High-Order Perturbation Theory for a One-Electron Ion in a Uniform Electric Field

Large-$Z$ expansion theory is applied to the nonrelativistic ground state of a one-electron ion and the subsequent first-order differential equations are solved recursively. For each order of perturbation theory, a general expression for the wave function is found. The substitution of this form into the perturbation-theory equations reduces the problem for the $K$th-order wave function and energy to solving recursively a set of linear algebraic equations of approximately $2K$th order for the coefficients of certain radial polynomial terms appearing in the wave function. The asymptotic-series results for both the energy and the wave function are obtained explicitly through tenth order. For hydrogen a calculation of the ionization field (ignoring tunneling) gives a value of 3.36 × ${10}^8$ V/cm. This agrees well with experimental results on field ionization and a recent accurate variational treatment and is found to be much superior to a WKB result.