Electronically excited and ionized states in condensed phase: theory and applications

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Content ELECTRONICALLY EXCITED AND IONIZED STATES IN CONDENSED PHASE: THEORY AND APPLICATIONS by Arman Sadybekov A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMISTRY) December 2017 Copyright 2017 Arman Sadybekov Acknowledgements I would like to express sincere gratitude to my advisor, Professor Anna I. Krylov, for invaluable scientific and life lessons. Her mentorship helped me to grow as a scientist and a person. Experience gained in Anna’s group will be essential for my future carrier as a scientist. I am honored to be a part of her group, which is a great and successful community of wonderful people. I would like to thank all the members of Anna’s group, Dr. Atanu Acharya, Dr. Anastasia Gunina, Dr. Xintian Feng, Professor Samer Gozem, Dr. Kaushik Nanda, Dr. Ilya Kaliman, for being a great support and help throughout my years in graduate school. They helped me to progress in the field of quantum chemistry and served as beacons of knowledge and wisdom to reach to. I especially thank Natalie Orms, Bailey Qin, Daniel Kwasnewsky, Han Wool, Richard Li, for being great friends and helping make the US my new home. I am grateful to our collaborators on the benzene excimers project, Professor Sand- ford Ruhman and Dr. Siva S. Iyer, for fruitful discussions and advice. I thank Professor Karl Christe and Professor Aiichiro Nakano for invaluable support and advice through- out my PhD career. ii Last but definitely not the least, I would like to thank my wife Anastasiia and my mother Maiia for help and psychological support, without which it would be impossible for me to get through the challenging years of graduate school. I also thank my family in Russia, Ukraine and Kazakhstan for definitive support in my thrive towards scientific career and all the love. iii Table of contents Acknowledgements ii List of tables vi List of figures viii Abbreviations xiii Abstract xv Chapter 1: Introduction and overview 1 1.1 Chapter 1 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2: Rewriting the Story of Excimer Formation in Liquid Benzene 7 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Experimental design . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.2 Theoretical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.3 Properties of different model dimer structures representing solid and liquid benzene . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.6 Chapter 2 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Chapter 3: Coupled-cluster based approach for core-ionized and core-excited states in condensed phase: Theory and application to different protonated forms of aqueous glycine 52 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Theoretical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 Modeling solvent effects for core-ionized and core-excited states . . . . 62 iv 3.4 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4.1 Benchmark calculations . . . . . . . . . . . . . . . . . . . . . . 63 3.4.2 Calculations of different forms of aqueous glycine . . . . . . . . 64 3.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.5.1 Benchmark calculations . . . . . . . . . . . . . . . . . . . . . . 67 3.5.2 Core-ionized states of different protonation forms of glycine . . 72 3.5.3 Core IEs of different protonation forms of glycine computed using equilibrium averaging with explicit solvent models . . . . 80 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.7 Chapter 3 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Chapter 4: Future work 91 4.1 Chapter 4 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Appendices 96 Chapter A: Supplementary information for Chapter 2. 97 A.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 A.2 Structure of solid benzene and model dimer structures . . . . . . . . . . 100 A.3 Exciton delocalization: Dimers versus trimers . . . . . . . . . . . . . . 101 A.4 Molecular orbitals framework . . . . . . . . . . . . . . . . . . . . . . . 103 A.5 Spectra of different model structures . . . . . . . . . . . . . . . . . . . 105 A.6 Structure of liquid benzene . . . . . . . . . . . . . . . . . . . . . . . . 107 Chapter B: Supplementary information for Chapter 3. 109 B.1 Programmable expressions for EOM-IP/EE-CCSD/MP2-S(D) . . . . . 109 B.2 Benchmark calculations using EOM-IP-CCSD-S(D) and EOM-IP-MP2- S(D) methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.3 Calculations of core-ionized states of different protonation forms of aque- ous glycine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Bibliography 125 v List of tables 2.1 Excitation energies (eV), oscillator strength, and wave function analy- sis of the relevant excited states of benzene monomer and model dimer structures. EOM-EE-CCSD/6-31+G(d). . . . . . . . . . . . . . . . . . 34 3.1 Core ionized states of model ammonia systems a . The chemical shifts relative to the reference NH 3 system are shown in parentheses. . . . . . 69 3.2 Experimental core IEs and chemical shifts against gas-phase values of different forms of glycine (in eV). . . . . . . . . . . . . . . . . . . . . 74 3.3 Computed gas-phase IEs and chemical shifts against Gly can of different forms of glycine (in eV). EOM-IP-CCSD-S(D)/cc-pVTZ. . . . . . . . 75 3.4 Core IEs and chemical shifts (eV) relative to Gly can of different proto- nation forms of glycine computed using representative single snapshots from equilibrium trajectory. IP-CCSD-S(D)/cc-pVTZ. . . . . . . . . . 79 3.5 Chemical shifts (eV) relative to Gly can of different protonation forms of glycine a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A.1 Fractional bleach of BE at various stages of its formation . . . . . . . . 97 A.2 Angular distribution in the first solvation shell forR<5 ˚ A. . . . . . . . 108 B.1 Valence ionized and excited states for formaldehyde. Errors against EOM-IP-CCSD are shown in parenthesis. All energies are in electron- volts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.2 Valence IEs (eV) of C 2 H 4 . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.3 Valence IEs (eV) of C 2 H 6 . . . . . . . . . . . . . . . . . . . . . . . . . 113 B.4 Carbon (1s) ionized states for selected molecules (IEs in eV). . . . . . 114 B.5 Chemical shifts of carbon (1s) ionized states for selected molecules against methane (IEs in eV). . . . . . . . . . . . . . . . . . . . . . . . 115 vi B.6 Oxygen (1s) ionized states for selected molecules (IEs in eV). . . . . . 116 B.7 Chemical shifts of oxygen (1s) ionized states for selected molecules against water (IEs in eV). . . . . . . . . . . . . . . . . . . . . . . . . . 117 B.8 Nitrogen (1s) ionized states for selected molecules and chemical shifts (IEs in eV). For chemical shifts, error against experiment is given in parenthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 B.9 Nitrogen (1s) ionized states for selected molecules and chemical shifts (IEs in eV). For chemical shifts, error against experiment is given in parenthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 B.10 Core-excited states for selected molecules. Energies (eV) against the respective edges are shown. . . . . . . . . . . . . . . . . . . . . . . . 120 B.11 Chemical shifts (eV) of different protonation forms of isolated glycine. . 122 B.12 Computed core IEs and chemical shifts relative Gly can of different proto- nation forms of glycine using model structures from Ref. 43. IP-CCSD- S(D)/cc-pVTZ. All energies are in electron-volts. . . . . . . . . . . . . 123 B.13 Core IEs and chemical shifts (eV) relative to Gly can of the anionic form of glycine computed using representative single snapshots from equilib- rium trajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 vii List of figures 1.1 An expanded Jablonski diagram illustrating singlet fission (SF) follow- ing electronic excitation. A coherent (TT) state dissociates to form two independent triplets. Adapted from Ref. 8. . . . . . . . . . . . . . . . 2 1.2 Excited-state processes in fluorescent proteins. The main relaxation channel is fluorescence. Radiationless relaxation, a process in which the chromophore relaxes to the ground state by dissipating electronic energy into heat, reduces the quantum yield of fluorescence. Other competing processes, such as transition to a triplet state via intersystem crossing (not shown), excited-state chemistry, and electron transfer, alter the chemical identity of the chromophore, thus leading to temporary or permanent loss of fluorescence (blinking and bleaching) or changing its color (photoconversion). Adapted from Ref. 9. . . . . . . . . . . . . . 3 1.3 HOMO and core 1s orbitals of zwitterionic glycine. On the left, valence p orbital of glycine is shown that is delocalized over carboxyl group and aC. On the right, five core orbitals are shown. The core orbitals are localized on the corresponding atoms. . . . . . . . . . . . . . . . . . . 4 2.1 Structures of benzene dimer. The left panel shows the definition of struc- tural parameters from Ref. 55: R is the distance between centers-of- mass of the two fragments, q and f characterize their relative orienta- tion, and r 1 and r 2 quantify the offset between the two rings. Center and right panels show sandwich (q=f=0 ), Y-shaped (q=90 ,f=0 ), and T-shaped (q=90 ,f=0 ) benzene dimers. . . . . . . . . . . . . . . . . 9 2.2 Two-dimensional probability distribution function,g(R;q), of liquid ben- zene (see Fig. 2.1 for the definition of structural parameters). The main maximum is at 5.65 ˚ A, and the shoulder seen at 4.25 ˚ A for parallel molecules indicates displaced (PD) structures. Reproduced with per- missionfromRef. 54. CopyrightACS2010. . . . . . . . . . . . . . . . 10 viii 2.3 Top: Absorption spectrum of benzene excimer. Bottom: Kinetics of excimer formation probed at 530 nm (2.34 eV). The plot shows the rise of the excimer absorption near the peak. The inset shows the first picosecond of delay. For details, see text. . . . . . . . . . . . . . . . . 11 2.4 Top: The schematics of the transient absorption measurements of BE formation using 2- (left) and 3- (right) pulse experiments. Bottom: Experimental setup. Liquid benzene is excited with 266 nm UV light (pump) generated by third harmonic generation (THG) of the amplifier fundamental, which initiates the excited-state dynamics. The transient excited-state absorption is measured by a supercontinuum generated by white light generation (WLG) at various time delays (t). The right panel shows 3-pulse (pump-dump-probe) experiment concept. In these exper- iments, the dump pulse (500 nm) bleaches the excited-state population. The polarizations of dump and probe are controlled by a waveplate that rotates the fundamental generating the white light. In VV experiment, the polarizations of dump and probe are the same, whereas in a VH experiment they are cross-polarized. The signals are reported in units of DOD (the difference in optical density of the sample, in the presence and absence of pump/dump). Note that in 2-pulse experiments the chopper is positioned in the pump arm. . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Potential energy surface (q versus R, f=0 , see Fig. 2.1) of benzene dimer computed using CHARMM27 forcefield. The black lines repre- sent the contour line of 0 kcal/mol. The points on the grid correspond to model structures used to represent liquid benzene. Reproduced with permissionfromRef. 55. CopyrightACS2011. . . . . . . . . . . . . . 19 2.6 Normalized transient absorption spectra at pump-probe delays ranging from 1 to 400 ps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.7 Magic angle pump-dump-probe transient spectra for relaxed and fully formed excimers. Spectra discontinuity is due to scattering interference from the dump pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.8 Spectral cuts in the pump-dump-probe data showing dependence ofDOD at various dispersed probe wavelengths as a function of dump-probe delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.9 Fractional dump-induced bleach of the excimer absorbance at various stages of formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.10 Anisotropy decay of population bleached during formation of excimer. 29 ix 2.11 Natural transition orbitals and the respective weights for the S 0 ! S 1 and S 0 ! S x transitions at the equilibrium BE geometry. Orbitals are computed with EOM-EE-CCSD/6-31+G(d). . . . . . . . . . . . . . . 30 2.12 Potential energy curves of the three relevant states (ground state of the dimer (S 0 ), excimer’s state (S 1 ), and the state that has the largest oscilla- tor strength for the BE absorption, (S x )) of the benzene sandwich along inter-fragment distance, R, at perfectly stacked orientation (q=0, f=0, r 1 =r 2 =0). The values above the S 1 (1B 3g ) curve (excimer state) show oscillator strengths of theS 1 (1B 3g )!S x (4B 2u ) transition along the scan. 32 2.13 Bright electronic transitions in various model dimer structures. Exc denotes relaxed BE, S is sandwich at R=4.3 ˚ A, T g is T-shaped dimer at the gas-phase equilibrium structure, T x is the T-shaped dimer from the x-ray structure, Y d is a displaced T-shaped dimer, L is L-shaped dimer, HtT is head-to-tail dimer, PD g is parallel-displaced dimer gas- phase equilibrium structure, M is benzene monomer. Structures of dimer configurations are shown in Fig. S7 in SI. Total oscillator strength of the transitions corresponding to different dimer configurations and are given relative to oscillator strength of the relaxed BE transition and is calculated in 0.2 eV range centered at the excitation energy given on the x-axis. The experimental maximum value of BE absorption band is 2.5 eV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.14 Oscillator strengths of model structures from liquid benzene. . . . . . . 38 3.1 Core-ionized states are Feshbach resonances. They can autoionize via a two-electron process in which one electron fills the core hole providing enough energy to eject the second electron. To describe this process, the wave-function ansatz needs to include 2h1p excitations, such as three right-most configurations. . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Target core-ionized (left) and core-excited (right) states are generated by the EOM-IP and EOM-EER 1 -operators from a closed-shell reference of the neutral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Errors in chemical shifts (eV) of carbon (1s) ionized states for selected molecules. Shifts are computed relative to methane. Stars mark errors that are too large and do not fit on the plot. . . . . . . . . . . . . . . . 71 3.4 Errors in chemical shifts (eV) of oxygen (1s) ionized states for selected molecules. Shifts are computed relative to water. . . . . . . . . . . . . 71 x 3.5 Structures of different protonation forms of glycine: Canonical (Gly can ), zwitterionic (Gly ZI ), deprotonated (Gly ), and protonated (Gly + ). . . . 73 3.6 Distribution of hydrogen bonds between solute and water molecules from the first solvation shell for different forms of glycine. . . . . . . . 77 3.7 Representative snapshots for Gly ZI , Gly , and Gly + from equilibrium MD simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.8 Chemical shifts of glycine computed with different levels of theory: nitrogen 1s edge (top), carbon (1s) from carboxyl (middle), and carbon (1s) from methyl group. . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.1 Near edge X-ray absorption event. The initial photon absorption by inner shell electron is shown in red, the system is excited into a metastable state. The following relaxation of the state through Auger decay is shown in blue. On the left, chemical notation of the atomic orbitals is given. On the right, the XAS notation for the same electronic shells is depicted for comparison. The resulting photoelectron spectra con- tain the information about core orbitals and the local environment of the respective atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 A.1 Transient absorption spectrum of benzene excimer in VH (left) and VV (right) configurations between the pump and probe in the three pulse experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 A.2 Recovery of bleach of populations excited at different times in the course of excimer formation. . . . . . . . . . . . . . . . . . . . . . . . . . . 99 A.3 Anisotropy decay of bleach of at 3 ps and 66 ps. . . . . . . . . . . . . 99 A.4 X-ray structure of solid benzene. Unit cell, P bca space group. b=9.660 ˚ A, a=7.460 ˚ A, c=7.030 ˚ A. Highlighted in color are the structures of the dimers with the shortest distance between the fragments. Blue is the central molecule. Blue-red pairs are Y-shaped displaced (Y d ) dimers, blue-yellow pairs are the T-shaped (T x ) dimers, blue-green pairs are L- shaped (L) dimers, and blue-purple pairs are head-to-tail (HtT) dimers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 A.5 The NTOs for the S 0 !S 1 transition in model stacked trimer structures. Top: The distances between the central and edge molecules are equal 3 ˚ A. Bottom: the distances between the central and edge molecules are 3 and 4 ˚ A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 xi A.6 Molecular orbitals diagram of benzene sandwich dimer at the equilib- rium BE geometry. D 2h point group symmetry labels are used. Orbitals are computed with HF/6-31+G* and shown with isovalue of 0.005. . . 103 A.7 Model benzene dimer structures. Top left: head-to-tail dimer, C 1 sym- metry. Top right: L-shaped dimer, C 1 symmetry. Middle row, left: T- shaped dimer, C 2v symmetry. Middle row, right: T-shaped displaced dimer, C 2v symmetry. Bottom left: PD dimer at the gas-phase equilib- rium geometry,C 2h symmetry. The definition of structural parameters is given in Fig. 1 in the main manuscript. . . . . . . . . . . . . . . . . . 105 A.8 Spectra of various model dimer structures (S 1 absorption). The stick spectra are convoluted with Gaussians (half-width is 0.05 eV). . . . . . 106 A.9 The angular distribution functions plotted for different values ofR. Blue line: isotropic distribution. Orange line: from the MD simulations using CHARMM27 force-field (results from Ref. 55). At R < 5 ˚ A, parallel arrangement is preferred, whereas at larger R, perpendicular structures are more prevalent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 A.10 Radial distribution function, g(r), of liquid benzene. The number of molecules in the first solvent shell is R 8:0 0 g(r)r 2 dr=12.5. Reproduced from Ref. 55. a data from Ref. 54. . . . . . . . . . . . . . . . . . . . . 107 B.1 Structures of model hexahydrated glycine complexes from Ref. 43. From left to right: zwitterionic form Gly ZI , Gly complex with sodium ion, and protonated Gly + complex with chloride. . . . . . . . . . . . . 121 B.2 The number of water molecules in the first solvation shell for different protonation states of glycine using cutoff radius of 3.0 ˚ A (left) and 2.7 ˚ A (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 B.3 The radial distribution function, g(R), of the distance between glycine and counterions: Gly -Na + (left) and Gly + -Cl (right). . . . . . . . . 122 xii Abbreviations BE benzene excimer CC coupled-cluster CCSD coupled-cluster with single and double substitutions CD Cholesky decomposition DFT density functional theory EA electron attachment EE excitation energy EFP effective fragment potential FRET F¨ orster resonance energy transfer GFP green fluorescent protein EOM equation-of-motion EOM-CCSD equation-of-motion coupled-clusters single and double substitutions HOMO highest occupied molecular orbital IP ionization potential LUMO lowest-unoccupied molecular orbital xiii MD molecular dynamics MO molecular orbital NBO natural bond orbital NEXAFS near-edge X-ray absorption fine structure PES potential energy surface QM/MM quantum mechanics/molecular mechanics RI resolution-of-identity XPS X-ray photoelectron spectroscopy xiv Abstract Predictive modeling of chemical processes in silico is a goal of XXI century. While robust and accurate methods exist for ground-state properties, reliable methods for excited states are still lacking and require further development. Electronically exited states are formed by interactions of matter with light and are responsible for key processes in solar energy harvesting, vision, artificial sensors, and photovoltaic applications. The greatest challenge to overcome on our way to a quantitative description of light-induced pro- cesses is accurate inclusion of the effect of the environment on excited states. All above mentioned processes occur in solution or solid state. Yet, there are few methodologies to study excited states in condensed phase. Application of highly accurate and robust methods, such as equation-of-motion coupled-cluster theory EOM-CC, is limited by a high computational cost and scaling precluding full quantum mechanical treatment of the entire system. In this thesis we present successful application of the EOM-CC family of methods to studies of excited states in liquid phase and build hierarchy of models for inclusion of the solvent effects. In the first part of the thesis we show that a simple gas- phase model is sufficient to quantitatively analyze excited states in liquid benzene, while the latter part emphasizes the importance of explicit treatment of the solvent molecules in the case of glycine in water solution. xv In chapter 2, we use a simple dimer model to describe exciton formation in liquid and solid benzene. We show that sampling of dimer structures extracted from the liq- uid benzene is sufficient to correctly predict exited-state properties of the liquid. Our calculations explain experimentally observed features, which helped to understand the mechanism of the excimer formation in liquid benzene. Furthermore, we shed light on the difference between dimer configurations in the first solvation shell of liquid benzene and in unit cell of solid benzene and discussed the impact of these differences on the formation of the excimer state. In chapter 3, we present a theoretical approach for calculating core-level states in condensed phase. The approach is based on EOM-CC and effective fragment poten- tial (EFP) method. By introducing an approximate treatment of double excitations in the EOM-CCSD (EOM-CC with single and double substitutions) ansatz, we addressed poor convergence issues that are encountered for the core-level states and significantly reduced computational costs. While the approximations introduce relatively large errors in the absolute values of transition energies, the errors are systematic. Consequently, chemical shifts, changes in ionization energies relative to the reference systems, are reproduced reasonably well. By using different protonation forms of solvated glycine as a benchmark system, we showed that our protocol is capable of reproducing the experimental chemical shifts with a quantitative accuracy. The results demonstrate that chemical shifts are very sensitive to the solvent interactions and that explicit treatment of solvent, such as EFP, is essential for achieving quantitative accuracy. In chapter 4, we outline future directions and discuss possible applications of the developed computational protocol for prediction of core chemical shifts in larger sys- tems. xvi Chapter 1: Introduction and overview Electronically excited states are ubiquitous in chemistry, biophysics, and biochemistry. They are at the heart of numerous phenomena such as energy and electron transfer, catalysis, and photosignaling. In the context of photovoltaic, excited states govern solar energy harvesting and electron transfer processes. In biological systems, photoinduced structural transformations, redox reactions, and fluorescence occur via excited states. The environment often strongly perturbs the electronic structure of the system by local and long-range electric fields, which is very important in photovoltaics and biology. While photovoltaic materials are solids, biological molecules are dissolved in water or embedded in lipidic membranes. Stabilization of some electronic states by the environment may lead to solvent- induced shifts of excitation energies and can be as large as 1 eV 1, 2 . The effect on ion- ization and electron attachment is even larger reaching several electron-volts 3–5 . Thus, it is important to include proper treatment of solvent effects into modeling excited-state properties. The assemblies of chromophores serve as building blocks for photovoltaic mate- rials. The energy flow and delocalization of the excited states in such systems are controlled by the electronic structure of the individual chromophores and interactions between them. Delocalization of the excited states in multichromophoric systems could 1 be described as linear combinations of excitonic configurations, also called local exci- tations, charge-resonance configurations, also known as ionic configurations, and mul- tiexcitons, configurations in which more than one chromophore is excited 6 . The charac- ter of the excited state arising in extended chromophoric systems is important because it defines the physics of the interactions between the chromophore and surroundings. Consider, for example, siglet fission materials that exploit a process in which one sin- glet excited state is converted into two coupled triplet states, as shown in Figure 1.1 7, 8 . Through this convertion the quantum yield doubles, thus increasing the efficiency solar cells. To control this process, one wants to tweak the properties of the bright locally excited state and a dark multiexcitonic state to maximize the singlet fission rate. Figure 1.1: An expanded Jablonski diagram illustrating singlet fission (SF) follow- ing electronic excitation. A coherent (TT) state dissociates to form two independent triplets. Adapted from Ref. 8. To understand the physics of this process, one needs to start with a small model to study excited states. The simplest model system representing molecular solids is ben- zene dimer. It is small enough to apply state-of-art methodologies for excited states such as EOM-CCSD and some results obtained for this model system could be generalized for more complex systems. Returning to biological applications, such photoinduced processes as photosynthe- sis, photoswitches, and fluorescence in the family of fluorescent proteins, are controlled 2 by the excited states. Interactions between chromophoric groups that absorb the pho- tons and the complex protein environment play a crucial role in these processes. In the case of the family of fluorescent proteins, there are various excited-state processes that follow the initial photoexitation, as shown in Figure 1.2 9 . Excited-state pathways are highly dependent on the local protein environment near the chromophore. Figure 1.2: Excited-state processes in fluorescent proteins. The main relaxation channel is fluorescence. Radiationless relaxation, a process in which the chro- mophore relaxes to the ground state by dissipating electronic energy into heat, reduces the quantum yield of fluorescence. Other competing processes, such as transition to a triplet state via intersystem crossing (not shown), excited-state chem- istry, and electron transfer, alter the chemical identity of the chromophore, thus leading to temporary or permanent loss of fluorescence (blinking and bleaching) or changing its color (photoconversion). Adapted from Ref. 9. Dependence of the transition energies on the local electronic structure enables appli- cation of spectroscopy to study local chemical environment in complex systems. For example, application of F¨ orster resonance energy transfer (FRET) reveals distance between chromophoric groups 10 . 3 A novel trend in spectroscopy of biomolecules is application of X-ray photoelectron spectroscopy (XPS) to study biomolecules in gas and condensed phases. Instead of ion- ization happening from delocalized valence orbitals, in the case of XPS, the ionization occurs from 1s orbitals that are highly localized on a single atom. This is illustrated in Figure 1.3. Figure 1.3: HOMO and core 1s orbitals of zwitterionic glycine. On the left, valence p orbital of glycine is shown that is delocalized over carboxyl group and aC. On the right, five core orbitals are shown. The core orbitals are localized on the corre- sponding atoms. The energy of such transitions are in range of hundreds electron-volts. Perturbations by the environment lead to about 1% shifts of the transition energies, which is several electron-volts, making them well resolved by modern synchrotron radiation sources. Thus, XPS spectra carry information about local chemical environment of specific atoms and may help study the local structure of biomolecules. This thesis consists of two parts. The first part (chapter 2) presents the application of highly accurate quantum chemistry method, EOM-EE-CCSD, to study excited states in liquid benzene. In particular we were interested in the nature of the lowest excited state in liquid and solid benzene and modeling excited states in condensed phase using simple gas phase dimer model. The results helped to understand the extent of delocalization of 4 the S 1 state in liquid benzene and shed light on differences between short-range order of liquid and solid benzene. The second part (chapter 3) presents method development and testing of perturbation theory based EOM-IP-S(D) approach to study core ionized states of molecules in con- densed phase. The developed approach was validated on a test set of small molecules in gas phase, and then extended to condensed phase using QM/MM methodology for treatment of solvent effects. The two parts of this thesis are connected by the main chal- lenge of these projects: an accurate description of condensed phase and accounting for interactions that affect excited states. This study resulted in the development of robust and scalable methodology to model X-ray absorption spectroscopy (XAS) that is both accurate and quantitative. 5 1.1 Chapter 1 references [1] C. Reichardt.SolventsandSolventEffectsinOrganicChemistry. VCH: Weinheim, 2nd edition, 1990. [2] A. DeFusco, N. Minezawa, L. V . Slipchenko, F. Zahariev, and M. S. Gordon. Mod- eling solvent effects on electronic excited states. J. Phys. Chem. Lett., 2:2184, 2011. [3] D. Ghosh, O. Isayev, L. V . Slipchenko, and A. I. Krylov. The effect of solvation on vertical ionization energy of thymine: from microhydration to bulk. J. Phys. Chem.A, 115:6028, 2011. [4] K. B. Bravaya, M. G. Khrenova, B. L. Grigorenko, A. V . Nemukhin, and A. I. Krylov. Effect of protein environment on electronically excited and ionized states of the green fluorescent protein chromophore. J.Phys.Chem.B, 115:8296, 2011. [5] D. Ghosh, A. Roy, R. Seidel, B. Winter, S. Bradforth, and A. I. Krylov. A first- principle protocol for calculating ionization energies and redox potentials of sol- vated molecules and ions: theory and application to aqueous phenol and phenolate. J.Phys.Chem.B, 116:7269, 2012. [6] David Casanova and Anna I. Krylov. Quantifying local exciton, charge resonance, and multiexciton character in correlated wave functions of multichromophoric sys- tems. JournalofChemicalPhysics, 144(1), 2016. [7] Millicent B. Smith and Josef Michl. Singlet fission. Chemical Reviews, 110(11):6891–6936, 2010. [8] M.B. Smith and J. Michl. Recent advances in singlet fission. Annual review of physicalchemistry, 64:361–86, 2013. [9] Pradeep Kumar Gurunathan, Atanu Acharya, Debashree Ghosh, Dmytro Kosenkov, Ilya Kaliman, Yihan Shao, Anna I. Krylov, and Lyudmila V . Slipchenko. Extension of the Effective Fragment Potential Method to Macro- molecules. JournalofPhysicalChemistryB, 120(27):6562–6574, 2016. [10] D.C. Harris. QuantitativeChemicalAnalysis. W. H. Freeman, 2010. 6 Chapter 2: Rewriting the Story of Excimer Formation in Liquid Benzene 2.1 Introduction Excimers (or exciplexes 1 ) formed by association of an electronically excited aromatic molecule with an identical (or different) one in its ground electronic state have intrigued photochemists for decades 2–5 . They are important reactive intermediates in photochemi- cal transformations 5–10 . Excimers are involved in the dissipation of electronic excitation in DNA oligomers by base pair exciplex formation, serving to protect our genome from radiation damage 11–15 . Excimer formation in photovoltaic materials may lead to exciton trapping, interfering with the generation of photocurrent in solar energy harvesting 16, 17 . Excimers can also promote desired excited-state processes, such as endothermic singlet fission 16, 18 . Full stabilization of an excimer is achieved at a specific intermolecular geometry (sandwich-like) and large changes in absorption and emission spectra accompany asso- ciation. This connection between excimers’ structure and the spectra can be exploited to probe orientation and transport in various micro-environments by time-resolving excimer formation through pulsed photoexcitation 19–23 . 7 As the simplest excimer, the benzene excimer (BE) serves as a model for such com- plexes 3 . Electronically, BE correlates with S 0 and S 1 monomers. It forms readily upon photoexcitation in the neat liquid and in solutions, fluoresces weakly 24 , but is easily detected by its intense broad absorption in the mid-visible 25–28 . Classic studies con- ducted in the mid-20th century determined a bond strength of0.3 eV (8.1 kcal/mol) for BE 29, 30 . The nature of bonding in BE is explained by molecular-orbital (MO) the- ory. In the ground state, the highest occupied MOs of the two benzene moieties form a bonding and an antibonding orbital, both of them doubly occupied, leading to zero bond order and, consequently, weak interaction (2.6 kcal/mol) between the two moieties 31 . The promotion of an electron from the antibonding orbital to higher MOs reduces the anti-bonding interactions resulting in partially covalent bonding. Because the splitting between the bonding and antibonding MOs is proportional to the orbital overlap, paral- lel arrangement of the two benzenes, as in the second panel of Figure 2.1, is required for BE formation. However, in the ground electronic state, the most favorable structure of the dimer is T-shaped (Figure 2.1, right panel) 31 . Thus, the formation of BE in the gas phase requires significant structural reorganization. Likewise, in solid benzene 32 the closest neighbors form a T-shaped pattern, as shown in Figure S4, which is not favorable for BE formation. A more detailed theoretical analysis of the electronic structure of BE is based on the excimer theory 9, 33–36 ; it shows that mixing of the local excitations (A AAA ) with charge resonance (A A + A + A ) terms is responsible for the excimer binding, which further explains why parallel stacking is required to afford full stabilization 37–45 . The required large structural reorganization is a likely reason for the absence of excimers in solid benzene, which is packed in a pair-wise perpendicular configuration of nearest neighbors 32 (Fig. S4), and for the slow BE formation in neat liquid benzene, as detailed 8 Figure 2.1: Structures of benzene dimer. The left panel shows the definition of structural parameters from Ref. 55: R is the distance between centers-of-mass of the two fragments, q and f characterize their relative orientation, and r 1 and r 2 quantify the offset between the two rings. Center and right panels show sandwich (q=f=0 ), Y-shaped (q=90 ,f=0 ), and T-shaped (q=90 ,f=0 ) benzene dimers. below 27 . Excitation spectra of BE formation in cold isolated dimers, which are initially arranged in a T-shaped geometry 46 , exhibit similar behavior. Selective vibronic exci- tation of isolated benzene dimers shows 47–51 that T-shaped dimers undergo evolution to a sandwich structure on a time scale of18 ps. The existence 52 of a shallow T- shaped minimum separated from a deeper minimum corresponding to BE structure, on the excited state PES has been confirmed by high-level electronic structure calculations by Roos and co-workers 53 . Here we revisit BE formation in neat liquid benzene. Owing to thermal fluctua- tions, structure of the liquid benzene 54, 55 differs considerably from that of the solid. Early experiments pointed towards mostly perpendicular arrangements of nearest neigh- bors 56, 57 , but a more recent study 54 revealed that the orientational structure of liquid benzene is more complex. High resolution neutron diffraction experiments with isotopic substitutions 54 have found that the first solvation shell extends from 3.9-7.25 ˚ A (g(r) maximum is at 5.5 ˚ A) and contains12 isotropically oriented molecules. The multi- dimensional analysis of the data has shown that at short separations (<5 ˚ A), parallel 9 structures are dominant. This is illustrated by two-dimensional probability distribution function,g(R;q), shown in Fig. 2.2. R θ Figure 2.2: Two-dimensional probability distribution function, g(R;q), of liquid benzene (see Fig. 2.1 for the definition of structural parameters). The main maxi- mum is at 5.65 ˚ A, and the shoulder seen at 4.25 ˚ A for parallel molecules indicates displaced (PD) structures. Reproduced with permission from Ref. 54. Copyright ACS2010. Molecular dynamics (MD) simulations 55 identified force-fields capable of repro- ducing structural properties of liquid benzene and provided more detailed data. The analysis 55 of the instantaneous orientations of the dimers from the first solvation shell have shown that parallel structures (defined asq<40 ) constitute about 24% of the total population and that the majority of the structures corresponds to perpendicular arrange- ments (T and Y-shaped). (The definitions of different structures are shown in Figure 2.1.) However, both experiment and theory agree that this overall distribution results from the averaging over two rather different subpopulations within the first solvation shell: at short distances (R<5 ˚ A), parallel structures dominate, whereas atR>5 ˚ A, per- pendicular orientations are prevalent. This structural inhomogeneity of liquid benzene is a key for understanding the mechanism of BE formation. 10 Figure 2.3: Top: Absorption spectrum of benzene excimer. Bottom: Kinetics of excimer formation probed at 530 nm (2.34 eV). The plot shows the rise of the excimer absorption near the peak. The inset shows the first picosecond of delay. For details, see text. The upper panel of Figure 2.3 shows the absorption spectrum of BE and the kinet- ics of its rise following UV excitation of neat liquid benzene. The characteristic BE absorption band centered at 505 nm (2.46 eV) has been associated with a symmetry forbidden B 1g !E 1u transition. Its substantial extinction coefficient of approximately 30,000 Lmol 1 cm 1 has been attributed to vibronic interactions and dynamic symme- try lowering 2, 26, 27, 29, 46 . The spectral breadth of BE absorption (5000 cm 1 or 0.62 eV) has been explained by fluctuations of the relatively floppy dimer structure. This characteristic spectrum has served for probing the dynamics of BE formation in neat 11 benzene. Pioneering studies employing one- and two-photon excitation were conducted by Mataga and coworkers with20 ps time resolution 2 ; they found that visible BE absorption rises gradually over30 ps. This slow rise was assigned to the timescale of reorientation of nearest neighbors from an initial perpendicular arrangement to the ultimate sandwich structure of the fully relaxed BE. To explain their data, the authors suggested a kinetic scheme which assumed initial instantaneous excitation of a benzene molecule to S 1 followed by pseudo first order formation of the BE. Accordingly, the delayed visible absorption observed to rise from zero upon irradiation was ascribed to an increasing concentration of a well-defined BE species. Later, in 1993, an ultrafast spectroscopic study Waldmanetal. used a three pulse pump-dump-probe method to see if transient hole burning could uncover inhomogeneous contributions to the BE absorp- tion 58 . While no inhomogeneity was detected with50 fs time resolution, the observed dump-induced bleach anisotropy was consistent with a BE transition dipole oriented in the excimer plane (perpendicular to the axis connecting the centers of the two benzene moieties). Several experimental and theoretical investigations have questioned some aspects of these findings. In 1997 Yoshihara and coworkers investigated the appearance of visible BE absorbance in the neat liquid at room temperature with100 fs resolution 59 . Contrary to the scenario presented by Miyasakaetal. 28 , nearly a quarter of the ultimate mid visible absorbance rose instantaneously, with the rest rising with a time constant of20 ps. Tentatively assigning this instantaneous rise to residual absorbance of the monomer S 1 state, this finding was not pursued further. This assignment contradicts earlier work by the same group where the absorbance of diluted solutions of S 1 benzene was characterized and no residual absorbance was observed 27 . Quantum chemical calculations by Diri and Krylov, which characterized the states involved in BE 12 absorption more precisely, showed that the BE absorption transition dipole is oriented along the intermolecular axis, and not in the sandwich plane 45 . The goal of this study is to revisit the process of BE formation in the neat liquid and to clarify its mechanism. The main question is the nature of the early time excimer absorption and its time evolution. Specifically, we want to distinguish between the two mechanistic possibilities: (i) initial excitation is localized on a single benzene molecule, which eventually finds a partner forming a delocalized exciton; (ii) initial exciton is delocalized on two neighboring molecules that happened to be in a favorable orientation; the excitation triggers attractive interactions and initiates structural dynamics leading to fully formed excimer. In case (i) the gradual rise of BE absorption would be due to the increased concentration of absorbers, whereas in case (ii) the rise of BE absorption would be due to the increased cross-sections of the individual absorbers. Thus, changes in polarization of the BE absorption would be different in the two cases. To distin- guish between the two scenarios, we employ improved experimental and theoretical tools. Experiments employ higher time resolution and broadband ultra-sensitive prob- ing capabilities. On the theory side, we compute electronic spectra of BE at different geometries by using equation of motion-coupled cluster method with single and double excitations (EOM-EE-CCSD) 60–63 . In addition to Franck-Condon and optimized BE structures characterized in Ref. 45, we performed scans along different inter-fragment displacements and considered representative structures from solid and liquid benzene. By combining theory and experiment, we show that, contrary to conclusions of Waldman et al. 58 , the relaxed BE bleach anisotropy induced with20 fs visible pulses starts at a value of0.3, indicating that the BE absorption transition dipole moment is indeed directed along the intermolecular axis. We ascribe the instantaneous visible 13 absorption first observed by Inokuchi et al. 59 , to transitions leading to excited states delocalized over nearby benzene molecules at the arrangements at which the benzene moieties are sufficiently close and nearly parallel so that there is a sufficient overlap between thep-systems of the two molecules. Such low-symmetry structures, which dif- fer considerably from the optimized structures of isolated benzene dimer and solid ben- zene, are sufficiently abundant in liquid benzene due to thermal fluctuations 55 . Finally, we assign the gradual rise in BE absorption to an increase in transition moment of the absorbing pairs as they relax and assume the ideal sandwich structure. 2.2 Methods 2.2.1 Experimental design Figure 2.4 sketches the design of the experiment. BE are generated by exciting room- temperature liquid benzene at 266 nm (4.6 eV) (pump pulse). The pump pulse initiates excited-state dynamics ultimately leading to BE formation. The excited-state dynamics is monitored by a white light probe pulse at different time delays giving rise to time- resolved excited-state absorption spectra. At large time delays, the probe pulse yields the spectrum of fully formed BE, which is shown in Fig. 2.3 (this spectrum was obtained by a UV pump — visible probe measurement at 50 ps time delay). BE has an absorption maximum at 505 nm (2.46 eV). In the second set of experi- ments, we introduced a third (dump) pulse centered at 500 nm. We call the secondary green excitation pulse a “dump” for consistency of terminology with similar excitation schemes. Here the population of excimer is indeed depleted by excitation to higher electronic states, not by stimulated emission. In these 3-pulse experiments, the excited- state absorption is bleached by the dump pulse. The dump pulse is introduced at several 14 Figure 2.4: Top: The schematics of the transient absorption measurements of BE formation using 2- (left) and 3- (right) pulse experiments. Bottom: Experimen- tal setup. Liquid benzene is excited with 266 nm UV light (pump) generated by third harmonic generation (THG) of the amplifier fundamental, which initiates the excited-state dynamics. The transient excited-state absorption is measured by a supercontinuum generated by white light generation (WLG) at various time delays (t). The right panel shows 3-pulse (pump-dump-probe) experiment con- cept. In these experiments, the dump pulse (500 nm) bleaches the excited-state population. The polarizations of dump and probe are controlled by a waveplate that rotates the fundamental generating the white light. In VV experiment, the polarizations of dump and probe are the same, whereas in a VH experiment they are cross-polarized. The signals are reported in units of DOD (the difference in optical density of the sample, in the presence and absence of pump/dump). Note that in 2-pulse experiments the chopper is positioned in the pump arm. delays ranging from 1 to 66 ps (see dots in the kinetic trace, bottom of Figure 2.3). As in the first set of experiments, the resulting population evolution is probed by the white light. The signals are reported in units ofDOD, the difference in optical density of the sample in the presence and absence of the pump (in the 2 pulse experiments) or dump (in the 3 pulse experiments).DOD represents the difference in absorbance of probe due 15 to the presence of pump/dump and gives the measure of change in absorbing popula- tions. By using different relative polarizations of dump and probe pulses (i.e., VV and VH), one can obtain information about the orientational dynamics of the excimers and the the direction of the transition dipole for BE absorption. The magic angle traces and bleach anisotropy were calculated from the VV and VH experiments, according to the following: 64 DOD MA = DOD VV + 2DOD VH 3 (2.1) r = DOD VV DOD VH DOD VV + 2DOD VH (2.2) The anisotropy results from the preferential bleaching of dimers whose transition dipole is aligned along the direction of dump pulse polarization. The probe then follows the return to a random orientation of the dimers which were not excited by the dump. A detailed description of the experimental setup is given below. Experimental details Transient absorption spectra were recorded on a home-built multi-pass amplified Ti:Sapphire apparatus, which produces 30 fs, 0.8 mJ pulses at the rate of 700 Hz and centered at 800 nm (1.55 eV). Pump pulses of 266 nm were produced by frequency tripling of the fundamental, and 25 fs dump pulses of were produced by TOPAS (Light conversion) at 500 nm (2.48 eV), by mixing the signal with the fundamental. Another fraction of the fundamental was focused on a 3 mm BaF 2 plate to generate broadband white light for probing. The pump-(dump)-probe signal were detected by focusing the probe on a fiber, which directed the beam to an imaging spectrograph (Oriel-Newport MS260i) assembled with a CCD (Andor technologies). 0.05 μJ dump pulses were 16 focused on the sample, flowing through a home built cell with quartz windows 120 μm in thickness and 0.5 mm path length. For the 3-pulse experiments, excimers were gen- erated to give an initial peak OD of0.2. This required a pump fluence of610 15 photons per square cm. We observed that BE generation is linear in the pump intensity. Accumulation of soot on the cell entrance wall over time required occasional translation of the cell. Polarization-dependent probing of dump-induced bleach was measured to obtain information about orientation of the nascent excimers. The rotation of polarization was achieved by using a half-wave plate before the BaF 2 crystal which rotated the continuum generating fundamental. 2.2.2 Theoretical framework To identify species responsible for the early time BE absorption, we computed electronic spectra of several model dimer structures representative of liquid benzene. While in molecular solids excitons are often delocalized over multiple chromophores 65 , in liquid benzene initial excitation is unlikely to extend beyond the two nearest neighbors because of: (i) low local symmetry and (ii) the delocalization in trimers and larger aggregates is less prominent that in dimers even in highly symmetric structures. The latter point has been recently quantified by Reid and co-workers in a combined experimental and theo- retical study of cofacially arrayed polyfluorene 66 . To test that delocalization in benzene follows similar trends, we computed excited states in two model trimer structures, one symmetric sandwich with the distance between the benzene equal 3 ˚ A and one in which the third molecule is further away, at 4 ˚ A. The results are shown in SI. We observe that in the symmetric trimer structure the extend of delocalization is noticeably smaller than in 17 the dimer and that in a lower symmetry structure the excited state is localized entirely on the dimer moiety. Thus, in the rest of the calculations we focus exclusively on dimers. We considered the following model dimer structures: (i) optimized structure of BE (oscillator strength of this structure is taken as absorption intensity of the fully formed BE); (ii) T-shaped and displaced sandwich structures of the gas-phase dimers 31 ; (iii) scans along inter-fragment distance of the sandwich structure; (iv) representative struc- tures from the x-ray structure 32 of solid benzene; (v) representative dimer structures from the first solvation shell of liquid benzene 55 . For each model structure, we computed electronic transitions (excitation energies and respective oscillator strength) between the lowest electronic state and a dense manifold of higher-lying states (up to 3 eV above the lowest excited state). We identified bright electronic transitions in the region of excimer absorption (2.5-3 eV). To account for homogeneous broadening arising from the inten- sity sharing between closely lying states, we computed integrated oscillator strength by summing the oscillator strengths of all electronic transitions within 0.2 eV from the brightest transition. For each model structure, we computed the ratio of the total oscil- lator strength to the oscillator strength of the fully formed excimer (structure (i)). This ratio represents a contribution of each model structure to the total zero-time absorption. To estimate zero-time absorption of solid and liquid benzene, we performed averaging over relevant molecular structures, as described below. To better understand the nature of the relevant states, we analyzed the respective electronic wave functions using wave function analysis tools 67, 68 . The two relevant parameters characterizing the transitions are DR 2 (the change in the size of electron density) and PR NTO (the number of natural transition orbitals involved in the transi- tion).DR 2 is the difference between the sizes of excited-state and ground-state electronic 18 Figure 2.5: Potential energy surface (q versus R, f=0 , see Fig. 2.1) of benzene dimer computed using CHARMM27 forcefield. The black lines represent the con- tour line of 0 kcal/mol. The points on the grid correspond to model structures used to represent liquid benzene. Reproduced with permission from Ref. 55. Copyright ACS2011. densities (computed as expectation value ofR 2 operator using the respective wave func- tions). Large values correspond to Rydberg or delocalized target states 69 , while small values (on the order of 1 bohr 2 ) correspond to local valence transitions. PR NTO stands for the participation ratio of natural transition orbitals 67 . It quan- tifies the number of individual one-electron transitions describing the excitation. As discussed in Ref. 68, this quantity can be used to identify the evolution of excited states from purely local excitations to charge-resonance ones. For example, a pure p!p transition in benzene hasPR NTO =2 because the two sets of degeneratep andp orbitals are involved in the transition. For two equivalent non-interacting benzenes (i.e., at infi- nite separation), this excited state will have PR NTO =4 corresponding to the four sets of degeneratep orbitals (depending on the symmetry, the orbitals can be either delocalized or localized, but PR NTO is always 4). This is the case of local excited states, i.e., a superposition of excitations on the individual fragments (A AAA ). However, if the two non-interacting benzenes are not equivalent due their local environment, the excited states are split and the excitations are localized on each moiety giving rise toPR NTO =2. 19 For the interacting monomers, the wave function acquires charge-resonance character, which is manifested by lowering ofPR NTO of the transition. The value ofPR NTO along with the character of the respective NTOs can be used to distinguish between local exci- ton and interacting excimer systems: excited states of the strongly integrating fragments with substantial charge-resonance character can be identified by low values of PR NTO and delocalized NTOs. Quantitatively, the amount of charge-resonance character in dimer’s wave function (with delocalized NTOs) can be computed as follows: w CR = 4PR NTO 2 100%; (2.3) wherePR NTO is the participation ratio for the dimer. If NTOs are localized on the same moiety, thenw CR =0. Computational details All calculations were carried out using the Q-Chem electronic structure package 70, 71 . Orbitals were visualized using IQMol and Gabedit 72 . The calculations of excited states were performed using the EOM-EE-CCSD/6-31+G(d) level of theory with the Cholesky decomposed two-electron integrals 73 , with threshold of 10 4 . Core electrons were frozen in correlated calculations. We note that excited-state wave functions in bichro- mophoric systems are heavily multi-configurational and include a mixture of valence and Rydberg local excitations and charge-resonance configurations 74 . EOM-EE-CCSD method is capable of treating all these configurations in a balanced and flexible way, since they all appear as single excitations from the closed-shell ground-state refer- ence 60–63 . In particular, EOM-EE-CCSD can describe interactions between Rydberg and valence states 69 and reproduce exact and near-degeneracies which are common in Jahn-Teller systems 45, 75, 76 ; this makes EOM-EE-CCSD a reliable choice for modeling 20 excited states in multi-chromophoric systems 77 (provided, of course, it can handle the size of an aggregate). Model dimer and trimer structures were constructed using monomers frozen at their equilibrium geometries (R CC =1.3915 ˚ A and R CH =1.0800 ˚ A) taken from Ref. 78 , where they were optimized using CCSD(T)/cc-pVQZ. Fig. 2.1 shows sandwich and T-shaped dimer structures. Inter-fragment distance R is defined as the distance between the centers-of-mass of the two monomers. Relative positions of the benzene monomers in the BE, parallel displaced configuration and the ground-state T-shaped dimer were taken from the literature 31, 45 . Structures of dimers representing solid benzene were constructed by taking relative positions of the monomers from the x-ray structure 32 , with the structures of the individ- ual molecules frozen at their equilibrium geometries. We only considered the closest neighbors from the crystal structure. Fig. S4 shows the structure of unit cell in which the structures of model dimers are highlighted in color. Blue is the central molecule, red molecules form the Y-shaped displaced (Y d ) dimers, yellow molecules form the T- shaped (T x ) dimers, green molecules form L-shaped dimers (L), purple molecules form head-to-tail (HtT) dimers. Dimer structures representing relevant configurations from the first solvation shell of liquid benzene were taken at the selectedR andq values (see Fig. 2.1 for the definition of structural parameters) corresponding to most abundant configurations. Figure 2.5 shows the (R/q) grid and the values of the selected structures superimposed on the potential energy surface (PES) of the dimer taken from Ref. 55; the values off, r 1 , and r 2 were taken as zeros. We do not consider a full grid ofq and R values, because some of these configurations (such as smallR and largeq values) lie in a strongly repulsive parts of the 21 PES and are, therefore, are not sampled in the course of equilibrium dynamics of liquid benzene. Following Ref. 45, the calculations of oscillator strength for all model structures were carried out at slightly distorted geometries to account for the effect of symmetry lowering on the oscillator strength. In the dimer, the symmetry was broken by manip- ulating four hydrogens lying in the plane containing the main symmetry axis. Those hydrogens were displaced such that they get closer to the horizontal plane (s h ) by 0.001 ˚ A, which corresponds to an angle of about 0.1 away from the planes of the respective monomers. The results reported in this study are not corrected for basis set super-position error (BSSE). As illustrated by Roos and co-workers, BSSE can noticeably affect binding energies, especially when using compact basis sets 53 . BSSE is mitigated by increasing the basis set and,especially, by including diffuse functions. The effect of basis set on electronic states of benzene dimer has been discussed Ref. 45, where it was shown that for spectroscopic properties (such as oscillator strength using the 6-31G+(d) basis is adequate. The Cartesian geometries of all the systems studied in this paper are given in the SI. The symmetry labels of the electronic states and the MOs correspond to the standard molecular orientation used in Q-Chem, which differs for some irreducible representa- tions from the Mulliken convention 79, 80 . 22 2.3 Results 2.3.1 Experimental The characteristic absorption spectrum of BE is shown in the upper panel of Figure 2.3. The absorbance maximum peaks at 505 nm (2.46 eV). The spectrum was obtained by exciting room-temperature liquid benzene at 266 nm and by probing changes in the opti- cal density of the sample (DOD) 50 ps later by using a broadband supercontinuum pulse. As shown below this delay is sufficient for full formation and equilibration of BE. The bottom of Figure 2.3 showsDOD at 530 nm as a function of pump-probe delay. DOD exhibits an extremely fast appearance of roughly a quarter of the ultimate absorbance, followed by a slower rise with a30 ps timescale. The timescales and relative ampli- tudes of the two components match those reported by Inokuchi et al. 59 In our exper- iments the fast rise is convoluted with a coherent artifact indicating a nearly instanta- neous rise within the experimental time resolution of100 fs. The spectra immediately after UV excitation are broadened and slightly red shifted relative to the equilibrium absorbance, which is established on the same timescale as the rise in the excimer band (Figure 2.6). Beyond this delay, there are no apparent changes in the shape of the BE absorption spectrum. We observe only a very gradual reduction in intensity, most likely due to singlet-singlet annihilation 27 . The nascent benzene excimers were interrogated by pump-dump-probe experiments. Figure 2.7 shows magic angle transient difference spectra calculated from separate VV and VH dump-probe relative polarization experiments upon 500 nm photolysis, as described in Section 2.2.1. Figure 2.8 shows spectral cuts of the data for a series of probe wavelengths. The salient effect of the re-excitation (dump) pulse is depletion of absorbance across the BE band. At short delay times this bleach appears to be narrower 23 Figure 2.6: Normalized transient absorption spectra at pump-probe delays ranging from 1 to 400 ps. than the full BE band, and even exhibits induced absorbance at the red and blue edges of the probed range (420-740 nm). This early narrowing decays rapidly leaving a residual bleach spectrum, which resembles the relaxed excimer absorption. The initial ultrafast spectral changes are most apparent in the cuts presented in Figure 2.8. This brief stage of spectral change is tentatively assigned to absorption and/or emission from the reactive excited state of the dissociating excimers. The excited-state absorption is prominent in the VV polarization data and absent in VH relative polarization between dump and probe pulses (see Figure S1). This assignment for fast spectral change indicates a dissociation time of a few hundred fs of the excimers after excitation in the visible. The residual bleach recovers partially on a timescale of30 ps, which matches the slow formation time of the excimers in two pulse experiments, suggesting that the same process is rate determining in both cases. Clearly, a large portion of photolyzed excimers revert after dissociation to the initial electronic state (S 1 ), but dissociation to the S 0 +S 1 pairs does not account for all photolyzed excimers. The initial BE absorbance is not restored even after hundreds of ps following the dump pulse, and about 25% of the bleach is irreversible (Fig. S2). We assign this persistent bleach to undetermined 24 Figure 2.7: Magic angle pump-dump-probe transient spectra for relaxed and fully formed excimers. Spectra discontinuity is due to scattering interference from the dump pulse. photochemical pathways 5 competing with the relaxation to S 1 . We note that even with- out secondary excitation, the UV excitation into S 1 of liquid benzene slowly produces soot, which accumulates on the Teflon filter diaphragm in our flow system. Adding another 2.5 eV of electronic energy must open new and more efficient photochemical pathways. To gauge the cross-section of BE absorption and its variation over the course of excimer formation, the dump-induced bleach was measured at different delays (1, 3, 6, 9, 20, 40, 66 ps) after the 266 nm pump (see dots in bottom panel of Figure 2.3). A quantitative analysis of the peak bleach intensity induced after the excimer band reaches its maximum (66 ps delay), normalized to the density of excitation and assuming 100% 25 Figure 2.8: Spectral cuts in the pump-dump-probe data showing dependence of DOD at various dispersed probe wavelengths as a function of dump-probe delay. bleaching efficiency, results in an extinction coefficient for the relaxed BE of 30-4010 4 (cm 1 m 1 ), in reasonable agreement with earlier reports 28 . This observation verifies that the long-lived 505 nm band 28 belongs to the BE, but it does not explain the underlying dynamics responsible for the gradual rise. This gradual rise in the visible absorbance can be explained by the two limiting mechanistic scenarios outlined in the introduction. Scenario (i), similar to that proposed by Miyasaka et al. 28 , is kinetic in nature; it entails a gradual transformation of excited monomers into BE according to the two-state scheme: S 1 +S 0 BE (2.4) In this picture, the rise in absorbance reflects an increase in the concentration of excimers, assuming that the contribution of each to the total absorption remains con- stant. In contrast, scenario (ii) involves a continuous evolution of the initially excited species over tens of ps, giving rise to a constant population whose transition dipole is 26 evolving over time, not its concentration. Such a continuous evolution can be roughly represented by a consecutive kinetic scheme with a very large number of intermediates: S 1 +S 0 !BE 1 !BE 2 !:::!BE eq ; (2.5) where BE eq denotes fully formed BE and BE i denote BE at different stages of structural relaxation. 0.00 0.04 0.08 0.12 0.00 0.03 0.06 0.09 Fractional bleach OD (530 nm) Figure 2.9: Fractional dump-induced bleach of the excimer absorbance at various stages of formation. To distinguish between the two scenarios, we recorded the fractional bleach, (DOD(530 nm, t=1 ps)/OD(530 nm), obtained with a constant dump fluence, at the delays cited above and marked by dots in the bottom panel of Figure 2.3. Figure 2.9 shows that the fractional bleach increases linearly with the instantaneous 530 nm absorp- tion at the dumping delay. This result perfectly matches the second scenario, a contin- uous rise in the absorption dipole strength of a constant number of absorbing dimers. To clarify this, assume that scenario (i) was correct, and that each excimer excited by the dump pulse is entirely bleached in the visible. The fractional bleach is simply the 27 absorption probability given by the dump fluenceF times the 500 nm absorption cross- section of the dimerss(500 nm). Since the excimer’ss(500 nm) in a two-state model is constant and the sample is approximately optically thin at all delays, so should the frac- tional bleach be. We investigate the fractional bleach at 530 nm instead of absorbance maximum, as the 500 nm region is overshadowed by the scatter of the dump pulse. The fractional bleach have been tabulated in Table S1 In practice, as we interrogate the sam- ple at different delays during the excimer band rise, the stronger the sample absorbs, the larger the fractional bleach induced by the dump, in perfect agreement with scenario (ii). It is surprising that one of these scenarios describes the dynamics so closely, as the reality might lie in between. What this finding implies about the nature of the absorption appearing immediately with the UV excitation will be discussed below. We also quantified the reorientational dynamics of the absorbing species throughout the rise in transient absorption by calculating the bleach anisotropy from VV and VH data according to Eq. (2.2). This anisotropy results from the preferential bleaching of dimers whose transition dipole is aligned along the direction of dump pulse polarization. The probe then follows the return to a random orientation of the dimers which were not excited by the dump. This experiment further allows us to test the symmetry of the transition dipole relative to the inter-fragment axis of the absorbing pairs. The resulting r(t) for the various dump delays is presented in Figure 2.10. Anisotropy of the bleach immediately after the dump has a relatively high value of0.3 for pump-dump delays of 10 ps and onwards. For the populations bleached earlier, we observe a significantly lower value of anisotropy (0.15). Nevertheless, in both cases we obtain a long-lived anisotropy with a component characterized by a time constant >5 ps (Figure 2.10 and S3), consistent with the same absorbing species throughout the rise in absorbance. This timescale clearly excludes benzene monomers as the absorbing species, since benzene 28 reorientation in the pure liquid takes place on a much shorter timescale. This initial value of0.3 is close to the theoretical value expected for a fixed orientation of the transition dipole in the molecular frame (0.4) and is much too high to agree with a transition oriented in a plane which should produce a value of 0.1 for r(t). Thus, at least at longer delays the absorbing species have a well-defined axial orientation of the transition dipole. Even at dump delays below 10 ps the initial anisotropy, if assigned to a single absorber is sufficiently above the theoretical value for in-plane polarization that we must conclude that the earlier study by Ruhman and co-workers erroneously assigned the green absorption to such a transition 58 . Figure 2.10: Anisotropy decay of population bleached during formation of excimer. 29 2.3.2 Theoretical Hole NTOs Particle NTOs S1 Sx Sx =0.4 =0.2 =0.3 Figure 2.11: Natural transition orbitals and the respective weights for the S 0 !S 1 and S 0 ! S x transitions at the equilibrium BE geometry. Orbitals are computed with EOM-EE-CCSD/6-31+G(d). Fig. A.6 shows NTOs for the S 0 ! S 1 and S 0 ! S x transitions at the geometry of the fully formed BE. The corresponding canonical MOs are shown in SI (Fig. S4). The transition corresponding to the excimer absorption (S 1 ! S x ) can be identified by its large oscillator strength. The S 1 state (1B 3g ) can be described as valencepp transition (DR 2 =0.78 bohr 2 ). As one can see from orbital shapes (Fig. A.6), the two leading excitations (l=0.4) of S 1 correspond to the excitation from orbitals that are antibonding with respect to the inter-fragment interaction to the orbitals that have bonding character. This is why the S 1 state of the dimer is bound. This dimer state correlates with the S 1 of benzene monomer (the correlation between dimer’s and monomer’s states is discussed in detail in Refs. 45 and 53). PR NTO of S 1 state of the monomer is 2.01. If the S 1 state 30 of the dimer had locally excited character (A AAA ), PR NTO would be equal to 4. PR NTO = 2.41 at the equilibrium geometry of BE reveals charge resonance contributions between the two moieties, which contributes to bonding. The S x state (4B 2u state) can be described as excitation from the p orbitals to diffuse Rydberg orbitals (its DR 2 is equal to 10.75 bohr 2 ). In contrast to the S 1 state, which is described by 2 pairs of NTOs, there are 4 pairs of NTOs with large weights in the S x state. The first two pairs of NTOs (l=0.3) can be described as excitation to diffuse antibonding (with respect to the two fragments) orbitals. The second pair (l=0.2) corresponds to excitation from bonding to diffuse bonding orbitals. These characters of leading NTOs result in more repulsive inter-fragment interactions in the S x state. We note that there is a shallow minimum on the S x PES at around 4.3 ˚ A(to be compared with the S 1 minimum at 3.1 ˚ A). The estimates of binding energies using unrelaxed PES scans from Fig. 2.12 yield 0.36 eV and 0.24 eV for the S 1 and S x minima, respectively (we note that for more accurate estimate, full geometry optimization and counterpoise corrections are necessary 45, 53 ). Figure 2.12 shows potential energy curves of the ground state (S 0 ), the lowest excited state (S 1 ), and the target excited state corresponding to the excimer absorption (S x ) along inter-fragment distance (R) for the sandwich structure. Fully formed excimer corre- sponds toR3 ˚ A. The S x curve is mostly repulsive; thus, excitation to this state should lead to the dissociation of BE. As one can see, the oscillator strength of the S 1 ! S x transition (shown along the S 1 potential energy curve) depends strongly on the inter- fragment separation. At the ground-state sandwich geometry (R4 ˚ A), the S 1 ! S x transition is four times less bright then transition from relaxed geometry of the excimer (oscillator strengths are 0.37 and 0.09 for the BE structure and for the ground state equi- librium structure, respectively). This means that if the experiments were initiated from the gas-phase ground-state sandwich dimer structure, one would observe the growth of 31 absorption cross-section in the course of relaxation and the instantaneous absorption would be equal 0.09/0.3725% of the absorption of the fully formed BE. Likewise, if the sandwiches were the most abundant structures in liquid benzene, their absorption would explain the observed zero-time absorption of BE. However, crystal structure of the solid benzene and simulations of liquid benzene show that these structures are not very common; thus, other types of structures need to be considered for explaining the instantaneous BE absorption in liquid benzene. 3 3.5 4 4.5 5 5.5 6 0 2 4 6 8 0.37 0.37 0.27 0.16 0.09 0.04 0.02 0.001 R, ˚ A Energy, eV S 0 S 1 S x 1 Figure 2.12: Potential energy curves of the three relevant states (ground state of the dimer (S 0 ), excimer’s state (S 1 ), and the state that has the largest oscillator strength for the BE absorption, (S x )) of the benzene sandwich along inter-fragment dis- tance,R, at perfectly stacked orientation (q=0,f=0,r 1 =r 2 =0). The values above the S 1 (1B 3g ) curve (excimer state) show oscillator strengths of theS 1 (1B 3g )!S x (4B 2u ) transition along the scan. 2.3.3 Properties of different model dimer structures representing solid and liquid benzene The unit cell of solid benzene 32 is shown in Figure S4 in SI. Representative dimer struc- tures, which are highlighted in color in Figure S4, are shown in Fig. S7 in SI: head-to-tail structure (HtT), L-shaped structure (L), T-shaped structure (T x ), and T-shaped displaced structure (Y d ). In these structures, the distance between the two rings is 4.9-5.8 ˚ A. We 32 did not consider parallel or parallel-displaced sandwich structures from the crystal struc- ture because the inter-fragment distance in these structures is too large (more then 7 ˚ A) to give rise to noticeable oscillator strength (cf. Fig. 2.12). For comparison, Figure S7 also shows structures of the parallel displaced gas-phase structure (PD g ), in which the distance between the monomer’s planes is 3.6 ˚ A and the shift between the main axes of the monomers is 1.6 ˚ A (Ref. 31). Table 2.1 summarizes the analysis of the S 0 -S 1 and the (S 1 -S x ) transitions (the latter corresponds to the excimer absorption). For the reference, relevant electronic transitions of the monomer (BZ) are also given. The top part of the table contains the results for model dimers shown in Fig. S7, the middle part shows the results for model structures representative of liquid benzene (Fig. 2.5), and the bottom shows the results for parallel- displaced structures with the same interfragment distance as in the fully formed BE. To account for homogeneous broadening arising from the intensity sharing between closely lying states, we computed integrated oscillator strength by summing up the oscillator strengths of all electronic transitions within 0.2 eV from the brightest transition. As one can see, the excitation energy of the S 1 state in all model dimers is almost the same as in benzene monomer. Moreover, this transition remains dark, despite lower symmetry and inter-fragment interactions: the computed oscillator strength for S 0 !S 1 transition does not exceed 0.0001. In the case of non-equivalent fragments, the S 1 state is split due to lower symmetry. In the L-shaped and head-to-tail dimers, PR NTO is close to 4, indicating the LE character of the S 1 transition in these structures. The value of PR NTO is close to 2 for T-shaped dimers, in which the excited states are localized on individual monomers (as one can see from the respective NTOs). The amount of charge-resonance in the S 1 wave function of BE is80%. As one can see, of all model structures, only 33 dimers with R = 4 ˚ A show substantial amount of charge resonance (16-19%). Charge- resonance persists upon small sliding displacements (about 1 ˚ A) and drops to zero for sliding distances above 2 ˚ A. Table 2.1: Excitation energies (eV), oscillator strength, and wave function analysis of the relevant excited states of benzene monomer and model dimer structures. EOM-EE-CCSD/6-31+G(d). Structure E S 0 !S 1 ex DR S 1 2 PR NTO (S 0 !S 1 ) a w CR (S 1 ) E S 1 !S x ex Osc. Str. b BZ 5.25 1.89 2.03 NA 2.76 0.015 (0.04) BE 4.20 0.76 2.41 (deloc) 79.5% 3.01 0.365 (1.00) T x 5.25 1.68 2.09 (loc) 0% 2.92 0.037 (0.10) Y d 5.25 1.71 2.03 (loc) 0% 2.93 0.044 (0.12) L 5.23 1.69 3.74 (loc) 0% 2.68 0.023 (0.06) HtT 5.25 1.72 3.61 (loc) 0% 2.74 0.006 (0.02) PD g 5.25 1.57 3.90 (deloc) 5.0% 2.52 0.038 (0.10) R=4,q=0,f=0 5.25 1.46 3.68 (deloc) 16.0% 2.79 0.091 (0.25) R=4,q=15,f=0 5.25 1.46 3.67 (deloc) 16.5% 2.82 0.075 (0.21) R=4,q=30,f=0 5.26 1.50 3.62 (deloc) 19.0% 2.84 0.051 (0.14) R=4.5,q=0,f=0 5.25 1.62 3.95 (deloc) 1.2% 2.71 0.035 (0.10) R=4.5,q=15,f=0 5.25 1.61 3.93 (deloc) 3.5% 2.71 0.028 (0.08) R=4.5,q=30,f=0 5.24 1.63 3.63 (loc) 0% 2.72 0.026 (0.07) R=4.5,q=45,f=0 5.25 1.60 2.15 (loc) 0% 2.74 0.018 (0.05) PD structures c R=4.0,q=0,d=1 5.20 1.54 3.85 (deloc) 7.5% 2.75 0.069 (0.19) R=4.0,q=0,d=2 5.23 1.70 4.05 (deloc) 0% 2.80 0.028 (0.08) R=4.0,q=0,d=3 5.25 1.74 4.06 (deloc) 0% 2.80 0.017 (0.05) R=4.0,q=0,d=4 5.25 1.75 4.06 (deloc) 0% 2.81 0.012 (0.03) R=4.0,q=0,d=5 5.25 1.75 4.06 (deloc) 0% 2.81 0.007 (0.02) a In parentheses, it is shown whether NTOs are localized or delocalized. b Integral oscillator strength for the S 1 absorption of the excitations in 0.2 eV range. The ratio to the oscillator strength of fully formed BE is given in parenthesis. c Parallel-displaced structures (d denotes the sliding distance, in ˚ A). The excited-state absorption (S 1 ! S x transition) in the benzene monomer is red- shifted relative to the BE by 0.25 eV and is weak (the oscillator strength is 4% of that 34 of BE). We see that all model structures from Table 2.1 show some absorption in the energy range 2.7-3.0 eV and that the relative intensity of these transitions varies from 0.02-0.25. The transition dipole moments of the bright transitions in all of the analyzed structures are aligned along the line connecting the centers-of-mass of the monomers. In the monomer, the dipole moment of the bright transition is perpendicular to the plane of the molecule. The oscillator strength corresponding to the absorption from the S 1 state of different model dimers is visualized in Figure 2.13. The y-axis shows the integrated oscillator strength for each dimer. The spectra of the individual model structures are shown in SI (Fig. S8). As one can see, the T x and Y d dimers, which are representative of the solid ben- zene, absorb within 0.1 eV from BE, with the relative intensity of about 10%. Other structures taken from the crystal structure of the solid benzene (e.g., L and HtT) show more monomer-like absorption (weaker and more blue-shifted). On the basis of these data, one can estimate that solid benzene might show some instantaneous absorption (<10%) originating from the T-shaped structures, however, the formation of the BE will be impeded by large structural reorganization required to form BE. To further strengthen this point, we approximated absorption of solid benzene by considering all possible dimers in the first coordination shell of benzene molecule. The 12 dimers arising from the shell are shown in Figs S4 and S7 in SI. Assuming equal probability of each dimer to absorb a photon, the total excimer absorption is just the average of instantaneous oscil- lator strengths of individual dimers (S 1 ! S x ) normalized by the maximum excimer absorption. Taking values of f l from the Table 2.1 the absorption of solid benzene at the initial time delay is: f l (Solid) f l (BE) = (0:037 2+ 0:044 2+ 0:023 4+ 0:006 4) 0:365 12 100%= 6:3% (2.6) 35 Model structures, which are representative of liquid benzene (structures of the near- est neighbors from the first solvation shell), may show substantial absorption (up to 25% of the fully formed BE). As expected, large values of f l are observed for structures with short R and small q and small parallel displacement (< 2 ˚ A); these are the structures that show substantial charge-resonance character. We note that this amount of charge resonance does not have a noticeable effect on theS 0 !S 1 transition (neither its energy nor its brightness). Thus, the existence of a proximal and nearly parallel neighboring monomer does not effect the absorption spectrum relevant to the pump, but significantly effects absorption of another photon in the mid visible. The results for S 1 !S x absorp- tion for model liquid structures are visualized in Figure 2.14. In order to connect these results with the observed early time absorption of liquid benzene, one needs to esti- mate relative abundance of different dimer structures in the first solvation shell. For that purpose, we rely on the results of the previous structural studies of liquid benzene 54, 55 . Figure S6 shows experimental and computed radial distribution function of liquid benzene 54, 55 . Following the conclusions from Ref. 55, we focus on the MD results obtained with the CHARMM27 force-field. The first solvation shell extends from 3.5 ˚ A to 7.0 ˚ A and contains 12.5 molecules. By integrating the radial distribution function from 0 to 5.2 ˚ A, we determine that the number of molecules at R<5.2 ˚ A is 2. Further analysis reveals that there is approximately one molecule at R=3.5-4.9 ˚ A. Since strong S 1 absorption corresponds to short intermolecular distances, we focus on the closest neighbors from the first liquid shell to estimate the absorption (as discussed in Sec. 2.2.2, the delocalization of excitons in liquid benzene is unlikely to extend beyond dimers). An important finding of structural studies of liquid benzene 54, 55 is that the angular distribution of benzene molecules is different at short and long ranges. This is illustrated in Fig. S6 in SI. When averaged over the first solvation shell, the angular distribution is 36 isotropic. However, at short distances, there is a strong preference for parallel configu- rations (q<45 ). This can be rationalized by analyzing dimer’s PES shown in Fig. 2.5. As one can see, at short distances T-shaped arrangement gives rise to repulsive interac- tions. By using angular distributions from Fig. S6, we can estimate relative populations of different orientations; the results are compiled in Table S2. As one can see, approxi- mately 50% of the nearest neighbors in liquid benzene have nearly parallel arrangement (q<30 ). Given that there are 2 molecules at R<5.1 ˚ A, we estimate that the solvation shell contains approximately one dimer atq<30 and R<5 ˚ A. We estimate the early time absorption of liquid benzene by averaging the absorption of the structures with R=4.0 ˚ A andq=0 , 15 , and 30 (see Table 2.1) weighted by the populations from Table S2: f l (liquid) f l (BE) = 2 (0:091 0:12+ 0:075 0:18+ 0:051 0:18) 0:365 100%= 18:4% (2.7) Since there are total of two molecules at R <5.1 ˚ A, this number provides an average instantaneous absorption of the S 1 state. 2.4 Discussion Concurrence of experiment and theory is a key point of our study. The crucial exper- imental advance necessary for this insight was the application of improved time reso- lution UV pump-VIS probe experiments. The theoretical modeling included quantum- chemical calculations of excited states of model dimer structures. These calculations clarified the implications of the structure of liquid benzene on its optical properties. Our main experimental findings can be summarized as follows: 37 100 BE 2.5 2.6 2.7 2.8 2.9 3 0 5 10 15 20 S T g T x T d L HtT PD g M E ex ,eV f l ,% 1 Figure 2.13: Bright electronic transitions in various model dimer structures. Exc denotes relaxed BE, S is sandwich at R=4.3 ˚ A, T g is T-shaped dimer at the gas-phase equilibrium structure, T x is the T-shaped dimer from the x-ray structure, Y d is a displaced T-shaped dimer, L is L-shaped dimer, HtT is head-to-tail dimer, PD g is parallel-displaced dimer gas-phase equilibrium structure, M is benzene monomer. Structures of dimer configurations are shown in Fig. S7 in SI. Total oscillator strength of the transitions corresponding to different dimer configurations and are given relative to oscillator strength of the relaxed BE transition and is calculated in 0.2 eV range centered at the excitation energy given on the x-axis. The experimen- tal maximum value of BE absorption band is 2.5 eV . 3.5 4.0 4.5 5.0 5.5 6.0 0 0.05 0.1 0.15 0.2 0 15 30 45 60 75 90 , ᴼ f l Figure 2.14: Oscillator strengths of model structures from liquid benzene. 38 1. Contrary to earlier literature, in the neat liquid ultrafast one-photon excitation into the S 1 state of benzene gives rise to a prompt mid-visible absorption band, which cannot be assigned to transitions from isolated excited monomers. 2. In agreement with earlier reports, the transient visible band rises gradually within tens of ps with only minor changes in peak position, leading to the characteristic BE absorption. On the basis of 3-pulse experiments, we assign this rise primarily to a growth in oscillator strength of a constant population and not an increase over time in the concentration of a well-defined absorbing species. 3. The initial value of bleach anisotropy in three-pulse scans indicates that the tran- sition moment of BE absorption is oriented along the intermolecular axis and not in the plane of the excimer. 4. Assuming the visible absorption is due to evolving excimers at all stages of BE formation, dynamics of the bleach anisotropy decay shows that already 2 ps after UV excitation, most absorbing pairs have forged a well-defined and significant association, not a fleeting interaction. This is clarified both by the initial value as well as the long lifetime for decay ofr(t) in all curves. Theory supports the conclusions drawn from the experiment. Calculations show that the excitation of liquid benzene results in significant instantaneous absorption in the mid-visible due to the abundance of dimer structures conducive to electronic delo- calization and, consequently, having noticeable excimer-like absorption at zero time. One can think of these structures, which are abundant in the liquid because of preferred parallel arrangements at short distances, as pre-formed excimers. The analysis of the wave function of these structures shows substantial amount of charge-resonance (16- 19%). Charge-resonance configurations are absent in locally excited states (even if these 39 states are delocalized over 2 benzenes) and reach 80% in fully formed BE. In the ground state, where interactions between all molecules are weak van der Waals ones, such pairs of molecules that happened to come close to each other, do not stay in this configuration for long and rapidly exchange partners. The analysis of g(r) of liquid benzene shows that in the first solvation shell there is approximately one molecule at R=3.5-4.9 ˚ A and this molecule is likely to be in a parallel arrangement to the central moiety. Once exci- tation selects a pre-formed excimer, the interaction between the two fragments is instan- taneously switched to a stronger attractive one and the two partners become locked. The attractive force pulls the two fragments together ultimately resulting in the fully formed sandwich structure. In the course of this structural reorganization, the electronic delocalization increases leading to the growth of the BE band. Experiment and theory also agree on the orientation of the transition dipole for absorption in the excimer frame (point 3). To elaborate on point 4, consider a situation when initial excitation is localized on a single benzene molecule rather than on a partially formed excimer. The score of nearest neighbors present prospective excimer partners and excited monomers would be unde- cided as to who to pair off with at least for a short time after excitation. Switching partners within the first solvation shell would be equivalent to rapid excimer reorienta- tion, and would add a rapid decay component to the anisotropy, whose amplitude would reflect the weight of this population. In addition, large initial variations in the structure of an absorbing pair could also cause significant reorientation of the dipole, contributing to such a rapid decay as well. From a pump-dump delay of 7 ps and throughout the rise in BE absorption, dump-induced bleach anisotropy does not change in amplitude or in decay kinetics, thus supporting point 4. 40 As stated in point 2 above, we assign the slow stage of BE formation to a buildup in the absorption cross-sections of an essentially constant concentration of pairs as they evolve towards fully relaxed excimer structure. Experimentally, this conclusion is based on the fractional bleach measurements. Yet, the structure of liquid is highly inhomo- geneous and instantaneous configurations of nearest neighbors are not the same, i.e., the starting points for the evolution of nascent excited states towards perfect sandwich geometry are different. Thus, an important question is how this structural inhomogene- ity should manifest itself spectroscopically and why we do not see its impact on the partial bleach results. The alternative mechanisms for explaining the gradual rise in BE absorbance, i.e., increased excimer concentration versus continuous variation in dipole strength are ideal extreme cases, neither of which is expected to perfectly describe obser- vations. It is true that the linear fit presented in Figure 2.9 involves significant uncer- tainty, but matches the latter within error. In particular, a positive deviation in the graph for low values of BE OD would be expected if the rise were due to a rising number den- sity of excimers since the impact of structural inhomogeneity should have its main effect at early delays. As the excited dimers relax towards fully formed BE on the attractive excimer PES, they become less likely to be disrupted by neighboring molecules. Theory provides qualitative explanation of the above observation. We show that the oscillator strength of the S 1 !S x transition of the BE in its equilibrium geometry is sig- nificantly higher than the oscillator strength at displaced and not perfectly parallel struc- tures of nearest neighbors abundant in liquid benzene. The analysis of the structure of liquid benzene suggests that a benzene molecule has at least one nearest neighbor within R=3.5-4.9 ˚ A and in a nearly parallel configuration. Thus, photoexcitation mainly results in excitons delocalized over pairs of neighboring molecules that happened to be in a nearly parallel arrangement. Once such pair is excited, the interaction between the two 41 molecules is switched to the attractive one, locking them together. These delocalized excitons feature 16-19 % of charge-resonance character, which gives rise to noticeable S 1 !S x absorption. To fully reconcile inherently dynamic insights from pump-probe experiments and the static perspective of the electronic structure calculations, dynamic simulations of the process of BE formation are needed. The only two measures which attest to liquid disorder effects in our experiments is the relatively mild spectral evolution in the visible absorption band during BE for- mation, and the concurrent moderate changes in bleach anisotropy scans over the same delay range. Both are complete within the first 10 ps of delay from the UV pump. The same initial disorder in the absorbing pairs both broadens their absorption band and adds uncertainty and rapid fluctuations in the orientation of the underlying transition dipole. It is still remarkable that throughout this significant restructuring, the peak of absorp- tion changes by only a few hundred cm 1 , and its width by only20%. The former proves that during restructuring and relaxation of excimers, the intermolecular coordi- nates involved have little impact on the energy gap between S 1 and the higher electronic states. This raises the question of which broadening mechanism is responsible for the full 5,000 cm 1 width of BE transition? Clearly it is not intermolecular motions of the two constituents of the excimer, as suggested in early studies of these species. If it were, then the structural reorganization taking place during excimer formation would cause significant changes in the peak wavelength of the visible absorption band, contrary to observation. This is also demonstrated by the bleach spectra in Figure 2.7. Only on sub-ps timescales does the transient bleach spectrum differ from a mirror image of the BE absorption, and we tentatively assign this to transient bands related to the excited state of the freshly excited excimers. The electronic structure calculations would sug- gest that absorption is heavily broadened by the dense manifold of states contributing at 42 all stages of evolution, providing an efficient mechanism of dephasing for the electronic transition dipole. It would also be interesting to know if intra-molecular vibrations con- tribute to the spectral width of the BE absorption as well. Experiments are ongoing in our lab to properly assign the extensive broadening active in this unusual chromophore and answer these open questions. Another interesting question which arises from this study pertains to the significant duration of the slow rise in BE absorption demonstrated in Figure 2.3. If the promptly absorbing pairs are already closely spaced and nearly parallel, one might expect that the final refinement of the sandwich structure should not be a 30 ps exponential rise as observed. To answer this full dynamic simulations of this process in the liquid combined with insights from electronic structure described above will be required. 2.5 Conclusions In this work, we have shown that excimer formation in neat liquid benzene at room temperature does not follow the simple two-state kinetic mechanism proposed based on classic picosecond laser investigations and involving reorientation from a T-shaped to a sandwich structure. Instead, we observe an immediate rise in excimer absorption, which we assign to excitation of two nearby benzenes in a nearly parallel arrangement that are already in a configuration facilitating sharing of the electronic excitation. Such struc- tures are abundant in liquid benzene at short distances. Such pre-formed excimers are capable of absorbing immediately after UV excitation in the same spectral region as that of relaxed sandwich structured excimers. Following the initial excitation, the extinction coefficient of this population continuously increases up to 4-fold, as the structures are relaxing towards fully formed BE. Reorientational dynamics of the absorbing excimers, measured by polarization-selective probing of a bleach signal induced by a secondary 43 ultrafast dumping pulse, proves that the BEs transition moment is directed along the inter-fragment axis. These insights were facilitated by enhanced time resolution and extension to multi-pulse excitation schemes, as well as by electronic structure calcula- tions comparison to theoretical calculations based on recent studies of local structure in the liquid benzene. These findings are discussed in relation to possible mechanisms underlying the extensive width of the mid-visible absorption spectrum of BEs. The proposed mechanism of BE formation should apply to other neat aromatic liq- uids favoring short-range stacked configurations (e.g., naphtalene, etc). Furthermore, this revised mechanistic picture of excimer formation in liquid benzene has implica- tions for singlet fission. While delocalization of the initial exciton over 2 (or more) chromophores is especial for promoting singlet fission due to entropic effects 81 , the full stabilization of the excimers results in the trap states and suppresses the yield of tiplets 16, 17, 82 . Thus, when considering the morphology of molecular solids, which is conducive of singlet fission, one should consider both the initial delocalization of the excitons and the their ability to form fully relaxed excimers. 44 2.6 Chapter 2 references [1] The term ”excimer” refers to the excited-state complex formed by identical chro- mophores, whereas the term ”exciplex” refers to the complex formed by different chromophores. [2] N. Mataga, M. 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J.Phys.Chem.C, 120:157–165, 2016. 51 Chapter 3: Coupled-cluster based approach for core-ionized and core-excited states in condensed phase: Theory and application to different protonated forms of aqueous glycine 3.1 Introduction By providing tunable high-energy radiation, advanced light sources enable a variety of X-ray based spectroscopies. Among those, x-ray photoelectron spectroscopy (XPS) and near-edge X-ray absorption spectroscopy (NEXAS/XANES) can be described as higher-energy analogues of the VUV-based techniques 1 . XPS determines ionization energies (IEs) of core electrons, whereas NEXAS probes electronic transitions from the core orbitals to low-lying valence or Rydberg orbitals. Owing to the localized nature of the core orbitals, these transitions enable probing local environment thus providing complementary information to valence-based techniques. 52 As in the case of VUV-based techniques, theoretical modeling is required to unambiguously relate experimental measurements to molecular structures. Thus, the advances in XPS and NEXAS have stimulated the development of theoretical tech- niques aiming at describing core-level transitions 2–17 . These theoretical studies have revealed that accurate description of core-level transitions is more challenging than that of valence-based ones. Superficially, ionization or electronic excitations of core electrons appear to be sim- ilar to valence transitions. That is, qualitatively, core-ionized states can be described by Koopmans theorem, whereas near-edge transitions correspond to single excitations of the core electrons. Thus, one may expect that electronic-structure methods developed for valence transitions can be trivially extended to core levels. Yet, numerical experiments have shown that direct application of standard approaches to core-level transitions leads to unsatisfactory results. For example, application of TDDFT to core excitations leads to errors of tens of electron-volts (typical errors are 10 eV for the second row atoms and are much larger for heavier elements). The errors are often systematic but functional- dependent 3 . Recently, Verma and Bartlett have shown 16 that some long-range corrected functionals 18, 19 tuned 20–22 to yield Kohn-Sham IEs consistent with the DSCF values perform very well for core-ionized and core-excited states: they reported impressive mean absolute errors (MAEs) of less than 1 eV for core-ionized and core-excited states when using their IP-tuned QTP(0,0) functional 22 . DSCF approaches show somewhat better performance 6, 23 , but they require a sepa- rate calculation for each core state and are often formulated in a spin-incomplete way. These features complicate evaluation of the spectra and, especially, transition properties. The experimentation with low-order wave function methods have been disappoint- ing. The application of CIS(D) method gives unsystematic errors (MAE of 2.6 eV , with 53 errors as large as 8 eV reported), especially for Rydberg states, whose energies are con- siderably underestimated 5 . The resulting spectra are, therefore, qualitatively incorrect. The situation is somewhat improved by using the scaled-opposite-spin version (MAE of 1-2 eV). Application of coupled-cluster methods to core states 9–15, 24 have shown promise, but also revealed some challenges, as elaborated below. Let us examine essential features of core-ionized and core-excited states. First, these states have open-shell character 25 . Thus, similarly to their valence counterparts, appli- cation of ground-state methods (such as HF, MP2, CCSD, or KS-DFT) would lead to spin-contaminated solutions. Spin-contamination is relatively moderate in the case of core-ionized states, which can be well represented by a single-determinantal ansatz. More serious issues can arise in molecules with symmetry-equivalent atoms due to sym- metry breaking. In the case of open-shell excited states, at least two determinants are needed for spin-completeness; consequently, a single-determinant ansatz leads to severe spin-contamination. Second, the core-levels states lie very high in energy. For example, the IEs of 1s elec- trons in C, N and O are around 300 eV , 400 eV , and 540 eV , respectively. Thus, in order to find the corresponding solutions, one needs to modify the SCF or iterative diagonal- ization algorithms. In the SCF procedure, constrained optimization is possible in which the algorithm locks the desired orbital occupation 6, 23 ; one robust algorithm of this type is called maximum overlap method (MOM) 23 . While at the SCF level these algorithms perform well, using these high-energy solutions in post-HF calculations (e.g., CCSD) often leads to numerically unstable behavior: i.e., CC ansatz would attempt to rotate the orbitals and find a lower-energy solution. In the methods involving diagonalization of an effective Hamiltonian (TDDFT, CIS, EOM-CC, ADC), the Davidson procedure can be modified to solve for the eigenstates dominated by the desired transition (MOM-like) 54 or lying within the desired energy range 12, 13 . Even more drastic reduction of the target space can be achieved by employing core-valence separation (CVS) approximation 26 in which pure valence excitations are filtered out 7, 8, 14 . Third, these high-lying states are metastable with respect to electron detach- ment 27–29 . More specifically, they are Feshbach resonances that can autoionize via two-electron transition in which one valence electron fills the core hole and the sec- ond valence electron is ejected. Thus, they are embedded in ionization continuum and their description within Hermitian quantum mechanics is problematic. Owing to their metastable nature, an attempt to compute such states often leads to finding pseudo- continuum states in which one electron occupies the most diffuse orbital 30 . 1h 2h1p (cont) 2h1p Figure 3.1: Core-ionized states are Feshbach resonances. They can autoionize via a two-electron process in which one electron fills the core hole providing enough energy to eject the second electron. To describe this process, the wave-function ansatz needs to include 2h1p excitations, such as three right-most configurations. Fourth, orbital relaxation effects are much more important for core states than for valence states because the outer orbitals are more delocalized and better shielded from the nuclear charge than the tight and localized core orbitals. This means that removal of an electron from a valence orbital does not perturb other orbitals to the same extent as removing an electron from a core orbital. 55 Relativity has the effect of lowering the energy of core orbitals, but for the second row elements the effect is small (Evangelista 6 and Belsey 5 reported corrections of 0.1, 0.2 and 0.3 eV for C, N, and O respectively). Here we focus on light atoms and, there- fore, we are not concerned about relativistic effects. In this paper we report a reduced-cost extension of equation-of-motion coupled- cluster (EOM-CC) theory to core-ionized and core-excited states. Our aim here is to develop an economical and robust approach that can be applied for modeling core states in moderate-size molecules in condensed phase. Since many experiments are conducted in condensed phase, it is important to include the effect of the environment on the electronic structure. Due to different electronic distributions, solvent leads to preferential stabilization of some electronic states relative to others, which results in solvent-induced shifts of transition energies. Shifts in excitation energies can be as large as 1 eV 31, 32 . The effect on ionization/electron attachment energies is even more pronounced — shifts of several electron-volt in polar solvents and in proteins are rather common 33–35 . For valence-ionized and excited states, the combination of EOM-CC with electrostatic embedding models 36 , such as QM/MM 37 or QM/EFP 38 , lead to accu- rate results 33–35, 39–41 . Unfortunately, using these explicit solvent models requires an extensive sampling of equilibrium solvent configurations (e.g., at least 20-50 snap- shots needed to be included in averaging for the converged results 42 ), which makes the requirements for the cost and the robustness of the underlying QM method even more stringent. Interestingly, despite the localized nature of the core states, the effect of solvent is critically important for interpreting the experimental core-level spectra 43 . Below we present a low-order approximation to EOM-CCSD that reduces the cost of the calculations and, more importantly, eliminates any convergence issues arising due to the metastable nature of the core states. The errors in absolute excitation/ionization 56 energies are relatively large but systematic. Our benchmarks illustrate that chemical shifts, i.e., changes in core IEs due to the local chemical environment, can be reliably reproduced. The resulting method is inexpensive and robust; thus, it can be easily com- bined with explicit solvent models. By considering different protonation states of aque- ous glycine, we show that our approach can accurately account for solvent effects and reproduce experimental chemical shifts. 3.2 Theoretical approach The EOM-CC family of methods enables a robust and reliable description of vari- ous electronic states including ionized, electron-attached, and electronically excited states 44–50 . The target states are described by using the following ansatz: Y= ˆ Re ˆ T F 0 ; (3.1) whereF 0 is the reference determinant, operator ˆ R is a general excitation operator and ˆ T is coupled-cluster operator for the reference state. As a less computationally expensive alternative to full EOM-CCSD, T 2 amplitudes from an MP2 calculation can be used to construct ¯ H, giving rise to the EOM-MP2 family of methods 51–54 . The amplitude of the operator ˆ R are found by solving the following eigen-problem: ¯ HR k =W k R k ; (3.2) where ¯ H = e T He T . Here we are considering the variant of the theory that includes single and double excitations (EOM-CCSD). In this case, the operator ˆ T includes hole-particle (1h1p) and two-holes-two-particles (2h2p) configurations and satisfies the 57 CCSD equations for the reference state. The form of the EOM operators ˆ R depends on the nature of the target state. To describe open-shell ionized state, the operator ˆ R includes single (1h) and double (2h1p) ionizing configurations: ˆ R IP = å i å r i i+ 1 2 å ija r a ij a † ji (3.3) giving rise to the EOM-IP-CCSD method. Electronically excited states can be described by using electron-conserving operator ˆ R: ˆ R EE =r 0 + å ia r a i a † i+ 1 4 å ijab r ab ij a † b † ji (3.4) giving rise to the EOM-EE-CCSD method. Here a † ;b † ;::: denote creation operators and i; j;::: denote annihilation operators. The reference state in EOM-EE and EOM-IP calculations of the ionized and excited states corresponds to the closed-shell ground state of the system (see Fig. 3.2). The reference determines the separation of orbital space into the occupied and virtual subspaces. Here we use indexes i; j;k;::: and a;b;c;::: to denote the orbitals from the two subspaces. R EE R IP Figure 3.2: Target core-ionized (left) and core-excited (right) states are generated by the EOM-IP and EOM-EE R 1 -operators from a closed-shell reference of the neutral. 58 Because of linear parameterization of the target states, EOM-CC methods are capa- ble of describing multiple electronic states in a balanced way. For valence states domi- nated by one-electron transitions, such as Koopmans-like ionized states or singly excited states, the errors of EOM-CCSD methods are less than 0.3 eV . The errors in the rela- tive energies of the target states are usually smaller. The EOM-CC ansatz is capable of describing states of different nature (e.g., valence and Rydberg) as well as of mixed character on the same footing. Importantly, EOM-CC provides a consistent descrip- tion of ionized and excited states, i.e., the onsets of ionization continua in EOM-EE calculations correspond to the EOM-IP IEs. Thus, Rydberg states series in EOM-EE calculations converge to the correct correlated ionization thresholds. The single excita- tion part of the operator ˆ R (1h or 1h1p) describes leading configurations of the target state, whereas the doubly excited part (2h1p and 2h2p) describes correlation and orbital relaxation effects of the target sates. The similarity transformation of the Hamiltonian wraps in electronic correlation effects of the reference states and insures size extensivity, which leads to improved accuracy of EOM-CC or EOM-MP2 relative to the respective CI counterparts (in which the target states are described by the same linear ansatz, but the operator ˆ T is zero). Can one extend EOM-CCSD methodology to describe core-ionized and core-excited states? The algorithm for solving eigen-problem (3.2) can be trivially extended 12, 13 to look for the eigenstates of desired character (e.g., 1s(N) hole or 1s(N)-p ) or states lying within a specified energy range. In some cases, the equations converge smoothly to the desired states yielding reasonable energies 12 . However, applications to a larger set of molecules revealed that, more often than not, the EOM-CC equations fail to converge. Problematic convergence (100 Davidson iterations) can be seen in the reported data in Ref. 13 ; it was also mentioned in Ref. 14 . 59 The analysis of the EOM-CC wave functions in the cases of problematic conver- gence allows us to attribute this behavior to the resonance nature of these states. That is, in EOM-IP calculations of a core-ionized state, the amplitudes of 2h1p operator keep increasing, in an attempt to converge to a detached state (see Fig. 3.1). This behavior is akin to the well-documented case of shape resonances, metastable states that can decay via one-electron process (see, for example Fig. 2 in Ref. 55 ). Upon increas- ing one-electron basis set, the shape resonances dissolve in the discretized continuum. Only by invoking special techniques, such as complex-variable extensions of standard approaches, these states can be reliably computed as eigenstates of a modified non- hermitian Hamiltonian 27, 28 . EOM-CC methods have recently been extended to describe resonances by using complex-scaled and CAP-augmented approaches 55–58 . Such calcu- lations of resonances are rigorous and provide not only energies of the resonances, but also their lifetimes. However, such calculations are much more expensive than regular bound-state calculations. Thus, our aim here is to introduce a less expensive approx- imate method for finding energies of core ionized and core-excited states. From the methodological point of view, there is an important difference between the two types types of resonances. Since the shape resonances can decay via one-electron detach- ment, their metastable nature manifest itself at the CIS level (as in the PYP example from Ref. 55 ). Thus, the attempt to compute these states in a reasonably large basis set will always be complicated by their mixing with discretized pseudo-continuum states. Contrary to that, the Feshbach resonances can only couple to the continuum by two- electron excitations. Thus, at the CIS level, they behave as bound states. Here we exploit this difference as follows. Our approach is based on an approximation to EOM-CCSD eigen-problem, which is similar to CC2 or CIS(D) approaches. We consider EOM-CCSD (or EOM-MP2) ¯ H 60 as our full Hamiltonian. We then introduce zero-order wave functions, which contain only 1h and 1h1p configurations. The amplitudes of the respective R 1 operators are found by diagonalizing the singles block of ¯ H. We then employ 2nd order Raileigh- Schr¨ odinger perturbation theory to evaluate energy correction due to double excitations. We follow the same formalism as in Ref. 59 . The programmable expressions for find- ing zero-order states and energy correction can be easily derived from the EOM-CCSD expressions fors-vectors 60, 61 ; they are given in the SI. The resulting methods can be called EOM-IP/EE-CCSD-S(D). A similar approach can be applied to the EOM-MP2 ansatz giving rise to EOM-IP/EE-MP2-S(D) methods. We note that since double excita- tions are described perturbatively, the most expensive EOM step is non-iterative. Thus, the scaling of the EOM-IP-MP2-S(D) calculation is N 5 and the N 5 step is non-iterative, which greatly reduces storage requirements. We note that the CVS approximation 7, 8, 14 effectively mitigates the problem of coupling with the continuum states by removing pure valence excitations (such as 3 rightmost configurations in Fig. 3.1) and, therefore, reducing the density of states in the continuum. However, straightforward implemen- tation 14 of CVS using a projector operator that zeros out parts of the EOM amplitudes that do not involve core electrons has the same computational scaling and cost as regular EOM-CCSD. Below we illustrate numeric performance of our approximate models for valence and core states by considering a set of small molecules. We then show that, when com- bined with explicit solvent models, these models can accurately reproduce experimental chemical shifts for core-ionized states. We note that our approach for core states is sim- ilar, in some aspects, to the methodology developed by Ghosh 54 for valence ionized and excited states. 61 3.3 Modeling solvent effects for core-ionized and core- excited states Including solvent is critically important for interpreting experimental spectra. For exam- ple, one exciting applications of core-level spectra is for distinguishing between differ- ent protonation states of amino-acids in realistic environments (protein, solution) 43, 62 . As was illustrated by Winter and co-workers 43 , although the protonation can change gas-phase IEs by several eV (e.g., 1-9 eV for glycine) the solvent screening results in much smaller chemical shifts (0.2-2 eV). Calculations on model systems 43 have shown that although the implicit solvent models capture the effect qualitatively, they are not sufficient for quantitative predictions of chemical shifts in solutions. Here we system- atically investigate the effect of the solvent on core IEs of glycine using the following approaches: 1. Non-equilibrium polarizable continuum model 63 (PCM). We use the zero-order approach in which no state-specific solvent response is included (see Refs. 64, 65 for detailed benchmarks of different versions of PCM using ADC wave functions). Following Ref. 43 , we consider two sets of models: bare glycine with PCM solvent and an optimized glycine molecule clustered with six explicit water molecules embedded in PCM solvent (note that the structure of the cluster is static and no equilibrium averaging is performed in this calculation). 2. Explicit solvent represented by either point charges (QM/MM) or by EFP (QM/EFP). In these approaches, the spectra are computed by averaging IEs over snapshots obtained from molecular dynamics (MD) simulations. For valence states, such protocols have been shown to yield very accurate spectroscopic 62 and thermochemical results 33, 35, 42, 66, 67 . We compare two approaches: (i) only glycine is included in the QM part and (ii) the QM part comprises glycine and several explicit water molecules. The structures of the QM and MM parts are taken from the MD snapshots. 3. To better understand different contributions to solvent shifts, we also considered model structures taken from representative snapshots from the MD trajectory. For these snapshots, we compare the magnitude of shifts computed by various solvent models without performing equilibrium averaging. 3.4 Computational details All electronic structure and QM/MM calculations were performed using the Q- Chem 68, 69 electronic structure package. QM/EFP calculations were performed with libefp 70, 71 . MD simulations were performed using the NAMD package 72 . 3.4.1 Benchmark calculations As a benchmark set, we considered several small molecules for which gas phase core IEs and core excitation energies are available. The structures for N 2 , CO, C 2 H 2 are taken from Ref. 59 ; all other structures were optimized with RI-MP2/cc-pVTZ. All Cartesian geometries are given in SI. Several Dunning basis sets were considered: cc-pVTZ, aug- cc-pVTZ, and cc-pCVTZ. We found that the cc-pVTZ basis is sufficient for core-ionized and core-valence excited states. For core-Rydberg states, augmented bases are neces- sary. Using cc-pCVTZ did not result in better performance. For comparison purposes, we also report DSCF calculations with DFT using wB97X-D functional 73, 74 and the cc-pVTZ basis set. In these calculations, core IEs 63 are computed as total energy differences between the neutral and core-ionized systems. The SCF solutions for core-ionized states are found using the MOM algorithm 23 , as implemented in Q-Chem 68 . 3.4.2 Calculations of different forms of aqueous glycine To describe equilibrium sampling of solvated glycine, we performed molecular dynam- ics (MD) simulations using NAMD software 72 with CHARMM forcefield 75 , using NPT ensemble (T=300 K, P=1 atm, density equals 1.0 g/ml 3 ) with periodic boundary condi- tions, rigid TIP3P water molecules, and simulation time step of 2 fs. Prior to running production trajectories, the water box was allowed to equilibrate for 2 ns. We first per- formed nanosecond relaxation of solvent box with solute frozen and then allowed the full system to equilibrate. After that we performed production run of 500 ps, taking the snapshots every 5 ps. In the simulation of the zwitterionic form of glycine, the box contained 1 glycine molecule dissolved in 1486 water molecules. In the case of the deprotonated form, the box contained a single deprotonated glycine dissolved in 1828 water molecules and a sodium atom added to neutralize the charge of the system. The parameters for the deprotonated glycine N-terminus atom were taken from the deprotonated form of lysine residue. N(NH 2 -terminal)-C(CH 2 )-C(COO-terminal) angle and dihedral C(COO- terminal)-C(CH 2 )-N(NH 2 -terminal)-H(CH 2 ) angle were treated using the parameters from the corresponding angles of the neutral (canonical) glycine. In the simulation of the protonated form of glycine, the simulation box contained one glycine molecule dissolved 1877 water molecules and a single chloride ion added to neutralize the charge of the system. In the MD simulation, the charge of C in the CH 2 group was adjusted to 64 balance the total charge of the molecule. The size of the water box was 35-40 ˚ A, which was shown 33, 76 to be sufficient for obtaining converged valence IEs. For the definition of the first solvation shell and for the hydrogen-bond analysis, we employed the following constraints. We defined cutoff for the first solvation shell at 3.0 ˚ A: if any atom of a water molecule was within this distance from either oxygen or nitrogen, this water was assigned to the solvation shell. The average number of water molecules assigned to the first solvation shell using 3 ˚ A cutoff was 8.2 for the zwitter- ionic form, 7.2 for the anionic form, and 5.1 for the cationic form. The respective distributions are shown in Fig. S2 in SI. This definition of solvation shell is used in selecting the QM subsystem in the QM/MM calculations. We also tested a shorter cutoff radius (2.7 ˚ A) and found that using shorter cutoff radius significantly affect the number of molecules assigned to the first solvation shell (see Fig. S2) and leads to large errors in IEs. We defined hydrogen bonds between the solute and the water molecules in the first solvation shell using bond length of 3.0 ˚ A or less and hydrogen bond angle of less than 30 degrees. The resulting distribution of the hydrogen bonds is discussed below. Running averages of the total energy and RMSD of glycine, which are shown in SI, confirm that the simulation length was sufficient to fully relax both the solvent box and the solute. The radial distribution function of the distance between the solute and the counter ion for simulations of protonated and deprotonated states are shown in Figure S3. The solute and counter ion are separated by several solvation shells during roughly 99% of the simulation time. Due to the screening effect of water, the resulting effect of the counterion on the electronic structure of the solute is relatively small. 65 Model structures and QM/MM and QM/EFP calculations As described in Section 3.3, we compared several approached to model solvent effects. The structure of the canonical form of glycine (Gly can ), which is representative of the gas-phase glycine, was optimized by RI-MP2/cc-pVTZ. All chemical shifts were com- puted with respect to this structure. The structures representing other forms of glycine (Gly ZI , Gly + and Gly ) were optimized using CPCM/wB97x-D/cc-pVTZ. All four structures are shown in Fig. 3.5; the respective Cartesian geometries are given in SI. To compare with the approach of Winter and co-workers 43 , we computed solvent shifts using CPCM model using the above optimized structures of different forms of glycine (these results are denoted by Gly PCM XX ) as well as model structures 43 of glycine with six water molecules embedded in PCM (these calculations are denoted by Gly PCM 6w;XX ). To compute IEs with explicit solvent models, we followed protocols similar to those developed earlier for valence states 33, 35, 42, 66, 67 . Specifically, we performed QM/MM and QM/EFP calculations using two choices of the QM system. In small QM calcula- tions, only glycine was included, similar to our previous calculations of valence IEs 33, 35 ; these calculations are denoted by Gly XX /MM or Gly XX /EFP. In large QM calculations, glycine and water molecules from the first solvation shell (defined using cutoff of 3 ˚ A, as explained above) were included in the QM part. The number of water molecules in these calculations (denoted as Gly XX;w /MM or Gly XX;w /EFP) varied from 2 to 8; the distributions for different protonation forms are shown in Fig. S2 in SI. For the EFP calculations, we report IEs computed with and without solvent polarization response correction 39 ; these are denoted by EFP and EFP fz , respectively. The QM/MM and QM/EFP IEs were computed by averaging over 100 snapshots taken every 5 ps from 66 500 ps equilibrium trajectories. We note that prior to the QM/MM and QM/EFP calcu- lations, each snapshot was moved such that glycine is always in the center of the box. For detailed analysis of different solvent models, we also considered model clusters from selected snapshots from our equilibrium MD simulations (the structures of these snapshots are described below; the respective Cartesian geometries are given in SI). 3.5 Results and discussion 3.5.1 Benchmark calculations Before proceeding to calculations of core-ionized and core-excited states, we consider the performance of these approximations by looking at selected valence ionized and excited states. Table S1 in SI shows results for the valence ionized and lowest excited states of formaldehyde. As one can see, the errors in IEs introduced by perturbative account of double excitations range between 0.36-0.72 eV . The behavior appears to be systematic: the EOM-CCSD-S(D) values are blue-shifted with respect to full EOM-IP- CCSD, which is expected since the reference state is described by the full EOM-CCSD and the target states by an approximate S(D) ansatz. We note that the EOM-MP2 approximation to EOM-IP-CCSD performs really well, both in the case of full EOMIP- MP2 (errors of 0.01-0.04 eV) and in the case of EOM-IP-S(D) (errors of 0.42-0.83 eV). We observe similar errors in ethylene and ethane (Tables S2 and S3 in SI). Thus, on the basis of these benchmark calculations, the MAEs for EOMIP-CCSD-S(D) and EOM-IP-MP2-S(D) are 0.23 and 0.51 eV , respectively, with standard deviations 0.23 and 0.18 eV . The errors for excited states are also systematically blue-shifted, but they are larger, e.g., 1.00-1.31 eV for EOM-EE-CCSD-S(D), suggesting that correlation and 67 orbital relaxation effects are more significant for excited states. We now proceed to the core-ionized states. The modified Davidson procedure 12 converges very fast for EOMIP-CCSD-S(D) and EOMIP-MP2-S(D). The core states have clear Koopmans character. The IEs obtained from diagonalizing the SS block of ¯ H are very close to the respective orbital energies (Koopmans IEs). The magnitude of the (D) correction is large: it varies between 10-20 eV . For the full EOM-IP-CCSD/MP2 methods, we were able to obtain results only for a subset of molecules. We begin by analyzing the results for selected small molecules. Table 3.1 shows the results for model ammonia clusters in different protonation states. For this system, we can also compare our results with previously reported ADC(4) calculations 77 . For the cases when converged EOMIP-CCSD results are avail- able, the errors of IP-CCSD-S(D) range between 0.4-0.7 eV (again, the errors are sys- tematic). As one can see, the protonation state and clustering have noticeable effect on core IEs. These chemical shifts relative to the reference NH 3 system are shown in parentheses; they range from -0.7 to +11.6 eV . The errors of approximate S(D) methods in chemical shifts relative to the full EOM-IP-CCSD are much smaller: the largest error of IP-CCSD-S(D) is 0.4 eV for NH + 4 , which has a chemical shift of 11.61 eV . Thus, one can expect that due to the systematic nature of the errors, IP-CCSD-S(D) might be able to reliably reproduce chemical shifts. We note that EOM-IP-MP2 versions perform sim- ilarly to the EOM-IP-CCSD; the differences in the computed chemical shifts between the two models are less than 0.2 eV . Tables S4-S9 in SI show the results for core-ionized states of selected molecules. Tables S4, S6, and S8 show absolute values of IE for carbon, oxygen, and nitrogen 1s levels, respectively. The chemical shifts computed against reference molecules are 68 Table 3.1: Core ionized states of model ammonia systems a . The chemical shifts relative to the reference NH 3 system are shown in parentheses. Molecule ADC(4) b IP-CCSD IP-CCSD-S(D) IP-MP2-SD IP-MP2-S(D) NH 3 406 406.44 (0.00) 406.97 (0.00) 406.87 (0.00) 407.59 (0.00) NH + 4 418 417.64 (+11.2) 418.58 (+11.61) 418.04 (+11.17) 419.09 (+11.5) NH 2 395 395.52 (-10.92) 395.89 (-11.08 ) 396.18 (-10.69) 396.78 (-10.81) NH + 4 NH 3 413.97 414.89 (+8.45) 415.38 (+8.41) 415.33 (+8.46) 415.97 (+8.38) 411.12 411.97 (+5.53) 412.61 (+5.64) DNC 413.20 (+5.61) (NH 3 ) 2 405.72 DNC 406.31 (-0.66) DNC 406.94 (-0.65) 404.65 DNC 406.05 (-0.92) DNC 406.69 (-0.90) NH 2 NH 3 399.16 DNC 400.15 (-6.82) DNC 400.80 (-6.79) 395.23 DNC 396.80 (-10.17) DNC 397.69 (-9.90) a ri-MP2/cc-pVTZ geometry. All EOM calculations were performed with the cc-pVTZ basis. Calculations marked with DNC did not converge in 60 cycles. b ADC(4) results from Ref. 77. ADC(4) calculations were performed with the DZP basis; the structures were not reported. collected in Tables S5, S7, and S9. The experimental values were taken from the syn- chrotron studies 78, 79 in the case of methane, ethene, ethyne and carbon monoxide and from the compilation 80 by Jolly et al. for other species. The results in Tables S4-S9 were obtained with the cc-pVTZ basis. For selected molecules, we also performed cc- pCVTZ and aug-pCVTZ calculations and found that the effect of the basis set on core ionized states is rather small: chemical shifts were affected by less than 0.1 eV , vertical IEs by+0.5 eV . Let us first consider errors in absolute values against the experimental IEs. For a small subset of molecules for which we were able to obtain full EOM-IP-CCSD results, the errors against the experimental values are larger than for valence IEs and for most cases are systematic (IEs are overestimated). The standard deviation (non- systematic errors) are about 0.2 eV . For carbon and oxygen edges in CH 4 , C 2 H 6 , H 2 O, CH 3 OH, oxygen edges in CO, HCOOH, CH 3 COOH, CH 2 O and nitrogen edges in N 2 69 and CH 3 NH 2 , the errors of the full EOM-IP-CCSD method are less than 1 eV . The inspection of the results obtained with the EOM-IP-CCSD-S(D) and EOM-IP-MP2- S(D) reveals more substantial errors in the absolute values of IEs. Relative to full EOM-IP-CCSD, the average errors increase by about 1 eV , however, the increase in the standard deviation is less. When comparing against the experimental values, we note that except for CH 3 OH, the sign of error is always positive, which is similar to the valence states. The errors increase up to 1.5 eV when using S(D) approximation. We observe larger errors for carbon edges in CO, C 2 H 4 , C 2 H 2 , CH 3 OH, CH 2 O, CH 3 COO and nitrogen edge in NH 3 : errors are less than 1.5 and 3 eV for full IP-CCSD and IP- CCSD-S(D), respectively. MAE against the experimental values are 1.96 eV and 0.55 eV for C and O edges, respectively. The respective standard deviations are 0.38 and 0.3 eV . For localized core states (i.e., in non-symmetric molecules),DSCF methodology reproduces experiment remarkably well, because the relaxation effects are described explicitly in these calculations. However, the DSCF errors are much larger in C 2 H 6 , C 2 H 4 , C 2 H 2 , and N 2 , where this single-determinantal approach fails to represent the delocalized hole. The chemical shifts relative to the reference systems are given in Tables S5, S7, and S9. Here, we computed chemical shifts for carbon 1s, nitrogen 1s, and oxygen 1s edges against the following reference systems: CH 4 , NH 3 , and H 2 O. The errors in chemical shifts against the experimental values are shown in Figures 3.3 and 3.4. Despite relatively large errors in absolute IEs, chemical shifts are reproduced by EOM- CC rather well. For the C edge, the full EOM-IP-CCSD and the S(D) approximation show errors of less than 0.5 eV and 1 eV , respectively. The errors are larger for O edge. The performance of EOM-MP2 variants is very similar to the respective CCSD variants. DSCF approach fails in the same cases as for the absolute energies. 70 Figure 3.3: Errors in chemical shifts (eV) of carbon (1s) ionized states for selected molecules. Shifts are computed relative to methane. Stars mark errors that are too large and do not fit on the plot. Figure 3.4: Errors in chemical shifts (eV) of oxygen (1s) ionized states for selected molecules. Shifts are computed relative to water. Overall, the results confirm the importance of orbital relaxation and show that perturbative treatment of double excitations in the S(D) approximation leads to larger errors relative to the full CCSD. On the basis of excellent agreement between full EOM-CCSD and the CVS approximation reported by Coriani and Koch 14, 15 , it appears 71 that the relaxation effect is captured extremely well within CVS. For comparison, we also computed core IEs for selected molecules using frozen valence approximation (in such calculations, all valence occupied orbitals are frozen). These calculations, which describe the relaxation of the core orbitals but neglect the effect of core electrons on the valence states, lead to much larger (10-20 eV) and non-systematic errors in IEs. Importantly, despite relatively large errors for absolute values of IEs, the performance of IP-CCSD/MP2-S(D) for chemical shifts is satisfactory, which suggest that our approximate methods might be useful for studying core-ionization spectra in large molecules and complex environments. We anticipate even smaller errors in applications where chemical shifts are computed within a similar class of compounds, such as in different protonation states of amino-acids. Our results for glycine presented below confirm this. Table S10 shows results for core-excited states for selected molecules. We report the position of the core-excited states relative to the respective edges. As one can see, in all molecules from Table S10 core-p states are located about 8 eV below the edge whereas core-3s Rydberg states are located at about 4 eV below the respective edges. EOM-IP-CCSD-S(D) results can be considered semi-quantitative at best: it reproduces relative positions of core-valence and core-Rydberg bands, but leads to errors of about 3-4 eV . Larger errors for core-excited states than for core-ionized states are consistent with the results for valence states. 3.5.2 Core-ionized states of different protonation forms of glycine Figure 3.5 shows structures of glycine in different protonation states. In the gas-phase, glycine exist in the canonical form (Gly can ), but in solution it assumes zwitter-ionic 72 Figure 3.5: Structures of different protonation forms of glycine: Canonical (Gly can ), zwitterionic (Gly ZI ), deprotonated (Gly ), and protonated (Gly + ). form (Gly ZI ) in which a proton is transferred from the carboxyl group to the amino group. At low pH, both the carboxy and the amino groups are protonated, giving rise to the cationic form (Gly + ) and at high pH it exists in deprotonated, anionic form (Gly ). Table 3.2 presents experimental core IEs of glycine measured in the gas-phase 81 and in liquid jet synchrotron experiments 43 . Table 3.2 also shows chemical shifts of core IEs computed as the difference betwen condensed-phase IEs and gas-phase IEs of the canonical form: Positive chemical shifts correspond to blue-shifted IEs and negative chemical shifts correspond to red-shifted IEs. As one can see, changes in protonation states have distinct spectroscopic signatures. The main trends can be readily explained by electrostatics: protonation of the amino group in Gly ZI and Gly + results in the higher IE of N(1s) (positive chemical shift), whereas the deprotonation of carboxy group in Gly ZI and Gly + results in the red-shifted carbon edge. 73 Table 3.2: Experimental core IEs and chemical shifts against gas-phase values of different forms of glycine (in eV). Core IEs Gly a can Gly b ZI Gly b Gly b + 1s (N) 405.40 406.81 404.30 406.91 1s (COO) 295.20 293.55 293.22 294.53 1s (CH 2 ) 292.30 291.43 290.67 291.88 Chemical shifts 1s (N) 1.18 -1.10 1.51 1s (COO) -1.65 -1.98 -0.67 1s (CH 2 ) -0.87 -1.63 -0.42 a From Ref. 81. b From Ref. 43. Table 3.3 shows IEs of the canonical form and chemical shifts against Gly can of other protonation forms of glycine computed with IP-CCSD-S(D)/cc-pVTZ (the results with IP-MP2-S(D) and with cc-pVDZ are given in SI (Table S11). Consistently with the benchmark calculations, we observe errors of 1-2 eV in the IEs of the canonical form. The computed chemical shifts for other protonation forms follow general trends expected from the electrostatic considerations, but their magnitudes are grossly exagger- ated (by up to 7 eV). Moreover, for some IEs even the sign of the shift is reversed rela- tive to the experiment. The computed shifts and large differences from the experimental values are in line with the results of DFT calculations from Ref. 43 . On the basis of PCM calculations and calculations on micro-solvated model clusters, the authors have attributed the large discrepancy between the gas-phase calculations and the measured aqueous IEs to solvent effects. The solvent effectively screens the charges thus result- ing in much smaller shifts due to protonation/deprotonation. Table S12 in SI shows calculated IEs and chemical shifts computed with IP-CCSD-S(D) and PCM. Follow- ing Ref. 43 , we compare the results of bare glycine embedded in PCM versus model clusters (glycine with six water molecules) surrounded by PCM solvent (we use the 74 same structures as in Ref. 43 ). Similarly to the DFT-based calculations 43 , we observe that PCM greatly reduces the magnitude of the shifts, bringing them to a better (but not perfect) agreement with the experiment. The authors of Ref. 43 reported better agree- ment when six explicit waters were included in the PCM calculations. Our results from Table S12 support this finding. Significant effect due to the inclusion of the explicit water molecules can be explained by the inability of PCM models to describe specific solvent-solute interactions such as hydrogen bonding. While this effect is captured by considering small optimized clusters (such as various Gly 6w structures from Ref. 43 ), these calculations do not account for the fluctuation of structures due to equilibrium thermal motions. More appropriate calculations should include averaging over snap- shots from equilibrium MD simulations. Our results below illustrate that this approach results in much better agreement with the experiment. Table 3.3: Computed gas-phase IEs and chemical shifts against Gly can of different forms of glycine (in eV). EOM-IP-CCSD-S(D)/cc-pVTZ. Gly can Gly ZI Gly Gly + 1s (N) 406.58 4.06 -6.32 9.09 1s (COO) 296.91 -1.43 -6.84 6.25 1s (CH 2 ) 294.70 0.58 -6.18 6.30 Analysis of hydrogen-bonding interactions of different protonation forms of glycine The distributions of hydrogen bonds between the solvated glycine and nearby water molecules along equilibrium MD trajectory are shown in Figure 3.6. In the zwitterionic form, in which both protic groups are charged, we observe on average about 2 hydro- gen bonds formed by the amino group. This number of hydrogen bonds formed by the carboxyl group, which has two O centers is four. In the deprotonated form, the amino 75 group is no longer charged resulting in the smaller average number of hydrogen bonds (one), but the number of hydrogen bonds formed by the carboxyl group increases from four to five. In the protonated form, the carboxyl group is neutral and forms only two hydrogen bonds (the number of hydrogen bonds formed by the amino group is the same as in zwitter-ionic form). As illustrated by the calculations below, these different hydro- gen bond patters are partially responsible for the observed trends in the core IEs. If a group acts as a hydrogen-bond donor (which is the case for the carboxyl group in both forms and the deprotonated amino group), we expect a blue shift in IEs, as the electron density on the donating group is depleted. Conversely, if a group acts as a hydrogen- bond acceptor (which is the case of protonated form of carboxyl group and both forms of the neutral and protonated amino group), we expect red-shifted IEs. Comparison of different solvent treatments by using a single snapshot from equi- librium MD trajectory Before proceeding to computing averages using equilibrium MD trajectories, we ana- lyze the effects of different solvent treatment using a single representative snapshot for each structure. We selected snapshots that have the number of hydrogen bonds equal to the average number of hydrogen bonds for each type of structure (see Fig. 3.6). The structures of glycine and nearest solvent molecules from each snapshot are shown in Fig. 3.7. We compare QM/MM and QM/EFP treatments using two different setups for the QM part: bare glycine and glycine with the nearest hydrogen-bonding waters. The results of IP-CCSD-S(D) calculations are collected in Table 3.4. In this set of calcu- lations, QM XX;w /EFP represents the highest level of theory (in this calculation, nearest hydrogen-bonded water molecules are included into the QM part and the rest of water 76 0 1 2 3 4 #ofHbondswithNH + 3 groupofGlyZI Occurrence,Arb. Units 0 1 2 3 4 #ofHbondswithNH2 groupofGly− Occurrence,Arb. Units 0 1 2 3 4 #ofHbondswithNH + 3 groupofGly+ Occurrence,Arb. Units 0 1 2 3 4 5 6 7 #ofHbondswithCOO − groupofGlyZI Occurrence,Arb. Units 0 1 2 3 4 5 6 7 #ofHbondswithCOO − groupofGlyZI Occurrence,Arb. Units 0 1 2 3 4 5 6 7 #ofHbondswithCOO − groupofGlyZI Occurrence,Arb. Units Figure 3.6: Distribution of hydrogen bonds between solute and water molecules from the first solvation shell for different forms of glycine. molecules is described by EFP including state-specific solvent polarization correction). We observe that even when nearest waters are included into the QM part, polarization correction is important and can contribute up to 0.7 eV to the chemical shift of core IEs. The differences between the QM XX;w /MM and QM XX;w /EFP shifts can be as large as 0.5 eV , which is substantial, given the magnitude of the shifts. Contrary to our calcula- tions of valence states 33, 35 , we observe noticeable differences between EFP calculations with small and extended QM, i.e., the largest observed difference between QM XX /EFP 77 Figure 3.7: Representative snapshots for Gly ZI , Gly , and Gly + from equilibrium MD simulation. and QM XX;w /EFP is 0.7 eV . Even larger differences (up to 1.1 eV) are observed between QM XX /MM and QM XX;w /MM. Higher sensitivity of QM/MM compared to QM/EFP can be explained by the lack of polarization in the former, where the electron distribution in nearby waters cannot adjust to the charges in QM. Along the same lines, we observe larger differences between small and extended QM parts for EFP calculations without polarization correction. In sum, the results in Table 3.4 indicate high sensitivity of the core IEs to the solvent-specific interactions and highlight the importance of using polar- izable solvent models, such as EFP. But even with EFP, the inclusion hydrogen-boded water molecules into the QM part is necessary for accurate results. In order to reduce computational costs when performing equilibrium averaging, we compared the magnitude of chemical shifts computed with EOM-IP-CCSD-S(D) and 78 Table 3.4: Core IEs and chemical shifts (eV) relative to Gly can of different proto- nation forms of glycine computed using representative single snapshots from equi- librium trajectory. IP-CCSD-S(D)/cc-pVTZ. QM ZI / QM ZI / QM ZI / QM ZI;w / QM ZI;w / QM ZI;w = MM EFP fz EFP MM EFP fz EFP 1s N 2.39 1.90 1.14 1.26 1.11 1.33 1s C(COO) -1.02 0.12 -0.55 -1.69 -1.92 -1.22 1s C(CH 2 ) 0.01 0.77 0.21 -0.65 -0.21 -0.93 QM / QM / QM / QM ;w / QM ;w / QM ;w / MM EFP fz EFP MM EFP fz EFP 1s N -0.29 -0.79 -0.44 -0.54 -0.21 -0.76 1s C(COO) -0.63 -0.48 -0.99 -1.25 -1.25 -1.72 1s C(CH 2 ) -0.36 -0.22 -0.76 -0.68 -0.33 -0.98 QM + / QM + / QM + / QM +;w / QM +;w / QM +;w / MM EFP fz EFP MM EFP fz EFP 1s N 3.61 0.64 0.89 1.69 0.95 1.16 1s C(COO) 1.26 -0.02 -0.26 -0.23 -0.47 -0.47 1s C(CH 2 ) 1.21 -0.66 -1.13 -0.41 -0.75 -0.21 EFP fz denotes EFP calculations without solvent polarization correction. EOM-IP-MP2-S(D). We also quantified the effect of using a smaller basis set, cc-pVDZ, instead of cc-pVTZ. The results are given in Table S13 in SI. Contrary to the strong dependence of the shifts on the solvent model, we observe that chemical shifts are rather insensitive to the basis set and to using MP2 rather than CCSD amplitudes. Among all solvent models, the largest difference between EOM-IP-CCSD/cc-pVTZ and EOM-IP- MP2/cc-pVDZ calculations is0.1 eV . For our best model, QM XX;6w /EFP the differ- ences for most states are 0.01-0.05 eV , and the largest difference of 0.1 eV is observed for the state with the largest chemical shift, C(1s) from the carboxy group. Thus, in tra- jectory averaging calculations one can safely employ EOM-IP-MP2-S(D)/cc-pVDZ. On our machine, the EOM-IP-MP2-S(D)/cc-pVDZ calculation with large QM is an order of magnitude faster than EOM-IP-CCSD-S(D)/cc-pVTZ. 79 3.5.3 Core IEs of different protonation forms of glycine computed using equilibrium averaging with explicit solvent models Table 3.5 and Figure 3.8 show chemical shifts of different protonation states of glycine computed by averaging over snapshots from the equilibrium MD trajectory. We observe that QM ZI;w /EFP results are in excellent agreement with the experiment, i.e., the signs and the relative values of the shifts are correctly reproduced and the errors in shifts range from 0.05 to 0.22 eV (again, the largest error is observed for the largest-magnitude shift). By inspecting the data in Table 3.5 one can clearly see that this agreement is not coincidental, as the differences between other, lower-level treatments are larger. This observation is in agreement with the behavior observed for the representative snapshots. For example, the errors of EFP calculations with small QM, which does not include nearby waters, can be as large as 1 eV and the largest error of QM XX;w /MM is0.6 eV . 3.6 Conclusion We presented a new computational approach based on the EOM-CC theory for modeling core-level spectroscopy. By using an approximate EOM-CC ansatz in which doubly excited amplitudes are treated perturbatively, we circumvent the convergence problems arising due to metastable (resonance) character of the core ionized/core-excited state. The benchmark calculations on small gas-phase molecules show that this approximation leads to errors of about 1-2 eV in core-ionized states, however, chemical shifts computed relative to a reference compound can be computed with higher accuracy. We applied this method, in combination with different solvent models, to model chemical shifts in core IEs of different protonation forms of glycine. We showed that explicit solvent treatment by EFP method (with solvent polarization correction) leads to the best results and that, 80 Table 3.5: Chemical shifts (eV) relative to Gly can of different protonation forms of glycine a . QM ZI / QM ZI / QM ZI / QM ZI;w / QM ZI;w / QM ZI;w = Exp ZI;aq MM EFP fz EFP MM EFP fz EFP 1s N 2.96 2.74 2.02 1.73 1.78 1.52 1.41 1s C(COO) -0.19 1.11 -0.53 -0.99 -0.93 -1.43 -1.65 1s C(CH 2 ) 0.48 1.00 0.50 -0.39 -0.22 -0.75 -0.87 QM / QM / QM / QM ;w / QM ;w / QM ;w / Exp ;aq MM EFP fz EFP MM EFP fz EFP 1s N -0.09 0.08 -0.20 -1.05 -0.83 -1.29 -1.10 1s C(COO) -0.45 -0.16 -0.82 -1.48 -1.38 -1.89 -1.98 1s C(CH 2 ) -0.52 0.34 -0.29 -1.72 -1.53 -1.69 -1.63 QM + / QM + / QM + / QM +;w / QM +;w / QM +;w / Exp +;aq MM EFP fz EFP MM EFP fz EFP 1s N 3.85 1.89 1.10 1.47 1.30 1.60 1.51 1s C(COO) 1.59 0.34 -0.09 -1.06 -1.12 -0.82 -0.67 1s C(CH 2 ) 1.58 0.01 -0.44 -0.30 -0.05 -0.47 -0.42 a The results with small QM were computed with IP-CCSD-S(D)/cc-pVTZ. The calculations with large QM part, QM XX;w /MM and QM XX;w /EFP were computed with IP-MP2-S(D)/cc-pVDZ. even with EFP, the nearest water molecules should be included in the QM part to achieve a quantitative agreement with the experimental shifts. These results once again highlight the high sensitivity of core states to local environment and specific interactions with the surrounding solvent. Our best protocol, QM/EFP calculation with the extended QM part including glycine and the nearest water molecules that are hydrogen-bonded to glycine, results in excellent agreement with the experiment, i.e., the signs and the relative values of the shifts are correctly reproduced and the errors do not exceed 0.22 eV . 81 GlyZI Gly− Gly+ −1 1 2 3 Chemicalshift 1sN(NH),eV Exp QM w /EFP QM PCM ΔSCF(B3LYP 6w ) GlyZI Gly− Gly+ −2 −1 Chemicalshift 1sC*OO,eV Exp QM w /EFP QM PCM ΔSCF(B3LYP 6w ) GlyZI Gly− Gly+ −2 −1 1 Chemicalshift 1sC*H 2 ,eV Exp QM w /EFP QM PCM ΔSCF(B3LYP 6w ) Figure 3.8: Chemical shifts of glycine computed with different levels of theory: nitrogen 1s edge (top), carbon (1s) from carboxyl (middle), and carbon (1s) from methyl group. 82 3.7 Chapter 3 references [1] O. Kostko, B. Bandyopadhyay, and M. Ahmed. Vacuum ultraviolet photoioniza- tion of complex chemical systems. Annu.Rev.Phys.Chem., 67:19–40, 2016. [2] N.A. Besley, M.J. G. Peach, and D.J. Tozer. 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However, the authors mention that the temperature may have significant effects; in the experiment, nanoparticles are observed after evaporative cooling and temperature of the system may be much lower than the standard tempera- ture. However, this effect has never been studied. Our QM/EFP approach could help to determine temperature effects on the microenvironment inside the nanodroplets and accurately predict the structural effects on core IEs, providing important missing infor- mation on the evaporative cooling technique at the point of photoionization. Another important direction is the study of core IE spectra of unnatural aminoacids. Such aminoacids are widely used in biology to gain understanding of the interactions and microenvironments of certain residues in proteins by mutating these residues to unnatural aminoacids 2–4 . Usually, the signal monitored is the fluorescence of these molecules when they are close to some other groups (e.g., tyrosine) 5–7 . This informa- tion is especially important for transmembrane proteins for many of which the crystal structure is unavailable. Structural models of such proteins are usually obtained by 91 methods such as homology modeling and may lack required predictive power without additional information on the protein microenvironment. X-ray photoelectron spec- troscopy (XPS) experiments applied to such proteins with certain aminoacids being mutated would give researchers yet another tool for studying protein structure. Some of the unnatural aminoacids may contain non C-O-N atoms like selenocysteine and 3- iodo-L-tyrosine, which make them promising for XPS studies. Another direction is to evaluate the perfomance of EOM-EE-S(D) to study core excited states. This would allow one to model near-edge X-ray photoelectron spec- troscopy (NEXAFS) experiments, powerful tool for studying the local electronic struc- ture environment. The physical principles behind NEXAFS are described in Figure 4.1. 1s K 2s L 1 2p L 2 ,L 3 vacuum 3+s,p,d,... M,... hν <IP 1s Augere − 1 Figure 4.1: Near edge X-ray absorption event. The initial photon absorption by inner shell electron is shown in red, the system is excited into a metastable state. The following relaxation of the state through Auger decay is shown in blue. On the left, chemical notation of the atomic orbitals is given. On the right, the XAS notation for the same electronic shells is depicted for comparison. The resulting photoelectron spectra contain the information about core orbitals and the local environment of the respective atoms. Since 1s orbitals of individual atoms (except for hydrogen, in which 1s is valence orbital) are well localized on the atomic centers, the K-edge of individual atoms and the same atom in some molecule usually differ not more then 1-2%, and the near-edge 92 spectrum contains the information about transitions from 1s to valence orbitals of the molecule and is highly dependent on the chemical structure of the molecule and its sur- roundings. This information is crucially important in drug discovery where the mapping of interactions between residues and ligands is one of the greatest challenges. To pur- sue this project one should start with benchmarking the methodology on a similar set of small molecules that we used to characterize EOM-IP-S(D) and then move to condensed phase simulations of small biomolecules. Gas phase NEXAFS spectra of many of aminoacids have already been calculated, though not all of them are covered and solution spectra were either not studied at all or crude models that do not account for explicit solute-solvent interactions were used 8–11 . The quality of the computational methods employed often is not optimal and applica- tion of more accurate computational methods is desirable. No analysis of the possible conformations and their impact on the spectrum was done apart from a few methods that used over simplified treatments of solute-solvent configurations. Common strategy is to sample out a limited number of lowest-lying configurations from the full configu- ration space, and to compose the final spectrum according to the equilibrium distribu- tion of those few isomers, i.e., as an average weighted by the Boltzmann factor. How- ever, this requires a thorough study of the ground-state potential energy surface and is highly dependent on the method used to obtain the PES. This is the case for studying pH profiles of NEXAFS spectra of the aminoacids, where the computational methods for calculation X-ray absorption spectrum were not accurate enough also the treatment of conformational dynamics in gas phase was crude 8 . That is why we feel confident that the application of the developed QM/EFP treatment of the solute-solvent systems to extract accurate XAS spectra of the aminoacids is very promising. 93 4.1 Chapter 4 references [1] B. Xu, M.I. Jacobs, O. Kostko, and M. Ahmed. Guanidinium group is proto- nated in a strongly basic arginine solution. Comp. Phys. Comm., 2017. in press; doi:10.1002/cphc.201700197. [2] Hong Xue and J. Wong. Future of the Genetic Code. Life, 7(1):10, 2017. [3] K. Sakamoto. Site-specific incorporation of an unnatural amino acid into proteins in mammalian cells. NucleicAcidsResearch, 30(21):4692–4699, 2002. [4] Kathrin Lang and Jason W. Chin. Cellular incorporation of unnatural amino acids and bioorthogonal Labeling of Proteins. Chemical Reviews, 114(9):4764–4806, 2014. [5] Amy Grunbeck, Thomas Huber, Pallavi Sachdev, and Thomas P. Sakmar. Mapping the ligand-binding site on a g protein-coupled receptor (gpcr) using genetically encoded photocrosslinkers. Biochemistry, 50(17):3411–3413, 2011. [6] Dany Fillion, J´ erˆ ome Cabana, Ga´ etan Guillemette, Richard Leduc, Pierre Lavigne, and Emanuel Escher. Structure of the human angiotensin II type 1 (AT1) receptor bound to angiotensin II from multiple chemoselective photoprobe contacts reveals a unique peptide binding mode. Journal of Biological Chemistry, 288(12):8187– 8197, 2013. [7] Ieva Vasiliauskait´ e-Brooks, Remy Sounier, Pascal Rochaix, Ga¨ etan Bellot, Math- ieu Fortier, Franc ¸ois Hoh, Luigi De Colibus, Ch´ erine Bechara, Essa M. Saied, Christoph Arenz, C´ edric Leyrat, and S´ ebastien Granier. Structural insights into adiponectin receptors suggest ceramidase activity. Nature, 544(7648):120–123, 2017. [8] Hongbao Li, Weijie Hua, Zijing Lin, and Yi Luo. First-principles study on core-level spectroscopy of arginine in gas and solid phases. J. Phys. Chem. B, 116(42):12641–12650, 2012. [9] Vitaliy Feyer, Oksana Plekan, Robert Richter, Marcello Coreno, Kevin C. Prince, and Vincenzo Carravetta. Core level study of alanine and threonine.J.Phys.Chem. A, 112(34):7806–7815, 2008. [10] B.M. Messer, C.D. Cappa, J.D. Smith, W.S. Drisdell, C.P. Schwartz, R.C. Cohen, and R.J. Saykally. Local hydration environments of amino acids and dipeptides studied by X-ray spectroscopy of liquid microjets. J. Phys. Chem. B, 109:21640– 21646, 2005. 94 [11] O. Plekan, V . Feyer, R. Richter, M. Coreno, M. de Simone, K.C. Prince, and V . Car- ravetta. Investigation of the amino acids glycine, proline, and methionine by pho- toemission spectroscopy. J.Phys.Chem.A, 111:10998–11005, 2007. 95 Appendices 96 Appendix A: Supplementary information for Chapter 2. A.1 Experimental Table A.1: Fractional bleach of BE at various stages of its formation Time delay, ps DOD Bleach intensity Fractional absorption 1 0.03963 9.35E-4 0.02359 3 0.05135 0.00245 0.04771 6 0.06467 0.00428 0.06618 9 0.076 0.0066 0.08679 20 0.102 0.00869 0.08524 40 0.1173 0.01062 0.0905 66 0.1209 0.01214 0.10037 97 -0.01 0.00 0.01 0.02 -0.010 -0.005 0.000 400 500 600 700 -0.010 -0.005 0.000 0fs 50fs 100fs 200fs 300fs 400fs 500fs ΔA 500fs 700fs 1ps 5ps 10ps 25ps 15ps 20ps BENZENE EXCIMER VH Wavelength/nm 25ps 40ps 70ps 100ps -0.04 -0.02 0.00 0.02 0.04 0.06 -0.02 -0.01 0.00 400 500 600 700 -0.009 -0.006 -0.003 0.000 0 50fs 100fs 200fs 300fs 400fs 500fs ΔA 500fs 700fs 1ps 5ps 10ps 25ps Wavelength/nm 25ps 40ps 70ps 100ps BENZENE EXCIMER VV Figure S2: Transient absorption spectrum of benzene excimer in VV pump-probe configuration 0 20 40 60 -0.012 -0.006 0.000 Time/ps 1 3 6 9 20 40 66 ΔOD Figure S3: Recovery of bleach of populations excited at different times during the formation of excimer Figure A.1: Transient absorption spectrum of benzene excimer in VH (left) and VV (right) configurations between the pump and probe in the three pulse experiments. 98 -0.04 -0.02 0.00 0.02 0.04 0.06 -0.02 -0.01 0.00 400 500 600 700 -0.009 -0.006 -0.003 0.000 0 50fs 100fs 200fs 300fs 400fs 500fs ΔA 500fs 700fs 1ps 5ps 10ps 25ps Wavelength/nm 25ps 40ps 70ps 100ps BENZENE EXCIMER VV Figure S2: Transient absorption spectrum of benzene excimer in VV pump-probe configuration 0 20 40 60 -0.012 -0.006 0.000 Time/ps 1 3 6 9 20 40 66 ΔOD Figure S3: Recovery of bleach of populations excited at different times during the formation of excimer Figure A.2: Recovery of bleach of populations excited at different times in the course of excimer formation. 0.0 0.1 0.2 0.3 0.4 0 10 20 30 40 50 0.0 0.1 0.2 0.3 0.4 C Equation y = A1*exp(-x/t1) + y0 Adj. R-Square 0.98208 Value 3ps y0 -0.00367 3ps A1 0.19687 3ps t1 6303.14811 r(t) Time/ps Equation y = A1*exp(-x/t1) + A2*exp(-x/t 2) + y0 Adj. R-Square 0.9748 Value 66ps y0 0 66ps A1 0.22307 66ps t1 8726.11417 66ps A2 0.0635 66ps t2 1434.11839 Figure S4: Anisotropy decay of bleach of at 3 ps and 66 ps Figure A.3: Anisotropy decay of bleach of at 3 ps and 66 ps. 99 A.2 Structure of solid benzene and model dimer struc- tures Figure A.4: X-ray structure of solid benzene. Unit cell, P bca space group. b=9.660 ˚ A,a=7.460 ˚ A,c=7.030 ˚ A. Highlighted in color are the structures of the dimers with the shortest distance between the fragments. Blue is the central molecule. Blue- red pairs are Y-shaped displaced (Y d ) dimers, blue-yellow pairs are the T-shaped (T x ) dimers, blue-green pairs are L-shaped (L) dimers, and blue-purple pairs are head-to-tail (HtT) dimers. 100 A.3 Exciton delocalization: Dimers versus trimers To assess whether the dimers represent a suffiicent model for understanding exciton delocalization and excimer formation in liquid benzene, we carried out calculations of excited states in model trimer structures. The NTOs for the S 0 !S 1 transition are shown in Fig. A.5. We begin with the perfectly stacked symmetric trimer in which the distance between the central and edge molecules is 3.0 ˚ A. In this symmetric structure, in which the benzene moieties are sufficiently close, we expect to observe maximum delocalization. The NTOs for the S 0 !S 1 show some degree of delocalization, but the delocalization is less pronounced relative to the dimer with the same distance between the rings. The PR NTO for the trimer equals 2.4, which signifies that the excitation is mostly localized on the central molecule. If the symmetry is distorted by moving one edge molecule further away, the excitation becomes perfectly localized on the dimer. 101 Hole NTOs Particle NTOs d 1 = d 2 = 3Å d 1 = 3Å d 2 = 4Å Figure A.5: The NTOs for the S 0 !S 1 transition in model stacked trimer struc- tures. Top: The distances between the central and edge molecules are equal 3 ˚ A. Bottom: the distances between the central and edge molecules are 3 and 4 ˚ A. 102 A.4 Molecular orbitals framework Figure A.6: Molecular orbitals diagram of benzene sandwich dimer at the equi- librium BE geometry. D 2h point group symmetry labels are used. Orbitals are computed with HF/6-31+G* and shown with isovalue of 0.005. Relevant MOs of the BE are shown in Figure A.6. The excimer state (1B 3g ), which is the lowest excited singlet state at the fully relaxed BE structure, is derived from the HOMO! LUMO+8 and HOMO-1! LUMO+7 transitions. HOMO and HOMO-1 correlate with the degenerate HOMOs of the monomers; they are out-of-phase combi- nations of thep-orbitals of the monomers. LUMO+7 and LUMO+8 are in-phase com- binations of the valencep orbitals of the monomers mixed with diffuse Rydberg-type orbital. The 4B 2u state is formed by HOMO!LUMO+12, HOMO-1! LUMO+11, 103 HOMO-2! LUMO+8 and HOMO-3! LUMO+7 transitions and has Rydberg char- acter, as one can tell from the corresponding MOs. The NTOs for these states are shown in main text, they provide a more compact description of the states’ characters. 104 A.5 Spectra of different model structures Head-to-Tail (HtT) R = 5.805 Å = 35.9 o φ = 43.1 o r 1 = 2.99 Å r 2 = 3.59 Å L-shaped (L) R = 4.905 Å = 61.45 o φ = 44.6 o r 1 = 3.85 Å r 2 = 2.64 Å T-shaped (T x ) R = 5.126 Å = 90 o φ = 0 o r 1 = 3.74 Å r 2 = 4.43 Å T-shaped displaced (T d ) R = 5.027 Å = 90 o φ = 0 o r 1 = 3.52 Å r 2 = 3.52 Å Parallel displaced gas phase (PD g ) R = 3.940 Å = 0 o φ = 30.5 o r 1 = 3.60 Å r 2 = 3.60 Å Figure A.7: Model benzene dimer structures. Top left: head-to-tail dimer,C 1 sym- metry. Top right: L-shaped dimer,C 1 symmetry. Middle row, left: T-shaped dimer, C 2v symmetry. Middle row, right: T-shaped displaced dimer, C 2v symmetry. Bot- tom left: PD dimer at the gas-phase equilibrium geometry, C 2h symmetry. The definition of structural parameters is given in Fig. 1 in the main manuscript. 105 Figure A.8: Spectra of various model dimer structures (S 1 absorption). The stick spectra are convoluted with Gaussians (half-width is 0.05 eV). 106 A.6 Structure of liquid benzene 0 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 ·10 −2 θ, degrees N(θ) isotropic benzene 0-5 ˚ A 1 0 10 20 30 40 50 60 70 80 90 0 1 2 3 4 5 ·10 −2 θ, degrees N(θ) isotropic benzene 5-6 ˚ A 1 0 10 20 30 40 50 60 70 80 90 0.03 0.06 0.09 0.12 θ, degrees N(θ) isotropic benzene in the first shell 1 Figure A.9: The angular distribution functions plotted for different values of R. Blue line: isotropic distribution. Orange line: from the MD simulations using CHARMM27 force-field (results from Ref. 55). At R< 5 ˚ A, parallel arrangement is preferred, whereas at largerR, perpendicular structures are more prevalent. 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 R, ˚ A g(R) exp. a CHARMM27 1 Figure A.10: Radial distribution function, g(r), of liquid benzene. The number of molecules in the first solvent shell is R 8:0 0 g(r)r 2 dr=12.5. Reproduced from Ref. 55. a data from Ref. 54. 107 Table A.2: Angular distribution in the first solvation shell forR<5 ˚ A. q population, % 0-10 12 10-20 18 20-30 18 30-40 14 40-50 13 50-60 11 60-70 6 70-80 5 80-90 3 0-90 100 108 Appendix B: Supplementary information for Chapter 3. B.1 Programmable expressions for EOM-IP/EE- CCSD/MP2-S(D) The equations are derived from full EOM-IP-CCSD equations from Refs. 60, 61. For EOM-MP2 variant, theT 2 amplitudes are computed by MP2 andT 1 amplitudes are zero. s a i = å b F 2 ab r b i å j F 2 ij r a j å jb I 1 ibja r b j (B.1) ˜ s a i = å c F 2 ca l c i å k F 2 ki l a k å kc I 1 kaic l c k (B.2) s ab ij =P (ab) å k I 2 ijkb r a k P (ij) å c I 3 jcab r c i + P (ij) å l T 1 il t ab jl +P (ab) å d T 2 ad t bd ij (B.3) ˜ s ab ij =P (ij)P (ab)l a i F 2 jb +P (ab) å k I 6 ijka l b k +P (ij) å c I 7 icab l c j (B.4) 109 s i = å j F 2 ij r j (B.5) ˜ s i = å j F 2 ji l j (B.6) s a ij = å k r k I 2 ijka (B.7) ˜ s a ij =P (ij)l i F 2 ja å k l k I 6 ijka (B.8) 110 Intermediates: F 2 ki = f ki + å c f kc t c i (B.9) F 2 bc = f bc å k f kc t b k å kld hkljjcdit b k t d l (B.10) F 2 ia = f ia + å jb t b j hijjjabi (B.11) I 1 iajb =hiajjjbi å c hibjjacit c i å kc hjkjjacit bc ik + å k ( å c hjkjjacit c i hjkjjiai)t b k (B.12) I 2 ijka =hijjjkai å l I 4 ijkl t a l ++ 1 2 å cd hkajjcdi ˜ t cd ij P (ij) å b (hkajjjbi å lc hkljjbcit ac jl )t b i + å lc hkljjjcit ac il + å c å lb hkljjbcit b l t ac ij å c t ac ij f kc (B.13) I 3 iabc =hiajjbci å d I 5 bcad t d i + 1 2 å jk hjkjjiait bc jk +P (bc) å j hjbjjiai 1 2 å k hjkjjiait b k å kd hjkjjadit bd ik t c j å kd hkcjjadit bd ik å k å jd hkjjjadit d j t bc ik å k t bc ik f ka (B.14) I 6 klic =hkljjici å d hkljjcdit d i (B.15) I 7 kacd =hkajjcdi å l hkljjcdit a l (B.16) T (1) ij = å kc r c k I (6) jkic (B.17) T (4) ab = 1 2 å ijc hijjjbcir ac ij (B.18) ˜ t ab ij =t ab ij +P (ab)t a i t b j (B.19) 111 B.2 Benchmark calculations using EOM-IP-CCSD- S(D) and EOM-IP-MP2-S(D) methods Table B.1: Valence ionized and excited states for formaldehyde. Errors against EOM-IP-CCSD are shown in parenthesis. All energies are in electron-volts. State EOM-IP-CCSD EOM-IP-MP2 EOM-IP-CCSD-S(D) EOM-IP-MP2-S(D) 1 2 A 1 16.04 16.06 (0.02) 16.51 (0.47) 16.55 (0.51) 2 2 A 1 21.76 21.80 (0.04) 22.48 (0.72) 22.59 (0.83) 1 2 B 1 10.76 10.78 (0.02) 11.24 (0.48) 11.25 (0.49) 2 2 B 1 17.45 17.47 (0.02) 17.81 (0.36) 17.90 (0.45) 1 2 B 2 14.55 14.56 (0.01) 14.95 (0.40) 14.97 (0.42) 2 1 A 1 9.82 9.71 (-0.11) 10.96 (+1.14) 10.81 (+0.99) 3 1 A 1 10.57 10.42 (-0.15) 11.58 (+1.01) 11.44 (+0.87) 1 1 A 2 4.06 3.98 (-0.08) 5.16 (+1.10) 5.06 (+1.00) 2 1 A 2 10.65 10.61 (-0.04) 11.96 (+1.31) 11.88 (+1.23) 1 1 B 1 8.36 8.29 (-0.07) 9.53 (+1.17) 9.44 (+1.08) 2 1 B 1 10.29 10.27 (-0.02) 11.29 (+1.00) 11.26 (+0.97) 1 1 B 2 9.36 9.30 (-0.06) 10.48 (+1.12) 10.42 (+1.06) 2 1 B 2 12.03 11.98 (-0.05) 13.07 (+1.05) 13.00 (+0.97) RI-MP2/cc-pVTZ geometry, cc-pVTZ basis. Table B.2: Valence IEs (eV) of C 2 H 4 . State CCSD-SD CCSD-S(D) MP2-SD MP2-S(D) 1A g 14.92 15.35 14.99 15.44 1B 1g 13.14 13.56 13.09 13.53 1B 1u 10.69 10.98 10.68 10.98 1B 2u 16.36 16.97 16.32 16.96 1B 3u 19.62 20.52 19.57 20.51 RI-MP2/cc-pVTZ geometry, cc-pVTZ basis. 112 Table B.3: Valence IEs (eV) of C 2 H 6 . State CCSD-SD CCSD-S(D) MP2-SD MP2-S(D) 1A g /1B g 12.72 13.09 12.65 13.03 1A u /1B u 15.49 15.92 15.43 15.87 RI-MP2/cc-pVTZ geometry, cc-pVTZ basis. 113 Table B.4: Carbon (1s) ionized states for selected molecules (IEs in eV). Molecule Edge CCSD-SD CCSD-S(D) MP2-SD MP2-S(D) DSCF-DFT Exp CH 4 1s(C) 291.4 292.89 291.78 293.33 291.01 290.76 C 2 H 6 1s(B u ) 291.31 292.23 291.69 292.68 286.22 290.71 1s(A g ) 291.33 292.25 291.71 292.7 286.23 NA C 2 H 4 1s(B 3u ) 291.58 293.27 292.03 292.71 287.04 290.82 1s(A g ) 291.64 293.33 292.08 292.77 287.07 NA C 2 H 2 1s(B 1u ) 291.94 293.17 292.56 293.96 287.82 291.25 1s(A g ) 292.08 293.28 292.69 294.08 287.87 NA CO 1s(C) 297.32 298.91 297.78 299.77 296.89 296.2 CH 3 OH 1s(C) 293.09 294.44 293.49 294.93 292.6 292.42 C 2 H 5 OH 1s(CH 2 ) DNC 293.96 DNC 294.46 DNC 292.5 1s(CH 3 ) DNC 292.35 DNC 292.81 DNC 291.1 CH 3 OCH 3 1s(C) DNC 293.83 DNC 294.31 DNC 292.3 CH 2 O 1s(C) 295.27 296.99 295.85 297.68 294.8 294.97 CH 3 COH 1s(CH 3 ) DNC 293.16 DNC 293.7 DNC 291.35 1s(C*O) DNC 296.19 DNC 296.92 DNC 294 CHOOH 1s(C*OOH) DNC 298.24 DNC 299.09 296.05 295.8 CH 3 COOH 1s(aC*) DNC 293.54 DNC 294.1 DNC 291.55 1s(C*OOH) DNC 297.49 DNC 298.34 DNC 295.38 CHOOCH 3 1s(CH 3 ) DNC 294.68 DNC 295.19 DNC 292.78 CH 3 NH 2 1s(C) DNC 293.4 DNC 293.87 DNC 291.6 MAE 0.68 (0.24) 1.96 (0.38) 1.15 (0.24) 2.49 (0.51) 1.65 (1.89) MSE 0.68 (0.24) 1.96 (0.38) 1.15 (0.24) 2.49 (0.51) -1.31 (2.17) Max. AE 1.12 2.71 1.58 3.57 4.49 MAE 1.38 (0.31) 0.47 (0.10) 1.77 (0.49) vs. CCSD MSE 1.38 (0.31) 0.47 (0.10) 1.77 (0.49) vs. CCSD Max. AE 1.72 0.62 2.45 vs. CCSD MAE:jx i x exp i j=n MSE, (x i x exp i )=n MAX: max absolute error 114 Table B.5: Chemical shifts of carbon (1s) ionized states for selected molecules against methane (IEs in eV). Molecule Edge CCSD-SD CCSD-S(D) MP2-SD MP2-S(D) DSCF-DFT Exp CH 4 1s(C) 0 0 0 0 0 0 C 2 H 6 1s(B u ) -0.09 -0.66 -0.09 -0.65 -4.79 -0.05 1s(A g ) -0.07 -0.64 -0.07 -0.63 -4.78 NA C 2 H 4 1s(B 3u ) 0.18 0.38 0.25 -0.62 -3.97 0.06 1s(A g ) 0.24 0.44 0.3 -0.56 -3.94 NA C 2 H 2 1s(B 1u ) 0.54 0.28 0.78 0.63 -3.19 0.49 1s(A g ) 0.68 0.39 0.91 0.75 -3.14 NA CO 1s(C) 5.92 6.02 6 6.44 5.88 5.44 CH 3 OH 1s(C) 1.69 1.55 1.71 1.6 1.59 1.66 C 2 H 5 OH 1s(CH 2 ) DNC 1.07 DNC 1.13 DNC 1.74 1s(CH 3 ) DNC -0.54 DNC -0.52 DNC 0.34 CH 3 OCH 3 1s(C) DNC 0.94 DNC 0.98 DNC 1.54 CH 2 O 1s(C) 3.87 4.1 4.07 4.35 3.79 4.21 CH 3 COH 1s(CH 3 ) 0.27 DNC 0.37 DNC 0.59 1s(C*O) DNC 3.3 DNC 3.59 DNC 3.24 CHOOH 1s(C*OOH) DNC 5.35 DNC 5.76 5.04 5.04 CH 3 COOH 1s(aC*) DNC 0.65 DNC 0.77 DNC 0.79 1s(C*OOH) DNC 4.6 DNC 5.01 DNC 4.62 CHOOCH 3 1s(CH 3 ) DNC 1.79 DNC 1.86 DNC 2.02 CH 3 NH 2 1s(C) DNC 0.51 DNC 0.54 DNC 0.84 MAE 0.18 (0.19) 0.34 (0.25) 0.21 (0.19) 0.43 (0.30) 1.91 (2.12) MSE 0.05 (0.26) -0.18 (0.39) 0.15 (0.25) -0.08 (0.53) -1.79 (2.25) Max. AE 0.48 0.88 0.56 1.00 4.74 MAE 0.28 (0.17) 0.10 (0.10) 0.44 (0.29) vs. CCSD MSE -0.12 (0.32) 0.10 (0.10) -0.18 (0.52) vs. CCSD Max. AE 0.57 0.24 0.80 vs. CCSD MAE:jx i x exp i j=n MSE, (x i x exp i )=n MAX: max absolute error 115 Table B.6: Oxygen (1s) ionized states for selected molecules (IEs in eV). Molecule Edge CCSD-SD CCSD-S(D) MP2-SD MP2-S(D) DSCF-DFT Exp H 2 O 1s(O) 540.75 540.53 541.19 541.23 539.79 539.91 CO 1s(O) 543.78 543.01 544.22 543.98 542.54 542.57 CH 3 OH 1s(O) 540.23 538.75 540.69 539.51 539.03 539.06 C 2 H 5 OH 1s(O) DNC 537.83 DNC 538.64 DNC 538.6 CH 3 OCH 3 1s(O) DNC 538.37 DNC 538.7 DNC 538.59 CH 2 O 1s(O) 540.89 539.71 541.29 540.61 539.28 539.48 CH 3 COH 1s(O) DNC 538.36 DNC 539.28 DNC 538.62 CHOOH 1s(CO*OH) DNC 538.59 DNC 539.58 538.62 538.96 1s(COO*H) DNC 540 DNC 541.03 540.69 540.56 CH 3 COOH 1s(CO*OH) DNC 537.78 DNC 538.73 537.05 538.33 1s(COO*H) DNC 539.08 DNC 540.11 537.48 540.12 CHOOCH 3 1s(O*H) DNC 538.69 DNC 539.79 DNC 539.88 1s(CO*) DNC 537.85 DNC 538.91 DNC 538.45 MAE 1.16 (0.24) 0.55 (0.30) 1.59 (0.22) 0.55 (0.47) 0.60 (0.92) MSE 1.16 (0.24) -0.35 (0.53) 1.59 (0.22) 0.54 (0.49) -0.56 (0.95) Max.AE 1.41 1.19 1.81 1.41 2.64 MAE 0.91 (0.55) 0.44 (0.03) 0.42 (0.23) vs.CCSD MAE -0.91 (0.55) 0.44 (0.03) -0.08 (0.53) vs.CCSD Max.AE 1.48 0.46 0.72 vs.CCSD MAE:jx i x exp i j=n MSE, (x i x exp i )=n MAX: max absolute error 116 Table B.7: Chemical shifts of oxygen (1s) ionized states for selected molecules against water (IEs in eV). Molecule Edge CCSD-SD CCSD-S(D) MP2-SD MP2-S(D) DSCF-DFT Exp H 2 O 1s(O) 0 0 0 0 0 0 CO 1s(O) 3.03 2.48 3.03 2.75 2.75 2.66 CH 3 OH 1s(O) -0.52 -1.78 -0.5 -1.72 -0.76 -0.85 C 2 H 5 OH 1s(O) DNC -2.7 DNC -2.59 DNC -1.31 CH 3 OCH 3 1s(O) DNC -2.16 DNC -2.53 DNC -1.32 CH 2 O 1s(O) 0.14 -0.82 0.1 -0.62 -0.51 -0.43 CH 3 COH 1s(O) DNC -2.17 DNC -1.95 DNC -1.29 CHOOH 1s(CO*OH) DNC -1.94 DNC -1.65 -1.17 -0.95 1s(COO*H) DNC -0.53 DNC -0.2 0.9 0.65 CH 3 COOH 1s(CO*OH) DNC -2.75 DNC -2.5 -2.74 -1.58 1s(COO*H) DNC -1.45 DNC -1.12 -2.31 0.21 CHOOCH 3 1s(O*H) DNC -1.84 DNC -1.44 DNC -0.03 1s(CO*) DNC -2.68 DNC -2.32 DNC -1.46 MAE 0.42 (0.13) 1.05 (0.47) 0.42 (0.10) 0.86 (0.42) 0.63 (0.92) MSE 0.60 (0.24) -0.18 (1.29) 0.99 (0.72) 0.01 (1.13) 0.76 (0.1.04) Max.AE 0.57 1.81 0.53 1.41 2.52 MAE 0.92 (0.36) 0.02 (0.02) 0.76 (1.04) vs.CCSD MSE -0.92 (0.36) -0.01 (0.03) -0.75 (0.46) vs.CCSD Max.AE 1.26 0.04 1.2 vs.CCSD MAE:jx i x exp i j=n MSE, (x i x exp i )=n MAX: max absolute error 117 Table B.8: Nitrogen (1s) ionized states for selected molecules and chemical shifts (IEs in eV). For chemical shifts, error against experiment is given in parenthesis. Molecule Edge CCSD-SD CCSD-S(D) MP2-SD MP2-S(D) DSCF-DFT Exp NH 3 1s(N) 406.44 406.97 406.87 407.59 405.57 405.52 N 2 1s(u) 410.54 411.41 411.43 412.63 405.75 409.9 1s(g) 410.65 411.51 411.54 412.74 405.78 NA CH 3 NH 2 1s(N) DNC 405.92 DNC 406.55 DNC 405.07 MAE 0.78 (0.20) 1.27 (0.36) 1.44 (0.13) 2.09 (0.63) 2.10 (2.90) MSE 0.78 (0.20) 1.27 (0.36) 1.44 (0.13) 2.09 (0.63) -2.05 (2.97) Max. AE 0.92 1.51 1.53 2.73 4.15 MAE 0.75 (0.19) 0.55 (0.43) 1.78 (0.54) vs. CCSD MSE 0.75 (0.19) 0.55 (0.43) 1.78 (0.54) vs. CCSD Max. AE 0.87 0.89 2.09 vs. CCSD MAE:jx i x exp i j=n MSE, (x i x exp i )=n MAX: max absolute error 118 Table B.9: Nitrogen (1s) ionized states for selected molecules and chemical shifts (IEs in eV). For chemical shifts, error against experiment is given in parenthesis. Molecule Edge CCSD-SD CCSD-S(D) MP2-SD MP2-S(D) DSCF-DFT Exp NH 3 1s(N) 0 0 0 0 0 0 N 2 1s(u) 4.1 (-0.28) 4.44 (0.06) 4.56 (0.18) 5.04 (0.66) 0.18 (-4.2) 4.38 1s(g) 4.21 4.54 4.67 5.15 0.21 NA CH 3 NH 2 1s(N) DNC -1.05 (-0.6) DNC -1.04 (-0.59) DNC -0.45 MAE 0.57 (0.41) 0.46 (0) 0.97 (0.06) vs. CCSD MSE -0.13 (0.80) 0.46 (0) 0.28 (1.14) vs. CCSD Max. AE 1.05 0.46 1.04 vs. CCSD MAE:jx i x exp i j=n MSE, (x i x exp i )=n MAX: max absolute error 119 Table B.10: Core-excited states for selected molecules. Energies (eV) against the respective edges are shown. State Exp, abs Exp,DE EOM-CCSD-S(D)/ EOM-CCSD-S(D)/ cc-pVTZ aug-cc-pVTZ CO 1s(C)-p 287.4 -8.8 -8.18 -8.33 1s(C)-3s(Ry) 292.4 -3.8 - -1.98 1s(O)-p 534.2 -8.4 -3.45 -3.66 1s(O)-3s(Ry) 538.9 -3.6 - -0.44 N 2 1s(N)-p 401.0 -8.9 -5.69 - 1s(N)-3s(Ry) 406.2 -3.7 - -0.52 C 2 H 2 1s(C)-p 284.7 -6.55 -2.83 - 1s(C)-3s(Ry) 287.1 -4.15 - -1.08 CH 2 O 1s(C)-p 286.0 -8.47 -7.04 -7.27 1s(C)-3s(Ry) 290.2 -4.27 - -0.33 1s(O)-p 530.8 -8.68 -3.52 -3.69 1s(O)-3s(Ry) 535.4 -4.08 - -0.36 120 B.3 Calculations of core-ionized states of different pro- tonation forms of aqueous glycine. Figure B.1: Structures of model hexahydrated glycine complexes from Ref. 43. From left to right: zwitterionic form Gly ZI , Gly complex with sodium ion, and protonated Gly + complex with chloride. 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 Number of waters in the first solvation shell Occurance, arb. units) GlyZI Gly+ Gly− 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 Number of waters in the first solvation shell Occurance, arb. units GlyZI Gly+ Gly− Figure B.2: The number of water molecules in the first solvation shell for different protonation states of glycine using cutoff radius of 3.0 ˚ A (left) and 2.7 ˚ A (right). 121 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 DistancebetweenGly − andNa + ,Å Probabilitydensity 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 DistancebetweenGly + andCl − ,Å Probabilitydensity Figure B.3: The radial distribution function, g(R), of the distance between glycine and counterions: Gly -Na + (left) and Gly + -Cl (right). Table B.11: Chemical shifts (eV) of different protonation forms of isolated glycine. EOM-IP-MP2-S(D)/cc-pVDZ EOM-IP-MP2-S(D)/cc-pVTZ Gly can Gly ZI Gly Gly + Gly can Gly ZI Gly Gly + 1s N(NH) 408.94 4.11 -5.99 9.04 1s N(NH) 407.32 4.06 -5.91 8.98 1s C(COO) 299.12 -1.46 -6.91 6.05 1s C(COO) 297.79 -1.44 -6.80 6.13 1s C(CH 2 ) 295.92 0.62 -6.28 6.30 1s C(CH 2 ) 294.70 0.61 -6.14 6.24 EOM-IP-CCSD-S(D)/cc-pVDZ EOM-IP-CCSD-S(D)/cc-pVTZ Gly can Gly ZI Gly Gly + Gly can Gly ZI Gly Gly + 1s N(NH) 408.59 4.09 -6.03 9.15 1s N(NH) 406.58 4.06 -6.32 9.09 1s C(COO) 298.41 -1.52 -6.96 6.07 1s C(COO) 296.91 -1.43 -6.84 6.25 1s C(CH 2 ) 295.55 0.60 -6.33 6.34 1s C(CH 2 ) 294.08 0.58 -6.18 6.30 122 Table B.12: Computed core IEs and chemical shifts relative Gly can of different pro- tonation forms of glycine using model structures from Ref. 43. IP-CCSD-S(D)/cc- pVTZ. All energies are in electron-volts. Gly ZI Gly PCM ZI Gly ZI;6w Gly PCM ZI;6w 1s (N) 410.64 409.01 408.46 407.32 1s (COO) 295.48 296.05 295.95 296.13 1s (CH 2 ) 294.66 293.93 294.23 293.53 1s (N) 4.06 2.43 1.88 0.74 1s (COO) -1.43 -0.86 -0.96 -0.77 1s (CH 2 ) 0.58 -0.15 0.15 -0.55 Gly Gly PCM Gly ;6w Gly PCM ;6w 1s (N) 400.26 404.81 404.49 404.55 1s (COO) 290.07 295.20 294.43 295.58 1s (CH 2 ) 287.90 292.30 292.19 292.53 1s (N) -6.32 -1.77 -2.10 -2.03 1s (COO) -6.84 -1.71 -2.47 -1.33 1s (CH 2 ) -6.18 -1.78 -1.89 -1.55 Gly + Gly PCM + Gly +;6w Gly PCM +;6w 1s (N) 415.67 409.91 409.70 407.49 1s (COO) 303.16 298.36 298.56 297.52 1s (CH 2 ) 300.38 295.12 295.63 294.04 1s (N) 9.09 3.33 3.12 0.91 1s (COO) 6.25 1.45 1.65 0.61 1s (CH 2 ) 6.30 1.04 1.55 -0.04 123 Table B.13: Core IEs and chemical shifts (eV) relative to Gly can of the anionic form of glycine computed using representative single snapshots from equilibrium trajectory. QM / QM / QM / QM ;w / QM ;w / QM ;w / MM EFP fz EFP MM EFP fz EFP IP-CCSD-S(D)/cc-pVDZ 1s N -0.29 -0.63 -0.20 -0.79 -0.27 -0.75 1s C(COO) -0.71 -1.24 0.31 -0.18 -1.21 -1.62 1s C(CH 2 ) -0.46 -0.63 0.41 -0.21 -0.41 -0.93 Wall time 7 35 35 69 795 795 IP-CCSD-S(D)/cc-pVTZ 1s N -0.29 -0.54 -0.27 -0.79 -0.21 -0.76 1s C(COO) -0.63 -1.25 0.26 -0.18 -1.25 -1.72 1s C(CH 2 ) -0.36 -0.68 0.41 -0.22 -0.33 -0.98 Wall time 101 640 640 566 6172 6172 IP-MP2-S(D)/cc-pVDZ 1s N -0.38 -0.45 -0.16 -0.76 -0.14 -0.89 1s C(COO) -0.71 -1.39 0.37 -0.14 -1.31 -1.63 1s C(CH 2 ) -0.33 -0.58 0.44 -0.37 -0.26 -1.09 Wall time 2 21 21 93 237 237 IP-MP2-S(D)/cc-pVTZ 1s N -0.24 -0.49 -0.30 -0.83 -0.25 -0.82 1s C(COO) -0.60 -1.29 0.24 -0.11 -1.28 -1.75 1s C(CH 2 ) -0.32 -0.64 0.47 -0.25 -0.34 -1.01 Wall time 10 615 615 271 1351 1351 Wall times in seconds on 16-core Intel Xeon E5-2690. 124 Bibliography [1] Local Hydration Environments of Amino Acids and Dipeptides Studied by X-ray Spectroscopy of Liquid Microjets. 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Creator Sadybekov, Arman (author)

Core Title Electronically excited and ionized states in condensed phase: theory and applications

Contributor Electronically uploaded by the author (provenance)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Publication Date 09/19/2017

Defense Date 05/25/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag condensed phase,EOM-CC,NEXAFS,OAI-PMH Harvest,QM/MM

Language English

Advisor Krylov, Anna I. (committee chair), Nakano, Aiichiro (committee member), Prezhdo, Oleg (committee member)

Creator Email sadybeko@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-429618

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Document Type Dissertation

Rights Sadybekov, Arman

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Abstract (if available)

Abstract Predictive modeling of chemical processes in silico is a goal of XXI century. While robust and accurate methods exist for ground-state properties, reliable methods for excited states are still lacking and require further development. Electronically exited states are formed by interactions of matter with light and are responsible for key processes in solar energy harvesting, vision, artificial sensors, and photovoltaic applications. The greatest challenge to overcome on our way to a quantitative description of light-induced processes is accurate inclusion of the effect of the environment on excited states. All above mentioned processes occur in solution or solid state. Yet, there are few methodologies to study excited states in condensed phase. Application of highly accurate and robust methods, such as equation-of-motion coupled-cluster theory EOM-CC, is limited by a high computational cost and scaling precluding full quantum mechanical treatment of the entire system. In this thesis we present successful application of the EOM-CC family of methods to studies of excited states in liquid phase and build hierarchy of models for inclusion of the solvent effects. In the first part of the thesis we show that a simple gasphase model is sufficient to quantitatively analyze excited states in liquid benzene, while the latter part emphasizes the importance of explicit treatment of the solvent molecules in the case of glycine in water solution. ❧ In chapter 2, we use a simple dimer model to describe exciton formation in liquid and solid benzene. We show that sampling of dimer structures extracted from the liquid benzene is sufficient to correctly predict exited-state properties of the liquid. Our calculations explain experimentally observed features, which helped to understand the mechanism of the excimer formation in liquid benzene. Furthermore, we shed light on the difference between dimer configurations in the first solvation shell of liquid benzene and in unit cell of solid benzene and discussed the impact of these differences on the formation of the excimer state. ❧ In chapter 3, we present a theoretical approach for calculating core-level states in condensed phase. The approach is based on EOM-CC and effective fragment potential (EFP) method. By introducing an approximate treatment of double excitations in the EOM-CCSD (EOM-CC with single and double substitutions) ansatz, we addressed poor convergence issues that are encountered for the core-level states and significantly reduced computational costs. While the approximations introduce relatively large errors in the absolute values of transition energies, the errors are systematic. Consequently, chemical shifts, changes in ionization energies relative to the reference systems, are reproduced reasonably well. By using different protonation forms of solvated glycine as a benchmark system, we showed that our protocol is capable of reproducing the experimental chemical shifts with a quantitative accuracy. The results demonstrate that chemical shifts are very sensitive to the solvent interactions and that explicit treatment of solvent, such as EFP, is essential for achieving quantitative accuracy. ❧ In chapter 4, we outline future directions and discuss possible applications of the developed computational protocol for prediction of core chemical shifts in larger systems.