Nonlinear local-field corrections to the optical second-harmonic susceptibility of insulating crystals

There are two kinds of local-field corrections to the optical second-harmonic susceptibility in insulating crystals: those linear and those nonlinear In the macroscopic electric field originating from the linearly and nonlinearly induced microscopic charge densities, respectively. An algebraic relation is established between these two local-field corrections, which obviates the need to calculate the nonlinearly induced density. There is a hierarchy of local-field corrections consisting of first-, second-, and third-order corrections containing one, two, and three matrix elements of the linear local field, respectively. Our calculations show that the first-order local-field correction gives the leading correction. We demonstrate that the first-order correction from the previously neglected nonlinear local fields is exactly one half of the linear-local-field correction in the static limit. The newly computed total local-field corrections range from -21% to +30% for the 15 semiconductors and insulators studied. The expression recently obtained for the second-harmonic susceptibility using the (2n+1) theorem [A. Dal Corso et al., Phys. Rev. B 53, 15 638 (1996)] is shown to be equivalent to the expression we obtained using a sum-over-states method.