Setting of wave-number grid and band range

In the calculation with this package, there are many kinds of wave-number grids and band ranges; 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 it 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 all cases, they are equal to the lower and the upper limit of bands that contain the Fermi level (these limits can be obtained by fermi_velocity.x). For materials that have extremely large phonon frequencies, this band range must be wider than in ordinary cases.

The lower and the upper limit for the electron-electron Coulomb term in scdft_tc : fbee, lbee

The default value [fbee=1, lbee=nbnd ($K^{el}$)] is recommended. When we check the convergence with respect to the number of bands, we reduce them from the default value.

The lower and the upper limit of bands printed by deltaf : fbfs, lbfs

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

The relation of magnitude of these bands becomes as follows:
1 ≤ fbee ≤ elph_nbnd_min ≤ fbfs ≤ lbfs ≤ elph_nbnd_max ~ nbnd (scf) ≤ lbee ≤ nbnd ($K^{el}$)

Wave-number grid

The k grid for the electronic state in scf, ph

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


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

The dependence of the numerical cost of each program on $N_{\bf k}^{\rm smooth}$ (the number of k points in this grid) is as follows:

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 q grid for phonon, the k grid for twin

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 dependence of the numerical cost of each program on $N_{\bf q}$ (the number of q in this grid) is as follows:

The numerical cost of pw.x in twin is proportional to $N_{\bf q}$.
The numerical cost for all q in Calculation of phonon frequency and deformation potential is proportional to $N_{\bf q}$.
The numerical cost for all q in Calculation of electron-phonon interaction is proportional to $N_{\bf q}^2$.
The numerical cost for all q in sctk.x is proportional to $N_{\bf q}^2$.

The k grid in Non-SCF calculation with a dense k grid [1]

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

Numerical costs for scdft and sctk.x are not so affected by $N_{\bf k}^{\rm dense}$.
The numerical cost of deltaf is proportional to $N_{\bf k}^{\rm dense}$.

The relation of these k grids becomes as follows:
\[ \begin{align} N_{\bf q} \leq N_{\bf k}^{\rm smooth} \leq N_{\bf k}^{\rm dense} \end{align} \]