Essentials of Computational Chemistry

13.3 BOUNDARIES THROUGH BONDS 473
Of course, Eq. (13.6) admits to further generalization. Rather than dividing a system into
large and small models, there may be instances where a division into large, medium, and
small models may be advantageous, with increasingly smaller regions treated with increasingly
higher levels of theory. Svensson et al. (1996) generalized their geometry optimization
scheme to this more general case, demonstrating the method for MO/MO/MM combinations,
and refer to it as ONIOM, where the acronym, representing ‘our own n-layered integrated
molecular orbital molecular mechanics’ scheme, is meant to emphasize the typical inwardto-outward,
near-spherical layering of models that is typically chosen and is reminiscent
of the almost eponymous lachrymatory bulb. To provide yet another layer of modeling
when condensed-phase effects are of interest, the combination of ONIOM with the PCM
continuum solvation model has been described (Vreven et al. 2001) as has a model for
permitting explicit solvent molecules to morph from MM to QM while passing through a
buffer region surrounding the QM subsystem (Kerdcharoen and Morokuma 2002).
13.3.2 Link Atoms
Situations arise where the influence of the MM region on the QM region to which it is
bonded cannot be regarded simply as steric. In a large protein, for instance, polar and
possibly charged residues in an MM region inevitably will polarize a QM region in the same
protein. The only way to eliminate such QM/MM coupling is to include the entire protein
in the QM region, and such an approach is extremely impractical for anything other than a
possible single-point calculation at a fairly low level of electronic structure theory.
Of course, the strong coupling invoked here between the two regions is in no manner
different than that dealt with in Section 13.2.2. What is different is that now there are
interaction energy terms between the QM and MM regions that are not non-bonded terms,
these new terms being associated with the bonds cut by the QM/MM boundary. In practice,
coupled QM/MM calculations involving link atoms tend to adopt the following protocols
for computation of the various terms.
1. HQM is computed for the QM region capped with hydrogen atoms at every bond cut
by the QM/MM boundary. The Fock operator may be like that defined in Eq. (13.5).
However, since the capping hydrogen atom is not really a part of the system, the third
term on the r.h.s. is not evaluated when µ or ν is a basis function on a capping hydrogen;
similarly, no nuclear repulsion between the capping hydrogen nucleus and the MM atoms
is computed.
2. The energies of bonds cut by the QM/MM boundary are evaluated using the
standard MM bond-stretching term (i.e., as though the QM atom were an MM
atom). In addition, a very large force constant is applied to the fictitious bond angle
MM–atom–QM–atom–capping–H so that it remains essentially zero (note that this
connectivity choice avoids the difficulty of working with bond angles near π radians).
3. Angle bending energies involving two MM atoms and one QM atom are computed using
the standard force-field formulation. Angle bending terms involving one MM atom and