In an earlier paper, I, the authors presented a theory of the low-energy tail of an electron band in a semiconductor in the presence of a high density of impurities. This theory was based on the approximation that all the local potential fluctuations binding states of a given energy have the same shape. We now discuss corrections to the density of states $\rho \mathrm{}\left(E\right)\mathrm{}$ and the spectral density $A(\mathbf{k},E)$ by treating the difference between the actual random potential and the average well shape as a small perturbation. An average higher-order energy correction to the density of states is proposed, which displaces all energies by the average difference between the perturbed and unperturbed energies, and the calculation of this correction is discussed for the limiting case of a random potential obeying Gaussian statistics. We also discuss the complete second-order correction proposed by Zittartz and Langer, which leads to the exact asymptotic form of $\rho \mathrm{}\left(E\right)\mathrm{}$ in a Gaussian potential, and we show how the formulas can be extended to impurity potentials of nonzero range. For three-dimensional Gaussian models, the approximate density of states of our earlier paper had the form $\rho \mathrm{}\left(E\right)\mathrm{}\sim {\xi }^{-2\mathrm{}}A\left(E\right)\mathrm{} \exp \left[\frac{-B\left(E\right)\mathrm{}}{\left(2\mathrm{}\xi \right)}\right]$, where $\xi$ is a parameter proportional to the density of impurities and to the square of the strength of the individual impurity potentials. The corrections described in the present paper modify the function $A\left(E\right)\mathrm{}$, but not $B\left(E\right)\mathrm{}$. We also discuss briefly corrections to the momentum dependence of our earlier approximation to the spectral density $A(\mathbf{k},E)$.

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