From quantum to classical molecular dynamics: Reduced models and numerical analysis

✍ Scribed by Lubich C.

• Publisher: EMS
• Year: 2008
• Tongue: English
• Leaves: 154
• Series: Zurich Lectures in Advanced Mathematics
• Category: Library

✦ Synopsis

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