A basis set is a set of functions combined linearly to model molecular orbitals. Basis functions can be considered as representing the atomic orbitals of the atoms and are introduced in quantum chemical calculations because the equations defining the molecular orbitals are otherwise very difficult to solve.

Many standard basis sets have been carefully optimized and tested over the years. In principle, a user would employ the largest basis set available in order to model molecular orbitals as accurately as possible. In practice, the computational cost grows rapidly with the size of the basis set so a compromise must be sought between accuracy and cost. If this is systematically pursued, it leads to a “theoretical model chemistry” [9], that is, a well-defined energy procedure (*e.g.*, Hartree-Fock) in combination with a well-defined basis set.

Basis sets have been constructed from Slater, Gaussian, plane wave and delta functions. Slater functions were initially employed because they are considered “natural” and have the correct behavior at the origin and in the asymptotic regions. However, the two-electron repulsion integrals (ERIs) encountered when using Slater basis functions are expensive and difficult to evaluate. Delta functions are used in several quantum chemistry programs. However, while codes incorporating delta functions are simple, thousands of functions are required to achieve accurate results, even for small molecules. Plane waves are widely used and highly efficient for calculations on periodic systems, but are not so convenient or natural for molecular calculations.

The most important basis sets are contracted sets of atom-centered Gaussian functions. The number of basis functions used depends on the number of core and valence atomic orbitals, and whether the atom is light (H or He) or heavy (everything else). Contracted basis sets have been shown to be computationally efficient and to have the ability to yield chemical accuracy (see Appendix B). The Q-Chem program has been optimized to exploit basis sets of the contracted Gaussian function type and has a large number of built-in standard basis sets (developed by Dunning and Pople, among others) which the user can access quickly and easily.

The selection of a basis set for quantum chemical calculations is very important. It is sometimes possible to use small basis sets to obtain good chemical accuracy, but calculations can often be significantly improved by the addition of diffuse and polarization functions. Consult the literature and review articles [9, 520, 521, 522, 11] to aid your selection and see the section “Further Reading” at the end of this Chapter.