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.EQ
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.EN
NOTES ON
.sp
Chapter 1 
.sp
BARRIER CROSSINGS: CLASSICAL THEORY OF RARE BUT IMPORTANT EVENTS
.sp
by David Chandler
.sp2
in "Classical and Quantum Dynamics in Condensed Phase Simulations:
proceedings
of the International School of Physics, Lerici, Villa Marigola, 7 July-18
July 1997", B. J. Berne, G.
Ciccotti, and D. F. Coker, eds., pp 583-616, World Scientific, Singapore,
1998.

.SK
1. - INTRODUCTION\ \ \ \ \ (Excellent read!)
.sp 2
a. Rare events:
.sp
.in +.5i
Bottleneck (a high barrier or pass) separates stable states
.in -.5i
.sp
b. Importance sampling:
.sp
.in +.5i
Chain folding: $tau ~approx~ 10 sup -6 s$
.sp
Movie of this event runs at 1 minute/ps of trajectory.
.sp
Thus a display of this event requires 2 years of film for $10 sup 6 ps$.
.sp
Therefore want to explore only the relevant region of phase space, the
bottleneck or barrier region, and to avoid watching the system when it is in the
region of a stable state (where it is most of the time).
.in -.5i
.SK
2. - CYCLOHEXANE ISOMERIZATION
.sp
.ce
\ ... <--> chair <--> boat <--> chair <--> ...\ \ \ \ \ (1 = 2.1)
.sp
a. Time scales:
.sp
.in +.5i
Slow reaction:
.sp
.in +.5i
$tau (isomerization) ~approx~ 0.1 s$
.sp
.in -.5i
Fast microscopic processes - 
.br
.sp
Brownian, coarse-grained in time:
.in +.5i
$tau (diffusion) ~approx~ 10 sup -11 s$
.sp
$tau (velocity~~autocorrelation) ~approx~ 10 sup -12 s$
.in -.5i
.sp
Very fast collisional events:
.in +.5i
.sp
$tau (collision) ~approx~ 10 sup -14 s$
.in -.5i
.in -.5i
.SK
b. Properties of isomerization process:
.sp
.in +.5i
i. Pressure dependence - Fig. 1
.in +.5i
Analyze and seek to understand through MD simulations; 
.br
note that the rate
increases with increasing pressure.
.in -.5i
.sp
ii. Potential energy function - Fig. 2
.in +.5i
$V( theta , phi )$
.in -.5i
.sp
iii. Free energy function - Fig. 3
.in +.5i
.DS
.EQ (2\ =\ 2.2)
- beta F( theta ) ~=~ 
ln ~ int from 0 to {2 pi} d phi ~ e sup {- beta V( theta , phi )}
.EN
.DE
where $F( theta )$ is the free energy (the work function; the potential
of mean force) along the reaction coordinate $theta$.
.in -.5i
.in -.5i
.SK
3. - HIGH BARRIERS, RARE EVENTS
.sp
a. Free energy function - Fig. 4:
.sp
.in +.5i
$q~=~q*$, the transition state, 
.br
must be passed for A <--> B.
.sp
$F* ~>>~ kT$: passages rare.
.sp
System committed to stable state at small displacement 
$l$ from $q*$.
.sp
NOTE: The picture and symbols of Fig. 4 are used throughout the 
discussion of this Chapter.
.in -.5i
.sp
b. Dynamics vs equilibrium:
.sp
.in +.5i
Configurations near $q*$ rare and not important for equilibrium
properties.
.sp
$q*$ region governs dynamics.
.in -.5i
.SK
c. Kinetic energy ($T sub k$):
.sp
.in +.5i
$T sub k$ small when system near $q*$.
.sp
$T sub k$ large after system passes $q*$:
.br
large $- partial V / partial q$ => large acceleration.
.sp
Deactivation and trapping by energy flow from $q$ to other degrees of
freedom.
.in -.5i
.sp
d. Frequency of passage, $nu$:
.sp
.in +.5i
passage through $q*$ fast, ca. $10 sup -13 $:
.DS
.EQ (3-4\ =\ 3.1-2)
tau sub mol ~=~ nu sup -1 ~approx~ l / <| q dot |>
.EN
.DE
.in -.5i
.sp
e. Reaction rate:
.in +.5i
.DS
.EQ (5\ =\ 3.3)
k ~approx~ nu ~ e sup {- beta F*}
.EN
.sp
.EQ (6\ =\ 3.4)
k sup -1 ~=~ tau sub rxn ~approx~ tau sub mol ~ e sup { beta F*}
.EN
.DE
Arrhenius behavior - hallmark of a barrier crossing: 
characteristic time $tau sub rxn$ grows exponentially with $F* /kT$.
.in -.5i
.SK
4. - REACTION COORDINATE, $q(t)$
.sp
a. Trajectory - Fig. 5:
.in +.5i
"instanton" - passage between stable states.
.sp
Note dithering at $q*$.
.in -.5i
.sp
b. Probability distribution:
.sp
.in +.5i
Histogram:
.DS
.EQ (7\ =\ 4.1)
P(q) mark ~=~ < delta ( q(t) ~-~ q ) >
.EN
.sp
.EQ
lineup ~=~ 1 over T int from 0 to T dt ~ delta ( q(t) ~-~ q )
.EN
.sp
.EQ
lineup ~=~  
int sub GAMMA d GAMMA ~ delta ( q( GAMMA ) ~-~ q ) e sup {- beta H( GAMMA )}
~/~ Q
.EN
.DE
.in -.5i
.SK
c. Importance sampling:
.sp
.in +.5i
For statistical significance, $T ~>>~ tau sub rxn$, 
which generally is not possible in a simulation.
.sp
Need to control $q$ in order to examine rare states, near $q*$.
.sp
Use umbrella sampling.
.sp
For cyclohexane in liquid $bold {CS sub 2}$ solvent, find barrier to
isomerization increases slightly with increased pressure; 
thus isomerization rate should decrease with pressure, counter to experiment
(Fig. 1).  The explanation must reside in the non-Arrhenius prefactor, $nu$.
.in -.5i
.SK
5. - RATE CONSTANTS AND TIME CORRELATION FUNCTIONS
.sp
a. Time evolution of population operator:
.br
.in +.5i
.sp
Fig. 6.
.sp
The trajectory is for a time of 15 $tau sub rxn$.
.sp
The population operator is:
.in +.5i
.DS
.EQ (10\ =\ 5.1)
h(q) mark ~=~ 1 ,~~~~q > q*
.EN
.EQ
lineup ~=~ 0 ,~~~~q < q*
.EN
.DE
.in -.5i
or equivalently,
.sp
.in +.5i
$h(q) ~=~ theta (q ~-~ q* )$
.in -.5i
.sp
The population operator is $1$ for state B.
.sp
The fraction time spent in state B
.sp
.in +.5i
$x ~=~ < h >$
.in -.5i
.in -.5i
.SK
b. Various relationships:
.in +.5i
.sp
$h(t) ~=~ h(q(t))$
.sp
$h ~=~ h(0) ~=~ h(q(0))$
.sp
$h sup 2 ~=~ h$
.sp
$delta h ~=~ h ~-~ <h>$
.DS
.EQ
<h~ delta h> mark ~=~ <h sup 2 > - <h> sup 2 ~=~ <h> - <h> sup 2
.EN
.EQ
lineup ~=~ x~(1-x)
.EN
.EQ
lineup ~=~ x~{delta h} bar (0)~e sup -kt
.EN
.DE
$h dot ~=~ dh over dt ~=~ dh over dq dq over dt ~=~ delta (q - q* )~q dot$
.sp
$- {< h~h dot (t) >} ~=~ < h dot~h (t) >$
.sp
.DS
.EQ
P( q* ) ~=~
(1~-~ x sub B )~
{e sup {- DELTA W* /kT}} over
{Q sub A}
.EN
.sp
.EQ (Hynes-4)
k sub f  sup TST mark ~=~
{omega sub A} over {2 pi} {e sup {- DELTA W* /kT}}
.EN
.DE
.in -.5i
.SK
c. Time correlation of $h$ with its fluctuation:
.in +.5i
.DS
.EQ (11\ =\ 5.2)
delta h(t) ~=~ h(q(t)) - <h>
.EN
.DE
.DS
.EQ (12\ =\ 5.3)
<h~ delta h(t)> mark ~=~ 1 over T int from 0 to T dt' ~ h(t') delta h(t'+t)
.EN
.EQ
lineup ~=~ left "" x
1 over T int from 0 to T dt' ~ 1 cdot delta h(t'+t) right ] sub h(t')=1 ~+~
.EN
.sp
.EQ
lineup ~~~~~~ left "" (1~-~x)
1 over T int from 0 to T dt' ~ 0 cdot delta h(t'+t) right ] sub h(t')=0
.EN
.sp
.EQ
lineup ~=~ x ~ left ""
1 over T int from 0 to T dt' ~ delta h(t'+t) right ] sub h(t')=1
.EN
.EQ
lineup ~=~ x ~ {delta h} bar (t)
.EN
.DE
${delta h} bar (t)$ is a \fInonequilibrium\fP average, over trajectories
started in state B, with $h ~=~ 1$.
.in -.5i
.SK
d. Relaxation of $<h~ delta h(t)>$:
.sp
.in +.5i
Relaxation of the time correlation function is determined 
by the relaxation of ${delta h} bar (t)$.
Correlation (memory of the initial state B) is lost as the system
undergoes sequential transitions A<-->B.
The transitions are rare events and are statistically independent.
They constitute a Poisson stochastic process.
The relaxation, like radioactive decay, 
is exponential in the time, i.e., a first order process:
.in +.5i
.DS
.EQ (13\ =\ 5.4)
{delta h} bar (t) ~approx~ {delta h} bar (0)~e sup -kt
.EN
.DE
.in -.5i
.sp
The constant $k$ is the average transition frequency,
or equivalently, $k sup -1$ is the average lifetime of state B.
.sp
Both $<h~ delta h(t)>$ and ${delta h} bar (t)$ go to zero in the limit of
large time, $t ~>>~ 1/k$.
.in -.5i
.SK
e. Phenomenology:
.sp
.in +.5i
Consider now a macroscopic isomerization process, A<-->B, 
between species that we 
can understand correspond to the microscopic states A and B.
.sp
We can assert a rate law and test it by experiment.
.sp
A two-state isomerization is characterized by a first-order rate constant 
that is the sum of the forward and reverse reactions:
.in +.5i
.DS
.EQ (14a\ =\ 5.5a)
k ~=~ k sub A->B ~+~ k sub B->A
.EN
.DE
.in -.5i
Also, by detailed balance,
.in +.5i
.DS
.EQ (14b\ =\ 5.5b)
x / (1 ~-~ x) ~=~ k sub A->B / k sub B->A
.EN
.DE
.in -.5i
where $x~=~ X sub B$, the mole fraction of the species B in the isomerization
process.
.in -.5i
.SK
f. Onsager's Hypothesis:
.in +.5i
.sp
Onsager asserted that for a system near equilibrium,
on average the microscopic dynamics of the decay of large fluctuations in a
variable (decay of the time correlation function), 
are the same as the dynamics of relaxation of the variable to its equilibrium
value in a macroscopic system.
.sp
By Onsager's Hypothesis, we can identify the macroscopic rate constant $k$ of
Eqn. (14\ =\ 5.5) with the microscopic relaxation constant $k$ of Eqn. 
(13\ =\ 5.4)
.sp
For the following discussion, the constant $k$ is identified
with both the microscopic and macroscopic processes.
.in -.5i
.SK
6. - REACTIVE FLUX CORRELATION FUNCTION:
.sp
a. Definition:
.in +.5i
.sp
Combining Eqns. (12\ =\ 5.3) and (13\ =\ 5.4):
.in +.5i
.DS
.EQ 
< h~ delta h(t) > mark ~=~ x ~ {delta h} bar (0)~e sup -kt
.EN
.EQ 
lineup ~=~ < h~ delta h > e sup {-kt}
.EN
.DE
.in -.5i
to give
.in +.5i
.DS
.EQ
{< h~ delta h(t) >} over {< h~ delta h >}  ~=~ e sup -kt
.EN
.DE
.in -.5i
.sp
We define the reactive flux correlation function, $k(t)$,
as the time derivative 
of the preceding expression:
.in +.5i
.DS
.EQ (15a\ =\ 5.6a)
k(t) mark ~=~ 
- d over dt {< h~ delta h(t) >} over {< h~ delta h >} 
.EN
.sp
.EQ (15b\ =\ 5.6b)
lineup ~approx~ k e sup {-kt}
.EN
.DE
.SK
.in -.5i
Carrying out the differentiation of Eqn. (15a\ =\ 5.6a),
.in +.5i
.DS
.EQ 
k(t) mark ~=~ 
- d over dt {left [ < h~h(t) > ~-~ < h <h> > right ]} over {< h~ delta h >} 
.EN
.sp
.EQ 
lineup ~=~
- {< h~h dot (t) >} over {< h~ delta h >}
.EN
.sp
.EQ 
lineup ~=~
{< h dot~h (t) >} over {< h~ delta h >}
.EN
.DE
.in -.5i
to give $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(16~=~6.1)$
.in +.5i
.DS
.EQ 
k(t) ~=~
1 over {x~(1~-~x)}  < q dot~delta (q ~-~ q* )~h (q(t)) >
.EN
.DE
.in -.5i
.sp
where we have used the chain rule and several identities and definitions,
.sp
.in +.5i
$h dot ~=~ dh over dt ~=~ dh over dq dq over dt ~=~ delta (q - q* )~q dot$
.br
$- {< h~h dot (t) >} ~=~ < h dot~h (t) >$
.br
$< h~ delta h > ~=~ x~(1~-~x)$
.br
$h(t)~=~h(q(t))$
.in -.5i
.in -.5i
.SK
b. Computable form of reactive flux correlation function:
.sp
.in +.5i
Eqn. (16\ =\ 6.1) can be rewritten for computation with importance sampling
of the barrier top configurations:
.in +.5i
.DS
.EQ (17\ =\ 6.2)
~
.EN
.EQ
k(t) ~=~
1 over {x~(1~-~x)} P(q * ) < q dot~h (q(t)) > sub {q *}
.EN
.DE
.in -.5i
where $P(q*)~=~< delta (q~-~q*)>$ and $<...> sub q*$ means averaging
with $q$ initially (for $t=0$) constrained to $q*$.
.sp
$P(q*)$ can be evaluated with umbrella sampling, 
and the conditional average, with trajectories starting at the barrier top.
.sp
Eqn. (17\ =\ 6.2) is the central result of this Chapter.
.in -.5i
.SK
c. How does this function behave? Limit of very small $t -> 0+$:
.sp
.in +.5i
Since $q(0)~=~q*$, a very short time later, 
.sp
.in +.5i
$h(q(t))~=~1,~~~~~~~ q dot > 0$
$h(q(t))~=~0,~~~~~~~ q dot < 0$
.sp
.in -.5i
Thus
.in +.5i
.DS
.EQ
k(0 sup + ) mark ~=~
1 over {x~(1~-~x)} P(q * ) < q dot~theta (q dot ) > sub {q *}
.EN
.EQ (18\ =\ 6.3)
~
.EN
.EQ
lineup ~=~
1 over {x~(1~-~x)} P(q * ) 1 over 2 < | q dot | > sub {q *}
.EN
.DE
.sp
.in -.5i
Eqn. (18\ =\ 6.3) is the Transition State Theory (TST) approximation:
all trajectories at the barrier top are in the product direction and do not
recross ("dither" at) the barrier top.
.SK
Evaluating $P(q*)$ gives a more traditional TST equation.
.sp
For isomerization and a symmetrical two-well potential:
.in +.5i
.DS
.EQ
P( q* ) ~=~  x ~ {e sup {- DELTA W* /kT}} over {Q}
.EN
.sp
.EQ
k sub B->A sup TST mark ~=~ (1~-~x) ~ k(0 sup + )
.EN
.sp
.EQ
lineup ~=~ 1 over 2 < | q dot | > sub {q *} {e sup {- DELTA W* /kT}} over {Q}
.EN
.DE
.in -.5i
Compare above with Ghosh and McCammon (1987).
.sp
Evaluating $Q$ and $< | q dot | > sub {q *} $,
.in +.5i
.DS
.EQ
k sub B->A  sup TST ~=~
{omega sub A} over {2 pi} {e sup {- DELTA W* /kT}}
.EN
.DE
.in -.5i
Compare above with Hynes (1988).
.in -.5i
.SK
c. How does this function behave? 
.br
For $tau sub mol ~<~ t ~<<~ tau sub rxn = 1/k$:
.sp
.in +.5i
[Read page 15.]
.sp
Any barrier recrossing reduces $k(t)$: 
.br
Examine Eqn. (17\ =\ 6.2).
.sp
After a time $>= tau sub mol$, trajectories initiated at the barrier top will
be committed to one of the stable states.
.sp
At times  $tau sub mol ~<~ t ~<<~ tau sub rxn = 1/k$,
$exp(-kt) ~approx~ 1$ and
$k(t)$ will have relaxed to the stable (constant) value $k$:
.br
Examine Fig. 7.
.sp
The factor $kappa$ is evaluated as:
.in +.5i
.DS
.EQ
kappa ~=~ k( tau sub mol ) / k(0 sup + ) ~=~ k / k(0 sup + ) ~=~ k / k sup TST
.EN
.DE
.in -.5i
.in -.5i
.SK
7. - CHOOSING THE TRANSITION STATE
.sp
Choose transition state $q*$ that minimizes $k(0 sup + )$.
.SK
8. - LINDEMANN-HINSHELWOOD AND KRAMERS REGIMES - Fig. 8
.sp
a. Weak coupling (Lindemann-Hinshelwood) regime:
Coupling (inter- or intramolecular) of reaction coordinate 
to bath is necessary for deactivation
after barrier crossing.  Without coupling, barrier recrossed repeatedly.
Reaction rate increases with increased coupling.
.sp
b. Strong coupling (Kramers) regime:
Strong coupling of reaction coordinate to bath results in diffusive motion at
barrier top.
Expected for reaction in a condensed phase.
Increased coupling (e.g., greater viscosity) reduces reaction rate.
.SK
Kramers equation for strong friction (high viscosity $gamma$):
.in +.5i
.DS
.EQ
k sub A->B sup overdamped mark ~=~ 
{omega sub A omega sub q*} over {2 pi gamma} e sup {- beta DELTA W*}
.EN
.sp
.EQ
lineup ~=~ {omega sub q*} over {gamma} k sub A->B sup TST
.EN
.DE
.in -.5i
.SK
9. - CYCLOHEXANE REACTIVE FLUX
.sp
[Read pages 18-21.]
.sp
Central point: A short but not negligible time is required to commit a
molecule to a stable state.  Within this time, transient trajectories can
recross the transition state.  Since coupling to a solvent can accelerate the
equililbration, the coupling will lower the number of recrossing trajectories
and thus increase the net rate.  For this to be the case, the isomerization
of cyclohexane in $bold CS sub 2$ must be between the low and high coupling
regimes.
