Density matrices

Density matrices provide a compact way of storing “information” that is otherwise only accessible by examining the wave function itself. They can be thought of as wave function proxies.

Given a normalized reference state \(\require{physics} \ket{\Psi}\), expressed as a linear combination of ONVs \[\begin{equation} \ket{\Psi} = \sum_{\vb{k}} c_{\vb{k}} \ket{\vb{k}} \thinspace , \end{equation}\] the expectation values of an arbitrary operator \(\hat{\Omega}\) (that contains at most two-electron interactions) can be calculated by using density matrices: \[\begin{equation} \ev{\hat{\Omega}}{\Psi} = \sum_{PQ}^M D_{PQ} \Omega_{PQ} + \frac{1}{2} \sum_{PQRS}^M d_{PQRS} \Omega_{PQRS} + \Omega_0 \thinspace , \end{equation}\] where \(\Omega_{PQ}\) and \(\Omega_{PQRS}\) are the one- and two-electron integrals in the given orthonormal spinor basis. \(\vb{D}\) is then called the one-electron density matrix and \(\vb{d}\) is called the two-electron density matrix, respectively.

As an example, using these density matrices leads to the following expression for the energy, i.e. the expectation value of the Hamiltonian: \[\begin{equation} E = \sum_{PQ}^M h_{PQ} D_{PQ} + \frac{1}{2} \sum_{PQRS}^M g_{PQRS} d_{PQRS} + h_{\text{nuc}} \thinspace . \end{equation}\]