Variational calculation of the single-particle density matrix and momentum density for the helium ground-state isoelectronic sequence

A recently developed variational formalism for the determination of the reduced single-particle density matrix, correct to second order, is applied to the ground state of the helium isoelectronic sequence. For a Slater-determinant-type trial wave function the method requires the initial determination of either the charge density or equivalently its cosine Fourier transform for spherically symmetric systems. The trial wave function employed is a one-parameter Hartree product of hydrogenic functions and use is made of the highly accurate analytic expressions derived elsewhere for the Fourier transform of the charge density for the helium sequence. Analytic expressions for the single-particle density matrix are obtained and the internal self-consistency of the technique with regard to the Kato cusp condition is discussed. These ex pressions are then employed to obtain closed form analytic expressions for the momentum density valid for the entire isoelectronic sequence. These results are subsequently employed to obtain expressions for the expectation values of the operators $p^4$, $p^2$, $\mathrm{}\left|p\right|\mathrm{}$, and ${\mathrm{}\left|p\right|\mathrm{}}^{-1\mathrm{}}$ and the Compton profile in the impulse approximation. Analytic Hartree-Fock calculations for these properties are also performed, and the results of the variational calculation are compared with these results and those of many-parameter correlated wave-function calculations wherever possible. It is observed that the results of the single-parameter variational calculation for helium are highly accurate and improve further for each heavier element of the isoelectronic sequence.