Natural amplitudes of the ground state of the helium atom: Benchmark calculations and their relevance to the issue of unoccupied natural orbitals in the H2 molecule

Contactium: A strongly correlated model system

At the limit of an infinite confinement strength ω, the ground state of a system that comprises two fermions or bosons in harmonic confinement interacting through the Fermi-Huang pseudopotential remains strongly correlated. A detailed analysis of the one-particle description of this "contactium" reveals several peculiarities that are not encountered in conventional model systems (such as the two-electron harmonium atom, ballium, and spherium) involving Coulombic interparticle interactions. First of all, none of the natural orbitals (NOs) {ψn(ω;r)} of the contactium is unoccupied, which implies nonzero collective occupancies for all the angular momenta. Second, the NOs and their non-ascendingly ordered occupation numbers {νn} turn out to be related to the eigenfunctions and eigenvalues of a zero-energy Schrödinger equation with an attractive Gaussian potential. This observation enables the derivation of their properties, such as the n-4/3 asymptotic decay of νn at the n→∞ limit (which differs from that of n-8/3 in the Coulombic systems), the independence of the confinement energy vn=⟨ψn(ω;r)|12ω2r2|ψn(ω;r)⟩ of n, and the n-2/3 asymptotic decay of the respective contribution νntn to the kinetic energy. Upon suitable scaling, the weakly occupied NOs of the contactium turn out to be virtually identical to those of the two-electron harmonium atom at the ω → ∞ limit, despite the entirely different interparticle interactions in these systems.

ERRORS IN APPROXIMATE IONIZATION ENERGIES DUE TO THE ONE-ELECTRON SPACE TRUNCATION OF THE EKT EIGENPROBLEM

Unless the approximate wavefunction of the parent system is expressed in terms of explicitly correlated basis functions, the finite size of the generalized Fock matrix is unlikely to be the leading source of the truncation error in the ionization energy E produced by the EKT (extended Koopmans' theorem) formalism. This conclusion is drawn from a rigorous analysis that involves error partitioning into the parent- and ionized-system contributions, the former being governed by asymptotic power laws when the underlying wavefunction is assembled from a large number of spinorbitals and the latter arising from the truncation of the infinite-dimensional matrix V whose elements involve the 1-, 2- and 3-matrices of the parent system. Quite surprisingly, the decay of the second contribution with the number $n$ of the natural spinorbitals (NOs) employed in the construction of the truncated V turns out to be strongly system-dependent even in the simplest case of the 1S states of two-electron systems, following the n-5 power law for the helium atom while exhibiting an erratic behavior for the H- anion. This phenomenon, which stems from the presence of the so-called solitonic natural spinorbitals among the NOs, renders the extrapolation of the EKT approximates of E to the complete-basis-set limit generally unfeasible. However, attaining that limit is not contingent upon attempted reproduction of the ill-defined one-electron function known as 'the removal orbital', which does not have to be invoked in the derivation of EKT and whose expansion in terms of the NOs diverges.

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 ψ_n(r⃗) with the nth largest occupation number ν_n approaches the solution ψ̃_n(r⃗) of the zero-energy Schrödinger equation that reads T̂([ρ₂(r⃗, r⃗)]^-1/8 ψ̃_n(r⃗)) - (π²/ṽ_n)^1/4 [ρ₂(r⃗, r⃗)]^1/4 ([ρ₂(r⃗, r⃗)]^-1/8 ψ̃_n(r⃗)) = 0 (where T̂ is the kinetic energy operator), whereas ν_n approaches ν̃_n. The resulting formalism, in which the "on-top" two-electron density ρ₂(r⃗, r⃗) solely controls the asymptotic behavior of both ψ_n(r⃗) and ν_n 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 ν_n and ⟨ψ_n(r⃗)|T̂|ψ_n(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 ¹Σ_g⁺ 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.

Uniform description of the helium isoelectronic series down to the critical nuclear charge with explicitly correlated basis sets derived from regularized Krylov sequences

An efficient computational scheme for the calculation of highly accurate ground-state electronic properties of the helium isoelectronic series, permitting uniform description of its members down to the critical nuclear charge Z_c, is described. It is based upon explicitly correlated basis functions derived from the regularized Krylov sequences (which constitute the core of the free iterative CI/free complement method of Nakatsuji) involving a term that introduces split length scales. For the nuclear charge Z approaching Z_c, the inclusion of this term greatly reduces the error in the variational estimate for the ground-state energy, restores the correct large-r asymptotics of the one-electron density ρ(Z; r), and dramatically alters the manifold of the pertinent natural amplitudes and natural orbitals. The advantages of this scheme are illustrated with test calculations for Z = 1 and Z = Z_c carried out with a moderate-size 12th-generation basis set of 2354 functions. For Z = Z_c, the augmentation is found to produce a ca. 5000-fold improvement in the accuracy of the approximate ground-state energy, yielding values of various electronic properties with between seven and eleven significant digits. Some of these values, such as those of the norms of the partial-wave contributions to the wavefunction and the Hill constant, have not been reported in the literature thus far. The same is true for the natural amplitudes at Z = Z_c, whereas the published data for those at Z = 1 are revealed by the present calculations to be grossly inaccurate. Approximants that yield correctly normalized ρ(1; r) and ρ(Z_c; r) conforming to their asymptotics at both r → 0 and r → ∞ are constructed.