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e consider the lowest-order nonlinear contributions to the electric dipole moment induced in closed-shell atomic systems by intense electromagnetic fields. These contributions are comprised of two terms: (i) an intensity-dependent refractive-index coefficient X««(-(d;~,~, -~) , and (ii) athird-harmoniccoefficientx~»(-3~; cu,~, cu). Theproblem is formulated within the framework of time-dependent Hartree-Fock perturbation theory. The expressions forx««(-&; (d, (d, -~) and X«(-3+; cu,~,~) contain thirdand lower-order frequency-dependent wave functions. It is found possible to eliminate the third-order terms from the expressions for the X's. A variational method for solving the required second-order integrodifferential equations is proposed. Numerical results for helium are obtained. A zero-frequency Hartree-Fock hyperpolarizability for helium of 35.8 a. u. is obtained which agrees reasonably weil with the previous Hartree-Fock calculations of Sitter and Hurst. The Hartree-Fock wave functions give static hyperpolarizability results which are about 17% smaller than more accurate calculations. A "static second-order function" approximation for calculating the X's is developed and is shown to be useful for obtaining the y's at low frequencies with a substantial saving in computing effort.

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