Chemistry Reference
In-Depth Information
Here we are using a
basis set
to approximate the unknown function
f
ð
x
Þ
. The
x
i
basis functions are
. The expansion coefficients,
c
i
, are deter-
mined by some sort of procedure that adjusts their values in order to obtain
the best fit to the function
f
f
:
i
¼
0
;
1
;
2
;
3
g
ð
x
Þ
. The approximation can generally be improved
by using a larger basis set,
c
2
x
2
c
3
x
3
c
4
x
4
c
5
x
5
f
ð
x
Þ
c
0
þ
c
1
x
þ
þ
þ
þ
½
39
and it becomes exact in the limit of an infinitely large or complete basis set
(CBS):
1
c
2
x
2
c
3
x
3
c
i
x
i
f
ð
x
Þ¼
c
0
þ
c
1
x
þ
þ
þ¼
½
40
i
¼
0
In quantum chemistry we are concerned with approximating a molecular
wave function,
c
, rather than a simple function of a single variable,
f
ð
x
Þ
. In the
Hartree-Fock approximation, the many-electron wave function,
, is approxi-
mated with the antisymmetrized product of one-electron molecular orbitals
(MOs). As you might expect, powers of
x
are not necessarily the best choice
for a basis set in which to expand these one-electron functions. It does not
require too much chemical intuition to recognize that the analytical wave func-
tions for one-electron atoms (i.e., the
s
,
p
, and
d
orbitals shown in general chem-
istry textbooks) might provide a good set of basis functions in which to expand
the molecular orbitals. After all, molecules are made of atoms. So, why not build
molecular orbitals out of atomic orbitals? This is, of course, the familiar linear
combination of atomic orbitals to form molecular orbitals (LCAO-MO)
approximation. Both Slater and Gaussian atomic orbitals (AOs) provide fairly
convenient basis functions for electronic structure computations. Of course, not
all basis sets need to have a chemically motivated origin. For example, plane
wave basis sets owe their success to computational efficiency.
Unfortunately, a bigger AO basis set does not necessarily give better
results. Bigger is better only if the basis sets are properly constructed. In
1989, Dunning introduced the correlation-consistent family of basis sets,
121-
123
which was a huge advance in the field of convergent quantum chemistry.
They were designed to converge systematically to the complete basis set (CBS)
or 1-particle limit. These basis sets are typically denoted cc-pV
X
Z where
X
denotes the maximum angular momentum of the Gaussian atomic orbitals
in the basis set (2 for
d
functions, 3 for
f
functions, etc.) and is also referred
to as the cardinal number of the basis (D for double-
c
,
etc.). Because of the convergence properties of these basis sets, we can expect the
larger basis sets to be more reliable. [
X
basis set, T for triple-
¼
4 (or Q) is better than
X
¼
3 (or T),
which is better than
X
¼
2 (or D).] The same is not true of other families of basis
sets [e.g., 6-311G(2
df
;
2
pd
) vs. 6-31G(
d
;
p
) or TZ2P vs. DZP].
