Gaussian type orbitals

In quantum chemistry, a Gaussian type orbital (GTO) is a basis function in a linear combination of atomic orbitals that forms a molecular orbital.

A GTO is a real-valued function of a 3-dimensional vector r, 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 real parameter. The parameter α is usually taken from tables of atomic orbital basis sets, which are often contained in quantum chemical computer programs, or can be downloaded from the web. 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

Cartesian GTOs are defined by an angular part that is a homogeneous polynomial in the components x, y, and z of the position vector r. That is,

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

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,

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

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 nine out of the ten Cartesian GTOs of degree n = 3 (recall that r² = x² + y² + z²):

${\begin{aligned}x{\big [}r^{2}e^{-\alpha r^{2}}{\big ]}&=(x^{3}+xy^{2}+xz^{2})e^{-\alpha r^{2}}\\y{\big [}r^{2}e^{-\alpha r^{2}}{\big ]}&=(x^{2}y+y^{3}+yz^{2})e^{-\alpha r^{2}}\\z{\big [}r^{2}e^{-\alpha r^{2}}{\big ]}&=(x^{2}z+y^{2}z+z^{3})e^{-\alpha r^{2}}\\\end{aligned}}$

Observe that the expressions between square brackets only depend on r and hence are spherical-symmetric. The angular parts of these functions are eigenfunctions of the orbital angular momentum operator with quantum number l = 1.

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 could conceivably 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, i.e., are an improvement of the basis, but often they are not. They are prone to give rise to linear dependencies. The spherical kind of GTOs are less plagued by this problem.

Spherical GTOs

(To be continued)