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Physical Review A

Hartree-Fock-like partitioning of the two-matrix for an exactly solvable two-electron model atom

I. Nagy, Jānos Pipek

Published Mar 14, 2011• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.83.034502

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Using a Hartree-Fock-like partitioning of the two-matrix in terms of the true correlated one-matrix, the corresponding approximate ground-state energy E0{E}_{0}E0​ is calculated for an exactly solvable closed-shell system proposed earlier by Moshinsky [Am. J. Phys. 36, 52 (1968)]. Comparisons of E0{E}_{0}E0​ with the Hartree-Fock ground-state energy EHF{E}_{\mathrm{HF}}EHF​ and the exact ground-state energy Eex{E}_{\mathrm{ex}}Eex​ 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 E0\ensuremath⩾EHF{E}_{0}\ensuremath{\geqslant}{E}_{\mathrm{HF}}E0​\ensuremath⩾EHF​ with unbounded oscillator potentials is given.
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.
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.
Holas, Howard and March [Phys. Lett. A {\bf 310}, 451 (2003)] have obtained analytic solutions for ground-state properties of a whole family of two-electron spin-compensated harmonically confined model atoms whose different members are characterized by a specific interparticle potential energy u(r12r_{12}r12​). Here, we make a start on the dynamic generalization of the harmonic external potential, the motivation being the serious criticism levelled recently against the foundations of time-dependent density-functional theory (e.g. [J. Schirmer and A. Dreuw, Phys. Rev. A {\bf 75}, 022513 (2007)]). In this context, we derive a simplified expression for the time-dependent electron density for arbitrary interparticle interaction, which is fully determined by an one-dimensional non-interacting Hamiltonian. Moreover, a closed solution for the momentum space density in the Moshinsky model is obtained.
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.
In recent developments on the pair density needed to treat the non-Hartree-Fock-like part of interparticle repulsion, the natural orbitals and sign-correct expansion coefficients play a central role. Since, in principle, an infinite number of natural orbitals must be included, the convergence of expectation values due to finite-term approximations is an important issue. Here we discuss quantitatively this convergence problem based on an exactly solvable two-electron model atom, where the Schr\"odinger wave function for the ground state is expressible in terms of L\"owdin's natural orbitals and sign-correct expansion coefficients. Using properly renormalized truncated series expansions for such an exact decomposition, the corresponding expectation values of the Schr\"odinger Hamiltonian are calculated analytically. A rapid and uniform convergence is found in these expectation values at given values of the coupling in the interparticle repulsion.
In pair density-functional theory, the only unknown piece of the energy is the kinetic energy TTT as a functional of the pair density P(x1,x2)P({x}_{1},{x}_{2})P(x1​,x2​). Although T[P]T\phantom{\rule{0.1em}{0ex}}[P]T[P] has a simpler structure than the Hohenberg-Kohn functional of conventional density-functional theory, computational requirements are still moderate. In the present work, a particularly convenient model system to represent many-electron pair densities is introduced. This ``boson pair model'' (BPM) approximately treats electron pairs as noninteracting bosons. The resulting explicit model for the kinetic energy T2[P]{T}_{2}[P]T2​[P] is shown to be exact for two-electron systems and a lower bound to T[P]T\phantom{\rule{0.1em}{0ex}}[P]T[P] for more than two electrons. The one- and two-particle density matrices obtained from the BPM yield upper bounds for the corresponding many-electron quantities. This suggests a partitioning T[P]=T2[P]+Teff[P]T\phantom{\rule{0.1em}{0ex}}[P]={T}_{2}[P]+{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]T[P]=T2​[P]+Teff​[P], where only the remainder ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]\ensuremath{\geqslant}0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schr\"odinger equation with an effective local two-particle potential results; the latter is identified as a sum of the bare Coulomb interaction and the functional derivative of Teff[P]{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]Teff​[P]. This self-consistent scheme to minimize the energy with respect to PPP is more efficient than previous procedures. Further information on the functional derivative of Teff[P]{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]Teff​[P] is derived from a contracted Schr\"odinger equation. Since Teff[P]{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]Teff​[P] is explicitly known in the two-electron and noninteracting (Hartree-Fock) limits, the present method provides an alternative to density-matrix functional theories, which can be exact in the same limits and are similar in computational cost.