5.33 Lecture Notes,A Classical Description of Absorption
Why is there light absorption?
What is your picture? Quantum mechanical?
Molecule in
ground state
BEFORE
hν
Molecule in
excited state
AFTER
(No more photon)
This quantum mechanical picture may satisfy conservation of energy (when you
quantize light and molecular energies),but it has lots of problems and doesn’t tell
you much,
We can learn a lot from classical models of light absorption
We need to describe three things,
(1) Light,An oscillating electric (and magnetic) field
(2) Matter,Treat as a harmonic oscillator
(3) Interactions,Oscillating external force field driving harmonic oscillator
1)
An oscillating electromagnetic field,which oscillates in time and space,
( ) ( ) ( )
o
Er,t r E cos t k r= ε φ
Polarization vector
Amplitude
Frequency (rad/sec)
Wavevector defines
direction of propagation
8
2
k
c
c2.998 10 m / s
πω
=
λ
=
Light
ω
=
×
To simplify,
(1) Propagate along x? ; (2) E polarized along z? ; (3) φ= 0
Ex,t )=ε?
z
E
o
cos (ωt? kx )(
2π
λ=
oscillations in space
x
E
E
0
k
π 12
:time =
oscillations in time
propagating at c
ω ν
Now for the time being we will drop the polarization,and we will drop wave vector
(since λ>>x and consider only molecules at x=0),
c 2
Et()= E
o
cos ωt I =
4π
E
o
(2) Molecules
→ treat as harmonic oscillator
Why should we be able to call molecules harmonic oscillators? i.e.,a mass on a spring?
Molecules feel a restoring force when pushed from equilibrium,
5.33 Lecture Notes,A Classical Model for Spectroscopy
Page 2
The covalent bond can be thought of as a spring,
The equilibrium length is a balance between attractive and repulsive forces,If we
push/pull on this bond,there is a restoring force that pushes the system back to
equilibrium,
This analogy works for other systems also,
→ Electronic states—think of pushing electron clouds away from equilibrium
distribution,(for instance,benzene pi orbitals)
→ Magnetic resonance—In a magnetic field,magnetic spin moments—nuclear
spins—align with field,
If we push a spin away from field,it will want to relax back,
Classical Equation of Motion for Harmonic Oscillation
Potential Energy for Harmonic Oscillator
() (
2
2
2
kQ
2
1
q q
q
V
2
1
q V
=
=
For molecules,we expect
anharmonic curves that reflect
attractive and repulsive forces
V(q)
0
)
2
q
0
q
Q=0
(Set Q = q? q
0
)
5.33 Lecture Notes,A Classical Model for Spectroscopy
Page 3
k is force constant
V
Linear restoring force,F
res
=?
Q
=?kQ
2
Q
=Equation of Motion from Newton Fma m
t
2
2
Q
m
t
2
= F
res
+ F
damp
+ F
ext
driving force
=?kQ? b
Q
+ F
ext
t
2
Q
+
b?Q
+
kQ F
ext
=
t
2
m?t m m
with light field
F
ext
(t)= F
o
cos ωt
Solutions,
a) Harmonic oscillator with no damping/no external force
stretch spring and let it go,,,
m
2
Q
+ kQ = 0
t
2
Solution,
Qt
()
= A sin
(
ω
o
t
)
+
( )
B cos t ω
o
drop for now
sin,impulse—kick
resonance frequency,ω=
k
m
cos,stretch/hold/release
o
given by k
5.33 Lecture Notes,A Classical Model for Spectroscopy
Page 4
Oscillations continue forever at ω
0
,
Q
t
A
2π
ω
0
b) Now we add damping—H.O,may feel friction that reduces the amplitude of
oscillation
m
2
Q
= F
res
+ F
damp
=?kQ? b
Q
t
2
t
2
Q
+
b?Q
+
kQ
= 0
t
2
m?t m
Qt
( )
= Ae
γt
sin?
o
t
Now oscillation decays away exponentially
γ= b /2m?
o
2 2
=ω? γ
o
≈ω
o
reduced for weak
Q
frequency damping
2π
t
eA
γ?
0
5.33 Lecture Notes,A Classical Model for Spectroscopy
Page 5

Rewrite equation of motion,
for Ft
()
= 0
2
Q
+ 2γ
Q
+ω
2
Q = F
ext
()
Qt
t
2
t
0
t
()= e
γt
sin?
ot
c) Apply external driving force → interaction of molecule with EM field
You know what to expect,
The effect depends on the frequency of the driving field,,, like pushing someone on
a swing,
The most efficient way to push someone higher is to push at a frequency
corresponding to their swinging frequency
This leads to a big displacement,
If you push with arbitrary frequency,nothing will happen,
So you know that we should have a,resonance”,
When you drive the system with frequency ω ≈ω
o
there will be an efficient
transfer of power and the displacement of the H.O,will increase
Indeed,that is what an absorption spectrum is! Measure the power absorbed by the
system from the field,
Now let’s solve the equation,
Set,F
0
∝ E
0
tF
ext
()=
F
o
cos ωt
m
Q t()= Asin (ωt +β) =
Fm
o
1
2
sin (ωt +β)
2
2
(
ω?ω
2
)
+ 4γ
2
ω
2
o
Fm
o
≈
2
2ω
o
(ω
o
ω) +γ
2
2
ω
2
ω
1
2
o
tanβ=
2γω
5.33 Lecture Notes,A Classical Model for Spectroscopy
Page 6
Notice that the coordinate oscillates at the driving frequency ω !
Notice that it oscillates 90° out of phase with field when driven on resonance,→
absorbing power from field,
If γ <<ω
o
and near resonance ω
o
≈ω
2
2
2
(
ω?ω
2
)
=(ω?ω)
2
(ω
o
+ ω)
2
≈ 4ω
o
(ω?ω)
2
o o o
Qt
()
=
Fm
o
1
2
sin
(
ωt +β
)
2
2ω
o
(
ω
o
ω
)
+γ
2
Now we can calculate the absorption spectrum → power absorbed,
power = force × velocity
F
2
γω
2
P
avg
=
F t()
Q
t
=
o
2
)
2
avg
m
(
ω
2
ω
o
+ 4γ
2
ω
2
≈
F
o
2
γ
)
4m
(ω?ω
o
2
+γ
2
Let’s plot the power absorbed as a function of frequency,
LORENTZIAN LINESHAPE
peak is at ω=ω
o
At what frequency is
the power absorbed
P
av
(ω)
ω
0
γ
reduced by
ω?ω
o
= ±γ
ω
The full-width at half-maximum intensity (FWHM) is 2γ,
5.33 Lecture Notes,A Classical Model for Spectroscopy
Page 7
peak → ω
o
width →γ
So what are we learning about molecules?
Well,you can see from the spectrum that we have been able to determine ω
o
and γ,
Take HCl,We take an IR absorption spectrum in solution,and we see,
ω
0
ω
2γ
A
ω
0
3000cm
1
γ? 15cm
1
2πc
ω
o
,We know the frequency of oscillation of the H-Cl bond,
2π
=τ = 1 × 10
14
s 10 fs
ω
0
How do we know that’s what it is? We predict an isotope effect!
ω
0
=
k
m
Now for the mass m we want to use the reduced mass μ or m
R
,
A
μ= m
R
=
mm
B
for a diatomic molecule A-B,
m
A
+ m
B
(This is the effective mass,we need this for 2 masses on a spring with a fixed
center of mass.)
35
μ
(
HCl
)
= 0.97
35
μ
(
DCl
)
= 1.89
2
Therefore,we predict,
ω
0
( DCl)→ 2150cm
1
5.33 Lecture Notes,A Classical Model for Spectroscopy
Page 8
k,We know the curvature of the HCl potential!
γ,Energy dissipates from HCl bond in roughly
τ
damp
2 × 10
12
s → 2ps
What about F
0

We said the driving force is proportional to Et(),
F
ext
= F
0
cosωt = a E
0
cosωt
What is a?
An electric field interacts with charges on molecules,
A neutral body should feel no force,but molecules are made of many charges species and
have positive and negative charges,
This leads to a dipole moment,
μ=Σ ql
H
Cl
μ
δ
-
δ
+
distance
charge
In an electric field,this dipole will feel a force,
V =?μ? E
F =?
V
=
μ
E
Q
Q
If there is a large displacement of charge → Strong absorption!
How does the field act to stretch or compress bond?
The E field oscillates along the polarization direction,
5.33 Lecture Notes,A Classical Model for Spectroscopy
Page 9
If the bond is aligned with the electric field,then the oscillating field can act to stretch or
compress the bond through attractive and repulsive electrostatic interactions with
the charges,
The ability of the field to act on Q depends on the alignment of the dipole moment with
the field,If there is an angle θ between μ and E,then
F
0
=
μ
E
0
cos θ (…for a single molecule)
Q
So spectroscopy can tell you about the orientation of molecules!
So we have also been able to predict,
μ, The strength of the absorption peak is related to how much charge is moved when
you displace a coordinate,
H-Cl will absorb light,
→ dipole changes when you stretch
What about N
2
μ? E, Absorption will tell you about alignment of molecules,
You can study a crystal where all molecules have the same orientation,and how
by rotating the light polarization,you can tell how different vibrations are
oriented,
→ also look at reorientational motion!
These principles can be applied to electronic spectroscopy — higher frequency → for
instance,m
e
<< m
w
,
We can model e
held to molecule by harmonic restoring force,
freq,+ force constant depend on electronic structure → Q.M,
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