No abstract

Effective Hamiltonians and effective operators produce, respectively, exact energies and matrix elements of a time-independent operator A for a finite number of eigenstates of a time-independent Hamiltonian H. We obtain degenerate and quasidegenerate perturbative expressions for the particularly useful canonical effective operator ÂC through second order in perturbation theory. The corresponding ÂC diagrammatic expressions are derived for the case where ÂC acts in a complete finite space. Our first order results have been used previously for ab initio computations of dipole and transition dipole moments in diatomic hydrides and for testing the assumptions in semiempirical methods for dipole properties. A brief discussion is also provided on the computational labors required by first and second order ÂC many-body calculations, the derivation of ÂC diagrams when ÂC acts in an incomplete finite space, and on the derivation of diagrammatic rules for ÂC in arbitrary perturbation order.

We extend to finite orders of perturbation theory our previous analysis of effective Hamiltonians h and effective operators a which produce, respectively, exact energies and matrix elements of a time-independent operator A for a finite number of eigenstates of a time-independent Hamiltonian H. The validity of various properties is examined here for perturbatively truncated h and a, particularly, the preservation upon transformation to effective operators of commutation relations involving H and/or constants of the motion, of symmetries, and of the equivalence between dipole length and velocity transition moments. We compare formal and computational features of all a definitions and of the more limited Hellmann–Feynman theorem based ‘‘effective operators,’’ which provide only diagonal matrix elements of A in special cases. Norm-preserving transformations to effective operators are found to yield a simpler effective operator formalism from both formal and computational viewpoints.

Perturbative and complete model space linked diagrammatic expansions for the canonical effective operatorWe 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 I. Both these mapping solutions and new identities involving h, a, k, and I 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 muItireference coupled cluster approaches.which is equivalent to the canonical case of Eq. (4.5), Le., KeHe = HKe . Mukherjee and co-workers also apply special versions of Eq. (4.5) to open-shell CC formalisms.44-46