Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • 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
Foundation of Density Functionals in the Presence of Magnetic Field
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains four articles related to mathematical aspects of Density Functional Theory.

In Paper A, the theoretical justification of density methods formulated with current densities is addressed. It is shown that the set of ground-states is determined by the ensemble-representable particle and paramagnetic current density. Furthermore, it is demonstrated that the Schrödinger equation with a magnetic field is not uniquely determined by its ground-state solution. Thus, a wavefunction may be the ground-state of two different Hamiltonians, where the Hamiltonians differ by more than a gauge transformation. This implies that the particle and paramagnetic current density do not determine the potentials of the system and, consequently, no Hohenberg-Kohn theorem exists for Current Density Functional Theory formulated with the paramagnetic current density. On the other hand, by instead using the particle density as data, we show that the scalar potential in the system's Hamiltonian is determined for a fixed magnetic field. This means that the Hohenberg-Kohn theorem continues to hold in the presence of a magnetic field, if the magnetic field has been fixed.

Paper B deals with N-representable density functionals that also depend on the paramagnetic current density. Here the Levy-Lieb density functional is generalized to include the paramagnetic current density. It is shown that a wavefunction exists that minimizes the "free" Hamiltonian subject to the constraints that the particle and paramagnetic current density are held fixed. Furthermore, a convex and universal current density functional is introduced and shown to equal the convex envelope of the generalized Levy-Lieb density functional. Since this functional is convex, the problem of finding the particle and paramagnetic current density that minimize the energy is related to a set of Euler-Lagrange equations.

In Paper C, an N-representable Kohn-Sham approach is developed that also include the paramagnetic current density. It is demonstrated that a wavefunction exists that minimizes the kinetic energy subject to the constraint that only determinant wavefunctions are considered, as well as that the particle and paramagnetic current density are held fixed. Using this result, it is then shown that the ground-state energy can be obtained by minimizing an energy functional over all determinant wavefunctions that have finite kinetic energy. Moreover, the minimum is achieved and this determinant wavefunction gives the ground-state particle and paramagnetic current density.

Lastly, Paper D addresses the issue of a Hohenberg-Kohn variational principle for Current Density Functional Theory formulated with the total current density. Under the assumption that a Hohenberg-Kohn theorem exists formulated with the total current density, it is shown that the map from particle and total current density to the vector potential enters explicitly in the energy functional to be minimized. Thus, no variational principle as that of Hohenberg and Kohn exists for density methods formulated with the total current density.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. , x, 40 p.
Series
TRITA-MAT-A, 2014:10
Keyword [en]
Current density functional theory, Hohenberg-Kohn theorems, paramagnetic current density functionals, Kohn-Sham theory, Levy-Lieb functional, variational principle, N-representable, degeneracy
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-145546ISBN: 978-91-7595-169-0 (print)OAI: oai:DiVA.org:kth-145546DiVA: diva2:718653
Public defence
2014-06-13, D2, Lindstedtsvägen 5, KTH, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20140523

Available from: 2014-05-23 Created: 2014-05-21 Last updated: 2014-05-23Bibliographically approved
List of papers
1. Hohenberg-Kohn Theorems in the Presence of Magnetic Field
Open this publication in new window or tab >>Hohenberg-Kohn Theorems in the Presence of Magnetic Field
2014 (English)In: International Journal of Quantum Chemistry, ISSN 0020-7608, E-ISSN 1097-461X, Vol. 114, no 12, 782-795 p.Article in journal (Refereed) Published
Abstract [en]

In this article, we examine Hohenberg-Kohn theorems for Current Density Functional Theory, that is, generalizations of the classical Hohenberg-Kohn theorem that includes both electric and magnetic fields. In the Vignale and Rasolt formulation (Vignale and Rasolt, Phys. Rev. Lett. 1987, 59, 2360), which uses the paramagnetic current density, we address the issue of degenerate ground states and prove that the ensemble-representable particle and paramagnetic current density determine the degenerate ground states. For the formulation that uses the total current density, we note that the proof suggested by Diener (Diener, J. Phys.: Condens. Matter. 1991, 3, 9417) is unfortunately not correct. Furthermore, we give a proof that the magnetic field and the ensemble-representable particle density determine the scalar and vector potentials up to a gauge transformation. This generalizes the result of Grayce and Harris (Grayce and Harris, Phys. Rev. A 1994, 50, 3089) to the case of degenerate ground states. We moreover prove the existence of a positive wavefunction that is the ground state of infinitely many different Hamiltonians.

Place, publisher, year, edition, pages
John Wiley & Sons, 2014
Keyword
current density functional theory, Hohenberg– Kohn theorem, degeneracy, magnetic field
National Category
Mathematics Chemical Sciences
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-145534 (URN)10.1002/qua.24668 (DOI)000335202500004 ()2-s2.0-84899983619 (Scopus ID)
Note

QC 20140523

Available from: 2014-05-21 Created: 2014-05-21 Last updated: 2017-12-05Bibliographically approved
2. Density functionals in the presence of magnetic field
Open this publication in new window or tab >>Density functionals in the presence of magnetic field
2014 (English)In: International Journal of Quantum Chemistry, ISSN 0020-7608, E-ISSN 1097-461X, Vol. 114, no 21, 1445-1456 p.Article in journal (Refereed) Published
Abstract [en]

In this article, density functionals for Coulomb systems subjected to electric and magnetic fields are developed. The density functionals depend on the particle density rho and paramagnetic current density j(p). This approach is motivated by an adapted version of the Vignale and Rasolt formulation of current density functional theory, which establishes a one-toone correspondence between the nondegenerate groundstate and the particle and paramagnetic current density. Definition of N-representable density pairs (rho,j(p)) is given and it is proven that the set of v-representable densities constitutes a proper subset of the set of N-representable densities. For a Levy-Lieb-type functional Q(rho,j(p)), it is demonstrated that (i) it is a proper extension of the universal Hohenberg-Kohn functional F-HK (rho,j(p)) to N-representable densities, (ii) there exists a wavefunction psi(0) such that Q(rho; j(p)) = (psi(0); H-0 psi(0))(rho), where H-0 is the Hamiltonian without external potential terms, and (iii) it is not convex. Furthermore, a convex and universal functional F(rho, j(p)) is studied and proven to be equal the convex envelope of Q(rho, j(p)). For both Q and F, we give upper and lower bounds.

Place, publisher, year, edition, pages
John Wiley & Sons, 2014
Keyword
current density functional theory, paramagnetic current density functionals, Levy-Lieb density functional, convexity, Euler-Lagrange equations
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-145539 (URN)10.1002/qua.24707 (DOI)000344331100005 ()2-s2.0-84908085960 (Scopus ID)
Note

QC 20141205. Updated from accepted to published.

Available from: 2014-05-21 Created: 2014-05-21 Last updated: 2017-12-05Bibliographically approved
3. Kohn-Sham Theory in the Presence of Magnetic Field
Open this publication in new window or tab >>Kohn-Sham Theory in the Presence of Magnetic Field
2014 (English)In: Journal of Mathematical Chemistry, ISSN 0259-9791, E-ISSN 1572-8897, Vol. 52, no 10, 2581-2595 p.Article in journal (Refereed) Published
Abstract [en]

In the well-known Kohn-Sham theory in Density Functional Theory, a fictitious non-interacting system is introduced that has the same particle density as a system of N electrons subjected to mutual Coulomb repulsion and an external electric field. For a long time, the treatment of the kinetic energy was not correct and the theory was not well-defined for N-representable particle densities. In the work of [Hadjisavvas and Theophilou, Phys. Rev. A, 1984, 30, 2183], a rigorous Kohn-Sham theory for N-representable particle densities was developed using the Levy-Lieb functional. Since a Levy-Lieb-type functional can be defined for Current Density Functional Theory formulated with the paramagnetic current density, we here develop a rigorous N-representable Kohn-Sham approach for interacting electrons in magnetic field. Furthermore, in the one-electron case, criteria for N-representable particle densities to be v-representable are given.

Keyword
Density Functional Theory, Kohn-Sham theory, Levy-Lieb functional, Current Density Functional Theory, N-representable
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-145543 (URN)10.1007/s10910-014-0400-7 (DOI)000343754600010 ()2-s2.0-84919933174 (Scopus ID)
Note

Updated from manuscript to article.

QC 20140523

Available from: 2014-05-21 Created: 2014-05-21 Last updated: 2017-12-05Bibliographically approved
4. Non-existence of a Hohenberg-Kohn Variational Principle in Total Current Density Functional Theory
Open this publication in new window or tab >>Non-existence of a Hohenberg-Kohn Variational Principle in Total Current Density Functional Theory
2015 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 91, no 3, 032508Article in journal (Other academic) Published
Abstract [en]

For a many-electron system, whether the particle density rho and the total current density j are sufficient to determine the one-body potential V and vector potential A, is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional E_{V_0,A_0}(rho,j) = <psi(rho,j),H(V_0,\A_0)psi(rho,j)> can be minimal for densities that are not the ground-state densities of the fixed potentials V_0 and A_0. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with rho and j, we show that a variational principle for Total Current Density Functional Theory as that of Hohenberg-Kohn for Density Functional Theory does not exist. The reason is that the assumed map from densities to the vector potential, written (rho,j) -> A(rho,j;x), enters explicitly in E_{V_0,A_0}(rho,j).

Keyword
Current density functional theory, Hohenberg-Kohn variational principle, total current density
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-145544 (URN)10.1103/PhysRevA.91.032508 (DOI)000351507900005 ()2-s2.0-84927534389 (Scopus ID)
Note

Updated from manuscript to article.

QC 20150430

Available from: 2014-05-21 Created: 2014-05-21 Last updated: 2017-12-05Bibliographically approved

Open Access in DiVA

Thesis(471 kB)86 downloads
File information
File name FULLTEXT01.pdfFile size 471 kBChecksum SHA-512
b68ec682b1c209ada0031347ab0410c82474150488d395281b98151dd1bf8715971e9c228a3368c416971094664a14d0eb4e39d157d00fb152978532df6b3d33
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Laestadius, Andre
By organisation
Mathematics (Div.)
Mathematical Analysis

Search outside of DiVA

GoogleGoogle Scholar
Total: 86 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

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 314 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • 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