Differences

This shows you the differences between two versions of the page.

--- package:faq [2024/03/04 08:35] – pbloechl
+++ package:faq [2024/03/04 08:42] (current) – pbloechl
@@ Line 38: / Line 38: @@
 ==== I obtain negative occupations. Is that wrong? ====
-No. This occurs normally for the tetrahedron method using the corrections from  [[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.49.16223|Blöchl et al. Phys. Rev. B 49, 16223 (1994)]]. It reflects that the Brillouin-zone integration is not approximated by a sampling over a discrete k-point set, but energies and matrix elements are interpolated between the discrete k-point set. What is printed as occupations are actually integration weights, which include the interpolation between the discrete k-point set. Using these elaborate integration weights, the integral of the interpolated matrix elements and energies can still be expressed as weighted sum over the discrete k-point set. The interpolation is effectively "hidden" from the user at a price that the weights have a complicated form an take on values that appear, at first sight, unphysical. When the interpolation between the discrete k-points is non-linear, which is the case with the "correction formula" from [[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.49.16223|Blöchl et al. Phys. Rev. B 49, 16223 (1994)]], the integration weights may also be negative or be larger than one. See for example Eq.22 and Fig.7 of [[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.49.16223|Blöchl et al. Phys. Rev. B 49, 16223 (1994)]].
+No. This occurs normally for the tetrahedron method using the corrections from  [[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.49.16223|Blöchl et al. Phys. Rev. B 49, 16223 (1994)]]. It reflects that the Brillouin-zone integration is not approximated by a sampling over a discrete k-point set, but energies and matrix elements are interpolated in between. What is printed as occupations are actually integration weights, which include the interpolation between the discrete k-points. Using these elaborate integration weights, the integral of the interpolated matrix elements and energies can still be expressed as weighted sum over the discrete k-point set. The interpolation is effectively "hidden" from the user at the price of complicated integration weights, which take on values that appear, at first sight, unphysical. When the interpolation between the discrete k-points is non-linear, which is the case with the "correction formula" from [[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.49.16223|Blöchl et al. Phys. Rev. B 49, 16223 (1994)]], the integration weights may also be negative or larger than one. See for example Eq.22 and Fig.7 of [[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.49.16223|Blöchl et al. Phys. Rev. B 49, 16223 (1994)]].