1549380323-Statistical Mechanics Theory and Molecular Simulation

358 Free energy calculations

H=

∑n

α=1

p^2 α

2 mα

+

∑N

i=1

p^2 i

2 mi

+U(r 1 ,...,rN) +

∑n

α=1

1

2

k(sα−fα(r))^2 (8.11.9)

for sampling the distribution in eqn. (8.11.8). The exact probability distribution in
eqn. (8.6.4) is recovered in the limitκ→ ∞. The extended variabless 1 ,...,snare
coupled via a harmonic potential to thencollective variables defined by the trans-
formation functionsqα=fα(r). Note that the physical variables are in their normal
Cartesian form in eqn. (8.11.9). In this scheme, which is known as “temperature-
accelerated molecular dynamics” or driven adiabatic free-energy dynamics (d-AFED),
we apply adiabatic conditions of Section 8.10 on the extended phase-space variables
rather than directly on the collective variables. In doing so, we circumvent the need

RG

NH

(^0) 3.5 4 4.5 5 5.5
1
2
3
4
5
0
2
4
6
8
10
12
0
1
2
3
4
5
3.5 4.0 4.5 5.0 5.5
N
H
R G(A)
o
0
2
4
6
8
10
12
Fig. 8.9Free energy surface of an alanine hexamer generated using the d-AFED method.
The energy scale on the right is in kcal/mol.
for explicit variable transformations. Thus, we assign the variabless 1 ,...,sna tem-
peratureTs≫Tand massesms≫mi. The harmonic coupling is used in much the
same way as in umbrella sampling, except that the dynamics of the extended vari-
abless 1 ,...,sneffectively “drag” the collective variables of interest over the full range
of their values, thereby sampling the free energy hypersurface.As in the method of
Section 8.10, the equations of motion need to be coupled to thermostats at the two
different temperatures in order to ensure proper canonical sampling. The free energy
surface is then approximated by the adiabatic probability distribution generated in
the extended variabless 1 ,...,sn
A(q 1 ,..,qn)≈A(s 1 ,...,sn) =−kTslnPadb(s 1 ,...,sn) (8.11.10)
and becomes exact in the limitκ→ ∞. Because the temperature-accelerated scheme
does not require explicit transformations, it improves on the flexibility of the adiabatic