The one-body reduced density matrix can be succinctly represented as \(\langle \Psi | c_i^\dagger c_j | \Psi \rangle.\) Its diagonal is simply the density of electrons. Its off-diagonal represents a measure of the correlation between the occupation of two single-particle orbitals. This is often interpreted as “hopping” (although there is no dynamical aspect to it) because large elements between two atoms implies there is bonding between those atoms. A tight binding model is a model that is expressed in terms of the one-body density matrix.

The two body density matrix, \(\langle \Psi | c_i^\dagger c_k^\dagger c_j c_l | \Psi \rangle,\) can represent correlations in electron density (like Hubbard-interactions), as well as other correlations.

These notes discuss the construction and interpretation of the density matrix, and some nice tricks you can do with them.