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SCDM-k: Localized orbitals for solids via selected columns of the density matrix
The recently developed selected columns of the density matrix (SCDM) method
[J. Chem. Theory Comput. 11, 1463, 2015] is a simple, robust, efficient and
highly parallelizable method for constructing localized orbitals from a set of
delocalized Kohn-Sham orbitals for insulators and semiconductors with
point sampling of the Brillouin zone. In this work we generalize the SCDM
method to Kohn-Sham density functional theory calculations with k-point
sampling of the Brillouin zone, which is needed for more general electronic
structure calculations for solids. We demonstrate that our new method, called
SCDM-k, is by construction gauge independent and is a natural way to describe
localized orbitals. SCDM-k computes localized orbitals without the use of an
optimization procedure, and thus does not suffer from the possibility of being
trapped in a local minimum. Furthermore, the computational complexity of using
SCDM-k to construct orthogonal and localized orbitals scales as O(N log N )
where N is the total number of k-points in the Brillouin zone. SCDM-k is
therefore efficient even when a large number of k-points are used for Brillouin
zone sampling. We demonstrate the numerical performance of SCDM-k using systems
with model potentials in two and three dimensions.Comment: 25 pages, 7 figures; added more background sections, clarified
presentation of the algorithm, revised the presentation of previous work,
added a more high level overview of the new algorithm, and mildly clarified
the presentation of the results (there were no changes to the numerical
results themselves
Compressed representation of Kohn-Sham orbitals via selected columns of the density matrix
Given a set of Kohn-Sham orbitals from an insulating system, we present a
simple, robust, efficient and highly parallelizable method to construct a set
of, optionally orthogonal, localized basis functions for the associated
subspace. Our method explicitly uses the fact that density matrices associated
with insulating systems decay exponentially along the off-diagonal direction in
the real space representation. Our method avoids the usage of an optimization
procedure, and the localized basis functions are constructed directly from a
set of selected columns of the density matrix (SCDM). Consequently, the only
adjustable parameter in our method is the truncation threshold of the localized
basis functions. Our method can be used in any electronic structure software
package with an arbitrary basis set. We demonstrate the numerical accuracy and
parallel scalability of the SCDM procedure using orbitals generated by the
Quantum ESPRESSO software package. We also demonstrate a procedure for
combining SCDM with Hockney's algorithm to efficiently perform Hartree-Fock
exchange energy calculations with near linear scaling.Comment: 7 pages, 4 figures; short example code for computing the SCDM;
parallel scaling results; slightly restructured introduction and
clarification of the input needed to compute the SCD
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