Explicitly Correlated Functions in Variational Calculations

Abstract

In this Chapter we will describe wave functions that are variational solutions of the electronic Schrodinger equation and depend explicitly on interelectronic distance, r 12. Such functions with the correlation term r 12 u have been proposed for the first time in 1929 by Hylleraas to solve the Schrodinger equation for helium atom [1]. The Hylleraas wave function can be described as composed of three factors: exponential (Slater type), power expansion of the coordinates and correlation factor. Therefore this function is not based on the one-electron approximation. The same approach has been extended to the hydrogen molecule by James and Coolidge in 1933 [2]. Explicit inclusion of an r 12 dependent term into the wave function improves significantly convergence of energy as compared with other functions not having such correlation term. Today, the methods based on explicitly correlated wave functions are able to achieve the spectroscopic accuracy in atomic and molecular energy calculations (errors of the order of one μhartree). Several methods using different expressions of r 12 dependence have been developed. They can be divided into two groups depending on the form of the correlation factor used. In the first group the correlation factor has the form of r 12 u whereas in the second one, the correlation factor has the exponential form of exp(−αr 12 2 ) [3] or less often exp(−αr 12) [4]. From the first group functions we will consider in this Chapter: Hylleraas-type functions for atoms, James-Coolidge and Kolos-Wolniewicz functions, Hylleraas-CI functions based on the orbital approach and Hylleraas-CI functions based on the general valence bond functions. From the second group we will consider only the functions with the Gaussian correlation factor. All these methods give very accurate wave functions and energies, however, the area of application of these methods is rather limited to a few-electron atoms and molecules.