Setting of wave-number grid and band range

In the calculation with this package, there are many kind of the wave-number grid and the band range; It may confuse us. In this chapter, the relation and the difference between them are described.

Range of bands

• The upper limit in SCF calculation of the charge density, Calculation of phonon frequency and deformation potential : nbnd(scf)

  We should use the number specified automatically by pw.x. Therefore, we do not have to write explicitly in the input file.

• The upper limit in Calculation of screened Coulomb interaction/Spin-fluctuation: nbnd(\(K^{el}\))

  Typically, nbnd(\(K^{el}\)) should be roughly the double of nbnd(scf).

  The numerical cost of sctk.x is proportional to the square of nbnd(\(K^{el}\)).

• elph_nbnd_min, elph_nbnd_max in Calculation of electron-phonon interaction

  In almost cases, they are equal to the lower- and the upper limit of bands that contain the Fermi level (These limit can be obtained by fermi_velocity.x ). For materials that have extremely large phonon frequencies, this band range must be wider than ordinary cases.

• The lower- and the upper- limit for the electron-electron Coulomb term in scdft_tc : Calculation of transition temperature with bisection method : fbee, lbee

  The default value [fbee=1, lbee=nbnd(\(K^{el}\))] is recommended. When we check the convergence about the number of \({\bf k}\) point, we reduce them from the default value.

• The lower- and the upper limit of bands printed by deltaf : Output FermiSurfer file of gap function : fbfs, lbfs

  They are the lower- and the upper limit of bands that contain the Fermi level (with non-crossing approximation). They are computed automatically.

The relation of magnitude of these bands becomes as follows:

1 \(\leq\) fbee \(\leq\) elph_nbnd_min \(\leq\) fbfs \(\leq\) lbfs \(\leq\) elph_nbnd_max \(\approx\) nbnd(scf) \(\leq\) lbee \(\leq\) nbnd(\(K^{el}\))

Wave-number grid

• The \({\bf k}\) grid for the electronic state in SCF calculation of the charge density, Calculation of phonon frequency and deformation potential

  It is specified in the input file of pw.x as follows:

  K_POINTS automatic
  {nk1} {nk2} {nk3} 0 0 0
  

  The numerical cost for SCF calculation of the charge density and Calculation of phonon frequency and deformation potential is proportional to \(N_{\bf k}^{\rm smooth}\) (the number of \({\bf k}\) points in this grid).

• The \({\bf q}\) grid for Calculations of phonon and electron-phonon interaction, the \({\bf k}\) grid for Calculation of wave functions for the screened Coulomb interaction

  nq1, nq2, nq3 in the input of ph.x, arguments of twingrid.x, and nk1, nk2, nk3 in the input of Calculation of electron-phonon interaction must be the same.

  The \(N_{\bf q}\) (the number of \({\bf q}\) in this grid) dependence of each program becomes as follows:

  • The numerical cost of pw.x in Calculation of wave functions for the screened Coulomb interaction is proportional to \(N_{\bf q}\).

  • The numerical cost for all \({\bf q}\) in Calculation of phonon frequency and deformation potential is proportional to \(N_{\bf q}\).

  • The numerical cost for all \({\bf q}\) in Calculation of electron-phonon interaction is proportional to \(N_{\bf q}^2\).

  • The numerical cost for all \({\bf q}\) in sctk.x is proportional to \(N_{\bf q}^2\).

• The \({\bf k}\) grid in Non-SCF calculation with a dense k grid [1]

  In this calculation, the \({\bf k}\) grid should be as dense as that for the calculation of the density of states. The \(N_{\bf k}^{\rm dense}\) (the number of \({\bf k}\) in this grid) dependence of each program becomes as follows:

  • Numerical costs for scdft : SCDFT calculation at specific temperature and sctk.x are not so affected by \(N_{\bf k}^{\rm dense}\).

  • The numerical cost of deltaf : Output FermiSurfer file of gap function is proportional to \(N_{\bf k}^{\rm dense}\).

The relation of these \({\bf k}\) grid becomes as follows:

\(N_{\bf q} \leq N_{\bf k}^{\rm smooth} \leq N_{\bf k}^{\rm dense}\)