304 Monte Carlo
one is forced to use a finer temperature “grid” in order to have a reasonable average
acceptance probability, thereby increasing the overhead of the method considerably.
Several improvements to the algorithm have been suggested to alleviate this problem.
For example, for biomolecules in aqueous solution, Berne and coworkers introduced a
modification of the algorithm in which attempted exchanges are madebetween coor-
dinates of the solute only rather than between the complete set ofcoordinates (Liu
et al., 2005).
7.6 Wang–Landau sampling
In this section, we will consider a rather different approach to Monte Carlo calcula-
tions pioneered by Wang and Landau (2001). Until now, we have focused on Monte
Carlo methods aimed at generating the canonical distribution of thecoordinates, mo-
tivated by the fact that the canonical partition function can be written in the form of
eqn. (7.3.35). However, let us recall that the canonical partition function can also be
expressed in terms of the microcanonical partition function Ω(N,V,E) as
Q(N,V,T) =
1
E 0
∫∞
0dEe−βEΩ(N,V,E) (7.6.1)(see eqn. (4.3.16)), whereE 0 is an arbitrary reference energy. Since we know thatN
andVare fixed, let us simplify the notation by droppingNandV in this section, set
E 0 = 1, and write eqn. (7.6.1) as
Q(β) =∫∞
0dEe−βEΩ(E). (7.6.2)Eqn. (7.6.2) suggests that we can calculate the partition function,and hence all ther-
modynamic quantities derivable from it, if we can devise a method to generate the
unknown function Ω(E) for a wide range of energies. Ω(E), in addition to being the
microcanonical partition function, is also referred to as thedensity of states, since it is
a measure of the number of microscopic states available to a systemat a given energy
E.^2 The probability, therefore, that a microscopic state with energyEwill be visited
is proportional to 1/Ω(E).
The approach of Wang and Landau is to sample the function Ω(E) directly and,
once known, calculate the partition function via eqn. (7.6.2). The rub, of course, is that
we do not actually know Ω(E)a priori, and trying to generate it using ordinary M(RT)^2
sampling and eqn. (7.3.36) is extremely inefficient. The Wang–Landau algorithm is a
simple yet elegant approach that can generate Ω(E) with impressive efficiency. We
begin by assuming that Ω(E) = 1 for all values ofE. Numerical implementation
requires that Ω(E) be discretized into a number of energy bins. Now imagine that
a trial move, such as a uniform displacement as in eqn. (7.3.37), is attempted. The
move changes the energy of the system fromE 1 toE 2. For such a move, the energy
(^2) In fact, if theE 0 is left off the prefactor in eqn. (3.2.20), then Ω(N,V,E) has units of inverse
energy.