magnitudes of various atomic charges Za); as a result and because the integral over the
electronic wavefunctions <ef  ei> vanishes, this contribution yields zero. The first and
second integrals need not vanish by symmetry because the wavefunction derivat
<ei   ei> = (R)
is the dipole moment of the initial electronic state (which is a function of the internal
geometrical degrees of freedom of the molecule, denoted R); and
< vi  (R)  vi> = ave
is the vibrationally averaged dipole moment for the particul
Chapter 15
The tools of timedependent perturbation theory can be applied to transitions among
electronic, vibrational, and rotational states of molecules.
I. Rotational Transitions
Within the approximation that the electronic, vibrational, and rotational
to the thermodynamic result:
Nf
gf
Ni = gi exp(h /kT),
and using the above black body g() expression and the identity
(Bi,f ) gf
(Bf,i ) = gi ,
one can solve for the Af,i rate coefficient in terms of the Bf,i coefficient. Doing so yields:
Af,i = Bf,i
2(h
At steady state, the populations of these two levels are given by setting
dNi
dN
= dt f = 0:
dt
Nf
(gBi,f )
= (A +gB ) .
Ni
f,i
f,i
When the light source's intensity is so large as to render gBf,i > Af,i (i.e., when the rate
of spontaneous emission is sma
separate out this intensity factor by defining an intensity independent rate coefficient Bi,f for
this process as:
gf Ri,f = g(f,i ) Bi,f .
Clearly, Bi,f embodies the finallevel degeneracy factor gf, the perturbation matrix
elements, and the 2 factor in
Quite often, the states between which transitions occur are members of levels that
contain more than a single state. For example, in rotational spectroscopy a transition
between a state in the J = 3 level of a diatomic molecule and a state in the J = 4 le
contains the ycomponent of the angular momentum operator. Hence, the following
contribution to f,i (E2+M1) arises:
A 2 e
f,i (M1) = 0
<f  j Lyj /me + a Za Lya /ma  i>.
2ch
The magnetic dipole moment of the electrons about the y axis is
y ,electrons =
Before closing this chapter, it is important to emphasize the context in which the
transition rate expressions obtained here are most commonly used. The perturbative
approach used in the above development gives rise to various contributions to the overall
To further analyze the above E2 + M1 factors, let us label the propagation direction
of the light as the zaxis (the axis along which k lies) and the direction of A0 as the xaxis.
These axes are socalled "labfixed" axes because their orientation is det
where is the electric dipole moment operator for the electrons and nuclei:
= j e rj + a Za e Ra .
The fact that the E1 approximation to f,i contains matrix elements of the electric dipole
operator between the initial and final states makes it clear why t
the factors exp [ikrj] and exp[i kRa] can be expanded as:
exp [ikrj] = 1 + (ikrj) + 1/2 (ikrj)2 + .
exp[i kRa] = 1 + (i kRa) + 1/2 (i kRa)2 + . .
Because k = 2/, and the scales of rj and Ra are of the dimension of the molecule, krj
and kRa are l
which will not be treated in detail here, gives rise to two distinct types of contributions to
the transition probabilities between i and f:
i. There will be matrix elements of the form
<f  j cfw_ (e2/2mec2) A(rj,t )2 + a cfw_ (Za2e2/2mac2) A(Ra,t )2
sin2(1/2(  f,i )T)
= 2  f,i 2 T g()
d T/2 .
1/4T2(  f,i )2

If the lightsource function is "tuned" to peak near = f,i , and if g() is much broader (in
sin2(1/2(  f,i )T)
space) than the
function, g() can be replaced by its value at the
(  f,i )2
grows with T. Physically, this means that when the molecules are exposed to the light
source for long times (large T), the sinc function emphasizes values near f,i (i.e., the
onresonance values). These properties of the sinc function will play important
The modulus squared Cf1(T)2 gives the probability of finding the molecule in the final
state f at time T, given that it was in i at time t = 0. If the light's frequency is tuned
close to the transition frequency f,i of a particular transition, the term
The rotational part of the <f   i> integral is not of the expectation value form
because the initial rotational function ir is not the same as the final fr. This integral has the
form:
<ir  ave  fr> = (Y*L,M (,)
ave Y L',M' (,) sin d d )
for linear mo
with m = 0 being the Z axis integral, and the Y and X axis integrals being combinations
of the m = 1 and m = 1 results. Doing so yields:
(Y*L,M (,) Y 1,m (,) Y L',M' (,) sin d d )
=
2L+1 2L'+1 3
(DL,M,0 D*1,m,0 D* L',M',0 sin d d d /2) .
4
4 4
The l
<ir  ave  fr>z = (Y*L,M (,) cos Y L',M' (,) sin d d ) ,
where is the magnitude of the averaged dipole moment. If the molecule has no
dipole moment, all of the above electric dipole integrals vanish and the intensity of E1
rotational transitions is zero
C. Vibronic Effects
The second term in the above expansion of the transition dipole matrix element a
f,i /Ra (Ra  Ra,e) can become important to analyze when the first term fi(Re) vanishes
(e.g., for reasons of symmetry). This dipole derivative term, when
For example, if the initial and final states have very similar geometries and
frequencies along the mode that is excited when the particular electronic excitation is
realized, the following type of FranckCondon profile may result:
2
<i f>
vf= 0 1 2 3
The => * transition thus involves ground (1A1) and excited (1A1) states whose
direct product (A1 x A1) is of A1 symmetry. This transition thus requires that the electric
dipole operator possess a component of A1 symmetry. A glance at the C2v point group's
"FranckCondon factors". Their relative magnitudes play strong roles in determining
the relative intensities of various vibrational "bands" (i.e., peaks) within a particular
electronic transition's spectrum.
Whenever an electronic transition causes a larg
When electronic transitions are involved, the initial and final states generally differ
in their electronic, vibrational, and rotational energies. Electronic transitions usually require
light in the 5000 cm1 to 100,000 cm1 regime, so their study lies wi
It should be noted that the spacings between the experimentally observed peaks in
HCl are not constant as would be expected based on the above P and R branch formulas.
This is because the moment of inertia appropriate for the v = 1 vibrational level is
These selection rules are derived by realizing that in addition to k = 1, one has:
(i) a linearmolecule rotational wavefunction that in the v = 0 vibrational level is described
in terms of a rotation matrix DL',M',0 (,) with no angular momentum along the
and
<ir  trans  fr> = (DL,M,K (,) trans D* L',M',K' (,) sin d d d )
that determine the rotational selection rules appropriate to vibrational transitions produce
similar, but not identical, results as in the purely rotational transition case.
The derivat
2. Linear Molecules
When the above analysis is applied to a diatomic species such as HCl, only k = 0 is
present since the only vibration present in such a molecule is the bond stretching vibration,
which has symmetry. Moreover, the rotational functions ar
In the special case in which L = L' =0 (and hence with M = M' =0 = K = K', which means
that m = 0 = k), these3j symbols again vanish. Therefore, transitions with
L = L' =0
are again forbidden. As usual, the fact that the labfixed quantum number m can ra
In summary then, vibrations for which the molecule's dipole moment is modulated
as the vibration occurs (i.e., for which (/Ra) is nonzero) and for which v = 1 tend
to have large infrared intensities; overtones of such vibrations tend to have smaller
inte