Transcript Lecture XVIII_XIX

Harmonic Oscillator
Harmonic Oscillator
Selections rules
Permanent Dipole moment
An electric dipole consists of two electric charges q and -q
separated by a distance R. This arrangement of charges is represented
by a vector, the electric dipole moment  with a magnitude:

+q
-q
Re
= Re q
Unit: Debye, 1D = 3.33×10-30Cm
When the molecule is at its equilibrium position, the dipole moment is
called “permanent dipole moment” 0.
Selections rules
Electric dipole moment operator
 The
probability for a vibrational transition to occur, i.e. the intensity of
the different lines in the IR spectrum, is given by the transition dipole
moment fi between an initial vibrational state i and a vibrational final
state f :
 fi    f ˆ i d   f ˆ i
1  2  2
  
 ( x)   0    x   2  x  ...
2  x 0
 x 0
The electric dipole moment operator depends on the location of all electrons
and nuclei, so its varies with the modification in the intermolecular distance “x”.
0 is the permanent dipole moment for the molecule in the equilibrium position
Re
1  2 
  
 fi  0   f i d      f xi d   2    f x 2i d  ...
2  x 0
 x 0
0
The two states i and f are
orthogonal.
Because they are solutions of the
operator H which is Hermitian
The higher terms can
be neglected for small
displacements of the
nuclei
  
 fi      f xi d
 x 0
First condition: fi= 0, if ∂/ ∂x = 0
In order to have a vibrational
transition
visible
in
IR
spectroscopy: the electric dipole
moment of the molecule must
change when the atoms are
displaced relative to one another.
Such vibrations are “ infrared
active”. It is valid for polyatomic
molecules.
Second condition:

f
x i d  0
By
introducing
the
wavefunctions of the
initial state i and final
state f , which are the
solutions of the SE for an
harmonic oscillator, the
following selection rules is
obtained:
 = ±1
Note 1: Vibrations in homonuclear diatomic molecules do not
create a variation of   not possible to study them with IR
spectroscopy.
Note 2: A molecule without a permanent dipole moment can be
studied, because what is required is a variation of  with the
displacement. This variation can start from 0.
IR Stretching Frequencies of two bonded atoms:
What Does the Frequency, , Depend On?
E  h clas
h

2
k

 = frequency
k = spring strength (bond stiffness)
 = reduced mass (~ mass of largest atom)
 is directly proportional to the strength of the bonding between
the two atoms (  k)
 is inversely proportional to the reduced mass of the two atoms (v  1/)
51
Stretching Frequencies
• Frequency decreases with increasing atomic weight.
• Frequency increases with increasing bond energy.
52
IR spectroscopy is an important tool
in structural determination of
unknown compound
IR Spectra: Functional Grps
Alkane
-C-H
C-C
Alkene
Alkyne
11
IR: Aromatic Compounds
(Subsituted benzene “teeth”)
C≡C
12
IR: Alcohols and Amines
O-H broadens with Hydrogen bonding
CH3CH2OH
C-O
Amines similar to OH
N-H broadens with Hydrogen bonding
13
CO2, A greenhouse gas ?
Electromagnetic Spectrum
Near Infrared
Thermal Infrared
•
•
Over 99% of solar radiation is in the UV, visible, and near infrared bands
Over 99% of radiation emitted by Earth and the atmosphere is in the thermal IR
band (4 -50 µm)
What are the Major Greenhouse Gases?
N2 = 78.1%
O2 = 20.9%
H20 = 0-2%
Ar + other inert gases = 0.936%
CO2 = 370ppm
CH4 = 1.7 ppm
N20 = 0.35 ppm
O3 = 10^-8
+ other trace gases
Molecular vibrations
• The lowest vibrational transitions of
diatomic molecules approximate
the quantum harmonic oscillator
and can be used to imply the bond
force
constants
for
small
oscillations.
• Transition occur for v = ±1
• This potential does not apply to
energies close to dissociation
energy.
• In fact, parabolic potential does not
allow molecular dissociation.
• Therefore
more
anharmonic oscillator.
consider
PY3P05
Vibrational modes of CO2
Anharmonic oscillator
• A molecular potential energy curve can
be approximated by a parabola near the
bottom of the well. The parabolic
potential leads to harmonic oscillations.
• At high excitation energies the parabolic
approximation is poor (the true
potential is less confining), and does not
apply near the dissociation limit.
• Must therefore use a asymmetric
potential. E.g., The Morse potential:
a(R R )
 hcDdepth
where De isV the
 potential
e 1 e of the
minimum and
e

  2 1/ 2
a  

2hcDe 
2
PY3P05
Anharmonic oscillator
•
The Schrödinger equation can be solved for the Morse potential, giving permitted energy
levels:
2
1
1


E     hc~     hcxe~ ;   0,1,2,... max
2
2


a 2
~
xe 

2meff 4 De
where xe is the anharmonicity constant:
•
•
The second term in the expression for E increases
with v => levels converge at high quantum numbers.
•
The number of vibrational levels for a Morse
oscillator is finite:
v = 0, 1, 2, …, vmax
PY3P05
Energy Levels: Basic Ideas
Basic Global Warming: The C02 dance …
About 15 micron radiation