Efficient algorithm for band connectivity resolution

An efficient algorithm for band connectivity (BC) resolution is presented. The method uses only readily available band coefficients and the overlap matrix, and has a low computational cost. The accuracy of the BC resolution is such that the method is practical for meshes of $\mathbf{k}$ points typically used in systems with small unit cells (e.g., $16\times 16\times 16$ mesh for a 3 Å unit cell). We establish that the errors in the linear tetrahedron (LT) method due to the undetected crossings have ${\Delta }^2$ dependence with respect to the characteristic spacing between $\mathbf{k}$ points. The intrinsic error of the LT method is proportional to ${\Delta }^2,$ while for the “improved” LT method (iLT) it is proportional to ${\Delta }^4.$ Thus, the BC error is in fact the leading error of the iLT method. Our benchmarks demonstrate that the resolution of band connectivity restores the high accuracy of the “improved” LT method $\left({\Delta }^4\right)$ in systems with band crossings near or at the Fermi level.