Key Concepts for Final

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Nov 15, 2013 (3 years and 4 months ago)

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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.