Basis set (chemistry)

In theoretical and computational chemistry, a basis set is a set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer.

The use of basis sets is equivalent to the use of an approximate resolution of the identity: the orbitals ${\displaystyle |\psi _{i}\rangle }$ are expanded within the basis set as a linear combination of the basis functions ${\textstyle |\psi _{i}\rangle \approx \sum _{\mu }c_{\mu i}|\mu \rangle }$, where the expansion coefficients ${\displaystyle c_{\mu i}}$ are given by ${\textstyle c_{\mu i}=\sum _{\nu }\langle \mu |\nu \rangle ^{-1}\langle \nu |\psi _{i}\rangle }$.

The basis set can either be composed of atomic orbitals (yielding the linear combination of atomic orbitals approach), which is the usual choice within the quantum chemistry community; plane waves which are typically used within the solid state community, or real-space approaches. Several types of atomic orbitals can be used: Gaussian-type orbitals, Slater-type orbitals, or numerical atomic orbitals. Out of the three, Gaussian-type orbitals are by far the most often used, as they allow efficient implementations of post-Hartree–Fock methods.