From electron configuration to crystals and bands

Many electrons fill the lowest-energy rooms first (aufbau), at most two per room (Pauli), singly before pairing within a level (Hund). The effective charge Z_eff, screened by inner electrons, sets radius and order.

Mixing s and p orbitals makes directed hybrid lobes. sp spreads to 180 degrees, sp² to 120, sp³ to 109.5, and the fraction of s character sets the bond angle.

Electron pairs push as far apart as they can, which sets the molecular shape (VSEPR). A lone pair squeezes the bond angle harder, bending water to 104.5 degrees. The vector sum of bond dipoles sets molecular polarity.

Radius, ionization energy, and electronegativity change regularly across the table. Moving across a period the effective charge grows and pulls harder; going down a group adds shells and grows the radius.

A crystal is a unit cell repeated in three dimensions. Counting corner atoms as 1/8, face as 1/2, body as 1 gives 1 for SC, 2 for BCC, 4 for FCC. Miller indices name planes inside the lattice.

The packing fraction (FCC and HCP 0.74, BCC 0.68, SC 0.52) and coordination number (12, 8, 12, 6) tell the lattices apart. FCC and HCP stack the same close-packed plane as ABCABC and ABAB, yet share the same packing.

As many atoms gather, discrete levels broaden into wide bands. A band gap E_g of zero is a metal, small a semiconductor, large an insulator. Doping places a level in the gap and shifts the Fermi level.