Spectrum function of G.

⚠️ Binaries read ctrlg.<sname>.toml only. Use Legacy2toml.py <sname> to convert legacy ctrl.<sname> / GWinput. See TOML migration.

How to calculate

We calculate the diagonal elements by the program hsfp0 in the mode --job=4, after a one-shot GW calculation by gw_lmfh (gw_lmfh leaves in the files __WVR.* and __WVI.*, which hsfp0 --job=4 reads). If you have sigm.*, it is read as in the case of gwsc.

We need to set and band index for which we calcualte. These are keys of [gw] in ctrlg.<sname>.toml, the same ones as for gw_lmfh.

(A) q points

The q points are those of QforGW, one q per line in the unit of 2pi/alat:

[gw]
QforGW = """
 0.0 0.0 0.0
 0.5 0.0 0.0
 1.0 0.0 0.0
"""

Without QforGW, or with QforGWIBZ = true, you will have self-energy for all irreducible k points of the mesh n1n2n3. This may be needed for A(omega). Because of the shifted-mesh method, q points which are not on the mesh can be given in QforGW.

(B) Band index

The bands are chosen by their energies. EMINforGW and EMAXforGW (eV, relative to the Fermi energy) give the range of the bands for which the self-energy is calculated:

[gw]
EMINforGW = -2.0
EMAXforGW =  3.0

(C) energy mesh

The mesh of the self-energy is (), up to twice the highest energy of the mesh of along the real axis.

Note that imaginary part of Sigma is given as the comvolution of ImW(omega) and the pole of Green's function ([gw].t_sigmaw (K) gives the width of the Fermi-Dirac smearing of the pole). Resolution for Im W (near omega=0) is set by [gw].HistBin_dw.

Run

gw_lmfh si -np 24
mpirun -np 24 hsfp0 si --job=4 > lsc_spec

Then we have SEComg.UP (DN) files. A line of the file is written in main_hsfp0.f90 by

    write(ifoutsec,"(4i5,3f10.6,3x,f10.6,2x,2f16.8,x,3f16.8)")
         iw,itq(i),ip,is, q(1:3,ip), wibz(ip), eqx(i,ip,is),
         (omega(i,iw)-ef)*rydberg(),  hartree*zsec(iw,i,ip)

This means we use energy in eV.

Table 1. Columns of SEComg.UP (SEComg.DN)

column	variable	meaning
1	iw	omega index
2	itq(i)	band index
3	ip	q point index (in the order of QforGW)
4	is	spin index
5-7	q	q vector (cartesian in 2pi/alat)
8	wibz	weight of the q point in the irreducible BZ
9	eqx	eigenvalue in eV, relative to the Fermi energy
10	(omega(i,iw)-ef)*rydberg()	omega relative to the Fermi energy
11, 12	hartree*zsec(iw,i,ip)	Self energy. real and imaginary part

The data of a band at a q point make a block; the blocks are divided by two blank lines. When you change QforGW, run gw_lmfh again (it makes the eigenfunctions at the q points of QforGW).

Cautions for q points of QforGW that are not on the mesh:

• Set EMAXforGW. At such q the eigenvalue solver returns fewer states than nband and pads the rest with 1d20; without an upper limit these padded states enter and hsfp0 stops with sxcf 222: |w-e| out of range.
• When you change EMINforGW or EMAXforGW, redo the exchange part as well (gw_lmfh from the start). The exchange files (SEXU, XCU) and the correlation file (SECU) must have the same bands; otherwise hqpe stops at the end of SEXU.
• The standard output of hgw and hx0fp0 has lines epsWVR: iq iw omg eps(wFC) eps(woLFC) for the small q around : there shows the plasmon (Re ) without an extra run. A level that falls on a sharp plasmon structure is sensitive to the real-axis mesh; check it with a finer HistBin_ratio (e.g. 1.03 → 1.015).

• Example.

  plot 'SEComg.UP' u ($10):($11) w l,'' u ($10):($12) w l

can give a plot for Re (Sigma_c(omega)) and Im(Sigma_c) (columns of Table 1).

4

To get integrated spectrum function (DOS), we need to superpose all the spectrum function (All q points and all band index). Be careful about the degeneracy (multiplicity) for each q points. You have to build it from SEComg file. The weight of a q point is in the 8th column (Table 1); see also the lines with the keyword Multiplicity in the console output of qg4gw (lqg4gw).

Anyway, consider about ``is it worth to do?'' To confirm your result, use sum rule (sum of spectrum weight). And pay attention to the relation between real and imag parts (Hilbert transformation).