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

Constraints upon Functionals of the 1-Matrix, Universal Properties of Natural Orbitals, and the Fallacy of the Collins "Conjecture"

Reliability of quantum-chemical calculations based upon the density functional theory and its 1-matrix counterpart hinges upon minimizing the extent of empirical parameterization in the approximate energy expressions of these formalisms while imposing as many rigorous constraints upon them as possible. The recently uncovered universal properties of the natural orbitals facilitate the construction of such constraints for the 1-matrix functionals. The benefits of their employment in the validation of these functionals are vividly demonstrated by a critical review of the three incarnations of the so-called Collins conjecture. Although the incorporation of rigorous definitions of the correlation energy and entropy, and the identification of individual potential energy hypersurfaces as probable domains of its applicability turn the originally published unsubstantiated claim into a proper conjecture, the resulting formalism is found to be merely a conduit for incorporation of static correlation effects in electronic structure calculations that is unlikely to allow attaining chemical accuracy.

Symmetry Equiincidence of Natural Orbitals

The symmetry equiincidence principle quantifies the apportionment of the natural orbitals (NOs), ordered according to their nonascending occupation numbers, among the irreducible representations (irreps) of the point group pertaining to the underlying on-top two-electron density. This principle, which is rigorously proven for the resolvable Cs, C2v, C3v, C4v, C6v, D2h, D3h, D4h, D6h, and Oh point groups, states that the symmetry incidences, i.e., the asymptotic probabilities with which the NOs belonging to different irreps occur, are proportional to the squares of irreps' dimensions. Since its proof hinges upon a sufficient number of planes of symmetry among the elements of a given point group, it yields only linear combinations of the symmetry incidences for the quasiresolvable groups with too few such planes and fails for the unresolvable C1, Ci, Cn, Dn, S2n, T, O, and I groups whose nontrivial elements comprise only symmetry axes and/or the center of inversion.

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.

1-Matrix functional for long-range interaction energy of two hydrogen atoms

The leading terms in the large-R asymptotics of the functional of the one-electron reduced density matrix for the ground-state energy of the H₂ molecule with the internuclear separation R are derived thanks to the solution of the phase dilemma at the R → ∞ limit. At this limit, the respective natural orbitals (NOs) are given by symmetric and antisymmetric combinations of "half-space" orbitals with the corresponding natural amplitudes having the same amplitudes but opposite signs. Minimization of the resulting explicit functional yields the large-R asymptotics for the occupation numbers of the weakly occupied NOs and the C₆ dispersion coefficient. The highly accurate approximates for the radial components of the p-type "half-space" orbitals and the corresponding occupation numbers (that decay like R^-6), which are available for the first time thanks to the development of the present formalism, have some unexpected properties.

Correcting Models with Long-Range Electron Interaction Using Generalized Cusp Conditions

Sources of energy errors resulting from the replacement of the physical Coulomb interaction by its long-range erfc(μr)/r approximation are explored. It is demonstrated that the results can be dramatically improved and the range of μ giving energies within chemical accuracy limits significantly extended if the generalized cusp conditions are used to represent the wave function at small r. The numerical results for two-electron harmonium are presented and discussed.

A Universal Power Law Governing the Accuracy of Wave Function-Based Electronic Structure Calculations

A universal power law governing the accuracy of wave function-based electronic structure calculations is derived from first principles. The resulting expression ΔE(N,N)N≳19π2gNN, where g is a system-specific factor assuming values between zero and one and ≳ stands for asymptotic inequality at the limit of N→∞, allows facile estimation of the error ΔE(N,N) in the electronic energy of a singlet state of an N-electron system computed with a basis set of N one-electron functions. Several approaches to the estimation of the factor g, which depends on the on-top two-electron density, are presented.

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.