The intensity of spectral lines
depends on:
• the transition probability
between the two states
(selection rules)
• population of states
Boltzmann
constant
• in absorption, the sample path length l
and concentration c (Beer-Lambert law)
extinction
coefficient
Nils Walter: Chem 260
Rotational spectroscopy:
The rotation of molecules
z (IC)
symmetric:
IA=IB≠IC≠0
x (IA)
y (IB)
Rotor in 3-D
linear: IA=0; IB=IC
asymmetric:
IA≠IB≠IC≠0
spherical: IA=IB=IC ≠0 Nils Walter: Chem 260
The moment of inertia for a
diatomic (linear) rigid rotor
r
definition: r = r1 + r2
m1
m2
C(enter of mass)
r1
Balancing equation: m1r1 = m2r2
Þ
m1m2 2
r = µr 2
I=
m1 + m2
r2
Moment of inertia:
I = m1r12 + m2r22
µ = reduced mass
Nils Walter: Chem 260
Solution of the Schrödinger equation for
the rigid diatomic rotor
h2 2
−
∇ Ψ = EΨ
2µ
d
d
d
2
=
dx
2
2
+
dy
2
+
2
dz 2
(No potential,
only kinetic
energy)
Þ EJ = hBJ(J+1);
rot. quantum number J = 0,1,2,...
D
B=
4πI
[Hz]
Þ EJ+1 - EJ = 2hB(J+1)
Þ microwave spectroscopy (GHz range)
Nils Walter: Chem 260
Selection rules for the diatomic rotor:
1. Gross selection rule
Light is a transversal
electromagnetic wave
Þ a molecule must be polar to
be able to interact with light
a polar rotor appears to
have an oscillating
electric dipole
Nils Walter: Chem 260
Selection rules for the diatomic rotor:
2. Specific selection rule
rotational quantum number J = 0,1,2,…
describes the angular momentum of a molecule
(just like electronic orbital quantum number l=0,1,2,...)
and
light behaves as a particle: photons
have a spin of 1, i.e., an angular
momentum of one unit
and
the total angular momentum upon
absorption or emission of a photon has to
be preserved
Þ ∆J = ± 1
Nils Walter: Chem 260
Okay, I know you are dying for it:
Schrödinger also can explain the
∆J = ± 1 selection rule
Transition dipole
moment
µ fi = ò Ψ f µΨi dτ
initial state
final state
Þ only if this integral is nonzero, the
transition is allowed (for ∆J = ± 1) ; if
it is zero, the transition is forbidden
Nils Walter: Chem 260
The angular momentum
1 2
E = Iω
2
rotational frequency
P = Iω
classical rotational
energy
and
with
Þ
E J = hBJ ( J + 1)
D
B=
4πI
Þ
P = 2 EI
classical angular momentum
mJ
J=1
P = J ( J + 1) D
But P is a vector, i.e.,
has magnitude and
direction
Nils Walter: Chem 260