Comment on “breakdown of the hellmann-feynman theorem: Degeneracy is the key”

Abstract

In a recent paper, Zhang and George [Phys. Rev. B 66, 033110 (2002)] have made an application of the Hellmann-Feynman theorem (HFT) to calculate forces acting on an atom of the buckyball C60 when subjected to small displacement. They noticed inconsistent results (as compared with that by other method) for forces connected with states belonging to a degenerate-energy level. However a sum of all such HFT forces for a given level was found consistent. Therefore they proposed the average force to be the representative of all forces connected with a degenerate level. Our analysis of these difficulties, based on degenerate perturbation theory, leads to modification of the HFT in application to degenerate-level states. By solving a matrix eigenequation of the extended HFT, as many forces are obtained as degeneracy of the level amounts. The diagonalized matrix involves unperturbed states only. Our considerations are illustrated by a simple system-a particle moving in a harmonic confinement, where forces are connected with anisotropic deformation of an isotropic confinement. © 2004 The American Physical Society.