Making the dynamical GW self energy
Table of Contents
 Table of Contents
 Preliminaries
 Command summary
 Introduction
 Theory
 Make the GW dynamical selfenergy
 spectral, the selfenergy translator
 Dynamical selfenergy editor lmfgws
 Compare interacting and independentparticle densityofstates in Fe
 Spectral Function of Fe near the H point
 Interacting joint DensityofStates and Optics
 Interacting band structure
Preliminaries
This tutorial assumes you have completed a QSGW calculation for Fe, following this tutorial, which requires that the GW script lmgwsc is in your path, along with the executables it requires. In addition it requires spectral and lmfgws.
This tutorial assumes the your build directory is ~/lm, and the executables are in ~/bin and ~/bin/code2.
Command summary
Repeat the steps for LDA selfconsistency and QSGW selfconsistency in the Fe tutorial; see Command summary.
If you have already done so without removing any files, you can skip those steps.
If on the other hand you have retained the QSGW selfenergy $\Sigma^0$ (file sigm), make sure your charge density is selfconsistent.
lmf fe > out.lmf
If you also retained the density restart file rst.fe this step is not necessary. Make all the inputs (e.g. screened coulomb interaction) up to the selfenergy step:
lmgwsc wt code2 sym metal tol=1e5 getsigp stop=sig fe
This will give you everything you need to make $\Sigma(\mathbf{k},\omega)$.
Make the spectral function *files SEComg.UP (and SEComg.DN)
env OMP_NUM_THREADS=8 MKL_NUM_THREADS=8 ~/bin/code2/hsfp0_om job=4 > out.hsfp0
Postprocessing step translating SEComg.{UP,DN} into lmfgwsreadable form
spectral ws nw=1
ln s se se.fe
... to be finished
Introduction
This tutorial starts after a QSGW calculation for Fe has been completed, in this tutorial.
The QSGW static selfenergy was made with the following command:
$ lmgwsc wt code2 sym metal tol=1e5 getsigp fe
Note: until that tutorial is written, perform the setup as follows (where ~/lm is your Questaal source directory)
~/lm/gwd/test/test.gwd mpi=#,# fe 4
This tutorial will do the following:
 Generate spectral functions directly from files SEComg.UP and SEComg.DN. (For nonmagnetic calculations, only SEComg.UP is made).
 Use lmfgws to generate the interacting densityofstates (DOS) from Im G, compare it to the noninteracting DOS from Im G_{0} and to the noninteracting DOS generated as an output of an lmf band calculation.
 Use lmfgws to generate to calculate the spectral function A(k,ω) for k near the H point, and also simulate photoemission spectra
Theory
Z factor renormalization
Begin with a noninteracting Green’s function $G_0$, defined through an hermitian, energyindependent exchangecorrelation potential $V^j_{xc}(k)$. $j$ refers to a particular QP state (pole of $G_0$). There is also an interacting Green’s function, $G$.
The contribution to $G_0$ from QP state $j$ is
$G_0^j(k,\omega ) = \frac{1}{\omega  \omega^j(k)}$where $\omega^j(k)$ is the pole of $G_0$.
Write the contribution to $G$ from QP state $j$ as
$G^j(k,\omega) = \frac{1}{\omega  \omega^j  \Sigma (k,\omega ) + V^j_{xc}(k)}$Note that this equation is only true if $\Sigma$ is diagonal in the basis of noninteracting eigenstates. We will ignore the nondiagonal elements of $\Sigma(k,\omega)$. Note that if $V^j_{xc}(k)$ is constructed by QSGW, this is a very good approximation, since ${\mathrm{Re}\Sigma (k,\omega ){=}V^j_{xc}(k)}$ at $\omega{=}\omega^j(k)$. Approximate G by its coherent part:
$G^{j,\mathrm{coh}}(k,\omega) = \frac{1}{\omega  \omega^j  \mathrm{Re} \Sigma (k,\omega^j) + V^j_{xc}(k)  (\omega  \omega^j)(1  1/Z^j)  i\mathrm{Im} \Sigma (k,\omega )}$where
$1  1/{Z^j} = \left. {\partial \Sigma (k,\omega )/\partial \omega } \right_{\omega ^j } .$defines the $Z$ factor. The dependence of $\omega^j$ and $Z^j$ on $k$ is suppressed.
Define the QP peak as the value of $\omega$ where the real part of the denominator vanishes.
$({\omega^*}  \omega^j)/Z^j = \mathrm{Re} \Sigma(k,\omega^j)  V^j_{xc}(k)$and so
${\omega^*} = \omega^j + Z^j\left( {\mathrm{Re} \Sigma (k,\omega^j)  V^j_{xc}(k)} \right)$Note that in the QSGW case, the second term on the r.h.s. vanishes by construction: the noninteracting QP peak corresponds to the (broadened) pole of G.
The group velocity is $d\omega/dk$. For the interacting case it reads
$\frac{d\omega^*}{dk} = \frac{d\omega ^j }{dk} + \frac{d}{dk}Z^j \left( {\text{Re}\Sigma(k,\omega^j)  V_{xc}^j (k)} \right)$Use the ratio of noninteracting and interacting group velocities as a definition of the ratio of inverse masses. From the chain rule
$\frac{m_0}{m^*} \equiv \frac{d\omega^*}{dk}/\frac{d\omega^j}{dk} = 1 + Z^j \left( \frac{\partial}{\partial\omega} \left. \text{Re}\Sigma (k,\omega ^j ) \right_{\omega^j} \frac{d\omega ^j }{dk} + \frac{\partial}{\partial k}\left. \text{Re}\Sigma(k,\omega^j) \right_{\omega ^j }  \frac{\partial }{\partial k}V_{xc}^j (k) \right) + \left(\frac{dZ^j}{dk}\right) \left(\text{Re}\Sigma(k,\omega^j)  V_{xc}^j (k) \right)$Ignore the dependence of Z^{j} on k. Write dω^{j}/dk as $v_0^j$, and use the definition of Z to get
$\frac{m_0}{m^*} = 1 + \frac{1}{v_0^j}Z^j \left( {\left( {1  1/{Z^j}} \right) v_0^j + \frac{\partial}{\partial k} \left.\text{Re}\Sigma(k,\omega^j) \right_{\omega^j}  \frac{\partial }{\partial k}V_{xc}^j(k)} \right)$So
$\frac{m_0}{m^*} = Z^j + \frac{Z^j}{v_0^j }\left( {\frac{\partial}{\partial k}\left. \text{Re}\Sigma (k,\omega^j) \right_{\omega ^j }  \frac{\partial }{\partial k}V_{xc}^j (k)} \right)$In the QSGW case the quantity in parenthesis vanishes. Thus QSGW there is no “mass renormalization” from the ωdependent selfenergy, Σ(ω).
Coherent part of the spectral function
Write $G^{j,\mathrm{coh}}(k,\omega)$ as
$G^{j,\mathrm{coh}}(k,\omega) = \left[(\omega  \omega^j){Z^j}^{1}  \mathrm{Re} \Sigma (k,\omega^j) + {V_{xc}}(k)  i\mathrm{Im} \Sigma (k,\omega )\right]^{1}$Rewrite as
$G^{j,\mathrm{coh}}(k,\omega) = \frac{Z^j}{\omega  \omega^*  iZ\mathrm{Im} \Sigma (k,\omega )} = Z^j\frac{\omega  \omega^* + iZ\mathrm{Im} \Sigma (k,\omega )}{(\omega  {\omega^*})^2 + (Z^j\mathrm{Im} \Sigma (k,\omega ))^2}$Using the standard definition of the spectral function, e.g. Hedin 10.9:
$A(\omega ) = {\pi ^{  1}}\left {\mathrm{Im} G(\omega )} \right$the approximate spectral function is
which shows that the spectral weight of the coherent part is reduced by Z.
$A_k^{j,\mathrm{coh}}(\omega ) = \frac{Z^j}{\pi}\frac{Z^j\mathrm{Im} \Sigma (k,\omega )}{(\omega  {\omega^*})^2 + (Z^j\mathrm{Im} \Sigma (k,\omega ))^2}$Simulation of Photoemission
(needs cleaning up)
Energy conservation : requires (see Marder, p735, Eq. 23.58)
$\hbar\omega=E_{kin}+{\varphi_s}E_b$where E_{b} is the binding energy and $E_{kin}+{\varphi_s}$ is the energy of the electron after being ejected. (Marder defines $E_{b}$ with the opposite sign, making it positive).
Momentum conservation : The final wave vector k_{f} of the ejected electron must be equal to its initial wave vector, apart from shortening by a reciprocal lattice vector to keep k_{f} in the first Brillouin zone.
Let $E_{kin}$ be the energy on exiting the crystal, $\varphi_s$ the work function and $E_b$ and $V_0$ are called the electron binding energy and “inner potential.”
Then
$\frac{\hbar^2}{2m}(k_\parallel^2 + k_\bot^2) = E_{kin} + V_0, \text{ where } E_{kin} = \hbar \omega  \varphi_s + E_b$The total momentum inside the crystal, $\mathbf{k}_\parallel{+}\mathbf{k}_\bot$, is linked to the kinetic energy measured outside the crystal through Eq.(1). The kinetic energy is linked to the binding energy through the equation ${E_{kin}}=\hbar\omega{E_b}{\varphi_a}$ where ${\varphi_a}$ is the work function of the analyzer. Usually ${\varphi_a}{=}{\varphi_s}$. The Fermi level is defined such that $E_b{=}0$. The inner potential is defined by scanning the range of photon energy under the constraint of normal emission: then the $\Gamma$point can be identified and by using Eq.(1), and the inner potential experimentally determined.
The momentum of the particle in free space is
$\frac{\hbar ^2 k_0^2 }{2m} = E_{kin}$Resolve $\mathbf{k}_f$ into components parallel and perpendicular to the surface
$\mathbf{k}_f = \mathbf{k}_\parallel + \mathbf{k}_\bot$After passing through the surface, $\mathbf{k}_f$ is modified to $\overline{\mathbf{k}}_f$; this is what is actually measured.
The conservation condition requires
$k_0^2 = \bar k_\parallel^2 + \bar k_\bot^2$$\mathbf{k}_\parallel$ is conserved on passing through the surface; thus $\bar k_\parallel{=} k_\parallel$. $\mathbf{k}_\bot$ is not conserved; therefore
$\bar k_\bot = \sqrt{k_0^2{}k_\parallel^2}$The wave number shift is then
$\Delta{\mathbf{k}} = (\overline{k}_\bot{k}_\bot)\hat{\mathbf{k}}_\bot$and the crystal momentum actually being probed by the experiment is
${\mathbf{k}}_f = \overline{\mathbf{k}}_f  \Delta{\mathbf{k}}$Make the GW dynamical selfenergy
The 1shot GW selfenergy maker, hsfp0, has a mode (job=4) make the dynamical Σ(k,ω). Some changes to GWinput are needed. lmfgwd will automatically make these changes if you used switch sigw in the QSGW tutorial.
With your text editor, modify GWinput. Change these two lines:
 Specify qp and band indices at which to evaluate Sigma
into these four lines:
***** Specify the q and band indices for which we evaluate the omega dependence of selfenergy 
0.01 2 (Ry) ! dwplot omegamaxin(optional) : dwplot is mesh for plotting.
: this omegamaxin is range of plotting omegamaxin to omegamaxin.
: If omegamaxin is too large or not exist, the omegarange of W by hx0fp0 is used.
Also change these lines
*** Sigma at all q >1; to specify q >0. Second arg : up only >1, otherwise 0
0 0
to
*** Sigma at all q >1; to specify q >0. Second arg : up only >1, otherwise 0
1 0
If you have removed intermediate files, you must remake them up to the point where the selfenergy is made. Do:
$ lmgwsc wt code2 sym metal tol=1e5 getsigp stop=sig fe
This step is not necessary if you have completed the QSGW Fe tutorial without removing any files.
The next step will make Σ(k_{n},ω) on a uniform energy mesh −2 Ry < ω < 2 Ry, spaced by 0.01 Ry at irreducible points k_{n}, for QP levels specified in GWinput. This is a fairly fine spacing so the calculation is somewhat expensive.
 Run hsfp0 (or better hsfp0_om) in a special mode job=4 to make the dynamical selfenergy.
export OMP_NUM_THREADS=8
export MPI_NUM_THREADS=8
~/bin/code2/hsfp0_om job=4 > out.hsfp0
This step should create SEComg.UP and SEComg.DN. These files contain Σ(k,ω), albeit in a not particularly readable format.
spectral, the selfenergy translator
spectral is a postprocessor that reads SEComg.UP (and SEComg.DN in the spin polarized case).
Its main purpose is to translate these files into file se.ext which lmfgws can read.
SEComg.UP and SEComg.DN contain the diagonal matrix element $\Sigma_{jj}(\mathbf{k},\omega)$ for each QP level j, for each irreducible point k_{n} in the Brillouin zone, on a uniform mesh of points ω as specified in the GWinput file of the last section. If the absence of interactions, $\Sigma_{jj}(\mathbf{k},\omega)=0$ so the spectral function would be proportional to δ(ω−ω^{*}), where ω* is the QP level (see Theory section).
Interactions give $\Sigma_{jj}(\mathbf{k},\omega)$ an imaginary part which broadens out the level, and in general, $\mathrm{Re}\Sigma_{jj}(\mathbf{k},\omega)$ shifts and renormalizes the quasiparticle weight by Z. As noted in the Theory section, there is no shift if $V_\mathrm{xc}^j$ is the QSGW selfenergy $\Sigma_{jj}^0(\mathbf{k},\omega)$; there remains, however, a reduction in the quasiparticle weight. This will be apparent when comparing the interacting and noninteracting DOS.
1. Setup for lmfgws
Starting from SEComg.UP (and SEComg.DN in the magnetic case) generated by hsfp0, use spectral to generate the se file as described in the lmfgws tutorial below.
2. Use spectral to directly generate spectral functions for q=0
spectral also has a limited ability to directly generate spectral functions from raw output SEComg.{UP,DN} which this section demonstrates.
Do the following:
$ spectral eps=.005 domg=0.003 'cnst:iq==1&eqp>10&eqp<30'
Commandline arguments are described here. In this context they have the following meaning:

eps=.005 : 0.005 eV is added to the imaginary part of the selfenergy. This is needed because as ω→0, ImΣ→0. Peaks in A(k,ω) become infinitely sharp for QP levels near the Fermi level.

domg=.003 : interpolates Σ(k_{n},ω) to a finer frequency mesh. ω is spaced by 0.003 eV. The finer mesh is necessary because Σ varies smoothly with ω, while A will be sharply peaked around QP levels.

cnst:expr : acts as a constraint to exclude entries in SEComg.{UP,DN} for which expr is zero.
expr is an integer expression using standard Questaal syntax for algebraic expressions. It can that can include the following variables: ib (band index)
 iq (kpoint index)
 qx,qy,qz,q (components of q, and amplitude)
 eqp (quasiparticle energy, in eV)
 spin (1 or 2)
The expression in this example, iq==1&eqp>10&eqp<30, does the following:
generates spectral functions only for the first k point (the first k point is the Γ point)
eliminates states below the bottom of the Fe s band (i.e. shallow core levels included in the valence through local orbital)
eliminates states 30 eV or more above the Fermi level.
spectral writes files sec_ibj_iqn.up and sec_ibj_iqn.dn which contain information about the G for band j and the k point k_{n}. A sec files takes the following format:
# ib= 5 iq= 1 Eqp= 0.797925 q= 0.000000 0.000000 0.000000
# omega omegaEqp Re sigmvxc Im sigmvxc int A(w) int A0(w) A(w) A0(w)
0.2721160D+02 0.2641368D+02 0.6629516D+01 0.1519810D+02 0.2350291D04 0.6897219D08 0.7774444D02 0.2281456D05
0.2720858D+02 0.2641065D+02 0.6629812D+01 0.1520157D+02 0.4701215D04 0.1379602D07 0.7776496D02 0.2281979D05
...
spectral also makes the kintegrated DOS. However, the k mesh is rather coarse and a better DOS can be made with lmfgws. See below.
spectral: read 29 qp from QIBZ
Dimensions from file(s) SEComg.(UP,DN):
nq=1 nband=9 nsp=2 omega interval (27.2116,27.2116) eV with (200,200) points
Energy mesh spacing = 136.1 meV ... interpolate to target spacing 3 meV. Broadening = 5 meV
Spectral functions starting from band 1, spin 1, for 9 QPs
file Eqp int A(G) int A(G0) rat[G] rat[G0]
sec_ib1_iq1.up 8.743948 0.8473 0.9999 T T
sec_ib2_iq1.up 1.674888 0.8251 0.9999 T T
sec_ib3_iq1.up 1.674819 0.8251 0.9999 T T
sec_ib4_iq1.up 1.674753 0.8251 0.9999 T T
sec_ib5_iq1.up 0.795683 0.8278 0.9999 T T
sec_ib6_iq1.up 0.795556 0.8278 0.9999 T T
sec_ib7_iq1.up 24.572881 0.7374 0.9994 T T
sec_ib8_iq1.up 24.572882 0.7374 0.9994 T T
sec_ib9_iq1.up 24.572884 0.7374 0.9994 T T
writing qintegrated dos to file dos.up ...
Spectral functions starting from band 1, spin 2, for 9 QPs
file Eqp int A(G) int A(G0) rat[G] rat[G0]
sec_ib1_iq1.dn 8.458229 0.8447 0.9998 T T
sec_ib2_iq1.dn 0.015703 0.8718 0.9999 T T
sec_ib3_iq1.dn 0.016072 0.8700 0.9999 T T
sec_ib4_iq1.dn 0.016437 0.8688 0.9998 T T
sec_ib5_iq1.dn 1.552938 0.8363 0.9998 T T
sec_ib6_iq1.dn 1.553722 0.8364 0.9999 T T
sec_ib7_iq1.dn 24.695801 0.7317 0.9994 T T
sec_ib8_iq1.dn 24.695832 0.7317 0.9994 T T
sec_ib9_iq1.dn 24.695865 0.7317 0.9994 T T
writing qintegrated dos to file dos.dn ...
Dynamical selfenergy editor lmfgws
lmfgws is the dynamical selfenergy editor, which peforms a variety of postprocessing of the GW or DMFT selfenergy $\Sigma(\mathbf{k}_n,\omega)$ for different purposes. The collection of points k_{n} are typically supplied for a regular mesh. This need not be the case, but when the points do not correspond to a regular mesh the parts of the editor that require k interpolation are not allowed. You must tell the editor that you are not using a uniform mesh (see irrmesh in the instructions below).
lmfgws requires the same files lmf needs to compute the QSGW band structure, e.g. ctrl.ext and sigm.ext. In addition it requires the dynamical selfenergy se.ext in special a format written by the spectral utility.
Setting up lmfgws with the spectral utility
For definiteness this section assumes that ext is fe. Starting from SEComg.UP (and SEComg.DN in the magnetic case) generated by hsfp0, use spectral to generate se.fe:
$ spectral ws nw=1
$ ln s se se.fe
 ws tells spectral to write the selfenergy to file se for all k points, in a special format designed for lmfgws. Individual files are not written. It must be renamed to se.ext for use by lmfgws.
 nw=1 tells spectral to write the selfenergy on the frequency mesh it was generated; no interpolation takes place.
Try starting the dynamical selfenergy editor:
$ lmfgws ctrl.fe `cat switchesforlm` sfuned
You should see:
Welcome to the spectral function file editor. Enter '?' to see options.
Option :
The editor operates interactively. It reads a command from standard input, executes the command, and returns to the Option prompt waiting for another instruction. The editor will print a short summary of instructions if you type ? <RET>.
Editor instructions
This sections documents the instruction set of the dynamical selfenergy editor. Codes that can generate the input for this editor are spectral and lmfdmft.
 readsek[flags]  readsekb[flags] [fn]
reads the selfenergy from an ASCII (or binary) file. In the absence fn, the file name defaults to the ASCII file se.ext (readsek), or the binary seb.ext (readsekb).
The structure of the file is documented here. Data is read in the basis of 1particle eigenfunctions for whatever states are supplied in the file. Some points of note: Data is stored for a collection of k points; the list of points is written in the file. These points may, or may not constitute a uniform mesh of points.
 QP levels are stored relative to the chemical potential (which may, but need not, be written in the header).
 Only the diagonal elements of the potentials are read. The full complement of static potentials consist of the static QSGW selfenergy $\Sigma^0$, the Fock exchange $V_\mathrm{xx}$, and $V_\mathrm{xc}^\mathrm{LDA}$.
 The se file may, but need not, contain these potentials. For example, none are supplied by lmfdmft.
 Optional flags are strung together and separated by a delimiter, taken is the first character, e.g. @ .
Note: if you are operating the editor in batch mode, be sure to distinguish this delimiter from the batch mode delimiter. @fn=nam use nam for selfenergy file name
 @useef file chemical potential becomes Fermi level
 @irrmesh points are not on a regular k mesh : no k interpolations allowed
 @ib=list after reading data from file, pare bands read from file to those in list
 @minmax print minimum and maximum QP levels for each band
 @dc=# subtract double counting # from Re sigma(omega) after reading
 @makeqpse Not documented yet

units Ry  units eV
Select Rydberg units or electron volt units (default=Ry).
Note: the se file can store data in either eV or Ry units; lmfgws internally converts it to whatever units you select. 
evsync
replace quasiparticle levels read from se.ext by recalculating them with the same algorithm lmf uses. 
eps val
add a constant val to Im Σ, needed to broaden spectral functions so that integrations are tractable. 
ef Ef  ef=Ef
Use Ef for the Fermi level, overriding the internally calculated value.
Note: the order in which you use this switch is important. If you use the ef switch before readsek, QP levels are shifted by μ−Ef when they are subsequently read (provided the chemical potential μ is supplied in the se file). If you use this switch after readsek, no shifts are added. In such a case you likely want to realign the QP levels with evsync after readsek. Always enter Ef in Ry units.
 dosjdosimeps [nq=#1,#2,#3] ib=list [getev[=#1,#2,#3]] [nw=#domg=#] [range=#1,#2] [isp=#]
dos integrates the spectral function to make both the QP and spectrum DOS.
jdos integrates either the QP or interacting spectral function to make the joint DOS.
imeps integrates either the QP or interacting spectral function to make Imε.
Options are: ib=lst restrict contribution to spectra from QP states in list.
 nq=#1,#2,#3 Interpolate Σ_{j}(k_{n},ω) to a new uniform mesh of k points, defined by (#1,#2,#3) divisions. Use between 1 and 3 numbers.
 nw=n Refine the given energy mesh by interpolating Σ to an n multiple of the given energy mesh. n must be an integer.
 range=#1,#2 Generate DOS in a specified energy window (#1,#2), in eV.
 kT Temperature, units of omega (applies only to jdos and imeps).
 a0 Spectra for noninteracting spectral function (only for jdos and imeps).
 isp=# Generate DOS for spin # (1 or 2). Default value is 1.
 se iq=nq=#1,#2,#3allqband[~args] ib=listibx=list [getev[=#1,#2,#3]] [nw=ndomg=#] [isp=#] [range=#1,#2]
Make Σ(ω) and A(ω) for given q and range of bands.
Required arguments are: iq=n index to q_{n}, from list in QIBZ. Alternatively specify q by:
 q=#1,#2,#3 qpoint in units of 2π/alat. lmfgws will interpolate Σ(q_{n}) to any q.
 allq Σ(ω) is made for all q in the irreducible BZ and written to disk
 band A(ω), Σ(ω) are made for qp along symmetry lines and written to disk.
Use this mode to draw interacting energy bands, in conjunction with plbnds −sp
Optional ~args are parsed like options of the band switch  ib=list Sum together A^{j}(ω) derived from QP states j in list.
ibx=list is similar to ib=list, but A^{j}(ω) is resolved by band, writing each A^{j}(ω) in succession.
Options are:  getev Do not interpolate energy but calculate it at q.
 getev=#1,#2,#3 Generate evals on independent mesh with #1,#2,#3 divisions of uniformly spaced points.
 nw=n Refine the given energy mesh by interpolating Σ to an n multiple of the given energy mesh. n must be an integer.
 range=#1,#2 Generate spectral function in a specified energy window (#1,#2)
 pepeqp iq=nq=#1,#2,#3 ib=# [getev[=#1,#2,#3]] [nw=#domg=#] [nqf=#] [ke0=#] [isp=i] [range=#1,#2]
Model ARPES for given q and band(s).
pe uses the spectrum selfenergy, while peqp uses just the quasiparticle hamiltonian. Finalstate effects are folded into both. Only the latter works with SO coupling now.
Required arguments are: iq=n index to q_{n}, from list in QIBZ. Alternatively specify q by:
 q=#1,#2,#3 qpoint in units of 2π/alat. lmfgws will interpolate Σ(q_{n}) to any q.
 ib=list Sum together PE spectrum derived from QP states j in list. See here for the syntax of integer lists.
Options are:  getev Do not interpolate energy but calculate it at q.
 getev=#1,#2,#3 Generate evals on independent mesh with #1,#2,#3 divisions of uniformly spaced points.
 nw=n Refine the given energy mesh by interpolating Σ to an n multiple of the given energy mesh. n must be an integer.
 isp=i Generate spectra for spin i (1 or 2). Default value is 1.
 nqf=n number of mesh points for final state integration. Default is 200.
 ke0=# kinetic energy of emitted electron. KE+V0=ℏω−φ_{s}+V_{0}
 range=#1,#2 Generate spectral function in a specified energy window (#1,#2)

qpse
Generates the Quasiparticle “selfenergy” (in practice the QP levels relative to the Fermi level). 
savesea [fn]
saves spectrum DOS or selfenergy + spectral function, in ASCII format. In the absence fn, the file name defaults to seia.ext or seia2.ext when writing band and kresolved spectral functions (se or pe) and to sdos.ext or sdos2.ext when writing spectrum dos (dos). 
savese [fn]
saves qinterpolated selfenergy + spectral function in binary format. In the absence fn, the file name defaults to seib.ext.  q
quits the editor unless information has generated that has not been saved. Program terminates.  a
(abort) unconditionally quits the editor. Program terminates.
You can also run the editor in batch mode by stringing instructions together separated by a delimiter:
$ lmfgws ctrl.fe `cat switchesforlm` 'sfuned~first command~second command~...'
The delimiter ( ~ in this case), is the first character following sfuned. lmfgws will parse through all the commands sequentially until it encounters “quit” instruction ( ~a or ~q ) which causes it to exit. If no exit instructions are encountered, lmfgws returns to the interactive mode and prompts you with Option : .
Compare interacting and independentparticle densityofstates in Fe
This section uses the selfenergy editor, lmfgws, to interpolate Σ(k_{n},ω) to a fine k and ω mesh to obtain a reasonably well converged densityofstates.
$ lmfgws fe `cat switchesforlm` 'sfuned~units eV~readsek~eps .030~dos isp=1 range=10,10 nq=32 nw=30~savesea~q'
This invocation runs lmfgws in batch mode, and writes the spectral and noninteracting DOS to file sdos.fe. The editor’s instructions do the following (documented here):
 units eV
Set units to eV; spectrum DOS will be written in eV.  readsek
Read se.fe  eps .030
Add 30 meV smearing to Im Σ  dos isp=1 range=10,10 nq=32 nw=30
Make the DOS for spin 1, in the energy range (10,10) eV, interpolating Σ to a k mesh 32×32×32 divisions, and refining the energy mesh by a factor of 30. The asgiven k mesh is 8×8×8 divisions.  savesea
Write the DOS.  q
Exit the editor.
Notes:
 The mesh is very fine, so the interpolation takes a little while (2 to 3 minutes). The frequency and k meshes are both pretty fine and the DOS is rather well converged, as the figure below demonstrates.
 The spectrum DOS is written to file sdos.fe. Columns 1,2,3 are ω, A(ω), and A_{0}(ω), respectively.
 A_{0}(ω) should compare directly to the DOS calculated as a byproduct of lmf.
You can make the QP DOS yourself, but to speed things up just copy it from the build directory to your working directory.
cp ~/lm/gwd/test/fe/dosp.fe dosp.fe
The following script draws a figure comparing the DOS generated the three different ways, using the fplot utility. Cut and paste the contents of the box below into script file plot.dos.
% char0 ltdos="1,bold=3,col=0,0,0"
% var ymax=1.4 dy=0.4 dw=.00 ymax+=dy emin=10 emax=5 ef=0
fplot
% var ymax=dy+dw dy=0.4 dmin=0 dmax=3
frme 0,1,{ymaxdy},{ymax} p0 x {emin},{emax} y {dmin},{dmax} tmy 1 1p
colsy 3 lt 1,bold=3,col=.5,.5,.5 sdos.fe
colsy 2 lt {ltdos} ord y qr dosp.fe
colsy 2 lt 1,bold=3,col=1,0,0 sdos.fe
lt 2,bold=3,col=0,0,0,2,.5,.05,.5 tp 2~{ef},{dmin},{ef},{dmax}
$ fplot f plot.dos
$ open fplot.ps [choose your postscript file viewer]
Notes on the figure:
 The black line (col=0,0,0) is the noninteracting DOS generated by lmf.
 The grey line (col=.5,.5,.5) is the noninteracting DOS A_{0}(ω), generated by lmfgws
 The red line (col=1,0,0) is the interacting DOS A(ω), generated by lmfgws
 Grey and black lines nearly coincide, as they should if the DOS is well converged. Note that the black line was generated from energy bands with the tetrahedron method, the other effectively by integrating G_{0}(k,ω) by sampling with a smearing of 30 meV.
 The noninteracting DOS at the Fermi level is D(E_{F})≅1/eV (one spin). The Stoner criterion for the onset of ferromagnetism is I×D(E_{F})>1, where I is the Stoner parameter, which DFT predicts to be approximately 1 eV for 3d transition metals. Combining DOS for the two spins would indicate that the Stoner criterion is well satisfied.
 The interacting DOS is smoothed out, and is roughly half the amplitude of the noninteracting DOS. This is also expected: the Z factor for the d states is about 0.5.
Spectral Function of Fe near the H point
This example computes the selfenergy for a q point near the H point. It is calculated from band 2 for the majority spin and bands 2,3 for the minority spin. These bands were chosen because of their proximity to the Fermi level.
$ lmfgws fe `cat switchesforlm` 'sfuned~units=eV~eps .01~readsek~evsync~se q=1.05,2.91,1.01 ib=2 nw=10 getev=12 isp=1~savesea~q'
$ lmfgws fe `cat switchesforlm` 'sfuned~units=eV~eps .01~readsek~evsync~se q=1.05,2.91,1.01 ib=2,3 nw=10 getev=12 isp=2~savesea~q'
The first command writes a file seia.fe, the second seia2.fe
The following makes a picture comparing A (solid lines) and A_{0} (dashed lines), majority spin (black) and minority spin (red)
$ fplot x 9,5 y 0,1 colsy 2 lt 1,col=0,0,0 seia.fe colsy 3 lt 2,col=0,0,0 seia.fe colsy 2 lt 1,col=1,0,0 seia2.fe colsy 3 lt 2,col=1,0,0 seia2.fe
$ open fplot.ps [choose your postscript file viewer]
You should see a weak plasmon peak in the majority spin band near −8 eV.
Interacting joint DensityofStates and Optics
lmfgws can make the joint densityofstates (JDOS) and the macroscopic dielectric function. The joint DOS is given by
$D(\omega ) = \sum\nolimits_{\mathbf{k}} {\int {d\omega'[f(\omega)  f(\omega' + \omega )]\sum\nolimits_{ij} {\{ A_i ({\mathbf{k}},\omega' + \omega )A_j ({\mathbf{k}},\omega')\} } } }$Note that $D$ is a (weak) function of temperature since the Fermi function $f(\omega)$ contains temperature.
In the limit of noninteracting particles and $k_BT{\rightarrow}0$ this expression reduces to the standard expression for joint densityofstates
$D(\omega) \rightarrow \sum\limits_{\mathbf{k}} {\sum\limits_{i=\mathrm{occ}\atop j=\mathrm{unocc}} {\delta (\omega _i  \omega _j  \omega )} }$The following computes joint DOS (noninteracting case) using the lmf optics package. It renames the file for future comparison.
lmf vnk=32 fe `cat switchesforlm` vlteto=0 voptmod=1 quit=rho
cp jdos.fe jdoslmf.fe
The following computes joint DOS for both static and interacting QS_GW_ selfenergies, using lmfgws.
lmfgws fe `cat switchesforlm` 'sfuned~units eV~readsek~eps .040~jdos range=10,10 nq=32 a0 nw=5~savesea~q'
lmfgws fe `cat switchesforlm` 'sfuned~units eV~readsek~eps .040~jdos range=10,10 nq=32 nw=5~savesea~q'
The first command makes file jdosni.fe, the second jdos.fe.
fplot ab 'x1*13.6' colsy 2,3 ord y/13.6 jdoslmf.fe lt 2,col=1,0,0 colsy 2,3 jdosni.fe lt 3,bold=5,col=0,1,0 colsy 2,3 jdos.fe
In the absence of a vertex, $\mathrm{Im} \epsilon(\omega)$ is proportional to the joint DOS, decorated by the matrix elements of velocity operator, $\lvert\langle {v_{ij}}\rangle\rvert^2$. The latter is usually calculated in terms of the momentum operator $\lvert\langle {p_{ij}}\rangle\rvert^2$. In Ry units $\mathrm{Im} \epsilon(\omega)$ reads
$\mathrm{Im} \epsilon(\omega ) = \frac{32\,\pi^2}{\omega^2 V} \sum\nolimits_{\mathbf{k}} {\sum\nolimits_{ij} \lvert\left< p_{ij}\right>\rvert^2 \int {d\omega'[f(\omega)  f(\omega' + \omega )] {\{ A_i ({\mathbf{k}},\omega' + \omega )A_j ({\mathbf{k}},\omega')\} } } }$The following makes $\mathrm{Im}\epsilon(\omega)$ using lmf with gaussian sampling integration.
lmf vnk=32 fe `cat switchesforlm` vlteto=0 voptmod=1 quit=rho
cp opt.fe optlmf.fe
The following computes $\mathrm{Im}\,\epsilon(\omega)$ for both static and interacting QSGW selfenergies, using lmfgws.
lmf vnk=32 fe `cat switchesforlm` vlteto=0 voptmod=1 vmefac=0 quit=rho opt:woptmc
lmfgws fe `cat switchesforlm` 'sfuned~units eV~readsek~eps .040~imeps range=10,10 nq=32 a0 nw=5~savesea~q'
lmfgws fe `cat switchesforlm` 'sfuned~units eV~readsek~eps .040~imeps range=10,10 nq=32 nw=5~savesea~q'
The lmf instructions generates stores matrix elements of the velocity operator in file optdatac.fe for lmfgws. The latter commands generate optni.fe and opt.fe. They are calculated in the same way, but for spectral functions from the static and interacting QSGW selfenergies.
Draw a picture of the three independent calculations of $\mathrm{Im}\,\epsilon$:
fplot frme 0,1,0,.7 frmt th=3,1,1 xl "~{w} (eV)" x 0,10 y 0,32 ab 'x1*13.6' colsy 2,5 optlmf.fe lt 2,col=1,0,0 colsy 2,5 optni.fe lt 3,bold=5,col=0,1,0 colsy 2,5 opt.fe
Note that lmf uses Ry units; we specified eV in the lmfgws instruction. Thus when comparing $\mathrm{Im}\,\epsilon$, the abscissa from for optlmf.fe must be scaled (alternatively tell lmfgws to use Ry units).
You should see something similar to the figure shown below (the figure shown is smoother because nq=64 divisions were used for the k mesh). For all three data, contributions are resolved into majority and minority parts. The physically relevant $\mathrm{Im}\,\epsilon$ is the sum of the two.
Black ($\mathrm{Im}\,\epsilon$ computed by lmf) and Red ($\mathrm{Im}\,\epsilon$ computed by lmfgws, noninteracting case) are very similar. Dotted green is the corresponding $\mathrm{Im}\,\epsilon$ computed in the RPA with the dynamical selfenergy. There is a strong reduction of order $Z^2$ because of loss of quasiparticle weight in the coherent part of $A(\mathbf{k},\omega)$
Interacting band structure
This block uses lmfgws to generates the band structure of the interacting Green’s function, i.e. the kresolved spectral function along symmetry lines similar to a band plot for a noninteracting $G_0$. Peaks in the spectral function correspond to the band structure; the plot can be compared directly to the bands of the noninteracting G_{0}. Use syml.fe from that tutorial, or use file syml2.fe, which contain the symmetry lines as appear in Figure 1 of this Phys. Rev. B paper. Invoke lmfgws in batch mode as follows:
$ cp ~/lm/gwd/test/fe/syml2.fe .
$ lmfgws fe `cat switchesforlm` 'sfuned~units=eV~readsek~eps .01~evsync=6~se band:fn=syml2 ib=1:10 nw=10 getev=12 isp=1 range=10,10'
The selfenergy editor carries out the following
 units eV
Set units to eV  readsek
Read se.fe  eps .01
Add 10 meV smearing to Im Σ  evsync
refresh quasiparticle levels read from se.fe by recalculating them.  se band:fn=syml2 ib=1:10 nw=10 getev=12 isp=1 range=10,10
Generate the selfenergy and spectral function $A(\mathbf{k},\omega)$ along symmetry lines given in file syml2.fe. Include bands 110, and generate $A(\mathbf{k},\omega)$ on a frequency mesh 10× finer than the one in se.fe. getev refines the kmesh to a 12×12×12 mesh, and using that mesh to interpolate bands along symmetry lines in syml2.fe. Genearte bands in an energy window [−10,10] eV.
lmfgws writes a file, spq.fe.
Invoke plbnds in “spectral function mode:”
$ plbnds sp~atop=10~window=4,4 spq.fe
It will generate a gnuplot script file gnu.plt together with a data file spf.fe.
Run gnuplot
$ gnuplot gnu.plt
$ open spf.ps [choose your postscript file viewer]
to generate and view postscript file spf.ps.
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