Key Concepts for Final
Heterogeneous Broadening of Lines

different molecules in the sample may have
instantaneously different environments (or be subject to different perturbations) leading to a
distribution of values for molecular eigenenergies.
Lif
etime broadening

If on average a system survives in a state for a time
, then its energy
levels are blurred by
E
, where
2
h
E
and
is the lifetime of the state. The unc
ertainty of
molecular eigenenergy is due to the Heisenberg Uncertainty Principle. Short

lived states have a
large
E
, while long

lived states have a small
E
. Excited

state lifetimes are determined by the
intrinsic
transition frequency (energy), intrinsic transition strength (probability), extrinsic
deactivation processes (e.g. non

radiative energy transfers such as intermolecular collisions).
Doppler Line Broadening

Radiation is shifted in frequency when the sou
rce is moving
towards or away from the observer. This causes different sub

populations of molecules in the
sample to "see" different radiation frequencies even though the radiation entering the sample is
monochromatic. For molecules moving parallel to (tow
ards) the direction of light propagation,
c
v
1
f
f
. For molecules moving anti

parallel to (away from) the direction of light
propagation,
c
v
1
f
f
where
f
is the actual frequency of light,
f '
is the apparent frequency of
lig
ht, v is the molecular speed, and c is the speed of light.
Electric Dipole Selection Rules
for radiative transitions in molecules
The electric dipole transition moment must be non

zero.
Note: The electric dipole transition moment =
d
μ
μ
i
f
if
ˆ
*
where
μ
ˆ
is the electric dipole
operator and
r
e
μ
ˆ
ˆ
.
Magnetic Dipole Selection Rules
for radiative transitions in molecules
The magnetic dipole transition moment must be non

zero.
Note: The magnetic dipole transit
ion moment =
d
m
m
i
f
if
ˆ
*
where
m
ˆ
is the magnetic
dipole operator.
Absorption Spectroscopy
I
is proportional to
if
if
i
D
N
where
I
is the integrated absorption

line intensity,
N
i
is the number
of molecules in
the initial state,
if
is the transition frequency, and
D
if
is the transition dipole
strength.
Luminescence Spectra
(spontaneous emission)
I
is proportional to
if
if
i
D
N
4
where
I
is the integrated absorption

line intensit
y,
N
i
is the number
of molecules in the initial state,
if
is the transition frequency,
D
if
is the transition dipole
strength, and the initial state (i) is the emitting state.
Absorption: the transition raises the energy of the atom or
molecule
Stimulated (or induced) emission: the transition lowers the energy of the atom or molecule.
The emission is stimulated by the radiation field. The radiation that is emitted has the same
wavelength as the incident radiation, moves in the same dire
ction, and is in phase with the
incident radiation. This type of radiation is also called coherent radiation and is emitted by
lasers.
Spontaneous emission: occurs in the absence of stimulating radiation. It is emitted in all
directions and is not coherent
.
Hydrogen Atom Selection Rules
ns
restrictio
no
1
1
,
0
n
l
m
Multielectron Atom Selection Rules
)
0
for
forbidden
0
to
(0
1
,
0
forbidden)
0
to
(0
1
,
0
0
1
J
M
J
S
L
J
Translation Motion of Molecules in a Confined Volume
In a big box, the spacings between energy levels are small enough that we can approxi
mate a
continuum of energy. Classically, the average translational energy
T
k
E
B
2
3
where k
B
is
Boltzman's constant.
In a confined volume (e.g. electrons in the
system of benzene), the energy levels must be
treated as
quantized because the spacings between energy levels are large.
2
2
2
2
2
2
2
8
c
n
b
n
a
n
m
h
E
z
y
x
where
,...
3
,
2
,
1
,
,
z
y
x
n
n
n
Rigid Rotor
The object rotates, but does not vibrate
I
h
J
J
E
J
2
2
8
1
where J = 0,1,2…
The degeneracy of the energy level correspon
ding to each J is given by
1
2
J
g
J
Rotational energy levels are NOT equally spaced
Selection Rules: The molecule must have a permanent electric

dipole moment and
1
J
;
1
,
0
J
M
;
0
J
K
.
Ty
pes of Rigid Rotor
Linear ……………………………………………. I
c
= I
b
and I
a
= 0
Spherical Top …………………………………… I
a
= I
b
= I
c
Symmetric Top (oblate)………………………….. I
a
= I
b
< I
c
Symmetric Top (prolate)………………………….. I
a
< I
b
= I
c
Asymmetric Top …………………………………. I
a
< I
b
< I
c
Where I
a
, I
b
, and I
c
are the moments of inertia about the principal axes of inertial a, b, and c.
These axes are mutually perpendicular. The unique principal axis (or figure axis) of a symmetric
top is always an axis of symmetry of order great
er than or equal to 3.
Moment of Inertia of a Diatomic
2
2
1
2
1
R
m
m
m
m
I
where
I
is the moment of inertia,
R
is the bond length, and the reduced mass =
2
1
2
1
m
m
m
m
.
Rotational Spectra of Diatomic Molecules
Rotational constant =
cI
h
B
2
8
when
B
is expressed in cm
1
; and
I
h
B
2
8
when
B
is
expressed in Hz.
Rotational Energy levels
Linear:
I
h
J
J
E
J
2
2
8
1
with a degeneracy of
1
2
J
g
J
, wher
e J = 0,1,2,3,… and
,...
2
,
1
,
0
J
M
Linear; Spherical Top; Symmetric Top … expressions for energy, total angular momentum,
and angular momentum about lab z

axis for each
Eigenfunctions are spherical harmonic functions
Vibrational Motion
Solutions
are Hermite Polynomials
0
v
2
1
v
h
E
where the fundamental frequency
k
2
1
0
and v = 0,1,2,…
Vibrational energy levels are equally spaced (
0
h
)
Anharmonicity: deviation from harmonic behavior
Selection Rules:
1
v
for harmonic behavior; and
...
3
,
2
,
1
v
for anharmonic
behavior. There must also be a change in dipole moment when the molecule is distorted
along the normal vibration coordinate.
Vibration

Rotation Absorption Spectra
Selection rules:
1
v
,
1
,
0
J
;
1
,
0
J
M
;
0
J
K
. There must also be a change in
dipole moment when the molecule is distorted along the normal vibration coordinate; and the
molecule must ha
ve a permanent electric

dipole moment. (
0
J
only in special cases.
Typicaly,
1
J
).
Raman Spectroscopy
Selection Rules: Molecule must have anisotropic electric polarizablity;
2
,
0
J
for linear
molecu
les,
2
,
1
,
0
J
for non

linear molecules; and
1
,
0
v
.
A molecule has anisotropic polarizability if its polarizability is different in different
directions
Anti

Stokes lines correspond to transitions from higher to lower energy
levels. In this case,
the molecule makes a transition with
2
J
and the scattered photon emerges with
increased energy (and therefore higher frequency than the incident radiation).
Spectral lines corresponding to transitions from a lowe
r to a higher molecular energy levels
are Stokes lines. The molecule makes a transition with
2
J
and the lines appear at lower
frequency than the incident radiation.
Lasers

L
ight
A
mplification by
S
timulated
E
mission
Must disturb an e
quilibrium population to obtain an inverted population
Stimulated vs. spontaneous emission
3

level laser, 4

level laser
Types of lasers: solid

state lasers, liquid lasers, gas lasers
General characteristics of lasers: high photon flux, beam collimation, ph
ase coherence
(spatial and temporal), monochromatic light output
Other Concepts
Band Theory: insulators vs. semiconductors (p

type, n

type semiconductors)
Classical vs. quantum descriptions of light
Degrees of freedom: translational (3 per molecule); rota
tional (2 per linear molecule, 3 per
non

linear molecule); vibrational (
5
3
N
per linear molecule,
6
3
N
per non

linear
molecule) where N is the number of atoms in the molecule
Einstein transitions probabilities; einstei
n corefficients
Rotational Stark Effect
Resonance condition (also known as the Bohr frequency rule):
JK
K
J
E
E
h
'
'
'
Centrifugal Distortion
Normal modes of vibration: specify the atomic trajectories associated with distinct (and
independent) modes
of vibrational motion
Franck

Condon principle states that for an electronic transition (optical excitation), the nuclei
do not respond immediately, and a non

equilibrium state results. This is followed by nuclear
relaxation.
Circular Dichroism
Fluorescence
Phosphorescence
Internal Conversion
Intersystem Crossing
Chromophore: a molecule or part of a molecule that exhibits a characteristic absorption
P, Q, and R branches
Note
: This list is only a guide to help you study. It is NOT comprehensive, and the exa
m may
cover any topics discussed in class.
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