^{1,a)}, Tao Zeng

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

^{1,c)}

### Abstract

We present a new methodological procedure, based on Post-Quantization Constraints (PQC), to obtain approximate density matrices and energy estimators for use in path integral molecular dynamics and Monte Carlo simulations. The approach serves as a justification of the use of “RATTLE & SHAKE” type methods for path integrals. A thorough discussion of the underlying geometrical concepts is given. Two standard model systems, the particle on a ring and the three-dimensional linear rotor, are used to illustrate and benchmark the approach. In these two cases, matrix elements of the newly defined propagator are explicitly computed in both “angular coordinate” and “angular momentum” bases. A detailed analysis of the convergence properties of the density matrix, and energy estimator with respect to their “exact” counterparts, is presented along with numerical illustrations. We conclude that the use of a PQC-type propagator is justified and practical.

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

A. Particle on a ring

B. Linear rigid rotor

III. RESULTS

A. Numerical accuracy of the method for the free rotational propagators

1. Particle on a ring

2. Linear rigid rotor

B. Illustration in an actual path integral simulation for the particle on a ring with potential

IV. CONCLUSIONS

### Key Topics

- Manifolds
- 9.0
- Few body systems
- 6.0
- Molecular dynamics
- 6.0
- Monte Carlo methods
- 5.0
- Polymers
- 5.0

## Figures

Upper panel: Difference between PQC and exact (E) particle on a ring density m-basis diagonal matrix element as a function of τ (in units of inverse Kelvin). The plot is focusing on very small values of τ. Values of m from 0 to 8 are shown. Lower panel: Same quantity as in the upper panel with a logarithmic scale for both axis. The dashed line corresponds to a model power law of exponent .

Upper panel: Difference between PQC and exact (E) particle on a ring density m-basis diagonal matrix element as a function of τ (in units of inverse Kelvin). The plot is focusing on very small values of τ. Values of m from 0 to 8 are shown. Lower panel: Same quantity as in the upper panel with a logarithmic scale for both axis. The dashed line corresponds to a model power law of exponent .

Difference between PQC and exact (E) particle on a ring energy estimator (in energy units of Kelvin) obtained directly from the m-basis diagonal matrix element as a function of τ (in units of inverse Kelvin). The plot is focusing on very small values of τ. Values of m from 0 to 8 are shown.

Difference between PQC and exact (E) particle on a ring energy estimator (in energy units of Kelvin) obtained directly from the m-basis diagonal matrix element as a function of τ (in units of inverse Kelvin). The plot is focusing on very small values of τ. Values of m from 0 to 8 are shown.

Upper panel: Difference between PQC and exact (E) linear rigid rotor density (ℓ, m)-basis diagonal matrix element as a function of τ (in units of inverse Kelvin). The plot is focusing on very small values of τ. Values of ℓ from 0 to 8 are shown. Lower panel: Same quantity as in the upper panel with a logarithmic scale for both axis. The dashed line corresponds to a model power law of exponent 2.

Upper panel: Difference between PQC and exact (E) linear rigid rotor density (ℓ, m)-basis diagonal matrix element as a function of τ (in units of inverse Kelvin). The plot is focusing on very small values of τ. Values of ℓ from 0 to 8 are shown. Lower panel: Same quantity as in the upper panel with a logarithmic scale for both axis. The dashed line corresponds to a model power law of exponent 2.

Difference between PQC and exact (E) linear rigid rotor energy estimator (in energy units of Kelvin) obtained directly from the (ℓ, m)-basis diagonal matrix element as a function of τ (in units of inverse Kelvin). The plot is focusing on very small values of τ. Values of ℓ from 0 to 8 are shown.

Difference between PQC and exact (E) linear rigid rotor energy estimator (in energy units of Kelvin) obtained directly from the (ℓ, m)-basis diagonal matrix element as a function of τ (in units of inverse Kelvin). The plot is focusing on very small values of τ. Values of ℓ from 0 to 8 are shown.

Path integral energies (in units of K) obtained with a SOS thermal propagator for a planar rotor with a periodic potential as a function of the imaginary time step (in units of reciprocal energy K−1) and their quadratic fit, at a temperature T = 0.37 K. The blue star corresponds to the exact FBR average energy at the same temperature.

Path integral energies (in units of K) obtained with a SOS thermal propagator for a planar rotor with a periodic potential as a function of the imaginary time step (in units of reciprocal energy K−1) and their quadratic fit, at a temperature T = 0.37 K. The blue star corresponds to the exact FBR average energy at the same temperature.

Path integral energies (in units of K) obtained with a PQC thermal propagator for a planar rotor with a periodic potential as a function of the imaginary time step (in units of reciprocal energy K−1) and their linear fit, at a temperature T = 0.37 K. The blue star corresponds to the exact FBR average energy at the same temperature.

Path integral energies (in units of K) obtained with a PQC thermal propagator for a planar rotor with a periodic potential as a function of the imaginary time step (in units of reciprocal energy K−1) and their linear fit, at a temperature T = 0.37 K. The blue star corresponds to the exact FBR average energy at the same temperature.

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