Atomic Force Constants of Copper from Feynman's Theorem

In the Born-Oppenheimer approximation atomic force constants determine the elastic and vibrational behavior of crystals for small nuclear displacements. In this paper the first- through third-neighbor force constants in the copper crystal are found as the components of the change in force produced on the nuclei when in equilibrium positions by the infinitesimal unit displacements of one nucleus. By Feynman's theorem quantum-mechanical forces can be calculated directly from the electronic and nuclear charge distribution by Coulomb's law. The Slater-Koster formalism developed for localized perturbations in crystals is used to find the change in conduction electron charge density resulting from the infinitesimal unit displacement of a nucleus. Free-electron wave functions are used, and most of the perturbation energy matrix elements are neglected. Calculations are made for small crystals with up to 2048 atoms and two different shapes (Bornvon Kármán boundary conditions): First-order perturbation theory gives exact results for atomic force constants. An approximate Thomas-Fermi calculation is also carried out for the perturbation in conduction electron density and simple approximations of this perturbation are discussed. The ion cores are assumed to move nearly rigidly, and their closed-shell repulsion is chosen so that the calculated atomic force constants lead to the values of the elastic constants found experimentally. The nine calculated atomic force constants are quite different from the values Jacobsen inferred from thermal diffuse x-ray scattering from a copper crystal.