Identities, perturbative expansions, and recursion relations for mapping operators, effective Hamiltonians, and effective operators

Hurtubise, Vincent
July 1993
Journal of Chemical Physics;7/1/1993, Vol. 99 Issue 1, p265
Academic Journal
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 l. 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.


Related Articles

  • Nonperturbative calculation of energies and widths of predissociative states of diatomic molecules. Gilbert, Richard D.; Porter, Richard N. // Journal of Chemical Physics;9/1/1988, Vol. 89 Issue 5, p3057 

    A nonperturbative theory of predissociation based upon an effective Hamiltonian is presented. For a model diatomic system in which a (bound-state) Morse potential-energy curve is crossed by an (unbound-state) exponential potential-energy curve, the exact effective Hamiltonian may be obtained in...

  • Weighted Supermembrane Toy Model. Lundholm, Douglas // Letters in Mathematical Physics;May2010, Vol. 92 Issue 2, p125 

    A weighted Hilbert space approach to the study of zero-energy states of supersymmetric matrix models is introduced. Applied to a related but technically simpler model, it is shown that the spectrum of the corresponding weighted Hamiltonian simplifies to become purely discrete for sufficient...

  • Ground states of the massless Derezinski–Gérard model. Ohkubo, Atsushi // Journal of Mathematical Physics;Nov2009, Vol. 50 Issue 11, p113511 

    We consider the massless Derezinski–Gérard model introduced by Derezinski and Gérard in 1999. We give a sufficient condition for the existence of a ground state of the massless Derezinski–Gérard model without the assumption that the Hamiltonian of particles has compact...

  • From Noncommutative Phase Space to Hilbert Space. Muthukumar, B. // AIP Conference Proceedings;10/3/2007, Vol. 939 Issue 1, p359 

    In the Hamiltonian formulation, classical mechanics employs the commutative algebra of functions that are defined on phase space a point of which could be represented using Dirac delta distributions. In the absence of such a concrete existence of the notion of point in the quantum domain, we...

  • Averaging principle for a class of stochastic reaction�diffusion equations. Cerrai, Sandra; Freidlin, Mark // Probability Theory & Related Fields;May2009, Vol. 144 Issue 1/2, p137 

    We consider the averaging principle for stochastic reaction�diffusion equations. Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, we calculate limiting slow motion. The study of solvability of Kolmogorov equations in...

  • Diagrammatic complete active space perturbation theory. Finley, James P. // Journal of Chemical Physics;1/15/1998, Vol. 108 Issue 3, p1081 

    Formulates a diagrammatic complete active space second-order perturbation theory. Basis of the theory formulation; Importance of wave operators in all Hilbert space methods; Limitation of the multireference perturbation theory; Computation of the final energy in the same manner as with state...

  • Quasilevels of a two-particle Schr�dinger operator with a perturbed periodic potential. Chuburin, Yu. // Theoretical & Mathematical Physics;Jan2009, Vol. 158 Issue 1, p96 

    We consider a two-dimensional periodic Schr�dinger operator perturbed by the interaction potential of two one-dimensional particles. We prove that quasilevels (i.e., eigenvalues or resonances ) of the given operator exist for a fixed quasimomentum and a small perturbation near the band...

  • Perturbative approaches to highly excited molecular vibrations of H2O, D2O, and HDO. McCoy, Anne B.; Silbert, Edwin L. // Journal of Chemical Physics;2/1/1990, Vol. 92 Issue 3, p1893 

    Molecular vibrations of water are studied using Van Vleck perturbation theory. In these calculations, the OH stretches are expressed in terms of the Morse coordinate, yi =[1-exp(-αri)]/α, and its conjugate momentum, while the bend is treated in a traditional manner. Nearly degenerate...

  • Effects of symmetry-breaking perturbations on excitonic states bound to systems of reduced symmetry. Francoeur, S.; Marcet, S. // Journal of Applied Physics;Sep2010, Vol. 108 Issue 4, p043710 

    Using an invariant expansion, we build an Hamiltonian describing the influence of the crystal-field, the electron-hole exchange interaction, and any symmetry-breaking perturbations on the fine structure of excitons bound to systems of reduced symmetry: D2d, C3v, and C2v. Several perturbations...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics