# *G*{sup}`0`*W*{sup}`0` quasiparticle band structure from FHI-aims data This tutorial calculates the one-shot *GW* quasiparticle band structure of silicon. FHI-aims performs the PBE calculation and writes the GreenX/LibRPA dataset; the LibRPA driver then evaluates the quasiparticle energies on the regular k-grid and along the requested band path. ```{note} The numerical settings below are intended to demonstrate the workflow. Check basis-set, k-point, and frequency-grid convergence before using the results in production calculations. ``` ## 1. Prerequisites You need: - FHI-aims built with GreenX support (`USE_GREENX=ON`); - LibRPA built with the driver and [`LIBRPA_USE_LIBRI=ON`](); and - an MPI launcher compatible with both executables. ## 2. Generate the FHI-aims dataset The example uses PBE, a 4 × 4 × 4 k-grid, 16 minimax frequency points, and two band-path segments, L–Γ and Γ–X. The relevant part of `control.in` is: ```text # Basic model xc pbe k_grid 4 4 4 occupation_type gaussian 0.001 # GW switches qpe_calc gw_expt frequency_points 16 anacon_type 1 # Band k-paths output band 0.50000 0.50000 0.50000 0.00000 0.00000 0.00000 13 L G output band 0.00000 0.00000 0.00000 0.50000 0.00000 0.50000 13 G X # Export dataset for LibRPA output librpa [light species default for Si] ``` Most settings are the same as in a standard periodic *GW* calculation in FHI-aims; `output librpa` is the key addition. The corresponding `geometry.in` is: ```text lattice_vector 3.8301668167 0.0000000000 0.0000000000 lattice_vector 1.9150834084 3.3170217640 0.0000000000 lattice_vector 1.9150834084 1.1056739213 3.1273181102 atom_frac 0.0000000000 0.0000000000 0.0000000000 Si atom_frac 0.2500000000 0.2500000000 0.2500000000 Si ``` Place both files in a `dataset` directory and run FHI-aims there: ```bash cd dataset mpirun -np 4 /path/to/aims.x > aims.out 2> aims.err ``` The `output librpa` directive writes the structure, basis, eigenstate, RI coefficients, Coulomb matrices, dielectric head, and band-path data required by the LibRPA driver. The generated dataset for this example is about 500 MiB. Its size and generation time depend strongly on the basis and numerical grids. ## 3. Configure LibRPA Create a sibling `librpa` directory containing this `librpa.in`: ```text task = g0w0 input_dir = ../dataset nfreq = 16 option_dielect_func = 0 replace_w_head = t parallel_routing = libri ``` `task = g0w0` calculates both regular-k-grid and band-path quasiparticle energies when band-path data are present. `nfreq` must match the FHI-aims frequency grid. `replace_w_head` uses the macroscopic dielectric function written by FHI-aims to correct the head of the dielectric matrix and improve k-point convergence. ## 4. Run LibRPA Run the driver from the `librpa` directory so that `input_dir = ../dataset` resolves correctly: ```bash cd librpa export OMP_NUM_THREADS=1 mpirun -np 4 /path/to/LibRPA/build/chi0_main.exe > librpa.out 2> librpa.err ``` A successful run ends with `libRPA finished successfully`. The run used to refresh this tutorial employed four MPI ranks and one OpenMP thread per rank and took about 1.4 hours. Runtime is hardware-dependent, and additional MPI ranks or OpenMP threads may reduce it. ## 5. Inspect the quasiparticle energies The first regular-k-grid block in the refreshed `librpa.out` includes: ```text Printing quasi-particle energy [unit: eV] spin 1, k-point 1: (0.00000, 0.00000, 0.00000) ---------------------------------------------------------------------------------------------------------------------------- State occ e_mf v_xc v_exx ReSigc ImSigc e_qp ---------------------------------------------------------------------------------------------------------------------------- 1 2.00000 -1789.50277 -153.07760 -244.87703 8.90361 3.15363 -1872.39859 ... 11 2.00000 -17.67291 -12.19859 -19.21812 6.69555 -0.86361 -17.99689 12 2.00000 -5.63619 -13.27559 -14.75075 1.24875 -0.00015 -5.86260 13 2.00000 -5.63548 -13.27912 -14.75356 1.24872 -0.00015 -5.86120 14 2.00000 -5.63533 -13.27943 -14.75386 1.24867 -0.00015 -5.86108 15 0.00000 -3.07993 -11.57831 -7.20939 -3.90057 -0.01063 -2.61157 16 0.00000 -3.07901 -11.57980 -7.21005 -3.90144 -0.01064 -2.61070 17 0.00000 -3.07776 -11.58158 -7.21096 -3.90159 -0.01066 -2.60874 18 0.00000 -2.14654 -14.88107 -9.67057 -4.69196 -0.01374 -1.62800 ... ``` The columns contain: - `State`: one-based band index $n$; - `occ`: input occupation $f_{nk}$; - `e_mf`: input Kohn–Sham eigenvalue $E^{\mathrm{KS}}_{nk}$; - `v_xc`: input exchange-correlation potential $v^{\mathrm{xc}}_{nk}$; - `v_exx`: exact-exchange potential $v^{\mathrm{exx}}_{nk}$; - `ReSigc` and `ImSigc`: real and imaginary parts of the correlation self-energy $\Sigma^{\mathrm{c}}_{nk}$; and - `e_qp`: quasiparticle energy $$ E^{\mathrm{QP}}_{nk} = E^{\mathrm{KS}}_{nk} - v^{\mathrm{xc}}_{nk} + v^{\mathrm{exx}}_{nk} + \Re\Sigma^{\mathrm{c}}_{nk}. $$ All energies in this table are in eV. ## 6. Plot the band structure When band-path data are present, `task = g0w0` writes: - `KS_band_spin_.dat`: input Kohn–Sham band structure; - `EXX_band_spin_.dat`: non-self-consistent exact-exchange band structure; and - `GW_band_spin_.dat`: *GW* quasiparticle band structure. These files follow the FHI-aims band-output format. Their raw energies use LibRPA's internal energy zero, so choose and document a common reference when comparing calculations. The plot below aligns the PBE and *GW* results to their respective valence-band maxima and omits the five highest bands, which exhibit large errors from the localized resolution-of-identity approximation. For this example, both calculations yield an indirect fundamental gap, with the valence-band maximum at Γ and the conduction-band minimum along Γ–X. The gap is 0.57 eV for PBE and 1.08 eV for *GW*. These values should not be treated as fully converged. :::{figure-md} fig-band-si ![PBE and one-shot GW band structures of silicon](band.svg){align=center} PBE and one-shot *GW* band structures of silicon. ::: The [result archive](solution.tar.gz) contains the complete FHI-aims and LibRPA inputs, the refreshed outputs, the three band files, the generated figure, and the plotting script. The script uses NumPy and Matplotlib.