Atomic Many-Body Problem. III. The Calculation of Hylleraas-Type Correlated Wave Functions for the Beryllium Atom

By applying the theory of correlated wave functions, Hylleraas-type functions are computed for the beryllium atom. The theory is applied in a form in which only two-electron correlations are considered. Several different correlation factors as well as several different combinations of orbital functions are considered. It is shown that the Hylleraas method gives better results for the energy than the superposition of configurations in the $1s$ shell as well as in the $2s$ shell. The problem of the additivity of the correlation energy in pairs is analyzed and it is shown that the sum of pair-correlation energies is different from the correlation energy computed taking into account correlation in both shells simultaneously. The difference can be positive or negative depending on the wave functions, and it is not negligible if accurate values are required for the correlation energy. Comparing the computed values for the correlation energy resulting from two-electron correlations with the experimental value of the correlation energy, the size of the many-electron (more than two-electron) correlation effects is estimated. It is shown that these effects can not be neglected if accurate values are required for the correlation energy.