(1) |

where

The smallest possible basis set is called the *minimal basis set*,
and it contains one orbital (which may be contracted) for every
orbital we usually think of for an atom (including unoccupied
orbitals). For example, hydrogen has just one orbital, but carbon has
5 (1s, 2s, 2p_{x}, 2p_{y}, and 2p_{z}) even though one of the p
orbitals for carbon atom will be unoccupied. The STO-3G basis is a
very well-known minimal basis set which contracts 3 Gaussian functions
to approximate the more accurate (but more difficult to compute)
Slater type orbitals. Although a contracted GTO might give a good
approximation to an atomic orbital, it lacks any flexibility to expand
or shrink in the presence of other atoms in a molecule. Hence, a
minimal basis set such as STO-3G is not capable of giving highly
accurate results.

The solution is to add extra basis functions *beyond* the minimum number
required to describe each atom. Then, the Hartree-Fock procedure (below)
can weight each atomic orbital basis function more or less to get a better
description of the wave function. If we have *twice* as many basis
functions as in a minimum basis, this is called a ``double zeta'' basis set
(the zeta, ,
comes from the exponent in the GTO). Hence, a
double-zeta basis set for hydrogen would have two functions, and a true
double-zeta basis set for carbon would have 10 functions. However,
sometimes people ``cheat'' and use only a single orbital for the core (1s),
giving 9 functions for carbon. Such basis sets are said to be ``double-zeta
*in the valence*'' space; they are also called ``split-valence'' basis
sets. Double-zeta basis sets are often denoted DZ. Often additional
flexibility is built in by adding higher-angular momentum basis functions.
Since the highest angular momentum orbital for carbon is a p orbital, the
``polarization'' of the atom can be described by adding a set of d
functions. A hydrogen atom would use a set of 3 p functions as polarization
functions. A double-zeta plus polarization basis set might be designated
DZP. The most famous example of a split-valence double-zeta plus
polarization basis set is Pople's so-called 6-31G* basis. This obscure
notation means that the core orbital is described by a contraction of 6
Gaussian orbitals, while the valence is described by two orbitals, one made
of a contraction of 3 Gaussians, and one a single Gaussian function. Just
to confuse you, the star (*) indicates polarization functions on non-hydrogen
atoms. If polarization was added to hydrogen atoms also, this basis would
be called 6-31G**. The confusing nature of this nomenclature has caused
some chemists to start switching to slightly improved notation such as
6-31G(d,p), where the polarization functions are listed explicitly.

This lab will not investigate the nuances of basis set design, but you need to have a basic grasp of what basis sets are, since you'll be using them!