# Density matrix solver

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

,

where are the KS eigenstates and their occupancies, which at zero temperature are restricted to 0 or 1. The corresponding operator is defined such that

.

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]}.

## References

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