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Molecular Monte Carlo Simulation
What is Molecular Monte Carlo
Simulation?
We already learned how random numbers could be used to
solve problems such as difficult multiple integrals. This
relatively straight-forward problem solving strategy is only
one aspect of Monte Carlo. More generally, Monte Carlo can
be used to simulate natural processes that are intractable by
conventional means.
All natural phenomena, such as the boiling point of a fluid,
the viscosity or hardness of a material etc, are caused by
interatomic interactions. Monte Carlo can be used to simulate
these interactions.
In some disciplines, the word ‘simulate’ is synonymous with
‘crude approximation.’ A ‘simulated’ solution is obtained
because the exact solution is intractable.
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Generalized MMC Algorithm
At the outset it is useful to have a generalized framework
for the MMC algorithm. MMC simulation works by
generating transitions between different states. This
involves :
– (a) generating a random trial configuration;
– (b) evaluating an ‘acceptance criterion’ by calculating
the change in energy and other properties in the trial
configuration; and
– (c) comparing the acceptance criterion to a random
number and either accepting or rejecting the trial
configuration
Generalized MMC Algorithm
(contd)
A general MMC Algorithm is illustrated below:
Part 1 Create an initial configuration (config).
Part 2 Generate a Markov Chain for Ncycles:
loop i 1 ... NCycles
Create a new configuration (configTrial).
Find the transition probability w(config, configTrial).
Generate a uniform random number (R) between 0 and 1.
if (w > R) then
Accept move (config = configTrial).
else
Reject move (config is unaltered).
end if
end loop
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Generalized MMC Algorithm
(contd)
After an initial configuration is generated (Part 1), the
algorithm works (Part 2) by repeatedly either accepting or
rejecting new configurations. The algorithm introduces the
concept of a ‘Markov chain.’
It is important to realize that not all states will make a
significant contribution to the configurational properties of
the system. We must confine our attention to those states
that make the most significant contributions in order to
accurately determine the properties of the system in a
finite time. This is achieved via a Markov chain which is a
sequence of trails for which the outcome of successive
trails depend only on the immediate predecessor.
In a Markov chain, a new state will only be accepted if it is
more ‘favorable than the existing state.
Generalized MMC Algorithm
(contd)
In the NVT-ensemble, the only MC move is atom
displacement, only consider the configurational or potential
energy of state.
It is important to realize that, in contrast to molecular
dynamics, the kinetic energy has no role to play in an MMC
simulation. The acceptance criterion only involves
‘configurational’ properties. Therefore, we are only interested
in calculating the potential energy of the system.
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Metropolis Sampling
The limitations of random sampling can be avoided by
generating configurations that make a large contribution to
the thermodynamic average. This is the philosophy behind
Metropolis sampling.
Metropolis sampling biases the generation of configurations
towards those that make the most significant contribution to
the integral.
It generates states with a probability of exp[−b ( )] and
counts each of them equally.
In contrast, simple Monte Carlo integration generates states
with equal probability assigning them a weight of
exp[− b ( )].
Metropolis Sampling (contd)
Metropolis sampling generates a Markov chain which
satisfies the conditions that:
(a) the outcome of each trial depends only on the
proceeding trail; and
(b) each trail belongs to a finite set of possible outcomes.
The former condition highlights the distinction between
Monte Carlo and molecular dynamics simulation methods.
In a molecular dynamics simulation, all the states are
connected in time.
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Metropolis Sampling (contd)
( ) = exp(−D ( ) / ) … … (∗)
The Metropolis scheme is implemented by attempting a
trial displacement and calculating the change in energy .
If D < 0, then (∗) > 1 and the move is accepted.
Otherwise, if D > 0, (∗) < 1 and the move is accepted
with a probability of (∗) = exp(−bD ). A random number
R is generated and the move is accepted if exp −bD ³ .
Metropolis Sampling (contd)
Typically, this procedure is summarized by defining the
probability of acceptance as:
= min[1 , exp(−b D ) ]
It should be noted that the above equation only
represents the acceptance criteria resulting from
molecular displacement. A different criterion is required as
a result of either volume change or a change in a number
of particles.
Lets now revisit the NVT-ensemble algorithm in greater
detail because ALL other ensemble algorithms follow its
basic approach.
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Detailed NVT-Ensemble
Algorithm
Part 1 loop cycle 1 ... maxCycle
Part 2 Attempt trial displacements:
loop atom 1 ... maxAtom
Part 2.1 Generate a trial displacement:
rxTrial pbc(rxatom + (2 * rand() - 1) * dMax)
ryTrial pbc(ryatom + (2 * rand() - 1) * dMax)
rzTrial pbc(rzatom + (2 * rand() - 1) * dMax)
Part 2.2 Calculate change in energy:
Calculate energy of the atom (ETrial) at trial position.
DE ETrial - Eatom
Detailed NVT-Ensemble
Algorithm
Part 2.3 Apply Metropolis acceptance criterion:
if (D < 0 or exp(−b D )³ ())
rxatom rxTrial //accept move
ryatom ryTrial
rzatom rzTrial
Eatom ETrial
E E + DE
numAccept numAccept + 1
end if
end atom loop
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Detailed NVT-Ensemble
Algorithm
Part 2.4 Update periodically the maximum displacement:
if (mod(cycle, updateCycle) = 0)
acceptRatio numAccept/(cycle * atomMax)
if (acceptRatio > 0.5)
dMax 1.05 * dMax
else
dMax 0.95 * dMax
end if
end if
end cycle loop
Detailed NVT-Ensemble
Algorithm (contd)
Typically, a Monte Carlo simulation is performed in cycles
(Part 1). During each cycle, a displacement is attempted for
every atom (Part 2). The atom to be displaced can be chosen
randomly or alternatively, each atom can be considered in
turn as illustrated above. It is also possible to displace every
atom and apply the acceptance criterion to the combined
move.
New trial coordinates (Part 2.1) are obtained randomly up to
a maximum value of dMax. This involves generating random
numbers on the interval from 0 to 1 and applying periodic
boundary conditions (pbc).
The change in energy resulting from the trial displacement
(Part 2.2) is simply the energy experienced by the atom at
the trial position (Etrial) minus the energy of the atom at its
existing position (Eatom).
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Detailed NVT-Ensemble
Algorithm (contd)
This change in energy is evaluated and the Metropolis
acceptance (Algorithm Part 2.3) rule is applied. If the move is
accepted, the atom’s coordinates, the atom’s energy and the
total energy (E) are updated.
The maximum allowed displacement (dMax) affects the
acceptance rate. Experience indicates that an acceptance rate
of approximately 50% is often desirable for a Monte Carlo
simulation.
A small value of dMax will generally improve the acceptance
rate but the phase space may be sampled too slowly.
Conversely, increasing the value of dMax will reduce the
acceptance rate. It is common to adjust the value of dMax
during the course of the simulation to maintain a 50%
acceptance rate.
Detailed NVT-Ensemble
Algorithm (contd)
In part 2.4, the value of dMax is adjusted at intervals
specified by updateCycle. If acceptRatio > 0.5, the value
of dMax is increased, whereas it is decreased if
acceptRatio < 0.5.
It should be noted that there is no theoretical basis for
using an acceptance rate of 50% and in some cases it
may be actually detrimental to efficient sampling.
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Averages & Error Estimates
The averages of a Monte Carlo simulation are only
accumulated after an equilibration period. The number of
cycles required for equilibrium is not known exactly
beforehand and it should ideally be probed by test runs.
Alternatively, it is common to discard a large number of
early runs to be on the safe side. For example, if the
simulation runs for 20000 cycles, the first 10000 are
discarded and averages are only accumulated for cycles
10001 to 20000.
The average quoted is always the average of the entire
postequilibration period.
Averages and Errors (contd)
It is always important to provide an error estimate of a
simulation quantity. The error represents the deviation
from the average value over the post-equilibrium cycles. If
the error estimate is large it may indicate that equilibrium
has not been reached or that the simulation is unreliable
for some reason. The reasons for large errors include both
errors in the code or an actual physical limitation.
There are alternative ways of estimating errors. Some
simulators average over independent runs commencing
from different initial configurations. However, more
commonly errors are estimated from a single run by
dividing the post-equilibrium run into several blocks
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Averages and Errors (contd)
The average of a property is determined from:
1
=
− −1
The post-equilibrium run is divided into blocks of
length such that = . The mean value of is
calculated for each block
1
=
The standard deviation is then determined by
averaging over all of the blocks:
Averages and Errors (contd)
1
= −
This standard deviation that is commonly quoted as the
error associated with a simulation quantity.
It should be noted, that some authors quote the
uncertainty as:
which will always be much less than the standard
deviation.
Therefore, some caution is required when comparing
error estimates from different workers.
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