Harmonium as a laboratory for mathematical chemistry

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 model” 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; however, 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 given here, made plain by working with Wigner quasiprobabilities on phase space. It allows in principle for thorough discussions of the merits of several approximate functionals popular in the theoretical chemical physics literature; in this respect, at the end we focus on Gill’s “Wigner intracule” method for the correlation energy.