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Title: Particle-Number Projection and the Density Functional Theory

Abstract

In the framework of the Density Functional Theory for superconductors, we study the restoration of the particle number symmetry by means of the projection technique. Conceptual problems are outlined and numerical difficulties are discussed. Both are related to the fact that neither the many-body Hamiltonian nor the wave function of the system appear explicitly in the Density Functional Theory. Similar obstacles are encountered in self-consistent theories utilizing density-dependent effective interactions.

Authors:
 [1];  [2];  [2];  [3]
  1. Warsaw University
  2. ORNL
  3. Universitat Erlangen, Germany
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Holifield Radioactive Ion Beam Facility
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
932099
DOE Contract Number:
DE-AC05-00OR22725
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review C; Journal Volume: 76; Journal Issue: 5
Country of Publication:
United States
Language:
English

Citation Formats

Dobaczewski, J., Stoitsov, Mario, Nazarewicz, Witold, and Reinhard, P.-G. Particle-Number Projection and the Density Functional Theory. United States: N. p., 2007. Web. doi:10.1103/PhysRevC.76.054315.
Dobaczewski, J., Stoitsov, Mario, Nazarewicz, Witold, & Reinhard, P.-G. Particle-Number Projection and the Density Functional Theory. United States. doi:10.1103/PhysRevC.76.054315.
Dobaczewski, J., Stoitsov, Mario, Nazarewicz, Witold, and Reinhard, P.-G. Mon . "Particle-Number Projection and the Density Functional Theory". United States. doi:10.1103/PhysRevC.76.054315.
@article{osti_932099,
title = {Particle-Number Projection and the Density Functional Theory},
author = {Dobaczewski, J. and Stoitsov, Mario and Nazarewicz, Witold and Reinhard, P.-G.},
abstractNote = {In the framework of the Density Functional Theory for superconductors, we study the restoration of the particle number symmetry by means of the projection technique. Conceptual problems are outlined and numerical difficulties are discussed. Both are related to the fact that neither the many-body Hamiltonian nor the wave function of the system appear explicitly in the Density Functional Theory. Similar obstacles are encountered in self-consistent theories utilizing density-dependent effective interactions.},
doi = {10.1103/PhysRevC.76.054315},
journal = {Physical Review C},
number = 5,
volume = 76,
place = {United States},
year = {Mon Jan 01 00:00:00 EST 2007},
month = {Mon Jan 01 00:00:00 EST 2007}
}
  • In the framework of the density functional theory for superconductors, we study the restoration of the particle-number symmetry by means of the projection technique. Conceptual problems are outlined and numerical difficulties are discussed. Both are related to the fact that neither the many-body Hamiltonian nor the wave function of the system appear explicitly in the density functional theory. Similar obstacles are encountered in self-consistent theories utilizing density-dependent effective interactions.
  • The nature of the explicit dependence on the particle number N and on the spin number N{sub s} of the Lieb definition for the energy density functional is examined both in spin-independent and in spin-polarized density functional theory. It is pointed out that the nonuniqueness of the external magnetic field B(r{yields}) corresponding to a given pair of ground-state density n(r{yields}) and spin density s(r{yields}) in spin-polarized density functional theory implies the nonexistence of the total derivative of the SDFT Lieb functional F{sub N,N{sub s}{sup L}[n,s] with respect to N{sub s}}. By giving a suitable extension of F{sub N}{sup L}[n] andmore » F{sub N,N{sub s}{sup L}[n,s] for N{ne}{integral}n(r{yields})dr{yields} and N{sub s{ne}{integral}}}s(r{yields})dr{yields}, it is then shown that their derivatives with respect to N and N{sub s} are equal to the derivatives, with respect to N and N{sub s}, of the total energies E[N,v] and E[N,N{sub s},v,B] minus the external-field energy components, respectively.« less
  • A particle-number projection technique is used to calculate transfer probabilities in the {sup 16}O+{sup 208}Pb reaction below the fusion barrier. The time evolution of the many-body wave function is obtained with the time-dependent Hartree-Fock (TDHF) mean-field theory. The agreement with experimental data for the sum of the proton-transfer channels is good, considering that TDHF has no parameter adjusted on the reaction mechanism. Some perspectives for extensions beyond TDHF to include cluster transfers are discussed.
  • The Lipkin-Nogami method is generalized to deal with finite range density dependent forces. New expressions are derived and realistic calculations with the Gogny force are performed for the nuclei {sup 164}Er and {sup 168}Er. The sharp phase transition predicted by the mean field approximation is washed out by the Lipkin-Nogami approach. A much better agreement with the experimental data is reached with the new approach than with the Hartree-Fock-Bogoliubov one, especially at high spins. {copyright} {ital 1996 The American Physical Society.}
  • A spin projected double-hybrid density functional theory is presented that accounts for different scaling of opposite and same spin terms in the second order correction. This method is applied to three dissociation reactions which in the unprojected formalism exhibit significant spin contamination with higher spin states. This gives rise to a distorted potential surface and can lead to poor geometries and energies. The projected method presented is shown to improve the description of the potential over unprojected double hybrid density functional theory. Comparison is made with the reference states of the two double hybrid functionals considered here (B2PLYP and mPW2PLYP)more » in which the projected potential surface is degraded by an imbalance in the description of dynamic and static correlation.« less