Rates of convergence and error estimation formulas for the Rayleigh–Ritz variational method

A general theory of rates of convergence for the Rayleigh–Ritz variational method is given for the ground states of atoms and molecules. The theory shows what functions should be added to the basis set to improve the rate of convergence, and gives explicit formulas for estimating corrections to variational energies and wave functions. An application of this general theory to a CI calculation on the ground state of the helium atom yields an explicit large L asymptotic formula for the ‘‘L’’-limit energies EL. The increments are found to obey the formula EL−EL−1 =−3C1(L+ 1/2 )−4−4C2(L+ 1/2 )−5+O(L−6), where the constants C1 and C2 are given by explicit integrals over the exact wave function evaluated at r12=0. Numerical evaluation of these integrals yields 3C1≅0.0741 and 4C2≅0.0309, in excellent agreement with the empirical results 3C1≅0.0740 and 4C2≅0.031 found by Carroll, Silverstone, and Metzger.