Natural Science Forum / Physics / Research / August 2007

To be a bit more precise, here the sum index i should range over all
eigenstate's of the quantum mechanical Hamiltonian of the system, while
E(i) gives the corresponding eigenvalue. An equivalent way to write Z is
in terms of the Hamiltonian operator

 Z = tr [exp(-H/kT)] = sum_n <n|exp(-H/kT)|n>,

where |n> represent any complete set of states, not necessarily
eigenstates of H. This is the most general way to write the canonical
partition function.

The step from the expression of the partition function as a trace over
quantum states to that of integration over the classical phase space is
an approximation. This approximation is justified if all degrees of
freedom behave classically. But, as in your examples, there are
situations where this approximation is justified for some, but not all,
degrees of freedom. In this case, the classical approximation can be
applied selectively. The key is to separate the classical motions from
the quantum ones.

I'll illustrate with the simple example of two particles in an external
potential. From there, perhaps you can figure out how to apply the same
idea to your molecular system.

Consider two particles of equal mass m. They are both subject to an
external potential U(x), in addition to their mutual interaction
potential V(x). The total Hamiltonian is

 H = (p_1^2 + p_2^2)/2m + U(x_1) + U(x_2) + V(x_1-x_2).

Now, suppose that the potential V(x) is strongly confining, and the two
particles are expected to spend most of the time in the bound ground
state. Suppose further that the external potential varies slowly on the
scale that typically separates x_1 and x_2 in the bound ground state.
Under these assumptions, it should be clear that the center of mass
motion, for large enough ambient temperature, can be treat classically,
while the relative motion of x_1 and x_2 cannot.

However, we can make a canonical change of coordinates that reflects
this information:

 X = (x_1 + x_2)/2,  P =  p_1 + p_2,
 x =  x_1 - x_2,     p = (p_1 - p_2)/2.

The Hamiltonian takes the new form

 H = P^2/4m + p^2/m + U(X+x/2) + U(X-x/2) + V(x)
   = [ P^2/4m + 2U(X) ] + [ p^2/m + V(x) ]
       + [ U(X+x/2) - 2U(X) + U(X-x/2) ]
   = H_0 + H_1 + K,

where H_0 and H_1 correspond to the first two square brackets, while K
corresponds to the last one. If we consider only the H_0 + H_1
Hamiltonian, then it describes a system where the center of mass and
relative motions, X and x, are decoupled. Since the first one can be
treated classically, we can make the approximation

Z_01 = tr[exp(-(H_0+H_1)/kT)]
     = integral (dP dX)/h tr[exp(-(E_0(X,P)+H_1)/kT)]
     = integral (dP dX)/h exp(-E_0(X,P)/kT) tr[exp(-H_1/kT)],

where the last trace is taken ove only states describing the relative
motion x and E_0(X,P) is the classical energy of the center of mass
motion. So, X is treated classically, while x quantum mechanically.

Once we reintroduce K, the expression becomes

 Z = integral (dP dX)/h exp(-E_0(X,P)/kT) tr[exp(-(H_1+K)/kT)].

Above, X and P are everywhere treatd classically and x and p is
everywhere a quantum operators. If K is expected to be small in a
typical state at the given temperature, then it can be treated as a
perturbation. In particular, you can choose |n> to be the set of H_1
eigenstates and use those to compute the remaining trace. If K is a
perturbation, only a small number of |n> states above the kT energy
should suffice.

This is especially useful when H_1 itself is (or can be approximated by)
a harmonic oscillator Hamiltonian. Then, K can be expanded in a power
series in x and H_1+K becomes a perturbed harmonic Hamiltonian. There
are various tricks for approximating eigenstates and the partition
function for a perturbed harmonic oscillator. The details can be found
in many books on quantum and statistical mechanics. In particular the
Statistical Mechanics books by Pathria or Huang should be useful.

Hope this helps.

Igor