# Clustered Sampling Interpolation (CSI)

The CSI allows you to obtain converged optical absorption spectra of quasi-2D materials with a relatively coarse k-point grid. Before you proceed with the CSI calculation, make sure you understand the formalism and the first mean-field calculation that has to be performed, as documented in the overview of the subsampling approach. You may also want to first compute the electronic self-energy using the NNS method.

## Pre-setup: kernel on a coarse grid

First, run kernel.x as usual on a coarser, uniform grid. Note that the converged grid for the kernel calculation is typically finer than the one used to converge epsilon and sigma calculations as in a NNS calculation. Use epsmat and esp0mat files from a usual, uniform epsilon calculation.

## Setup: setup_subsampling_csi.x

Analogous to the NNS, run setup_subsampling_csi.x format WFN_co WFN nk_fi_x nk_fi_y nk_fi_z, where format is either ASCII, BIN, or HDF5, WFN_co is the coarse wave function used to calculate the BSE kernel, and nk_fi_x nk_fi_y nk_fi_z are the find kgrid to be used in BSE. This must be an integer multiple of the coarse grid.

The output files are:

• epsilon_q0s.inp: contains part of the epsilon.inp file needed to generate epsmat for the clustered points.
• kpoints_wfnq.dat: contains a list of kpoints needed for the WFNq used to calculate epsmat.
• kpoints_sub_*.dat: contains a list of subsampled k-points surrounding each coarse point. There is one file per coarse point.
• subsample.inp: contains the header for an input file needed during absorption.

Use the k-points in kpoints_wfnq.dat to generate a WFNq file with the mean-field code.

## Mean-field and epsilon calculations

• Run epsilon using epsilon_q0s.inp as the basis for epsilon.inp and link to the WFNq file generated above. The output file is called eps0mat.h5, but it is referred to as epsmat_sub.h5 for clarity in the rest of this document.
• Generate a WFN file containing the k-points in each kpoints_sub_*.dat file. For example, for a 2x2 coarse grid, you will have 4 kpoints_sub_*.dat files, so you will need to generate 4 wavefunctions, WFN_1, WFN_2, WFN_3, WFN_4.
• Run kernel for every WFN_1, WFN_2, etc. The recommended way to do this is to have a separate directory for each WFN_* file and then set up symbolic links to the correct input files. For example: In directory_1/ link eps0mat.h5 to the same eps0mat.h5 used in the coarse kernel calculation from step 1 with ln -s epsmat_sub.h5 epsmat.h5, ln -s WFN_1 WFN_co. In directory_2/, link eps0mat.h5 to the same eps0mat.h5 used in the coarse kernel calculation, and ln -s epsmat_sub.h5 epsmat.h5 and ln -s WFN_2 WFN_co, and so forth.

## Kernel calculation

In kernel.inp, make sure to add the following flags: no_symmetries_coarse_grid, and patched_sampling_co.

Warning

When you run the kernel calculation, make sure that $n_k^2$ > number of processors, where $n_k$ is the number of k-points. This is a current limitation of how the CSI is implemented.

## Absorption calculation

• Move the generated subsample.inp file to the directory where you will run your absorption calculation.
• Open and edit subsample.inp. You can find a sample subsample.inp file that explains the file format in BSE/subsample.inp in your BerkeleyGW source directory.
• Run absorption. Link the bsemat coarse file, WFN_co is same as in the coarse kernel calculation, and WFN_fi is as usual, with a size that has to be converged. In absorption.inp include the following flag: subsample_line cutoff, where cutoff is your coarse grid spacing. For instance, if your coarse grid is 30x30x1, the cutoff is 0.0333. Everything else in absorption.inp should be the same as your previous calculation. Make sure that subsample.inp is in the same directory as absorption.inp.