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@PHDTHESIS{Zhang:878578,
      author       = {Zhang, Qian},
      title        = {{B}uilding {E}ffective {M}odels for {C}orrelated {E}lectron
                      {S}ystems},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      reportid     = {FZJ-2020-02921},
      pages        = {227 pages},
      year         = {2020},
      note         = {Dissertation, RWTH Aachen University, 2020},
      abstract     = {To understand strongly correlated systems, we must confront
                      the many-body problem. This is practically impossible for
                      the ab-initio Hamiltonian. To make such studies feasible it
                      is, thus, crucial to construct model Hamiltonians that are
                      as simple as possible, so they can be solved, while
                      containing still enough details to be material-specific.Our
                      starting point is density functional theory for individual
                      atoms and ions to obtain re- alistic basis functions and the
                      corresponding matrix elements. For the open-shell orbitals,
                      which show the strongest correlation effects due to the
                      degeneracy of the multiplets, we calculate the Slater-Condon
                      and spin-orbit parameters from the resulting self-consistent
                      radial wave functions and potentials. We study the trends of
                      the parameters systemati- cally across the periodic table,
                      develop an intuitive parametrization, and calculate atomic
                      open-shell spectra in LS-, intermediate-, and jj-coupling
                      schemes.The comparison of the interaction strengths of
                      different coupling schemes gives rise to the study of the
                      moment formulas, which reduce the calculation from the
                      “impossible” many-electron Hilbert space to a one- or
                      two-electron space. We derive the analytic moment formulas
                      for the general one- and two-body Hamiltonians. The moment
                      formulas provide us a new approach to handle the
                      many-electron Hamiltonians without the need of working with
                      a many-electron basis, but only with matrix representations
                      under the one- or two-electron basis.To model the
                      Hamiltonians for realistic materials, orthonormal basis
                      orbitals are preferred. However, while the atomic orbitals
                      are mutually orthonormal within a single atom, they are, in
                      general, non-orthogonal for atoms on different lattice
                      sites. We study and develop efficient multi-center integral
                      techniques for evaluating orbital overlaps, which are essen-
                      tial for performing the orbital orthogonalization. To
                      orthogonalize the basis orbitals, we apply the Löwdin
                      symmetric orthogonalization scheme, which minimizes the
                      orbital modification. To generalize the multi-center
                      integrals, we introduce the re-centering method, which is a
                      spherical harmonic expansion that requires the Gaunt
                      coefficients with large angular quantum numbers. To compute
                      the Gaunt coefficients, the previously known numerical
                      methods are, however, inaccurate for the coefficients that
                      involve large quantum numbers. Therefore, we provide a
                      numerically stable algorithm for computing the Gaunt
                      coefficients efficiently and accurately. The re-centering
                      method enables us to compute general multi-center integrals
                      including the hopping matrix elements and the long-range
                      Coulomb matrix elements. After performing the basis
                      orthonormalization, we study the deformation of the
                      resulting orbitals and investigate the modification of the
                      corresponding multi-center matrix elements under changes of
                      the bond lengths or lattice constants.},
      cin          = {JSC},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {511 - Computational Science and Mathematical Methods
                      (POF3-511)},
      pid          = {G:(DE-HGF)POF3-511},
      typ          = {PUB:(DE-HGF)11},
      url          = {https://juser.fz-juelich.de/record/878578},
}