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Scalable Localizable Density Functional Theory
K.S. Kang, J.W. Davenport, D. Volja, J. Zheng, D. Keyes,
and J. Glimm
Nanoscale science and technology and biology are driving a search for
new ways to calculate the electronic properties of clusters containing
many thousands of atoms. Density Functional Theory (DFT) has been shown
to yield accurate total energies, charge, and spin densities in
molecules and crystalline solids but has been limited in the size system
that can be treated.
The density functional equations consist of coupled Schrodinger and
Poisson equations, which must be solved self consistently. Usually one
chooses a basis set (for example Gaussian type orbitals or plane waves)
thereby converting the Schrodinger equation into an algebraic eigenvalue
problem. The number of such basis functions will scale linearly with N,
the number of atoms in the system. Solving the eigenvalue problem, which
is formally dense, will then scale as N3. Hence for large
systems, there is a premium on efficiency, or reducing the number of
basis functions required per atom. Gaussian type orbitals (GTOs) have
the advantage of being highly localized in space, decaying like exp(-ar2).
However, they are not maximally efficient, in the sense that many
Gaussians are required to accurately represent the wave functions. Plane
waves have the disadvantage that they are not localized at all, but
rather extend over the whole system. In addition their use generally
requires a “supercell” in which the system is repeated periodically in
space, leading to inaccurate treatment of surface and edge effects in
some cases.
We solve these problems by using a mixed numerical and localized
analytical basis set. We partition the space into nonoverlapping spheres
surrounding each atom and an extra-atomic region outside the spheres.
Inside the spheres the basis consists of numerical solutions of the
Schrodinger (or Dirac) equation for the spherical part of the potential.
Outside the spheres, the basis is given by Slater type orbitals, which
have the form
where the Ylm are spherical harmonics. These functions are not overly localized
as GTOs and achieve the same accuracy more efficiently, typically 5x
fewer basis elements per atom.
At each sphere boundary these functions are matched onto the numerical
solutions inside the sphere. Such a scheme has already been implemented
for periodic systems and is known as LASTO, the Linear Augmented Slater
Type Orbital method [1]. It is a local orbital version of the Linear
Augmented Plane Wave (LAPW) method, considered the most accurate method
for solving the DFT equations.
The Poisson equation is solved on a numerical grid using sparse matrix
techniques and the hypre library [2]. Hypre is a set of highly parallel
preconditioners and solvers suitable for large sparse systems. The
eigenvalue problem is solved using ScaLAPAK [3], a set of codes for the
solution of large eigenproblems.
Figure 1 shows the density of states
in a 13-atom cobalt-nickel cluster, CoNi12. The bond lengths were chosen
to be the same as in bulk nickel. The discrete eigenvalues were
broadened with a Gaussian.
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Figure 1. Density of states versus
energy for CoNi12. |
For the largest systems we use an O(N) technique known as “divide and
conquer” [5]. In this method, a large cluster is partitioned into
subsystems, and the charge density is calculated for each. The
computational effort scales like SNs3, where S is the number of
subsystems and Ns is the number of atoms in the subsystem. This scheme
is possible because the charge density (more generally, the density
matrix) is localized in space even when the eigenfunctions are not [6].
Using divide and conquer, along with the localized basis set provided by
STOs, we expect to be able to calculate the charge and spin density of
clusters containing up to 5000 atoms.
References
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[1] Fernando, G.W., Davenport, J.W., Watson, R.E., and Weinert, M.
Full-potential linear augmented-Slater type orbital method. Phys. Rev. B
40: 2757 (1989).
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[2] hypre, high performance preconditioners,
http://www.llnl.gov/CASC/linear_solvers/
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[3] ScaLAPAK, http://netlib2.cs.utk.edu/scalapack/
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[4] PETSc, Portable, Extensible Toolkit for Scientific Computing.
http://www.mcs.anl.gov/petsc/
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[5] Yang, W. Direct calculation of electron density in
density-functional theory. Phys. Rev. Lett. 6: 1438 (1991).
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[6] Kohn, W. Density functional and density matrix method scaling
linearly with the number of atoms. Phys. Rev. Lett. 76: 3168 (1996).
Last Modified: January 31, 2008
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Claire Lamberti
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