Basis set (chemistry)

In <a href="/facts/Theoretical_chemistry/uqcyS5KH">theoretical</a> and <a href="/facts/Computational_chemistry/KyuCgRV3">computational chemistry</a>, a basis set is a set of <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> (called <a href="/facts/Basis_function/Bt7jqjY2">basis functions</a>) that is used to represent the <a href="/facts/Wave_function/KgM5ykjY">electronic wave function</a> in the <a href="/facts/Hartree%E2%80%93Fock_method/U9mlRWVR">Hartree–Fock method</a> or <a href="/facts/Density_functional_theory/wwz4Gqdg">density-functional theory</a> in order to turn the <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a> 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 <a href="/facts/Atomic_orbital/tKhgl6mL">orbitals</a> 
 
 
 
 
 |
 
 
 ψ
 
 i
 
 
 ⟩
 
 
 {\displaystyle |\psi _{i}\rangle }
 
 are expanded within the basis set as a <a href="/facts/Linear_combination/CVjL6kuN">linear combination</a> 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 <a href="/facts/Atomic_orbital/tKhgl6mL">atomic orbitals</a> (yielding the <a href="/facts/Linear_combination_of_atomic_orbitals/K7mcdv2L">linear combination of atomic orbitals</a> approach), which is the usual choice within the quantum chemistry community; <a href="/facts/Plane_wave/0aDUbNhQ">plane waves</a> which are typically used within the solid state community, or real-space approaches. Several types of atomic orbitals can be used: <a href="/facts/Gaussian_orbital/AUbqkDtC">Gaussian-type orbitals</a>, <a href="/facts/Slater-type_orbital/jPPIkkUQ">Slater-type orbitals</a>, or numerical atomic orbitals. Out of the three, Gaussian-type orbitals are by far the most often used, as they allow efficient implementations of <a href="/facts/Post-Hartree%E2%80%93Fock/SEG7bvRn">post-Hartree–Fock</a> methods.

Basis set (chemistry) open-in-new

Basis set (chemistry)