The most successful study of unperturbed helium in its normal state has been made by Hylleraas, using the Ritz method. The Schrödinger partial differential equation is reduced to an equivalent problem in the calculus of variations and this problem is then solved by a so-called direct method.

In the present paper an extension is developed to the Ritz method, forming a theory which is applicable to perturbation problems. This theory is applied to a study of the normal helium atom under the influence of an electric field. As in the usual perturbation theory, the characteristic energy parameter $E$ and the characteristic function $\psi$ are assumed developable in power series in the field strength $F$, but now the coefficients in these expansions are given in terms of problems in the calculus of variations. The work of Hylleraas furnishes a knowledge of the terms in both expansions independent of the field strength. Then the minimization of a single integral furnishes values of ${\psi }_1$ and $E_2$, the subscript of each denoting the power of $F$ for which it forms the coefficient. Values of these coefficients are calculated to three approximations by the direct method. From $E_2$ we may calculate the dielectric constant in the usual way giving $\epsilon =1.0000665\mathrm{}$ a value 5 percent below the accepted value of $\epsilon =1.000070\mathrm{}$. One should note that it is a characteristic of the Ritz method that further approximations can only increase the value of $\epsilon$, bringing it into better agreement with the experimental value.

• Received 11 August 1930

DOI:https://doi.org/10.1103/PhysRev.36.1232