1549380323-Statistical Mechanics Theory and Molecular Simulation

Wang–Landau sampling 347

simulation in the window. Because we have assumed a specific form forthe distribution,
we can calculate the unbiased approximation to the derivative dAk/dq in thekth
window
dAk
dq

=

1

βσ^2 k

(q−q ̄k)−κ(q−s(k)). (8.8.21)

Letnwandnrepresent the number of umbrella windows and thermodynamic inte-
gration points, respectively. We now obtain the full derivative profile dA/dqat each
integration pointq(i)by “gluing” the windows together as in the original WHAM
procedure:
dA
dq(i)

=

∑nw

k=1

Ck(q(i))

dAk

dq

∣

∣

∣

∣

q=q(i)

. (8.8.22)

The coefficientsCk(q) satisfy eqn. (8.8.7). Eqns. (8.8.20), (8.8.21), and (8.8.22) are the
starting points for the development of a weighted histogram method that is consider-
ably simpler than the one developed previously as it eliminates the global constantA 0.
Once the values of dA/dq(i)are obtained (see Problem 8.12), a numerical integration
is used to obtain the full free energy profileA(q).

8.9 Wang–Landau sampling

In Section 7.6 of Chapter 7, we introduced the Wang–Landau approach for obtaining
a flat density of energy statesg(E). There we showed that in addition to a move in
configuration space fromr 0 torleading to an energy change fromE 0 toEwith a
Metropolis acceptance probability

acc(r 0 →r) = min

[

1 ,

g(E 0 )

g(E)

]

, (8.9.1)

the density of statesg(E) is scaled at each energyEvisited by a factorf:g(E)→
fg(E). Herer 0 is a complete set of initial Cartesian coordinates, andris the complete
set of trial Cartesian coordinates. Initially, we takeg(E) to be 1 for all possible energies.
The Wang–Landau sampling scheme has been extended to reaction coordinates
by F. Calvo (2002). The idea is to use the Wang–Landau scalingf(see Section 7.6)
to generate a function that approaches the probabilityP(s) in eqn. (8.7.1) over many
Monte Carlo passes. Letg(s) be a function that we initially take to be 1 over the entire
range ofs, i.e. over the entire range of the reaction coordinateq 1 =f 1 (r 1 ,...,rN)≡
f(r). Leth(s) = lng(s) so thath(s) is initially zero everywhere. A Monte Carlo
simulation is performed with the Metropolis acceptance rule:

acc(r 0 →r) = min

[

1 ,

exp (−βU(r))

exp (−βU(r 0 ))

g(s 0 )

g(s)

]

= min

[

1 ,

exp (−βU(r))

exp (−βU(r 0 ))

exp (−h(s))

exp (−h(s 0 ))

]

. (8.9.2)

Heres 0 =f 1 (r 0 ) ands=f 1 (r). In addition to this acceptance rule, for each values
of the reaction coordinateq 1 =f 1 (r) visited, the functionh(s) is updated according