Hightlight Paper:
Rotations of occupied invariant subspaces in self–consistent field calculations
Preliminaries
The last twenty years, electronic structure calculations using
Kohn–Sham Density Functional Theory have become very
popular. For insulators, it is possible to perform electronic
structure calculations with a computational effort that increases only
linearly with system size. Linear scaling is achieved by the use of
computational approximations. Examples include Cauchy–Schwarz
screening of integrals, multipole approximations, approximate
numerical integration, and removal of small matrix elements to keep
matrices sparse. These approximations are usually governed by one or
several threshold values. Although tightening the threshold values
typically gives more accurate results at a higher computational cost,
a direct relation between threshold value and accuracy in the electron
density is usually not known. Therefore, linear scaling Density
Functional calculations are plagued by suboptimal ad hoc
selected threshold values.
Mathematical Framework
In this paper, a mathematical framework to directly relate
computational approximations to the accuracy in the electron density
is proposed. The self–consistent field procedure as used in
Hartree–Fock and Kohn–Sham calculations is viewed as a
sequence of rotations of the so–called occupied invariant
subspace of the potential and density matrices. Computational
approximations are characterized as erroneous rotations of this
subspace. Differences between subspaces are measured and controlled by
the canonical angles between them.
No more ad hoc threshold values
With this approach, a first step is taken toward a method where errors
from computational approximations are rigorously controlled and
threshold values are directly related to the accuracy of the current
trial density, thus eliminating the use of ad hoc selected
threshold values. Then, the use of computational resources can be kept
down as much as possible without impairment of the self–consistent
field convergence.
Figures:
In the calculations presented in this figure, perturbations of
the density matrix and the Fock matrix are controlled by the use of
the mathematical framework of this paper. The rotation of the occupied
invariant subspace of these two matrices due to the perturbation, as
measured by the canonical angles, is plotted. The line at 10-4
indicates the requested accuracy. The results are for a hydride
molecule test set and a water droplet test set.
Publication reference >>
Emanuel H. Rubensson, Elias Rudberg, and Paweł Sałek
Journal of Mathematical Physics 49, 032103 (2008)
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