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

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 $E_0⩾E_{\mathrm{HF}}$ with unbounded oscillator potentials is given.

Figure 1

Difference functions $d_i(\Lambda )$ for $p=q=1$, defined in Eqs. (16) and (17), as a function of $\Lambda \in [0,0.5]$. The solid curve refers to $d_1$, and the dashed curve to $d_2$. The inset illustrates the slow tendency of $d_1$ to its limiting value $-0.5$ at $\Lambda =0.5$.

Figure 2

Overlap ratio $D_{p,q}/(1/2)$, defined in the text, as a function of $(p=q)\in [0.5,1]$.