IDENTITIES, PERTURBATIVE EXPANSIONS, AND RECURSION-RELATIONS FOR MAPPING OPERATORS, EFFECTIVE-HAMILTONIANS, AND EFFECTIVE OPERATORS

We derive perturbation expansions for the mapping operators (k,l) that transform a full Hilbert space time-independent Hamiltonian H and operator A into, respectively, a finite (multidimensional) space effective Hamiltonian h and effective operator a. The eigenvalues of h are identical to some of those of H, and a produces exact matrix elements of A for the corresponding states. Our derivations are substantially both more general and simpler than most literature ones and yield simple linear recursive expressions for k and 1. Both these mapping solutions and new identities involving h, a, k, and l straightforwardly produce new recursive relations for h and the first known recursive a expressions. We apply our results to the Bloch, Kato, and all norm-preserving formalisms, including the canonical one. The new h and a identities are also shown to be suitable for iterative and multireference coupled cluster approaches.