Essentials of Computational Chemistry

462 13 HYBRID QUANTAL/CLASSICAL MODELS
where i and j run over N QM electrons, k and l run over the K nuclei in the QM fragment,
and m runs over the M molecular mechanics atoms. The second equality in Eq. (13.4)
simply expands the QM Hamiltonian into its usual individual terms and uses Eq. (13.3) to
expand the QM/MM component of the Hamiltonian. The terms having no dependence on
the electronic coordinates – HMM, the QM-nuclei/MM-atom electrostatic interactions, and
the LJ interactions – may be taken outside of expectation value integrals, which are then
simply one by normalization of the wave function. The third equality simply collects terms
together in a convenient fashion.
Note that the only operator acting on the electronic wave function for the QM/MM system
that would not be present in the isolated QM system is that involving the charges of the MM
atoms. In operator formalism, these atoms behave exactly like QM nuclei, except that they
bear partial atomic charges instead of atomic-number-based charges. As such, they enter
into the standard Fock operator just as nuclear charges do, i.e., as part of the one-electron
operator. Elements of the QM/MM Fock matrix that minimize the energy computed from
Eq. (13.4) are thus calculated from the generalization of Eq. (4.54) as
Fµν =
µ
−1 2 ∇2
ν
−
+
λσ
Pλσ
QM
nuclei
k
Zk
µ
1
ν
−
rk
(µν|λσ ) − 1
2 (µλ|νσ)
MM
atoms
m
qm
µ
1
ν
rm
(13.5)
where only the third term on the r.h.s. is different from the usual QM expression. The
third term involves the computation of M one-electron integrals. Insofar as the bottlenecks
in HF theory tend to be assembly of the two-electron integrals or diagonalization of the
Fock matrix, the actual increase in computational time required for a QM/MM calculation
compared to a purely QM calculation on the same fragment can be quite small.
DFT equations analogous to Eqs. (13.4) and (13.5) can be derived in a similarly straightforward
way. Again, the ultimate influence of the MM system on the KS orbitals is made
manifest only by the appearance of additional one-electron integrals associated with the MM
atoms in the KS operator.
Of course, the simplicity of the QM/MM operator does not imply that it has only a small
effect. Large atomic partial charges placed near the QM fragment would be expected to
polarize the system strongly. Table 13.2 compares the dipole moments of the standard nucleic
acid bases at the AM1 level evaluated in the gas phase and in a QM/MM calculation carried
out modeling aqueous solvation with a periodic box of TIP3P water molecules. For comparison,
results from the AM1-SM2 aqueous continuum solvation model are also provided.
It is important to recognize how a QM/MM calculation like that for the nucleic acid base
solvated dipole moments is accomplished. We outline here a typical series of steps
1. Choose the particular QM and MM levels to be used.
2. Given those QM and MM levels, select a set of LJ parameters for the QM fragment. One
option is to use the same parameters for atoms in the QM fragment as those that would