Basis Set

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A basis set in theoretical and computational chemistry 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 [math]\displaystyle{ |\psi_i\rangle }[/math] are expanded within the basis set as a linear combination of the basis functions [math]\displaystyle{ |\psi_i\rangle \approx \sum_\mu c_{\mu i} |\mu\rangle }[/math], where the expansion coefficients [math]\displaystyle{ c_{\mu i} }[/math] are given by [math]\displaystyle{ c_{\mu i} = \sum_{\nu} \langle \mu|\nu \rangle^{-1} \langle \nu |\psi_i \rangle }[/math]. 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.

atomic orbitals computational chemistry quantum chemistry

References

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