Preface ........................................................ ix
Chapter 1 Introduction ......................................... 1
Chapter 2 One-Dimensional Theory ............................... 5
2.1 Exactly Solvable Cases ..................................... 5
2.1.1 Case of Delta-Function Barrier ...................... 5
2.1.2 Case of Parabolic Potential Barrier ................. 6
2.1.3 Case of Eckart Potential Barrier .................... 8
2.2 WKB Approximation and Connection Formula .................. 10
2.3 Comparison Equation Method ................................ 11
2.4 Diagrammatic Technique .................................... 13
2.5 Instanton Theory and Modified WKB Method .................. 16
2.5.1 Instanton Theory ................................... 16
2.5.2 Modified WKB Method ................................ 24
2.6 Energy Levels in a Double Well Potential .................. 26
2.6.1 Asymmetric Double Well Potential ................... 26
2.6.2 Symmetric Double Well Potential .................... 28
2.7 Decay of Metastable State ................................. 29
Chapter 3 Two-Dimensional Theory .............................. 33
3.1 WKB Theory ................................................ 33
3.2 Instanton Theory .......................................... 40
Chapter 4 Multidimensional Effects: Peculiar Phenomena ........ 43
4.1 Effects of Vibrational Excitation on Tunneling Splitting .. 43
4.1.1 Adiabatic and Sudden Approximations ................ 43
4.1.2 Case of Symmetric Mode Coupling Potential .......... 44
4.1.3 Case of Antisymmetric Mode Coupling Potential ...... 49
4.1.4 Case of Squeezed (Sqz) Double Well Potential ....... 50
4.2 Insufficiency of Two-Dimensional Model .................... 54
4.3 Proton Tunneling in Tropolone ............................. 54
4.3.1 Available Experimental Data ........................ 54
4.3.2 Tunneling Dynamics in the Ground X State ........... 56
4.3.3 Analysis of Tunneling Dynamics of the
Excited A State .................................... 59
Chapter 5 Nonadiabatic Tunneling .............................. 61
5.1 Definition and Qualitative Explanation .................... 61
5.2 One-Dimensional Theory .................................... 64
5.2.1 Caseof E ≤ Et ...................................... 67
5.2.2 Case of Et ≤ E ≤ Eb ................................ 68
5.2.3 Case of Eb ≤ E ..................................... 68
Chapter 6 Multidimensional Theory of Tunneling Splitting ...... 75
6.1 General Formulation ....................................... 75
6.1.1 Multidimensional Extension of the Instanton
Theory ............................................. 75
6.1.2 WKB Approach in Cartesian Coordinates .............. 82
6.1.3 WKB Approach in the Case of General Hamiltonian
in Curved Space .................................... 85
6.2 How to Find Instanton Trajectory .......................... 89
6.3 How to Use the Theory ..................................... 92
6.3.1 Evaluation of the Pre-Exponential Factor ........... 92
6.3.2 Incorporation of High Level of ab initio Quantum
Chemical Calculations .............................. 95
6.4 Case of Low Vibrationally Excited States .................. 96
6.4.1 One- and Two-Dimensional Cases ..................... 96
6.4.2 Multidimensional Case in Terms of Cartesian
Coordinates ........................................ 99
6.4.3 Case of General Multidimensional Curved Space ..... 103
Chapter 7 Numerical Applications to Polyatomic Molecules ..... 109
7.1 N-Dimensional Separable Potential Model .................. 109
7.2 Hydroperoxy Radical HO2 .................................. 111
7.3 Vinyl Radical C2H3 ....................................... 120
7.4 Malonaldehyde C3O2H4 ..................................... 128
7.5 Formic Acid Dimer (DCOOH)2 ............................... 139
Chapter 8 Decay of Metastable States ......................... 149
8.1 General Formulation ...................................... 149
8.1.1 Determination of Instanton Trajectory ............. 149
8.1.2 Formulation in Terms of Cartesian Coordinates ..... 151
8.1.3 General Canonically Invariant Formulation ......... 154
8.2 Numerical Application .................................... 158
Chapter 9 Tunneling in Chemical Reactions .................... 163
9.1 Determination of Caustics and Propagation in
Tunneling Region ......................................... 163
9.1.1 Caustics in Chaotic Henon-Heiles System ........... 166
9.1.2 Caustics in Chemical Reaction Dynamics ............ 167
9.2 Direct Evaluation of Reaction Rate Constant .............. 174
9.2.1 Adiabatic Chemical Reaction ....................... 174
9.2.2 Nonadiabatic Chemical Reaction .................... 178
Chapter 10 Concluding Remarks and Future Perspectives ......... 183
Appendix A Proofs of Equation (2.95) and Equation (2.110) .... 185
Appendix В Derivation of Equation (6.80) ..................... 187
Appendix С Herring Formula in Curved Space ................... 189
Appendix D Derivation of Equation (6.97) ..................... 191
Appendix E Computer Code to Calculate Instanton Trajectory ... 193
Appendix F Derivation of Some Equations in Section 6.4.2 ..... 201
Bibliography .................................................. 205
Index ......................................................... 213
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