Generalized Path Integral Energy and Heat Capacity Estimators of Quantum Oscillators and Crystals Using Harmonic...

Area of Science:

• Computational Quantum Chemistry
• Materials Science
• Statistical Mechanics

Background:

• Imaginary-time path integral (PI) methods are essential for accurately modeling nuclear quantum effects in static properties.
• Standard PI estimators face high computational demands, necessitating the development of more precise and efficient estimation techniques.
• Existing methods like centroid virial (CVir) provide a baseline but can be improved for specific system types.

Purpose of the Study:

• To introduce novel generalized path integral estimators, termed harmonically mapped averaging (HMA), for enhanced precision in calculating energy and heat capacity.
• To leverage the harmonic characteristics of quantum oscillators and crystals through coordinate mapping to improve computational efficiency.
• To evaluate the performance of different HMA variants against standard estimators in various quantum systems.

Main Methods:

• Development of generalized PI estimators utilizing coordinate mapping, including harmonically mapped averaging (HMAc, HMAq-NM, HMAq-SG).
• Application of path integral molecular dynamics (PIMD) simulations to a 1D anharmonic oscillator and a 3D Lennard-Jones crystal.
• Examination of estimator performance based on precision, computational cost, anharmonicity, quantum effects, and Trotter number.

Main Results:

• HMA estimators consistently yielded more precise results than the standard centroid virial (CVir) estimator.
• The HMAq-NM (normal mode coordinates) variant demonstrated the highest precision, followed by HMAq-SG (harmonic oscillator staging) and HMAc (decoupled modes).
• The study confirmed the effectiveness of HMA for systems with harmonic character, while noting its inapplicability to fluids or systems with imaginary modes.

Conclusions:

• The developed HMA estimators provide a significant improvement in precision for path integral simulations, particularly for systems exhibiting harmonic behavior.
• HMA offers a computationally efficient framework for tackling more complex systems, including those requiring *ab initio* calculations.
• The availability of forces and Hessian matrix is assumed, but finite difference alternatives can be employed when these are not accessible.