^{1,a)}, Tao Zeng

^{1,b)}and Pierre-Nicholas Roy

^{1,c)}

### Abstract

In this paper, we extend the previously introduced Post-Quantization Constraints (PQC) procedure [G. Guillon, T. Zeng, and P.-N. Roy, J. Chem. Phys.138, 184101 (2013)] to construct approximate propagators and energy estimators for different rigid body systems, namely, the spherical, symmetric, and asymmetric tops. These propagators are for use in Path Integral simulations. A thorough discussion of the underlying geometrical concepts is given. Furthermore, a detailed analysis of the convergence properties of the density as well as the energy estimators towards their exact counterparts is presented along with illustrative numerical examples. The Post-Quantization Constraints approach can yield converged results and is a practical alternative to so-called sum over states techniques, where one has to expand the propagator as a sum over a complete set of rotational stationary states [as in E. G. Noya, C. Vega, and C. McBride, J. Chem. Phys.134, 054117 (2011)] because of its modest memory requirements.

This research has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), Ministry of Research and Innovation (MRI), Ontario, and the Canada Foundation for Innovation (CFI).

I. INTRODUCTION

II. METHODOLOGICAL DEVELOPMENT

III. NUMERICAL RESULTS AND DISCUSSION

A. Spherical top

B. Symmetric top

C. Asymmetric top

IV. ENERGY ESTIMATORS

V. CONCLUDING REMARKS

### Key Topics

- Speed of sound
- 10.0
- Rotation constants
- 7.0
- Interpolation
- 5.0
- Manifolds
- 4.0
- Numerical analysis
- 3.0

## Figures

Difference between PQC and Exact (E) spherical top density operator (thermal propagator) (j, k, m)-basis diagonal matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 3 are shown. The dashed line corresponds to a model power law of exponent 5/2.

Difference between PQC and Exact (E) spherical top density operator (thermal propagator) (j, k, m)-basis diagonal matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 3 are shown. The dashed line corresponds to a model power law of exponent 5/2.

Difference between PQC and Exact (E) spherical top energy estimator (j, k, m)-basis diagonal matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 3 are shown. The linear convergence towards the Exact SOS estimator, up to a constant shift (not included in the PQC energy), is explicitly apparent.

Difference between PQC and Exact (E) spherical top energy estimator (j, k, m)-basis diagonal matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 3 are shown. The linear convergence towards the Exact SOS estimator, up to a constant shift (not included in the PQC energy), is explicitly apparent.

Difference between PQC and Exact (E) symmetric top density operator (thermal propagator) (j, k, m)-basis diagonal (j, |k|)-matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 3 are shown. The dashed line corresponds to a model power law of exponent 5/2.

Difference between PQC and Exact (E) symmetric top density operator (thermal propagator) (j, k, m)-basis diagonal (j, |k|)-matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 3 are shown. The dashed line corresponds to a model power law of exponent 5/2.

Difference between PQC and Exact (E) symmetric top energy estimator (j, k, m)-basis (j, |k|)-diagonal matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 3 are shown. The linear convergence towards the Exact SOS estimator, up to a constant shift, is explicitly apparent.

Difference between PQC and Exact (E) symmetric top energy estimator (j, k, m)-basis (j, |k|)-diagonal matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 3 are shown. The linear convergence towards the Exact SOS estimator, up to a constant shift, is explicitly apparent.

Difference between PQC and Exact (E) asymmetric top density operator (thermal propagator) (j, k, m)-basis -matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 2 are shown. The dashed line corresponds to a model power law of exponent 5/2.

Difference between PQC and Exact (E) asymmetric top density operator (thermal propagator) (j, k, m)-basis -matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 2 are shown. The dashed line corresponds to a model power law of exponent 5/2.

Difference between PQC and Exact (E) asymmetric top energy estimator (j, k, m)-basis -matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 2 are shown. The linear convergence towards the Exact SOS estimator, up to a constant shift, is explicitly apparent.

Difference between PQC and Exact (E) asymmetric top energy estimator (j, k, m)-basis -matrix element as a function of τ (in units of inverse Kelvin) in logarithmic scale for both axis. The plot is focusing on very small values of τ. Values of j from 0 to 2 are shown. The linear convergence towards the Exact SOS estimator, up to a constant shift, is explicitly apparent.

Path integral Monte Carlo energies (in units of K) obtained with a PQC thermal propagator for an asymmetric top as a function of the imaginary time step (in units of reciprocal energy K^{−1}) and their linear fit, at a temperature T = 10 K. The blue star corresponds to the exact FBR average energy at the same temperature.

Path integral Monte Carlo energies (in units of K) obtained with a PQC thermal propagator for an asymmetric top as a function of the imaginary time step (in units of reciprocal energy K^{−1}) and their linear fit, at a temperature T = 10 K. The blue star corresponds to the exact FBR average energy at the same temperature.

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