Gaussian Type Orbitals

The Gaussian-type orbitals (GTOs, German "Gaussian orbitals") are Gaussian-shaped approximation functions (contracted Gaussian functions) of atomic orbitals to the correct Slater orbitals ("Slater-type orbitals", STOs). As with Slater orbitals are also here to wave functions , which in the LCAO - approximation as atomic orbitals are used.

Spherical coordinates

The Gaussian basis functions can be broken down into a radial and an angle component using the usual radial angle decomposition:

${\displaystyle \ \Phi (\mathbf {r} )=R_{l}(r)Y_{lm}(\theta ,\phi )}$,

${\displaystyle Y_{lm}(\theta ,\phi )}$represents the angular component and the radial component, and are the corresponding angular momenta and their z-components. are accordingly the spherical coordinates. ${\displaystyle R_{l}(r)}$ ${\displaystyle l}$ ${\displaystyle m}$ ${\displaystyle r,\theta ,\phi }$

The radial component for the Slater orbitals looks like this:

${\displaystyle \ R_{l}(r)=A(l,\alpha )r^{l}e^{-\alpha r},}$

${\displaystyle A(l,\alpha )}$ as a normalization constant, for primitive GTOs the radial component is represented as follows:

${\displaystyle \ R_{l}(r)=B(l,\alpha )r^{l}e^{-\alpha r^{2}},}$

${\displaystyle B(l,\alpha )}$ is the normalization constant for the Gaussian orbital.

Cartesian coordinates

The Cartesian Gaussian functions are often used, as these are particularly easy to handle for derivations and integrations:

${\displaystyle \ \Phi _{GTO}(\mathbf {r} )=\underbrace {x^{i}y^{j}z^{k}} _{Y_{l,m}}\cdot \underbrace {e^{-\alpha r^{2}}} _{R_{l}}}$The prefactors x, y and z and their exponents are intended to "simulate" the angle component . ${\displaystyle Y_{lm}(\theta ,\phi )}$

GTOs as an STO approach

In the STO-NG basic sets , GTOs are used to approximate STOs. The STO-3G is the most frequently used basic set, here the GTOs are represented by a linear combination of three primitive Gaussian functions.

Errors of GTOs compared to STOs

When using GTOs instead of STOs, two qualitative mistakes are made:

1. GTOs do not have a peak (the derivative at is ). ${\displaystyle r=0}$ ${\displaystyle 0}$
2. The function curve of the GTOs is too steep (in the exponent of the Euler number, and not just like with Slater orbitals ) ${\displaystyle r^{2}}$ ${\displaystyle r}$

As a rule, these errors can be neglected, as they have a strong effect on the absolute energies but less on the relative energies.

Advantages of GTOs compared to STOs

Compared to Slater orbitals, calculations with Gaussian orbitals are 4–5 orders of magnitude faster, which means that they are used by almost all quantum chemistry programs, even if a larger base set is required.

Individual evidence

1. ^ Peter MW Gill: Molecular integrals Over Gaussian Basis Functions . In: Advances in Quantum Chemistry . Elsevier, 1994, ISBN 978-0-12-034825-1 , pp. 141–205 , doi : 10.1016 / s0065-3276 (08) 60019-2 ( elsevier.com [accessed July 10, 2018]).
2. ^ ^{A b} H. Bernhard Schlegel, Michael J. Frisch: Transformation between Cartesian and pure spherical harmonic Gaussians . In: International Journal of Quantum Chemistry . tape 54 , no. 2 , April 15, 1995, ISSN  0020-7608 , p. 83-87 , doi : 10.1002 / qua.560540202 ( wiley.com [accessed July 10, 2018]).