[Back]

R. Hammerling, O. Koch, C. Simon, E. Weinmüller:
"Numerical solution of singular ODE eigenvalue problems in electronic structure computations";
in: "ASC Report 07/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

We put forward a new method for the solution of eigenvalue problems for (systems
of) ordinary differential equations, where our main focus is on eigenvalue problems
for singular Schr¨odinger equations arising for example in electronic structure computations.
In most established standard methods, the generation of the starting values
for the computation of eigenvalues of higher index is a critical issue. Our approach
comprises two stages: First we generate rough approximations by a matrix method,
which yields several eigenvalues and associated eigenfunctions simultaneously, albeit
with moderate accuracy. In a second stage, these approximations are used as starting
values for a collocation method which yields approximations of high accuracy efficiently
due to an adaptive mesh selection strategy, and additionally provides reliable error estimates.
We successfully apply our method to the solution of the quantum mechanical
Kepler, Yukawa and the coupled ODE Stark problems.

electronic structure computation, polynomial collocation, fullpotential core

http://www.asc.tuwien.ac.at/preprint/2010/asc07x2010.pdf