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arXiv:1002.3396 · Physical Review A

Hooke’s law correlation in two-electron systems

Pierre‐François Loos

Published Mar 10, 2010• 34 citations
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Abstract

We study the properties of the Hooke's law correlation energy (${E}_{\mathrm{c}}$), defined as the correlation energy when two electrons interact via a harmonic potential in a $D$-dimensional space. More precisely, we investigate the ${}^{1}S$ 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 ${E}_{\mathrm{c}}$ 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 ($D=1 \text{and} 3$) and that the exact wave function is based on Mathieu functions.

DOI
10.1103/physreva.81.032510

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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.
We consider the high-density-limit correlation energy E(c) in D > or = 2 dimensions for the (1)S ground states of three two-electron systems: helium (in which the electrons move in a Coulombic field), spherium (in which they move on the surface of a sphere), and hookium (in which they move in a quadratic potential). We find that the E(c) values are strikingly similar, depending strongly on D but only weakly on the external potential. We conjecture that for large D, the limiting correlation energy E(c) approximately -delta(2)/8 in any confining external potential, where delta = 1/(D-1).
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.
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.
The system of two interacting electrons in one-dimensional harmonic potential or Hooke’s atom is considered, again. On one hand, it appears as a model for quantum dots in a strong confinement regime, and on the other hand, it provides us with a hard test bench for new methods with the “space splitting” arising from the one-dimensional Coulomb potential. Here, we complete the numerous previous studies of the ground state of Hooke’s atom by including the excited states and dynamics, not considered earlier. With the perturbation theory, we reach essentially exact eigenstate energies and wave functions for the strong confinement regime as novel results. We also consider external perturbation induced quantum dynamics in a simple separable case. Finally, we test our novel numerical approach based on real-time path integrals (RTPIs) in reproducing the above. The RTPI turns out to be a straightforward approach with exact account of electronic correlations for solving the eigenstates and dynamics without the conventional restrictions of electronic structure methods.
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.
Two interacting electrons that are confined to the surface of a sphere have a uniform ground-state (surface) density. The Schr\"odinger equation of this helium-type two-electron system is solved here accurately for different values $\ensuremath{\alpha}∊\mathbb{R}$ of a constant that is multiplied to the electron-electron repulsion ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{\mathit{ee}}$. The correlation structure in the resulting wave functions is analyzed for different values of $\ensuremath{\alpha}$. The asymptotic limits $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ and $\ensuremath{\alpha}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ are treated analytically. Using these results, the ISI (interaction-strength interpolation) model for the density-functional ${E}_{\mathit{xc}}[\ensuremath{\rho}]$ of the exchange-correlation energy in the real system with $\ensuremath{\alpha}=1$ is tested against the exact adiabatic connection in density-functional theory.