Sparse Matrices in Self–Consistent Field Methods

This thesis is part of an effort to enable large–scale Hartree–Fock/Kohn–Sham (HF/KS) calculations. The objective is to model molecules and materials containing thousands of atoms at the quantum mechanical level. HF/KS calculations are usually performed with the Self-Consistent Field (SCF) method. This method involves two computationally intensive steps. These steps are the construction of the Fock/Kohn–Sham potential matrix from a given electron density and the subsequent update of the electron density usually represented by the so-called density matrix. In this thesis the focus lies on the representation of potentials and electron density and on the density matrix construction step in the SCF method. Traditionally a diagonalization has been used for the construction of the density matrix. This diagonalization method is, however, not appropriate for large systems since the time complexity for this operation is O(n). Three types of alternative methods are described in this thesis; energy minimization, Chebyshev expansion, and density matrix purification. The efficiency of these methods relies on fast matrix–matrix multiplication. Since the occurring matrices become sparse when the separation between atoms exceeds some value, the matrix–matrix multiplication can be performed with complexity O(n). A hierarchic sparse matrix data structure is proposed for the storage and manipulation of matrices. This data structure allows for easy development and implementation of algebraic matrix operations, particularly needed for the density matrix construction, but also for other parts of the SCF calculation. The thesis addresses also truncation of small elements to enforce sparsity, permutation and blocking of matrices, and furthermore calculation of the HOMO–LUMO gap and a few surrounding eigenpairs when density matrix purification is used instead of the traditional diagonalization method.