We present a mathematical and numerical analysis of the rates of convergence of variational calculations and their impact on the issue of the convergence or divergence of expectation values obtained from variational wave functions. The rate of convergence of a variational calculation is critically dependent on the ability of finite linear combinations of basis functions to simulate the nonanalyticities (cusps) in the exact wave function being approximated. A slow rate of convergence of the variational energy can imply that the corresponding variational wave functions will yield divergent expectation values of physical operators not relatively bounded by the Hamiltonian. We illustrate the sorts of problems which can arise by examining Gausshyphen;type approximations to hydrogenic orbitals. Since all manyhyphen;electron wave functions have cusps similar to those in hydrogenic wave functions, this simple example is relevant to variational calculations performed on atoms and molecules. Finally, we offer suggestions on what types of variational wave functions are likely to yield rapid rates of convergence for the energy and reasonable rates of convergence for physical operators such as the dipole moment operator.
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