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The matrix diagonalization routine which is used to compute the KS orbitals and their eigenvalues does not give exactly the boundary conditions desired.
dirichlet
condition, it fixes the value of the eigenfunctions to zero at thengrid +1
grid point; we want them equal to zero at thengrid
grid pointneumann
condition, it correctly sets the derivative to zero at thengrid
point, but is unable to resolve the eigenfunction at that point and returns a junk valueAlthough these only produced very small numerical errors, it is better to fix them. Their behaviour might also be more critical when we consider higher-order properties and more advanced features.
This PR introduces several fixes to ensure the correct behaviour of both bcs. Notably, the dirichlet grid is modified to first ensure the zero occurs at the
ngrid
point. For both bcs, the final point is determined via manual integration of the numerov formula.