Premium
Characteristics of numerical realization via stochastic partial differential equation: An application to density matrix calculation
Author(s) -
Saito Yoshihiro,
Nagaoka Masataka
Publication year - 1999
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/(sici)1097-461x(1999)74:6<653::aid-qua6>3.0.co;2-c
Subject(s) - density matrix , mathematics , realization (probability) , quantum , stochastic matrix , partial differential equation , stochastic differential equation , matrix (chemical analysis) , convergence (economics) , mathematical analysis , quantum mechanics , physics , chemistry , chromatography , statistics , markov chain , economics , economic growth
For calculating the transition amplitude, especially the density matrix, important for the quantum‐mechanical chemical reaction dynamics, the stochastic path‐integral method (SPIM) was developed and then realized by solving the stochastic partial differential equation in use of a corresponding set of difference equations (Nagaoka, M. Int J Quantum Chem Quantum Chem Symp 1996, 30, 91). Three kinds of numerical schemes, that is, the Euler–Maruyama, the Heun, and the Explicit 1.5 schemes, whose strong orders of convergence are 0.5, 1.0, and 1.5, respectively, are examined by analyzing the averaged coordinate, the averaged coordinate correlation function, and the density matrix. Comparing the latter with the analytical result obtained for a typical quantum harmonic system at temperature 100 K, it was curiously found that the Heun scheme brings about a rather worse transition amplitude than do the other two schemes in spite of its strong order of convergence 1.0. In conclusion, it is conjectured that there should be some unknown profound properties that characterizes the numerical schemes for the stochastic partial differential equations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 653–660, 1999