Canonical Potentials and Spectra within the Born–Oppenheimer Approximation

Abstract

A generalized formulation of canonical transformations and spectra are used to investigate the concept of a canonical potential strictly within the Born-Oppenheimer approximation. Data for the most accurate available ground electronic state pairwise intermolecular potentials in H2, HD, D2, HeH(+), and LiH are used to rigorously evaluate such transformations. The corresponding potentials are generated explicitly using parameters calculated with algebraic functions from that of the single canonical potential of the simplest molecule, H2(+). The efficacy of this approach is further tested by direct comparison of the predicted eigenvalues of all vibrational states in the selected molecular systems considered with the corresponding most accurately known Born-Oppenheimer eigenvalues currently available. Deviations are demonstrated to be less than 2 cm(-1) for all vibrational states in H2, HD, D2, HeH(+), and LiH, with an average standard deviation of 0.27 cm(-1) for the 87 states considered. The implications of these results for molecular quantum chemistry are discussed.

A canonical approach to forces in molecules

“…Correspondingly, for R < R e , the sequence R rj , j = 0,1, …, is defined by F(R rj ) = F m 2 j where R r0 is the separation distance at which the repulsive force equals the magnitude of the largest attractive force, etc. It was observed previously [12,14] that for given j, sections of the potential curves for two different molecules in the considered classes defined by R a(j-1) < R < R aj or R rj < R < R r(j-1) are canonical.…”

“…(11) and (12). A prescribed degree of accuracy across the class of molecules is also required for the dimensionless force to be considered canonical.…”

“…In constructing canonical representations of potentials for diatomic molecules [12,14], we developed an approach based upon a method for decomposing a 1-dimensional potential curve into a finite numbers of canonical sections that have the same scale invariant "shape" across a broad class of molecules. The notion of scale invariant shape utilized in this approach asserts that each designated section of the potential curve for one molecule has a unique counterpart in another molecule for which there exits an affine transformation to a single dimensionless curve.…”

“…For example, it was shown in Ref. [12] that by appealing to both the piecewise affine transformation to canonical form and its inverse, one can construct a representation of the potential for any molecule in the considered class in terms of the potential for one of the molecules chosen as reference. In Ref.…”

“…These investigations explored the concept of a canonical potential strictly within the Born-Oppenheimer approximation for the most accurate available ground electronic state pairwise intermolecular potentials in H 2 , HeH + , and LiH [12]. Explicit canonically-based transformations have also been developed for transformation to a canonical potential for both these diatomics as well as two body intermolecular interactions [13,14].…”