10.34 – Fall 2006
Homework #10 Solution
Monte Carlo Sampling within a molecular potential
The purpose of this problem was to take a representative sampling of molecular configurations at
a given temperature for the given molecular potential.
Proper sampling from the probability of
the molecule having a certain amount of energy will allow for average quantities to be estimated,
such as the <1/R
HH
6
> and <R
HH
> asked for in the problem statement.
This could be done using internal coordinates of the molecules, but was essentially given to you
in terms of Cartesian coordinates in the problem.
The pertinent equation for the average of
interest was given by the following equation.
In this equation, several degrees of freedom have
been removed by assuming a fixed position of one oxygen atom (O1), the other oxygen (O2)
only moves along one dimension, and that one H atom (H1) is confined to a 2D plane.
This
leaves a 6D phase space to be probed, which in Cartesian coordinates can be written as the
following variables: x
O2
, x
H1
, y
H1
, x
H2
, y
H2
, and z
H2
.
⎛
1
⎞
∫∫∫
⎜
⎝
[
R
HH
]
q
⎟
⎠
dx
dx
dy
dx
dz
H
2
⎜
6
⋅
w
(
)
⎟
O
2
H
1
H
1
H
2
dy
H
2
1
=
6
∫∫∫
w
(
)
q
⋅
dx
O
2
dx
H
1
dy
H
1
dx
H
2
dy
H
2
dz
H
2
R
HH
⎡
−
(
)
⎤
V
q
2
where
:
w
(
)
⋅
q
=
x
O
⋅
exp
⎢
⎥
2
y
H
1
k T
⎢
B
⎥
⎣
⎦
It is easy to see that the weighting function in this case is a modified Boltzmann distribution,
which was modified because the degrees of freedom were reduced.
This weighting function will
then be used in the Metropolis algorithm to accept or reject steps. The acceptance criterion for
an uphill step in energy is given by:
2
x
⋅
y
H
1
rand
[
]
w
new
(
)
q
=
(
O
2
0,1
<
)
new
⋅
exp
⎡
⎢
−
{
V
new
−
V
current
}
⎤
⎥
⇒
Accept
step
q
x
2
)
k T
⎦
w
(
)
(
O
2
⋅
⎣
B
current
y
H
1
current
Generally, a downhill step in energy is always accepted and this formula is not used (you can see
that for a pure Boltzmann factor, a downhill step will always be accepted with the above
formula).
However, this criterion could also be used in this case for a downhill step, and for very
small downhill steps in energy, there could be a small chance of rejecting the step based the
geometry ratio prefactor.
Now that the acceptance criterion is defined, we can begin to take steps.
In this problem, each
coordinate could be changed independently by +/ 0.10 Å.
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to
Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu),
Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Part A:
Equilibrium value at 0 K:
<R_HH> = 2.5836 (Angstroms)
<1/R_HH^6> = 0.0033623 (1/A^6)
Number of MC steps taken: 100000
One would expect that the equilibrium 0 K
distance and the <distance> at elevated are
different, though the difference may be small.
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 Fall '04
 ClarkColton
 Thermodynamics, Atom, Mole, Energy, Statistical Mechanics

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