Some bilinear convergence characteristics of the solutions of dissymmetric secular equations

Abstract

If an approximate eigenfunction is obtained by the solution of a dissymmetric set of secular equations formed by the use of different expansion functions to precede and follow the operator, it is shown here that the error in the eigenvalue is proportional to $\mu^\dagger\mu$ where $\mu^\dagger$ and $\mu$ are measures of the amounts by which the pre- and post-expansion functions are not able to fit the adjoint and direct eigenfunctions of the operator. This replaces the $\mu^2$ error in Rayleigh-Ritz variation theory. This result is of considerable value for the circumstances where the use of different sets makes the integrals evaluable for specially desirable post-expansion functions. The introduction of direct electronic correlation into wavefunctions is a case where the integrals can be evaluated with different sets of functions but not with the same set. Further, these results show how a particular use of numerical integration gives eigenvalues with errors of lower order than those associated with the same integration procedure in normal integrals.