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.EQ
delim $$
.EN
.ce
EQUATIONS AND RELATIONSHIPS 
.ce
FROM VARIOUS THEORIES OF REACTION RATES
.sp2
I. TRANSITION STATE THEORY (Eyring, 1935)
.sp
Fowler and Guggenheim (1952): Eqn. (1208.7) for a bimolecular reaction,
.DS I
.EQ (I.1)
k sub 2 ~=~ kappa {phi sub T (T)} over {phi sub 1 (T) phi sub 2 (T)}
kT over h e sup {- epsilon sub 0 / kT}
.EN
.DE
.sp
Hynes (1988): Eqn. (2) for an isomerization,
.DS I
.EQ (I.2)
k sup TST ~=~ 
kT over h {Q sup \(dd } over {Q sub R} 
e sup {- DELTA E sup \(dd / k sub B T}
.EN
.DE
.sp
Hynes (1988): Eqn. (4) for an isomerization,
.DS I
.EQ (I.3)
k sup TST ~=~ 
{omega sub R} over {2 pi}
e sup {- DELTA G sup \(dd / k sub B T}
.EN
.DE
where $omega sub R$ is the vibrational frequency along the reaction
coordinate for the reactant well. 
.sp
TST was known to give a high estimate for a reaction rate.
The true rate of reaction was smaller by a factor $kappa$:
.DS I
.EQ (I.4)
k ~=~ kappa ~ k sup TST
.EN
.DE
The factor $kappa$ could not be calculated by TST.
It has been the object of many other theoretical treatments.
.SK
II. KRAMERS THEORY (Kramers, 1940)
.sp
Hynes (1988): Eqn. (9) for Kramers calculation of the factor $kappa$,
for the strong coupling regime,
.DS I
.EQ (II.1)
kappa sup Kramers ~=~ [ 1 ~+~ alpha sup 2 ] sup 1/2 ~-~ alpha
.EN
.DE
.in +1.0i
.DS
.EQ
alpha ~=~ zeta over {2 omega sub b}
.EN
.DE                                                         
.in -1i
where $zeta$ is the solvent viscosity, 
and $omega sub b$ is the frequency corresponding to the curvature of the
reaction coordinate at the top of the barrier, analogous to $omega sub R$
which corresponds to the curvature of the reactant well bottom.
The time of passage over the barrier top (time spent near the barrier top)
is inversely proportional to $omega sub b$.
.sp
For limiting high viscosity, $zeta ~>>~ 2 omega sub b$,
.DS I
.EQ (II.2)
kappa sup {Kramers, overdamped} ~=~ {omega sub b} over zeta
.EN
.DE
and the reaction rate for a two-well potential is
.DS I                                                       
.EQ (II.3)
k sup {Kramers, overdamped} ~=~ {omega sub b} over zeta
{omega sub R} over {2 pi}
e sup {- DELTA G sup \(dd / k sub B T}
.EN
.DE
.sp
For low viscosity, $zeta ~<~ 2 omega sub b$,
$kappa sup Kramers ~=~ 1$ and the TST estimate of the reaction rate is
correct, i.e., the low viscosity limit of Kramers theory 
(for the high coupling regime) is the TST rate.
.SK
III. REACTIVE FLUX (Chandler, 1978)
.sp
Chandler (1998): Eqn. (16-17 = 6.1-6.2) for the reactive flux,
.DS I
.EQ (III.1a)
k(t) mark ~=~ -~d over dt  C sub {delta h} ~=~ 
-~d over dt  {< delta h delta h(t)>} over {< ( delta h) sup 2 >}
.EN
.EQ (III.1b)
lineup ~=~
1 over {x~(1~-~x)}  < q dot~delta (q ~-~ q* )~h (q(t)) >
.EN
.DE
.DS I
.EQ (III.2)
lineup ~=~
1 over {x~(1~-~x)} P(q * ) < q dot~h (q(t)) > sub {q *}
.EN
.DE
.sp
Chandler (1998): Eqn. (18 = 6.3) for the TST rate,
.DS I
.EQ (III.3)
k sup TST ~=~ k(0 sup + ) ~=~
1 over {x~(1~-~x)} P(q * ) < q dot~theta (q dot ) > sub {q *}
.EN
.DE
which for a symmetrical two-well potential evaluates as
.DS I
.EQ (III.4)
k sub A->B sup TST mark ~=~ 
1 over 2 < | q dot | > sub {q *} {e sup {- DELTA W* /kT}} over {Q}
.EN
.DE
.DS I
.EQ (III.5)
lineup ~=~
{omega sub A} over {2 pi} {e sup {- DELTA W* /kT}}
.EN
.DE
Eqn. (III.5) is identical with Eqn. (I.3).
.sp
Chandler (1998): Eqn. (21 = 6.6) for $kappa$,
.DS I
.EQ (III.6)
.EQ
kappa ~=~ k over {k sup TST} ~=~ k(t*) over {k(0 sup + )}
.EN
.DE
where the time $t*$ is in the range 
$tau sub mol < t* << tau sub rxn = 1/k$;
$tau sub mol$ is the characteristic time of transient processes at the
barrier top.
.SK
IV. OTHER THEORETICAL TREATMENTS OF RATES OF REACTION IN SOLUTION
.sp
A. Grote-Hynes theory (1980): 
treats the case of high, sharp barriers, for which,
owing to short residence time at the top of the barrier, the simple Langevin
description with constant friction fails, and thus Kramers theory fails
also. Grote-Hynes theory for solution reactions is based on the Generalized
Langevin equation, with memory friction.
.sp
B. Multidimensional transition state theory: treats a multidimensional
reaction coordinate.  This can be complicated coupling 
between the reaction coordinate
and other degrees of freedom, such as solvent, 
as treated by Grote-Hynes and Kramers theories.
It has been shown that Multidimensional transition state theory and
Grote-Hynes theory are equivalent expressions of the same underlying model.
Kramers theory and TST theory are limiting cases of each.
.sp
C. Lowest eigenvalue for the dynamical equation governing evolution of the
coarse-grained probability density, e.g., equations of the Fokker-Planck type.
.sp
D. Mean-first-passage time: another approach to 
solution of stochastic dynamical equations.
