Electric dipole hyperpolarizabilities are calculated for a large number of 2- to 20-electron $S$-state atoms and ions. The calculations are carried out within the framework of a Hartree-Fock perturbation procedure. This approximation is developed to high order and it is shown explicitly that the total Hartree-Fock energy is obtained to fifth-order from a knowledge of the Fock orbitals to second-order. The first- and second-order coupled Hartree-Fock orbital equations are "uncoupled" in a direct manner and the resulting approximate equations solved by minimizing associated one-electron functionals. The resulting hyperpolarizabilities for the He, Ne, and Ar series are all positive. The calculated values for the inert gases agree with experiment within a factor of two. Some of the high $Z$ hyperpolarizabilities for the Na, Mg, and K series are negative, while for the Li and Be series the values are large and positive. These interesting signs are discussed. Finally, it is shown that the hyperpolarizabilities are quite sensitive to the choice of zeroth-order Hartree-Fock function used in the calculation.

• Received 18 October 1965

DOI:https://doi.org/10.1103/PhysRev.148.18