Gaussian-type orbitals primitive GTOs

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... 

The next step was to represent each Slater atomic orbital as a fixed linear combination of Gaussian orbitals so a Slater-type orbital with exponent f is written as a sum of GTOs with exponents a, q 2, and so on. For example, in the case of three primitive GTOs we might write... 

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