Essentials of Computational Chemistry

196 6 AB INITIO HARTREE–FOCK MO THEORY
interaction energy as
E CP
bind = Ea∪b HF (AžB)AžB − E a∪b
HF (A)AžB − E a∪b
HF (B)AžB
+ [E a HF (A)AžB − E a HF (A)A] + [E b HF (B)AžB − E b HF (B)B]
(6.13)
where the subscripts appearing after the molecular species describe the geometry employed.
Thus, in the first line on the r.h.s., the energy of bringing the two monomers together, each
monomer already having the geometry it has in the complex, is computed using a consistent
basis set. Thus, in the monomer calculations, basis functions for the missing partner are
included in the calculation, even though the nuclei on which those functions are centered
are not actually there – such basis functions are sometimes called ghost functions. Since the
ghost functions slightly lower the energies of the monomers, the overall binding energy is
less than would be the case if they were not to be used. The second line on the r.h.s. of
Eq. (6.13) then accounts for the energy required to distort each monomer from its preferred
equilibrium structure to the structure found in the complex. Since it is not obvious where to
put the ghost functions when the monomer adopts its equilibrium geometry, the geometrydistortion
energies are computed using only the nuclei-centered monomer basis sets.
However, it must be noted that the borrowing of basis functions is only partly a mathematical
artifact. To the extent that some charge transfer and charge polarization take place as part
of forming the bimolecular complex, some of the borrowing simply reflects chemical reality.
Thus, CP correction always overestimates the BSSE, and there is no clear way to correct for
this overestimation. Indeed, Masamura (2001) has found from analysis of ion-hydrate clusters
that interaction energies computed with basis sets of augmented-polarized-double-ζ quality
or better were in closer agreement with complete basis-set results before CP correction than
after. As a result, there tend to be two schools of thought on how best to deal with BSSE.
Some researchers prefer to spend the time that would be required for CP correction instead on
the evaluation of Eq. (6.12) with a more saturated basis set. Since, in the limit of an infinite
basis, Eqs. (6.12) and (6.13) are equivalent, a demonstration of convergence of Eq. (6.12)
with respect to basis-set size is a reasonable indication of accuracy, at least at the HF level.
6.4.2 Geometries
Optimization of the molecular geometry at the HF level appears at first sight to be a daunting
task because of the difficulty of obtaining analytic derivatives (see Section 2.4.1). To take
the first derivative of Eq. (4.54) with respect to the motion of an atom, we can exhaustively
apply the chain rule term by term. Thus, we must determine derivatives of basis functions
and operators with respect to a particular coordinate, and this is not so hard, but we also
need to know the derivatives of the density matrix elements with respect to atomic motion,
and these derivatives are not obvious at all. However, Pulay (1969) discovered an elegant
connection between these very complicated derivatives and the much simpler derivatives of
the overlap matrix (which depend only on analytically known basis function derivatives).
This breakthrough led to rapid developments in computing higher-order derivatives and
optimization algorithms, and as a result, HF geometries are now quite efficiently available.