CALCULATION OF THE COHESIVE ENERGY OF METALS BY THE FERMI-THOMAS METHOD

Abstract

Slater and Krutter have attempted to solve the problem of cohesive energy of metals by using the Fermi-Thomas method. They treated the valence electrons and the electrons in the ion core on the same footing. In a way similar to Seitz's cellular method, a sphere with a volume equal to that of a unit cell was drawn and periodic boundary conditions were applied on the surface. Furthermore another boundary condition was imposed in their calculation, namely, on the ground that the sphere as a whole is electrically neutral, the potential energy of an electron on the surface should be equal to zero. Then it was expected that there existed a minimum of the total energy versus the interatomic distance. However, actual computation gave no such a minimum at all. This discrepancy was ascribed to the neglect of the correlation energy of the electrons.It now appears to the present author that the failure to find a minimum is the result of an incorrect boundary condition imposed by the previous authors. In view of the fact that each electron is only acted by all the other electrons except the one being considered, obviously an electron on the surface on the average receives a potential energy-e2/R. Using this boundary condition we can calculate the total energy by numerical integration of the equation, in which the data of Feynman, Metropolis and Teller are adopted. A minimum of the energy versus the interatomic distances is found. The calculated interatomic distances for the minimum for various elements together with the observed ones are tabulated and compared. For the multivalent heavy elements the agreement is remarkably good. For the monovalent light elements, e.g. for the alkali metals, the agreement is not so satisfactory, giving only an order of magnitude. This, however, is expected from the statistical nature of the present method, which certainly is not adquate for the description of the single valence electrons, e.g. those in the alkali metals.For the cohesive energy of the metals, the total energy of the system is subtracted by the energy of the free atoms. We use the Fermi-Thomas model for the free atoms also. In view of the fact that for the free atoms the Fermi-Thomas model is accurate only within 1%, the result of the subtraction would introduce an error which is about hundred times that of the total cohesive energy. Hence the absolute value of the cohesive energy up to the present stage is not reliable. The author believes that this method can be extended to the calculation of the energy of formation of alloys, where the total energy of a free atom is not necessarily known. It is hoped that this programm will be carried out in the near future.