Exponentially correlated Hylleraas-configuration interaction non-relativistic energy of the ¹S ground state of the helium atom

A generalization of the Hylleraas-Configuration Interaction method (Hy-CI) first proposed by Wang, et al., the Exponentially Correlated Hylleraas-Configuration Interaction method (E-Hy-CI) in which the single r _ij of an Hy-CI wave function is generalized to a form of the generic type $r_{ij}^{{\nu }_{ij}}{e}^{-{\omega }_{ij}{r}_{ij}}$ , is explored. This type of correlation, suggested by Hirshfelder in 1960, has the right behavior both in the vicinity of the r_ij cusp as r_ij goes to 0 and as r_ij goes to infinity; this work explores whether wave functions containing both linear and exponential r _ij factors converge more rapidly than either one alone. The method of calculation of the two-electron E-Hy-CI kinetic energy and electron repulsion integrals in a stable and efficient way using recursion relations is discussed, and the relevant formulas are given. The convergence of the E-Hy-CI wave function expansion is compared with that of the Hy-CI wave function without exponential correlation factors, demonstrating both convergence acceleration and an improvement in the accuracy for the same basis. This makes the application of the E-Hy-CI method to systems with N > 4, for which this formalism with at most a single $r_{ij}^{{\nu }_{ij}}{e}^{-{\omega }_{ij}{r}_{ij}}$ factor per term leads to solvable integrals, very promising. E-Hy-CI method variational calculations with up to 10080 expansion terms are reported for the ground ¹ S state of the neutral helium atom, with a resultant nonrelativistic energy of -2.9037 2437 7034 1195 9831 1084 hartree for the best expansion.