New Computational Frameworks and Perspectives for Ab Initio Electronic-structure Calculations in Quantum Many-body Theory
Year: 2026
Type: dissertation
Abstract: More than a century after the 1925 foundations of quantum mechanics, quantum theory has become the standard formulation of atomic physics. Yet practical computation remains difficult because the many-body problem scales exponentially with system size. With today’s far more powerful computing resources and advances in mathematical algorithms, these calculations can increasingly be carried out more systematically and with greater completeness. In this thesis, we present two main contributions. In the first contribution, we introduce an alternative real-space framework, termed the Embedded Unit-Cell (EUC) method, for accurate band structure calculations in solid-state physics and electronics. Conventional k-space plane-wave approaches making use of pseudo-potentials for solving Kohn–Sham Density Functional Theory (DFT) (or beyond, such as GW approximation, etc.), have achieved remarkable success. However, their accuracy relies heavily on k-point sampling (Brillouin-zone integration), and moreover, establishing a reliable and accurate vacuum reference level within periodic k-space frameworks remains challenging. In contrast, the EUC method departs fundamentally from Brillouin-zone-based formulations. It performs self-consistent DFT calculations on enlarged unit-cell embedded finite atomistic systems, where the electron density and the effective potential are obtained directly from discrete finite energy states, thereby bypassing k-point sampling and related numerical complications. After solving the unit-cell embedded finite system, periodic wave-functions and their corresponding energy levels are recovered within the unit cell in the same manner as in the conventional k-space plane-wave approach. Although the method was initially developed using a real-space grid/mesh formulation based on the finite element technique, it is, in principle, compatible with other basis representations, including atomic orbitals and plane waves. The EUC framework opens a new perspective for electronic band structure calculations. It enables intrinsic treatment of surface effects on band structures and, importantly, provides a direct route to obtaining absolute vacuum-referenced band energies. Furthermore, the method is applicable to both local and non-local exchange–correlation potentials and can be extended beyond standard DFT to incorporate non-local exact exchange, hybrid functionals, GW, Coupled Cluster, DMFT, etc.. The second contribution addresses the nonlinear eigenvalue problems that arise in Green’s function formulations of quantum many-body theory. Due to their intrinsic energy dependence, such formulations, most notably the quasiparticle equation in the GW approximation, lead to challenging nonlinear diagonalization tasks. To overcome this difficulty, the recent contour-integral–based eigenvalue algorithm (NL-FEAST) is employed to directly solve these Green’s-function-induced nonlinear eigenvalue problems. The approach computes both quasiparticle energies and the associated wavefunctions simultaneously, improving accuracy while reducing computational cost compared to conventional graphical or spectral-function methods. Results for representative molecules confirm that Kohn–Sham (and Hartree–Fock) HOMOs are already close to G0W0 values, whereas LUMOs exhibit significant corrections. More broadly, this approach provides a general and efficient framework for tackling nonlinear eigenvalue problems arising in quantum many-body physics.
DOI: 10.7275/58663
Author: Li, Dongming
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213989599
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