Gaussian orbital

In molecular physics, Gaussian orbitals are a mathematical technique used for the computation of electron orbitals in molecules.

Definition and mathematical form

Gaussian orbitals, a.k.a. Gaussian type orbitals (GTOs), a.k.a. Gaussians for short, are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method.

They are distinguished by the form of their radial component, which is given by the Gaussian function

<math>e^{-\alpha{x^2}}.<math>

A general 'Cartesian Gaussian type orbital' (CGTO) is given as:

<math>f(x,y,z)=N_c x^l y^m z^n e^{-\alpha r^2}<math>

where l,m,n are three nonnegative integers (n not to be confused with the principal quantum number), <math>r^2=x^2+y^2+z^2<math> and <math>N_c<math> is a normalization constant given by

<math>N_c = \left(\frac{2\alpha}{\pi}\right)^{3/4}\sqrt{\frac{(4\alpha)^{l+m+n}}{(2l-1)!!(2m-1)!!(2n-1)!!}} <math>

The normalization constant is chosen such that the overlap integral over all space of the function with itself equals unity. The sum <math>l+m+n<math> is said to be the order of the function: in an atomic calculation it is equal to the principal quantum number minus one. The set of all functions with like <math>\{alpha,l,m,n\}<math> is said to be a 'shell'.

The <math>e^{-\alpha x^2}<math> is said to be the 'radial' part, and the <math>x^l y^m z^n<math> factor is said to be the 'angular' part of the basis function. In 'spherical Gaussian type orbitals', the angular part is replaced by true spherical harmonics <math>Y_{lm}(\theta,\phi)<math>, where <math>l<math> is the angular momenta (see also electron configuration). Cartesian and spherical Gaussian shells relate as follows:

• zero-order: a single Cartesian Gaussian corresponding to an <math>l=0<math> <math>(1s)<math> type orbital
• first order: three Cartesian Gaussians with angular parts <math>x,y,z<math> that correspond to the three spatially distinct <math>(2p)<math> type orbitals (with <math>l=1<math>)
• second order: six Cartesian Gaussians, symmetry-adapted linear combinations of which yield the five spatially distinct <math>(3d)<math> type orbitals with angular parts <math>\{xy,xz,yz,x^2-y^2,3z^2-r^2\}<math> (with <math>l=2<math>), plus a single <math>(3s)<math> type 'spherical contaminant' with (no longer) angular part <math>r^2<math> (<math>l=0<math>)
• third order: ten Cartesian Gaussians, seven linear combinations of which correspond to the seven spatially distinct <math>(4f)<math> functions (<math>l=3<math>), plus three <math>(4p)<math> type spherical contaminants (<math>l=1<math>)
• fourth order: fifteen Cartesian Gaussians, nine linear combinations of which correspond to the nine spatially distinct <math>(5g)<math> functions (<math>l=4<math>), plus five <math>(5d)<math> type spherical contaminants (<math>l=2<math>), plus a single <math>(5s)<math> type contaminant (<math>l=0<math>)
• and so forth.

Rationale

The principal reason for the use of Gaussian basis functions in molecular quantum chemical calculations is the 'Gaussian Product Theorem', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them. In this manner, four-center integrals can be reduced to finite sums of two-center integrals, and in a next step to finite sums of one-center integrals. The speedup by 4--5 orders of magnitude compared to Slater orbitals more than outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.

For reasons of convenience, many Gaussian integral evaluation programs work in a basis of Cartesian Gaussians even when spherical Gaussians are requested: the 'contaminants' are deleted a posteriori.

External links

• A visualization of all common and uncommon atomic orbitals, from 1s to 7g (http://www.shef.ac.uk/chemistry/orbitron/) Note that the radial part of the expressions given corresponds to Slater orbitals rather than Gaussians. The angular parts, and hence their shapes as displayed in figures, are the same as those of spherical Gaussians.