Implementace kvantově-mechanických výpočtů elektronové struktury a vlastností neperiodických materiálových struktur

Abstract

A new ab-inito method and the related computer code has been developed within this work --- in accordance with its aims --- for calculating electronic states and structural properties of (primarily) nonperiodic, possibly electrically charged, material structures. This method has been implemented into the software package FENNEC.

The new method fills a gap among the current well-established methods and represents a counterpart of what the plane-wave method is within the class of the methods employing the Bloch's theorem (for materials with translational symmetry). The newly developed method is based on the density functional theory (DFT), the finite element method, or isogeometric analysis as an option, and environment-reflecting separable pseudopotentials, and it provides an excellent convergence control within a general, in principle arbitrarily precise basis.

During the development of the method, a new precise and computationally efficient formula for evaluating Hellmann-Feynman forces has been derived, implemented, and used for geometry optimizations. Its advantages and very good convergence properties have been demonstrated in sample calculations.

Among other originals results of this work, a purely algebraic description of separable pseudopotentials should be mentioned. This description allows constructing a projection basis to achieve any required accuracy.

Besides that, an adaptive version of the Anderson iteration scheme, which solves the problem of choosing the right mixing parameter (an approximation of the Jacobian) in the DFT loop, has been proposed, implemented, and analyzed.

Discretization of the Kohn-Sham equations using the finite element methods leads to a rank-$k$-update generalized eigenvalue problem, for solving of which an efficient method had to be developed: to find the appropriate eigenvalue solvers, integrate them into the FENNEC software package, and make them capable to solve the rank-k-update problem.

The convergence properties of that method in dependence on various technical parameters --- i.e. the finite element shape, basis type, and the order and number of projectors for separable pseudopotentials --- have been examined and analyzed to determine the best options for the new method.

The newly created method has been implemented using the Python finite element framework SfePy (and many other libraries) into the software package FENNEC. This software package is characterized by a modular plugin architecture which significantly simplifies its further development and makes its usage easy. FENNEC has been used, among other tasks, e.g. for geometry optimization of a deformed graphene fragment and calculations of forces in it as a preparation for the ab-initio construction of force-interaction molecular dynamics potentials.