Hartree Fock and GW
Contents
Basic concepts of the GW approximation by the Yambo Team
In this tutorial you will learn the basic concepts of the HartreeFock and of the GW[[[6]]] approximations. In particular we will illustrate how to calculate QuasiParticle energies with a single shot G_{0}W_{0} approximation. Different lecture notes on the GW approach are available in the "Lecture Notes" section.
The tutorial is split in different sections. In the first part we will deal with HartreeFock (HF) and the Coulombhole and screenedexchange (COHSEX) approximation. Finally in the last section we will discuss dynamical correlation with a Plasmon Pole Approximation (PPA).
The material: Silicon
Bulk Silicon

The Tutorial structure
Once the tutorial zip file is unzipped the following folder structure will appear
COPYING README Solid_Si/
with the Solid_Si folder containing
>ls Solid_Si/
Pwscf/ YAMBO/
 In the Pwscf folder information on the ground state generation are provided. In particular the student will find an input/output directory with input/output files for pw.x. Also a psp folder is provided with the silicon pseudo files.
 The YAMBO folder contains the Yambo input/output files and core databases.
HartreeFock
Now we will study the convergence of the HartreeFock self energy, respect to the number of Gvectors and kpoints. Yambo is a planewave code, therefore all the operators and wavefunctions are expanded as:
 File:Orbital2.png  Test SimpleMathJax: [math]\Phi_i({\bf r1}) = \sum_{\bf G}^{N_G} c_i({\bf G}) e^{i{\bf G}\cdot{\bf r}}[/math] 
Before you start any calculations we can decide how many Gvectors you want use to represent your wavefunction. By default Yambo will use all of them, and for the present tutorial this is fine, however if you are studying larger systems you would like to reduce them in order to speedup calculations, you can do it with yambo i V RL and then changing the variable MaxGvecs that referees to N_{g} in the previous formula.
Now we will proceed in the calculation of the HartreeFock(HF) exchange. This is composed of two terms, that Hartree and the Fock (or exchange) one:
In this tutorial we will calculate only the first order correction to the KohnSham Hamiltonian due to the exchange term. Because of we are working in periodic system, the most appropriate basis to represent the Coulomb potential, appearing in the Fock term, is a planewave basis:
In order to generate the input file for the exchange type yambo x. The variable that governs the number of G vectors N_{ex} in the Coulomb potential is EXXRLvcs. In addition notice that in the previous formula for V(rr') we have an integral on the q vector, that in the code is discretized on a regular qgrid generated from the kpoint one, q=kk'.
Kpoints convergence
Enter the YAMBO folder
>cd YAMBO/
>ls
Convergence_Plots 2x2x2 4x4x4 6x6x6 8x8x8 Gamma
Enter each grid folder and run >yambo without options. Then type
yambo x F INPUTS/01HF_corrections
and edit the 01HF_corrections file
HF_and_locXC # [R XX] HartreeFock Selfenergy and Vxc EXXRLvcs= 7 Ry # [XX] Exchange RL components %QPkrange # [GW] QP generalized Kpoint/Band indices 1 8 1 15 % %QPerange # [GW] QP generalized Kpoint/Energy indices 1 8 0.01.0 %
setting EXXRLvcs=7 Ry
. Then run in each kpoint folder
yambo F INPUTS/01HF_corrections J HF_7Ry
Note that after the each run Yambo produces three output file, one starting with r_... , another with l_... and a third one starting with o_.... The first one is a report of the actual calculation with all the details of the system, input, and results, while the second is a log with the running time of each process.
The file we are most interested in is the output file, oHF_7Ry.hf. This file contains several columns. The fourth, labelled with
Ehf represents the HartreeFock energy. This is the quantitu we have to look at.
Gvectors convergence
Enter the 4x4x4 folder and edit the 01HF_corrections file changing the field EXXRLvcs=3,6,7 and 15 Ry. For each case run
yambo F INPUTS/01HF_corrections J HF_???Ry
with ???=3,6,7 and 15
The results of this series of calculations are summarized in the following plots. Note the rapid convergence with the number of kpoints.
To obtain the plots of the following graphs just plot the columns #4#3 versus #3 of the o.hf files.
COHSEX without empty bands
The HartreeFock selfenergy just described in the previous section, although successful on some molecular systems, miserably fails in extended systems. The reason of this failure lies in the fact that electrons can screen the Coulomb potential, and therefore one electron feels a screened potential instead of the bare one. A common approximation for this screened potential is the socalled Random Phase Approximation:
where the bare Coulomb potential V(r) is replaced by a nonlocal and frequency dependent one W(r,r',ω). Subsequently it is possible to redefine all the perturbation theory in term of this screened potential, and disregarding the additional corrections coming from the vertex part, one obtains the socalled GW approximation.
However the substantial complexity associated with calculating the nonlocal, energy dependent Σ=GW operator inspired early efforts to find simplifying approximations as the static
COHSEX approximation prosed by Hedin[[[4]]]. The Coulomb Hole plus Screened Exchange (COHSEX) approximation[[[2]],3] eliminates the summation over
empty states for the self energy operator and has the added benefit of being a static operator, a particular simplification for self consistent calculations. With the COHSEX approximation the selfenergy is
composed of two parts:
where W_{p}=W  V.
In order to calculate the COHSEX selfenergy with yambo you need first the static screened interaction yambo b k hartree. As for the V(r) case, also W(r,r',E=0) will
depend from the number of Gvectors, and the kgrid, but this time also the number of conduction bands XsBndsRn will enter in the screening calculation through the polarization function,
see The Interacting response function: ManyBody and TDDFT section.
Notice however that there are not any additional dependence of the selfenergy operator from the conduction bands.
After you obtained the screened interaction you are ready to build the selfenergy operator and solve the corresponding Dyson equation, have
a look to the Dyson Equation solvers in Yambo. In order to get the quasiparticle energies, just do yambo b g n k hartree.
After calculation are completed Yambo will produce an output file o.qp which contains the values of the bare and renormalized energy levels.
Kpoints convergence
Follow the same strategy of the HF case. Enter each kpoint folder and type
> yambo b k hartree g n p c F INPUTS/02Cohsex
edit the input file and change the values of the fields
EXXRLvcs=7 Ry
%BndsRnXs
1  10  # [Xs] Polarization function bands
%
NGsBlkXs= 1 RL # [Xs] Response block size
 UseEbands # [GW] Force COHSEX to use empty bands
%QPkrange # [GW] QP generalized Kpoint/Band indices
1 8 1 10
%
leaving the other parameters unchanged. Now, in each kpoint folder, run
>yambo F INPUTS/02Cohsex J Cohsex_HF7Ry_X0Rynb10
W size convergence
Enter the 4x4x4 folder and edit the 02Cohsex_corrections file changing the field NGsBlkXs=3,6,7 Ry. Run
>yambo F INPUTS/02Cohsex J Cohsex_HF7Ry_X???Rynb10
with ???=3,6,7 and 15
W bands
Open your input file and change only the input variable NGsBlkXs=1 Ry. Then change
%BndsRnXs
1  20  # [Xs] Polarization function bands
%
and run yambo F INPUTS/02Cohsex J Cohsex_HF7Ry_X1Rynb20. Repeat the calculations using 30 and 40 bands changing nb???
in the job string identifier. Finally check how the Chosex Direct and Indirect band gap behave as a function of the number of bands in the
screening.
COHSEX with empty bands
While the Screened Exchange(SEX) part of the COHSEX selfenergy has a structure similar to the HartreeFock exchange term, the Coulomb Hole(COH) acts as static external potential on the electrons. In the COH part the delta function δ(rr') comes from the completeness relation:
Now we avoid the use of this relation in such a way to have a selfenergy that depends from the conduction bands too. This can be done uncommenting flags UseEbands and setting the number of bands in the Green's function:
% GbndRnge
1  10  # [GW] G[W] bands range
%
Repeat the previous calculations with different GbndRnge for instance, 10 20 30 40, and check how the COHSEX direct and indirect band gap behave as a function of the number of bands in the Green's function.
Recently an enhanced version of the COHSEX approximation has been proposed, if you want learn more have a look to the reference[[[4]]].
The plasmon pole approximation
Even the static COHSEX approximation is very appealing, it was clear from the first calculation that dynamical effects cannot be disregarded in solids[[[6]]]. This fact motivated the research for approximated way to deal with a frequency dependent interaction, and one of the first proposal was the socalled PlasmonPole Approximation (PPA) (see ref. [[[7]]]).
Now we will proceed to the calculation of the silicon band gap in G_{0}W_{0} within PPA. In each section you will be asked to perform several calculations
varying the value of the relevant variables involved. More specifically let's consider a typical Yambo input file to calculate GW corrections in the PPA.
gw0 # [R GW] GoWo Quasiparticle energy levels
ppa # [R Xp] Plasmon Pole Approximation
HF_and_locXC # [R XX] HartreeFock Selfenergy and Vxc
em1d # [R Xd] Dynamical Inverse Dielectric Matrix
EXXRLvcs= 7 Ry # [XX] Exchange RL components
% QpntsRXp
1  8  # [Xp] Transferred momenta
%
% BndsRnXp
1  30  # [Xp] Polarization function bands
%
NGsBlkXp= 7 Ry # [Xp] Response block size
% LongDrXp
1.000000  0.000000  0.000000  # [Xp] [cc] Electric Field
%
PPAPntXp= 27.21138 eV # [Xp] PPA imaginary energy
% GbndRnge
1  30  # [GW] G[W] bands range
%
GDamping= 0.10000 eV # [GW] G[W] damping
dScStep= 0.10000 eV # [GW] Energy step to evalute Z factors
DysSolver= "n" # [GW] Dyson Equation solver (`n`,`s`,`g`)
%QPkrange # [GW] QP generalized Kpoint/Band indices
1 8 1 10
%
%QPerange # [GW] QP generalized Kpoint/Energy indices
1 8 0.01.0
%
The variables are the same of the COHSEX case plus a new one PPAPntXp that describes the imaginary energy used to fit the Plasmon Pole model.
Follow the same strategy of the COHSEX case. Enter each kpoint folder and type:
yambo d k hartree g n p p F INPUTS/03GoWo_PPA_corrections
otherwise you can use the existent file Input/03GoWo_PPA_corrections.
Now you have to study the convergence of the G_{0}W_{0} gap versus the number of kpoints, the number of
bands in χ BndsRnXs, the size of the dielectric constant NGsBlkXs and the number of bands in the Green's function GbndRnge.
Please follow the same strategy of the COHSEX case running yambo using
yambo F INPUTS/03GoWo_PPA_corrections J GoWo_PPA_HF7Ry_X???Rynb???_nb???
The final convergence plots should look like these.
Final runs!! Now you know how to converge a G_{0}W_{0} calculation so you can decide which are the parameters needed for full convergence. First do a COHSEX and then a GoWo (PPA) run at convergence, and you should obtain results similar to the following graph:
Curiosity: What's happen if the PPA fails? Have a look to the tutorial on the RealAxis Integration.
References
 Silicon on wikipedia.
 Electron Correlation in the Solid State, Norman H. March (1999).
 GW approach to the calculation of electron selfenergies in semiconductors, B. Farid et al., Phys. Rev. B 38, 7530 (1988).
 New Method for Calculating the OneParticle Green's Function with Application to the ElectronGas Problem L. Hedin, Phys. Rev. 139, A796 (1965).
 Enhanced Static Approximation to the Electron SelfEnergy Operator for Efficient Calculation of Quasiparticle Energies, Wei Kang, Mark S. Hybertsen, Phys. Rev. B 82, 195108 (2010).
 Dynamical Correlation Effects on the Quasiparticle Bloch States of a Covalent Crystal, G. Strinati, H. J. Mattausch, and W. Hanke , Phys. Rev. Lett. 45, 290 (1980).
 Spacetime method for ab initio calculations of selfenergies and dielectric response functions of solids, H. N. Rojas, R. W. Godby, and R. J. Needs, Phys. Rev. Lett. 74, 1827 (1995) .