The computation of the density matrix in Kohn--Sham (KS) type density functional theory plays an important role in quantum chemistry. Since standard solution algorithms are based on the diagonalisation of non-local operators, the computational complexity scales cubically. Our goal is to develop fast algorithms for computing these matrices by using $\mathcal{H}$-matrix representations.

The KS equations for a system of $N$ electrons

can be solved using a self-consistent algorithm where the update of the density matrix $P$ is calculated solving an eigenvalue problem.

Another approach is the direct computation of the matrix $P$ using the sign function

To efficiently solve this $n^2$-dimensional problem $\mathcal{H}$-matrix arithmetic is used. The $\mathcal{H}$-matrix approach allows to multiply, add, and invert matrices with almost linear complexity at the cost of small errors due to truncations.

We will present $\mathcal{H}$-matrix representations for $H$ and $P$ that can be used together with algorithms for the calculation of the sign function to provide accurate and fast solutions for the KS equations. Numerical experiments will demonstrate the efficiency of the new algorithm.