### Abstract

In this paper we report the initial steps in the development of a Monte Carlo method for evaluation of real-time Feynman path integrals for many-particle dynamics. The approach leads to Gaussian importance sampling for real-time dynamics in which the sampling function is short ranged due to the occurrence of Gaussian factors. These Gaussian factors result from the use of a generalization of our new discrete distributed approximating functions (DDAFs) to continuous distributed approximating functions (CDAFs) so as to replace the exact coordinate representation free-particle propagator by a "CDAF-class, free-particle propagator" which is highly banded. The envelope of the CDAF-class free propagator is the product of a "bare Gaussian", exp[-(x′ - x)^{2}σ^{2}(0)/(2σ^{4}(0) + ℏ^{2}τ^{2}/m^{2})], with a "shape polynomial" in (x′ - x)^{2}, where σ(0) is a width parameter at zero time (associated with the description of the wavepacket in terms of Hermite functions), τ is the time step (τ = t/N, where t is the total propagation time), and x and x′ are any two configurations of the system. The bare Gaussians are used for Monte Carlo integration of a path integral for the survival probability of a Gaussian wavepacket in a Morse potential. The approach appears promising for real-time quantum Monte Carlo studies based on the time-dependent Schrödinger equation, the time-dependent von Neumann equation, and related equations.

Original language | English (US) |
---|---|

Pages (from-to) | 9622-9630 |

Number of pages | 9 |

Journal | Journal of Physical Chemistry |

Volume | 96 |

Issue number | 24 |

State | Published - 1992 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physical and Theoretical Chemistry

### Cite this

*Journal of Physical Chemistry*,

*96*(24), 9622-9630.

**Monte Carlo evaluation of real-time Feynman path integrals for quantal many-body dynamics : Distributed approximating functions and Gaussian sampling.** / Kouri, Donald J.; Zhu, Wei; Ma, Xin; Pettitt, Bernard; Hoffman, David K.

Research output: Contribution to journal › Article

*Journal of Physical Chemistry*, vol. 96, no. 24, pp. 9622-9630.

}

TY - JOUR

T1 - Monte Carlo evaluation of real-time Feynman path integrals for quantal many-body dynamics

T2 - Distributed approximating functions and Gaussian sampling

AU - Kouri, Donald J.

AU - Zhu, Wei

AU - Ma, Xin

AU - Pettitt, Bernard

AU - Hoffman, David K.

PY - 1992

Y1 - 1992

N2 - In this paper we report the initial steps in the development of a Monte Carlo method for evaluation of real-time Feynman path integrals for many-particle dynamics. The approach leads to Gaussian importance sampling for real-time dynamics in which the sampling function is short ranged due to the occurrence of Gaussian factors. These Gaussian factors result from the use of a generalization of our new discrete distributed approximating functions (DDAFs) to continuous distributed approximating functions (CDAFs) so as to replace the exact coordinate representation free-particle propagator by a "CDAF-class, free-particle propagator" which is highly banded. The envelope of the CDAF-class free propagator is the product of a "bare Gaussian", exp[-(x′ - x)2σ2(0)/(2σ4(0) + ℏ2τ2/m2)], with a "shape polynomial" in (x′ - x)2, where σ(0) is a width parameter at zero time (associated with the description of the wavepacket in terms of Hermite functions), τ is the time step (τ = t/N, where t is the total propagation time), and x and x′ are any two configurations of the system. The bare Gaussians are used for Monte Carlo integration of a path integral for the survival probability of a Gaussian wavepacket in a Morse potential. The approach appears promising for real-time quantum Monte Carlo studies based on the time-dependent Schrödinger equation, the time-dependent von Neumann equation, and related equations.

AB - In this paper we report the initial steps in the development of a Monte Carlo method for evaluation of real-time Feynman path integrals for many-particle dynamics. The approach leads to Gaussian importance sampling for real-time dynamics in which the sampling function is short ranged due to the occurrence of Gaussian factors. These Gaussian factors result from the use of a generalization of our new discrete distributed approximating functions (DDAFs) to continuous distributed approximating functions (CDAFs) so as to replace the exact coordinate representation free-particle propagator by a "CDAF-class, free-particle propagator" which is highly banded. The envelope of the CDAF-class free propagator is the product of a "bare Gaussian", exp[-(x′ - x)2σ2(0)/(2σ4(0) + ℏ2τ2/m2)], with a "shape polynomial" in (x′ - x)2, where σ(0) is a width parameter at zero time (associated with the description of the wavepacket in terms of Hermite functions), τ is the time step (τ = t/N, where t is the total propagation time), and x and x′ are any two configurations of the system. The bare Gaussians are used for Monte Carlo integration of a path integral for the survival probability of a Gaussian wavepacket in a Morse potential. The approach appears promising for real-time quantum Monte Carlo studies based on the time-dependent Schrödinger equation, the time-dependent von Neumann equation, and related equations.

UR - http://www.scopus.com/inward/record.url?scp=0001259668&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001259668&partnerID=8YFLogxK

M3 - Article

VL - 96

SP - 9622

EP - 9630

JO - Journal of Physical Chemistry

JF - Journal of Physical Chemistry

SN - 0022-3654

IS - 24

ER -