Refractory to Improvement: A Cautionary Tale of Two-Electron Densities Computed with One-Electron Basis Sets and of Their Employment in Quantum-Chemical Formalisms

A report on detailed investigations of the properties of the approximate on-top two-electron densities Φ(r⃗) computed with one-electron basis functions is presented. The key findings of these investigations, which also concern the reduced variant of Φ(r⃗), i.e., ϕ(r→)=4Φ(r→)[ρ1(r→)]2 [where ρ1(r⃗) is the one-electron density], are as follows: 1) Φ(r⃗) converge extremely slowly to the complete basis set (CBS) limits, their errors asymptotically scaling ∼N-1/3 with the number N of basis functions (as contrasted with the ∼N-1 error scaling of the electronic energy); 2) this slow convergence results in gross inaccuracies of the approximate Φ(r⃗) computed with standard basis sets; 3) the errors in these Φ(r⃗) are, in general, refractory to reduction with standard convergence acceleration techniques such as the CBS-limit extrapolation; 4) it is essential to make a distinction between the physically meaningful genuine Φ(r⃗) (such as those produced by highly accurate calculations) and the contrived ones (such as those appearing in some variants of the Kohn-Sham formalism and in overly simplistic models) that are merely auxiliary quantities to which no physical meaning should be attached; 5) due to the presence of spurious features in the contrived Φ(r⃗), their employment in the analysis of electronic structure leads to false and misleading conclusions on chemical bonding and electron correlation; and 6) the MC-PDFT formalism suffers from a hidden "double-counting" problem, whose presence is masked for sufficiently small active spaces by its use of the contrived Φ(r⃗).