Gaussian-type orbitals contracted GTOs

The situation is quite similar in chemistry. Due to decades of experience with Hartree-Fock and Cl calculations much is known about the construction of basis functions that are suitable for molecules. Almost all of this continues to hold in DFT — a fact that has greatly contributed to the recent popularity of DFT in chemistry. Chemical basis functions are classified with respect to their behaviour as a function of the radial coordinate into Slater type orbitals (STOs), which decay exponentially far from the origin, and Gaussian type orbitals (GTOs), which have a gaussian behaviour. STOs more closely resemble the true behaviour of atomic wave functions [in particular the cusp condition of Eq. (19)], but GTOs are easier to handle numerically because the product of two GTOs located at different atoms is another GTO located in between, whereas the product of two STOs is not an STO. The so-called contracted basis functions , in which STO basis functions are reexpanded in... 

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.

H. Tatewaki, Y. Sakai, and S. Huzinaga, /. Comput. Chem., 2, 96 (1981). A Systematic Preparation of New Contracted Gaussian-Type Orbital Sets. IV. The Effect of Additional 3s Functions Introduced by the Use of the Six-Membered 3d GTO s. 

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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