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20th GAMM-Seminar Leipzig on
Numerical Methods for Non-Local Operators

Max-Planck-Institute for Mathematics in the Sciences
Inselstr. 22-26, D-04103 [O->]Leipzig
Phone: +49.341.9959.752, Fax: +49.341.9959.999


     
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  20th GAMM-Seminar
January, 22th-24th, 2004
 
     
  Winterschool on hierarchical matrices  
     
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  Abstract Thomas Kastl, Sat, 10.00-10.30 Previous Contents Next  
  Hierarchical matrices in Density Functional Theory
Thomas Kastl (University Zürich)

The computation of the density matrix in Kohn-Sham (KS) type density functional theory plays an important role in quantum chemistry. Since standard solution algorithms are based on the diagonalisation of non-local operators, the computational complexity scales cubically. Our goal is to develop fast algorithms for computing these matrices by using H-matrix representations.

The KS equations for a system of N electrons

H(P) P - P H(P) = 0    Tr(P) = N    P2 = P
can be solved using a self-consistent algorithm where the update of the density matrix P is calculated solving an eigenvalue problem.

Another approach is the direct computation of the matrix P using the sign function

P = 0.5 ( I - sign H' ) ,
where H' = (H - μ I) is the KS matrix H shifted by the chemical potential μ.

To efficiently solve this n2-dimensional problem H-matrix arithmetic is used. The H-matrix approach allows to multiply, add, and invert matrices with almost linear complexity at the cost of small errors due to truncations.

We will present H-matrix representations for H and P that can be used together with algorithms for the calculation of the sign function to provide accurate and fast solutions for the KS equations. Numerical experiments will demonstrate the efficiency of the new algorithm.
 

 
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Last updated:
30.11.2004 Impressum
 
Concept, Design and Realisation
[O->]Jens Burmeister (Uni Kiel), Kai Helms (MPI Leipzig)
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