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 atomic orbitals | ψ i ⟩ {\displaystyle |\psi _{i}\rangle } are expanded within the basis set as a linear combination of the basis functions | ψ i ⟩ ≈ ∑ μ c μ i | μ ⟩ {\textstyle |\psi _{i}\rangle \approx \sum _{\mu }c_{\mu i}|\mu \rangle } , where the expansion coefficients c μ i {\displaystyle c_{\mu i}} are given by c μ i = ∑ ν ⟨ μ | ν ⟩ − 1 ⟨ ν | ψ i ⟩ {\textstyle c_{\mu i}=\sum _{\nu }\langle \mu |\nu \rangle ^{-1}\langle \nu |\psi _{i}\rangle } .