Gaussian type orbitals

In quantum chemistry, a Gaussian type orbital (GTO) is a basis function used in a linear combination of atomic orbitals forming a molecular orbital. A GTO is a real-valued function of a 3-dimensional vector r, which is the position vector of an electron with respect to an origin. Usually this origin is centered on a nucleus in a molecule, but in principle the origin can be anywhere in, or outside, a molecule.

The defining characteristic of Gaussian type orbital is its radial part, which is given by a Gaussian function ${\scriptstyle \exp[-\alpha r^{2}]}$, where r is the length of r and α is a free parameter. In practice, α is often taken from tables of atomic orbital basis sets. The tables may be prepared by energy minimizations, or by fitting to other (known) orbitals, for instance to Slater type orbitals.

Angular parts of Gaussian type orbitals

There are two kinds of GTOs in common use.

Cartesian GTOs

Here the angular part is a homogeneous polynomial in the components x, y, and z of the position vector r. Hence

$G_{ijk,\alpha }\equiv x^{i}\,y^{j}\,z^{k}\,e^{-\alpha r^{2}}\quad {\hbox{with}}\quad i+j+k=n.$

In general there are ${\scriptstyle {n+2 \choose n}}$ homogeneous polynomials of degree n in three variables. For instance, for n = 3 we have the following ten Cartesian GTOs,

${\begin{aligned}x^{3}e^{-\alpha r^{2}},&\;\;y^{3}e^{-\alpha r^{2}},\;\;z^{3}e^{-\alpha r^{2}},\;\;x^{2}ye^{-\alpha r^{2}},\;\;x^{2}ze^{-\alpha r^{2}},\;\;\\y^{2}xe^{-\alpha r^{2}},&\;\;y^{2}ze^{-\alpha r^{2}},\;\;z^{2}xe^{-\alpha r^{2}},\;\;z^{2}ye^{-\alpha r^{2}},\;\;xyze^{-\alpha r^{2}}\end{aligned}}$

Note that a set of three p-type (l = 1) atomic orbitals (see hydrogen-like atom for the meaning of p and l) can be found as linear combinations of three out of the ten Cartesian GTOs of degree n = 3:

${\begin{aligned}x(x^{2}+y^{2}+z^{2})e^{-\alpha r^{2}}&=x\,r^{2}e^{-\alpha r^{2}}\\y(x^{2}+y^{2}+z^{2})e^{-\alpha r^{2}}&=y\,r^{2}e^{-\alpha r^{2}}\\z(x^{2}+y^{2}+z^{2})e^{-\alpha r^{2}}&=z\,r^{2}e^{-\alpha r^{2}}\\\end{aligned}}$

Likewise, a single s-orbital is "hidden" in a set of six orbitals of degree n = 2. The 15-dimensional Cartesian set of order n = 4 "hides" one s- and five d-orbitals. It may be assumed that these "hidden" orbitals of angular momentum quantum number l with

$l={\begin{cases}&1,3,\ldots ,n-2\quad \mathrm {odd} \;n\\&0,2,\ldots ,n-2\quad \mathrm {even} \;n\\\end{cases}}$

are an asset, but they are not. They often give rise to linear dependencies. The following type of GTOs is less plagued by this.

Spherical GTOs

(To be continued)