Generalized Huzinaga building-block equations for nonorthogonal electronic groups: Relation to the Adams–Gilbert theory

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Abstract
We show in this article that the Huzinaga building-block equations for many-electron systems, derived under the hypothesis of strong orthogonality among the system groups, can be also deduced as a particular case of the Adams–Gilbert formalism, in which intergroup overlapping is recognized from the outset. This viewpoint is more realistic because the strong orthogonality cannot be fully achieved in most practical cases due to basis-set truncation. Incomplete orthogonality leads to nonzero expectation values of the projection operators appearing in the Fock Hamiltonian (projection energy). This fact raises some conceptual problems since the projection energy has been used to interpret several important physical effects in recent investigations. The present work clarifies several practical aspects relative to the incomplete fulfillment of the strong-orthogonality hypothesis, in particular (a) the analytical relationship between the projection energy and the intergroup overlap energy involving weakly overlapping groups; (b) the required values for the projection constants in cases of incomplete orthogonality; and (c) how the effects of basis-set truncation can be analyzed unambiguously.