Basis expansion Cartesian Gaussians

Appendix B Expansion of Cartesian Gaussian Basis Functions Using Spherical Harmonics... 

APPENDIX B EXPANSION OF CARTESIAN GAUSSIAN BASIS FUNCTIONS USING SPHERICAL HARMONICS... 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

For the expansion of the Hartree-Fock molecular orbitals we have used either Slater or Cartesian Gaussian functions. In addition to these basis functions we can also include spherical Gaussian functions in the initial scattering basis. A detailed discussion of the single-center expansion of Slater and Cartesian Gaussian functions has been given by Harris and Michels (19) and by Fliflet and McKoy (20), respectively. Spherical Gaussian functions, i.e.. 

Since the Hermite Gaussians are to be used as one-centre basis functions for the expansion of Cartesian overlap distributions (over which the integration wiU eventually be carried out), it is important to determine the integrals over these Gaussians. The integral over the x component of the Hermite Gaussians is given by... 

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