# 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 222-8585 (Japan)
- (Japan)
- School of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192 (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, E-mail: kawatsu@fukui.kyoto-u.ac.jp, E-mail: smiura@mail.kanazawa-u.ac.jp, School of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192, and Miura, Shinichi, E-mail: kawatsu@fukui.kyoto-u.ac.jp, E-mail: smiura@mail.kanazawa-u.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, E-mail: kawatsu@fukui.kyoto-u.ac.jp, E-mail: smiura@mail.kanazawa-u.ac.jp, School of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192, & Miura, Shinichi, E-mail: kawatsu@fukui.kyoto-u.ac.jp, E-mail: smiura@mail.kanazawa-u.ac.jp.
```*Efficient algorithms for semiclassical instanton calculations based on discretized path integrals*. United States. doi:10.1063/1.4885437.

```
Kawatsu, Tsutomu, E-mail: kawatsu@fukui.kyoto-u.ac.jp, E-mail: smiura@mail.kanazawa-u.ac.jp, School of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192, and Miura, Shinichi, E-mail: kawatsu@fukui.kyoto-u.ac.jp, E-mail: smiura@mail.kanazawa-u.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, E-mail: kawatsu@fukui.kyoto-u.ac.jp, E-mail: smiura@mail.kanazawa-u.ac.jp and School of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192 and Miura, Shinichi, E-mail: kawatsu@fukui.kyoto-u.ac.jp, E-mail: smiura@mail.kanazawa-u.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}

}