Comment on “Breakdown of the Hellmann-Feynman theorem: Degeneracy is the key”

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 ${\mathrm{C}}_{60}$ 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.