  The $\mathbf{k \cdot p}$ method is a successful approach to obtain band
structure, optical and transport properties of semiconductors, and it depends
on external parameters that are obtained either from experiments, tight binding
or ab initio calculations. Despite the widespread use of the $\mathbf{k \cdot
p}$ method, a systematic analysis of the stability and the accuracy of its
parameters is not usual in the literature. In this work, we report a
theoretical framework to determine the $\mathbf{k \cdot p}$ parameters from
state-of-the-art hybrid density functional theory including spin-orbit
coupling, providing a calculation where the gap and spin-orbit energy splitting
are in agreement with the experimental values. The accuracy of the set of
parameters is enhanced by fitting over several directions at once, minimizing
the overall deviation from the original data. This strategy allows us to
systematically evaluate the stability, preserving the accuracy of the
parameters, providing a tool to determine optimal parameters for specific
ranges around the ${\Gamma}$-point. To prove our concept, we investigate the
zinc blende GaAs that shows results in excellent agreement with the most
reliable data in the literature.

Materials Science