Computational Methodology

Adventures in (Real) Space

Some thirty years ago, Chelikowsky, Saad, and co-workers suggested solving the Kohn-Sham equations of density functional theory using a high-order finite difference approach on a real-space grid. For much of that time, our group has been a major partner in developing and applying this approach. It is now expressed in a massively parallel software suite, which we call PARSEC - the pseudopotential algorithm for real-space electronic structure calculations. This approach has many advantages. First and foremost, it produces Hamiltonian matrices that are very sparse. Therefore, the Hamiltonian is never computed or stored explicitly, but rather only its action on a wave function vector is computed. As a direct consequence, massively parallel calculations scale extremely well. Put in simpler language, when performing a parallel computation on a number of processors (from several to tens of thousands and more), for a large problem the computation time will decrease almost linearly with the increase in processor number. This, together with advances various algorithmic advances, allows us to attack problems with tens of thousands of electrons.

The approach has other advantages as well:

With respect to localized basis sets, it is an approach without an explicit basis so that convergence is trivial (just decrease the grid spacing), and all those pesky basis set issues are avoided. This also means that no recurring basis set-up and no spurious forces are associated with atom movement and that localized and delocalized electrons are treated on the same footing.
With respect to planewave approaches, we can treat periodic and non-periodic systems equally. This means that "super-cells", which introduce spurious periodicity, are not employed. Consequently, problems "inherited" from the super-cell, e.g., difficulties with treating monopolar or dipolar systems (e.g, in the study of charged defects or polar surfaces, respectively) are not encountered.
The code is well-structured and physically transparent. This makes the implementation of new ideas (relatively) easy!
Last but not at all least, the approach serves as a basis for all-electron real-space calculations. Over the years, we have developed two such codes:

DARSEC – the diatomic algorithm for real-space electronic structure calculations: This program is a “cousin” of PARSEC, which employs a prolate-spheroidal grid to allow for all-electron solutions of atoms and diatomic molecules.

CARMA – the concurrent all-electron real-space multigrid algorithm: This program uses a locally-refined multi-grid approach to obtain an all-electron solution for an arbitrary molecular system.

Recently, we have focused on improving the precision of energies and forces even at coarser grids, thereby accelerating the computation. Specifically, we showed that aliasing errors can be minimized through a scheme that we call finite difference interpolation.

Computational Methodology

Adventures in (Real) Space

The approach has other advantages as well:

Recent articles in this research direction are: