Linear coupled-cluster method. I. Exchange-correlation effects in atoms

The linear coupled-cluster method and a hierarchy of its approximations are employed to investigate the Slater $X\alpha$ method and to study the exchange-correlation effects in the beryllium atom. An analysis of the various contributions to the exchange-correlation corrections to the total energy is made using the approximations to the coupled-cluster model. The total dominance of the valenceshell exchange-correlation corrections, the role of the single-particle excitation terms, and the relative importance of the ring- and ladder-diagram terms are gleaned from these calculations. The exchange parameter $\alpha$, for the calculation of the reference state, i.e., the $X\alpha$ orbitals, is chosen using the criterion that the total energy computed by the $X\alpha$ method be equal to the Hartree-Fock total energy for the atom. For this reference state, the total energy of the beryllium atom, with the inclusion of the exchange-correlation correction, is found to be - 14.663 598 hartrees, a value very close to the experimental energy of - 14.669 hartress. The $X\alpha$ method is seen to provide good single-particle reference states for many-body calculations.