How Does the Free Complement Wave Function Become Accurate and Exact Finally for the Hydrogen Atom Starting From the Slater and Gaussian Initial Functions and for the Helium Atom on the Cusp Conditions?

摘要

The free complement (FC) method, or the free iterative-complement-interaction (ICI) method, for generating the exact wave function from an approximate initial wave function has been applied to the hydrogen atom starting from the Slater and Gaussian functions for comparison. The process of improvement was followed by checking the wave function itself and other quantities that have definite exact values. Because the exact wave function is simple in this case, we could make clear analyses for many aspects of the wave function. We examined the energy, the wave function itself, the wave function error, the H-square error, the local energy near the nucleus, and the cusp. Both the Slater and Gaussian functions gave similar convergence rates to the exact function with respect to the order of the FC method, but the number of complement functions at a particular order is three times larger for the Gaussian case than for the Slater case. Although the cusp value of the Gaussian initial function is zero, it grows as the FC calculation proceeds and finally becomes essentially exact at convergence. The same was true for all the quantities studied here, irrespective of the type of the initial wave function. For the helium atom, the cusp conditions including the electron-electron cusp were also examined with the FC wave function calculated before and shown to converge to the exact values.