Partner und Internationale Organisationen
(Englisch)
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AT, BE, BG, HR, FI, MK, FR, DE, EL, HU, IL, IT, LT, NL, NO, PL, SK, ES, SE, CH, UK
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Abstract
(Englisch)
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Electronic structure calculations based on single particle Hamiltonians, such as the Hartree-Fock and Kohn-Sham density functional theory play a central role in computational quantum chemistry and physics and have proven their usefulness for a wide range of systems. However, the computational demands of these methods prohibit their application to systems containing more than a few hundred atoms. Since many systems of practical interest, e.g. biomolecules and nanostructures, exceed by far this regime, it is crucial to develop new algorithms which reduce the computational cost. Algorithms that reduce the scaling dependence on the number of electrons/size of basis set included in the calculation have been developed. These algorithms are based on the assumption that the one particle density matrix vanishes outside some interaction region. Numerical experiments show that this assumption is not fulfilled in accurate calculation on many systems. The use of hierarchical H-matrices for the density operators allow for linear scaling algorithms based on the weaker condition of uniform decay of the density matrix with distance. The development of H-matrices for the sparse representation of non-local operators is an important field of numerical analysis.The notion hierarchical indicates that a tree-like block partitioning of the matrix is used, where each block is represented by a low-rank matrix. H-matrix representation have been developed for the approximation of the inverses of sparse finite element matrices, for the representation of the matrix exponential function and of boundary integral operators. Further applications are in the fields of control problems and process simulation. The goal of this project was to integrate this field of mathematics with computational quantum chemistry for the development of efficient linear scaling methods.
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