Essentials of Computational Chemistry

476 13 HYBRID QUANTAL/CLASSICAL MODELS
usually taken to be that atom’s contribution to a localized orbital from a fully QM calculation
on a slightly expanded region (that might itself have been capped at some boundary if
necessary). The population of the orbital – as fully paired spin density – may either be treated
as a free variable, or computed from the density matrix of the original QM calculation. At the
NDDO level, once the spatial orientation of the orbital and its s to p ratio have been set, its
orthogonality to all other orbitals may be very simply enforced in QM/MM calculations. To
maintain an overall zero charge on the QM + auxiliary regions, it is necessary that the total
number of electrons in the auxiliary orbitals be equal to the total number of such orbitals, so
some care must be taken to ensure excess charge does not introduce problems. The LSCF
method, then, looks very much like the link atom method, except that the orbitals describing
QM-atom–capping-atom bonds are not optimized as part of the SCF, but are instead treated
as frozen throughout. Extension of the LSCF formalism to ab initio HF and DFT levels of
QM theory has been described by Philipp and Friesner (1999).
In comparison, a larger auxiliary region is employed in the generalized hybrid orbital (GHO)
approach described by Gao et al. (1998). In this case, it is better to think of the QM/MM
boundary as passing through an atom instead of through bonds, as certain carbon atoms are
assigned both QM and MM character. On those atoms, three sp 3 orbitals are held frozen with
paired-spin-density populations equal to one minus one-third of the partial atomic charge the
atom would carry for MM purposes, i.e., there is an attempt to spread out the character of the
boundary atom over its frozen orbitals. The remaining orbital, pointing to the QM region, is
frozen in shape by orthogonality constraints, but its population and contribution to the various
MOs is free to vary according to the SCF procedure. Thus, there is again a similarity to the
link atom procedure, in that there is a fully optimized MO representing each bond at the
QM/MM frontier, but in this case the orbital is surrounded by a much more realistic charge
environment from the hybrid atom nucleus and its three frozen auxiliary orbitals. The three
different approaches are compared schematically in Figure 13.6.
A subtle but key difference in the methodologies is that the orbital containing the two
electrons in the C–X bond is frozen in the LSCF method, optimized in the context of an
X–H bond in the link atom method, and optimized subject only to the constraint that atom
C’s contribution be a particular sp hybrid in the GHO method. In the link atom and LSCF
methods, the MM partial charge on atom C interacts with some or all of the quantum system;
in the GHO method, it is only used to set the population in the frozen orbitals.
The GHO approach has been designed in such a way that the QM/MM atoms at the
boundary are intended to be transferable. Thus, hybrid atoms have modified semiempirical
parameters and force-field parameters for use in computing the QM and MM portions of
the QM/MM energy according to Eq. (13.4), supplemented by MM bond stretching, angle
bending, and torsional terms whenever any one atom in the relevant linkage is a purely MM
atom. The modifications have been made for the combination of the AM1 Hamiltonian and
the CHARMM force field so as best to reproduce structural, energetic, and charge results
from fully AM1 calculations for a spectrum of molecules bearing functional groups similar to
those found in proteins. A CHARMM/PM3 implementation has also been reported (Garcia-
Viloca and Gao 2004) as has the formalism for an ab initio Hartree–Fock GHO method (Pu,
Gao, and Truhlar 2004).