Determination of the size-consistency error in the single and double excitation configuration interaction model

The ''size-consistency'' or ''unlinked-diagram'' problem in a configuration interaction calculation employing all double excitations (DCI) from a Hartree-Fock determinant is investigated numerically. By using many-body perturbation theory and the ''linked-cluster'' expansion, the contribution of all double-excitation diagrams in all orders is obtained for the correlation energy of H2O without approximation. Comparison with DCI gives a value of 14 mhartree (9 kcal/mol) for the unlinked, pathological terms remaining in the DCI, emphasizing the importance of quantum chemical methods built upon the linked-cluster expansion for larger molecules. A formula proposed by Davidson to estimate these terms is derived from a consideration of the primary ''unlinked'' diagrams remaining in the DCI. The estimates of the corrections given by Davidson's formula and one suggested by Pople are compared with the exact value. The agreement is within 0.3 mhartree for Davidson's formula and 1.6 mhartree for that of Pople. Another value for the correction is given by the fourth-order MBPT result for double-excitation diagrams, which is 1.2 mhartree higher than the infinite order result