# 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.