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
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.