| In this thesis new numerical techniques are described for the solution of a three dimensional pseudo-spectral Self-Consistent Field Approximation (SCF). The self-consistent system solution is stabilized and accelerated by the application of a Jacobian-free version of Newton's method implemented with the GMRES algorithm. A sufficiently accurate initial guess to the system solution is provided as needed for this version of Newton's method.; Trigonometric approximation of point charge singularities present in the model is avoided by a carefully designed analytical charge smoothing method. Enhanced accuracy and mesh-independence are achieved by the translation invariance property of such procedure for a reduce-lengthed solution vector. An accurate potential reference is also implemented in the spectral method employed.; For the eigenvalue problem, a novel variant of Davidson's method, or spectrally localized Arnoldi's method, is used. The full-size problem is projected onto Krylov subspaces of limited size with an explicitly orthogonalized basis. The smaller-size projected problem is solved using LAPACK library functions and then the solution is transformed back to the original space. In Arnoldi's method, vectors in the selected range of the energy spectrum are amplified by a specially designed energy range selector, which amounts to tempered inverse iteration. The starting vectors for Arnoldi are enriched with new randomized components in order to decrease the iteration number and to reduce the chance of missing eigenvectors. The linear systems in the Krylov space generation are solved by a spectrally preconditioned conjugate gradient (CG) method. The energy range selector contains two parameters, which let us select a limited range in the eigenvalue spectrum. With the greatly improved adaptive algorithm for controlling the parameters and the more flexible adjustable subspace size, the solver has become highly efficient, robust and accurate. |
