| About This Book |
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xix | |
| Introduction: Electron Multistate Molecular Dynamics |
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xxv | |
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1 | (198) |
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1 Ab Initio Theory of Electronic Structure |
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3 | (42) |
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4 | (8) |
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1.1.1 Molecular and atomic orbitals |
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9 | (2) |
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1.1.2 Expectation values from molecular orbitals: The example of spin--orbit coupling |
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11 | (1) |
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1.2 The Minimal LCAO Model |
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12 | (4) |
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16 | (7) |
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1.3.1 The Hartree--Fock equations |
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17 | (4) |
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21 | (1) |
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1.3.3 The Hartree--Fock Hamiltonian |
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22 | (1) |
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1.4 The Restricted and the Unrestricted Hartree--Fock Formalism |
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23 | (7) |
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1.4.1 The restricted Hartree--Fock method |
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24 | (1) |
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1.4.2 The unrestricted Hartree--Fock method |
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25 | (2) |
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1.4.3 The Roothaan formalism |
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27 | (3) |
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1.5 Post-Hartree--Fock Methods |
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30 | (6) |
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1.5.1 Configuration interaction |
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31 | (3) |
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1.5.2 Many-body perturbation theory |
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34 | (2) |
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1.6 Excited Electronic States |
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36 | (5) |
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1.7 Appendix: The Functional Derivative |
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41 | (4) |
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2 The Adiabatic and the Diabatic Representation |
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45 | (42) |
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2.1 The Born--Oppenheimer Approximation |
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46 | (4) |
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2.2 Harmonic Vibrational Modes |
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50 | (4) |
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2.3 Adiabatic and Diabatic Frames |
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54 | (5) |
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2.3.1 The diabatic approximation |
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58 | (1) |
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2.4 Gauge Theoretical Form of the Nuclear Equation |
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59 | (4) |
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2.5 Avoided Crossings, Degeneracies, Conical Intersections |
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63 | (15) |
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63 | (2) |
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2.5.2 Conical intersections and the Jahn--Teller effect |
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65 | (3) |
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2.5.3 Jahn---Teller distortion |
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68 | (3) |
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2.5.4 The molecular Aharonov--Bohm effect |
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71 | (4) |
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2.5.5 The geometric phase in molecular pseudorotation |
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75 | (3) |
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78 | (9) |
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2.6.1 Intersection-adapted coordinates |
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79 | (3) |
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2.6.2 Determination of the seam space |
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82 | (2) |
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2.6.3 Seam subspaces by Lagrangian minimization |
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84 | (3) |
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3 Basic Concepts of Scattering Theory |
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87 | (34) |
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3.1 The Time-Dependent and the Time-Independent View of Scattering Processes |
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88 | (1) |
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3.2 Quantum Mechanical Equations of Motion |
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89 | (5) |
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3.3 The Scattering Matrix |
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94 | (10) |
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3.3.1 The Møller operators |
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96 | (2) |
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3.3.2 The Lippmann--Schwinger equations |
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98 | (2) |
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3.3.3 Unitarity of the S-matrix |
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100 | (4) |
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3.4 Elastic Scattering by a Spherical Potential |
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104 | (9) |
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3.4.1 The asymptotic scattering solution |
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104 | (3) |
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3.4.2 T-, S-, and K-matrix boundary conditions |
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107 | (4) |
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3.4.3 The elastic cross section |
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111 | (2) |
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113 | (8) |
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121 | (40) |
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4.1 Path Integrals and the Quantum Propagator |
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122 | (14) |
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4.1.1 The quantum and the semiclassical propagator |
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123 | (5) |
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4.1.2 The Van Vleck propagator |
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128 | (5) |
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4.1.3 The monodromy matrix |
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133 | (3) |
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4.2 The WKB Approximation |
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136 | (10) |
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4.2.1 The WKB wave function |
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136 | (8) |
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4.2.2 The Bohr--Sommerfeld quantization rules for bound WKB states |
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144 | (2) |
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4.3 The Wigner Function: A Quantum Mechanical Phase Space Distribution |
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146 | (8) |
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4.3.1 Defining properties of the Wigner function |
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146 | (3) |
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4.3.2 Time dependence of the Wigner function |
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149 | (4) |
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4.3.3 The Moyal formalism |
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153 | (1) |
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154 | (7) |
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4.4.1 Coherent and particle number states |
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155 | (3) |
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4.4.2 Coherent states as minimal uncertainty solutions |
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158 | (1) |
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4.4.3 The nuclear coherent state |
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159 | (2) |
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5 Open Systems: Elements of Rate Theory |
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161 | (38) |
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5.1 Classical Rate Theory |
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163 | (6) |
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5.2 Quantum Transition State Theory |
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169 | (5) |
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5.2.1 The quantum transition state approximation |
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171 | (3) |
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5.3 The Euclidean Path Integral |
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174 | (7) |
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5.3.1 Classical polymer isomorphism |
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178 | (3) |
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181 | (6) |
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5.5 The Path Integral Form of the Golden Rule Rate Constant |
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187 | (3) |
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5.6 Beyond the Golden Rule: Reduced Density Matrix Theory |
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190 | (9) |
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5.6.1 A two-state problem |
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195 | (4) |
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199 | (462) |
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6 Time-Independent Theory of Molecular Collisions I: Multichannel Scattering |
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201 | (44) |
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6.1 The Multichannel Problem |
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201 | (5) |
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6.2 The Lippmann---Schwinger Equation for Inelastic Scattering |
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206 | (4) |
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6.3 The Born Approximation |
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210 | (3) |
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6.3.1 The distorted-wave Born approximation (DWBA) |
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212 | (1) |
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213 | (4) |
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6.5 R-matrix and Log Derivative Propagation |
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217 | (6) |
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6.5.1 The log derivative method |
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220 | (3) |
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6.6 Reactive Scattering I: The Differential Equation Approach |
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223 | (6) |
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224 | (3) |
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6.6.2 Hyperspherical coordinates |
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227 | (2) |
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6.7 Space-Fixed and Body-Fixed Frames of Reference |
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229 | (9) |
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6.7.1 Space-fixed representation |
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230 | (2) |
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6.7.2 Body-fixed representation |
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232 | (6) |
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6.8 Reactive Scattering II: The Integral Equation Approach |
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238 | (7) |
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7 Time-Independent Theory of Molecular Collisions II: The Electronic Problem |
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245 | (46) |
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7.1 Inclusion of the Electronic System |
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246 | (9) |
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247 | (4) |
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251 | (2) |
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253 | (2) |
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7.2 Case Study: The F + H2 Reaction |
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255 | (8) |
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7.3 Variational Procedures |
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263 | (8) |
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7.3.1 The Kohn variational principle |
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267 | (2) |
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269 | (2) |
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7.4 Case Study: Quenching of the Sodium Atom 3p State by Interaction with Hydrogen Molecules |
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271 | (11) |
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277 | (2) |
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7.4.2 Algebraic realization of the outgoing wave variational principle |
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279 | (1) |
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7.4.3 Exciplex funnel dynamics |
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280 | (2) |
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7.5 The Landau--Zener--Stuckelberg Model of Nonadiabatic Transitions |
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282 | (9) |
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8 The Time-Dependent Self-Consistent Field Theory |
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291 | (50) |
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8.1 Time-Dependent Variational Principles |
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292 | (8) |
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8.1.1 Time-dependent perturbations |
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295 | (4) |
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8.1.2 Free and forced oscillations |
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299 | (1) |
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8.2 The Time-Dependent Hartree--Fock Theory: Application to Molecules |
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300 | (5) |
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8.3 Wave-Function-Based Ab Initio Molecular Dynamics |
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305 | (17) |
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8.3.1 Direct molecular dynamics in the time-dependent Hartree--Fock framework |
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305 | (2) |
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8.3.2 Classical trajectories within TDHF dynamics |
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307 | (5) |
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8.3.3 The Hellmann--Feynman theorem |
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312 | (3) |
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315 | (3) |
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8.3.5 Car--Parrinello dynamics |
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318 | (4) |
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8.4 Time-Dependent Hartree--Fock Dynamics in the Eikonal Approximation |
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322 | (11) |
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8.4.1 The eikonal approximation |
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324 | (3) |
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8.4.2 TDHF approach to the electronic problem within the eikonal approximation |
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327 | (2) |
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8.4.3 The Liouville--von Neumann equation in a traveling orbital basis |
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329 | (4) |
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8.5 Case Study: Light Emission in Slow Proton--Hydrogen Collisions |
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333 | (8) |
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9 Evolution of Coherent Molecular States: Electron Nuclear Dynamics Theory |
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341 | (42) |
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9.1 The Thouless Representation |
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343 | (5) |
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348 | (9) |
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9.2.1 Derivation of the END equations |
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349 | (5) |
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9.2.2 Interpretation of the END equations |
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354 | (3) |
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9.3 Two Special Cases: The Boosted Self-Consistent Field and the Linearized END Equations |
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357 | (4) |
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9.3.1 The boosted electronic system |
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357 | (3) |
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9.3.2 The linear version of the electronic END equations |
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360 | (1) |
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9.4 Inclusion of Nuclear Quantum Effects |
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361 | (15) |
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9.4.1 Trajectory interference |
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362 | (4) |
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9.4.2 Case study: H impact on molecular and atomic targets by END theory |
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366 | (6) |
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9.4.3 Rovibrational analysis of the nuclear system |
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372 | (4) |
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9.5 Nonadiabatic Effects in Bound Systems by END Theory: The Pseudorotation of H+3 |
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376 | (7) |
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10 The Classical Electron Analog |
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383 | (20) |
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10.1 Critique of the Ehrenfest Representation |
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384 | (2) |
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10.2 The Classical Electron Analog |
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386 | (9) |
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10.2.1 The CEA equations of motion |
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388 | (1) |
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10.2.2 Adiabatic representation of the Hamilton function |
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389 | (2) |
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10.2.3 The classical analog of the electronic two-state problem |
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391 | (4) |
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10.3 CEA Theory Applied to a Conical Intersection Problem |
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395 | (8) |
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403 | (34) |
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11.1 The Trajectory Surface Hopping Method |
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404 | (4) |
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11.2 The Fewest Switches Algorithm |
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408 | (11) |
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413 | (5) |
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11.2.2 Complex-valued trajectories |
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418 | (1) |
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419 | (10) |
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11.3.1 Applications to model problems |
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425 | (4) |
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11.4 Case Study: The Dynamics of Na*-Quenching by Collision with Hydrogen Molecules |
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429 | (4) |
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11.5 Comparison with Other Methods |
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433 | (4) |
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12 Semiclassical Propagator Techniques |
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437 | (54) |
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12.1 The Path Integral Approach to Molecular Dynamics |
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438 | (4) |
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12.2 Semiclassical Propagation and Surface Hopping |
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442 | (5) |
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12.3 The Initial Value Representation |
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447 | (4) |
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12.4 The Mapping Approach to Electronic Degrees of Freedom |
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451 | (6) |
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12.4.1 The Schwinger mapping formalism |
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452 | (2) |
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12.4.2 Extension to general N-level systems |
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454 | (3) |
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12.5 The Mapping Technique Applied to Nonadiabatic Dynamics |
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457 | (9) |
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12.5.1 The SC-IVR approach applied to nonadiabatic model cases |
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460 | (4) |
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12.5.2 Comparison with the Ehrenfest model |
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464 | (2) |
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12.6 Case Study: The S1--S2 Transition in Pyrazine: SC-IVR Treatment of a Conical Intersection Problem |
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466 | (5) |
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12.7 Numerical Procedures for Semiclassical Propagation Methods |
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471 | (11) |
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12.7.1 Monte Carlo integration |
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473 | (3) |
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476 | (3) |
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12.7.3 The forward-backward initial value representation |
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479 | (3) |
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482 | (9) |
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13 Quantum Hydrodynamics I: Coupled Trajectories in Bohmian Mechanics |
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491 | (26) |
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13.1 Elements of the Quantum Theory of Motion |
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492 | (7) |
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13.1.1 Quantum trajectories |
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495 | (3) |
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13.1.2 The pilot wave and the guided particle |
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498 | (1) |
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13.2 Lagrangian Quantum Hydrodynamics |
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499 | (6) |
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13.2.1 Assembling the wave function |
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502 | (1) |
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13.2.2 Technical challenges for quantum trajectory propagation |
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503 | (2) |
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13.3 Nonadiabatic Lagrangian Quantum Hydrodynamics |
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505 | (7) |
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13.4 The Classical Limit of the Quantum Theory of Motion |
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512 | (5) |
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14 Quantum Hydrodynamics II: The Semiclassical Liouville--Von Neumann Equation |
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517 | (30) |
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14.1 The Semiclassical Liouville Formalism for Multistate Problems |
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518 | (10) |
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14.1.1 Two coupled states: A model problem |
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524 | (4) |
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14.2 Phase Space Trajectory Implementation |
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528 | (6) |
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14.3 Generalized Quantum Hydrodynamics: Mixed States |
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534 | (8) |
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538 | (2) |
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540 | (2) |
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14.4 Coupled Electronic States |
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542 | (5) |
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15 Wave Packet Propagation Methods |
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547 | (40) |
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15.1 The Grid Representation |
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548 | (9) |
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15.1.1 The discrete variable representation (DVR) |
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553 | (2) |
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15.1.2 The fast Fourier transform (FFT) |
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555 | (2) |
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15.2 Numerical Wave Packet Propagation Techniques |
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557 | (7) |
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15.2.1 The Crank--Nicolson scheme |
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557 | (2) |
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15.2.2 Split operator propagation |
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559 | (1) |
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15.2.3 Propagator expansion techniques |
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560 | (4) |
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15.3 The Multiconfiguration Time-Dependent Hartree (MCTDH) Method |
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564 | (10) |
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15.3.1 The time-dependent Hartree (TDH) approach |
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565 | (2) |
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15.3.2 The multiconfiguration time-dependent Hartree (MCTDH) approach |
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567 | (2) |
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15.3.3 The MCTDH equations |
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569 | (5) |
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15.4 Case Study: Photostability of Biologically Relevant Molecules |
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574 | (13) |
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15.4.1 Ultrafast deexcitation by passage through conical intersections in nucleic acid bases and base pairs |
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575 | (4) |
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15.4.2 Dynamics at the 1πσ*-S0 conical intersection of pyrrole |
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579 | (8) |
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16 Density Functional Dynamics |
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587 | (26) |
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16.1 Fundamentals of Density Functional Theory |
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588 | (7) |
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16.1.1 Exchange-correlation potentials |
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593 | (2) |
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16.2 Excited Electronic States in DFT |
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595 | (3) |
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16.3 Time-Dependent Density Functional Theory |
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598 | (7) |
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16.3.1 TDDFT in the linear response domain |
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601 | (2) |
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16.3.2 Time-dependent current density functional theory |
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603 | (2) |
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16.4 Direct Molecular Dynamics Based on DFT |
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605 | (8) |
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16.4.1 Calculating molecular photoabsorption spectra |
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606 | (2) |
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16.4.2 Molecular bonding properties analyzed by the electron localization function |
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608 | (1) |
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16.4.3 Combining TDDFT with standard methods of nonadiabatic dynamics |
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609 | (4) |
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613 | (48) |
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17.1 The Dissipative Liouville--von Neumann Equation |
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615 | (11) |
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17.2 Evaluating Decoherence Times in a Semiclassical Framework |
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626 | (10) |
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17.2.1 Ensemble average of the decoherence function |
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632 | (4) |
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17.3 Case Study: The Dynamics of Electron Hydration |
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636 | (8) |
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17.3.1 Isotope effects in hydrated electron relaxation |
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638 | (6) |
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17.4 Continuous Surface Switching: A Compromise between Mean-Field and Individual Surface Propagation |
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644 | (4) |
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648 | (13) |
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657 | (1) |
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17.5.2 Determining the decoherent state |
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658 | (3) |
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661 | (180) |
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18 Ultrafast Optical Spectroscopy |
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663 | (52) |
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18.1 Linear and Nonlinear Polarization |
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664 | (5) |
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18.1.1 Deriving the pump--probe signal |
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666 | (3) |
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18.2 Theory of Nonlinear Polarization in Femtosecond Molecular Spectroscopy |
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669 | (13) |
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18.2.1 The perturbative approach |
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672 | (9) |
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18.2.2 The non-perturbative approach |
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681 | (1) |
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18.3 Polarization Studies of cis-trans Isomerization |
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682 | (8) |
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18.3.1 Adiabatic formulation |
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688 | (2) |
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18.4 The Density Matrix Approach to Simulating Pump---Probe Signals |
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690 | (10) |
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18.4.1 The pump--probe signal |
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698 | (2) |
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18.5 Case Study: Ultrafast Spectroscopy on Non-Stoichiometric Alkali-Halide Clusters |
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700 | (11) |
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18.5.1 Effective single-electron systems of the form NanFn-1 |
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701 | (5) |
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18.5.2 Extension to nonadiabatic dynamics |
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706 | (5) |
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18.6 Appendix: Derivation of the Pump--Probe Signal S(td) |
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711 | (4) |
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19 Optical Control of Electron Multistate Molecular Dynamics |
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715 | (34) |
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19.1 Interaction of a Molecule with a Pulse of Light |
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716 | (3) |
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19.2 The Tannor--Rice Scheme: Optimal Control |
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719 | (6) |
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19.3 The Brumer--Shapiro Scheme: Coherent Control |
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725 | (5) |
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19.4 Case Study: Coherent Control of ICN Photodissociation |
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730 | (6) |
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19.5 Optimal Control in Pump--Probe Spectroscopy |
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736 | (13) |
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19.5.1 Case study: Application to Na3F2 |
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743 | (6) |
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20 Electron Transfer in Condensed Media |
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749 | (68) |
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20.1 The Electronic Hamiltonian |
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753 | (3) |
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20.2 Electronic--Vibronic Coupling: The Spin-Boson Hamiltonian |
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756 | (4) |
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20.3 Adiabatic versus Nonadiabatic Electron Transfer |
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760 | (3) |
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20.4 Thermally Activated Transfer |
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763 | (4) |
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20.5 Inclusion of Nuclear Tunneling |
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767 | (7) |
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20.5.1 The continuous limit of nuclear frequencies |
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771 | (3) |
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20.6 Effects of Polar Solvents on Electron Transfer |
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774 | (16) |
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20.6.1 The dielectric displacement field |
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776 | (2) |
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20.6.2 Polarization and polarizability |
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778 | (5) |
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20.6.3 The free energy functional |
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783 | (3) |
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20.6.4 The electron transfer rate in a polar environment |
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786 | (4) |
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20.7 Ultrafast Electron Transfer |
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790 | (3) |
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20.8 Case Study: Aqueous Ferrous--Ferric Exchange |
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793 | (12) |
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20.8.1 Monte Carlo modeling |
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793 | (5) |
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20.8.2 Euclidean path integral simulations |
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798 | (6) |
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20.8.3 Recent quantum dynamical extensions |
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804 | (1) |
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20.9 Appendix: Formulae Relevant for Electron Transfer Theory within the Marcus Model |
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805 | (12) |
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20.9.1 Electron transfer in a vibrational bath: Formal procedures used in the derivation of the rate constant |
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806 | (4) |
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20.9.2 Derivation of the effective free energy functional Eq. (20.109) |
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810 | (3) |
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20.9.3 The density of states for electron transfer in a solvent: Calculating the trace Eq. (20.118) |
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813 | (4) |
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21 Electronic Friction in Molecule--Surface Interactions |
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817 | (24) |
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21.1 Langevin Formulation of Ehrenfest Dynamics |
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820 | (4) |
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21.2 An Ab Initio Model for Electronic Friction |
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824 | (4) |
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21.3 Case Study: Nonadiabatic Effects in the Interaction between the Cu(100) Surface and a CO Molecule |
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828 | (10) |
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21.3.1 Vibrational relaxation of CO on the Cu(100) surface: The impact of electronic friction |
|
|
828 | (7) |
|
21.3.2 Vibrational excitation and hot diffusion |
|
|
835 | (3) |
|
21.4 Beyond Langevin Theory |
|
|
838 | (3) |
| Epilogue |
|
841 | (6) |
| Bibliography |
|
847 | (28) |
| Index |
|
875 | |