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Kernel code overview

This code constructs the direct and exchange Kernel matrix on the coarse grid. This is done essentially by computing Eqs 34, 35 and 42-46 of Rohlfing and Louie, PRB 62, 4927 (2000).

Summary of input and output files

Required input files

  • kernel.inp: Input parameters.

  • WFN_co: Wavefunctions on coarse grid. Recommended: use an unshifted grid of the same size as the q-grid in epsmat. Shift will increase number of q-vectors needed in epsmat.

  • epsmat[.h5]: Inverse dielectric matrix (q\ne 0), created using epsilon. Must contain the all q=k-k' generated from WFN_co, including with symmetry if use_symmetries_coarse_grid is set. The file has a .h5 extension if the code was built with HDF5 support.

  • eps0mat[.h5]: Inverse dielectric matrix (q\rightarrow 0), created using epsilon The file has a .h5 extension if the code was built with HDF5 support. Note Kernel does not require quasiparticle eigenvalues. It may be run in parallel with sigma

Additional input

  • WFNq_co: Coarse-grid wavefunctions for finite-Q calculations. Currently not supported

Output files

  • bsedmat: Direct kernel matrix elements on unshifted coarse grid.

  • bsexmat: Exchange kernel matrix elements on unshifted coarse grid.

  • bsemat.h5: Includes data from both of above if compiled with HDF5 support. For specification, see bsemat.h5.spec.

Details: wings of epsilon

For semiconductors, the wings of \chi have terms of the following form:

\langle vk | e^{i(G+q)r} | ck+q \rangle\langle ck+q | e^{-iqr} | vk \rangle

The matrix element on the right is \langle u_{ck+q} | u_{vk} \rangle where u is the periodic part of the Bloch function. From k.p perturbation theory, this matrix element is proportional to q. The matrix element on the left with a non-zero G is typically roughly a constant as a function of q for small q (q being a small addition to G).

Thus for a general G-vector, \chi_\mathrm{wing}(G,q) \propto q. This directly leads to the wings of the screened untruncated Coulomb interaction being proportional to 1/q. Note that this function changes sign as q \rightarrow -q. Thus, when treating the q=0 point, we set the value of the wing to zero (the average of its value in the mini-Brillouin zone (mBZ).

For G-vectors on high-symmetry lines, some of the matrix elements on the left of the equation above will be zero for q=0, and therefore proportional to q. For such cases, \chi_\mathrm{wing}(G,q) \propto q^2, and the wings of the screened Coulomb interaction are constant as a function of q. However, setting the q\rightarrow 0 wings to zero still gives us, at worst, linear convergence to the final answer with increased k-point sampling, because the q\rightarrow 0 point represents an increasingly smaller area in the BZ. Thus, we still zero out the q\rightarrow 0 wings, as discussed in A Baldereschi and E Tosatti, Phys. Rev. B 17, 4710 (1978).

In the future, it may be worthwhile to have the user calculate \chi / \epsilon at two q-points (a third q-point at q=0 is known) in order to compute the linear and quadratic coefficients of each \chi_\mathrm{wing}(G,q) so that all the correct analytic limits can be taken. This requires a lot of messy code and more work for the user for only marginally better convergence with respect to k-points (the wings tend to make a small difference, and this procedure would matter for only a small set of the G-vectors).

It is important, as always, for the user to converge their calculation with respect to both the coarse k-point grid used in sigma and kernel as well as with the fine k-point grid in absorption.

Tricks and hints

  • To optimize distribution of work among MPI processors, choose the number of processors to divide:

    • n_k^2, if n_\mathrm{pes} \le n_k^2; or
    • n_k^2 n_c^2, if n_\mathrm{pes} \le n_k^2 n_c^2; or
    • n_k^2 n_c^2 n_v^2, if n_\mathrm{pes} \le n_k^2 n_c^2 n_v^2,

    where n_\mathrm{pes} is the number of MPI ranks, n_k is the number of symmetry-unfolded k-points, and n_v (n_c) is the number of valence (conduction) bands included in the kernel calculation.

  • Converters from old versions of file formats to current version are available in version 2.4.