Linear coupled-cluster method. II. Analysis of local exchange-correlation potentials in beryllium and its isoelectronic series

The Slater $X\alpha$ method and its local-potential modifications are examined with reference to the many-electron exchange-correlation effects in the beryllium atom and its isoelectronic series. The linear coupled-cluster method and a hierarchy of approximations to it are employed for this purpose. The role of the exchange parameter $\alpha$ in providing an accurate description of the exchange-correlation effects is analyzed in the light of the electron-gas model. It is found that for Be atoms an $\alpha$ value of 0.768, that which causes the local potential to mimic the Hartree-Fock potential, is the best suited reference state for many-body calculations. The impact of the.$\alpha$ variation on the exchange-correlation corrections in the Be isoelectronic series is assessed. With increase in the nuclear charge $Z$, exchange-correlation corrections favor the use of $\alpha$ values closer to $\frac{2}{3}$, the Gaspar-Kohn-Sham limit, in the $X\alpha$ model. The instabilities in the cluster equations induced by ringdiagram terms are also noted. The futility of using gradient corrections to the $X\alpha$ model to account for exchange-correlation effects is brought out in the calculations. It is found that a simple scaling of the electron-gas potential results in excellent single-particle reference states for many-body calculations.