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




|


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 }

.
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.

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