Solitonic natural orbitals

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.

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.

Wigner localization in two and three dimensions: An ab initio approach

In this work, we investigate the Wigner localization of two interacting electrons at very low density in two and three dimensions using the exact diagonalization of the many-body Hamiltonian. We use our recently developed method based on Clifford periodic boundary conditions with a renormalized distance in the Coulomb potential. To accurately represent the electronic wave function, we use a regular distribution in space of Gaussian-type orbitals and we take advantage of the translational symmetry of the system to efficiently calculate the electronic wave function. We are thus able to accurately describe the wave function up to very low density. We validate our approach by comparing our results to a semi-classical model that becomes exact in the low-density limit. With our approach, we are able to observe the Wigner localization without ambiguity.

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.

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

Employment of exact numerical quadratures in the evaluation of matrix elements involving highly accurate wavefunctions of helium (and its isoelectronic congeners) generated with the help of the regularized Krylov sequences of Nakatsuji results in an efficient algorithm for the calculation of natural orbitals and the corresponding natural amplitudes {λ_nl}. The results of such calculations are presented for the total of 600 natural orbitals pertaining to the ground state of the helium atom. The benchmark-quality values of {λ_nl} computed for 1 ≤ n ≤ 100 and 0 ≤ l ≤ 5 reveal gross inaccuracies in the previously published data. In particular, the dependence of λ_nl on n is found to follow very closely a simple power-scaling law λ_nl≈-A_l (n+B_l)^-4 with A_l that, contrary to previous claims, varies only weakly with l. Even more importantly, the numerical trends observed in the present calculations strongly suggest that in the case of the ground state of the helium atom, the only positive-valued natural amplitude is that pertaining to the strongly occupied orbital, i.e., λ₁₀. The relevance of this finding to the existence of unoccupied natural orbitals pertaining to the ground state wavefunction of the H₂ molecule is discussed.

Diverging Exchange Force and Form of the Exact Density Matrix Functional

For translationally invariant one-band lattice models, we exploit the ab initio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered. First, within each symmetry sector, the interaction functional F depends only on the natural occupation numbers n. The respective sets P_{N}^{1} and E_{N}^{1} of pure and ensemble N-representable one-matrices coincide. Second, and most importantly, the exact functional is strongly shaped by the geometry of the polytope E_{N}^{1}≡P_{N}^{1}, described by linear constraints D^{(j)}(n)≥0. For smaller systems, it follows as F[n]=[under ∑]i,i^{'}V[over ¯]_{i,i^{'}}sqrt[D^{(i)}(n)D^{(i^{'})}(n)]. This generalizes to systems of arbitrary size by replacing each D^{(i)} by a linear combination of {D^{(j)}(n)} and adding a nonanalytical term involving the interaction V[over ^]. Third, the gradient dF/dn is shown to diverge on the boundary ∂E_{N}^{1}, suggesting that the fermionic exchange symmetry manifests itself within RDMFT in the form of an "exchange force." All findings hold for systems with a nonfixed particle number as well and V[over ^] can be any p-particle interaction. As an illustration, we derive the exact functional for the Hubbard square.