Density matrices

The density matrix operator is defined as

\[\begin{equation} \hat{\rho}_{ij} \defd \ketbra{i}{j} \end{equation}\]

If the wavefunction is approximated using Slater determinants, where one Slater determinant of $N$ electrons is defined as

\[\begin{equation} \Phi(1,2,3,...,N) = \frac{1}{\sqrt{N!}} \left|\begin{matrix} \chi_1(1)& \chi_2(2)& ...& \chi_N(N) \end{matrix}\right| \end{equation}\]

where $1,2,3...,N$ denote the $N$ different electronic coordinates (spatial and spin) and $\chi_i$ are the $N$ different spin-orbitals, the transition density matrix for all $N$ electrons, between two Slater determinants $\Phi_A$ and $\Phi_B$ is given by

\[\begin{equation} \begin{aligned} \rho_N^{AB}(1,...N;1',...,N') &= N!\Phi_A(1,...,N)\conj{\Phi_B}(1',...,N')\\ &= \left|\begin{matrix} \rho_1(1,1')&...&\rho_1(1,N')\\ \vdots&\ddots&\vdots\\ \rho_1(N,1')&...&\rho_1(N,N') \end{matrix}\right|. \end{aligned} \end{equation}\]

References

• Per-Olov Löwdin (1955). Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction. Physical Review, 97(6), 1474–1489. 10.1103/physrev.97.1474

• Per-Olov Löwdin (1955). Quantum Theory of Many-Particle Systems. II. Study of the Ordinary Hartree-Fock Approximation. Physical Review, 97(6), 1490–1508. 10.1103/physrev.97.1490

• Per-Olov Löwdin (1955). Quantum Theory of Many-Particle Systems. III. Extension of the Hartree-Fock Scheme to Include Degenerate Systems and Correlation Effects. Physical Review, 97(6), 1509–1520. 10.1103/physrev.97.1509