Handbook of Solvents - George Wypych - ChemTech - Ventech!

474 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
Sometimes, instead of using the ρ(Γ), a “weight” function w(Γ) is used:
()
ρΓ
() Γ
= w
Q
Q =∑w Γ
Γ
A
ensemble =
()
∑ w()() Γ A Γ
Γ
∑ w()
Γ
Γ
[8.101]
[8.102]
[8.103]
Q is the partition function of the system.
From Eq. [8.102], it is possible to derive an approach to the calculation of thermodynamics
properties by direct evaluation of Q for a particular ensemble. Q is not directly estimated,
but the idea of generating a set of states in phase space sampled in accordance with
the probability density ρ(Γ) is the central idea of MC technique. Proceeding exactly as done
for MD, replacing an ensemble average as in Eq. [8.103] with a trajectory average as in Eq.
[8.99], a succession of states is generated in accordance with the distribution function ρNVE for the microcanonical NVE ensemble. The basic aim of the MC method (so-called because
of the role that random numbers play in the method), which is basically a technique for performing
numerical integration, is to generate a trajectory in phase space that samples from a
chosen statistical ensemble. It is possible to use ensembles different from the
microcanonical: the only request is to have a way (physical, entirely deterministic or stochastic)
of generating from a state Γ(τ) a next state Γ(τ + 1). The important point to be
stressed is that some conditions have to be fulfilled: these are that the probability density
ρ( Γ) for the ensemble should not change as the system evolves, any starting distribution
ρ( Γ) should tend to a stationary solution as the simulation proceeds, and the ergodicity of
the systems should hold. With these recommendations, we should be able to generate from
an initial state a succession of points that in the long term are sampled with the desired probability
density ρ( Γ ) . In this case, the ensemble average will be the same as a “time average”
(see Eq. [8.99]). In a practical simulation, τ runs over the succession of states generated following
the previously mentioned rules, and is a large finite number. This approach is exactly
what is done in an MC simulation, in which a trajectory is generated through phase
space with different recipes for the different ensembles. In other words, in a MC simulation
a system of particles interacting through some known potential is assigned a set of initial coordinates
(arbitrarily chosen) and a sequence of configurations of the particles is then generated
by successive random displacements (also called ‘moves’).
If f(R) is an arbitrary function of all the coordinates of the molecule, its average value
in the canonical ensemble is:
f
()
∫
V R
fre dR
=
V R
e dR
∫
−β
( )
−β
( )
where β = 1/kT and V(R) is the potential energy.
[8.104]