^{1,a)}, Mohammadhasan Dinpajooh

^{1}, J. Ilja Siepmann

^{1,a)}and Donald G. Truhlar

^{1,a)}

### Abstract

We present a procedure to calculate ensemble averages, thermodynamic derivatives, and coordinate distributions by effective classical potential methods. In particular, we consider the displaced-points path integral (DPPI) method, which yields exact quantal partition functions and ensemble averages for a harmonic potential and approximate quantal ones for general potentials, and we discuss the implementation of the new procedure in two Monte Carlo simulation codes, one that uses uncorrelated samples to calculate absolute free energies, and another that employs Metropolis sampling to calculate relative free energies. The results of the new DPPI method are compared to those from accurate path integral calculations as well as to results of two other effective classical potential schemes for the case of an isolated water molecule. In addition to the partition function, we consider the heat capacity and expectation values of the energy, the potential energy, the bond angle, and the OH distance. We also consider coordinate distributions. The DPPI scheme performs best among the three effective potential schemes considered and achieves very good accuracy for all of the properties considered. A key advantage of the effective potential schemes is that they display much lower statistical sampling variances than those for accurate path integral calculations. The method presented here shows great promise for including quantum effects in calculations on large systems.

This work was supported in part by the National Science Foundation (NSF) through Grant Nos. CHE-09 56776 and CHE-1051396.

I. INTRODUCTION

II. BACKGROUND AND THEORY

A. Feynman path integrals

B. Effective classical potential methods

C. DPPI methods

D. Treatment of imaginary frequencies

E. Ensemble averages in ECP schemes

F. Multidimensional DPPI calculations

III. IMPLEMENTATION OVERVIEW

IV. CALCULATIONS

A. Accurate path integral and classical calculations

B. ECP calculations

V. RESULTS

VI. DISCUSSION

VII. CONCLUDING REMARKS

### Key Topics

- Heat capacity
- 13.0
- Quantum effects
- 9.0
- Oscillators
- 8.0
- Monte Carlo methods
- 7.0
- Eigenvalues
- 5.0

## Figures

Comparison of DPPI-AS calculations as implemented in two different Monte Carlo codes (cyan and magenta triangles) to the results of classical calculations (black circles); variables with asterisks are results of accurate path integral calculations.

Comparison of DPPI-AS calculations as implemented in two different Monte Carlo codes (cyan and magenta triangles) to the results of classical calculations (black circles); variables with asterisks are results of accurate path integral calculations.

HOH angle distributions: Accurate path integral at 300 K (solid red line), DPPI-AS at 300 K (dashed black line), accurate path integral at 1000 K (solid green line), DPPI-AS at 1000 K (dashed purple line).

HOH angle distributions: Accurate path integral at 300 K (solid red line), DPPI-AS at 300 K (dashed black line), accurate path integral at 1000 K (solid green line), DPPI-AS at 1000 K (dashed purple line).

## Tables

Numerical results for *Q*(*T*) for a 1D quadratic-quartic potential *V*(*q*) = *aq* ^{4} + *bq* ^{2} with *a* = 0.01 a.u. and *b* = 0.02 a.u. for a particle with *μ* = 1224.259 *m* _{e}.

Numerical results for *Q*(*T*) for a 1D quadratic-quartic potential *V*(*q*) = *aq* ^{4} + *bq* ^{2} with *a* = 0.01 a.u. and *b* = 0.02 a.u. for a particle with *μ* = 1224.259 *m* _{e}.

Parameters for the H_{2}O calculations for various *T* ranges.

Parameters for the H_{2}O calculations for various *T* ranges.

Partition functions, *Q*(*T*), for H_{2}O calculated by various methods, with uncertainties given as 95% confidence intervals.

Partition functions, *Q*(*T*), for H_{2}O calculated by various methods, with uncertainties given as 95% confidence intervals.

Calculations of *C* _{ V }/*k* _{B} for H_{2}O, with uncertainties given as 95% confidence intervals.

Calculations of *C* _{ V }/*k* _{B} for H_{2}O, with uncertainties given as 95% confidence intervals.

Calculations of ⟨*E*⟩ in kcal/mol for H_{2}O, with uncertainties given as 95% confidence intervals.

Calculations of ⟨*E*⟩ in kcal/mol for H_{2}O, with uncertainties given as 95% confidence intervals.

Calculations of ⟨*V*⟩ in kcal/mol for H_{2}O, with uncertainties given as 95% confidence intervals.

Calculations of ⟨*V*⟩ in kcal/mol for H_{2}O, with uncertainties given as 95% confidence intervals.

Calculations of ⟨*R* _{OH}⟩, in Å, for H_{2}O, with uncertainties given as 95% confidence intervals.

Calculations of ⟨*R* _{OH}⟩, in Å, for H_{2}O, with uncertainties given as 95% confidence intervals.

Calculations of ⟨*θ* _{HOH}⟩, in deg, for H_{2}O, with uncertainties given as 95% confidence intervals.

Calculations of ⟨*θ* _{HOH}⟩, in deg, for H_{2}O, with uncertainties given as 95% confidence intervals.

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