Computational Physics

68 The Hartree–Fock method

φ1s

0.25

0.5

0 0.5 1 1.5 2

r

STO-2G

STO-3G

STO-4G

STO

Figure 4.3. Approximation of a 1s Slater orbital (STO) by STO-2G up to STO-4G

basis functions.

and the basis functions themselves is the reason why the GTOs were calledprimitive
basis functions above: they are used to build contracted basis functions. If allN
primitive basis functions are contracted into one normal basis function, the basis set
is called a STO-NG basis, denoting thatNGTOs have been used to fit a Slater type
orbital. Figure 4.3 shows how a Slater orbital is approximated by STO-NG basis
functions forN=2, 3, 4. Note that deviations from the exact Slater orbital are only
noticeable for smallr, and these are usually suppressed, as in volume integrals the
integrands are weighted with a factorr^2.
In most basis sets, an intermediate approach is taken, in which the set of primitive
Gaussians is split into two (or more) parts. The primitive Gaussian functions in
each such part are contracted into one basis function. A common example is the
STO-31 basis set, in which the three basis functions with shortest range (i.e. highest
αp) are contracted (using the correspondingDps) into one basis function and the
remaining one (with the longest range) is taken as a second basis function. The
reason for splitting the basis in this way is that the perturbation of the orbitals by
the environment will affect primarily the long-range tails, and leave the short-range
part essentially unchanged.
There exist many other basis sets, like split-basis sets, in which from every
primitive Gaussian two new ones are constructed, one with a range slightly shorter
than in the original function (αslightly larger) and another with a slightly longer
range (αslightly smaller), so that it is possible for an orbital in a molecule to assume
a slightly contracted or expanded form with respect to the atomic orbital. For more
detailsconcerningthevariousbasissets,seeRef.[12].
An important consideration is the symmetry of the orbitals which should be
taken into account. As an example, consider a HF calculation for H 2. The starting