THE CALCULATION OF THE ELECTRON–ION MATRIX ELEMENT BY THE SPHERICAL CELLULAR METHOD: Na

An accurate method is developed for the calculation of the "Schrodinger" part (in the ilotation of Vosko et al. 1965) of the electron–ion matrix element in the Wigner–Seitz spherical cell model. The rigid-ion model is used and the potential in the zero-order Hamiltonian is the sum of the free-ion potentials and the Hartree field of the conduction electrons. The Bloch wave function in a single cell is expanded in spherical waves and the matrix element is expressed as the product of an angular factor and an integral involvirlg the radial parts of the wave function, which for energy-conserving transitions may be expressed in terms of their values at the cell boundary. The radial parts of the wave functions are obtained by two methods: (1) the Kohn (1954) variational method and (2) A power series in k (Brooks 1958). The methods are applied to Na and converge to the same values, though the former converges considerably faster. The results indicate that the single orthogonalized plane-wave method gives a good representation of this matrix element in Na, while the Bardeen (1937) approximation does not, as is also demonstrated by explicit evaluation of the higher-order (in k) contributions neglected by Bardeen. The effective electron–ion matrix element is also calculated. In the course of the calculations a new theoretical value of the Fermi electron density at the nucleus (P F = 0.439a 0 −3 ) is obtained which is in closer agreement with recent experiments than previous theoretical estimates.