Comparison of TFD and Xα methods for molecules and solids

Abstract

The Thomas–Fermi–Dirac (TFD) and multiple-scattering Xα (MS-Xα) methods are compared, for single atoms and for molecules and solids. Both methods are derived from Hamiltonians by variation of the charge density or spin orbitals, respectively, and the Hamiltonians for the two methods differ only in the kinetic energy term. It is pointed out that the main errors in the TFD method arise from the statistical version of the kinetic energy term. The Xα method, which uses the exact kinetic energy term, can be regarded as a modification of the TFD method which removes all the major errors in that method. The Coulomb energy and the exchange-correlation term are identical in both methods. By the example of the TFD method for the carbon atom we illustrate the various features of both methods, and it is pointed out that many features of the calculation of total energy by the Xα method, as described by the overlapping-sphere approach, can be understood better by studying the same problem from the TFD point of view.