A derivation of the GLE valid for a general bath can be worked out. The details of the derivation are given in the book by Berne and Pecora called Dynamic Light Scattering. The system coordinate q and its conjugate momentum p are introduced as a column vector:
and, in addition, one introduces statistical projection operators P
and Q that project onto subspaces in phase space parallel and orthogonal to
. These operators take the form
These operators are Hermitian and satisfy the property of idempotency:
Also, note that
The time evolution of is given by application of the classical propagator:
Note that the evolution of is unitary, i.e., it preserves the norm of :
Differentiating both sides of the time evolution equation for gives:
Then, an identity operator is inserted in the above expression in the form I=P+Q:
The first term in this expression defines a frequency matrix acting on :
where
In order to evaluate the second term, another identity operator is inserted directly into the propagator:
Consider the difference between the two propagators:
If this difference is Laplace transformed, it becomes
which can be simplified via the general operator identity:
Letting
we have
or
Now, inverse Laplace transforming both sides gives
Thus, multiplying fromthe right by 
gives
Define a vector
so that
Because 
is completely orthogonal to 
, it is straightforward to
show that
Then,
Thus,
Finally, we define a memory kernel matrix:
Then, combining all results, we find, for 
:
which equivalent to a generalized Langevin equation for a particle subject to a harmonic potential, but coupled to a general bath. For most systems, the quantities appearing in this form of the generalized Langevin equation are
It is easy to derive these expressions for the case of the
harmonic bath Hamiltonian when 
.
For the case of a harmonic bath Hamiltonian, we had shown that the friction kernel was related to the random force by the fluctuation dissipation theorem:
For a general bath, the relation is not as simple, owing to the fact that
is evolved using a modified propagator 
. Thus, the
more general form of the fluctuation dissipation theorem is
so that the dynamics of R(t) is prescribed by the
propagator . This more general relation illustrates the difficulty of
defining a friction kernel for a general bath. However, for the special case of
a stiff harmonic diatomic molecule interacting with a bath for which all the modes
are soft compared to the frequency of the diatomic, a very useful approximation
results. One can show that
where 
is the Liouville operator for a system in which the
diatomic is held rigidly fixed at some particular bond length (i.e., a constrained
dynamics). Since the friction kernel is not sensitive to the details of the
internal potential of the diatomic, this approximation can also be used for
diatomics with stiff, anharmonic potentials. This approximation is
referred to as the rigid bond approximation
(see Berne, et al, J. Chem. Phys. 93, 5084 (1990)).