Download 10. Molecules and Solids

CHAPTER 9
Molecules
Rotations
Spectra
Complex planar molecules
Homework due Wednesday Nov. 4th
Only 5 problems: Krane Chapter 9
Johannes Diderik van der Waals
(1837 – 1923)
8, 14, 17, 20, 22
“Life ... is a relationship between molecules.”
Linus Pauling
Vibrational Motion: A Simple Harmonic
Oscillator
The Schrödinger Equation can be
separated into equations for the positions
of the electrons and those of the nuclei.
The simple harmonic oscillator accurately
describes the nuclear positions of a
diatomic molecule, as well as more
complex molecules.
Vibrational States
The energy levels are
those of a quantummechanical oscillator.
Evibr  (n  1 2) 
n is called the
vibrational quantum
number. Don’t
confuse it for n, the
principal quantum
number of the
electronic state.
Vibrational-transition
selection rule:
Dn = ±1
The only spectral line is  !
However, deviations from a perfect
parabolic potential allow other
transitions (~2, ~3, …), called
overtones, but they’re much weaker.
Vibrational Frequencies for Various Bonds
Different bonds have different vibrational frequencies (which are
also affected by other nearby atoms).
← Higher energy (frequency)
Wavenumber (cm-1)
Notice that bonds containing Hydrogen vibrate faster because H is
lighter.
Water’s Vibrations
Rotational States
Consider diatomic molecules.
A diatomic molecule may be thought of as two atoms held together
with a massless, rigid rod (rigid rotator model).
In a purely rotational system, the kinetic energy is expressed in
terms of the angular momentum L and rotational inertia I.
Erot
L2

2I
Rotational States
L is quantized.
L  (  1) 
where ℓ can be any integer.
The energy levels are
Erot 
2
(  1)
2I
Erot varies only as a function of the
quantum number ℓ.
= ħ2/I
Rotational transition energies
And there is a selection
rule that Dℓ = ±1.
Erot 
2
(  1)
2I
Transitions from ℓ +1 to ℓ :
Emitted photons have energies at regular intervals:
E ph 
2
2I

(
 1)(  2)  (  1) 
2
 2  3  2 
2I
2
2
   (  1)
I
Vibration and
Rotation
Combined
Note the difference
in lengths (DE) for
larger values of ℓ.
E  Erot  Evib 
2
(  1) 
1
n  
2I
2

DE increases
linearly with ℓ.
Most transitions are
forbidden by the
selection rules that
require Dℓ = ±1 and
Dn = ±1.
Note the
similarity in
lengths (DE)
for small
values of ℓ.
Vibration and Rotation Combined
The positions and intensities of the observed bands are ruled by
quantum mechanics. Note two features in particular:
1) The relative intensities of the bands are due to different
transition probabilities.
2) Some transitions are forbidden by the selection rule that
requires Δℓ = ±1.
Absorption spectra:
Within Δℓ = ±1 rotational state changes, molecules can absorb
photons and make transitions to a higher vibrational state when
electromagnetic radiation is incident upon a collection of a particular
kind of molecule.
Vibrational/Rotational Spectrum
In the absorption spectrum of HCl, the spacing between the peaks
can be used to compute the rotational inertia I. The missing peak
in the center corresponds to the forbidden Dℓ = 0 transition.
ℓi ℓf = 1
ℓi ℓf = 1
ni nf = 1
Studying Vibrations and Rotations
Infrared spectroscopy allows the study of vibrational and
rotational transitions and states.
But it’s often difficult to generate and detect the required IR light.
It’s easier to work in the visible or near-IR.
Input light
DE
Output light
Raman scattering:
If a photon of energy greater than DE
is absorbed by a molecule, another
photon with ±DE additional energy
may be emitted.
The selection rules become:
Δn = 0, ±2 and Δℓ = 0, ±2
Frequencies in Atoms and Molecules
Electrons vibrate in their motion around nuclei
High frequency:
~1014
-
1017
cycles per second.
Nuclei in molecules vibrate
with respect to each other E~0.1eV
Intermediate frequency:
~1011 - 1013 cycles per second.
Nuclei in molecules rotate
E~0.01eV
Low frequency: ~109 - 1010 cycles per second.
E~1eV
Including Electronic Energy Levels
A typical large molecule’s
energy levels:
E = Eelectonic + Evibrational + Erotational
2nd excited
electronic state
Energy
1st excited
electronic state
Lowest vibrational and
rotational level of this
electronic “manifold.”
Excited vibrational and
rotational level
Transition
Ground
electronic state
There are many other
complications, such as
spin-orbit coupling,
nuclear spin, etc.,
which split levels.
As a result, molecules generally have very complex spectra.
Modeling Very Complex Molecules
Sometimes more complex is
actually easier!
Many large organic (carbonbased) molecules are planar, and
the most weakly bound electron
is essentially free to move along
the perimeter. We call this model
the Perimeter Free-Electron
Orbital model.
plus
inner
electrons
This is just a particle in a one-dimensional box! The states are just
sine waves. The only difference is that x = L is the same as x = 0.
So y doesn’t have to be zero at the boundary, and there is another
state, the lowest-energy state, which is a constant:
y 0 ( x)  1/ L
Auroras
Intensity
Typical Aurora Emission Spectrum
Species Present in the Atmosphere
Constituents Contributing to Auroras
+
+
O
N
O
+
2
H
2
N
O2