# 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

${\displaystyle \rho \left(\mathbf {r} ,\mathbf {r} '\right)=\sum _{i}f_{i}\psi _{i}\left(\mathbf {r} \right)\psi _{i}^{*}\left(\mathbf {r} '\right)}$,

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

${\displaystyle {\hat {\rho }}\left|\xi \right\rangle =\sum _{i}f_{i}\left|\psi _{i}\right\rangle \left\langle \psi _{i}|\xi \right\rangle }$.

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].

## References

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