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Cited by 25

The DOI matches its Crossref record; 50 of the 50 references it cites were found in a registry or an index.

Efficient geometric integrators for nonadiabatic quantum dynamics. I. The adiabatic representation

Choi, Seonghoon, Vaníček, Jiří·2019·The Journal of Chemical Physics
DOI 10.1063/1.5092611
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Abstract

Geometric integrators of the Schrödinger equation conserve exactly many invariants of the exact solution. Among these integrators, the split-operator algorithm is explicit and easy to implement but, unfortunately, is restricted to systems whose Hamiltonian is separable into kinetic and potential terms. Here, we describe several implicit geometric integrators applicable to both separable and nonseparable Hamiltonians and, in particular, to the nonadiabatic molecular Hamiltonian in the adiabatic representation. These integrators combine the dynamic Fourier method with the recursive symmetric composition of the trapezoidal rule or implicit midpoint method, which results in an arbitrary order of accuracy in the time step. Moreover, these integrators are exactly unitary, symplectic, symmetric, time-reversible, and stable and, in contrast to the split-operator algorithm, conserve energy exactly, regardless of the accuracy of the solution. The order of convergence and conservation of geometric properties are proven analytically and demonstrated numerically on a two-surface NaI model in the adiabatic representation. Although each step of the higher order integrators is more costly, these algorithms become the most efficient ones if higher accuracy is desired; a thousand-fold speedup compared to the second-order trapezoidal rule (the Crank-Nicolson method) was observed for a wavefunction convergence error of 10−10. In a companion paper [J. Roulet, S. Choi, and J. Vaníček, J. Chem. Phys. 150, 204113 (2019)], we discuss analogous, arbitrary-order compositions of the split-operator algorithm and apply both types of geometric integrators to a higher-dimensional system in the diabatic representation.

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Citations

Cites

Research works printed but not linked: 18

  1. 2018, The Semiclassical Way to Dynamics and Spectroscopy
  2. 2006, Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections
  3. 2012, Nonadiabatic Transition: Concepts, Basic Theories and Applications
  4. 2015, Chemical Theory Beyond the Born-Oppenheimer Paradigm: Nonadiabatic Electronic and Nuclear Dynamics in Chemical Reactions
  5. 2008, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis
  6. 2006, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations
  7. 2002, Understanding Molecular Simulation
  8. 2007, Introduction to Quantum Mechanics: A Time-Dependent Perspective
  9. 2004, Simulating Hamiltonian Dynamics
  10. 2002, Quantum Simulation of Complex Many-Body Systems: From Theory to Algorithms, Lecture Notes, 459
  11. 1964, Bol. Soc. Mat. Mex., 9, 55
  12. 2006, Dynamical Systems: Stability Theory and Applications
  13. 1996, Matrix Computations
  14. 2007, Numerical Recipes, The Art of Scientific Computing
  15. 2003, Iterative Methods for Sparse Linear Systems
  16. 1942, Finite Dimensional Vector Spaces
  17. 1951, Introduction to Hilbert Space and the Theory of Spectral Multiplicity
  18. 1998, Matrix Algorithms

Metadata

Authors
Choi, S., & Vaníček, J.
Venue
The Journal of Chemical Physics
Publisher
AIP Publishing
Published
2019-05-28
Volume
150
Issue
20
ISSN
  • 0021-9606
  • 1089-7690
DOI
10.1063/1.5092611
Cited by
25
References
50
Language
en
Licence
Funders
H2020 European Research Council (683069)

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