GENERAL CRITERIA FOR ASSESSING THE ACCURACY OF APPROXIMATE WAVE-FUNCTIONS AND THEIR DENSITIES

By means of examples, Lowdin showed that L(2) convergence of approximate wave functions psi(n) to the exact psi using the single limit lim(n-->infinity)[psi(n), A psi(n)] = [psi, A psi] is not sufficient to compute accurate expectation values. It is shown that L(2) convergence is indeed a sufficient condition to compute accurate expectation values using iterated limits lim(m-->x) lim(n-->x) [psi(n), A psi(m)] = [psi, A psi] instead of a single limit. Practical conditions that guarantee the stability of single-limit calculations are given. It is also shown that the L(2) convergence of wave functions implies the convergence in the L(1)(R(3))-norm of their corresponding densities. This permits us to prove Weinhold's conjecture that the rate of convergence of densities are greater than that of wave functions. The results are extended to the momentum space, and their equivalence with those of position space is shown. Properties of L(p) spaces are used to introduce the Cauchy criterion that permits us to check the convergence in norm of approximate wave functions and their densities, as well as to estimate exact errors. This is illustrated by a numerical example. (C) 1995 John Wiley and Sons, Inc.