Essentials of Computational Chemistry

13.3 BOUNDARIES THROUGH BONDS 469
of theory with a double-ζ basis set, reducing the system by 35 heavy atoms and 30 hydrogen
atoms substantially reduced the total number of basis functions. The necessary MM energies
were then computed with the UFF force field. Application of the model in this fashion has
been especially attractive within the organometallic community, where large ligands can often
be regarded as having a core portion that is electronically important, and remaining regions
that are not. Thus, for example, Matsubara et al. (1996) have used combined DFT/MM
models to study dihydrogen activation by platinum with different phosphine ligands, and
Deng et al. (1997) have used other DFT/MM models to study the role of bulky substituents
in Brookhart-type Ni(II) diimine-catalyzed olefin polymerization.
The alternative motivation for the second equality of Eq. (13.6) arises in cases where a
force field may be regarded as being reasonably accurate except perhaps for some specific
quantum mechanical effect(s) not well accounted for in the functional form of the force field.
For example, French et al. (2000) constructed a (φ,ψ) potential energy surface for the torsions
about the anomeric linkages in sucrose by adjusting an MM3 surface for the full molecule
based on the difference between HF/6-31G(d) and MM3 surfaces for a tetrahydropyrantetrahydrofuran
ether model (i.e., sucrose without any hydroxyl groups, Figure 13.4). The
MM3 force field exhibits a weakness in accounting for the so-called ‘anomeric effect’ in
sugars (see Section 2.2.3). By correcting for this weakness using the QM results, French
et al. were able to demonstrate that a sizable number of crystal structures containing sucrose
moieties that had previously been assumed to be adopting abnormally high-energy conformations
were instead in low-energy regions of the surface.
Note that the embedding philosophy of Eq. (13.6) may be applied more generally than
simply in the context of QM/MM calculations. For example, one can imagine situations
where the importance of a high-level accounting for electron correlation effects may be
restricted to a small region of a large system, but the full system still requires an overall QM
treatment. In such an instance, two different QM levels might be used in Eq. (13.6) instead
of one QM and one MM level; obviously, the more efficient QM level is the one applied to
the large system. For example, Sherer and Cramer (2001) studied the context dependence of
the pKa of the cytosine:2-aminopurine base pair in different double-helical RNA trimers by
taking the base pair itself to be the small system and the trimer to be the large system, and
choosing as the high and low levels of theory MP2/6-31G(d) and PM3, respectively, each
augmented with an aqueous continuum solvation model (Figure 13.5).
Note that Eq. (13.6) is written in terms of energies and not Hamiltonian operators. That
is because there is a certain ambiguity about how to define a wave function that would
be simultaneously appropriate for all of the Hamiltonian operators that would otherwise
appear on the r.h.s. This is not purely a notational issue, since it leaves open the question
of the geometries used for the different energy terms. For instance, one approach would
be to consider each energy on the r.h.s. to refer to complete geometry optimization at the
appropriate level. This is clearly the simplest method, since every energy determination may
be carried out completely independently of the others. However, if there are large differences
between the corresponding regions of any pair of geometries, it calls into question the validity
of the overall energy expression.