Visualizing the zero order basis of the spectroscopic Hamiltonian

Abstract

Recent works have shown that a generalization of the spectroscopic effective Hamiltonian can describe spectra in surprising regions, such as isomerization barriers. In this work, we seek to explain why the effective Hamiltonian is successful where there was reason to doubt that it would work at all. All spectroscopic Hamiltonians have an underlying abstract zero-order basis (ZOB) which is the "ideal" basis for a given form and parameterization of the Hamiltonian. Without a physical model there is no way to transform this abstract basis into a coordinate representation. To this end, we present a method of obtaining the coordinate space representation of the abstract ZOB of a spectroscopic effective Hamiltonian. This method works equally well for generalized effective Hamiltonians that encompass above-barrier multiwell behavior, and standard effective Hamiltonians for the vicinity of a single potential minimum. Our approach relies on a set of converged eigenfunctions obtained from a variational calculation on a potential surface. By making a one-to-one correspondence between the energy eigenstates of the effective Hamiltonian and those of the coordinate space Hamiltonian, a physical representation of the abstract ZOB is calculated. We find that the ZOB basis naturally adjusts its complexity depending on the underlying nature of phase space, which allows spectroscopic Hamiltonians to succeed for systems sampling multiple stationary points.

Simulation of the Raman spectra of CO2: Bridging the gap between algebraic models and experimental spectra

“…112 In phenomenological algebraic models a test of the quality of the wavefunctions is recommended since a good energy fit does not necessarily guarantee the wavefunctions quality. 19,113 Indeed, the computation of line intensities with the resulting wavefunctions is a sensitive indicator of the quality of the description. This situation will be discussed in detail in Sec.…”

“…2,3,[11][12][13][14][15][16] It has recently been shown how a polyadbreaking effective Hamiltonian is capable of describing the spectrum of systems with a reaction barrier (e.g., dissociation or isomerization systems). [17][18][19] The traditional description of molecular vibrations in terms of normal modes 20 involves harmonic collective nuclear motions, and works reasonably well for molecules without large mass differences in their constituents. However, when such mass differences are present, an approach based on local modes may be more convenient to explain the patterns appearing in the energy spectrum.…”

“…[68][69][70][71][72][73][74][75] Recently, an alternative approach to solve the problem of linking with a correct coordinate space representation in the case of a spectroscopic effective Hamiltonian has been applied to the case of a polyad-breaking Hamiltonian. 19 The polyad pseudo quantum number implies a linear combination of normal mode quantum numbers multiplied by positive weights. 7 Thus, the natural way to define a system's polyad is from a normal mode scheme, and a translation into a local scheme may not be possible.…”

Molecular spectroscopy and dynamics: a polyad-based perspective

“…Here, we thus consider almost exclusively global polyad Hamiltonians, including rotation whenever worked out, usually addressing small to medium-size molecules up to moderate levels of vibrational excitation in their ground electronic state. A number of related topics, including multiple well problems, 156 torsional ladders, 66 highly vibrationally excited van der Waals species, 106,[157][158][159] and isomerization processes, [160][161][162] have received considerable attention in the literature, both from spectroscopic and dynamical points of view, but despite their interest, they will only be mentioned briefly in this review.…”