The algebra of effective Hamiltonians and operators: Truncated operators and computational aspects

Abstract

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.

Gustavson’s procedure and the dynamics of highly excited vibrational states

“…Actually, CVPT and LOPT are closely related, and classical LOPT's, 35,[37][38][39][40][41][42][43] specially Dragt-Finn perturbation theory, 35,[39][40][41] mostly involve replacing quantum commutators by classical Poisson brackets in quantum CVPT. [44][45][46][47][48][49] Also, a new method, 50 which can be viewed as a specific case of a more general work 51,52 and is called mixed diagonalization, has recently been introduced, in which an effective Hamiltonian operator acting on a reduced dimensional space is constructed using the same similarity transformations of CVPT. Detailed descriptions of these methods will not be presented here, the interested reader being referred to the articles listed above, but it is noted that strong similarities also exist between LOPT or CVPT and BGPT: indeed, the null space ⌫ (s) in BGPT plays the same role as the transformed Hamiltonian K n (n) (nϭsϪ2) in CVPT, and the remainder R (s) as the operator S (n) associated with the unitary transformation exp(i n ͓S (n) ,͔) ͑the notations here are those of Sibert 48,49 ͒.…”

Calculation of single-beam two-photon absorption rate of lanthanides: Effective operator method and perturbative expansion

“…These techniques have been extensively developed in the literature. 25,26,27,28,29,30 The basic idea is to transform the exact time-independent (usually many-body) Hamiltonian H into an effective Hamiltonian H eff acting on a restricted model space of manageable dimension. The exact eigenvalues and model space eigenvectors (not the exact eigenvectors) can be obtained by diagonalizing H eff .…”

Non-Hermitian perturbative effective operators: Connectivity and derivation of diagrammatic representation

“…For a general operator O, there are also many choices of O eff , which have been summarized by Hurtubise and Freed. Hurtubise and Freed 15,16 have stated that, for a general operator O, the Hermitian effective operator O eff,C is the only known connected effective operator. We shall refer to this property as ''connected.''…”