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Non-existence of a Hohenberg-Kohn Variational Principle in Total Current Density Functional Theory
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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).

Place, publisher, year, edition, pages
2015. Vol. 91, no 3, 032508
Keyword [en]
Current density functional theory, Hohenberg-Kohn variational principle, total current density
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-145544DOI: 10.1103/PhysRevA.91.032508ISI: 000351507900005Scopus ID: 2-s2.0-84927534389OAI: oai:DiVA.org:kth-145544DiVA: diva2:718636
Note

Updated from manuscript to article.

QC 20150430

Available from: 2014-05-21 Created: 2014-05-21 Last updated: 2017-12-05Bibliographically approved
In thesis
1. Foundation of Density Functionals in the Presence of Magnetic Field
Open this publication in new window or tab >>Foundation of Density Functionals in the Presence of Magnetic Field
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
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:nbn:se:kth:diva-145546 (URN)978-91-7595-169-0 (ISBN)
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

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