Gaussian basis sets primitive GTOs

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),... 

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. 

The essential features of this approach is this. Let each unit cell be divided into molecules (real ones or not), and we devise a basis set to represent the MO s of these molecules. The primitive basis set used are spherical-harmonic Gaussian Type Orbitals (GTO s) centered about different origins, and have the form... 

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