DISPERSION FORMULAS FOR REAL-FREQUENCY-DEPENDENT AND IMAGINARY-FREQUENCY-DEPENDENT HYPERPOLARIZABILITIES

The dynamic second hyperpolarizability for real frequencies, gamma(par allel to)(-omega(sigma);omega(1),omega(2),omega(3)) in the limit omega (i) --> 0 can be expressed as gamma(parallel to)(0) + A(parallel to)(0 ) omega(L)(2), where omega L(2) = omega(sigma)(2) + omega(1)(2) + omeg a(2)(2) + omega(3)(2) and gamma(parallel to)(0) is the frequency-indep endent (static) quantity; the parallel subscript (parallel to) indicat es that the polarization and electric fields all lie along the same ax is. In this paper the coefficient A(parallel to)(0) is evaluated exact ly for the H atom and very accurately for H-, He, and Li+. A similar a nalysis is carried out for gamma(parallel to)(-i omega; i omega, 0, 0) in the limit 0 --> infinity.