We 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 ${\chi }_{\mathrm{zzzz}}(-\omega ;\omega ,\omega ,-\omega )$, and (ii) a third-harmonic coefficient ${\chi }_{\mathrm{zzzz}}(-3\mathrm{}\omega ;\omega ,\omega ,\omega )$. The problem is formulated within the framework of time-dependent Hartree-Fock perturbation theory. The expressions for ${\chi }_{\mathrm{zzzz}}(-\omega ;\omega ,\omega ,-\omega )$ and ${\chi }_{\mathrm{zzzz}}(-3\mathrm{}\omega ;\omega ,\omega ,\omega )$ contain third- and lower-order frequency-dependent wave functions. It is found possible to eliminate the third-order terms from the expressions for the $\chi \text{'}\mathrm{s}$. 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 well 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 $\chi \text{'}\mathrm{s}$ is developed and is shown to be useful for obtaining the $\chi \text{'}\mathrm{s}$ at low frequencies with a substantial saving in computing effort.

• Received 6 July 1971

DOI:https://doi.org/10.1103/PhysRevA.4.1760