Variational Principles for Single-Particle Expectation Values in the Hartree Approximation

The application of Delves's variational principle for the calculation of the expectation value of single-particle operators $W$ is investigated in the Hartree approximation. In this approximation the auxiliary function is taken to be of the form ${\psi }_{1T}={\Sigma }_i{f}_i\left({\overrightarrow{\mathrm{r}}}_i\right){\psi }_{0T}$, where ${\psi }_{0T}$ is the trial ground-state wave function. For a given ${\psi }_{0T}$, a set of coupled integrodifferential equations satisfied by the $f_i\left({\overrightarrow{\mathrm{r}}}_i\right)$ is derived by minimizing an auxiliary functional containing ${\psi }_{0T}$, ${\psi }_{1T}$, $W$, and the Hamiltonian of the system. The resulting equations are uncoupled in two different approximations each valid for one- and two-particle systems and are compared to those employed by others and to those previously suggested by us on the basis of certain self-consistency considerations. The uncoupled equations are solved exactly for single-particle operators that depend on the radial distance only. The utility of the technique is demonstrated by showing that for a model hydrogen-atom problem it leads to highly accurate results for the electron density at the origin and the Fourier transform of the electron density even when the calculation of these quantities employing ${\psi }_{0T}$ alone is substantially in error. The results of applying the technique to the helium atom for the calculation of $〈{\mathcal{r}}^n〉$, $n=-2, -1, 1, 2$, and the electron density at the origin employing an energy-minimized product of hydrogenic wave functions for ${\psi }_{0T}$ are reviewed and compared with those of Pekeris and found to have an accuracy equivalent to a numerical Hartree-Fock calculation. Finally, the decoupling approximations are extended to systems containing more than two particles.