# Low variance energy estimators for systems of quantum Drude oscillators: Treating harmonic path integrals with large separations of time scales

## Abstract

In the effort to develop atomistic models capable of accurately describing nanoscale systems with complex interfaces, it has become clear that simple treatments with rigid charge distributions and dispersion coefficients selected to generate bulk properties are insufficient to predict important physical properties. The quantum Drude oscillator model, a system of one-electron pseudoatoms whose 'pseudoelectrons' are harmonically bound to their respective 'pseudonuclei', is capable of treating many-body polarization and dispersion interactions in molecular systems on an equal footing due to the ability of the pseudoatoms to mimic the long-range interactions that characterize real materials. Using imaginary time path integration, the Drude oscillator model can, in principle, be solved in computer operation counts that scale linearly with the number of atoms in the system. In practice, however, standard expressions for the energy and pressure, including the commonly used virial estimator, have extremely large variances that require untenably long simulation times to generate converged averages. In this paper, low-variance estimators for the internal energy are derived, in which the large zero-point energy of the oscillators does not contribute to the variance. The new estimators are applicable to any system of harmonic oscillators coupled to one another (or to the environment) via an arbitrarymore »

- Authors:

- IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598 (United States)

- Publication Date:

- OSTI Identifier:
- 20991221

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 126; Journal Issue: 7; Other Information: DOI: 10.1063/1.2424708; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; ANHARMONIC OSCILLATORS; ATOMS; CHARGE DISTRIBUTION; ELECTRONS; EQUATIONS OF STATE; HARMONIC OSCILLATORS; INTERACTION RANGE; NANOSTRUCTURES; OSCILLATORS; PATH INTEGRALS; PHYSICAL PROPERTIES; SIMULATION; XENON

### Citation Formats

```
Whitfield, Troy W., and Martyna, Glenn J.
```*Low variance energy estimators for systems of quantum Drude oscillators: Treating harmonic path integrals with large separations of time scales*. United States: N. p., 2007.
Web. doi:10.1063/1.2424708.

```
Whitfield, Troy W., & Martyna, Glenn J.
```*Low variance energy estimators for systems of quantum Drude oscillators: Treating harmonic path integrals with large separations of time scales*. United States. doi:10.1063/1.2424708.

```
Whitfield, Troy W., and Martyna, Glenn J. Wed .
"Low variance energy estimators for systems of quantum Drude oscillators: Treating harmonic path integrals with large separations of time scales". United States.
doi:10.1063/1.2424708.
```

```
@article{osti_20991221,
```

title = {Low variance energy estimators for systems of quantum Drude oscillators: Treating harmonic path integrals with large separations of time scales},

author = {Whitfield, Troy W. and Martyna, Glenn J.},

abstractNote = {In the effort to develop atomistic models capable of accurately describing nanoscale systems with complex interfaces, it has become clear that simple treatments with rigid charge distributions and dispersion coefficients selected to generate bulk properties are insufficient to predict important physical properties. The quantum Drude oscillator model, a system of one-electron pseudoatoms whose 'pseudoelectrons' are harmonically bound to their respective 'pseudonuclei', is capable of treating many-body polarization and dispersion interactions in molecular systems on an equal footing due to the ability of the pseudoatoms to mimic the long-range interactions that characterize real materials. Using imaginary time path integration, the Drude oscillator model can, in principle, be solved in computer operation counts that scale linearly with the number of atoms in the system. In practice, however, standard expressions for the energy and pressure, including the commonly used virial estimator, have extremely large variances that require untenably long simulation times to generate converged averages. In this paper, low-variance estimators for the internal energy are derived, in which the large zero-point energy of the oscillators does not contribute to the variance. The new estimators are applicable to any system of harmonic oscillators coupled to one another (or to the environment) via an arbitrary set of anharmonic interactions. The variance of the new estimators is found to be much smaller than standard estimators in three example problems, a one-dimensional anharmonic oscillator and quantum Drude models of the xenon dimer and solid (fcc) xenon, respectively, yielding 2-3 orders of magnitude improvement in computational efficiency.},

doi = {10.1063/1.2424708},

journal = {Journal of Chemical Physics},

number = 7,

volume = 126,

place = {United States},

year = {Wed Feb 21 00:00:00 EST 2007},

month = {Wed Feb 21 00:00:00 EST 2007}

}