Accuracy Diagnostics for Natural Orbitals and Their Occupation Numbers from the Contracted Schrödinger Equation

Although calculations of electronic properties from wave functions obtained with explicitly correlated basis sets pose no particular difficulties in general, diagonalization of the resultant one-electron reduced density matrices, required for the computation of the natural orbitals (NOs) and their occupation numbers (ONs), is far more complicated, as these matrices are not amenable to direct construction of spectral representations and thus have to be projected onto finite sets of one-electron basis functions. The characteristics of these basis sets (which cannot be retrieved from the standardized libraries), such as their cardinalities and the explicit forms of their elements, are unknown a priori. Their construction and the evaluation of the overall accuracy of the resultant NOs and ONs hinge upon the availability of accuracy diagnostics capable of reliable estimation of the deviations of these quantities from their exact counterparts and partitioning of these deviations into the contributions due to the inexactness of the wave function and the inaccuracy of the projection itself. Diagnostics derived from the contracted Schrödinger equation provide these data, making it possible to identify a given combination of the wave function Ψ and the one-electron basis set {ϕj} as that producing Ψ-bound, ϕ-bound, or balanced errors. Extensive tests of these new accuracy diagnostics are carried out for the ground states of several few-electron systems.