In the theory of chemical reactions, it is often possible to isolate a small number
or even a single degree of freedom in the system that can be used to characterize
the reaction. This degree of freedom is coupled to other degrees of freedom
(for example, reactions often take place in solution). Isomerization or
dissociation of a diatomic molecule in solution is an excellent example of
this type of system. The degree of freedom of paramount interest is the
distance between the two atoms of the molecule - this is the degree of freedom
whose detailed dynamics we would like to elucidate. The dynamics of the
``bath'' or environment to which is couples is less interesting, but still
must be accounted for in some manner. A model that has maintained a certain level
of both popularity and success is the so called ``harmonic bath'' model, in which
the environment to which the special degree(s) of freedom couple is
replaced by an effective set of harmonic oscillators. We will examine this
model for the case of a single degree of freedom of interest, which we will
designate q. For the case of the isomerizing or dissociating diatomic,
q could be the coordinate 
, where r is the distance between
the atoms. The particular definition of q ensures that 
.
The degree of freedom q is assumed to couple to the bath linearly, giving
a Hamiltonian of the form
where the index 
runs over all the bath degrees of freedom,
are the harmonic bath frequencies, 
are the harmonic bath masses,
and 
are the coupling constants between the bath and the coordinate q.
p is a momentum conjugate to q, and m is the mass associated with
this degree of freedom (e.g., the reduced mass 
in the case of a diatomic).
The coordinate q is assumed to be subject to a potential 
as well
(e.g., an internal bond potential). The form of the coupling between the
system (q) and the bath ( 
) is known as bilinear.
Below, using a completely classical treatment of this Hamiltonian, we will derive an equation for the detailed dynamics of q alone. This equation is known as the generalized Langevin equation (GLE).