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14Jan2011
Chemistry 21b – Spectroscopy
Lecture # 6 – Nuclear Motion in Diatomic Rotors
Once the electronic potential energy surfaces have been computed the motion of the
nuclei can be determined by solving eq. (4.10). Here for simplicity we will consider the
case of a diatomic molecule, over the next couple of weeks we’ll build up the quantum
mechanical and group theoretical machinery needed to handle polyatomic systems.
If we transform to the centerofmass coordinates and deFne the internuclear
coordinate
−→
R
=
−→
R
1
−
−→
R
2
, the nuclear kinetic energy term reduces to a term containing the
centerofmass motion, which is not of spectroscopic interest (being a constant, you’ll look
more carefully at these degrees of freedom in Ch 21c), and a term describing the relative
motion. This last term can be separated further into a part governing the radial motion
along
R
=

−→
R

, and a part containing the angular coordinates, described by the nuclear
angular momentum operator
J
:
T
N
=
−
1
2
μR
2
∂
∂R
p
R
2
∂
∂R
P
+
J
2
2
μR
2
,
(6
.
1)
where
μ
=
M
1
M
2
/
(
M
1
+
M
2
) is the reduced nuclear mass. If we write
Ψ
nuc
(
−→
R
) =
Y
JM
(
ˆ
R
)
F
(
R
)
/R
(6
.
2)
where
ˆ
R
denotes the angular part of
−→
R
and use
J
2
Y
JM
(
ˆ
R
) =
J
(
J
+ 1)
Y
JM
(
ˆ
R
)
(6
.
3)
we obtain
b
−
1
2
μ
d
2
dR
2
+
E
el
(
R
) +
J
(
J
+ 1)
2
μR
2
−
E
B
F
(
R
) = 0
.
(6
.
4)
a) Vibration
If the electronic state is bound, that is, stable with respect to dissociation, it has a
minimum
−
D
e
at a certain distance
R
e
. The potential can then be expanded in powers of
(
R
−
R
e
) about
R
e
(writing
V
=
E
el
):
V
(
R
) =
V
(
R
e
) +
p
dV
dR
P
R
e
(
R
−
R
e
) +
1
2
p
d
2
V
DR
2
P
R
e
(
R
−
R
e
)
2
+
.
(6
.
5)
At
R
=
R
e
, the Frst derivative vanishes, and up to quadratic terms:
V
(
R
)
≃ −
D
e
+
1
2
k
(
R
−
R
e
)
2
(6
.
6)
47
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View Full DocumentFigure 6.1 a) Harmonic approximation to a molecular potential curve
E
el
(
R
): b) Energy
levels of a harmonic oscillator (dashed lines), compared with those for an anharmonic
potential (solid line).
with
k
= (
d
2
V/dR
2
)
R
e
. Thus, if rotation is neglected (
J
= 0), the purely radial equation
(6.4) becomes the equation for the harmonic oscillator. Thus, the solution to eq. (6.4) can
in that case be written as
E
=
−
D
e
+
E
vib
(6
.
7)
with
E
vib
=
ω
e
(
v
+ 1
/
2)
(6
.
8)
where
ω
e
= (
k/μ
)
1
/
2
.
In the harmonic oscillator approximation, the vibrational levels are equidistant. As Figure
6.1 shows, this is a good approximation for the lowest few vibrational levels, but the higher
levels will lie much closer together due to “anharmonicities” in the potential curve. The
harmonic oscillator wave functions can be written in terms of Hermite polynomials. Some
characteristic wave functions are illustrated in Figure 6.2. Note that the quantum number
v
of the wave function is the same as the number of nodes.
Eq. (6.4) can also be solved analytically for the case that the potential
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