Gaussian basis sets contracted GTOs

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 starting point to obtain a PP and basis set for sulphur was an accurate double-zeta STO atomic calculation4. A 24 GTO and 16 GTO expansion for core s and p orbitals, respectively, was used. For the valence functions, the STO combination resulting from the atomic calculation was contracted and re-expanded to 3G. The radial PP representation was then calculated and fitted to six gaussians, serving both for s and p valence electrons, although in principle the two expansions should be different. Table 3 gives the numerical details of all these functions. 

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),... 

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. 

The very first application of the GCHF method was for the construction of universal atomic basis sets [17], culminating with very accurate Gaussian (GTO) and the construction of Slater (STO) bases for neutral and charged, ground and excited states for atoms H to Xe (see Ref. [18] and references therein). Contracted GTO sets were also introduced [19,20]. The extension of integral transforms other than for Is functions (Section 3) was also presented [21]. 

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.

Although the Gaussian-type orbitals (contracted or not) are not atomic orbitals but just basis functions, one still keeps the nomenclature and distinguishes between valence orbitals, which are meant to describe the electrons in the outermost shell, e.g. the 2s and 2p electrons in carbon or the Is electron in hydrogen, and core orbitals, which are meant to describe the inner electrons, e.g. the Is electrons in carbon. If each core and valence orbital of an atom is represented by a single primitive or contracted GTO one speaks of a minimal basis set. 

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