Harmonium as a laboratory for mathematical chemistry

TL;DR

This paper uses the harmonium model and phase space Wigner quasiprobabilities to analyze energy functionals and solve the phase dilemma in quantum chemistry, providing new exact results and insights.

Contribution

It introduces a phase space approach to exactly determine energy and state functionals for the harmonium model, addressing unresolved issues in the theory.

Findings

Exact energy and full state functionals derived for harmonium

Analysis of the phase dilemma in quantum chemistry

Evaluation of Wigner intracule method for correlation energy

Abstract

Thanks to an algebraic duality property of reduced states, the Schmidt best approximation theorems have important corollaries in the rigorous theory of two-electron moleculae. In turn, the "harmonium mode" or "Moshinsky atom" constitutes a non-trivial laboratory bench for energy functionals proposed over the years (1964--today), purporting to recover the full ground state of the system from knowledge of the reduced 1-body matrix. That model is usually regarded as solvable, but some important aspects of it, in particular the exact energy and full state functionals ---unraveling the "phase dilemma" for the system--- had not been calculated heretofore. The solution is made plain here by working with Wigner quasiprobabilities on phase space. It allows in principle for a thorough discussion of the (de)merits of several approximate functionals popular in the theoretical chemical physics literature. We focus on Gill's "Wigner intracule" method for the correlation energy.