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Internationa Research Training Group Diffusion
in Porous Materials Workshop Leipzig 9th-12th May
2006
Transition State Theory Basing on the Concept of
the Local Free Energy
Siegfried Fritzsche Leipzig 11th May 2006
University of Leipzig, Institut of Theoretical
Physics Group Molecular Dynamics / Computer
Simulations
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Contents Introduction rare events - purpose of
TST Alternative possibilities Origin and basic
idea of TST Examples for systems with transition
states TST transition rate for a bottleneck TST
in terms of the local free energy The minimum
energy path (MEP) Reduction to an one dimensional
problem The harmonic approximation A simple
example Methane in an model potential Variational
transition state theory Methods to find the
MEP Recrossing events and dynamical
corrections Some examples for advanced simulation
technics An illustrative example for the use of
TST
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What is the purpose of TST?
TST is used when the time controlling step for a
slow process is a bottleneck that must be passed
One example is the passage from the
potential minimum A to the potential minimum B
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Is it possible to use simulations instead of TST?
In principle yes. But, the barrier crossing is a
rare event. You can easily make a very long
simulation run without to see any one passage.

• To improve this situation you can e. g. make use
  of
• parallel computing
• boost potentials

Parallel computing Start many simulation runs at
once on a multiprocessor machine with optimized
parallel computing. In an ergodic system the
result is equivalent to one single long
trajectory because the time average is equal to
an ensemble average.
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The method of a boost potential
This method makes also use of the equivalence of
time average and ensemble average.
In the canonical ensemble the probability ?W to
find the system in a subvolume ?V of the phase
space can be expressed by use of the local free
energy F
C is a normalization constant for the whole
system. Because of the equivalence of time
average and ensemble average the average time
that the system will spend within this subvolume
is proportional to ?W.
At high dilution F can be replaced by the
external potential ? and hence
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If a potential ?b, that is constant in ?V and
zero elsewhere, is added to ? then
That means that the time that the system spends
in ?V is enhanced by the factor
in comparison to other regions.
If a negative potential ?b is introduced for the
transition region, then transitions will be
observed more frequently.
To avoid a change of measured averages, the
quantities measured in the transition regions
must be divided by
or, a modified time must be used in the
evaluation (not in the simulation) that is
accelerated by this factor for periods
during which the system is in the boosted
region. In practice of course, a continuously
varying boost potential will be used.
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Where does TST come from and what is the
principle? The first paper about the basic
ideas of TST was
H. Eyring and M. Polanyi, Z. Phys. Chem. B 12,
279(1931).
The rate (relative probability per time) k for a
passage is calculated as the product of the
probability T to reach the transition state and
the probability f to reach state B after
crossing the transition state
k Tf
T is calculated from equilibrium statistical
mechanics
f is a dynamical quantity that can be obtained
only from microscopic considerations
(simulations). It is often replaced by an
estimated value.
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Examples for systems with transition states
Two adsorption centers separated by a barrier
The saddle point is called the transition state
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Center of mass positions of a thermalized
dumbbell molecule in the model potential at T
350 K.
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More complicated arrangements of adsorption
centers
In general there are more than two adsorption
sites and the resulting diffusion coefficient is
also dependent upon their geometrical arrangement
Example silicalite - 1
The sites with lowest potential energy are
situated in the channels not in the intersections.
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Isopotential lines for methane in the straight
channel of silicalite-1
Isopotential lines for methane in the zig zag
channel of silicalite-1
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TST transition rate for a
bottleneck We explain the general method for the
example of a flux in a zeolite from cavity A to
cavity B through a window with cross section O
Let us assume that we have only spherical
particles in 3D space.
Number of crossing particles per unit time
Let N be the number of all particles in state A
Then the relative number of particles leaving
state A per unit time is
is the integration variable in the volume or in
the plane O
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The velocity at each site is replaced by the
average velocity in positive x-direction as
obtained from the canonical ensemble.
This is an approximation. But, it is also based
on an assumption that is crucial for TST
Equilibrium in the velocity space for all
regions of the system. This is approximately
true in most cases. But, not always.
Then the transition rate T becomes
This is the standard formula of the Transition
State Theory
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This formula can be written in another way. For
an arbitrary function ? it is
is the x coordinate of the plane O
Defining a normalized density
we get
where the average means
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TST in terms of the local free energy
Expressing n by the local free energy using the
formula
we find for the transition rate T
How to find the local free energy needed for the
use of this formula?

• From the particle density
• b) At low density from the barometric law
• c) From Monte Carlo algorithms e.g. umbrella
  sampling or
• particle insertion algorithms

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The minimum energy path (MEP)
A path connecting states A and B Is called the
minimum energy path (MEP) if it consists only of
points that are local minima in all planes
perpendicular to this path. The MEP is so to say
the most probable of all the improbable paths
connecting states A and B. In some versions of
the TST it is assumed approximately that
all transitions from A to B follow this path. A
better approximation is the projection on the MEP.
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Reduction to a one dimensional problem
Example In the example of the first
transparency the minimum energy path is simply
the x axis.
Conversion to a one dimensional problem by
projection
From the density we get a one dimensional
probability distribution
The corresponding local free energy is
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Note that in regions where the the potential is
infinite the integrand is zero. Therefore, in
the one dimensional description the formula for
the transition rate becomes simply
and its version written in terms of the local
free energy becomes
This is the simplest way of TST at least in cases
where p(x) is availeable from MD or MC data.
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The harmonic approximation
Starting again from
The local free energy around the minimum of state
A is approximated by a Taylor expansion up to
second order
Using
The denominator in the formula for the rate
becomes
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If the saddle point is marked by a similar
expansion in the plane O yields
The rate T becomes
The factor in front of the exponential is often
called frequency factor. This can be understood
in the following way.
we have
In the special case
with
This is obviously the frequency formula for an
harmonic oscillator.
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A crude but sometimes used approximation is to
identify
with a vibrational frequency of the system in
state A that can be measured. The transition
rate becomes in this simple approximation
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A simple example Random walk of a spherical
molecule in a model potential
U( x,y,z) A x4 - Bx2 C( y2z2)
Parameters of the simulation run m 16 g/mol T
500 Kelvin, time step h 10-15 s A 5
10-3 kJ/mol B 0.8 kJ/mol C 20.0 kJ/mol
The single particle is thermalised by collisions
with light heat bath particles (Kast algorithm).
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Potential energy along the x - axis
minimum at -32.0 kJ/mol, saddle point at 0.0
kJ/mol
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Potential energy of the test particle at
different x coordinates
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The average local temperature
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The normalized density in the one dimensional
description
hence
The dashed line is the analytical solution
(barometric law)
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The normalized density near the transition state
From MD p(0) (2.5 0.8) 10-4 A-1 Real value
p(0) 1.784 10-4 A-1
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The average velocity in positive x direction is
2.034 A/ps ( 203.4 m/s)
The normalized probability from MD was p(0)
(2.50.8) 10-4 A-1
Hence, the rate constant from TST using the
density distribution from MD is T (5.11.6)
10-4 1/ps. Using the density distribution from
the barometric law the rate constant from TST is
3.63 10-4 1/ps.
Direct evaluation from the MD run yields the rate
constant 3.9 10-4 1/ps
For the harmonic approximation the model
potential yields Umin -32 kJ/mol, Umax 0.0
kJ/mol ax 3.2 kJ/mol, ayazbybz40.0
kJ/mol We find a rate constant of 3.2 10-4 1/ps
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Variational transition state theory
Along the MEP p(x) has a minimum in the
transition state. If the transition state is not
known exactly then using formally
will give the smallest value for T, if x00.
Variational TST means that trials are made with
different possible locations of the dividing
surface. The smalles rate constant is assumed to
be closest to truth.
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Methods to find the MEP
The direct drag method the most intuitive and
simplest way Choose any one coordinate (dashed
line in the picture) that has different values
for state A and B. This coordinate is called the
drag coordinate (a line in the d-dimensiomnal
space).
Step along this coordinate from A to B in small
steps. After each step find in an d-1
dimensional hyperplane, that meets the drag
coordinate only in this one point, the minimum
of the local free energy.
a
The path connecting these minimum points is
assumed to be the MEP.
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There exist more sophisticated versions of the
drag method
An MD docking algorithm (Auerbach)
A straight line from state A to state B is
introduced by
? is a parameter that varies between 0 and 1.
An additional docking potential that acts on the
center of mass of a test molecule
Is added to the interactionin the
system. Heating up and cooling down the system
in MD runs will cause the Test molecule to move
to a position of lowest energy for each given
value of ?. This movement includes all degrees
of freedom.
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The nudget elastic band
A first arbitrary guess for the MEP is made. Then
M points i1,...,M on this path are chosen. Let
and correspond to the stable states A
and B of the system. These two points are fixed.
Let be the local free energy that the
molecule would have at the site .
Then a function ? is defined that is
If we understand the negative gradient of this
function as a force then the points are moved in
direction of these forces to minimize ?. To
improve the algorithm the true forces, coming
from F are put eqal to zero in direction of the
path and the elastic spring forces are put
equal to zero perpendicular to the path. This
avoids artefacts and improves the accuracy close
to the saddle point.
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Recrossing events the transmission factor
In reality the TST yields only an upper limit of
the transmission rate because not each of the
trajectories crossing the saddle point
will really end in state B. That means that the
rate from TST has to be multiplied by the
probability that a molecule crossing the
transition state will really reach state B.
is called the dynamical correction factor or the
transmission factor.
can be obtained from MD simulations
(Benett-Chandler approach)
Starting at the transition state with a velocity
randomly chosen from the Boltzmann distribution
the trajectory will be followed forward and
backward. In this way it can be found how many
of the trajectories reaching the transition
state, coming from state A, will end in state B.
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Importance of the position of the dividing surface
We learned that in the TST without dynamical
correction the rate has the smallest value when
the dividing surface is in the correct
position at the transition state.
Reason particles crossing a dividing surface
closer to state A will often turn back to state A
instead of crossing the saddle point.
In the Bennet-Chandler approach this effect is
corrected by the dynamical correction factor.
The problem is shifted to the calculation of this
factor that needs in such a case more trials to
get a reasonable statistics.
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Some examples for advanced simulation technics
1. The method of overlapping regions
The probability that a molecule starting in state
A will reach state B is very small. Instead, the
probability to reach state B is much higher if
the molecule starts at a position already close
to the transition state. The probability to
reach state B can be calculated as the product
of The probability to reach the intermediate
state and the conditional probability to reach
the state B if this intermediate state has
already been reached. The whole configuration
space between state A and B can be divided into
small overlapping regions and the single
probabilities of transfer from one region to the
next one can be evaluated as a function of
time (propagator). The regions must be
overlapping to enable normalization of the
number of starting positions from each single
subregion.
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2. Transition path sampling (TPS)
TPS is a generalization of well known Monte Carlo
methods. Instead of sampling moves in the space
of configurations as it is done in MC in TPS
whole trajectories are sampled.
After having found one trajectory leading from
state A to state B others can be found by
slightly varying the positions and momenta. If
the new trajectory starts at state A and ends in
state B it will be accepted. Otherwise it will
be rejected. The statistical weight of a given
trajectory starting at phase space point x0 is
hA( x0 ) is a step function that is equal to 1 if
x0 is a point in state A and zero
otherwise hB( xt ) is a step function that is
equal to 1 if xt (the positionat time t)
is a point in state B and zero otherwise p(
x0 ) is the probability to be at x0 in state A
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An illustrative example for the use of TST
Ethane in the cation free LTA zeolite
The structure of the LTA zeolite
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Isopotential lines for a single methane molecule
Steep increase
weak increase
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Intuitive interpretation
With increasing temperature the phase space in
the large cavity (denominator of the rate
coefficient formula) increases more rapidly than
the phase space in the transition region
(numerator of the rate coefficient
formula). Even at higher temperatures rotation
in the window is prevented by the steep wall
potential. The rotation in the large cavity
increases with increasing temperature more
easily.
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Distribution of the angle between molecule axis
and window axis at the entrance to the window
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Local Free Energy and Entropy
Local free energy
Average force
Local entropy
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• The numerical evaluation of the local
• free energy, the local internal energy
• and the local entropy during a long
• MD run shows
• The free energy has a threshold
• in the window at higher temperatures
• b) The threshold arises from the
• entropic part of the free energy
• This is in agreement with the
• former interpretations.

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Adsorption sites for ethane in cation free LTA
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The jump sites
3
5
2
1
4
6
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Comparison of D values from MD and from
the random walk treatment