Essentials of Computational Chemistry

82 3 SIMULATIONS OF MOLECULAR ENSEMBLES
volume, constant temperature ensemble (NVT ensemble), the probability p of ‘accepting’
point q2 is
p = min 1, exp(−E2/kBT)
exp(−E1/kBT)
(3.34)
Thus, if the energy of point q2 is not higher than that of point q1, the point is always accepted.
If the energy of the second point is higher than the first, p is compared to a random number
z between 0 and 1, and the move is accepted if p ≥ z. Accepting the point means that the
value of A is calculated for that point, that value is added to the sum in Eq. (3.33), and the
entire process is repeated. If second point is not accepted, then the first point ‘repeats’, i.e.,
the value of A computed for the first point is added to the sum in Eq. (3.33) a second time
and a new, random perturbation is attempted. Such a sequence of phase points, where each
new point depends only on the immediately preceding point, is called a ‘Markov chain’.
The art of running an MC calculation lies in defining the perturbation step(s). If the steps
are very, very small, then the volume of phase space sampled will increase only slowly over
time, and the cost will be high in terms of computational resources. If the steps are too large,
then the rejection rate will grow so high that again computational resources will be wasted
by an inefficient sampling of phase space. Neither of these situations is desirable.
In practice, MC simulations are primarily applied to collections of molecules (e.g., molecular
liquids and solutions). The perturbing step involves the choice of a single molecule,
which is randomly translated and rotated in a Cartesian reference frame. If the molecule is
flexible, its internal geometry is also randomly perturbed, typically in internal coordinates.
The ranges on these various perturbations are adjusted such that 20–50% of attempted moves
are accepted. Several million individual points are accumulated, as described in more detail
in Section 3.6.4.
Note that in the MC methodology, only the energy of the system is computed at any given
point. In MD, by contrast, forces are the fundamental variables. Pangali, Rao, and Berne
(1978) have described a sampling scheme where forces are used to choose the direction(s)
for molecular perturbations. Such a force-biased MC procedure leads to higher acceptance
rates and greater statistical precision, but at the cost of increased computational resources.
3.5 Ensemble and Dynamical Property Examples
The range of properties that can be determined from simulation is obviously limited only
by the imagination of the modeler. In this section, we will briefly discuss a few typical
properties in a general sense. We will focus on structural and time-correlation properties,
deferring thermodynamic properties to Chapters 10 and 12.
As a very simple example, consider the dipole moment of water. In the gas phase, this
dipole moment is 1.85 D (Demaison, Hütner, and Tiemann 1982). What about water in
liquid water? A zeroth order approach to answering this problem would be to create a
molecular mechanics force field defining the water molecule (a sizable number exist) that
gives the correct dipole moment for the isolated, gas-phase molecule at its equilibrium