Efficient algorithms for semiclassical instanton calculations based on discretized path integrals
Abstract
Path integral instanton method is a promising way to calculate the tunneling splitting of energies for degenerated two state systems. In order to calculate the tunneling splitting, we need to take the zero temperature limit, or the limit of infinite imaginary time duration. In the method developed by Richardson and Althorpe [J. Chem. Phys. 134, 054109 (2011)], the limit is simply replaced by the sufficiently long imaginary time. In the present study, we have developed a new formula of the tunneling splitting based on the discretized path integrals to take the limit analytically. We have applied our new formula to model systems, and found that this approach can significantly reduce the computational cost and gain the numerical accuracy. We then developed the method combined with the electronic structure calculations to obtain the accurate interatomic potential on the fly. We present an application of our ab initio instanton method to the ammonia umbrella flip motion.
 Authors:
 Institute for Molecular Science, National Institute of Natural Science, 38 Nishigonaka, Myodaiji, Okazaki 2228585 (Japan)
 (Japan)
 School of Mathematics and Physics, Kanazawa University, Kanazawa 9201192 (Japan)
 Publication Date:
 OSTI Identifier:
 22308765
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Chemical Physics; Journal Volume: 141; Journal Issue: 2; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; ACCURACY; ALGORITHMS; AMMONIA; ELECTRONIC STRUCTURE; PATH INTEGRALS; SEMICLASSICAL APPROXIMATION; TUNNEL EFFECT
Citation Formats
Kawatsu, Tsutomu, Email: kawatsu@fukui.kyotou.ac.jp, Email: smiura@mail.kanazawau.ac.jp, School of Mathematics and Physics, Kanazawa University, Kanazawa 9201192, and Miura, Shinichi, Email: kawatsu@fukui.kyotou.ac.jp, Email: smiura@mail.kanazawau.ac.jp. Efficient algorithms for semiclassical instanton calculations based on discretized path integrals. United States: N. p., 2014.
Web. doi:10.1063/1.4885437.
Kawatsu, Tsutomu, Email: kawatsu@fukui.kyotou.ac.jp, Email: smiura@mail.kanazawau.ac.jp, School of Mathematics and Physics, Kanazawa University, Kanazawa 9201192, & Miura, Shinichi, Email: kawatsu@fukui.kyotou.ac.jp, Email: smiura@mail.kanazawau.ac.jp. Efficient algorithms for semiclassical instanton calculations based on discretized path integrals. United States. doi:10.1063/1.4885437.
Kawatsu, Tsutomu, Email: kawatsu@fukui.kyotou.ac.jp, Email: smiura@mail.kanazawau.ac.jp, School of Mathematics and Physics, Kanazawa University, Kanazawa 9201192, and Miura, Shinichi, Email: kawatsu@fukui.kyotou.ac.jp, Email: smiura@mail.kanazawau.ac.jp. Mon .
"Efficient algorithms for semiclassical instanton calculations based on discretized path integrals". United States.
doi:10.1063/1.4885437.
@article{osti_22308765,
title = {Efficient algorithms for semiclassical instanton calculations based on discretized path integrals},
author = {Kawatsu, Tsutomu, Email: kawatsu@fukui.kyotou.ac.jp, Email: smiura@mail.kanazawau.ac.jp and School of Mathematics and Physics, Kanazawa University, Kanazawa 9201192 and Miura, Shinichi, Email: kawatsu@fukui.kyotou.ac.jp, Email: smiura@mail.kanazawau.ac.jp},
abstractNote = {Path integral instanton method is a promising way to calculate the tunneling splitting of energies for degenerated two state systems. In order to calculate the tunneling splitting, we need to take the zero temperature limit, or the limit of infinite imaginary time duration. In the method developed by Richardson and Althorpe [J. Chem. Phys. 134, 054109 (2011)], the limit is simply replaced by the sufficiently long imaginary time. In the present study, we have developed a new formula of the tunneling splitting based on the discretized path integrals to take the limit analytically. We have applied our new formula to model systems, and found that this approach can significantly reduce the computational cost and gain the numerical accuracy. We then developed the method combined with the electronic structure calculations to obtain the accurate interatomic potential on the fly. We present an application of our ab initio instanton method to the ammonia umbrella flip motion.},
doi = {10.1063/1.4885437},
journal = {Journal of Chemical Physics},
number = 2,
volume = 141,
place = {United States},
year = {Mon Jul 14 00:00:00 EDT 2014},
month = {Mon Jul 14 00:00:00 EDT 2014}
}

We present and discuss a detailed derivation of an analytical method that systematically improves the convergence of path integrals of a generic Nfold discretized theory. We develop an explicit procedure for calculating a set of effective actions S{sup (p)}, for p=1,2,3,... which have the property that they lead to the same continuum amplitudes as the starting action, but converge to that continuum limit ever faster. Discretized amplitudes calculated using the plevel effective action differ from the continuum limit by a term of order 1/N{sup p}. We obtain explicit expressions for the effective actions for levels p{<=}9. We end by analyzingmore »

A new approach to Gaussian path integrals and the evaluation of the semiclassical propagator. [Charged particle constant magnetic field]
The expansion of path variations in terms of solutions of Morse's boundary problem is applied in order to evaluate Gaussian path integrals. Together with a recently discovered theorem on infinite products of eigenvalues of SturmLiouvilletype operators, this yields an expression for the most general semiclassical propagator. The properties of the latter are investigated in the light of the Morse theory. The general methods developed here are illustrated by the example of a charged particle moving in a homogeneous magnetic field.