A boundary condition method is developed for deriving the coefficient ${E}_{2n}$ 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 ${\ensuremath{\psi}}_{k}(\mathbf{r})={e}^{{i}^{\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{r}}}({u}_{0}+{u}_{1}k+{u}_{2}{k}^{2}+\ensuremath{\cdots})$ it is shown that the boundary condition $[(\frac{\ensuremath{\partial}}{\ensuremath{\partial}r})(s \mathrm{part}\mathrm{of} {u}_{2n})]r={r}_{s}=0$ leads naturally to an evaluation of ${E}_{2n}$ in terms of values at ${r}_{s}$ of homogeneous solutions of the Schr\"odinger equation and their derivatives with respect to energy and radius. In this way, a simple expression for ${E}_{4}$ is obtained analogous to that derived by Bardeen for ${E}_{2}$. For the case of metallic lithium, this expression leads to the value ${E}_{4}=\ensuremath{-}0.031$, 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 $k$.