166
5 Complex Reactive Applications: A Forward Look to Open Science
outgoing body is replaced by its clone incoming from the corresponding position of
one of the surrounding cells.
Important for a proper molecular dynamics study is the correct definition of the
initial conditions, of the nature of the ensemble and of the collective indicators of
the ensemble of bodies considered. As already mentioned, this is particularly simple
for bimolecular collisions of simple molecules for which quantum-like states of the
(vibrational and rotational energies) reactants and products can be associated with
atom positions and momenta. However, this is not so for very large ensembles of
interacting bodies and large molecules. In this case, in order to calculate the average
value P of the observable property P rather than using the corresponding quantum
expression
P =
i e
−E i /k B T
i |P| i
i e −E i /k B T
(5.19)
one should adopt its classical statistical mechanics equivalent for which the property
to be calculated is formulated as
P =
P({W})F({W})d{W}
(5.20)
where
F({W}) =
ex p [−H ({W})/k B T ]
ex p [−H ({W})/k B T ] d{W}
(5.21)
is the Boltzmann distribution, U ({W}) the internal energy and {W} the r
N (coordinates) and p
N (momenta) of the N particles system. The value of these integrals
are usually estimated using the Monte Carlo technique (i.e., by sampling randomly
the various configurations of the system) possibly associated with an importance
sampling factor (i.e., by giving a weight to each sampled point). The Monte Carlo
technique has been already considered for electronic structures at the beginning of
Chap. 3 where it was pointed out that if one needs only the ratio of two integrals (of
which one is referred to a given reference configuration like in the case of a transition between two states). the Metropolis method can be adopted. In the Metropolis
method a random walk is constructed moving out of the region of space associated
with a given configuration (where the integrand is nonnegligible). In the random walk
we introduce subsequent random displacements according to some ad hoc criteria
(like the one requiring that at equilibrium the number of accepted moves from a state
to any other state is exactly canceled by the reverse move).
5.2.2 Some Popular Molecular Dynamics Codes
As already mentioned the clear advantage of classical mechanics techniques is their
easy extensibility to large numbers of atoms (of the order of several millions or more).
5 Complex Reactive Applications: A Forward Look to Open Science
outgoing body is replaced by its clone incoming from the corresponding position of
one of the surrounding cells.
Important for a proper molecular dynamics study is the correct definition of the
initial conditions, of the nature of the ensemble and of the collective indicators of
the ensemble of bodies considered. As already mentioned, this is particularly simple
for bimolecular collisions of simple molecules for which quantum-like states of the
(vibrational and rotational energies) reactants and products can be associated with
atom positions and momenta. However, this is not so for very large ensembles of
interacting bodies and large molecules. In this case, in order to calculate the average
value P of the observable property P rather than using the corresponding quantum
expression
P =
i e
−E i /k B T
i |P| i
i e −E i /k B T
(5.19)
one should adopt its classical statistical mechanics equivalent for which the property
to be calculated is formulated as
P =
P({W})F({W})d{W}
(5.20)
where
F({W}) =
ex p [−H ({W})/k B T ]
ex p [−H ({W})/k B T ] d{W}
(5.21)
is the Boltzmann distribution, U ({W}) the internal energy and {W} the r
N (coordinates) and p
N (momenta) of the N particles system. The value of these integrals
are usually estimated using the Monte Carlo technique (i.e., by sampling randomly
the various configurations of the system) possibly associated with an importance
sampling factor (i.e., by giving a weight to each sampled point). The Monte Carlo
technique has been already considered for electronic structures at the beginning of
Chap. 3 where it was pointed out that if one needs only the ratio of two integrals (of
which one is referred to a given reference configuration like in the case of a transition between two states). the Metropolis method can be adopted. In the Metropolis
method a random walk is constructed moving out of the region of space associated
with a given configuration (where the integrand is nonnegligible). In the random walk
we introduce subsequent random displacements according to some ad hoc criteria
(like the one requiring that at equilibrium the number of accepted moves from a state
to any other state is exactly canceled by the reverse move).
5.2.2 Some Popular Molecular Dynamics Codes
As already mentioned the clear advantage of classical mechanics techniques is their
easy extensibility to large numbers of atoms (of the order of several millions or more).