Download Rotational and Vibrational Spectroscopy

Spectroscopy
Advanced Physical Chemistry
Chemistry 5350
R OTATIONAL AND V IBRATIONAL
S PECTROSCOPY
Molecular spectroscopy is a powerful tool for learning about molecular
structure and molecular energy levels.
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Professor Angelo R. Rossi
The study of rotational spectra gives us information about
moments of inertia, interatomic distances, and bond angles.
Vibrational spectra yield fundamental vibrational frequencies and
force constants.
Electronic spectra provide electronic energy levels and
dissociation energies.
http://homepages.uconn.edu/rossi
Department of Chemistry, Room CHMT215
The University of Connecuticut
Fall Semester 2013
[email protected]
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The Basic Ideas of Spectroscopy
Spectroscopic Regions of the Electromagnetic Spectrum
When an isolated molecule undergoes a transition from one quantum
eigenstate with energy E1 to another with energy E2 , energy is
conserved by the absorption or emission of a photon.
The energy eigenvalues of a molecule can be written as
The frequency ν of the photon is related to the difference in energy of
the two states by the relation
where Erot is the rotational energy, Evib is the vibrational energy, and
Eelec is the electronic energy.
E = Erot + Evib + Eelec
When a molecule undergoes a transition to another state with the
emission or absorption of a single photon of frequency, ν, then
hν = hcν̃ = |E1 − E2 |
where ν̃ = λ1 and is the transition energy in wave numbers (the
number of waves per unit length).
′′
′
′′
′
′′
′
)
− Eelec
) + (Eelec
− Evib
) + (Evib
− Erot
hν = (Erot
where the primes refer to the state of higher energy and double primes
to the state of lower energy.
The SI unit is m−1 , but usually cm−1 is used.
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2
If E1 > E2 . the process is photon emission.
If E1 < E2 . the process is photon absortion.
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Spectroscopic Regions of the Electromagnetic Spectrum
Spectroscopic Regions of the Electromagnetic Spectrum
The classification of the various regions of the electromagnetic by the
type of transition involved because
The frequency range of photons, or the electromagnetic spectrum, is
classified into different regions according to custom and experimental
methods outlined in the table below.
′′
′
′′
′
′′
′
− Eelec
<< Eelec
− Evib
<< Evib
− Erot
Erot
Wavelength, λ
Electronic energy level differences are much greater that vibrational
energy differences, which in turn are much greater than rotational
energy level differences.
γ rays
X-rays
Vacuum UV
Near UV
Visible
Near IR
Mid IR
Far IR
Microwaves
Radio Waves
Electronic transitions are often in the visible and ultraviolet part of the
spectrum.
Vibrational transitions are in the infrared, and rotational transitions are
in the far infrared and microwave regions.
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30.00 EHz
30.00 PHz
1.50 PHz
789 THz
384 THz
120 THz
6.00 THz
300 GHz
3.00 GHz
300 MHz
Photon
Energy, hν
19.9 x 10−15
19.9 x 10−18
993 x 10−21
523 x 10−21
255 x 10−21
79.5 x 10−21
3.98 x 10−21
199 x 10−24
1.99 x 10−24
0.199 x 10−24
J
J
J
J
J
J
J
J
J
J
Molar
Energy, NA hν
12.0 GJ/mol
12.0 MJ/mol
598 kJ/mol
315 kJ/mol
153 kJ/mol
47.9 kJ/mol
2.40 kJ/mol
120 J/mol
12.0 J/mol
1.2 J/mol
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The electromagnetic spectrum is shown schematically below:
The frequency of the photon in the absorption or emission often
indicates the kinds of molecular that are involved:
•
Frequency, ν
The Electromagnetic Spectrum Depicted on a Logarithmic
Wavelength Scale
Spectroscopic Regions of the Electromagnetic Spectrum
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10 pm
10 pm
200 nm
380 nm
780 nm
2.5 µm
50 µm
1 mm
100 mm
1000 mm
Wave
Number, ν̃
1 x 109 cm−1
1 x 106 cm−1
50.0 x 103 cm−1
26.3 x 103 cm−1
12.8 x 103 cm−1
4.00 x 103 cm−1
200 cm−1
10 cm−1
0.1 cm−1
0.01 cm−1
In the radio-frequency region (very low energy), transitions
between nuclear spin states can occur.
In the microwave region, transitions in molecules with unpaired
electrons and transitions between rotational states take place.
In the infrared region, transitions between vibrational states take
place with and without transitons between rotational states.
In the visible and ultraviolet regions, the transitons occur between
electronic states accompanied by vibrational and rotational changes.
Finally, in the far ultraviolet and X-ray regions, transitions occur
that can ionize or dissociate molecules.
The above figure shows that visible light is a very small part of the
electromagnetic spectrum.
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Electric and Magnetic Fields Associated with a
Traveling Light Wave
Spectroscopic Selection Rules
Light is an electromagnetic traveling wave that has perpendicular
magnetic and electic field components:
Atoms and molecules possess a set of discrete energy levels which is
an essential feature of all spectroscopies.
If all molecules had a continuous energy spectrum, it would be very
difficult to distinguish one from another on the basis of their absorption
spectra.
Selection rules indicate which transitions will be experimentally
observed.
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Energy Transfer from the Electromagnetic Field to a Molecule
Leading to Vibrational Excitation
Spectroscopic Selection Rules
Because spectroscopies involve transitions between quantum states, it
is important to describe how electromagnetic radiation interacts with
molecules.
The principal interactions of molecules with electromagnetic radiation
are of the electric dipole type and will be the focus of study.
Magnetic dipole transitions are about 105 weaker than electric dipole
transitions, and electric quadrupole transitions are about 108 weaker.
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Energy Transfer from the Electromagnetic Field to a Molecule
Leading to Vibrational Excitation
Interaction of a Rigid Rotor with an Electromagnetic Field
Imagine a sinusoidally varying
electric field applied between a
pair of capacitor places.
Consider the effect of a time-dependent electric field on a classical
polar molecule constrained to move in one dimension:
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The spring allows the two masses to oscillate about their equilibrium
position generating a a periodically varying dipole moment.
If the electric field and oscillation of the dipole moment have the
same frequency and if they are in phase, the molecule can absorb
energy from the field.
For a classical molecule, any amount of energy can be absorbed,
and the spectrum is continuous.
For a quantum mechanical molecule, the interaction with the
electromagnetic field is similar but only discrete amounts of energy
can be absorbed.
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The arrows indicate the direction of force on each of the two
charged masses.
If the frequencies of the field and
rotation are equal, the rotor will
absorb from the electric field.
The amount of energy absorbed
from the electromagnetic field will
be controlled the quantum mechanical energy for a rigid rotor.
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Spectroscopic Selection Rules
Absorption, Spontaneous Emission, and Stimulated Emission
The types of transitions that can occur are limited by selection rules.
The basic processes by which photon-assisted transitions between
energy levels occur are by absorption, spontaneous emission, an
stimulated emission as shown in the figure below for a two
level-system.
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As with atoms, the principle interactions of molecules with
electronmagnetic radiation are of the dipole type, and will be our
principle focus.
Although the selection rules limit radiative transitions that can occur,
molecular collisions can cause many additional kinds of transitions.
Because of molecular collisions, the populations of the various
molecular energy levels are in thermal equilibrium.
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In absorption, the incident photon induces a transition to a higher
level.
In emmision, a photon is emitted as an excited state relaxes to one
of lower energy.
Spontaneous emission is a random event, and its rate is related to
the lifetime of the excited state.
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Absorption, Spontaneous Emission, and Stimulated Emission
Schrödinger Equation for Nuclear Motion
Previously we showed how the Schrödinger equation can be treated in
the Born-Oppenheimer approximation so that the electronic
Hamiltonian is for fixed nuclei.
These three processes are not independent in a system
at equilibrium where the overall rate for level 1 to 2 must
be the same as that for 2 to 1.
B12 ρ(ν)N1 = B21 ρ(ν)N2 + A21 N2
The Hamiltonian for nuclear motion contains the kinetic energy
operator and the electronic energy as a function of the nuclear
coordinates as the potential energy operator
Spontaneous emission is independent of the radiation
density at a given frequency, ρ(ν), but the rates of stimulated absorption and emission are directly proportional
to ρ(ν).
Each of these rates is directly proportional to the number of molecules (N1 or N2 ) in the state from which the
transition originates.
This means that unless the lower state is populated, a
signal will not be seen in the absorption experiment, and
unless the upper state is populated, a signal will not be
observed in the emission experiment.
ρ(ν) is the blackbody spectral density which provides the
distribution of frequencies at equilibrium for a given temperature.
B12 = B21 and
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Ĥ = −
~2 2
∇ + E(R)
2µ R
The potential energy term E(R) depends only on the relative positions
and not where the molecule is placed or on the orientation of the
molecule in space.
16π 2 ~ν 3
A21
=
B21
c3
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Schrödinger Equation for Nuclear Motion
Schrödinger Equation for Nuclear Motion
The kinetic energy operator consists of the kinetic energy of the center
of mass (translational energy), kinetic energy associated with rotational
motion, and the kinetic energy of the vibrational motion
If the Hamiltonian is the sum of three terms, one for each kind of motion,
then the wave function ψ can be written as a product of wave functions:
ψ = ψtrans × ψrot × ψvib
Ĥ = Ĥtrans + Ĥrot + Ĥvib
The Schrödinger equations for the three terms are
where the translational and rotatational Hamiltonians contain only
kinetic energy terms, and the vibrational Hamiltonian contains E(R)
depending on the internuclear distances.
Ĥtrans ψtrans = Etrans ψtrans
Ĥrot ψrot
=
Erot ψrot
Ĥvib ψvib
=
Evib ψvib
These internuclear distances are the vibrational coordinates of the
molecule.
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The translational wavefunction is that for a free particle with a mass
equal to the mass of a molecule.
The translational eigenvalues are very closely spaced and cannot
be probed in molecular spectroscopy.
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Schrödinger Equation for Nuclear Motion
Schrödinger Equation for Nuclear Motion
The total number of coordinates required to describe a polyatomic
molecule with N atoms in a molecule is 3N .
For a diatomic molecule,
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However, to describe the internal motions in a molecule, we are not
interested in its location in space.
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For polyatomic molecules,
So three coordinates required to specify the position of the center of
mass of a molecule can be subtraced leaving 3N − 3 coordinates.
To describe rotational motions in a molecule, we are interested in its
orientation in a coordinate system.
The orientation of a linear molecule with respect to a coordinate
system requires two angles leaving 3N − 5 coordinates to describe
the internal motions whereas a nonlinear polyatomic molecule
requires three angles leaving 3N − 6 coordinates.
These 3N − 3 or 3N − 6 internal motions are referred to as
vibrational degrees of freedom.
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Ĥrot depends only on two angles, θ and ψ.
Ĥvib depends only on R, the internuclear separation.
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Ĥvib is more complex, depending on 3N − 6 coordinates for
nonlinear molecules and 3N − 5 coordinates for linear molecules.
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Introduction to Vibrational Spectroscopy
Introduction to Vibrational Spectroscopy
Spectroscopy is an important chemical tool.
The table below shows the number of diatomic molecules in the first
vibrational state (N1 ) relative to those in the ground state (N0 ) at 300 K
and 1000 K calculated using the Boltzmann distribution.
Two features have enabled vibrational spectroscopy to achieve the
importance that it has as a tool in Chemistry.
1.
2.
Vibrational State Populations for Selected Diatomic Molecules
N1
N1
for 300 K N
for 100 K
Molecule ν̃(cm−1 ) ν(s−1 )
N0
0
14
−10
H-H
4400 1.32 x 10
6.88 x 10
1.78 x 10−3
H-F
4138 1.24 x 1014 2.42 x 10−9 2.60 x 10−3
H-Br
2649 7.94 x 1013 3.05 x 10−6 2.21 x 10−2
N-N
2358 7.07 x 1013 1.23 x 10−5 3.36 x 10−2
C-O
2170 6.51 x 1013 3.03 x 10−5 4.41 x 10−2
Br-Br
323 9.68 x 1012 0.213
0.628
The vibrational frequency depends primarily on the identity of the
two vibrating atoms on either end of the bond and to a much lesser
degree on the presence of atoms farther away from the bond.
This property generates characteristic frequencies for atoms joined
by a bond known as group frequencies.
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Introduction to Vibrational Spectroscopy
Vibrational Spectra of Diatomic Molecules
For absorption by a quantum mechanical harmonic oscillator,
∆v = vf inal − vinitial = +1.
The potential energy curve for diatomic molecules are not exactly
parabolic but is approximately parabolic in the vicinity of the equilibrium
internuclear distance Re as indicated by the dashed line.
Nearly all molecules are in the ground state even at 1000 K because
N1 /N0 << 1 except for Br2 .
This means that absorption of light will occur from molecules in the
v = 0 state.
It will be shown that in most cases only the v = 0 → v = 1 transition is
observed in vibrational spectroscopy.
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Vibrational Spectra of Diatomic Molecules
Vibrational Spectra of Diatomic Molecules
The potential energy indicated by the dashed line is given by the
parabola (harmonic potential)
For larger internuclear separations, the potential curve becomes
anharmonic.
It is difficult to solve for the exact form of E(R), but to a good
approximation, a realisitc anharmonic potential can be described in
analytical form by the Morse potenital:
1
E(R) = k(R − Re )2
2
where k is the force constant.
V (R) = De [1 − e−α(R−Re ) ]2
The energy levels for a simple harmonic oscillator are given by
1
Ev = (v + )hν
2
where
ν=
and µ is the reduced mass.
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1
2π
•
v = 0, 1, 2, . . .
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k
µ
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De isqthe dissociation energy relative to the bottom of the well.
α = 2Dk e
q
2 V (R)
k
.
The force constant, k is defined by k = ( d dR
2 )R=Re and ν =
µ
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Vibrational Spectra of Diatomic Molecules
Vibrational Spectra of Diatomic Molecules
The spectroscopic dissociation energy D0 is defined with respect to
the lowest allowed vibrational energy level, rather than to the bottom of
the potential as shown in the figure below
The energy levels for the Morse potential are given by
2
(hν)2
1
1
−
v+
Ev = hν v +
2
4De
2
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Values of Molecular Constants for Selected Diatomic Molecules
The second term gives the anharmonic correction to the energy
levels.
Measurements of the frequencies of the overtone vibrations allow
the parameter De in the Morse potential to be determined for a
specific molecule.
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The Origin of Selection Rules
Not all diatomic molecules have an infrared (vibrational) absorption
spectrum.
To determine which transitions are possible, an equation for a transition
dipole momemt is used.
According to the Born-Oppenheimer approximation, the wavefunction
for a molecule in the electronic state ψelec , the vibrational state ψvib , and
having a particular set of rotational quantum numbers, can be written as
a product of ψelec × ψvib × ψrot .
The permanent dipole moment µelec
of a molecule in this electronic
0
state is equal to the expectation value of the operator µ̂ over the
wavefunction of the electronic state:
Z
elec
∗
µ0 = ψelec
µ̂ ψelec dτelec
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The Origin of Selection Rules
The Origin of Selection Rules
R
Since the dipole moment for a diatomic molecule depends on the
internuclear distance, the electronic dipole moment can be expanded in
a Taylor series about R = Re :
∂µ
1 ∂ 2µ
= µe +
(R − Re )2 + · · ·
(R − Re ) +
µelec
0
∂R Re
2 ∂R2 Re
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•
For a molecule in a given electronic state, the transition dipole moment
for a vibrational transition is given by
R ∗
R
R ∗
e
ψn′′ (R − Re ) ψn′ dτ +
ψn′′ µ ψn′ = µe ψn∗ ′′ ψn′ dτ + ∂µ
∂R Re
2 R
∂ µe
2
∗
ψn′′ (R − Re ) ψn′ dτ + · · ·
∂R2
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•
Re
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R ∗
R
e
ψn′′ (R − Re ) ψn′ dτ +
ψn∗ ′′ µ ψn′ = µe ψn∗ ′′ ψn′ dτ + ∂µ
∂R Re
2 R
∂ µe
ψn∗ ′′ (R − Re )2 ψn′ dτ + · · ·
∂R2
Re
The first term is zero because the vibrational wavefunctions for
different v are orthogonal.
The second term is nonzero if the dipole moment depends on the
internuclear distance R.
The integral in the second term vanishes unless v ′ = v ′′ + 1 for
harmonic oscillator wavefunctions.
The second and higher derivatives of the dipole moment with
respect to internuclear distance are small but do give rise to
overtone transitions with ∆v = ±2, ±3, · · · , with rapidly
diminishing intensities.
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The Origin of Selection Rules
Vibrational Overtones
The selection rule for a diatomic is that a molecule will show a
vibrational spectrum only if the dipole moment changes with
internuclear distance.
The vibrational absorption spectrum of HCl is shown schematically
below in a ”stick” representation:
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Homonuclear diatomic molecules such as H2 and N2 have zero dipole
moments for all lengths and therefore do not show vibrational spectra.
In general, heteronuclear diatomic molecules do have dipole moments
that depend on internuclear distance, so they exhibit vibrational spectra.
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The strongest absorption band is at 3.46 µm.
There is a much weaker band at 1.76 µm and a very much weaker
one at 1.198 µm. These are overtone transitions v = 0 → v = 2 and
v = 0 → v = 3.
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Group Frequencies
Group Frequencies
For a molecule such as
Group frequencies are very valuable in determining the structure of
molecules, and a few values are given in the table below:
O
C
R′
Selected Group Frequencies
Group
Frequency (cm−1
Group
Frequency (cm−1 )
O-H stretch
3600
C=O stretch
1700
N-H stretch
3350
C=C stretch
1650
C-H stretch
2900
C-C stretch
1200
C-H bend
1400
C-Cl stretch
700
R
the vibrational frequency of the C and O atoms is determined by the
force constant for the C=O bond.
This force constant is primarily determined by the chemical bond
between the C and O atoms and to a much lesser degree by the
adjacent R and R′ groups.
For this reason, the carbonyl group (C=O) has a characteristic
frequency (group frequency) at which it absorbs infrared radiation in a
narrow range for different molecules.
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Experimental Vibrational Spectra
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The Td Character Table
Vibrational spectra for gas-phase CO and CH4 are shown in the figure
below:
A1
A2
E
T1
T2
E
1
1
2
3
3
8C3
1
1
-1
0
0
3C2
1
1
2
-1
-1
6S4
1
-1
0
1
-1
6σd
1
-1
0
-1
1
x2 + y 2 + z 2
(Rx , Ry , Rz )
(x, y, z)
(2z 2 − x2 − y 2 , x2 − y 2 )
(xy, xz, yz)
Because CO and CH4 are linear and nonlinear molecules, we expect
one (3×2 -5) and nine (3×5 -6) vibrational modes, respectively.
However, the spectrum for CH4 shows two rather than nine peaks as
well as several unexpected broad peaks.
Γreducible = A1 + E + 2T2
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Normal Mode Vibrations of Molecules
Normal Mode Vibrations of Molecules
Normal mode characteristics of molecules.
Normal mode characteristics of molecules.
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During a vibrational period, the center of mass of the molecule
remains fixed, and all atoms undergo in-phase periodic motion
about their equilibrium positions.
All atoms in a molecule reach their minimum and maximum
amplitudes at the same time.
These collective motions are called normal modes, and the
frequencies are called the normal mode frequencies.
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The frequencies measured in vibrational spectroscopy are the
normal mode frequencies.
All normal modes are independent in the harmonic approximation;
the excitation of one normal mode does not transfer vibrational
energy into another normal mode.
All motions of the atoms in a molecule can be expressed as a linear
combination of the normal modes of that molecule.
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Normal Mode Vibrations of Molecules
Normal Mode Vibrations of Molecules
To see what normal modes of vibration are, we first consider the
vibration of polyatomic molecules from a classical mechanical point of
view.
The qi are independent of time, and the kinetic energy becomes
3N
T =
The kinetic energy T of a polyatomic molecule is given by
"
2 2 2 #
N
1X
dyk
dzk
dxk
T =
mk
+
+
2 k=1
dt
dt
dt
1
q4 = m2 (x2 − x2e )
...
1
q3N = mN (zN − zN e )
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2
N
1 X X ∂ 2V
qi qj
2 i=1 j=1 ∂qi ∂qj
A new set of vibrational coordinates can be found Qj (q1 , q2 , . . . , qN ) that
simplify the above equation to
1
2
N
where xie are the values of the coordinates at the equilibrium geometry
of the molecule.
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dqi
dt
N
V (q1 , q2 , . . . , qN ) =
q1 = m12 (x1 − x1e ) q2 = m12 (y1 − y1e ) q3 = m12 (z1 − z1e )
1
2
For a molecule with N vibrational degrees of freedom, the potential
energy is given by
The equation can be simplified by introducing mass-weighted Cartesian
displacement coordinates q1 , . . . , q 3N :
1
1X
2 i=1
V (Q1 , Q1 , . . . , QN ) =
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1X
2 i=1
∂ 2V
∂Q2i
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Q2i
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The Transformation Coordinate System for the Atoms in H2 O
Under the C2v Point Group
Normal Mode Vibrations of Molecules
The Qj (q1 , q2 , . . . , qN ) are known as the normal coordinates of the
molecule.
Because there are no cross terms of the type Qi Qj in the potential
energy, the vibrational modes are independent in the harmonic
approximation
ψvibrational (Q1 , Q2 , . . . QN ) = ψ1 (Q1 )ψ1 (Q1 )ψ2 (Q2 ) . . . ψN (QN )
N
P
Evibrational =
vj + 21 hνj
C2 Transformation Matrix for H2 O Molecule
i=1







Ĉ2 = 





Because of the transformation to normal coordinates, each of the
normal modes contributes independently to the energy, and the
vibrational motions of different normal coordinates are coupled.
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Transformations of the Coordinate Systems on the Atoms in H2 O
Under Symmetry Operations of the C2v Group




x1
−x3
 y1 
 −y3 




 z1 
 z3 




 x2 
 −x2 







Ĉ2 
 y2  =⇒  −y2 
 z2 
 z2 




 x3 
 −x1 




 y3 
 −y1 
z3
z1
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0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
+1
0
0
0
−1
0
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
+1
0
0
0
−1
0
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
+1
0
0
0
0
0
0













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Guidelines for Calculating Characters in Reducible
Representations for Coordinate Transformations
Only the diagonal elements contribute to the character. Therefore, only
atoms that are not shifted by an operation contribute to the character.
•
•
•
47
If the atoms remains in the same position under the transformation,
and the sign of x, y, or z is not changed, the value of +1 is
associated with each unchanged coordinate.
If the sign of x, y, or z is changed, the value of -1 is associated with
each changed coordinate.
If the coordinate system is exchanged with the position of another
coordinate system, the value of 0 is associated with each of the
coordinates.
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The C2v Character Table
A1
A2
B1
B2
Reducible Representation into an Irreducible Representation
E C2 σv σv′
1
1 1 1 z
1
1 -1 -1
Rz
1 -1 1 -1 x, Ry
1 -1 -1 1 y, Rx
2
2
x ,y ,z
xy
xz
yz
2
Γreducible
E C2 σv σv′
= 9 −1 1 3
The number of times, ap , any irreducible representation, p occurs in
any reducible representation is given by
X
1
nR χ(R)χp (R)
ap =
h
sum
The leftmost column shows the symbol for each irreducible
representation.
over classes
By convention, a representation that is symmetric (+1) with respect to
rotation about the principle axis (Ĉ2 in this case) is given the symbol A.
Γreduciible = 3A1 + A2 + 2B1 + 3B2
Remove translations and rotations
A representation that is antisymmetric (-1) with respect to rotation
about the principle axis is given the symbol B.
Γreduciible = 3A1 + A2 + 2B1 + 3B2 − (B1 + B2 + A1 ) − (B2 + B1 + A2 )
then
Γreduciible = 2A1 + B2
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The Normal Modes of H2 O
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Rotational Spectroscopy
To a first approximation, the rotational spectrum of a diatomic molecule
can be understood in terms of the Schrödinger equation for rotational
motion of a rigid rotor.
The wavefunctions are the spherical harmonics YJM (θ, φ), and there are
two quantum numbers J and M for molecular rotation.
The normal modes of H2 O:
•
•
•
bond bending
O-H symmetric stretch
O-H antisymmetric stretch.
The energy eigenvalues are given by
E=
The vectors indicate atomic
displacements.
~2
h2
J(J + 1) = 2 2 J(J + 1) = hcBJ(J + 1)
2
2µR0
8π µR0
h
−1
and
where B = 8π2 cµR
2 , the rotational constant, has units of cm
0
constants specific to the molecule.
Since the energy does not depend on M = −J, . . . , 0, . . . , +J, the
rotational levels are (2J + 1)-fold degenerate.
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Rotational Spectroscopy
Rotational Spectroscopy
The energies levels and transitions allowed are given by the selection
rule ∆J = Jf inal − Jinitial = ±1
The larger J the value of the orginating energy level, the more energetic
the photon must be to promote excitation to the next highest level.
For successive initial values of J, the ∆E associated with the transitions
increases in such a way that the difference between these frequencies
(∆ν), i.e. ∆(∆ν) is constant.
The energies corresponding to rotational transitions of ∆J = +1
correspond to absorption
∆E+ = hcB(J + 1)(J + 2) − hcBJ(J + 1) = hcBJ(J + 1)
J → J′
0→1
1→2
2→3
3→4
4→5
while those transitions of ∆J = −1 correspond to emission.
∆E− = hcB(J − 1)(J) − hcBJ(J + 1) = hcBJ
The |∆E+ | 6= |∆E− | because the rotational energy levels are not
equally spaced.
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∆
2cB
4cB
6cB
8cB
10cB
∆(∆ν)
2cB
2cB
2cB
2cB
2cB
This means that the spectrum will show a series of equally spaced
lines separated in frequency by 2cB.
53
Rotational Spectroscopy
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54
Vibration-Rotation Spectra of Diatomic Molecules
Up to this point, rotation and vibration have been considered separately.
•
•
The energy levels for the rigid rotor are shown on the left with the
allowed transitions between levels shown as vertical bars.
The spectrum observed through absorption of microwave radiation
is shown on the right.
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55
•
In the microwave region of the electromagnetic spectrum, the
photon energy is sufficient to excite rotational transitions but not to
excite rotational transitions.
•
For infrared radiation, diatomic molecules can make transitions in
which both v and J change according to the selection rules
∆v = +1 and ∆J = ±1.
•
Because of this structure, molecular spectra are often referred to a
band spectra.
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Vibration-Rotation Spectra of Diatomic Molecules
Vibration-Rotation Spectra of Diatomic Molecules
The fundamental vibration band for HCl (v = 0 → 1) is shown below:
Schematic Representation of Rotational and Vibrational Levels
The double peaks are due to the presence of H35 Cl (75% abundance)
and H37 Cl (25% abundance).
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Vibration-Rotation Spectra of Diatomic Molecules
Vibration-Rotation Spectra of Diatomic Molecules
Vibrational Spectroscopy
Rotational Spectroscopy
For vibrational spectroscopy, one intense peak is expected because the
energy level spacing is the same for all quantum numbers so that for
∆v = ±1, all transitions have the same frequency.
Because the rotational energy levels are not equally spaced in energy,
different transitions give rise to separate peaks.
58
∆Erotation << kT under most conditions so that many rotational energy
levels will be populated.
Only the v = 0 energy level has significant population so that even
taking anharmonicity into account will not generate additional peaks
originating from v > 0.
Many peaks are generated in the rotational spectrum.
Although the overtone frequencies differ from the fundamental
frequency, these peeks will have a low intensity.
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Relative Intensities of Peaks in Vibration-Rotation Spectra
Relative Intensities of Peaks in Vibration-Rotation Spectra
The intensity of a spectral line is determined by the mumber of molecules in the
energy level from which it originates.
Relative Number of Molecules
The ratio of the number of molecules in a state for a given value of J relative to the
number in the ground state (J = 0) can be calculated using the Boltzmann distribution:
~2 J(J+1)
gJ − (eJ −e0 )
nJ
kT
=
e
= (2J + 1)e− 2IkT
n0
g0
The 2J + 1 term in front of the exponential gives the degeneracy of the energery level
of J and generally dominates nnJ0 for small J and large T .
The exponential term causes
nJ
n0
to decrease rapidly with increasing J
For a molecule such as HD with a small momentum of intertia, the rotational energy
levels can be far enough apart that few rotational states are populated.
For molecules with a large moment of inertia such as CO, the exponential term does
not dominate until J is quite large yielding many occupied rotational energy levels.
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Vibrational and Rotational energy Levels for a
Diatomic Molecule
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62
Simulated Absorption Spectrum for HCl at 300K
The transitions with ∆J = +1 give rise to the lines
in the R Branch, and the transitions with ∆J =
−1 give rise to the lines in the P Branch of the
spectrum.
The intensities of the lines in these branches reflect the thermal populations of the initial rotational
states.
The Q Branch, when it occurs, consists of lines corresponding to ∆J = 0. Generally, these transitions
are forbidden, except for molecules such as NO
which have orbital angular momentum about their
axis.
P Branch
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R Branch
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64
High-Resolution Spectrum for CO In Which P and R Branches are
resolved into the individual rotational transitions
Return to Experimental Vibrational Spectra for CO and CH4
•
•
•
The broad resolved peaks seen for CO between 2000 cm−1 - 2250
cm−1 are the P and R branches corresponding to
rotational-vibrational transitions.
The minimum near 2200 cm−1 corresponds to the forbidden ∆J = 0
transition.
The broad and only partially resolved peaks for CH4 seen around
the sharp peaks centered near 1300 cm−1 and 3000 cm−1 are again
the P and R branches.
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Raman Spectra
Raman Spectra
When a sample is irradiated with monochromatic light, the incident
radiation may be absorbed, may stimulate emission, or may be
scattered.
The intepretation of Raman spectra is based on the conservation of
energy.
•
A part of the scattered radiation radiation is referred to as the
Raman spectrum.
•
It is found that some photons lose energy in scattering from a
molecule and emerge with a lower frequency. These photons
produce Stokes lines in the spectrum of scattered radiation.
•
A smaller fraction of the scattered photons gains energy in striking
a molecule and emerges with higher frequency. These photons
produce anti-Stokes lines in the spectrum of scattered radiation.
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This requires that when a photon of frequency ν is scattered by a
molecule in a quantum state with energy Ei and the outgoing photon
has a frequency ν ′ , the molecule ends up in quantum state f with
energy Ef :
hν + Ei = hν ′ + Ef
or
h(ν ′ − ν) = Ei − Ef = h∆νR = hc∆ν̃R
where the shift in frequency is labeled ∆νR , and the shift in wave
number is labeled ∆ν̃R .
67
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Raman Spectra
Raman Spectra
Raman spectroscopy is different from absorption or emission
spectroscopy in that the incident light need not coincide with the
quantized energy difference in the molecule.
The Raman effect arises from the induced polarization of scattering
molecules that is caused by the electric vector of the electromagnetic
radiation.
Any frequency of light can be used.
An isotropic molecule is one that has the same optical properties in all
directions, e.g. the CH4 molecule.
Since many final states consisting of both higher and lower energy than
the initial state are possible, many Raman spectral lines can be
observed.
A dipole moment µ is induced in the molecule by an electric field E .
The frequency shifts seen in Raman experiments correspond to
vibrational or rotational energy differences, so this kind of frequency
gives us information on the vibrational and rotational states of
molecules.
where α is the polarizability which has units of dipole moment divided
2
2
by electric field strength, i.e. VCmm−1 = C Jm .
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µ = αE
For an isotropic molecule the vectors µ and E point in the same
direction, and the polarizability is a scalar.
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Raman Spectra
Raman Spectra
The polarizability α of a rotating or vibrating molecule is not constant
but varies with some frequency, νvib or νrot .
As shown earlier, the dipole moment is related to the polarizability and
the magnitude of the electric field.
Consider a molecule with a characteristic vibrational frequency νvib in a
time-dependent electromagnetic field:
The polarizability is an anisotropic quantity, and its value depends on
the direction of the electric field relative to the molecular axes:
70
µinduced (t) = αE0 cos πνt
E = E0 cos 2πνt
The polarizability depends on bond length xe + x(t), where xe is the
equilibrium value. The polarizability can be expanded
dα
α(x) = α(xe ) + x
+ ···
dx x=xe
The electric field distorts the molecule slightly because the negative
valence electrons and positive nuclei experience forces in opposite
directions.
This induces a time-dependent dipole moment of magnitude µinduced of
the same frequency as the electric field.
The vibration of a molecule is time dependent and is given by
x(t) = xmax cos 2πνvib t
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Raman Spectra
Raman Spectra
Combining the equations from the previous slide gives
The equation on the previous slide demonstrates that the intensity of
the Stokes and anti-Stokes peaks is zero unless dα
6= 0.
dx
Raleigh
}|
{
z
µinduced = α(xe )E0 cos eπνt


"
#
+ dα
xmax E0 cos(2πν + 2πνvib )t + cos(2πν − 2πνvib )t
dx x=xe
|
{z
} |
{z
}
anti-Stokes
The polarizability of a molecule must change as it vibrates. This
condition is satisfied for many vibrational modes including the stretching
vibration of a homonuclear diatomic molecule.
Thus, the stretching vibration of a homonuclear molecule is infrared
inactive but Raman active.
Stokes
The time-varying dipole moment radiates light at the same frequency
(Raleigh), at a higher frequency (anti-Stokes), or at lower frequency
(Stokes).
Not all vibrational modes that are active for the absorption of light are
Raman active and vice versa demonstrating that infrared and Raman
spectroscopies complement each other.
In addition to scattered light at the incident frequency, light will also be
scattered at frequencies corresponding to vibrational excitation and
de-excitation.
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Raman Spectra
Raman Spectra
Are the intensities of the Stokes and anti-Stokes peaks equal?
The relative intensitiy is governed by the relative number of molecules
in the originating states
For the Stokes line, the transition originates from v = 0 state, whereas
for the anti-Stokes line, the transition originates from the v = 1 state.
The relative intensities of the Stokes and anti-Stokes peaks can be
calculated from the Boltzmann distribution:
In order for a molecular motion to be Raman active, the polarizability α must change
when that motion occurs ( dα
dx 6= 0 ).
In order for a vibrational mode to be active, the polarizability α must change during a
vibration, and for a rotation to be Raman active, the polarizability must change as the
molecule rotates in an electric field.
•
•
3hν
− 2kT
hν
Ianti−Stokes
nexcited
e
− kT
=
=
hν = e
−
IStokes
nground
e 2kT
•
Last Updated: November 13, 2013 at 6:15pm
The polarizability of both homonuclear and heteronuclear diatomic molecules
changes as the distance changes because this alters the electronic structure.
The polarizability of a spherical molecule does not change in a rotation. Thus,
spherical rotors do not have a rotational Raman effect.
All other molecules are anisotropically polarizable which means that the
polarization is dependent on the orientation of the molecule in the electric field.
The mutual exclusion rule states that for molecules with a center of symmetry,
fundamental transitions that are active in the infrared, are forbidden in Raman
scattering and vice versa.
For vibrations for which ν̃ is in the range 1000 - 3000 cm−1 , this ratio
ranges between 8 x 10−3 and 5 x 10−7 at 300K demonstrating that the
intensities of the Stokes and anti-Stokes lines will be different.
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Raman Active Modes
Rotational Raman Spectrum for a Linear Molecule
•
To be Raman active, a vibrational mode must produce a change in molecular
polarizability of the molecule.
•
The Raman modes are those that have the same symmetry as the molecular
polarizability (α) as the binary functions (x2 , y 2 , z 2 , xy, xz, yz).
•
In addition, Raman active modes will leave plane polarized light polarized, if they
are totally symmetric, otherwise the light will be depolarized.
The theoretical rotation-vibration Raman spectrum for ∆v = +1, 0, −1 and ∆J = +2, 0, −1 is shown below
for a linear molecule.
The lines which appear at higher frequencies are referred to as the O branch. In addition, there is a Q
branch for ∆J = 0.
The S, Q, and O branches correspond to the P , Q, and R branches of infrared spectroscopy.
Example: H2 O
For H2 O, the Raman active modes must be a1 , a2 , b1 , b2 symmetry. The Raman activity
of a1 modes is identified by adding pol in the brackets after the listed modes. and the
activity of the other modes is identified by adding depol.
Γvib (H2 O) = 2A1 (IR, pol) + B2 (IR, depol)
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Raman Spectra and Selection Rules
Symmetry Selection Rules for Infrared Spectra
The specific selection rule for the vibrational Raman effect is ∆v = ±1.
A Fundamental Transition consists of a transition from a molecule in a vibrational ground state (initial
vibrational state wave function, ψi ) to a vibrationally excited state (final vibrational state wave function, ψf )
where the molecule absorbs one quantum of energy in one vibrational mode.
A vibrational transition in the infrared occurs when the molecular dipole moment (µ) interacts with incident
radiation which occurs with a probability which is proportional to the transition moment:
The vibrational transitions are accompanied by rotational Raman transitions with the specific selection rules
∆J = 0, ±2.
•
•
Z
78
ψi µψf dτ
A transition is said to be forbidden in the infrared if the value of this integral is zero because the
probability of that transition is zero and no absorption will be observed.
The integral will be zero unless the direct product of ψi µψf contains the totally symmetric representation
which has the character +1 for all symmetry operations for the molecule under consideration.
The vector µ can be split into three components, µx , µy , and µz along the Cartesian coordinate axes, and
only one of the three integrals needs to be non-zero:
Z
 
µx
ψi µy  ψf dτ
µz
The pure rotational Raman spectrum of CO2
The intense peak at 488 nm is due to elastic scattering.
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Symmetry Selection Rules for Infrared Spectra of RuO4
Symmetry Selection Rules for Raman Spectra
Consider the vibrations of the tetrahedral molecule, ruthenium tetroxide (Td symmetry),
The probability of a vibrational transition occurring in Raman scattering is proportional to:
Z
where α is the polarizability of the molecule.
The Raman effect depends on a molecular dipole induced by the electromagnetic field of the incident
radiation and is proportional to the polarizability of the molecule which is a measure of the ease with which
the molecular electron distribution can be distorted.
α is a tensor, i.e. a 3 x 3 array of components


αx2 αxy αxz
αyx αy2 αyz 
αzx αzy αz2
where there are vibrations of A1 , E, and T2 , and deduce the infrared activity of each of them.
1. ψi has A1 symmetry.
2. The character table for Td shows that Tx , Ty , Tz together have T2 symmetry.
The direct products are then
A1 vibration:
E vibration:
T2 vibration:
A1 ⊗ T2 ⊗ A1
A1 ⊗ T2 ⊗ E
A1 ⊗ T2 ⊗ T2
=
=
=
T2
T2 ⊗ E
T2 ⊗ T2
=
=
T1 + T2
A1 + E + T1 + T2
so there will be six distinct components
An important result of the above analysis is that if an excited vibrational mode has the same symmetry as
the translation vectors, Tx , Ty , Tz , for that point group, then the totally symmetric irreducible representation
is present and a transition from the vibrational ground state to that excited vibrational mode will be infrared
active.
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81
For the ruthenium tetroxide molecule with Td symmetry, the components of polarizability have the following symmetries:
x2 + y 2 + z 2
2z 2 − x2 − y 2 , x2 − y 2
xy, yz, zx
The vibrations are A1 , E, and T2 , and it is possible to deduce the infrared activity of each of them. The direct products are then
 
A1
=
A1 , E, T2 ;
A1 vibration is possible.
A1 ⊗  E  ⊗ A1
T2 
A1
A1 ⊗  E  ⊗ E
=
E, (A1 + A2 + E), (T1 + T2 );
E vibrations are possible.
T2 
A1
A1 ⊗  E  ⊗ T2
=
T2 , (T1 + T2 ), (A1 + E + T1 + T2 );
T2 vibrations are possible.
T2
Thus, all of the vibrations in the RuO4 molecule are Raman active.
A summary of the infrared and Raman activity is given as
A1 :
E:
T2 :
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x2 + y 2 + z 2
2z 2 − x2 − y 2 , x2 − y 2
Tx , Ty , Tz ), (xy, yz, zx)
(T
Raman Only
Raman Only
Infrared and Raman Active
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x2

where one non-zero integral is needed to have an allowed Raman transition.
Symmetry Selection Rules for Raman Spectra of RuO4
A1
E
T2
α
 αy 2 
Z


α 2 
ψi  z  ψf dτ
αxy 


αyz
αzx
Thus, the T2 vibrations are infrared active because the direct products produce an A1 representation, but
the E and T2 vibrations will not appear in the infrared spectrum.
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ψi αψf dτ
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