The above approach to obtain the average of the product ( R ) v considers the ground state potential (electronic) energy surface with no vibrational and rotational structure. To get the true microscopic rate, we need the state to state reaction cross section R ( nR nP ) where nR and nP are the complete set of reactant and product quantum numbers. The state to state rate constant can then be obtained from:
k {( nRnP ); T } = < v R ( nR nP )>
(35.6)
The averaging (which is commonly denoted by < > ) is usually done by using the Maxwell-Boltzmann velocity distribution f ( v ) at temperature T
k ( nR nP; T ) =
dv f ( v ) v R ( nR nP )
(35.7)
To obtain R (nR nP) we need to study the evolution of the reactant wavefunction by solving the time dependent Schrodinger equation. This gives us the probabilities of obtaining the products in various states (i.e., through the "extent" of the reactant wave function going over to those various product wave functions). From the state to state rate constants, the macroscopic rate constant k can be obtained by the procedure outlined in Fig (35.5). There is an averaging over all the reactant and product quantum states. We have also introduced here the concept of the differential cross section, R (nR nP,
, v) where
is the relative orientation of the products. The integral cross section R ( nR nP, v ) is obtained by integrating the differential cross sections over all possible orientations ( ) in three dimensions.
Figure 35.5 Different stages in going from the differential cross section [ R (nR nP, , v) ] and the the microscopic rate constant [ k ijlm (T) ] to the macroscopic rate constant k.
( R ) v considers the ground state potential (electronic) energy surface with no vibrational and rotational structure. To get the true microscopic rate, we need the state to state reaction cross section 
nP ) where nR and nP are the complete set of reactant and product quantum numbers. The state to state rate constant can then be obtained from: 
dv f ( v ) v 
, v) where
is the relative orientation of the products. The integral cross section 
