Beyond Gaussian fluctuations of quantum anharmonic nuclei

TL;DR

This paper introduces nonlinear SCHA (NLSCHA), a novel method that extends the Self-Consistent Harmonic Approximation to accurately describe non-Gaussian quantum fluctuations in anharmonic nuclei, enabling direct free energy and entropy calculations.

Contribution

NLSCHA employs an invertible nonlinear transformation to improve the variational description of non-Gaussian fluctuations, providing direct access to free energy and entropy in quantum anharmonic systems.

Findings

NLSCHA accurately describes non-Gaussian quantum fluctuations.

The method provides direct free energy and entropy calculations.

It extends the applicability of SCHA to systems with tunneling and rotational modes.

Abstract

The Self-Consistent Harmonic Approximation (SCHA) describes atoms in solids, including quantum fluctuations and anharmonic effects, in a non-perturbative way. It computes ionic free energy variationally, constraining the atomic quantum-thermal fluctuations to be Gaussian. Consequently, the entropy is analytical; there is no need for thermodynamic integration or heavy diagonalization to include finite temperature effects. In addition, as the probability distribution is fixed, SCHA solves all the equations with Monte Carlo integration without employing Metropolis sampling of the quantum phase space. Unfortunately, the Gaussian approximation breaks down for rotational modes and tunneling effects. We show how to describe these non-Gaussian fluctuations using the quantum variational principle at finite temperatures, keeping the main advantage of SCHA: direct access to free energy. Our method, nonlinear SCHA (NLSCHA), employs an invertible nonlinear transformation to map Cartesian coordinates into an auxiliary manifold parametrized by a finite set of variables. So, we adopt a Gaussian \textit{ansatz} for the density matrix in this new coordinate system. The nonlinearity of the mapping ensures that NLSCHA enlarges the SCHA variational subspace, and its invertibility conserves the information encoded in the density matrix. We evaluate the entropy in the auxiliary space, where it has a simple analytical form. As in the SCHA, the variational principle allows for optimizing free parameters to minimize free energy. Finally, we show that, for the first time, NLSCHA gives direct access to the entropy of a crystal with non-Gaussian degrees of freedom.