Chemistry Reference
In-Depth Information
by partially occupying the other molecule's basis functions. The counterpoise
(CP) correction of Boys and Bernardi
49,50
is a technique to remove this kind of
basis set superposition error, and it is widely employed in studies of weakly
interacting complexes. In the absence of the CP correction, interaction energies
such as those in Figure 3 can be computed by simply subtracting the sum of the
monomer energies from the dimer energy. For a dimer, the CP correction is
determined by evaluating the energy of each monomer in the
dimer
basis:
that is, a monomer's energy is evaluated using all the basis functions of that
monomer
and the basis functions of the other monomer of the dimer
. When
one wishes to include basis functions at a particular point in space, but with-
out the associated nuclear charge and electrons, this is called using a
ghost
atom
. Most electronic structure programs have the ability to use ghost atoms
to facilitate CP correction computations. The difference between a monomer's
energy in isolation and the monomer's energy in the presence of the ghost
functions of the other monomer constitutes the CP correction for that mono-
mer. Of course, one must evaluate the CP correction for each monomer in a
dimer (unless the monomers are symmetry equivalent, in which case only one
computation is required because the other monomer would give the same
results, and so the correction could be multiplied by 2).
It is helpful to look at some simple equations to see exactly how the CP
correction is applied. We may write the interaction energy of a bimolecular
complex consisting of molecules A and B as
E
A
AB
E
A
E
B
E
int
¼
½
1
where
E
A
AB
represents the energy of the bimolecular complex (subscript AB)
using its usual basis set (the union of the basis sets on A and B, denoted by
superscript AB), and
E
A
and
E
B
represent the energies of the isolated molecules
A and B using their own usual molecular basis sets. The degree of ''artificial''
stabilization, or CP error, that molecule A experiences due to the presence of
the basis functions of molecule B present in the dimer computation is
E
CP err
A
E
A
A
E
A
¼
½
2
Of course, a similar equation holds for molecule B:
E
CP err
B
E
AB
B
E
B
¼
½
3
If one then subtracts the total CP error,
E
CP err
A
E
CP er
B
, from the equation for
the interaction energy,
E
int
, in Eq. [1], one obtains an equation for the CP-
corrected interaction energy:
þ
E
CP corr
int
E
A
AB
E
A
A
E
AB
B
¼
½
4
