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## Content of this lesson

- Patching a pseudopodential with a PAW correction:

- Generation of ptildes and psitildes to increase the psp energy range.
- Fitting ptildes with complex gaussians.

- Generating an SCF potential for a small unit cell.
- Generating the spectra:

- Dependence on nkpts, number of Chebyshev components, polarisation, charge ...
- Comparison to experimental data.

## Material : files are generated by `make exercise` in each subdir

### Xabs/pawpatchgen/

- atom.dat
- input.dat
- psppar.P
- pawpatchfit.py

### Xabs/scfpotgen/

- posinp.xyz
- input.dft
- psppar.O
- psppar.Si
- psppar.P

optional bigger cell

- posinp.xyz_72p

### Xabs/spectra/

- input.abscalc
- input.dft
- input.kpt
- psppar.O
- psppar.Si
- posinp.xyz

optional bigger cells

- posinp_72X8p.xyz
- posinp_72X27p.xyz

# Xanes: Calculating the spectra of quartz

The purpose of this lesson is to get familiar with the steps required to generate an x-ray absorption spectra, in particular in the Xanes region. These steps are three :

- the first step consists in generating a PAW patch for the pseudopotential in order to extend its validity over the Xanes region. This region corresponds to photo-electron energy from zero to few Hartree. We will do this using atom and pseudo programs which are distributed with the BigDFT project.
- The potential felt by the photoelectron can be generated by a standard BigDFT run. A non SCF potential can also be generated summing atomic densities. This can be done using abscalc with the appropriate option.
- We will calculate the spectra using a paw-patched pseudopotential and the local potential from a BigDFT calculation. Using several k-points is equivalent to calculate the spectra in a larger cell whose size is a multiple of the unit cell. The number of k-points is important to get smoother spectra free from spurious band discrete sampling effects. We will explore the importance of this parameters and of other parameters like the charge of the system, and the light polarisation.

This lesson may take XX minutes for the basic point and we give track for further testing. We suggest to coordinate your runs with your neighbours so that the test of the dependancies on different choices of certain parameters can be parallelised.

## Patching a pseudopodential with a PAW correction

### Generating the potential of the AE reference

Our goal is to calculate the Xanes spectra of Quartz at the K edge of Silicon. In order to do so we need to construct a pseudopotential, with the program pseudo, which describes a Silicon atom with one hole in the 1s shell. Asimple way to do this, that we will follow here, consistes in starting from the pseudopotential of the Z+1 element, (Phosphorus in this case), that we will patch with a PAW correction that we are going to build. The Z+1 approximation consists in replacing a Z nuclear charge plus on 1s electron, with a Z+1 nuclear charge plus two electron. This keeps the total charge unchanged. It would possible, with the program pseudo, to generate a new pseudopotential for Silicon with one hole, and then patch it. We focus here instead on building only the PAW correction which is the original part of this method, and we start therefore directly from the Z+1 pseudopotential which is already given.

The calculation proceed in two steps. First we generate with the program atom the very same all-electron (AE) reference that has been used to create the pseudopotential. This first step is used to generate a radial potential that will be used, in the second step, to generate AE eigenfunctions over a wide energy range.

In the second step we create the PAW correction such that the patched pseudo potential can fit the AE energy over a wider range.

The first step is quickly accomplished with the following command

Exercise: run ``../../../../pseudo/src/atom atom.dat``

where file `atom.dat` is the one used as reference for the `psppar.P` pseudopotential.

      P
      -20
      relativistic
           25.0      10.0 60.0       rmax,aa,bb
           2.09d0   4.19d0		 rcov,rprb
        3    13			 number of core and valence orbitals
        3    0     2.00      0.00
        4    0     0.00      0.00
        5    0     0.00      0.00
        6    0     0.00      0.00
        3    1     3.00      0.00
        4    1     0.00      0.00
        5    1     0.00      0.00
        6    1     0.00      0.00
        3    2     0.00      0.00
        4    2     0.00      0.00
        5    2     0.00      0.00
        4    3     0.00      0.00
        5    3     0.00      0.00

the format of `atom.dat` is quite simple. On the first line we find the atomic symbol(P). Then the XC functional code ( here -20=LDA). The fourth line contains some information on how atom must build the radial logarithmic grid. The fifth line contains the covalent radius and the radius of the parabolic potential. On the sixth line the first number that we find indicates how many core bottom orbitals are fully occupied. Here 3 means that 10 electron occupies 1s, 2s and 2p. The second number is the number of valence orbitals that have been calculated by atom when the reference for pseudo was generated. Here we are concerned only by those orbital which have non zero occupancies. These are the one which have influence on the local potential.

At the end of the run you should get in your working directory the file ae.pot.conf.0.plt which is the potential for configuration 0 ( pseudo can fit a pseudo potential against a set of configurations, here we have just one ).

### Generation of ptildes and psitildes to increase the psp energy range.

You are ready to generate the paw-patch :

Exercise: run ``../../../../pseudo/src/pseudo input.dat``

This program, besides input.dat, reads several files generated by atom, and the file psppar containing the initial psp parameters. In out case we dont fit the psp parameter, so psppar must be coherent with the pseudopotential psppar.P that we want to patch.

It is important to explain the option used in input.dat because you will have to fine tune them

    -c 0 -plot -paw 0  -noflpaw 3  -nchannelspaw 2
    -pawstatom ae.pot.conf.0.plt -pawstn 1 -pawstl 0 -pawstp 1

- the first two option :`-c0 -plot` tell pseudo not to fit the psp parameters, we already have that we already know : `psppar.P`
- -paw 0 means : do the patch for configuration number 0
- -noflpaw 3 means that the correction are calculated of three values of angular moment : s, p and d waves. We suppose here that the correction for f waves would be here negligeable.
- -nchannelspaw 2 means that for each angular moment two projectors are generated. The more projectors one has the bigger the energy range where the correction is good.
- -pawstatom ae.pot.conf.0.plt reads the local potential from file ae.pot.conf.0.plt that we generated with atom.

the following items concern the generation of the initial photoelectron wavefuction

- -pawstn 1 we want to generate a wavefunction from 1s. Quantum number n is therefore 1 here
- -pawstl 0 For the same reason above, we indicate here that quantum number l is 0
- -pawstp 1 This is the power of r by which we multiply the radial part of the initial wave fuction. The angular part is take care of by BigDFT ( abscalc) when the polarisation is known. For dipolar interaction pawstp is 1, for quadrupolar it is 2 and so on.

The result of the fit can be monitored looking at the output. Here below we report the output concerning the `l=0` space

     ===============================================================
     ========== now CALCULATING PAWpatch correction  for l  =             0
     ===============================================================
     now calculating          100  function of the AE basis for LPaw=           0
     doing the fit
      routine gatom_modified  ,  comparaison between  first 5 energies real and  pseudo-not_fitted
               1  -75.952756404876709
               2  -6.2910273671150208
               3  0.41157764650931183       0.37085166222413934
               4   5.9902891679064592        2.6090777062770307
               5   14.400375561474663        6.4202517989703054
     >>  Comparaison betwenn first 5 Egrid and   Egrid_pseudo
      -75.952756404876709
      -6.2910273671150208
      0.41157764650931183       0.41157765025060822
       5.9902891679064592        5.9902845552977215
       14.400375561474663        14.399645486591949
    DUALITY : CONDITION NUMBER FROM DPOSVX  0.5501E+00
      first  eigenvalues
      0.41116489582251731
       2.9722717862947752
       7.7169956612536588
       13.735301520083759
     routine pawpatch  , PROJECT  initial wf*r**pawstP on pseudos
     From initial wave  subtract psigrid_bigger(           1 ) with coeff   0.10433231170909278
     From initial wave  subtract psigrid_bigger(           2 ) with coeff  -3.29213674528539627E-002
     Initial Projected wf at igrid=100,500,2000
     -1.19502133768229197E-007
     -2.11513353895853788E-007
     -1.97394835887812979E-006

We can see here, below the line `comparaison between first 5 energies real and pseudo-not_fitted` that, without the PAW patch, the eigenvalues of the radial solution calculated in the AE potential diverges at high energies from those calculated in the pseudo potential, while in the valence region the used psppar.P pseudopotential fits well the AE solution within some thousandth of Hartree. Nota Bene that the radial solution is calculated from zero to the covalent radius, and this pushes the first pseudo eigenvalue at positive energy where the difference with the AE solution is already of the order of some hundreth of Hartree.

The lines below `Comparaison between first 5 Egrid and Egrid_pseudo` show the corrected eigenvalues for a modified pseudo potential where, for each eigenvalue, we add a short range gaussian at the nucleus with an appropriate factor such that the AE eigenvalue is recovered. These eigensolution are our psitildes which are not orthonormal but outside the core region they coincide with the AE solutions.

By solving a simple linear equation we get the ptildes as the dual of the psitildes (line `DUALITY : CONDITION NUMBER FROM DPOSVX 0.5501E+00` ). Once the psitildes and ptildes are know, the PAW patch is calculated as a correction in the nchannelspaw-dimensional space spanned by psitildes.

Such correction is tested in the space spanned by the pseudo solutions ( which has a dimension of the order of 100) and the results are plotted for comparaison.

Exercise: increase nchannespaw to improve the fit of the second pseudo eigenvalue

Exercise: plot AE solutions (file ae.wfs.L=0.plt ) and psitildes ( file psitildes.L=0.plt) to verify that they really coincide outside of the core

For future use:: -pawrcovfact0.8 or another factor can be used to change the psitildes/ptildes support

### Fitting ptildes with complex gaussians.

Finally the PAW patch is generated with the following command for L from 0 to 2 and using 30 complex gaussians

Exercise: run ``python pawpatchfit.py 0 2 30 ``

This script create an output : file pseudopaw containing a description of the ptildes , of the initial wave function projection in the space of pseudo-functions, for different values of L, and the correction in the form of a numerical matrix. The quality of the fit can be observed plotting these files:

    fitresult_l_2_initial
    fitresult_l_2_channel_1
    fitresult_l_2_channel_0
    fitresult_l_1_initial
    fitresult_l_1_channel_1
    fitresult_l_1_channel_0
    fitresult_l_0_initial
    fitresult_l_0_channel_1
    fitresult_l_0_channel_0

each file contains four columns : the radial position, the original numerical function, the fit to function as a sum of complex gaussian, the error.

Now the last step. Create the patched pseudopotential file ready for the spectra!!. This is done postpending file pseudopaw to psppar.P

Exercise: run ``cat psppar.P pseudopaw > psppar.P_1s``

## Generating an SCF potential for a small unit cell

This is done with a BigDFT run in directory Xabs/scfpotgen . The unit cell contains 18 atoms, corresponding to the structure of Quartz, with one Silicon substituted by Phosphorus. In input.dft we have set a charge of 1. The run is fast enough so that, using different parameters, you may eventually generate alternative potentials to be used for the additional spectra calculation. You will do this if you have enough cpus to run multiple spectra in parallel or it you have enough time. To generate an SCF potential, when you are in directory Xabs/scfpotgen

Exercise: run ``mpirun -n XXXX ../../../../src/bigdft``

the potential is written to the disk in the cube format when in file input.dft output_grid is set to 2. The created potential is data/local_potential.cube .

There is also the possibility to generate a potential from straightforward sum of atomic charges. This is done running abscalc program. The abscalc program reads input.abscalc whose key pot_shortcut is set to option 1 ( linear superposition of charges ) Option pot_shortcut=2 means instead that the local potential is read from file.

Optional exercise: run ``../../../../src/abscalc``

the local potential is then written to local_potentialb2B.cube

## Generating the spectra

The directory Xabs/spectra is almost already set-up with input files to run a spectra calculation. To finalize the set-up you must copy the patched pseudopotential, that you created, into the work directory and copy or create a link of Xabs/scfpotgen/data/local_potential.cube to a file named b2B_xanes.cube. The prefix b2B stands for box to box which is the name of the routine which interpolate a potential from a given unit cell (box) to another unit cell. The unit cells must either have the same shape and same atomic positions or the target cell can be a multiple of the input cell.

In out set-up we use the same posinp.xyz used in SCF calculation except that we have changed the name of the absorbing atom from P to P_1s so that the absorbing atom ( that the absorbing atom is the first is indicated in input.abscalc ) will use a patched pseudo-potential.

The unit cell that we use is quite small to keep the tutorial fast and, to increase the number of states in the continuous region, we specify in input.kpt the inverse of the k-lenght of the reciprocal sampling which is larger than the unit cell diameter.

You can dry-check how many k-points you have with the memguess program. The given setup generates 6 kpoints. You can run it :

Exercise: run ``mpirun -n 4 ../../../../src/abscalc``

which should take XXX minutes. As explained in the lesson the method does a systematic refinement of the spectra, adding at each iteration two Chebyshev components. The spectra is plotted every 100 iterations on file Zb_cheb_spectra_XXXX. Where XXXX is the number of components. We show here the spectra with 4000 components for a kptrlenght of 22 Bohr (6 kpoints) and 50Bohr (X kpoints ), and another spectra done using the bigger posinp_72p.xyz (72 atoms) cell and 50Bohr (X kpoints ). We see that for our small cell the spectra is not yet converged because the absorbing Phosphorus site has in the SCF a deformation of the potential respect to the rest of the lattice, and this deformation is reproduced at periodic intervals in our calculation which is based on periodic boundary conditions and k-points.. In the case of a small cell the oscillations correspond to scattering of the photolectron on other Phosphorus sites.

The obtained spectra can be compared with figure 2 of http://arxiv.org/abs/cond-mat/0207733v1 . We produce here the analogous picture where the experimental data has been digitalized from that figure, and the calculation is done with posinp_72p.xyz and and a 50Bohr lenght for the reciprocal space.

As optional exercise you can test different parameters for the polarisation. If you have enough computing power you can increase the number of k-points by increasing the lenght given in input.kpt, or play with bigger cells.