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The Journal of Chemical Physics

Wigner molecules: Natural orbitals of strongly correlated two-electron harmonium

Jerzy Ciosłowski, Marcin Buchowiecki

Published Aug 10, 2006• 29 citations
Publisher

Abstract

Explicit asymptotic expressions for natural orbitals and their occupancies are derived for the harmonium atom at the strong-correlation limit at which the confinement strength omega tends to zero. Unlike in systems with moderate correlation effects, the occupancies at the omega-->0 limit (derived from occupation amplitudes with alternating sign patterns) are vanishingly small and asymptotically independent of the angular momentum, forming a geometric progression with the scale factor proportional to omega(1/3) and the common ratio of ca. 0.0186. The radial components of the natural orbitals are given by products of polynomials and Gaussian functions that, as expected, peak at approximately half of the equilibrium interelectron distance.

DOI
10.1063/1.2222360

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Journal of Physics B Atomic Molecular and Optical Physics•Aug 31, 2017•9 citations
Explicit asymptotic expressions for natural orbitals and their occupancies are derived for the harmonium atom at the strong-correlation limit at which the confinement strength omega tends to zero. Unlike in systems with moderate correlation effects, the occupancies at the omega-->0 limit (derived from occupation amplitudes with alternating sign patterns) are vanishingly small and asymptotically independent of the angular momentum, forming a geometric progression with the scale factor proportional to omega(1/3) and the common ratio of ca. 0.0186. The radial components of the natural orbitals are given by products of polynomials and Gaussian functions that, as expected, peak at approximately half of the equilibrium interelectron distance.
Explicit asymptotic expressions are derived for the reduced one-electron density matrices (the 1-matrices) of strongly correlated two- and three-electron harmonium atoms in the ground and first excited states. These expressions, which are valid at the limit of small confinement strength ω, yield electron densities and kinetic energies in agreement with the published values. In addition, they reveal the ω(5/6) asymptotic scaling of the exchange components of the electron-electron repulsion energies that differs from the ω(2/3) scaling of their Coulomb and correlation counterparts. The natural orbitals of the totally symmetric ground state of the two-electron harmonium atom are found to possess collective occupancies that follow a mixed power/Gaussian dependence on the angular momentum in variance with the simple power-law prediction of Hill's asymptotics. Providing rigorous constraints on energies as functionals of 1-matrices, these results are expected to facilitate development of approximate implementations of the density matrix functional theory and ensure their proper description of strongly correlated systems.
In the harmonium atom, the collective occupancies {n(l)} of natural orbitals with different angular momenta l can be rigorously studied for those values of the confinement strength omega that lead to ground-state wave functions with correlation factors given by polynomials of kth order. Analysis of two such cases, with omega equal to 12 (k=1) and 110 (k=2), uncovers some interesting properties of {n(l)}, which turn out to be given by rational functions of pi(12) with simple coefficients. In particular, contrary to naive expectations, the collective occupancies for l>2 actually decrease upon strengthening of correlation. The explicit formulas for l-matrices and {n(l)} can be readily utilized in testing, calibration, and benchmarking of approximate implementations of the density matrix functional theory (DMFT).
Asymptotically exact wave functions of the four-electron harmonium atom at the strong-correlation limit are given by Slater determinants multiplied by a Jastrow factor in the form of a single exponential function with the argument bilinear in the interelectron distances. The spin orbitals that enter the determinants set the multiplicities and angular momenta of individual states whereas the Jastrow factor fully accounts for the electron correlation. Asymptotically exact expressions for the corresponding electron densities, which can be computed from such wave functions with the help of the saddle-point approximation, have the form of products of Gaussian functions and polynomials in the Cartesian coordinates.
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.
Abstract A formalism enabling computation of the one-particle density of a freely rotating assembly of identical particles that vibrate about their equilibrium positions with amplitudes much smaller than their average distances is presented. It produces densities as finite sums of products of angular and radial functions, the length of the expansion being determined by the interplay between the point-group and permutational symmetries of the system in question. Obtaining from a convolution of the rotational and bosonic components of the parent wavefunction, the angular functions are state-dependent. On the other hand, the radial functions are Gaussians with maxima located at the equilibrium lengths of the position vectors of individual particles and exponents depending on the scalar products of these vectors and the eigenvectors of the corresponding Hessian as well as the respective eigenvalues. Although the new formalism is particularly useful for studies of the Wigner molecules formed by electrons subject to weak confining potentials, it is readily adaptable to species (such as ʻballiums’ and Coulomb crystals) composed of identical particles with arbitrary spin statistics and permutational symmetry. Several examples of applications of the present approach to the harmonium atoms within the strong-correlation regime are given.