Generating functions for effective hamiltonians. Symmetry, topology, combinatorics

Abstract

Recent developments associated with old technique of generating functions and invariant theory which I have started to apply to molecular problems due to my collaboration with Yu.F. Smirnov about 25 years ago are discussed. Three aspects are presented: the construction of diagonal in polyad quantum number effective resonant vibrational Hamiltonians using the symmetrized Hadamard product; the decomposition of the number of state generating function into regular and oscillatory contributions and its relation with Todd polynomials and topological characterization of energy bands; qualitative aspects of resonant oscillators and fractional monodromy as one of new generalizations of Hamiltonian monodromy.

Molecular spectroscopy and dynamics: a polyad-based perspective

“…Consequently, the initial physical meaning of the parameters is lost, or at least blurred. As briefly highlighted in the Introduction, perturbation and ab initio or variational theories can help to decode the effective parameters in terms of the original ones, 83,[163][164][165][166][167][168] as illustrated by the determination of the acetylene geometry in ref. 169.…”

“…Note that in addition to the quantum mechanical treatments not included in this review (see warning in the Introduction), there are also in the literature other, or complementary, successful approaches to global polyad Hamiltonians that are not further reviewed here. Some are based on algebraic methods; [43][44][45][46]52,225 others employ generating functions 83,226 or tensorial [227][228][229] formalisms.…”