Essentials of Computational Chemistry

434 12 EXPLICIT MODELS FOR CONDENSED PHASES
where λ is broken up into individual intervals of length dλ (they need not all have identical
widths, but in typical practice they do). Thus, for instance, in the HCN case we might decide
to divide λ into 20 segments having a width of 0.05 each. In the first ensemble, generated
for λ = 0, the energy difference associated with interaction with atom HD would no longer
be computed using Eq. (12.14), but instead according to
EHD = 0.05
−
= 0.05
aHH
aHH
r12 HAHD
r12 −
HBHD
bHH
r6 +
HBHD
qHBqHD εrHBHD
aHH
− bHH
r 6 HAHD
+ qHA qHD
εrHAHD
r12 −
HBHD
bHH
r6 +
HBHD
qHBqHD εrHBHD
+ 0.95
−
aHH
r12 HAHD
aHH
− bHH
r 6 HAHD
+ qHA qHD
εrHAHD
r12 HAHD
− bHH
r6 HAHD
+ qHAqHD
εrHAHD
(12.17)
The effect is to remove 95 percent of the unfavorable consequences of materializing the
HB atom, making the ensemble hopefully more relevant for the chimeric molecule than for
‘full’ HNC. Once sufficient statistics have been collected for this window, a fresh ensemble
is generated using a simulation for the chimera with λ = 0.05, and evaluating the energy
difference between λ = 0.10 and λ = 0.05. This process is repeated, interval by interval,
until λ reaches 1, at which point all of the Helmholtz free-energy changes for each interval
are summed together to give the total for HCN to HNC.
By creating new ensembles with each increase in λ, potentially offending water molecules
in the region of the nitrogen atom like the one mentioned above are ‘eased’ out of the way,
since in each new ensemble the presence of HB becomes more manifest. The cost, however,
is that now 20 simulations need to be undertaken instead of one (assuming an interval width
of 0.05 as in the example).
When one is generating an ensemble for a fractional value of λ, it is equally easy to
evaluate the energy change for λ − dλ as it is for λ + dλ. The former is equivalent to
imagining the reaction not as HCN → HNC but rather HNC → HCN. Evaluation in this
fashion thus simultaneously determines the forward and reverse free-energy changes from
the identical ensemble. In principle the free-energy change computed for the interval [λ →
λ + dλ] should be exactly the opposite of that computed for the interval [λ + dλ → λ].
In practice, however, this is rarely true, and the variations provide some indication of the
potential error in the FEP process. For instance, in Figure 12.2 the reverse mutation predicts a
negative free-energy change slightly larger in magnitude than the positive free-energy change
for the forward mutation. This difference is sometimes reported as the error in the simulation.
Because the free-energy change should be linear in λ if Eq. (12.15) is used (dotted line), the
hysteresis of the FEP diagram is sometimes used as a more conservative estimate of the error.
An alternative procedure is known as ‘double-wide sampling’. In this case, the ensemble
is generated by MC or MD methods for the Hamiltonian corresponding to a given value of
λ, but the evaluation of the free energy change is for the interval [λ − 0.5dλ → λ + 0.5dλ].
Thus, the total interval width is still dλ, but the evaluation is over half-step changes left
and right in the Hamiltonian parameters. In principle, this may lead to improved sampling