This title appears in the Scientific Report :
2020
Please use the identifier:
http://hdl.handle.net/2128/25556 in citations.
Building Effective Models for Correlated Electron Systems
Building Effective Models for Correlated Electron Systems
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...
Saved in:
Personal Name(s): | Zhang, Qian (Corresponding author) |
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Contributing Institute: |
Jülich Supercomputing Center; JSC |
Imprint: |
2020
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Physical Description: |
227 pages |
Dissertation Note: |
Dissertation, RWTH Aachen University, 2020 |
Document Type: |
Dissertation / PhD Thesis |
Research Program: |
Computational Science and Mathematical Methods |
Link: |
OpenAccess OpenAccess |
Publikationsportal JuSER |
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. |