View source for Next steps: RPA calculations (standalone)

==Optical absorption in hBN: independent particle approximation ==

=== Background ===
The dielectric function in the long-wavelength limit, at the independent particle level (RPA without local fields), is essentially given by the following:
[[File:CH1.png|none|x50px|Yambo tutorial image]]
In practice, Yambo does not use this expression directly but solves the Dyson equation for the susceptibility ''X'', which is described in the [[Local fields]] module.

=== Choosing input parameters ===
Enter the folder for bulk hBN that contains the ''SAVE'' directory, run the initialization and generate the input file.
From <code>yambo -h</code> you should understand that the correct option is <code>yambo -optics c</code> (or <code>yambo -o c</code>). Let's add some command line options:

 $ cd YAMBO_TUTORIALS/hBN/YAMBO
 $ yambo               ''(initialization)''
 $ yambo -F yambo.in_IP -o c
This corresponds to optical properties in G-space at the independent particle level: in the input file this is indicated by (<code>Chimod= "IP"</code>).

===Optics runlevel===
For optical properties we are interested just in the long-wavelength limit ''q = 0''. This always corresponds to the ''first'' ''q''-point in the set of possible ''q=k-k' ''-points. 
Change the following variables in the input file to:
 % [[Variables#QpntsRX|QpntsRXd]]
  1 |  '''1''' |                   # [Xd] Transferred momenta
 %
 [[Variables#ETStpsX|ETStpsXd]]= '''1001'''               # [Xd] Total Energy steps
in order to select just the first ''q''. The last variable ensures we generate a smooth spectrum. 
Save the input file and launch the code, keeping the command line options as before (i.e., just remove the lower case options):
 $ yambo -F yambo.in_IP -J Full
 ...
 <---> [05] Optics
 <---> [LA] SERIAL linear algebra
 <---> [DIP] Checking dipoles header
 <---> [x,Vnl] computed using 4 projectors
 <---> [M  0.017 Gb] Alloc WF ( 0.016)
 <---> [WF] Performing Wave-Functions I/O from ./SAVE
 <01s> Dipoles: P and iR (T): |########################################| [100%] 01s(E) 01s(X)
 <01s> [M  0.001 Gb] Free WF ( 0.016)
 <01s> [DIP] Writing dipoles header
 <01s> [X-CG] R(p) Tot o/o(of R)  :   5501   52992     100
 <01s> Xo@q[1] |########################################| [100%] --(E) --(X)
 <01s> [06] Game Over & Game summary
  
 $ ls
 Full   SAVE  yambo.in_IP   r_setup
 o-Full.eel_q1_ip  o-Full.eps_q1_ip  r-Full_optics_chi
Let's take a moment to understand what Yambo has done inside the Optics runlevel:
# Compute the ''[x,Vnl]'' term
# Read the wavefunctions from disc [WF]
# Compute the ''dipoles'', i.e. matrix elements of '''p'''
# Write the dipoles to disk as ''Full/ndb.dip*'' databases. This you can see in the report file:
 $ grep -A20 "WR" r-Full_optics_chi 
 [WR./Full//ndb.dip_iR_and_P]
 Brillouin Zone Q/K grids (IBZ/BZ):  14   72   14   72
 RL vectors                   (WF): 1491
 Electronic Temperature        [K]: 0.0000000
 Bosonic    Temperature        [K]: 0.0000000
 X band range           :   1  100
 RL vectors in the sum  : 1491
 [r,Vnl] included       :yes
 ...
<ol start="5">
<li>Finally, Yambo computes the non-interacting susceptibility ''X<sub>0</sub>'' for this ''q'', and writes the dielectric function inside the ''o-Full.eps_q1_ip'' file for plotting</li>
</ol>

===Energy cut off===

Before plotting the output, let's change a few more variables. The previous calculation used ''all'' the G-vectors in expanding the wavefunctions, up to 1491 (~1016 components). This corresponds roughly to the cut off energy of 40Ry we used in the DFT calculation. Generally, however, we can use a smaller value. We use the verbosity to switch on this variable, and a new ''-J'' flag to avoid reading the previous database:
 $ yambo -F yambo.in_IP '''-J 6Ry -V RL''' -o c
Change the '''value''' of <code>[[Variables#FFTGvecs|FFTGvecs]]</code> and also its '''unit''' from <code>RL</code> (number of G-vectors) to <code>Ry</code> (energy in Rydberg):
 [[Variables#FFTGvecs|FFTGvecs]]= '''6'''           '''Ry'''    # [FFT] Plane-waves
Save the input file and launch the code again:
  $ yambo -F yambo.in_IP '''-J 6Ry '''
and then plot the ''o-Full.eps_q1_ip'' and ''o-6Ry.eps_q1_ip'' files:
 $ gnuplot
 gnuplot> plot "o-Full.eps_q1_ip" w l,"o-6Ry.eps_q1_ip" w p

[[File:CH-hBN-6Ry.png|none|500px|Yambo tutorial image]]

Clearly there is very little difference between the two spectra. This highlights an important point in calculating excited state properties: generally, fewer G-vectors are needed than what are needed in DFT calculations. Regarding the spectrum itself, the first peak occurs at about 4.4eV. This is consistent with the minimum direct gap reported by Yambo: 4.28eV. The comparison with experiment (not shown) is very poor however. 

If you make some mistake, and cannot reproduce this figure, you should check the value of <code>[[Variables#FFTGvecs|FFTGvecs]]</code> in the input file, delete the ''6Ry'' folder, and try again - taking care to plot the right file! (e.g. ''o-6Ry.eps_q1_ip_01'').

===q-direction===
Now let's select a different component of the dielectric tensor:
 $ yambo -F yambo.in_IP -J 6Ry -V RL -o c
 ...
 % [[Variables#LongDrXd|LongDrXd]]
 '''0.000000''' | 0.000000 | '''1.000000''' |        # [Xd] [cc] Electric Field
 %
 ...
 $ yambo -F yambo.in_IP -J 6Ry
This time yambo reads from the ''6Ry'' folder, so it does not need to compute the dipole matrix elements again, and the calculation is fast. Plotting gives:
 $ gnuplot
 gnuplot> plot "o-6Ry.eps_q1_ip" t "q || x-axis" w l,"o-6Ry.eps_q1_ip_01" t "q || c-axis" w l

[[File:CH-hBN-ac.png|none|500px|Yambo tutorial image]]
The absorption is suppressed in the stacking direction. As the interplanar spacing is increased, we would eventually arrive at the absorption of the BN sheet (see [[Local fields]] tutorial).

===Non-local commutator===
Last, we show the effect of switching off the non-local commutator term (see ''[Vnl,r]'' in the equation at the top of the page) due to the pseudopotential. As there is no option to do this inside yambo, you need to hide the database file. Change back to the ''q || (1 0 0)'' direction, and launch yambo with a different <code>-J</code> option:
 $ mv SAVE/ns.kb_pp_pwscf SAVE/ns.kb_pp_pwscf_'''OFF'''
 $ yambo -F yambo.in_IP -J '''6Ry_NoVnl''' -o c             ''(change to q || 100)''
 $ yambo -F yambo.in_IP -J 6Ry_NoVnl
Note the warning in the output:
 <---> [WARNING] Missing non-local pseudopotential contribution
which also appears in the report file, and noted in the database as <code>[r,Vnl] included       :no</code>. The difference is tiny:
[[File:CH-hBN-Vnl.png|none|500px|Yambo tutorial image]]

However, when your system is larger, with more projectors in the pseudopotential or more k-points (see the BSE tutorial), the inclusion of ''Vnl'' can make a huge difference in the computational load, so it's always worth checking to see if the terms are important in your system.

==Optical absorption in 2D BN: local field effects ==

=== Background ===
[[File:Yambo-handbook-v4.1.2-p-4.png|thumb|Cheatsheet on LFE|150px]]
The macroscopic dielectric function is obtained by including the so-called local field effects (LFE) in the calculation of the response function. Within the time-dependent DFT formalism this is achieved by solving the Dyson equation for the susceptibility ''X''. In reciprocal space this is given by:
[[File:Yambo-CH7.png|none|x50px|Yambo tutorial image]]
The microscopic dielectric function is related to ''X'' by:
[[File:Yambo-CH5.png|none|x30px|Yambo tutorial image]]
and the macroscopic dielectric function is obtained by taking the (0,0) component of the inverse microscopic one:
[[File:Yambo-CH6.png|none|x60px|Yambo tutorial image]]
Experimental observables like the optical absorption and the electron energy loss can be obtained from the macroscopic dielectric function:
[[File:Yambo-CH8.png|none|x60px|Yambo tutorial image]]

In the following we will neglect the ''f <sup> xc</sup>'' term: we perform the calculation at the RPA level and consider just the Hartree term (from ''v<sub>G</sub>'') in the kernel. If we also neglect the Hartree term, we arrive back at the independent particle approximation, since there is no kernel and ''X = X<sub>0</sub>''.

=== Choosing input parameters ===
Enter the folder for 2D hBN that contains the SAVE directory, and generate the input file. From <code>yambo -H</code> you should understand that the correct option is <code>yambo -o c -k hartree</code>. Let's start by running the calculation for light polarization ''q'' in the plane of the BN sheet:
 $ cd YAMBO_TUTORIALS/hBN-2D/YAMBO
 $ yambo        ''(Initialization)''
 $ yambo -F yambo.in_RPA -V RL -J q100 -o c -k hartree
We thus use a new input file ''yambo.in_RPA'', switch on the <code>FFTGvecs</code> variable, and label all outputs/databases with a ''q100'' tag. Make sure to set/modify all of the following variables to:
 [[Variables#FFTGvecs|FFTGvecs]]=     '''6        Ry'''    # [FFT] Plane-waves
 [[Variables#Chimod|Chimod]]= "Hartree"            # [X] IP/Hartree/ALDA/LRC/BSfxc
 [[Variables#NGsBlkXd|NGsBlkXd]]= '''    3        Ry'''    # [Xd] Response block size
 % [[Variables#QpntsRXd|QpntsRXd]]
  1 |  1 |                   # [Xd] Transferred momenta
 %
 % [[Variables#EnRngeXd|EnRngeXd]]
  0.00000 | 20.00000 | eV    # [Xd] Energy range
 %
 % [[Variables#DmRngeXd|DmRngeXd]]
 0.200000 | 0.200000 | eV    # [Xd] Damping range
 %
 [[Variables#ETStpsXd|ETStpsXd]]= 2001               # [Xd] Total Energy steps
 % [[Variables#LongDrXd|LongDrXd]]
 1.000000 | 0.000000 | 0.000000 |        # [Xd] [cc] Electric Field
 %
In this input file, we have:
* We evaluate the q-->0 response function choosing the direction for the limit parallel in the plane of the hBN sheet 
* A wider energy range than before, and more broadening
* Selected the Hartree kernel, and expanded G-vectors in the screening up to 3 Ry (about 85 G-vectors)

===LFEs in periodic direction===
Now let's run the code with this new input file (CECAM in serial: about 2mins; parallel 4 tasks: 50s)
 $ yambo -F yambo.in_RPA  -J q100 
and let's compare the absorption with and without the local fields included. By inspecting the ''o-q100.eps_q1_inv_rpa_dyson'' file we find
that this information is given in the 2nd and 4th columns, respectively:
 $ head -n30 o-q100.eps_q1_inv_rpa_dyson
 # Absorption @ Q(1) [q->0 direction] : 1.0000000  0.0000000  0.0000000
 #  E/ev[1]     EPS-Im[2]   EPS-Re[3]   EPSo-Im[4]  EPSo-Re[5]
Plot the result:
 $ gnuplot
gnuplot> plot "o-q100.eps_q1_inv_rpa_dyson" u 1:2 w l t "RPA-LFA","o-q100.eps_q1_inv_rpa_dyson" u 1:4 w l t "noLFE", "o-q100.eel_q1_inv_rpa_dyson" u 1:4 w l ls 7 dt 2 t "EELS"
[[File:CH-LFE4.png|none|600px|Yambo tutorial image]]
It is clear that there is little influence of local fields in this case. This is generally the case for semiconductors or materials with a smoothly varying electronic density. We have also shown the EELS spectrum (''o-q100.eel_q1_inv_rpa_dyson'') for comparison.

===LFEs in non-periodic direction===
Now let's switch to ''q'' perpendicular to the BN plane:
 $ yambo -F yambo.in_RPA -V RL -o c -k hartree        ''and set''
 ...
 % [[Variables#LongDrXd|LongDrXd]]
 0.000000 | 0.000000 | '''1.000000''' |        # [Xd] [cc] Electric Field
 %
 
You can try out the default parallel usage now, or run again in serial, i.e.
 $ yambo -F yambo.in_RPA  -J '''q001'''       ''(serial)''
 $ mpirun -np 4 yambo -F yambo.in_RPA  -J '''q001''' &      ''(parallel, MPI only, 4 tasks)''
As noted previously, the ''log'' files in parallel appear in the LOG folder, you can follow the execution with <code>tail -F LOG/l-q001_optics_chi_CPU_1</code> .

Plotting the output file:
 $ gnuplot
 gnuplot> plot "o-q001.eps_q1_inv_rpa_dyson" u 1:2 w l,"o-q001.eps_q1_inv_rpa_dyson" u 1:4 w l
[[File:CH-LFE6.png|none|600px|Yambo tutorial image]]
In this case, the absorption is strongly blueshifted with respect to the in-plane absorption. Furthermore, the influence of local fields is striking, and quenches the spectrum strongly. This is the well known depolarization effect. Local field effects are much stronger in the perpendicular direction because the charge inhomogeneity is dramatic. Many G-vectors are needed to account for the sharp change in the potential across the BN-vacuum interface.

===Absorption versus EELS===
In order to understand this further, we plot the electron energy loss spectrum for this component and compare with the absorption:
 $ gnuplot
 gnuplot > plot "o-q001.eps_q1_inv_rpa_dyson" w l,"o-q001.eel_q1_inv_rpa_dyson" w l
[[File:CH-LFE5.png|none|600px|Yambo tutorial image]]
The conclusion is that the absorption and EELS coincide for isolated systems. 
To understand why this is, you need to consider the role of the ''macroscopic'' screening in the response function and the long-range part of the Coulomb potential. 
See e.g.<ref>TDDFT from molecules to solids: The role of long‐range interactions, F. Sottile et al, International journal of quantum chemistry 102 (5), 684-701 (2005)</ref>

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