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=Σi^W(→rᵢ) 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 ³S and 2 ¹S isoelectronic sequences to obtain analytic expressions for the expectation value of the operators rⁿ, n=2,1,-1, and δ(→r). 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.