Essentials of Computational Chemistry

12.2 COMPUTING FREE-ENERGY DIFFERENCES 443
12.2.6 Technical Issues and Error Analysis
Free-energy simulations are extremely demanding in a technical sense, and it is well beyond
the scope of this book to fully prepare readers to apply the technology without further
instruction. Nevertheless, there are a few technical issues that arise (on top of those already
discussed for simulations in general in Section 3.6) that merit attention insofar as they
affect many published free-energy simulation results. Much more authoritative treatments
are available in the bibliography and suggested reading.
When perturbations from one molecule to another are carried out, there are two distinct
approaches that may be taken. The ‘single topology’ approach involves a single solute species
that is smoothly transformed from the first molecule to the second as a function of λ. In
the HCN/HNC example above, the single topology approach would involve not only the
steady disappearance of the carbon-bound hydrogen and the appearance of the nitrogenbound
hydrogen, but also any change in the C–N equilibrium length and force constant as it
transforms from a nitrile to an isonitrile bond type. In addition, if the atomic partial charges
on C and N were to be different for nitriles and isonitriles, these too would change as a
function of λ. The solute molecule at intermediate values of λ is thus truly chimeric.
The ‘dual topology’ approach, on the other hand, involves having the distinct initial and
final solutes simultaneously present, but no force-field interactions between the two are ever
calculated. The interactions of both are calculated with the surrounding medium in the
normal way, but at intermediate values of λ the total energy of the system will be derived as
a λ-dependent function of the two. The dual topology approach is simpler in implementation
but problems can arise if the two topologies drift away from one another during the course
of the simulation (for instance, if one solute were to leave the active site of an enzyme
while the other stayed in it, obviously the difference in binding free energies would not be
calculable). Both single and dual topology calculations continue to see about equal use.
As already mentioned above, the sudden appearance of atoms at positions in space occupied
by solvent molecules as the result of a mutation can lead to severe sampling problems.
As a rule, changes in van der Waals interactions must be introduced much more slowly
than changes in charge in order to maintain good equilibrium in ensemble averages. Since a
free-energy change is independent of the mutation path (assuming perfect sampling), paths
that carry out changes in charges more quickly than changes in van der Waals interactions
are not uncommon.
The discussion in this chapter has focused almost exclusively on computing changes in
the Helmholtz free energy A. However, most experimental measurements are carried out at
constant pressure, not constant volume, so the majority of thermochemical data is in the
form of Gibbs free energies G. As long as the total number of particles in a free-energy
simulation remains constant, almost all simulations assume that P V is zero, in which case
the Gibbs and Helmholtz free energy changes are identical (this is readily derived from
Eqs. (12.2)–(12.7)). When this is not the case, the additional contributions to G must be
explicitly accounted for.
Of the three methods discussed above, FEP, TI, and slow growth, the first two see far
more application than the third. The slow-growth condition, that the system is constantly
at, or at least very, very near equilibrium, is quite hard to maintain over the course of a