1. CHAPTER 10 Molecules Why do molecules form? Molecular bonds Rotations
Vibrations
Spectra
Complex planar molecules
Johannes Diderik van der Waals (1837 – 1923)
“Life ... is a relationship between molecules.” 
Linus Pauling
Prof. Rick Trebino, Georgia Tech,

2. Molecules are combinations of atoms.
When more than one atom is involved, the potential and the wave function are functions of way more than one position (a position vector for each nucleus and electron):
Electrons’ positions
Nuclei positions
Solving the Schrodinger Equation in this case is even harder than for multi-electron atoms.
Serious approximation methods are required.
This is called Chemistry!

3. 10.1: Molecular Bonding and Spectra
Nucleus
Electron cloud
Nucleus
Electron cloud
The only force that binds atoms together in molecules is the Coulomb force.
But aren’t most atoms electrically neutral? Yes!
Indeed, there is no attraction between spherically symmetrical molecules—the positive and negative charges both behave like point sources and so their fields cancel out perfectly!
So how do molecules form?

4. Why Molecules Form + - Most atoms are not spherically symmetrical.
For example, these two “atoms” attract each other: + -
This is because the distance between opposite charges is less than that between charges of the same sign.
The combination of attractive and repulsive forces creates a stable molecular structure.
Force is related to the potential energy surface, F = −dV/dr, where r is the position.

5. Charge is distributed very unevenly in most atoms.
The probability density for the hydrogen atom for three different electron states.

6. Closed shells of electrons are very stable.
Atoms with closed shells (noble gases) don’t form molecules.
Ionization energy (eV)
Atomic number (Z)
Noble gases (difficult to remove an electron)

7. Atoms like closed electron shells.
Atoms with closed shells (noble gases) also have the smallest atomic radii.
Atomic number (Z)
Atomic radius (nm)
But add an extra electron, and it’s weakly bound and far away.

8. An extra electron or two outside a closed shell are very easy to liberate.
Atoms with one or two (or even more) extra electrons will give them up to another atom that requires one or two to close a shell.
Atoms with extra electrons are said to be electropositive.
Those in need of electrons are electronegative.

9. Ionic Bonds
An electropositive atom gives up an electron to an electronegative one. Example: Sodium (1s22s22p63s1) readily gives up its 3s electron to become Na+, while chlorine (1s22s22p63s23p5) easily gains an electron to become Cl−.

10. Covalent Bonds
Two electronegative atoms share one or more electrons. Example: Diatomic molecules formed by the combination of two identical electronegative atoms tend to be covalent. Larger molecules are formed with covalent bonds.
Diamond

11. Metallic Bonds
In metals, in which electrons are very weakly bound, valence electrons are essentially free and may be shared by a number of atoms. The Drude model for a metal: a free-electron gas!

12. Molecular Potential Energy Curve
The potential depends on the charge distributions of the atoms involved, but there is always an equilibrium separation between two atoms in a molecule. The energy required to separate the two atoms completely is the binding energy, roughly equal to the depth of the potential well.
Vibrations are excited thermally, that is, by collisions with other molecules, or by light, creating superpositions of ground plus an excited state(s).

13. Molecular Potential
An approximation of the force felt by one atom in the vicinity of another atom is: where A and B are positive constants. Because of the complicated shielding effects of the various electron shells, n and m are not equal to 1. One example is the Lennard-Jones potential in which n = 12 and m = 6.
The shape of the curve depends on the parameters A, B, n, and m.

14. 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.

15. 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.

16. 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.

17. Water’s Vibrations
Movies from

18. 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.

19. 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

20. 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:

21. 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.

22. 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

23. 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

24. 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.

25. 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.

26. 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

27. 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:

28. Auroras

29. Typical Aurora Emission Spectrum
Intensity

30. Species Present in the Atmosphere

31. Constituents Contributing to AurorasH N