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从量子论到经典分子动力学：简化模型及数值分析（英文版）
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From quantum to classical molecular dynamics: Reduced models and numerical analysis Quantum dynamics of molecules poses a variety of computational challenges that are presently at the forefront of research efforts in numerical analysis in a number of application areas: high-dimensional partial differential equations, multiple scales, highly oscillatory solutions, and geometric structures such as symplecticity and reversibility that are favourably preserved in discretizations. This text addresses such problems in quantum mechanics from the viewpoint of numerical analysis, illustrating them to a large extent on intermediate models between the Schrodinger equation of full many-body quantum dynamics and the Newtonian equations of classical molecular dynamics. The fruitful interplay between quantum dynamics and numerical analysis is emphasized. Table of contents : Preface......Page 5 Table of Contents......Page 7 I.1.1 ClassicalMechanics......Page 11 I.1.2 Quantum Mechanics......Page 12 I.2.1 Dispersion Relation......Page 13 I.2.2 Solution by Fourier Transformation......Page 14 I.2.3 Propagation of HeavyWave Packets......Page 16 I.3.1 Self-Adjoint Operators and Existence of Dynamics......Page 17 I.3.3 Lie–Trotter Product Formula......Page 19 I.4.2 Heisenberg Picture and Ehrenfest Theorem......Page 21 I.4.3 Heisenberg Uncertainty Relation......Page 23 I.5.1 Distinguishable Particles......Page 24 I.5.2 Indistinguishable Particles......Page 25 I.5.3 The Molecular Hamiltonian......Page 26 II.1.1 Abstract Formulation......Page 29 II.1.2 Interpretation as an Orthogonal Projection......Page 30 II.1.3 Interpretation as a Symplectic Projection......Page 31 II.1.5 Conservation Properties......Page 33 II.1.6 An A Posteriori Error Bound......Page 34 II.2.2 Schrödinger Equation for the Nuclei on an Electronic Energy Surface......Page 35 II.2.3 Semi-Classical Scaling......Page 37 II.2.4 Spectral Gap Condition......Page 38 II.2.5 Approximation Error......Page 39 II.3 Separating the Particles: Self-Consistent Field Methods......Page 43 II.3.1 Time-Dependent Hartree Method (TDH)......Page 44 II.3.2 Time-Dependent Hartree–Fock Method (TDHF)......Page 47 II.3.3 Multi-Configuration Methods (MCTDH, MCTDHF)......Page 52 II.4 Parametrized Wave Functions: Gaussian Wave Packets......Page 56 II.4.1 Variational Gaussian Wave-Packet Dynamics......Page 57 II.4.2 Non-Canonical Hamilton Equations in Coordinates......Page 58 II.4.3 Poisson Structure of Gaussian Wave-Packet Dynamics......Page 61 II.4.4 Approximation Error......Page 63 II.5.1 Mean-Field Quantum-Classical Model......Page 64 II.5.2 Quantum Dressed Classical Mechanics......Page 66 II.5.3 Swarms of Gaussians......Page 67 II.6 Quasi-Optimality of Variational Approximations......Page 68 III.1 Space Discretization by Spectral Methods......Page 73 III.1.1 Galerkin Method, 1D Hermite Basis......Page 74 III.1.2 Higher Dimensions: Hyperbolic Cross and Sparse Grids......Page 80 III.1.3 Collocation Method, 1D Fourier Basis......Page 85 III.1.4 Higher Dimensions: Hyperbolic Cross and Sparse Grids......Page 89 III.2.1 Chebyshev Method......Page 95 III.2.2 Lanczos Method......Page 101 III.3.1 Splitting Between Kinetic Energy and Potential......Page 106 III.3.2 Error Bounds for the Strang Splitting......Page 108 III.3.3 Higher-Order Compositions......Page 111 III.4 Integrators for Time-Dependent Hamiltonians......Page 112 IV.1.1 Abstract Formulation......Page 115 IV.1.2 Space Discretization of the Hartree and MCTDH Equations......Page 116 IV.1.3 Discretization Error......Page 118 IV.2.1 Splitting the Variational Equation......Page 119 IV.2.2 Error Analysis......Page 120 IV.3 Variational Splitting for MCTDH......Page 126 IV.4 Variational Splitting for Gaussian Wave Packets......Page 128 V.1 Hagedorn’s Parametrization of Gaussian Wave Packets......Page 133 V.2 Hagedorn’s Semi-Classical Wave Packets......Page 137 V.3 A Numerical Integrator for Hagedorn Wave Packets......Page 141 Bibliography......Page 147 Index......Page 152 |
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