Correlation Effects in Two- and Three-Electron Systems

Correlation effects in the lithium isoelectronic series ${\mathrm{}\left(1s\right)\mathrm{}}^{2}2s{}_{}{}^2{}_{}{}^{}S$ have been studied by computing the expectation values of a number of one- and two-electron operators. Calculations were performed using both Hartree-Fock and configuration interaction wave functions. The operators considered are $\Sigma {r_i}^N$ ($N=-2, -1, 2, 4$), ${\delta }^3\left({\mathrm{r}}_i\right)$, ${p_i}^2$, $\Sigma {i>j}^{}{\delta }^3\left({\mathrm{r}}_{\mathrm{ij}}\right)$, $\Sigma {i>j}^{}\left({\mathrm{r}}_i·{\mathrm{r}}_j\right)$, and $\Sigma {i>j}^{}\left({\mathrm{p}}_i·{\mathrm{p}}_j\right)$. Computations were performed for $Z=3, 4, 5, \mathrm{and} 8$. For comparison purposes the same expectation values were computed from configuration interaction wave functions of comparable accuracy for the two-electron systems ${\mathrm{}\left(1s\right)\mathrm{}}^2{}_{}{}^1{}_{}{}^{}S$ with $Z=2, 3, 4, 5, \mathrm{and} 8$. The results of the two-electron computations show reasonable agreement with more accurate computations and with empirical estimates. For the three-electron systems the main conclusions are: (a) Correlation has little effect on the one-electron expectation values, (b) the expectation values of $\Sigma {i>j}^{}\left({\mathrm{r}}_i·{\mathrm{r}}_j\right)$ and $\Sigma {i>j}^{}\left({\mathrm{p}}_i·{\mathrm{p}}_j\right)$ are proportionately larger for three- than for two-electron systems, and (c) the configuration interaction approach probably gives poor estimates of the expectation value of $\Sigma {i>j}^{}{\delta }^3\left({\mathrm{r}}_{\mathrm{ij}}\right)$ even though this value appears to converge as the number of terms in the wave function is increased.