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Lecture 14: Molecular structure
o Rotational transitions
o Vibrational transitions
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o Electronic transitions
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Bohn-Oppenheimer Approximation
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Born-Oppenheimer Approximation is the assumption that the electronic motion and the
nuclear motion in molecules can be separated.
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This leads to molecular wavefunctions that are given in terms of the electron positions (ri) and
the nuclear positions (Rj):
 molecule(rˆi , Rˆ j )   electrons( rˆi , Rˆ j ) nuclei (Rˆ j )
o
Involves the following assumptions:

o Electronic wavefunction depends on nuclear positions but not upon their velocities, i.e.,
the nuclear motion is so much slower than electron motion that they can be considered to
be
fixed.
o The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the fastmoving electrons.
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Molecular spectroscopy
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Electronic transitions: UV-visible
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Vibrational transitions: IR
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Rotational transitions: Radio
E
Electronic
Vibrational
Rotational
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Rotational motion
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Must first consider molecular moment of inertia:
I   mi ri2
i
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At right, there are three identical atoms bonded to
“B” atom and three different atoms attached to “C”.

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Generally specified about three axes: Ia, Ib, Ic.
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For linear molecules, the moment of inertia about the
internuclear axis is zero.
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See Physical Chemistry by Atkins.
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Rotational motion
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Rotation of molecules are considered to be rigid rotors.
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Rigid rotors can be classified into four types:
o Spherical rotors: have equal moments of intertia (e.g., CH4, SF6).
o Symmetric rotors: have two equal moments of inertial (e.g., NH3).
o Linear rotors: have one moment of inertia equal to zero (e.g., CO2, HCl).
o Asymmetric rotors: have three different moments of inertia (e.g., H2O).
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Quantized rotational energy levels
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The classical expression for the energy of a rotating body is:
E a  1/2Ia a2
where a is the angular velocity in radians/sec.

E  1/2Ia a2  1/2Ib b2  1/2Ic c2
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For rotation about three axes:
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In terms of angular momentum (J = I):

o
o
J a2
J b2
J c2
E


2Ia 2Ib 2Ic
We know from QM that AM is quantized:

J(J 1)
Therefore, E J 
2I
J  J(J  1)
2
, J = 0, 1, 2, …
, J = 0, 1, 2, …


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Quantized rotational energy levels
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Last equation gives a ladder of energy levels.
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Normally expressed in terms of the rotational constant,
which is defined by:
hcB 
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2I
 B 
4cI
Therefore, in terms of a rotational term:

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2
F(J)  BJ(J 1) cm-1
The separation between adjacent levels is therefore

F(J) - F(J-1) = 2BJ
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As B decreases with increasing I =>large molecules
have closely spaced energy levels.
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Rotational spectra selection rules
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Transitions are only allowed according to selection rule for
angular momentum:
J = ±1
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Figure at right shows rotational energy levels transitions and
the resulting spectrum for a linear rotor.
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Note, the intensity of each line reflects the populations of the
initial level in each case.
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Molecular vibrations
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Consider simple case of a vibrating diatomic molecule,
where restoring force is proportional to displacement
(F = -kx). Potential energy is therefore
V = 1/2 kx2
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Can write the corresponding Schrodinger equation as
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d 2
 [E  V ]  0
2 dx 2
2
d 2
 [E 1/2kx 2 ]  0
2
2 dx
2
where
o

m1m2
m1  m2

The SE results in allowed energies
Ev  (v 1/2) 

k 1/ 2
   
 
v = 0, 1, 2, …
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
Molecular vibrations
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The vibrational terms of a molecule can therefore
be given by
G(v)  (v  1/2)v˜

1/ 2
1 k 
v˜ 
 
2c  
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Note, the force constant is a measure of the
curvature of the potential energy close to the

equilibrium extension of the bond.
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A strongly confining well (one with steep sides, a
stiff bond) corresponds to high values of k.
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Molecular vibrations
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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.
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Transition occur for v = ±1
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This potential does not apply to energies
close to dissociation energy.
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In fact, parabolic potential does not allow
molecular dissociation.
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Therefore
oscillator.
more
consider
anharmonic
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Anharmonic oscillator
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A molecular potential energy curve can be
approximated by a parabola near the bottom of the
well. The parabolic potential leads to harmonic
oscillations.
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At high excitation energies the parabolic
approximation is poor (the true potential is less
confining), and does not apply near the
dissociation limit.
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Must therefore use a asymmetric potential. E.g.,
The Morse potential:
V  hcDe 1 ea(R R e ) 
2
where De is the depth of the potential minimum
and
1/ 2
  2 

a  

2hcDe 
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
Anharmonic oscillator
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The Schrödinger equation can be solved for the Morse potential, giving permitted energy
levels:
G(v)  (v  1/2) v˜  (v˜  1/2) 2 x e v˜
where xe is the anharmonicity constant:
o
o

a2
xe 
2
The second term in the expression for G 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
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Vibrational-rotational spectroscopy
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Molecules vibrate and rotate at the same time =>
S(v,J) = G(v) + F(J)
S(v,J)  (v  1/2)v˜  BJ(J  1)
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Selection rules obtained by combining rotational
selection rule ΔJ = ±1 with vibrational rule Δv = ±1.

o
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When vibrational transitions of the form v + 1  v
occurs, ΔJ = ±1.
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Transitions with ΔJ = -1 are called the P branch:
v˜P (J)  S(v 1,J 1)  S(v,J)  v˜  2BJ
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
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
Transitions with ΔJ = +1 are called the R branch:
v˜R (J)  S(v 1,J 1)  S(v,J)  v˜  2B(J 1)
Q branch are all transitions with ΔJ = 0
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Vibrational-rotational spectroscopy
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Molecular vibration spectra consist of bands of lines in IR region of EM spectrum (100 –
4000cm-1 0.01 to 0.5 eV).
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Vibrational transitions accompanied by rotational transitions. Transition must produce a
changing electric dipole moment (IR spectroscopy).
Q branch
P branch
R branch
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Electronic transitions
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Electronic transitions occur between molecular
orbitals.
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Must adhere to angular momentum selection
rules.
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Molecular orbitals are labeled, , , , …
(analogous to S, P, D, … for atoms)
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For atoms, L = 0 => S, L = 1 => P
For molecules,  = 0 => ,  = 1 => 
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Selection rules are thus
 = 0, 1, S = 0, =0,  = 0, 1
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Where  =  +  is the total angular momentum
(orbit and spin).
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The End!
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All notes and tutorial set available from
http://www.physics.tcd.ie/people/peter.gallagher/lectures/py3004/
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Questions? Contact:
o [email protected]
o Room 3.17A in SNIAM
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