Gaussian Functions Basis Set Preliminaries Direct SCF

The electron distribution around an atom can be represented in several ways. Hydrogenlike functions based on solutions of the Schrodinger equation for the hydrogen atom, polynomial functions with adjustable parameters, Slater functions (Eq. 5.95), and Gaussian functions (Eq. 5.96) have all been used [34]. Of these, Slater and Gaussian functions are mathematically the simplest, and it is these that are currently used as the basis functions in molecular calculations. Slater functions are used in semiempirical calculations, like the extended Hiickel method (Section 4.4) and other semiempirical methods (Chapter 6). Modem molecular ab initio programs employ Gaussian functions. 

The product of gA and gB is the Gaussian gc, centered at rc. Now consider the general electron-repulsion integral 

If each basis function (f were a single, real Gaussian, then from Eq. 5.155 this would reduce to 

The STO-3G basis function in Fig. 5.12 is a contracted Gaussian consisting of three primitive Gaussians each of which has a contraction coefficient (0.4446, 

5353 and 0.1543). Typically, an ab initio basis function consists of a set of primitive Gaussians bundled together with a set of contraction coefficients. Now consider the two-electron integral (rsltu) (Eq. 5.156 = 5.73). Suppose each basis function is an STO-3G contracted Gaussian, i.e. 

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