Essentials of Computational Chemistry

13.4 EMPIRICAL VALENCE BOND METHODS 481
follows a coordinate from one VB optimum to another. The full technical details are not
examined here, but an outline is provided – further details are available in the additional
reading at the end of the chapter.
To implement the reaction path following scheme in the EVB formalism, one defines a
λ-dependent mapping
Hλ = (1 − λ)H11 + λH22
(13.15)
where λ is the usual coupling parameter for FEP that runs from 0 to 1 in discrete steps.
Assuming sufficiently small steps in λ, one can define a free energy change using
G ∗ =
−RT ln〈e −(Hλ+1−Hλ)/RT
〉λ
(13.16)
λ
which has the usual appearance for FEP when molar energies are used (cf. Eq. (12.16)).
However, for every value of λ Eq. (13.15) defines an energy surface that is different from
the true PES, unless by coincidence at one or several points. This situation, of evaluating
free energy changes from trajectories moving on adjusted free energy surfaces, is reminiscent
of umbrella sampling (see Section 12.2.5), where energy modifications are introduced as a
function of certain geometric coordinates in order to force trajectories into otherwise low
probability regions of space. Just as is the case with umbrella sampling, the correct free
energy change for the transition from one VB minimum to the other must be determined
from an evaluation of each trajectory, using in particular the difference between the correct
potential and the one used to generate the trajectory (see Eq. (12.26)). In the EVB case, that
is the difference at any point between Hλ and the lowest eigenvalue of H as defined by
Eq. (13.14).
The results of such simulations can be used to further refine the various parameters
appearing in the VB energy expressions. In the case presented here, for instance, the parameters
would be adjusted to provide the proper pKa for water. The motivation for the extensive
parameter validation is typically then to move the EVB system into an environment where
experimental data are not well understood, e.g., an enzyme active site. Thus, the procedure
outlined above was used by ˚Aqvist and Warshel (1992) to model the initial deprotonation
of water that is the rate-determining step in the hydration of carbon dioxide to bicarbonate
catalyzed by carbonic anhydrase. After optimizing the EVB parameters on gas-phase and
aqueous water–water proton transfers, the surrounding medium was changed to that of the
solvated enzyme (represented by a force field and thus interacting with the VB Hamiltonian
just as described above for surrounding water). Following this approach they obtained
very close agreement with the experimentally measured catalytic effect of the enzyme. By
inspecting simulation snapshots from the portion of the free-energy curve corresponding
to the transition state region, they were able to gain structural insight into the catalytic
mechanism of proton translocation.
13.4.3 Generalization to QM/MM
The EVB method as outlined above is not formally a QM/MM method during the course of
any simulation. Instead, the connection to QM is that the parameters required for the EVB