# Molecules and Condensed Matter

Molecular bonds and molecular spectra:  The principal types of molecular bonds are ionic, covalent, van der Waals, and hydrogen bonds.  In a diatomic molecule the rotational energy levels are given by equation: $E_l=l(l+1)\frac{\hbar^2}{2I}$ $(l=0,1,2,\ldots)$, where $I=m_{\mathrm{r}}r_0^2$ is the moment of inertia of the molecule, $m_{\mathrm{r}}=\frac{m_1m_2}{m_1+m_2}$ is its reduced mass, and $r_0$ is the distance between the two atoms.  The vibrational energy levels are given by equation $E_n=(n+\frac12)\hbar\omega=(n+\frac12)\hbar\sqrt{\frac{k'}{m_{\mathrm{r}}}}$ $n=(0,1,2,\ldots)$, where $k'$ is the effective force constant of the interatomic force.

Solids and energy bands:  Interatomic bonds in solids are of the same types as in molecules plus one additional type, the metallic bond.  Associating the basis with each lattice point gives the crystal structure.

When atoms are bound together in condensed matter, their outer energy levels spread out into bands.  At absolute zero, insulators and semiconductors have a completely filled valence band separated by an energy gap from an empty conduction band.  Conductors, including metals, have partially filled conduction bands.

Free-electron model of metal:  In the free-electron model of the behavior of conductors, the electrons are treated as completely free particles within the conductor.  In this model the density of states is given by equation $g(E)=\frac{(2m)^{2/3}V}{2\pi^2\hbar^3}E^{1/2}$.  The probability that an energy state of energy $E$ is occupied is given by the Fermi-Dirac distribution, $f(E)=\frac1{e^{(E-E_{\mathrm{F}})/kT}+1}$ ($E_{\mathrm{F}}$ is the Fermi energy), which is a consequence of the exclusion principle.

Semiconductors:  A semiconductor has an energy gap of about 1 eV between its valence and conduction bands.  Its electrical properties can be drastically changed by the addition of small concentrations of donor impurities, giving an $n$-type semiconductor, or acceptor impurities, giving a $p$-type semiconductor.

Semiconductor devices:  Many semiconductor devices, including diodes, transistors, and integrated circuits use one or more $p\text{-}n$-junctions.  The current-voltage relationship for an ideal $p\text{-}n$-junction diode is given by equation $I=I_S(e^{eV/kT}-1)$.