TD3 Statistical Physics (M1)
Diatomic Molecules
We will consider in the following a perfect gas of N identical diatomic molecules, at the temperature T in the
volume V. We will first study the case of a single isolated diatomic molecule composed by two atoms with
different masses mA and mB.
A simple model of heteronuclear diatomic molecule consists in writing the total Hamiltonian of the molecule
as the sum of 5 contributions
H=Htrans+Hvib+Hrot+Hélec+Hnucl where
Htrans describes the motion of the center of mass of the molecule
Hvib is due to the vibration of the molecule around its equilibrium bond length
Hrot characterizes the rotational motion of the molecule around its center of mass
Hélec describes the electronic motion in the field created by the nuclei
Hnucl describes the motion of the nucleons inside nuclei
1) Motion of the center of Mass:
Write the classical Hamiltonian for the classical motion of the center of mass of the isolated molecule.
Calculate Z classical
the partition function corresponding to this classical motion.
trans
2) Classical Vibrational Motion.
The Hamiltonian of a 1D classical harmonic oscillator is:
H classical

vib
p2 1
 Kx 2
2 2
where  is the “reduced mass”  
mAmB
mA  mB
Calculate Z classical
the partition function of this classical oscillator.
vib
3) Quantum Vibrational Motion.
If the vibration energy corresponds to those of a 1D quantum harmonic oscillator with the frequency
w, the vibration energies are  vib  (n  1 / 2) .
Show that the corresponding partition function can be written:
Z vib
  

exp  
2k B T 


  

1  exp  
 k BT 
Give the temperature Tvib above which the vibrational behavior of the diatomic molecule is classical.
4) Classical Rotational Motion.
The classical kinetic energy of the rotational part can be written in spherical coordinates as:
H classical

rot

1 2 2
r   sin 2  .2
2

where r  rA  rB is the distance between the 2 particles (rA is the distance of particle A to the center
of mass, rB the distance of particle B to the center of mass) and   mA .mB /( mA  mB ) is the
effective mass of the molecule.
Prove that r 2  m A rA2  mB rB2 .
I  r 2 is the momentum of inertia of the molecule around its center of mass.
Give the expression of the effective momenta p  and p  .
Calculate the corresponding partition function Z classical
by a proper integration in phase-space.
rot
5) Quantum Rotational Motion.
In quantum mechanics, the rotational energy levels are given by:
2
J ( J  1)
2I
Where J (0  J  ) is the rotational quantum number giving the discrete values of the angular
 rot 
momentum L   J ( J  1) and I is the momentum of inertia of the molecule around its center of
mass. We remember that, for a given J , there are 2J  1 accessible quantum states referenced by
the quantum number m( J  m  J ) associated to the eigenvalues of the L z operator.
T
2
and x  rot .
2.I.k B
T
Write the partition function Z rot characterizing the rotational motion of the molecule.
We note Trot 
Give the temperature above which the vibrational behavior of the diatomic molecule is classical.
In the high temperature regime (x<<1), write the contribution of the rotational motion to the free
energy per molecule f rot , to the internal energy u rot , and to the Heat Capacity c V , rot .
Idem in the low temperature regime (x>>1).
6) Electrons and nucleons
For the last two contributions to the total partition function, Z élec and Z nucl , we suppose that only the
fundamental electronic and nuclear energy levels contribute.
We note  elec the rotational quantum number of electrons, Selec the electronic spin, SA the nuclear spin
of molecule A and SB the nuclear spin of molecule B, A and B the nuclear energy levels. Write Z élec
and Z nucl .
Assembly of molecules.
7) What would be the total partition function for N indistinguishable and “non-interating” diatomic
heteronuclear molecules (dilute limit) in the classical limit?
8) Compute the corresponding free energy F, the internal energy U, the entropy S and the heat capacity
at constant volume Cv in the classical limit.
Appendix
Euler – Mac Laurin Formula :

 f (n )  
n 0

0
f (u )du 
1
1
1 ( 3)
f 0   f ' 0  
f 0   ...
2
12
720