Efficient Bosonic and Fermionic Sinkhorn Algorithms for Non-Interacting Ensembles in One-body Reduced Density Matrix Functional Theory in the Canonical Ensemble

TL;DR

This paper develops efficient algorithms based on Sinkhorn methods to approximate non-interacting ensembles in one-body reduced density matrix functional theory, enabling improved modeling of quantum systems at finite and zero temperatures.

Contribution

It introduces bosonic and fermionic Sinkhorn algorithms for inverting NOONs and orbital energies, advancing non-interacting ensemble approximations in 1-RDMFT.

Findings

Algorithms accurately reproduce NOONs of simulated and real molecules.

Non-interacting entropy and energy approximations vary with bond length.

Connections to entropically-regularized optimal transport are established.

Abstract

We introduce 1-RDMFT in the canonical ensemble and then proceed to approximate the interacting ensemble by a non-interacting ensemble that maximizes the entropy, independently of temperature. Bosonic and Fermionic Sinkhorn algorithms are derived and used to invert the relationship between the Natural Orbital Occupation Numbers (NOONs) and the effective orbital energies of the non-interacting ensemble. Both the Bosonic and Fermionic Sinkhorn algorithms are shown to perform well in reproducing the NOONs of simulated distributions and the ground-state NOONs of H$_2$O and H$_2$. In the case of H$_2$ we also study the resulting non-interacting entropy and non-interacting approximation to the interaction energy within several wavefunction subspaces as the bond length varies. This provides several new starting points for approximations of the interaction energy, also at zero-temperature. Connections to entropically-regularized Multi-Marginal Optimal Transport (MMOT) are highlighted that may prove interesting for future research.