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Physical Review A

Exact density matrix for a two-electron model atom and approximate proposals for realistic two-electron systems

C. Amovilli, Norman Henry March

Published Feb 28, 2003• 46 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.67.022509

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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 \ensuremathρ(r)\ensuremath{\rho}(r)\ensuremathρ(r) is related to the (also analytically known) Hartree-Fock density \ensuremathρHF(r).{\ensuremath{\rho}}_{\mathrm{HF}}(r).\ensuremathρHF​(r). The generalization to the off-diagonal matrix \ensuremathγ(r,r\ensuremath′)\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})\ensuremathγ(r,r\ensuremath′) is then effected, this being related to the idempotent \ensuremathγHF(r,r\ensuremath′)/2.{\ensuremath{\gamma}}_{\mathrm{HF}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})/2.\ensuremathγHF​(r,r\ensuremath′)/2. This exact information on this ``model atom'' prompts us to propose an approximate form of \ensuremathγ(r,r\ensuremath′)\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})\ensuremathγ(r,r\ensuremath′) for the He-like ions, the H2{\mathrm{H}}_{2}H2​ molecule and, in general, all two-electron systems. \ensuremathγ(r,r\ensuremath′)\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})\ensuremathγ(r,r\ensuremath′) is constructed solely from the exact \ensuremathρ(r)\ensuremath{\rho}(\mathbf{r})\ensuremathρ(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.
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.
The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction u(r12)u({r}_{12})u(r12​) is studied. One representation with contracted information is first explored by constructing the momentum density $\ensuremath{\rho}(\mathbf{p})$ in terms of the wave function of the relative motion, say ${\ensuremath{\Psi}}_{R}({r}_{12})$, which naturally depends on the choice of u(r12)u({r}_{12})u(r12​). For u(r12)=e2∕r12u({r}_{12})={e}^{2}∕{r}_{12}u(r12​)=e2∕r12​, the so-called Hookean atom, and for the inverse square law $u({r}_{12})=\ensuremath{\lambda}∕{r}_{12}^{2}$, plots are presented of the above density $\ensuremath{\rho}(\mathbf{p})$ in momentum space. The correlated kinetic energy is recovered from averaging p2∕2m{p}^{2}∕2mp2∕2m, mmm denoting the electron mass, with respect to $\ensuremath{\rho}(\mathbf{p})$. The second method developed is in coordinate space and expands the density matrix $\ensuremath{\gamma}({\mathbf{r}}_{1},{\mathbf{r}}_{2})$ in Legendre polynomials, using relative coordinate ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}$, center-of-mass coordinate (r1+r2)∕2({\mathbf{r}}_{1}+{\mathbf{r}}_{2})∕2(r1​+r2​)∕2 and the angle, $\ensuremath{\theta}$ say, between these two vectors. For the Moshinsky atom in which u(r12)=12kr122u({r}_{12})=\frac{1}{2}k{r}_{12}^{2}u(r12​)=21​kr122​ only the sss term (l=0)(l=0)(l=0) contributes to the Legendre polynomial expansion. The specific example we present of the inverse square law model is shown to be characterized by the low-order terms (s+d)(s+d)(s+d) of the Legendre expansion. The Wigner function is finally calculated analytically for both Moshinsky and inverse square law models.
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 ${E}_{0}\ensuremath{\geqslant}{E}_{\mathrm{HF}}$ with unbounded oscillator potentials is given.
Abstract The recent study by Joubert on effects of Coulomb repulsions in a many‐electron system has focused attention on an integral identity involving the pair density. This has motivated the derivation presented here of a vectorial differential form related to this integral result. Our differential identity is then illustrated explicitly by using (i) an exact ground‐state wave function for the so‐called Hookean atom having external potential energy (1/2)kr2, with k = 1/4, and (ii) Moshinsky's model in which both the interparticle interaction and the external potential are of harmonic type. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
A system of NNN antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.