The interaction between a laser field and a crystalline solid is considered. The electron wave function in the crystal is obtained from an appropriately chosen pseudopotential. The dipole approximation is introduced, and only vertical transitions are taken into account. We present a nonperturbative method based on the numerical computation of solutions of the Schr\"odinger equation, to obtain the generation of harmonics of the incident laser. Particular numerical results are presented for the case of pure silicon. The mechanisms to enhance harmonic generation without excessive excitation of electrons to the conduction band are analyzed.