5 Basics of Molecular Modeling and Molecular Simulation
223
for the reason that it is based on the importance sampling of the equilibrium Boltzmann distribution. The simulation step can be arbitrarily defined for the integration in Eq. (5.8). The MD simulations are better to be used if we want to study
non-equilibrium systems, or to reproduce microscopic dynamics of particles in the
systems.
The default MC simulation implements the NVT ensemble. However, in practice, it
is necessary for us to implement different ensembles, for which a series of techniques
are often applied. As we have previously suggested, the key of the algorithm is to use
a Monte Carlo procedure that introduces a random walk in the regions of phase space
that contribute to the ensemble average. The acceptance rules are determined to attain
the needed probability distribution, where the detailed balance condition is fundamental. Thus, to sample different distributions according to different ensembles, we
can use the following procedure:
1. Determine the distribution that needed to be sampled.
2. Impose the detailed balance condition:
M (o → n) = M (n → o)
(5.5.17)
where M (o → n) refers to the flow from configuration o to n, which is the product
of the probability of the emergence of configuration n from configuration o, which
is α(o → n), and the acceptance probability acc(o → n):
M (o → n) = α(o → n) × acc(o → n)
(5.5.18)
3. Determine the probability for a particular configuration
4. Come up with the condition that acceptance rules should meet.
This procedure is highly general, the details of which may vary in accordance with
the ensemble that researchers are interested in. The readers should be clear that one
should be extremely cautious when treating different ensembles with the MC method,
since errors may easily be introduced. The choice of ensembles for Monte Carlo simulations is very broad. Isobaric-isothermal, constant-stress-isothermal, grand canonical, and microcanonical ensembles are all suitable. For further information, one may
refer to Refs. [8–11].
5.6 Molecular Dynamics (MD) Simulation
5.6.1 Idea of Molecular Dynamics
The substantial idea of MD is to solve classical many-body Newton’s equations
of motion numerically. The typical procedure of a MD simulation is presented in
Fig. 5.3. It should be noted that how to obtain the empirical force field
F ij
is vital
223
for the reason that it is based on the importance sampling of the equilibrium Boltzmann distribution. The simulation step can be arbitrarily defined for the integration in Eq. (5.8). The MD simulations are better to be used if we want to study
non-equilibrium systems, or to reproduce microscopic dynamics of particles in the
systems.
The default MC simulation implements the NVT ensemble. However, in practice, it
is necessary for us to implement different ensembles, for which a series of techniques
are often applied. As we have previously suggested, the key of the algorithm is to use
a Monte Carlo procedure that introduces a random walk in the regions of phase space
that contribute to the ensemble average. The acceptance rules are determined to attain
the needed probability distribution, where the detailed balance condition is fundamental. Thus, to sample different distributions according to different ensembles, we
can use the following procedure:
1. Determine the distribution that needed to be sampled.
2. Impose the detailed balance condition:
M (o → n) = M (n → o)
(5.5.17)
where M (o → n) refers to the flow from configuration o to n, which is the product
of the probability of the emergence of configuration n from configuration o, which
is α(o → n), and the acceptance probability acc(o → n):
M (o → n) = α(o → n) × acc(o → n)
(5.5.18)
3. Determine the probability for a particular configuration
4. Come up with the condition that acceptance rules should meet.
This procedure is highly general, the details of which may vary in accordance with
the ensemble that researchers are interested in. The readers should be clear that one
should be extremely cautious when treating different ensembles with the MC method,
since errors may easily be introduced. The choice of ensembles for Monte Carlo simulations is very broad. Isobaric-isothermal, constant-stress-isothermal, grand canonical, and microcanonical ensembles are all suitable. For further information, one may
refer to Refs. [8–11].
5.6 Molecular Dynamics (MD) Simulation
5.6.1 Idea of Molecular Dynamics
The substantial idea of MD is to solve classical many-body Newton’s equations
of motion numerically. The typical procedure of a MD simulation is presented in
Fig. 5.3. It should be noted that how to obtain the empirical force field
F ij
is vital