Basis Sets for the Calculation of Molecular Properties

In all the quantum chemical methods described in Chapters 10 to 12 the approximate electronic wavefunctions are built from molecular orbitals, which are the solutions of the Hartree-Fock equations Eq. (9.11) or their multiconfigurational extension. In all modern methods these molecular orbitals are expanded in a set of one-electron functions x ) called the basis set 

In most cases these basis functions are what is called Gaussian-type orbitals (GTO) 

Mathematically, the set of functions only has to be complete, apart from fulfilling the same boundary conditions as the molecular orbitals. However, this is not possible in practice, because it would require infinitely large basis sets. Often, one chooses therefore a more physical approach, where the basis functions are placed at the atoms 

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. 

All these basis sets are essentially optimized for the calculation of electronic energies and are therefore able to represent the operators included in the field-free electronic Hamiltonian reasonably well. However, in the calculation of molecular electromagnetic properties it is necessary also to represent other operators such as the electric dipole operator, the electronic angular momentum operator, the Fermi-contact operator and more. Most of these basis sets are a priori not optimized for this and have to be extended. 

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