o  Rotational transitions
o  Vibrational transitions
o  Electronic transitions
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o 
Born-Oppenheimer Approximation is the assumption that the electronic motion and the
nuclear motion in molecules can be separated.
o 
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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o 
Electronic transitions: UV-visible
o 
Vibrational transitions: IR
o 
Rotational transitions: Radio
E
Electronic
Vibrational
Rotational
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o 
Must first consider molecular moment of inertia:
I = " mi ri2
i
o 
At right, there are three identical atoms bonded to
“B” atom and three different atoms attached to “C”.
!
o 
Generally specified about three axes: Ia, Ib, Ic.
o 
For linear molecules, the moment of inertia about the
internuclear axis is zero.
o 
See Physical Chemistry by Atkins.
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o  Rotation of molecules are considered to be rigid rotors.
o  Rigid rotors can be classified into four types:
o  Spherical rotors: have equal moments of inertia (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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o 
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.
o 
For rotation about three axes:!
o 
In terms of angular momentum (J = I!):
!
o 
E = 1/2Ia" a2 + 1/2Ib" b2 + 1/2Ic" c2
E=
J a2
J2
J2
+ b + c
2Ia 2Ib 2Ic
We know from QM that AM is quantized:
J = J(J + 1)! 2
!
J(J + 1)!
o  Therefore, E J =
, J = 0, 1, 2, …
2I
, J = 0, 1, 2, …
!
!
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o 
Last equation gives a ladder of energy levels.
o 
Normally expressed in terms of the rotational constant,
which is defined by:
!2
!
hcB =
=> B =
2I
4 "cI
o 
Therefore, in terms of a rotational term:
!
o 
F(J) = BJ(J + 1) cm-1
The separation between adjacent levels is therefore
!
F(J) - F(J-1) = 2BJ
o 
As B decreases with increasing I =>large molecules
have closely spaced energy levels.
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o  Transitions are only allowed according to selection
rule for angular momentum:
"J = ±1
o  Figure at right shows rotational energy levels
transitions and the resulting spectrum for a linear
rotor.
o  Note, the intensity of each line reflects the populations
of the initial level in each case.
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o 
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
o 
Can write the corresponding Schrodinger equation as
! 2 d 2"
+ [E # V ]" = 0
2µ dx 2
! 2 d 2"
+ [E #1/2kx 2 ]" = 0
2µ dx 2
where
o 
µ=
m1m2
m1 + m2
!
The SE results in allowed energies
E v = (v + 1/2)!"
!
# k &1/ 2
" =% (
$µ'
v = 0, 1, 2, …
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!
o 
!
The vibrational terms of a molecule can therefore
be given by
G(v) = (v + 1/2)v˜
v˜ =
!
1/ 2
1 #k&
% (
2"c $ µ '
o 
Note, the force constant is a measure of the
curvature of the potential energy close to the
!
equilibrium extension of the bond.
o 
A strongly confining well (one with steep sides, a
stiff bond) corresponds to high values of k.
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o 
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.
o 
Transition occur for "v = ±1
o 
This potential does not apply to energies
close to dissociation energy.
o 
In fact, parabolic potential does not allow
molecular dissociation.
o 
Therefore
oscillator.
more
consider
anharmonic
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o 
A molecular potential energy curve can be
approximated by a parabola near the bottom of the
well. The parabolic potential leads to harmonic
oscillations.
o 
At high excitation energies the parabolic
approximation is poor (the true potential is less
confining), and does not apply near the dissociation
limit.
o 
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
# µ" 2 &1/ 2
!
a =%
(
$ 2hcDe '
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!
o 
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 
xe =
!
a2!
2µ"
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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o 
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)
o 
Selection rules obtained by combining rotational
selection rule !J = ±1 with vibrational rule !v = ±1.
!
o 
When vibrational transitions of the form v + 1 ! v
occurs, !J = ±1.
o 
Transitions with !J = -1 are called the P branch:
v˜ P (J) = S(v + 1,J "1) " S(v,J) = v˜ " 2BJ
o 
!
o 
!
Transitions with !J = +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 !J = 0
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o 
Molecular vibration spectra consist of bands of lines in IR region of EM spectrum (100 –
4000cm-1 0.01 to 0.5 eV).
o 
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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o 
Electronic transitions occur between molecular
orbitals.
o 
Must adhere to angular momentum selection rules.
o 
Molecular orbitals are labeled, ", #, $, …
(analogous to S, P, D, … for atoms)
o 
o 
For atoms, L = 0 => S, L = 1 => P
For molecules, % = 0 => ", % = 1 => #
o 
Selection rules are thus
$% = 0, ±1, $S = 0, $"=0, $& = 0, ±1
o 
Where & = % + " is the total angular momentum
(orbit and spin).
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