# Frozen Gaussian approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms

@article{Lu2016FrozenGA,
title={Frozen Gaussian approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms},
author={Jianfeng Lu and Zhen-jiang Zhou},
journal={Math. Comput.},
year={2016},
volume={87},
pages={2189-2232}
}
• Published 20 February 2016
• Computer Science
• Math. Comput.
We develop a surface hopping algorithm based on frozen Gaussian approximation for semiclassical matrix Schr\"odinger equations, in the spirit of Tully's fewest switches surface hopping method. The algorithm is asymptotically derived from the Schr\"odinger equation with rigorous approximation error analysis. The resulting algorithm can be viewed as a path integral stochastic representation of the semiclassical matrix Schr\"odinger equations. Our results provide mathematical understanding to and…

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## References

SHOWING 1-10 OF 37 REFERENCES

• Physics
Multiscale Model. Simul.
• 2015
In the paper we derive a semiclassical model for surface hopping allowing quantum dynamical nonadiabatic transition between different potential energy surfaces in which cases the classical
• Mathematics
• 2007
This article presents and evaluates a surface hopping algorithm for time-dependent two-level Schrodinger systems with conically intersecting eigenvalues. The algorithm implements an asymptotic
We present a unified derivation of the mean-field (Ehrenfest) and surface-hopping approaches to mixed quantum–classical dynamics that elucidates the underlying approximations of the methods and their
• Physics
• 2002
In mixed quantum-classical molecular dynamics few but important degrees of freedom of a molecular system are modeled quantum-mechanically while the remaining degrees of freedom are treated within the
• Physics
The Journal of chemical physics
• 2011
This paper proposes an inexpensive correction to standard FSSH dynamics wherein it explicitly model the decoherence of nuclear wave packets on distinct electronic surfaces to provide a new and natural approach for rescuing nuclear momenta after a surface hop.
• Mathematics
SIAM J. Math. Anal.
• 2008
A surface hopping semigroup, which asymptotically describes nuclear propagation through crossings of electron energy levels, is constructed and convergence to the true solution is proved with an error of the order of $\varepsilon^{1/8}$, where $\varpsilon$ is the semiclassical parameter.
• Mathematics
Multiscale Model. Simul.
• 2011
The surface hopping method for the Schrodinger equation with conical crossings in the Eulerian formulation is developed, based on the semiclassical approximation governed by the Liouville equations, which are valid away from the conical crossing manifold.
A stochastic mean-field (SMF) approach to nonadiabatic molecular simulations is introduced. Based on the quantum-classical mean-field approximation, SMF extents the classical model of the environment
• Mathematics
• 2007
A class of Fourier Integral Operators which converge to the unitary group of the Schrödinger equation in the semiclassical limit ε → 0 in the uniform operator norm is constructed. The convergence
• Physics
The Journal of chemical physics
• 2007
Quantum mechanical analysis and numerical calculations presented in this paper show that the h2 terms involving the interstate coupling functions have significant effects on the quantum transition probabilities.