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American Journal of Physics

How good is Hartree–Fock? Calculation of the static electric polarizability in an exactly soluble model

Joseph B. Krieger, S. E. Kaufman

Published Jun 1, 1976• 1 citation

Abstract

The exactly soluble model, originally considered by Moshinsky, consisting of two spin-1/2 particles interacting through harmonic-oscillator forces and moving in a common harmonic-oscillator potential is extended to include the effects of an external electric field. We solve for the energy exactly, exactly in the Hartree–Fock approximation, and also compare to the results of Hartree–Fock perturbation theory. We find that the Hartree–Fock method yields the same result for the electric polarizability as the exact solution even though the field-independent part of the energy is not the same. However, when second-order Hartree–Fock perturbation theory is employed, the results for the polarizability can be substantially in error. Similar results obtain for the calculation of the matrix element for optical absorption in the dipole approximation.

DOI: 10.1119/1.10380

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The exactly soluble model, originally considered by Moshinsky, consisting of two spin-1/2 particles interacting through harmonic-oscillator forces and moving in a common harmonic-oscillator potential is extended to include the effects of an external electric field. We solve for the energy exactly, exactly in the Hartree–Fock approximation, and also compare to the results of Hartree–Fock perturbation theory. We find that the Hartree–Fock method yields the same result for the electric polarizability as the exact solution even though the field-independent part of the energy is not the same. However, when second-order Hartree–Fock perturbation theory is employed, the results for the polarizability can be substantially in error. Similar results obtain for the calculation of the matrix element for optical absorption in the dipole approximation.

Moshinsky introduced an exactly soluble model of a two-electron atom consisting of two spin-

\frac{1}{2}

particles interacting via harmonic forces and moving in a harmonic-oscillator potential. Here, the exact ground-state density $\ensuremath{\rho}(r)$ is related to the (also analytically known) Hartree-Fock density ${\ensuremath{\rho}}_{\mathrm{HF}}(r).$ The generalization to the off-diagonal matrix $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ is then effected, this being related to the idempotent ${\ensuremath{\gamma}}_{\mathrm{HF}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})/2.$ This exact information on this ``model atom'' prompts us to propose an approximate form of $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ for the He-like ions, the

{\mathrm{H}}_{2}

molecule and, in general, all two-electron systems. $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ is constructed solely from the exact $\ensuremath{\rho}(\mathbf{r})$ and its Hartree-Fock counterpart. Some detailed treatment of the two-electron Hookean atom with spring constant

k=1/4

(atomic units) is also presented.

Using a Hartree-Fock-like partitioning of the two-matrix in terms of the true correlated one-matrix, the corresponding approximate ground-state energy

{E}_{0}

is calculated for an exactly solvable closed-shell system proposed earlier by Moshinsky [Am. J. Phys. 36, 52 (1968)]. Comparisons of

{E}_{0}

with the Hartree-Fock ground-state energy

{E}_{\mathrm{HF}}

and the exact ground-state energy

{E}_{\mathrm{ex}}

are made. These comparisons provide valuable insight into the usefulness of a Hartree-Fock-like partitioning of the two-matrix with repulsive interparticle potentials. In particular, an explicit quantification for the applicability limit of Lieb's [Phys. Rev. Lett. 46, 457 (1981)] bound ${E}_{0}\ensuremath{\geqslant}{E}_{\mathrm{HF}}$ with unbounded oscillator potentials is given.

In both molecular physics and condensed matter theory, deeper understanding of the correlation energy density

{ϵ}_{c}(\mathbf{r})

remains a high priority. By adopting L\"owdin's definition of correlation energy as the difference between the exact and the Hartree-Fock values, here we propose two alternative routes to define this. One of these involves both exact and Hartree-Fock (HF) wave functions, while the second requires a coupling constant integration. As an exact analytical example of the first route, we treat the two-electron model atom of Moshinsky, for which both confinement potential and interactions are harmonic. Though the correlation energy density

{ϵ}_{c}(\mathbf{r})

is known analytically, we also investigate numerically its relation to the exact ground-state density in this example.