The ``good'' quantum numbers associated with the transition states
discussed above in Sec. iiB are in general not globally
conserved quantum numbers.
If they were, the molecular Hamiltonian would be integrable, and
thus certainly not chaotic, strongly mixed, etc.
Most of the ``good'' quantum numbers of the transition state
-- e.g., the vibrational quantum numbers
-- are ``good'' only in the
transition state region and thus only relevant for approximating
the eigenvalues of 
.
The total angular momentum J (in field-free space),
however, is a globally
conserved quantum number; consequently
states of different J are non-interacting.
In applying any statistical theory one should thus take cognizance of all
globally conserved quantum numbers -- e.g., total angular momentum
(in field-free space), global discrete symmetries (the molecular
symmetry group), etc. -- and invoke the statistical assumption of strong
mixing only within each manifold of states labeled by the globally
conserved quantum numbers.
How the globally conserved symmetries are included into RM/TST is
the subject of the present analysis.
The role of symmetry in statistical theories has been treated in
other contexts by several authors.[17, 18]
Therefore suppose that 
are the globally conserved quantum
numbers, and that 
are the quantum numbers conserved only
locally in the transition state region.
The eigenvalues, 
,
of the matrix
are labeled by the complete set of quantum numbers arising
from the direct product of and .
One then applies the RM/TST theory separately for each set of the
conserved quantum numbers : the distribution of unimolecular
decay rates for the -manifold of strongly mixed states is
thus given by
(noting Eqs. (2.13a) and (2.35))
where 
is the average decay rate for the states in
the -manifold and
is the cumulative reaction probability for the -manifold.
The combined or total distribution is the sum
over all the -distributions weighted by the density of states,
i.e.,
with
where
is the density of states of the -manifold and
is the total density of states.
This is rewritten in terms of the reduced
distributions, Eq. (2.13), as
Note that this distribution yields the usual transition state
(or RRKM) expression for the average rate:
The total distribution can also be written, in terms of the cumulative
reaction probabilities of the different -manifolds, as
Furthermore, the moments of the distribution can be written analytically in terms of the moments for each manifold as
where 
denotes an average with respect to
.
A simple example of these expressions results when the only
underlying symmetry divides the states into two uncoupled
manifolds, each with an equal density of states.
This does not necessarily imply that the corresponding CRP's are
equal as the states in a given manifold access only the transition
states labeled by the corresponding global symmetry of the
manifold.
(This case is actually physically relevant as it can arise if
the molecular symmetry group is 
.)
Eq. (2.41) becomes
and Eq. (2.42) becomes
This expression with n=2 can be used to obtain the effective number of channels,
where 
is the effective number of channels
for the i-manifold.
Thus in order to predict a decay rate probability distribution for
a given system, one first searches for any conserved symmetries or quantum
numbers.
Eq. (2.36) is used to obtain the distribution for each of the symmetry
blocks.
These are combined using Eq. (2.41) to obtain
the final result.
Note that if only the moments are desired then one first uses
the 's to obtain
analytically
(e.g. Eq. (2.11)
for the second moment) and then use of Eq. (2.42)
provides the symmetry adapted RM/TST moments directly.
This can also provide a useful check on the numerical evaluation of the
probability distribution.
