**1.
Introduction**

Each molecular orbital (one electron
function) y_{i} is expressed as a
linear combination of n basis functions F_{m}. By convention, molecular
orbitals are abbreviated by the greek letter y (psi), while the basis functions are symbolized by the greek letter F (phi).

The coefficients c_{mi }are called ** molecular orbital expansion coefficients** or
simply

The basis functions used in
molecular orbital calculations are usually described through an abbreviation or
acronym such as "6-31G(d)". The complete description of the
theoretical model used in MO calculations is given in the format **"theoretical
method/basis set"**. A calculation using Hartree-Fock theory in
combination with the 6-31G(d) basis set
would then be referred to as "HF/6-31G(d)". Often the molecular
geometries are optimized at one level of theory (method 1) and relative
energies are then recomputed using these geometries, but a different
theoretical method (method 2). The conventional description of this level of
theory is "**method
2//method 1**". If, for example, geometry optimization has been
performed at the HF/6-31G(d) level of theory and relative energies have then
been calculated at the MP2/6-31G(d) level, this approach would be described as
"MP2/6-31G(d)//HF/6-31G(d)".

In principle, any mathematical
function can be used to construct molecular orbitals. There are, however, only
a few solutions, which have found widespread application.

**1.1 Slater type orbitals (STOs) **

STOs are constructed
from a radial part describing the radial extend of the orbital and an angular
part describing the shape of the orbital (equ. 3).

The **radial part r**^{n-1} exp(-**z****r)** depends on the distance r
from the origin of the basis function (usually the location of the nucleus),
the orbital exponent z (greek: zeta),
and the principal quantum number n. The **spherical part Y**

Slater type
orbitals are not very useful for electronic structure calculations, because two
electron integrals over STOs are difficult to calculate.

**1.2 Gaussian type orbitals (GTOs)**

GTOs are also
constructed from a radial and a spherical part, but the radial part now has a
different dependence on *r* (equ. 5):

The radial extend
is proportional to *exp(-**a**r***^{2})**,

Gaussian type orbitals are very useful for electronic structure calculations, because two
electron integrals over GTOs are NOT difficult to calculate. GTOs have the
advantage that all the intergrals needed for the LCAO calculations can be done
by using simple formulae. (S. F. Boys, *Proceedings of the Royal Society of
London Series A*, **1950**, *200*, 542-554.)

Finally, a comparison between STOs and GTOs: There are two major differences (i) the behaviour for r = 0 and for (ii) r = ∞. The STO displays a cusp at r = 0 while the GTO has an incorrect behaviour at the nucleus. Moreover GTOs fall off too quickly as the electron gets further from the nucleus.

last changes: 01.04.2008, AS questions & comments to: axel.schulz@uni-rostock.de