# Communication: Relation of centroid molecular dynamics and ring-polymer molecular dynamics to exact quantum dynamics

## Abstract

We recently obtained a quantum-Boltzmann-conserving classical dynamics by making a single change to the derivation of the “Classical Wigner” approximation. Here, we show that the further approximation of this “Matsubara dynamics” gives rise to two popular heuristic methods for treating quantum Boltzmann time-correlation functions: centroid molecular dynamics (CMD) and ring-polymer molecular dynamics (RPMD). We show that CMD is a mean-field approximation to Matsubara dynamics, obtained by discarding (classical) fluctuations around the centroid, and that RPMD is the result of discarding a term in the Matsubara Liouvillian which shifts the frequencies of these fluctuations. These findings are consistent with previous numerical results and give explicit formulae for the terms that CMD and RPMD leave out.

- Authors:

- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22415782

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 142; Journal Issue: 19; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9606

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; CORRELATION FUNCTIONS; FLUCTUATIONS; MEAN-FIELD THEORY; MOLECULAR DYNAMICS METHOD; POLYMERS; QUANTUM MECHANICS

### Citation Formats

```
Hele, Timothy J. H., Willatt, Michael J., Muolo, Andrea, and Althorpe, Stuart C., E-mail: sca10@cam.ac.uk.
```*Communication: Relation of centroid molecular dynamics and ring-polymer molecular dynamics to exact quantum dynamics*. United States: N. p., 2015.
Web. doi:10.1063/1.4921234.

```
Hele, Timothy J. H., Willatt, Michael J., Muolo, Andrea, & Althorpe, Stuart C., E-mail: sca10@cam.ac.uk.
```*Communication: Relation of centroid molecular dynamics and ring-polymer molecular dynamics to exact quantum dynamics*. United States. doi:10.1063/1.4921234.

```
Hele, Timothy J. H., Willatt, Michael J., Muolo, Andrea, and Althorpe, Stuart C., E-mail: sca10@cam.ac.uk. Thu .
"Communication: Relation of centroid molecular dynamics and ring-polymer molecular dynamics to exact quantum dynamics". United States. doi:10.1063/1.4921234.
```

```
@article{osti_22415782,
```

title = {Communication: Relation of centroid molecular dynamics and ring-polymer molecular dynamics to exact quantum dynamics},

author = {Hele, Timothy J. H. and Willatt, Michael J. and Muolo, Andrea and Althorpe, Stuart C., E-mail: sca10@cam.ac.uk},

abstractNote = {We recently obtained a quantum-Boltzmann-conserving classical dynamics by making a single change to the derivation of the “Classical Wigner” approximation. Here, we show that the further approximation of this “Matsubara dynamics” gives rise to two popular heuristic methods for treating quantum Boltzmann time-correlation functions: centroid molecular dynamics (CMD) and ring-polymer molecular dynamics (RPMD). We show that CMD is a mean-field approximation to Matsubara dynamics, obtained by discarding (classical) fluctuations around the centroid, and that RPMD is the result of discarding a term in the Matsubara Liouvillian which shifts the frequencies of these fluctuations. These findings are consistent with previous numerical results and give explicit formulae for the terms that CMD and RPMD leave out.},

doi = {10.1063/1.4921234},

journal = {Journal of Chemical Physics},

issn = {0021-9606},

number = 19,

volume = 142,

place = {United States},

year = {2015},

month = {5}

}