  Implicit summation is a technique for the conversion of sums over
intermediate states in multiphoton absorption and the high-order susceptibility
in hydrogen into simple integrals. Here, we derive the equivalent technique for
hydrogenic impurities in multi-valley semiconductors. While the absorption has
useful applications, it is primarily a loss process; conversely, the non-linear
susceptibility is a crucial parameter for active photonic devices. For Si:P, we
predict the hyperpolarizability ranges from $\chi^{(3)}/n_{\text{3D}}=2.9 $ to
$580 \times 10^{-38}$ $\text{m}^5/\text{V}^2$ depending on the frequency, even
while avoiding resonance. Using samples of a reasonable density,
$n_{\text{3D}}$, and thickness, $L$, to produce third-harmonic generation at 9
THz, a frequency that is difficult to produce with existing solid-state
sources, we predict that $\chi^{(3)}$ should exceed that of bulk InSb and
$\chi^{(3)}L$ should exceed that of graphene and resonantly enhanced quantum
wells.
