liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Theoretical prediction of properties of atomistic systems: Density functional theory and machine learning
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0001-5439-711X
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The prediction of ground state properties of atomistic systems is of vital importance in technological advances as well as in the physical sciences. Fundamentally, these predictions are based on a quantum-mechanical description of many-electron systems. One of the hitherto most prominent theories for the treatment of such systems is density functional theory (DFT). The main reason for its success is due to its balance of acceptable accuracy with computational efficiency. By now, DFT is applied routinely to compute the properties of atomic, molecular, and solid state systems.

The general approach to solve the DFT equations is to use a density-functional approximation (DFA). In Kohn-Sham (KS) DFT, DFAs are applied to the unknown exchangecorrelation (xc) energy. In orbital-free DFT on the other hand, where the total energy is minimized directly with respect to the electron density, a DFA applied to the noninteracting kinetic energy is also required. Unfortunately, central DFAs in DFT fail to qualitatively capture many important aspects of electronic systems. Two prime examples are the description of localized electrons, and the description of systems where electronic edges are present.

In this thesis, I use a model system approach to construct a DFA for the electron localization function (ELF). The very same approach is also taken to study the non-interacting kinetic energy density (KED) in the slowly varying limit of inhomogeneous electron densities, where the effect of electronic edges are effectively included. Apart from the work on model systems, extensions of an exchange energy functional with an improved KS orbital description are presented: a scheme for improving its description of energetics of solids, and a comparison of its description of an essential exact exchange feature known as the derivative discontinuity with numerical data for exact exchange.

An emerging alternative route towards the prediction of the properties of atomistic systems is machine learning (ML). I present a number of ML methods for the prediction of solid formation energies, with an accuracy that is on par with KS DFT calculations, and with orders-of-magnitude lower computational cost.

Abstract [sv]

Att kunna förutsäga egenskaper hos atomistiska system utgör en viktigdel av vår teknologiska utveckling, samt spelar en betydande roll i defysikaliska vetenskaperna. Sådana förutsägelser bygger på en kvantmekaniskbeskrivning av mångelektronsystem. En av de mest framståendeteorierna för att behandla den här typen av system är täthetsfunktionalteorin(DFT). Den främsta orsaken till dess framgång är attden lyckas kombinera skaplig noggrannhet med en bra beräkningseffektivitet.DFT används numera rutinmässigt för att beräkna storheterhos atomer, molekyler, och fasta kroppar.

Generellt sett löses ekvationerna inom DFT genom att man inför entäthetsfunktionalapproximation (DFA). I Kohn-Sham (KS) DFT, användsDFAer för att approximera utbytes-korrelationsenergin. Inom orbitalfriDFT, där målet är att direkt minimera den totala energin med avseendepå elektrontätheten, så approximerar man också den icke-interageranderörelseenergin hos elektronerna. Dessvärre så fallerar många centralaDFAer att kvalitativt beskriva många viktiga aspekter hos elektronsystem.Två viktiga exempel är beskrivningen av lokaliserade elektroner,samt beskrivningen av system där det förekommer elektronytor.

I denna avhandling använder jag modellsystem för att konstruera enDFAför elektronlokaliseringsfunktionen (ELF). Samma tillvägagångssättappliceras sedan för att studera den kinetiska energitätheten i gränsen avlångsamt varierande elektrontätheter, där effekten av elektronytor effektivtinkluderas. Förutom arbetet som berör modellsystem, så presenterasen utökad variant av en utbytes-energifunktional med en förbättrad KSorbitalbeskrivning: ett schema för att förbättra dess energiegenskaperför solida material, samt en jämförelse av dess beskrivning av en viktigegenskap hos den exakta utbytesenergin, vilket utgörs av diskontinuiteteri dess derivata.

Ett mera nyligen uppkommet samt alternativt sätt att kunna förutsägaegenskaper hos atomistiska system utgörs av maskinlärning (ML).Jag presenterar ett antal ML-modeller för att kunna förutsäga formeringsenergierhos fasta material med en noggrannhet som är i linje medresultat som uppnås av beräkningar med hjälp av KS DFT, och med enberäkningseffektivitet som är flera storleksordningar snabbare.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. , p. 68
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1868
National Category
Theoretical Chemistry Other Physics Topics Condensed Matter Physics Control Engineering Other Engineering and Technologies not elsewhere specified
Identifiers
URN: urn:nbn:se:liu:diva-139767DOI: 10.3384/diss.diva-139767ISBN: 978-91-7685-486-0 (print)OAI: oai:DiVA.org:liu-139767DiVA, id: diva2:1131686
Public defence
2017-09-08, Planck, Fysikhuset, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2017-08-15 Created: 2017-08-15 Last updated: 2017-08-18Bibliographically approved
List of papers
1. Quantum oscillations in the kinetic energy density: Gradient corrections from the Airy gas
Open this publication in new window or tab >>Quantum oscillations in the kinetic energy density: Gradient corrections from the Airy gas
2014 (English)In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 90, no 7, p. 075139-Article in journal (Refereed) Published
Abstract [en]

We derive a closed-form expression for the quantum corrections to the kinetic energy density in the Thomas-Fermi limit of a linear potential model system in three dimensions (the Airy gas). The universality of the expression is tested numerically in a number of three-dimensional model systems: (i) jellium surfaces, (ii) confinement in a hydrogenlike potential (the Bohr atom), (iii) particles confined by a harmonic potential in one and (iv) all three dimensions, and (v) a system with a cosine potential (the Mathieu gas). Our results confirm that the usual gradient expansion of extended Thomas-Fermi theory does not describe the quantum oscillations for systems that incorporate surface regions where the electron density drops off to zero. We find that the correction derived from the Airy gas is universally applicable to relevant spatial regions of systems of types (i), (ii), and (iv), but somewhat surprisingly not (iii). We discuss possible implications of our findings to the development of functionals for the kinetic energy density.

Place, publisher, year, edition, pages
American Physical Society, 2014
National Category
Condensed Matter Physics
Identifiers
urn:nbn:se:liu:diva-110971 (URN)10.1103/PhysRevB.90.075139 (DOI)000341238200002 ()
Note

Funding Agencies|Swedish Research Council (VR) [621-2011-4249]; Linnaeus Environment at Linkoping on Nanoscale Functional Materials (LiLi-NFM) - VR; U.S. Department of Energys National Nuclear Security Administration [DE-AC04-94AL85000]

Available from: 2014-10-02 Created: 2014-10-01 Last updated: 2017-12-05
2. Energetics of the AK13 semilocal Kohn-Sham exchange energy functional
Open this publication in new window or tab >>Energetics of the AK13 semilocal Kohn-Sham exchange energy functional
2016 (English)In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 94, no 15, article id 155143Article in journal (Refereed) Published
Abstract [en]

The recent nonempirical semilocal exchange functional of Armiento and Kummel [Phys. Rev. Lett. 111, 036402 (2013)], AK13, incorporates a number of features reproduced by higher-order theory. The AK13 potential behaves analogously with the discontinuous jump associated with the derivative discontinuity at integer particle numbers. Recent works have established that AK13 gives a qualitatively improved orbital description compared to other semilocal methods, and reproduces a band structure closer to higher-order theory. However, its energies and energetics are inaccurate. The present work further investigates the deficiency in energetics. In addition to AK13 results, we find that applying the local-density approximation (LDA) non-self-consistently on the converged AK13 density gives very reasonable energetics with equilibrium lattice constants and bulk moduli well described across 13 systems. We also confirm that the attractive orbital features of AK13 are retained even after full structural relaxation. Hence, the deficient energetics cannot be a result of the AK13 orbitals having adversely affected the quality of the electron density compared to that of usual semilocal functionals; an improved orbital description and good energetics are not in opposition. This is also confirmed by direct calculation of the principal component of the electric field gradient. In addition, we prove that the non-self-consistent scheme is equivalent to using a single external-potential-dependent functional in an otherwise consistent, nonvariational Kohn-Sham density-functional theory (KS DFT) scheme. Furthermore, our results also demonstrate that, while an internally consistent KS functional is presently missing, non-self-consistent LDA on AK13 orbitals works as a practical nonempirical computational scheme to predict geometries, bulk moduli, while retaining the band structure features of AK13 at the computational cost of semi-local DFT.

Place, publisher, year, edition, pages
AMER PHYSICAL SOC, 2016
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:liu:diva-133756 (URN)10.1103/PhysRevB.94.155143 (DOI)000390031400002 ()
Note

Funding Agencies|Swedish Research Council (VR) [621-2011-4249]; Linnaeus Environment at Linkoping on Nanoscale Functional Materials (LiLi-NFM) - VR; Swedish e-Science Research Centre (SeRC)

Available from: 2017-01-09 Created: 2017-01-09 Last updated: 2017-11-29
3. Crystal structure representations for machine learning models of formation energies
Open this publication in new window or tab >>Crystal structure representations for machine learning models of formation energies
2015 (English)In: International Journal of Quantum Chemistry, ISSN 0020-7608, E-ISSN 1097-461X, Vol. 115, no 16, p. 1094-1101Article in journal (Refereed) Published
Abstract [en]

We introduce and evaluate a set of feature vector representations of crystal structures for machine learning (ML) models of formation energies of solids. ML models of atomization energies of organic molecules have been successful using a Coulomb matrix representation of the molecule. We consider three ways to generalize such representations to periodic systems: (i) a matrix where each element is related to the Ewald sum of the electrostatic interaction between two different atoms in the unit cell repeated over the lattice; (ii) an extended Coulomb-like matrix that takes into account a number of neighboring unit cells; and (iii) an ansatz that mimics the periodicity and the basic features of the elements in the Ewald sum matrix using a sine function of the crystal coordinates of the atoms. The representations are compared for a Laplacian kernel with Manhattan norm, trained to reproduce formation energies using a dataset of 3938 crystal structures obtained from the Materials Project. For training sets consisting of 3000 crystals, the generalization error in predicting formation energies of new structures corresponds to (i) 0.49, (ii) 0.64, and (iii) 0.37eV/atom for the respective representations.

Place, publisher, year, edition, pages
Wiley, 2015
Keywords
machine learning; formation energies; representations; crystal structure; periodic systems
National Category
Physical Sciences
Identifiers
urn:nbn:se:liu:diva-120326 (URN)10.1002/qua.24917 (DOI)000357606000010 ()
Note

Funding Agencies|Swedish Research Council (VR) [621-2011-4249]; Linnaeus Environment at Linkoping on Nanoscale Functional Materials - VR; Swiss National Science foundation [PP00P2_138932]; Office of Science of the U.S. DOE [DE-AC02-06CH11357]; Air Force Office of Scientific Research, Air Force Material Command, USAF [FA9550-15-1-0026]

Available from: 2015-07-31 Created: 2015-07-31 Last updated: 2017-11-01
4. Machine Learning Energies of 2 Million Elpasolite (AB2D6) Crystals
Open this publication in new window or tab >>Machine Learning Energies of 2 Million Elpasolite (AB2D6) Crystals
2016 (English)In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, ISSN 031-9007, Vol. 117, no 13, article id 135502Article in journal (Refereed) Published
Abstract [en]

Elpasolite is the predominant quaternary crystal structure (AlNaK2F6 prototype) reported in the Inorganic Crystal Structure Database. We develop a machine learning model to calculate density functional theory quality formation energies of all ∼2×106 pristine ABC2D6 elpasolite crystals that can be made up from main-group elements (up to bismuth). Our model’s accuracy can be improved systematically, reaching a mean absolute error of 0.1  eV/atom for a training set consisting of 10×103 crystals. Important bonding trends are revealed: fluoride is best suited to fit the coordination of the D site, which lowers the formation energy whereas the opposite is found for carbon. The bonding contribution of the elements A and B is very small on average. Low formation energies result from A and B being late elements from group II, C being a late (group I) element, and D being fluoride. Out of 2×106 crystals, 90 unique structures are predicted to be on the convex hull—among which is NFAl2Ca6, with a peculiar stoichiometry and a negative atomic oxidation state for Al.

Place, publisher, year, edition, pages
American Physical Society, 2016
Keywords
Machine learning, AI, Elpasolite, Materials
National Category
Condensed Matter Physics
Identifiers
urn:nbn:se:liu:diva-131473 (URN)10.1103/PhysRevLett.117.135502 (DOI)000383849400010 ()
Funder
Swedish Research Council, 621-2011-4249,
Note

Funding agencies: Swiss National Science Foundation [PP00P2_138932]; Air Force Office of Scientific Research, Air Force Material Command, USAF [FA9550-15-1-0026]; NCCR MARVEL - Swiss National Science Foundation; Swiss National Supercomputing Centre (CSCS) [mr14]; Swedish R

Available from: 2016-09-22 Created: 2016-09-22 Last updated: 2017-11-21Bibliographically approved

Open Access in DiVA

fulltext(1279 kB)108 downloads
File information
File name FULLTEXT01.pdfFile size 1279 kBChecksum SHA-512
a3ba0008b8892c7fd6f912ce3738a1545b46ab2f1c880f441c64dddc6475115c7f59ade5d3d1bc67b3d7f4d8c5b3bcecf448b5d112ee6cf117f4e50f1b2a1f7e
Type fulltextMimetype application/pdf
omslag(2718 kB)7 downloads
File information
File name COVER01.pdfFile size 2718 kBChecksum SHA-512
dcc789efeed610d4a19716eb311bc8c103c9428e38787721433ab8183b01fab257ef51bb5cdac7d96ba93921d090b82baa6e8b871cd86662adaa30d90d51e8d2
Type coverMimetype application/pdf

Other links

Publisher's full text

Search in DiVA

By author/editor
Lindmaa, Alexander
By organisation
Theoretical PhysicsFaculty of Science & Engineering
Theoretical ChemistryOther Physics TopicsCondensed Matter PhysicsControl EngineeringOther Engineering and Technologies not elsewhere specified

Search outside of DiVA

GoogleGoogle Scholar
Total: 108 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

doi
isbn
urn-nbn

Altmetric score

doi
isbn
urn-nbn
Total: 641 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf