Essentials of Computational Chemistry

478 13 HYBRID QUANTAL/CLASSICAL MODELS
That is, these two wave functions are taken as the basis functions that are linearly combined
to describe the system at an arbitrary point along the proton-transfer coordinate (the proton
that is transferred has been labeled with an asterisk and the two oxygen atoms labeled a and
b for ease of subsequent discussion).
Most chemists are quite comfortable thinking of chemical structure and reactivity in terms
of valence bond notions – the resonance structures so often invoked in organic chemistry
are one example of this phenomenon – so this approach has conceptual appeal. From a
computational standpoint, the issue is how to derive a Hamiltonian operator that will act on
VB wave functions so as to deliver useful energies.
13.4.1 Potential Energy Surfaces
VB wave functions like those in Eqs. (13.8) and (13.9) are in some sense MM-like representations
of a chemical system. We insist, for instance, that the system described by Eq. (13.8)
always has H ∗ bound to O a , irrespective of the length of the bond at any given moment.
Obviously, however, if the separation between those two atoms is large, it is absurd to
imagine that there is a bond between them. Put more quantum mechanically, one would
say that the contribution of that VB basis function to the ‘correct’ adiabatic ground state is
very small. Thinking of the adiabatic wave function as a linear combination of the VB basis
functions, we would say that the coefficient of 1 in a CI-like expansion should be small,
which is equivalent to saying that it must be at rather high energy relative to states making
larger contributions.
All of these qualitative considerations suggest that a first step to designing a Hamiltonian
for the VB system would be to use a simple force field where the making and breaking O–H ∗
bonds are described by a Morse potential, the other OH bonds and bond angles in each of
the two molecular fragments are described by harmonic potential functions, and interactions
between the two fragments are modeled by standard electrostatic and LJ potential functions.
That is, we would have for the uncharged VB Hamiltonian
−αOH(rOaH∗ −rOH,eq)
H1 = DOH 1 − e 2
+
H=H ∗
1
2 kOH(rOH − rOH,eq) 2
+ 1
2 kHOH(θHO a H ∗ − θHOH,eq) 2 + 1
2 kHOH(θ HO b H − θHOH,eq) 2
+
qiqj
εrij
i∈a j∈b
+
4εij
i∈a j∈b
σij 12 rij
−
σij
rij
6
(13.10)
where the fragments a and b are the molecules containing the oxygen atoms having the same
label. Although the bond involving H ∗ is unique in using a Morse potential, all MM terms
are otherwise standard and assume a defined connectivity consistent with Eq. (13.8).