The development of a method for calculating the frequency-dependent second harmonic generation coefficient of insulators and semiconductors based on the self-consistent linearized muffin-tin orbitals band structure method is reported. The calculations are at the independent particle level and are based on the formulation introduced by Aversa and Sipe [Phys. Rev. B 52, 14 636 (1995)]. The terms are rearranged in such a way as to exhibit explicitly all required symmetries including the Kleinman symmetry in the static limit. Computational details and convergence tests are presented. The calculated frequency-dependent ${\ensuremath{\chi}}^{(2)}(\ensuremath{-}2\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega})$ for the zinc-blende materials GaAs, GaP and wurtzite GaN and AlN are found to be in excellent agreement with that obtained by other first-principles calculations when corrections to the local density approximation are implemented in the same manner, namely, using the ``scissors'' approach. Similar agreement is found for the static values of ${\ensuremath{\chi}}^{(2)}$ for zinc-blende GaN, AlN, BN, and SiC. The strict validity of the usual ``scissors'' operator implementation is, however, questioned. We show that better agreement with experiment is obtained when the corrections to the low-lying conduction bands are applied at the level of the Hamiltonian, which guarantees that eigenvectors are consistent with the eigenvalues. Results are presented for the frequency-dependent ${\ensuremath{\chi}}^{(2)}(\ensuremath{-}2\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega})$ for $3C\ensuremath{-}\mathrm{S}\mathrm{i}\mathrm{C}$. The approach is found to be very efficient and flexible, which indicates that it will be useful for a wide variety of material systems including those with many atoms in the unit cell.