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The Journal of Chemical Physics

Collective natural orbital occupancies of harmonium

Jerzy Ciosłowski, Marcin Buchowiecki

Published Feb 11, 2005• 14 citations
Publisher

Abstract

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).

DOI
10.1063/1.1846655

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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).
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.
For a quantum system of NNN identical, harmonically interacting particles in a one-dimensional harmonic trap we calculate for the bosonic and fermionic ground state the corresponding one-particle reduced density operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ analytically. In case of bosons ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ is a Gibbs state for an effective harmonic oscillator. Hence the natural orbitals are Hermite functions and their occupation numbers obey a Boltzmann distribution. Intriguingly, for fermions with not too large couplings the natural orbitals coincide up to just a small error with the bosonic ones. In case of strong coupling this still holds qualitatively. Moreover, the decay of the decreasingly ordered fermionic natural occupation numbers is given by the bosonic one, but modified by an algebraic prefactor. Significant differences to bosons occur only for the largest occupation numbers. After all, the ``gap'' at the ``Fermi level'' decreases with increasing coupling strength but remains well pronounced even for strong interaction.
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.
Calculations of sub-μhartree accuracy employing explicitly correlated Gaussian lobe functions produce comprehensive data on the energy E(ω), its components, and the one-electron properties of the two lowest-energy states of the three-electron harmonium atom. The energy computations at 19 values of the confinement strength ω ranging from 0.001 to 1000.0, used in conjunction with a recently proposed robust interpolation scheme, yield explicit approximants capable of estimating E(ω) and the potential energy of the harmonic confinement within a few tenths of μhartree for any ω ≥ 0.001, the respective errors for the kinetic energy and the potential energy of the electron-electron repulsion not exceeding 2 μhartrees. Thanks to the correct ω → 0 asymptotics incorporated into the approximants, comparable accuracy is expected for values of ω smaller than 0.001. Occupation numbers of the dominant natural spinorbitals and two different measures of electron correlation are also computed.
The ground-state entanglement of a single particle of the N-harmonium system (i.e., a completely-integrable model of NNN particles where both the confinement and the two-particle interaction are harmonic) is shown to be analytically determined in terms of NNN and the relative interaction strength. For bosons, we compute the von Neumann entropy of the one-body reduced density matrix by using the corresponding natural occupation numbers. There exists a critical number NcN_cNc​ of particles so that below it, for positive values of the coupling constant, the entanglement grows when the number of particles is increasing; the opposite occurs for N>NcN > N_cN>Nc​. For fermions, we compute the one-body reduced density matrix for the closed-shell spinned case. In the strong coupling regime, the linear entropy of the system decreases when NNN is growing. For fixed NNN, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.