Contracted GTOs contraction coefficients

To overcome the primary weakness of GTO fimetions (i.e. their radial derivatives vanish at the nucleus whereas the derivatives of STOs are non-zero), it is coimnon to combine two, tliree, or more GTOs, with combination coefficients which are fixed and not treated as LCAO-MO parameters, into new functions called contracted GTOs or CGTOs. Typically, a series of tight, medium, and loose GTOs are multiplied by contraction coefficients and suimned to produce a CGTO, which approximates the proper cusp at the nuclear centre. 

The values of the orbital exponents ( s or as) and the GTO-to-CGTO contraction coefficients needed to implement a particular basis of the kind described above have been tabulated in several journal articles and in computer data bases (in particular, in the data base contained in the book Handbook of Gaussian Basis Sets A. Compendium for Ab initio Molecular Orbital Calculations, R. Poirer, R. Kari, and I. G. Csizmadia, Elsevier Science Publishing Co., Inc., New York, New York (1985)). 

The larger active space was then used in a new set of calculations which used a considerably larger AO basis. The 13s, 8p basis of van Duijneveldt was contracted (7,1,1,1,1,1,1/4,1,1,1,1). Additional diffuse functions were added, one of s-type (exponent 0.04) and of one p-type (exponent 0.025). Three 3d polarization functions were also included, contracted from four primitive GTOs with exponents 2.179, 0.865, 0.362 and 0.155. The exponents and the contraction coefficients were chosen according to the polarized basis-set... 

Optimization of the orbital exponents ( s or as) and the GTO-to-CGTO contraction coefficients for the kind of bases deseribed above have undergone explosive growth in reeent years. As a result, it is not possible to provide a single or even a few literature referenees from whieh one ean obtain the most up-to-date bases. However, the theory group at the Paeific Northwest National Laboratories (PNNL) offer a webpage [45] from whieh one ean find (and even download in a form prepared for input to any of several eommonly used eleetronie strueture eodes) a wide variety of Gaussian atomie basis sets. 

Figure 11. Graph of the H Is Slater-type orbital (STO) and its approximation using a contracted gaussian-type orbital (GTO). Three gaussian functions, centered at r = 0, are allowed to vary in a fitting routine that adjusts their relative heights (leading to the contraction coefficient) and widths (exponents) until their sum best matches the STO behavior. The best fit is found with the exponents 0.11, 0.41, and 2.2 for gaussians 1-3 respectively, giving the STO-3G basis set. Although in this case the decay of the STO is well approximated by the STO-3G basis set, the cusp near r = 0 is not.

Fixed linear combinations such as (8.2.16) are known as contracted GTOs, and the individual Gaussians from which the contracted GTOs are constructed are referred to as primitive GTOs. The coefficients in (8.2.16) are called the contraction coefficients. 

The atomic unit of wavefunction is. The dashed plot is the primitive with exponent 2.227 66, the dotted plot is the primitive with exponent 0.405 771 and the full plot is the primitive with exponent 0.109 818. The idea is that each primitive describes a part of the STO. If we combine them together using the expansion coefficients from Table 9.5, we get a very close fit to the STO, except in the vicinity of the nucleus. The full curve in Figure 9.4 is the contracted GTO, the dotted curve the STO. 

---