STO-nG basis sets are minimal basis sets used in computational chemistry, more specifically in ab initio quantum chemistry methods, to calculate the molecular orbitals of chemical systems within Hartree-Fock theory or density functional theory. The basis functions are linear combinations of n {\displaystyle n} primitive Gaussian-type orbitals (GTOs) that are fitted to single Slater-type orbitals (STOs). They were first proposed by John Pople and n {\displaystyle n} originally took the values 2 – 6. A minimal basis set is where only sufficient orbitals are used to contain all the electrons in the neutral atom. Thus, for the hydrogen atom, only a single 1s orbital is needed, while for a carbon atom, 1s, 2s and three 2p orbitals are needed.
General definition STO- n {\displaystyle n} G basis sets consist of one STO for each orbital in the neutral atom (with suitable parameter ζ {\displaystyle \zeta } ) for each atom in the system to be described (e.g. molecule). The STOs assigned to a particular atom are centered around its nucleus. Therefore, the number of basis functions for each atom depends on its type. The STO- n {\displaystyle n} G basis sets are available for all atoms from hydrogen up to xenon.
Each STO (both core and valence orbitals) ψ m l {\displaystyle \psi _{ml}} , where m {\displaystyle m} is the principal quantum number and l {\displaystyle l} is the angular momentum quantum number, is approximated by a linear combination of n {\displaystyle n} primitive GTOs ϕ l α m j {\displaystyle \phi _{l\alpha _{mj}}} with exponents α m j {\displaystyle \alpha _{mj}} :
ψ m l STO − n G = ∑ j = 1 n c m l j ϕ l α j . {\displaystyle \psi _{ml}^{{\text{STO}}-n{\text{G}}}=\sum _{j=1}^{n}c_{mlj}\phi _{l\alpha _{j}}.}
The expansion coefficients c m l j {\displaystyle c_{mlj}} and exponents α m j {\displaystyle \alpha _{mj}} are fitted with the least squares method (this differs from the more common procedure, where they are chosen to give the lowest energy) to all STOs within the same shell m {\displaystyle m} simultaneously. Note that all ψ m l STO − n G {\displaystyle \psi _{ml}^{{\text{STO}}-n{\text{G}}}} within the same shell m {\displaystyle m} (e.g. 2s and 2p) share the same exponents, i.e. they do not depend on the angular momentum, which is a special feature of this basis set and allows more efficient computation. The fit between the GTOs and the STOs is often reasonable, except near to the nucleus: STOs have a cusp at the nucleus, while GTOs are flat in that region. Extensive tables of parameters have been calculated for STO-1G through STO-6G for s orbitals through g orbitals and can be downloaded from the Basis Set Exchange.
STO-2G basis set The STO-2G basis set is a linear combination of 2 primitive Gaussian functions. The original coefficients and exponents for first-row and second-row atoms are given as follows (for ζ = 1 {\displaystyle \zeta =1} ).
For general values of ζ {\displaystyle \zeta } , one can use the scaling law ψ m l ζ ( r ) = ζ 3 / 2 ψ m l 1 ( ζ r ) {\displaystyle \psi _{ml}^{\zeta }(\mathbf {r} )=\zeta ^{3/2}\psi _{ml}^{1}(\zeta \mathbf {r} )} to approximate general STOs with ζ ≠ 1 {\displaystyle \zeta \neq 1} .
STO-3G basis set The STO-3G basis set is the most commonly used among the STO- n {\displaystyle n} G basis sets and is a linear combination of 3 primitive Gaussian functions. The coefficients and exponents for first-row and second-row atoms are given as follows (for ζ = 1 {\displaystyle \zeta =1} ).
Accuracy The exact energy of the 1s electron of H atom is −0.5 hartree, given by a single Slater-type orbital with exponent 1.0. The following table illustrates the increase in accuracy as the number of primitive Gaussian functions increases from 3 to 6 in the basis set.
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