Numerical stability and efficiency of response property calculations in density functional theory

The goal of quantum-chemical calculations is the simulation of materials and molecules. In density-functional theory (DFT) the first step along this line is obtaining the electron density minimising an energy functional. However, since energies and the density are usually not very tractable quantities in an experimental setup, comparison to experiment and scientific intuition also requires the computation of properties. Important properties include the forces (i.e. the energetic change due to a displacement of the structure) polarisabilities (change in dipole moment due to an external electric field) or phonon spectra (which can be measured using infrared spectroscopy). Therefore an efficient and reliable property computation is crucial to make quantum-chemical simulations interpretable and to close the loop back to experimentalists.

In DFT property calculations are done using density-functional perturbation theory (DFPT), which essentially computes the linear response of the electronic structure to the aforementioned changes in external conditions (external field, nuclear displacements etc.). Solving the equations underlying DFPT can become numerically challenging as (especially for metallic systems) the equations are ill-conditioned.

In a collaboration with my former PostDoc advisor Benjamin Stamm and my old group at the CERMICS at École des Ponts, including Eric Cancès, Antoine Levitt, Gaspard Kemlin, we just published an article, where we provide a more mathematical take on DFPT. In our work we provide an extensive review of various practical setups employed in main-stream codes such as ABINIT and QuantumEspresso from a numerical analysis point of view, highlighting the differences and similarities of these approaches. Moreover we develop a novel approach approach to solve the so-called Sternheimer equations (a key component of DFPT), which allows to make better use of the byproducts available in standard SCF schemes (the algorithm used to obtain the DFT ground state). With our approach we show savings up to 40% in the number of matrix-vector products required to solve the response equations. Since these are the most expensive step in DFPT this implies a similar saving in computational cost overall. Naturally our algorithm has been implemented as the default response solver in our DFTK code, starting from version 0.5.9.

Most of this work was done during a two-month visit of Gaspard Kemlin with Benjamin and myself here in Aachen. I think I speak for the both of us when I say that it has been a great pleasure to have Gaspard around, both on a professional as well as a personal level.

The full abstract of the paper reads

Response calculations in density functional theory aim at computing the change in ground-state density induced by an external perturbation. At finite temperature these are usually performed by computing variations of orbitals, which involve the iterative solution of potentially badly-conditioned linear systems, the Sternheimer equations. Since many sets of variations of orbitals yield the same variation of density matrix this involves a choice of gauge. Taking a numerical analysis point of view we present the various gauge choices proposed in the literature in a common framework and study their stability. Beyond existing methods we propose a new approach, based on a Schur complement using extra orbitals from the self-consistent-field calculations, to improve the stability and efficiency of the iterative solution of Sternheimer equations. We show the success of this strategy on nontrivial examples of practical interest, such as Heusler transition metal alloy compounds, where savings of around 40% in the number of required cost-determining Hamiltonian applications have been achieved.

A robust and efficient line search for self-consistent field iterations

In an ongoing effort with Antoine Levitt our aim is to develop reliable density-functional theory (DFT) methods for computational materials design. Recently we looked into a strategy to automatically select the damping parameter for the self-consistent field iterations (SCF). Our adaptive damping approach is based on a theoretically sound quadratic …
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Posted on Fr 15 Oktober 2021 in Publications.
Tags: electronic structure theory theoretical chemistry DFTK Julia DFT numerical analysis and Kohn-Sham
SIAM LA: Robust and efficient accelerated methods for density-functional theory

Just one day after my talk at the SIAM Materials Science conference (blog article) I gave another talk at a SIAM meeting, this time at SIAM Linear Algebra. I was very much looking forward to participate in SIAM LA, firstly because it was the first time I attended this conference …
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Posted on So 30 Mai 2021 in Research.
Tags: talk electronic structure theory Julia DFTK numerical analysis Kohn-Sham high-throughput DFT and solid state
SIAM MS: Using the density-functional toolkit to design black-box DFT methods

After being moved by one year due to the pandemic, the last two weeks (from 17th to 28th May) the SIAM materials science conference finally took place in virtual form. Unfortunately this meant that the conference was scheduled in parallel to the SIAM Linear Algebra virtual conference, where I also …
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Posted on So 30 Mai 2021 in Research.
Tags: talk electronic structure theory Julia DFTK theoretical chemistry numerical analysis Kohn-Sham high-throughput DFT and solid state
A novel black-box preconditioning strategy for high-throughput density-functional theory

A couple of weeks ago, from 15th to 19th March, I participated in the virtual annual meeting of the German Association of Applied Mathematics and Mechanics (GAMM). For me this meeting was the first time I presented my work to an audience of applied mathematicians with a broad background and …
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Posted on Fr 09 April 2021 in Research.
Tags: talk electronic structure theory Julia DFTK theoretical chemistry numerical analysis Kohn-Sham high-throughput DFT and solid state
CESMIX TST meeting: DFTK.jl: A multidisciplinary Julia code for density-functional theory development

These past two days I have participated in the Tri-Lab Support Team (TST) meeting of the CESMIX, the newly founded Center for the Exascale Simulation of Material Interfaces in Extreme Environments at the Massachusetts Institute of Technology. Within the next few years the idea of the CESMIX is to develop …
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Posted on Do 04 Februar 2021 in Research.
Tags: talk electronic structure theory Julia DFTK theoretical chemistry numerical analysis Kohn-Sham high-throughput invited talk DFT and solid state
High-throughput density-functional theory calculations: An interdisciplinary challenge

Last Thursday I was invited to give a virtual talk at the Scientific Computing Seminar of working group of Prof. Nicolas Gauger at TU Kaiserslautern. Since the research in Prof. Gauger's group mostly concerns topics which are not directly related to electronic structure theory and density-functional theory (DFT), I chose …
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Posted on Mi 23 Dezember 2020 in Research.
Tags: talk electronic structure theory Julia DFTK theoretical chemistry numerical analysis Kohn-Sham high-throughput invited talk DFT and solid state
Challenges and prospects of a posteriori error estimation in density-functional theory

Last week Wednesday I was invited to give a talk at the group seminar of the AG Christoph Jacob at TU Braunschweig, Germany. Christoph was especially interested in our recent publication on a posteriori error estimation in Kohn-Sham problems and so I decided to use the opportunity to give a …
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Posted on Mi 21 Oktober 2020 in Research.
Tags: talk electronic structure theory Julia DFTK theoretical chemistry error estimates numerical analysis Kohn-Sham high-throughput and invited talk
Faraday Discussions: New horizons in density functional theory

Following the submission of our paper on a posteriori error estimation in the Kohn-Sham equations a few months ago I was recently invited to present our work at the Faraday Discussions on New horizons in density functional theory. Being amongst speakers such as Kieron Burke, Andreas Savin or Weitao Yang …
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Posted on Di 15 September 2020 in Research.
Tags: talk electronic structure theory Julia DFTK theoretical chemistry error estimates numerical analysis Kohn-Sham and high-throughput
Black-box inhomogeneous preconditioning for self-consistent field iterations in density-functional theory

For the past half a year or so Antoine Levitt and myself have been looking at a particular tricky busyness for solid-state density-functional theory (DFT) calculations, namely how to design efficient self-consistent field (SCF) schemes for large inhomogeneous systems. I have already previously reported on this matter in a short …
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Posted on Fr 04 September 2020 in Publications.
Tags: electronic structure theory theoretical chemistry DFTK Julia DFT numerical analysis and Kohn-Sham
A posteriori error estimation for the non-self-consistent Kohn-Sham equations

After about a year of work on the density-functional toolkit (DFTK) we finally started using the code for some new science. Given the mathematics background of the CERMICS the first two DFTK-related articles were not so much about large-scale applications, but rather deal some fundamental questions of Kohn-Sham density-functional theory …
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Posted on Di 28 April 2020 in Publications.
Tags: electronic structure theory theoretical chemistry DFTK Julia DFT numerical analysis Kohn-Sham and error estimates