View source for STO-nG basis sets

{{Short description|Basis sets used in quantum chemistry}}
{{Use dmy dates|date=January 2020}}
{{Italic title|string=n}}
'''STO-''n''G basis sets''' are minimal [[Basis set (chemistry)|basis sets]] used in [[computational chemistry]], more specifically in [[ab initio quantum chemistry methods]], to calculate the [[Molecular orbital|molecular orbitals]] of chemical systems within [[Hartree–Fock method|Hartree-Fock theory]] or [[density functional theory]]. The basis functions are linear combinations of <math>n</math> primitive [[Gaussian orbital|Gaussian-type orbital]]s (GTOs) that are fitted to single [[Slater-type orbital|Slater-type orbitals]] (STOs). They were first proposed by [[John Pople]] and <math>n</math> 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-<math>n</math>G basis sets consist of one STO for each orbital in the neutral atom (with suitable parameter <math>\zeta</math>) 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-<math>n</math>G basis sets are available for all atoms from hydrogen up to xenon.<ref>''Computational Chemistry'', David Young, Wiley-Interscience, 2001. pg 86.</ref><ref name=":1">{{Cite journal |last=Pritchard |first=Benjamin P. |last2=Altarawy |first2=Doaa |last3=Didier |first3=Brett |last4=Gibson |first4=Tara D. |last5=Windus |first5=Theresa L. |year=2019 |title=A New Basis Set Exchange: An Open, Up-to-date Resource for the Molecular Sciences Community |url=https://www.basissetexchange.org/ |journal=Journal of Chemical Information and Modeling |volume=59 |issue=11 |pages=4814-4820 |doi=10.1021/acs.jcim.9b00725|url-access=subscription }}</ref>
{| class="wikitable"
|+
!element
!number of STOs
!STOs
|-
|H, He
|1
|1s
|-
|Li, Be
|2
|1s, 2s
|-
|B, C, N, O, F, Ne 
|5
|1s, 2s, 2p 
|-
|Na, Mg
|6
|1s, 2s, 2p, 3s 
|-
|Al, Si, P, S, Cl, Ar 
|9
|1s, 2s, 2p, 3s, 3p 
|-
|K, Ca
|10
|1s, 2s, 2p, 3s, 3p, 4s 
|-
|Sc-Zn
|15
|1s, 2s, 2p, 3s, 3p, 4s, 3d 
|-
|Ga-Kr
|18
|1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p
|-
|Rb, Sr
|19
|1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s
|-
|Y-Cd
|24
|1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d
|-
|In-Xe
|27
|1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p
|}
Each STO (both core and valence orbitals) <math>\psi_{ml}</math>, where <math>m</math> is the principal quantum number and  <math>l</math> is the angular momentum quantum number, is approximated by a linear combination of <math>n</math> primitive GTOs <math>\phi_{l\alpha_{mj}}</math> with exponents <math>\alpha_{mj}</math>:<ref name=":2">{{Cite book |last=Helgaker |first=Trygve |title=Molecular Electronic‐Structure Theory |last2=Jørgensen |first2=Poul |last3=Olsen |first3=Jeppe |publisher=John Wiley & Sons, LTD |year=2000 |isbn=9780471967552 |location=Chichester}}</ref>

<math>\psi_{ml}^{\text{STO}-n\text{G}} = \sum_{j=1}^n c_{mlj} \phi_{l\alpha_j} .</math>

The expansion coefficients <math>c_{mlj}</math> and exponents <math>\alpha_{mj}</math> 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 <math>m</math> simultaneously. Note that all <math>\psi_{ml}^{\text{STO}-n\text{G}}</math> within the same shell <math>m</math> (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.<ref name="pople">{{Cite journal |last=Hehre |first=W. J. |author2=R. F. Stewart |author3=J. A. Pople |year=1969 |title=Self-Consistent Molecular-Orbital Methods. I. Use of Gaussian Expansions of Slater-Type Atomic Orbitals |journal=Journal of Chemical Physics |volume=51 |issue=6 |pages=2657–2664 |bibcode=1969JChPh..51.2657H |doi=10.1063/1.1672392}}</ref>

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.<ref>''Chemical Modeling From Atoms to Liquids'', Alan Hinchliffe, John Wiley & Sons, Ltd., 1999. pg 294.</ref><ref>''Molecular  Modelling'', Andrew R. Leach, Longman, 1996. pg 68 – 73.</ref> Extensive tables of parameters have been calculated for STO-1G through STO-6G for s orbitals through g orbitals<ref name=":0">{{cite journal |last1=Stewart |first1=Robert F. |date=1 January 1970 |title=Small Gaussian Expansions of Slater‐Type Orbitals |journal=The Journal of Chemical Physics |volume=52 |issue=1 |pages=431–438 |doi=10.1063/1.1672702}}</ref> and can be downloaded from the Basis Set Exchange.<ref name=":1" />

==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 <math>\zeta=1</math>).<ref name="pople" />
{| class="wikitable"
|-
|-
|STO-2G
|''α''<sub>1</sub>
|''c''<sub>1</sub>
|''α''<sub>2</sub>
|''c''<sub>2</sub>
|-
| 1s
|0.151623
|0.678914
|0.851819
|0.430129
|-
| 2s
|0.0974545
|0.963782
|0.384244
|0.0494718
|-
| 2p
|0.0974545
|0.61282
|0.384244
|0.511541
|}
For general values of <math>\zeta</math>, one can use the scaling law <math>\psi_{ml}^\zeta(\mathbf r) = \zeta^{3/2}\psi_{ml}^1(\zeta\mathbf r)</math> to approximate general STOs with <math>\zeta\neq 1</math>.

== STO-3G basis set ==
The STO-3G basis set is the most commonly used among the STO-<math>n</math>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 <math>\zeta=1</math>).<ref name=":2" />
{| class="wikitable"
|-
|-
|STO-3G
|''α''<sub>1</sub>
|''c''<sub>1</sub>
|''α''<sub>2</sub>
|''c''<sub>2</sub>
|''α3''
|''c3''
|-
| 1s
|2.22766
|0.154329
|0.405771
|0.535328
|0.109818
|0.444635
|-
| 2s
|0.994203
| -0.0999672
|0.231031
|0.399515
|0.0751386
|0.700115
|-
| 2p
|0.994203
|0.155916
|0.231031
|0.607684
|0.0751386
|0.391957
|}

==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.<ref name="pople" />
{| class="wikitable"
|-
|-
|Basis set
|Energy [hartree]
|-
|STO-3G
| −0.49491
|-
|STO-4G
| −0.49848
|-
|STO-5G
| −0.49951
|-
|STO-6G
| −0.49983
|-
|}

==Use of STO-''n''G basis sets==
The most widely used basis set of this group is STO-3G, which is used for large systems and for preliminary geometry determinations. However, they are not suited for accurate ab-initio calculations due to their lack of flexibility in radial direction. For such tasks, larger basis sets are needed, such as the [[Basis set (chemistry)|Pople basis sets]].

==See also==
*[[List of quantum chemistry and solid state physics software]]

==References==
{{Reflist}}

{{DEFAULTSORT:Sto-Ng Basis Sets}}
[[Category:Quantum chemistry]]