AB INITIO MOLECULAR DYNAMICS: APPLICATIONS TO DEFECTIVE SILICON NANOCRYSTALS AND DEVELOPMENTS TOWARD DENSE MANIFOLD SYSTEMS By Wei-Tao Peng A DISSERTATION Michigan State University in partial fulfillment of the requirements Submitted to for the degree of Chemistry – Doctor of Philosophy 2019 ABSTRACT AB INITIO MOLECULAR DYNAMICS: APPLICATIONS TO DEFECTIVE SILICON NANOCRYSTALS AND DEVELOPMENTS TOWARD DENSE MANIFOLD SYSTEMS By Wei-Tao Peng Ab initio molecular dynamics (AIMD) methods consider the nuclear motions under the potential generated by electronic wavefunctions which are determined from ab initio quantum mechanical calculations on-the-fly. AIMD methods allow researchers to investigate chemical processes without prior knowledge or assumptions about the shape of the potential energy surface (PES). In this thesis, we applied AIMD methods to study silicon nanocrystals with dangling bond defects (DB-SiNCs). DB defects on SiNCs have been known as nonradiative (NR) decay centers. However, the atomistic mechanism for the decay process is unclear. Previously, researchers considered a pyramidalization mode surrounding the DB site involved in the process. Based on our AIMD calculations on the first excited state and the static analysis of the PESs of SiNC systems, we discovered that asymmetrical Si-Si bond stretching modes surrounding DB sites are important, in addition to pyramidalization. Most importantly, we found a low-lying defect- induced conical intersection (DICI) in the neutral DB system. The minimum energy conical intersection (MECI) is estimated to be 1.74 eV above the ground state minimum energy geometry by application of multi-state complete active space second-order perturbation theory (MS-CASPT2) to a small cluster model system. In addition, the roles of charged DBs on NR decay process are investigated. We found DICIs for both positively and negatively charged DB systems. The MECI energies are 2.10 eV and 2.65 eV respectively. The rationalization of the existence of conical intersections and detailed dynamics after excitation of these systems are discussed in the thesis. Additionally, to study the possible defect-defect interactions during the NR recombination process, we considered slab models with two DB defects at short (~4 Å) and long (~10 Å) separations. According to our simulations, the NR recombination process is localized on a single DB site, regardless the defect-defect distances. However, energy transfer between defect sites with short separations is possible. For the defective SiNC systems, we demonstrated the power of the AIMD method to investigate the dynamics after excitations. However, the applications of AIMD to high-lying states are much more challenging, due to the dense manifold of states that cause immense computational effort. In the thesis, we developed several methods toward the application to such systems. First, we developed a time-dependent configuration interaction (TD-CI) method that can simulate the electron dynamics under a strong field efficiently. The method is based on the direct scheme to form the σ vector, σ= Hc , which can be accelerated by a graphical processing unit. A TD-CI calculation with 853776 determinants requires only 20.1 hours to propagate to 100 fs with 1 attosecond (10-18 second) time steps. On the other hand, when the field is strong enough, the electrons can be driven to the boundary of the basis set, which would cause unphysical effects such as reflection. To account for this, we developed an analytical expression for a molecule-centered complex absorbing potential which can be evaluated efficiently to remove the unwanted effects. Finally, for the nuclear dynamics, we developed an Ehrenfest dynamics method based on the TD-CI wavefunction. In this approach, the nuclear motions are propagated under the averaged potential generated by TD-CI wave function, thus the approach is promising for application to systems with dense manifolds of states. ACKNOWLEDGEMENTS I would like to thank my advisor, Benjamin G. Levine, for his guidance, patience, and assistance throughout my PhD studies. I am grateful to him for offering the opportunity to join his scientific group. Besides the research, he taught me how to communicate effectively to other scientists. I am also inspired by his enthusiasm in science. I could not have imagined having a better advisor and mentor. I would also like to thank my committee members, Katherine C. Hunt, Marcos Dantus, and Piotr Piecuch, for their invaluable comments. I am thankful to my fellow labmates: Yinan, Garrett, Scott, Dmitry, Mike, Dylan, Andy, and Fangchun, for all the discussions, collaborations, and advices. I greatly appreciate the continued support and warm love from my beloved family and friends. This dissertation would not have been possible without them. iv TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES CHAPTER 1 INTRODUCTION 1.1 Description of Chapters REFERENCES CHAPTER 2 DYNAMICS OF RECOMBINATION VIA CONICAL INTERSECTION IN A SEMICONDUCTOR NANOCRYSTAL 2.1 Introduction 2.2 Results and Discussion 2.3 Conclusions APPENDIX REFERENCES CHAPTER 3 AB INITIO MOLECULAR DYNAMICS STUDY OF THE INTERACTION BETWEEN DEFECTS DURING NONRADIATIVE RECOMBINATION 3.1 Introduction 3.2 Method 3.3 Results and Discussions 3.4 Conclusions APPENDIX REFERENCES CHAPTER 4 NONADIABATIC DYNAMICS STUDY OF SILICON DANGLING 4.1 Introduction 4.2 Method 4.3 Results and Discussion BOND DEFECTS AT DIFFERENT CHARGED STATES 4.3.1 Dynamical Simulations of the Neutral Si-DB 4.3.2 Static Analysis of the Potential Energy Surface on the Neutral Si-DB 4.3.3 Dynamical Simulations of the Cationic Si-DB 4.3.4 Static Analysis of the Potential Energy Surface on the Cationic Si-DB 4.3.5 Dynamical Simulations of the Anionic Si-DB 4.3.6 Static Analysis of the Potential Energy Surface on the Anionic Si-DB 4.3.7 Discussion 4.4 Conclusions v vii ix 1 3 5 8 8 10 19 20 44 52 52 54 58 68 70 91 97 97 99 101 101 104 107 109 112 115 118 120 APPENDIX REFERENCES CHAPTER 5 SIMULATING ELECTRON DYNAMICS OF COMPLEX 5.1 Introduction 5.2 Method 5.3 Results and Discussion MOLECULES WITH TIME-DEPENDENT COMPLETE ACTIVE SPACE CONFIGURATION INTERACTION 5.3.1 Absorption Spectrum 5.3.2 Rabi Oscillations 5.3.3 Control by Shaped Pulses 5.3.4 Performance APPENDIX REFERENCES CHAPTER 6 ANALYTICAL EXPRESSION FOR A MOLECULE-CENTERED 5.4 Conclusions 121 135 141 141 143 148 148 150 154 161 161 163 173 COMPLEX ABSORBING POTENTIAL USING GAUSSIAN BASIS SETS 6.1 Introduction 6.2 Method REFERENCES CHAPTER 7 THE EHRENFEST MOLECULAR DYNAMICS IN CONJUNCTION WITH 181 181 183 188 TIME-DEPENDENT CONFIGURATION INTERACTION ELECTRONIC WAVEFUNCTIONS 7.1 Introduction 7.2 Method 194 194 195 199 REFERENCES vi LIST OF TABLES 12 The Si-Si bond lengths (Å), Si-Si-Si bond angles (θ as indicated in Figure 1a; in degrees), and D1 energies of differently sized silicon clusters at the Franck-Condon point (FC) and MECI. All energies are relative to the D0 minimum energy. Energies are computed at the CASCI level of theory as described in the APPENDIX. When available, CASPT2 energies at CASCI geometries are presented in parenthesis. Relevant geometric parameters surrounding the DB defect sites at several important geometries in each model. Parameters include Si-Si bond lengths (RSi-Si in Å) and Si-Si-Si bond angles (q in degrees) as illustrated in Figure 3.1. Si-Si bond lengths greater than 2.5 Å are bolded. Result for the 1DB system (Si72H63) from reference 36 is also listed for comparison. 58 The vertical excitation energies (Eex in eV) and excitation characters at the Franck-Condon point of the NN and TNN systems. The energies (in eV) of the MECIs of the NN, TNN, and 1DB systems. Results for 1DB are drawn from reference 36. As the 1DB system has only one dangling bond, only one energy is listed. All energies are relative to ground state energy of the T0-optimized structure. The minimum energy geometries of NN and TNN systems at the singlet ground state. The relative energies (R.E.; unit meV) are the energies relative to corresponding triplet ground state. 60 64 70 Optimized neutral DB cluster model FC point, D1/D0 MECI, and D1min energies (eV). Optimized positive DB cluster model FC point, S1/S0 MECI, and S1min energies (eV). Optimized negative DB cluster model FC point, S1/S0 MECI, and S1min energies (eV). The excitation energies at Franck-Condon point (FC). The FC geometries are optimized with the B3LYP/6-31G(d,p) method. The IP-EOM- CCSD(2h,1p)/6-31G(d,p), EOM-CCSD/6-31G(d,p) and EOM-CCSD/6- 31+G(d,p) energies are calculated for the neutral, cationic and anionic DB systems respectively. The optimized cluster model FC, MECI, and D1min energies for the neutral DB system. 105 110 115 121 121 vii Table 2.1. Table 3.1. Table 3.2. Table 3.3. Table B1. Table 4.1. Table 4.2. Table 4.3. Table C1. Table C2. Table C3. Table C4. Table 5.1. Table 5.2. Table 5.3. Table D1. Table D2. The optimized cluster model FC, MECI, and S1min energies for the positive DB system. The optimized cluster model FC, MECI, and S1min energies for the negative DB system. 122 123 Comparison of the excitation energies (eV) of decacene calculated via TD-CISNO-CASCI and TI-CISNO-CASCI. Only the states with non- zero transition dipole moment in the polarization direction (μy; atomic units) are listed. Absolute errors are reported in eV. The error in the population (Pop. Err.), error in the Rabi oscillation period ( ) relative to the exact value (21.506 fs), and norm of the wave ΔT function after 100 fs of propagation for simulations of ethylene in a resonant CW field as a function of the integration time step. Simulation details are presented in the text. The wall time to solution for several of the calculations. The propagation time is 100 femtosecond and the time step is 1 as for all simulations shown. 149 153 160 163 Comparison of the wave function norm at the end of 100 fs simulations of decacene after excitation by a δ-function pulse calculated using different time steps. Calculations are performed at the TD-CISNO-CASCI /6-31G(d) basis set with a (10,10) active space. The first excitation energies (in eV) of decacene obtained with TI- CISNO-CASCI and TD-CISNO-CASCI methods with 6-31G(d) basis set and a (2,2) active space. TD-CISNO-CASCI calculations were performed with a series of time steps from 1 to 500 as. The norm at the end of the 100 fs simulations is also presented. 163 viii LIST OF FIGURES Figure 2.1. The Pb-containing silicon clusters studied in this work. (a) Si10H15 (sila- adamantane cluster), (b) Si22H27, (c) Si26H31, (d) Si47H49 (1.3-nm SiNC), and (e) Si72H63 (1.7-nm SiNC). One of the three symmetry-equivalent Si-Si bond lengths (RSi-Si) and bond angles (q) discussed herein are indicated in red in (a). The red arrows indicate the positions of dangling bond defects. Figure 2.2. Orbitals representative of the D0→D1 (2A1→2E) transition of the Si72H63 system. (a) The singly occupied molecular orbital (SOMO) is the nonbonding (n) orbital of the Pb center, and (b) the highest doubly occupied orbital (HDOMO) has Si-Si σ bonding character in the vicinity of the Pb defect (σSi-Si). Note that this HDOMO is approximately degenerate due to the local symmetry of the defect. We show only one of the two nearly degenerate orbitals. The red arrows indicate the locations of the Pb center. (a) The potential energies of the D1 (red) and D0 (black) electronic states as a function of time from the D1 AIMD simulation of Si72H63. (b) The three Si-Si bond lengths (RSi-Si) adjacent to the Pb defect as a function of time from the same AIMD calculation. (c) The three Si-Si-Si angles (q, illustrated in Figure 1a) as a function of time from the same AIMD calculation. Each color represents one of the three symmetry equivalent bond lengths or angles in (b) and (c), respectively. (a) Schematic illustration of the dynamics of nonradiative recombination of an excitation at a Pb center. The PESs are plotted as a function of an asymmetric stretching coordinate about the Pb center (illustrated along the x-axis with the dangling bond site represented by a filled circle and the three adjacent silicon atoms represented by open circles). Insets show the orbital occupations of D0 and D1 and relative orbital energies at the FC point (left) and MECI (right). The n and σSi-Si orbitals of the smallest (sila-adamantane) system are shown on the bottom left. 15 11 13 16 Figure 2.3. Figure 2.4. Figure A1. The potential energies of the first excited (red line) and ground states ( black line) change with the time following the first excited state trajectory from an AIMD simulation where the force comes from the first excited state in Si10H15. 20 Figure A2. The potential energies of the first excited (red line) and ground states ( black line) change with the time following the first excited state trajectory from an AIMD simulation where the force comes from the first excited state in Si22H27. 21 ix Figure A3. The potential energies of the first excited (red line) and ground states ( black line) change with the time following the first excited state trajectory from an AIMD simulation where the force comes from the first excited statein Si26H31. 22 Figure A4. The potential energies of the first excited (red line) and ground states ( black line) change with the time following the first excited state trajectory from an AIMD simulation where the force comes from the first excited state in Si47H49. Figure A5. The three Si-Si bond lengths (RSi-Si) of Si10H15 change with time from the AIMD calculation. Each color represents one Si-Si bond length. Figure A6. The three Si-Si bond lengths (RSi-Si) of Si22H27 change with time from the AIMD calculation. Each color represents one Si-Si bond length. 23 24 25 26 27 55 59 Figure A7. The three Si-Si bond lengths (RSi-Si) of Si26H31 change with time from the AIMD calculation. Each color represents one Si-Si bond length. Figure A8. The three Si-Si bond lengths (RSi-Si) of Si47H49 change with time from the AIMD calculation. Each color represents one Si-Si bond length. Figure 3.1. The Si70H68 slab model. The Si and H atoms are shown in blue and white, respectively. The left and right DB sites are labeled L and R, respectively. Site R is the same for both models. The nearest neighbor (NN), and third nearest neighbor (TNN) models are created by removing the capping H atoms from the yellow circled positions labeled L(NN) and L(TNN), respectively. The right panel illustrates the geometric parameters, θ and RSi-Si, reported in Table 3.1 and discussed in the text . Figure 3.2. The singly occupied natural orbitals of the triplet ground state (T0) of the TNN system. These are both nDB orbitals, as described in the text. Figure 3.3. Natural orbitals with occupation numbers 1.00 (top) and 2.00 (bottom) of state T1 at the NN FC geometry. The top and bottom orbitals are examples of σSi-Si and nDB orbitals, respectively, as described in the text. 61 Figure 3.4. A simplified picture of the orbitals arrangement in the systems studied here. All excitations investigated in this work involve promotion of an electron from a σSi-Si orbital to a nDB orbital. In the low-lying excited states (T1-T4) the majority of the excitation character is local to a single DB site. The dashed arrow indicates a possible charge-transfer excitation between two sites (e.g. T5). 62 x Figure 3.5. Top panel: The T1-T0 energy gap as a function of time for the NN system. Bottom panel: the T1-T0 energy gap as a function of time for the TNN system. The light grey curves depict the individual trajectories and the red curves depict the average gap over 20 trajectories for each system. Figure 3.6. The MECI(L) geometries and natural orbitals for the NN and TNN Figure 3.7. Top panel: the averaged Si-Si bond lengths around the DB sites for the Figure B1. Figure B2. Figure B3. Figure B4. Figure B5. Figure B6. Figure B7. systems. (a) The top view of the MECI(L) in NN. (b) The side view of MECI(L) in NN. (c) A natural orbital with occupation number 1.50 (nDB) for MECI(L) in NN. (d) A natural orbital with occupation number 1.50 (σSi-Si) for MECI(L) in NN. (e) The top view of the MECI(L) in TNN. (f) The side view of MECI(L) in TNN. (g) A natural orbital with occupation number 1.50 (nDB) for MECI(L) in TNN. (h) A natural orbital with occupation number 1.50 (σSi-Si) for MECI(L) in TNN. Panels c and d are adapted from reference 40 with permission from the PCCP Owner Societies. NN system. Bottom panel: the averaged Si-Si bond lengths around the DB sites for the TNN system. The bond lengths are sorted as described in the main text. On-site (more distorted) and off-site (less distorted) bonds are shown on the left and right, respectively. Longest, middle, and shortest lengths are shown by yellow, orange, and blue lines, respectively. The ground state natural orbitals with occupation number 1.00 in the NN system. The T1 state natural orbitals with occupation number 1.00 and sigma bonding character in the TNN system. The T1 state natural orbitals with occupation number 2.00 and (majority) DB nonbonding character in the TNN system. The T5 state (CT state) natural orbitals with occupation number 1.06 and sigma bonding character in the NN system. The T5 state (CT state) natural orbitals with occupation number 1.94 and DB nonbonding character in the NN system. The T5 state (CT state) natural orbitals with occupation number 1.00 and sigma bonding character in the TNN system. The T5 state (CT state) natural orbitals with occupation number 2.00 and DB nonbonding character in the TNN system. 63 65 66 71 72 72 73 73 74 74 xi Figure 4.1. (a) The Si8H13 cluster model studied in this work. One of the three Si-Si bond lengths (RSi-Si) and Si-Si-Si bond angles (q) surrounding DB site are indicated in red. (b) The top view of the one nonbonding and two sSi-Si bonding orbitals at the FC point in the neutral state system. (c) The side view of the diffused LUMO and nonbonding HOMO at the FC point in the negatively charged DB system. 102 Figure 4.2. The time evolutions of the D1-D0 energy gap (grey lines: 20 trajectories; red line: average) for the neutral DB system. Figure 4.3. The time evolutions of the population of the first excited and ground states for the neutral DB system. Cyan lines: excited state populations of 20 trajectories; magenta lines: ground state populations of 20 trajectories; thick blue line: averaged excited state population; thick red line: averaged ground state population. Figure 4.4. The time evolutions of (a) the averaged Si-Si bond lengths surrounding 102 103 108 108 Figure 4.5. DB (RSi-Si) and (b) the averaged Si-Si-Si bond angles surrounding DB (q) for the neutral DB system. The bond lengths (bond angles) are sorted as the longest (largest), the middle, and the shortest (smallest) RSi-Si distances (q degree) for each trajectory at each time step, then averaged, and are shown by yellow, orange and blue lines respectively. (a) The D1/D0 MECI geometry of the neutral DB cluster optimized by SA-CASSCF. (b) The D1/D0 MECI geometry of the neutral DB cluster optimized by MS-CASPT2. (c) and (d) Top view and side view of D0-D1 state-averaged natural orbitals with occupation numbers 1.50 for the neutral DB at the MECI geometry. 104 106 Figure 4.6. The time evolutions of (a) the averaged Si-Si bond lengths surrounding DB (RSi-Si) and (b) the averaged Si-Si-Si bond angles surrounding DB (q) for the positive DB system. The bond lengths (bond angles) are sorted as the longest (largest), the middle, and the shortest (smallest) RSi-Si distances (q degree) for each trajectory at each time step, then averaged, and are shown by yellow, orange and blue lines respectively. Figure 4.7. The time evolutions of the S1-S0 energy gap (grey lines: 20 trajectories; red line: average) for the positive DB system . Figure 4.8. The time evolutions of the population of the first excited and ground states for the positive DB system. Cyan lines: excited state populations of 20 trajectories; magenta lines: ground state populations of 20 trajectories; thick blue line: averaged excited state population; thick red line: averaged ground state population. 109 xii Figure 4.9. (a) The S1/S0 MECI geometry of the positive DB cluster optimized by SA-CASSCF. (b) The S1/S0 MECI geometry of the positive DB cluster optimized by MS-CASPT2. (c) and (d) Top view and side view of S0-S1 state-averaged natural orbitals with fractional occupation numbers 1.48 and 0.53 for the positive DB at the MECI geometry. 112 Figure 4.10. The time evolutions of (a) the averaged Si-Si bond lengths surrounding DB (RSi-Si) and (b) the averaged Si-Si-Si bond angles surrounding DB (q) for the negative DB system. The bond lengths (bond angles) are sorted as the longest (largest), the middle, and the shortest (smallest) RSi-Si distances (q degree) for each trajectory at each time step, then averaged, and are shown by yellow, orange and blue lines respectively. 113 114 114 116 118 Figure 4.11. The time evolutions of the S1-S0 energy gap (grey lines: 20 trajectories; red line: average) for the negative DB system. Figure 4.12. The time evolutions of the population of the first excited and ground states for the negative DB system. Cyan lines: excited state populations of 20 trajectories; magenta lines: ground state populations of 20 trajectories; thick blue line: averaged excited state population; thick red line: averaged ground state population. Figure 4.13. (a) The S1/S0 MECI geometry of the negative DB cluster optimized by SA-CASSCF. (b) The S1/S0 MECI geometry of the negative DB cluster optimized by MS-CASPT2. (c) and (d) Top view and side view of S0-S1 state-averaged natural orbitals with fractional occupation numbers 1.50 and 0.60 for the negative DB at the MECI geometry. Figure 4.14. The plot of three Si-Si bond lengths surrounding DB site changes with time for (a) the simulation where the one Si-Si bond stretch occurred at 200 fs, (b) the simulation where the one Si-Si bond stretch occurred at 900 fs. Figure C1. Optimized cluster model FC, MECI and S1min energies. The geometry optimizations were performed at B3LYP (dotted lines at FC point), SA- CASSCF (dash lines at MECI, S1min and D1min) and MS-CASPT2 ( solid lines at MECI, S1min and D1min) levels of theory. Energy calculations were performed at SA-CASSCF (blue lines), MS-CASPT2 (red lines) and MRCI (yellow lines) levels of theory. (a) The neutral DB system. (b) The positive DB system. (c) The negative DB system. The orbitals of positively charged DB system at the ground state minimum energy geometry. The state-averaged natural orbitals with fractional occupation numbers for a negative DB at the S1min geometry. Figure C3. Figure C2. 124 124 125 xiii Figure 5.1. Molecule of decacene. Dark gray: carbons. Light gray: hydrogens. Figure 5.2. The absorption spectrum of decacene computed at the TD-CISNO-CASCI 148 149 level as described in the text. Figure 5.4. Figure 5.3. The populations of the S0 and S1 states as a function of time from calculations of the dynamics of ethylene in a resonant CW field, as described in the text. Calculations are performed with time steps of 1 as (solid) and b) 10 as (dashed). The final state populations obtained from 100 fs simulations of decacene (TD-CISNO-CASCI(10/10)/6-31G(d)) in a series of laser pulses with chirp ranging from β = -0.342 fs-2 to β = 0.342 fs-2 All pulses have a FWHM of 10 fs and a maximum intensity of 3*1012 W/cm2. The populations of bright and dark states up to S8 are shown with solid lines and dashed lines, respectively. The populations in all states above S8 are summed into a single (purple) line. Populations of states with negligible population in all simulations (S3-S5) are not shown. A Jablonski diagram on the right shows the energies of the populated states up to S8. Figure 5.5. The electronic state populations as a function of time for decacene (TD- 152 154 CISNO-CASCI(10,10)/6-31G(d)) in 10 femtosecond (FWHM) TL pulses with maximum intensities of 1*1012 W/cm2 (middle panel) or 3*1012 W/cm2 (bottom panel). The electric field as a function of time is shown in the top panel (arbitrary units). This pulse is resonant with the S0→S2 transition. Only states up to S8 that contain non-negligible population at the end of the simulations are shown. The sum of all population in states higher than S8 is represented by a dotted purple line. 156 Figure 5.6. The electronic state populations as a function of time for decacene (TD- CISNO-CASCI(10,10)/6-31G(d)) in 10 fs (FWHM) chirped pulses with β = 0.256 fs-2 and an intensity of 3x1012 W/cm2 are shown in the bottom panel. The electric field as a function of time is illustrated in the top panel. Only populations of states up to S8 that contain non-negligible population at the end of the simulation are shown. The sum of all population in states higher than S8 is represented by a dotted purple line. Figure 5.7. The electronic state populations as a function of time for decacene (TD- CISNO-CASCI(10,10)/6-31G(d)) in 10 fs (FWHM) chirped pulses with β = -0.256 fs-2 and an intensity of 3x1012 W/cm2 are shown in the bottom panel. The electric field as a function of time is illustrated in the top panel. Only populations of states up to S8 that contain non-negligible population at the end of the simulation are shown. States that are dark with respect to the electronic ground state (S1, S6, and S7) are shown with dashed lines. The sum of all population in states higher than S8 is represented by a dotted purple line. 157 158 xiv Figure D1. The electronic state populations as a function of time for decacene (TD- CISNO-CASCI(12,12)/6-31G(d)) in 10 fs (FWHM) chirped pulses with β = -0.256 fs-2 and an intensity of 3x1012 W/cm2. Only populations of states up to S9 that contain non-negligible population at the end of the simulation are shown. Note that S9 in this active space corresponds to S8 in the (10,10) active space used throughout this work. States that are dark with respect to the electronic ground state (S1, S6, and S7) are shown with dashed lines. The sum of all population in states higher than S9 is represented by a dotted purple line. Figure D2. The final population on S0 for a simulation of decacene with a (10,10) 164 active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 165 Figure D3. The final population on S1 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 166 Figure D4. The final population on S2 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 167 Figure D5. The final population on S6 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 168 Figure D6. The final population on S7 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 169 Figure D7. The final population on S8 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 170 Figure 6.1. A representation of a molecule-centered complex absorbing potential developed in this work. The radius r0 from the center of the molecule is defined for the boundary (show in red). Beyond the boundary, a quadratic form of absorbing potential is applied (shown in the green line and in the inserted figure). Within the boundary, the potential is zero. 183 xv Figure 7.1. The schematic representation of the Ehrenfest dynamics algorithm. The steps (i)-(vi) mentioned in the text are labeled. The orange and blue arrows indicate the velocity-Verlet algorithm with time steps for propagating momentum and position respectively. The smaller green arrows indicate the time steps TD-CI calculations. For the feedback between electronic and nuclear dynamics, the force calculation and orbital diabatizing (both indicated in red words) are performed. for propagating electronic dynamics in Δτ Δt 198 xvi CHAPTER 1 INTRODUCTION Ab initio molecular dynamics (AIMD) methods simulate the nuclear motions on the potential energy surface (PES) generated by electronic wavefunctions which are determined from quantum mechanical calculations on-the-fly. Such methods are extensively applied to understand various chemical processes in which the transient elementary steps are difficult (if possible) to investigate experimentally. In the 1970s, Karplus, Warshel and Levitt applied a combined quantum mechanical and molecular mechanical (QM/MM) approach, where the chemically relevant part of the system is treated quantum mechanically, to study the PESs of a conjugated organic molecule1 and an enzyme.2 Ever since, the QM/MM method has gradually gained popularity for understanding complex chemical and enzymatic reactions, and eventually Karplus, Warshel and Levitt shared the Nobel Prize in Chemistry 2013. On the other hand, Car and Parrinello proposed a method that employed density functional theory for the electronic structure and an extended Lagrangian scheme for the dynamics in 1985,3 which later became the most prevalent AIMD approach and is known as the CP method. For considering systems where multiple electronic states are involved, many nonadiabatic ab initio molecular dynamics (NAMD) methods have been developed. The ab initio multiple spawning (AIMS) method,4-5 surface hopping method6 and Ehrenfest dynamics7-9 are three popular NAMD methods being used. Starting from pioneering work by the Prezhdo10 and Batista11 groups, growing numbers of studies are now applying NAMD methods to elucidate photochemical and photophysical processes in materials science. In this thesis, we applied AIMD and NAMD methods to semiconductor nanocrystals, specifically silicon nanocrystals (SiNCs), to investigate the photophysics of such systems when defects are present. Semiconductor nanocrystals are promising in many applications such as light emitting diodes,12 light harvesting materials13 and bioimaging applications,14 just to name a few. As one may notice, many applications take advantage of nanocrystals’ ability to efficiently absorb and emit light, as well as the tunability of their band gaps. These optical properties are achieved by quantum confinement effects, which occur when the size of a semiconductor nanocrystal is smaller than its exciton Bohr radius. As the semiconductor nanocrystals have high surface-to- volume ratio, another route to tune their optical properties is based on surface modification. 1 Hence, it is important to understand how the composition of the surface of semiconductor nanocrystals affects their properties. In addition to artificially modified surface composition, many types of defects may exist on the surface as well. These defects generally cause unwanted effects to the devices. Understanding the effects is essential for device optimization and rational design of high-performance materials. It is well-known that surface defects play important roles in nonradiative recombination (NRR) processs which convert excitation energy to useless heat. However, it is generally difficult to identify a specific defect as an NRR center by experimental means, not to mention to investigate the microscopic mechanism of the NRR process. Toward this aim, theoretical calculations are performed to complement the experiments and to give guidance toward new discoveries. A semiconductor nanocrystal usually contains more than a hundred atoms. Thus, theoretical study of photophysics based on quantum mechanics often relies on single-particle pictures (e.g. orbitals or band structures), and nuclear motions are often simplified or totally ignored. However, electron-hole interactions and local distortions of nuclear structure can be crucial, due to the localized nature of the defect states. Following advances in computer hardware, AIMD methods are now feasible for unprecedentedly large systems. AIMD explores the potential energy surface (PES), including many particle interactions and the full set of nuclear degrees of freedom. In this thesis, we apply state-of-the-art AIMD methods to understand the photophysics of defective semiconductor nanocrystals. With the aid of graphic processing units (GPUs), we are able to run Born-Oppenheimer AIMD in conjunction with the complete active space configuration interaction (CASCI-AIMD) level of theory. The CASCI- AIMD method allows us to explore the excited state PES, and representative points on the PES such as excited state minimum energy geometries and conical intersections (CIs) may be discovered and can be further optimized. Defect-induced conical intersections (DICIs) 15-16 provide efficient pathways for NRR processes and microscopic mechanisms can be deduced from CASCI-AIMD simulations. On the other hand, methods that are suitable for studying photophysics with population transfer between adiabatic electronic states require going beyond Born-Oppenheimer approximation, e.g. NAMD methods. Since the local nature of DICIs supports the use of small cluster models,17 we also apply the AIMS method to include 2 nonadiabatic effects in cluster models to analyze the NRR process involving several electronic states in semiconductor nanocrystals. In contrast to NAMD methods such as AIMS that are applicable to low-lying excited states, the tools for high-lying excited states including Rydberg and continuum states are less mature, due to the dense manifold of adiabatic electronic states. Conventional NAMD is not applicable since the computational cost grows immensely with the number of states involved and the need for considering the numerous nonadiabatic couplings between states. New developments for NAMD methods are required for these systems. Toward this goal, first we developed real-time time-dependent configuration interaction (TD-CI) methods to account for the electronic dynamics under different external fields. Then, in conjunction with the TD-CI method for computing the electronic wavefunction, we developed an Ehrenfest approach for ab initio nonadiabatic molecular dynamics. This approach avoids explicitly calculating each state in a dense manifold situation, in which the electronic wavefunction is expressed as a time- dependent CI vector. On the other hand, since the electronic wavefunction in TD-CI is expanded with Gaussian basis sets which are finite in the space, we developed a complex absorption potential (CAP) to eliminate unphysical effects that may appear from the finite boundary when an intense field is applied. The other advantage of the CAP is that the ionization rate from the intense field-matter interaction can be derived without explicitly simulating the continuum states. We endeavor to develop efficient methods that allow us to consider large systems. The TD-CI method is based on a direct CI scheme that can be greatly accelerated using GPUs. Thus, the computational cost of the most time-consuming step of Ehrenfest dynamics (electronic dynamics) is much reduced. In addition, analytical force calculations in the molecular dynamics portion can be completed in GPUs as well.18-19 For the CAP, we have identified an analytical expression of the potential that can be evaluated efficiently. 1.1 Description of Chapters Dangling bond defects in silicon nanocrystals (DB-SiNCs) have been known to work as nonradiative (NR) decay center. However, the microscopic detail is unclear. In Chapter 2, we studied the photophysics of DB-SiNCs via AIMD in conjunction with a complete active space 3 configuration interaction (CASCI) electronic structure method to elucidate the NR decay mechanism. The extension to a situation with multiple DB defects and the study of possible interactions between defects during the NR decay process is presented in Chapter 3. To include nonadiabatic effects and consider how the different charge states of the DB affect the photophysical properties of silicon clusters, we present an AIMS study of charged DB systems in Chapter 4. Toward theoretical studies of the photophysics of systems with dense manifolds of electronic states, method developments are required. We first present the development of an efficient time-dependent complete active space configuration interaction (TD-CASCI) approach for describing electronic dynamics in Chapter 5. An Ehrenfest dynamics method based on the TD-CASCI wavefunction (TD-CASCI-Ehrenfest) is presented in Chapter 6. We note that the extension of Ehrenfest dynamics to time-dependent multireference configuration interaction singles (TD-MRCIS-Ehrenfest) is straightforward if the force generated by a TD-MRCIS wavefunction can be evaluated efficiently. To consider the electron dynamics under intense laser fields, a molecule-centered CAP is developed in conjunction with the TD-MRCIS method as presented in Chapter 7. 4 REFERENCES 5 REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Warshel, A.; Karplus, M., Calculation of Ground and Excited-State Potential Surfaces of Conjugated Molecules .1. Formulation and Parametrization. J Am Chem Soc 1972, 94, 5612-5625. Warshel, A.; Levitt, M., Theoretical Studies of Enzymic Reactions - Dielectric, Electrostatic and Steric Stabilization of Carbonium-Ion in Reaction of Lysozyme. J Mol Biol 1976, 103, 227-249. Car, R.; Parrinello, M., Unified Approach for Molecular-Dynamics and Density- Functional Theory. Phys Rev Lett 1985, 55, 2471-2474. Ben-Nun, M.; Quenneville, J.; Martinez, T. J., Ab initio multiple spawning: Photochemistry from first principles quantum molecular dynamics. J Phys Chem A 2000, 104, 5161-5175. Levine, B. G.; Coe, J. D.; Virshup, A. M.; Martinez, T. J., Implementation of ab initio multiple spawning in the MOLPRO quantum chemistry package. Chem Phys 2008, 347, 3-16. Tully, J. C., Molecular-Dynamics with Electronic-Transitions. J Chem Phys 1990, 93, 1061-1071. Andrade, X.; Castro, A.; Zueco, D.; Alonso, J. L.; Echenique, P.; Falceto, F.; Rubio, A., Modified Ehrenfest Formalism for Efficient Large-Scale ab initio Molecular Dynamics. J Chem Theory Comput 2009, 5, 728-742. Hack, M. D.; Truhlar, D. G., Nonadiabatic trajectories at an exhibition. J Phys Chem A 2000, 104, 7917-7926. Li, X. S.; Tully, J. C.; Schlegel, H. B.; Frisch, M. J., Ab initio Ehrenfest dynamics. J Chem Phys 2005, 123, 084106. Stier, W.; Prezhdo, O. V., Nonadiabatic molecular dynamics simulation of light-induced, electron transfer from an anchored molecular electron donor to a semiconductor acceptor. J Phys Chem B 2002, 106, 8047-8054. Rego, L. G. C.; Batista, V. S., Quantum dynamics simulations of interfacial electron transfer in sensitized TiO2 semiconductors. J Am Chem Soc 2003, 125, 7989-7997. Dai, X. L.; Deng, Y. Z.; Peng, X. G.; Jin, Y. Z., Quantum-Dot Light-Emitting Diodes for Large-Area Displays: Towards the Dawn of Commercialization. Adv Mater 2017, 29, 1607022. 6 13. 14. 15. 16. 17. 18. 19. Kundu, S.; Patra, A., Nanoscale Strategies for Light Harvesting. Chem Rev 2017, 117, 712-757. Jaque, D.; Richard, C.; Viana, B.; Soga, K.; Liu, X. G.; Sole, J. G., Inorganic nanoparticles for optical bioimaging. Adv Opt Photonics 2016, 8, 1-103. Shu, Y. N.; Fales, B. S.; Levine, B. G., Defect-Induced Conical Intersections Promote Nonradiative Recombination. Nano Lett 2015, 15, 6247-6253. Shu, Y. N.; Fales, B. S.; Peng, W. T.; Levine, B. G., Understanding Nonradiative Recombination through Defect-Induced Conical Intersections. J Phys Chem Lett 2017, 8, 4091-4099. Levine, B. G.; Peng, W. T.; Esch, M. P., Locality of Conical Intersections in Semiconductor Nanomaterials. Phys Chem Chem Phys 2019, 21, 10870. Hohenstein, E. G.; Bouduban, M. E. F.; Song, C. C.; Luehr, N.; Ufimtsev, I. S.; Martinez, T. J., Analytic first derivatives of floating occupation molecular orbital-complete active space configuration interaction on graphical processing units. J Chem Phys 2015, 143, 014111. Fales, B. S.; Shu, Y. N.; Levine, B. G.; Hohenstein, E. G., Complete active space configuration interaction from state-averaged configuration interaction singles natural orbitals: Analytic first derivatives and derivative coupling vectors. J Chem Phys 2017, 147, 094104. 7 CHAPTER 2 DYNAMICS OF RECOMBINATION VIA CONICAL INTERSECTION IN A SEMICONDUCTOR NANOCRYSTAL Conical intersections (CIs) are points of degeneracy between the potential energy Reproduced with permission from “Dynamics of Recombination via Conical Intersection in a Semiconductor Nanocrystal”, W.-T. Peng, B. S. Fales, Y. Shu, B. G. Levine, Chem. Sci. 9, 681, 2018. Published by The Royal Society of Chemistry. 2.1 Introduction surfaces (PESs) of two or more adiabatic electronic states.1-6 It is now well established that molecules that undergo efficient, ultrafast nonradiative transitions between electronic states of the same spin often do so by passing through CIs connecting those states. CIs can therefore be thought of in analogy to transition states; transition states are representative of paths that connect reactant and product wells on the same PES, whereas CIs are representative of nonradiative pathways connecting different electronic states. Theoretical predictions of CIs have provided insights into many important photochemical phenomena, e.g photoisomerization,4, 7 photodissociation,8-10 vision,11, 12 and nonradiative decay of nucleic acids.13, 14 Identification of CIs is now a routine part of the computational molecular photochemistry toolbox.15 However, only recently and through the use of advanced computing technology16 has the role that CIs play in the nonradiative recombination of excitations in semiconductors come to light.17, 18 Here we investigate the role of CIs in the photophysics of silicon nanocrystals (SiNCs) with dangling bond defects. SiNCs and other low-dimensional silicon systems have received intense attention due to their unique photophysical properties. Unlike bulk silicon, which has an indirect band gap, low-dimensional silicon materials can efficiently emit visible light19 with a wavelength that can be tuned via quantum confinement20-23 or surface modification.24-27 This tunable emission enables their application in optoelectronic devices,28-30 biological imaging,31 and silicon lasers.32, 33 Under ambient conditions, however, SiNCs are prone to oxidize quickly upon exposure to O2 and/or H2O. Oxidation generates various defects on the surfaces of SiNCs, including dangling bonds (DBs) and some silicon oxide species (Si-O-Si bridges and Si-OH). 8 Among them, silicon dangling bond defects have been studied extensively.34-38 In general, there are two common types of dangling bond defects: Pb centers, which are dangling bonds on a three-coordinated silicon atom on the surface, and D centers, which are DB defects located in amorphous silicon.39 DB defects have been known to degrade the performance of silicon-based devices for both photovoltaic40, 41 and light emission38, 42-44 applications. Thus, it is well-known that silicon DB defects are nonradiative centers in SiNCs. It is also established that electronic movement at DB centers is strongly coupled to local vibrational motions.45, 46 The widely accepted mechanism for recombination involves the sequential capture of electron and hole into the non-bonding orbital of the Pb center. Each change in the oxidation state of the Pb center is accompanied by nuclear relaxation along a bending mode that maintains local C3v symmetry; reduction results in the defect silicon atom taking on a less planar structure (i.e. less sp2-like) while oxidation results in a more planar structure. The line of thinking that yields this mechanism arises from the assumption that each change in charge state is instantaneous, however. It neglects a) the fact that electrons and holes are strongly confined in SiNCs and therefore may interact strongly with one another and with the defect site before localization to the Pb center, and b) the interaction of electron and hole with the defect is not instantaneous and may, instead, involve complex electron-nuclear dynamics. The CI theory of recombination considers these interactions and dynamics explicitly, therefore it would be instructive to reinvestigate this recombination process from a CI point of view. In this work we a) investigate whether nonradiative dynamics at Pb centers can be attributed to CIs between the ground and first excited electronic states, and b) inform our fundamental physical intuition for recombination processes in general by analysis of the CIs associated with the dangling bond defect. To these ends we will bring to bear novel ab initio molecular dynamics (AIMD) tools capable of modeling the dynamics of electronically excited SiNCs as they approach CIs with the ground electronic state. In AIMD simulations, the nuclear dynamics are computed on PESs that are solved on the fly via electronic structure calculations. AIMD has recently become a tool of choice for the theoretical study of the photophysics of nanomaterials when either direct knowledge of excited state dynamics or extensive thermodynamic sampling are required, shedding light on various aspects of the charge carrier 9 dynamics of SiNCs.47-56 The current study is the first to apply an AIMD approach based on a multireference description of the electronic structure to a true nanocrystal (diameter 1.7 nm). The advantage of multireference electronic structure approaches such as the complete active space configuration interaction (CASCI) approach used here57, 58 is that they can accurately describe the PES in the vicinity of CIs between the ground and first excited electronic states. This is in contrast to single reference electronic structure methods—such as time-dependent density functional theory—which cannot accurately describe the potential energy surface near CIs involving the ground electronic state.59 The below study illustrates how applying CASCI- AIMD to model the dynamics of a semiconductor nanocluster from excitation to the neighborhood of a conical intersection can inform our fundamental understanding of nonradiative recombination. 2.2 Results and Discussion The methodological and computational specifics of our CASCI-AIMD simulations are presented in APPENDIX, but here we outline our study. We have performed a single CASCI- AIMD simulation for each of a series of five silicon clusters. Each cluster has a single Pb defect on the surface. The clusters (pictured in Figure 2.1) range in size from a single sila-adamantine unit (Si10H15) to a 1.7-nm particle (Si72H63). All simulations were run on the first excited electronic PES starting from a structure in the Franck-Condon region. (Structures are presented in APPENDIX.) All surface silicon atoms aside from the defect site were capped with hydrogen atoms. We emphasize that these simulations are performed in the Born-Oppenheimer approximation. Geometries of near-zero energy gap were drawn from the AIMD trajectories and the minimal energy CIs (MECIs; the local minima on the CI seam) were optimized. Because CASCI lacks dynamic electron correlation, complete active space second-order perturbation theory (CASPT2) calculations60 were performed on the smallest cluster to estimate errors in our CASCI energies. Static CASCI and CASCI-AIMD calculations were performed in the TeraChem software package,16, 61-63 which enables these demanding calculations through the use of graphics processing units—high performance computer processors designed for graphical applications 10 such as video games. CASPT2 calculations were performed in MolPro,64-68 coupled cluster calculations were performed in GAMESS,69-71 and conical intersection optimizations were performed with CIOpt.72 Through this work we prefer adiabatic state labels: D0 and D1 to indicate the ground and first excited spin doublet electronic states of the clusters, respectively. However, when useful and appropriate we also include term symbols 2A1 and 2E to reflect the approximate symmetry of the states with respect to the local C3v symmetry of the defect site. Figure 2.1. The Pb-containing silicon clusters studied in this work. (a) Si10H15 (sila-adamantane cluster), (b) Si22H27, (c) Si26H31, (d) Si47H49 (1.3-nm SiNC), and (e) Si72H63 (1.7-nm SiNC). One of the three symmetry-equivalent Si-Si bond lengths (RSi-Si) and bond angles (q) discussed herein are indicated in red in (a). The red arrows indicate the positions of dangling bond defects. 11 Table 2.1. The Si-Si bond lengths (Å), Si-Si-Si bond angles (θ as indicated in Figure 2.1a; in degrees), and D1 energies of differently sized silicon clusters at the Franck-Condon point (FC) and MECI. All energies are relative to the D0 minimum energy. Energies are computed at the CASCI level of theory as described in the APPENDIX. When available, CASPT2 energies at CASCI geometries are presented in parenthesis. Si10H15 Si22H27 Si26H31 Si47H49 Si72H63 FC MECI RSi-Si / Å q / Degree 2.43 2.45 2.94 2.46 2.46 2.89 2.43 2.51 2.83 2.43 2.54 2.81 2.44 2.48 2.82 123.8 123.7 124.0 122.6 121.9 123.5 122.0 123.2 123.6 122.4 123.6 123.6 122.4 122.9 124.3 D1 Energy / eV 2.67 (2.69) 2.52 2.52 2.44 2.38 RSi-Si / Å q / Degree 2.34 2.34 2.34 2.36 2.36 2.36 2.36 2.36 2.36 2.36 2.36 2.36 2.35 2.35 2.35 107.2 107.2 107.2 105.2 105.2 105.2 104.9 104.9 104.9 105.6 105.6 105.6 105.7 105.7 105.7 D1 Energy / eV 4.23 (3.53) 4.25 4.29 3.99 4.00 12 Figure 2.2. Orbitals representative of the D0→D1 (2A1→2E) transition of the Si72H63 system. (a) The singly occupied molecular orbital (SOMO) is the nonbonding (n) orbital of the Pb center, and (b) the highest doubly occupied orbital (HDOMO) has Si-Si σ bonding character in the vicinity of the Pb defect (σSi-Si). Note that this HDOMO is approximately degenerate due to the local symmetry of the defect. We show only one of the two nearly degenerate orbitals. The red arrows indicate the locations of the Pb center. 13 The D0 (2A1) minimum energy structures of our five clusters are presented in Figure 2.1. Bond lengths and bond angles around the Pb defect (as indicated in Figure 2.1a) at these structures are listed in the Table 2.1, labeled FC (Franck-Condon point). In these initial structures the three Si-Si bond lengths surrounding the DB and the associated bond angles are symmetric. The vertical excitation energies (Table 2.1) of the five clusters are very similar to one another, varying by only 0.3 eV, and there is not a strong trend with system size. This absence of quantum confinement effects suggests the locality of the excitation. This locality can be seen in the orbitals involved in the excitation (pictured in Figure 2.2). The excitation occurs from the highest doubly occupied molecular orbital (HDOMO) to the singly occupied molecular orbital (SOMO). The HDOMO is a Si-Si σ bonding orbital (σSi-Si; e), which is relatively local to the region of the defect, while the SOMO is the dangling bond itself: the nonbonding (n; a1) sp3 orbital of the defect silicon atom. Note that there is a nearly degenerate pair of HDOMOs due to the local C3v symmetry of the defect. We present only one of the two degenerate σSi-Si orbitals in Figure 2.2. (2E; σSi-Si→n) state. The time-dependent D1 and D0 potential energies, Si-Si bond lengths (RSi-Si as defined in Figure 2.1a), and bond angles (θ as illustrated in Figure 2.1a) of the 1.7-nm SiNC are presented in Figure 2.3. The dynamics of the smaller clusters were nearly identical; similar graphs for these cases are reported in Figures A1-A12. In all five cases the D1/D0 energy gap approaches zero (<0.1 eV) in the first 40-60 fs after excitation (Figure 2.3a). The vanishing energy gaps strongly suggest the existence of low-lying D1/D0 CIs.73 Now we consider the results of the excited state AIMD simulations initiated on the D1 14 Figure 2.3. (a) The potential energies of the D1 (red) and D0 (black) electronic states as a function of time from the D1 AIMD simulation of Si72H63. (b) The three Si-Si bond lengths (RSi- Si) adjacent to the Pb defect as a function of time from the same AIMD calculation. (c) The three Si-Si-Si angles (q, illustrated in Figure 2.1a) as a function of time from the same AIMD calculation. Each color represents one of the three symmetry equivalent bond lengths or angles in (b) and (c), respectively. 15 Figure 2.4. (a) Schematic illustration of the dynamics of nonradiative recombination of an excitation at a Pb center. The PESs are plotted as a function of an asymmetric stretching coordinate about the Pb center (illustrated along the x-axis with the dangling bond site represented by a filled circle and the three adjacent silicon atoms represented by open circles). Insets show the orbital occupations of D0 and D1 and relative orbital energies at the FC point (left) and MECI (right). The n and σSi-Si orbitals of the smallest (sila-adamantane) system are shown on the bottom left. 16 Using low-gap structures from the AIMD trajectories as starting guesses, MECIs were optimized in all five systems. The energies and structural details of these MECIs are reported in Table 2.1. Full structures are presented in APPENDIX. Comparing the structures of MECIs to the Franck-Condon points, one can see that both bond lengths and bond angles around the DB defects increase at the MECIs. The three Si-Si bonds surrounding the DB are asymmetrically stretched in all five clusters; one Si-Si bond grows longer (2.81-2.94 angstrom) than the other two (2.43-2.54 angstrom). Similar asymmetric stretching is observed in the AIMD simulations of all five systems (Figures 2.3b and Figure A5-A8) on the same 40-60 fs time scale on which the D1/D0 energy gap approaches zero. Much smaller changes in bond length are observed for Si-Si bonds not immediately adjacent to the Pb defect. Consistent with past work on dangling bond defects,45 symmetric bending motion is also observed to be important; the θ angles are observed to increase significantly both in the AIMD simulations (Figures 2.3c and Figure A9- A12) and in the optimized MECI structures (Table 2.1). Taken together, these calculations suggest that upon excitation of the lowest defect-localized excited state, the Pb defect moves ballistically to the CI region in 40-60 fs. It is also noteworthy that the trajectories remain in a region of small energy gap after 40-60 fs, suggesting that it may pass over the intersection multiple times, enabling efficient decay. The role that symmetry breaking plays in these nonradiative dynamics can be intuitively understood through analysis of the orbitals occupied during excitation. Figure 2.4 is a schematic diagram summarizing these dynamics. As described above, excitation to D1 involves the promotion of an electron from one of the degenerate σSi-Si orbitals to the n orbital. This reduces the Si-Si bond order, resulting in a lengthening of one of the Si-Si bonds (as observed in the AIMD simulations of all five clusters). In the locally C3v-symmetric FC structure the D1 (2E) state is doubly degenerate by symmetry, thus this symmetry-breaking motion is a Jahn-Teller distortion. The lengthening of a single Si-Si bond (moving from left to right along the D1 PES in Figure 2.4) brings the molecule towards the MECI structure. This symmetry breaking raises the energy of one of the σSi-Si orbitals into near degeneracy with the n orbital, bringing about a CI between the D0 and D1 states. That a Jahn-Teller distortion in D1 drives the molecules directly towards the D1/D0 CI provides a straightforward explanation for the ultrafast nonradiative process that follows creation of the defect-localized excitation. 17 The MECIs of the defective silicon clusters studied here are all accessible at energies in the visible range (2.38-2.67 eV above the ground state minimum structure; Table 2.1) and therefore capable of quenching visible light emission. The MECIs show a slight decrease in energy with increasing system size; the energy decreases from 2.67 eV for the small sila- adamantane cluster to 2.38 eV for the 1.7-nm SiNC. This small energy decrease of 0.29 eV is consistent with the localized nature of the defect-localized excited state. Calculations at the dynamically correlated CASPT2 level confirm the accuracy of the MECI energies predicted by CASCI, though vertical excitation energies are somewhat overestimated. All five clusters have D1 minimum energy structures distinct from the MECIs. In all cases this minimum is 0.05-0.06 eV below the MECI, compared to the 1.6-1.9 eV released during relaxation on the excited states. Thus the MECI is energetically accessible upon excitation. Energies and structures of D1 minima are presented in APPENDIX. The existence of defect-induced CIs with energies in the 2.4-2.7 eV range is consistent with several experimental observations of the photoluminescence (PL) of SiNCs after oxidation. The MECI energies suggest that the PL of SiNCs with emission energies larger than ~2.4-2.7 eV is likely to be quenched by the DB defects. Indeed the quantum yield of PL from oxidized SiNCs is observed to drop with increasing energy,74 and single particle experiments on oxidized SiNCs show no emissive particles with PL maxima above 2.5 eV.75 In addition, the PL lifetime of oxidized SiNCs decreases with increasing PL energy, and a dramatic decrease is observed in the 2.0-2.2 eV energy range.76 This decreasing PL lifetime suggests the existence of an efficient nonradiative recombination pathway accessible above these energies, consistent with our computed CI energies. Finally, as noted in our previous studies of oxygen-containing defects, the unusual size-insensitive orange (S-band) emission of oxidized SiNCs observed in ensemble PL measurements77 is consistent with the presence of CIs accessible in this energy range. We argue that the size-insensitivity of the observed ensemble emission arises not because the emission energy of individual oxidized SiNCs becomes insensitive to particle size, but instead because the rate of nonradiative recombination is strongly size sensitive, dramatically reducing the PL yields of smaller SiNCs with shorter wavelengths. This argument reconciles the observation of size-insensitive emission with PL lifetime, linewidth, and polarization measurements suggesting that the S-band arises from quantum-confined excitons.75, 78, 79 18 2.3 Conclusions Thus, we have elucidated the mechanism of nonradiative recombination via a dangling bond defect by application of AIMD simulations based on a multireference description of the electronic structure to SiNCs up to 1.7-nm in diameter. Within 40-60 fs after excitation of a defect-localized electronic excited state, a CI between the D0 and D1 states is accessed. This CI is accessible at energies in the 2.4-2.7 eV range, and thus is detrimental to visible PL, consistent with the fact that DBs are well-known nonradiative centers. The ultrafast recombination process is driven both by Jahn-Teller distortion in the D1 state and by totally symmetric bending of the DB center. The role that symmetry-breaking plays in this mechanism underlines the importance of treating coupled electron-nuclear dynamics in the study of recombination. 19 APPENDIX SUPPORTING INFORMATION FOR: DYNAMICS OF RECOMBINATION VIA CONICAL INTERSECTION IN A SEMICONDUCTOR NANOCRYSTAL Si10H15 Figure A1. The potential energies of the first excited (red line) and ground states (black line) change with the time following the first excited state trajectory from an AIMD simulation where the force comes from the first excited state in Si10H15. 20 Si22H27 Figure A2. The potential energies of the first excited (red line) and ground states (black line) change with the time following the first excited state trajectory from an AIMD simulation where the force comes from the first excited state in Si22H27. 21 Si26H31 Figure A3. The potential energies of the first excited (red line) and ground states (black line) change with the time following the first excited state trajectory from an AIMD simulation where the force comes from the first excited state in Si26H31. 22 Si47H49 Figure A4. The potential energies of the first excited (red line) and ground states (black line) change with the time following the first excited state trajectory from an AIMD simulation where the force comes from the first excited state in Si47H49. 23 Si10H15 Figure A5. The three Si-Si bond lengths (RSi-Si) of Si10H15 change with time from the AIMD calculation. Each color represents one Si-Si bond length. 24 Si22H27 Figure A6. The three Si-Si bond lengths (RSi-Si) of Si22H27 change with time from the AIMD calculation. Each color represents one Si-Si bond length. 25 Si26H31 Figure A7. The three Si-Si bond lengths (RSi-Si) of Si26H31 change with time from the AIMD calculation. Each color represents one Si-Si bond length. 26 Si47H49 Figure A8. The three Si-Si bond lengths (RSi-Si) of Si47H49 change with time from the AIMD calculation. Each color represents one Si-Si bond length. 27 Ground State Minimum Energy Geometries (in Angstrom) Si10H15 Si -0.5174205979 -0.8348349814 -0.7170299444 Si 1.8349944728 -0.8348264311 -0.7034572298 Si -1.2975571211 1.2346984879 0.0845084533 Si -1.3052009638 -2.5548459701 0.6728075910 Si 2.6315781484 -0.4801811161 1.4813624363 Si -0.5177122134 1.6008056015 2.2734868246 Si -0.4846934236 -2.1343753473 2.8284603389 Si 1.8346837138 1.5893845329 2.2660259352 Si -1.3047721410 -0.1085476518 3.6768972784 Si 1.8582635743 -2.1980831303 2.8819602955 H -0.9982771294 -1.0362895976 -2.0995241051 H 2.3315062053 0.2372717958 -1.5870674988 H 2.3323357458 -2.1296624310 -1.2075209264 H -0.8158399009 2.3161803939 -0.7958276067 H -2.7733368421 1.2426787832 0.0773677090 H -0.8177833037 -3.8633496961 0.1999671071 H -2.7792581411 -2.5648306177 0.6939505559 H -0.9987009985 2.9137881883 2.7507928354 H 4.1089148884 -0.4613933574 1.4662466180 H 2.3329439541 -3.5078878963 2.3998373351 H 2.3309456347 -1.9888831636 4.2626165112 H 2.3362682403 1.8278307131 3.6333053374 H 2.3268252019 2.6684318023 1.3883697280 H -2.7788325440 -0.1308235029 3.6830320283 H -0.8167544595 0.0893645920 5.0538623924 Si22H27 Si -0.3972569683 -0.5830501301 -0.8321377189 Si 1.9547937034 -0.5351843517 -0.9261372969 Si -1.2673419317 1.4207876463 0.0460120336 Si -1.0925993707 -2.3994500329 0.5183564137 Si 2.7958091258 -0.1644944637 1.2417357013 Si -0.3752694178 1.7597885391 2.1984031749 Si -0.1842263064 -1.9641860638 2.6552086261 Si 1.9772606786 1.8458715325 2.1539087435 Si -1.0702013941 0.0005215740 3.6226425378 Si 2.1782576455 -1.9705589033 2.6428534318 Si 2.7795135709 -2.5536488064 -1.7967200647 Si -3.6115798856 1.3295809584 0.1346450351 Si -0.2244978466 -4.4218911388 -0.3137364596 Si -3.4407387476 -2.4674922803 0.6496570015 Si 2.9663271290 -4.0003233853 1.7513439153 28 Si 2.9839993170 -1.6272021648 4.8265803683 Si 2.8285758589 2.1711593733 4.3176421018 Si -3.4187750187 -0.1309944319 3.6768518851 Si -0.1810162833 0.2952618485 5.7809228431 Si -4.2716600504 -0.4512847041 1.5148443519 Si 2.1615665176 0.3909193270 5.6953479926 Si 2.1178154977 -4.3296505758 -0.4110542982 H -0.9337269855 -0.7896112302 -2.1955426474 H 2.3691472072 0.5899615504 -1.7909698734 H -0.8415282220 2.5392852707 -0.8219118306 H -0.8977627692 3.0310669206 2.7461811431 H 4.2733117579 -0.1072635696 1.1868895734 H 2.3912495084 2.9635134242 1.2792119398 H -3.9472753343 1.1236982662 4.2472869753 H -3.8212391577 -1.2485943134 4.5520456397 H -3.8462215043 -3.5953227101 1.5100842802 H -3.9802870975 -2.6951354357 -0.7054811742 H -4.1531920071 1.1338546260 -1.2239303759 H -4.1379983402 2.5986143590 0.6729439203 H -5.7471969680 -0.5132427112 1.5733383063 H 4.3019464522 2.2360897950 4.2671192792 H 2.3201495689 3.4431803692 4.8659126820 H -0.6127148144 -0.8236447730 6.6402960867 H -0.7170649358 1.5506740166 6.3428263104 H 2.6949058479 0.5920344669 7.0590176458 H 4.4592270578 -1.5942712095 4.7840164407 H 2.5658146727 -2.7557007535 5.6801035554 H 2.5562723179 -5.1101149082 2.6328987217 H 4.4413386301 -3.9540895487 1.7126277108 H -0.7772667381 -4.6525748767 -1.6630173745 H -0.6526344611 -5.5288472442 0.5627333459 H 2.2482628661 -2.7601367084 -3.1579086455 H 2.6373504254 -5.6026008335 -0.9535717420 H 4.2524071997 -2.5002016047 -1.8694102124 Si26H31 Si -0.4398719991 -0.7667028184 -0.7199895311 Si 1.9169358250 -0.7102511493 -0.8312161401 Si -1.2842117817 1.2742724470 0.1134434767 Si -1.1519899237 -2.5663199225 0.6420641284 Si 2.7467564604 -0.3525062221 1.3493626084 Si -0.4030484240 1.6007804485 2.2788258343 Si -0.2411305565 -2.1134912524 2.7735430173 Si 1.9549530372 1.6951550095 2.2168084688 Si -1.1143735472 -0.1348854308 3.7214863010 Si 2.1205673899 -2.1406940490 2.7673489807 29 Si 2.5808280743 1.0741374168 -2.2171839705 Si 2.7206385134 -2.7580946669 -1.6618975200 Si -0.5758336726 3.0366740558 -1.2791141000 Si -3.6319488280 1.1820443817 0.2162064221 Si -0.2954349561 -4.6001904717 -0.1743329304 Si -3.5014485152 -2.6102368358 0.7637577222 Si 2.8953657268 -4.1835792702 1.8939246403 Si 2.9336981691 -1.7727609553 4.9446198572 Si 2.7933177311 2.0180727279 4.3902694973 Si 2.6249274101 3.4471592811 0.7924887629 Si -3.4640844554 -0.2442985623 3.7712029892 Si -0.2241290923 0.1787678782 5.8764729334 Si -4.3127756716 -0.5794774241 1.6108681831 Si 2.1180184114 0.2574945304 5.7888243180 Si 2.0465897874 -4.5213280144 -0.2660075568 Si 1.7682817395 3.1040968674 -1.3651261280 H -0.9710306912 -0.9743469029 -2.0862548834 H -0.9125347485 2.8871119676 2.8058286964 H 4.2250326162 -0.2987073289 1.2882133424 H -3.9815046875 1.0217478641 4.3266353671 H -3.8782921805 -1.3483119622 4.6582253502 H -3.9239118392 -3.7293332623 1.6276877574 H -4.0350424388 -2.8398208301 -0.5934380726 H -4.1745620243 0.9810230975 -1.1417621168 H -4.1499058706 2.4610358238 0.7401823545 H -5.7892884662 -0.6267279777 1.6662030424 H 4.2673825755 2.0803816423 4.3433956334 H 2.2896985737 3.2989120615 4.9235788132 H -0.6650489031 -0.9268836631 6.7483800987 H -0.7519640874 1.4446426595 6.4226048641 H 2.6526019450 0.4719867603 7.1502497667 H 4.4087895903 -1.7412117524 4.8934142895 H 2.5199654199 -2.8906287069 5.8143326491 H 2.4749943262 -5.2824727349 2.7844520881 H 4.3707110737 -4.1490727793 1.8578148437 H -0.8470104930 -4.8355329683 -1.5231914466 H -0.7338434828 -5.6984209044 0.7083249666 H 2.1950252878 -2.9729417343 -3.0243594431 H 2.5597543707 -5.8007898857 -0.8000298153 H 4.1941915726 -2.7136854510 -1.7327245565 H 2.0535348997 0.8514849939 -3.5776806851 H 4.0546489524 1.1107737335 -2.2886280827 H 4.0994803269 3.4969360510 0.7470832498 H 2.1324462611 4.7286577853 1.3346293894 H -1.0982839833 4.3110636678 -0.7480806102 H -1.1253410139 2.8371986564 -2.6344226252 H 2.1962402666 4.2121840809 -2.2460944904 30 Si47H49 Si -0.4818420248 -0.8450923780 -0.7475587158 Si 1.8949966387 -0.7996570386 -0.8162480058 Si -1.3303573489 1.1989091972 0.1229132491 Si -1.1755107650 -2.6314669539 0.6580713694 Si 2.7614439979 -0.4119190912 1.3642498020 Si -0.4519139512 1.5792950376 2.3000190220 Si -0.2666834627 -2.1565037493 2.7818217331 Si 1.9249267731 1.6337651392 2.2426335061 Si -1.1507491115 -0.1861076156 3.7293841351 Si 2.0914212410 -2.1910813469 2.7903429621 Si 2.5944697213 0.9792177906 -2.2208695306 Si 2.6807903266 -2.8867062136 -1.6183781784 Si -0.6396160161 2.9597268145 -1.3085314684 Si -3.6934319714 1.0790379800 0.2576922038 Si -0.3592695126 -4.6827916917 -0.1669116664 Si -3.5194482202 -2.7278756784 0.8443536220 Si 2.8187629272 -4.2757194561 1.9759099962 Si 2.9301101245 -1.8520950816 4.9666858546 Si 2.7611918722 1.9254265007 4.4408754043 Si 2.5933201778 3.4148528825 0.8257341579 Si -3.5040148649 -0.2787133612 3.8207359208 Si -0.2819005257 0.0623623586 5.9022623346 Si -4.3339422246 -0.6953247292 1.6627258444 Si 2.0523344301 0.1543695273 5.8169624287 Si 1.9890879841 -4.6167839264 -0.1827316245 Si 1.7198854433 2.9780875672 -1.3223194444 Si -1.2293428774 -1.2217303113 -2.9608951612 Si -1.3483726738 3.6343239864 3.0567823960 Si 5.1213861172 -0.2397435357 1.3416418125 Si -4.3098523751 1.7613445134 4.6729575549 Si -4.6890509186 3.0164105966 1.1418196883 Si 5.1086338144 1.9596228770 4.5621460617 Si 5.2805681378 -1.7903940128 4.8715997533 Si -1.2243641113 -5.0282761474 -2.3278736369 Si 1.8776437223 -3.4401584042 -3.7572951465 Si 1.7262588436 0.8642730044 -4.4080798826 Si 4.9482378499 1.0368641185 -2.2746888332 Si 4.9242128824 3.5630397462 0.5161719393 Si 1.7495958374 5.4389799618 1.6845076927 Si -1.2604001518 5.1026231602 -0.5512120553 Si -1.4934024518 2.5742142430 -3.4692906229 Si -3.6959958783 3.4970237055 3.2138417689 Si -0.4661705185 -3.3110225378 -3.7401762309 Si 5.9458367901 0.0225087435 3.5339836191 Si -0.6022266357 0.5836609780 -4.3350375041 31 Si -0.5963768059 5.3900298442 1.6809182656 Si 5.7841121117 1.4765960150 -0.1266699457 H -3.8859362855 -1.3863572271 4.7222133972 H -3.8751739213 -3.8373381830 1.7492600550 H -4.1024051055 -2.9992501783 -0.4831471239 H -4.2257575946 0.8246175845 -1.0975283260 H -5.8114485808 -0.7338762132 1.7199904631 H 2.2201947541 3.1900623677 4.9819150730 H -0.7316548092 -1.0779557287 6.7230349423 H -0.8074568981 1.3055273298 6.4973343607 H 2.5953744091 0.3739229220 7.1751601098 H 2.4903655432 -2.9835896413 5.8103323133 H 2.3456272198 -5.3308164031 2.8918611470 H 4.2934703939 -4.3011898528 1.9519397105 H -0.8198921900 -5.7579406416 0.7371122698 H 2.4978651041 -5.8980430385 -0.7187408229 H 4.1583325695 -2.8521895154 -1.6392676432 H 2.1484598895 4.0858135310 -2.2077360361 H 5.8188261508 -1.6159927180 6.2346614169 H 5.7881138508 -3.0612517576 4.3228916287 H 7.4217518662 0.0823595037 3.4814031858 H 5.6723371414 3.1793836881 3.9643513991 H 5.4854232127 1.9284875125 5.9895040901 H 7.2611244374 1.5332093712 -0.1601782368 H 5.1514627665 4.5188882624 -0.5866263415 H 5.5993602751 4.0868526713 1.7143465869 H 2.2204062097 6.5560069793 0.8417857946 H 2.2662621744 5.6382663286 3.0518565379 H -1.1300152265 6.6670161657 2.2009631224 H -2.7008934004 5.3455417253 -0.7285938053 H -0.5298639645 6.0806816764 -1.3827483960 H 5.3854025508 2.1001812319 -3.2007537064 H 5.4533967566 -0.2546809505 -2.7779471062 H 2.0179429905 2.1596918784 -5.0549396069 H 2.3796383149 -0.1920363209 -5.1973426781 H -1.1206506992 0.3484369495 -5.6995239532 H -2.9654398079 2.5027405608 -3.4032652102 H -1.1211729008 3.7027132892 -4.3453394285 H -2.7078395095 -1.2758026474 -2.9411079642 H -0.9888588572 -3.5273706027 -5.1057061256 H -0.7499465958 -6.3283618900 -2.8404151937 H -2.6976049557 -5.0513664116 -2.2743048301 H 5.6675240594 -1.5113586086 0.8181947097 H 2.4616213138 -2.5854439499 -4.8020046494 H 2.2644308634 -4.8378826380 -4.0348252662 H -4.2125441458 4.7821908724 3.7297934077 H -5.7812647245 1.7014364729 4.7701212832 32 H -3.7568532536 1.9887112059 6.0206386615 H -6.1186061319 2.7239426360 1.3678727238 H -4.5935550529 4.1551233989 0.2159800954 H -0.8325825741 3.8833259734 4.4210475184 Si72H63 Si 0.0142241203 -0.0252846567 0.0768646407 Si 0.0346997444 -0.0401227583 2.4414756915 Si 2.2575291124 0.0123854838 -0.7170080344 Si 2.2269845101 0.0275667081 -3.0942123496 Si -1.1397905465 -1.9564352762 -0.6992361571 Si -1.1169672063 1.9175074904 -0.7219432065 Si -1.1576391126 -1.9337091445 -3.0764343523 Si -1.1324828590 1.9214860268 -3.1022953316 Si -0.0089272882 -3.9048106254 0.0565840655 Si 3.3922990567 -1.9333970945 0.0397311433 Si 0.1940503141 -3.9783251531 2.4053931311 Si 3.3725782305 -2.1323839650 2.3904206130 Si 1.1645852039 -1.9837872048 3.1851727340 Si 2.1719975613 -3.7553835232 -0.8169293028 Si -2.2920927412 0.0014049753 -3.8884208992 Si 1.1039942636 1.9690430258 -3.9065620857 Si 1.0810055341 -1.9030507914 -3.8870353468 Si -2.2562654975 3.8547204996 -3.8945861082 Si -2.2523561113 3.8298838715 -6.2620365959 Si -3.2746325565 1.9127465058 -7.1625716490 Si -0.0873905138 3.7590490879 -7.1778926271 Si -2.2720068559 -0.0211748796 -6.2610696976 Si 1.0957859215 1.9290465522 -6.2793755974 Si -0.0320338637 -0.0131251069 -6.9886978399 Si 2.2152395250 -3.8524750271 -3.1670272271 Si 1.0921088668 -1.9443224955 -6.2520793138 Si 4.4427814737 -3.8277688209 -3.9601626326 Si 1.0801913498 -5.7764320480 -3.9430256700 Si -0.0277910630 -3.8859502178 -7.0011095015 Si 3.3277638978 -1.9428613762 -7.0186636634 Si 1.1244517982 -5.8027701264 -6.2915397414 Si 4.4240250368 -3.8917365542 -6.3083633081 Si -2.1977898930 -0.0016194395 3.2279762350 Si 1.1169638634 1.9200254425 3.2102560129 Si -3.3323125511 1.9406687411 0.0998412176 Si -0.0299565104 3.8550378699 0.0823035009 Si -4.4656082230 3.8534117935 -3.0446885343 Si -1.1513717485 5.7750655685 -3.0612606269 Si -1.1762957101 5.7675193883 -0.7060102251 Si -0.0357924735 3.8276198904 2.4463205355 33 Si -3.2877847392 1.9422087721 2.4634526806 Si -4.4280516160 3.8823805864 -0.6896083457 Si -2.2404689841 3.8899128591 3.2469016364 Si -3.3632035610 5.8002105993 0.1424363872 Si -3.3561099228 5.8082547919 2.4877000092 Si -3.2346696447 -1.9598306240 -7.1869488927 Si -4.5119858403 0.0107269519 -3.0502515018 Si -2.2855978847 -3.8606541916 -3.8567065167 Si -2.2513791794 -3.8642965448 -6.2236791040 Si -0.8955075372 -5.9242935790 -0.7746374599 Si -1.1208204174 -5.7807103101 -3.1080003619 Si -3.3716697116 -1.9319512400 0.1174679077 Si -4.4188517815 0.0157331836 -0.6965260507 Si -5.7050631671 -1.9219747662 -3.6746701917 Si -4.6082953956 -3.7695972942 -0.6839547211 Si -4.5055505891 -3.8415351567 -3.0323898522 Si 3.3305469068 -5.8146847986 -7.0861842545 Si 3.3516360758 1.9656919225 0.0810170439 Si 2.2048992291 3.9044161170 -3.0867247941 Si 2.1740340752 3.8374616935 -0.7321670002 Si 5.5467840461 -1.9157290608 -3.1431640868 Si 4.4204942138 0.0008111951 -6.2597663805 Si 4.4530841503 0.0452328946 -3.8933023715 Si 5.5530377927 2.1292009520 -0.7417784001 Si 4.4668496462 3.9741396258 -3.7395437411 Si 5.5449466650 1.9867753683 -3.0894829012 Si 3.3181220300 2.1616378953 2.4267966383 Si -3.5049464308 -1.7968665710 2.4637278514 Si -5.7141847553 1.9520801125 -3.6304626877 Si 1.1144103948 5.9111350362 -3.6642925358 Si 5.5792623203 -2.1661673277 -0.8079868935 Si 3.2457820888 1.7901203204 -7.2287832961 H 3.3294258772 -5.8252786239 -8.5612795450 H 4.0307907851 -7.0197495786 -6.6024741603 H 0.4050367645 -7.0063137169 -6.7596789923 H 5.8219188057 -3.8683799275 -6.7884697769 H 5.1240361038 -5.0278012291 -3.4280182341 H 3.2963233770 -1.9090342418 -8.4975059835 H 1.7881197862 -6.9612758834 -3.4116902747 H -0.0536240787 -3.8489242016 -8.4800230480 H 5.8167341193 -0.0151380906 -6.7475452777 H 6.9331944217 -1.9182992981 -3.6588470167 H -1.8125536084 -6.9866563461 -3.6131456451 H -2.9350844591 -5.0857433756 -6.7016869713 H -2.1557407259 -6.2670192120 -0.0989349870 H 0.0871465362 -6.9843394534 -0.4760635219 H 6.0213639124 -3.5406422807 -0.5015065070 34 H 6.5032141846 -1.2316573450 -0.1481688338 H 6.9356944424 1.9835493131 -3.5927156454 H 3.9780637396 3.0617076432 -7.1306394516 H 3.0675395064 1.4785931668 -8.6614519288 H -0.0411831528 -0.0134110108 -8.4687141404 H -4.6999164117 -1.9622589549 -7.0598473957 H -2.9039391886 -1.9676912736 -8.6263084957 H -5.1969421500 -5.0532431888 -3.5235604103 H -4.7423551943 5.8140120304 2.9928249244 H -2.6674336406 7.0156105574 2.9826317342 H -4.0607410847 6.9988572255 -0.3698020210 H -2.2171476938 3.8637572827 4.7248940991 H 0.6998669372 5.0226699930 2.9150575824 H -0.4382281706 6.9596104243 -0.2336173924 H -5.8260067600 3.8380805016 -0.2071350442 H -4.6871330116 1.8988753859 2.9420646976 H -5.1509848247 5.0727391089 -3.5265212821 H -1.8734201042 6.9726348975 -3.5439614291 H 1.1062881603 1.9122852253 4.6893452109 H -2.1725010493 0.0108612683 4.7068002061 H -5.8108147169 0.0085511911 -0.1934183682 H 1.1762166089 -1.9931735808 4.6641121260 H 2.8760695207 5.0435095895 -0.2390188285 H -2.9545763086 5.0395554241 -6.7433972762 H -6.9459872919 1.9565329808 -2.8154325006 H -6.1104039164 1.9502023130 -5.0469435022 H -3.1458340514 -3.0703875536 3.1066082790 H -4.9125327652 -1.5013802083 2.7997225986 H 3.7592771113 3.5334495751 2.7510222232 H 4.2523940966 1.2240063210 3.0689281187 H 1.3031452370 6.2513511597 -5.0826892336 H 1.7300593092 6.9826652672 -2.8553742254 H -6.9727762757 -1.9227277101 -2.9166639727 H -6.0367596186 -1.9134730067 -5.1072716468 H -6.0192835367 -3.5491531073 -0.3072968675 H -4.1663172785 -5.0244648727 -0.0568752474 H -1.1082197360 -4.2653518755 3.0242925262 H 1.1108720070 -5.0877008497 2.7332951953 H 4.2684436566 -1.1365254850 2.9963154923 H 3.8878706450 -3.4748230174 2.7238099150 H 6.0576241624 3.4696070179 -0.3803899101 H 6.4348812122 1.1326402387 -0.1150591964 H 4.6026455996 4.2472009580 -5.1780569907 H 5.1086291622 5.0838120453 -3.0059758344 H -0.2259383858 3.5249706989 -8.6295609701 H -4.7322154567 1.8993213135 -6.9687363699 H 0.6525473782 5.0161682626 -6.9926420676 35 H -3.0110230539 1.9082580693 -8.6158982183 Minimum Energy Conical Intersection Geometries (in Angstrom) Si10H15 C -0.66500 0.00000 0.00000 C 0.66500 0.00000 0.00000 H -1.23780 0.92380 0.00000 H -1.23780 -0.92380 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-3.4713343033 -2.2243944500 H -1.1738164171 0.0618354369 -4.0103226054 H 2.2233553657 1.4609977905 -3.6174114922 H -1.0820628018 3.4851591718 -2.5921224151 H -1.1311402304 3.7742085309 0.9937682765 H 4.0018056655 0.6405725429 -0.5772747866 H 2.1293714308 3.7505192376 -0.5264611276 H -4.1254793892 2.0894975839 2.4583237127 H -4.1668349098 -0.2920070314 2.7764718682 H -3.8383227001 -2.5917138873 0.0476790966 36 H -4.1193916952 -2.0109926956 -2.2573476752 H -4.4356372698 2.0805794118 -3.0109482601 H -4.3364463599 3.5202373987 -1.1044861996 H -5.9877693936 0.3535617047 -0.2719996540 H 4.0030027383 3.1287090971 2.5078782075 H 1.9425424050 4.2228044921 3.0585507558 H -0.9253699345 0.0416400900 4.8396980552 H -0.9556362606 2.4037068863 4.4380001433 H 2.4287965302 1.3521745496 5.2300637492 H 4.2176666815 -0.7304585330 2.8561902564 H 2.4024285512 -1.9715876941 3.8136684006 H 2.5335709937 -4.3015705297 0.7756058785 H 4.2865059047 -2.9763295786 -0.1977124321 H -1.1439095982 -3.8178919403 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2.7401201315 0.2670210574 Si 4.4570073061 1.6408014720 3.7311366905 Si 4.4377669585 -2.1416630484 3.9257200319 Si -1.9192139135 -5.2443957890 -3.1752871020 Si 1.1823430219 -3.6304848912 -4.6768476540 Si 1.0536428443 0.6843412662 -5.2949068780 Si 4.3173762793 0.8901516243 -3.1395463639 Si 4.2763493778 3.3357630484 -0.2907762346 Si 1.0888224525 5.2398163413 0.7928676092 Si -1.9335359852 4.8526776727 -1.3982258975 Si -2.1844718619 2.3162024613 -4.2934647126 Si -4.3399731709 3.2795803135 2.3258884482 Si -1.1598151429 -3.5402635970 -4.5974557428 Si 5.2293212478 -0.3104764429 2.6811935871 Si -1.2672353588 0.3455136613 -5.1679846886 Si -1.2531340657 5.1447482067 0.8263630714 Si 5.1283771411 1.2546600126 -0.9636777148 H -4.5410739400 -1.5527447654 3.9414566464 H -4.4132789808 -4.1170622451 0.8835354701 H -4.5443977790 -3.1928530189 -1.3233510096 H -4.8489659278 0.5087864537 -1.9299454459 H -6.4110597236 -1.0611176823 0.8637645288 H 1.6019677579 2.9674342594 4.0976149243 H -1.4217452843 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-5.8757488699 -6.2611360495 H 4.0575605465 -6.9974131540 -4.2653154696 H 0.4228055487 -6.9830750383 -4.4318051355 H 5.8422355555 -3.8476807973 -4.5691446044 H 5.2102235349 -4.9867614124 -1.1908601692 H 3.2799795304 -1.8593258373 -6.2154633257 H 1.8291913638 -6.9283667253 -1.1067286987 H -0.0154167020 -3.8158962458 -6.1729377676 H 5.8327736596 0.0170082342 -4.4530938889 H 6.9723057679 -1.8441731863 -1.3124276253 H -1.7585878451 -6.9531071087 -1.2812313213 H -2.9032559171 -5.0686477947 -4.3807009684 H -2.1844335052 -6.0562979790 2.2059284518 H -0.0150511567 -6.9901711361 1.9069712129 H 5.8870691918 -3.5932552577 1.7485806756 H 6.3022436936 -1.2959688665 2.2096049735 H 6.9452124135 2.0191709680 -1.3015674417 H 4.0005769785 3.1068692531 -4.7317832803 H 3.1327486158 1.6046783296 -6.3575156110 H -0.0016345125 -0.0220192205 -6.1575893667 H -4.6671523728 -1.9458092559 -4.7508424050 H -2.8772262187 -1.9489619377 -6.3169131662 H -5.1555624499 -5.0268163064 -1.2509614564 H -4.7102404438 5.8521220266 5.3196531593 H -2.6175022376 7.0185907091 5.2819160604 H -4.0566770432 7.0082452598 1.9460349560 H -2.2052634557 3.8626155354 7.0317085293 H 0.7059525339 5.0379451000 5.2204600205 H -0.4382938287 6.9799115442 2.0708599351 H -5.8250762583 3.8570148611 2.0895292391 42 H -4.6828899720 1.9096361080 5.2445886612 H -5.1430227770 5.0896711229 -1.2222828390 H -1.8732032858 6.9925601329 -1.2366616102 H 1.1242799976 1.9194311872 6.9886279421 H -2.1651284030 0.0089185832 7.0020961362 H -5.7634210639 -0.0330270519 2.1141907802 H 1.2504730830 -1.9317926481 6.9711231104 H 2.8891912936 5.0488410902 2.0776013812 H -2.9546652615 5.0520111482 -4.4453260526 H -6.8950611115 1.9573002189 -0.4079777109 H -6.1781871239 1.9567266320 -2.6774575875 H -3.0851761039 -3.0750013580 5.4004318088 H -4.8856594205 -1.5534437255 5.0779006530 H 3.9015265915 3.3834234507 5.1201352677 H 4.1801146605 1.0226093507 5.3116647741 H 1.3022243533 6.2609355322 -2.7832235299 H 1.7278151306 7.0146696742 -0.5620914299 H -6.9431511311 -1.8982149961 -0.6415467730 H -5.9946246761 -1.8668044024 -2.8305640124 H -6.0076561482 -3.4240960396 1.9446632936 H -4.2523785059 -5.0110984204 2.2369987743 H -1.1261220549 -4.2683107540 5.3193262742 H 1.1095911072 -5.0671351093 5.1264859133 H 4.3286681026 -1.3456454741 5.1233086827 H 3.7317896234 -3.6251365369 4.8298737253 H 6.0826344429 3.4463483278 1.9415354394 H 6.4345468528 1.0977060015 2.1602235847 H 4.6428605522 4.3511312951 -2.8268045320 H 5.1054058241 5.1007843207 -0.6119526705 H -0.2471736554 3.5114448810 -6.3440957393 H -4.7330203240 1.8914639066 -4.5237388865 H 0.6437048613 5.0321223019 -4.7344943837 H -3.1272290280 1.9148404136 -6.2833083349 43 REFERENCES 44 REFERENCES J. 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Lett., 2005, 94, 087405. 51 CHAPTER 3 AB INITIO MOLECULAR DYNAMICS STUDY OF THE INTERACTION BETWEEN DEFECTS DURING NONRADIATIVE RECOMBINATION Reproduced with permission from “Ab Initio Molecular Dynamics Study of the Interaction Between Defects During Nonradiative Recombination”, W.-T. Peng, B. G. Levine, J. Phys. Chem. C., 123, 16588, 2019. Copyright 2019 American Chemical Society. 3.1. Introduction As a result of quantum confinement effects, low-dimensional silicon materials such as porous silicon and silicon nanocrystals (SiNCs) have desirable optoelectronic properties, including efficient emission of visible light.1-3 As such, SiNCs have gained attention as potential materials for light emitting diodes,4-7 solar cells,8-9 and semiconductor lasers.10-11 Though they have desirable optical properties, hydrogen-terminated SiNCs are prone to oxidation under ambient conditions, dramatically changing both the color and yield of photoluminescence (PL).12-13 Several defects arising during oxidation have been experimentally characterized, including dangling bonds (DB), silanol groups (Si-OH), bridging oxygen atoms, and silicon epoxide rings.14-18 DB defects, in particular, have been determined to be important nonradiative (NR) decay centers.14-15 The certainty of this determination was possible because DB defects are easily observable in electron paramagnetic resonance experiments, and this signal shows a strong inverse correlation with PL yields. However, due to the complex heterogeneous nature of the interface and the lack of experimental techniques that can unambiguously connect atomistic structure to function in nanomaterials, the role of many other defects remains unclear, as does the detailed NR decay mechanism in real nanoparticles. Theoretical and computational methods offer the opportunity to gain microscopic insights into the decay process that can guide future experiments and material design. Theoretically, nonradiative recombination is most often viewed through the lens of electronic band theory. In this theory, the exciton is generated by creation of an electron and a hole in the conduction and valence bands, respectively. Most often interactions between charge carriers and with the lattice are neglected, though techniques for incorporation of both of these 52 effects exist and have been applied in the study of silicon nanomaterials.19-22 Neglect of these interactions is a good approximations when one considers spatially delocalized excitations (e.g. quantum-confined excitons). However, in band theory recombination involving defects (Shockley-Read-Hall or SRH recombiation23-24) requires the localization of charge carriers to mid- gap states centered on the defect site. One might expect interactions between charge carriers and with the lattice to be stronger in these more localized states, requiring a many-body treatment of the electronic structure and an explicit treatment of local geometric distortions to obtain an accurate picture of recombination dynamics. Indeed the importance of lattice relaxation during NR recombination at DB defects has been known for several decades.19 Recently, our group has proposed the notion of defect induced conical intersections (DICIs): points of degeneracy between the potential energy surfaces (PESs) of materials that arise through local distortions of the nuclear and electronic structures at defect sites.25-27 Conical intersections (CIs) are a textbook concept in molecular photochemistry, where they are known to facilitate ultrafast transitions between electronic states.28-31 In the context of silicon PL, identification of DICIs has yielded insights into the effect of surface chemistry on both yield and color of luminescence. Our group has identified intersections associated with various silicon surface defects, including silicon epoxide rings,25, 32-33 silicon-oxygen double bonds,25, 34 and hypervalent silyl defects.35 Most recently, we identified DICIs associated with the NR decay process occurring in SiNCs with surface DB defects.36 We found that upon excitation to the first excited electronic state, energetically favorable geometric distortions bring the system near a CI with the ground state, resulting in an ultrafast NR decay process. The predicted geometric relaxations observed include both a long-known symmetric pyramidalization motion (sp2↔sp3 hybridization) of the DB site19 and a Jahn-Teller distortion arising from the approximate C3v symmetry of the DB site.36 The theoretical investigation of DICIs associated with DB and epoxide defects shed light on several important experimental findings. Most notably, the existence of low-lying DICIs in defective SiNCs reconciles two seemingly contradictory experimental observations regarding the slow- (S-) band PL in oxidized SiNCs; the observed PL energy is insensitive to particle size,12 although various evidence suggests that it arises from a quantum-confined exciton.37-39 53 With the exception of a preliminary report of the present work included in a recent perspective article,40 to date our studies of DICIs have focused exclusively on idealized particles containing a single defect site. Nanocrystals are likely to contain more than one surface defect after oxidation, however. Herein we will investigate how neighboring defects interact with each other during recombination. Specifically, we ask whether interactions between neighboring defect modify the mechanism of recombination. At the same time, we will examine the possibility of energy transfer between dangling bond sites. Pairs (or larger sets) of interacting silicon DB sites with net charge have been proposed as potential quantum bits (qubits) for quantum information processing.41-44 In principle, an extra electron can sit in one or the other of a pair of degenerate DB defect states or it can be in a quantum mechanical superposition of these two states. However, it is known that charge transfer between surface defects can be strongly influenced by interactions with local lattice vibrations.42, 45 As a natural consequence of the simulations reported below, we can comment here on the possibility of a related process: energy transfer (exciton hopping) between DB sites, taking interactions with the lattice fully into account. In this study, we applied ab initio molecular dynamics simulations in conjunction with a multireference electronic structure method to nanoscale slab models of the silicon (111) surface containing pairs of interacting dangling bonds at varying distance. The results shed light both on the effect of defect-defect interactions on the nonradiative recombination mechanism and the prospect of exciton transfer between neighboring defects. 3.2 Method In this work we studied slab models of the silicon (111) surface with dimension 2.5×1.0 nm (Si70H68; Figure 3.1). This slab consists of a single layer of sila-adamantane units with two DB defects on the surface. These models allow us to study the effect of distance on the interaction between two defects. Here we studied two different configurations in which the DB defects are either nearest neighbors (NN) or third nearest neighbors (TNN), with defect-defect distances of 3.90 Å and 10.25 Å, respectively, in their ground state minimum energy structures. 54 Figure 3.1. The Si70H68 slab model. The Si and H atoms are shown in blue and white, respectively. The left and right DB sites are labeled L and R, respectively. Site R is the same for both models. The nearest neighbor (NN), and third nearest neighbor (TNN) models are created by removing the capping H atoms from the yellow circled positions labeled L(NN) and L(TNN), respectively. The right panel illustrates the geometric parameters, θ and RSi-Si, reported in Table 3.1 and discussed in the text. To investigate the photophysics of these models, we applied a floating occupation molecular orbital complete active space configuration interaction (FOMO-CASCI) description of the electronic structure.46-48 A multireference electronic structure approach was chosen because it can accurately describe dynamics in the vicinity of CIs.49 The FOMO-CASCI method has several advantages over the more widely used state-averaged complete active space self-consistent field (SA-CASSCF)50 approach. First, unlike SA-CASSCF, FOMO-CASCI provides a size-intensive description of electronic excitations.51 That is, in systems composed of multiple excitable subsystems the error does not increase with the number of subsystems. This property is especially important in the present application, because we will compare the present results for slabs with two interacting DB defects with our previous work on systems containing a single isolated DB defect. In addition, FOMO-CASCI alleviates many of the convergence issues arising from the simultaneous optimization of the orbitals and configuration interaction coefficients in SA- CASSCF and in most cases is less computationally expensive. It is noteworthy that the scaling of complete active space calculations with the size of the one-electron basis is, in general, the same 55 as Hartree-Fock (O(N4)). Thus extension to nanoscale systems does not require extraordinary effort unless the system necessitates a large active space. In our previous work on an isolated DB defect36 we found that an active space of 5 electrons in 3 orbitals provided an accurate description of the low-lying excited states of interest when compared to accurate multi-state complete active space second order perturbation theory (CASPT2)52 calculations. Here we chose an active space of 10 electrons in 6 orbitals by simply doubling this active space so that we can accurately describe the excitations of two DB defects. The LANL2DZ basis set and effective core potentials53 were used throughout this study. The ground state geometries were optimized for both singlet and triplet states. The ground state was found to be the triplet, 40 and 0.1 meV below the lowest singlet state for the NN and TNN systems, respectively. Neglecting spin orbit effects, we have only modeled dynamics on the excited triplet states in this work. (See Table B.1 for optimized geometric parameters and energies of the singlet state.) We applied Born-Oppenheimer (adiabatic) ab initio molecular dynamics (AIMD) to model the nuclear dynamics of the electronically excited system. The PES was computed on-the-fly at the FOMO-CASCI level described above (CASCI-AIMD). Initial nuclear positions and momenta were sampled from the ground state vibrational Wigner distribution computed in the harmonic approximation. Trajectories were propagated with the velocity Verlet algorithm on the first triplet excited state (T1) PES using a 0.5 fs time step. Twenty trajectories were propagated for both the NN and TNN systems. All quantum chemical and AIMD calculations were performed with the TeraChem software package.46, 54-57 Points where the energy gap between the ground (T0) and T1 states approaches zero were chosen as the starting points for minimal energy conical intersection (MECI) optimization calculations. For this purpose, the CIOpt software package58 was applied in conjunction with TeraChem. All MECI calculations were performed using the same FOMO-CASCI PES described above. 56 In one out of twenty of the trajectories for each system (NN and TNN), the excitation was observed to transfer between DB sites. In both cases this energy transfer occurred at structure where the T1 state became nearly degenerate with the next higher energy triplet state (T2). Our AIMD simulations neglect nonadiabatic effects, which may result in the diabatic passage through such a crossing, therefore this transfer may or may not be physical. To quantify the probability of energy transfer in these trajectories, we applied the norm-preserving interpolation (NPI) method to compute time derivative nonadiabatic couplings (TDC) and population transfer probabilities at the crossing points associated with these two trajectories.59 The NPI method is designed to accurately evaluate the population transfer probability from the overlap between adiabatic electronic wave functions at different times, even when the TDC spikes very sharply, e.g. at trivial unavoided crossings that arise during long-range energy transfer.60-61 Time steps ranging from 0.5 to 4.0 fs were used to ensure convergence of the computed probabilities. 57 Table 3.1. Relevant geometric parameters surrounding the DB defect sites at several important geometries in each model. Parameters include Si-Si bond lengths (RSi-Si in Å) and Si-Si-Si bond angles (q in degrees) as illustrated in Figure 3.1. Si-Si bond lengths greater than 2.5 Å are bolded. Result for the 1DB system (Si72H63) from reference 36 is also listed for comparison. MECI (R) MECI (L) FC NN La Ra TNN La Ra 1DBb RSi-Si/Å 2.35 2.36 2.36 2.37 2.37 2.36 2.36 2.36 2.36 2.35 2.36 2.37 2.35 2.35 2.35 q/degree 106.7 106.4 107.2 106.4 106.5 106.6 FC FC 106.4 106.3 106.3 106.1 106.5 106.4 105.7 105.7 105.7 RSi-Si/Å 2.44 2.45 3.00 2.39 2.36 2.41 q/degree 123.5 123.2 121.4 108.7 106.5 106.5 RSi-Si/Å 2.34 2.36 2.35 2.43 2.74 2.52 q/degree 106.7 106.7 106.3 123.0 126.0 125.2 MECI (L) MECI (R) 2.90 2.44 2.50 2.35 2.36 2.37 2.44 2.48 2.82 124.6 123.1 123.3 105.7 106.4 106.3 122.4 122.9 124.3 MECI 2.36 2.35 2.36 2.47 2.44 2.86 --- --- --- 106.4 106.1 106.4 122.7 123.5 125.0 --- --- --- a The local geometries of left and right DB sites as oriented in Figure 3.1 b See Si72H63 in reference 36 3.3. Results and Discussions First, we optimized the geometries of both slab models in their triplet ground states. Hereafter the optimized structures will be referred to as the Franck-Condon (FC) points. The most relevant geometric parameters (illustrated in the inset of Figure 3.1) can be found in Table 3.1. Although the two defects are not equivalent on our slab models because the model is asymmetric, the differences in local structure surrounding the left (L) and right (R) DB defects are very small at the FC point. (L and R labels are shown in Figure 3.1.) In both the NN and TNN cases the Si- Si bond distances (RSi-Si) differ by only 0.02 Å and the Si-Si-Si bond angles (θ) are within 1 degree. At this minimum energy structure, natural orbital analysis demonstrates the unpaired electrons to 58 reside in the nonbonding orbitals at the dangling bond sites, nDB (Figure 3.2 for TNN and Figure B1 for NN). Figure 3.2. The singly occupied natural orbitals of the triplet ground state (T0) of the TNN system. These are both nDB orbitals, as described in the text. 59 Table 3.2. The vertical excitation energies (Eex in eV) and excitation characters at the Franck- Condon point of the NN and TNN systems. Eex / eV 3.89 4.08 4.11 4.30 5.19 NN Character Local Local Local Local CT Eex / eV 4.00 4.22 4.32 4.41 5.53 TNN Character Local Local Local Local CT The excitation energies and chemical characters of the lowest five electronic excited states (T1-T5) at the FC point are reported in Table 3.2. States T1 to T4 are characterized by local excitations on an individual DB site. In all four cases, an electron is excited from a partially localized Si-Si s bonding orbital, sSi-Si, to the nDB orbital on the same DB site. (Natural orbitals are presented in Figure 3.3 for NN and Figures B2 and B3 for TNN. An orbital scheme is shown in Figure 3.4.) Four excitations of this type (T1-T4) exist because each DB site has two nearly degenerate sSi-Si orbitals from which an electron can be excited. The energies of these excitations vary over a range of 0.41 eV, indicating coupling between DB sites at the FC point. On the other hand, the highest excitations computed (T5) involve charge transfer (CT) between the DB sites, with an electron excited from a sSi-Si orbital on one DB site to the nDB orbital on the other site (as depicted by the blue arrow in Figure 3.4; see Figures B4-B7 for natural orbitals). The CT states are higher in energy due to the larger spatial separation and thus smaller Coulombic interaction between the electron and hole. It is noteworthy that for the NN system a nonnegligible CT contribution can be mixed with lower-lying excited states (T1-T4 states) due to the shorter distance between the two DB sites and the partially delocalized nature of the sSi-Si orbitals (as can be seen in the natural orbitals in Figures 3.3, B2, and B3). Nevertheless, the excited electron is always localized on one of the nDB orbitals at the FC point. 60 Figure 3.3. Natural orbitals with occupation numbers 1.00 (top) and 2.00 (bottom) of state T1 at the NN FC geometry. The top and bottom orbitals are examples of σSi-Si and nDB orbitals, respectively, as described in the text. 61 Figure 3.4. A simplified picture of the orbitals arrangement in the systems studied here. All excitations investigated in this work involve promotion of an electron from a σSi-Si orbital to a nDB orbital. In the low-lying excited states (T1-T4) the majority of the excitation character is local to a single DB site. The dashed arrow indicates a possible charge-transfer excitation between two sites (e.g. T5). Adiabatic CASCI-AIMD calculations were performed on the T1 state, as described in the previous section. Although the low-lying excited states are close in energy in the FC region, the gap between T1 and the higher energy states increases rapidly in all simulations. This splitting was previously observed in our study of an isolated DB defect (referred to as 1DB hereafter).36 It can be understood as the Jahn-Teller splitting between the nearly degenerate states that arise from the two sSi-Si→nDB states localized to a particular DB site. 62 Figure 3.5. Top panel: The T1-T0 energy gap as a function of time for the NN system. Bottom panel: the T1-T0 energy gap as a function of time for the TNN system. The light grey curves depict the individual trajectories and the red curves depict the average gap over 20 trajectories for each system. 63 Table 3.3. The energies (in eV) of the MECIs of the NN, TNN, and 1DB systems. Results for 1DB are drawn from reference 36. As the 1DB system has only one dangling bond, only one energy is listed. All energies are relative to ground state energy of the T0-optimized structure. NN TNN 1DB MECI(L) 2.33 2.41 2.38 MECI(R) 2.42 2.42 --- The T0-T1 energy gap as a function of time is presented in Figure 3.5. This gap approaches zero for both the NN and TNN systems 40-60 fs after excitation, with all trajectories exploring regions of near-zero energy gap. This suggests that, as in the 1DB case,36 there are CIs between the ground and first excited electronic states in these systems. To explore this possibility, MECI optimization calculations have been performed starting from points of near-zero energy gap drawn from the AIMD simulations. The optimized MECI geometries are shown in Figures 3.6a-b and e- f with important geometric parameters reported in Table 3.1. The energies of the MECIs are presented in Table 3.3. Two different MECIs were located for each of the model systems (NN and TNN). These MECIs are labeled MECI(L) and MECI(R), indicating that the nuclear distortions bringing about degeneracy are local to either the left or right DB site. Taking as an example MECI(L) of the NN system, one of the Si-Si bond lengths adjacent to the left DB site is stretched to 3.00 Å, while all Si-Si bond lengths surrounding the right DB site (2.41, 2.39, and 2.36 Å) remain near their ground state minimum values. The asymmetric bond stretching and increased θ angles surrounding the left DB site are consistent with those observed in previous work for 1DB (see Table 3.1) and described in the introduction. Similarly localized bond length distortions are observe in the other three MECI structures as well (see bold values in Table 3.1). The MECI energies for all four intersections reported here are very similar to one another, ranging from 2.33 to 2.42 eV above the ground state minimum energy (Table 3.3). These energies are, again, similar to that previous reported for 1DB (2.38 eV). 64 (a) (b) (c) (d) On-site Off-site (e) On-site Off-site (f) (g) (h) Figure 3.6. The MECI(L) geometries and natural orbitals for the NN and TNN systems. (a) The top view of the MECI(L) in NN. (b) The side view of MECI(L) in NN. (c) A natural orbital with occupation number 1.50 (nDB) for MECI(L) in NN. (d) A natural orbital with occupation number 1.50 (σSi-Si) for MECI(L) in NN. (e) The top view of the MECI(L) in TNN. (f) The side view of MECI(L) in TNN. (g) A natural orbital with occupation number 1.50 (nDB) for MECI(L) in TNN. (h) A natural orbital with occupation number 1.50 (σSi-Si) for MECI(L) in TNN. Panels c and d are adapted from reference 40 with permission from the PCCP Owner Societies. At each MECI, T0-T1 state-averaged natural orbitals were computed to investigate the nature of the electronic transition enabled by these intersections. In all cases two natural orbitals with occupation numbers ~1.5 are observed. It is transitions of a single electron between these pairs of orbitals that are facilitated by the MECIs. In all cases, these orbitals are localized to a 65 single DB site, as can be seen in Figures 3.6c-d and g-h, which show the partially occupied natural orbitals for MECI(L) of NN and TNN, respectively. Figure 3.7. Top panel: the averaged Si-Si bond lengths around the DB sites for the NN system. Bottom panel: the averaged Si-Si bond lengths around the DB sites for the TNN system. The bond lengths are sorted as described in the main text. On-site (more distorted) and off-site (less distorted) bonds are shown on the left and right, respectively. Longest, middle, and shortest lengths shown by yellow, orange, and blue lines, respectively. We now return to the AIMD data to assess whether the optimized MECIs accurately reflect the predicted dynamics. The Si-Si bond lengths, averaged over all 20 trajectories for NN and TNN respectively, are reported as a function of time in Figure 3.7. For each trajectory, the DB sites are sorted as either on-site or off-site, with the on-site DB defined as the one with the more distorted Si-Si bond lengths. Then, within each site, the three adjacent Si-Si bonds are sorted by length. Then, six averages are performed over the sorted bonds (that is, the longest on-site bond lengths from the twenty trajectories are averaged to produce the yellow curve on the left, etc.). Clearly the on-site bonds distort significantly in the first 50 fs after excitation, while the off-site bonds remain very near to their ground state minimum values. This emphasizes the local nature of the dynamics; the coupled electron-nuclear motions of NR recombination occur largely at a single DB site, with the other DB site relatively unperturbed. This is even true for the NN configuration where DB sites are separated by only two Si-Si bonds (though very modest oscillations in the off- site bond lengths are observed in this case). The ratio of trajectories with larger distortions on the 66 left DB to those with larger distortions on the right DB is 11:9 for both the NN and TNN systems, which is statistically indistinguishable from 50%. Thus, the exciton appears to have roughly equal probability of undergoing NR recombination at either DB site. To briefly summarize, we have found that during the first 40-60 fs after excitation of a defect-localized excitation in our model slabs the bond lengths surrounding the DB defects undergo asymmetric stretching and the energy gap between T1 and T0 approaches zero, suggesting a pathway for NR recombination via DICI. Optimization of MECIs confirms this suggestion. Several factors suggest that neighboring DB defects behave essentially independently: a) The geometric distortions observed in the AIMD simulations are local to a single defect site (Figure 3.7), b) the electronic transitions facilitated by the MECIs are local to a single defect site (Figure 3.6), and c) the observed dynamics and MECI energetics are very similar to those predicted in our previous work on a single isolated DB site.36 The localized nature of the DICI supports the extrapolation of the NR decay mechanism and dynamics from the simulations of idealized models containing a single isolated defect to realistic systems with multiple defects. Though in all forty simulations reported here NR recombination occurred upon dynamics that were local to a single defect site, we did observe communication between defect sites in two simulations (one out of twenty for each of NN and TNN). Specifically, within the first ~10 fs after excitation we observed crossing of the T1 and T2 excited states. In our adiabatic AIMD simulations, this resulted in the transfer of the exciton between DB sites. Because nonadiabatic effects are neglected, however, it is unclear whether this energy transfer process (an adiabatic process) is physical or not. In order to study this nonadiabatic event, we applied NPI59 to evaluate the TDC and population transfer probability associated with these PES crossings. Below we present the computed energy transfer probabilities, defined such that 0% corresponds to the excitation remaining on the same DB site and 100% corresponds to the transfer of the excitation between dangling bonds with unit probability. First, we evaluate the probability of energy transfer in the NN trajectory. We expect that the exciton transfer is more likely to occur in this system due to the shorter DB-DB distance. The energy transfer probability is computed to be 98.5%. It is important to remember that this is the 67 probability of transfer for only one out of twenty trajectories. No evidence of transfer was observed in the other nineteen trajectories. This suggests a significant probability of energy transfer between DB defects separated by two Si-Si bonds (nearest neighbors) on the Si (111) surface in the first femtoseconds after excitation. However, without a larger sample size we hesitate to quantify the probability of such transfer. Now we consider whether energy transfer can occur over a longer distance (~10 Å; the TNN case). An energy transfer probability of 0% is predicted by NPI in this case. Not surprisingly, the probability of energy transfer between DB defects separated by ~10 Å is much lower than in the NN case. Given the sampling of our present study (twenty trajectories), the probability cannot be discerned from zero. Further insights into the probability of energy transfer could also be gained through full nonadiabatic molecular dynamics simulations. 3.4. Conclusions In this study, we applied CASCI-AIMD to investigate NR recombination in slab models of the silicon (111) surface containing two dangling bond defects. All trajectories, initiated on the lowest triplet electronic excited state, explored regions of near-zero energy gap with the triplet ground electronic state within 40-60 fs after excitation. Conical intersections were identified in these regions, suggesting that these systems decay to the electronic ground state via DICI. The nuclear distortions associated with recombination were found to be nearly completely localized to a single defect site, as were the electronic transitions associated with the MECIs. The mechanism, timescale, and energetics of recombination were all in nearly perfect agreement with our past work on a single isolated dangling bond defect.36 Thus, even when separated by a mere two Si-Si bonds, the presence of a second defect has negligible effect on the NR decay mechanism and dynamics. This is consistent with a growing body of evidence that even in large systems, energetically accessible conical intersections tend to involve relatively local distortions of electronic and geometric structure.40 On the other hand, based on the NPI calculations of the nonadiabatic coupling, we found that a localized exciton on one defect site can transfer to a nearest-neighbor DB site (separated by two Si-Si bonds, ~4 Å) with significant probability. This process occurs within the first 10 fs after 68 excitation. After this short window, Jahn-Teller distortion of the excited DB stabilizes the localized excitation, breaking the degeneracy of the states involved and disfavoring transfer. In fact, this distortion appears to disfavor electronic communication of any kind, as evidenced by the local mechanism of NR recombination observed in our AIMD simulations. Nuclear relaxation of the defect site thus plays a crucial role in determining the degree of communication between defect sites. 69 APPENDIX SUPPORTING INFORMATION FOR: AB INITIO MOLECULAR DYNAMICS STUDY OF THE INTERACTION BETWEEN DEFECTS DURING NONRADIATIVE RECOMBINATION Table B1. The minimum energy geometries of NN and TNN systems at the singlet ground state. The relative energies (R.E.; unit meV) are the energies relative to corresponding triplet ground state. NN L R TNN L R R.E./meV 40.5 0.1 RSi-Si/Å 2.35 2.36 2.36 2.37 2.37 2.36 RSi-Si/Å 2.36 2.36 2.36 2.35 2.36 2.37 q/degree 106.9 106.6 106.8 106.0 106.5 106.6 q/degree 106.4 106.3 106.3 106.1 106.5 106.4 70 Figure B1. The ground state natural orbitals with occupation number 1.00 in the NN system. 71 Figure B2. The T1 state natural orbitals with occupation number 1.00 and sigma bonding character in the TNN system. Figure B3. The T1 state natural orbitals with occupation number 2.00 and (majority) DB nonbonding character in the TNN system. 72 Figure B4. The T5 state (CT state) natural orbitals with occupation number 1.06 and sigma bonding character in the NN system. Figure B5. The T5 state (CT state) natural orbitals with occupation number 1.94 and DB nonbonding character in the NN system. 73 Figure B6. The T5 state (CT state) natural orbitals with occupation number 1.00 and sigma bonding character in the TNN system. Figure B7. The T5 state (CT state) natural orbitals with occupation number 2.00 and DB nonbonding character in the TNN system. 74 Ground State Minimum Energy Geometries (in Angstrom) NN System Si -0.5576323185 -0.4696373504 -0.5920927866 Si 1.7982390365 -0.4053398800 -0.6923731798 Si -1.3882497620 1.5214595289 0.3701750119 Si -1.2003660670 -2.2968807090 0.7710967586 Si 2.7173630701 -0.0880106730 1.4569051027 Si -0.4763805609 1.8269212341 2.5275452825 Si -0.2786724268 -1.9805410724 2.9206367854 Si 1.8814725610 1.9029682772 2.4165588267 Si -1.1189093405 0.0012536655 3.8939505339 Si 2.0776452179 -1.9138966149 2.8227829129 Si 2.6415292250 -2.3941707900 -1.6707089170 Si -3.7366811353 1.4614762050 0.5208380030 Si -0.3626797296 -4.2860183150 -0.1833238623 Si -3.5464471644 -2.3773792303 0.9456521969 Si 2.9037944108 -3.9048424014 1.8607442295 Si 2.9302289348 -1.6234465941 4.9976737530 Si 2.7524891647 2.2166515582 4.5808543680 Si -3.4764822610 -0.0641955087 4.0129501094 Si -0.2154120340 0.3149323036 6.0532480155 Si -4.3593719693 -0.3726185292 1.8547087982 Si 2.1315931173 0.3832053206 5.9162064436 Si 1.9936334351 -4.2153625381 -0.2940768224 Si -1.4075070276 -0.7565346349 -2.7688484997 Si -1.3125710618 3.8198073673 3.4936178410 Si 5.0629271458 0.0001101856 1.2799415000 Si -4.2763144538 1.9291161430 4.9741008362 Si -4.5450946429 3.4623058112 1.4449741377 Si 1.9401980985 4.2171572139 5.5032025071 Si -1.0628042904 2.3030513314 6.9848411982 Si 5.2490416295 -3.8410275138 1.7206068239 Si -1.2399322676 -4.6021421023 -2.3416343407 Si 1.7287816830 -2.7114489087 -3.8226895601 Si 4.9971015644 -2.3259029052 -1.7771136286 Si -0.4059388753 4.1369106062 5.6516568587 Si -3.4103580532 2.2375656600 7.1407680258 Si -3.6711092086 3.7572841617 3.6088468836 Si -1.2471164516 6.1283088965 6.5878541095 Si -4.4780071726 5.7525079399 4.5662494995 Si -4.2278638526 4.2317891021 8.0767151627 Si -3.5891890630 6.0373409174 6.7202937516 Si -0.6186298447 -2.7700746853 -3.6811378161 Si 2.5571324453 -4.6943659362 -4.7900243331 Si 2.8252055923 -6.2078585836 -1.2698629294 75 Si 5.8368727134 -4.3043818329 -2.7414682383 Si 5.8770241451 -2.0106771597 0.3823343804 Si 4.9144119023 -4.6363397590 -4.8935381648 Si 1.7089649397 -4.9943167450 -6.9670099521 Si 1.9164935926 -6.5251401342 -3.4273936374 Si 5.1860822102 -6.1253893522 -1.3642700019 Si 8.1865772037 -4.2420546785 -2.9145066326 Si 5.7589957025 -6.6067363966 -5.8776706610 Si 2.7391957410 -8.5198508731 -4.3779360615 Si -0.4305348654 -6.6103001264 -3.2424394666 Si 6.0437936056 -5.8520304613 0.8140231237 Si 6.0384950426 -8.0721329346 -2.3689642716 Si 1.8824390046 -8.8353761344 -6.5516086567 Si 5.0938183318 -8.4293272900 -4.4979872935 Si 8.1185561160 -6.5632894354 -5.9795625201 Si 4.8499671066 -6.9351539396 -8.0324447258 Si 8.3955804019 -8.0832834301 -2.4845307107 Si 5.9617609270 -10.3838492954 -5.5116091091 Si 2.5051407308 -6.9992956164 -7.8859603824 Si 2.6824743731 -10.8459319379 -7.4607099382 Si 5.7057951090 -8.9111943531 -8.9846913032 Si 9.0055701737 -6.2433466499 -3.8257289182 Si 8.9341893308 -8.5437837372 -6.9538611383 Si 8.3255527972 -10.3525051971 -5.5715515702 Si 8.0496359933 -8.8239770055 -9.1120547260 Si 5.0233458608 -10.7290979016 -7.6479289929 Si 9.1573662458 -10.1152198844 -3.3850550543 H 2.2001335667 0.7360245708 -1.5455014028 H -0.9816770730 2.6662277540 -0.4744106725 H 2.2727295033 3.0454472848 1.5617507574 H -3.8666939418 -1.2050822360 4.8698987011 H -3.9298092492 -3.5076632351 1.8141844735 H -4.1411519133 -2.5947676003 -0.3873964147 H -4.2995922878 1.2843203605 -0.8350249369 H -5.8347961530 -0.4204646210 1.9472034751 H 4.2278454712 2.2765493879 4.5008344911 H -0.6237710338 -0.8281446857 6.8987422351 H 2.6828782165 0.5704603009 7.2759196757 H 4.4053662762 -1.5994068007 4.9596039536 H 2.5070179777 -2.7588351152 5.8406664321 H 2.5019206059 -5.0493628477 2.7081258975 H -0.7543509857 -5.4286605224 0.6715190550 H 2.3340934675 5.3570870406 4.6521483024 H 2.5121138241 4.4185095282 6.8490031991 H -0.4908202058 2.4853547393 8.3372971077 H -0.9114702631 4.9612371618 2.6407864224 H -5.7532520736 1.8734442045 5.0451234968 76 H -6.0192505415 3.4266196499 1.5107898261 H -4.1421134494 4.6034076531 0.5997173702 H -3.8031007814 1.0875473818 7.9826919403 H -3.6778751698 4.4053640989 9.4351835082 H -5.6995140515 4.1701033676 8.1698199675 H -0.8331521112 7.2749573280 5.7557014399 H -0.6750140319 6.3019962449 7.9377859788 H -5.9512988836 5.6886680249 4.6366715294 H -4.1018826616 6.8949930667 3.7106201361 H -0.6767516118 -3.1243023916 3.7719546699 H -4.0992418482 7.2975827836 7.3018247284 H -2.7146449053 -4.6684845300 -2.2621626262 H 2.1363982577 -1.5737572936 -4.6768375556 H -0.9666456780 0.3717261557 -3.6128734602 H -2.8825589451 -0.7652495304 -2.7402760248 H -1.1800995540 -2.9547605316 -5.0365762350 H 5.4406572592 1.1197078644 0.3948282949 H 5.6667264578 0.2330507438 2.6057496156 H 5.8180207153 -3.6728725369 3.0746722574 H 5.3988397550 -1.1898031762 -2.6357695408 H 7.3526476399 -1.9598092334 0.3027177591 H 2.4413704354 -7.3586811063 -0.4228194773 H 5.3232273773 -3.5019031494 -5.7518133290 H -0.7950323986 -7.7386914549 -2.3627483483 H -1.0231029269 -6.8491257821 -4.5724890292 H 7.5182816968 -5.8507165600 0.7651055016 H 5.6034526590 -6.9921989442 1.6407294255 H 2.3684814930 -9.6704876043 -3.5274940655 H 2.1390156994 -3.8663419802 -7.8160777206 H 0.2344738536 -5.0191460617 -6.9342550812 H 0.4083951655 -8.8983403833 -6.4715768617 H 5.2555750973 -5.7888917733 -8.8744987772 H 8.5118132963 -5.4214214052 -6.8335009029 H 8.9385412519 -7.9133494092 -1.1222454397 H 8.7844458953 -4.0255371025 -1.5836435562 H 8.5778376055 -3.1215191311 -3.7912126136 H 1.9512693950 -7.1967380072 -9.2427724652 H 10.4811063744 -6.1958232822 -3.9179143530 H 10.4103589846 -8.4847452129 -7.0160214701 H 5.1356172052 -9.0900147871 -10.3368997656 H 8.7869177860 -11.6242768900 -6.1650494012 H 8.4561438895 -7.6846713673 -9.9558894073 H 8.5676733133 -10.0690202845 -9.7095783170 H 5.5695396847 -11.9889181176 -8.1930065681 H 2.1102862447 -11.0674258434 -8.8021508748 H 2.3229042234 -11.9823711446 -6.5931469846 H 10.6320766370 -10.1198007873 -3.4290076038 77 H 8.6976407448 -11.2437879293 -2.5569553201 TNN System Minimum Energy Conical Intersection Geometries (in Angstrom) NN System: MECI-L Si -2.5494918535 2.3235668108 -0.0549987860 Si -0.1914314307 2.3883635899 -0.1554164091 Si -3.3669027310 4.3165608112 0.9164746452 Si -3.1669471560 0.4786940695 1.2934084170 Si 0.7250925855 2.6722030321 2.0022133404 Si -2.4560477814 4.5994036279 3.0754027150 Si -2.2650610336 0.7789074480 3.4527830952 Si -0.0991431039 4.6602024772 2.9663381062 Si -3.1026733591 2.7662487965 4.4280306862 Si 0.0952199915 0.8290701649 3.3459502368 Si 0.6470279101 0.4155919359 -1.1620426143 Si -5.7163610498 4.2701782875 1.0670872664 Si -2.3117380917 -1.5086861341 0.3249884882 Si -5.5213639660 0.4180485870 1.4433837818 Si 0.9512693088 -1.1465499949 2.3651867483 Si 0.9401941436 1.1000582858 5.5285236991 Si 0.7760526156 4.9476982891 5.1322322392 Si -5.4618376068 2.7192068332 4.5483669845 Si -2.2050646539 3.0648048062 6.5902551803 Si -6.3288699695 2.4178148482 2.3812231030 Si 0.1422854123 3.1069928399 6.4516518737 Si 0.0579219822 -1.4280501094 0.1987730939 Si -3.4160891956 2.0387091444 -2.2286051006 Si -3.2778985118 6.5923855118 4.0470320935 Si 3.0702817020 2.7816446962 1.8256369625 Si -6.2506487087 4.7175844311 5.5207090751 Si -6.5382470390 6.2561658208 2.0173397516 Si -0.0177862386 6.9489888659 6.0658090705 Si -3.0347270496 5.0593921621 7.5273973315 Si 3.2955371135 -1.0541438496 2.2057684440 Si -3.2181823099 -1.8022439013 -1.8244373445 Si -0.2719954401 0.0996153136 -3.3188597927 Si 2.9973146695 0.5331228703 -1.2757632595 Si -2.3643303443 6.8958864368 6.2041953474 Si -5.3822786162 5.0162211049 7.6884764155 Si -5.6366490747 6.5515062968 4.1688312687 Si -3.1791092503 8.9010553042 7.1329152246 78 Si -6.4145231972 8.5563935450 5.1328733573 Si -6.1612726081 7.0198171519 8.6407964836 Si -5.5198111738 8.8264004827 7.2868095954 Si -2.6185164848 0.0393153436 -3.1571279484 Si 0.5570422467 -1.8781518916 -4.2850550974 Si 0.9238245155 -3.4302387849 -0.8171632460 Si 3.8622805620 -1.4112856595 -2.2868800346 Si 3.8863497183 0.7981828923 0.8843346658 Si 2.9091971304 -1.8152953262 -4.4024835169 Si -0.2858800123 -2.2020558426 -6.4625268186 Si -0.0689164520 -3.7193401570 -2.9348858934 Si 3.3357050813 -3.2582355105 -0.9383696597 Si 6.1999307232 -1.2715869772 -2.4459799062 Si 3.7300034833 -3.8441294484 -5.2576050667 Si 0.6897219828 -5.7356018630 -3.8866682088 Si -2.4095689415 -3.8040295131 -2.7268406967 Si 4.1595846675 -3.0205672397 1.2512928415 Si 3.7408857035 -5.8608523472 -1.5516160498 Si -0.1415107714 -6.0543563293 -6.0593107788 Si 3.0366641869 -5.6352791614 -3.8610181700 Si 6.0810492998 -3.7547722054 -5.3564316351 Si 2.8675845843 -4.1667991396 -7.4369531582 Si 6.1897069470 -5.0603512606 -1.7596967360 Si 3.9060064180 -7.5744416406 -4.8998897282 Si 0.5181381104 -4.2136416448 -7.3640512730 Si 0.6946841100 -8.0727419470 -6.9181061764 Si 3.7387469952 -6.1393126452 -8.3748361816 Si 6.9181863623 -3.3572259019 -3.1978603391 Si 6.9301055093 -5.7436936107 -6.2720830208 Si 6.2755792821 -7.4960326774 -4.8405548042 Si 6.0844354461 -6.0479514479 -8.4469918730 Si 3.0375378933 -7.9556025604 -7.0510141870 Si 6.9991186234 -7.1240882273 -2.6327116189 H 0.2225624500 3.5425605266 -0.9841947267 H -2.9436170875 5.4640186084 0.0838677245 H 0.3050630494 5.7957013285 2.1079437103 H -5.8625625790 1.5807197776 5.4033349618 H -5.9426447437 -0.7294039356 2.2701684465 H -6.0963833733 0.2548989976 0.0940661979 H -6.2790907192 4.1073317078 -0.2905535658 H -7.8046272879 2.3599085867 2.4576791185 H 2.2514908275 4.9911921471 5.0517448231 H -2.6330591101 1.9238237024 7.4284685200 H 0.6980497986 3.2819799570 7.8106251125 H 2.4153672515 1.1164003114 5.4938166289 H 0.5067613537 -0.0426842967 6.3562852853 H 0.5509367361 -2.3117196565 3.1843911830 79 H -2.6946428678 -2.6536896500 1.1799814353 H 0.3951830411 8.0904056769 5.2262837082 H 0.5494008218 7.1325169911 7.4161225673 H -2.4562922368 5.2228996252 8.8793984108 H -2.8676751299 7.7295607171 3.1928941435 H -7.7280913642 4.6661445445 5.5874886784 H -8.0094755391 6.1821120324 2.1174684057 H -6.1846025447 7.4110710659 1.1692989884 H -5.7846904860 3.8709722500 8.5319835578 H -5.5812430519 7.1754854756 9.9891286515 H -7.6311915856 6.9757550093 8.7628426308 H -2.7721415246 10.0318409049 6.2755779082 H -2.5863252785 9.0919301574 8.4712687331 H -7.8889788994 8.5243024263 5.1968976879 H -6.0133656018 9.6919480042 4.2792208674 H -2.6730433056 -0.3671535571 4.2963990515 H -6.0175575884 10.0899514143 7.8717941989 H -4.6904113900 -1.8705288526 -1.7122351219 H 0.1313410029 1.2465696545 -4.1624715981 H -2.9953525510 3.1761955846 -3.0684940635 H -4.8909869704 2.0186116717 -2.1774713307 H -3.1819726509 -0.1438955192 -4.5120355672 H 3.4404729574 3.9247013156 0.9692276617 H 3.6704720437 2.9792208676 3.1585390305 H 3.8713316920 -0.8827541513 3.5571251808 H 3.3776139821 1.6977240175 -2.1060714413 H 5.3600019432 0.8620965683 0.7866748512 H 0.5848652839 -4.5755949432 0.0541161731 H 3.3119658323 -0.7261667509 -5.3192822189 H -2.7784315614 -4.9308373562 -1.8486985032 H -2.9924009990 -4.0419891211 -4.0613398678 H 5.6312079519 -2.9353398557 1.2339833729 H 3.7634092376 -4.1991648110 2.0438142033 H 0.2641811273 -6.8526570783 -3.0161723615 H 0.1410662633 -1.0754561190 -7.3146108856 H -1.7607484194 -2.2239204695 -6.4212050627 H -1.6171189873 -6.1221394016 -6.0081669301 H 3.2949929510 -3.0128120662 -8.2580732801 H 6.4680414001 -2.6157764968 -6.2179999532 H 6.7234329762 -4.8673604693 -0.3947297596 H 6.7828152979 -0.9847317303 -1.1221147469 H 6.5949143798 -0.2083573148 -3.3867193074 H 0.0001132116 -4.3862696022 -8.7381788206 H 8.3933786752 -3.4566882151 -3.2139884423 H 8.4069334888 -5.6799583200 -6.3122426586 H 3.1976251910 -6.3041485836 -9.7407714680 H 6.8183725899 -8.7801560598 -5.3277315026 80 H 6.5065298128 -4.9098770022 -9.2851430332 H 6.6244935495 -7.2909730049 -9.0303494809 H 3.5911151547 -9.2151723334 -7.5895842241 H 0.1436749275 -8.3362876257 -8.2602140779 H 0.3226692427 -9.1833152287 -6.0213001763 H 8.4684350997 -6.9864996270 -2.6447179622 H 6.6262859502 -8.2487058553 -1.7569559627 NN System: MECI-R Si -2.5661827184 2.3238226837 -0.0440001497 Si -0.2060317901 2.4043199539 -0.1605815227 Si -3.4191313529 4.3007570081 0.9211827851 Si -3.1974388685 0.4916271772 1.3166452717 Si 0.7136522709 2.7203831543 1.9882673639 Si -2.4890877036 4.6224747041 3.0672978859 Si -2.2686125718 0.8156268621 3.4684353195 Si -0.1304554761 4.7112069977 2.9333654463 Si -3.1017738946 2.7973617066 4.4460676533 Si 0.0863910247 0.8928534652 3.3543872366 Si 0.6536940573 0.4238206171 -1.1430971152 Si -5.7662077708 4.2105473779 1.1066338751 Si -2.3555779962 -1.4810343303 0.3384292274 Si -5.5440751432 0.3995116131 1.4988971520 Si 0.9038788867 -1.0953783317 2.3866263544 Si 0.9667114585 1.1999569514 5.5190916865 Si 0.7648785476 5.0410973052 5.0857917091 Si -5.4550302630 2.7031361007 4.6056974027 Si -2.1738621124 3.1273747245 6.5934606988 Si -6.3584152568 2.3796007248 2.4575378352 Si 0.1697968538 3.2086747418 6.4350226382 Si -0.0020460895 -1.3953154583 0.2278020650 Si -3.3861015090 2.0594143772 -2.2364465852 Si -3.3346419573 6.6085293306 4.0338110711 Si 3.0613502856 2.8167262399 1.8046958739 Si -6.2571296520 4.6947875140 5.5717252948 Si -6.5921342247 6.2007405622 2.0334187645 Si -0.0483202382 7.0454198301 5.9977875348 Si -3.0169991147 5.1182603187 7.5253005534 Si 3.2500173971 -1.0227418673 2.2615597963 Si -3.2286762176 -1.7797475960 -1.8284934326 Si -0.2468625372 0.1190800149 -3.3027962529 Si 3.0148396375 0.4920925739 -1.2424841704 Si -2.3926837613 6.9484635545 6.1726792127 Si -5.3597112941 5.0375028119 7.7212977846 Si -5.6914161567 6.5185716986 4.1833248627 Si -3.2364220849 8.9465590925 7.0900812411 81 Si -6.5217285286 8.5030030287 5.1423941317 Si -6.1806393597 7.0367357736 8.6455579371 Si -5.5743291101 8.8341669813 7.2643625455 Si -2.5939696202 0.0529440906 -3.1597205191 Si 0.5918225642 -1.8674405973 -4.2659816434 Si 0.8265723258 -3.3848091038 -0.7435666642 Si 3.8596913930 -1.5025077087 -2.1885254601 Si 3.8808502749 0.8033164568 0.9192646664 Si 2.9496580808 -1.8354322835 -4.3385481313 Si -0.2163542662 -2.1589195231 -6.4558261213 Si -0.0704478775 -3.6861600724 -2.9045405365 Si 3.1807757791 -3.3048690915 -0.8068757356 Si 6.2140939437 -1.4923635869 -2.3171174064 Si 3.7703837841 -3.8377746653 -5.2959397081 Si 0.7451798010 -5.6786216377 -3.8379553116 Si -2.4146500314 -3.7769022604 -2.7494219449 Si 4.0488251517 -3.0360165594 1.3637695131 Si 3.9874969505 -5.2586423842 -1.8040481003 Si -0.0428885093 -6.0244089726 -6.0339705699 Si 3.1097752487 -5.6694513688 -3.9449306198 Si 6.1215566859 -3.7197203054 -5.4261751993 Si 2.9093900137 -4.1253737991 -7.4742520414 Si 6.3403253057 -5.3676851432 -1.9886897378 Si 3.5665018678 -7.9818069642 -4.5372985122 Si 0.5629743326 -4.1805352375 -7.3639133652 Si 0.8449577492 -8.0212104361 -6.9008057041 Si 3.7859487385 -6.0927624133 -8.4176698740 Si 7.0085388543 -3.4994911371 -3.2606705736 Si 6.9234217745 -5.6491855634 -6.4940744486 Si 6.1673354244 -7.4824608864 -5.2521490106 Si 6.1024038435 -5.9012826253 -8.6697610889 Si 3.2276032518 -7.9061981979 -7.0295101971 Si 6.8473707574 -7.4390803098 -2.9792934714 H 0.1815408808 3.5552280871 -1.0075656673 H -3.0386439496 5.4491861632 0.0706493562 H 0.2500815949 5.8467554628 2.0641018281 H -5.8140971331 1.5584984990 5.4711739153 H -5.9262368765 -0.7684444486 2.3162130974 H -6.1344242996 0.2320762241 0.1569222382 H -6.3423226245 4.0135562174 -0.2403224393 H -7.8324909556 2.3178427722 2.5607961725 H 2.2385586958 5.1057784103 4.9854125814 H -2.5678976019 1.9888332919 7.4510020257 H 0.7320127997 3.4019928877 7.7896285762 H 2.4404997076 1.2387714076 5.4516764356 H 0.5684593768 0.0672954903 6.3769552947 H 0.4915686803 -2.2415379127 3.2271332090 82 H -2.7367249108 -2.6387655955 1.1772153636 H 0.3297414198 8.1785159497 5.1308881208 H 0.5354868926 7.2668011071 7.3358249464 H -2.4231226148 5.3077593425 8.8673322813 H -2.9629985548 7.7506782597 3.1690906321 H -7.7318355037 4.6237684882 5.6669485823 H -8.0635675570 6.1297247629 2.1265568466 H -6.2268414177 7.3444297027 1.1752294401 H -5.7263031883 3.8962879225 8.5859534269 H -5.6205479325 7.2228980767 9.9981668295 H -7.6509767503 6.9651903708 8.7484046715 H -2.8471146696 10.0794266196 6.2277286746 H -2.6454404653 9.1563976845 8.4268800059 H -7.9877568847 8.3822606339 5.2735408700 H -6.2222125456 9.6448792544 4.2571060850 H -2.6600886245 -0.3230768881 4.3296215111 H -6.0849155790 10.0950500293 7.8437153491 H -4.7044139897 -1.8356315655 -1.7602051538 H 0.1607850428 1.2639051721 -4.1474732749 H -2.9164649428 3.1959460673 -3.0544815152 H -4.8603462833 2.0597952386 -2.2329753068 H -3.1541534999 -0.1224593654 -4.5171958932 H 3.4288997171 3.9315449624 0.9091262332 H 3.6696995163 3.0596661216 3.1256511739 H 3.8062313865 -0.8332419228 3.6176574407 H 3.4170571585 1.6260421229 -2.1038525247 H 5.3570323972 0.8534619233 0.8463362803 H 0.4240767033 -4.5280621266 0.1045934683 H 3.3842047029 -0.7145398634 -5.2034446450 H -2.7933897672 -4.9253802910 -1.9039588449 H -2.9844081220 -3.9815208753 -4.0956456635 H 5.5228965852 -3.0220198749 1.3088418585 H 3.6221712332 -4.1728759295 2.2017525609 H 0.3332263073 -6.8206468926 -2.9963325570 H 0.2678134347 -1.0438450374 -7.2937611156 H -1.6915369568 -2.1471201139 -6.4649296390 H -1.5158879655 -6.1456829192 -6.0020349090 H 3.3439365986 -2.9715390499 -8.2903381216 H 6.4891192959 -2.5311962689 -6.2247224267 H 6.9548320019 -5.3184592426 -0.6462341012 H 6.8010405714 -1.3153877920 -0.9756427116 H 6.6617904129 -0.3742007289 -3.1700537250 H 0.0261934365 -4.3699162405 -8.7282293599 H 8.4860416762 -3.4807796225 -3.3356184619 H 8.4009946059 -5.6785568073 -6.4520947886 H 3.1370536864 -6.3329000653 -9.7253682529 H 6.5027900001 -8.7660898652 -5.9019892926 83 H 6.4554095140 -4.7439418591 -9.5104618949 H 6.6662922523 -7.1260688264 -9.2656103253 H 3.7508348494 -9.1615846512 -7.5994264703 H 0.3535894879 -8.2490654724 -8.2743845353 H 0.4611462352 -9.1621686321 -6.0534603829 H 8.2997413196 -7.6939998779 -2.9749847472 H 6.1695113579 -8.5182506463 -2.2442934527 TNN System: MECI-L Si -2.5669511322 2.3588724045 -0.1039824540 Si -0.2068270253 2.4304001037 -0.2222780248 Si -3.3738896706 4.3441631451 0.8775815437 Si -3.2305189881 0.5260126410 1.2369977139 Si 0.7382260785 2.7841501906 1.9012892757 Si -2.4620642587 4.6222010053 3.0398749090 Si -2.2660257003 0.8029933374 3.3570501901 Si -0.1107851083 4.7288236526 2.9301033045 Si -3.0963474243 2.7738533086 4.3789765678 Si 0.0905378264 0.9397455297 3.1794511427 Si 0.6166126598 0.3900096743 -1.0549971279 Si -5.7215336890 4.2907351379 1.0182502188 Si -2.4381069102 -1.4874382376 0.3291135093 Si -5.5727371896 0.4374829063 1.4581830496 Si 0.8373613910 -1.8059074834 2.6100046726 Si 0.9486448079 1.0708549509 5.3863497435 Si 0.7717701641 4.9568497214 5.0993510227 Si -5.4603495332 2.7311410752 4.4980817856 Si -2.2011273014 3.0448852769 6.5478325235 Si -6.3584551679 2.4579476719 2.3448211131 Si 0.1484801442 3.0734970920 6.3652743924 Si -0.0952765027 -1.3856402171 0.3309975683 Si -3.4354548747 2.0489072477 -2.2718503423 Si -3.2798036709 6.6028432664 4.0320604163 Si 3.0871782916 2.8205713276 1.8109199886 Si -6.2527372659 4.7218310523 5.4688830457 Si -6.5077695729 6.2962605724 1.9469617978 Si -0.0301482047 6.9285127803 6.0864805915 Si -3.0500711261 5.0264590910 7.5052433677 Si 3.1517497581 -1.0683966993 2.4100816256 Si -3.2719225873 -1.8005501277 -1.8399523123 Si -0.2687230032 0.0846763456 -3.2272185629 Si 2.9744587898 0.4448437944 -1.1302665039 Si -2.3793823855 6.8776499076 6.2020882120 Si -5.4042775579 5.0054693596 7.6459330365 Si -5.6396324751 6.5612051782 4.1187365372 Si -3.2086315601 8.8681324528 7.1501981625 84 Si -6.4031717298 8.5799467636 5.0582973822 Si -6.2494405861 7.0066103350 8.5487082791 Si -5.5538026237 8.8210679037 7.2322843509 Si -2.6169797136 0.0351328804 -3.1543025126 Si 0.5575764974 -1.8943952819 -4.1906309378 Si 0.8192518387 -3.3983009716 -0.6583553306 Si 3.8229781052 -1.5085490948 -2.1303980489 Si 3.8124874763 0.7350855332 1.0504027648 Si 2.9072952917 -1.8147067596 -4.2820383189 Si -0.3002759893 -2.1656644330 -6.3628784450 Si -0.0967422344 -3.7119276838 -2.8219099275 Si 3.1814350562 -3.3293051240 -0.7569271973 Si 6.1681553724 -1.4175969412 -2.3197187662 Si 3.7396125931 -3.7839064236 -5.2782271392 Si 0.7345653677 -5.7053115261 -3.7875047378 Si -2.4442301610 -3.8135362474 -2.7045231187 Si 4.0572995071 -3.0202035964 1.4081148990 Si 4.0072068223 -5.3210062092 -1.7320547213 Si -0.1311804806 -6.0084195376 -5.9579549672 Si 3.0904071884 -5.6021006527 -3.8936030797 Si 6.0976670383 -3.7257542993 -5.4008654150 Si 2.8313700130 -4.1043135442 -7.4359100582 Si 6.3560452328 -5.2584674773 -1.9132510999 Si 3.9512535030 -7.5284609937 -4.9271861997 Si 0.4837243636 -4.1711485659 -7.2930506840 Si 0.6696886785 -8.0039688095 -6.8967985752 Si 3.6849169998 -6.0736040986 -8.4045771226 Si 6.9796917264 -3.4171655171 -3.2412724884 Si 6.9165090151 -5.6986991491 -6.3867242391 Si 6.3132281557 -7.5150361141 -5.0156677782 Si 6.0288824613 -5.9692226276 -8.5454710008 Si 3.0113696664 -7.8927805074 -7.0688991026 Si 7.1667753943 -7.2574313182 -2.8401263516 H 0.1913300610 3.5275229660 -1.1309188780 H -2.9540715876 5.4916865934 0.0434529436 H 0.2735806900 5.8986768279 2.1087964369 H -5.8628714928 1.5838864515 5.3402906921 H -5.9093968108 -0.6667565419 2.3779394077 H -6.1893213403 0.1599137247 0.1473843493 H -6.2722355412 4.1158787964 -0.3423650556 H -7.8331244651 2.4323641758 2.4516649298 H 2.2456325016 5.0168417726 5.0107913721 H -2.6132378570 1.8885793992 7.3729652145 H 0.7377535940 3.1914716758 7.7179490699 H 2.4206869018 1.0752478676 5.3743397381 H 0.4759382335 -0.0844121117 6.1708856711 H -2.8715914975 -2.6008963753 1.1977672957 85 H 0.4027499871 8.0861695977 5.2804153756 H 0.5243217069 7.0710050595 7.4469750848 H -2.4795142925 5.1861642569 8.8611385912 H -2.8513787389 7.7484088582 3.1986544245 H -7.7299432670 4.6603397443 5.5251563273 H -7.9826673802 6.2855509570 1.9983809674 H -6.0781567176 7.4339291661 1.1103246111 H -5.8351123869 3.8619095198 8.4785112851 H -5.7893834659 7.1821643419 9.9400343253 H -7.7235019779 6.9391123099 8.5478426420 H -2.7535859407 10.0148966252 6.3412742873 H -2.6656081257 9.0146540328 8.5155250301 H -7.8787992814 8.5879338549 5.0798236109 H -5.9450310664 9.6981685310 4.2101198055 H -2.6050373309 -0.3551236393 4.2111662762 H -6.0553721326 10.0828808659 7.8169688021 H -4.7469169839 -1.8578595427 -1.7725045249 H 0.1646239096 1.2319614262 -4.0553360903 H -3.0322025039 3.1737070505 -3.1376712995 H -4.9094073163 2.0121827253 -2.2095419656 H -3.1343059409 -0.1428958439 -4.5277267301 H 3.5244803284 3.8833956529 0.8868727823 H 3.6241277430 3.1197060487 3.1522037453 H 3.7797708977 -0.9367160417 3.7450595457 H 3.3692289681 1.6170402951 -1.9426659639 H 5.2907821349 0.7347429897 0.9845154225 H 0.4079038568 -4.5391863000 0.1891903983 H 3.3002410038 -0.6676212276 -5.1308940493 H -2.8492590695 -4.9386044503 -1.8398457669 H -2.9969984640 -4.0460920870 -4.0531378942 H 5.5244971827 -2.8780761725 1.3249400301 H 3.7472203310 -4.2052551261 2.2312553976 H 0.3625822824 -6.8630962665 -2.9456632719 H 3.6027147403 -6.4795400696 -0.9071623713 H 0.1369217710 -1.0325752657 -7.2009678181 H -1.7747200993 -2.1777042560 -6.3285870246 H -1.6056966723 -6.0814539745 -5.8781560641 H 3.2343086783 -2.9550224359 -8.2749374442 H 6.4680232453 -2.5802014819 -6.2597847671 H 6.9351385076 -5.0961971783 -0.5629566626 H 6.7886204920 -1.1966825025 -1.0003572200 H 6.5355608590 -0.2917507241 -3.2007450993 H -0.0741497151 -4.3568591985 -8.6497673675 H 8.4553368743 -3.3792569123 -3.3312797821 H 8.3928042705 -5.6310065172 -6.4504763653 H 3.1135114108 -6.2460529197 -9.7583795228 H 6.7735732984 -8.7798252021 -5.6239421389 86 H 6.4130673907 -4.8179214730 -9.3822237371 H 6.5525646401 -7.2036456996 -9.1618612127 H 3.5591650778 -9.1535344048 -7.6078725754 H 0.1135862402 -8.1826848852 -8.2519690843 H 0.2874701688 -9.1605456381 -6.0666195872 H 8.6391970083 -7.2146530585 -2.9234350639 H 6.7650491470 -8.4106203675 -2.0142726972 TNN System: MECI-R Si -2.5723092236 2.3222655077 -0.0500938233 Si -0.2122571571 2.3765305788 -0.1641646118 Si -3.3965441807 4.3058861496 0.9254849318 Si -3.2098556510 0.4915084585 1.3104341606 Si 0.7095007280 2.6939324364 1.9867610139 Si -2.4695074955 4.6136672908 3.0757359504 Si -2.2922378257 0.7965687093 3.4644976604 Si -0.1096742984 4.6875214755 2.9448152879 Si -3.1040581462 2.7871443560 4.4459688172 Si 0.0642343755 0.8758004407 3.3611467945 Si 0.6316238096 0.3958191770 -1.1556379299 Si -5.7432000430 4.2447552695 1.1025666305 Si -2.3915913357 -1.4965625135 0.3439329946 Si -5.5567220789 0.4091135924 1.5154891870 Si 0.9034392882 -1.0686420564 2.3274481460 Si 0.9535488737 1.1466786686 5.5299970496 Si 0.7892350693 5.0040120010 5.0989175236 Si -5.4643685767 2.7148794357 4.5947505031 Si -2.1698280069 3.1027789191 6.5914325902 Si -6.3590658842 2.4099914967 2.4399270011 Si 0.1752639664 3.1691262119 6.4352887077 Si -0.0326018276 -1.4384916914 0.1963723615 Si -3.4169084949 2.0460326936 -2.2338909770 Si -3.2944432104 6.6012095065 4.0499037165 Si 3.0587510450 2.7583110713 1.8311468748 Si -6.2486710277 4.7096840499 5.5612637757 Si -6.5359672777 6.2461975031 2.0348952270 Si -0.0182470954 7.0005092666 6.0359306362 Si -3.0105047778 5.0904108015 7.5321419043 Si 3.2549792230 -1.0894988553 2.2378129864 Si -3.2611858861 -1.7937195225 -1.8255063409 Si -0.2780983120 0.0980112602 -3.3155724260 Si 2.9928845201 0.4499238524 -1.2513750306 Si -2.3653137219 6.9306830193 6.1961872266 Si -5.3567726366 5.0179916636 7.7163187269 Si -5.6501948496 6.5338666205 4.1926963532 Si -3.2093558932 8.9310627688 7.1141043546 87 Si -6.4746129623 8.5104353835 5.1637499394 Si -6.1426253122 7.0128866178 8.6759684296 Si -5.5480093700 8.8160680043 7.2987436131 Si -2.6244966450 0.0387377788 -3.1549044764 Si 0.5495637536 -1.8854934670 -4.2912787851 Si 0.8080909118 -3.4257070938 -0.7799283931 Si 3.8244101775 -1.5399275230 -2.2134549541 Si 3.8735052558 0.7566248508 0.9105062212 Si 2.9101136802 -1.8504691856 -4.3660504013 Si -0.2529127531 -2.1548950189 -6.4906788957 Si -0.1178738045 -3.7042394669 -2.9359290531 Si 3.1676033093 -3.3707510310 -0.8700017873 Si 6.1694522103 -1.4566027115 -2.4038227595 Si 3.7409153788 -3.8498959322 -5.3310355290 Si 0.6980988809 -5.6932449575 -3.8802488292 Si -2.4583900619 -3.8021134418 -2.7290658045 Si 3.9915107381 -3.1201951430 1.3202951736 Si 3.9919780722 -5.3671677108 -1.8334363813 Si -0.0928381701 -6.0130981046 -6.0710175524 Si 3.0402814696 -5.6777599634 -4.0017685557 Si 6.0961746278 -3.7968190645 -5.4490988564 Si 2.8804184691 -4.0649748206 -7.5114896372 Si 6.3495723371 -5.3117977420 -1.9754137953 Si 3.9329128778 -7.9642938990 -4.2981922876 Si 0.5343491306 -4.1729310604 -7.4048752549 Si 0.8484671077 -8.0330630278 -6.8026543596 Si 3.7810581927 -6.0041722983 -8.4721630682 Si 6.9773447415 -3.4696852183 -3.2959381053 Si 6.8911561585 -5.8002179209 -6.4020195732 Si 6.2267838349 -7.6157112134 -5.0522062044 Si 6.1144886886 -5.9768162247 -8.6029953684 Si 3.1683420989 -7.7471723982 -7.0474675930 Si 7.2132605328 -7.2833863991 -2.9281774910 H 0.1915304316 3.5207565604 -1.0116315566 H -3.0006835404 5.4546596796 0.0820529129 H 0.2841361898 5.8195080185 2.0766319615 H -5.8435319667 1.5731711403 5.4557410404 H -5.9205252903 -0.7288274370 2.3821889479 H -6.1612749234 0.2016798041 0.1860624802 H -6.3155971549 4.0651008315 -0.2487087827 H -7.8348472217 2.3626656217 2.5329192339 H 2.2639711843 5.0652162808 5.0036150948 H -2.5596508625 1.9554705850 7.4386534316 H 0.7346381857 3.3441891556 7.7928697002 H 2.4274535506 1.1508860628 5.4665890219 H 0.5246553799 0.0177916074 6.3777967711 H -2.7743033774 -2.6452352184 1.1936705750 88 H 0.3872570506 8.1468756637 5.1988586880 H 0.5558549761 7.1789754572 7.3834875792 H -2.4279399909 5.2609956001 8.8816038396 H -2.9097860317 7.7371692894 3.1825213839 H -7.7252216254 4.6631451093 5.6429702520 H -8.0096457130 6.2257586385 2.1110065999 H -6.1215450962 7.3833529828 1.1904996745 H -5.7341421778 3.8657074916 8.5620070174 H -5.5373331400 7.1857981441 10.0105228249 H -7.6091882363 6.9555575110 8.8256657415 H -2.8344050818 10.0607384979 6.2409444173 H -2.6076807782 9.1562168659 8.4430390287 H -7.9449341763 8.4110490803 5.2599943553 H -6.1375828333 9.6587592027 4.3008725139 H -2.6882807608 -0.3494037099 4.3124731713 H -6.0602713166 10.0733946578 7.8840278804 H -4.7365908514 -1.8368413760 -1.7436160240 H 0.1257226429 1.2481491478 -4.1530271978 H -2.9724604728 3.1831526915 -3.0634774618 H -4.8922159152 2.0361459610 -2.2124649250 H -3.1880656116 -0.1416833071 -4.5093815617 H 3.4543055393 3.8946948670 0.9770476770 H 3.6392711983 2.9595978253 3.1724661965 H 3.7789781032 -0.9232539605 3.6089449320 H 3.4016532524 1.5918205168 -2.0978834140 H 5.3488150232 0.8187494903 0.8330421663 H 0.3870570611 -4.5729175596 0.0537398471 H 3.3347772847 -0.7150878077 -5.2170881525 H -2.8048551127 -4.9229578534 -1.8330698118 H -3.0726883600 -4.0503299897 -4.0479977188 H 5.4664358243 -3.1481879396 1.3173979773 H 3.5012700075 -4.2348444144 2.1529194742 H 0.2762794078 -6.8388318135 -3.0484862700 H 3.5749291887 -6.5077683861 -0.9904052114 H 0.2603347361 -1.0419914294 -7.3133473467 H -1.7273907631 -2.1148754064 -6.5182795211 H -1.5649639561 -6.1364180038 -6.0758542197 H 3.2925744561 -2.8861025729 -8.3018023866 H 6.4796110401 -2.6609460226 -6.3167685577 H 6.8928272730 -5.1333163158 -0.6119001409 H 6.7805447459 -1.2074736503 -1.0838113455 H 6.5238440901 -0.3413586674 -3.3027386604 H -0.0018181131 -4.3499005217 -8.7717211250 H 8.4525571781 -3.4226370572 -3.3851544083 H 8.3698310940 -5.7548695546 -6.3995889792 H 3.1402483571 -6.2637169912 -9.7807948769 H 6.7463241337 -8.8745543858 -5.6298951347 89 H 6.5678867497 -4.8359908308 -9.4187574252 H 6.5789679511 -7.2313389440 -9.2237997540 H 3.8735349027 -9.0094652895 -7.3515427269 H 0.3228028070 -8.4513310034 -8.1177670783 H 0.5843788080 -9.0982207508 -5.8203560830 H 8.6732608324 -7.1324011383 -3.0735420578 H 6.9305561184 -8.4339798853 -2.0513334176 90 REFERENCES 91 REFERENCES Canham, L. 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R., A COMPLETE ACTIVE SPACE SCF METHOD (CASSCF) USING A DENSITY-MATRIX FORMULATED SUPER-CI APPROACH. Chem Phys 1980, 48, 157-173. Shu, Y. A.; Hohenstein, E. G.; Levine, B. G., Configuration interaction singles natural orbitals: An orbital basis for an efficient and size intensive multireference description of electronic excited states. J Chem Phys 2015, 142, 024102. 95 Ufimtsev, I. S.; Martinez, T. J., Quantum Chemistry on Graphical Processing Units. 3. Analytical Energy Gradients, Geometry Optimization, and First Principles Molecular Dynamics. J Chem Theory Comput 2009, 5, 2619-2628. Fales, B. S.; Levine, B. G., Nanoscale Multireference Quantum Chemistry: Full Configuration Interaction on Graphical Processing Units. J Chem Theory Comput 2015, 11, 4708-4716. Song, C. C.; Wang, L. P.; Martinez, T. J., Automated Code Engine for Graphical Processing Units: Application to the Effective Core Potential Integrals and Gradients. J Chem Theory Comput 2016, 12, 92-106. Fales, B. S.; Hohenstein, E. G.; Levine, B. G., Robust and Efficient Spin Purification for Determinantal Configuration Interaction. J Chem Theory Comput 2017, 13, 4162-4172. Levine, B. G.; Coe, J. D.; Martinez, T. J., Optimizing conical intersections without derivative coupling vectors: Application to multistate multireference second-order perturbation theory (MS-CASPT2). J Phys Chem B 2008, 112, 405-413. 59. Meek, G. A.; Levine, B. G., Evaluation of the Time-Derivative Coupling for Accurate Electronic State Transition Probabilities from Numerical Simulations. J Phys Chem Lett 2014, 5, 2351-2356. 52. 54. 55. 56. 57. 58. 60. 61. Finley, J.; Malmqvist, P. A.; Roos, B. O.; Serrano-Andres, L., The multi-state CASPT2 method. Chem Phys Lett 1998, 288, 299-306. 53. Wadt, W. R.; Hay, P. J., Abinitio Effective Core Potentials for Molecular Calculations - Potentials for Main Group Elements Na to Bi. J Chem Phys 1985, 82, 284-298. Fernandez-Alberti, S.; Roitberg, A. E.; Nelson, T.; Tretiak, S., Identification of unavoided crossings in nonadiabatic photoexcited dynamics involving multiple electronic states in polyatomic conjugated molecules. J Chem Phys 2012, 137, 014512. Nelson, T.; Fernandez-Alberti, S.; Roitberg, A. E.; Tretiak, S., Artifacts due to trivial unavoided crossings in the modeling of photoinduced energy transfer dynamics in extended conjugated molecules. Chem Phys Lett 2013, 590, 208-213. 96 CHAPTER 4 NONADIABATIC DYNAMICS STUDY OF SILICON DANGLING BOND DEFECTS AT DIFFERENT CHARGED STATES 4.1 Introduction Dangling bond (DB) defects on the surfaces of SiNCs have been an intriguing subject to study. The low-dimensional silicon materials such as SiNCs were found to emit light as a result of quantum confinement effects.1-2 However, DB defects in such systems were deemed detrimental and unwanted since they have the ability to quench photoluminescent (PL) effectively.3-6 Besides, the fluorescence intermittency in SiNCs has recently been connected to the existence and the charging/discharging of surface DBs.7 Both effects impair the potential employment of SiNCs for light emitting applications, in spite of the great advantage of its compatibility to modern silicon technology and infrastructure.8-11 In addition, DBs work as deep- level charge traps in photovoltaic materials, which impose limits on the performance of SiNCs in optoelectronic devices as well.12 However, there is renewed interest among researchers who are now taking advantages to make useful applications out of DBs, based on the unique electronic properties and chemical reactivity as well as the experimental techniques to precisely pattern DBs on surfaces.13-14 For example, DBs can be engineered to work as coupled quantum wells for quantum computing qubits.15 Besides, the DB on the surface of SiNCs can initialize and direct chain reactions of organic molecules,16 and different mechanisms for a chemical reaction occur on surfaces with DBs of different charge states.17 Our group is particularly interested in the photophysics of SiNCs (along with other semiconductor nanocrystals18) with various kinds of defects.19-25 Although it is well known that defects play an important role in nonradiative (NR) decay processes, it is generally hard to identify a specific defect as a NR decay center, and the atomistic details of NR decay processes are not completely understood. Knowledge of such mechanisms is the key for rational design and device optimization. In view of this, we applied ab initio molecular dynamics (AIMD) to simulate electronic and nuclear dynamics after excitation. Without assuming a particular reaction coordinate coupled to the NR decay processes, our studies bring new insights into the photophysics of defective SiNCs via the notion of defect-induced conical intersections (DICIs) that facilitate NR 97 recombination.26 For the neutral DB-SiNCs, we reveal a previously unknown NR decay mechanism in DB-SiNCs. In addition to the conventional wisdom that a pyramidalization mode where Si-Si bonds surrounding DB undergo symmetrical stretching is involved in the NR decay process,7, 27 we found asymmetrical stretching modes (one Si-Si bond or two Si-Si bonds stretching) surrounding the DB site that are important to the NR recombination process. These vibrational modes are driven by the Jahn-Teller effect on the doubly degenerate first excited state, which drives the system toward a DICI between the first excited and the ground states. Due to the accessibility of the DICI, the excitation energy will dissipate to heat efficiently when the vertical excitation energy of an SiNC is larger than the energy of its DICI (estimated 2.4 eV by a complete active space configuration interaction method in a SiNC model). The discovery of conical intersections in defective SiNCs is consistent with several experimental findings. Most importantly, the existence of energetically and kinetically accessible DICIs between the first excited and ground states in defective SiNCs reconciles the experimental observations of slow (S-) band in SiNCs after oxidation. Some experiments found the emission of S-band is independent of the size of SiNCs,28 but others suggested that the size-independent red-shifted PL (S-band) is from quantum- confined states rather than defect states.29-31 Based on our DICI concept, when an SiNC has an energy gap larger than the energy of DICI, the PL of oxidized SiNCs becomes susceptible to the CI since it is energetically accessible. Thus, the unusual features of the S-band originates from the quenching of smaller wavelength emitters in an ensemble of SiNCs after oxidation, which causes an overall red-shifted and size-insensitive PL spectra. Silicon DBs can exist in positive (DB+) and negative (DB-) charged states other than the neutral (DB0) state, adjusting to the environment (e.g. electrochemical potential). The attachment/removal of an electron to/from the DB alters its electronic structure, that in turn changes the properties and achievable applications as introduced in the first paragraph. To obtain a thorough picture for the photophysics of SiNCs with DB defects, we consider different charged states of DBs, including nonadiabatic effects to the simulations. We apply the ab initio multiple spawning (AIMS) method to study the dynamics of the neutral as well as charged silicon DBs after excitation using small cluster models. As mentioned above, the dynamical simulations give us valuable insights into the atomistic details of photophysical processes. The important points on the potential energy surfaces (PESs), such as minimum energy geometries on the excited state 98 and conical intersections (CIs), can be explored by the dynamical simulations, and further extraction of the energies and locations are performed by geometry optimization with higher order methods that include both static and dynamical electron correlation. It is worth mentioning that most of our previous simulations examine the NR recombination processes between the defect localized state and ground state. For the energy transfer process from the delocalized excitonic state to the defect localized state, we have estimated the transfer rate based on a Marcus-like theory.24 The consideration of negatively charged DB will give us some insight of this process from direct dynamics simulations for the first time. Based on our results, experimental means to observe NR decay via DB is briefly discussed. We have shown in previous studies that CIs are generally localized in nature,25, 32 thus a small cluster model can give us a reasonable estimate of the geometry and energy of first excited/ground states MECI, and the results can be applied to understand the NR decay process via DICIs in defective SiNCs. The other merit of using small cluster models is that by reducing the dimensionality of the SiNC the important reaction coordinates related to the NR decay process are easier to extract. When the full dimensionality is applied, the analysis of structural dynamics can be projected to the reaction coordinates extracted from the cluster models to understand the process. The caveat is that the estimated rate might be affected by the overestimation of the excitation energy and the reduced rate of energy dissipation due to the reduced dimensionality of the small cluster model. 4.2 Method The cluster model (Si8H13) for dynamic study is shown in Figure 4.1a. All silicon atoms are passivated to full valency with hydrogen atoms except one silicon with only trivalent coordination (surrounded by three silicon atoms) to represent the dangling bond defect. Thus, the defect can be deemed as a Pb center, which commonly appears on the interfaces of bulk silicon and SiNCs. In this study, we investigate the DB defect in different charged states, which are the neutral, the positively charged (the cationic), and the negatively charged (the anionic) states. The geometry of the clusters at their ground state (Frank-Condon point) are optimized via B3LYP33-34 with the 6-31G(d,p) basis set for the neutral and cationic states and the 6-31+G(d,p) basis set for 99 the anionic state. We applied the ab initio multiple spawning (AIMS) method,35-36 which allows us to calculate full time-dependent wave functions of molecules after excitation, to study the photophysics of the Si clusters with DB defects. The electronic wave functions are described at state averaged complete active space self-consistent field (SA-CASSCF)37-39 level with LANL2DZdp basis sets.40 It is worth mentioning that the LANL2DZdp basis contains diffuse gaussian functions for properly describing negatively charged states. For example, the most diffuse gaussian exponent on the p shell of the Si atom is 0.0237 for LANL2DZdp basis, where it is 0.0331 for the 6-31+G(d) basis set. For the neutral state, we averaged three states to optimize the orbitals and used an active space with five electrons and three orbitals in the SA-CASSCF calculations (abbreviated as SA3-CASSCF(5,3)). For the charged states, the SA3-CASSCF(4,3) and SA2-CASSCF(6,4) have been used for the cationic and anionic states, respectively. These active spaces are chosen because of their agreement to the excitation energies and excitation characters predicted by high level methods, such as (ionization potential) equation-of-motion coupled cluster with single and double excitations (EOM-CCSD41-43 for charged states and IP- EOM-CCSD(2h,1p)44-47 for the neutral state), multi-state complete active space second order perturbation theory (MS-CASPT2)48-50 and multireference configuration interaction method (MRCI) with the Davidson correction51-53 using the same state average and active space as above (the energies can be found below and in APPENDIX). Twenty simulations are conducted in the AIMS calculations with each simulation initiated with one nuclear basis function. The initial positions and momenta for each nuclear wavepacket are sampled from the ground state vibrational Wigner distribution. The time-derivative coupling is evaluated via the norm- preserving interpolation method developed by Meek et al.54-55 In this paper, we adopted adiabatic state labels, in which D0 (S0) denotes the doublet (singlet) ground state, D1 (S1) is the doublet (singlet) first excited state, etc. The important points on the potential energy surfaces (PES), which include the energy minimum on the first excited state (denote as D1min and S1min) and the minimal energy point on the CI seam (MECI), are optimized at the SA-CASSCF and MS- CASPT2 levels of theory. The SA-CASSCF, MS-CASPT2, MRCI and EOM-CCSD electronic structure calculations are performed in the Molpro software package,56 and the AIMS dynamic simulations are perform using FMS-Molpro.36 The IP-EOM-CCSD calculations are performed in the GAMESS program.44-47, 57-58 On the other hand, the MECI geometry optimization is 100 performed using the CIOpt software package, which uses the sequential penalty method to optimize the minimum on the seam of conical intersections.59 4.3 Results and Discussion 4.3.1 Dynamical Simulations of the Neutral Si-DB At the ground state, the three Si-Si bond lengths, RSi-Si, and Si-Si-Si bond angles, q, surrounding the DB site (as illustrated in Figure 4.1a) are practically identical for all three charge states. For the DB neutral state, the bond lengths RSi-Si are 2.36 Å and bond angles q are 109°. We run AIMS in conjunction with the SA3-CASSCF(5,3) electronic structure method to study the photodynamics after excitation in the neutral DB system. The dynamics after excitation are essentially the same as our previous work on real nanometer-sized models of DB-SiNCs.25 To shortly summarize, the averaged (over twenty trajectories) D1-D0 energy gap approaches zero at 40-60 fs with many (if not all) trajectories exploring regions of near-zero energy gap (Figure 4.2), suggesting an accessible D1/D0 CI in the system. The ultrafast population transfer from the D1 to the D0 state (Figure 4.3) further suggests the existence of CIs. More than 60% of the initially prepared D1 state population is transferred to the ground state within the first hundred fs. For the structural dynamics, all three averaged bond lengths, RSi-Si, and bond angles, q, increase within 50 fs (Figure 4.4), where one bond length is significantly elongated more than the other two, resulting in an asymmetric bond stretching motion surrounding the DB. The structural dynamics after excitation can be attributed to the Jahn-Teller effect. As the orbital arrangements show in Figure 4.1b, the first excited state at the FC point is a doubly degenerate state corresponding to the excitation of one electron from one of the two sSi-Si bonding orbitals surrounding the DB site to the DB nonbonding orbital (nDB). As a result, the structure undergoes Jahn-Teller distortion on the first excited state at the FC point, in which the asymmetric bond stretching of RSi-Si breaks the degeneracy and stabilizes the structure. 101 (a) (b) (c) Rsi-si θ nDB σSi-Si σSi-Si nDB σSi-Si σSi-Si Figure 4.1. (a) The Si8H13 cluster model studied in this work. One of the three Si-Si bond lengths (RSi-Si) and Si-Si-Si bond angles (q) surrounding the DB site are indicated in red. (b) The top view of the one nonbonding and two sSi-Si bonding orbitals at the FC point in the neutral state system. (c) The side view of the diffuse LUMO and nonbonding HOMO at the FC point in the negatively charged DB system. Figure 4.2. The time evolutions of the D1-D0 energy gap (grey lines: 20 individual trajectories; red line: average) for the neutral DB system. 102 Figure 4.3. The time evolution of the population of the first excited and ground states of the neutral DB system. Cyan lines: excited state populations of 20 trajectories; magenta lines: ground state populations of 20 trajectories; thick blue line: averaged excited state population; thick red line: averaged ground state population. 103 (a) (b) Figure 4.4. The time evolutions of (a) the averaged Si-Si bond lengths surrounding the DB (RSi- Si) and (b) the averaged Si-Si-Si bond angles surrounding the DB (q) for the neutral DB system. The bond lengths (bond angles) are sorted as the longest (largest), the middle and the shortest (smallest) RSi-Si distances (q degree) for each trajectory at each time step, then averaged, and are shown by yellow, orange and blue lines, respectively. 4.3.2 Static Analysis of the Potential Energy Surface on the Neutral Si-DB To calibrate and qualify the accuracy of SA-CASSCF as well to gain chemical insight through analysis of the PES, we applied higher order methods and geometry optimizations for important points on the PES. At the FC point, we compared the vertical excitation energies to those computed at the IP-EOM-CCSD level of theory (see Table C1) at the minimum energy geometry of the ground state, optimized by B3LYP/6-31G(d,p). IP-EOM-CCSD predicts a doubly-degenerated first excited state with a 3.08 eV excitation energy at the FC point, and the SA3-CASSCF(5,3) excitation energy for the doubly-degenerated D1 state is 3.63 eV (Table 4.1). The discrepancy between the SA-CASSCF and IP-EOM-CCSD is most likely due to the lack of dynamical correlation in the SA-CASSCF method. After inclusions of dynamical correlation, MS-CASPT2 and MRCI methods with the same active space gave excitation energies of 3.12 and 3.29 eV, respectively, as shown in Table 4.1 and APPENDIX, which are in good agreement with the IP-EOM-CCSD result. 104 Table 4.1. Optimized neutral DB cluster model FC point, D1/D0 MECI, and D1min energies (eV). FC D1/D0 MECI D1min SA3-CASSCF(5,3) D0 Energy 0 2.35 1.87 D1 Energy 3.63 2.35 2.33 MS-CASPT2 D0 Energy 0 1.73 --- D1 Energy 3.12 1.74 --- The AIMS trajectories exploring near-zero gap region and the population transfer from the D1 to D0 state suggest that there is a CI in the neutral DB system. To characterize the decay pathways, we applied geometry optimization using SA-CASSCF and MS-CASPT2 methods to search for the D1/D0 MECI as well as the D1min. Single point energy calculations are performed at these optimized geometries with the MRCI method and are presented in the APPENDIX. With the initial guess geometries drawn from the near-zero gap region, we found the D1/D0 MECI at both level of theories. The geometry of the MECI (Figure 4.5) shows the asymmetrically stretched Si-Si bonds surrounding the DB site (RSi-Si = 2.38 Å, 2.52 Å, 2.53 Å; q = 122°, 128°, 117° optimized at SA-CASSCF level of theory; RSi-Si = 2.37 Å, 2.45 Å, 2.46 Å; q = 123°, 129°, 119° optimized at MS-CASPT2 level of theory). The MECI energy is estimated to be 2.35 eV and 1.74 eV above the D0 state minimum at the SA-CASSCF and MS-CASPT2 levels, respectively (Table 4.1). The nature of the electronic transition between the D1 and D0 states via MECI can be examined by computing state-averaged natural orbitals (NOs) at the D1/D0 MECI geometry. As shown in the Figure 4.5, the NOs with occupation number 1.5 have the characters of the nDB and one of the sSi-Si orbitals, thus these two orbitals become near-degenerated, introducing the MECI. For the D1min, we found a minimum energy structure at the SA-CASSCF level of theory, which is very close to the D1/D0 MECI with geometric differences less than 0.02 Å for RSi-Si (RSi-Si = 2.38 Å, 2.50 Å, 2.51 Å) and 3° for q (q = 121°, 125°, 117°), and a 0.02 eV difference for the D1 state energy (Table 4.1). We cannot find a distinct D1min geometry near the D1/D0 MECI at the MS-CASPT2 level. The results suggest that the Jahn-Teller distortion at the FC point bring the system toward the D1/D0 CI directly, which facilitates the ultrafast NR recombination for the neutral DB system. The accuracy of the dynamical simulations with the SA-CASSCF PES can be qualified by comparing the difference between the D1 state energies at FC and MECI points to the results from higher level of theories. SA-CASSCF predicts an 1.28 105 eV energy difference, where MS-CASPT2 and MRCI (with an MS-CASPT2-optimized geometry) gave 1.38 eV and 1.33 eV, respectively. Hence, agreement is very good and the dynamical simulations are trustworthy. Also, it is worth mentioning that the results for the neutral DB system support the use of a small cluster model. The structural dynamics after excitation, the time scale for reaching the near-zero gap region, and the electronic and geometric structures in the study of this small cluster model agree well with the previous DB-SiNCs results.25 (a) (c) D1/D0 MECI CASSCF HOMO (n = 1.50) (b) (d) D1/D0 MECI CASPT2 LUMO (n = 1.50) Figure 4.5. (a) The D1/D0 MECI geometry of the neutral DB cluster optimized by SA-CASSCF. (b) The D1/D0 MECI geometry of the neutral DB cluster optimized by MS-CASPT2. (c) and (d). Top view and side view of D0-D1 state-averaged natural orbitals with occupation numbers 1.50 for the neutral DB at the MECI geometry. 106 4.3.3 Dynamical Simulations of the Cationic Si-DB After removing one electron from the neutral DB, the positively charged DB cluster has smaller bond lengths (RSi-Si = 2.34 Å) and a more planar geometry (q = 99°) surrounding the DB site at the minimum energy geometry on its ground state. We applied AIMS in conjunction with the SA3-CASSCF(4,3) electronic structure method to study the photophysical dynamics in the DB+ system after excitation. Similar to the DB0 system, the first excited state at the FC point is a doubly degenerated excited state characterized by an electronic excitation from one of the sSi-Si orbitals to the nDB orbital (see Figure C2). Thus, the dynamics after excitation are also driven by the Jahn-Teller effect at FC geometry, where the asymmetric Si-Si bond stretching modes surrounding the DB defect break the degeneracy on the first excited state. As shown in Figure 4.6, the averaged bond lengths, RSi-Si, as well as bond angles, q, increase to their maximum at ~50 fs, where one of the Si-Si bonds grows significantly longer than the other two. The time evolution of the energy gap is shown in the Figure 4.7, in which a few trajectories (gray lines) occasionally explore near-zero gap region suggesting an accessible CI in the positive DB system. Although the structural dynamics after excitations are similar to the DB0 system and the trajectories explore near-zero gap region in both cases, the nonadiabatic transition rate to the ground state is much slower in the DB+ system as estimated by AIMS simulations (Figure 4.8). Only 3% of the averaged population is transferred to the ground state after 1 ps based on the simulations. Nonetheless, the time-scales are not directly relevant to realistic nanocrystals, but the location and energy of the MECI play more important roles in our interpretation of our simulations. 107 (a) (b) Figure 4.6. The time evolutions of (a) the averaged Si-Si bond lengths surrounding the DB (RSi- Si) and (b) the averaged Si-Si-Si bond angles surrounding DB (q) for the positive DB system. The bond lengths (bond angles) are sorted as the longest (largest), the middle and the shortest (smallest) RSi-Si distances (q degree) for each trajectory at each time step, then averaged, and are shown by yellow, orange and blue lines respectively. Figure 4.7. The time evolutions of the S1-S0 energy gap (grey lines: 20 trajectories; red line: average) for the positive DB system. 108 Figure 4.8. The time evolutions of the population of the first excited and ground states for the positive DB system. Cyan lines: excited state populations of 20 trajectories; magenta lines: ground state populations of 20 trajectories; thick blue line: averaged excited state population; thick red line: averaged ground state population. 4.3.4 Static Analysis of the Potential Energy Surface on the Cationic Si-DB The PES of SA-CASSCF method for the positive DB system is calibrated with higher order methods. In addition, chemical insight can be obtained through static analysis of the important points on the PES. Thus, we applied EOM-CCSD, MS-CASPT2 and MRCI levels of theories for these purposes. At the FC point, EOM-CCSD method is performed at the minimum energy geometry (see Table C1) of the ground state (B3LYP/6-31G(d,p) optimized). The first excited state is a doubly-degenerated state with excitation energy 2.89 eV calculated by EOM- CCSD. SA3-CASSCF(4,3) predicts the excitation energy to be 3.32 eV (Table 4.2). After inclusion of dynamical correlation, MS-CASPT2 and MRCI with the same active space predict 109 excitation energies of 2.61 and 3.00 eV, respectively, as shown in Table 4.2 and APPENDIX. These are in good agreement with the EOM-CCSD result. Table 4.2. Optimized positive DB cluster model FC point, S1/S0 MECI, and S1min energies (eV). FC S1/S0 MECI S1min SA3-CASSCF(4,3) S0 Energy 0 2.43 1.40 S1 Energy 3.32 2.45 2.26 MS-CASPT2 S0 Energy 0 2.08 1.16 S1 Energy 2.61 2.10 1.83 The AIMS trajectories for the DB+ cluster occasionally explore regions of near-zero gap suggesting the existence of a S1/S0 CI in the system. Geometry optimizations at the SA-CASSCF and MS-CASPT2 levels are applied to search for the S1/S0 MECI as well as the S1min. MRCI single point calculations are performed at these geometries and are presented in the APPENDIX. Using a geometry from the region of near-zero gap from the AIMS simulations as an initial guess, we discovered a low-lying S1/S0 MECI for the DB+ cluster at both the SA-CASSCF and MS-CASPT2 levels of theories. As shown in Figure 4.9, the MECI geometry shows asymmetrical stretching of the Si-Si bonds surrounding the DB defect, with three RSi-Si as 2.37 Å, 2.39 Å, 2.75 Å for the SA-CASSCF method and 2.34 Å, 2.35 Å, 2.51 Å for the MS-CASPT2 method. The three q are 111°, 113°, 110° and 112°, 119°, 111°, respectively. The MECI energies are estimated to be 2.45 eV and 2.10 eV at the SA-CASSCF and MS-CASPT2 levels, respectively (Table 4.2). The S0-S1 state-averaged NOs show the nature of the electronic transition facilitated by the MECI, in which the transition is between the nDB orbital (occupation number 0.53) and one of the sSi-Si orbitals surrounding the DB (occupation number 1.48), as shown in Figure 4.9. For the S1min, we found the minimum energy geometries at both level of theories, in which the three RSi-Si are estimated 2.37 Å, 2.38 Å, 2.67 Å at SA-CASSCF and 2.33 Å, 2.41 Å, 2.42 Å at MS-CASPT2, and the energies are 2.26 and 1.83 eV respectively (Table 4.2). As can be seen from the results of both methods, the S1/S0 MECI geometry exhibits significant stretching of one Si-Si bond surrounding the DB from S1min geometry. The S0-S1 energy gap at the S1min is large, estimated to be 0.86 eV at the SA-CASSCF level and 0.67 eV at the MS-CASPT2 level. As a result, the Jahn-Teller distortion at the FC point in the DB+ cluster drives the system toward the S1min, which has a large energy gap between the first excited and 110 ground states. The S0-S1 energy gap remains large near the S1min region, with no driving force toward the MECI, resulting in a much slower NR decay rate compared to the DB0 system. In addition, the energy difference between the S1min and S1/S0 MECI (0.19 eV for SA-CASSCF and 0.27 eV for MS-CASPT2) can be deemed as an energy barrier for the system to reach the near-zero energy gap region. Nevertheless, the estimated NR decay rate is still orders of magnitudes faster than the typically time scale for radiative decay process in SiNCs (~µs), and we expect that SiNCs with energy gaps larger than the MECI energy do not efficiently emit light when positively charged DBs are presente. The energy difference between the S1 energies at the FC and MECI points is 0.87 eV for SA-CASSCF, and it is 0.51 eV and 0.62 eV for MS-CASPT2 and MRCI (at the MS-CASPT2 optimized geometry), respectively. The agreement between methods are reasonable, thus the dynamical simulations can be deemed as reliable. 111 (a) (c) S1/S0 MECI CASSCF HOMO (n = 1.48) (b) (d) S1/S0 MECI CASPT2 LUMO (n = 0.53) Figure 4.9. (a) The S1/S0 MECI geometry of the positive DB cluster optimized by SA-CASSCF. (b) The S1/S0 MECI geometry of the positive DB cluster optimized by MS-CASPT2. (c) and (d) Top view and side view of the S0-S1 state-averaged natural orbitals, with fractional occupation numbers 1.48 and 0.53, for the positive DB at the MECI geometry. 4.3.5 Dynamical Simulations of the Anionic Si-DB Opposite to the DB+, the minimum energy structure of the ground state of the negatively charged DB system has the longest bond lengths (RSi-Si = 2.39 Å) and the largest bond angles (q = 121°) surrounding the DB site, due to charge congestion. The excitation character of the DB- cluster at the FC point is one electron excited from the nDB orbital to a diffuse orbital (Figure 4.1c). Because the excitation character is different from that in the DB0 and DB+ systems, the DB- system has distinct dynamics after excitation. In the first 30 fs, both the averaged bond lengths, RSi-Si, and the averaged bond angles, q, decrease with time, as shown in the Figure 4.10. 112 After this decreasing stage, the averaged bond lengths, RSi-Si, shows a prominent stretching in one Si-Si bond with the other two bonds remaining relatively unchanged. The averaged energy gap between the first excited and ground states never approach zero (always > 0.7 eV as the red line in Figure 4.11), but many trajectories (gray lines in Figure 4.11) explore regions of near-zero gap. The AIMS simulations predict population transfer between first excited and ground states (Figure 4.12), which suggests an accessible S1/S0 CI in the DB- system as well. (a) (b) Figure 4.10. The time evolutions of (a) the averaged Si-Si bond lengths surrounding the DB (RSi- Si) and (b) the averaged Si-Si-Si bond angles surrounding the DB (q) for the negative DB system. The bond lengths (bond angles) are sorted as the longest (largest), the middle, and the shortest (smallest) RSi-Si distances (q degree) for each trajectory at each time step, then averaged, and are shown by yellow, orange and blue lines respectively. 113 Figure 4.11. The time evolutions of the S1-S0 energy gap (grey lines: 20 trajectories; red line: average) for the negative DB system. Figure 4.12. The time evolutions of the population of the first excited and ground states for the negative DB system. Cyan lines: excited state populations of 20 trajectories; magenta lines: ground state populations of 20 trajectories; thick blue line: averaged excited state population; thick red line: averaged ground state population. 114 4.3.6 Static Analysis of the Potential Energy Surface on the Anionic Si-DB To calibrate the SA-CASSCF method and gain chemical insight through static analysis of PES, we applied higher order methods such as EOM-CCSD, MS-CASPT2 and MRCI level of theories to the negative DB system. The SA-CASSCF method for the DB- system is calibrated by the EOM-CCSD level of theory (Table C1) on the ground state minimum energy geometry (B3LYP/6-31+G(d,p) optimized). EOM-CCSD predicted a 3.54 eV excitation energy for the first excited state, and SA2-CASSCF(6,4), MS-CASPT2 and MRCI methods gave 3.51 eV, 3.79 eV and 3.79 eV, respectively (Table 4.3 and APPENDIX). These are in good agreement with EOM- CCSD result. Thus, we applied SA2-CASSCF(6,4) as the electronic structure method for the AIMS dynamical simulations. Table 4.3. Optimized negative DB cluster model FC point, S1/S0 MECI, and S1min energies (eV). FC S1min-1 S1/S0 MECI S1min SA2-CASSCF(6,4) S0 Energy 0 0.54 2.70 1.84 MS-CASPT2 S0 Energy 0 0.57 2.63 1.57 S1 Energy 3.79 2.91 2.65 2.38 S1 Energy 3.51 2.64 2.72 2.65 Many of the AIMS trajectories explore regions of near-zero gap and exhibit population transfer between the first excited and ground statesm suggesting the existence of a low-lying S1/S0 CI in the DB- system. We applied geometry optimization to search for the S1/S0 MECI in this system with both the SA-CASSCF and MS-CASPT2 electronic structure methods. The single point energy calculations on the optimized geometries are carried out at the MRCI level of theory and presented in APPENDIX. The low-lying MECI has been found at both levels of theory, where the estimated energies are 2.72 eV and 2.65 eV for SA-CASSCF and MS-CASPT2 levels of theory, respectively (Table 4.3). The important geometry parameters, RSi-Si, are 2.32 Å, 2.32 Å, 2.77 Å for SA-CASSCF and 2.31 Å, 2.33 Å, 2.56 Å for MS-CASPT2, and the three q are estimated 103°, 98°, 91° and 105°, 98°, 94°, respectively (see the MECI geometry also in Figure 4.13). Thus, the geometry at the S1/S0 MECI of the negative DB system is a local distortion of one of the Si-Si bonds surrounding the DB. The electronic structure at the MECI 115 geometry can be examined by calculating the state-averaged NOs. The NR recombination can be characterized as a transition between a sSi-Si* orbital and the nDB orbital, as shown in Figure 4.13. The sSi-Si* orbital is an antibonding orbital for one of the Si-Si bonds surrounding the DB site. Thus, the significant elongation of one Si-Si bond is caused by the repulsive potential from the antibonding electronic character in this bond. (a) (c) S1/S0 MECI CASSCF HOMO (n = 1.50) (b) (d) S1/S0 MECI CASPT2 LUMO (n = 0.60) Figure 4.13. (a) The S1/S0 MECI geometry of the negative DB cluster optimized by SA- CASSCF. (b) The S1/S0 MECI geometry of the negative DB cluster optimized by MS-CASPT2. (c) and (d) Top view and side view of the S0-S1 state-averaged natural orbitals with fractional occupation numbers 1.50 and 0.60 for the negative DB cluster at the MECI geometry. For the excited state PES in the DB- system, we searched for the minimum energy geometries at two regions: region where the bond lengths, RSi-Si and bond angles, q, are relaxed and the region nearby the S1/S0 MECI structure. Thus, the initial structures for the two regions are taken from the AIMS trajectories at ~30 fs and the optimized MECI geometry. Distinct 116 minimum energy geometries are found in both regions at both the SA-CASSCF and MS- CASPT2 levels. In the relaxed region (denoted as S1min-1), both bond lengths, RSi-Si, and bond angles, q, are smaller than in the structure at the FC point. The RSi-Si are estimated to be 2.34 Å, 2.34 Å, 2.35 Å by SA-CASSCF and 2.31 Å, 2.31 Å, 2.36 Å by MS-CASPT2, and the three q are estimated 107°, 108°, 106° and 105°, 107°, 101°, respectively. Thus, the exploration from the FC point to the nearby S1min-1 region is responsible for the first ~30 fs of dynamics after excitation in the AIMS simulations, where the geometry is relaxed from the charge-congested DB structure. The energy at S1min-1 is estimated to be 2.64 eV at the SA-CASSCF level and 2.91 eV at the MS-CASPT2 level (Table 4.3). On the other hand, the minimum energy geometry near the MECI region (denoted as S1min) contains similar electronic character to the S1/S0 MECI (see Figure C3). The geometry surrounding the DB is 2.32 Å, 2.33 Å, 2.63 Å for RSi-Si and 106°, 103°, 97° for q at the SA-CASSCF level, and is 2.31 Å, 2.39 Å, 2.41 Å for RSi-Si and 106°, 99°, 102° for q at the MS-CASPT2 level. Thus, in the MECI geometry one Si-Si bond surrounding DB site is stretched further relative to the S1min region. The antibonding character of sSi-Si* provides the driving force for the system to reach the S1min region, which is near the S1/S0 CI. To qualify the SA-CASSCF PES used in the AIMS simulations, the difference in S1 energy between the FC and MECI points is considered. It is 0.79 eV at the SA-CASSCF level, and 1.14 eV and 0.93 eV for MS-CASPT2 and MRCI (at the MS-CASPT2 optimized geometry), respectively. Thus, the agreement between methods are reasonable and the dynamical simulations can be deemed as reliable. The fact that the dynamics explore two regions in negative DB system can also be seen upon examining the individual trajectories. Unlike the DB0 and DB+ systems, where the individual trajectories follow the averaged time evolution of the energy gap (red line in Figure 4.2 and Figure 4.7) reasonably well, the individual trajectories diverge from averaged value after the first few tensof fs (relaxation process) and span the range 0-2 eV throughout the simulations in the DB- system. In addition, although the averaged structural dynamics show a prominent one Si-Si bond (RSi-Si) stretching mode after the relaxation process (Figure 4.10), this one bond stretching mode occurs at different times for individual trajectories. For instance, Figure 4.14a presents a trajectory where a single RSi-Si stretches at 200 fs, which is similar to the averaged 117 dynamics. However, in Figure 4.14b the other trajectory has three RSi-Si oscillating roughly in between 2.2-2.5 Å from 0-800 fs, followed by the stretching of a single RSi-Si at 900 fs. (a) (b) Figure 4.14. The plot of three Si-Si bond lengths surrounding DB site changes with time for (a) the simulation where the one Si-Si bond stretch occurred at 200 fs, (b) the simulation where the one Si-Si bond stretch occurred at 900 fs. 4.3.7 Discussion According to the results, DBs with different charged states all undergo NR decay via DICIs. We found low-lying MECIs between the first excited and ground states for all three systems, with MS-CASPT2 energies 1.74 eV, 2.10 eV and 2.65 eV above the minimum energy of the ground state for the neutral, positively and negatively charged states respectively. The estimated MECI energies are within the visible light range, thus the DB DICIs are capable of quenching visible photoluminescence. Along with our dynamical simulations and static analysis of the PESs, we investigated the NR decay mechanisms between the first and ground states for all three charged states. Although the microscopic mechanisms are related, with the asymmetric bond stretching mode(s) surrounding DB site appear to be important for the NR decay processes, the detailed mechanisms and dynamics are distinct. For the DB0 and DB+ systems, the asymmetrical stretching modes can be either elongations of one or two Si-Si bonds surrounding the DB. Although similar structural dynamics are observed after excitation, the recombination rates of these cluster models are estimated DB0 >> DB+ based on the dynamical simulations as well as 118 the static analysis of the PES. For the DB- system, it is stretching of a single Si-Si bond mode launched by the repulsive potential of an occupied sSi-Si* orbital surrounding the DB site that brings the system toward the DICI. It is interesting to compare our results to the related study by Brawand et al.,7 where the authors proposed that the changing charge state of the DB might be responsible for the fluorescence intermittency in SiNCs. The authors considered four elementary pathways. First, the DB0 à DB+ pathway corresponds to the electron transitioning from a singly occupied nDB nonbonding orbital to the valence band (s bonding orbital in our study), which we denoted as (s)1(nDB)1à(s)2(nDB)0. Second, the DB0 à DB- pathway corresponds to the electron transition from the conduction band (s* antibonding orbital in our study) to the singly occupied nDB, which can be denoted as (nDB)1(s*)1à( nDB)2(s*)0. Third, the DB- à DB0 pathway corresponds to the electron transition from doubly occupied nDB nonbonding orbital to the s bonding orbital, which we denoted as (s)1(nDB)2à(s)2(nDB)1. And finally, the DB+ à DB0 pathway considers the electron transition from s* antibonding orbital to an unoccupied nDB, thus we denoted as (nDB)0(s*)1à( nDB)1(s*)0. They applied Fermi’s golden rule to evaluate both the nonradiative and radiative recombination rates for these four processes. Our simulations of neutral, positive and negative DB systems correspond to (s)1(nDB)2à(s)2(nDB)1, (s)1(nDB)1à(s)2(nDB)0 and (nDB)1(s*)1à( nDB)2(s*)0 pathways respectively, in which we found low-lying DICIs for all three systems. Brawand, et al. predicted these pathways to be NR processes as well. Thus, the two studies are in agreement with each other regarding the identification of the NR decay pathways in DB-SiNCs. We do not consider the radiative pathway (nDB)0(s*)1à( nDB)1(s*)0 they suggested because it is not the lowest energy mechanisms in our cluster model. For the neutral DB cluster at the FC point, the low-lying excitations predicted by the IP-EOM-CCSD level of theory compose of excitations from valence electron to the nDB orbital. Thus, the excitations of one electron to virtual orbitals of the neutral DB cluster are higher in energy. The merit of our method is that we use dynamical simulations to explore the PES of different charged DB directly, where Brawand et al. assumed the same symmetric stretching mode (pyramidalization) surrounding the DB for NR decay processes in all pathways. We found that for the NR decay processes the excitation energies of differently charged DB systems are dumped to similar 119 vibrational mode (asymmetric stretching mode), but occur via different mechanisms. These local distortion mode(s) can work as the signature for the NR decay processes via DB DICIs. If a time- resolved spectra for monitoring the asymmetrically vibrational mode surrounding DB is available, the NR decay via differently charged DB sites may be identifiable. In turn, it may provide a chance to validate the proposed blinking mechanism as well. One can imagine that the four pathways investigated by Brawand et al. can be distinguished from the structural dynamics of the processes ((s)1(nDB)2à(s)2(nDB)1, (s)1(nDB)1à(s)2(nDB)0 and (nDB)1(s*)1à( nDB)2(s*)0) and the radiative signal ((nDB)0(s*)1à( nDB)1(s*)0). 4.4 Conclusions In this study, we investigated the nonadiabatic dynamics occurring upon excitation of differently charged tsilicon DB defects using the ab initio multiple spawning method in conjunction with a SA-CASSCF description of the electronic structure. In addition, energy calculations at the Franck-Condon geometries, the excited state minimum energy geometries and the minimal energy conical intersection geometries are performed with higher order levels of theory such as EOM-CCSD, MS-CASPT2 and MRCI methods. We found low-lying conical intersections which can facilitate nonradiative decay in all three systems. The nonradiative decay processes involve signature asymmetrical Si-Si bond stretching mode(s) surrounding the DB site in all three cases. However, the microscopic mechanisms are distinct. For the neutral DB, the excited state force drives the system toward the low-lying conical intersection directly, thus result to ultrafast NR decay. The cationic DB has similar structural dynamics after excitation as in the neutral state DB. However, the excited state force does not drive the system to the conical intersection directly but to the S1min region with larger S0/S1 energy gap. As a result, the NR decay is much slower in the positive DB system than the neutral DB system. For the anionic DB, the dynamics explore two regions after excitation. First, a relaxation process brings the system to approach an excited state minimun S1min-1. Then, the other region of the S1min can be accessed, with the S0/S1 MECI located nearby. The signature asymmetric Si-Si bond stretching mode(s) and different microscopic mechanisms provide opportunities to examine the NR decay processes via DB DICIs, if the experimental means are available. 120 APPENDIX SUPPORTING INFORMATION FOR: NONADIABATIC DYNAMICS STUDY OF SILICON DANGLING BOND DEFECT AT DIFFERENT CHARGED STATES Table C1. The excitation energies at Franck-Condon point (FC). The FC geometries are optimized with the B3LYP/6-31G(d,p) method. The IP-EOM-CCSD(2h,1p)/6-31G(d,p), EOM- CCSD/6-31G(d,p) and EOM-CCSD/6-31+G(d,p) energies are calculated for the neutral, cationic and anionic DB systems respectively. State ES1(FC) ES2(FC) ES3(FC) ES4(FC) Neutral IP-EOM- CCSD(2h,1p) 3.08 3.08 3.96 3.96 Cationic EOM-CCSD Anionic EOM-CCSD 2.89 2.89 3.40 3.40 3.54 3.78 3.78 3.94 Table C2. The optimized cluster model FC, MECI and D1min energies for the neutral DB system. geometry optimization method B3LYP SA- CASSCF MS- CASPT2 SA- CASSCF FC D0/D1 MECI D1min SA- CASSCF D0 energy 0 2.35 SA- CASSCF D1 energy 3.63 2.35 MS- CASPT2 D0 energy 0 1.78 MS- CASPT2 D1 energy 3.12 1.81 MRCI D0 energy MRCI D1 energy 0 1.96 3.29 1.98 2.39 2.41 1.73 1.74 1.93 1.96 1.87 2.33 1.40 1.83 1.55 1.99 121 Table C3. The optimized cluster model FC, MECI and S1min energies for the positive DB system. geometry optimization MRCI S0 energy MRCI S1 energy 0 2.18 3.00 2.36 method B3LYP SA- CASSCF MS- CASPT2 SA- CASSCF MS- CASPT2 FC S0/S1 MECI S1min CASSCF S0 energy CASSCF S1 energy 0 2.43 3.32 2.45 MS- CASPT2 S0 energy 0 1.87 MS- CASPT2 S1 energy 2.61 2.20 2.62 2.78 2.08 2.10 2.36 2.38 1.40 2.26 1.23 1.98 1.26 2.22 1.54 2.41 1.16 1.83 1.25 2.19 122 Table C4. The optimized cluster model FC, MECI and S1min energies for the negative DB system. geometry optimization MRCI S0 energy MRCI S1 energy 0 2.87 3.79 2.90 method B3LYP SA- CASSCF MS- CASPT2 SA- CASSCF MS- CASPT2 SA- CASSCF MS- CASPT2 FC S0/S1 MECI S1min S1min-1 CASSCF S0 energy CASSCF S1 energy 0 2.70 3.51 2.72 MS- CASPT2 S0 energy 0 2.54 MS- CASPT2 S1 energy 3.79 3.05 2.69 2.86 2.63 2.65 2.71 2.86 1.84 2.65 1.98 2.66 2.00 2.77 1.57 2.78 1.57 2.38 1.58 2.65 0.54 2.64 0.50 2.99 0.53 3.02 0.75 2.74 0.57 2.91 0.66 3.01 123 (a) (b) (c) Figure C1. Optimized cluster model FC, MECI and S1min energies. The geometry optimizations were performed at B3LYP (dotted lines at FC point), SA-CASSCF (dash lines at MECI, S1min and D1min) and MS-CASPT2 (solid lines at MECI, S1min and D1min) levels of theory. Energy calculations were performed at SA-CASSCF (blue lines), MS-CASPT2 (red lines) and MRCI (yellow lines) levels of theory. (a) The neutral DB system. (b) The positive DB system. (c) The negative DB system. nDB σSi-Si σSi-Si Figure C2. The orbitals of positively charged DB system at the ground state minimum energy geometry. 124 HOMO (n = 1.50) LUMO (n = 0.56) Figure C3. The state-averaged natural orbitals with fractional occupation numbers for a negative DB at the S1min geometry. 125 Geometries of The Neutral DB System (in Angstrom) Ground State Minimum Energy Geometry Si -0.0579090190 0.7238324132 0.1042865142 Si 0.3657420222 -0.8333928145 -1.6145610716 Si 2.6985762443 -1.3099096917 -1.6746996734 Si 3.7153711667 -0.5297483840 0.3137181686 Si 3.6315299358 1.8341432481 0.3258389525 Si 1.4094732879 2.5527180190 -0.1410316088 Si 0.3331325220 -0.3007616982 2.1919290247 Si 2.4778581283 -1.3358763777 2.1614726856 H 5.1314729465 -0.9998314378 0.3915796047 H 1.3378027850 3.0624622057 -1.5409615497 H 1.0349641645 3.6706953612 0.7708006873 H 4.5826563416 2.4027536847 -0.6728957720 H 4.0343098133 2.3377239517 1.6717453983 H 3.3079552061 -0.6173102136 -2.8477691588 H 2.9256051279 -2.7743777580 -1.8451517078 H -0.0796091432 -0.3084103746 -2.9365345764 H -0.3899971294 -2.0908103723 -1.3456338329 H 3.2014073091 -1.0703849883 3.4387352567 H 0.2982215441 0.7336236063 3.2657474441 H -0.7159858705 -1.3131632297 2.5007487475 H 2.3051386170 -2.8128031495 2.0366824672 Minimum Energy Conical Intersection Geometry: CASSCF Si -0.5369052956 0.6502415455 -0.1137860198 Si 0.4590676533 -1.0255215720 -1.4823532049 Si 2.7859718583 -1.2174490250 -1.7482536524 Si 3.8110726273 -0.6542633577 0.2817785920 Si 3.6747317923 1.6683911455 0.4786009756 Si 1.4215225806 2.2197591462 0.0850534681 Si 0.6261955238 0.1381125419 2.0747493690 Si 2.4207790981 -1.3761064322 2.0165675671 H 5.1964466062 -1.1558776039 0.4110567331 H 1.2666090529 2.5648258830 -1.3497436667 H 0.9501378971 3.3398031955 0.9164149807 H 4.4899945098 2.3852419454 -0.5246774166 H 4.0686911914 2.1189380324 1.8303703906 H 3.2757994047 -0.2884427222 -2.7925585018 H 3.1527066130 -2.5946001502 -2.1455220075 H -0.1486328997 -0.9308697896 -2.8305102272 H 0.0181273471 -2.3216679853 -0.9050379698 H 3.0992192601 -1.3307851473 3.3302077654 H 1.0162170223 1.3457988569 2.8299347822 126 H -0.5697873436 -0.4352380661 2.7301638770 H 1.8986307275 -2.7395219757 1.7916145788 Minimum Energy Conical Intersection Geometry: CASPT2 Si -0.5081918375 0.6672491083 -0.0984449772 Si 0.4847916399 -1.0027820640 -1.4547331259 Si 2.7882479577 -1.2014513494 -1.7337089611 Si 3.7943264514 -0.6471272260 0.2828840516 Si 3.6551514451 1.6557010895 0.4861964297 Si 1.4130108979 2.1748832711 0.1016775182 Si 0.6332993341 0.1432755088 2.0151306468 Si 2.4084398266 -1.3696713755 1.9923772480 H 5.1854180650 -1.1483794634 0.4111019122 H 1.2682508494 2.5094348531 -1.3517254472 H 0.9453524082 3.3194639199 0.9208268066 H 4.4727209013 2.3704171882 -0.5273664981 H 4.0561591864 2.1057124187 1.8430585012 H 3.2719213377 -0.2702509209 -2.7870399192 H 3.1535628183 -2.5873633648 -2.1287883816 H -0.1497954911 -0.9220269927 -2.8009611465 H 0.0347236759 -2.3007040691 -0.8662436901 H 3.0965913366 -1.3189092679 3.3099703421 H 1.0425907715 1.3441974475 2.7920649800 H -0.5626288165 -0.4186188236 2.7110024145 H 1.8926524361 -2.7422814270 1.7667922895 Excited State Minimum Geometry (D1min): CASSCF Si -0.4778264450 0.6607938151 -0.1050761809 Si 0.4540660846 -1.0278833102 -1.4917040415 Si 2.7815234224 -1.2312312852 -1.7472312054 Si 3.7998183232 -0.6468741346 0.2806128438 Si 3.6717071614 1.6782541859 0.4698697246 Si 1.4301480369 2.2656215181 0.0624495104 Si 0.6037389549 0.0987162480 2.0899315876 Si 2.4237092852 -1.3831489673 2.0217752393 H 5.1877964847 -1.1426154160 0.4077915710 H 1.2908637403 2.6599841815 -1.3607406950 H 0.9569670838 3.3586419437 0.9285043996 H 4.5070452457 2.3819861875 -0.5262147301 H 4.0620181094 2.1243940206 1.8244034904 H 3.2749381741 -0.3156403461 -2.8014605312 H 3.1471892021 -2.6140039317 -2.1260058301 H -0.1545814922 -0.9183670679 -2.8374715632 H -0.0042935553 -2.3167625745 -0.9130677850 127 H 3.1103328570 -1.3318105710 3.3308844853 H 0.9717541566 1.3148825928 2.8418708976 H -0.5829344943 -0.4998285672 2.7401316983 H 1.9226148916 -2.7543400568 1.7948175275 Geometries of The Positive DB System (in Angstrom) Ground State Minimum Energy Geometry Si 0.2912724992 0.6092868581 0.1227163020 Si 0.3723970070 -0.8207057892 -1.7307374400 Si 2.7092926620 -1.4021395927 -1.6375485506 Si 3.7368292360 -0.5347982428 0.3164006413 Si 3.6815030572 1.8352372844 0.2348353554 Si 1.4328881324 2.6467195394 -0.0660803788 Si 0.2915558143 -0.4097158743 2.2320051507 Si 2.5391556052 -1.2777146046 2.2237552877 H 5.1469103166 -1.0024419380 0.3919814623 H 1.2034442963 3.2165354095 -1.4152676150 H 1.0072408483 3.5653943914 1.0162433169 H 4.4714634268 2.3610431729 -0.9077911283 H 4.1525770040 2.4167629909 1.5183441700 H 3.3190843458 -0.8101471002 -2.8562707673 H 2.7738929080 -2.8854070012 -1.6782484017 H 0.0212029655 -0.0731324232 -2.9614981176 H -0.5199416722 -1.9814551194 -1.4982639828 H 3.1728719505 -0.8122310339 3.4836650017 H 0.0753909061 0.6181999945 3.2783874205 H -0.7323389511 -1.4810910309 2.2721418946 H 2.4010236422 -2.7570278907 2.2152763789 Minimum Energy Conical Intersection Geometry: CASSCF Si -0.1153874180 0.8005373208 -0.2822600633 Si 0.4996714569 -1.1356114803 -1.5050225459 Si 2.8352908845 -1.3308836549 -1.6664646127 Si 3.7846655368 -0.5646574152 0.3388539822 Si 3.6711654536 1.7807656044 0.3202003739 Si 1.4311430657 2.5077204370 0.3536858835 Si 0.4811343269 -0.2941915689 2.1711063155 Si 2.4700071937 -1.5200852081 2.0369069577 H 5.1761612108 -1.0249236742 0.5186803905 H 1.1506237671 3.4639658331 -0.7404761737 H 0.9831367218 3.0642992848 1.6371043652 H 4.2355491729 2.2669325294 -0.9528281003 H 4.3876603824 2.3782895406 1.4579998391 128 H 3.2697038989 -0.4705588564 -2.7830584349 H 3.1733846883 -2.7409005171 -1.9221719511 H -0.1565926905 -1.0120004370 -2.8188676458 H -0.1053688357 -2.2413291972 -0.7371552652 H 3.0240260247 -1.4564238971 3.3986103675 H 0.4427096769 0.8861851376 3.0346445856 H -0.8600307937 -0.8905247532 1.9978636678 H 2.0681687724 -2.8922552555 1.6869845811 Minimum Energy Conical Intersection Geometry: CASPT2 Si -0.1192242854 0.7364505877 -0.0811724165 Si 0.4032810387 -1.0578970230 -1.4879158019 Si 2.7084686557 -1.3330712252 -1.7390193333 Si 3.7422512478 -0.6432139294 0.2313855973 Si 3.7670616545 1.6691455692 0.4650839694 Si 1.5778996290 2.3637886258 0.0317513004 Si 0.5609836394 -0.0262854832 2.2076964414 Si 2.5153149546 -1.3076414614 2.1036212770 H 5.1037031003 -1.2141658462 0.3662786456 H 1.3934553491 2.7414669651 -1.4088778949 H 1.0871291571 3.4358968699 0.9227306675 H 4.6871023782 2.3737976283 -0.4536194346 H 4.0501939097 2.0147397847 1.8796550198 H 3.1988096313 -0.4994144211 -2.8611251836 H 2.9820426933 -2.7670386272 -1.9869845725 H -0.3522223701 -0.9081025827 -2.7502572773 H -0.1567305919 -2.1776160481 -0.6907728979 H 3.2305277194 -1.0044233646 3.3672386735 H 0.3885099101 1.0571774858 3.1906315545 H -0.7476323310 -0.6951986692 1.8839049231 H 2.0440058357 -2.7092686409 2.0582267568 Excited State Minimum Geometry (S1min): CASSCF Si -0.1430278800 0.2773864267 0.1754255847 Si 0.5830825619 -0.7082461165 -1.8595022388 Si 2.8212425513 -1.4210230618 -1.6234058600 Si 3.7553899757 -0.5334049719 0.3395411730 Si 3.5967500109 1.8085956517 0.2996212355 Si 1.3630288466 2.4672597160 -0.0551048629 Si 0.5438897516 -0.2305817452 2.3889155589 Si 2.6056346016 -1.3538262520 2.2169023868 H 5.1878751152 -0.8799275606 0.4187962709 H 0.9422375749 2.8301825819 -1.4146074247 H 0.6562182711 3.2392945063 0.9752630010 H 4.3390822843 2.4104871529 -0.8215645601 129 H 3.9755516251 2.4217259015 1.5840956297 H 3.5706992419 -0.9657074971 -2.8079232860 H 2.8176332641 -2.8921743554 -1.5399153987 H 0.4127565899 0.3368555359 -2.8812412456 H -0.3674093247 -1.8136557070 -2.0743645117 H 3.3673341625 -1.1263984508 3.4571556163 H 0.6741171850 1.0741951583 3.0608328986 H -0.5449317915 -1.0216694201 2.9843662665 H 2.3234913005 -2.7887182209 2.0283753916 Excited State Minimum Geometry (S1min): CASPT2 Si 0.0064923337 0.6284062532 0.3976189538 Si 0.5494716864 -0.7155730996 -1.5273516725 Si 2.8469212123 -1.1099260268 -1.6693456725 Si 3.7627264812 -0.4733503573 0.3763141303 Si 3.4800424502 1.8311897475 0.5508800557 Si 1.4623190207 2.3154315382 -0.5332334762 Si 0.6613442453 -0.2220249513 2.4597732352 Si 2.5288780136 -1.5588547423 2.0290537265 H 5.2012784706 -0.8281875580 0.4201998189 H 1.5391845712 2.1356455839 -2.0011938135 H 0.7161067868 3.5486812984 -0.1746164103 H 4.5256505690 2.5877516728 -0.1788824372 H 3.3327595856 2.2852036455 1.9523063240 H 3.3548997765 -0.2933773838 -2.7956818575 H 3.0041120172 -2.5633910914 -1.9098316949 H -0.1335207801 -0.1009826108 -2.6834481898 H -0.2495118516 -1.8297555311 -0.9256456357 H 3.3044180427 -1.7422453561 3.2774316584 H 1.0391092839 1.0028215842 3.2105080775 H -0.4881651648 -0.9140709795 3.0813850614 H 2.0361291678 -2.8527423636 1.4954214436 Geometries of The Negative DB System (in Angstrom) Ground State Minimum Energy Geometry Si -0.4385050473 0.8554082689 0.0845600986 Si 0.3606141735 -0.7184353268 -1.5285870368 Si 2.6754190644 -1.3002997274 -1.6482335406 Si 3.7497268922 -0.5457877982 0.3171928626 Si 3.6090581589 1.8127435360 0.3078763780 Si 1.3392783390 2.4438999879 -0.0949363810 Si 0.3122065537 -0.2724730685 2.0551757030 Si 2.4649284824 -1.3051698825 2.1490410088 130 H 5.1718560784 -1.0211728910 0.3947950334 H 1.3654732528 3.0385553509 -1.4763097397 H 1.1124797069 3.6306479236 0.8002742814 H 4.5634602879 2.3627372270 -0.7121880312 H 4.1049550163 2.3257433264 1.6281672944 H 3.3497385465 -0.6671076323 -2.8298816806 H 2.9055606940 -2.7752573051 -1.8074875060 H 0.0015076419 -0.3033948799 -2.9285430424 H -0.3385999029 -2.0389685900 -1.3562465330 H 3.2016424719 -1.0155035816 3.4246621975 H 0.3104198591 0.6718884414 3.2256545643 H -0.6459990541 -1.3553314676 2.4680684259 H 2.3724947843 -2.8015499109 2.0809916435 Minimum Energy Conical Intersection Geometry: CASSCF Si 0.1193919937 0.6563068509 0.1438079796 Si 0.3541821729 -0.7638414154 -1.6775449345 Si 2.6038816367 -1.4731685017 -1.5199495280 Si 3.5282937904 -0.3785341491 0.3421204295 Si 3.5519769613 1.9269493618 -0.1539618223 Si 1.4817222092 3.0485176031 -0.1712877589 Si 0.0217678708 -0.6616874446 2.0440970439 Si 2.3447194865 -1.1084309875 2.2503046803 H 4.9435197298 -0.8302626692 0.4973134795 H 1.8897669643 4.4247599183 -0.7276285296 H 1.1454694199 3.3915901171 1.2364504757 H 4.2245937341 2.0083556856 -1.4841170325 H 4.4918883715 2.6057754921 0.7830646551 H 3.3761110016 -1.1133486515 -2.7384926100 H 2.7227592283 -2.9499797699 -1.3934662563 H 0.2390967504 -0.0103778703 -2.9540061427 H -0.5260182978 -1.9527835868 -1.8386259558 H 2.9003730223 -0.4361356129 3.4566060697 H -0.3855553431 0.0817478875 3.2666930774 H -0.7409844363 -1.9416656870 2.1021673585 H 2.6455186318 -2.5591398061 2.4172096329 Minimum Energy Conical Intersection Geometry: CASPT2 Si 0.1373945108 0.7735781544 0.1771909391 Si 0.3776059456 -0.6821695406 -1.6058162596 Si 2.5916771479 -1.4356604029 -1.4887878530 Si 3.5312695211 -0.3679223555 0.3441293232 Si 3.5336706122 1.9191323106 -0.1130536074 Si 1.4334202049 2.9500768253 -0.2112889366 Si 0.0402461470 -0.7223918127 1.9595725726 131 Si 2.3454961812 -1.0832677284 2.2160060026 H 4.9382843514 -0.8472406843 0.5015937404 H 1.9129353302 4.2893425751 -0.8649566928 H 1.1654233896 3.4233208519 1.1902285108 H 4.1997032499 2.0155230326 -1.4528751155 H 4.5017713947 2.5539245567 0.8361063832 H 3.3547068268 -1.0775184516 -2.7202757548 H 2.6749364982 -2.9208184631 -1.3472532010 H 0.2393256241 0.0368587394 -2.9074043019 H -0.5166605813 -1.8750962064 -1.6887290727 H 2.8854011396 -0.4065962526 3.4319470471 H -0.3843697242 0.0160994871 3.1875151043 H -0.6606795100 -2.0514589321 2.0426065342 H 2.6309167787 -2.5430690614 2.3942982833 Excited State Minimum Geometry (S1min-1): CASSCF Si -0.0158721454 0.6489973166 0.1317444333 Si 0.3948077623 -0.8869043167 -1.5789545724 Si 2.7269463925 -1.2419872146 -1.6656679167 Si 3.6757895587 -0.5096145219 0.3543331703 Si 3.5714829729 1.8361961058 0.3922006421 Si 1.3810373372 2.4928820207 -0.1910027979 Si 0.3880640569 -0.2220483460 2.2737600170 Si 2.4013071660 -1.4303382119 2.1099012277 H 5.0916355368 -0.9509902083 0.4245074160 H 1.3808381210 2.9084452454 -1.6146814004 H 0.9573069893 3.6552871774 0.6200926009 H 4.5508591745 2.4254104046 -0.5543731363 H 3.9035462530 2.3359140232 1.7458119628 H 3.2884861374 -0.4792590983 -2.8082985699 H 3.0323785422 -2.6736560525 -1.8974637773 H -0.0742011599 -0.3892725436 -2.8937181290 H -0.2932868145 -2.1669050597 -1.3049118878 H 3.1778166724 -1.4126228313 3.3634889668 H 0.6111024595 0.9483152450 3.1459955982 H -0.7289751757 -1.0317741301 2.7699627714 H 2.1266461627 -2.8349030035 1.7313193810 Excited State Minimum Geometry (S1min-1): CASPT2 Si -0.0158721454 0.6489973166 0.1317444333 Si 0.3948077623 -0.8869043167 -1.5789545724 Si 2.7269463925 -1.2419872146 -1.6656679167 Si 3.6757895587 -0.5096145219 0.3543331703 Si 3.5714829729 1.8361961058 0.3922006421 Si 1.3810373372 2.4928820207 -0.1910027979 132 Si 0.3880640569 -0.2220483460 2.2737600170 Si 2.4013071660 -1.4303382119 2.1099012277 H 5.0916355368 -0.9509902083 0.4245074160 H 1.3808381210 2.9084452454 -1.6146814004 H 0.9573069893 3.6552871774 0.6200926009 H 4.5508591745 2.4254104046 -0.5543731363 H 3.9035462530 2.3359140232 1.7458119628 H 3.2884861374 -0.4792590983 -2.8082985699 H 3.0323785422 -2.6736560525 -1.8974637773 H -0.0742011599 -0.3892725436 -2.8937181290 H -0.2932868145 -2.1669050597 -1.3049118878 H 3.1778166724 -1.4126228313 3.3634889668 H 0.6111024595 0.9483152450 3.1459955982 H -0.7289751757 -1.0317741301 2.7699627714 H 2.1266461627 -2.8349030035 1.7313193810 Excited State Minimum Geometry (S1min): CASSCF Si 0.1218980737 0.8040695986 0.2199092664 Si 0.3436900369 -0.7490432711 -1.4742126684 Si 2.6305766366 -1.2085843430 -1.6540091873 Si 3.5804857938 -0.4973020621 0.3447470942 Si 3.5578016381 1.8332389254 0.2905028597 Si 1.5037056833 2.6152246241 -0.4977152893 Si 0.2009292615 -0.5435404971 2.2142948081 Si 2.3957081616 -1.3501668648 2.1632417999 H 4.9966134873 -0.9773589271 0.4125264960 H 1.8022219753 3.3139975432 -1.8248226460 H 1.1036062759 3.7545256338 0.3860102226 H 4.6908951647 2.2894521157 -0.5798821493 H 3.8844546519 2.3217349598 1.6637651252 H 3.2415427565 -0.4467119876 -2.7828926310 H 2.9102542588 -2.6538869338 -1.9106885812 H -0.1726809550 -0.3191666887 -2.8075560902 H -0.3331151226 -2.0401721889 -1.1421555609 H 3.1976478143 -1.0192758898 3.3816469735 H 0.1247244471 0.4549622586 3.3240890110 H -0.6192594102 -1.7155211295 2.7462259748 H 2.3860153707 -2.8453028756 2.0810211720 Excited State Minimum Geometry (S1min): CASPT2 Si 0.1218980737 0.8040695986 0.2199092664 Si 0.3436900369 -0.7490432711 -1.4742126684 Si 2.6305766366 -1.2085843430 -1.6540091873 Si 3.5804857938 -0.4973020621 0.3447470942 Si 3.5578016381 1.8332389254 0.2905028597 133 Si 1.5037056833 2.6152246241 -0.4977152893 Si 0.2009292615 -0.5435404971 2.2142948081 Si 2.3957081616 -1.3501668648 2.1632417999 H 4.9966134873 -0.9773589271 0.4125264960 H 1.8022219753 3.3139975432 -1.8248226460 H 1.1036062759 3.7545256338 0.3860102226 H 4.6908951647 2.2894521157 -0.5798821493 H 3.8844546519 2.3217349598 1.6637651252 H 3.2415427565 -0.4467119876 -2.7828926310 H 2.9102542588 -2.6538869338 -1.9106885812 H -0.1726809550 -0.3191666887 -2.8075560902 H -0.3331151226 -2.0401721889 -1.1421555609 H 3.1976478143 -1.0192758898 3.3816469735 H 0.1247244471 0.4549622586 3.3240890110 H -0.6192594102 -1.7155211295 2.7462259748 H 2.3860153707 -2.8453028756 2.0810211720 134 REFERENCES 135 REFERENCES Canham, L. 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J Phys Chem B 2008, 112, 405-413. 140 CHAPTER 5 SIMULATING ELECTRON DYNAMICS OF COMPLEX MOLECULES WITH TIME-DEPENDENT COMPLETE ACTIVE SPACE CONFIGURATION INTERACTION Reproduced with permission from “Simulating Electron Dynamics of Complex Molecules with Time-Dependent Complete Active Space Configuration Interaction”, W.-T. Peng, B. S. Fales, B. G. Levine, J. Chem. Theory Comput., 14, 4129, 2018. Copyright 2018 American Chemical Society. 5.1 Introduction Ultrafast, strong, and shaped laser pulses have been shown to drive a wealth of interesting and often still poorly understood physical phenomena.1-5 Experiments based on these pulses motivate the development of computational methods for modeling electronic dynamics in real time. (For a nice recent review see ref 6.) Such methods enable ultrafast experiments to be modeled directly, providing detailed information about the motions of individual electrons which cannot be directly observed experimentally. Because the time-dependent wave function or electron density is explicitly modeled, interactions with the field are included to infinite order. Time-dependent electronic structure methods have also seen use in conjunction with nonadiabatic molecular dynamics schemes7-9 and for efficient computation of linear spectra.10 Among real-time electronic structure methods, real-time time-dependent density functional theory (RT-TDDFT) is the most popular choice for medium to large systems due to its favorable balance between cost and accuracy.11-14 RT-TDDFT has enabled a microscopic, time- dependent view of electronic motions in light harvesting complexes, plasmonic materials, and other applications.10, 15-20 Though exact in principle, in practice RT-TDDFT suffers from a number of issues arising from the approximate functional used to describe exchange and correlation effects, in particular the adiabatic approximation (i.e. neglect of the history dependence of the functional) and the use of the ground state functional to describe nonstationary densities.21-29 For example, RT-TDDFT predicts physically incorrect two-electron Rabi oscillations.22 The nonlinear dependence of the Kohn-Sham operator on the current state of the 141 system also causes the method to fail to predict full population inversion during Rabi oscillations and results in unphysical energy peak shifting.21, 24 In principle the exact history-dependent functional would correct these issues, but practical approaches to develop approximate functionals that solve these problems remain elusive. Real-time time-dependent configuration interaction (TD-CI) methods, on the other hand, do not suffer these problems because the electronic Hamiltonian, a linear operator which is independent of the current state of the system, drives the dynamics.22, 24 In recent years, TD-CI and related Green’s function approaches have enabled the computation of nonlinear response properties and dynamics of molecules in strong or shaped laser pulses.30-45 Time-dependent multiconfigurational self-consistent field (TD-MCSCF) calculations, in which both CI coefficients and orbitals are propagated in time, show promise for modeling high-harmonic generation, as well.46-48 Time-dependent equation-of-motion coupled cluster methods offer similar advantages to TD-CI, but within the faster converging framework of coupled cluster theory.36, 49-53 Though promising, TD-CI calculations have limitations. Within the often employed configuration interaction singles (CIS) approximation, TD-CI is limited to the singly excited electronic states of systems without strong electron correlation. TD-CI approaches based on higher-order expansions have typically been limited to small systems and/or short propagation times by computational cost. In addition, many schemes cannot practically be scaled to large configuration spaces because they require full diagonalization of the CI Hamiltonian prior to propagation, which becomes intractable for large configuration spaces. A high-performance TD-CI implementation would open opportunities to accurately model electronic dynamics in medium-to-large molecular systems over long time scales. In this study, we present an implementation of TD-CI based on a complete active space expansion54 of the electronic wave function (TD-CASCI). Rather than requiring diagonalization of the Hamiltonian prior to propagation, it is based on the on-the-fly calculation of configuration interaction strategies used to solve the time-independent electronic structure problem.55-56 Graphics processing unit (GPU) acceleration enables extension to large systems and configuration spaces. In the next section, we describe our numerical procedure for integrating the TDSE. In the Results and Discussion section we analyze the stability and , akin to direct Hc 142 accuracy of our implementation and demonstrate its utility by predicting the dynamics of a large, strongly correlated molecule (decacene) in a series of shaped laser pulses. Finally, we conclude and discuss future prospects for GPU-accelerated TD-CASCI. 5.2 Method Here we describe our graphics processing unit (GPU-) accelerated implementation of TD-CASCI. In TD-CI, the time-dependent electronic wave function, Ψ(t) , may be represented as a linear combinations of Slater determinants, {ΦI} , with time-dependent expansion coefficients CI (t) , Ψ(t) = ∑ I CI (t)ΦI . (1) A complete active space expansion, familiar from time-independent quantum chemistry,54 is used in this work, though the propagation scheme described below is general to any TD-CI expansion. The wave function is propagated by numerically solving the time-dependent Schrodinger equation, i !C(t) = H(t)C(t) where the overdot indicates differentiation with respect to time. The formal solution of equation (2) can be written as (2) t∫ 0 . (3) C(0) C(t) = e−i H(t)dt However, solving the equation in this form requires exponentiating the CI Hamiltonian matrix, which is prohibitively expensive for large configuration spaces. Thus, different propagation schemes to approximate the exponentiation steps have been proposed.57-58 In this study, we used a second-order symplectic split operator integrator.59 It is well-known that the time-dependent Schrodinger equation can be recast in symplectic form, !q = Hp !p = −Hq where the expansion coefficients have been split into their real and imaginary parts according to C(t) = q(t) + ip(t) (4) (5) (6) , . 143 Propagation proceeds as follows. First we obtain initial conditions. In this work, all simulations start from the ground state solution of the field-free Hamiltonian matrix, H0 . The (10) h . (7) (8) (9) ( ( is the integration time step. At the first step: propagation occurs in a leap-frog fashion analogous to the velocity-Verlet integrator familiar in classical dynamics. Below !p 0( ) = −H 0( )q 0( ) ) = p 0( ) + 1 2 h( 2 h !p 0( ) p 1 )p 1 2 h( ) = H 1 2 h( 2 h( ) !q 1 ) q h( ) = q 0( ) + h !q 1 2 h( Then, for each subsequent time step the following four assignments are repeated: !p t( ) = −H t( )q t( ) ) + h !p t( ) ) = p t − 1 ( p t + 1 2 h 2 h )p t + 1 ( ) = H t + 1 ( !q t + 1 2 h 2 h 2 h ) ( ) = q t( ) + h !q t + 1 ( . q t + h 2 h For the final step: !p t( ) = −H t( )q t( ) p t( ) = p t − 1 ) + 1 2 h !p t( ) 2 h The symplectic symmetry of the integrator will in many cases result in long-time norm conservation—though the norm may fluctuate on short timescales. Beyond the advantages of symplectic symmetry, this integrator also offers minimal storage requirements, enabling propagation in large CI spaces. Extension to higher order is straight forward, but comes at the expense of additional storage. Chebyshev propagation60 and other symplectic integrators61 would both likely also provide robust and efficient integration. (11) (16) (14) (12) ) (13) (15) ( . The most time-consuming step of this algorithm is the evaluation of matrix-vector Hp Hq and multiplications operation in the iterative Davidson scheme for solution of the time-independent Schrodinger equation (TISE). In this work, this operation is efficiently performed via the algorithm at every time step. This step is equivalent to the s-vector formation 144 developed by Knowles and Handy,62 one of several fast algorithms that have been proposed.54, 63- 65 To achieve high performance, we will use the GPU-accelerated implementation of this algorithm recently reported by two of the authors66 to evaluate equations (7), (9), (11), (13), and (15). We implemented our algorithm into a development version of the TeraChem GPU- accelerated electronic structure package.67-68 All floating point calculations are performed in double (64-bit) precision. It is important to note that only the CI vector is propagated in this scheme; the orbitals remain frozen. Others have implemented propagation schemes that allow both the CI coefficients and orbital coefficients to evolve (TD-MCSCF).46-48 Though orbital propagation is certainly of benefit in applications where the time-dependent electronic wave function would not be representable in the initial active space at all times, it comes at a significant cost. Many of the disadvantages of RT-TDDFT relative to TD-CI arise from the orbital-dependence of the Kohn- Sham matrix. Orbital propagation necessarily involves a similar Fock matrix, and therefore would likely exhibit similar difficulties. For example, we suspect that the detuning of Rabi oscillations observed in RT-TDDFT as discussed in the introduction will be observed in time- dependent multi-configurational self-consistent field calculations as well.21, 24 The extent to which this behavior is a serious problem in the context of multi-configurational wave function theory would be an interesting topic for further study. Given this limitation, we choose to eschew orbital propagation and judiciously apply our method to model dynamics that can reasonably be expected to be well described in an active space of orbitals. We expect, for example, that quantum control of the population of low-lying excited states can be modeled in a typical active space calculation, but ionization and high- harmonic generation, which would require population of high energy virtual orbitals, cannot. Extension of the TD-CI ansatz to include single excitations to the virtual space will enable modeling of these higher energy processes. This extension will be described in a subsequent publication. The electric field is represented in the electric dipole approximation, ˆH(t) = ˆH0 − ˆm ⋅dE(t) , 145 (17) ˆH0 E(t) , , ˆm d is the unit vector in the field is the field-free CI Hamiltonian, is the molecular dipole operator, the scalar , is the time-dependent external field strength, and in which function, polarization direction. Magnetic field effects, which are neglected here, become prominent in ultra-intense fields,69-70 and under such conditions one must also consider a relativistic treatment.71 Four different pulse shapes are used in this work: 1) δ-function pulse, E(t) = E0δ(t) 2) continuous wave (CW), E(t) = E0 sin(ωt) 3) transform-limited (TL) pulse, (18) (19) E(t) = E0 exp ⎛ ⎝⎜ t − t0 2σ2 ⎞ ⎠⎟ sin ω(t − t0) ( ) , and 4) chirped pulse, t − t0 ⎞ ⎠⎟ sin ω+ 1 2σ2 E(t) = E0 exp ⎛ ⎝⎜ ( ( 2β(t − t0) )(t − t0) ) , (20) (21) where ω is the carrier frequency, σ is the pulse width, and β is the chirp parameter. For numerical convenience, is shifted such that the ground state energy of the system is zero. ˆH0 The energy spectrum of the electronic wave function following excitation can be , ε R(t) (22) is a time immediately after the end of the pulse. Given a δ-function pulse, the Fourier extracted from the time correlation function, R(t) = C(ε)†C(ε+ t) where transform of yields the electronic absorption spectrum. When a large number of excited states are of interest, this approach yields an appealing alternative to the solution of the TISE, which may become intractable due to the memory requirements associated with the storage of a large number of electronic wave functions. In this case, TD-CI can provide a memory-efficient means of computing the energy spectrum, because only a single complex electronic wave function need be stored. This is similar to the way that RT-TDDFT can outperform linear response (LR-) TDDFT in obtaining excitation energies in large systems with high densities of states.10 146 In this work, the population of a given adiabatic electronic state is often reported as a function of time. This is computed by projection of the time-dependent wave function onto the electronic eigenstates as computed by solution of the TISE, Pi(t) = ψi †C(t) 2 , where ψi is the electronic eigenstate of interest. (23) Calculations are reported for two systems. We model the absorption spectrum and the ultrafast dynamics of decacene (C42H24; picture in Figure 5.1). Decacene was chosen because a) it is larger than most molecules that have previously been studied via TD-CI, b) it is known to possess strong electron correlation and thus benefits from a multireference description of the electronic structure,72-74 and c) with its large conjugated system, it is relatively rigid in its low- lying electronic excited states and thus electronic dynamics alone can be expected to provide a reasonable description of its response to a light field. The decacene calculations were performed using an orbital basis of state-averaged CIS natural orbitals (CISNO).75-76 CISNO-CASCI has been shown to reproduce the low-lying excited states of various molecules with semiquantitative accuracy (similar to state-averaged CASSCF) and, unlike state-averaged CASSCF, offers a rigorously size-intensive description of excitations in large systems. An active space of 10 electrons in 10 orbitals and the 6-31G(d) basis set were used. This active space was chosen to include all orbitals that contribute to strong correlation as predicted by density matrix renormalization group calculations.72 The decacene geometry was taken from ref. 72 where it was optimized at the UB3LYP/6-31G* level. In order to assess the accuracy of population transfer probabilities computed from our integration scheme, we have computed Rabi oscillations in a simpler system: ethylene (C2H4). Ethylene was optimized at the B3LYP/6- 31G(d,p) level. A 2-electron, 2-orbital active space, STO-3G basis, and Hartree-Fock (HF) orbitals were used for the TD-CI calculations. A future study will investigate the accuracy of various possible orbital choices for TD-CASCI. 147 Figure 5.1. Molecule of decacene. Dark gray: carbons. Light gray: hydrogens. Throughout this work, time-dependent calculations will be abbreviated TD--CASCI (e.g. TD-CISNO-CASCI). Analogous time-independent calculation (traditional electronic structure calculations) will be abbreviated with the prefix TI for clarity (e.g. TI-CISNO-CASCI). Active spaces with M electrons and N orbitals are abbreviated (M,N). Various integration time steps are used in different calculations and are mentioned with each result. Throughout this work adiabatic state labels are used (S0 for the singlet ground state, S1 for the first singlet excited state, etc.). 5.3 Results and Discussion 5.3.1 Absorption Spectrum To demonstrate the applicability and numerical accuracy of our TD-CI scheme, we have computed the electronic absorption spectrum of decacene, a large molecule with a strongly correlated electronic ground state. This spectrum was computed as described above using a δ- function pulse polarized along the short in-plane molecular axis (the vertical direction in Figure 5.1). The resulting spectrum up to 6 eV is shown in Figure 5.2. The spectrum is based on 100 fs of propagation using a 1 attosecond (as) integration time step. The excitation energies are taken to be the points of maximum absorbance. These excitation energies are reported in Table 5.1. Given accurate numerical propagation, these energies should closely match those obtained by solution of the analogous time-independent electronic structure problem, TI-CISNO-CASCI, with the same basis and active space. 148 Figure 5.2. The absorption spectrum of decacene computed at the TD-CISNO-CASCI level as described in the text. Table 5.1. Comparison of the excitation energies (eV) of decacene calculated via TD-CISNO- CASCI and TI-CISNO-CASCI. Only the states with non-zero transition dipole moment in the polarization direction (μy; atomic units) are listed. Absolute errors are reported in eV. TD-CASCI TI-CASCI |Error| μy S0→S2 2.357 2.378 0.021 1.04 S0→S8 4.012 4.007 0.005 1.51 S0→S9 4.218 4.217 0.001 0.51 S0→S13 4.591 4.592 0.001 0.52 S0→S23 5.418 5.435 0.017 0.49 S0→S25 5.542 5.566 0.024 0.13 S0→S32 5.955 5.940 0.015 0.96 As shown in Table 5.1, our TD-CASCI implementation reproduces the excitation energies with only minor discrepancies. The maximum error is 0.024 eV (S0→S25). This discrepancy is of the order expected due to the Fourier relationship between the dynamics and 149 the energy spectrum; upon Fourier transformation of the correlation function computed from this 100 fs dynamic simulation, the resulting spectrum is discretized with an energy interval of 0.041 eV. That the maximum error is roughly half of this quantity suggests that the errors are due to the finite length of the dynamic simulations. The dependence of the computed spectrum on the integration time step has been analyzed The smallest time step for which such divergence is observed is likely determined by the by considering the peak energies as a function of time step. We find that the energies of the peaks below 6 eV are invariant to the time step (to within the 0.041 eV accuracy for a 100 fs simulation) for time steps from 1 to 19 as. A 20 as time step results in catastrophic failure; the norm of the wave function diverges to infinity. This is in contrast to the 19 as time step, in which the norm of the wave function remains 1.006 after 100 fs of simulation. Details can be found in Table D1. energy of the highest-energy state populated during the dynamics. This is because the high energy states rotate rapidly in the complex plane, and this rotation requires a short time step to integrate accurately. Obviously the choice of a large (e.g. (10,10)) active space creates the opportunity for some very high energy states to be populated. Here we consider the same system (decacene) but with a smaller active space (2,2) to investigate how the size of the active space influences the choice of integration time step. Note that there is only one bright transition available. The energy of this transition remains identical for time steps ranging from 1 to 100 as (2.151 eV, compared to a TI-CISNO-CASCI value of 2.157). Error increases with increasing time step (see Table D2), but the norm is well conserved even for a 450 as time step. Thus, the time step required to accurately predict the absorption spectrum depends strongly on the choice of active space, or more generally on the range of energies of the populated eigenstates. We also note that the norm is not a sensitive measure of the accuracy of integration. 5.3.2 Rabi Oscillations A well-known failing of RT-TDDFT is that Rabi oscillations are not correctly described, both because energies detune as the Kohn-Sham matrix evolves21, 24 and because the mean-field 150 nature of RT-TDDFT results in electrons transitioning between orbitals in pairs (two-electron Rabi oscillations).22 Due to the many-body nature of CI and the linearity of the Hamiltonian driving the dynamics, these problems do not arise in TD-CI.22, 24 Here we again demonstrate this advantage of TD-CI and use the well-defined Rabi oscillations as a model to further investigate the accuracy of our integration scheme. The period of Rabi oscillations between two states, i and j, is Tij = 2π Ωij with Rabi oscillation frequency mij i E Ωij = ! mij (24) (25) where is the transition dipole vector and E is the electric field strength vector. Here we study ethylene, with simulation parameters as described above. A continuous wave field polarized along the C=C bond is applied with an intensity of 1012 W/cm2 and a field frequency of 3.444x1015 Hz. This corresponds to the energy difference between the S0 and S1 states (π→π* transition) as calculated at the TI-HF-CASCI level. The exact Rabi oscillation period computed for this system according to equations (24) and (25) is 21.506 fs. The state populations as a function of time drawn from a simulation with a 1 as time step are shown in Figure 5.3a. Qualitatively correct one-electron Rabi oscillations are observed. The observed oscillation frequency (21.507 fs) is in excellent agreement with the exact value (21.506 fs) 151 Figure 5.3. The populations of the S0 and S1 states as a function of time from calculations of the dynamics of ethylene in a resonant CW field, as described in the text. Calculations are performed with time steps of 1 as (solid) and b) 10 as (dashed). Because Rabi oscillations are an exactly solvable problem, they provide us with an opportunity to examine the influence of time step on the accuracy of population dynamics. Three different measures of the integration error are reported in Table 5.2. The population error (“Pop. Err.” in Table 5.2) measures whether full population inversion is achieved. We define the population error as the minimum population of S0 observed during the Rabi oscillation cycle. Each simulation includes multiple such minima. The largest error observed in each simulation is reported. The error in the period ( in Table 5.2) is defined as the difference between the exact Rabi oscillation period, 21.506 fs, and the period as predicted by TD-CASCI (determined by measuring the peak-to-peak distance for the first oscillation). Finally, the norm of the wave function after 100 fs of propagation is reported. ΔT 152 We find that the accurate calculation of population dynamics requires a finer time step than was needed to reproduce the absorption spectrum. As shown in the Table 5.2, the error in the population dynamics increases from 0.0002 to 0.0292 as the time step is increased from 1 to 10 as. Integration error results in population dynamics comparable to interaction with a field that is slightly off resonance (Figure 5.3). In addition, the calculated period of the Rabi oscillations increases by 0.316 fs as the time step is increased from 1 to 10 as. Note that the norm is well conserved in all simulations (Table 5.2), and therefore is not a sensitive measure of the accuracy of the population dynamics. Table 5.2. The error in the population (Pop. Err.), error in the Rabi oscillation period ( relative to the exact value (21.506 fs), and norm of the wave function after 100 fs of propagation for simulations of ethylene in a resonant CW field as a function of the integration time step. Simulation details are presented in the text. ΔT ) 1 as 2 as 4 as 6 as 8 as 10 as Pop. Err. 0.0002 0.0004 0.0013 0.0042 0.0110 0.0292 ΔT (fs) 0.001 0.011 0.020 0.062 0.162 0.316 norm 1.000 1.000 1.001 1.001 1.000 0.997 153 Figure 5.4. The final state populations obtained from 100 fs simulations of decacene (TD- CISNO-CASCI(10/10)/6-31G(d)) in a series of laser pulses with chirp ranging from β = -0.342 fs-2 to β = 0.342 fs-2 All pulses have a FWHM of 10 fs and a maximum intensity of 3*1012 W/cm2. The populations of bright and dark states up to S8 are shown with solid lines and dashed lines, respectively. The populations in all states above S8 are summed into a single (purple) line. Populations of states with negligible population in all simulations (S3-S5) are not shown. A Jablonski diagram on the right shows the energies of the populated states up to S8. 5.3.3 Control by Shaped Pulses electronic dynamics of decacene in a series of chirped laser pulses. All pulses are polarized along the short in-plane molecular axis and have a 10 fs FWHM, a maximum intensity of 3*1012 W/cm2, and a carrier frequency matching the gap between the S0 and S2 states (2.378 eV; the lowest allowed transition for decacene for the chosen polarization direction). Figure 5.4 shows the populations of all significantly populated electronic states after the pulse decays for chirp parameters ranging from β = -0.342 fs-2 to β = 0.342 fs-2 (-0.0002 a.u. to 0.0002 a.u.). Clearly the populations depend strongly on the chirp parameter. Specifically, positive chirp rates (meaning that the frequency increases with time) result in a significant increase in the population of higher energy states. This includes S8, which is bright with respect to the ground state, and states higher than S8, many of which are dipole-forbidden with respect to the ground state. For Here we demonstrate our implementation of TD-CISNO-CASCI by modeling the 154 less chirped pulses (β ≈ 0.1 fs-2) more population is seen in states above S8, whereas for more chirped pulses (β ≈ 0.3 fs-2) more population is observed in S8 itself. In contrast, negatively chirped pulses result in either more population in S2 (for β ≈ -0.05 fs-2) or in low energy states that are dark relative to the ground electronic state (S1, S6, and S7; for β < -0.1 fs-2). For pulses that are nearly transform limited (β ≈ 0.0 fs-2) the ground states is more strongly depleted than for pulses with larger positive or negative chirps. In order to understand the dependence of the electron state populations on pulse shape, we examine the population dynamics driven by several different pulses. First we consider TL pulses at two different intensities to investigate the probability of multiphoton processes. The middle and bottom panels of Figure 5.5 present the electronic state populations as a function of time for decacene in TL pulses with maximum intensities of 1*1012 W/cm2 and 3*1012 W/cm2, respectively. Both pulses have FWHM of 10 fs. For the weaker pulse (1*1012 W/cm2; middle panel), most of the population (75 %) transfers from the ground state to the S2 state with less than 8% of the population on other (higher) states after the pulse. When the intensity is tripled, however, maintaining the same duration and carrier frequency, only 41.5% of the population stays on the S2 state. A similarly large fraction of the population (44%) is promoted to states higher in energy than S8. Examination of the populations shows that in this more intense field the molecule tends to absorb multiple photons, climbing a ladder of states: S0→S2→S18→S57. Each of these transitions has a large transition dipole (>1 a.u.) and the transition energies are roughly resonant with the carrier frequency (2.378 eV for S0→S2, 2.684 eV for S2→S18, and 2.151 eV for S18→S57). Note that the S0→S18 transition is dipole forbidden. The S0→S57 transition is dipole allowed, but the energy of this transition is 7.213 eV, large enough so as to preclude direct excitation from S0. These higher states are not populated until after S2 is significantly populated, consistent with the proposed ladder-climbing behavior. Thus, at this intensity strong nonlinear effects are observed. 155 Figure 5.5. The electronic state populations as a function of time for decacene (TD-CISNO- CASCI(10,10)/6-31G(d)) in 10 femtosecond (FWHM) TL pulses with maximum intensities of 1*1012 W/cm2 (middle panel) or 3*1012 W/cm2 (bottom panel). The electric field as a function of time is shown in the top panel (arbitrary units). This pulse is resonant with the S0→S2 transition. Only states up to S8 that contain non-negligible population at the end of the simulations are shown. The sum of all population in states higher than S8 is represented by a dotted purple line. 156 Figure 5.6. The electronic state populations as a function of time for decacene (TD-CISNO- CASCI(10,10)/6-31G(d)) in 10 fs (FWHM) chirped pulses with β = 0.256 fs-2 and an intensity of 3x1012 W/cm2 are shown in the bottom panel. The electric field as a function of time is illustrated in the top panel. Only populations of states up to S8 that contain non-negligible population at the end of the simulation are shown. The sum of all population in states higher than S8 is represented by a dotted purple line. 157 Figure 5.7. The electronic state populations as a function of time for decacene (TD-CISNO- CASCI(10,10)/6-31G(d)) in 10 fs (FWHM) chirped pulses with β = -0.256 fs-2 and an intensity of 3x1012 W/cm2 are shown in the bottom panel. The electric field as a function of time is illustrated in the top panel. Only populations of states up to S8 that contain non-negligible population at the end of the simulation are shown. States that are dark with respect to the electronic ground state (S1, S6, and S7) are shown with dashed lines. The sum of all population in states higher than S8 is represented by a dotted purple line. 158 Now we consider the detailed dynamics of decacene in chirped pulses. Figures 5.6 and 5.7 show the electronic state populations as a function of time when decacene is exposed to chirped pulses with a carrier frequency resonant with the S0→S2 transition and chirp rates of 0.256 fs-2 and -0.256 fs-2 (0.00015 to -0.00015 a.u.), respectively. First we consider the positively chirped pulse. As noted above, the S2 and S8 states are the lowest-lying dipole- allowed transitions from S0. Significant population is observed in the two dipole allowed states (S2 and S8), the ground state, and states above S8. The S2 state accumulates populations first, followed by the S8 state. This is not surprising given a positively chirped pulse. The frequency increases with time, so the S0→S2 transition is resonant first, and the S0→S8 transition is resonant later. Population in states above S8 largely arises due to ladder-climbing from S2, similar to that observed in the TL pulse above. As can be seen in Figure 5.4, larger positive chirp results in more population in S8, consistent with the fact that a pulse with a larger chirp parameter reaches resonance with the S0→S8 transition while its intensity remains large and before the ground state population is depleted by the S0→S2 transition. positively chirped pulse. Population transfer to S8 is observed before transfer to S2, as one would expect for a negatively chirped pulse. Subsequent population transfer is observed to several states that are dark relative to S0: S1, S6, and S7. In all three cases population transfer begins once the population of S2 becomes significant. The S2→S1, S2→S6 and S2→S7 transitions energies (0.791 eV, 0.945 eV, and 1.116 eV, respectively) are all lower than the energy of S0→S2 and these three transitions all have a significant transition dipole moment. As the frequency of the negatively chirped field decreases with time, it reaches resonance with these transitions. Thus, these states are populated via a two-photon process: resonant two-photon absorption in the case of S6 and S7 and stimulated electronic Raman scattering in the case of S1. Population of these dark states is not observed for positively chirped pulses because the lower frequencies resonant with the S2→Sn (n=1,6,7) transitions precede population of S2. As can be seen in Figure 5.4, the populations of all of these states depends intricately on the value of the chirp parameter (β). Less negatively chirped pulses are so low in intensity by the time the frequency reaches resonance with the S2→Sn transitions that these two-photon processes are not observed with significant probability. The dependence of the dynamics on active space was The dynamics in the negatively chirped pulse (Figure 5.7) are more complex than in the 159 To investigate the convergence of the population dynamics with respect to time step, we investigated by running an identical calculation with an expanded (12,12) active space. Qualitatively similar dynamics were observed, with small quantitative differences in the final populations of the various states. See Figure D1 for details. have modeled the dynamics of decacene in a pulse with a chirp rate of -0.256 fs-2, a (10/10) active space, and all other parameters as above using time steps ranging from 1 to 16 as. We find that the final population as predicted with a 16 as time step is in error by less than 0.001 for all states. As the time step decreases, the population converges nicely toward the zero time step limit, suggesting that future studies could utilize time step extrapolation to minimize integration errors. Data can be found in Figures D2-D7. only valence electronic dynamics are observed and processes involving high energy virtual orbitals (e.g. ionization and high harmonic generation) are not. As mentioned above, the use of a TD-CASSCF approach could circumvent this limitation because the occupied orbitals can rotate into the virtual space, but TD-CASSCF would suffer from many of the difficulties associated with RT-TDDFT. As will be discussed in the conclusion, application to higher energy processes can also be achieved by extension of the configuration space. In all of the above cases, the dynamics observed are limited to the active space. Thus, Table 5.3. The wall time to solution for several of the calculations. The propagation time is 100 femtosecond and the time step is 1 as for all simulations shown. Field Type Molecule Method Active Space Wall Time (seconds) CW ethylene TD-HF-CASCI/STO-3G (2,2) δ-function decacene TD-CISNO-CASCI/6-31G(d) (10,10) TL decacene TD-CISNO-CASCI/6-31G(d) (10,10) Chirped pulse decacene TD-CISNO-CASCI/6-31G(d) (10,10) Chirped pulse decacene TD-CISNO-CASCI/6-31G(d) (12,12) 160 146 (2.43 min) 8328 (2.31 hr) 8612 (2.39 hr) 8536 (2.37 hr) 72473 (20.1 hr) 5.3.4 Performance Table 5.3 reports the wall time to solution for a representative set of the calculations mentioned above. All calculations were performed on a single NVIDIA K40 GPU. For the smaller ethylene system (TD-HF-CASCI(2,2)/STO-3G) the CW field calculation took less than 2.5 minutes to propagate for 100 femtosecond using a 1 as time step. The larger decacene simulations with δ-function, TL, and chirped fields (TD-CISNO-CASCI(10,10)/6-31G(d)) required less than 2.5 hours to propagate for 100 fs using a 1 as time step. Of the 2.5 hours, approximating one hour is required to compute the TI-CISNO-CASCI(10,10)/6-31G(d) ground state for use as the initial wave function and higher energy eigenstates for subsequent population projections. Extrapolating from these benchmarks, the 17 simulations of decacene used to create Figure 5.4 require less than 2 days on a single GPU. One additional calculation based on a (12,12) active space is shown. Even with this large active space (853776 determinants) integration for 100 fs requires only 20.1 hr. 5.4 Conclusions In this study, we implemented a new algorithm to efficiently perform TD-CASCI calculations with the aid of GPUs. We found that when employed with a sufficiently short time step this algorithm provided robust predictions of absorption spectra and population dynamics (i.e. Rabi oscillations). The dynamics of decacene with a large (10,10) active space can be simulated for 100 fs in less than 2.5 hours on an NVidia K40 GPU. This new tool was employed to explore how shaped laser pulses can be used to control the population of dipole-forbidden states in decacene. Because excitations to the virtual orbitals outside the CAS space are not possible in TD- CASCI, our current method is limited to dynamics that do not leave the set of valence excited electronic states. In future work we will extend our method to high-energy processes by inclusion of single excitations to the virtual orbital space. We will also investigate the choice of orbital determination methods75, 77-79 on the accuracy of TD-CASCI calculations and couple our 161 GPU-accelerated TD-CI method to a nonadiabatic molecular dynamics scheme in order to model full molecular dynamics in dense manifolds of states. 162 APPENDIX SUPPORTING INFORMATION FOR: SIMULATING ELECTRON DYNAMICS OF COMPLEX MOLECULES WITH TIME-DEPENDENT COMPLETE ACTIVE SPACE CONFIGURATION INTERACTION Table D1. Comparison of the wave function norm at the end of 100 fs simulations of decacene after excitation by a δ-function pulse calculated using different time steps. Calculations are performed with the TD-CISNO-CASCI/6-31G(d) basis set with a (10,10) active space. Time Step 1 as 10 as 15 as 19 as 20 as Norm 1.000 1.002 1.003 1.006 Inf Table D2. The first excitation energies (in eV) of decacene obtained with TI-CISNO-CASCI and TD-CISNO-CASCI methods with 6-31G(d) basis set and a (2,2) active space. TD-CISNO- CASCI calculations were performed with a series of time steps from 1 to 500 as. The norm at the end of the 100 fs simulations is also presented. 1 as 100 as 200 as 300 as 400 as 450 as 500 as 2.151 1.000 2.151 1.001 2.192 1.002 2.303 1.005 2.358 1.012 2.426 1.014 --- Inf 163 TI-CISNO- CASCI 2.157 --- Eex norm Figure D1. The electronic state populations as a function of time for decacene (TD-CISNO- CASCI(12,12)/6-31G(d)) in 10 fs (FWHM) chirped pulses with β = -0.256 fs-2 and an intensity of 3x1012 W/cm2. Only populations of states up to S9 that contain non-negligible population at the end of the simulation are shown. Note that S9 in this active space corresponds to S8 in the (10,10) active space used throughout this work. States that are dark with respect to the electronic ground state (S1, S6, and S7) are shown with dashed lines. The sum of all population in states higher than S9 is represented by a dotted purple line. 164 Figure D2. The final population on S0 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 165 Figure D3. The final population on S1 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 166 Figure D4. The final population on S2 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 167 Figure D5. The final population on S6 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 168 Figure D6. The final population on S7 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 169 Figure D7. The final population on S8 for a simulation of decacene with a (10,10) active space in a pulse with a chirp rate of -0.256 fs-2 as a function of time step. All other simulation parameters are as described in the main text. 170 Geometries (in Angstrom) Ethylene, B3LYP/6-31G** C -0.66500 0.00000 0.00000 C 0.66500 0.00000 0.00000 H -1.23780 0.92380 0.00000 H -1.23780 -0.92380 0.00000 H 1.23780 0.92380 0.00000 H 1.23780 -0.92380 0.00000 Decacene, UB3LYP/6-31G* singlet (from Hachmann, J.; Dorando, J. J.; Aviles, M.; Chan, G. K. L., The radical character of the acenes: A density matrix renormalization group study. J. Chem. Phys. 2007, 127 (13), 134309.) C -12.286459 0.713204 0.000000 C -12.286459 -0.713204 0.000000 H -13.232858 -1.247136 0.000000 H -13.232858 1.247136 0.000000 C -11.104552 1.407441 0.000000 C -11.104552 -1.407441 0.000000 C -9.848699 0.724254 0.000000 C -9.848699 -0.724254 0.000000 H -11.102098 2.494931 0.000000 H -11.102098 -2.494931 0.000000 C -8.626957 1.405881 0.000000 C -8.626957 -1.405881 0.000000 C -7.395009 0.727344 0.000000 C -7.395009 -0.727344 0.000000 H -8.627386 2.494068 0.000000 H -8.627386 -2.494068 0.000000 C -6.158121 1.408266 0.000000 C -6.158121 -1.408266 0.000000 C -4.934211 0.730243 0.000000 C -4.934211 -0.730243 0.000000 H -6.158802 2.496330 0.000000 H -6.158802 -2.496330 0.000000 C -3.693580 1.409318 0.000000 C -3.693580 -1.409318 0.000000 C -2.468192 0.731552 0.000000 C -2.468192 -0.731552 0.000000 H -3.693840 2.497387 0.000000 H -3.693840 -2.497387 0.000000 C -1.231033 1.409431 0.000000 C -1.231033 -1.409431 0.000000 C -0.000005 0.731841 0.000000 171 C -0.000005 -0.731841 0.000000 H -1.231077 2.497521 0.000000 H -1.231077 -2.497521 0.000000 C 1.231025 1.409431 0.000000 C 1.231025 -1.409431 0.000000 C 2.468184 0.731552 0.000000 C 2.468184 -0.731552 0.000000 H 1.231052 2.497521 0.000000 H 1.231052 -2.497521 0.000000 C 3.693573 1.409317 0.000000 C 3.693573 -1.409317 0.000000 C 4.934206 0.730244 0.000000 C 4.934206 -0.730244 0.000000 H 3.693823 2.497385 0.000000 H 3.693823 -2.497385 0.000000 C 6.158118 1.408263 0.000000 C 6.158118 -1.408263 0.000000 C 7.395010 0.727345 0.000000 C 7.395010 -0.727345 0.000000 H 6.158798 2.496327 0.000000 H 6.158798 -2.496327 0.000000 C 8.626961 1.405879 0.000000 C 8.626961 -1.405879 0.000000 C 9.848706 0.724254 0.000000 C 9.848706 -0.724254 0.000000 H 8.627398 2.494066 0.000000 H 8.627398 -2.494066 0.000000 C 11.104563 1.407439 0.000000 C 11.104563 -1.407439 0.000000 C 12.286471 0.713205 0.000000 C 12.286471 -0.713205 0.000000 H 11.102124 2.494928 0.000000 H 11.102124 -2.494928 0.000000 H 13.232869 1.247141 0.000000 H 13.232869 -1.247141 0.000000 172 REFERENCES 173 REFERENCES Zewail, A. 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For example, laser-induced electron diffraction (LIED) experiments have reached a few femtosecond and sub-angstrom resolutions, allowing the real-time observations of the structural change of a single chemical bond.1-5 Another example is orbital tomography, which reaches similar resolutions for electron dynamics.6 In addition, nonlinear optical phenomena such as high harmonic generation (HHG)7-8 and above-threshold ionization (ATI)9 are now routinely explored. With HHG, extreme ultraviolet (XUV) or soft x-ray attosecond pulses have been made possible and have been applied to study electron dynamics in atoms, molecules and solids.10-14 Despite the advancements in the experimental methods, the theoretical description of the microscopic mechanisms for ultrafast dynamics driven by intense lasers remain challenging. Various real-time time-dependent methods for electron dynamics under external fields have been developed by numerically propagating electronic wavefunctions or electron densities.15 Among them, real-time time-dependent density functional theory (RT-TDDFT) is one of the most popular methods because of its balance treatment of electron correlation with favorable computational cost. However, the exact density functional form is unknown and approximated functionals are used, which make them inaccurate or inapplicable in many applications. On the other hand, wave function-based methods can be systematically improved to include both dynamical and static electron correlation. There are many wave function-based time-dependent electronic structure methods that have been developed including time-dependent configuration interaction based methods (TD-CIS, TD-CISD, … etc.),16-29 time-dependent multiconfigurational self-consistent field methods (TD-MCSCF),30-32 and time-dependent equation-of-motion coupled cluster methods (TD-EOM-CC).33-34 Although higher accuracy for a higher tier method, the computational cost limits their applications to large systems. Recently, we have developed a TD- CASCI method based on a direct CI scheme, which can be accelerated by graphics processing 181 units (GPUs).35 The approach is promising for large systems and systems where static electron correlation is important. By extending to consider all single excitations from a multiconfigurational reference state (a CASCI state),36 the method becomes the time-dependent multireference configuration interaction singles (TD-MRCIS) method, which is able to describe ionization processes via high single excitations. Another advantage of the TD-MRCIS method is that it restores part of the dynamical electron correlation missing in CASCI. In this study, we are looking to further develope a method in conjunction with the TD- MRCIS approach for simulating electron dynamics under intense laser radiation. The electrons in a system can be driven to move far away from their original locations under a strong laser field. Since finite Gaussian basis sets are used in our TD-MRCIS method, the electrons may reach the boundary of the basis in a simulation, resulting in unphysical events such as reflection due to the insufficiency of the basis sets. To include adequate basis sets for all high-lying Rydberg states as well as the continuum states is computationally infeasible. One route to remedy these unphysical events and to treat the ionization processes without adding excessive diffuse basis functions is to place an absorbing potential on a defined boundary of the system. When an electron arrives at the boundary, it will be absorbed by the potential and considered ionized. The complex absorbing potential (CAP) has been developed to remove unphysical effects from finite boundaries when numerically solving the time-dependent Schrodinger equation.37-43 CAP has been applied to study the photoionization processes with TD-CI methods.19, 44-45 For atom- centered Gaussian basis sets, a similar heuristic model that treats electrons in the ionizing states which have energies above the ionization potential with an energy-dependent lifetime to account for the loss of electrons.46-47 In addition, the equation-of-motion coupled cluster methods have been coupled to CAP for studying resonance states.48-49 Recently, Schlegel et al. developed a numerical scheme to add CAP to atom-centered gaussian basis sets.44 These methods have been used for studying HHG47, 50 and charge resonance-enhanced ionization (CREI)44 among other processes. The goal of this study is to develop an analytical form of the CAP which can be evaluated efficiently. In conjunction with the TD-MRCIS method, we aim at an efficient approach for simulating electron dynamics under intense laser fields in large systems. In order to reach this 182 goal, we consider a molecular-centered CAP as shown in Figure 6.1. The CAP is zero within the boundary and has quadratic form beyond the boundary. The derivations of the CAP are presented in the method section. r0 VCAP Vabs 0 0 r0 r0 r Figure 6.1. A representation of the molecule-centered complex absorbing potential developed in this work. The radius, r0, from the center of the molecule is defined for the boundary (show in red). Beyond the boundary, a quadratic form of absorbing potential is applied (shown in the green line and in the inserted figure). Within the boundary, the potential is zero. 6.2 Method As in the TD-CASCI method (Chapter 5), our goal is to solve the time-dependent electronic Schrodinger equation with TD-CI expansion, i !C(t) = H(t)C(t) (1) , 183 is the time-dependent Hamiltonian. To C(t) H(t) is the time-dependent CI vector and where include the CAP, we added a complex potential to the Hamiltonian H(t) = H0 − ˆµ⋅dE(t) − iVabs where the is the time-independent CI Hamiltonian, , H0 (1) is the applied electric field − ˆµ⋅dE(t) −iVabs described in the dipole approximation, and the last term is the complex absorbing 0 . (2) ∞∫ j = χi(r) ˆVabs(r)χj(r) dr potential, the matrix element of which can be written as Vij = i ˆVabs By adding the CAP, the Hamiltonian is no longer a Hermitian matrix, thus the conservation of the norm of the wave functions will not be guaranteed. When the electron travels to the defined boundary, the CAP will annihilate the electronic wavefunction that touches it. In this study, we are looking for a CAP with a quadratic form beyond the boundary, ˆVabs(r) = b(r − r0)2 In equation (4), the parameter r0 is the radius of the CAP from the center of the molecule and the parameter b controls the curvature of the CAP. Thus, we define a molecule-centered CAP with a in equation (3) are molecule-centered diffused spherical boundary (Figure 6.1). r > r0 Vabs = 0 r ≤ r0 for for (3) . χi(r) and χj(r) , Gaussian functions which we introduce to the system to describe ionization processes. The low- lying valence electrons that are bounded to the molecules are treated with normal atom-centered basis sets. Given their limited spatial extent, the valence electrons are not annihilated by the CAP. The radius can be adjusted to make sure all of the atom-centered basis functions decay to near zero at the boundary of CAP. Molecule-centered diffuse functions can then be added to describe the higher-lying states. With the gaussian basis functions, equation (3) can be written as 184 r0 Vij = ∞∫ π∫r0 2π∫0 0 b(r − r0)2 xA i yA j z A ke−ar2r 2 sinθdr dθdφ . (4) To integrate equation (5), we rewrite the Cartesian components in spherical coordinates, xA i yA Then the integration of equation (5) can be separated into three contributions, k = (r sinθcosφ)i(r sinθsinφ) j(r cosθ)k j z A . π∫ 0 (sinθ)i+ j+1 (cosθ)k dθ 2π∫ 0 (cosφ)i(sinφ) j dφ Fθ = Fφ = Fr = ∞∫ r0 r i+ j+k+2(r − r0)2e−ar2 dr . These three terms can be integrated analytically, and the results can be expressed as Fθ = ( 1+ (−1)k )Γ ⎡ 2Γ ⎣⎢ ⎡ ⎣⎢ 1 2 1 2 ( ( ) 2 + i + j ⎤ ⎦⎥Γ ⎤ ) 3+ i + j + k ⎦⎥ 1+ k 2 ⎡ ⎣⎢ ⎤ ⎦⎥ ( 1+ (−1)i Fφ = ) 1+ (−1)i+ j ( ( 2Γ ⎡ )Γ ⎣⎢ 2 + i + j 1 2 ⎡ ⎣⎢ 1+ i 2 ⎤ ) ⎦⎥ ⎤ ⎦⎥Γ ⎡ ⎣⎢ 1+ j 2 ⎤ ⎦⎥ 185 (5) (6) (7) (8) (9) (10) ( ) −i− j−k−5 2 a ( ) ( 2 2 i + j + k + 3 ( 2 2ar0 ⎧ ⎪ ) −2a ⎨ ⎩⎪ )Γ 2 + i + j + k + 3 ( ) i+ j+k+5 2 i+ j+k+5 − 2 ar0ear0 r0 2 ( )Γ i + j + k + 3 ( 1 2 ⎡ ⎢ ⎣ i + j + k + 4 ),ar0 2 ⎤ ⎥ ⎦ . ( 1 2 ⎡ ⎢ ⎣ i + j + k + 5 ),ar0 2 ⎤ ⎥ ⎦ ⎫ ⎬ ⎭ (11) Fr = ear0 +ear0 We implemented equations (10), (11), and (12) for the evaluation of the matrix elements of the CAP into the TeraChem software package. In addition, since the imaginary potential breaks the symplectic symmetry of the Schrodinger equation for the TD-CI method, the previously implemented symplectic split operator integrator becomes inappropriate for applications requiring the CAP. Instead, we implemented a fourth order Runge-Kutta (RK4) integrator for TD-CI propagation with the CAP. The RK4 algorithm is implemented in its standard form, as shown in the following, k1 = Δt H(t) i! H(t + Δt 2 (12) C(t) ) (13) k 2 = Δt k 3 = Δt i! H(t + Δt 2 i! (C(t) + k1 2 ) ) (C(t) + k 2 2 ) k 4 = Δt H(t + Δt) i! (C(t) + k 3) C(t + Δt) = C(t) + k1 6 + k 2 3 + k 3 3 + k 4 6 . (14) (15) (16) In order to verify our derivations and the implementation of the evaluation of matrix elements Vij , we considered an s shell Gaussian basis for a molecule-centered diffuse function. In this situation, the integration of the equation (7) and the equation (8) can be done by hand (where . We can compare to the values ), giving values i = j = k = 0 Fθ = 2 and Fφ = 2π 186 evaluating from our implementation in the TeraChem package using equation (10) and equation (11) to verify the derivations and our implementations. For the term expressed in equation Fr (12), we evaluate using Matlab and compare it to our implementations. We used a diffuse function with exponent , and the CAP boundary is set to r0 = 20 a.u. . For the angular a = 0.0256 FθFφ = 12.566371= 4π parts, we got a value of from our TeraChem implementation, which suggests our derivations and implementations are correct for Fθ and Fφ . 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I., Complex absorbing potential equation-of-motion coupled- cluster method yields smooth and internally consistent potential energy surfaces and lifetimes for molecular resonances. J Phys Chem Lett 2014, 5, 3078-3085. Ghosh, A.; Vaval, N.; Pal, S.; Bartlett, R. J., Complex absorbing potential based equation-of-motion coupled cluster method for the potential energy curve of CO2− anion. J Chem Phys 2014, 141, 4113. 192 50. White, A. F.; Heide, C. J.; Saalfrank, P.; Head-Gordon, M.; Luppi, E., Computation of high-harmonic generation spectra of the hydrogen molecule using time-dependent configuration-interaction. Mol Phys 2016, 114, 947-956. 193 CHAPTER 7 THE EHRENFEST MOLECULAR DYNAMICS IN CONJUNCTION WITH TIME-DEPENDENT CONFIGURATION INTERACTION ELECTRONIC WAVE FUNCTIONS 7.1 Introduction Ab initio molecular dynamics (AIMD) methods model nuclear motion under the forces generated by electronic wave functions which are determined from ab initio quantum mechanical calculations on-the-fly. AIMD methods allow researchers to study chemical processes without prior knowledge of or extensive assumptions about the shape of the potential energy surfaces (PES). When the PES of multiple states are involved, the nonadiabatic effects need to be considered in the simulations. The ab initio multiple spawning (AIMS) method,1-2 surface hopping method3 and Ehrenfest dynamics4-6 are prevalent approaches for the study of such nonadiabatic dynamics. On the other hand, although the nonadiabatic AIMD methods for the low-lying excited states are well-developed, tools for calculating dynamics in high-lying excited states such as Rydberg and continuum states are less mature. Due to the dense manifold of adiabatic electronic states in these cases, conventional nonadiabatic dynamical methods are not applicable as the computational cost generally grows with the number of states involved and the need to evaluate numerous nonadiabatic couplings between each state. The mean field nature of the Ehrenfest dynamics method provides a route to sidestep these hurdles. With high intensity, ultrafast laser pulses, many intriguing dynamics in this dense manifold regime are discovered experimentally. We aim at developing a time-dependent configuration interaction (TD-CI)7 -based Ehrenfest dynamics method in the present work for simulating such systems. In order to account for light-matter interactions in nonadiabatic dynamics simulations, the time-dependent electronic wave functions are propagated explicitly with an applied external fields in our approach. The time-dependent electronic wave functions are represented as linear combinations of configurational states, where the expansion coefficients are evolved in time as the responses to the external field. As a result, the TD-CI wave functions can be deemed as the weighted sums of adiabatic electronic states, avoiding the explicit calculations of the dense manifold of electronic states. The intramolecular coordinates are 194 subjected to the mean field potential generated by the TD-CI wavefunction, hence the ions move according to the mean force acting on them. In this work, we develop Ehrenfest dynamics in conjunction with time-dependent complete active space configuration interaction method (TD- CASCI-Ehrenfest) as a first step toward modeling dense manifold systems. Although excitations from active space to high-lying virtual orbitals are required for processes including high-lying states, e.g. ionization process, we noted that the extension to Ehrenfest dynamics with time- dependent multireference configuration interaction singles (TD-MRCIS-Ehrenfest) is straightforward when the force generated from the TD-MRCIS electronic wave function can be evaluated efficiently. The detailed algorithms of the TD-CASCI-Ehrenfest method are explained in the method section. 7.2 Method In this work, we developed a nonadiabatic Ehrenfest dynamics method that can be applied to elucidate laser-induced dynamics. The electronic dynamics are propagated with the TD-CASCI method, and the molecular dynamics are simulated using the velocity Verlet (VV) algorithm, where the forces experienced by the molecules are generated from the TD-CASCI wave functions (TD-CASCI-Ehrenfest). Noting that the electrons move much faster than the nuclei, we employed two time step sizes for the electronic and nuclear dynamics, . Given an initial geometry R0, an initial momentum p0, and an respectively, in which electronic wavefunction , the algorithm for the TD-CASCI-Ehrenfest approach can be outlined in the following and is summarized in the Figure 7.1. The detailed descriptions of the algorithm are presented in the following paragraphs and the cited references therein. Δτ≪ Δt Ψ Δτ and Δt (i). Calculate the force F = − d dR Ψ H Ψ . (ii). Propagate the momentum with = p0 + pΔt 2 1 2 FΔt (or p t+ Δt 2 = p (iii). Propagate the position with RΔt = R0 + (iv). Obtain the orbitals at new geometry RDt (or Rt+Dt). 195 + FΔt for t ≠ 0) t− Δt 2 p t+ Δt m Δt for t ≠ 0) 2 pΔt m Δt (or Rt+Δt = Rt + 2 (describe below). (v). Diabatize the orbitals to have largest overlap with orbitals from previous molecular time step Δt (vi). Use the new set of orbitals to build the electronic Hamiltonian and start TD-CASCI propagation for the electronic wavefunction . (vii). Repeat steps (i)-(vi) until the desired propagation time reached. with time steps Δτ Ψ The initial condition for the position and momentum (R0, p0) can be sampled from a Wigner distribution corresponding to the ground vibrational state in the harmonic approximation or from classical molecular dynamics simulations on the ground state. With the initial R0, the initial electronic wave function is obtained by a user-selected CASCI approach such as Hartree- Fock CASCI (HF-CASCI), the floating occupation molecular orbital CASCI (FOMO-CASCI)8-9 or the configuration interaction singles natural orbital CASCI (CISNO-CASCI),10-11 in which the HF, FOMO and CISNO methods are the orbital determination approaches for the CASCI method. To calculate the force at each nuclear coordinate (step (i)), F = − d dR Ψ H Ψ . (1) . Notice that the wave function is a complex vector in the TD-CASCI, Ψ = q + ip As a result, the separation of the calculations of the force which are contributed from the real and imaginary parts of the wave function (CI vector) is performed, (2) F = − d dR Ψ H Ψ = − ⎛ ⎝⎜ d dR q H q + d dR p H p ⎞ ⎠⎟ . (3) The analytical energy gradients for the HF-CASCI, FOMO-CASCI, and CISNO-CASCI methods can be obtained from the GPU-accelerated implementations, 8, 10 where the equation (42) in reference 8 or the equation (56) in reference 10 are employed, depends on which CASCI method is applied. Note that the real and imaginary parts of the CI vectors along with other parameters (parameters that define the wave functions and depend on R, but are not propagated for all three CASCI methods by the TD-CASCI method, e.g. the orbital coefficients matrix Cµp or the configuration interaction singles eigenvectors for the CISNO-CASCI method) are used for the analytical energy gradient calculations. After the force is calculated, the VV algorithm (steps (ii) and (iii)) is performed to update the momentum and position for the nucleus.12 With the new 196 nuclear geometry Rt+Δt , the selected orbital determination scheme (HF, FOMO or CISNO) is applied for a new set of orbitals (step (iv)). The electronic structure changes when the nuclear geometry is updated. In order to account for the electronic structure which is coupled to the nuclear motion, an orbital diabatization scheme that is described as following equations is performed (step (v)). , ψi,t+Δt c j iψj,t+Δt (4) = ' ∑ j in which the orbitals ψj,t+Δt are rotated to a new set of orbitals ' ψi,t+Δt with the following constrain ψi,t+Δt ' ψj,t = 0 for i ≠ j . (5) As a result, the coefficients i c j can be obtained from µν . , . i, j (7) † (t + Δt)S (6) C and the c j i = ψj,t+Δt ψi,t Overall, the orbital diabatization scheme can be written with the coefficient matrices overlap matrix between basis functions at time t + Δt and time t denote for (t + Δt |t)Cνi(t) denote for the molecular orbitals in the active space and Sµν(t + Δt |t) Cµi ' (t + Δt) = Cµj(t + Δt)Cµj In equation (4), the the atomic basis sets. With the new set of molecular orbitals and applied external field, the electronic Hamiltonian is built and the electronic degrees of freedom are propagated via the TD- CASCI method (step (vi)). Steps (i)-(vi) are repeated until the desired propagation time is reached. The TD-CASCI-Ehrenfest dynamics method is implemented in the TeraChem software package.7-14 The extension of this approach to the TD-MRCIS-Ehrenfest will be straightforward when the analytical energy gradient of the MRCIS method (or other efficient means for retrieving the forces from the MRCIS method) is available. In addition, schemes for correcting the Ehrenfest dynamics method for decoherence in the dense manifold regime are being developed in the Levine’s group. µ,ν 197 Rt pt! (i) Force (iii) (ii) Δ! (vi) (iv) and (v) Diabatize orbitals Next (i) Force t t + (1/2)Δt t + Δt Δt Δτ for propagating electronic dynamics in the TD-CI for propagating momentum and position respectively. The smaller Figure 7.1. The schematic representation of the Ehrenfest dynamics algorithm. The steps (i)-(vi) mentioned in the text are labeled. The orange and blue arrows indicate the velocity-Verlet algorithm with time steps green arrows indicate the time steps calculations. For the feedback between electronic and nuclear dynamics, the force calculation and orbital diabatization (both indicated in red words) are performed. 198 REFERENCES 199 REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Ben-Nun, M.; Quenneville, J.; Martinez, T. J., Ab initio multiple spawning: Photochemistry from first principles quantum molecular dynamics. J Phys Chem A 2000, 104, 5161-5175. Levine, B. G.; Coe, J. D.; Virshup, A. M.; Martinez, T. J., Implementation of ab initio multiple spawning in the MOLPRO quantum chemistry package. Chem Phys 2008, 347, 3-16. Tully, J. C., Molecular-Dynamics with Electronic-Transitions. J Chem Phys 1990, 93, 1061-1071. Parandekar, P. V.; Tully, J. C., Mixed quantum-classical equilibrium. 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