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Physical Review A

Wave functions and two-electron probability distributions of the Hooke’s-law atom and helium

D.P. O'Neill, Peter M. W. Gill

Published Aug 21, 2003• 52 citations
Publisher

Abstract

The Hooke's-law atom (hookium) provides an exactly soluble model for a two-electron atom in which the nuclear-electron Coulombic attraction has been replaced by a harmonic one. Starting from the known exact position-space wave function for the ground state of hookium, we present the momentum-space wave function. We also look at the intracules, two-electron probability distributions, for hookium in position, momentum, and phase space. These are compared with the Hartree-Fock results and the Coulomb holes (the difference between the exact and Hartree-Fock intracules) in position, momentum, and phase space are examined. We then compare these results with analogous results for the ground state of helium using a simple, explicitly correlated wave function.

DOI
10.1103/physreva.68.022505

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We study the properties of the Hooke's law correlation energy (Ec{E}_{\mathrm{c}}Ec​), defined as the correlation energy when two electrons interact via a harmonic potential in a DDD-dimensional space. More precisely, we investigate the 1S{}^{1}S1S ground-state properties of two model systems: the Moshinsky atom (in which the electrons move in a quadratic potential) and the spherium model (in which they move on the surface of a sphere). A comparison with their Coulombic counterparts is made that highlights the main differences of the Ec{E}_{\mathrm{c}}Ec​ in both the weakly and strongly correlated limits. Moreover, we show that the Schr\"odinger equation of the spherium model is exactly solvable for two values of the dimension (D=1and3D=1 \text{and} 3D=1and3) and that the exact wave function is based on Mathieu functions.
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Moshinsky introduced an exactly soluble model of a two-electron atom consisting of two spin-12\frac{1}{2}21​ 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 H2{\mathrm{H}}_{2}H2​ 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=1/4k=1/4k=1/4 (atomic units) is also presented.
A formulation is proposed to treat the ground-state properties of an assembly of electrons, interacting mutually through the Coulomb potential and confined by an external potential V whose maximum is finite so that both bound and extended continuum states appear as eigen states for the one-body Hamiltonian composed of the electron kinetic energy and V . These extended states are found to play a crucial role in incorporating the effect of electron correlation into the localized ground-state many-electron wave function. As an illustration of our formalism, the ground-state energy of a He atom is calculated in its simplest version and compared rather successfully with experiment as well as the results in other serious numerical methods. We also calculate the density and pair distribution functions in the same version. Special attention is paid to the latter function in this system in order to elucidate a qualitative difference from that in the homogeneous electron gas. This function shows clearly that the two electrons with opposite spins in He correlate strongly over quite a long range in contrast to the common expectation that the ground-state wave function for the He atom can be described accurately in the Hartree-Fock approximation.
In both molecular physics and condensed matter theory, deeper understanding of the correlation energy density ϵc(r){ϵ}_{c}(\mathbf{r})ϵc​(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 ϵc(r){ϵ}_{c}(\mathbf{r})ϵc​(r) is known analytically, we also investigate numerically its relation to the exact ground-state density in this example.
We present analytical solutions to a quantum-mechanical three-body problem in three dimensions, which describes a heliumlike two-electron atom. Similarly to Hooke's atom, the Coulombic electron-nucleus interaction potentials are replaced by harmonic potentials. The electron-electron interaction potential is taken to be both screened (decaying faster than the inverse of the interparticle separation) and regularized (in the limit of zero separation). We reveal the exactly solvable few-electron ground state, which explicitly includes electron correlation, for certain values of the harmonic containment.