Essentials of Computational Chemistry

466 13 HYBRID QUANTAL/CLASSICAL MODELS
about a QM anion has the additional benefit of substantially reducing any likely instabilities
associated with charge penetration (see Section 11.4.1.3).
13.2.3 Fully Polarized Interactions
Allowing the QM system to be polarized by the MM charges without at the same time
accounting for polarization of the molecules comprising the MM system may be regarded
as being possibly unbalanced. One approach for including polarizability in the MM system
has already been described in Chapter 12, and its extension to a QM/MM system is algorithmically
trivial. Thus, each MM molecule or atom is assigned a polarizability tensor α,
and the induced dipole at each polarizable center is determined from Eq. (12.30); in the
QM/MM system, the electric field E has the same components from the MM partial charges
and induced dipoles as in a fully classical system, and an additional component deriving
from the nuclei and electronic wave function of the QM system that is straightforward to
calculate. The interaction of the induced dipoles with the MM partial charges (Eq. (12.31))
and with one another (Eq. (2.23)) are added in the HMM term of Eq. (13.1). In addition,
the induced dipoles interact with the nuclei of the QM system according to Eq. (12.31),
and with the electronic wave function as the expectation value of the operator equivalent of
Eq. (12.31) (thereby adding additional one-electron integrals to the Fock operator, one for
each induced dipole).
The evaluation of all of these terms must proceed iteratively until self-consistency is
reached, since the induced dipoles and the relaxing QM wave function modify the electric
field on which the induced dipoles are dependent. Thus, the increase in computational
resources required to include MM polarizability can be quite significant – one order of
magnitude is not uncommon. Comparisons between QM/MM systems modeled with and
without MM polarizability have been largely equivocal on the utility of its inclusion (adding
alternative three-body correction terms has also been examined for the hydrated manganous
ion (Loeffler, Yague, and Rode 2002) and was similarly found to lead to no significant
improvement in describing hydration structure). Given its very high cost of implementation,
there seems to be little point in carrying the model to this degree of physicality. However,
the lack of improvement in many cases may be attributable to the polarizability being added
post facto to an already existing force field. By virtue of fitting to experiment, formally nonpolarizable
force fields must include polarization in some average way into their parameters,
making it less likely that additional explicit accounting for polarization will show dramatic
effects. It is likely that only ongoing efforts aimed at developing fully polarizable force fields
from scratch will prove definitive in determining the level of additional physical insight that
may be gained from having polarization present in explicit form (see, for instance, Banks
et al. 1999).
Although complete, fully polarizable QM/MM schemes are computationally demanding, a
simplified version of this formalism was arguably the first QM/MM approach to be described
(Warshel and Levitt 1976), and the method still sees some use today. The simplification
involves replacing explicit, polarizable MM molecules with a three-dimensional grid of
fixed, polarizable dipoles – each a so-called Langevin dipole (LD) as it is required to obey