Exploiting Sparsity in Matrix Functions, with Applications to Electronic Structure Calculations
In this talk, some general results on analytic decay bounds for the off-diagonal entries of functions of large, sparse (in particular, banded) Hermitian matrices will be presented. These bounds will be specialized to various approximations to density matrices arising in Hartree-Fock and Density Functional Theory (assuming localized basis functions are used), proving exponential decay (i.e., `nearsightedness') for gapped systems at zero electronic temperature and thus providing a rigorous justification for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. The case of density matrices for disordered systems (Anderson model) and for metallic systems at finite temperature will also be considered. Additional applications, possible extensions to the non-Hermitian case, and some open problems will conclude the talk.
This is joint work with Paola Boito and Nader Razouk.
Michele Benzi, Emory University