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@ARTICLE{Winkelmann:889651,
      author       = {Winkelmann, Miriam and Di Napoli, Edoardo and Wortmann,
                      Daniel and Blügel, Stefan},
      title        = {{S}olution to the modified {H}elmholtz equation for
                      arbitrary periodic charge densities},
      journal      = {Frontiers in physics},
      volume       = {8},
      issn         = {2296-424X},
      address      = {Lausanne},
      publisher    = {Frontiers Media},
      reportid     = {FZJ-2021-00283},
      pages        = {618142},
      year         = {2021},
      abstract     = {We present a general method for solving the modified
                      Helmholtz equation without shape approximation for an
                      arbitrary periodic charge distribution, whose solution is
                      known as the Yukawa potential or the screened Coulomb
                      potential. The method is an extension of Weinert's
                      pseudo-charge method [M. Weinert, J. Math. Phys. 22, 2433
                      (1981)] for solving the Poisson equation for the same class
                      of charge density distributions. The inherent differences
                      between the Poisson and the modified Helmholtz equation are
                      in their respective radial solutions. These are polynomial
                      functions, for the Poisson equation, and modified spherical
                      Bessel functions, for the modified Helmholtz equation. This
                      leads to a definition of a modified pseudo-charge density
                      and modified multipole moments. We have shown that Weinert's
                      convergence analysis of an absolutely and uniformly
                      convergent Fourier series of the pseudo-charge density is
                      transferred to the modified pseudo-charge density. We
                      conclude by illustrating the algorithmic changes necessary
                      to turn an available implementation of the Poisson solver
                      into a solver for the modified Helmholtz equation.},
      cin          = {JSC / IAS-1 / PGI-1},
      ddc          = {530},
      cid          = {I:(DE-Juel1)JSC-20090406 / I:(DE-Juel1)IAS-1-20090406 /
                      I:(DE-Juel1)PGI-1-20110106},
      pnm          = {521 - Quantum Materials (POF4-521) / 5111 - Domain-Specific
                      Simulation $\&$ Data Life Cycle Labs (SDLs) and Research
                      Groups (POF4-511) / MaX - MAterials design at the eXascale.
                      European Centre of Excellence in materials modelling,
                      simulations, and design (824143) / Simulation and Data
                      Laboratory Quantum Materials (SDLQM) (SDLQM)},
      pid          = {G:(DE-HGF)POF4-521 / G:(DE-HGF)POF4-5111 /
                      G:(EU-Grant)824143 / G:(DE-Juel1)SDLQM},
      typ          = {PUB:(DE-HGF)16},
      pubmed       = {12947202},
      UT           = {WOS:000632399300001},
      doi          = {10.3389/fphy.2020.618142},
      url          = {https://juser.fz-juelich.de/record/889651},
}