This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems — the particle in a box, the harmonic oscillator, the rigid rotor and the ...
VIBRATIONAL SPECTROSCOPY
As we’ve emphasized many times in this course, within the Born
Oppenheimer approximation,
the nuclei move on a potential Harmonic
energy surface (PES) Approximation
determined by the electrons. R
For example, the potential R0 A + B separated atoms
felt by the nuclei in a V(R)
diatomic molecule is shown in
cartoon form at right. At low
energies, the molecule will sit
near the bottom of this
potential energy surface. In
this case, no matter what the equilibrium bond length
detailed structure of the potential is, locally the nuclei will “feel” a nearly
harmonic potential. Generally, the motion of the nuclei along the PES is
called vibrational motion, and clearly at low energies a good model for the
nuclear motion is a Harmonic oscillator.
Simple Example: Vibrational Spectroscopy of a Diatomic
If we just have a diatomic molecule, there is only one degree of freedom
(the bond length), and so it is reasonable to model diatomic vibrations using a
1D harmonic oscillator:
P̂ 2 1 ˆ 2 P̂ 2 1
Ĥ = + ko R = + mωo 2 Rˆ 2
2µ 2 2µ 2
where ko is a force constant that measures how stiff the bond is and can be
approximately related to the second derivative to the true (anharmonic) PES
near equilibrium:
∂ 2V
ko ≈ 2
∂R R
0
Applying Fermi’s Golden Rule, we find that when we irradiate the molecule,
the probability of a transition between the ith and fth Harmonic oscialltor
states is:
2
V fi
W fi ∝ ⎡δ ( Ei − E f − �ω ) + δ ( Ei − E f + �ω )⎤
4� 2 ⎣ ⎦
, 5.61 Physical Chemistry Lecture #34 2
where ω is the frequency of the light (not to be confused with the
frequency of the oscillator, ωo). Because the vibrational eigenstates involve
spatial degrees of freedom and not spin, we immediately recognize that it is
the electric field (and not magnetic) that is important here. Thus, we can
write the transition matrix element as:
2 2 2 2
V fi = ∫ φ f *µ̂
µ iE 0φi dτ = E 0 i ∫ φ f *µ̂
µφi dτ = E 0 i ∫ φ f * eR̂φi dτ
Now, we define the component of the electric field, ER, that is along the
bond axis which gives
2 2 2 2
V fi = E R ∫ φ f * eR̂φi dτ = e2 E R ∫ φ f R̂φi dτ
*
So the rate of transitions is proportional to the square of the strength of
the electric field (first two terms) as well as the square of the transition
dipole matrix element (third term). Now, because of what we know about
the Harmonic oscillator eigenfunctions, we can simplify this. First, we re
write the position operator, R, in terms of raising and lowering operators:
2
2 2 � �e 2 2 2
∫ φf ∫ φ f ( aˆ + + aˆ − ) φi dτ
2 * *
V fi = e ER ( aˆ + + aˆ − ) φi dτ = ER
2 µω 2 µω
2 �e 2 2
⇒ V fi =
2 µω
ER ( (i + 1) δ f ,i +1 + i δ f ,i −1 ) i + 1 φi +1 iφi−1
where above it should be clarified that in this expression “i" never refers to
√1 – it always refers to the initial quantum number of the system. Thus, we
immediately see that a transition will be forbidden unless the initial and
final states differ by one quantum of excitation. Further, we see that
the transitions become more probable for more highly excited states. That
is, Vfi gets bigger as i gets bigger.
Combining the explicit expression for the transition matrix element with
Fermi’s Golden Rule again gives:
e2 2
W fi ∝
8�µω
( ) (
ER ( i + 1) δ f ,i +1 + i δ f ,i −1 ⎡⎣δ Ei − E f − �ω + δ Ei − E f + �ω ⎤⎦ ) ( )
2
⇒ W fi ∝ E R {(i + 1) δ f ,i +1
⎡⎣δ ( Ei − Ei+1 − �ω ) + δ ( Ei − Ei+1 + �ω )⎤⎦
+ i δ f ,i −1 ⎡⎣δ ( Ei − Ei−1 − �ω ) + δ ( Ei − Ei−1 + �ω )⎤⎦ }
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