Eigenvalues of uncorrelated, density-difference matrices and the interpretation of Δ-self-consistent-field...

Two theorems on the eigenvalues of differences of idempotent matrices determine the natural occupation numbers and orbitals of electronic detachment, attachment, or excitation that pertain to transitions between wavefunctions that each consist of a single Slater determinant. They are also applicable to spin density matrices associated with Slater determinants. When the ranks of the matrices differ, unit eigenvalues occur. In addition, there are ±w pairs of eigenvalues where |w| ≤ 1, whose values are related to overlaps, t, between the corresponding orbitals of Amos and Hall, and Löwdin by the formula w=±1-t^{2 12}. Generalized overlap amplitudes, including Dyson orbitals and their probability factors, may be inferred from these eigenvalues, which provide numerical criteria for: classifying transitions according to the number of holes and particles in final states with respect to initial states, identifying the most important effects of orbital relaxation produced by self-consistent fields, and the analysis of Fukui functions. Two similar theorems that apply to sums of idempotent matrices regenerate formulae for the natural orbitals and occupation numbers of an unrestricted Slater determinant that were published first by Amos and Hall.