Chemistry Reference
In-Depth Information
the
augmented
correlation-consistent basis sets. The aug- prefix denotes that a
set of diffuse functions has been added for every angular momentum present in
the basis set. This makes the aug-cc-pVXZ basis sets truly enormous for large-
basis-set cardinal numbers X. For example, the cc-pVQZ basis set for carbon
contains 5 (contracted)
s
functions, 4 sets of
p
functions, 3 sets of
d
functions,
2 sets of
f
functions, and a set of
g
functions, for a total of 55 functions. The
augmented version, aug-cc-pVQZ, adds a single set of diffuse
s
,
p
,
d
,
f
, and
even
g
functions, or an additional 25 functions, to yield a total of 80 con-
tracted basis functions for a single carbon atom! Note that our counting of
basis functions assumed so-called pure angular momentum or spherical har-
monic functions (somewhat misnamed because they are not actually complex
functions), meaning that there are 5
d
functions, 7
f
functions, 9
g
functions,
etc. We note for the sake of pedagogy that some programs default to using
these pure angular momentum functions, while others default to using Carte-
sian functions — e.g., 6
d
functions,
x
2
,
y
2
,
z
2
,
xy
,
xz
, and
yz
. It is important to
know whether Cartesian functions or pure angular momentum functions are
being used and to be aware of this when attempting to compare your results to
published results or to results from other programs that might have different
defaults.
The aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets thus
contain 384, 828, and 1512 contracted Gaussian functions, respectively,
for the benzene dimer. Straightforward application of the MP2 method
with these basis sets to the sandwich benzene dimer yields the dotted curves
in Figure 3. Compared to the largest and most complete basis set, aug-cc-
pVQZ, the aug-cc-pVDZ basis greatly overestimates the binding energy
near equilibrium (by more than 2 kcal mol
1
, or 50%). Improving the basis
from aug-cc-pVDZ to aug-cc-pVTZ yields results much closer to the aug-cc-
pVQZ values, but the binding energy remains overestimated by 0.7 kcal
mol
1
. Given the huge differences between MP2 energies with these basis
sets, Figure 3 gives us no confidence that even the enormous aug-cc-pVQZ
basis has yet reached the complete basis set (CBS) limit.
Basis Set Superposition Errors and the Counterpoise
Correction
The reason that the MP2/aug-cc-pVXZ results approach the MP2/CBS
limit from below is that weakly bound complexes feature large basis set super-
position errors (BSSEs). A tutorial on BSSEs, in theory and in practice, was
presented earlier in this topic series.
48
Qualitatively, each molecule in the com-
plex desires more basis functions to describe electron correlation effects more
accurately. Each molecule can gain access to more basis functions, and can
thus be stabilized, by moving closer to another molecule. This creates an arti-
ficial stabilization that lowers the energy when the molecules come closer
together—simply because the electrons around each molecule can be stabilized
