transitions to: (i) the transitions within the OH moiety, (ii) the transitions from the H 3 O + to the OH moiety leading to the formation of OH − ··· H 3 O 2+ . The two groups can be related to the excited states of OH and H 3 O 2+ molecules, respectively. In contrast, excitations at the neutral vertical geometry always correspond to H 2 O + ··· H 2 O, the charge being located either on the H-bond donor or the acceptor. Thus, one needs to consider only the states of H 2 O + . The excited states of monomers are given in Table 5.2. There are two possible excitations in H 2 O + in the experimentally relevant energy range (1 – 6eV). Theb 1 ← a 1 transition is optically allowed and occurs at 2.3 eV , while 138 Table 5.1: Ionization energies (eV) of H 2 O, OH − , and H 3 O + . EOM-IP-CCSD EOM-IP-CC(2,3) Exp. 6-311++G** aug-cc-pVTZ 6-311++G** aug-cc-pVTZ H 2 O 1b 1 12.32 12.61 12.29 12.51 12.6 a , 12.62 b 3a 1 14.61 14.87 14.57 14.78 14.8 a , 14.64 b 1b 2 18.84 18.94 18.78 18.83 18.6 a , 18.6 b OH − π 1.37 1.77 1.17 1.48 1.83 c σ 5.60 5.90 5.37 5.59 - H 3 O + a 1 24.38 24.64 24.41 24.61 - e 30.16 30.25 30.17 30.21 - a Ref. 50 b Ref. 51 c Ref. 52, 53 the dipole forbidden b 1 ← b 2 excitation is at 6.5 eV . The character of the electronic states does not change upon structural relaxation from the vertical neutral to the cation equilibrium geometry. The OH radical has only one excitation, π ← σ , at 4.2 eV . Addition of a proton to H 2 O + produces H 3 O 2+ , which features a weaka 1 ← e transition at 5.8 eV . Overall, addition/subtraction of a proton to/from H 2 O + dramatically changes the nature of the MOs and the electronic transitions. Thus, each species has unique electronic states that cannot be described as perturbed H 2 O + states. The EOM-IP-CCSD method treats accurately the dimer cation states derived by sin- gle ionization of the neutral, while the cation states corresponding to excitation to a vir- tual orbital of the neutral are not well described owing to their significant two-electron character. Neglecting these cation excitations is acceptable only if these states are out- side the experimentally relevant spectral region. In the context of the water dimer cation at the geometry of the neutral species, we need to consider the excited states of H 2 O + 139 Table 5.2: Excitation energies (eV) and transition properties (a.u.) of H 2 O + , OH, and H 3 O 2+ . A 6-311++G** basis set was used throughout. EOM-IP-CCSD EOM-EE-CCSD EOM-IP-CC(2,3) exp. E ex μ 2 f E ex μ 2 f E ex E ex H 2 O + at the geometry of the neutral 3a 1 -1b 1 2.29 0.0229 0.00128 2.29 0.0225 0.00126 2.28 2.02 a 1b 2 -1b 1 6.52 - - 6.48 - - 6.49 6.0 a H 2 O + at the geometry of the cation 3a 1 -1b 1 2.02 0.0179 0.000885 2.01 0.0177 0.000874 2.01 1.89 b 1b 2 -1b 1 6.53 - - 6.89 - - 6.49 - OH at the geometry of the neutral σ -π a 4.22 0.0301 0.00312 4.21 0.0311 0.00321 4.20 4.12 c H 3 O 2+ at the geometry of H 3 O + a 1 -e b 5.78 6.7 10 − 6 9.5 10 − 7 5.74 2.5 10 − 5 3.4 10 − 6 5.75 - a Computed as a difference of vertical IEs from Ref. 51 b Ref. 54 c Ref. 55 140 and H 2 O. The lowest excited state of water monomer in the gas phase is the ˜ A 1 B 1 state at 7.4-7.5 eV 56 . It is derived by excitation from the 1b 1 to the 4a 1 /Rydberg 3s orbital. The first two excitations (2 and 6 eV) in H 2 O + are excitations to the SOMO, and are well described by EOM-IP-CCSD. The next excited state, 2 2 B 1 , occurs at 13.7 eV as calcu- lated by EOM-EE-CCSD/6-311++G**. At the proton-transferred configuration states of H 3 O + and OH are relevant. The lowest excited state of H 3 O + , 2 1 A 1 , occurs at 11.8 eV (EOM-EE-CCSD/6-311++G**). The lowest OH excitation is σ ← π at 4 eV and is described by EOM-IP-CCSD. The next one, to the B 2 Σ + state, is at 8.7 eV . Thus, monomer states not described by EOM-IP-CCSD are well outside the region probed in the experiment, both at the neutral and at the proton-transferred geometry. 5.4.2 Dimers Ground state geometry The structures of (H 2 O) + 2 considered in this work are shown in Fig. 5.2. The neutral structure hasC s symmetry with the oxygen atoms separated by 2.91 ˚ A. This geometry is not a stationary point on the cation PES and provides time-zero reference for the struc- ture prepared by time-resolved spectroscopy. We refer to it as the vertical geometry. IEs and MOs at this geometry are shown in Fig. 5.3. EOM-IP-CCSD and EOM-IP-CC(2,3) predict the vertical IE to be 11.4 and 11.3 eV , respectively. This is lower than the experi- mental value of 12.1 eV 18 , assuming that the absorption maximum in the recorded spec- trum corresponds to a vertical transition, which depends on the population of different conformers in the experiment. The adiabatic IE was determined experimentally to lie in the range 10.8 - 11.2 eV 16–18 . The equivalent energy gap (not including zero-point energy differences) computed using EOM-IP-CCSD and EOM-IP-CC(2,3) is 10.55 and 10.40 eV , respectively. 141 (a) 2.91 Å (b) 2.47 Å (c) 2.02 Å Figure 5.2: Geometries of (a) vertical neutral (H 2 O) 2 , (b) proton-transferred (H 2 O) + 2 , and (c) hemibonded(H 2 O) + 2 . Oxygen-oxygen distance has been marked on the plots. . Ionization facilitates the downhill barrierless proton transfer from the H-bond donor to the acceptor, leading to the formation of OH··· H 3 O + (Fig. 5.2b). Electronically, these products correlate with the OH − ··· H 3 O + state of the neutral. There is a large geometric change associated with the proton transfer as the oxygen-oxygen distance decreases to 2.47 ˚ A and the symmetry is lowered toC 1 , due to an out-of-plane rotation of the OH fragment by 40 ◦ . However, theC 1 structure is lower by only 0.1 kcal/mol relative 142 b 1 /0 11.41 eV (b 2 /a 1 ) * 15.41 eV a 1 /b 1 13.66 eV (a 1 /b 1 ) * 12.91 eV b 2 /a 1 17.98 eV 0/b 2 19.48 eV -0.003 0.034 0.898 0.870 0.116 0.124 0.126 0.363 0.970 0.965 0.785 0.578 Figure 5.3: Six highest occupied MOs of the neutral water dimer. Ionization energies were calculated using EOM-IP-CCSD/6-311++G**. The numbers on the left are the NBO charge on the H-bond donor fragment calculated using EOM-IP-CCSD (upper number) and CCSD/EOM-EE-CCSD (lower number) wave functions. The geometric frame is rotated for each orbital to best show orbital character. The NBO analysis of the reference CCSD/6-311++G** wave function showed -0.012 charge on the H-bond donor. to the C s proton-transferred configuration. In the C s symmetry saddle, the oxygen- oxygen distance is also 2.47 ˚ A. 143 Table 5.3: Energies (kcal/mol) of (H 2 O) + 2 relative to the proton-transferred geometry calculated using wave function based methods. A 6-311++G** basis set was used throughout. hemibonded vertical neutral HF 26.8 20.1 MP2 7.4 21.4 CCSD 9.9 21.4 CCSD(T) 8.2 21.6 EOM-IP-CCSD 5.3 20.0 EOM-IP-CC(2,3) 7.4 21.8 To benchmark the DFT methods, we have investigated how different models repro- duce the relative energies of the isomers. In view of SIE, we also included the hemi- bonded structure predicted to be the lowest isomer by some DFT calculations (Table 5.4). It has (near) C 2h symmetry with the charge equally distributed between the frag- ments. Such systems are problematic for both HF and DFT methods. The former tends to destabilize them due to symmetry breaking, while the latter over-stabilize them due to SIE. The EOM-IP-CC methods based on the neutral reference are free from these arti- facts. Table 5.3 compares the energies of the (H 2 O) + 2 isomers obtained at different levels of theory using wave function methods. These methods consistently predict the proton- transferred isomer to be lower than the hemibonded form. At the HF level, the difference between the proton-transferred and hemibonded structures is 26 kcal/mol, and electron correlation brings it down to 7-8 kcal/mol. The large difference at the HF level is due to the symmetry breaking problem at the hemibonded geometry. The energy difference between the proton-transferred and vertical configurations only weakly depends on the level of theory and is approximately 20 kcal/mol. This stable behavior is due to the charge-localized character of the wave functions, which are free from the HF instability, and are well described using both open-shell and closed-shell references. 144 Table 5.4: Energies (kcal/mol) of (H 2 O) + 2 relative to the proton-transferred geometry calculated using DFT methods. hemibonded vertical neutral EOM-IP-CCSD and MP2 geometry UBLYP/DZVP -8.3 6.7 ROBLYP/DZVP -8.3 6.6 SIC(0.2/0.0)-ROBLYP/DZVP 7.3 21.1 SIC(0.2/0.0)-ROBLYP/TZVDD 6.6 19.9 SIC(0.2/0.2)-ROBLYP/DZVP 1.0 15.0 optimized geometry UBLYP/DZVP -8.8 - ROBLYP/DZVP -8.7 - SIC(0.2/0.0)-ROBLYP/DZVP (cp2k) 7.7 - SIC(0.2/0.0)-ROBLYP/TZVDD 8.0 - SIC(0.2/0.2)-ROBLYP/DZVP 0.8 - The set of (H 2 O) + 2 structures was re-optimized using DFT. The results are summa- rized in Table 5.4. The DFT calculations without SIC show the following trends. First, there is virtually no energy difference between the unrestricted and restricted open-shell BLYP results. Second, compared to the benchmark EOM-IP-CCSD calculations, BLYP overstabilizes both the vertical and the hemibonded dimer structures by -12 and -13 kcal/mol, respectively. The error for the hemibonded structure is comparable in abso- lute value to that of HF. However, the sign is opposite and, as was shown before 28 , the SIE uncorrected DFT erroneously places the hemibonded minimum below the proton- transferred one. This is a well-known signature of SIE observed in many systems with charge separation 47, 57, 58 . A simple empirical SIC correction for doublet states based on removing SI of the unpaired electron implemented for restricted open shell BLYP 14, 48 improves the results significantly. For the recommended choice of the two scaling parameters (0.2 and 0), the error in the above relative energies drops to 2-3 kcal/mol (compared to EOM-IP-CCSD) 145 and is further reduced to less than 2 kcal/mol upon moving from DZVP to TZVDD basis set. This is encouraging and justifies considering the use of DFT/BLYP with the present SIC for condensed phase AIMD calculations 12 . Although, it is theoretically more appealing to employ the same value for both scaling parameters (i.e., 0.2 and 0.2, as in other approaches 59 ), this produces less accurate results: the correct order of minima is still preserved, but the error increases to about -4 kcal/mol. Finally, DFT reoptimization of the proton-transfer and hemibonded structures results in minor geometry and, consequently, minor energy changes (Table 5.4). The geometry of the neutral dimer, which is not a minimum on the cationic surface, collapses to the proton-transfer minimum upon minimization (as for EOM-IP-CCSD). The use of restricted open-shell formalism with Kohn-Sham DFT deserves an addi- tional comment. As shown by Pople and coworkers 60 , in open-shell systems that have excess α electrons, regions of negative spin density exist. That means, that the local density of β electrons may be higher than the density of α electrons. The local excess ofβ electrons is reproduced by correlated calculations, as well as confirmed experimen- tally. In the DFT framework, it can only be reproduced within UKS formalism, simply because the electron density is a sum of the MO densities, which are the same for both spins in ROKS. However, the SIC applied here is stable only within ROKS formalism, as it requires one to identify the unpaired electron. Density functionals containing non- local operators, such as long range Hartree-Fock exchange, reduce the SIE in a more fundamental way 61–65 . However, symmetric radical cations are particularly difficult sys- tems, and performance is not yet fully satisfactory 65 . DMO-LCFMO framework The dimer spectroscopy can be explained in terms of the individual fragment contri- butions. The theoretical framework is provided by the DMO-LCFMO theory, which 146 was applied previously to the electronic structure of the benzene dimer cation 15 , and is described in Section 2.3. It allows one to correlate the dimer and monomer transi- tions based on the degree of mixing of the FMOs in the dimer. We employ a (H-bond donor orbital)/(H-bond acceptor orbital) notation, allowing one to quickly discern the FMOs that contribute to a given DMO. If there is no component on the given fragment, a symbol 0 is used. An asterisk signifies antibonding character of the dimer MO with respect to the fragments interaction. The extent to which a given FMO participates in a given DMO can be quantified using the NBO analysis. NBO yields the charges of the fragments, which are related to the square of the diabatic wave function defined here as the charge-localized state. Below we discuss the DMOs at the vertical and at the proton-transferred geometry. The orbitals at the vertical configuration are shown in Fig. 5.3. The SOMO is the b 1 orbital of the H-bond donor molecule, and is called b 1 /0. It is antisymmetric with respect to the plane of symmetry. The delocalization of this orbital requires mixing with an antisymmetric orbital of the H-bond acceptor molecule. The only such orbital is 0/b 2 , which is high in energy, and, therefore, the ground state hole is localized. The two lower orbitals are linear combinations of the acceptor b 1 and the donor a 1 MOs. The higher energy combination is antibonding with respect to the monomers and is called (a 1 /b 1 ) ∗ . The bonding combination is (a 1 /b 1 ). Lower in energy we find the (b 2 /a 1 )* and (b 2 /a 1 ) pair. The antibonding DMO is located mostly on the acceptor, while the donor hosts the bonding component. Finally, the acceptor 0/b 2 is the lowest DMO considered here. States corresponding to ionizing the H-bond donor are lower in energy (the cor- responding FMOs are higher in energy), which can be easily rationalized in terms of electrostatic interaction. A hole on the H-bond donor fragment is stabilized by the neg- atively charged oxygen of the acceptor, while a hole on the acceptor is destabilized by the positively charged proton. 147 Orbitals at the C s and C 1 proton-transferred configurations are shown in Fig. 5.4. The proton transfer drastically changes the character of the MOs and it is no longer pos- sible to diabatically correlate them with those of the neutral. The MOs clearly separate into the H 3 O + and OH fragments. Orbitals of the hydroxyl part are higher in energy than those of H 3 O + , as follows from the difference of IEs. Under C s symmetry, the HOMO is the out-of-plane component of the degenerate π -pair on the OH − fragment, π oop /0. Below it are theπ inp /0 andσ /0 DMOs. The degenerateπ system follows the out- of-plane rotation of the hydroxyl group. H 3 O + orbitals appear in the following order: 0/a 1 , 0/e 0 , and 0/e 00 . The OH σ and H 3 O + e 0 orbitals are slightly mixed. Overall, only minor differences are observed in theC s andC 1 structure orbitals. The NBO charges reveal stronger delocalization than suggested by visual inspection of the MOs. The delocalization is due the wave function having significant amplitude between the two fragments at close distances. Additionally, for the excitations from H 3 O + to OH one might expect to see -1 charge on the hydroxyl fragment. However, the large positive charge on H 3 O 2+ will polarize the OH − and thus decrease its charge. This is confirmed by the calculation, revealing -0.771 charge on the OH moiety. Spectroscopy at the vertical configuration Table 5.5 and Fig. 5.5 present excitation energies and transition properties of (H 2 O) + 2 at the geometry of the neutral species. All excitations involve transfer of an electron to the SOMO, i.e., theb 1 /0 orbital. Overall, all theoretical methods are in good agreement in terms of energetics. A state involving excitation to a virtual orbital of the neutral appeared in the EOM-EE-CCSD calculation at 8 eV and was disregarded. At low energies, up to ca. 2 eV , we find excitations from the (a 1 /b 1 ) ∗ and (a 1 /b 1 ) pair into the SOMO. Their intensity can originate both from the intramolecular and intermolecular terms and reflects the partitioning of the DMOs into FMOs, and can 148 C s C 1 8.17 eV 0.145 22.48 eV -0.629 8.63 eV 0.096 12.65 eV 0.082 16.56 eV -0.610 21.20 eV -0.564 S oop /0 S inp /0 V /0 0/a 1 0/e ’ 0/e’’ 8.15 eV 0.145 8.64 eV 0.098 12.67 eV 0.084 16.50 eV -0.614 21.20 eV -0.564 22.49 eV -0.629 Figure 5.4: Six highest occupied MOs of the neutral water dimer at the C s and C 1 proton-transferred configurations. Ionization energies were calculated using EOM-IP- CCSD/6-311++G**. The NBO charge on the hydroxyl radical calculated using EOM- IP-CCSD wave functions is given below. The geometric frame is rotated for each orbital to best show orbital character. The NBO analysis of the reference CCSD/6-311++G** wave function showed -0.771 and -0.770 charge on the hydroxyl radical in the C s and C 1 structures, respectively. 149 Table 5.5: Excitation energies (eV) and transition properties (a.u.) of the (H 2 O) + 2 cation at the geometry of the neutral. All transitions are to the 2a 00 (b 1 /0) orbital. A 6-311++G** basis set was used throughout. EOM-IP-CCSD EOM-EE-CCSD EOM-IP-CC(2,3) E ex μ 2 f E ex μ 2 f E ex 8a 0 (a 1 /b 1 ) ∗ 1.50 0.00379 1.39 10 − 4 1.81 0.00995 4.41 10 − 4 1.48 7a 0 a 1 /b 1 2.25 0.0181 9.98 10 − 4 2.37 0.0122 7.06 10 − 4 2.23 6a 0 (b 2 /a 1 ) ∗ 3.99 6.95 10 − 4 6.80 10 − 5 4.34 1.69 10 − 4 1.80 10 − 5 3.97 5a 0 b 2 /a 1 6.57 7.29 10 − 6 1.12 10 − 6 6.56 1.65 10 − 5 2.65 10 − 6 6.52 1a 00 0/b 2 8.07 0.00362 7.16 10 − 4 8.46 0.00183 3.80 10 − 4 8.04 150 (a) 0 2 4 6 8 0.000 0.005 0.010 0.015 0.020 (b) 0 2 4 6 8 0.000 0.004 0.008 0.012 Figure 5.5: Electronic states ordering and transition dipole moments of water dimer cation at the neutral configuration calculated using (a) EOM-IP-CCSD/6-311++G** and (b) EOM-EE-CCSD/6-311++G** . All transitions are to theb 1 /0 orbital. . 151 be explained by DMO-LCFMO. Referring to formalism summarized in Section 2.3.4, (a 1 /b 1 ) ∗ and (a 1 /b 1 ) are examples of DMOs in which both theα andβ coefficients (i.e., the weights of the FMOs) are significant. The weights of the FMOs are just the square roots of the EOM-IP-CCSD NBO charges, since both fragments are neutral in the refer- ence state: (a 1 /b 1 ) ∗ =0.355(a 1 /0)− 0.935(0/b 1 ) (5.3) (a 1 /b 1 )=0.886(a 1 /0)+0.464(0/b 1 ) (5.4) Neglecting the intermolecular contribution to the intensity, one can evaluate the transi- tion dipole moments for the two transitions according to Eq. (2.48). The corresponding squares are 0.0029 and 0.0180 a.u. 2 . This estimate is in an excellent agreement with the actual EOM-IP-CCSD dimer calculation, which yields 0.0038 and 0.0181 a.u. 2 , thus supporting the assumption that these two transitions draw their intensity from the H- bond donora 1 component of the DMO and that the intermolecular terms are negligible. The EOM-IP-CCSD and EOM-EE-CCSD methods predict different relative intensity of the two bands. While EOM-EE-CCSD predicts nearly equal intensities, EOM-IP-CCSD suggests 1:5 intensity ratio in favor of the higher energy band. The origins of this dis- crepancy can be traced back to different partitioning of the a 1 component between the (a 1 /b 1 ) ∗ and (a 1 /b 1 ) pair. Our benchmark study 66 demonstrated that EOM-IP-CCSD pro- vides a more accurate description of charge delocalization and more accurate transition properties. EOM-IP-CCSD predicts weak excitations from the (b 2 /a 1 ) ∗ and (b 2 /a 1 ) pair to be at 4 and 6.5 eV , respectively. EOM-EE-CCSD predicts the transition from the antibonding DMO to lie 0.3 eV higher and agrees as to the position of transition from the bonding DMO. The antibonding orbital is confined to the H-bond acceptor, while the bonding 152 one to the donor. Conceptually, the character of the transitions is similar to that discussed above. These two excitations correspond to the symmetry forbiddenb 1 ← a 1 excitation in the monomer. Consequently, the intramolecular term in Eq. (2.48) is zero. Orbital relaxation and intermolecular terms account for the intensity of these bands, which are still a hundred times weaker than the allowed transitions. Finally, around 8 eV we find the (b 1 /0)← (0/b 2 ) transition. It is a pure charge- transfer band (α =0), where the hole is moved from the donor to the acceptor molecule. Its considerable intensity is due to particularly favorable overlap between the initial and final orbitals. The transition is still an order of magnitude weaker than the sum of the (b 1 /0)← (a 1 /b 1 ) and (b 1 /0)← (a 1 /b 1 ) ∗ excitations. Spectroscopic signatures of proton transfer Excitation energies and transition properties at theC 1 andC s proton-transferred geome- tries are presented in Tables 5.6 and 5.7, respectively. The EOM-IP-CCSD data are shown in Fig. 5.6. We were able to obtain EOM-EE-CCSD results only for the three lowest excitations, as the states involving virtual orbitals of the neutral wave function are the next higher in energy. The lowest energy transition is at 9.6 eV , well beyond the energy range potentially probed in a pump-probe experiment. Additionally, the Rydberg character of these states means that they are likely to be significantly perturbed in the condensed phase. All the excitations considered here involve the transfer of an electron to the singly occupied (π oop /0) orbital. Due to the localized character of the DMOs, the spectrum partitions into two parts: transitions within the OH fragment and transitions from the H 3 O + to the OH fragment. In the language of Section 2.3.4, the lower-energy part of the spectrum is close to the β =0 limit, while the higher energy one approaches theα =0 limit, i.e., local and charge transfer excitations, respectively. 153 Table 5.6: Excitation energies (eV) and transition properties (a.u.) of the (H 2 O) + 2 cation at theC 1 proton-transferred geometry. All transitions are to the 10a 1 (π oop /0) orbital. A 6-311++G** basis set was used throughout. EOM-IP-CCSD EOM-EE-CCSD E ex μ 2 f E ex μ 2 f 9a 1 π inp /0 0.49 0.00190 2.28 10 − 5 0.49 0.00220 2.66 10 − 5 8a 1 σ /0 4.52 0.0136 0.00151 4.53 0.0129 0.00143 7a 1 0/a 1 8.35 0.0141 0.00289 8.87 0.00957 0.00208 6a 1 0/e 0 13.05 0.00194 6.21 10 − 4 - - - 5a 1 0/e 00 14.34 0.00224 7.86 10 − 4 - - - 154 Table 5.7: Excitation energies (eV) and transition properties (a.u.) of the (H 2 O) + 2 cation at theC s proton-transferred geometry. All transitions are to the 10a 1 (π oop /0) orbital. A 6-311++G** basis set was used throughout. EOM-IP-CCSD EOM-EE-CCSD E ex μ 2 f E ex μ 2 f 8a 0 π oop /0 0.46 0.00176 2.01 10 − 5 0.50 0.00217 2.50 10 − 5 7a 0 σ /0 4.48 0.0138 0.00151 4.49 0.0132 0.00145 6a 0 0/a 1 8.39 3.11 10 − 5 6.40 10 − 6 8.95 1.13 10 − 5 2.47 10 − 6 5a 0 0/e 0 13.03 7.07 10 − 4 2.26 10 − 4 - - - 4a 00 0/e 00 14.31 0.00593 0.00208 - - - 155 (a) 0 4 8 12 0.000 0.004 0.008 0.012 0.016 (b) 0 4 8 12 0.000 0.004 0.008 0.012 0.016 Figure 5.6: Electronic states ordering and transition dipole moments of water dimer cation at theC 1 (a) andC s symmetry proton-transferred configurations calculated using EOM-IP-CCSD/6-311++G**. All transitions are to theπ oop /0 orbital. . 156 The C s and C 1 configurations differ only slightly in positions and intensities of the transitions in the low-energy part of the spectrum. The transitions in this region are from the (π inp /0) and (σ /0) orbitals. The former excitation is not present in an isolated OH monomer, as the (π inp /0) - (π oop /0) pair is degenerate. In the dimer, it splits by 0.5 eV and acquires oscillator strength. The (π oop /0)← (σ /0) excitation is at 4.5 eV , very close to the monomer value of 4.2 eV , however, its intensity is less than half. The origin of the intensity decrease can be investigated using DMO-LCFMO, although its quantitative application is complicated by the large extent of charge transfer occurring in the neutral and in the cation states. We discuss the calculation for the C s structure, but the result for the C 1 geometry is essentially identical. To obtain the singly occupied orbital, one needs to consider the difference between the neutral and the cation states (the singly occupied orbital is the same as the orbital of the outgoing electron). The NBO analysis of the CCSD wave function of the OH − – H 3 O + system revealed a -0.771 charge on the hydroxyl moiety. In other words, 0.916 and 0.853 electron comes from this moiety when an electron is removed from the (π inp /0) and (σ /0) orbitals, respectively. Using these numbers to calculate the weights of the appropriate diabatic states and, subsequently, the transition dipole moment, one obtains 0.024 a.u. 2 , almost twice the ab initio calculated value. The source of the discrepancy appears to be the coupling with 0/e 0 and the arising intermolecular contribution to the intensity. Higher in energy there are excitations involving the transfer of an electron from H 3 O + to OH. The first one, at∼ 8 eV , is the transition from the lone pair (0/a 1 ) orbital. It is very weak in the C s structure, however, upon rotation of the hydroxyl group it acquires intensity comparable to the (π oop /0)← (σ /0) transition, presumably due to the favorable overlap of the two DMOs, which increases upon rotation. In theC s structure, the two orbitals resemble two orthogonal p orbitals. Hence, the positive and negative contributions to the transition dipole moment cancel out. In C 1 , they are akin to p z 157 orbitals form the π system in ethylene and all the contributions have the same sign. Above 13 eV we find excitations from (0/e 0 ) and (0/e 00 ). Proton transfer PES scan Fig. 5.7a presents the PES scan of the proton transfer reaction in the C s geometry. The coordinates are the oxygen-oxygen distance and the distance between the transfer- ring proton and the accepting oxygen. Points A and C correspond to the vertical and proton-transferred geometries, respectively. They were identified based solely on the two geometric parameters scanned. The energy of point A is 3.5 kcal/mol lower than the energy at the neutral geometry due to relaxation of unconstrained coordinates. This lowers the energy change due to the reaction from 20 kcal/mol to 16.5 kcal/mol. A two- step picture of the proton transfer reaction emerges from the graph. Between points A and B, the H-bond donor molecule moves towards the acceptor, as the oxygen-oxygen and hydrogen-oxygen distances change in unison. This motion lowers the energy by 11.4 kcal/mol. Subsequent dynamics is restricted to the proton, which transfers from the donor to the acceptor, accompanied by minor adjustments in the oxygen-oxygen distance. An energy change of 5.0 kcal/mol is associated with this motion. The PES allows us to compute the spectral changes along the steepest descent reac- tion path. Calculated excitation energies and transition properties are presented in Fig. 5.7b-d. We employ the C s symmetry labels to identify the states in this discussion, as the character of the orbitals changes along the reaction coordinate. All the transitions involve the transfer of an electron to SOMO, the2a 00 orbital, which is the out-of-plane orbital on the H-bond donor. Its character evolves from the H 2 O + lone pair b 1 orbital to the π orbital of OH. Of particular interest are the 6a 0 , 7a 0 , and 8a 0 transitions, as they are within the spectroscopic 1 - 6 eV window. At point A, two absorption bands 158 (a) (b) 2.4 2.6 2.8 3.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 C B r O-O, Å r O-H, Å 0 4 8 12 16 20 24 A kcal/mol 0 5 10 15 0 2 4 6 8 10 12 14 C B A 1a'' 5a' 6a' 7a' excitation energy, eV enery released, kcal/mol 8a' (c) (d) 0 5 10 15 0.000 0.005 0.010 0.015 C B A 5a' 7a' 6a' 1a'' 8a' μ 2 , a.u. 2 energy released, kcal/mol 0 5 10 15 0.0 0.4 0.8 1.2 C A B 5a' 6a' 1a'' x 0.5 7a' 8a' oscillator strength × 10 3 energy released, kcal/mol Figure 5.7: (a) The ground state PES scan for the proton transfer reaction. The x-axis is the oxygen-oxygen distance and the y-axis is the distance between the transferring proton and the accepting oxygen.At each point a constrained geometry optimization was conducted Points A and C correspond to the neutral and proton transferred geome- tries, respectively.Point B marks the start of the proton transfer. The black line is the steepest descent path. (b) -(d) Vertical excitation energies, transition dipole moments and oscillator strengths along the reaction coordinate. All calculations were done using EOM-IP-CCSD/6-311++G**. appear around 2 eV (8a 0 and7a 0 ). They are transitions from the bonding and antibond- ing combinations of the donora 1 and acceptorb 1 fragment MOs [(a 1 /b 1 ) ∗ and (a 1 /b 1 )]. The lower energy transition carries more intensity, which is different from the fully opti- mized neutral configuration, where the higher energy transition carries more intensity. 159 This is due to slightly different geometry and orbital mixing of H-bond acceptorb 1 and donor a 1 in the two states. 6a 0 and 5a 0 are bonding and antibonding combinations of donor b 2 and acceptor a 1 . They are at 6.2 and 4.3 eV , respectively. The intensities of both bands are small. Finally, at 8.5 eV we find the transition from the1a 00 (acceptorb 2 ) orbital. As the reaction proceeds to point B, the intensity of8a 0 and7a 0 changes as the bands move apart. In other words, the partitioning of H-bond donora 1 and acceptorb 1 in the (a 1 /b 1 ) ∗ and (a 1 /b 1 ) pair changes. The small magnitude of change is easily understood within the diabatic framework of (a 1 /0) and (0/b 1 ) states, i.e., the states with the charge localized on the H-bond donor and acceptor, respectively. The coupling and separation of the diabatic states increase at shorter distances. The two effects largely cancel out and no drastic changes in intensity are observed. There is a small increase in the intensity of the6a 0 (H-bond acceptora 1 ) band. At point B, the proton transfer step starts taking place and the character of MOs changes more dramatically. The SOMO becomes the out-of-planeπ orbital. The inten- sity of the8a 0 band drops significantly with a decrease in energy, as it becomes aπ − π transition on the OH fragment. The7a 0 excitation gains intensity as energy rises to 4.5 eV , and becomes the σ − π excitation. At the same time, the 6a 0 and 5a 0 excitations move to higher energy, becoming CT excitations from H 3 O + to OH. The already small intensity of excitation from6a 0 drops further as it becomes the apical orbital of H 3 O + . Both1a 00 and5a 0 transitions move to 12 – 14 eV , outside the experimental region. 5.5 Discussion Three geometries on the ground state PES of the water dimer cation are of prime impor- tance in the photoionization process. The first two are minima — the proton-transferred 160 and the hemibonded structures (Fig. 5.2b-c). The third one is the geometry of the neutral dimer reached by vertical ionization (Fig. 5.2a). The geometries and relative energet- ics of these structures calculated by correlated electronic structure methods are in good agreement with each other (Table 5.3) and previous ab initio results 24, 25, 28 . All these calculation correctly characterize the proton-transferred geometry as the global mini- mum lying 5-10 kcal/mol below the hemibonded local minimum and about 20 kcal/mol below the structure corresponding to vertical ionization of the neutral water dimer. This ordering is reproduced already at the HF level, which, however, grossly (by about 20 kcal/mol) destabilizes the hemibonded structure due to the tendency of HF to artificially localize the MOs in systems with symmetrically equivalent centers. In contrast, the proton-transferred structure has spin localized on one fragment and charge localized on the other and in the vertical geometry, corresponding to the neutral water dimer, the spin and charge are initially localized on one water. The behavior of the DFT methods is exactly opposite. Due to SIE, DFT/BLYP overstabilizes structures with delocalized charge, and erroneously predict the hemibonded structure to be the global minimum. However, a simple empirical a posteriori SIC 14, 48 , almost completely removes this arti- fact and the predictions of the SIC-corrected DFT methods are very close to those of MP2. This is a good news for the DFT-based AIMD studies of ionization in liquid water 12 . Nonetheless, further benchmark studies including the spin localization and cluster dynamics are needed. For ionization in the condensed phase, the issue of electronic localiza- tion/delocalization is of interest. We are interested in the question whether the charge is localized at one site immediately upon ionization or whether it will localize after being initially delocalized over many water molecules. This localization process can only be explored by considering clusters beyond the dimer. In the dimer, the hole forms on 161 the b 1 orbital of the H-donor fragment immediately upon ionization. This initial local- ization is due to the fact that the donor water is not acting as an acceptor to any other H-bond. Theb 1 orbital of the H-bond acceptor in fact couples with thea 1 orbital of the donor. Neutral water dimer thus represents the most asymmetric arrangement of water molecules. Already in the cyclic trimer the water molecules become equivalent, which means that upon ionization the hole must be initially delocalized. In the bulk phase, each water molecule is likely to serve simultaneously as a hydrogen bond acceptor and donor, thus more likely delocalizing the hole. However, an the initial period of delocalization, whose period is not yet known, the positive charge will localize thereby starting the proton transfer reaction. From this point onward the dimer is presumably an adequate model for the spectroscopy of the condensed phase proton transfer process. In general, small and medium-sized water clusters with different geometries will provide a natural laboratory to investigate the electronic and nuclear dynamics upon ionization with varying degree of initial localization/delocalization of the hole 67 . Charge localization is intimately related to the electronic spectroscopy of the sys- tem. Using formalism of Section 2.3.4, theα andβ coefficients may both be significant (delocalized charge), or one can dominate (localized). Thus, excitations will have a mixed charge transfer and local character. With no knowledge of the degree of charge delocalization, it is impossible to say anything about the intensity of mixed bands and reliable ab initio calculations are needed. In the case of dimer cation states, one may expect intermolecular contributions to be less significant than in their neutral counter- parts due to the more compact nature of MOs. This maybe counteracted by the decreased separation manifested in a charged species. Our calculations for water dimer cation reveal that the intermolecular terms are typically an order of magnitude smaller than the intramolecular terms for allowed transitions. Note, however, that the interfragment contribution varies exponentially with the distance (separately from α and β ) and may 162 change significantly with relative orientation of the two fragments, thus allowing one to monitor the molecular dynamics via intensity and/or position of those bands. Next, we discuss the nuclear motions along the reaction coordinate leading from the geometry corresponding to the neutral water dimer to the proton-transfer structure (Fig. 5.7a). The reaction, which is a downhill process without a barrier, proceeds in two steps. The first one involves heavy atom motions, i.e., the two water molecules move closer to each other with the oxygen-oxygen distance decreasing from 2.9 to 2.5 A. This step is responsible for the largest part of the energy gain (∼ 12 kcal/mol) along the reaction coordinate. Since it involves motion of heavy atoms, it is relatively slow compared to the second step, the rapid transfer of the proton, which can only happen when the water oxygens are sufficiently close to each other. This process involves a motion of a light particle and is, therefore, possibly as fast as a few femtoseconds. Precise time scales are under a detailed experimental and theoretical investigation in our labs. The energy gain associated with the proton hop is smaller, amounting to roughly 5 kcal/mol. In larger cluster and in bulk liquid water, this two-step mechanism should be preserved, although it might be preceded by initial fast charge localization 12 . This is exactly the dynamics that we wish to resolve in the condensed phase, with a spectroscopic handle on these events being provided by the changing excitation spectrum of the radical species along the proton transfer reaction. Let us consider how the electronic transitions, their band positions and their inten- sities, evolve as we move along the reaction coordinate. Femtosecond spectroscopy should be able to monitor the system evolution by recording the changing transient absorption spectrum. Formally, the H 2 O + cation is derived from the OH radical by the addition of a proton. One might thus expect the electronic structure and the spec- troscopy of the two species to be similar. However, the results from Table 5.2 show that the addition of a proton is not a benign perturbation to the electronic structure. The OH 163 radical has a characteristic absorption band around 4.2 eV corresponding to the transi- tion of the bondingσ electron into the non-bondingπ hole. Even if the proton is brought up along the O-H axis leading to a linear H-O-H + , the σ to π promotion is pushed up to higher energy (∼ 6 eV). More significantly, allowing the structure to adopt the lower energy bent configuration splits theπ orbital intob 1 anda 1 symmetry components. The former π ← σ transition becomes dipole forbidden for the ground state component (b 1 ← b 2 ). In C 2v , the transition between the two formerly degenerate b 1 and a 1 orbitals is dipole allowed with a transition energy of ca. 2.3 eV and one third of the oscillator strength of the OH transition. Simply put, the 4 eV band disappears and a weaker 2.3 eV band appears in its place. At the vertical geometry, the transitions for the ionized water dimer are perturbed and include some charge-transfer character, however, the wave functions can still be correlated with those of the monomers. Extracting the three points A, B and C from Fig. 5.7, we have replotted how the experimental spectrum can track the chemical reac- tion dynamics in Fig. 5.8. At A, the bands at 2 eV (b 1 /0)← (a 1 /b 1 ) ∗ and (b 1 /0)← (a 1 /b 1 ) [2a 00 ← 8a 0 and 2a 00 ← 7a 0 ] have almost the same oscillator strength as the monomer. The distribution of intensity between the two heavily depends on the system geometry. The 6 eV monomer-like transition is now not strictly symmetry forbidden, particularly the 4 eV (b 2 /0)← (b 2 /a 1 ) ∗ [2a 00 ← 6a 0 ] component with more CT character. It still is almost two orders of magnitude weaker than the 2 eV transition. Therefore, the dominant characteristic electronic absorption of the dimer cation at the Franck-Condon geometry is around 2 eV (620 nm) as for the gas phase monomer. Then at B, the (b 2 /0) ← (a 1 /b 1 ) ∗ and (b 2 /0)← (a 1 /b 1 ) [2a 00 ← 8a 0 and 2a 00 ← 7a 0 ] shift apart, the lower energy band carrying more intensity. The 4 eV transition shifts slightly to the blue. At this point proton transfer begins. The lower energy band becomes a weak π − π exci- tation, while the 4 eV band gains intensity to become theσ − π excitation. The shift is 164 a clear fingerprint of the reaction and the significant change corresponds to the charge transfer between the species, i.e., from B to C. Overall, the band positions resemble the monomers, however, the intensity pattern and fine structure is a strong function of the relative geometries in the cluster. For example, the comparison of the C s and C 1 geometries of the proton-transferred complexes show a dramatic variation in the 0/a 1 [6a 0 ] band because of the alignment of the p orbitals on the two fragments. Although C s is a saddle point between the two equivalent C 1 configurations of the product, it is only 0.1 kcal/mol above the minima and it allows us to symmetry label the spectroscopic state of the evolving system. However, the transition toC 1 does lead to this large inten- sity change and one should be wary of the role of conformational changes in the band intensities. We are now ready to discuss the effects of bulk water. Even if the dimer core is a good representation of the vertically prepared hole in water, there is a large range of local neutral donor-acceptor geometries populated in room temperature water. Although the configuration considered is the lowest energy cluster, other configurations, particularly with different orientations of the free hydrogen of the donor with respect to the acceptor σ v plane, should also be considered. Preliminary calculations have shown that, if the O- H group of the H-bond donor is aligned with one of the acceptor O-H bonds (the donor molecule is rotated by∼ 90 ◦ ), the transitions with significant charge transfer character can be significantly enhanced 67 . In particular, the band around 4 eV which involves CT (b 2 /0)→ (b 2 /a 1 ) ∗ [2a 00 ← 7a 0 ] acquires oscillator strength and can become comparable to the valence band near 2.3 eV . What if more solvating waters are included around the ionized core water? Preliminary EOM-IP-CCSD computations on a vertically ionized pentamer extracted from ice Ih show that excitations on a central water give rise to a similar spectrum to that of the monomer with an intense band near 2 eV and little oscillator strength at 4 eV . These results will be further quantified elsewhere 67 . 165 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.000 0.005 0.010 0.015 0.020 absorption, au energy, eV Figure 5.8: Evolution of the electronic spectrum of the water dimer cation along the reaction path at points A (red solid line), B (green dotted line), and C (blue dashed line). Points A and C correspond to vertical neutral and proton-transferred geometries. Point B marks the start of the proton transfer. 0.2 eV full width at half-maximum was assumed. See Fig. 5.7 for details. 5.6 Conclusions The water dimer cation is a prototypical system for the proton transfer process in the gas and condensed phases. The vertical structure formed immediately upon ioniza- tion is not a stationary point on the cation PES and the system follows the downhill gradient to OH··· H 3 O + . Our study demonstrates that this process can be monitored by femtosecond time-resolved electronic spectroscopy. At the simplest level, the ini- tial spectrum resembles that of H 2 O + . As the reaction proceeds, band positions and intensities change. The product of the reaction spectroscopically resembles the free OH radical. A more detailed look at how the electronic spectrum evolves along the proton 166 transfer coordinate shows things are more subtle. We observed strong coupling between the H-bond donor and acceptor orbitals, which dissolves the monomer states into more delocalized dimer states. This coupling is likely to be present in the condensed phase as well, and will lead to significantly delocalized states. Modeling of such states requires a full quantum treatment of the entire system. Hybrid quantum mechanical/molecular mechanical methods are not appropriate for this situation as the system cannot be par- titioned into a solvent and a chromophore, as, for example, in our study of electronic spectroscopy of the solvated CN radical 68 . AIMD, which is able to directly describe the electronic structure of the entire system, requires a fast electronic structure method. We have found that the energetics and structures of (H 2 O) + 2 are reasonably reproduced by the ROKS-BLYP method with the simple SIC correction 14, 48 , which thus can be employed in AIMD simulations of the bulk. 5.7 Postscript 5.7.1 Introduction In this Postscript we present preliminary results on the effect of dimer conformation and including additional water molecules on the character of the electronic wave function of water cation. In the first study we further investigated the water dimer. Specifically, the H-bond donor molecule was rotated along the oxygen-oxygen axis by 180 ◦ in 10 ◦ intervals. Then, by considering the water pentamer we assessed the extent to which a hole can be assigned to the central water molecule. 167 5.7.2 Dimer rotation Fig. 5.9 presents the natural charge on the H-bond donor fragment in the water dimer cation. Rotation lowers the symmetry of the system from C s to C 1 . Under the latter symmetry mixing of all states and significant changes in their character relative to the neutral configuration are formally possible. This turns out not to be the case. The lowest ionization, from the b 1 /0 orbital is limited to the H-bond donor molecule. Significant changes in mixing are observed for the (a 1 /b 1 )) and ((a 1 /b 1 ) ∗ ) pair of orbitals. The degree of mixing decreases with rotation. 0 60 120 180 0.00 0.25 0.50 0.75 1.00 Figure 5.9: Natural charge on the H-bond donor fragment in water dimer cation in six lowest electronic. Rotation is around the oxygen — oxygen axis, and the angle 0 corresponds to the neutral dimer configuration. All calculations were done using EOM- IP-CCSD/6-311++G**.. 168 Fig. 5.10 shows the excitation energy and transition dipole moment along the rota- tion angle for the five lowest transitions of the water dimer cation. The effect on excita- tion energies is small, but pronounced changes in the intensity are observed. In particu- lar the(b 1 /0)← (a 1 /b 1 ) and(b 1 /0)← (a 1 /b 1 ) ∗ transitions have much higher intensity at 90 ◦ than at 0 ◦ . This affect cannot be explained by a change in the nature of states, as shown by the analysis of the natural charge. This is further supported by a visual inspection of MOs. This signifies an increase in the intermolecular contribution to the intensity. At the neutral configuration theb 1 orbitals on the H-bond donor and acceptor fragments are perpendicular. Consequently the transition dipole moment will be small. Rotation of the donor fragment by 90 ◦ puts them parallel to each other. Large dipole moment coupling can be expected due to the close proximity of the orbitals. This study suggests that dimer and small cluster calculations may be useful in pre- dicting the excitation energies of ionized bulk water. Calculation of the associated inten- sities requires proper averaging over the thermally sampled configurations. 5.7.3 Pentamer The geometry of the water 17-mer was obtained from Ref. 69 . This is the smallest sta- ble water cluster, for which an interior water molecule is observed. The structure was reoptimized using MP2/6-311++G**. Subsequently an interior pentamer, shown in Fig. 5.11, was carved out and not reoptimized. For the 15 lowest states of the pentamer cation at the vertical geometry just defined the NBO analysis was conducted on the EOM-IP-CCSD/6-31+G* wave function and unrestricted density matrices. Based on this the charge on the central molecule was calculated. The central water molecule is a model for a water molecule in the condensed phase. Results are shown in Fig. 5.12. The analysis reveals that significant delocalization occurs in all states and ionization process involves removal of an electron from several molecules. The two lowest electronic states 169 (a) 0 60 120 180 2.0 4.0 6.0 8.0 (b) 0 60 120 180 0.000 0.025 0.050 0.075 0.100 Figure 5.10: (a) Excitation energies and (b) transition dipole moments in water dimer cation. Rotation is around the oxygen — oxygen axis, and the angle 0 corresponds to the neutral dimer configuration. All calculations were done using EOM-IP-CCSD/6- 311++G**. of the pentamer cation correspond to ionization of surface molecules. Not a single state can be completely assigned to the central fragment, but crude correlations can be estab- lished. States corresponding to the ionization from b 1 and b 2 orbitals localized on the 170 central water can be discerned, but not froma 1 70 . This is in agreement with the photo- electron and X-ray emission spectra of the liquid phase, where the ionization/emission from thea 1 orbital is broadened compared other orbitals 70, 71 . This effect is attributed to the strong participation in the hydrogen bonding and the concomitant delocalization of a 1 . Despite a large changes in character of the states compared to the dimer, the calcu- lated electronic absorption spectra of both species are very similar. Within the pentamer model, a spectrum corresponding to the spectrum of the water cation in the liquid phase can be calculated, by assuming that the third excited state of the cation is its ground state. Figure 5.11: MP2/6-311++G** optimized geometry of the water pentamer. 171 0.0 2.0 4.0 6.0 8.0 0.00 0.25 0.50 0.75 1.00 Figure 5.12: NBO charge on the central fragment in a water pentamer cation as a function of the excitation energy. All calculations were done using EOM-IP-CCSD/6- 31+G*. 172 5.8 Chapter 5 Bibliography [1] Sancar, A. Biochemistry 1994, 33, 2. [2] Bixon, M. ; Jortner, J. J. Phys. Chem. A 2001, 105, 10322. [3] Garrett, B.C. ; Dixon, D.A. ; Camaioni, D.M. ; Chipman, D.M. ; Johnson, M.A. ; Jonah, C.D. ; Kimmel, G.A. ; Miller, J.H. ; Rescigno, T.N. ; Rossky, P.J. ; Xanth- eas, S.S. ; Colson, S.D. ; Laufer, A.H. ; Ray, D. ; Barbara, P.F. ; Bartels, D.M. ; Becker, K.H. ; Bowen, H. ; Bradforth, S.E. ; Carmichael, I. ; Coe, J.V . ; Cor- rales, L.R. ; Cowin, J.P. ; Dupuis, M. ; Eisenthal, K.B. ; Franz, J.A. ; Gutowski, M.S. ; Jordan, K.D. ; Kay, B.D. ; JA, J.A. LaVerne ; Lymar, S.V . ; Madey, T.E. ; McCurdy, C.W. ; Meisel, D. ; Mukamel, S. ; Nilsson, A.R. ; Orlando, T.M. ; Petrik, N.G. ; Pimblott, S.M. ; Rustad, J.R. ; Schenter, G.K. ; Singer, S.J. ; Tokmakoff, A. ; Wang, L.S. ; Wittig, C. ; Zwier, T.S. Chem. Rev. 2005, 105, 355. [4] Elles, C.G. ; Jailaubekov, A.E. ; Crowell, R.A. ; Bradforth, S.E. J. Chem. Phys. 2006, 125, 44515. [5] Elles, C.G. ; Shkrob, I.A. ; Crowell, R.A. ; Bradforth, S.E. J. Chem. Phys. 2007, 126, 64503. [6] Schnitker, J¨ urgen ; Rossky, P.J. J. Chem. Phys. 1987, 86, 3471. [7] Boero, M. ; Parrinello, M. ; Terakura, K. ; Ikeshoji, T. ; Liew, C.C. Phys. Rev. Lett. 2003, 90, 226403. [8] Tachikawa, H. J. Phys. Chem. A 2004, 108, 7853. [9] Tachikawa, H. J. Phys. Chem. A 2002, 106, 6915. [10] Furuhama, A. ; Dupuis, M. ; Hirao, K. J. Chem. Phys. 2006, 124, 164310. [11] Novakovskaya, Y .V . Int. J. Quant. Chem. 2007, 107, 2763. [12] VandeV ondele, J. ; Pieniazek, P.A. ; Krylov, A.I. ; Bradforth, S.E. ; Jungwirth, P. to be published. [13] Nielsen, S. O. ; Michael, B.D. ; Hart, E.J. J. Phys. Chem. 1976, 80, 2482. [14] VandeV ondele, J. ; Sprik, M. Phys. Chem. Chem. Phys. 2005, 7, 1363. [15] Pieniazek, P.A. ; Krylov, A.I. ; Bradforth, S.E. J. Chem. Phys. 2007, 127, 044317. [16] Ng, C.Y . ; Trevor, D.J. ; Tiedemann, P.W. ; Ceyer, S.T. ; Kronebusch, P.L. ; Mahan, B.H. ; Lee, Y .T. J. Chem. Phys. 1977, 67, 4235. 173 [17] de Visser, S. P. ; de Koning, L.J. ; Nibbering, N.M.M. J. Phys. Chem. 1995, 99, 15444. [18] Tomoda, S. ; Achiba, Y . ; Kimura, K. Chem. Phys. Lett. 1982, 87, 197. [19] Moncrieff, D. ; Hillier, I.H. ; Saunders, V .R. Chem. Phys. Lett. 1982, 89, 447. [20] Sato, K. ; Tomoda, S. ; Kimura, K. Chem. Phys. Lett. 1983, 95, 579. [21] Curtiss, L.A. Chem. Phys. Lett. 1983, 96, 442. [22] Tomoda, S. ; Kimura, K. Chem. Phys. 1983, 82, 215. [23] Curtiss, L.A. Chem. Phys. Lett. 1984, 112, 409. [24] Gill, P.M.W. ; Radom, L. J. Am. Chem. Soc. 1988, 110, 4931. [25] Sodupe, M. ; Oliva, A. ; Bertran, J. J. Am. Chem. Soc. 1994, 116, 8249. [26] Barnett, R.N. ; Landman, U. J. Phys. Chem. 1995, 99, 17305. [27] Barnett, R.N. ; Landman, U. J. Phys. Chem. A 1997, 101, 164. [28] Sodupe, M. ; Bertran, J. ; Rodrguez-Santiago, L. ; Baerends, E.J. J. Phys. Chem. A 1999, 103, 166. [29] M¨ uller, I.B. ; Cederbaum, L.S. J. Chem. Phys. 2006, 125, 204305. [30] L¨ owdin, P.O. Rev. Mod. Phys. 1963, 35, 496. [31] Davidson, E.R. ; Borden, W.T. J. Phys. Chem. 1983, 87, 4783. [32] Russ, N.J. ; Crawford, T.D. ; Tschumper, G.S. J. Chem. Phys. 2005, 120, 7298. [33] Sinha, D. ; Mukhopadhyay, D. ; Mukherjee, D. Chem. Phys. Lett. 1986, 129, 369. [34] Sinha, D. ; Mukhopadhya, D. ; Chaudhuri, R. ; Mukherjee, D. Chem. Phys. Lett. 1989, 154, 544. [35] Chaudhuri, R. ; Mukhopadhyay, D. ; Mukherjee, D. Chem. Phys. Lett. 1989, 162, 393. [36] Stanton, J.F. ; Gauss, J. J. Chem. Phys. 1999, 111, 8785. [37] Sekino, H. ; Bartlett, R.J. Int. J. Quant. Chem. Symp. 1984, 18, 255. [38] Koch, H. ; Jensen, H.J.Aa. ; Jørgensen, P. ; Helgaker, T. J. Chem. Phys. 1990, 93, 3345. 174 [39] Stanton, J.F. ; Bartlett, R.J. J. Chem. Phys. 1993, 98, 7029. [40] NBO 5.0. Glendening, E.D. ; Badenhoop, J.K. ; Reed, A.E. ; Carpenter, J.E. ; Bohmann, J.A. ; Morales, C.M. ; Weinhold, F. Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, 2001. [41] Shao, Y . ; Molnar, L.F. ; Jung, Y . ; Kussmann, J. ; Ochsenfeld, C. ; Brown, S. ; Gilbert, A.T.B. ; Slipchenko, L.V . ; Levchenko, S.V . ; O’Neil, D. P. ; Distasio Jr., R.A. ; Lochan, R.C. ; Wang, T. ; Beran, G.J.O. ; Besley, N.A. ; Herbert, J.M. ; Lin, C.Y . ; Van V oorhis, T. ; Chien, S.H. ; Sodt, A. ; Steele, R.P. ; Rassolov, V . A. ; Maslen, P. ; Korambath, P.P. ; Adamson, R.D. ; Austin, B. ; Baker, J. ; Bird, E.F.C. ; Daschel, H. ; Doerksen, R.J. ; Drew, A. ; Dunietz, B.D. ; Dutoi, A.D. ; Furlani, T.R. ; Gwaltney, S.R. ; Heyden, A. ; Hirata, S. ; Hsu, C.-P. ; Kedziora, G.S. ; Khalliulin, R.Z. ; Klunziger, P. ; Lee, A.M. ; Liang, W.Z. ; Lotan, I. ; Nair, N. ; Peters, B. ; Proynov, E.I. ; Pieniazek, P.A. ; Rhee, Y .M. ; Ritchie, J. ; Rosta, E. ; Sherrill, C.D. ; Simmonett, A.C. ; Subotnik, J.E. ; Woodcock III, H.L. ; Zhang, W. ; Bell, A.T. ; Chakraborty, A.K. ; Chipman, D.M. ; Keil, F.J. ; Warshel, A. ; Herhe, W.J. ; Schaefer III, H.F. ; Kong, J. ; Krylov, A.I. ; Gill, P.M.W. ; Head-Gordon, M. Phys. Chem. Chem. Phys. 2006, 8, 3172. [42] Schuchardt, K.L. ; Didier, B.T. ; Elsethagen, T. ; Sun, L. ; Gurumoorthi, V . ; Chase, J. ; Li, J. ; Windus, T.L. J. Chem. Inf. Model. 2007, 47, 1045. [43] VandeV ondele, J. ; Krack, M. ; Mohamed, F. ; Parrinello, M. ; Chassaing, T. ; J, J. Hutter Comp. Phys. Comm. 2005, 167, 103. [44] Goedecker, S. ; Teter, M. ; Hutter, J. Phys. Rev. B 1996, 54, 1703. [45] Zhang, Y . ; Yang, W. J. Chem. Phys. 1998, 109, 2604. [46] Polo, V . ; Kraka, E. ; Cremer, D. Molecular Physics 2002, 100, 1771 . [47] Lundber, M. ; Siegbahn, P.E.M. J. Chem. Phys. 2005, 122, 1. [48] d’Avezac, M. ; Calandra, M. ; Mauri, F. Phys. Rev. B 2005, 71, 205210. [49] Mantz, Y .A. ; Gervasio, F.L. ; Laino, T. ; Parrinello, M. J. Phys. Chem. A 2007, 111, 105. [50] Banna, M.S. ; McQuaide, B.H. ; Malutzki, R. ; Schmidt, V . J. Chem. Phys. 1986, 84. [51] Reutt, J.E. ; Wang, L.S. ; Lee, Y .T. ; Shirley, D.A. J. Chem. Phys. 1986, 85, 6928. [52] Hotop, H. ; Patterson, T.A. ; Lineberger, W.C. J. Chem. Phys. 1974, 60, 1806. 175 [53] Smith, J.R ; Kim, J.B. ; Lineberger, W.C. Phys. Rev. A 1997, 55, 2036. [54] Das, B. ; Farley, J.W. J. Chem. Phys. 1991, 95, 8809. [55] Constants of diatomic molecules (data prepared by J.W. Gallagher and R.D. John- son, III). Huber, K.P. ; Herzberg, G. NIST Chemistry WebBook, NIST Standard Reference Database Number 69; Eds. P.J. Linstrom and W.G. Mallard, July 2001, National Institute of Standards and Technology, Gaithersburg MD, 20899 (http://webbook.nist.gov). [56] Cheng, B.-M. ; Chew, E.P. ; Liu, C.-P. ; Bahou, M. ; Lee, Y .-P. ; Yung, Y .L. ; Gerstell, M.F. Geophys. Res. Lett. 1999, 26, 3657. [57] Bally, T. ; Sastry, G.N. J. Phys. Chem. A 1997, 101, 7923. [58] Vanovschi, V . ; Krylov, A.I. ; Wenthold, P.G. Theor. Chim. Acta 2008, 120, 45. [59] Vydrov, O.A. ; Scuseria, G.E. J. Chem. Phys. 2006, 124, 191101. [60] Pople, J.A. ; Gill, P.M.W. ; Handy, N.C. Int. J. Quant. Chem. 1995, 56, 303. [61] Tawada, Y . ; Tsuneda, T. ; Yanagisawa, S. ; T, T. Yanai ; Hirao, K.K. J. Chem. Phys. 2004, 120, 8425. [62] Livshits, E. ; Baer, R. Phys. Chem. Chem. Phys. 2007, 9, 2932. [63] Cohenand, A.J. ; Mori-S´ anchez, P. ; Yang, W. J. Chem. Phys. 2007, 126, 191109. [64] Becke, A.D. ; Johnson, E.R. J. Chem. Phys. 2007, 127, 124108. [65] Chai, J.-D. ; Head-Gordon, M. J. Chem. Phys. 2008, 128, 084106. [66] Pieniazek, P.A. ; Arnstein, S.A. ; Bradforth, S.E. ; Krylov, A.I. ; Sherrill, C.D. J. Chem. Phys. 2007, 127, 164110. [67] Pieniazek, P.A. ; Sundstrom, E.J. ; Bradforth, S.E. ; Krylov, A.I. to be published 2008. [68] Pieniazek, P.A. ; Bradforth, S.E. ; Krylov, A.I. J. Phys. Chem. A 2006, 110, 4854. [69] Hartke, B. Phys. Chem. Chem. Phys. 2003, 5, 275. [70] Winter, B. ; Weber, R. ; Widdra, W. ; Dittmar, M. ; Faubel, M. ; Hertel, I.V . J. Phys. Chem. A 2004, 108, 2625. [71] Y ., J.-H Guo ; Luo; Augustsson, A. ; Rubensson, J.-E. ; S˚ athe, C. ; ˚ Agren, H. ; Siegbahn, H. ; Nordgren, J. Phys. Rev. Lett. 2002, 89, 137402. 176 Chapter 6: Spectroscopy of the cyano radical in an aqueous environment 6.1 Overview The effect of bulk water on the B 2 Σ + ← X 2 Σ + and A 2 Π ← X 2 Σ + electronic transitions of the cyano radical is investigated in this Chapter. First, the cyano radical - water dimer is characterized to understand the nature of the interactions and parameterize molecular mechanics (MM) potentials. The carbon atom, which hosts the unpaired electron, is found to have a smaller Lennard-Jones radius than typical force fields values. Classical molecular dynamics (MD) is then used to sample water configurations around the radical, employing two sets of MM parameters for the cyano radical and water. Subsequently, vertical excitation energies are calculated using time-dependent density functional theory (TD-DFT) and equation-of-motion coupled-cluster with single and double substitutions (EOM-CCSD). The effect of water is modeled by point charges used in the MD simulations. It is found that both bands blue-shift with respect to their gas phase position; the magnitude of the shift is only weakly dependent on the method and the MM parameter set used. The calculated shifts are analyzed in terms of the solute-solvent interactions in the ground and excited states. Significant contributions come from valence repulsion and electrostatics. Consequences for experiments on ICN photodissociation in water are discussed. 177 6.2 Introduction Radicals in aqueous solutions play an important role in radiation chemistry and biology, heterogeneous atmospheric processes and are often implicated as enzymatic intermedi- ates. However, the understanding of the electronic structure and spectroscopy of even simple radicals in water is very limited. This is due to the high reactivity of these species which precludes experimental measurement of their spectra and properties. Of interest are the specific effects of a strongly interacting solvent such as water on the electronic structure of a solute, over and above classical dielectric solvation, on both the ground and particularly the excited electronic states. For example, charge transfer between sol- vent and solute can introduce new absorption bands. Furthermore, the degree of valence and Rydberg character in the isolated chromophore electronic transition can change markedly when the solute is embedded in a solution environment. Ironically, from the perspective of gas phase and matrix isolation electronic spectroscopy 1 , diatomic radicals (and diatomic molecules in general) are some of the most poorly characterized species 2 in terms of their visible and ultraviolet absorption bands once dissolved in water. An excellent case in point is the cyano radical, CN, which is one of the most exten- sively studied radicals in the vapor phase due to its importance in combustion and astro- physics, as well as in fundamental reaction dynamics studies 3–10 . Its low lying electronic states are well known through the ”red” and ”violet” absorption band systems where all rovibronic transitions have been cataloged into spectral atlases 3, 4 . However, the elec- tronic absorption spectrum of aqueous CN has never been reported. This poses consid- erable problems for a new generation of reaction dynamics studies exploring the role of the solvent on benchmark chemical reactions, e.g., the ICN photodissociation. Numer- ous theoretical studies have addressed how the environment affects energy dissipation, non-adiabatic curve crossing probabilities and the caging dynamics for this reaction thus 178 making it a most attractive system for calibrating theory with experiment 11–16 . Exper- imental work on this system in solution has begun to appear 17–19 , but uncertainty in spectral assignment of reaction products is currently hampering our progress towards a unified understanding of the reaction dynamics of this model system in solution 20 . In particular, femtosecond studies probing reaction products near 390 nm have been assigned to CN (B 2 Σ + ← X 2 Σ + ) by analogy to the gas phase violet band 18 . On the other hand, Keiding’s group monitoring the evolution of the full broadband spectrum developing after the reaction, instead assigned absorption near 390 nm to the I* product and a broad absorption band near 550 nm to CN 21 . Support for the latter assignment is provided by ab initio calculations for the CN··· H 2 O dimer 21 . The dramatically con- flicting spectral assignments highlight the following: the magnitude and direction of the shift for the CN intramolecular B 2 Σ + ← X 2 Σ + band is not known and the existence and location of any intermolecular charge-transfer (CT) band formed involving a neighbor- ing water molecule, such as is observed for halogen radicals in water, has not yet been established. Despite the simplicity of this or other diatomic radicals, ab initio theory has yet to provide help in this area. As can be seen for the case of CN detailed above, even the direction of the shift from vacuum can be in dispute. Rigorous theoretical description of solvent effects is one of the greatest challenges faced by ab initio methodology 22–25 . Indeed, every solvent molecule considerably increases the number of the electronic and nuclear degrees of freedom. In view of the polynomial scaling of the electronic structure methods, even the least computationally demanding electronic structure calculations, such as Hartree-Fock or Density Functional ones, become prohibitively expensive long before the bulk phase limit is reached. Moreover, averaging over solvent nuclear degrees of freedom is required to compute experimentally relevant quantities. Fortunately, sol- vated species often retain their identity, which justifies the separate treatment of the 179 solvent and solute wavefunctions and describing the solvent effects as a perturbation. These ideas (based on the assumption that the solute electrons remain localized on the solute) has given rise to a plethora of solvation approaches, ranging from implicit solvent models to a variety of quantum mechanics/molecular mechanics (QM/MM) methods. The implicit solvent models (see Ref. 23 and references therein) are easily the least computationally expensive, as they do not include the solvent electronic or nuclear degrees of freedom. However, these methods suffer from two main drawbacks: (i) the inability to account for specific interactions, i.e. hydrogen bonding; and (ii) information on the distribution of a given property is not provided. If the solvent is included explicitly, as in a QM/MM calculations, averaging over the equilibrium solvent configurations, which can be performed using molecular dynam- ics (MD) or Monte Carlo techniques, requires many thousands of electronic structure calculations at different solvent configurations. Various levels of sophistication can be used to describe the solvent in QM/MM schemes. The simplest description of polar solvents is by a set of fixed point charges that emulate the electric field. Although very simple, this approach has been successfully used in the past to gain insight into the elec- tronic structure of molecules in solution 26–29 . At the next level of sophistication one can include the changes in the electronic polarization in the ground and excited states of the solvent, as pioneered by Warshel and Levitt, who accounted for electronic relaxation 30 . Other researchers have explored different ways of coupling a solute to a polarizable MM solvent 24, 26, 31, 32 . Although polarization effects may be dominant for non-polar solvents, their relative contributions to electronic spectra in aqueous solutions are likely to be somewhat smaller. For example, Christiansen and coworkers, who studied the effect of water on then→ π ∗ transition in closed-shell formaldehyde 33 , have found that includ- ing polarization changes solvent induced shifts by less than 0.1 eV . Similar observations were reported by Gao for then→π ∗ transition of pyrimidine 26 . Arguably, those effects 180 might be stronger for the cyano radical because of the larger dipole moment change upon excitation. However, we expect that the electrostatic effects will remain dominant, as their magnitude is also proportional to the dipole moments values. The QM/MM description can be further improved by using an ab initio derived set of parameters to describe electrostatic, induction, and exchange interactions, as in the Effective Frag- ment Potential method by Gordon 34, 35 , which was employed, among other applications, to study solvent-induced shifts in metalloprotein systems 36 . In this chapter, we apply a combination of MD and electronic structure calculations to predict and understand the first two valence absorption bands and their widths for the CN radical in water. The effect of water is modeled by a set of point charges. We chose this initial approach due to its simplicity and because the above mentioned studies indicate a modest effect of solvent’s electronic response on the spectral shift in aqueous solutions. 6.3 Methodology 6.3.1 Overview The electronic spectrum of the aqueous cyano radical was simulated in a two step pro- cedure. First, classical MD trajectories were launched to sample different thermally accessible configurations of water around the cyano radical. Intermolecular potentials were approximated by pairwise Lennard-Jones and charge-charge electrostatic terms, as detailed in section 6.3.3. Subsequently, vertical excitation energies of CN were com- puted by an electronic structure method, the solvent being described as point charges at positions from MD snapshots. The validity of this approach was tested on the CN··· H 2 O dimer, as is described in section 6.3.2. The Amber 37 package was used to perform MD 181 simulations, whereas quantum calculations employed Q-Chem 38 . Additional calcula- tions were performed usingACESII 39 . 6.3.2 The cyano radical - water dimer calculations Although seemingly simple, the gas phase cyano radical requires an advanced level of theory for accurate description, as demonstrated by the results for various computational models summarized in Table 6.1. The unrestricted Hartree-Fock (UHF) solution exhibits a large spin-contamination, i.e. hS 2 i UHF is 1.075 instead of 0.75, as calculated using the cc-pVTZ basis. On the other hand, the restricted open-shell (ROHF) solution of the Hartree-Fock equations may be unstable due to the low lying B 2 Σ + excited state at 3.195 eV vertically above X 2 Σ + . These problems with the reference adversely impact properties of the cyano radical at Hartree-Fock (HF) and second-order Møller-Plesset perturbation theory (MP2) levels. For instance, both dipole moment are above 2 Debye, as compared to the experimental value of 1.45 Debye 40 . The UHF-MP2 vibrational fre- quency is 2908 cm − 1 , which is more than 40% higher than the experimental value. Only at the coupled-cluster with single and double substitutions (CCSD) 41 and the CCSD with perturbative account of triple excitations [CCSD(T)] 42, 43 levels the experimental values for the gas phase bond length, vibrational frequency, and dipole moment 1 are closely matched. Noteworthy, at the CCSD level the reference has little impact on the quality of the ground state results. However, the effects of spin-contamination in the reference on the excited states can be more more pronounced 44 . To alleviate these problems, the ROHF references was used in all wave function calculations. Density functional theory (DFT) with the B3LYP 45 functional yields values comparable with CCSD results. In 182 Table 6.1: The ground state equilibrium properties of the cyano radical calculated at different levels of theory using cc-pVTZ basis set. r e , ˚ A ν e , cm − 1 μ , Debye UHF 1.1505 2013 2.185 ROHF 1.1279 2453 2.285 UHF-MP2 1.1243 2908 2.149 ROHF-MP2 1.1882 1838 1.493 UHF-CCSD 1.1640 2178 1.522 ROHF-CCSD 1.1650 2158 1.468 ROHF-CCSD(T) 1.1753 2074 1.301 EOM-IP-CCSD 1.1636 2167 1.751 B3LYP 1.1627 2153 1.322 experiment a 1.1718 2069 1.45 a Ref. 1 both cases, the agreement with experimental numbers is not quantitative. The aug-cc- pVTZ 46 and 6-311++G** 47, 48 basis sets were used in the dimer ground state calcula- tions. Only the latter basis set was used to compute excitation energies for the dimer. Pure angular momentum d and f functions were employed. The first part of this study focuses on the cyano radical - water interactions charac- terized by calculations on the CN··· H 2 O dimer. These calculations were used to derive Lennard-Jones parameters for the MD simulations, as well as assess the accuracy of the method for simulating bulk phase spectra. Potential energy curves of the the CN··· H 2 O dimer were calculated at the monomer gas phase geometries shown in Fig. 6.1. The dis- tance between the fragments was varied from 2.00 ˚ A and 2.25 ˚ A up to 6.50 ˚ A, for C 2v and C s symmetry structures, respectively. Binding energies were obtained at the CCSD and CCSD(T) levels of theory. Basis set superposition error (BSSE) was accounted for by the counterpoise correction (CP) 49 . 183 C N O H H C N H O H 1.1718 0.958 0.958 104.34 a) b) Figure 6.1: Geometries of the cyano radical - water dimer used for the PES scans. Monomers are frozen at their gas phase geometries. Bond lengths are in ˚ A and angles in degrees. (a) Planar C 2v structure, (b) C s symmetry structure, C-N··· H-O formation is collinear. The distance between the fragments was varied from 2.00 ˚ A and 2.25 ˚ A up to 6.50 ˚ A, for C 2v and C s symmetry structures, respectively. Vertical excitation energies from the X 2 Σ + state to the A 2 Π and B 2 Σ + states were calculated using EOM-CCSD 50, 51 as well as TD-DFT 52 in the Tamm-Dancoff approxi- mation 53 with the B3LYP functional 45 . Single-reference EOM-CC methods are particu- larly attractive for for describing multi-configurational excited states of the cyano radical because these methods (i) include both dynamical and non-dynamical correlation in one computational scheme; and (ii) describe several states of interest in a single calculation (not state-by-state). To validate our method of simulating bulk phase spectra two types of dimer calculations were performed: (i) full quantum treatment of the dimer, (ii) the water molecule replaced by point charges located at hydrogen and oxygen atoms. Two sets of charges were employed: (i) reproducing the gas phase dipole moment (q H = 0.328) 54 , (ii) those from the SPC/E water model (q H = 0.4238) 55 . 184 To elucidate the degree of charge transfer between the cyano radical and water, the natural population analysis 56 was performed on the ground CCSD and the excited EOM- CCSD wave functions obtained using 6-311++G** basis set. This will provide hints as to the existence of a CT band in the condensed phase and allow an assessment of the classical description of the system, which assumes that there is no electron delocaliza- tion between water and the cyano radical. 6.3.3 Bulk phase calculations The MD system consisted of 1000 water molecules (986 for TIP5P/E) and a cyano radical. SPC/E 55 and TIP5P/E 57 molecular mechanics models were used to simulate water. TIP5P/E is a version of Jorgensen’s TIP5P model 58 re-optimized for use with Ewald summation. Parameters of both water models used are given in Table 6.2. All simulations were performed in the isothermal-isobaric ensemble (constant T, P, N). Tem- perature was kept constant at 298 K using the weak-coupling algorithm 59 , with a time constant of 3 ps. A pressure variant of the weak-coupling algorithm with isotropic posi- tion scaling was used to keep pressure constant at 1 atm. Barostat time constant was set to 3 ps. All bonds were constrained during the dynamics using the SHAKE algo- rithm with tolerance of 10 − 6 ˚ A. Equations of motion were integrated using a 1 fs time step. Long range interactions were handled using Ewald summation and a cut-off of 8 ˚ A was applied to the Lennard-Jones interactions. Periodic boundary conditions were applied. Quantum mechanical calculations of the cyano radical in the electric field of point charges were performed in the cc-pVTZ basis set. The interaction of the cyano radical with water in the MD simulations was mod- eled by a sum of 6-12 Lennard-Jones and charge-charge electrostatic terms. SPC/E parameterization was used to model water in derivation of charges. The interaction sites of the cyano radical were placed on carbon and nitrogen. Lorentz-Berthelot mixing 185 Table 6.2: Basic SPC/E and TIP5P-E water model parameters. SPC/E TIP5P/E q H 0.4238 0.2410 σ , ˚ A 3.166 3.097 , kcal/mol 0.1553 0.178 r OH , ˚ A 1.0 0.9572 θ HOH 109.47 104.52 Table 6.3: The Lennard - Jones and electrostatic interaction parameters for CN··· H 2 O. Set A has been taken from the Amber force field and Set B has been developed in this work. C , kcal/mol σ C , ˚ A N , kcal/mol σ N , ˚ A q C Set A 0.086 3.40 0.170 3.25 0.3648 Set B 0.089 2.60 0.180 3.32 0.3632 rules were used to obtain interaction parameters between different atom types. Two sets of Lennard-Jones parameters used in calculations are given in Table 6.3. Set A is taken from the Amber force field 60 , Set B was derived from the dimer calculations (CN··· H 2 O). The atomic charges for CN were derived by the following self-consistent procedure. Starting with the initial guess of 0.400, with the carbon atom bearing the positive charge, and for each set of Lennard-Jones parameters, three trajectories were allowed to equilibrate for 500 ps, followed by a production run of 150 ps. Snapshots were taken every 1 ps. At each snapshot the dipole moment of cyano radical was calcu- lated using CCSD/cc-pVTZ. New charges were hence derived to reproduce the average dipole moment. Then the system was allowed to equilibrate for 150 ps with the new set of charges, followed by another production run. The procedure was repeated until the change in charge was less than 0.001. This required 5 iterations for Set A and 6 iterations for Set B. The obtained charges are listed in Table 6.3. The dipole moment increased from 1.45 Debye in the gas phase to 2.05 Debye in water. No re-optimization was performed for simulations employing the TIP5P/E model. 186 A total of three different bulk phase model systems was simulated: (i) Set A + SPC/E, (ii) Set B + SPC/E, and (iii) Set B + TIP5P/E. For each system, five MD tra- jectories were launched using random initial velocities and positions (molecules were placed on a simple cubic lattice and then rotated by the three Euler angles sampled from a uniform distribution). Each trajectory was allowed to equilibrate for 500 ps. This was followed by a 200 ps production run with snapshots taken every 1 ps. At each snapshot, vertical excitation energies were computed using EOM-CCSD/cc-pVTZ. Additionally, for model system (iii) the energies were computed using TD-DFT/B3LYP/cc-pVTZ. The local structure of water around the cyano radical was analyzed by means of radial and angular distribution functions. 6.4 Results and discussion 6.4.1 The cyano radical - water dimer In the X 2 Σ + ground state the cyano radical has a dipole moment of 1.45 Debye, the car- bon atom bearing the positive charge and the unpaired electron 40 . Based on electrostatic considerations, the two configurations of the cyano radical - water dimer presented in Fig. 6.1 were selected for potential energy surface (PES) scans. These scans represent the two limiting interaction scenarios between the fragments. One corresponds to a neg- atively charged oxygen atom interacting with the positively charged carbon atom. This gives rise to the planar C 2v complex. The C s structure represents a hydrogen bonding situation, in which the negatively charged nitrogen atom acts as a hydrogen bond accep- tor. Frequency calculations carried out near the the minima reveal that the C s structure is, in fact, a minimum, whereas the C 2v structure has three imaginary frequencies (160, 66, and 65 cm − 1 ), corresponding to out-of-plane hydrogen wag, out-of-plane CN wag, and in-plane antisymmetric pseudorotation of the monomers, respectively. We retained 187 this structure because it captures the essentials of the carbon - oxygen interaction and provides the simplicity needed in the analysis of dimer properties and force field param- eterization. Firstly, let us look at the strength of CN··· H 2 O interaction and the range of resulting intermolecular separations. Fig. 6.2 presents the CP corrected scans of PES, as well as a single curve without the CP correction. The general effect of the CP correction is an increase of the minimum energy distance (r e ) and a decrease of the complex disso- ciation energy (D e ). Surprisingly, perturbative inclusion of triple excitations results in only minor changes of CCSD PES. The basis set dependence is more pronounced — larger basis set yields stronger binding. The minimum along the C s curve is located around 3.25 ˚ A. The CCSD(T)/6-311++G ∗∗ binding energy is 1.70 kcal/mol. Increas- ing the basis set to aug-cc-pVTZ increases D e to 1.95 kcal/mol. The respective, before BSSE-correction, values are 2.20 and 3.46 kcal/mol. These can be compared to the corresponding value for the hydroxyl radical which is 2.40 kcal/mol at the CCSD(T)/6- 311++G(2d,2p)//B3LYP/6-311++G(2d,2p) 61 (without the BSSE correction). The C 2v structure exhibits a minimum at about 2.75 ˚ A. The CCSD(T) binding energy increases from 1.83 kcal/mol to 2.15 kcal/mol upon increasing the basis set size. Values without the BSSE correction are 2.82 and 2.93 kcal/mol, respectively. Again, the correspond- ing value for the hydroxyl radical is 5.87 kcal/mol 61 . The large difference cannot be accounted for by pure electrostatics, as the dipole moment of the hydroxyl radical is 1.668 Debye and that of the cyano radical 1.45 Debye 1 . Rather, this is a signature of hydrogen bonding, where the OH group is a hydrogen donor. Overall, although the cyano radical interacts quite strongly with water, it is a weaker hydrogen bond accep- tor than OH. The latter is very fortunate, as such an interaction is likely to affect the spectrum beyond electrostatics and is not easily incorporated into the bulk phase calcu- lations. 188 (a) 2.0 3.0 4.0 5.0 6.0 -4.0 -3.0 -2.0 -1.0 0.0 CN O H H binding energy, kcal/mol r C ⋅⋅⋅ O, Å (b) 2.0 3.0 4.0 5.0 6.0 -4.0 -3.0 -2.0 -1.0 0.0 CN HO H binding energy, kcal/mol r N ⋅⋅⋅ O, Å Figure 6.2: The CP corrected dimer potential energy curves along interfragment sepa- ration coordinate. (a) C 2v symmetry structure. (b) C s symmetry structure. The energies were obtained at the CCSD/6-311++G** (blue dotted line), CCSD(T)/6-311++G** (red dotted line), CCSD/aug-cc-pVTZ (blue solid line), and CCSD(T)/aug-cc-pVTZ (red solid line line) levels of theory. For comparison, the CCSD(T)/aug-cc-pVTZ scan with- out BSSE correction is shown (solid black line). The above ab initio PES scans can be used to derive interaction parameters and assess the quality of the MM CN··· H 2 O potential, which is approximated by pairwise 189 6-12 Lennard-Jones and charge-charge electrostatic energy terms. Polarization and elec- tron delocalization effects, which become important at short intermolecular separations, are absent in the MD model. To ensure an unambiguous comparison, charges for the gas phase potential were fitted so as to reproduce the gas phase dipole moments of the cyano radical and water: 1.45 and 1.855 Debye 54 respectively (the corresponding charges are 0.258 for carbon and 0.3285 for hydrogen). Individual fragments are kept neutral. Two sets of Lennard-Jones parameters were tested. Set A employs the values from theAmber force field 60 . Set B was derived by fitting to the ab initio scans of dimer PES. The param- eters are presented in Table 6.3 and the potentials are plotted on Fig. 6.3. The potential for C s PES scan is well reproduced using typical Lennard-Jones radii. Consequently, only minor adjustments were made to nitrogen parameters. The interaction on the car- bon side, where the unpaired electron is located, was more problematic. Typical force field radii make the carbon atom too big. On the other hand, a much smaller value leads to overestimation of potential well depth due to the electrostatic collapse. The position of the Lennard-Jones hard wall may have a significant effect on the calculated spectra, since at short distances water has the most dramatic effect on vertical excitation ener- gies, as can be seen in Fig. 6.4. For instance, in the C 2v structure the excitation energy changes by 0.3 eV when the intermolecular distance decreases from 2.50 ˚ A to 2.25 ˚ A. This distance change corresponds roughly to the shift in the carbon-oxygen radial dis- tribution functions obtained using the two parameterizations. Consequently, deriving a CN specific parameter set is justified. Having described the ground state intermolecular potential, we now proceed to the excited states of the cyano radical in the dimer. The two excited states of the cyano radical that are of interest here are the valence A 2 Π and B 2 Σ + states, corresponding to theπ → n C andn N → n C excitations, respectively. The oscillator strength associated with those transitions are 0.0034 and 0.033 respectively. 1 . Both excitations result in 190 (a) 2.0 3.0 4.0 5.0 6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 CN O H H binding energy, kcal/mol r C ⋅⋅⋅ O, Å (b) 2.0 3.0 4.0 5.0 6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 CN HO H binding energy, kcal/mol r N ⋅⋅⋅ O, Å Figure 6.3: Comparison of the ab initio and molecular mechanics dimer potential energy surfaces. (a) C 2v symmetry structure. (b) C s symmetry structure. The energies were obtained using: CCSD(T)/aug-cc-pVTZ (black solid line), Amber’s Set A and SPC/E Lennard-Jones parameters with charges reproducing gas phase dipole moments (red solid line) and condensed phase charges (red dotted line), newly derived Set B and SPC/E Lennard-Jones parameters with charges reproducing gas phase dipole moments (blue solid line) and condensed phase charges (blue dotted line). 191 (a) (b) 2.0 3.0 4.0 5.0 6.0 1.2 1.4 1.6 1.8 2.0 EOM-CCSD TD-DFT A 2 Π CN O H H vertical excitation energy, eV r C ⋅⋅⋅ O, Å 2.0 3.0 4.0 5.0 6.0 3.2 3.4 3.6 3.8 4.0 4.2 TD-DFT EOM-CCSD B 2 Σ + CN O H H vertical excitation energy, eV r C ⋅⋅⋅ O, Å (c) (d) 2.0 3.0 4.0 5.0 6.0 1.1 1.2 1.3 1.4 1.5 A'' CN HO H vertical excitation energy, eV r N ⋅⋅⋅ O, Å A 2 Π EOM-CCSD TD-DFT A' 2.0 3.0 4.0 5.0 6.0 3.2 3.3 3.4 3.5 3.6 3.7 3.8 TD-DFT EOM-CCSD B 2 Σ + CN HO H vertical excitation energy, eV r N ⋅⋅⋅ O, Å Figure 6.4: Vertical excitation energies as a function of monomer separation calculated by full EOM-CCSD (black solid line), full B3LYP (blue solid line). Results of calcu- lations with water replaced by point charges reproducing the water gas phase dipole moment (EOM-CCSD – black dotted line, B3LYP – blue dotted line) and charges used in SPC/E water model (EOM-CCSD – black dashed line, B3LYP – blue dashed line) are also shown. Panel (a) the A 2 Π state along the C 2v scan (only the in-plane B 1 component is shown, as the B 2 component is nearly degenerate), (b) the B 2 Σ + state along the C 2v scan, (c) the A 2 Π state along the C s scan (both the A’ and A” components are shown), (d) the B 2 Σ + state along the C s scan. a change of the direction of the dipole moment relative to ground state (1.45 Debye), which is 0.56 Debye in A 2 Π and 1.15 Debye in B 2 Σ + state (nitrogen atom bears the positive charge) 40, 62 . EOM-CCSD using 6-311++G** basis predicts dipole moments 192 of 0.33 and 1.45 Debye respectively. The calculated oscillator strength are 0.0032 and 0.039 for the A and B states, respectively. The accuracy of the bulk phase spectra can be assessed by considering the effect of replacing water by point charges in the dimer calculations. These and full quantum excitation energies computed as a function of the distance between the monomers using EOM-CCSD and TD-DFT B3LYP are presented in Fig. 6.4. The TD-DFT energies are systematically lower than the corresponding EOM-CCSD values for the transition to the A 2 Π state, and higher for the B 2 Σ + state transition. Further analysis considers the C 2v and C s scans separately. Water approaching the cyano radical from the carbon side increases the energy gap between the ground and excited states (Fig. 6.4a, 6.4b). The dependence on the distance is particularly strong in the region below 3 ˚ A. The splitting of A 2 Π into the 2 B 1 and 2 B 2 states in C 2v is rather small, e.g. 0.01 eV at 2.75 ˚ A, therefore only the higher energy component 2 B 1 is shown (in-plane). EOM-CCSD and TD-DFT are in qualitative agreement. When water is replaced by point charges slightly lower excitation energies are observed, but the trends are reproduced correctly. A more complicated behavior is observed when water is located on the nitrogen end of the cyano radical (Fig. 6.4c, 6.4d). Full EOM-CCSD calculations reveal that the B 2 Σ + ← X 2 Σ + transition energies fall dramatically after a small initial rise. The exci- tation energies to A 2 Π increase at shorter distances. Moreover, around 2.75 ˚ A splitting of this level into the A’ and A” components in the C s symmetry becomes significant. B3LYP incorrectly predicts that the B 2 Σ + ← X 2 Σ + transition energies will increase at short distances, but correctly describes the behavior of the A 2 Π state. When charges are used to represent water the separation between the ground and the B 2 Σ + states increases at short distances, in disagreement with the full EOM-CCSD treatment. The transi- tions to the A 2 Π state occur at lower energies relative to the full quantum mechanical 193 treatment, regardless whether EOM-CCSD or TD-DFT is used to treat the radical. For- tunately, the discrepancies between the quantum and electrostatic treatments of water become significant only in the region below 2.75 ˚ A. Nitrogen-oxygen radial distribu- tion functions calculated from MD trajectories essentially vanish at this distance, which means that these configurations are not sampled at equilibrium conditions. This is con- sistent with the condensed phase dimer potential, as it becomes strongly repulsive at this distance resulting in low Boltzmann weights of those configurations. To summarize, our model, in which water is described by point charges, can quan- titatively reproduce the excitation energies of the cyano radical in the CN··· H 2 O dimer at large and intermediate distances. Another interesting issue is the character of the electronic states of the dimer. For example, both the ground state and the excited state may acquire significant charge transfer character in water and will consequently require explicit water molecules in the quantum calculation. Natural population analy- sis allowed to assess the degree of charge transfer between the fragments. The natural charge on the cyano radical as a function of the monomer separation is shown in Fig. 6.5. For both geometries under scrutiny charge transfer occurs at distances below 3.25 ˚ A. In the ground state C s structure the cyano radical acquires a partial positive charge; charge is pulled from the negative nitrogen atom by the positive hydrogen atom. When the system is excited to the B 2 Σ + state (nitrogen becomes positive) the charge flows back from water to the radical making it negative, which is consistent with the dipole moment flip upon excitation. Components of the A 2 Π state are also characterized by a positive charge on CN. This is surprising, as in this state the dipole moment also flips relative to ground state and the nitrogen is positive, though its magnitude is smaller than in the B 2 Σ + state. This is possibly due to the concomitant delocalization of theπ cloud across the hydrogen bond. In the C 2v structure there is electron flow to the radical in the ground state, oxygen donates its electrons to the carbon atom. Excitation to the B 2 Σ + 194 state leads to depletion of the net charge on the cyano radical, however it remains nega- tive. The A 2 Π state exhibits only minor charge transfer and the monomers are neutral. Overall, appreciable amount of charge transfer occurs only at very short monomer sepa- rations, i.e. 2.5 ˚ A for C 2v structures and 2.75 ˚ A for C s structures. It is at those distances that the intermolecular potentials become strongly repulsive and the radial distribution functions essentially vanish. Thus, at configurations sampled in bulk water simulations, the valence electronic states of the cyano radical preserve their gas phase identity in the condensed phase and our method describes them correctly. However, it is reasonable to expect that a CT state may appear in water, in addition to the two valence excited states of the cyano radical. 6.4.2 Bulk phase calculations In this section, we first analyze the structure of water around the cyano radical, focusing mostly on differences between force field parameterizations. Then we proceed to the analysis of the calculated electronic excitation spectra. Local solvent structure around the CN radical General features of the solvent structure around the solute are given by the radial dis- tribution functions between the geometric center (GC) of the cyano radical and water oxygen (OW) and hydrogen (HW), which are plotted in Fig. 6.6. Overall, they exhibit little dependence on the water model employed. The GC-OW plot exhibits a shift to larger distance of the first peak on the distribution function obtained using Set A of Lennard-Jones parameters with respect to the newly derived Set B by ca. 0.2 ˚ A, consis- tent with relative sizes of the carbon atom in the two sets. A similar effect is observed in GC-HW distribution functions. This is surprising, since there is very little difference between the parameters of nitrogen atom, where one expects to find the closest hydrogen 195 (a) 2.0 3.0 4.0 5.0 6.0 -0.15 -0.10 -0.05 0.00 0.05 B 1 CN O H H CN charge, e r C ⋅⋅⋅ O, Å B 2 (b) 2.0 3.0 4.0 5.0 6.0 -0.2 -0.1 0.0 0.1 A'' A' CN HO H CN charge, e r N ⋅⋅⋅ O, Å Figure 6.5: Natural charge of the cyano radical in the ground and excited states of the cyano radical - water dimer. The CCSD/6-311++G** and EOM-CCSD/6-311++G** wave functions are analyzed along C 2v and C s scans, panels (a) and (b) respectively. The X 2 Σ + ground state (solid line), the B 2 Σ + state (dotted line), and the A 2 Π state (dashed line). atom. There is essentially no shift in the nitrogen - hydrogen radial distribution function (not shown), which suggests that this effect is due to hydrogen atoms located on the side of the cyano radical axis. 196 (a) 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0.0 0.5 1.0 1.5 2.0 g(r) r GC ⋅⋅⋅ O, Å (b) 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0.0 0.5 1.0 1.5 2.0 g(r) r GC ⋅⋅⋅ H, Å Figure 6.6: Radial distribution functions of oxygen (a) and hydrogen (b) around the geometric center of the cyano radical. The bin size is 0.2 ˚ A. The curves are: Set B + SPC/E (solid line), Set A + SPC/E (dotted line), Set B + TIP5P/E (dashed line). Due to the angular averaging, radial distribution functions provide a relatively crude description of the solvent structure. A closer look at the immediate vicinity of the cyano radical is desired, because it is those molecules that have the strongest effect on the electronic spectrum. One useful parameter is the coordination number as determined by 197 Table 6.4: Coordination number of cyano radical in water using different definitions of the first solvation shell. r a , ˚ A N b 1 N c 2 Set B + SPC/E 3.55 3.84 16.64 Set A + SPC/E 3.70 4.24 16.89 Set B + TIP5P/E 3.45 3.16 16.64 a maximum on the geometric center of CN - oxygen radial distribution function b number of water closer closer than than the maximum on radial distribution function c number of water molecules closer than 5.1 ˚ A the integration of the radial distribution function. Two upper integration bounds were used: (i) the first maximum on GC-OW radial distribution function, and (ii) 5.1 ˚ A, where the first minimum is located. The results are summarized in Table 6.4. Overall, it appears that 16 water molecules are needed to fully surround the cyano radical by a shell of water. This result is rather insensitive to the potential used for the cyano radical, because the long-range structure of solution is defined primarily by the water - water potential. There are ca. 3.5 nearest-neighbor water molecules. Amber’s Set A, with a slightly larger carbon radius, yields a larger number of nearest neighbors. The difference between the SPC/E and TIP5P/E results is largely due to the uncertainty in the determination of the maximum of the radial distribution function and a strong dependence on the cut-off radius. We can further investigate the positions of the four nearest neighbors by means of angular probability distribution functions shown in Fig. 6.7 and 6.8. The angular vari- able is the angle between N-C and GC-OW (HW) vectors; e.g. zero corresponds to the oxygen (hydrogen) atom located on the CN axis at the carbon atom end. The radial lim- its were chosen to include one, two, three, and four nearest oxygen (hydrogen) atoms. Overall, the angular probability distributions exhibit stronger dependence on the force field for the cyano radical than the radial distribution functions. The first oxygen atom 198 is located preferentially on the side of the cyano radical (ca. 90 degrees). The angle is smaller for newly derived Set B, meaning that the oxygen is closer to the carbon. This is a direct consequence of the smaller Lennard-Jones radius, which allows for a stronger electrostatic interaction. As the radial range is extended to include more water molecules, the distribution for Amber’s Set A and SPC/E water becomes broader, the maximum remaining at approximately 90 degrees. On the other hand, when the radial range is extended for Set B, featuring a smaller carbon atom, the probability distribution develops a shoulder at smaller angles, meaning that the successive oxygen atoms are located preferentially near carbon. The angular distribution of the first hydrogen show the presence of the hydrogen atom on the CN axis near nitrogen thus indicating a hydro- gen bond. Subsequent hydrogen atoms are likely to be found along side of the cyano radical. The dependence on both the water model and the cyano radical parameterization is stronger for the angular distribution of hydrogen than oxygen. Overall, the averaged structure of the first solvation shell of the cyano radical is barrel-like. The nearest water molecules are located along the side of the CN axisand the whole structure is capped on both ends. The position of the capping water on the nitrogen end is consistent with a hydrogen bond. Note that the definition of the coordina- tion sphere used here is biased towards water molecules located along side of the cyano radical, because it is defined with respect to the geometric center of CN. The observed structural differences between different parameterizations are not qualitative and can be easily understood based on the parameterization of the CN radical, although they may still lead to different dynamics of the cyano radical. Moreover, including water explicitly into the quantum calculation of electronic spectrum would yield stronger dependence on the angular distribution of water, because orbital overlap has a stronger directional char- acter than electrostatic interactions. 199 (a) (b) 0 30 60 90 120 150 180 0.0 0.2 0.4 0.6 0.8 1.0 probability α, degrees 0 30 60 90 120 150 180 0.0 0.2 0.4 0.6 0.8 1.0 probability α, degrees (c) (d) 0 30 60 90 120 150 180 0.0 0.2 0.4 0.6 0.8 1.0 probability α, degrees 0 30 60 90 120 150 180 0.0 0.2 0.4 0.6 0.8 1.0 probability α, degrees Figure 6.7: Angular distribution of oxygen around the cyano radical. The angular vari- able is the angle between CN and the geometric center - oxygen (hydrogen) vectors. Radial range is successively increased to include between (a) one and (d) four atoms. The curves are: Set B + SPC/E (solid line), Set A + SPC/E (dotted line), Set B + TIP5P/E (dashed line). The electronic spectrum of the aqueous cyano radical The interaction between water molecules and the cyano radical is different in the CN ground and excited states, the strength of the interaction varying with the geometry of the system. In the present study we sample different configurations of water, thus accounting for the inhomogeneous broadening part of the electronic spectrum. The 200 (a) (b) 0 30 60 90 120 150 180 0.0 0.2 0.4 0.6 0.8 1.0 probability α, degrees 0 30 60 90 120 150 180 0.0 0.2 0.4 0.6 0.8 1.0 probability α, degrees (c) (d) 0 30 60 90 120 150 180 0.0 0.2 0.4 0.6 0.8 1.0 probability α, degrees 0 30 60 90 120 150 180 0.0 0.2 0.4 0.6 0.8 1.0 probability α, degrees Figure 6.8: Angular distribution of hydrogen around the cyano radical. The angular variable is the angle between CN and the geometric center - oxygen (hydrogen) vectors. Radial range is successively increased to include between (a) one and (d) four atoms. The curves are: Set B + SPC/E (solid line), Set A + SPC/E (dotted line), Set B + TIP5P/E (dashed line). spectra are presented in Fig. 6.9, and the band parameters are listed in Table 6.5. Over- all, the centers of the bands are blue-shifted relative to the calculated gas phase position (1.327 and 3.347 eV for the A 2 Π ← X 2 Σ + and the B 2 Σ + ← X 2 Σ + transition respec- tively, at the EOM-CCSD/cc-pVTZ level), regardless of the MD and electronic structure model. The widths are ca. 0.4 and 0.14 eV for the B and A bands, respectively. The rela- tive insensitivity of the calculated spectra to the water model employed is consistent with 201 the radial distribution functions, which are very similar for both models, which means that the point charges are positioned in similar range. The particular choice of the water model has a greater impact on the A 2 Π ← X 2 Σ + than the B 2 Σ + ← X 2 Σ + transition, as observed in both the corresponding shifts and widths. Next, the spectra obtained using Amber’s Set A are systematically slightly narrower and less blue-shifted. This is because water in this model is farther away on average from the cyano radical. Over- all, the differences between the calculated spectra are much smaller than their widths. Thus, we conclude that our model is robust and the effect on the absorption spectrum is insensitive to the variation of the parameters within the uncertainty range. Table 6.5: A (A 2 Π ← X 2 Σ + ) and B (B 2 Σ + ← X 2 Σ + ) band parameters. Set B + Set A + Set B + Set B + SPC/E SPC/E TIP5P/E TIP5P/E + TD-DFT A band (A 2 Π ← X 2 Σ + ) ν 0 a 1.47 1.43 1.50 1.40 ν 1/2 b 0.30 0.24 0.37 0.41 Δ ν c 0.14 0.10 0.18 0.07 ν s d 0.061 0.057 0.059 0.061 B band (B 2 Σ + ← X 2 Σ + ) ν 0 a 3.61 3.57 3.62 3.61 ν 1/2 b 0.42 0.37 0.42 0.34 Δ ν c 0.26 0.22 0.27 0.27 a position of the band center, eV b full width at half maximum, eV c blue shift relative to gas phase position, eV d splitting of the A 2 Π level, eV As observed in the dimer calculations, TD-DFT yields reliable results (as compared against EOM-CCSD) when charges are used to model the effect of water. To test how big the difference between the two methods is in the bulk phase we performed TD-DFT calculations for the trajectory utilizing the newly derived Set B and the TIP5P/E water. 202 (a) 1.2 1.4 1.6 1.8 0 50 100 150 200 250 300 # of cases energy, eV (b) 3.2 3.4 3.6 3.8 4.0 0 40 80 120 160 200 # of cases energy, eV Figure 6.9: Simulated spectra of the cyano radical in water: (a) the A← X transition; (b) the B← X transition. The spectra are: Set B + SPC/E (solid line), Set A + SPC/E (dotted line), Set B + TIP5P/E (dashed line), Set B + TIP5P/E +TD-DFT/cc-pVTZ. Unless otherwise indicated the energies were computed using EOM-CCSD/cc-pVTZ (dash-dot line). Wiggles are a result of a finite sample of excitation energies. The density functional results for individual snapshots are, on average, shifted to the red by 0.102± 0.012 eV for A band and 0.005± 0.026 eV for B band as compared to the EOM-CCSD results. Thus, essentially identical shifts are obtained at a much lower cost, 203 which suggests that one could employ TD-DFT in calculations of spectroscopic correla- tion function necessary for computing homogeneous and inhomogeneous contributions to the bandwidth. Such calculations require calculating excitation energies along the equilibrium trajectories at very small time steps for extended periods to ensure appro- priate sampling. However, we note that TD-DFT is not a viable approach for including explicit water molecules into the quantum calculation, because charge transfer excita- tions are incorrectly described by TD-DFT. Those states artificially mix with valence excited states and neither type of states is described correctly 63 . Indeed, in a cluster calculation with four water molecules, it was impossible to assign any valence excited states of the cyano radical. Nature of CN-water interactions and their contributions in the spectral shifts Let us analyze the calculated spectra in the context of first principles solvation ideas, following the discussion by Bayliss and McRae 64 . We neglect here any geometrical changes in the CN radical. This seems justified due to high vibrational frequency of the bond. In solutions, the electronic states of a solute are modulated by the surrounding solvent molecules, making them as well as the excitation process more complex than in the gas phase. The main interactions are: (i) electrostatics, (ii) dispersion, and (iii) valence repulsion (referred to as packing strain in the original paper). Those interac- tions give rise to an equilibrium cage structure of a polar solvent around a polar solute. Upon excitation, a Franck-Condon excited state is formed, i.e. the electronic distribu- tion changes, whereas the nuclear degrees of freedom remain frozen. As a result the aforementioned interactions will be different in the excited and the ground states giving rise to a spectral shift. Below we discuss the relative importance of those factors for the cyano radical - water system and whether the observed shift is due to the ground state stabilization or the excited state destabilization. 204 Since for the problem at hand both the solute and the solvent are polar, one may expect that electrostatics will dominate the interactions in both the ground and the excited states, and that the dipolar solvation model will explain the spectral shifts. Indeed, the calculated direction of the shift is in agreement with predictions based on the dipole moment of the cyano radical in the ground and the excited states. In the ground state, the carbon atom bears a positive charge and water will stabilize this configuration, by appropriately orienting its dipoles and forming a hydrogen bond with nitrogen. Once the chromophore is excited from the ground state, either to the A 2 Π or the B 2 Σ + state, the CN dipole moment flips. Now the nitrogen bears the positive charge, and the initial water configuration is destabilizing due to the Coulomb repulsion between the positively charged nitrogen and hydrogens. Consequently, the energy gap between the ground and excited states increases with respect to the gas phase. The shift for the A state is smaller than for the B state because the dipole moment flip is smaller. Our simulations therefore quantify the simple expectation based on dipolar solvation ideas. As mentioned in the Introduction, our bulk simulations do not allow for electronic relaxation of water molecules upon excitation. Thus, the water charge distribution in our model is more unfavorable for the excited state than it would be in full quantum mechanical calculations. Our cluster studies indicate that this is not a major effect, but they include only a single water molecule. This effect may increase with the number of water molecules and result in a smaller blue-shift. On the other hand, real water has a larger dipole moment than the computational water used, e.g. 2.35 Debye for SPC/E model. Consequently, greater stabilization of the ground state can be expected leading to an increased blue-shift. The relative magnitude of both effects is unclear, however, the recent study by Christiansen et al. 33 demonstrated that the water electronic response effects are small. Their calculations of the n→ π ∗ transition of aqueous formalde- hyde employed both the non-polarizable TIP3P and the polarizable SPCpol potentials 205 for water in the classical MD simulations. The same water parameterizations, including polarization for SPCpol, were used in the subsequent quantum calculation of the elec- tronic spectrum, which therefore included both electrostatic and polarization terms. The difference in the calculated vertical excitation energies is mostly due to different struc- tures predicted in the MD simulations. When vertical excitation energies for the same set of configurations were calculated using both polarizable and non-polarizable water the difference was about 0.01 eV , which is much smaller than the calculated bandwidth for formaldehyde. Polarization effect may be larger in the case of the cyano radical due to the greater change of the dipole moment upon excitation. However, we expect that electrostatic effects will remain dominant, as its strength is also proportional to the mag- nitude of the dipole moment, and that the inclusion of the electronic response would not drastically change our results. The energies of valence repulsion and dispersion interactions depend on the size of the electronic cloud. The size of electronic density in different electronic states of CN, as defined by a gas phase value ofhR 2 i, is 11.24 ˚ A 2 , 11.78 ˚ A 2 , 11.57 ˚ A 2 for the X, A and B states respectively. Thus, weak dispersion interactions will stabilize the excited states more than the ground state. However, the valence repulsion in the excited states will be larger, due to the increased overlap between the cyano radical and water electronic density. For C 2v structures near the minimum, the destabilization of the excited states is almost twice as large as the stabilization of the ground state. The dipole moments in the excited states point in the opposite directions and are roughly equal to 2/3 and 1/3 of the ground state dipole moment (in the B 2 Σ + and A 2 Π states, respectively). If only electro- static interactions were present, the potential energies would exhibit the same ratios as shown in Fig. 6.10 by dashed lines. The remaining destabilization must, consequently, be assigned to valence repulsion, which is greater for the A 2 Π state than for the B 2 Σ + state. Again, the C s structures pose certain problems. Destabilization of the B 2 Σ + state 206 is roughly half the stabilization of the ground state. However, the A 2 Π state actually bound but by a little less than the ground state. Extra stabilization in the full calculation may originate in additional delocalization of the π cloud across the hydrogen bond, as suggested by the calculated natural charges. In the condensed phase molecules tend to approach each other closer than the min- imum of the dimer intermolecular potential. At short distances, the valence repulsion becomes very large. Let us quantify this effect and, for sake of simplicity, concentrate on C 2v structures. When one moves in by 0.25 ˚ A from the minimum on the dimer potential the ground state now is only stabilized by ca. 2.5 kcal/mol and the excited state destabi- lized by ca. 9 kcal/mol. Hence, we expect that the degree of excited state destabilization will increase in the condensed phase relative to the dimer. To conclude, for a diatomic molecule with a relatively large dipole moment in water, the electrostatic effects are dominant and, therefore, a bulk model that describes solvent by fixed point charges provides a reasonable description. Differential changes in valence repulsion and dispersion have opposing effects on the absorption shift direction; both are neglected in our approach and the magnitude of the error depends sensitively on the typ- ical solute-solvent distances sampled at equilibrium. In the case of CN B 2 Σ + ← X 2 Σ + , there is a fortuitous cancellation of errors, at least near the dimer minimum. 6.5 Conclusions In this chapter, we have described the effect of water on the electronic spectrum of the cyano radical in the bulk and in the cyano radical - water dimer. We found that, for the dimer configurations considered, the presence of water leads to a blue shift of the transi- tions, except for very short distances for the C s structures. In the range that is sampled in the ground state MD the effect of water can be reliably modeled by point charges. This 207 (a) 2.0 3.0 4.0 5.0 6.0 -5 0 5 10 15 CN O H H binding energy, kcal/mol r C ⋅⋅⋅ O, Å (b) 2.0 3.0 4.0 5.0 6.0 -5 0 5 10 15 CN HO H binding energy, kcal/mol r N ⋅⋅⋅ O, Å Figure 6.10: Potential energies of the cyano radical - water dimer in different elec- tronic states of the cyano radical in (a) the C 2v symmetry and (b) the C s symmetry structures. The lines are: X 2 Σ + CN··· X 1 A 1 H 2 O (black line), A 2 Π CN··· X 1 A 1 H 2 O (red line), B 2 Σ + CN··· X 1 A 1 H 2 O (violet line). Dotted curves were obtained by replac- ing water with point charges derived to reproduce its gas phase dipole moment. Only the B 1 and A’ components of the A 2 Π state in C 2v and C s structures are shown. The ground and excited state energies were calculated using CCSD/6-311++G** and EOM- CCSD/6-311++G** respectively. The CP correction is not included. 208 significantly simplifies calculations, as this interaction enters the Hamiltonian via the one-particle part and is thus inexpensive to compute. Binding energy curves show that the size of the carbon atom, which hosts the unpaired electron, is smaller as compared to a typical saturated carbon. Thus care must be exercised when adapting typical force field parameters to model radicals in solution. Calculations of the electronic spectrum in the condensed phase confirmed that indeed the two bands of interest exhibit a blue-shift. Satisfyingly, the magnitude of the effect depends only weakly on the parameterization used in the MD part, i.e. the model sensitivity is much smaller than the bandwidth. Let us revisit the experimental issue that motivated this work, namely the spectral assignment of the products of the ICN dissociation. Clearly, the electronic structure cal- culations and analysis support a moderate blue shift of the B 2 Σ + ← X 2 Σ + transition of CN in water. Both EOM-CCSD and TD-DFT suggest the band center at∼ 343 nm, with a wing extending to the probe wavelengths used in the experiments of Moskun et al. 18 . In addition, new experiments have been carried out in Aarhus jointly between the Brad- forth and Keiding groups to find independent evidence for an I* charge transfer band by photodetaching I − (aq) . The results, however, suggest that either I* does not absorb near 315 nm as previously anticipated 21 or that I* is deactivated by electronic and vibrational energy transfer on timescales much shorter than 1 ps, thus ruling out a contribution by I* detectable in the photodissociation of ICN 65 . Interestingly, the bandwidth of∼ 3300 cm − 1 and max = 2300 M − 1 cm − 1 suggested by this work 1 for the CN radical are consistent with experimental transient spectra between 220 and 380 nm that include all products from the ICN photodissociation reaction 21 , if ground state I atoms account for the blue side and CN accounts for all absorption on the red end of the indicated range. 1 The extinction coefficient has been calculated using the experimental oscillator strength of the cyano radical in the gas phase (f=0.033) and the bandwidth computed using Set B+TIP5P/E (ν 1/2 = 3300 cm − 1 ) by using 1 : max (M − 1 cm − 1 ) = 2.31· 10 8 · f· ν − 1 1 2 209 To unambiguously confirm that CN is responsible for the 320 - 400 nm absorption in the experimental studies, we are carrying out a similar study based on photodetachment of CN − (aq) . These will provide a rigorous calibration of the theoretical models applied in this work and information on charge transfer bands occurring in the spectrum of CN, in addition to the valence states considered here. 210 6.6 Chapter 6 Bibliography [1] Huber, K.P. ; Herzberg, G. Constants of diatomic molecules; Van Nostrand Rein- hold: New York, 1979. [2] Hug, G.L. Optical Spectra of Nonmetallic Inorganic Transient Species in Aqueous Solution; U.S. Department of Commerce, National Bureau of Standards: Wash- ington, D.C., 1981. [3] Brocklehurst, B. ; Hebert, G.R. ; Innanen, S.H. ; Seel, R.M. ; Nicholls, R.W. The identication atlas of molecular spectra. the CN A 2 Π ← X 2 Σ + Red System ; York University, Centre for Research in Experimental Space Science, Toronto, 1971. [4] Brocklehurst, B. ; Hebert, G.R. ; Innanen, S.H. ; Seel, R.M. ; Nicholls, R.W. The identication atlas of molecular spectra. the CN B 2 Σ + ← X 2 Σ + Red System ; York University, Centre for Research in Experimental Space Science, Toronto, 1972. [5] Nadler, I. ; Reisler, H. ; Wittig, C. Chem. Phys. Lett. 1984, 103, 451. [6] Nadler, I. ; Mahgerefteh, D. ; Reisler, H. ; Wittig, C. J. Chem. Phys. 1985, 82, 3885. [7] Casavecchia, P. ; Balucani, N. ; Cartechini, L. ; Capozza, G. ; Bergeat, A. ; V olpi, G.G. Faraday Discuss. 2001, 119, 27. [8] Ling, J.G. ; Wilson, K.R. J. Chem. Phys. 1975, 63, 101. [9] Goldfield, E.M. ; Houston, P.L. ; Ezra, G.S. J. Chem. Phys. 1986, 84, 3120. [10] Dantus, M. ; Rosker, M.J. ; Zewail, A.H. J. Chem. Phys. 1988, 89, 6128. [11] Benjamin, I. ; Wilson, K.R. J. Chem. Phys. 1989, 90, 4176. [12] Krylov, A.I. ; Gerber, R.B. J. Chem. Phys. 1994, 100, 4242. [13] Amatatsu, Y . ; Morokuma, K. Chem. Phys. Lett. 1995, 245, 469. [14] Benjamin, I. J. Chem. Phys. 1995, 103, 2459. [15] Vieceli, J. ; Chorny, I. ; Benjamin, I. J. Chem. Phys. 2001, 115, 4819. [16] Winter, N. ; Chorny, I. ; Vieceli, J. ; Benjamin, I. J. Chem. Phys. 2003, 119, 2127. [17] Wan, C. ; Gupta, M. ; Zewail, A.H. Chem. Phys. Lett. 1996, 256, 279. [18] Moskun, A.C. ; Bradforth, S.E. J. Chem. Phys. 2003, 119, 4500. 211 [19] Helbing, J. ; Chergui, M. J. Phys. Chem. A 2000, 104, 10293. [20] Jailaubekov, A.C. Moskun A.E. ; Bradforth, S.E. ; Tao, G. ; Stratt, R.M. Science 2006, 311, 1907. [21] Larsen, J. ; Madsen, D. ; Poulsen, J.-A. ; Poulsen, T.D. ; Keiding, S.R. ; Thogersen, J. J. Chem. Phys. 2002, 116, 7997. [22] Orozco, M. ; Luque, F.J. Chem. Rev. 2000, 100, 4187. [23] Cramer, C.J. ; Truhlar, D.G. Chem. Rev. 1999, 99, 2161. [24] Luzhkov, V . ; Warshel, A. J. Am. Chem. Soc. 1991, 113, 4491. [25] Smith, P.E. ; Pettitt, B.M. J. Phys. Chem. 1994, 98, 9700. [26] Gao, J. ; Byun, K. Theor. Chim. Acta 1997, 96, 151. [27] Bradforth, S.E. ; Jungwirth, P. J. Phys. Chem. A 2002, 106, 1286. [28] Winter, N. ; Chorny, I. ; Vieceli, J. ; Benjamin, I. J. Chem. Phys. 2003, 119, 2127. [29] Mercer, I.P. ; Gould, I.R. ; Klug, D.R. J. Phys. Chem. B 1999, 103, 7720. [30] Warshel, A. ; Levitt, M. J. Mol. Biol. 1976, 103, 227. [31] Thompson, M.A. ; Schenter, G.K. J. Phys. Chem. 1995, 99, 6374. [32] Kongsted, J. ; Osted, A. ; Mikkelsen, K.V . ; Christiansen, O. Mol. Phys. 2002, 100, 1813. [33] Kongsted, J. ; Osted, A. ; Mikkelsen, K.V . ; Astrand, P.O. ; Christiansen, O. J. Chem. Phys. 2004, 121, 8435. [34] Gordon, M.S. ; Freitag, M.A. ; Bandyopadhyay, P. ; Jensen, J.H. ; Kairys, V . ; Stevens, W.J. J. Phys. Chem. A 2001, 105, 293. [35] Bandyopadhyay, P. ; Gordon, M.S. ; Mennucci, B. ; Tomasi, J. J. Chem. Phys. 2002, 116, 5023. [36] Krauss, M. Computers Chem. 1995, 19, 33. [37] Case, D.A. ; Darden, T.A. ; III, T.E. Cheetham ; Simmerling, C.L. ; Wang, J. ; Duke, R.E. ; Luo, R. ; Merz, K.M. ; Wang, B. ; Pearlman, D.A. ; Crowley, M. ; Brozell, S. ; Tsui, V . ; Gohlke, H. ; Mongan, J. ; Hornak, V . ; Cui, G. ; Beroza, P. ; Schafmeister, C. ; Caldwell, J.W. ; Ross, W.S. ; Kollman, P.A. Amber 8, University of California, San Francisco 2004. 212 [38] Kong, J. ; White, C.A. ; Krylov, A.I. ; Sherrill, C.D. ; Adamson, R.D. ; Furlani, T.R. ; Lee, M.S. ; Lee, A.M. ; Gwaltney, S.R. ; Adams, T.R. ; Ochsenfeld, C. ; Gilbert, A.T.B. ; Kedziora, G.S. ; Rassolov, V .A. ; Maurice, D.R. ; Nair, N. ; Shao, Y . ; Besley, N.A. ; Maslen, P. ; Dombroski, J.P. ; Daschel, H. ; Zhang, W. ; Korambath, P.P. ; Baker, J. ; Bird, E.F.C. ; Van V oorhis, T. ; Oumi, M. ; S. Hirata, C.-P. Hsu ; Ishikawa, N. ; Florian, J. ; Warshel, A. ; Johnson, B.G. ; Gill, P.M.W. ; Head-Gordon, M. ; Pople, J.A. J. Comput. Chem. 2000, 21, 1532. [39] ACES II. Stanton, J.F. ; Gauss, J. ; Watts, J.D. ; Lauderdale, W.J. ; Bartlett, R.J. 1993. The package also contains modified versions of the MOLECULE Gaussian integral program of J. Alml¨ of and P.R. Taylor, the ABACUS integral derivative program written by T.U. Helgaker, H.J.Aa. Jensen, P. Jørgensen and P.R. Taylor, and the PROPS property evaluation integral code of P.R. Taylor. [40] Thomson, R. ; Dalby, F.W. Can. J. Phys. 1968, 46, 2815. [41] Purvis, G.D. ; Bartlett, R.J. J. Chem. Phys 1982, 76, 1910. [42] Raghavachari, K. ; Trucks, G.W. ; Pople, J.A. ; Head-Gordon, M. Chem. Phys. Lett. 1989, 157, 479. [43] Watts, J.D. ; Gauss, J. ; Bartlett, R.J. J. Chem. Phys. 1993, 98, 8718. [44] Cristian, A.M.C. ; Shao, Y . ; Krylov, A.I. J. Phys. Chem. A 2004, 108, 6581. [45] Becke, A.D. J. Chem. Phys. 1993, 98, 5648. [46] Kendall, R.A. ; Dunning Jr., T.H. ; Harrison, R.J. J. Chem. Phys. 1992, 96, 6796. [47] Krishnan, R. ; Binkley, J.S. ; Seeger, R. ; Pople, J.A. J. Chem. Phys. 1980, 72, 650. [48] Clark, T. ; Chandrasekhar, J. ; Schleyer, P.V .R. J. Comput. Chem. 1983, 4, 294. [49] Boys, S.F. ; Bernardi, F. Mol. Phys. 1970, 19, 65. [50] Sekino, H. ; Bartlett, R.J. Int. J. Quant. Chem. Symp. 1984, 18, 255. [51] Stanton, J.F. ; Bartlett, R.J. J. Chem. Phys. 1993, 98, 7029. [52] Runge, E. ; Gross, E.K.U. Phys. Rev. Lett. 1984, 52, 997. [53] Hirata, S. ; Head-Gordon, M. Chem. Phys. Lett. 1999, 314, 291. [54] Lovas, F.J. J. Phys. Chem. Ref. Data 1978, 7, 1445. [55] Berendsen, H.J.C. ; Grigera, J.R. ; Straatsma, T.P. J. Phys. Chem. 1987, 91, 6269. 213 [56] NBO 4.0. Glendening, E.D. ; Badenhoop, J.K. ; Reed, A.E. ; Carpenter, J.E. ; Weinhold, F. Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, 1996. [57] Rick, S.W. J. Chem. Phys. 2004, 120, 6085. [58] Mahoney, M.W. ; Jorgensen, W.L. J. Chem. Phys. 2000, 112, 8910. [59] Berendsen, H.J.C. ; Postma, J.P.M. ; van Gunsteren, W.F. ; DiNola, A. ; Haak, J.R. J. Chem. Phys. 1984, 81, 3684. [60] Cornell, W.D. ; Cieplak, P. ; Bayly, C.I. ; Gould, I.R. ; Merz, K.M. ; Ferguson, D.M. ; Spellmeyer, D.C. ; Fox, T. ; Caldwell, J.W. ; Kollman, P.A. J. Am. Chem. Soc. 1995, 117, 5179. [61] Wang, B. ; Hou, H. ; Gu, Y . Chem. Phys. Lett. 1999, 303, 96. [62] Cook, T.J. ; Levy, D.H. J. Chem. Phys. 1973, 59, 2387. [63] Tozer, D.J. ; Amos, R.D. ; Handy, N.C. ; Roos, B.O. ; Serrano-Andres, L. Mol. Phys. 1999, 97, 859. [64] Bayliss, N.S. ; McRae, E.G. J. Phys. Chem. 1954, 58, 1002. [65] Moskun, A.C ; Bradforth, S.E. ; Thogersen, J. ; Keiding, S. J. Phys. Chem. A 2006, 110, 10947. 214 Chapter 7: Future work 7.1 Overview This work largely focused on developing quantitative and qualitative methods for the description of dimer cation systems and then applications to model systems. They are interesting in their own right, but in the long run, they pave the way to understanding systems occurring in nature. In this Chapter we would like to explore several prospective directions of future research. Both the pertinent methods and the scientific questions are addressed. 7.2 New ideas The benzene dimer cation and water dimer cation projects merit further extensions. The big question is the nature of the ionized states in heterogeneous fluctuating environ- ments. The heterogeneity can be introduced both by the chemical nature of the chro- mophores, as well as their different local environment. In bulk water all molecules are the same, however because their local environment is different their energy levels and couplings are also different. In DNA, both the surroundings of neighboring nucleobases and their chemical composition are different. The hole can be either localized on a par- ticular chromophore, or spread over several of them. The extent of this effect cannot be quantified without accurate and reliable calculations. Also, delocalization will depend whether the nuclear degrees of freedom had the time to relax or not. The nuclear dynam- ics ensuing upon electron removal may lead to a period of charge delocalization or to 215 an immediate formation of a multimeric or monomeric core cation. The character of this dynamics is largely not known. In DNA, it is likely to involve internal degrees of freedom of the strand, as well as the solvent motion. Finally, the physical systems are typically at room temperature, meaning that the nature of the hole will evolve in time, i.e., it can be localized on one base for some time and than become delocalized. Also, the fluctuations of the surroundings my lead to the destabilization of the hole and its detrapping. Thus, the study of the amplitude of such fluctuations is warranted. Exploring those issues will require fast and reliable methods suitable for large sys- tems, as described in the next paragraph. Once those methods are in place, they can be combined with the classical molecular dynamics to gain insight into the charge transfer processes in DNA. It is widely recognized that the stacked dimers and trimers of guanine (G) residues are particularly efficient hole traps due to their low ionization energies 1–4 . Is the hole localized on one of them or is it distributed? If it is distributed, what is the fraction on each residue? How does it change with time? The need for a first-principles molecular level simulation is highlighted by the conflicting theoretical results, which rely on the estimate of the crucial coupling and solvation parameters. Depending on the choice of the method, the hole can encompass either a single nucleobase or five nucle- obases 5–7 . Bouvier combined the the valence bond Hamiltonian approach with molecu- lar dynamics simulations to study the ionization of small aromatic clusters 8, 9 . Despite the neglect of the intramolecular degrees of freedom, this approach is very promising. However, it has never been extended to large heterosystems, for which parametrization me prove significantly more difficult. We propose that a small solvated duplex containing the GG and GGG motifs [eg. ((TA)(GC) 3 (TA)] can be simulated with classical molecular mechanics methods. This will allow to explore the vast conformational space of the system. At select point along this trajectory quantum mechanics/molecular mechanics (QM/MM) calculations, where 216 guanine nucleobases are treated using quantum mechanics and the remaining part of the system using molecular mechanics, will be performed. The calculation would include population analysis aimed at validating the distribution of the hole. Several low-lying excited states could also be calculated, evaluating the possible spectroscopic probing of the system. Both neighboring bases, thymine and cytosine, have a higher ionization potential than guanine, which should favor the hole to stay within the G manifold. Trial calculations should nonetheless be performed aimed at evaluating this assumption. This part of the simulation will explore the nature of the vertical state, i.e. the state that is instantaneously formed upon the removal of an electron. For a representative, however limited due to the computational cost, set of trajectories QM/MM dynamics of the hole could be explored. In the dynamical calculation itself the positively charged guanine stack would be treated using quantum mechanics, while molecular mechanics would be used to describe the rest of the system. Thus this simulation would explore the cation potential energy surface. The nature of the state and its spectroscopic properties could be explored as described above, but additionally the distortion of the strand caused by ionization could be addressed. Also trajectory of the hole could be explored i.e., whether it stay localized or does it move around. Quantum treatment of the stacked trimer would account for the diabatic coupling and Jahn-Teller effects, while molecular mechanics part would account for solvation effects. In both simulations polarizable force fields should be used to describe both the DNA strand and water. As was shown, ionization can induce strong electronic and nuclear relaxation 10, 11 . Such force fields are readily available 12 . Dimer and small water cluster calculations suggest that initially the hole is indeed delocalized. After a period of time, it localizes at a given spot causing the proton transfer reaction. However, nothing is known about this initial period and the methods currently used to describe it may introduce artifacts in its description. In case of the density 217 functional theory (DFT), there is the tendency to delocalize the charge due to the self- interaction error (SIE), while in case of Hartree-Fock (HF) it is the opposite. A possible solution is Koopmans HF approach delineated below. Furthermore, it is reasonable to suggest the water molecules that are not hydrogen bond acceptors are particularly suit- able hole traps. This is based on the electrostatic considerations, which suggest the the hydrogen will destabilize the cation, thus increasing its ionization energy. Such defects can be introduced into the structure of bulk water by a solute. In particular nonpolar solutes are know to weaken the hydrogen bonds. It is thus possible that water in their immediate vicinity can be most easily ionized leading to chemistry. This hypothesis could be explored using Car-Parrinello dynamics of a box of water with methane or ethane. The equation-of-motion coupled-cluster with single and double substitutions for ion- ized states (EOM-IP-CCSD) method is computationally expensive and in practice lim- ited to medium-size systems, i.e. similar in size to the benzene dimer cation. Two avenues to extending its applicability should be explored. The bottleneck is the CC calculation, which scales as N 6 , where N denotes the number of orbitals. It may be accelerated using the resolution-of-identity methods, but they do not change the com- putational scaling. These approaches typically require a large basis set to be applicable, thus it is not obvious whether practical gains in the size of the system are possible. A more realistic approach is combining EOM-IP with the second-order approximate cou- pled cluster singles and doubles model (CC2) 13 . This model scales as N 5 . Treating even larger systems should be feasible using the Koopmans Hartree-Fock wave functions – by this we mean simply removing an orbital from the HF wave function. This approach lacks correlation, which may change the nature of the states in systems comprising many (nearly)-degenerate sites (like a box of water). Thus, careful benchmarking on systems 218 like helium clusters and small water clusters, which are also accessible to EOM-IP- CCSD, must be performed. Testing should not only include static, but also the dynamic properties of these clusters. The EOM-IP-CC methodology developed here is suitable for the description of cations. However, dimer anions are also of interest and can be described by the equation- of-motion coupled-cluster for electron attachment (EOM-EA-CC) methodology. For instance, the thymine dimer is a very common form of DNA photodamage 14, 15 . The mechanism of repair involves injection of an electron to a DNA strand, formation of a thymine dimer anion, and cleavage of the cyclobutane ring. The nature of the excess electron in the acetonitrile poses an interesting problem 16 . Two species, assigned as the solvated electron and radical anion have been observed. However, the nature of the presumably multimeric radical anion is not known. The EOM-EA-CC calculation also starts with the neutral reference, however electron is added to the system. The anion open-shell wave functions will suffer from analogous symmetry breaking problems and excessive charge localization. Developing working codes for this method would paral- lel the work done here. A possible problem may occur with benchmarking the method. Neutral closed-shell species typically do not bind an electron, unless they have a large dipole moment or are immersed in an electric field, Variational methods are are not suitable for obtaining these wave functions as they are buried in the continuum. One solution it is to use a basis set that does not support diffuse wave functions correspond- ing to electron detachment. Alternatively, a continuum solvation model may by used in the benchmarking to introduce the polarization effect needed to stabilize the anion. 219 7.3 Chapter 7 Bibliography [1] Saito, I. ; Nakamura, T. ; Nakatani, K. ; Yoshioka, Y . ; Yamaguchi, K. ; Sugiyama, H. J. Am. Chem. Soc. 1998, 120, 12686. [2] Nunez, M.E. ; Hall, D.B. ; Barton, J.K. Chem. Biol. 1999, 6, 85. [3] Giese, B. Acc. Chem. Res. 2000, 33, 631. [4] Lewis, F.D. Photochem. Photobiol. 2005, 81, 65. [5] Kurnikov, I.V . ; Tong, G.S.M. ; Madrid, M. ; Beratan, D.N. J. Phys. Chem. B 2002, 106, 7. [6] Basko, D.M. ; Conwell, E.M. Phys. Rev. Lett. 2002, 88, 098102. [7] V oityuk, A.A. J. Chem. Phys. 2005, 122, 204904. [8] Bouvier, B. ; Brenner, V . ; Millie, P. ; Soudan, J.M. J. Phys. Chem. A 2002, 106, 10326. [9] Bouvier, B. ; Millie, P. ; Mons, M. J. Phys. Chem. A 2004, 108, 4254. [10] Winter, B. ; Weber, R. ; Widdra, W. ; Dittmar, M. ; Faubel, M. ; Hertel, I.V . J. Phys. Chem. A 2004, 108, 2625. [11] Jagoda-Cwiklik, B. ; Slaviek, P. ; Cwiklik, L. ; Nolting, D. ; Winter, B. ; Jungwirth, P. J. Phys. Chem. B 2008, 112, 3499. [12] Baucom, J. ; Transue, T. ; Fuentes-Cabrera, M. ; Krahn, J.M. ; Darden, T. ; Sagui, C. J. Chem. Phys. 2004, 121, 6998. [13] Christiansen, O. ; Koch, H. ; Jorgensen, P. Chem. Phys. Lett. 1995, 243, 409. [14] Dumaz, N. ; van Kranen, H.J. ; de Vries, A. ; Berg, R.J.W. ; Wester, P.W. ; van Kreijl, C.F. ; Sarasin, A. ; Daya-Grosjean, L. ; de Gruijl, F.R. Carcinogenesis 1997, 18, 897. [15] Masson, F. ; Laino, T. ; Tavernelli, I. ; Rothlisberger, U. ; Hutter, J. J. Am. Chem. Soc. 2008, 130, 3443. [16] Takayanagi, T. ; Hoshino, T. ; Takahashi, K. Chem. Phys. 2006, 324, 679. 220 Bibliography [1] M.E. Akopian, F.I. Vilesov, and A.N. Terenin. A mass-spectrometer investiga- tion of how photoionization efficiency of benzene derivatives is related to their spectra. Sov. Phys. Doklady (Engl. Transl.), 6:490, 1961. [2] M.H. Alexander. Adiabatic and approximate diabatic potential energy surfaces for the B··· H 2 van der Waals molecule. J. Chem. Phys., 99:6014–6026, 1993. [3] S. Aloisio and J.S. Francisco. Radical-water complexes in Earth’s atmosphere. Acc. Chem. Res., 33:825–830, 2000. [4] Y . Amatatsu and K. Morokuma. A theoretical study on the photochemical reac- tion of ICN in liquid Ar. Chem. Phys. Lett., 245:469–474, 1995. [5] A. Amini and A. Harriman. Computational methods for electron-transfer sys- tems. J. Photochem. Photobiol. C, 4:155–177, 2003. [6] E. Arunan and H.S. Gutowsky. The rotational spectrum, structure and dynamics of a benzene dimer. J. Chem. Phys., 98:4294–4296, 1993. [7] B. Askew, P. Ballester, C. Buhr, K.S. Jeong, S. Jones, K. Parris, K. Williams, and J. Rebek Jr. Molecular recognition with convergent functional groups. VI. Syn- thetic and structural studies with a model receptor for nucleic acid components. J. Am. Chem. Soc., 111:1082–1090, 1989. [8] P.W. Atkins and R.S. Friedman. Molecular Quantum Mechanics. New York: Oxford University Press, 2005. [9] B. Badger and B. Brockleh. Absorption spectra of dimer cations. 2. Benzene derivatives. Trans. Farad. Soc., 65:2582–2587, 1969. [10] T. Bally and G.N. Sastry. Incorrect dissociation behavior of radical ions in density functional calculations. J. Phys. Chem. A, 101:7923–7925, 1997. [11] P. Baltzer, L. Karlsson, B. Wannberg, G. Ohrwall, D.M.P. Holland, M.A. Mac- Donald, M.A. Hayes, and D. von Niessen. An experimental and theoretical study of the valence shell photoelectron spectrum of the benzene molecule. Chem. Phys., 224:95–119, 1997. 221 [12] P. Bandyopadhyay, M.S. Gordon, B. Mennucci, and J. Tomasi. An integrated effective fragment polarizable continuum approach to solvation: Theory and application to glycine. J. Chem. Phys., 116:5023–5032, 2002. [13] M.S. Banna, B.H. McQuaide, R. Malutzki, and V . Schmidt. The photoelectron spectrum of water in the 30 to 140 eV photon energy range. J. Chem. Phys., 84, 1986. [14] R.N. Barnett and U. Landman. Pathways and dynamics of dissociation of ionized (H 2 O) 2 . J. Phys. Chem., 99:17305–17310, 1995. [15] R.N. Barnett and U. Landman. Structure and energetics of ionized water clusters: (H 2 O) + n , n=2-5. J. Phys. Chem. A, 101:164–169, 1997. [16] R.J. Bartlett and J.F. Stanton. Applications of post-Hartree-Fock methods: A tutorial. Rev. Comp. Chem., 5:65–169, 1994. [17] D.M. Basko and E.M. Conwell. Effect of solvation on hole motion in DNA. Phys. Rev. Lett., 88:098102, 2002. [18] J. Baucom, T. Transue, M. Fuentes-Cabrera, J.M. Krahn, T. Darden, and C. Sagui. Molecular dynamics simulations of the d(CCAACGTTGG) 2 decamer in crystal environment: Comparison of atomic point-charge, extra-point, and polarizable force fields. J. Chem. Phys., 121:6998–7008, 2004. [19] N.S. Bayliss and E.G. McRae. Solvent effects in organic spectra: dipole forces and the Franck-Condon principle. J. Phys. Chem., 58:1002–1006, 1954. [20] A.D. Becke. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys., 98:5648–5652, 1993. [21] A.D. Becke and E.R. Johnson. A unified density-functional treatment of dynam- ical, nondynamical, and dispersion correlations. J. Chem. Phys., 127:124108, 2007. [22] I. Benjamin. Photodissociation of ICN in liquid chloroform: Molecular dynamics of ground and excited state recombination, cage escape, and hydrogen abstraction reaction. J. Chem. Phys., 103:2459–2471, 1995. [23] I. Benjamin and K.R. Wilson. Proposed experimental probes of chemical reac- tion molecular dynamics in solution: ICN photodissociation. J. Chem. Phys., 90:4176–4197, 1989. [24] H.J.C. Berendsen, J.R. Grigera, and T.P. Straatsma. The missing term in effective pair potentials. J. Phys. Chem., 91:6269–6271, 1987. 222 [25] H.J.C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, A. DiNola, and J.R. Haak. Molecular dynamics with coupling to an external bath. J. Chem. Phys., 81:3684– 3690, 1984. [26] J.B. Birks. Photophysics of Aromatic Molecules. Wiley: New York, 1970. [27] E.R. Bittner. Lattice theory of ultrafast excitonic and charge-transfer dynamics in dna. J. Chem. Phys., 125:094909, 2006. [28] M. Bixon and J. Jortner. Hole trapping, detrapping, and hopping in DNA. J. Phys. Chem. A, 105:10322–10328, 2001. [29] M. Boero, M. Parrinello, K. Terakura, T. Ikeshoji, and C.C. Liew. First-principles molecular-dynamics simulations of a hydrated electron in normal and supercriti- cal water. Phys. Rev. Lett., 90:226403, 2003. [30] Y .J. Bomble, J.C. Saeh, J.F. Stanton, P.G. Szalay, M. K´ allay, and J. Gauss J. Equation-of-motion coupled-cluster methods for ionized states with an approxi- mate treatment of triple excitations. J. Chem. Phys., 122:154107, 2005. [31] B. Bouvier, V . Brenner, P. Millie, and J.M. Soudan. A model potential approach to charge resonance phenomena in aromatic cluster ions. J. Phys. Chem. A, 106:10326–10341, 2002. [32] B. Bouvier, P. Millie, and M. Mons. Investigation of the photoionization mecha- nism of small aromatic homoclusters. J. Phys. Chem. A, 108:4254–4260, 2004. [33] S.F. Boys and F. Bernardi. Calculation of small molecular interactions by dif- ferences of separate total energies - some procedures with reduced errors. Mol. Phys., 19:65–73, 1970. [34] S.E. Bradforth and P. Jungwirth. Excited states of iodide anions in water: A comparison of the electronic structure in clusters and in bulk solution. J. Phys. Chem. A, 106:1286–1298, 2002. [35] B. Brocklehurst, G.R. Hebert, S.H. Innanen, R.M. Seel, and R.W. Nicholls. The identication atlas of molecular spectra. the CN A 2 Π ← X 2 Σ + Red System . York University, Centre for Research in Experimental Space Science, Toronto, 1971. [36] B. Brocklehurst, G.R. Hebert, S.H. Innanen, R.M. Seel, and R.W. Nicholls. The identication atlas of molecular spectra. the CN B 2 Σ + ← X 2 Σ + Red System . York University, Centre for Research in Experimental Space Science, Toronto, 1972. [37] P.J. Bruna and S.D. Peyerimhoff. Excited-state potentials. In Ab initio methods in quantum chemistry, I, pages 1–98. John Wiley & Sons, 1987. 223 [38] S.K. Burley and G.A Petsko. Aromatic-aromatic interaction: a mechanism of protein structure stabilization. Science, 229:23–28, 1985. [39] C. Carra, N. Iordanova, and S. Hammes-Schiffer. Proton-coupled electron trans- fer in dna-acrylamide complexes. J. Phys. Chem. B, 106:8415–8412, 2002. [40] P. Casavecchia, N. Balucani, L. Cartechini, G. Capozza, A. Bergeat, and G.G. V olpi. Crossed beam studies of elementary reactions of N and C atoms and CN radicals of importance in combustion. Faraday Discuss., 119:27–49, 2001. [41] D.A. Case, T.A. Darden, T.E. Cheetham III, C.L. Simmerling, J. Wang, R.E. Duke, R. Luo, K.M. Merz, B. Wang, D.A. Pearlman, M. Crowley, S. Brozell, V . Tsui, H. Gohlke, J. Mongan, V . Hornak, G. Cui, P. Beroza, C. Schafmeister, J.W. Caldwell, W.S. Ross, and P.A. Kollman. Amber 8, University of California, San Francisco, 2004. [42] M.C. Castex. Experimental determination of the lowest excited Xe 2 molecular states from VUV absorption spectra. J. Chem. Phys., 74:759–771, 1981. [43] R.J. Cave and M.D. Newton. Generalization of the Mulliken-Hush treatment of the calculation of electron transfer matrix elements. Chem. Phys. Lett., 249:15– 19, 1996. [44] R.J. Cave and M.D. Newton. Calculation of electronic coupling matrix ele- ments for ground and excited state electron transfer reactions: Comparison of the generalized Mulliken-Hush and block diagonalization method. J. Chem. Phys., 106(22):9213–9226, 1997. [45] P. Celani and H.-J. Werner. Analytical energy gradients for internally contracted second-order multireference perturbation theory. J. Chem. Phys., 119:5044– 5057, 2003. [46] J.-D. Chai and M. Head-Gordon. Systematic optimization of long-range corrected hybrid density functionals. J. Chem. Phys., 128:084106, 2008. [47] R. Chaudhuri, D. Mukhopadhyay, and D. Mukherjee. Applications of open-shell coupled cluster theory using an eigenvalue-independent partitioning technique: Approximate inclusion of triples in IP calculations. Chem. Phys. Lett., 162:393, 1989. [48] V .J. Chebny, R. Shukla, and R. Rathore. Toroidal hopping of a single hole through the circularly-arrayed naphthyl groups in hexanaphthylbenzene cation radical. J. Phys Chem. A, 110:13003–13006, 2006. 224 [49] B.-M. Cheng, E.P. Chew, C.-P. Liu, M. Bahou, Y .-P. Lee, Y .L. Yung, and M.F. Gerstell. Photo-induced fractionation of water isotopomers in the martian atmo- sphere. Geophys. Res. Lett., 26:3657–3660, 1999. [50] O. Christiansen, H. Koch, and P. Jorgensen. The second-order approximate cou- pled cluster singles and doubles model CC2. Chem. Phys. Lett., 243:409, 1995. [51] C.G. Claessens and J.F. Stoddart. π -π interactions in self-assembly. J. Phys. Org. Chem., 10:254–272, 1997. [52] T. Clark, J. Chandrasekhar, and P.V .R. Schleyer. Efficient diffuse function- augmented basis sets for anion calculations. III. The 3-21+g basis set for first-row elements, Li-F. J. Comput. Chem., 4:294–301, 1983. [53] R.D. Cohen and C.D. Sherrill. The performance of density functional theory for equilibrium molecular properties of symmetry breaking molecules. J. Chem. Phys., 114:8257, 2001. [54] A.J. Cohenand, P. Mori-S´ anchez, and W. Yang. Development of exchange- correlation functionals with minimal many-electron self-interaction error. J. Chem. Phys., 126:191109, 2007. [55] T.J. Cook and D.H. Levy. Electric dipole moment of the A 2 Π state of CN as measured by level anticrossing spectroscopy. J. Chem. Phys., 59:2387–2393, 1973. [56] W.D. Cornell, P. Cieplak, C.I. Bayly, I.R. Gould, K.M. Merz, D.M. Ferguson, D.C. Spellmeyer, T. Fox, J.W. Caldwell, and P.A. Kollman. A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J. Am. Chem. Soc., 117:5179–5197, 1995. [57] C.J. Cramer and D.G. Truhlar. Implicit solvation models: Equilibria, structure, spectra, and dynamics. Chem. Rev., 99:2161–2200, 1999. [58] T.D. Crawford, C.D. Sherrill, E.F. Valeev, J.T. Fermann, R.A. King, M.L. Leininger, S.T. Brown, C.L. Janssen, E.T. Seidl, J.P. Kenny, and W.D. Allen. PSI3: An open-source ab initio electronic structure package. J. Comput. Chem., 28:1610–1616, 2007. [59] A.M.C. Cristian, Y . Shao, and A.I. Krylov. Bonding patterns in benzene trirad- icals from structural, spectroscopic, and thermochemical perspectives. J. Phys. Chem. A, 108:6581–6588, 2004. [60] L.A. Curtiss. Abinitio molecular-orbital calculations of the 1st 2 adiabatic ion- izations of the water dimer. Chem. Phys. Lett., 96:442–446, 1983. 225 [61] L.A. Curtiss. Theoretical investigation into the nature of the 2nd vertical ionic state of the water dimer. Chem. Phys. Lett., 112:409–411, 1984. [62] M. Dantus, M.J. Rosker, and A.H. Zewail. Femtosecond real-time probing of reactions. II. The dissociation reaction of ICN. J. Chem. Phys., 89:6128–6140, 1988. [63] B. Das and J.W. Farley. Observation of the visible absorption spectrum of H 2 O + . J. Chem. Phys., 95:8809–8815, 1991. [64] M. d’Avezac, M. Calandra, and F. Mauri. Density functional theory description of hole-trapping in SiO 2 : A self-interaction-corrected approach. Phys. Rev. B, 71:205210, 2005. [65] E.R. Davidson. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys., 17:87–94, 1975. [66] E.R. Davidson and W.T. Borden. Symmetry breaking in polyatomic molecules: Real and artifactual. J. Phys. Chem., 87:4783–4790, 1983. [67] S. P. de Visser, L.J. de Koning, and N.M.M. Nibbering. Reactivity and thermo- chemical properties of the water dimer radical cation in the gas phase. J. Phys. Chem., 99:15444–15447, 1995. [68] S. Delaney and J.K. Barton. Long-range DNA charge transport. J. Org. Chem., 68:6475–6482, 2003. [69] A.J. Dobbyn and P.J. Knowles. A comparative study of methods for describing non-adiabatic coupling: Diabatic representation of the 1 Σ + / 1 Π HOH and HHO conical intersections. Mol. Phys., 91(6):1107–1123, 1997. [70] M. Doscher, H. Koppel, and P.G. Szalay. Multistate vibronic interactions in the benzene radical cation. I. Electronic structure calculations. J. Chem. Phys., 117:2645–2656, 2002. [71] A. Dreuw and M. Head-Gordon. Single-reference ab initio methods for the cal- culation of excited states of large molecules. Chem. Rev., 105:4009–4037, 2005. [72] A. Dreuw and M. Head-Gordon. Comment on: ’Failure of time-dependent den- sity functional methods for excitations in spatially separated systems’ by Wolf- gang Hieringer and Andreas Gorling. Chem. Phys. Lett., 426:231–233, 2006. [73] N. Dumaz, H.J. van Kranen, A. de Vries, R.J.W. Berg, P.W. Wester, C.F. van Kreijl, A. Sarasin, L. Daya-Grosjean, and F.R. de Gruijl. The role of UV-B light in skin carcinogenesis through the analysis of p53 mutations in squamous cell carcinomas of hairless mice. Carcinogenesis, 18:897–904, 1997. 226 [74] A.L.L. East and E.C. Lim. Naphthalene dimer: Electronic states, excimers, and triplet decay. J. Chem. Phys., 113,:8981–8994, 2000. [75] W. Eisfeld and K. Morokuma. A detailed study on the symmetry breaking an its effect on the potential surface of NO 3 . J. Chem. Phys., 113:5587–5597, 2000. [76] C.G. Elles, A.E. Jailaubekov, R.A. Crowell, and S.E. Bradforth. Excitation- energy dependence of the mechanism for two-photon ionization of liquid H 2 O and D 2 O from 8.3 to 12.4 eV. J. Chem. Phys., 125:44515, 2006. [77] C.G. Elles, I.A. Shkrob, R.A. Crowell, and S.E. Bradforth. Excited state dynam- ics of liquid water: Insight from the dissociation reaction following two-photon excitation. J. Chem. Phys., 126:64503, 2007. [78] K. Emrich. An extension of the coupled-cluster formalism to excited states (I). Nucl. Phys., A351:379–396, 1981. [79] R.G. Endres, D.L. Cox, and R.R.P. Singh. Colloquium: The quest for high- conductance DNA. Rev. Mod. Phys., 76:195–214, 2004. [80] K. Enomoto, J.A. LaVerne, and M. S. Araos. Heavy ion radiolysis of liquid pyridine. J. Phys. Chem. A, 111:9–15, 2007. [81] E.E. Ferguson, F.C. Fehsenfield, and D.L. Albritton. In M. T. Bowers, editor, Gas Phase Ion Chemistry, volume 1, pages 45–82. Academic Press, 1979. [82] F.H. Field, P. Hamlet, and W.F. Libby. Effect of temperature on the mass spectra of benzene at high pressures. J. Am. Chem. Soc., 91:2839–2842, 1969. [83] L. Flamigni, N. Camaioni, P. Bortolus, F. Minto, and M. Gleria. Intramolecular naphthalene triplet excimers in solutions of phosphazene copolymers. J. Phys. Chem., 95:971–975, 1991. [84] M. Ford, R. Lindner, and K. M¨ uller-Dethlefs. Fully rotationally resolved zeke photoelectron spectroscopy of C 6 H 6 and C 6 D 6 : Photoionization dynamics and geometry of the benzene cation. Mol. Phys., 101:705–716, 2003. [85] F. Furche and R. Ahlrichs. Adiabatic time-dependent density functional methods for excited state properties. J. Chem. Phys., 117:7433–7447, 2002. [86] A. Furuhama, M. Dupuis, and K. Hirao. Reactions associated with ionization in water: A direct ab initio dynamics study of ionization in (H 2 O) 17 . J. Chem. Phys., 124:164310, 2006. [87] J. Gao and K. Byun. Solvent effects on the n→ π ∗ transition of pyrimidine in aqueous solution. Theor. Chim. Acta, 96:151–156, 1997. 227 [88] M.E. Garcia, G.M. Pastor, and K.H. Bennemann. Delocalization of a hole in van-der-Waals clusters - ionization-potential of rare-gas and small Hg n clusters. Phys. Rev. B, 48:8388, 1993. [89] B.C. Garrett, D.A. Dixon, D.M. Camaioni, D.M. Chipman, M.A. Johnson, C.D. Jonah, G.A. Kimmel, J.H. Miller, T.N. Rescigno, P.J. Rossky, S.S. Xantheas, S.D. Colson, A.H. Laufer, D. Ray, P.F. Barbara, D.M. Bartels, K.H. Becker, H. Bowen, S.E. Bradforth, I. Carmichael, J.V . Coe, L.R. Corrales, J.P. Cowin, M. Dupuis, K.B. Eisenthal, J.A. Franz, M.S. Gutowski, K.D. Jordan, B.D. Kay, J.A. LaVerne JA, S.V . Lymar, T.E. Madey, C.W. McCurdy, D. Meisel, S. Mukamel, A.R. Nils- son, T.M. Orlando, N.G. Petrik, S.M. Pimblott, J.R. Rustad, G.K. Schenter, S.J. Singer, A. Tokmakoff, L.S. Wang, C. Wittig, and T.S. Zwier. Role of water in electron-initiated processes and radical chemistry: Issues and scientific advances. Chem. Rev., 105:355–389, 2005. [90] J. Gauss and J.F. Stanton. The equilibrium structure of benzene. J. Phys. Chem. A, 104:2865–2868, 2000. [91] J. Geertsen, M. Rittby, and R.J. Bartlett. The equation-of-motion coupled-cluster method: Excitation energies of Be and CO. Chem. Phys. Lett., 164:57–62, 1989. [92] F.L. Gervasio, A. Laio, M. Parrinello, and M. Boero. Charge localization in dna fibers. Phys. Rev. Lett., 94:158103, 2005. [93] B. Giese. Long-distance charge transport in DNA: The hopping mechanism. Acc. Chem. Res., 33:631–636, 2000. [94] T.J. Giese and D.M. York. High-level ab initio methods for calculation of poten- tial energy surfaces of van der Waals complexes. Int. J. Quantum Chem., 98:388– 408, 2004. [95] P.M.W. Gill and L. Radom. Structures and stabilities of singly charged 3-electron hemibonded systems and their hydrogen-bonded isomers. J. Am. Chem. Soc., 110:4931–4941, 1988. [96] E.D. Glendening, J.K. Badenhoop, A.E. Reed, J.E. Carpenter, J.A. Bohmann, C.M. Morales, and F. Weinhold. NBO 5.0. Theoretical Chemistry Institute, Uni- versity of Wisconsin, Madison, WI, 2001. [97] E.D. Glendening, J.K. Badenhoop, A.E. Reed, J.E. Carpenter, and F. Weinhold. NBO 4.0. Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, 1996. [98] S. Goedecker, M. Teter, and J. Hutter. Separable dual-space Gaussian pseudopo- tentials. Phys. Rev. B, 54:1703–1710, 1996. 228 [99] E.M. Goldfield, P.L. Houston, and G.S. Ezra. Nonadiabatic interactions in the photodissociation of ICN. J. Chem. Phys., 84:3120–3129, 1986. [100] M.S. Gordon, M.A. Freitag, P. Bandyopadhyay, J.H. Jensen, V . Kairys, and W.J. Stevens. The effective fragment potential method: A QM-based MM approach to modeling environmental effects in chemistry. J. Phys. Chem. A, 105:293–307, 2001. [101] H.B. Gray and J.R. Winkler. Electron tunneling through proteins. Q. Rev. Bio- phys., 36:341–371, 2003. [102] J.R. Grover, E.A. Walters, and E.T. Hui. Dissociation energies of the benzene dimer and dimer cation. J. Phys. Chem., 91:3233–3237, 1987. [103] S.R. Gwaltney and R.J. Bartlett. Gradients for the partitioned equation-of-motion coupled-cluster method. J. Chem. Phys., 110:62–71, 1999. [104] N.C. Handy and H.F. Schaefer III. On the evaluation of analytic energy deriva- tives for correlated wave functions. J. Chem. Phys., 81:5031–5033, 1984. [105] M. Haque and D. Mukherjee. Application of cluster expansion techniques to open-shells: Calculation of difference energies. J. Chem. Phys., 80(10):5058– 5069, 1984. [106] B. Hartke. Size-dependent transition from all-surface to interior-molecule struc- tures in pure neutral water clusters. Phys. Chem. Chem. Phys., 5:275–284, 2003. [107] J. Helbing and M. Chergui. Spectroscopy and photoinduced dynamics of ICN and its photoproducts in solid argon. J. Phys. Chem. A, 104:10293–10303, 2000. [108] T. Helgaker, P. Jørgensen, and J. Olsen. Molecular electronic structure theory. Wiley & Sons, 2000. [109] P.C. Hiberty, S. Humbel, D. Danovich, and S. Shaik. What is physically wrong with the description of odd-electron bonding by Hartree-Fock theory? A simple nonempirical remedy. J. Am. Chem. Soc., 117:9003–9011, 1995. [110] K. Hirao and H. Nakatsuji. A generalization of the Davidson’s method to large nonsymmetric eigenvalue problems. J. Comput. Phys., 45:246–254, 1982. [111] K. Hiraoka, S. Fujimaki, K. Aruga, and S. Yamabe. Stability and structure of benzene dimer cation (C 6 H 6 ) + 2 in the gas phase. J. Chem. Phys., 95:8413–8418, 1991. [112] S. Hirata and M. Head-Gordon. Time-dependent density functional theory within the Tamm-Dancoff approximation. Chem. Phys. Lett., 314:291–299, 1999. 229 [113] S. Hirata, M. Nooijen, and R.J. Bartlett. High-order determinantal equation- of-motion coupled-cluster calculations for ionized and electron-attached states. Chem. Phys. Lett., 328:459–468, 2000. [114] T. Hirata, H. Ikeda, and H. Saigusa. Dynamics of excimer formation and relax- ation in the t-shaped benzene dimer. J. Phys. Chem. A, 103:1014–1024, 1999. [115] P. Hobza, H.L. Selzle, and E.W. Schlag. Potential energy surface for the benzene dimer. results of ab initio CCSD(T) calculations show two nearly isoenergetic structures: T-shaped and parallel-displaced. J. Phys. Chem., 100:18790–18794, 1996. [116] P. Hobza, R. Zahradnik, and K. M¨ uller-Dethlefs. The world of non-covalent interactions: 2006. Collect. Czech. Chem. Commun., 71:443–531, 2006. [117] H. Hotop, T.A. Patterson, and W.C. Lineberger. High resolution photodetachment study of OH − and OD − in the threshold region 7000-6450 a. J. Chem. Phys., 60:1806–1812, 1974. [118] K.P. Huber and G. Herzberg. Constants of diatomic molecules (data prepared by J.W. Gallagher and R.D. Johnson, III). NIST Chemistry WebBook, NIST Standard Reference Database Number 69. Eds. P.J. Linstrom and W.G. Mallard, July 2001, National Institute of Standards and Technology, Gaithersburg MD, 20899 (http://webbook.nist.gov). [119] K.P. Huber and G. Herzberg. Constants of diatomic molecules. Van Nostrand Reinhold: New York, 1979. [120] G.L. Hug. Optical Spectra of Nonmetallic Inorganic Transient Species in Aque- ous Solution. U.S. Department of Commerce, National Bureau of Standards, Washington, D.C., 1981. [121] C.A. Hunter. Meldola lecture. The role of aromatic interactions in molecular recognition. Chem. Soc. Rev., 23:101–109, 1994. [122] C.A. Hunter and J.K.M. Sanders. The nature of π -π interactions. J. Am. Chem. Soc., 112:5525–5534, 1990. [123] C.A. Hunter, J. Singh, and J.M. Thornton. π -π interactions: the geometry and energetics of phenylalanine-phenylalanine interactions in proteins. J. Mol. Biol., 218:837–846, 1991. [124] Y . Ibrahim, E. Alsharaeh, M. Rusyniak, Simon Watson, Michael Meot-Ner (Mautner), and M.S. El-Shall. Separation of isomers by dimer formation: isomer- ically pure benzene + and toluene + ions, and their dimers: ab initio calculations on (benzene)2+. Chem. Phys. Lett., 380:21–28, 2003. 230 [125] A.J. Illies, M.L. McKee, and H.B. Schlegel. Ab initio study of the carbon dioxide dimer and the carbon dioxide ion complexes [(co2)2+ and (co2)3+]. J. Phys. Chem., 91:3489, 1987. [126] Y . Inokuchi, Y . Naitoh, K. Ohashi, K. Saitow, K. Yoshihara, and N. Nishi. For- mation of benzene dimer cations in neat liquid benzene studied by femtosecond transient absorption spectroscopy. Chem. Phys. Lett., 269:298–304, 1997. [127] Y . Itagaki, N.P. Benetis, R.M. Kadam, and A. Lund. Structure of dimeric radical cations of benzene and toluene in halocarbon matrices: an EPR, ENDOR and MO study. Phys. Chem. Chem. Phys., 2:2683–2689, 2000. [128] R.L. Jaffe and G.D. Smith. A quantum chemistry study of benzene dimer. J. Chem. Phys., 105:2780–2788, 1996. [129] B. Jagoda-Cwiklik, P. Slaviek, L. Cwiklik, D. Nolting, B. Winter, and P. Jung- wirth. Ionization of imidazole in the gas phase, microhydrated environments, and in aqueous solution. J. Phys. Chem. B, 112:3499–3505, 2008. [130] A.C. Moskun A.E. Jailaubekov, S.E. Bradforth, G. Tao, and R.M. Stratt. Rota- tional coherence and a sudden breakdown in linear response seen in room- temperature liquids. Science, 311:1907–1911, 2006. [131] P. Jungwirth and T. Bally. The C 4 H 8 + potential energy surface. 2. The (C 2 H 4 ) + 2 complex cation and its reaction to the radical cations of cyclobutane and 1-butene. J. Am. Chem. Soc., 115:5783–5789, 1993. [132] M. Kamiya and S. Hirata. Higher-order equation-of-motion coupled-cluster methods for ionization processes. J. Chem. Phys., 125:074111, 2006. [133] M. Kamyia and S. Hirata. Higher-order equation-of-motion coupled-cluster methods for ionization processes. J. Chem. Phys., 125:074111–074125, 2006. [134] R.A. Kendall, T.H. Dunning Jr., and R.J. Harrison. Electron affinities of the first- row atoms revisited. systematic basis sets and wavefunctions. J. Chem. Phys., 96:6796–6806, 1992. [135] W. Klemperer and V . Vaida. Molecular complexes in close and far away. Proc. Nat. Acad. Sci., 103:10584–10588, 2006. [136] H. Koch, H.J.Aa. Jensen, P. Jørgensen, and T. Helgaker. Excitation energies from the coupled clusters singles and doubles linear response functions (CCSDLR). Applications to Be, CH + , CO, and H 2 O. J. Chem. Phys., 93(5):3345–3350, 1990. 231 [137] H. Koch, R. Kobayashi, A. S. de Mer´ as, and P. Jørgensen. Calculation of size- extensive transition moments from coupled cluster singles and doubles linear response function. J. Chem. Phys., 100:4393–4400, 1994. [138] J. Kong, C.A. White, A.I. Krylov, C.D. Sherrill, R.D. Adamson, T.R. Furlani, M.S. Lee, A.M. Lee, S.R. Gwaltney, T.R. Adams, C. Ochsenfeld, A.T.B. Gilbert, G.S. Kedziora, V .A. Rassolov, D.R. Maurice, N. Nair, Y . Shao, N.A. Besley, P. Maslen, J.P. Dombroski, H. Daschel, W. Zhang, P.P. Korambath, J. Baker, E.F.C. Bird, T. Van V oorhis, M. Oumi, C.-P. Hsu S. Hirata, N. Ishikawa, J. Flo- rian, A. Warshel, B.G. Johnson, P.M.W. Gill, M. Head-Gordon, and J.A. Pople. Q-Chem 2.0: A high performance ab initio electronic structure program package. J. Comput. Chem., 21(16):1532–1548, 2000. [139] J. Kongsted, A. Osted, K.V . Mikkelsen, P.O. Astrand, and O. Christiansen. Sol- vent effects on the n→ π ∗ electronic transition in formaldehyde: A combined coupled cluster/molecular dynamics study. J. Chem. Phys., 121:8435–8445, 2004. [140] J. Kongsted, A. Osted, K.V . Mikkelsen, and O. Christiansen. The QM/MM approach for wavefunctions, energies and response functions within self- consistent field and coupled cluster theories. Mol. Phys., 100:1813–1828, 2002. [141] H. K¨ oppel and L. S. Cederbaum. Interplay of Jahn-Teller and pseudo-Jahn-Teller vibronic dynamics in the benzene cation. J. Chem. Phys., 89:2023–2040, 1988. [142] H. Koppel, M. Doscher, I. Baldea, H.D. Meyer, and P.G. Szalay. Multistate vibronic interactions in the benzene radical cation. II. Quantum dynamical simu- lations. J. Chem. Phys., 117:2657–2671, 2002. [143] K. Kowalski and P. Piecuch. The active-space equation-of-motion coupled- cluster methods for excited electronic states: Full EOMCCSDt. J. Chem. Phys., 115(2):643–651, 2001. [144] H. Krause, B. Ernstberger, and H.J. Neusser. Binding energies of small benzene clusters. Chem. Phys. Lett., 184:411–417, 1991. [145] M. Krauss. Effective fragment potentials and spectroscopy at enzyme active-sites. Computers Chem., 19:33–204, 1995. [146] R. Krishnan, J.S. Binkley, R. Seeger, and J.A. Pople. Self-consistent molecular orbital methods. XX. A basis set for correlated wave functions. J. Chem. Phys., 72:650, 1980. [147] A.I. Krylov. Size-consistent wave functions for bond-breaking: The equation-of- motion spin-flip model. Chem. Phys. Lett., 338:375–384, 2001. 232 [148] A.I. Krylov. The spin-flip equation-of-motion coupled-cluster electronic struc- ture method for a description of excited states, bond-breaking, diradicals, and triradicals. Acc. Chem. Res., 39:83–91, 2006. [149] A.I. Krylov. Equation-of-motion coupled-cluster methods for open-shell and electronically excited species: The hitchhiker’s guide to Fock space. Annu. Rev. Phys. Chem., 59:433–462, 2008. [150] A.I. Krylov and R.B. Gerber. Photodissociation of ICN in solid and in liquid Ar: Dynamics of the cage effect and of excited state isomerization. J. Chem. Phys., 100:4242–4252, 1994. [151] A.I. Krylov, C.D. Sherrill, E.F.C. Byrd, and M. Head-Gordon. Size- consistent wavefunctions for non-dynamical correlation energy: The valence active space optimized orbital coupled-cluster doubles model. J. Chem. Phys., 109(24):10669–10678, 1998. [152] S.A. Kucharski, M. Włoch, M. Musiał, and R.J. Bartlett. Coupled-cluster theory for excited electronic states: The full equation-of-motion coupled-cluster single, double, and triple excitation method. J. Chem. Phys., 115:8263–8266, 2001. [153] I.V . Kurnikov, G.S.M. Tong, M. Madrid, and D.N. Beratan. Hole size and ener- getics in double helical dna: Competition between quantum delocalization and solvation localization. J. Phys. Chem. B, 106:7–10, 2002. [154] S.R. Langhoff and E.R. Davidson. Configuration interaction calculations on the nitrogen molecule. Int. J. Quant. Chem., 8:61–72, 1974. [155] J. Larsen, D. Madsen, J.-A. Poulsen, T.D. Poulsen, S.R. Keiding, and J. Thogersen. The photoisomerization of aqueous ICN studied by subpicosec- ond transient absorption spectroscopy. J. Chem. Phys., 116:7997–8005, 2002. [156] S.V . Levchenko and A.I. Krylov. Equation-of-motion spin-flip coupled-cluster model with single and double substitutions: Theory and application to cyclobu- tadiene. J. Chem. Phys., 120(1):175–185, 2004. [157] S.V . Levchenko, T. Wang, and A.I. Krylov. Analytic gradients for the spin- conserving and spin-flipping equation-of-motion coupled-cluster models with single and double substitutions. J. Chem. Phys., 122:224106–224116, 2005. [158] F.D. Lewis. DNA molecular photonics. Photochem. Photobiol., 81:65–72, 2005. [159] D. Leys, T.E. Meyer, A.S. Tsapin, K.H. Nealson, M.A. Cusanovich, and J.J. Van Beeumen. Crystal structures at atomic resolution reveal the novel concept of ”electron-harvesting” as a role for the small tetraheme cytochrome c. J. Biol. Chem., 277:35703–35711, 2002. 233 [160] I. Lindgren and D.Mukherjee. On the connectivity criteria in the open-shell coupled-cluster theory for general-model spaces. Phys. Rep., 151:93–127, 1987. [161] R. Lindh and L.A. Barnes. The fraternal twins of quartet O + 4 . J. Chem. Phys., 100:224–237, 1994. [162] R. Lindner, K. M¨ uller-Dethlefs, E. Wedum, K. Haber, and E.R. Grant. On the shape of C 6 H + 6 . Science, 271:1698–1702, 1996. [163] J.G. Ling and K.R. Wilson. Photofragment spectrum of ICN. J. Chem. Phys., 63:101–109, 1975. [164] E. Livshits and R. Baer. A well-tempered density functional theory of electrons in molecules. Phys. Chem. Chem. Phys., 9:2932–2941, 2007. [165] F.J. Lovas. Microwave spectral tables - 2. Triatomic molecules. J. Phys. Chem. Ref. Data, 7:1445–1750, 1978. [166] P.O. L¨ owdin. Discussion on the Hartree-Fock approximation. Rev. Mod. Phys., 35:496, 1963. [167] M. Lundber and P.E.M. Siegbahn. Quantifying the effects of the self-interaction error in DFT: When do the delocalized states appear? J. Chem. Phys., 122(224103):1–9, 2005. [168] V . Luzhkov and A. Warshel. Microscopic calculations of solvent effects on absorption spectra of conjugated molecules. J. Am. Chem. Soc., 113:4491–4499, 1991. [169] A. Mac´ ıas and A. Riera. Calculation of diabatic states from molecular properties. J. Phys. B, 11(16):L489–L492, 1978. [170] M.W. Mahoney and W.L. Jorgensen. A five-site model for liquid water and the reproduction of the density anomaly by rigid, nonpolarizable potential functions. J. Chem. Phys., 112:8910–8922, 2000. [171] E.J.P. Malar and A.K. Chandra. Intermolecular potentials in the dimer, the excimers, and the dimer ions of ethylene. J. Phys. Chem., 85:2190–2194, 1981. [172] Y .A. Mantz, F.L. Gervasio, T. Laino, and M. Parrinello. Charge localization in stacked radical cation DNA base pairs and the benzene dimer studied by self- interaction corrected density-functional theory. J. Phys. Chem. A, 111:105–112, 2007. [173] R.A. Marcus. Theory of oxidation-reduction reactions involving electron transfer. 4. A statistical-mechanical basis for treating contributions from solvent, ligands, and inert salt. Discussions Faraday Soc., 29:21–31, 1960. 234 [174] R.A. Marcus. On the theory of electron-transfer reactions. VI. Unified treatment of homogeneous and electrode reactions. J. Chem. Phys., 43:679–701, 1965. [175] R.A. Marcus and N. Sutin. Electron transfers in chemistry and biology. Biochim. Biophys. Acta, 811:265–322, 1985. [176] F. Masson, T. Laino, I. Tavernelli, U. Rothlisberger, and J. Hutter. Computational study of thymine dimer radical anion splitting in the self-repair process of duplex DNA. J. Am. Chem. Soc., 130:3443–3450, 2008. [177] C.A. Mead and D.G. Truhlar. Conditions for the definition of a strictly diabatic electronic basis for molecular systems. J. Chem. Phys., 77:6090–6098, 1982. [178] L. Meissner and R.J. Bartlett. Transformation of the Hamiltonian in excitation energy calculations: comparison between Fock-space multireference coupled- cluster and equation-of-motion coupled-cluster methods. J. Chem. Phys., 94(10):6670–6675, 1991. [179] M. Meot-Ner (Mautner), P. Hamlet, E.P. Hunter, and F.H. Field. Bonding energies in association ions of aromatic compounds. Correlations with ionization energies. J. Am. Chem. Soc., 100:5466–5471, 1978. [180] I.P. Mercer, I.R. Gould, and D.R. Klug. A quantum mechanical molecular mechanical approach to relaxation dynamics: Calculation of the optical prop- erties of solvated bacteriochlorophyll a. J. Phys. Chem. B, 103:7720–7727, 1999. [181] E.A. Meyer, R.K. Castellano, and F. Diederich. Interactions with aromatic rings in chemical and biological recognition. Angew. Chem. Int. Edit., 42:1210–1250, 2003. [182] Y . Miller and R.B. Gerber. Dynamics of vibrational overtone excitations of H 2 SO 4 , H 2 SO 4 -H 2 O: Hydrogen-hopping and photodissociation processes. J. Am. Chem. Soc., 128:9594–9595, 2006. [183] Y . Miller, R.B. Gerber, and V . Vaida. Photodissociation yields for vibrationally excited states of sulfuric acid under atmospheric conditions. Geophys. Res. Lett., 34:L16820, 2007. [184] S.A. Milosevich, K. Saichek, L. Hinchey, W.B. England, and P. Kovacic. Coor- dination in benzene dimer cation radical. J. Am. Chem. Soc., 105:1088–1090, 1983. [185] B. Minaev. Fine structure and radiative lifetime of the low-lying triplet states of the helium excimer. Phys. Chem. Chem. Phys., 5:2314–2319, 2003. 235 [186] M. Misawa and T. Fukunaga. Structure of liquid benzene and naphthalene studied by pulsed neutron total scattering. J. Chem. Phys., 93:3495–3502, 1990. [187] E. Miyoshi, T. Ichikawa, T. Sumi, Y . Sakai, and N. Shida. Ab initio CASSCF and MRSDCI calculations of the (C 6 H 6 ) + 2 radical. Chem. Phys. Lett., 275:404–408, 1997. [188] E. Miyoshi, N. Yamamoto, M. Sekiya, and K. Tanaka. Structure and bonding of the (C 6 H 6 ) + 2 radical. Mol. Phys., 101:227–232, 2003. [189] L. Mohanambe and S. Vasudevan. Aromatic molecules in restricted geometries: Pyrene excimer formation in an anchored bilayer. J. Phys. Chem. B, 110:14345– 14354, 2006. [190] D. Moncrieff, I.H. Hillier, and V .R. Saunders. Configuration-interaction calcu- lations of the valence ionization energies of the water dimer. Chem. Phys. Lett., 89:447–449, 1982. [191] A.C. Moskun and S.E. Bradforth. Photodissociation of ICN in polar solvents: Evidence for long lived rotational excitation in room temperature liquids. J. Chem. Phys., 119:4500–4515, 2003. [192] A.C Moskun, S.E. Bradforth, J. Thogersen, and S. Keiding. The spectroscopy and lifetime of I( 2 P 1/2 ) in water via the 200 nm photodetachment of I − aq . J. Phys. Chem. A, 110:10947–10955, 2006. [193] I.B. M¨ uller and L.S. Cederbaum. Ionization and double ionization of small water clusters. J. Chem. Phys., 125:204305, 2006. [194] K. M¨ uller-Dethlefs and P. Hobza. Noncovalent interactions: A challenge for experiment and theory. Chem. Rev., 100:143–167, 2000. [195] K. M¨ uller-Dethlefs and J.B. Peel. [196] R.S. Mulliken and W.B. Person. Molecular Complexes. Wiley-Interscience, 1969. [197] M. Musial and R.J. Bartlett. Equation-of-motion coupled cluster method with full inclusion of connected triple excitations for electron-attached states: EA-EOM- CCSDT. J. Chem. Phys., 119:1901–1908, 2003. [198] M. Musial, S.A. Kucharski, and R.J. Bartlett. Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for ionized states: IP-EOM-CCSDT. J. Chem. Phys., 118:1128–1136, 2003. [199] I. Nadler, D. Mahgerefteh, H. Reisler, and C. Wittig. The 266 nm photolysis of icn - recoil velocity anisotropies and nascent E, V , R, T excitations for the CN + I( 2 P 3/2 ) and CN + I( 2 P 1/2 ) channels. J. Chem. Phys., 82:3885–3893, 1985. 236 [200] I. Nadler, H. Reisler, and C. Wittig. Energy disposal in the laser photodissocia- tion of ICN and BrCN at 300 K and in a free jet expansion. Chem. Phys. Lett., 103:451–457, 1984. [201] H. Nakatsuji. Description of two- and many- electron processes by the SAC-CI method. Chem. Phys. Lett., 177(3):331–337, 1991. [202] H. Nakatsuji and K. Hirao. Cluster expansion of the wavefunction. Symmetry- adapted-cluster expansion, its variational determination, and extension of open- shell orbital theory. J. Chem. Phys., 68:2053–2065, 1978. [203] H. Nakatsuji, K. Ohta, and K. Hirao. Cluster expansion of the wave function. electron correlations in the ground state, valence and rydberg excited states, ion- ized states, and electron attached states of formaldehyde by SAC and SAC-CI theories. J. Chem. Phys., 75:2952–2958, 1981. [204] M.D. Newton. Quantum chemical probes of electron-transfer kinetics - the nature of donor-acceptor interactions. Chem. Rev., 91:767–792, 1991. [205] C.Y . Ng, D.J. Trevor, P.W. Tiedemann, S.T. Ceyer, P.L. Kronebusch, B.H. Mahan, and Y .T. Lee. Photoionization of dimeric polyatomic molecules: Proton affinities of H 2 O and HF. J. Chem. Phys., 67:4235–4237, 1977. [206] S. O. Nielsen, B.D. Michael, and E.J. Hart. Ultraviolet absorption spectra of hydrated electrons, hydrogen, hydroxyl, deuterium, and hydroxyl-d radicals from pulse radiolysis of aqueous solutions. J. Phys. Chem., 80:2482–2488, 1976. [207] M. Nooijen and R.J. Bartlett. Equation of motion coupled cluster method for electron attachment. J. Chem. Phys., 102:3629–3647, 1995. [208] M. Nooijen, K.R. Shamasundar, and D. Mukherjee. Reflections on size- extensivity, size-consistency, and generalized extensivity in many-body theory. Mol. Phys., 103:2277–2298, 2005. [209] J.R. Norris, R.A. Uphaus, H.L. Crespi, and J. Katz. Electron spin resonance of chlorophyll and origin of signal-I in photosynthesis. Proc. Nat. Acad. Sci., 68:625–628, 1971. [210] Y .V . Novakovskaya. Dynamics of water clusters upon UV-excitation leading to ionization: Nonempirical study. Int. J. Quant. Chem., 107:2763–2780, 2007. [211] M.E. Nunez, D.B. Hall, and J.K. Barton. Long-range oxidative damage to DNA: effects of distance and sequence. Chem. Biol., 6:85–97, 1999. 237 [212] M. Oana and A.I. Krylov. Dyson orbitals for ionization from the ground and electronically excited states within equation-of-motion coupled-cluster formal- ism: Theory, implementation, and examples. J. Chem. Phys., 127:234106, 2007. [213] K. Ohashi, Y . Inokuchi, and N. Nishi. Pump-probe photodepletion spectroscopy of (C6H6) + 2 . Identification of spectrum in the charge resonance band region. Chem. Phys. Lett., 263:167–172, 1996. [214] K. Ohashi, Y . Nakai, T. Shibata, and N. Nishi. Photodissociation spectroscopy of (C 6 H 6 ) + 2 . Laser Chem., 14:3–14, 1994. [215] K. Ohashi and N. Nishi. Photodissociation spectroscopy of benzene cluster ions:(C 6 H 6 ) + 2 and (C 6 H 6 ) + 3 . J. Chem. Phys., 95:4002–4009, 1991. [216] K. Ohashi and N. Nishi. Photodepletion spectroscopy on charge resonance band of (C 6 H 6 ) + 2 and (C 6 H 6 ) + 3 . J. Phys. Chem., 96:2931–2932, 1992. [217] Y . Ohtsuka, P. Piecuch P, J.R. Gour, M. Ehara, and H. Nakatsuji H. Active-space symmetry-adapted-cluster configuration-interaction and equation- of-motion coupled-cluster methods for high accuracy calculations of potential energy surfaces of radicals. J. Chem. Phys., 126:164111, 2007. [218] K. Okamoto, A. Saeki, T. Kozawa, Y . Yoshida, and S. Tagawa. Subpicosecond pulse radiolysis study of geminate ion recombination in liquid benzene. Chem. Lett., 32:834–835, 2003. [219] J. Olsen. The initial implementation and applications of a general active space coupled cluster method. J. Chem. Phys., 113(17):7140–7148, 2000. [220] M. Orozco and F.J. Luque. Theoretical methods for the description of the solvent effect in biomolecular systems. Chem. Rev., 100:4187–4225, 2000. [221] S. Pal, M. Rittby, R.J. Bartlett, D. Sinha, and D. Mukherjee. Multireference coupled-cluster methods using an incomplete model space — application to ionization-potentials and excitation-energies of formaldehyde. Chem. Phys. Lett., 137:273–278, 1987. [222] C. Petrongolo, G. Hirsch, and R.J. Buenker. Diabatic representation of the ˜ a 2 A 1 / ˜ b 2 B 2 conical intersection in NH 2 . Mol. Phys., 70(5):825–834, 1990. [223] P. Piecuch and R.J. Bartlett. EOMXCC: A new coupled-cluster method for elec- tronic excited states. Adv. Quantum Chem., 34:295–380, 1999. [224] P.A. Pieniazek, S.A. Arnstein, S.E. Bradforth, A.I. Krylov, and C.D. Sherrill. Benchmark full configuration interaction and EOM-IP-CCSD results for proto- typical charge transfer systems: Noncovalent ionized dimers. J. Chem. Phys., 127:164110, 2007. 238 [225] P.A. Pieniazek, S.E. Bradforth, and A.I. Krylov. Spectroscopy of the cyano radi- cal in an aqueous environment. J. Phys. Chem. A, 110:4854–4865, 2006. [226] P.A. Pieniazek, A.I. Krylov, and S.E. Bradforth. Electronic structure of the ben- zene dimer cation. J. Chem. Phys., 127:044317, 2007. [227] P.A. Pieniazek, E.J. Sundstrom, S.E. Bradforth, and A.I. Krylov. to be published, 2008. [228] R. Podeszwa, R. Bukowski, and K. Szalewicz. Potential energy surface for the benzene dimer and perturbational analysis of pi-pi interactions. J. Phys. Chem. A, 110:10345–10354, 2006. [229] V . Polo, E. Kraka, and D. Cremer. Electron correlation and the self-interaction error of density functional theory. Molecular Physics, 100:1771 – 1790, 2002. [230] J.A. Pople, P.M.W. Gill, and N.C. Handy. Spin-unrestricted character of Kohn- Sham orbitals for open-shell systems. Int. J. Quant. Chem., 56:303–305, 1995. [231] G.D. Purvis and R.J. Bartlett. A full coupled-cluster singles and doubles model: The inclusion of disconnected triples. J. Chem. Phys, 76:1910–1918, 1982. [232] W. Radloff, V . Stert, T. Freudenberg, I.V . Hertel, C. Jouvet, C. Dedonder- Lardeux, and D. Solgadi. Internal conversion in highly excited benzene and ben- zene dimer: femtosecond time-resolved photoelectron spectroscopy. Chem. Phys. Lett., 281:20–26, 1997. [233] K. Raghavachari, G.W. Trucks, J.A. Pople, and M. Head-Gordon. A fifth-order perturbation comparison of electron correlation theories. Chem. Phys. Lett., 157:479–483, 1989. [234] J. Rebek Jr. Assembly and encapsulation with self-complementary molecules. Chem. Soc. Rev., 25:255–264, 1996. [235] G.C. Reid. Ion chemistry in D-region. Adv. At. Mol. Phys., 12:375–413, 1976. [236] E.E. Rennie, C.A.F. Johnson, J.E. Parker, D.M.P. Holland, D.A. Shaw, and M.A. Hayes. A photoabsorption, photodissociation and photoelectron spectroscopy study of C 6 H 6 and C 6 D 6 . Chem. Phys., 229:107–123, 1998. [237] S. Rettrup. An iterative method for calculating several of the extreme eigensolu- tions of large real non-symmetric matrices. J. Comput. Phys., 45:100–107, 1982. [238] J.E. Reutt, L.S. Wang, Y .T. Lee, and D.A. Shirley. Molecular beam photoelectron spectroscopy and femtosecond intramolecular dynamics of H 2 O + and D2O + . J. Chem. Phys., 85:6928–6939, 1986. 239 [239] Y .M. Rhee, T.J. Lee, M.S. Gudipati, L.J. Allamandola, and M. Head-Gordon. Charged polycyclic aromatic hydrocarbon clusters and the galactic extended red emission. Proc. Nat. Acad. Sci., 104:5274–5278, 2007. [240] S.W. Rick. A reoptimization of the five-site water potential (TIP5P) for use with ewald sums. J. Chem. Phys., 120:6085–6093, 2004. [241] L. Rodriguez-Monge and S. Larsson. Conductivity in polyacetylene. 3. Ab initio calculations for a two-site model for electron transfer. J. Phys. Chem., 100:6298– 6303, 1996. [242] D.J. Rowe. Equations-of-motion method and the extended shell model. Rev. Mod. Phys., 40:153–166, 1968. [243] E. Runge and E.K.U. Gross. Density-functional theory for time-dependent sys- tems. Phys. Rev. Lett., 52:997–1000, 1984. [244] N.J. Russ, T.D. Crawford, and G.S. Tschumper. Real versus artifactual symmetry breaking effects in Hartree-Fock, density-functional, and coupled-cluster meth- ods. J. Chem. Phys., 120:7298–7306, 2005. [245] M. Rusyniak, Y . Ibrahim, E. Alsharaeh, M. Meot-Ner (Mautner), and M.S. El- Shall. Mass-selected ion mobility studies of the isomerization of the benzene radical cation and binding energy of the benzene dimer cation. Separation of iso- meric ions by dimer formation. J. Phys. Chem. A, 107:7656–7666, 2003. [246] W. Saenger. Principles of Nucleic Acid Structure. Springer-Verlag: New York, 1984. [247] I. Saito, T. Nakamura, K. Nakatani, Y . Yoshioka, K. Yamaguchi, and H. Sugiyama. Mapping of the hot spots for DNA damage by one-electron oxi- dation: Efficacy of GG doublets and GGG triplets as a trap in long-range hole migration. J. Am. Chem. Soc., 120:12686–12687, 1998. [248] A. Sancar. Structure and function of DNA photolyase. Biochemistry, 33:2–9, 1994. [249] F. Santoro, V . Barone, and R. Improta. Influence of base stacking on excited-state behavior of polyadenine in water, based on time-dependent density functional calculations. Proc. Nat. Acad. Sci., 104:9931–9936, 2007. [250] K. Sato, S. Tomoda, and K. Kimura. Electronic and molecular-structure of the water dimer cation - a theoretical-study. Chem. Phys. Lett., 95:579–583, 1983. [251] J¨ urgen Schnitker and P.J. Rossky. Quantum simulation study of the hydrated electron. J. Chem. Phys., 86:3471–3485, 1987. 240 [252] K.L. Schuchardt, B.T. Didier, T. Elsethagen, L. Sun, V . Gurumoorthi, J. Chase, J. Li, and T.L. Windus. Basis set exchange: A community database for computa- tional sciences. J. Chem. Inf. Model., 47:1045–1052, 2007. [253] H. Sekino and R.J. Bartlett. A linear response, coupled-cluster theory for excita- tion energy. Int. J. Quant. Chem. Symp., 18:255–265, 1984. [254] Y . Shao, L.F. Molnar, Y . Jung, J. Kussmann, C. Ochsenfeld, S. Brown, A.T.B. Gilbert, L.V . Slipchenko, S.V . Levchenko, D. P. O’Neil, R.A. Distasio Jr., R.C. Lochan, T. Wang, G.J.O. Beran, N.A. Besley, J.M. Herbert, C.Y . Lin, T. Van V oorhis, S.H. Chien, A. Sodt, R.P. Steele, V . A. Rassolov, P. Maslen, P.P. Koram- bath, R.D. Adamson, B. Austin, J. Baker, E.F.C. Bird, H. Daschel, R.J. Doerk- sen, A. Drew, B.D. Dunietz, A.D. Dutoi, T.R. Furlani, S.R. Gwaltney, A. Hey- den, S. Hirata, C.-P. Hsu, G.S. Kedziora, R.Z. Khalliulin, P. Klunziger, A.M. Lee, W.Z. Liang, I. Lotan, N. Nair, B. Peters, E.I. Proynov, P.A. Pieniazek, Y .M. Rhee, J. Ritchie, E. Rosta, C.D. Sherrill, A.C. Simmonett, J.E. Subotnik, H.L. Woodcock III, W. Zhang, A.T. Bell, A.K. Chakraborty, D.M. Chipman, F.J. Keil, A. Warshel, W.J. Herhe, H.F. Schaefer III, J. Kong, A.I. Krylov, P.M.W. Gill, and M. Head-Gordon. Advances in methods and algorithms in a modern quantum chemistry program package. Phys. Chem. Chem. Phys., 8:3172–3191, 2006. [255] T. Shida and W. H. Hamill. Molecular ions in radiation chemistry. 3. Absorption spectra of aromatic-hydrocarbon cations and anions in organic glasses. J. Chem. Phys., 44:4372–4377, 1966. [256] I.A. Shkrob. Ionic species in pulse radiolysis of supercritical carbon dioxide. 2. Ab initio studies on the structure and optical properties of (CO 2 ) + n , (CO 2 ) − 2 , and CO − 3 ions. J. Phys. Chem. A, 106:11871–11881, 2002. [257] I.A. Shkrob, M.C. Sauer Jr., C.D. Jonah, and K. Takahashi. Ionic and neutral species in pulse radiolysis of supercritical CO 2 . 1. Transient absorption spec- troscopy, electric field effect, and charge dynamics. J. Phys. Chem. A, 106:11855– 11870, 2002. [258] J. Simons and W. D. Smith. Theory of electron affinities of small molecules. J. Chem. Phys., 58:4899–4907, 1973. [259] D. Sinha, D. Mukhopadhya, R. Chaudhuri, and D. Mukherjee. The eigenvalue- independent partitioning technique in Fock space: An alternative route to open- shell coupled-cluster theory for incomplete model spaces. Chem. Phys. Lett., 154:544–549, 1989. [260] D. Sinha, D. Mukhopadhyay, and D. Mukherjee. A note on the direct calcula- tion of excitation-energies by quasi-degenerate MBPT and coupled-cluster the- ory. Chem. Phys. Lett., 129:369–374, 1986. 241 [261] M.O. Sinnokrot and C.D. Sherrill. Highly accurate coupled cluster potential energy curves for the benzene dimer: Sandwich, t-shaped, and parallel-displaced configurations. J. Phys. Chem. A, 108:10200–10207, 2004. [262] M.O. Sinnokrot and C.D. Sherrill. High-accuracy quantum mechanical studies of π -π interactions in benzene dimers. J. Phys. Chem. A, 110:10656–10668, 2006. [263] L.V . Slipchenko and A.I. Krylov. Spin-conserving and spin-flipping equation- of-motion coupled-cluster method with triple excitations. J. Chem. Phys., 123:84107–84120, 2005. [264] L.V . Slipchenko and A.I. Krylov. Efficient strategies for accurate calculations of electronic excitation and ionization energies: Theory and application to the dehydro-meta-xylylene anion. J. Phys. Chem. A, 110:291–298, 2006. [265] D.M.A. Smith, K.M. Rosso, M. Dupuis, M. Valiev, and T.P. Straatsma. Elec- tronic coupling between heme electron-transfer centers and its decay with dis- tance depends strongly on relative orientation. J. Phys. Chem. B, 110:15582– 15588, 2006. [266] J.R Smith, J.B. Kim, and W.C. Lineberger. High-resolution threshold photode- tachment spectroscopy of OH − . Phys. Rev. A, 55:2036–2043, 1997. [267] P.E. Smith and B.M. Pettitt. Modeling solvent in biomolecular systems. J. Phys. Chem., 98:9700–9711, 1994. [268] M. Sodupe, J. Bertran, L. Rodrguez-Santiago, and E.J. Baerends. Ground state of the (H 2 O) + 2 radical cation: DFT versus post-Hartree-Fock methods. J. Phys. Chem. A, 103:166–170, 1999. [269] M. Sodupe, A. Oliva, and J. Bertran. Theoretical-study of the ionization of the H 2 O-H 2 O, NH 3 -H 2 O, and FH-H 2 O hydrogen-bonded molecules. J. Am. Chem. Soc., 116:8249–8258, 1994. [270] J.F. Stanton. Separability properties of reduced and effective density matrices in the equation-of-motion coupled cluster method. J. Chem. Phys., 101(10):8928– 8937, 1994. [271] J.F. Stanton and R.J. Bartlett. The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties. J. Chem. Phys., 98:7029–7039, 1993. [272] J.F. Stanton and J. Gauss. Analytic energy derivatives for ionized states described by the equation-of-motion coupled cluster method. J. Chem. Phys., 101(10):8938–8944, 1994. 242 [273] J.F. Stanton and J. Gauss. A simple scheme for the direct calculation of ionization potentials with coupled-cluster theory that exploits established excitation energy methods. J. Chem. Phys., 111:8785–8788, 1999. [274] J.F. Stanton, J. Gauss, J.D. Watts, W.J. Lauderdale, and R.J. Bartlett. ACES II, 1993. The package also contains modified versions of the MOLECULE Gaussian integral program of J. Alml¨ of and P.R. Taylor, the ABACUS integral derivative program written by T.U. Helgaker, H.J.Aa. Jensen, P. Jørgensen and P.R. Taylor, and the PROPS property evaluation integral code of P.R. Taylor. [275] J.M. Steed, T.A. Dixon, and W. Klemperer. Molecular beam studies of ben- zene dimer, hexafluorobenzene dimer, and benzene-hexafluorobenzene. J. Chem. Phys., 70:4940–4946, 1979. [276] S. Sun and E.R. Bernstein. Aromatic van der Waals clusters: Structure and non- rigidity. J. Phys. Chem., 100:13348–13366, 1996. [277] P.G. Szalay. Analytic energy derivatives for coupled-cluster methods describing excited states: General formulas and comparison of computational costs. Int. J. Quant. Chem., 55:151–163, 1995. [278] H. Tachikawa. A Direct ab-Initio Trajectory Study on the Ionization Dynamics of the Water Dimer. J. Phys. Chem. A, 106(30):6915–6921, 2002. [279] H. Tachikawa. Ionization dynamics of the small-sized water clusters: A direct ab initio trajectory study. J. Phys. Chem. A, 108:7853–7862, 2004. [280] T. Takayanagi, T. Hoshino, and K. Takahashi. Electronic structure calculations of acetonitrile cluster anions: Stabilization mechanism of molecular radical anions by solvation. Chem. Phys., 324:679–688, 2006. [281] K. Takeshita. A theoretical study of the second band of the photoelectron spec- trum of benzene with an analysis of the vibrational structure. Theo. Chem. Acc., 103:64–69, 1999. [282] Y . Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai T, and K.K. Hirao. A long-range- corrected time-dependent density functional theory. J. Chem. Phys., 120:8425, 2004. [283] Y . Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai, and K. Hirao. A long-range- corrected time-dependent density functional theory. J. Chem. Phys., 120:8425– 8433, 2004. [284] M.A. Thompson and G.K. Schenter. Excited states of the bacteriochlorophyll b dimer of rhodopseudomonas viridis: A QM/MM study of the photosynthetic 243 reaction center that includes MM polarization. J. Phys. Chem., 99:6374–6386, 1995. [285] R. Thomson and F.W. Dalby. Experimental determination of the dipole moments of the X( 2 Σ + ) and B( 2 Σ + ) states of the CN molecule. Can. J. Phys., 46:2815– 2819, 1968. [286] S. Tomoda, Y . Achiba, and K. Kimura. Photoelectron spectrum of the water dimer. Chem. Phys. Lett., 87:197–200, 1982. [287] S. Tomoda and K. Kimura. Proton-transfer potential-energy surfaces of the water dimer cation (H 2 O) + 2 in the 12A” and 12A’ states. Chem. Phys., 82:215–227, 1983. [288] D.J. Tozer, R.D. Amos, N.C. Handy, B.O. Roos, and L. Serrano-Andres. Does density functional theory contribute to the understanding of excited states of unsaturated organic compounds? Mol. Phys., 97:859–868, 1999. [289] T. Tsuneda, M. Kamiya, and K. Hirao. Regional self-interaction correction of density functional theory. J. Comput. Chem., 24:1592–1598, 2003. [290] V . Vaida, H.G. Kjaergaard, and K.J. Feierabend. Hydrated complexes: Relevance to atmospheric chemistry and climate. Int. Rev. Phys. Chem., 22:203–219, 2003. [291] J. VandeV ondele, M. Krack, F. Mohamed, M. Parrinello, T. Chassaing, and J. Hut- ter J. QUICKSTEP: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comp. Phys. Comm., 167:103–128, 2005. [292] J. VandeV ondele, P.A. Pieniazek, A.I. Krylov, S.E. Bradforth, and P. Jungwirth. to be published. [293] J. VandeV ondele and M. Sprik. A molecular dynamics study of the hydroxyl radical in solution applying self-interaction-corrected density functional methods. Phys. Chem. Chem. Phys., 7:1363–1367, 2005. [294] V . Vanovschi, A.I. Krylov, and P.G. Wenthold. Structure, vibrational frequen- cies, ionization energies, and photoelectron spectrum of the para-benzyne radical anion. Theor. Chim. Acta, 120:45–58, 2008. [295] J. Vieceli, I. Chorny, and I. Benjamin. Photodissociation of ICN at the liq- uid/vapor interface of chloroform. J. Chem. Phys., 115:4819–4828, 2001. [296] E. V ohringer-Martinez, B. Hansmann, H. Hernandez, J.S. Francisco, J. Troe, and B. Abel. Water catalysis of a radical-molecule gas-phase reaction. Science, 315:497–501, 2007. 244 [297] A.A. V oityuk. Charge transfer in DNA: Hole charge is confined to a single base pair due to solvation effects. J. Chem. Phys., 122:204904, 2005. [298] O.A. Vydrov and G.E. Scuseria. A simple method to selectively scale down the self-interaction correction. J. Chem. Phys., 124:191101, 2006. [299] C. Wan, M. Gupta, and A.H. Zewail. Femtochemistry of ICN in liquids: dynam- ics of dissociation, recombination and abstraction. Chem. Phys. Lett., 256:279– 287, 1996. [300] B. Wang, H. Hou, and Y . Gu. Density functional study of the hydrogen bonding: H 2 O· HO. Chem. Phys. Lett., 303:96–100, 1999. [301] A. Warshel and M. Levitt. Theoretical studies of enzymatic reactions: Dielec- tric electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. J. Mol. Biol., 103:227–249, 1976. [302] J.D. Watts, J. Gauss, and R.J. Bartlett. Coupled-cluster methods with nonitera- tive triple excitations for restricted open-shell hartree-fock and other general sin- gle determinant reference functions. energies and analytical gradients. J. Chem. Phys., 98:8718–8733, 1993. [303] H.-G. Weikert and L.S. Cederbaum. On the satellite structure accompanying the ionization of benzene. Chem. Phys. Lett., 237:1–6, 1995. [304] F. Weinhold and C. R. Landis. Natural bond orbitals and extensions of localized bonding concepts. Chem. Ed.: Res.& Pract. Eur., 2:91–104, 2001. [305] H.-J. Werner, P.J. Knowles, R. Lindh, M. Sch¨ utz, et al. MOLPRO 2002.6, 2003. [306] H.-J. Werner and W. Meyer. MCSCF study of the avoided curve crossing of the two lowest 1 Σ + states of LiF. J. Chem. Phys., 74(10):5802–5807, 1981. [307] H.J. Werner and P.J. Knowles. An efficient internally contracted multiconfiguration-reference configuration interaction method. J. Chem. Phys., 89:5803–5814, 1988. [308] B. Winter, R. Weber, W. Widdra, M. Dittmar, M. Faubel, and I.V . Hertel. Full valence band photoemission from liquid water using euv synchrotron radiation. J. Phys. Chem. A, 108:2625–2631, 2004. [309] N. Winter, I. Chorny, J. Vieceli, and I. Benjamin. Molecular dynamics study of the photodissociation and photoisomerization of ICN in water. J. Chem. Phys., 119:2127–2143, 2003. 245 [310] N. Winter, I. Chorny, J. Vieceli, and I. Benjamin. Molecular dynamics study of the photodissociation and photoisomerization of ICN in water. J. Chem. Phys., 119:2127–2143, 2003. [311] J.-H Guo Y ., Luo, A. Augustsson, J.-E. Rubensson, C. S˚ athe, H. ˚ Agren, H. Sieg- bahn, and J. Nordgren. X-ray emission spectroscopy of hydrogen bonding and electronic structure of liquid water. Phys. Rev. Lett., 89:137402, 2002. [312] Y . Shao, L.F. Molnar, Y . Jung, J. Kussmann, C. Ochsenfeld, S. Brown, A.T.B. Gilbert, L.V . Slipchenko, S.V . Levchenko, D.P. O’Neil, R.A. Distasio Jr, R.C. Lochan, T. Wang, G.J.O. Beran, N.A. Besley, J.M. Herbert, C.Y . Lin, T. Van V oorhis, S.H. Chien, A. Sodt, R.P. Steele, V .A. Rassolov, P. Maslen, P.P. Koram- bath, R.D. Adamson, B. Austin, J. Baker, E.F.C. Bird, H. Daschel, R.J. Doerksen, A. Drew, B.D. Dunietz, A.D. Dutoi, T.R. Furlani, S.R. Gwaltney, A. Heyden, S. Hirata, C.-P. Hsu, G.S. Kedziora, R.Z. Khalliulin, P. Klunziger, A.M. Lee, W.Z. Liang, I. Lotan, N. Nair, B. Peters, E.I. Proynov, P.A. Pieniazek, Y .M. Rhee, J. Ritchie, E. Rosta, C.D. Sherrill, A.C. Simmonett, J.E. Subotnik, H.L. Wood- cock III, W. Zhang, A.T. Bell, A.K. Chakraborty, D.M. Chipman, F.J. Keil, A. Warshel, W.J. Herhe, H.F. Schaefer III, J. Kong, A.I. Krylov, P.M.W. Gill, M. Head-Gordon. Advances in methods and algorithms in a modern quantum chem- istry program package. Phys. Chem. Chem. Phys., 8:3172–3191, 2006. [313] C.H. Yang and C.P. Hsu. The dynamical correlation in spacer-mediated electron transfer couplings. J. Chem. Phys., 124:244507, 2006. [314] D.R. Yarkony. On the construction of diabatic bases using molecular proper- ties. Rigorous results in the vicinity of a conical intersection. J. Phys. Chem. A, 102(42):8073–8077, 1998. [315] A.J. Yencha, R.I. Hall L. Avaldi, G. Dawber, A.G. McConkey, M.A. MacDonald, and G.C. King. Threshold photoelectron spectroscopy of benzene up to 26.5 eV. Can. J. Chem., 82:1061–1066, 2004. [316] Z.Q. You, Y .H. Shao, and C.P. Hsu. Calculating electron transfer couplings by the spin-flip approach: Energy splittings and the dynamical correlation effects. Chem. Phys. Lett., 390:116–123, 2004. [317] L.Y . Zhang and R.A. Friesner. Ab initio calculation of electronic coupling in the photosynthetic reaction center. Proc. Nat. Acad. Sci., 95:13603–13605, 1998. [318] Y . Zhang and W. Yang. A challenge for density functionals: Self-interaction error increases for systems with a noninteger number of electrons. J. Chem. Phys., 109(7):2604–2608, 1998. 246 Appendix A: EOM-IP-CC jobs in Q-CHEM This chapter describes the practical aspects of running EOM-IP-CCSD and EOM-IP- CC(2,3) calculations in Q-CHEM based on the water example. The three lowest states of the water monomer cation are X 2 B 1 , A 2 A 1 , and B 2 B 2 . Both energy and property calculations are covered. The Q-Chem input consists of a single file specifying the molecular system and the type of the calculation. Lines beginning with! contain comments and are disregarded by the parser. They can be omitted from the examples give below. The input has the following general format: $molecule !molecular system specification: charge, multiplicity, geometry $end $rem !calculation specification $end The input is not case-sensitive. Lines beginning with ! contain comments and are disregarded by the parser. The molecular geometry can be specified either as a Z-matrix or aa Cartesian coordinates. The former is preferred as it is easier to ensure proper molecular symmetry. $molecule !z-matrix geometry input 0 1 H O 1 bond H 2 bond 1 angle bond=0.95849 angle=104.34 $end 247 This part of the input remains constant throughout the examples given here, and will thus be omitted. The third line specifies the charge and multiplicity of the reference state. In this example it is the neutral singlet ground state of water. The $rem cart specifies the type of calculation. We are interested in obtaining the first three ionization energies of water, corresponding to removing an electron from the b 1 , a 1 , and b 2 orbitals, respectively. The$rem section is the following: $rem !calculation of single-point energy jobtype sp !one-particle basis set basis 6-31++G** !level of correlation correlation ccsd !calculation is for ionized state cc_ip_proper true !transition symmetry cc_nlowspin [1,0,1,1] $end Each number in the cc_nlowspin specifies the number of states having a given transition symmetry. The transition symmetry is a product of the reference and final states symmetry. Since the reference state is totally symmetric (A 1 ), it is simply the symmetry of the final cation state. The standard order of irreducible representations is followed (A 1 , A 2 , B 1 , B 2 ). In case when it is not clear, a trial calculation should be performed. The next example is the calculation of the transition dipole moments between the ground state of the cation and its electronic excited states. $rem jobtype sp basis 6-31++G** correlation ccsd cc_ip_proper true cc_nlowspin [1,0,1,1] !calculation of the transition properties cc_trans_prop true !reference state symmetry cc_refsym 4 !number of the reference state 248 cc_state_deriv 1 $end The reference state in the transition properties calculation is specified by giving its irreducible representation (cc_refsym) and number (cc_state_deriv) . Two variants of state property calculation are supported: the relaxed and unrelaxed properties. The relaxed properties is the formally correct way of doing this, but they take more time and in may cases the results are very similar to the unrelaxed values. In the following example we calculate the fully relaxed properties: $rem jobtype sp basis 6-31++G** correlation ccsd cc_ip_proper true cc_nlowspin [1,0,1,1] !properties of the excited states cc_exstates_prop true !we want fully relaxed properties !for nonrelaxed properties specify false instead of true cc_eom_full_resp true $end The above input will calculate the dipole moments and the spatial extent of the wave function for all three cation states. Note, that the NBO analysis can be performed on any of the cation states: $rem jobtype sp basis 6-31++G** correlation ccsd cc_ip_proper true !do not need to calculate all states cc_nlowspin [1,0,0,0] cc_exstates_prop true cc_eom_full_resp true !specify nbo analysis nbo true !target state symmetry cc_refsym 1 !number of the target state cc_state_deriv 1 $end 249 The NBO analysis can be performed on a single state only in one calculation. The specification of the state is completely analogous to the reference state in the transition properties calculation. Geometry optimization is very similar, except for thejobtype specification: $rem !specify that we want to do optimization jobtype opt basis 6-31++G** correlation ccsd cc_ip_proper true cc_nlowspin [1,0,0,0] !target state symmetry cc_refsym 1 !target state number cc_state_deriv 1 $end The input for EOM-IP-CC(2,3) calculations is akin to EOM-IP-CCSD, except for the cc_do_triples. Due to the large cost of the calculations, it is recommended that additional keywords are also specified. $rem jobtype sp basis 6-31++G** correlation ccsd cc_ip_proper true cc_nlowspin [1,0,1,1] !requests the triple excitations to be included in the EOM part cc_do_triples true !number of preconverging EOM-IP-CCSD iterations !speeds up the calculations and 20 is generally a good number cc_preconv_sd 20 !reduce the dimension of cc_dmaxvectors $end Reducing the dimension of the Davidson iteration space drastically reduces the ten- sor storage requirements and helps convergence. It is not necessary in all cases. 250 Appendix B: TPA calculation using Dalton In this Appendix the calculation of the two-photon absorption cross-sections using the Dalton electronic structure package is described. The package is installed on the local Krylov group network and is accessible from this network. The main machine allow- ing access from the outside world is hogwarts.usc.edu. For the package to be accessible the setenv PATH ${PATH}:/home/qcsoftware/dalton-2.0/bin line must be added to the.cshrc file. The first five lowest electronic excited state of water serve as an example. The dalton program is invoked by the following command: dalton input.dal molecule.mol The input is separated into to two files. Fileinput.dal specifies the type of the calculation, while molecule.mol specifies the one-particle basis set and molecular geometry. Both files are case-sensitive. The order of lines must be preserved. A sample Z-matrix file looks like this: BASIS 6-311++G** TPA calculation water Atomtypes=2 Angstrom ZMAT O 1 8.0 H1 2 1 0.95849 1.0 H2 3 1 0.95849 2 104.34 1.0 The first two lines specify the one-particle basis set used, followed by two comment lines. Line 5 specifies the number of atom types and the units of the geometry. Water molecule consists of hydrogen and oxygen, thus there are two atom types. ZMAT line specifies that the geometry is in the Z-Matrix format. The subsequent lines are in the 251 standard format. The second number signifies the running number of the atom, while the last number is the atomic number. Alternatively the geometry can be specified as a Cartesian file: BASIS 6-311++G** comment line 1 comment line 2 Atomtypes=2 Angstrom Charge=8.0 Atoms=1 O 0.000000 0.000000 0.117572 Charge=1.0 Atoms=2 H -0.757048 0.000000 -0.470289 H 0.757048 0.000000 -0.470289 The structure of the input file is as follows: **DALTON INPUT .RUN WAVE FUNCTIONS **INTEGRAL .DIPLEN **WAVE FUNCTIONS .CC *CC INPUT .CCSD .NSYM 4 *CCEXCI .NCCEXCI 2 2 0 1 *CCTPA .DIPLEN .PRINT 4 **END OF DALTON INPUT Lines 9 and 10 specify the number of symmetry operations in the point group. For C 2v it is 4. Lines 13 and 14 give the symmetry of the desired states. Here, we request 2, 1, a 2 states in the A 1 , B 1 , and A 2 , respectively. This ordering of the irreducible representations is non-standard and in general it is a good idea to run a preliminary calculation to determine their ordering. Lines 15/16 specify the required printing level. Setting it to 4 results in the δ F , δ G , and δ H quantities being printed in the output. The output has the following format: 252 transition : X ˆ1A1 <-- 2ˆ1B2 excitation energy : 0.5100178907 a.u. 13.878 e.V. 111936. cmˆ-1 photon energy for A: 0.2550089453 a.u. 6.939 e.V. 55968. cmˆ-1 photon energy for B: 0.2550089453 a.u. 6.939 e.V. 55968. cmˆ-1 +-------------------+----------------+----------------+----------------+ | A B | M_0f( -0.2550) | M_f0( 0.2550) | Sˆ0f( 0.2550) | +-------------------+----------------+----------------+----------------+ | XDIPLEN XDIPLEN | --- | --- | --- | | XDIPLEN YDIPLEN | --- | --- | --- | | XDIPLEN ZDIPLEN | --- | --- | --- | | YDIPLEN XDIPLEN | --- | --- | --- | | YDIPLEN YDIPLEN | --- | --- | --- | | YDIPLEN ZDIPLEN | -4.41694507 | -2.24493598 | 9.91575892 | | ZDIPLEN XDIPLEN | --- | --- | --- | | ZDIPLEN YDIPLEN | -4.41694507 | -2.24493598 | 9.91575892 | | ZDIPLEN ZDIPLEN | --- | --- | --- | +-------------------+----------------+----------------+----------------+ delta_F = Sˆ0f_iijj/30 (length gauge) : 0.00000000 delta_G = Sˆ0f_ijij/30 (length gauge) : 0.66105059 delta_H = Sˆ0f_ijji/30 (length gauge) : 0.66105059 The above input will compute the cross-sections assuming that the two photons are degenerate. It is also possible to fix the energy of one of the photons. In the following input the energy of one of the photons is set to 0.1 hartree: **DALTON INPUT .RUN WAVE FUNCTIONS **INTEGRAL .DIPLEN **WAVE FUNCTIONS .CC *CC INPUT .CCSD .NSYM 4 *CCEXCI .NCCEXCI 2 2 0 1 *CCTPA .STATES 1 1 0.1 1 2 0.1 2 1 0.1 2 2 0.1 4 1 0.1 .DIPLEN **END OF DALTON INPUT 253 The lines following the .STATES card specify the number of the irreducible rep- resentation, the number of the state in this irreducible representation and the energy of one of the photons in atomic units. The actual cross-section is not computed in the program, only the transition tensor. The quantities found in the output can be easily converted into the rotationally averaged transition strengths, which depend on the relative polarization of the two photons: • parallel: δ TP =2δ F +2δ G +2δ H • perpendicular: δ TP =− 1δ F +4δ G − 1δ H The result is in atomic units. It can be converted into the cross-section using the formula: σ =8π 3 a 4 0 t 0 E 1 E 2 δ TP (B.1) The variables: • a 0 — Bohr radius,a 0 =0.5291772085910 − 10 m • t 0 —atomic unit of timet 0 =2.41888432650510 − 17 s • α — fine structure constant,α =7.2973525376 − 3 • E 1 ,E 2 — photon energies, in Hartree • δ TP — rotationally averaged transition strengths, in atomic units For example, ifδ TP = 50au thenσ = 4.23010 − 52 cm 4 s = 0.0423GM, assuming two degenerate photons and 10 eV excitation energy. It is also possible to use point charges in the input. In the following water dimer example the hydrogen-bond donor molecule is replaced by a set of point charges: 254 ATOMBASIS comment line 1 comment line 2 Atomtypes=4 Angstrom Charge=8.0 Atoms=1 Basis=6-311++G** O 1.346262 -0.089929 0.000000 Charge=1.0 Atoms=2 Basis=6-311++G** H 1.799371 0.288810 -0.756998 H 1.799371 0.288810 0.756998 Charge=-0.8 Atoms=1 Basis=pointcharge O -11.556302 0.124314 0.000000 Charge=0.4 Atoms=2 Basis=pointcharge H -11.882393 -0.776599 0.000000 H -10.597109 0.018188 0.000000 The last two entries specify the positions of the point charges. Noth that a label for each point charge needs to be specified. The molecular strusture hasC s symmetry, and the input file needs to be modifed accordingly: **DALTON INPUT .RUN WAVE FUNCTIONS **INTEGRAL .DIPLEN **WAVE FUNCTIONS .CC *CC INPUT .CCSD .NSYM 2 *CCEXCI .NCCEXCI 4 1 *CCTPA .DIPLEN .PRINT 4 **END OF DALTON INPUT The difference is in the specification of the excited state symmetries. 255 Appendix C: Programmable EOM-IP-CC expressions The Davidson procedure requires the computation ofσ -vectors. They are the product of multiplication of the ¯ H matrix and a trialR-vector. For the EOM-IP-CC(2,3) model the matrix equation assumes the following form: ¯ H SS − E CC ¯ H SD ¯ H ST ¯ H DS ¯ H DD − E CC ¯ H DT ¯ H TS ¯ H TD ¯ H TT − E CC R 1 R 2 R 3 = σ 1 σ 2 σ 3 (C.1) (C.2) The EOM-IP-CCSD model for is recovered by settingR 3 = 0. The left eigenvalue problem for EOM-IP-CCSD has the form: L 1 L 2 ! ¯ H SS − E CC ¯ H SD ¯ H DS ¯ H DD − E CC = ˜ σ 1 ˜ σ 2 ! (C.3) σ 1 =σ i = ([ ¯ H SS − E CC ]R 1 ) i +( ¯ H SD R 2 ) i +( ¯ H ST R 3 ) i σ 2 =σ a ij = ( ¯ H DS R 1 ) a ij +([ ¯ H DD − E CC ]R 2 ) a ij +( ¯ H DT R 3 ) a ij σ 3 =σ ab ijk = ( ¯ H TS R 1 ) ab ijk +( ¯ H TD R 2 ) ab ijk +([ ¯ H TT − E CC ]R 3 ) ab ijk 256 ˜ σ 1 = ˜ σ i = (L 1 [ ¯ H SS − E CC ]) i +(L 2 ¯ H DS ) i ˜ σ 2 = ˜ σ a ij = (L 1 ¯ H SD ) a ij +(L 2 [ ¯ H DD − E CC ]) a ij Density matrices follow Ref. 90 . 257 Table C.1: Programmable expressions for the right σ -vectors in EOM-IP-CC(2,3) model. σ i ([ ¯ H SS − E CC ]R 1 ) i =− P j r j F ij ( ¯ H SD R 2 ) i = P jb r b ij F jb + 1 2 P jkb r b jk I 6 kjib ( ¯ H ST R 3 ) i = 1 4 P jkab hjk||abir ab ijk σ a ij ( ¯ H DS R 1 ) a ij = P k r k I 2 ijka ([ ¯ H DD − E CC ]R 2 ) a ij =P(ij) P k r a jk F ik + P b r b ij F ab − P(ij) P kb r b ik I 1 jbka + P kl r a kl I 4 ijkl + P b T 4 b t ab ij ( ¯ H DT R 3 ) a ij = P kb F kb r ab ijk + 1 2 P(ij) P klb I 6 klib r ab klj − 1 2 P kbc I 7 kabc r bc ijk σ ab ijk ( ¯ H TS R 1 ) ab ijk =P(ij|k) h P l t ab kl H 2 ijl − P(ab) P d t ac ij H 3 kcb i ( ¯ H TD R 2 ) ab ijk =P(ij|k)[− P c r c ij I 3 kcba + P l t ab kl H 4 ijl − P(ab) P l r b kl I 2 ijla +P(ab) P c t ac ij H 5 kcb ([ ¯ H TT − E CC ]R 3 ) ab ijk =P(ij|k) h − P l F kl r ab ijl + P lm I 4 ijlm r ab lmk i + P(ab) P c F bc r ac ijk + P cd I 5 abcd r cd ijk − P(ab)P(i|jk) P cl I 1 icla r cb ljk + P(ij|k) h P l H 6 ijl t ab kl − P(ab) P c H 7 kbc t ac ij i 258 Table C.2: R-independenet intermediates used in the energy EOM-IP-CCSD and EOM- IP-CC(2,3) expressions. F ia =f ia + P jb t b j <ij||ab> F ij =f ij + P a t a i f ja + P ka t a k <jk||ia> + P kab t a i t b k <jk||ab> + 1 2 P kbc t bc ik <jk||bc> F ab =f ab − P i t a i f ib − P ic t c i <ia||bc> + P ijc t c i t a j <ij||bc>− 1 2 P jkc t ac jk <jk||bc> I 1 iajb =<ia||jb>− P k t b k <jk||ia>− P c t c i <jb||ac> + P kc t c i t b k <jk||ac>− P kc t bc ik <jk||ac> I 2 ijka =− <ij||ka> +2 P l t a l I 4 ijkl + P b t b i (<jb||ka>− P lc t ac jl <kl||bc>)− t b j (<ib||ka>− P lc t ac il <kl||bc>) − P bc t b i t c j <ka||bc>− P lbc t b l t ac ij <kl||bc> + P c t bc ij f kc − 1 2 P cd t cd ij <kb||cd> + P lc (t bc il <jc||kl >− t bc jl <ic||kl >) I 3 icab =− <ic||ab> +2 P d t d i I 5 bcad + P j t b j (<ia||jc>− P kd t cd ik <jk||ad>)− t c j (<ia||jb>− P kd t bd ik <jk||ad>) − P jk t b j t c k <jk||ia>− P jkd t d j t bc ik <jk||ad> + P k t ab ik f kc − 1 2 P kl t ab kl <ic||kl > + P kd (t ad ik <kb||cd>− t bd ik <ka||cd>) I 4 ijkl = 1 2 <ij||kl > + 1 2 P a (t a j <kl||ia>− t a i <kl||ja>)+ 1 2 P ab t a i t b j <kl||ab> + 1 4 P cd t cd ij <kl||cd> I 6 ijka =<ij||ka>− P c t c k <ij||ac> I 7 iabc =<ia||bc>− P j t a j <ij||bc> 259 Table C.3: R-dependent intermediates used in the energy EOM-IP-CCSD and EOM-IP- CC(2,3) expressions. H 2 ijk = 2 P l r l I 4 ijkl H 3 iab = P l r j I 1 iajb H 4 ijk =P(ij) P al r a il I 6 klja H 5 iab = 1 2 P jk r b jk I 6 jkia + P jc r c ij I 7 jbca H 6 ijk = 1 2 P abl r ab ijl hkl||abi H 7 iab = 1 2 P ckj r ac kji hkj||bci T 4 a =− 1 2 P klb r b kl <kl||ab> T 6 a = 1 2 P ijb l b ij t ab ij 260 Table C.4: Programmable expressions for unrelaxed EOM-IP-CCSD density matrices. γ 0 ij = ˜ γ ij +δ ij ˜ γ ij = 1 2 P + (ij)[− l i r j − ˜ l ij + P a Y 1 ia t a j ] γ 0 ab = 1 2 P + (ab)[ ˜ l ab − P i Y 1 ia t b i ] γ 0 ia = 1 2 [− Y 1 ia − P j l j (r a ij +r i t a j )− P jb Y 1 jb (t ab ij − t a j t b i ) − r i ˜ ˜ l a − P j ˜ l ji t a j − P b ˜ l ba t b i +t a i ] Γ 0 ijkl = ˜ Γ ijkl − δ li δ kj +δ ki δ lj − δ li ˜ γ jk +δ ki ˜ γ jl +δ lj ˜ γ ik − δ kj ˜ γ il ˜ Γ ijkl = 1 2 P a (l a kl ¯r a ij +l a ij ¯r a kl ) Γ 0 abcd = 0 Γ 0 ijka = ˜ Γ ijka − δ kj γ ia +δ ki γ ja ˜ Γ ijka =− 1 2 [l a ij r k +l k ¯r a ij + P b Y 1 kb ˜ t ab ij − P(ij) ˜ l kj t a i − P lb l b kl (¯r b ij t a l − P(ij)t b i r a jl − P(ij)t ab il r j )] Γ 0 iajb = ˜ Γ iajb +δ ji γ ab ˜ Γ iajb = 1 2 [(Y 1 ja t b i +Y 1 ib t a j )− P k (l b ik r a jk +l a jk r b ik )− P k (l a jk t b k r i +l b ik t a k r j )] Γ 0 iabc =− 1 2 [ P j Y 1 ja ˜ t bc ij − P(bc)t b i ˜ l ac + P jk l a jk ˜ t bc jk r i − P(bc) P jk l a jk t c j r b ik ] Γ 0 ijab = 1 2 [ ˜ t ab ij − P k l k (P(ij)t ab ik r j − P(ab)¯r a ij t b k +P(ij)P(ab)t a i r b jk )+P(ab) ˜ ˜ l b r a ij + P kc Y 1 kc (P(ab)t b k ˜ t ac ij − P(ij)t c i t ab jk − P(ij)P(ab)t b j t ac ik ) +P(ij)P(ab) ˜ ˜ l b t a j r i − P(ij)P(ab) P c ˜ l cb t c j t a i + P klc l c kl (− 1 2 P(ab)t ac ij r b kl + 1 2 t ab kl r c ij − 1 2 P(ij)t ab jk r c il +P(ij)P(ab)t ac ik r b jl +P(ij)P(ab)t b k t ac jl r i − 1 2 P(ij)t c i t ab kl r j − 1 2 P(ab)t b k t a l ¯r c ij +P(ij)P(ab)t b k t c i r a jl ) 261 Table C.5: Intermediate used in the unrealxed EOM-IP-CCSD density matrices calcula- tion. Y 1 ia = P j l a ij r j ˜ l ij = P ka l a ik r a jk ˜ l ab = 1 2 P ij l a ij r b ij ˜ ˜ l a = 1 2 P ijb l b ij t ab ij ¯r a ij =r a ij +P(ij)r i t a j ˜ t ab ij =t ab ij + 1 2 P(ij)P(ab)t a i t b j Table C.6: EOM-IP-CCSD amplitude responseξ -vector programmable expressions. ξ a i =F ia (1− P j r j l j )+T 4 a l i − P jkb l k r b jk <ij||ab> + − 1 2 P b Y 2 ab F ib + 1 2 P bc Y 2 bc I 7 ibac − P j Y 3 ij F ja + P jk Y 3 jk I 6 ikja + 1 2 P jklc l a ij r c kl I 6 kljc + P jb Y 4 iajb F jb − P jkb Y 4 jakb I 6 ikjb − P jbc Y 4 icjb I 7 jcab + 1 2 P jkl Y 5 ijkl I 6 lkja ξ ab ij =<ij||ab>− P(ij)(ab)Y 1 ia F jb − 1 2 P(ab) P c Y 2 bc <ij||ac> +P(ij) P k [(X 2 ik +Y 3 ik )<jk||ab>] − P(ab) P c l a ij T 4 b +P(ij)P(ab) P kc Y 4 iakc <jk||bc> + 1 2 P kl Y 5 ijkl <kl||ab> +P(ab) P k r k l a ij F kb − P(ij)P(ab) P klc r l l a ik I 6 jlkb − P kc r k l c ij I 7 kcab − P(ab) P k I 6 ijka Y 1 kb +P(ij) P c Y 1 ic I 7 jcab 262 Table C.7: Intermediates used in theξ -vector calculation. X 2 ij =l i r j Y 2 ab = P ijc l a ij r b ij Y 3 ij = P ka l a ik r a jk Y 4 iajb = P kc l a ik r b jk Y 5 ijkl = P a l a ij r a kl 263 Appendix D: Electron transfer rate calculation In this Appendix we address the calculation of the electron transfer rate using the Mar- cus theory. We consider the example of the benzene dimer cation in a sandwich con- figuration presented in Fig. D.1. Initially the positive charge is located on fragment B, which has the geometry of the cation. Fragment A has the geometry of the neutral. The electron hops from fragment A to fragment B, and they acquire the cation and neutral geometries, respectively. In the product state the positive charge is located on fragment A. + + + + fragment A fragment B fragment A fragment B Figure D.1: Electron transfer reaction in the sandwich benzene dimer cation. The elec- tron hops from fragment A to fragment B. The Marcus theory rate expression has the form: k ET = 2π ¯h H 2 RP 1 √ 4πλβ − 1 exp(− β 4λ (λ +Δ G 0 ) 2 ) (D.1) where the variables are: • H RP — diabatic coupling • λ — reorganization energy • Δ G 0 — reaction free energy 264 The variables are schematically shown in Fig. D.2. In our case the free energies of the reactant and product states is identical, thusΔ G 0 = 0. The only needed quantities are the reorganization energy and the diabatic coupling. The reorganization energy is the difference between the energy of the product state at the geometry of the reactants and the geometry of the products. It can be calculated using four quantities: • E neutral (neutral) — energy of the neutral species at the geometry of the neutral • E cation (neutral) — energy of the cation at the neutral geometry • E neutral (cation) — energy of the neutral at the cation geometry • E cation (cation) — energy of the cation at the cation geometry energy reaction coordinate 0 G ' † G ' O AB + A + B Figure D.2: Marcus theory picture of an electron transfer reaction. AB + and A + B are reactant and product potential energy surfaces, respectively. Δ G 0 r and Δ G† are reaction and activation free energies, respectively. λ is the reorganization energy. 265 The energy of the product state at the product geometry is equal to: E 1 =E neutral (neutral)+E cation (cation) (D.2) when the geometry is distorted to the reactant geometry the energy is E 2 =E cation (neutral)+E neutral (cation) (D.3) The reorganization energy is: λ =E 2 − E 1 = (E cation (neutral)− E neutral (neutral))− [E cation (cation)− E neutral (cation)] (D.4) Thus, the reorganization energy is the difference between ionization energies of benzene at the neutral and the cation geometries,λ = 0.265eV . In this calculation we replaced free energy with energy and assumed that the energy of the dimer is a sum of monomer energies. This is strictly true only at infinite separation. We also need the diabatic coupling. According to the Condon principle the coupling does not depend on the degrees of freedom that do not change the relative orientation of, or the distance between the fragments. We can exploit this principle, and carry out the calculation for the sandwich isomer with two identical rings (in this case neutral geometry rings). At this geometry the energy of the charge resonance transition is 0.51 eV . The diabatic states are degenerate, and the coupling is equal to half the splitting, i.e. H RP = 0.205eV . Substituting these numbers into Eq. D.1 we obtained the rate as k ET =2.70510 12 s − 1 . The corresponding time constant is 370 fs. In some cases the Generalized Mulliken-Hush model is needed to calculate the cou- pling. We will illustrate this calculation using the t-shaped isomer of the benzene dimer 266 at the vertical configuration, and theπ o g (stem)/π o g (top) pair of states. The first calcula- tion is the calculation of the permanent dipole moment and the transition dipole moment. The dipole moment operator, in the matrix representation, has the form: μ = − 3.176 2.647 2.647 3.320 (D.5) In this particular case the directions of all dipole moments are parallel. For systems, where this is not true, projections on the difference of permanent dipole moments vector should be considered. Similarly, we construct the Hamiltonian operator in the basis using the excitation energy. By definition, the are no off-diagonal elements: H = 0 0 0 3376 (D.6) The only entry in this matrix is the excitation energy. The next step is finding the matrix T that diagonalizes the dipole moment matrix. Thus, states with maximally sep- arated charge are obtained: T = − 0.942 − 0.335 0.335 − 0.942 (D.7) When T transformation is applied to μ , the dipole moment in the diabatic basis is obtained. It is diagonal in the diabatic representation: 267 T − 1 μT = − 4.118 0 0 4.262 (D.8) Similarly, applying the transformation to the Hamiltonian, we obtained the Hamil- tonian in the diabatic basis: T − 1 HT = 380 1067 1067 2996 (D.9) The off-diagonal element of the above matrix is the diabatic coupling. 268
Abstract (if available)
Abstract
Understanding radicals is central to understanding the mechanics of chemical change. In this work, quantitative and qualitative methods for the description of the electronic spectroscopy of open-shell species in the gas and condensed phases are developed.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Electronic structure and spectroscopy in the gas and condensed phase: methodology and applications
PDF
Electronic structure and spectroscopy of excited and open-shell species
PDF
Electronically excited and ionized states in condensed phase: theory and applications
PDF
Spectroscopic signatures and dynamic consequences of multiple interacting states in molecular systems
PDF
Computational spectroscopy in gas and condensed phases
PDF
Modeling x-ray spectroscopy in condensed phase
PDF
Development of predictive electronic structure methods and their application to atmospheric chemistry, combustion, and biologically relevant systems
PDF
Electronic states and photodissociation dynamics of hydroxyalkyl radicals
PDF
Ultrafast spectroscopy of aromatic amino acids and their chromophores in the condensed phase
PDF
Spectroscopy and photodissociation dynamics of hydroxyethyl radicals
PDF
Energy transfer pathways for NO2-rare gas complexes in helium nanodroplets
PDF
Liquid phase photoelectron spectroscopy as a tool to probe excess electrons in liquid ammonia along with Birch chemistry carbanions: from dilute electrolytic solutions to the metallic regime
PDF
Spectroscopy of hydrogen and water clusters in helium droplets
PDF
New electronic structure methods for electronically excited and open-shell species within the equation-of-motion coupled-cluster framework
PDF
Electronic structure of ionized non-covalent dimers: methods development and applications
PDF
The synthesis and characterization of [Ga]ZSM-5 by NMR spectroscopy and density functional theory
PDF
Two-photon absorption spectroscopy and excited state photochemistry of small molecules
PDF
Charge separation in transition metal and quantum dot systems
PDF
Kinetic modeling of high-temperature oxidation and pyrolysis of one-ringed aromatic and alkane compounds
PDF
Photoelectron and ion imaging investigations of spectroscopy, photoionization, and photodissociation dynamics of diazomethane and diazirine
Asset Metadata
Creator
Pieniazek, Piotr Adam
(author)
Core Title
Electronic structure and spectroscopy of radicals in the gas and condensed phases
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Chemistry
Publication Date
07/25/2008
Defense Date
06/06/2008
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
electronic spectroscopy,OAI-PMH Harvest,quantum chemistry
Language
English
Advisor
Bradforth, Stephen E. (
committee chair
), Dappen, Werner (
committee member
), Krylov, Anna I. (
committee member
), Wittig, Curt (
committee member
)
Creator Email
pap@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m1417
Unique identifier
UC1222530
Identifier
etd-Pieniazek-20080725 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-203123 (legacy record id),usctheses-m1417 (legacy record id)
Legacy Identifier
etd-Pieniazek-20080725.pdf
Dmrecord
203123
Document Type
Dissertation
Rights
Pieniazek, Piotr Adam
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
electronic spectroscopy
quantum chemistry