.ig
:!cat %|tbl|pic|eqn|troff |dpost >%.ps;ghostview %.ps&
:!cat %|tbl|pic|eqn|troff |dpost >%.ps
:g/omega hat sub size -2 alpha/s//Wa/g
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..
.ig
.in +1i
.DS I
.EQ (z)
.EN
.DE
.in
..
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.EQ
delim $$
define Angstrom '~ up 17 "\(oa" back 60 A'
.EN
.EQ
define xbar  ` chi bar `
define xhat  ` chi hat `
define lhat  ` "\fI" L hat "\fP"`
define qlhat  ` "\fI" Q hat L hat "\fP"`
define xpop  ` "\fI" P "\fP"`
define xqop  ` "\fI" Q "\fP"`
define tmic  ` tau sub MIC`
define trxn  ` tau sub RXN`
define t*    ` t sup * `
define qdag  ` q sup "\(dd" `
define dag   ` "\(dd" `
.EN
.SP 7
.SP 3
Raymond Kapral, Styliani Consta, and Liam McWhirter,
\fI"Chemical rate laws and rate constants"\fP,
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.
.SP
"... for systems close to equilibrium where linearized versions of mass
action laws apply, one may derive generalized forms of the rate laws from the
microscopic evolution equations."
.SP 3
[The following is keyed to section headings and equation numbers in the
Kapral et al. paper; the intent is to outline the logic of the development.]
.SP
2. - \fBDerivation of chemical rate laws.\fP
.P
2.1 \fIPhenomenological description.\fP  Define a progress variable,
.in +1i
.DS I
.EQ (6)
xbar (t) ~=~ ( nu sub i ~-~ nu bar sub i ) sup -1 ( N bar sub i (t) ~-~ N bar sub i sup eq )
.EN
.DE
.in
.SP
where $( nu sub i ~-~ nu bar sub i )$ is the difference in stoichiometry,
reactant - product, for chemical species \fIi\fP, 
and $( N bar sub i (t) ~-~ N bar sub i sup eq )$ is the difference from the
equilibrium value of the average number of molecules for species \fIi\fP at
time \fIt\fP.
.SP
For systems near equilibrium, 
the progress of the reaction,
a return to equilibrium,
is first order:
.in +1i
.DS I
.EQ (7)
{d xbar (t)} over dt ~=~ -~ k xbar (t)
.EN
.DE
.in
This expression is the phenomenological (macroscopic; experimental) rate
law.
The constant \fIk\fP is a relaxation rate constant.
It can be expressed as a function of the forward and reverse step constants,
with the form of the function depending 
on the particular chemical mechanism (see eq. 8 of Kapral et al. (1998)).
.SK
For example, for an isomerization reaction,
.in +1i
.PS
box invis "$A$"
line <-> right .5 "$k sub f$" above "$k sub r$" below
box invis "$B$"
move right 2.9 "(9)"
.SP
.PE
.DS I
.EQ
k ~=~ k sub f ~+~ k sub r
.EN
.DE
.SP 2
.in
.P
2.3 \fIRate law derivation.\fP 
The derivation of a rate law from microscopic theory, 
for systems close to
equilibrium,
is done by use of the Mori-Zwanzig formalism.
This  method gives an \fIexact\fP
description of the dynamics in the form of a Generalized Langevin equation
for a particular set of (slow) variables of a system.
Projection operators, $P$ and $Q=I-P$,
are used to separate the dynamics of $xhat$ into a part that evolves in the
subspace of phase space 
parallel with $xhat$ and projected by $P$
(more properly, 
this is a subspace of the Liouville space of all dynamical variables),
and another part that evolves in a subspace orthogonal to $xhat$,
projected by $Q$ and complementary to that projected by $P$.
$xhat$ is the only variable that evolves on the slow time
scale of the chemical reaction, $trxn$.
Variables of the subspace defined by $Q$ have fast dynamics; 
their dynamics are on the microscopic time scale,
$tmic$,
of atomic collisions, bond stretching, etc.
.SP
From this analysis,
the ensemble-averaged dynamics 
for the progress variable $xhat$ is
.in +1i
.DS I
.EQ (24)
{d xbar (t)} over dt ~=~ 
- integral from 0 to t d tau ~ k tilde ( tau ) xbar (t~-~ tau )
.EN
.DE
.SP
.in
The dynamics of $xhat$ can be expressed equivalently in terms of 
the time correlation function,
.DS I
.EQ (25)
{d C sub chi (t)} over dt ~=~
- integral from 0 to t d tau ~ k tilde ( tau ) C sub chi (t~-~ tau )
.EN
.DE
.SP
.in
where 
$C sub chi (t) ~=~ ( xhat (t), xhat ) ( xhat , xhat ) sup -1$,
the Kubo canonical correlation.
.SP
The kernel $k tilde ( t )$ is itself a time correlation function,
.in +1i
.DS I                                                       
.EQ (22)                                                    
k tilde ( tau ) ~=~
( e sup {i qlhat t} i lhat xhat , i lhat xhat ) ( xhat , xhat ) sup -1
.EN
.DE                                                         
.SP
.in
$k tilde ( t )$ is the time-dependent friction of the Generalized Langevin
equation.
It also is called the memory function.
.SP
Note that $k tilde ( t )$ evolves by unnatural dynamics, $i qlhat$,
denoted by the superscript ~,
as compared to ordinary dynamics, $i lhat$.
Thus $k tilde ( t )$,
evolving in the subspace defined by $Q$,
decays to zero on the time scale $tmic$.
.SP
.P
2.4 \fIStructure of the rate kernel.\fP
Comparison of eq. 24 with the phenomenological rate law of eq. 7
identifies eq. 24 as a rate law derived from microscopic theory.
Up to this point, no approximations have been made,
and eq. 24 is exact.
However, eq. 24 is unsatisfying, as the microscopic picture of
the rate contant $k$ of eq. 7 remains unclear:
eq. 24 is an integro-differential equation (a convolution); 
and $k tilde ( t )$ obeys unnatural dynamics.
Fortunately, these difficulties can be remedied by making several
approximations that introduce negligible error.
.SP
Because $k tilde ( t )$ decays on the time scale $tmic$
and $xhat$ evolves on the much longer time scale $trxn$,
for times $t*$, $tmic << t* << trxn$, $xbar (t)$ is effectively constant,
and we can write the rhs of eq. 24 as a product:
.in +1i
.DS I
.EQ (29)
{d xbar (t)} over dt ~approx~ 
- left [ integral from 0 to {fwd 100 t* } 
d tau ~ k tilde ( tau ) right ] xbar (t)
.EN
.DE
.in
Comparing eqs. 29 and 7 and noting that the upper limit $t*$ of the
integral of eq. 29 can be extended to infinity, since by time $t*$ the
integrand will have decayed to zero,
we can identify the phenomenological rate constant of eq. 7 as
.in +1i
.DS I
.EQ (30)
k ~approx~ 
integral from 0 to inf d tau ~ k tilde ( tau )
.EN
.DE
.SP
.in
.SP
.SK
It is convenient to define the function
.in +1i
.DS I
.EQ (27)
K tilde (t) ~=~
integral from 0 to t d tau ~ k tilde ( tau )
~=~ ( e sup {i qlhat t} xhat , i lhat xhat ) ( xhat , xhat ) sup -1
.EN
.DE
.SP
.in
For times greater than $tmic$, $K tilde (t)$ is constant and 
the rate constant
.in +1i
.DS I
.EQ (27)
k ~approx~ K tilde ( inf )
.EN
.DE
.SP
.in
.SP
We now pass from projected dynamics to ordinary dynamics,
$i qlhat ~->~ i lhat$, 
by defining the ordinary-dynamics variables
.in +1i
.DS I
.EQ (33)
K (t) ~=~
( xhat (t) , i lhat xhat ) ( xhat , xhat ) sup -1
.EN
.DE
.in
and
.in +1i
.DS I
.EQ (34)
k (t) ~=~
( i lhat xhat (t) , i lhat xhat ) ( xhat , xhat ) sup -1
.EN
.DE
.SP
.in
The tilde is dropped for ordinary dynamics.
.SP
Analysis of eqs 22, 27, 33 and 34 shows that for for times $t >> tmic$,
and for ordinary dynamics,
.in +1i
.DS I
.EQ (43)
K (t) ~approx~ k e sup -kt
.EN
.DE
.in
For times $t*$, $tmic << t* << trxn$, $K ( t* )$ is approximately constant,
and in this plateau region, extending from time $tmic$ to $trxn$,
we can identify the phenomenological rate constant as being
.in +1i
.DS I
.EQ (44)
k ~approx~ K ( t* ) 
~=~ ( xhat ( t* ) , i lhat xhat ) ( xhat , xhat ) sup
-1
.EN
.DE
.in
.SP
The principal result of all the foregoing analysis
is the expression of eq. 44 for the rate constant.
We have now identified the phenomenological rate constant with a 
microscopic time correlation function,
and the link between theory and experiment is complete.
The picture is now clear: 
the macroscopic rate constant $k$ is
equal to the derivative of the time correlation function for fluctuations
of the progress variable $xhat$,
evaluated for times $t*$ in the plateau region.
Because eq. 44 is for ordinary dynamics, the time correlation function
is accessible to analysis and to modeling by molecular dynamics.
.SP
The approximations made to this point, in passing from eq. 24 to 29, from
eq. 33 to 43, and from eq. 43 to 44, 
all depend on the time-scale separation of $trxn$ from $tmic$.
For reactions with $trxn >$ ps, the error has been shown to be negligible.
If there is no plateau region, then eq. 24 or eq 25 must be used.
.SP
We note that eq. 43 is an expression of the Onsager principle
relating microscopic and macroscopic dynamics:
for a system near equilibrium,
on average
the microscopic dynamics of the decay of large fluctuations in a variable
(the dynamics of decay of its time correlation function),
are the same as the dynamics of
relaxation of the variable to its equilibrium value in a macroscopic system.
.SP 2
3. - \fBClassical systems.\fP
.P
The above expressions hold when modified for classical dynamics and
averaging.
Specifically,
.in +1i
.DS I
.EQ (49)
K (t) mark ~=~ < chi dot chi ( t ) > < chi chi > sup -1
.EN
.SP
.EQ
lineup ~=~
-~ d over dt < chi chi ( t ) > < chi chi > sup -1
.EN
.DE
.in
.SP
As an example, 
consider a system with a single reactive degree of freedom $q$ 
and a double-well potential $W(q)$ (Fig. 2 of Kapral et al.).
For this system, the progress variable is determined by whether the value of
$q$ corresponds to well A or well B, to the left or the right of the barrier top at $q sup "\(dd" $:
.in +1i
.DS I
.EQ (50)
chi (q) ~=~ 
- ( theta ( q ~-~ qdag ) ~-~ < theta ( q ~-~ qdag ))>
.EN
.DE
.in
From eqs. 49 and 50, with the operations given in \fIgreat\fP detail,
.in +1i
.DS I
.EQ 
K ( t ) mark ~=~
-~ d over dt < chi chi ( t ) > < chi chi > sup -1 
.EN
.EQ
lineup ~=~
- d over dt
left [
< theta (q - qdag ) theta (q(t) - qdag ) > - < theta (q - qdag ) > sup 2
right ]
< chi chi > sup -1
.EN
.EQ 
lineup ~=~ 
- < theta (q - qdag ) theta dot (q(t) - qdag ) > < chi chi > sup -1
.EN
.EQ 
lineup ~=~ 
< theta dot (q - qdag ) theta (q(t) - qdag ) > < chi chi > sup -1
.EN
.EQ (51)
lineup ~=~ 
< q dot delta (q - qdag ) theta ( q ( t ) ~-~ qdag ) > < chi chi > sup -1
.EN
.EQ
lineup ~=~
< delta (q - qdag ) > < q dot theta ( q ( t ) ~-~ qdag  ) > sub {qdag}
< chi chi > sup -1
.EN
.EQ (51a)
lineup ~=~
1 over {{x sub B} ~ (1 ~-~ {x sub B})} ~
P ( qdag ) ~ < q dot theta ( q ( t ) ~-~ qdag  ) > sub {qdag} 
.EN
.DE
.in
where
.in +1i
.DS I
.EQ
< chi chi ( t ) > mark ~=~ 
< ( theta (q - qdag ) - < theta (q - qdag ) > ) cdot
( theta (q(t) - qdag ) - < theta (q - qdag ) > ) >
.EN
.SP
.EQ
lineup ~=~
< theta (q - qdag ) theta (q(t) - qdag ) > - < theta (q - qdag ) > sup 2
.EN
.SP
.EQ
d over dt theta ( q ( t ) ~-~ qdag  ) mark ~=~
d over dq theta ( q ( t ) ~-~ qdag  ) ~ d over dt ( q ( t ) ~-~ qdag  )
.EN
.EQ 
lineup ~=~
delta (q(t) - qdag ) q dot ( t ) 
.EN
.SP
.EQ
< chi chi > mark ~=~
< ( theta (q - qdag ) - < theta (q - qdag ) > ) sup 2 >
.EN
.EQ
lineup ~=~ 
< theta (q - qdag ) sup 2 > - < theta (q - qdag ) > sup 2 
.EN
.EQ
lineup ~=~ < theta (q - qdag ) > - < theta (q - qdag ) > sup 2
~=~ {x sub B} ~ (1 ~-~ {x sub B})
.EN
.SP
.EQ
theta (x) ~=~  "Heaviside function: 1 if x > 0, 0 otherwise "
.EN
.SP
.EQ
P ( qdag ) mark ~=~ < delta (q - qdag ) >
.EN
.EQ
lineup ~=~ "normalized probability density for " q= qdag
.EN
.SP
.EQ
<...> sub {qdag} ~=~  "averaging of ... with q initially constrained at " qdag
.EN
.DE
.in
.SK
Eq. 51a is eq. 17 of Chandler (1998). 
.SP
Eq. 18 of Chandler (1998) for the TST theory rate constant is obtained for 
$t -> 0 sup +$:
.in +1i
.DS I
.EQ (52)
k sup TST mark ~=~ lim from {epsilon -> 0 sup +} K( epsilon )
.EN
.EQ
lineup ~=~
lim from {epsilon -> 0 sup +} 
1 over {{x sub B} ~ (1 ~-~ {x sub B})} ~
P ( qdag ) ~ < q dot theta ( q ( epsilon ) ~-~ qdag  ) > sub {qdag}
.EN
.EQ
lineup ~=~
1 over {{x sub B} ~ (1 ~-~ {x sub B})} ~
P ( qdag ) ~ < q dot theta ( q dot ) > sub {qdag}
.EN
.EQ (54)
lineup ~=~
1 over {{x sub B} ~ (1 ~-~ {x sub B})} ~
P ( qdag ) ~ {< | q dot | >} over 2
.EN
.DE
.in
.SP
Eq. 54 for the TST rate constant can be expressed 
in a form more familiar from our previous treatment
of TST, by evaluating $P( qdag )$ and $< | q dot | >$.
For convenience, we assume that the work function (free energy) $W(q)$,
with minima at $q sub A$ and $q sub B$, has the same shape 
(same frequency, or force constant) for both wells and is harmonic,
i.e., near $q sub A$, $W(q) = W''(q - q sub A )$ 
with $W'' = m omega sub A sup 2 $, and similarly near $q sub B$.
$W(q sub A )$ need not equal $W(q sub B )$.
.in +1i
.DS I
.EQ
P( qdag ) mark ~=~ 
{e sup {- W ( qdag )/kT}} over
{int sub {A+B} dq ~ e sup {- W ( q )/kT}}
.EN
.SP
.EQ
lineup ~=~
{e sup {- (W ( qdag ) - W(q sub A ))/kT}} over
{Q sub A ~+~ Q sub B cdot  e sup {- (W ( q  sub B) - W ( q  sub A))/kT}}
.EN
.EQ
lineup ~=~
(1~-~ x sub B )~
{e sup {- DELTA W ( qdag )/kT}} over
{(1~-~ x sub B ) Q sub A ~+~ x sub B Q sub B }
.EN
.EQ
lineup ~=~
(1~-~ x sub B )~
{e sup {- DELTA W sup dag /kT}} over
{Q sub A}
.EN
.DE
.in
giving
.in +1i
.DS I
.EQ
P( qdag ) ~=~
(1~-~ x sub B )~{omega sub A} over {sqrt {2 pi }} 
left [ m over {k T} right ] sup 1/2 
{e sup {- DELTA W sup dag /kT}}
.EN
.DE
.in
for
.in +1i
.DS I
.EQ
Q sub A mark ~=~ Q sub B 
.EN
.SP
.EQ
lineup ~=~ 
int sub A dq ~ e sup {- (W ( q ) - W(q sub A ))/kT}
.EN
.SP
.EQ
lineup ~=~ 
int sub A dq ~ e sup {- m omega sub A sup 2 (q - q sub A) sup 2 /2kT}
~=~ 
left [ {2 pi k T} over m right ] sup 1/2 1 over {omega sub A}
.EN
.DE
.in
Also,
.in +1i
.DS I
.EQ
{< | q dot | >} over 2 mark ~=~
{int from {- inf} to {+ inf} dp ~ |p| over m e sup {{- {p sup 2}} over 2mkT}}
over
{int from {- inf} to {+ inf} dp ~ e sup {{- {p sup 2}} over 2mkT}}
~=~ left [ kT over {2 pi m} right ] sup 1/2
.EN
.DE
.in
Evaluating eq. 54,
.in +1i
.DS I
.ig
.EQ
k sup TST mark ~=~
1 over {{x sub B} ~ (1 ~-~ {x sub B})} ~
(1~-~ x sub B )~{omega sub A} over {sqrt {2 pi }} 
left [ m over {k T} right ] sup 1/2 
{e sup {- DELTA W sup dag /kT}}
left [ kT over {2 pi m} right ] sup 1/2
.EN
.SP
..
.EQ
k sup TST  ~=~ 
1 over {x sub B} {omega sub A} over {2 pi} {e sup {- DELTA W sup dag /kT}}
.EN
.DE
.in
Noting that for a double well potential
the relaxation rate constant 
$k = k sub f + k sub r = k sub f / x sub B$,
we obtain Hynes (1988) eq. 4:
.in +1i
.DS I
.EQ (Hynes-4)
k sub f  sup TST mark ~=~
{omega sub A} over {2 pi} {e sup {- DELTA W sup dag /kT}}
.EN
.DE
.in
.SK
COMMENT: Chandler's theory, the reactive flux method, is important:
.SP
.in +.25i
(1) It is firmly grounded in physical theory,
with precisely-stated approximations, 
the effect of which can be evaluated.
.br
.SP
(2) The expression for the rate constant (eq. 44 or 49)
can be evaluated by computer simulation.
.br
.SP
(3) Transition State theory is a limiting case of reactive flux theory
(see eq. Hynes-4).
TST is placed on a firm and clear theoretical footing,
without the need for tricks and special assumptions to solve the mathematical
model.
Kramers theory also does this, but perhaps with less elegance and
generality.
Note that TST gives always an upper bound on the true rate.
.br
.SP
(4) Reactive flux theory makes contact with Kramers theory and Grote-Hynes
theory through the Generalized Langevin equation, 
i.e., through
the memory function $k tilde (t)$ and the corresponding ordinary-dynamics 
function $k(t)$ of eqs. 22 and 34.
However,
I do not know of a derivation of an explicit statement of this relationship.
.in
.SP
The last point above points to the principal drawback of reactive flux
theory:
the correlation function expression for the dynamics of a progress
variable is easy to work with computationally but
difficult to work with analytically, as in reduction to the expressions
of Kramers theory or to those of the multi-dimensional extensions of
TST.
Kramers theory and most other theories work directly with the 
dynamics of phase-space variables
and lead to expressions for a rate constant that can be evaluated without the
need for numerical methods.
Reactive flux theory depends upon evaluation at a plateau time, $t*$, 
which is difficult to explore other than by numerical methods.
This difficulty is inherent in the nature of a chemical reaction:
the system changes in a finite time, $trxn$,
which prevents the extrapolation to infinite time/zero frequency that
is so effective in analysis of other transport properties
by use of the Kubo and Mori-Zwanzig methods.
