Hyperpolarizabilities ofH−, He, andLi+

The Ritz variational approximation is used to calculate the energy of the ground state of He, ${\mathrm{Li}}^{+}$, and ${\mathrm{H}}^{-}$ in the presence of a uniform external electric field. The interaction energies are expressed as power series in the electric-field intensity. From the coefficients of these expansions, the polarizabilities $\alpha$ and the hyperpolarizabilities $\gamma$ are obtained. In this study the hyperpolarizabilities are computed with Hylleraas-type wave functions consisting of 78, 96, 102, and 150 terms. The convergence of $\alpha$ and $\gamma$ are satisfactory for He and ${\mathrm{Li}}^{+}$, but not for ${\mathrm{H}}^{-}$. For ${\mathrm{H}}^{-}$ it is suggested that in order to obtain proper convergence one would need to include more terms in the wave function than would be tractable for our computer. Thus, a different type of function should be used for this ion. The 150-term wave functions gave 169, 1.383, and 0.1925 atomic units (a.u.) for the polarizabilities of ${\mathrm{H}}^{-}$, He, and ${\mathrm{Li}}^{+}$, respectively. This 150-term function also gave 1.74 × ${10}^7$, 42.8, and 0.244 a.u. for the hyperpolarizabilities of these same atoms. It is suggested from a study of the convergence of $\alpha$ and $\gamma$ as more terms are included in the wave function, of the accuracy of the computed $\alpha \text{'}\mathrm{s}$, and of the free atom energies that the computed $\gamma$ results are correct to within a few percent or better for He and ${\mathrm{Li}}^{+}$, but that the $\gamma$ computed for ${\mathrm{H}}^{-}$ is unreliable. The only available measurement of $\gamma$, for these ions, gave 51.6±7.9 a.u. for the helium atom.