Density matrix solver

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A density matrix solver is a routine for solving the Kohn-Sham (KS) eigenproblem by calculating only the single-particle density matrix for the system

\rho \left ( \mathbf{r}, \mathbf{r}' \right ) = \sum_i f_i \psi_i \left ( \mathbf{r} \right ) \psi_i^* \left ( \mathbf{r}' \right ),

where \left \{ \psi_i \left ( \mathbf{r} \right ) \right \} are the KS eigenstates and \left \{ f_i \right \} their occupancies, which at zero temperature are restricted to 0 or 1. The corresponding operator is defined such that

\hat{\rho} \left | \xi \right \rangle = \sum_i f_i \left | \psi_i \right \rangle \left \langle \psi_i | \xi \right \rangle.

Since the solver does not calculate individual KS eigenstate and eigenenergies, it is especially useful as a linear-scaling DFT method. This is made possible due to the decay of the density matrix elements far from the diagonal (exponential in the case of insulators). A review of the properties of the density matrix and linear-scaling density matrix solvers can be found here[1].


  1. S. Goedecker, Linear scaling electronic structure methods, Rev. Mod. Phys. 71, 1085 (1999). DOI: 10.1103/RevModPhys.71.1085