CHAPTER 9 Molecules Rotations Spectra Complex planar molecules Homework due Wednesday Nov. 5th Only 5 problems: 8, 14, 17, 20, 22 Johannes Diderik van der Waals (1837 – 1923) http://en.wikipedia.org/wiki/Image:Johannes_Diderik_van_der_Waals.jpg “Life ... is a relationship between molecules.”  Linus Pauling

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.

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

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:

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.

Vibration and Rotation Combined The emission (and absorption) spectrum spacing varies with ℓ. The higher the starting energy level, the greater the photon energy. Vibrational energies are greater than rotational energies. For a diatomic molecule, this energy difference results in band structure. The line strengths depend on the populations of the states and the vibrational selection rules, however. Weaker overtones Dn = 3 Dn = 0 Dn = 1 Dn = 2 Energy or Frequency → Dℓ = -1 Dℓ = 1

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

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 Intermediate frequency: ~1011 - 1013 cycles per second. Nuclei in molecules rotate Low frequency: ~109 - 1010 cycles per second.

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.

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

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:

Auroras http://www.itp.uni-hannover.de/~zawischa/ITP/atoms.html

Typical Aurora Emission Spectrum Intensity http://www.itp.uni-hannover.de/~zawischa/ITP/atoms.html

Species Present in the Atmosphere http://www.itp.uni-hannover.de/~zawischa/ITP/atoms.html

Constituents Contributing to Auroras H N http://www.itp.uni-hannover.de/~zawischa/ITP/atoms.html