A Numerical Variational Method

A procedure applicable to the solution of linear partial differential equations of second order is described with special reference to an application to the calculation of the wave function for the $1s2s$, ${}_{}{}^3{}_{}{}^{}S$ state of Li II. The procedure has features in common with Southwell's relaxation method but differs from it by being formulated as a special case of the Ritz variational method. The energy parameter for the ground state of a term system which is obtained in this manner is always higher than the true one, a circumstance having a well-known convenience in the comparison of calculation with experiment. For comparable accuracy of the wave function the numerical work is somewhat larger, however, than in the relaxation method. The errors caused by the finiteness of the number of numerical operations are studied in simple cases, and they are compared with those of Hartree's procedure for integrating ordinary differential equations of second order. In the application to Li II only a correction to the wave function is calculated numerically; the modifications of the general method required in this case are discussed.