Bulk Moduli of Simple Metals Calculated from the Method of Homogeneous Deformations and the Method of Long Waves

It is customary in the model-potential theory of simple metals to solve the Schrödinger equation for the conduction electrons by perturbation theory and to retain only terms to second order in the lattice structure factor. This method of solution leads to a simple expression for the total energy of a metal. Two values of the bulk modulus of the metal may be deduced from this expression: one ($B_D$) by direct differentation with respect to volume and the other ($B_{\mathrm{LW}}$) by an application of the method of long waves. These two approximations to the bulk modulus are not equal in a second-order theory. However, equality may be obtained, for local electron-ion potentials, if certain terms of third and fourth order are retained in the perturbation solution of the Schrödinger equation. The magnitude of these terms has been estimated for several potentials and metals and has often been found to be large. Results are also extremely sensitive to the potential used. In this paper, it is argued that the variation among existing results implies that the full nonlocality and energy dependence of the electron-ion potential must be retained if reliable estimates of ($B_D-B_{\mathrm{LW}}$) are to be made. Consistent and complete expressions for the total energy of a metal in terms of an optimized form of the Heine-Abarenkov model potential are presented and it is shown that these expressions give an adequate account of the cohesive energies and lattice parameters of several simple metals. From the expression for total energy, formulas for $B_D$ and $B_{\mathrm{LW}}$ are obtained which are qualitatively different from those of a local approximation. The nonlocal and energy-dependent contributions are found to be numerically important in calculations of $B_D$ and $B_{\mathrm{LW}}$ for three simple metals. The expressions obtained imply small differences between $B_D$ and $B_{\mathrm{LW}}$ for these metals, when one of the parameters of the theory is suitably adjusted. This parameter, which represents the spatial distribution of the depletion charge, is not obtainable within the framework of conventional model-potential theory. Thus the calculations suggest that further work is necessary before a complete statement about the influence of higher-order perturbation terms on ($B_D-B_{\mathrm{LW}}$) can be made.