Efficient Density Matrix Methods for Large Scale Electronic Structure Calculations

Efficient and accurate methods for computing the density matrix are necessary to be able to perform large scale electronic structure calculations. For sufficiently sparse matrices, the computational cost of recursive polynomial expansions to construct the density matrix scales linearly with increasing system size. In this work, parameterless stopping criteria for recursive polynomial expansions are developed. The proposed stopping criteria automatically adapt to a change in the requested accuracy, perform at almost no additional cost and do not require any user-defined tolerances. Compared to the traditional diagonalization approach, in linear scaling methods molecular orbitals are not readily available. In this work, the interior eigenvalue problem for the Fock/Kohn-Sham matrix is coupled to the recursive polynomial expansions. The idea is to view the polynomial, obtained in the recursive expansion, as an eigenvalue filter, giving large separation between eigenvalues of interest. An efficient method for computation of homo and lumo eigenvectors is developed. Moreover, a method for computation of multiple eigenvectors around the homo-lumo gap is implemented and evaluated. An original method for inverse factorization of Hermitian positive definite matrices is developed in this work. Novel theoretical tools for analysis of the decay properties of matrix element magnitude in electronic structure calculations are proposed. Of particular interest is an inverse factor of the basis set overlap matrix required for the density matrix construction. It is shown that the proposed inverse factorization algorithm drastically reduces the communication cost compared to state-of-the-art methods. To perform large scale numerical tests, most of the proposed methods are implemented in the quantum chemistry program Ergo, also presented in this thesis. The recursive polynomial expansion in Ergo is parallelized using the Chunks and Tasks matrix library. It is shown that the communication cost per process of the recursive polynomial expansion implementation tends to a constant in a weak scaling setting