# Fixed-Point Optimization of Atoms and Density in DFT

dataset

posted on 11.06.2013, 00:00 by L. D. MarksI describe
an algorithm for simultaneous fixed-point optimization
(mixing) of the density and atomic positions in Density Functional
Theory calculations which is approximately twice as fast as conventional
methods, is robust, and requires minimal to no user intervention or
input. The underlying numerical algorithm differs from ones previously
proposed in a number of aspects and is an autoadaptive hybrid of standard
Broyden methods. To understand how the algorithm works in terms of
the underlying quantum mechanics, the concept of algorithmic greed
for different Broyden methods is introduced, leading to the conclusion
that if a linear model holds that the first Broyden method is optimal,
the second if a linear model is a poor approximation. How this relates
to the algorithm is discussed in terms of electronic phase transitions
during a self-consistent run which results in discontinuous changes
in the Jacobian. This leads to the need for a nongreedy algorithm
when the charge density crosses phase boundaries, as well as a greedy
algorithm within a given phase. An ansatz for selecting the algorithm
structure is introduced based upon requiring the extrapolated component
of the curvature condition to have projected positive eigenvalues.
The general convergence of the fixed-point methods is briefly discussed
in terms of the dielectric response and elastic waves using known
results for quasi-Newton methods. The analysis indicates that both
should show sublinear dependence with system size, depending more
upon the number of different chemical environments than upon the number
of atoms, consistent with the performance of the algorithm and prior
literature. This is followed by details of algorithm ranging from
preconditioning to trust region control. A number of results are shown,
finishing up with a discussion of some of the many open questions.