Essentials of Computational Chemistry

13.3 BOUNDARIES THROUGH BONDS 467
the Langevin polarization law. Each dipole enters the Fock operator just as described above
(Luzhkov and Warshel 1992).
Much like the RISM method, the LD approach is intermediate between a continuum model
and an explicit model. In the limit of an infinite dipole density, the uniform continuum model
is recovered, but with a density equivalent to, say, the density of water molecules in liquid
water, some character of the explicit solvent is present as well, since the magnitude of the
dipoles and their polarizability are chosen to mimic the particular solvent (Papazyan and
Warshel 1997). Since the QM/MM interaction in this case is purely electrostatic, other nonbonded
interaction terms must be included in order to compute, say, solvation free energies.
When the same surface-tension approach as that used in many continuum models is adopted
(Section 11.3.2), the resulting solvation free energies are as accurate as those from ‘pure’
continuum models (Florián and Warshel 1997). Unlike atomistic models, however, the use
of a fixed grid does not permit any real information about solvent structure to be obtained,
and indeed the fixed grid introduces issues of how best to place the solute into the grid,
where to draw the solute boundary, etc. These latter limitations have curtailed the application
of the LD model.
13.3 Boundaries Through Bonds
All of the QM/MM models discussed this far, much like continuum models, envision partitioning
a chemical system into (at least) two distinct regions, where the boundary between
these regions is everywhere characterized by a very low level of electron density. That is, no
atoms on one side of the boundary are bonded to atoms on the other side. As a result, the
HQM/MM term in the Hamiltonian of Eq. (13.1) is restricted to non-bonded interactions.
The situation is vastly more complicated when the boundary between the QM and MM
regions passes across one or more chemical bonds. Somehow, the dangling valences from the
two separate regions must be joined in a chemically (and computationally) sensible fashion.
Developmental work is ongoing in this area; this section will focus on the current most
widely used procedures.
13.3.1 Linear Combinations of Model Compounds
Many efforts in molecular design make use of sterically demanding groups, e.g., t-butyl
groups, to enforce particular molecular geometries. Viewing the total molecule as some kind
of sum of its functional groups, the intent is for the interaction between the large groups and
the remainder of the molecule to be entirely steric in nature. In such a situation, the inclusion
of the bulky group(s) in a fully QM calculation may be regarded as pointlessly expensive,
since the size of the fragment(s) guarantees a large increase in the total number of QM basis
functions, but the non-polarity of the fragments also indicates little likelihood of perturbing
the electronic structure of the remainder of the molecule via electrostatic interactions (steric
interactions are, of course, fundamentally electronic exchange-repulsion interactions, but for
the moment we will ignore this level of detail and consider steric effects to be distinct
from more classical electrostatic interactions). Thus, there is a clear motivation for passing a