Correlation Effects on Hyperfine-Structure Expectation Values for Atoms with Open2pSubshells

The first-order (FO) wave functions for the second period of atoms have been analyzed in terms of natural spin orbitals (NSO). The FO function is constructed to account for the same polarization effects as are accounted for by the unrestricted Hartree-Fock method (UHF), but the FO function has the correct symmetry. The strongly occupied NSO and the UHF orbitals contain the same kind of symmetry admixtures. The agreement between the first NSO determinant and the UHF wave function is bad for the contact term but good for the other expectation values that contribute to hyperfine structure. The FO function accounts also for internal and semi-internal correlation effects. The direct contributions through pair correlations (PC) from the "NSO sea" have been calculated. They agree for B with the values calculated from the 187-term function in our earlier paper. The PC correction to the contact term is of the same magnitude as the contact term itself. Furthermore, there is a large PC correction to the electric field gradient at the nucleus for B and C connected with the internal $s^2\to p^2$ correlation. The remaining PC effects are rather small but lead almost always to a decrease of the $〈{{\mathcal{r}}_x}^{-3\mathrm{}}〉$ expectation values. Since the expected increase of the $〈{{\mathcal{r}}_x}^{-3\mathrm{}}〉$ expectation values due to all-external correlations is not taken into account by the FO function, this function yields, at least for B and C, too small $〈{{\mathcal{r}}_x}^{-3\mathrm{}}〉$ values. The FO function is also missing important self-consistency effects. The quadrupole moment of the ${\mathrm{C}}^{11}$ nucleus is estimated to be $\pm 0.0333b$ on the basis of our analysis and experimental results.