PROJECTS

Operator calculus of density matrices and sparse wavelet representations

PRINCIPAL INVESTIGATOR

TOGETHER WITH

ABSTRACT

The multi-scale character of the electronic structure of molecules and solids expresses itself in the energy- and length scales to be encountered, which extend over several orders of magnitude. Wavelet based multi-resolution analysis, a central topic of applied mathematics, proved to be a useful tool for applications in quantum chemistry. We suggest an alternative approach to linear scaling methods for Hartree-Fock and Kohn-Sham equations based on wavelet representations of density matrices. Such kind of approach might become competitive to conventional quantum chemistry methods, using atomic centered Gaussian-type basis sets, for systems where high numerical accuracy is required. Density matrices and other relevant operators in quantum chemistry belong to a general class of operators, where the sparsity with respect to a wavelet representation has been proved, nameley pseudo-differential and Calderón-Zygmund operators. Their operator algebra enables to control sparsity not only for individual operators but also for operator products and more general of iterative processes, which provide self-consistent solutions for these equations. So far these operators have been investigated on a pureley mathematical background. It is the purpose of the present project to exploit the potential of recent work in applied mathematics for realistic applications in quantum chemistry.