322
6 Molecular Systems
temperatures less than approximately 10 rot , perhaps an Euler–Maclaurin-type
expression) for calculating values of the usual thermodynamic state functions. This
procedure is appropriate largely because the thermodynamic state functions, which
can be referred to as ‘bulk properties’ of matter because they are associated with
macroscopic amounts of matter, are represented in terms of macroscopic thermal
averages over the equilibrium energy distributions of the microscopic molecular
states, they will hence be insensitive to the behaviours of individual molecules that
occupy the microscopic energy states.
Molecular spectra, however, provide direct probes of the behaviours of individual molecules, and will consequently be highly sensitive to actual equilibrium
distributions. The intensity of individual electronic or vibronic spectral transitions
involves two factors that depend upon aspects of the initial and/or final microscopic
molecular states. The first of these factors, known as the Franck–Condon factor,
is determined by the overlap of the molecular wavefunctions for the initial and
final electronic states of a spectral transition. The second factor is the population of
the initial molecular state. Moreover, because such spectroscopic transitions occur
on a very short time scale, typically fractions of a femtosecond, spectra can be
obtained both for equilibrium and nonequilibrium systems. We shall, however, only
be concerned here with spectra obtained under equilibrium conditions.
Because Franck–Condon factors vary widely from one vibronic transition to
another, the effect of initial-state populations is very difficult to disentangle for such
transitions: even for ultra-high-resolution spectra at elevated temperatures (of the
order of a few thousand kelvin), the role of the initial-state populations is effectively
masked. Fortunately, however, the Franck–Condon factors are almost constant for a
set of rotational fine-structure transitions within a specific vibronic transition. This
means that the population factors play a significant role in determining the intensity
patterns of the rotational fine-structure lines. For heteronuclear diatomic molecules,
we shall often focus upon infrared rotational fine-structure transitions associated
with the vibrational fundamental transition (i.e., the transition v = 0 −→ v = 1)
for molecules in their ground electronic states, while for homonuclear diatomic
molecules, which do not undergo infrared transitions, we shall instead focus upon
Raman transitions for molecules lying in their ground electronic states.
Heteronuclear Diatomic Molecular Spectra
We have seen that the rotational partition function for a heteronuclear diatomic
molecule is given by Eq. (6.2.59), viz.,
z rot (T ) =
∞
j =0
(2j + 1)e
−j (j+1)) rot /T ,
from which the probability for a given molecule to be found in the rotational state
characterized by the rotational quantum number j is simply given by
6 Molecular Systems
temperatures less than approximately 10 rot , perhaps an Euler–Maclaurin-type
expression) for calculating values of the usual thermodynamic state functions. This
procedure is appropriate largely because the thermodynamic state functions, which
can be referred to as ‘bulk properties’ of matter because they are associated with
macroscopic amounts of matter, are represented in terms of macroscopic thermal
averages over the equilibrium energy distributions of the microscopic molecular
states, they will hence be insensitive to the behaviours of individual molecules that
occupy the microscopic energy states.
Molecular spectra, however, provide direct probes of the behaviours of individual molecules, and will consequently be highly sensitive to actual equilibrium
distributions. The intensity of individual electronic or vibronic spectral transitions
involves two factors that depend upon aspects of the initial and/or final microscopic
molecular states. The first of these factors, known as the Franck–Condon factor,
is determined by the overlap of the molecular wavefunctions for the initial and
final electronic states of a spectral transition. The second factor is the population of
the initial molecular state. Moreover, because such spectroscopic transitions occur
on a very short time scale, typically fractions of a femtosecond, spectra can be
obtained both for equilibrium and nonequilibrium systems. We shall, however, only
be concerned here with spectra obtained under equilibrium conditions.
Because Franck–Condon factors vary widely from one vibronic transition to
another, the effect of initial-state populations is very difficult to disentangle for such
transitions: even for ultra-high-resolution spectra at elevated temperatures (of the
order of a few thousand kelvin), the role of the initial-state populations is effectively
masked. Fortunately, however, the Franck–Condon factors are almost constant for a
set of rotational fine-structure transitions within a specific vibronic transition. This
means that the population factors play a significant role in determining the intensity
patterns of the rotational fine-structure lines. For heteronuclear diatomic molecules,
we shall often focus upon infrared rotational fine-structure transitions associated
with the vibrational fundamental transition (i.e., the transition v = 0 −→ v = 1)
for molecules in their ground electronic states, while for homonuclear diatomic
molecules, which do not undergo infrared transitions, we shall instead focus upon
Raman transitions for molecules lying in their ground electronic states.
Heteronuclear Diatomic Molecular Spectra
We have seen that the rotational partition function for a heteronuclear diatomic
molecule is given by Eq. (6.2.59), viz.,
z rot (T ) =
∞
j =0
(2j + 1)e
−j (j+1)) rot /T ,
from which the probability for a given molecule to be found in the rotational state
characterized by the rotational quantum number j is simply given by