Estimation of the Minimizer of the Thomas-Fermi-Dirac-von Weizsäcker Functional of NaCl Crystal Lattice

Abstract

Abstract. Estimation of the minimizer of the Thomas-Fermi-Dirac-von Weizsäcker functional of NaCl crystal lattice is calculated by making use of a direct method we have developed recently. The method is referred to as direct method for the reason that in the course of the calculation of the estimation, we do not derive the Euler-Lagrange equation at all. By using the graph of the umbrella functions, the estimation of the minimizer function is represented. The minimizer is bounded from above by the umbrella functions. IntroductionThe Thomas-Fermi (TF) theory was proposed by Thomas and Fermi independently in 1927. Dirac introduced the exchange correction to the theory in 1930, so that the corrected model is called the Thomas-Fermi-Dirac (TFD). Then the gradient correction of the kinetic energy was added by von Weizsäcker in 1935, so that it is called the Thomas-Fermi-von Weizsäcker (TFW) model. Those theories were placed on a firm mathematical footing by Lieb [1]. Although the Thomas-Fermi-Dirac-von Weizsäcker (TFDW) theory has not been studied as extensively as other theories, some authors studied it, such as Engel and Dreizler [2], who presented an accurate numerical solution of the variational equation resulting from the standard TFDW energy functional plus fourth-order gradient terms of the kinetic energy.Chan et.al.[3] perform a numerical study of the TFDW theory in finite systems to gain an understanding of the variational behavior of kinetic energy functionals. Their findings indicate that the TFDW theory can give an approximate description of matter, with atomic energies, binding energies, and bond lengths of the correct order of magnitude, though not to the accuracy required of a qualitative chemical theory.Zhuravlev et.al.[4] calculate valence electron densities in series of crystal substances MA (M = Li, Na, K, Rb, Ag, Mg, Ca; A = F, Cl, Br, O, S) with a NaCl lattice using the ab initio local DFT method combined with the ab initio pseudopotential technique. They found that systematic variations of valence electron density are depending on the atomic number of anion and cation.