13,843 research outputs found
Artificial Neural Network Methods in Quantum Mechanics
In a previous article we have shown how one can employ Artificial Neural
Networks (ANNs) in order to solve non-homogeneous ordinary and partial
differential equations. In the present work we consider the solution of
eigenvalue problems for differential and integrodifferential operators, using
ANNs. We start by considering the Schr\"odinger equation for the Morse
potential that has an analytically known solution, to test the accuracy of the
method. We then proceed with the Schr\"odinger and the Dirac equations for a
muonic atom, as well as with a non-local Schr\"odinger integrodifferential
equation that models the system in the framework of the resonating
group method. In two dimensions we consider the well studied Henon-Heiles
Hamiltonian and in three dimensions the model problem of three coupled
anharmonic oscillators. The method in all of the treated cases proved to be
highly accurate, robust and efficient. Hence it is a promising tool for
tackling problems of higher complexity and dimensionality.Comment: Latex file, 29pages, 11 psfigs, submitted in CP
Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations
The iterative diagonalization of a sequence of large ill-conditioned
generalized eigenvalue problems is a computational bottleneck in quantum
mechanical methods employing a nonorthogonal basis for {\em ab initio}
electronic structure calculations. We propose a hybrid preconditioning scheme
to effectively combine global and locally accelerated preconditioners for rapid
iterative diagonalization of such eigenvalue problems. In partition-of-unity
finite-element (PUFE) pseudopotential density-functional calculations,
employing a nonorthogonal basis, we show that the hybrid preconditioned block
steepest descent method is a cost-effective eigensolver, outperforming current
state-of-the-art global preconditioning schemes, and comparably efficient for
the ill-conditioned generalized eigenvalue problems produced by PUFE as the
locally optimal block preconditioned conjugate-gradient method for the
well-conditioned standard eigenvalue problems produced by planewave methods
Spurious Modes in Dirac Calculations and How to Avoid Them
In this paper we consider the problem of the occurrence of spurious modes
when computing the eigenvalues of Dirac operators, with the motivation to
describe relativistic electrons in an atom or a molecule. We present recent
mathematical results which we illustrate by simple numerical experiments. We
also discuss open problems.Comment: Chapter to be published in the book "Many-Electron Approaches in
Physics, Chemistry and Mathematics: A Multidisciplinary View", edited by
Volker Bach and Luigi Delle Sit
B-Spline Finite Elements and their Efficiency in Solving Relativistic Mean Field Equations
A finite element method using B-splines is presented and compared with a
conventional finite element method of Lagrangian type. The efficiency of both
methods has been investigated at the example of a coupled non-linear system of
Dirac eigenvalue equations and inhomogeneous Klein-Gordon equations which
describe a nuclear system in the framework of relativistic mean field theory.
Although, FEM has been applied with great success in nuclear RMF recently, a
well known problem is the appearance of spurious solutions in the spectra of
the Dirac equation. The question, whether B-splines lead to a reduction of
spurious solutions is analyzed. Numerical expenses, precision and behavior of
convergence are compared for both methods in view of their use in large scale
computation on FEM grids with more dimensions. A B-spline version of the object
oriented C++ code for spherical nuclei has been used for this investigation.Comment: 27 pages, 30 figure
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