STO-NG basic rates

STO-NG basis sets are minimal basis sets in which Slater orbitals (“Slater-type orbitals”, STOs) are approximated by N primitive Gaussian orbitals (“Gaussian-type orbitals”, GTOs). N typically takes values ​​from 2 to 6. STO-NG basic sets were initially developed by John Pople . The minimum number of functions with which all electrons in a system can be described is called the minimum basis set. Only a single 1s orbital is required for the hydrogen atom , while one 1s, 2s and three 2p orbitals are required for each carbon atom . One STO is required for each orbital, three GTOs are usually well suited for approximating an STO, so an STO-3G-s orbital is given by:

${\displaystyle \mathbf {\psi } _{STO-3G}(s)=c_{1}\phi _{1}+c_{2}\phi _{2}+c_{3}\phi _{3}}$

${\displaystyle \mathbf {\phi } _{i}}$stands for the primitive Gaussian functions, c for the coefficients. The following applies to the primitive Gaussian functions:

${\displaystyle \mathbf {\phi } _{1}=\left({\frac {2\alpha _{1}}{\pi }}\right)^{3/4}e^{-\alpha _{1}r^{2}}}$

${\displaystyle \mathbf {\phi } _{2}=\left({\frac {2\alpha _{2}}{\pi }}\right)^{3/4}e^{-\alpha _{2}r^{2}}}$

${\displaystyle \mathbf {\phi } _{3}=\left({\frac {2\alpha _{3}}{\pi }}\right)^{3/4}e^{-\alpha _{3}r^{2}}}$

The resulting contracted Gaussian function (cGTO) is thus generally:

${\displaystyle \mathbf {\psi } _{STO-3G}(s)=c_{1}\cdot \left(\left({\frac {2\alpha _{1}}{\pi }}\right)^{3/4}e^{-\alpha _{1}r^{2}}\right)+c_{2}\cdot \left(\left({\frac {2\alpha _{2}}{\pi }}\right)^{3/4}e^{-\alpha _{2}r^{2}}\right)+c_{3}\cdot \left(\left({\frac {2\alpha _{3}}{\pi }}\right)^{3/4}e^{-\alpha _{3}r^{2}}\right)}$

The values ​​of the coefficients c ₁ , c ₂ , c ₃ and the exponents α ₁ , α ₂ and α ₃ have yet to be determined. For the STO-NG basic sets, this is achieved by adapting the least squares of the three primitive Gaussian functions to the individual Slater orbitals; the more common method would be to choose the coefficients (c) and the exponents (α) so that the lowest energy for a molecule is obtained. A typical feature of the STO-NG basis set is that common exponents are used for orbitals in the same shell (e.g. 2s and 2p) as this allows for a more efficient calculation.

STO-3G

The most widely used basic set of this group is STO-3G , which is used for large systems and for qualitative results. This basic set is available for all atoms from hydrogen to xenon.

STO-2G

The STO-2G basis set is a linear combination of two primitive Gaussian functions. The original coefficients and exponents for atoms of the first and second periods are given as follows.

STO-2G	α 1	c 1	α 2	c 2
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

The STO-2G basic set is rarely used because the quality of the results is often poor.

Higher STO-NG basic rates

Higher STO-NG basic rates are rarely used due to their lower flexibility.

The exact energy of the 1s electron of the H atom is −0.5 Hartree , given by a single Slater orbital with the exponent 1.0. The following table shows the increase in accuracy as the number of primitive Gaussian functions increases from 3 to 6 in the basis set:

Base rate	Energy [Hartree]
STO-3G	−0.49491
STO-4G	−0.49848
STO-5G	−0.49951
STO-6G	−0.49983

Ab initio calculations

The STO-NG basic sets are unsuitable for good quality ab initio calculations because their flexibility in the radial direction is poor. For sufficiently good calculations, more flexible basic sets, such as B. Pople bases necessary.

Individual evidence

1. ↑ ^{a b c} Kunz, Roland W .: Molecular Modeling for Users: Application of Force Field and MO Methods in Organic Chemistry . 1st edition. Teubner, Stuttgart 1991, ISBN 3-519-03511-1 , pp. 122 .
2. ↑ ^{a b c} W. J. Hehre, RF Stewart, JA Pople: Self ‐ Consistent Molecular ‐ Orbital Methods. I. Use of Gaussian Expansions of Slater ‐ Type Atomic Orbitals . tape 51 . The Journal of Chemical Physics, 1969, pp. 2657 , doi : 10.1063 / 1.1672392 .
3. ↑ ^{a b c} Young, David C .: Computational chemistry: a practical guide for applying techniques to real world problems . Wiley, New York 2001, ISBN 0-471-33368-9 , pp. 86 .