# Improved methods for Feynman path integral calculations and their application to calculate converged vibrational–rotational partition functions, free energies, enthalpies, entropies, and heat capacities for methane

## Abstract

We present an improved version of our “path-by-path” enhanced same path extrapolation scheme for Feynman path integral (FPI) calculations that permits rapid convergence with discretization errors ranging from O(P{sup −6}) to O(P{sup −12}), where P is the number of path discretization points. We also present two extensions of our importance sampling and stratified sampling schemes for calculating vibrational–rotational partition functions by the FPI method. The first is the use of importance functions for dihedral angles between sets of generalized Jacobi coordinate vectors. The second is an extension of our stratification scheme to allow some strata to be defined based only on coordinate information while other strata are defined based on both the geometry and the energy of the centroid of the Feynman path. These enhanced methods are applied to calculate converged partition functions by FPI methods, and these results are compared to ones obtained earlier by vibrational configuration interaction (VCI) calculations, both calculations being for the Jordan–Gilbert potential energy surface. The earlier VCI calculations are found to agree well (within ∼1.5%) with the new benchmarks. The FPI partition functions presented here are estimated to be converged to within a 2σ statistical uncertainty of between 0.04% and 0.07% for the givenmore »

- Authors:

- Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, 207 Pleasant St. S.E., Minneapolis, Minnesota 55455-0431 (United States)

- Publication Date:

- OSTI Identifier:
- 22416033

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 142; Journal Issue: 4; Other Information: (c) 2015 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; ATOMS; BENCHMARKS; COMPARATIVE EVALUATIONS; CONFIGURATION INTERACTION; ENTHALPY; ENTROPY; FEYNMAN PATH INTEGRAL; FREE ENERGY; METHANE; MOLECULES; PARTITION FUNCTIONS; POTENTIAL ENERGY; SPECIFIC HEAT; STRATIFICATION; SURFACES

### Citation Formats

```
Mielke, Steven L., E-mail: slmielke@gmail.com, E-mail: truhlar@umn.edu, and Truhlar, Donald G., E-mail: slmielke@gmail.com, E-mail: truhlar@umn.edu.
```*Improved methods for Feynman path integral calculations and their application to calculate converged vibrational–rotational partition functions, free energies, enthalpies, entropies, and heat capacities for methane*. United States: N. p., 2015.
Web. doi:10.1063/1.4905526.

```
Mielke, Steven L., E-mail: slmielke@gmail.com, E-mail: truhlar@umn.edu, & Truhlar, Donald G., E-mail: slmielke@gmail.com, E-mail: truhlar@umn.edu.
```*Improved methods for Feynman path integral calculations and their application to calculate converged vibrational–rotational partition functions, free energies, enthalpies, entropies, and heat capacities for methane*. United States. doi:10.1063/1.4905526.

```
Mielke, Steven L., E-mail: slmielke@gmail.com, E-mail: truhlar@umn.edu, and Truhlar, Donald G., E-mail: slmielke@gmail.com, E-mail: truhlar@umn.edu. 2015.
"Improved methods for Feynman path integral calculations and their application to calculate converged vibrational–rotational partition functions, free energies, enthalpies, entropies, and heat capacities for methane". United States.
doi:10.1063/1.4905526.
```

```
@article{osti_22416033,
```

title = {Improved methods for Feynman path integral calculations and their application to calculate converged vibrational–rotational partition functions, free energies, enthalpies, entropies, and heat capacities for methane},

author = {Mielke, Steven L., E-mail: slmielke@gmail.com, E-mail: truhlar@umn.edu and Truhlar, Donald G., E-mail: slmielke@gmail.com, E-mail: truhlar@umn.edu},

abstractNote = {We present an improved version of our “path-by-path” enhanced same path extrapolation scheme for Feynman path integral (FPI) calculations that permits rapid convergence with discretization errors ranging from O(P{sup −6}) to O(P{sup −12}), where P is the number of path discretization points. We also present two extensions of our importance sampling and stratified sampling schemes for calculating vibrational–rotational partition functions by the FPI method. The first is the use of importance functions for dihedral angles between sets of generalized Jacobi coordinate vectors. The second is an extension of our stratification scheme to allow some strata to be defined based only on coordinate information while other strata are defined based on both the geometry and the energy of the centroid of the Feynman path. These enhanced methods are applied to calculate converged partition functions by FPI methods, and these results are compared to ones obtained earlier by vibrational configuration interaction (VCI) calculations, both calculations being for the Jordan–Gilbert potential energy surface. The earlier VCI calculations are found to agree well (within ∼1.5%) with the new benchmarks. The FPI partition functions presented here are estimated to be converged to within a 2σ statistical uncertainty of between 0.04% and 0.07% for the given potential energy surface for temperatures in the range 300–3000 K and are the most accurately converged partition functions for a given potential energy surface for any molecule with five or more atoms. We also tabulate free energies, enthalpies, entropies, and heat capacities.},

doi = {10.1063/1.4905526},

journal = {Journal of Chemical Physics},

number = 4,

volume = 142,

place = {United States},

year = 2015,

month = 1

}