Today, there are hundreds of basis sets composed of Gaussian
Type Orbitals (GTOs). The smallest of these are called minimal
basis sets, and they are typically composed of the minimum number
of basis functions required to represent all of the electrons on
each atom. The largest of these can contain literally dozens to
hundreds of basis functions on each atom.
Minimum Basis sets
A minimum basis set is one in which a single
basis function is used for each orbital in a
Hartree-Fock calculation on the atom. However, for atoms such
as lithium, basis functions of p type are added to the basis
functions corresponding to the 1s and 2s orbitals of each atom. For
example, each atom in the first row of the periodic system (Li -
Ne) would have a basis set of five functions (two s functions and
three p functions).
In a minimum basis set, a single basis function is used for each
atomic orbital on each constituent atom in the system.
The most common minimal basis set is STO-nG, where n is an
integer. This \(n\) value represents the number GTOs used to
approximate the Slater Type orbital (STO) for both core and valence
orbitals. Minimal basis sets typically give rough results that are
insufficient for research-quality publication, but are much cheaper
(less calculations requires) than the larger basis sets discussed
below. Commonly used minimal basis sets of this type are: STO-3G,
STO-4G, and STO-6G.
Two is Often Better than One
Minimal basis sets are not flexible enough for accurate
representation of, which requires the use multiple functions to
represent each atomic orbital. The distribution of the electron
density of valence electrons is better represented by the sum of
two orbitals with different "effective charges". This is a
double-\(\zeta\) basis sets and includes split-valence set (inner
and valence) and linear combination of two orbitals of same type,
but with different effective charges (i.e., \(\zeta\)). This
flexibility can be used to generate atomic orbital of adjustable
sizes.
For example, the double-zeta basis set allows us to treat each
orbital separately when we conduct the Hartree-Fock
calculation.
\[ \phi_i = a_1 \phi_{2s}^{STO}(r,
\zeta_1) + a_2\phi_{2s}^{STO}(r, \zeta_2) \label{11.2.1} \]
The 2s atomic orbital approximated as a sum of two STOs. The two
equations are the same except for the value of \(\zeta\) which
accounts for how large the orbital is. The constants \(a_1\) and
\(a_2\) determines how much each STO contributes to the final
atomic orbital, which will vary depending on the type of atom that
the atomic orbit (i.e., hydrogen and lithium orbitals will have
different \(a_1\), \(a_2\), \(\zeta_1\), and \(\zeta_2\)
values).
Extended Basis Sets
The triple and quadruple-zeta basis sets work the same way,
except use three and four STOs instead of two like in
\(\ref{11.2.1}\). The typical trade-off applies here as well,
better accuracy, however with more expensive calculations. There
are several different types of extended basis sets including: n
split-valence, n polarized sets, n diffuse sets, and n correlation
consistent sets. The notation of this sort of basis set (with a
Gaussian basis) is
for describing split-valence basis set. \(N\) is the number of
Gaussian functions describing inner-shell orbitals, while the
hyphen denotes a split-valence set. \(M\) and \(P\) designate the
number of Gaussian functions used to fit the two orbitals of the
valence shell:
A minimal basis set is when one basis function for each atomic
orbital in the atom, while a double-\(\zeta\), has two two basis
functions for each atomic orbital. Correspondingly, a triple and
dquarupe-\(\zeta\) set had three and four basis functions for each
atomic orbital, respectively. Higher order basis set have been
constructed too, e.g., 5Z, 6Z,)..
There are hundreds of basis sets composed of Gaussian-type
orbitals (Figure Figure 11.3.1
). The smallest of these
are called minimal basis sets, and they are typically composed of
the minimum number of basis functions required to represent all of
the electrons on each atom. The largest of these can contain dozens
to hundreds of basis functions on each atom.
