From Fredholm to Schrödinger via Eikonal: A New Formalism for Revealing Unknown Properties of Natural Orbitals.

Previously unknown properties of the natural orbitals (NOs) pertaining to singlet states (with natural parity, if present) of electronic systems with even numbers of electrons are revealed upon the demonstration that, at the limit of n → ∞, the NO ψn(r⃗) with the nth largest occupation number νn approaches the solution ψ̃n(r⃗) of the zero-energy Schrödinger equation that reads T̂([ρ2(r⃗, r⃗)]-1/8 ψ̃n(r⃗)) - (π2/ṽn)1/4 [ρ2(r⃗, r⃗)]1/4 ([ρ2(r⃗, r⃗)]-1/8 ψ̃n(r⃗)) = 0 (where T̂ is the kinetic energy operator), whereas νn approaches ν̃n. The resulting formalism, in which the "on-top" two-electron density ρ2(r⃗, r⃗) solely controls the asymptotic behavior of both ψn(r⃗) and νn at the limit of the latter becoming infinitesimally small, produces surprisingly accurate values of both quantities even for small n. It opens entirely new vistas in the elucidation of their properties, including single-line derivations of the power laws governing the asymptotic decays of νn and ⟨ψn(r⃗)|T̂|ψn(r⃗)⟩ with n, some of which have been obtained previously with tedious algebra and arcane mathematical arguments. These laws imply a very unfavorable asymptotics of the truncation error in the total energy computed with finite numbers of natural orbitals that severely affects the accuracy of certain quantum-chemical approaches such as the density matrix functional theory. The new formalism is also shown to provide a complete and accurate elucidation of both the observed order (according to decreasing magnitudes of the respective occupation numbers) and the shapes of the natural orbitals pertaining to the 1Σg+ ground state of the H2 molecule. In light of these examples of its versatility, the above Schrödinger equation is expected to have its predictive and interpretive powers harnessed in many facets of the electronic structure theory.