Potential integrals

Real space integrators

Quadrature rule

The numerical integrals by creating a table of integrals by using the Gaussian quadrature method , which is designed for accurate results using a minimum number of evaluations for functions with singularities.

The potential of a single atom is localized, but in principle infinite in extent, hence we need to set a reasonable cutoff. The cutoff is calculated by solving the equation

\[ V(r) = V_{tol} \quad , \]

where \(V_{tol}\) is the error at the cut-off. The equation is solved for each species. The use of the cut-off radius creates a discontinuity; hence, abTEM uses a tapering near the cut-off. \(V_{cut}\) can be modified using the cutoff_tolerance argument of the Potential or AtomicPotential objects. abTEM uses a tapering cutoff starting at \(85 \ \%\) of the full cutoff.

from abtem.potentials.integrals import ProjectionQuadratureRule

integrator = ProjectionQuadratureRule()

integrator