### Abstract

In this paper we study numerical methods for calculating thermal rate coefficients using flux correlation functions, with the goal of determining optimal methods for producing values with a specified accuracy. In all cases we employ grid based methods for solving the time-dependent Schrödinger equation in one mathematical dimension for a simple barrier potential function. The solutions are used to determine the propagator matrix elements needed to evaluate the flux correlation functions. Within this framework, we examine (1) several time-dependent methods for propagating the wave packets, (2) several procedures for evaluating the action of the Hamiltonian on the wave function, (3) the choice of complex time contours for evaluating the rate coefficient expression, (4) alternatives for estimating the initial short-time evolution of the wave packet (which starts as a δ function), (5) quadrature methods for evaluating the spatial and time integrals appearing in the flux correlation function, and (6) special numerical strategies which can dramatically improve the accuracy of the calculation, particularly at low temperatures. We find that several methods yield rate coefficients accurate to 1% or 0.1% using about the same computational effort. These include (a) split-operator time propagators combined with fast-Fourier-transform evaluations of the wave-function derivatives, and (b) the Chebyshev time propagator combined with either an eleventh-order finite-difference or fifth-order spline evaluation of the wave-function derivatives. These finite-difference and spline methods can also be used competitively with the split-operator approach provided that a Crank-Nicholson approximation is utilized in evaluating the action of the kinetic-energy propagator. It was also found that inaccuracies in estimating the initial short-time behavior of the wave function could limit the effectiveness of the more accurate methods. A multigrid approach based on the split-operator/Fourier transform method is developed for treating this, and provides results with sufficient accuracy. However, for some choices of grid parameters this initialization process determines the overall computational efficiency of the calculation, independent of the numerical efficiency of methods used after the initialization.

Original language | English |
---|---|

Pages (from-to) | 7297-7313 |

Number of pages | 17 |

Journal | Journal of Chemical Physics |

Volume | 97 |

Issue number | 10 |

Publication status | Published - 1992 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*Journal of Chemical Physics*,

*97*(10), 7297-7313.

**Time-dependent methods for calculating thermal rate coefficients using flux correlation functions.** / Thachuk, Mark; Schatz, George C.

Research output: Contribution to journal › Article

*Journal of Chemical Physics*, vol. 97, no. 10, pp. 7297-7313.

}

TY - JOUR

T1 - Time-dependent methods for calculating thermal rate coefficients using flux correlation functions

AU - Thachuk, Mark

AU - Schatz, George C

PY - 1992

Y1 - 1992

N2 - In this paper we study numerical methods for calculating thermal rate coefficients using flux correlation functions, with the goal of determining optimal methods for producing values with a specified accuracy. In all cases we employ grid based methods for solving the time-dependent Schrödinger equation in one mathematical dimension for a simple barrier potential function. The solutions are used to determine the propagator matrix elements needed to evaluate the flux correlation functions. Within this framework, we examine (1) several time-dependent methods for propagating the wave packets, (2) several procedures for evaluating the action of the Hamiltonian on the wave function, (3) the choice of complex time contours for evaluating the rate coefficient expression, (4) alternatives for estimating the initial short-time evolution of the wave packet (which starts as a δ function), (5) quadrature methods for evaluating the spatial and time integrals appearing in the flux correlation function, and (6) special numerical strategies which can dramatically improve the accuracy of the calculation, particularly at low temperatures. We find that several methods yield rate coefficients accurate to 1% or 0.1% using about the same computational effort. These include (a) split-operator time propagators combined with fast-Fourier-transform evaluations of the wave-function derivatives, and (b) the Chebyshev time propagator combined with either an eleventh-order finite-difference or fifth-order spline evaluation of the wave-function derivatives. These finite-difference and spline methods can also be used competitively with the split-operator approach provided that a Crank-Nicholson approximation is utilized in evaluating the action of the kinetic-energy propagator. It was also found that inaccuracies in estimating the initial short-time behavior of the wave function could limit the effectiveness of the more accurate methods. A multigrid approach based on the split-operator/Fourier transform method is developed for treating this, and provides results with sufficient accuracy. However, for some choices of grid parameters this initialization process determines the overall computational efficiency of the calculation, independent of the numerical efficiency of methods used after the initialization.

AB - In this paper we study numerical methods for calculating thermal rate coefficients using flux correlation functions, with the goal of determining optimal methods for producing values with a specified accuracy. In all cases we employ grid based methods for solving the time-dependent Schrödinger equation in one mathematical dimension for a simple barrier potential function. The solutions are used to determine the propagator matrix elements needed to evaluate the flux correlation functions. Within this framework, we examine (1) several time-dependent methods for propagating the wave packets, (2) several procedures for evaluating the action of the Hamiltonian on the wave function, (3) the choice of complex time contours for evaluating the rate coefficient expression, (4) alternatives for estimating the initial short-time evolution of the wave packet (which starts as a δ function), (5) quadrature methods for evaluating the spatial and time integrals appearing in the flux correlation function, and (6) special numerical strategies which can dramatically improve the accuracy of the calculation, particularly at low temperatures. We find that several methods yield rate coefficients accurate to 1% or 0.1% using about the same computational effort. These include (a) split-operator time propagators combined with fast-Fourier-transform evaluations of the wave-function derivatives, and (b) the Chebyshev time propagator combined with either an eleventh-order finite-difference or fifth-order spline evaluation of the wave-function derivatives. These finite-difference and spline methods can also be used competitively with the split-operator approach provided that a Crank-Nicholson approximation is utilized in evaluating the action of the kinetic-energy propagator. It was also found that inaccuracies in estimating the initial short-time behavior of the wave function could limit the effectiveness of the more accurate methods. A multigrid approach based on the split-operator/Fourier transform method is developed for treating this, and provides results with sufficient accuracy. However, for some choices of grid parameters this initialization process determines the overall computational efficiency of the calculation, independent of the numerical efficiency of methods used after the initialization.

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UR - http://www.scopus.com/inward/citedby.url?scp=0000517401&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0000517401

VL - 97

SP - 7297

EP - 7313

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 10

ER -