STO-nG basis sets

STO-$n$G basis sets consist of one STO for each orbital in the neutral atom (with suitable parameter $\backslash zeta$) 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-$n$G basis sets are available for all atoms from hydrogen up to xenon.Computational Chemistry, David Young, Wiley-Interscience, 2001. pg 86.{{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}}

Each STO (both core and valence orbitals) $\backslash psi\_\{ml\}$, where $m$ is the principal quantum number and $l$ is the angular momentum quantum number, is approximated by a linear combination of $n$ primitive GTOs $\backslash phi\_\{l\backslash alpha\_\{mj\}\}$ with exponents $\backslash alpha\_\{mj\}$:{{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}}

$\backslash psi\_\{ml\}^\{\backslash text\{STO\}-n\backslash text\{G\}\}\; =\; \backslash sum\_\{j=1\}^n\; c\_\{mlj\}\; \backslash phi\_\{l\backslash alpha\_j\}\; .$

The expansion coefficients $c\_\{mlj\}$ and exponents $\backslash alpha\_\{mj\}$ 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 $m$ simultaneously. Note that all $\backslash psi\_\{ml\}^\{\backslash text\{STO\}-n\backslash text\{G\}\}$ within the same shell $m$ (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.{{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}}

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.Chemical Modeling From Atoms to Liquids, Alan Hinchliffe, John Wiley & Sons, Ltd., 1999. pg 294.Molecular Modelling, Andrew R. Leach, Longman, 1996. pg 68 – 73. Extensive tables of parameters have been calculated for STO-1G through STO-6G for s orbitals through g orbitals{{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}} and can be downloaded from the Basis Set Exchange.

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 $\backslash zeta=1$).

For general values of $\backslash zeta$, one can use the scaling law $\backslash psi\_\{ml\}^\backslash zeta(\backslash mathbf\; r)\; =\; \backslash zeta^\{3/2\}\backslash psi\_\{ml\}^1(\backslash zeta\backslash mathbf\; r)$ to approximate general STOs with $\backslash zeta\backslash neq\; 1$.

The STO-3G basis set is the most commonly used among the STO-$n$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 $\backslash zeta=1$).

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.

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 Pople basis sets.

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Category:Quantum chemistry