1. CHAPTER 9 Molecules Rotations Spectra Complex planar molecules
Homework due Wednesday Nov. 4th
Only 5 problems: Krane Chapter 9
8, 14, 17, 20, 22
Johannes Diderik van der Waals (1837 – 1923)
“Life ... is a relationship between molecules.” 
Linus Pauling

2. 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.

3. Vibrational States Dn = ±1
n is called the vibrational quantum number. Don’t confuse it for n, the principal quantum number of the electronic state.
The energy levels are those of a quantum-mechanical oscillator.
Vibrational-transition selection rule:
Dn = ±1
The only spectral line is w !
However, deviations from a perfect parabolic potential allow other transitions (~2w, ~3w, …), called overtones, but they’re much weaker.

4. Vibrational Frequencies for Various Bonds
Different bonds have different vibrational frequencies (which are also affected by other nearby atoms).
Wavenumber (cm-1)
← Higher energy (frequency)
Notice that bonds containing Hydrogen vibrate faster because H is lighter.

5. Water’s Vibrations
Movies from

6. 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.

7. Rotational States
L is quantized. where ℓ can be any integer. The energy levels are Erot varies only as a function of the quantum number ℓ.
= ħ2/I

8. Rotational transition energies
And there is a selection rule that Dℓ = ±1.
Transitions from ℓ +1 to ℓ : Emitted photons have energies at regular intervals:

9. Vibration and Rotation Combined
Note the difference in lengths (DE) for larger values of ℓ.
Note the similarity in lengths (DE) for small values of ℓ.
DE increases linearly with ℓ. Most transitions are forbidden by the selection rules that require Dℓ = ±1 and Dn = ±1.

10. 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.

11. 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

12. 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. DE
Input light
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

13. Frequencies in Atoms and Molecules
Electrons vibrate in their motion around nuclei
High frequency: ~ cycles per second.
Nuclei in molecules vibrate with respect to each other
Intermediate frequency: ~ cycles per second.
Nuclei in molecules rotate
Low frequency: ~ cycles per second.
E~1eV
E~0.1eV
E~0.01eV

14. Including Electronic Energy Levels
A typical large molecule’s energy levels:
E = Eelectonic + Evibrational + Erotational
2nd excited
electronic state
Lowest vibrational and rotational level of this electronic “manifold.”
Energy
1st excited
electronic state
Excited vibrational and rotational level
Transition
There are many other complications, such as spin-orbit coupling, nuclear spin, etc., which split levels.
Ground
electronic state
As a result, molecules generally have very complex spectra.

15. Modeling Very Complex Molecules
Sometimes more complex is actually easier!
Many large organic (carbon-based) 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 elec-trons
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:

16. Auroras

17. Typical Aurora Emission Spectrum
Intensity

18. Species Present in the Atmosphere

19. Constituents Contributing to AurorasH N