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

The formalism developed by the authors for the variational determination of the expectation value of single-particle operators $W=\Sigma i^{}W\left({\overrightarrow{\mathrm{r}}}_i\right)$ via Delves's principle in the Hartree approximation is extended here to the Hartree-Fock approximation by employing a Slater-determinant-type trial wave function and an appropriately antisymmetrized auxiliary function. A set of coupled integral-differential equations for the components of the auxiliary function are obtained by a subsidiary variational minimization of a functional involving the trial wave function, the auxiliary function, the Hamiltonian, and the operator $W$. Owing to the antisymmetric nature of both the trial and auxiliary functions, an exchange term involving the specific single-particle operator in question appears in these equations. Decoupling approximations are discussed and the equations solved exactly for single-particle operators that depend on the radial distance only. Employing a single-parameter appropriately antisymmetrized product of the $1S$ and $2S$ hydrogenic functions as the trial wave function, the formalism in this Hartree-Fock approximation is then applied to both the helium $2 {}_{}{}^3{}_{}{}^{}S$ and $2 {}_{}{}^1{}_{}{}^{}S$ isoelectronic sequences to obtain analytic expressions for the expectation value of the operators $r^n$, $n=2,1,-1\mathrm{}$, and $\delta \left(\overrightarrow{\mathrm{r}}\right)$. The results of these calculations are observed to approximate closely the results of a 2300-parameter calculation due to Accad et al. and in the orthohelium case to be also equivalent to those due to Hartree-Fock.