1,215 research outputs found
On the Analysis of the Discretized Kohn-Sham Density Functional Theory
In this paper, we study a few theoretical issues in the discretized Kohn-Sham
(KS) density functional theory (DFT). The equivalence between either a local or
global minimizer of the KS total energy minimization problem and the solution
to the KS equation is established under certain assumptions. The nonzero charge
densities of a strong local minimizer are shown to be bounded below by a
positive constant uniformly. We analyze the self-consistent field (SCF)
iteration by formulating the KS equation as a fixed point map with respect to
the potential. The Jacobian of these fixed point maps is derived explicitly.
Both global and local convergence of the simple mixing scheme can be
established if the gap between the occupied states and unoccupied states is
sufficiently large. This assumption can be relaxed if the charge density is
computed using the Fermi-Dirac distribution and it is not required if there is
no exchange correlation functional in the total energy functional. Although our
assumption on the gap is very stringent and is almost never satisfied in
reality, our analysis is still valuable for a better understanding of the KS
minimization problem, the KS equation and the SCF iteration.Comment: 29 page
Scalable computational approach to extract chemical bonding from real-space density functional theory calculations using finite-element basis: A projected orbital and Hamiltonian population analysis
We present an efficient and scalable computational approach for conducting
projected population analysis from real-space finite-element (FE) based
Kohn-Sham density functional theory calculations (DFT-FE). This work provides
an important direction towards extracting chemical bonding information from
large-scale DFT calculations on materials systems involving thousands of atoms
while accommodating periodic, semi-periodic or fully non-periodic boundary
conditions. Towards this, we derive the relevant mathematical expressions and
develop efficient numerical implementation procedures that are scalable on
multi-node CPU architectures to compute the projected overlap and Hamilton
populations. This is accomplished by projecting either the self-consistently
converged FE discretized Kohn-Sham eigenstates, or the FE discretized
Hamiltonian onto a subspace spanned by localized atom-centered basis set. The
proposed method is implemented in a unified framework within DFT-FE where the
ground-state DFT calculations and the population analysis are performed on the
same finite-element grid. We further benchmark the accuracy and performance of
this approach on representative material systems involving periodic and
non-periodic DFT calculations with LOBSTER, a widely used projected population
analysis code. Finally, we discuss a case study demonstrating the advantages of
our scalable approach to extract the chemical bonding information from
increasingly large silicon nanoparticles up to a few thousand atoms.Comment: 9 Figures, 5 Tables, 53 pages with references and supplementary
informatio
On the Convergence of the Self-Consistent Field Iteration in Kohn-Sham Density Functional Theory
It is well known that the self-consistent field (SCF) iteration for solving
the Kohn-Sham (KS) equation often fails to converge, yet there is no clear
explanation. In this paper, we investigate the SCF iteration from the
perspective of minimizing the corresponding KS total energy functional. By
analyzing the second-order Taylor expansion of the KS total energy functional
and estimating the relationship between the Hamiltonian and the part of the
Hessian which is not used in the SCF iteration, we are able to prove global
convergence from an arbitrary initial point and local linear convergence from
an initial point sufficiently close to the solution of the KS equation under
assumptions that the gap between the occupied states and unoccupied states is
sufficiently large and the second-order derivatives of the exchange correlation
functional are uniformly bounded from above. Although these conditions are very
stringent and are almost never satisfied in reality, our analysis is
interesting in the sense that it provides a qualitative prediction of the
behavior of the SCF iteration
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
Two-level Chebyshev filter based complementary subspace method: pushing the envelope of large-scale electronic structure calculations
We describe a novel iterative strategy for Kohn-Sham density functional
theory calculations aimed at large systems (> 1000 electrons), applicable to
metals and insulators alike. In lieu of explicit diagonalization of the
Kohn-Sham Hamiltonian on every self-consistent field (SCF) iteration, we employ
a two-level Chebyshev polynomial filter based complementary subspace strategy
to: 1) compute a set of vectors that span the occupied subspace of the
Hamiltonian; 2) reduce subspace diagonalization to just partially occupied
states; and 3) obtain those states in an efficient, scalable manner via an
inner Chebyshev-filter iteration. By reducing the necessary computation to just
partially occupied states, and obtaining these through an inner Chebyshev
iteration, our approach reduces the cost of large metallic calculations
significantly, while eliminating subspace diagonalization for insulating
systems altogether. We describe the implementation of the method within the
framework of the Discontinuous Galerkin (DG) electronic structure method and
show that this results in a computational scheme that can effectively tackle
bulk and nano systems containing tens of thousands of electrons, with chemical
accuracy, within a few minutes or less of wall clock time per SCF iteration on
large-scale computing platforms. We anticipate that our method will be
instrumental in pushing the envelope of large-scale ab initio molecular
dynamics. As a demonstration of this, we simulate a bulk silicon system
containing 8,000 atoms at finite temperature, and obtain an average SCF step
wall time of 51 seconds on 34,560 processors; thus allowing us to carry out 1.0
ps of ab initio molecular dynamics in approximately 28 hours (of wall time).Comment: Resubmitted version (version 2
Accelerating Atomic Orbital-based Electronic Structure Calculation via Pole Expansion and Selected Inversion
We describe how to apply the recently developed pole expansion and selected
inversion (PEXSI) technique to Kohn-Sham density function theory (DFT)
electronic structure calculations that are based on atomic orbital
discretization. We give analytic expressions for evaluating the charge density,
the total energy, the Helmholtz free energy and the atomic forces (including
both the Hellman-Feynman force and the Pulay force) without using the
eigenvalues and eigenvectors of the Kohn-Sham Hamiltonian. We also show how to
update the chemical potential without using Kohn-Sham eigenvalues. The
advantage of using PEXSI is that it has a much lower computational complexity
than that associated with the matrix diagonalization procedure. We demonstrate
the performance gain by comparing the timing of PEXSI with that of
diagonalization on insulating and metallic nanotubes. For these quasi-1D
systems, the complexity of PEXSI is linear with respect to the number of atoms.
This linear scaling can be observed in our computational experiments when the
number of atoms in a nanotube is larger than a few hundreds. Both the wall
clock time and the memory requirement of PEXSI is modest. This makes it even
possible to perform Kohn-Sham DFT calculations for 10,000-atom nanotubes with a
sequential implementation of the selected inversion algorithm. We also perform
an accurate geometry optimization calculation on a truncated (8,0)
boron-nitride nanotube system containing 1024 atoms. Numerical results indicate
that the use of PEXSI does not lead to loss of accuracy required in a practical
DFT calculation
Gradient type optimization methods for electronic structure calculations
The density functional theory (DFT) in electronic structure calculations can
be formulated as either a nonlinear eigenvalue or direct minimization problem.
The most widely used approach for solving the former is the so-called
self-consistent field (SCF) iteration. A common observation is that the
convergence of SCF is not clear theoretically while approaches with convergence
guarantee for solving the latter are often not competitive to SCF numerically.
In this paper, we study gradient type methods for solving the direct
minimization problem by constructing new iterations along the gradient on the
Stiefel manifold. Global convergence (i.e., convergence to a stationary point
from any initial solution) as well as local convergence rate follows from the
standard theory for optimization on manifold directly. A major computational
advantage is that the computation of linear eigenvalue problems is no longer
needed. The main costs of our approaches arise from the assembling of the total
energy functional and its gradient and the projection onto the manifold. These
tasks are cheaper than eigenvalue computation and they are often more suitable
for parallelization as long as the evaluation of the total energy functional
and its gradient is efficient. Numerical results show that they can outperform
SCF consistently on many practically large systems.Comment: 24 pages, 11 figures, 59 references, and 1 acknowledgement
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