Gaussian type orbitals

In quantum chemistry, a Gaussian type orbital (GTO) is an atomic orbital used in linear combinations forming molecular orbitals. GTOs were proposed by Boys as early as 1950, and are at present the basis functions most generally used in quantum chemical program packages.

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 may be 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 have been 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 and spherical GTOs. Both will be discussed.

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 (n + 1)(n + 2)/2 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 [eigenvalue 1(1+1)].

Likewise, a single s-orbital is "hidden" in a set of six orbitals of degree n = 2. The general rule is that powers r^2k can be factored out of the Cartesian GTOs, leaving homogeneous polynomials of order l, which are eigenfunctions of orbital angular momentum with eigenvalue l(l+1). The quantum number l runs as follows

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

which are to be multiplied with correponding factor r^2k, where

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

This result has an interesting group theoretical intepretation.

It is conceivable that these "hidden" orbitals of angular momentum quantum number l < n are an asset, i.e., are an improvement of the basis. However, often they are not. They are prone to give rise to linear dependencies. The spherical GTOs, to be discussed now, are less plagued by this problem.

Spherical GTOs

Cosine- and sine-type regular solid harmonics (normalized to unity) can be defined by the following unitary matrix/vector expression

${\begin{pmatrix}^{c}\!R_{\ell }^{|m|}\\^{s}\!R_{\ell }^{|m|}\end{pmatrix}}\equiv {\frac {1}{\sqrt {2}}}{\begin{pmatrix}(-1)^{m}&\quad 1\\-(-1)^{m}i&\quad i\end{pmatrix}}{\begin{pmatrix}r^{\ell }Y_{\ell }^{|m|}\\r^{\ell }Y_{\ell }^{-|m|}\end{pmatrix}},\qquad m\neq 0,$

and for m = 0:

$^{c}\!R_{\ell }^{0}\equiv r^{\ell }Y_{\ell }^{0},$

where ${\scriptstyle Y_{\ell }^{|m|}}$ is a spherical harmonic function. For instance the functions for l = 2 are explicitly,

${\begin{matrix}^{c}\!R_{2}^{0}={\sqrt {\frac {5}{4\pi }}}(3z^{2}-r^{2})/2,&^{c}\!R_{2}^{1}={\sqrt {\frac {5}{4\pi }}}{\sqrt {3}}\,xz,&^{c}\!R_{2}^{2}={\sqrt {\frac {5}{4\pi }}}{\frac {1}{2}}{\sqrt {3}}\,(x^{2}-y^{2}),\\&^{s}\!R_{2}^{1}={\sqrt {\frac {5}{4\pi }}}{\sqrt {3}}\,yz,&^{s}\!R_{2}^{2}={\sqrt {\frac {5}{4\pi }}}{\sqrt {3}}\,xy.\qquad \quad \\\end{matrix}}$

See solid harmonics for closed expressions of regular harmonics expressed in Cartesian coordinates. If one expresses regular solid harmonics in spherical polar coordinates, the cosine-type functions contain the factor cos |m| φ and the sine-type functions sin |m| φ. Hence the name. It is of some interest to recall that spherical and solid harmonics span equivalent irreducible representations of the rotation group SO(3).

After this preliminary a spherical GTO is defined by either

$G_{\ell mc,\alpha }(\mathbf {r} )\equiv {}^{c}\!R_{\ell }^{m}(\mathbf {r} )e^{-\alpha r^{2}},$

or

$G_{\ell ms,\alpha }(\mathbf {r} )\equiv {}^{s}\!R_{\ell }^{m}(\mathbf {r} )e^{-\alpha r^{2}}.$

It can be proved that the 2l + 1 (orthogonal) spherical GTOs of order l are linear combinations of the Cartesian GTOs of order n = l. That is, the spherical GTOs span a 2l+1 dimensional subspace of the (n+1)(n+2)/2-dimensional Cartesian GTO space spanned by the Cartesian GTOs of order n. In fact, the spherical GTOs span the subspace orthogonal to the subpaces of l < n that—as we have seen above—are subspaces of the Cartesian GTO space. As we have seen, the subspaces with l < n are characterized by the fact that powers of r² are factored out. For regular solid harmonics a power of r² cannot be factored out by forming some linear combination [since regular harmonics span an irreducible representation of the group SO(3)].

The quantum chemical methods, for which GTOs are designed, require the calculation of numerous integrals (one- and two-electron integrals, one- through four-center integrals). In practice these integrals are computed on basis of Cartesian GTOs and then transformed to spherical GTOs. This means that basis functions containing powers of r² are discarded when spherical GTOs are used. Although the CPU cycles required for the integration seem wasted, this discarding shortens the list of integrals. Moreover, spherical GTOs do not become linearly dependent as soon as Cartesian GTOs, and, because they are eigenfunctions of orbital angular momentum, lead to well-defined selection rules (a priori rules that tell whether certain integrals will vanish). This latter property makes results of calculations based on spherical GTOs more transparent and easier to interpret than those based on Cartesian GTOs. It is therefore not surprising that spherical GTOs have been gaining ground lately on Cartesian GTOs.

Contracted sets

(To be continued)