We next consider a more challenging problem, that of VER in neat liquid
oxygen (where now the solute and solvent molecules are both oxygen). This
is more challenging for two reasons: (1) Whereas in the last problem VER
was dominated by the vibration-translation mechanism, in this case since
the solvent is molecular, and one can anticipate that VER will have contributions from both vibration-translation and vibration-rotation channels. This
means that rotations of the solvent will have to be considered explicitly,
and, for example, we will not be able to use the analytical results discussed
above. (2) Whereas the last problem was nearly classical kT ' h¯ ω0 , in
this case, because of the much higher vibrational frequency of oxygen, one
is deeply in the quantum regime h¯ ω0 × kT , and so the issue of the most
appropriate semiclassical approximation scheme becomes very important.
The Hamiltonian for the vibration of a “tagged” (solute) oxygen
molecule is taken to be harmonic, as in Equation (22). The bath
Hamiltonian involves the translations and rotations of all (solute and
solvent) oxygen molecules (4):
L2
P2i
C i
2M
2I
Hb D
i
C
jEri˛
Erjˇ j
34
i
where i and j index the molecules (i D 0 is the solute and i D 1, 2, . . .
E i is the momentum of the ith molecule, M is the
are solvent molecules), P
E i is the angular momentum of the ith molecule, and I
molecular mass, L
is the moment of inertia. ˛, ˇ D 1, 2 index the two sites (atoms) of each
molecule, and Eri˛ is the position of site ˛ on atom i. Thus the potential
energy involves the site-site pairwise potential, r , which is taken to be
of the Lennard-Jones form in Equation (24). The oscillator-bath interaction
has the form V D qF, and F has two contributions, from the centrifugal
and potential forces on the oscillator, respectively (4):
FD
1
L20
C
Ire
2
E ˇ Ð ˆrˇ
F
35
ˇ
where re is the separation of the sites in a molecule (the bond length),
Eˇ D
F
˛
i
Eri˛
jEri˛
Er0ˇ
Er0ˇ j
0
jEri˛
Er0ˇ j
36
is the force on site ˇ of the tagged diatomic, and ˆrˇ is the unit vector
pointing from the center of mass of the tagged diatomic to site ˇ.