Primitive Cartesian Gaussian basis functions

Iyengar and Frisch have demonstrated the fundamental equivalence between the wavelet theory of multiresolution analysis and the translation and dilation operations on the primitive Cartesian Gaussian basis functions used in electronic structure theory ... 

Sometimes it turns out that we need to include a number of polarization functions, not just one of each type. The notation 4-31G(3d, 2p) indicates a standard 4-31G basis set augmented with three d-type primitive Cartesian Gaussians per centre and two p-type primitives on every hydrogen atom. Again, details of the... 

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

The Gaussian lobe function method was introduced by Preuss and developed for routine calculations by Whitten. The contraction of Gaussian lobe function (GLF) basis sets is made along the same lines as with Cartesian GTF s. As regards the primitives, the s-type functions are expressed in the usual way. But the primitives of p, d, f,. .. types are expressed as linear combinations of s-Gaussians (lobe functions) placed at different points so as to retain the proper symmetry (see Fig. 2,4 ). Thus, a p-type function on nucleus A may be... 

The d-type functions that are added to a 6-31G basis to form a 6-3IG basis are a single set of uncontracted 3d primitive Gaussians. For computational convenience there are six 3d functions per atom—3dyy, 3d , 3d,y, 3dy2, and 3d x- These six, the Cartesian Gaussians, are linear combinations of the usual five 3d functions—3d y, 3d 2 y2, 3dy, 3d, and 3d i and a 3s function + z ). The 6-3IG basis, in addition to adding... 

A typical basis function is a hxed linear combination of simpler, primitive functions. Such a composite function is termed a contracted basis function. Each primitive basis function is centered at an atomic nucleus and has a Gaussian dependence on distance from that nucleus. Except for s-functions, it also has a Cartesian factor to describe its angular dependence. Eor example, a p primitive function looks like xexp(—(r ), where ( (zeta) is the exponent of the Gaussian function. A p contracted function is a hxed linear combination of two or more primitive functions with different exponents. 

The primitive ERI is a six-dimensional integral over four basis functions. The integral as represented in Cartesian Gaussians can be written as... 

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