Gaussian functions contracted

In order to combine the best feature of GTOs (computational efficiency) with that of STOs (proper radial shape), most of the first basis sets developed with GTOs used them as building blocks to approximate STOs. That is, the basis functions p used for SCF calculations were not individual GTOs, but instead a linear combination of GTOs fit to reproduce as accurately as possible a STO, i.e., 

While the acronym STO-3G is designed to be informative about the contraction scheme, it is appropriate to mention an older and more general notation that appears in much of the earlier literature, although it has mostly fallen out of use today. In that notation, the STO-3G H basis set would be denoted (3s)/[ls], The material in parentheses indicates the number and type of primitive functions employed, and the material in brackets indicates the number and type of contracted functions. If first-row atoms are specified too, the notation for STCT-3G would be (6s3p/3s)/[2s 1 p/1 s]. Thus, for instance, lithium would require 3 each (since it is STO-3G) of Is primitives, 2s primitives, and 2p primitives, so the total primitives are 6s3p, and the contraction schemes creates a single Is, 2s, and 2p set, so the contracted functions are 

These are separated from the hydrogenic details by a slash in each instance. Extensions to higher rows follow by analogy. Variations on this nomenclature scheme exist, but we will not examine them here. 

The STO-3G basis set is what is known as a single- basis set, or, more commonly, a minimal basis set. This nomenclature implies that there is one and only one basis function 

Valence orbitals, on the other hand, can vary widely as a function of chemical bonding. Atoms bonded to significantly more electronegative elements take on partial positive charge 

In order to combine the best feature of GTOs (computational efficiency) with that of STOs 

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The contracted Gaussian functions are a linear combination of the primitive Gaussian functions. That is,... 

Here, n corresponds to the principal quantum number, the orbital exponent is termed and Ylm are the usual spherical harmonics that describe the angular part of the function. In fact as a rule of thumb one usually needs about three times as many GTO than STO functions to achieve a certain accuracy. Unfortunately, many-center integrals such as described in equations (7-16) and (7-18) are notoriously difficult to compute with STO basis sets since no analytical techniques are available and one has to resort to numerical methods. This explains why these functions, which were used in the early days of computational quantum chemistry, do not play any role in modem wave function based quantum chemical programs. Rather, in an attempt to have the cake and eat it too, one usually employs the so-called contracted GTO basis sets, in which several primitive Gaussian functions (typically between three and six and only seldom more than ten) as in equation (7-19) are combined in a fixed linear combination to give one contracted Gaussian function (CGF),... 

These basis sets consist of 34 (6-311+G(d,p)), 92 (aug-cc-pVTZ), 172 (aug-cc-pVQZ), and finally 287 (aug-cc-pV5Z) contracted Gaussian functions and include polarization functions up to d- (6-311+G(d,p)), f- (aug-cc-pVTZ), g- (aug-cc-pVQZ), and even h-character (aug-cc-pV5Z). 

Example 1. Dissociation of H O by symmetric stretch Reference calculations are first performed at equilibrium geometry, using a simple basis of contracted gaussian functions [8]. The molecule is then dissociated by symmetric stretch, energies being calculated at bond length intervals of AR = Q.2Re up to i = 6.0J e (where dissociation is effectively complete). 

In recent years a compromise has been found which presently dominates polyatomic calculations. Each function fj is expanded as a linear combination of gaussian orbitals (f is then called a contracted gaussian function). Since this is basically a numerical fitting procedure, various choices have been suggested for the contraction scheme. The most popular choices are presently Pople s approximations (15) to Slater orbitals and Dunning s approximations (16) to free atom Hartree-Fock orbitals. 

The basis sets described here in most detail are those developed by Pople3 and coworkers [40], which are probably the most popular now, but most general-purpose (those not used just on small molecules or on atoms) basis sets utilize some sort of contracted Gaussian functions to simulate Slater orbitals. A brief discussion of basis sets and references to many, including the widely-used Dunning... 

By choosing the contracting scheme, it is possible to determine the contracted gaussian functions which best approximate Slater functions,88 Hartree-Fock atomic orbitals,89-90 etc. It should be noted that the size and balance (revealed, for example, by the optimum ratio of the number of s-type and p-type functions) of the basis are definite factors which control the quality of the final result (e.g., ref. 90). 

Model C a single SO/ anion cluster, ab initio HF SCF calculations were performed by using the GAMESS [57] package. An extended split valence band sp basis set of contracted Gaussian functions were used with 3d polarization functions (6-3IG basis). 

Like many other approaches in quantum chemistry, the GAPW method uses a basis of contracted Gaussian functions to expand the Kohn-Sham orbitals i(r)... 

It is found that contracted Gaussian functions (CGFs) [27]... 

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