Fermi Energy of Metallic Lithium

A boundary condition method is developed for deriving the coefficient E₂₍ in the power series expansion of the energy of an electron of wave number k moving in the lattice of an alkali metal. (The entire calculation proceeds within the framework of the Wigner-Seitz atomic sphere approximation. ) If the electron wave function is expanded as ₊ (r) =e^{i^k} (u₀+u₁k+u₂k^2+) it is shown that the boundary condition () (s partof u₂₍) r=rₒ=0 leads naturally to an evaluation of E₂₍ in terms of values at rₒ of homogeneous solutions of the Schr\"odinger equation and their derivatives with respect to energy and radius. In this way, a simple expression for E₄ is obtained analogous to that derived by Bardeen for E₂. For the case of metallic lithium, this expression leads to the value E₄=-0. 031, which agrees with that obtained by the more tedious method of evaluating the expectation value of the Hamiltonian using a wave function correct to the second order in k.