General formulations of finite-field method classified by symmetry for molecular linear and nonlinear polarizabilities

The formulations of the finite-field approach to calculate the linear and n on-linear optical coefficients mu (i), alpha (ij), beta (ijk) and gamma (ij kl) of a molecular system with different symmetries have been deduced and s ummarized. The possible choices of the energy sets of the 48 frequent point groups have been optimized and categorized into 11 classes. With the restr iction of symmetry operators, a minimum of 9, no more than 21 energy points have to be calculated in order to determine the coefficients, except in th e case of the first class to which C-1 point group belongs and in which the 34 non-relative energy points selected in our uniform and general scheme a re all needed. The symmetric operators that cause some of the tensor compon ents to vanish have been demonstrated as well.