Matrix Elements for Explicitly-Correlated Atomic Wave Functions

Abstract

We refer to atomic wave functions that contain the interelectron distances as “explicitly correlated”; we consider here situations in which an explicit correlation factor \(r_{ij}\) can occur as a power multiplying an orbital functional form (a Hylleraas function) and/or in an exponent (producing exponential correlation ). Hylleraas functions in which each wave-function term contains at most one linear \(r_{ij}\) factor define a method known as Hylleraas-CI . This paper reviews the analytical methods available for evaluating matrix elements involving exponentially-correlated and Hylleraas wave functions; attention is then focused on computation of integrals needed for the kinetic energy. In contrast to orbital-product and exponentially-correlated wave functions, no general formulas have been developed by others to relate the kinetic-energy integrals in Hylleraas-CI (or its recent extension by the Nakatsuji group) to contiguous potential-energy matrix elements. The present paper provides these missing formulas, obtaining them by using relevant properties of vector spherical harmonics. Validity of the formulas is confirmed by comparisons with kinetic-energy integrals obtained in other ways.