Density functional approximation; Density functional techniques; Density Functional Calculations; Meta-GGA
en
Abstract:
In our recently published paper, we wrote that “If a radial equation (relativistic or not) is integrated using a nonmultiplicative potential, the solutions are, in general, not orthogonal.” This statement is not accurate. The orthogonality of the solutions to an eigenvalue problem is instead dependent on the Hermiticity of the operator (Formula Presented). Note also that this method solves the radial equation by diagonalization of an algebraic eigenvalue problem. Therefore, it produces radial solutions as eigenvectors at set eigenvalues. In APW based methods the radial functions are calculated at (chosen) energy parameters which are generally set close to the band energies, not to “atomlike” eigenvalues. Atomlike eigenvalues for spheres in a periodic solid can only be approximated by extending the spherical potential into a virtual confining potential, which might not correspond to optimal radial basis functions if the eigenvalues are not close to the middle of the band energies. Additionally, there is no relativistic radial equation treated in Ref. [2], and we are not aware of any work that does solve a relativistic radial gKS MGGA equation by integration, rather than by diagonalization. The incorrect statement does not have consequences for the results presented in the paper. It does imply the theoretical possibility of an implementation of gKS MGGA potentials in APW based methods that does not depend on the approximation of using an auxiliary KS potential to construct the radial basis functions. The development of such an implementation would require significant additional effort and might also have drawbacks in the sense that the treatment of some functionals (such as SCAN) may still be numerically intractable. We gratefully acknowledge N. Holzwarth for pointing out the error and engaging in a constructive discussion.
