# 2: Model Problems That Form Important Starting Points

- Page ID
- 11552

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

The model problems discussed in this Chapter form the basis for chemists’ understanding of the electronic states of atoms, molecules, nano-clusters, and solids as well as the rotational and vibrational motions and energy levels of molecules.

In this Chapter, you should have learned about the following things.

- Free particle energies and wave functions and their densities of states, as applied to polyenes, electron in surfaces, solids, and nanoscopic materials and as applied to bands of orbitals in solids.
- The tight-binding or Hückel model for chemical bonding.
- The hydrogenic radial and angular wave functions. These same angular functions occur whenever one is dealing with a potential that depends only on the radial coordinate, not the angular coordinates.
- Electron tunneling and quasi-bound resonance states.
- Angular momentum including coupling two or more angular momenta, and angular momentum as applied to rotations of rigid molecules including rigid rotors, symmetric, spherical, and asymmetric top rotations. Why half-integral angular momenta cannot be thought of as arising from rotational motion of a physical body.
- Vibrations of diatomic molecules including the harmonic oscillator and Morse oscillator models including harmonic frequencies and anharmonicity.

- 2.6: Electron Tunneling
- Tunneling is a phenomenon of quantum mechanics, not classical mechanics. It is an extremely important subject that occurs in a wide variety of chemical species including nano-scale electronic devices and protons moving through water.

## Contributors and Attributions

Jack Simons (Henry Eyring Scientist and Professor of Chemistry, U. Utah) Telluride Schools on Theoretical Chemistry

Integrated by Tomoyuki Hayashi (UC Davis)