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Molecular transitions and vibrations
Molecular spectra arise from
Electronic, vibrational, rotational transtitions
Erot < Evib < Eelec  hirarchy
Powerfull:
shapes sizes of molecules
strenght and stiffness of bonds
Information needed to account for chemical reactions
Gross selection rules
statements about the properties that a molecule must
possess to perform a specific transition
Specific selection rules
changes in quantum number
http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/molspecon.html#c1
Absorption and emission
Transitions are induced by the interaction of the electric
component of the electromagnetic field with the electric
dipole associated with the transition
 fi  f  i
 = Electric dipol moment operator
Physical interpretation: measure of dipolar migration of
charge that accompanies the transition.
When  is calculated it can be used for the Rates of
transitions:
Stimulated: W = Brad(E)
B
 fi
Spontaneous: W = A
2
6 0 
2
A
8h 3fi
c
3
B
Raman processes
Inelastic scattering of a photon when it is incident on a molecule
Selection rules for Raman transitions are based on aspects of
the polarizability of a molecule, the measure of its responce to
an electric field.
Classical argument
Consider time-variation of magnitude of the dipole moment
induced in a molecule by an electromagnetic field E(t):
(t) = (t)E(t)
(t) = polarizability,
Incident radiation with frequency 
 of molecule changes between min and max at frequency int
as a result of its rotation or vibration
1
 (t )  (   cos  intt ) E0 cos t
2
with a range of variation = min-max , the product expands to:
1
 (t )  E0 cos t  E0 cos(   int )t  cos(   int )t
4
Rayleigh
Stokes
Anti-Stokes
Molecular rotations
Rotational energy levels
The classical kinetic energy of a freely rotating molecule can
be expressed as the kinetic energy of rotation of a body of
moment of inertia Iqq about an axis q
q
J
2
q
1
2
T   I qq q  
2 q
q 2 I qq
m2
cm
m1
q is the angular frequency about the axis
R
Rotational Spectra
Rotational spectrum of diatomic molecules
Assume that diatomic molecules rotate as rigid rotors.
The energies can be modeled in a manner parallel to the
classical description of the rotational kinetik energy of a rigid
object.
From these descriptions, structural information can be
obtained (bond lengths and angles).
m2
cm
m1
R
Since the rotational kinetic energy of the rigid rotor can be
expressed in terms of the angular momentum, we can imply the
form for the Hamiltonian associated with the rotation around
a single principal axis.
For this limited case of rotation about a single axis, the
Schrödinger equation can be formulated in terms of the
total angular momentum and the form of the energy
eigenvalues implied.
Determining the rotational constant B enables you to calculate
the bond length R.
Centrifugal distortion
As the degree of rotational excitation increases the bonds
are stressed.
A diatomic molecule with reduced mass  rotating at an
angular velocity  will experience a centrifugal force.
Tends to stretch the bond acting like a spring with
restoring force obeying Hook’s law proportional to the
displacement from equilibrium R0 with
k(R – R0)
k = force constant.
The increase in moment of inertia that accompanies this
centrifugal distortion results in a lowering of the rotational
constant the energy levels are less far apart at high J than
expected on the basis of the rigid rotor assumption.
Pure rotational selection rules
use Born-Oppenheimer approximation
vibrations are much faster than rotations  can be
separated too.
J
The overall wavefunction
MJ
z
of the molecule can be written
K
, J, M
J
The transition matrix factorizes into:
Z
 , J ' , M J'   , J , M J  J ' , M J'    J , M J  J ' , M J'  J , M j
 = permanent electric dipole moment of the molecule in the state .
The transition element is the matrix element of the permanent
electric dipole moment between the two states connected by the
transition.
Only polar molecules (  0) can have a pure rotational spectrum.
The specific selection rules governing rotational transitions
can be established by investigating the eigenvalues of J’
and M’J for given eigenvalues of J and MJ for which the
matrix element
J ' , M J'   J , M J  0
Linear molecule: rotational wavefunctions are
eigenfunctions of the operators J2 and Jz (z = laboratory
axis).
In connection with orbital angular momentum the
eigenfunctions are the spherical harmonics YJMJ(,).
 2
J , M Q J , M J    Y
'
'
J
0 0

J ' M J'
QYJM sin dd
J
To evaluate the matrix elements we need to evaluate
 2
IM   Y

J ' M J'
QYJM sin dd
J
0 0
With M = 0, 1
Ideal for group theoretical arguments and the joint
selection rules are
J =  1
MJ = 0, 1
For a polar linear rotor.
Symmetric rotors can invole changes in quantum number K.
Any permanent electric dipole moment must lie parallel to Cn
axis, not perpendicular.
The electromagnetic field cannot couple to any transitions
that correspond to chages in the component of angular
momentum around the principal axis and to changes in K.
There is no handle perpendicular to the principal axis on
which an electric field can exert a torque. The selection
rules become
J =  1
MJ = 0, 1
K = 0
Spherical rotors do not have permanent dipole moments by
symmetry. They do not show pure rotational transitions.
Rotational Raman selection rules
Molecules with anisotropic electric polarizabilites can show
pure rotational Raman lines. The selection rules are
J =  2,  1 K = 0 but K = 0  0 is forbidden for J =  1
Rules out J =  1 for linear molecules.
Why a 2 for J? Raman effect depends on polarizability of
molecule changing with time, with an internal frequency.
1
 (t )  (   cos  intt )
2
For a rotation the polarizability returns to its original value twice
per revolution  int = 2rot.
Molecule seams to be rotating twice as fast as its mechanical
motion.
Idealized depiction of a Raman line produced by interaction
of a photon with a diatomic molecule for which the
rotational energy levels depend upon one moment of inertia
Establishing selection rules:
Recognize that the anisotropy of the polarizability has
components that vary with time with angle Y2M(,).
Consider diatomic molecule with polarizabilities  and an
electric field E applied in the laboratory z direction.
The induced dipole is parallel to z so z = zzE.
In the molecular frame the components of the dipole moment
will be x y
and
z
z = xsincos + xsinsin + z cos
Ex = Esincos
EY = Esinsin
Ez = Ecos
The molecular component of the induced electric dipole
moment is related to the molecular component of the electric
field by q = qqE
z = xxExsincos + yyEysinsin + zzEzcos
= Esin2cos2 + Esin2sin2 + ||Ezcos2
=Esin2 + ||Ezcos2
With  = xx = yy and || = zz the mean polarizability is  =
1/3( || + 2) and
1


4   2


 z      Y20  ,  E
3 5




The first term does not contribute to off-diagonal elements
but the second gives a contribution to the transition dipole
moment
1
J ' , M J'  Z J , M J
4   2
   E J ' , M J' Y20 J , M J
3 5
The integral that determins wether or not this matrix
element vanishes is
 2
I   Y

J ' M J'
 , Y20  , YJM  , sin dd
J
0 0
The integral is zero unless J’ = J  2.
Raman lines can be expected at the following wavenumbers:
Stokes lines (J = + 2 ): J = 0 – 4B(J + 3/2) J = 0,1,2,….
Anti-Stokes lines (J = - 2 ): J = 0 – 4B(J - 3/2) J = 2,3,….
Where 0 is the wavenumber of the incident radiation.
Nuclear statistics
Certain molecules show a peculiar alternation in intensity of
the rotational Raman spectra.
A linear molecule shows an alternation in intensity
due to the Pauli principle and the fact that the rotation of
a molecule may interchange identical nuclei having spin I
(analogue of s for electrons).
Spinn of nuclei can be integral or half integral depending
on specific nuclide.
According to the Pauli principle the interchange of
identical fermions (fractional spin particles, such as
protons or carbon-13 nuclei or) or bosons ( integral spin
particles like carbon-12 or oxygen-16 nuclei) must obey:
   (1,2)bosons
 (2,1)  
  (1,2) fermions
These symmetries are obeyed when a molecule rotates
through  or some other angle for symmetric rotors.
Molecular vibration
Diatomic molecules have only one degree of vibrational
freedom, namely the stretching of the bond.
The molecular energy of a diatomic molecule increases if
the nuclei are displaced from their equilibrium positions.
For small displacements (x = R - Re) the potential energy
can be expressed as the first few terms of a Taylor
series where the interesting term is
V(x) = ½ kx2 k=(d2V/dx2)0
The potential energy close to equilibrium is parabolic. The
hamiltonian for the two atoms of masses m1 and m2 is
2
2
2
2
 d
 d
1 2
H 

 kx
2
2
2m1 dx1 2m2 dx1 2
Vibrational Spectra of Diatomic Molecules
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.
Sampling of transition
frequencies from the
n=0 to n=1 vibrational
level for diatomic
molecules and the
calculated force
constants.
When the potential energy depends only on the separation of
the particles, the hamiltonian can be expressed as a sum, one
term referring to the motion of the center of mass of the
system and the other to the relative motion. The former is
of no concern and the latter is
2
2
 d
1 2
H 
 kx
2
2 dx
2
With  being the effective mass
1/  = 1/m1 + 1/m2
The motion is dominated by the lighter atom, when m1>>m2 m2,
A hamiltonian with a parabolic potential energy is
characteristic to a harmonic oscillator with:
1

Ev   v  
2

k
   

1
2
With  = 0, 1, 2,…. Uniform ladder with separation .
The corresponding wavefunctions are bell-shaped
gaussian functions multiplied by an Hermite polynomial.
Anharmonic oscillation
Solve Schrödinger equation with a potential energy
term that matches the true potential energy the Morse
potential
V(x) = hcDe{1-e-ax}2
a = (k/2hcDe)1/2
The parameter De is the depth of the minimum of the
curve. The Schrödinger equation becomes:
2
1
1


Ev   v     v   xe
2
2


k
   

1
2
2
a
xe 
2
Xe is the anharmonicity constant as v becomes large second term
becomes imporant, at high excitations the energy converges.
Vibrational selection rules
The transition matrix element
v v  v'  v
'
 = dipole moment of the molecule in electronic state ,
with bond lenght R.
 depends on R since the electronic wavefunction depends
parametrically on the internuclear separation.
The transition matrix element is
v v
'
2

1 d 
 d 
2
   v' x v   2  v' x v  ...
2  dx 0
 dx 0
The gross selection rule for the vibrational transitions of diatomic
molecules is that they must have a dipole moment that varies with
extension  homonuclear diatomic molecules do not undergo electric
dipole vibrational transitions
 of a molecule can vary linearly with the extention of the
bond for small displacements; true for a heteronuclear
molecule in which the partial charges on the two atoms are
independent of the internuclear distance, then the quadratic
and higher terms in the expansion can be ignored and
v v
'
 d 
   v' x v
 dx 0
When is the matrix element not zero?
Use the following property of Hermite polynomials:
2yHv(y) = Hv+1(y) + 2vHv-1(y)
The only nonzero contributions to v’v will be obtained when v’=
v1
The selection rule for the electronic dipole transition within
the harmonic approximation is v = 1
The wavenumbers of the transitions that can be observed
by electric dipole transitions in a harmonic oscillator are
Ev 1  Ev 

~
v


hc
hc 2c
The spectrum would consist of a single line regardless of
the initial vibrational states.
In real life anharmonicities cause different transitions to
occur with different wavenumbers.
Large displacements adjust the partial charges as the
internuclear distance changes  the electrical
anharmonicities permit transitions with v = 2 which are
the first overtones or second harmonics of the vibrational
spectrum.
Vibration-rotation spectra of diatomic molecules
The vibrational transition of a diatomic molecule is accompanied by a
simultaneous rotational transition with J = 1
The total energy changes and the frequency of the transition
depends on the rotational constant, B, of the molecule and the initial
value of J. The energy is:
2
1
1


E (v, J )   v     v   xe  hcBv J ( J  1)  hcDv J 2 ( J  1) 2  ...
2
2


The transition v= +1 and J = -1 give rise to P-branch of the
vibrational spectrum. The wavenumbers of the transitions are
v~ P (v, J )  v~  2(v  1)v~xe  ...  ( Bv 1  Bv ) J  ( Bv 1  Bv ) J 2  ....
A series of lines is obtained since many initial rotational states are occupied
Transitions with J = 0 give rise to the Q-branch of the
vibrational spectrum. This is only allowed when the molecule
possesses angular momentum parallel to the internuclear axis
 a diatomic molecule can possess a Q-branch only if the
total orbital angular momentum for the electrons around the
internuclear axis is nonzero.
The wavenumbers of this branch are:
v~ Q (v, J )  v~  2(v  1)v~xe  ...  ( Bv 1  Bv ) J  ( Bv 1  Bv ) J 2  ....
The transition with J = 1 give rise to the R branch of the
vibrational spectrum with the wavenumbers:
v~ R (v, J )  E (v  1, J  1)  E (v, J )/ hc
 v~  2(v  1)v~xe  ...  2 Bv 1  (3Bv 1  Bv ) J  ( Bv 1  Bv ) J 2  ...
Vibrational Raman transitions of diatomic molecules
The gross selection rule for the observation of vibrational
Raman spectra of diatomic molecules is that the molecular
polarizability should vary with internuclear separation.
That is generally the case with diatomic molecules regardless
of their polarity, so all diatomic molecules are vibrationally
Raman active.
The origin of the gross selection rule,
and the derivation of the particular:
Consider the the transition dipole moment without troubling
about the orientation dependence of the interaction
between the electromagnetic field and the molecule:
v'v   , v'   , v   , v'   , v E
The electronic and vibrational wavefunctions can be separated
in the Born-Oppenheimer approximation and evaluated
for a series of selected displacements, x, from equilibrium.
Expand the polarizability as a Taylor function in the
displacement
v 'v
 d 
 d 
 v'  (0)  
 x  ... v E  v' v  (0) E  
 v' x v E  ...
 dx 
 dx 0
The first matrix element is zero on account of the
orthogonality of the vibrational states when v’v:
v 'v
 d 

 v' x v E
 dx 0
The selection rule is
Stokes lines
Anti-Stokes lines
v = 1
v = +1
v = -1
Only Stokes lines are normaly observed since initially most
molecules have v = 0
In the gas phase the Stokes and anti-Stokes lines show branch
structure with the selection for diatomic molecules.
The selection rules are J = 0, 2.
In addition to the Q-branch, there are also O- and S-branches
for J = -2 and J = +2 respectively. A Q- branch is observed
for all diatomic molecules regardless of their orbital angular
momentum
Summary
Ineraction of electromagnetic field with electric dipole
Erot
Diatomic molecule
rigid rotor rotating around
single axis
Schrödinger equation in terms
of total angular momentum
Is good for calculating bond
lengths
Selection rule: only polar
molecules can have pure
rotational spectrum
change in quantum number k
for symmetries
Molecules with anisotropic
electric polarizabilities can
show pure rotational Raman
lines. Certain molecules show
alternations of intensity
Evib
Diatomic molecules have only one
degree of vibrational freedom, the
stretching of bonds (two beads on a
spring) approximate quantum
mechanical harmonic oscillator
Gross selection rules
Dipole moment must vary with
extension
homonuclear diatomic molecules do
not undergo electric dipole
vibrational transitions
Vibrational Raman transitions
the molecular polarizability varies
with internuclear separation
true for all diatomic molecules
regardless of polarity