# Electron-positron annihilation¶

## Calculation of positron lifetime and momentum distribution in silicon¶

This tutorial aims at showing how to perform **Two-Component Density-Functional
Theory (TCDFT)** calculations in the PAW framework to obtain the following
physical properties:

- the positron lifetime in the perfect material,
- the lifetime of a positron localized in a vacancy,
- the electron-positron momentum distribution.

For the description of the implementation of TCDFT in `ABINIT`

see [Wiktor2015].

The user should be familiar with the four basic tutorials of `ABINIT`

and the first PAW tutorial.

This tutorial should take about 2 hours.

Note

Supposing you made your own installation of ABINIT, the input files to run the examples
are in the *~abinit/tests/* directory where *~abinit* is the **absolute path** of the abinit top-level directory.
If you have NOT made your own install, ask your system administrator where to find the package,
especially the executable and test files.

In case you work on your own PC or workstation, to make things easier, we suggest you define some handy environment variables by executing the following lines in the terminal:

```
export ABI_HOME=Replace_with_absolute_path_to_abinit_top_level_dir # Change this line
export PATH=$ABI_HOME/src/98_main/:$PATH # Do not change this line: path to executable
export ABI_TESTS=$ABI_HOME/tests/ # Do not change this line: path to tests dir
export ABI_PSPDIR=$ABI_TESTS/Psps_for_tests/ # Do not change this line: path to pseudos dir
```

Examples in this tutorial use these shell variables: copy and paste
the code snippets into the terminal (**remember to set ABI_HOME first!**) or, alternatively,
source the `set_abienv.sh`

script located in the *~abinit* directory:

```
source ~abinit/set_abienv.sh
```

The ‘export PATH’ line adds the directory containing the executables to your PATH
so that you can invoke the code by simply typing *abinit* in the terminal instead of providing the absolute path.

To execute the tutorials, create a working directory (`Work*`

) and
copy there the input files of the lesson.

Most of the tutorials do not rely on parallelism (except specific tutorials on parallelism). However you can run most of the tutorial examples in parallel with MPI, see the topic on parallelism.

## Computing the positron lifetime in Si lattice¶

*Before beginning, you might consider to work in a different subdirectory as
for the other tutorials. Why not* `Work_positron`

*?*

```
cd $ABI_TESTS/tutorial/Input
mkdir Work_positron
cd Work_positron
cp ../tpositron_1.abi .
```

The tutorial begins with a calculation of the positron lifetime in a silicon lattice. In a perfect material the positron is delocalized. We can assume that its density approaches zero and that it cannot affect the electron density. We will perform a calculation in only two steps:

- Calculation of the ground-state electron density without the positron.
- Calculation of the ground-state positron density in the presence of the electron density from step 1.

The two densities are used to calculate the positron lifetime, which is
proportional to the inverse of the overlap of the electron and positron
densities. This 2-step calculation, considering the *zero-positron density
limit*, corresponds to the conventional scheme (CONV).

In the `tpositron_1.abi`

file, you will find two datasets.

# Input for Positron tutorial # First step of the tutorial on electron-positron annihilation # Positron lifetime calculation within PAW # Si, 2 atoms in the box #Define the different datasets ndtset 2 # 2 datasets #FIRST DATASET positron1 0 # Dataset 1 is a simple electronic GS calculation #SECOND DATASET positron2 1 # Dataset 2 is a positronic GS calculation getden2 1 # in presence of the previous electronic density kptopt2 0 # Use only k=gamma point ixcpositron2 1 # We are using the Boronski and Nieminen parametrization posocc2 1 # Occupation number for the positron # (should be set <1 for bulk calculation with a small cell). # Here the zero positron density limit is used, # so results do not depend on posocc. #------------------------------------------------------------------------------- #Definition of data common to all datasets #Definition of the unit cell acell 3*5.43 angstrom # Lengths of the primitive vectors (exp. param. in angstrom) rprim # 3 orthogonal primitive vectors (FCC lattice) 0.0 1/2 1/2 1/2 0.0 1/2 1/2 1/2 0.0 #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 14 # Atomic number of the possible type(s) of atom. Here silicon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "Pseudodojo_paw_pw_standard/Si.xml" # Name and location of the pseudopotential #Definition of the atoms natom 2 # There are two atoms typat 1 1 # They both are of type 1, that is, Silicon xred # Location of the atoms: 0.0 0.0 0.0 # Triplet giving the reduced coordinates of atom 1 1/4 1/4 1/4 # Triplet giving the reduced coordinates of atom 2 #Definition of bands and occupation numbers nband 6 # Compute 6 bands occopt 1 # Automatic generation of occupation numbers, as a semiconductor #Numerical parameters of the calculation : planewave basis set and k point grid ecut 8. # Maximal plane-wave kinetic energy cut-off, in Hartree pawecutdg 15. # Max. plane-wave kinetic energy cut-off, in Ha, for the PAW double grid kptopt 1 # Automatic generation of k points, taking into account the symmetry ngkpt 4 4 4 # This is a 4x4x4 grid based on the primitive vectors of the recip. space nshiftk 1 # We do not shift the grid in order to have Gamma point in it shiftk 0. 0. 0. #Parameters for the SCF procedure nstep 50 # Maximal number of SCF cycles tolvrs 1.0d-8 # Will stop when, twice in a row, the difference # between two consecutive evaluations of potential residual # differ by less than tolvrs #Miscelaneous parameters prtwf 0 # Do not print wavefunctions prtden 1 # Print density (electronic and/or positronic) prteig 0 # Do not print eigenvalues optforces 0 # Forces computation is not relevant here optstress 0 # Stress tensor computation is not relevant here ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpositron_1.abo, tolnlines= 0, tolabs= 0.0, tolrel= 0.0, fld_options= -easy #%% [paral_info] #%% max_nprocs = 10 #%% [extra_info] #%% authors = J. Wiktor #%% keywords = POSITRON,PAW #%% description = #%% Input for Positron tutorial #%% First step of the tutorial on electron-positron annihilation #%% Positron lifetime calculation within PAW #%% Si, 2 atoms in the box #%%<END TEST_INFO>

The first dataset is a standard ground-state calculation. The second one introduces a positron into the system. You can see that in this case we set:

```
positron2 1 # Dataset 2 is a positronic GS calculation
getden2 1 # in presence of the previous electronic density
kptopt2 0 # Use only k=gamma point
ixcpositron2 1 # We are using the Boronski and Nieminen parametrization
```

Here we set positron=1, which corresponds to a positronic ground-state
calculation, considering that the electrons are not perturbed by the presence
of the positron (*zero-positron density limit*). The electron density is read from the file resulting from
dataset 1. As we consider the positron to be completely delocalized, we only
consider the Γ point in the *Brillouin* zone. The keyword ixcpositron
selects the **electron-positron correlation functional** and **enhancement factor**.
In this calculation we use the functional parametrized by Boronski and Nieminen
[Boronski1986], using the data provided by Arponen and Pajanne [Arponen1979].

We can now run the calculation. In the working directory
`Work_positron`

, copy the file `tpositron_1.abi`

.

Then, issue:

```
abinit tpositron_1.abi >& log
```

This calculation should only take a few seconds.

You can look at the `tpositron_1.abo`

file.
We find the positron lifetime calculated in the RPA limit:

```
########## Lifetime computation 2
# Zero-positron density limit of Arponen and Pajanne provided by Boronski & Nieminen
Ref.: Boronski and R.M. Nieminen, Phys. Rev. B 34, 3820 (1986)
# Enhancement factor of Boronski & Nieminen IN THE RPA LIMIT
Ref.: Boronski and R.M. Nieminen, Phys. Rev. B 34, 3820 (1986)
Positron lifetime (ps) = 2.22879743E+02
```

The lifetime of 223 ps agrees well with the value of 225 ps calculated with the same number of valence electrons in [Wiktor2015] and with the experimental value of about 219 ps [Panda1997].

Important

If we had not used the “zero positron density limit” approximation (using, for example, another value of ixcpositron), we would theoretically have needed a box of infinite size for the positron to completely delocalise itself in the crystal. This can be avoided by the use of the posocc input parameter. Numerically, it is equivalent to calculate the density of 1 positron in a box of size N and that of x positron in a box of size N * x. Thus, we can calculate the lifetime of a positron in a primitive cell by setting posocc to a small value (0.0001 …). This value must obviously be tested…

## Computing the positron lifetime in a Si monovacancy¶

We will now perform a positron lifetime calculation for a monovacancy in
silicon in the conventional scheme (which we applied to the perfect lattice
previously). Note that when the positron localizes inside a vacancy, the
*zero-positron density limit* does not apply anymore. However, in some cases, the
conventional scheme proved to yield results in agreement with experiments.

For the purpose of this tutorial, we generate a defect in a cell containing only 16 atoms. This supercell is too small to get converged results, but the calculation is relatively fast.

# Input for Positron tutorial # Second step of the tutorial on electron-positron annihilation # Positron lifetime calculation within PAW # Si monovacancy, "conventional" scheme #Define the different datasets ndtset 2 # 2 datasets #FIRST DATASET positron1 0 # Dataset 1 is a simple electronic GS calculation #SECOND DATASET positron2 1 # Dataset 2 is a positronic GS calculation getden2 1 # in presence of the previous electronic density kptopt2 0 # Use only k=gamma point ixcpositron2 1 # We are using the Boronski and Nieminen parametrization posocc2 1 # Occupation number for the positron # (should be set <1 for bulk calculation with a small cell). # Here the zero positron density limit is used, # so results do not depend on posocc. #------------------------------------------------------------------------------- #Definition of data common to all datasets #Definition of the unit cell acell 3*5.43 angstrom # Lengths of the primitive vectors (exp. param. in angstrom) rprim # 3 orthogonal primitive vectors (FCC lattice, non primitive cell) 0.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 0.0 chkprim 0 # Do not stop if cell is not primitive #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 14 # Atomic number of the possible type(s) of atom. Here silicon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "Pseudodojo_paw_pw_standard/Si.xml" # Name and location of the pseudopotential #Definition of the atoms natom 15 # There are 15 atoms typat 15*1 # They all are of type 1, that is, Silicon xred # Location of the 15 atoms (one triplet per atom): 0.0 0.0 0.0 0.0 0.0 0.5 0.0 0.5 0.0 0.5 0.0 0.0 0.0 0.5 0.5 0.5 0.0 0.5 0.5 0.5 0.0 0.5 0.5 0.5 0.125 0.125 0.125 0.125 0.125 0.625 0.125 0.625 0.125 0.625 0.125 0.125 0.125 0.625 0.625 0.625 0.125 0.625 0.625 0.625 0.125 # 0.625 0.625 0.625 # We remove one Si atom #Definition of bands and occupation numbers nband 36 # Compute 36 bands occopt 1 # Automatic generation of occupation numbers, as a semiconductor #Numerical parameters of the calculation : planewave basis set and k point grid ecut 8. # Maximal plane-wave kinetic energy cut-off, in Hartree pawecutdg 15. # Max. plane-wave kinetic energy cut-off, in Ha, for the PAW double grid kptopt 1 # Automatic generation of k points, taking into account the symmetry ngkpt 2 2 2 # This is a 2 2 2 grid based on the primitive vectors of the recip. space nshiftk 1 # We do not shift the grid in order to have Gamma point in it shiftk 0. 0. 0. #Parameters for the SCF procedure nstep 50 # Maximal number of SCF cycles tolvrs 1.0d-8 # Will stop when, twice in a row, the difference # between two consecutive evaluations of potential residual # differ by less than tolvrs #Miscelaneous parameters prtwf 0 # Do not print wavefunctions prtden 1 # Print density (electronic and/or positronic) prteig 0 # Do not print eigenvalues optforces 0 # Forces computation is not relevant here optstress 0 # Stress tensor computation is not relevant here ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpositron_2.abo, tolnlines= 0, tolabs= 0.0, tolrel= 0.0, fld_options= -easy #%% [paral_info] #%% max_nprocs = 10 #%% [extra_info] #%% authors = J. Wiktor #%% keywords = POSITRON,PAW #%% description = #%% Input for Positron tutorial #%% Second step of the tutorial on electron-positron annihilation #%% Positron lifetime calculation within PAW #%% Si monovacancy, "conventional" scheme #%%<END TEST_INFO>

You can now, issue:

```
abinit tpositron_2.abi >& log
```

Once the calculation is finished, look at the `tpositron_2.abo`

file.
Again, we look at the reported lifetime:

```
########## Lifetime computation 2
# Zero-positron density limit of Arponen and Pajanne provided by Boronski & Nieminen
Ref.: Boronski and R.M. Nieminen, Phys. Rev. B 34, 3820 (1986)
# Enhancement factor of Boronski & Nieminen IN THE RPA LIMIT
Ref.: Boronski and R.M. Nieminen, Phys. Rev. B 34, 3820 (1986)
Positron lifetime (ps) = 2.46923233E+02
```

We observe that when the positron localizes inside the vacancy, its lifetime increases from 223 to 247 ps. This is because now the majority of the positron density is localized in the vacancy region, where the electron density is small. The overlap of the electron and positron densities is reduced, and the lifetime increased.

In the `Work_positron`

directory, you will also find a `tpositron_2o_DS2_DEN_POSITRON`

file, containing the positron density. Visualizing this file (using e.g.
*cut3d* and *XcrysDen* or *VMD*) you can see that the positron is localized
inside the vacancy. You can see below how the positron (in red, isodensity at
30% of the maximum density) localized the silicon monovacancy looks like:

## Performing a self-consistent electron-positron calculation for a Si vacancy¶

We will now perform a self-consistent calculation of the positron and electron
densities. As this calculation will take a few minutes, you can already issue, using
the `tpositron_3.abi`

input file:

```
abinit tpositron_3.abi >& log
```

# Input for Positron tutorial # Third step of the tutorial on electron-positron annihilation # Positron lifetime calculation within PAW # Si monovacancy, self-consistent scheme #To perform a self-consistent electron-positron calculation, we need only one dataset #------------------------------------------------------------------------------- #Definition of variables specific to electron-positron calculation #TC-DFT Self-consistent cycle positron -10 # We perform automatic calculation of electrons and positron densities # in the two-component DFT context (storing wavefunctions in memory) posnstep 20 # Maximum number of electon and positron steps postoldfe 1d-5 # We will repeat the electon and positron steps # until the energy difference is lower than 1d-5 ixcpositron 1 # We are using the Boronski and Nieminen parametrization posocc 1.0 # Occupation number for the positron # (we have only one positron in the cell) #------------------------------------------------------------------------------- #Definition of the unit cell acell 3*5.43 angstrom # Lengths of the primitive vectors (exp. param. in angstrom) rprim # 3 orthogonal primitive vectors (FCC lattice, non primitive cell) 0.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 0.0 chkprim 0 # Do not stop if cell is not primitive #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 14 # Atomic number of the possible type(s) of atom. Here silicon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "Pseudodojo_paw_pw_standard/Si.xml" # Name and location of the pseudopotential #Definition of the atoms natom 15 # There are 15 atoms typat 15*1 # They all are of type 1, that is, Silicon xred # Location of the 15 atoms (one triplet per atom): 0.0 0.0 0.0 0.0 0.0 0.5 0.0 0.5 0.0 0.5 0.0 0.0 0.0 0.5 0.5 0.5 0.0 0.5 0.5 0.5 0.0 0.5 0.5 0.5 0.125 0.125 0.125 0.125 0.125 0.625 0.125 0.625 0.125 0.625 0.125 0.125 0.125 0.625 0.625 0.625 0.125 0.625 0.625 0.625 0.125 # 0.625 0.625 0.625 # We remove one Si atom #Definition of bands and occupation numbers nband 36 # Compute 36 bands occopt 1 # Automatic generation of occupation numbers, as a semiconductor #Numerical parameters of the calculation : planewave basis set and k point grid ecut 8. # Maximal plane-wave kinetic energy cut-off, in Hartree pawecutdg 15. # Max. plane-wave kinetic energy cut-off, in Ha, for the PAW double grid kptopt 1 # Automatic generation of k points, taking into account the symmetry ngkpt 2 2 2 # This is a 2 2 2 grid based on the primitive vectors of the recip. space nshiftk 1 # We do not shift the grid in order to have Gamma point in it shiftk 0. 0. 0. #Parameters for the SCF procedure nstep 500 # Maximal number of SCF cycles. We increase it! toldfe 1.0d-8 # Will stop when, twice in a row, the difference # between two consecutive evaluations of energy # differ by less than toldfe #Miscelaneous parameters prtwf 0 # Do not print wavefunctions prtden 0 # Do not print density (electronic and/or positronic) prteig 0 # Do not print eigenvalues optforces 0 # Forces computation is not relevant here optstress 0 # Stress tensor computation is not relevant here ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpositron_3.abo, tolnlines= 30, tolabs= 1.6e-3, tolrel= 7.0e-1, fld_options= -easy #%% [paral_info] #%% max_nprocs = 10 #%% [extra_info] #%% authors = J. Wiktor #%% keywords = POSITRON,PAW #%% description = #%% Input for Positron tutorial #%% Third step of the tutorial on electron-positron annihilation #%% Positron lifetime calculation within PAW #%% Si monovacancy, self-consistent scheme #%%<END TEST_INFO>

This calculation is significantly longer than the previous one, because the electron and positron steps will be repeated until the convergence criterion is reached.

In `tpositron_3.abi`

we only have one dataset and we set
positron = -10 to perform an automatic calculation of electrons and positron
densities. The convergence is controlled by postoldfe = 1d-5. This means
that we will repeat the electron and positron steps until the energy
difference between them is lower than 1d-5 Ha. This value should always be
larger than toldfe. In this calculation we still use ixcpositron = 1,
which means that we are using the GGGC scheme (see [Gilgien1994] and [Wiktor2015]

Once the calculation is finished, look at the positron lifetime in `tpositron_3.abo`

.

```
########## Lifetime computation 2
# Zero-positron density limit of Arponen and Pajanne provided by Boronski & Nieminen
Ref.: Boronski and R.M. Nieminen, Phys. Rev. B 34, 3820 (1986)
# Enhancement factor of Boronski & Nieminen IN THE RPA LIMIT
Ref.: Boronski and R.M. Nieminen, Phys. Rev. B 34, 3820 (1986)
Positron lifetime (ps) = 2.55612619E+02
```

Including the self-consistency increases the positron lifetime, because its localization inside the vacancy becomes stronger when the positron and the electron densities are allowed to relax.

## Relaxing the vacancy according to forces due to electrons and the positron¶

In addition to the self-consistency, the lifetime of a positron inside a
vacancy can be strongly affected by the relaxation of the atoms due to the
forces coming from both the electrons and the positron. You can already start
the relaxation (with the `tpositron_4.abi`

input file) of the vacancy by issuing:

```
abinit tpositron_4.abi >& log
```

In this calculation we switched on the atomic relaxation by setting
ionmov = 2. We need to calculate forces to be able to move the atoms, so we
set optforces = 1. In the provided `tpositron_4.abi`

file, we only perform 4
relaxation steps (ntime = 4) to save time, but more steps would be needed to
converge the positron lifetime.

# Input for Positron tutorial # Fourth step of the tutorial on electron-positron annihilation # Positron lifetime calculation within PAW # Si monovacancy, relaxation effect #To perform a self-consistent electron-positron calculation, we need only one dataset #------------------------------------------------------------------------------- #Definition of variables specific to electron-positron calculation #TC-DFT Self-consistent cycle positron -10 # We perform automatic calculation of electrons and positron densities # in the two-component DFT context (storing wavefunctions in memory) posnstep 20 # Maximum number of electon and positron steps postoldfe 1d-5 # We will repeat the electon and positron steps # until the energy difference is lower than 1d-5 ixcpositron 1 # We are using the Boronski and Nieminen parametrization posocc 1.0 # Occupation number for the positron # (we have only one positron in the cell) #------------------------------------------------------------------------------- #Definition of variables specific to atomic relaxation ionmov 2 # We now include the effect of the atomic relaxation # (BFGS relaxation algorithm) ntime 4 # We will perform only 4 steps of relaxation # in reality more steps are required optforces 1 # Forces computation done at each (electronic or positronic) SCF step #------------------------------------------------------------------------------- #Definition of the unit cell acell 3*5.43 angstrom # Lengths of the primitive vectors (exp. param. in angstrom) rprim # 3 orthogonal primitive vectors (FCC lattice, non primitive cell) 0.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 0.0 chkprim 0 # Do not stop if cell is not primitive #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 14 # Atomic number of the possible type(s) of atom. Here silicon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "Pseudodojo_paw_pw_standard/Si.xml" # Name and location of the pseudopotential #Definition of the atoms natom 15 # There are 15 atoms typat 15*1 # They all are of type 1, that is, Silicon xred # Location of the 15 atoms (one triplet per atom): 0.0 0.0 0.0 0.0 0.0 0.5 0.0 0.5 0.0 0.5 0.0 0.0 0.0 0.5 0.5 0.5 0.0 0.5 0.5 0.5 0.0 0.5 0.5 0.5 0.125 0.125 0.125 0.125 0.125 0.625 0.125 0.625 0.125 0.625 0.125 0.125 0.125 0.625 0.625 0.625 0.125 0.625 0.625 0.625 0.125 # 0.625 0.625 0.625 # We remove one Si atom #Definition of bands and occupation numbers nband 36 # Compute 36 bands occopt 1 # Automatic generation of occupation numbers, as a semiconductor #Numerical parameters of the calculation : planewave basis set and k point grid ecut 8. # Maximal plane-wave kinetic energy cut-off, in Hartree pawecutdg 15. # Max. plane-wave kinetic energy cut-off, in Ha, for the PAW double grid kptopt 1 # Automatic generation of k points, taking into account the symmetry ngkpt 2 2 2 # This is a 2 2 2 grid based on the primitive vectors of the recip. space nshiftk 1 # We do not shift the grid in order to have Gamma point in it shiftk 0. 0. 0. #Parameters for the SCF procedure nstep 500 # Maximal number of SCF cycles. We increase it! toldfe 1.0d-8 # Will stop when, twice in a row, the difference # between two consecutive evaluations of energy # differ by less than toldfe #Miscelaneous parameters prtwf 0 # Do not print wavefunctions prtden 0 # Do not print density (electronic and/or positronic) prteig 0 # Do not print eigenvalues optstress 0 # Stress tensor computation is not relevant here ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpositron_4.abo, tolnlines= 55, tolabs= 3.0e-2, tolrel= 8.0e-1, fld_options= -easy #%% [paral_info] #%% max_nprocs = 10 #%% [extra_info] #%% authors = J. Wiktor #%% keywords = POSITRON,PAW #%% description = #%% Input for Positron tutorial #%% Fourth step of the tutorial on electron-positron annihilation #%% Positron lifetime calculation within PAW #%% Si monovacancy, relaxation effect #%%<END TEST_INFO>

Look at the positron lifetime in the RPA limit after each ionic step:

```
Positron lifetime (ps) = 2.55612619E+02
Positron lifetime (ps) = 2.56978378E+02
Positron lifetime (ps) = 2.82166606E+02
Positron lifetime (ps) = 2.82878399E+02
```

As the vacancy relaxes outwards, the positron lifetime increases. 4 steps were not enough to relax the defect completely, as the lifetime still changes. Indeed, setting ntime to 10 delivers:

```
Positron lifetime (ps) = 2.55612619E+02
Positron lifetime (ps) = 2.56978379E+02
Positron lifetime (ps) = 2.82166601E+02
Positron lifetime (ps) = 2.82878398E+02
Positron lifetime (ps) = 2.86515373E+02
Positron lifetime (ps) = 2.86983434E+02
Positron lifetime (ps) = 2.87266489E+02
Positron lifetime (ps) = 2.87359897E+02
Positron lifetime (ps) = 2.87313132E+02
```

Although the results at ionic steps 3 and 4 differ from each other by less than one percent, they differ by more from the final result. The one percent convergence is only reached at ionic step 5 and after.

Also, remember that the 16-atom supercell is not large enough to get converged results. In Table IV of [Wiktor2015] you can see converged results of the positron lifetime of Si monovacancy within various methods.

## Computing the electron-positron momentum distribution (Doppler spectrum) of a Si lattice¶

In the last part of the tutorial we will calculate the electron-positron
momentum distribution (*Doppler spectrum*) of a silicon lattice in the conventional
scheme. This type of calculation is much more time and memory consuming than
the *lifetime* calculation, as it is using the electron and positron
*wavefunctions* (not only *densities*).

You can already issue:

```
abinit tpositron_5.abi >& log
```

Now take a look at the input file `tpositron_5.abi`

.

# Input for Positron tutorial # Fifth step of the tutorial on electron-positron annihilation # Doppler spectrum calculation within PAW # Si, 2 atoms in the box #To perform a self-consistent electron-positron calculation, we need only one dataset #------------------------------------------------------------------------------- #Definition of variables specific to electron-positron calculation #TC-DFT Self-consistent cycle positron -10 # We perform automatic calculation of electrons and positron densities # in the two-component DFT context (storing wavefunctions in memory) # Automatic electron-positron loop has to be switched on in Doppler calculations # to have both electron and positron wavefunctions in memory posnstep 2 # Maximum number of electon and positron steps = 2 # We simulate a delocalized positron, so we only perform two steps # of electon-positron calculations. It means that the electronic # wavefunction is not affected by the positron. posdoppler 1 # Activation of Doppler broadening calculation ixcpositron 1 # We are using the Boronski and Nieminen parametrization posocc 1.0 # Occupation number for the positron # (should be set <1 for bulk calculation with a small cell). # Here the zero positron density limit is used, # so results do not depend on posocc. # Note about Brillouin zone sampling: # In Doppler calculation we need to have a uniform k-point grid # in the momentum space. Symmetries are not used, # so the full grid needs to be specified. #------------------------------------------------------------------------------- #Definition of the unit cell acell 3*5.43 angstrom # Lengths of the primitive vectors (exp. param. in angstrom) rprim # 3 orthogonal primitive vectors (FCC lattice) 0.0 1/2 1/2 1/2 0.0 1/2 1/2 1/2 0.0 #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 14 # Atomic number of the possible type(s) of atom. Here silicon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "Pseudodojo_paw_pw_standard/Si.xml" # Name and location of the pseudopotential #Definition of the atoms natom 2 # There are two atoms typat 1 1 # They both are of type 1, that is, Silicon xred # Location of the atoms: 0.0 0.0 0.0 # Triplet giving the reduced coordinates of atom 1 1/4 1/4 1/4 # Triplet giving the reduced coordinates of atom 2 #Definition of bands and occupation numbers nband 6 # Compute 6 bands occopt 1 # Automatic generation of occupation numbers, as a semiconductor #Numerical parameters of the calculation : planewave basis set and k point grid ecut 8. # Maximal plane-wave kinetic energy cut-off, in Hartree pawecutdg 15. # Max. plane-wave kinetic energy cut-off, in Ha, for the PAW double grid # In Doppler broadening calculation we need to have a uniform # k-point grid in the momentum space. Symmetries are not used, # so the full grid needs to be specified. kptopt 0 # - Option for manual setting of k-points istwfk *1 # - No time-reversal symmetry optimization nkpt 8 # - Corresponds to a 2x2x2 grid, denser grids may be needed to get converged spectra kpt # - K-point coordinates in reciprocal space: 0 0 0 0 0 0.5 0 0.5 0 0.5 0 0 0 0.5 0.5 0.5 0 0.5 0.5 0.5 0 0.5 0.5 0.5 #Parameters for the SCF procedure nstep 50 # Maximal number of SCF cycles. tolvrs 1.0d-8 # Will stop when, twice in a row, the difference # between two consecutive evaluations of potential/density # differ by less than tolvrs #Miscelaneous parameters prtwf 0 # Do not print wavefunctions prtden 0 # Do not print density (electronic and/or positronic) prteig 0 # Do not print eigenvalues optforces 0 # Forces computation is not relevant here optstress 0 # Stress tensor computation is not relevant here ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpositron_5.abo, tolnlines= 5, tolabs= 3.0e-2, tolrel= 4.3e-2, fld_options= -easy #%% [paral_info] #%% max_nprocs = 10 #%% [extra_info] #%% authors = J. Wiktor #%% keywords = POSITRON,PAW #%% description = #%% Input for Positron tutorial #%% Fifth step of the tutorial on electron-positron annihilation #%% Doppler spectrum calculation within PAW #%% Si, 2 atoms in the box #%%<END TEST_INFO>

The momentum distribution calculation is activated by posdoppler = 1. You can also notice that instead of having two datasets as in the first part of this tutorial, we now use the automatic electron-positron loop and set posnstep = 2. This is done because we need to have the full electron and positron wavefunctions in memory, which is only the case when positron <= -10. Additionally, the momentum distribution calculations require using a full k-point grid. In the input file we set:

```
kptopt 0 # Option for manual setting of k-points
istwfk *1 # No time-reversal symmetry optimization
nkpt 8 # Corresponds to a 2x2x2 grid
kpt # K-point coordinates in reciprocal space:
0 0 0
0 0 0.5
0 0.5 0
0.5 0 0
0 0.5 0.5
0.5 0 0.5
0.5 0.5 0
0.5 0.5 0.5
```

This grid is used in both electron and positron calculations, but only the
positron *wavefunction* at the first point is taken in the momentum distribution
calculation, so the \Gamma point should always be given first.

In the calculation of the momentum distribution, we need to include both *core*
and *valence* electrons. The *wavefunctions* of the core electrons are read from a
file (one per atom type), which needs to be provided. This *core WF file* should
be named <psp_file_name>.corewf.xml
(where `<psp_file_name>`

is the name of the PAW atomic dataset file, without `.xml`

suffix).
*Core WF files* can be obtained with the `atompaw`

tool
(see the tutorial on generating PAW datasets (PAW2) ) by the use of the
`prtcorewf`

keyword. You will find the core wavefunction file used in this calculation in
`$ABI_PSPDIR/Si.LDA-PW-paw.abinit.corewf`

.

Note

If you use a PAW dataset in *ABINIT legacy proprietary format* (with the `.abinit`

suffix),
the core wavefunction file has to be named `<psp_file_name>.corewf.abinit`

.
It also can be obtained with the `atompaw`

tool by the use of the `prtcorewf`

keyword.

Once the calculation is complete, you can find a `tpositron_5o_DOPPLER`

file
containing the *momentum distribution* on the FFT grid. You can use the
`$ABI_HOME/scripts/post_processing/posdopspectra.F90`

tool to generate 1D
projections (*Doppler spectra*) in (001), (011) and (111) directions and to
calculate the low- and high-momentum contributions to the
momentum distribution (so called `S`

and `W`

parameters, see [Wiktor2015]).

## Studying the effect of the PAW dataset completeness¶

The positron lifetime and momentum distribution calculations within the PAW
method are very sensitive to the number of valence electrons in the **PAW
dataset**. It is due to the fact that it is not easy to describe the positron
*wavefunction*, tending to zero at the nucleus, using the electron atomic
orbitals. The **PAW basis set** in this case needs to be more complete than only
for describing the electron *wavefunctions*.

The simplest way to make the **PAW
dataset** more complete is to include `semicore electrons`

. It is also possible to
add the `partial waves`

corresponding to the `semicore electrons`

in the basis
used only for the positron wave function description, while keeping the
initial number of valence electrons (as done in [Wiktor2015]).
However, this second method is less straightforward.

The previous calculations were done with only **4 valence electrons** (`3s`

and `3p`

).
We will now see what happens if we include the `2s`

and `2p`

states in the **PAW dataset**.
We use the `Si_paw_pw_12el.xml`

PAW dataset which includes 8 additional valence electrons.

Tip

To generate the new dataset we use the `atompaw`

tool.

To add `semicore states`

, the input file is modified
as follows:

- Replace
`c`

by`v`

for the selected orbitals in the*electronic configuration*section - Decrease the PAW augmentation radius (because semicore states are more localized)

Don’t forget to add `prtcorewf`

keyword to create the core orbital file.

We can now rerun the lifetime calculation with the new atomic dataset:

# Input for Positron tutorial # Sixth step (part 1) of the tutorial on electron-positron annihilation # Positron lifetime calculation within PAW - 12 valence electrons # Si, 2 atoms in the box # This input file is similar to tpositron_1.abi, except: # - the number of bands, because of the additional 8 electrons states #Define the different datasets ndtset 2 # 2 datasets #FIRST DATASET positron1 0 # Dataset 1 is a simple electronic GS calculation #SECOND DATASET positron2 1 # Dataset 2 is a positronic GS calculation getden2 1 # in presence of the previous electronic density kptopt2 0 # Use only k=gamma point ixcpositron2 1 # We are using the Boronski and Nieminen parametrization posocc2 1 # Occupation number for the positron # (should be set <1 for bulk calculation with a small cell). # Here the zero positron density limit is used, # so results do not depend on posocc. #------------------------------------------------------------------------------- #Definition of data common to all datasets #Definition of the unit cell acell 3*5.43 angstrom # Lengths of the primitive vectors (exp. param. in angstrom) rprim # 3 orthogonal primitive vectors (FCC lattice) 0.0 1/2 1/2 1/2 0.0 1/2 1/2 1/2 0.0 #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 14 # Atomic number of the possible type(s) of atom. Here silicon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "Si_paw_pw_12el.xml" # Name and location of the pseudopotential #Definition of the atoms natom 2 # There are two atoms typat 1 1 # They both are of type 1, that is, Silicon xred # Location of the atoms: 0.0 0.0 0.0 # Triplet giving the reduced coordinates of atom 1 1/4 1/4 1/4 # Triplet giving the reduced coordinates of atom 2 #Definition of bands and occupation numbers nband 14 # Compute 14 bands (we have semicore states) occopt 1 # Automatic generation of occupation numbers, as a semiconductor #Numerical parameters of the calculation : planewave basis set and k point grid ecut 8. # Maximal plane-wave kinetic energy cut-off, in Hartree pawecutdg 15. # Max. plane-wave kinetic energy cut-off, in Ha, for the PAW double grid kptopt 1 # Automatic generation of k points, taking into account the symmetry ngkpt 4 4 4 # This is a 4x4x4 grid based on the primitive vectors of the recip. space nshiftk 1 # We do not shift the grid in order to have Gamma point in it shiftk 0. 0. 0. #Parameters for the SCF procedure nstep 50 # Maximal number of SCF cycles tolvrs 1.0d-8 # Will stop when, twice in a row, the difference # between two consecutive evaluations of potential residual # differ by less than tolvrs #Miscelaneous parameters prtwf 0 # Do not print wavefunctions prtden 1 # Print density (electronic and/or positronic) prteig 0 # Do not print eigenvalues optforces 0 # Forces computation is not relevant here optstress 0 # Stress tensor computation is not relevant here ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpositron_6.abo, tolnlines= 0, tolabs= 0.0, tolrel= 0.0, fld_options= -easy #%% [paral_info] #%% max_nprocs = 10 #%% [extra_info] #%% authors = J. Wiktor #%% keywords = POSITRON,PAW #%% description = #%% Input for Positron tutorial #%% Sixth step (part 1) of the tutorial on electron-positron annihilation #%% Positron lifetime calculation within PAW - 12 valence electrons #%% Si, 2 atoms in the box #%%<END TEST_INFO>

```
abinit tpositron_6.abi >& log
```

We now find the positron lifetime calculated in the RPA limit:

```
########## Lifetime computation 2
# Zero-positron density limit of Arponen and Pajanne provided by Boronski & Nieminen
Ref.: Boronski and R.M. Nieminen, Phys. Rev. B 34, 3820 (1986)
# Enhancement factor of Boronski & Nieminen IN THE RPA LIMIT
Ref.: Boronski and R.M. Nieminen, Phys. Rev. B 34, 3820 (1986)
Positron lifetime (ps) = 2.11470610E+02
```

This value is significantly lower than 223 ps achieved with 4 valence
electrons in the first step. **It is, therefore, very important to always test
the PAW dataset completeness for positron calculations**.

The PAW dataset completeness is even more important in the *Doppler spectra*
calculations. We will now recalculate the momentum distribution including 12
*valence electrons* using `tpositron_7.abi`

:

# Input for Positron tutorial # Sixth step (part 2) of the tutorial on electron-positron annihilation # Doppler spectrum calculation within PAW - 12 valence electrons # Si, 2 atoms in the box # This input file is similar to tpositron_5.abi, except: # - the number of bands, because of the additional 8 electrons states # - the plane wave cut-off energy because additional PAW basis functions are localized #------------------------------------------------------------------------------- #Definition of variables specific to electron-positron calculation #TC-DFT Self-consistent cycle positron -10 # We perform automatic calculation of electrons and positron densities # in the two-component DFT context (storing wavefunctions in memory) # Automatic electron-positron loop has to be switched on in Doppler calculations # to have both electron and positron wavefunctions in memory posnstep 2 # Maximum number of electon and positron steps = 2 # We simulate a delocalized positron, so we only perform two steps # of electon-positron calculations. It means that the electronic # wavefunction is not affected by the positron. posdoppler 1 # Activation of Doppler broadening calculation ixcpositron 1 # We are using the Boronski and Nieminen parametrization posocc 1.0 # Occupation number for the positron # (should be set <1 for bulk calculation with a small cell). # Here the zero positron density limit is used, # so results do not depend on posocc. # Note about Brillouin zone sampling: # In Doppler calculation we need to have a uniform k-point grid # in the momentum space. Symmetries are not used, # so the full grid needs to be specified. # so results do not depend on posocc. #------------------------------------------------------------------------------- #Definition of data common to all datasets #Definition of the unit cell acell 3*5.43 angstrom # Lengths of the primitive vectors (exp. param. in angstrom) rprim # 3 orthogonal primitive vectors (FCC lattice) 0.0 1/2 1/2 1/2 0.0 1/2 1/2 1/2 0.0 #Definition of the atom types and pseudopotentials ntypat 1 # There is only one type of atom znucl 14 # Atomic number of the possible type(s) of atom. Here silicon. pp_dirpath "$ABI_PSPDIR" # Path to the directory were # pseudopotentials for tests are stored pseudos "Si_paw_pw_12el.xml" # Name and location of the pseudopotential #Definition of the atoms natom 2 # There are two atoms typat 1 1 # They both are of type 1, that is, Silicon xred # Location of the atoms: 0.0 0.0 0.0 # Triplet giving the reduced coordinates of atom 1 1/4 1/4 1/4 # Triplet giving the reduced coordinates of atom 2 #Definition of bands and occupation numbers nband 16 # Compute 16 bands (we have semicore states) occopt 1 # Automatic generation of occupation numbers, as a semiconductor #Numerical parameters of the calculation : planewave basis set and k point grid ecut 12. # Maximal plane-wave kinetic energy cut-off, in Hartree pawecutdg 15. # Max. plane-wave kinetic energy cut-off, in Ha, for the PAW double grid # In Doppler broadening calculation we need to have a uniform # k-point grid in the momentum space. Symmetries are not used, # so the full grid needs to be specified. kptopt 0 # - Option for manual setting of k-points istwfk *1 # - No time-reversal symmetry optimization nkpt 8 # - Corresponds to a 2x2x2 grid, denser grids may be needed to get converged spectra kpt # - K-point coordinates in reciprocal space: 0 0 0 0 0 0.5 0 0.5 0 0.5 0 0 0 0.5 0.5 0.5 0 0.5 0.5 0.5 0 0.5 0.5 0.5 #Parameters for the SCF procedure nstep 100 # Maximal number of SCF cycles tolvrs 1.0d-10 # Will stop when, twice in a row, the difference # between two consecutive evaluations of potential residual # differ by less than tolvrs # We need additional parameters to imporve the convergency: nline 8 # - increase the number of iterations of the minimization algorithm nnsclo 2 # - perform 2 non-self-consistent loop per SCF cycle nbdbuf 16 # This is to make the test portable (don't use this usually) #Miscelaneous parameters prtwf 0 # Do not print wavefunctions prtden 1 # Print density (electronic and/or positronic) prteig 0 # Do not print eigenvalues optforces 0 # Forces computation is not relevant here optstress 0 # Stress tensor computation is not relevant here ############################################################## # This section is used only for regression testing of ABINIT # ############################################################## #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% [files] #%% files_to_test = #%% tpositron_7.abo, tolnlines= 7, tolabs=9., tolrel= 1.0, fld_options= -easy #%% [paral_info] #%% max_nprocs = 10 #%% [extra_info] #%% authors = J. Wiktor #%% keywords = POSITRON,PAW #%% description = #%% Input for Positron tutorial #%% Sixth step (part 2) of the tutorial on electron-positron annihilation #%% Doppler spectrum calculation within PAW - 12 valence electrons #%% Si, 2 atoms in the box #%%<END TEST_INFO>

```
abinit tpositron_7.abi >& log
```

Before processing the new `tpositron_7o_DOPPLER file`

, you should copy files
`rho_001`

, `rho_011`

, `rho_111`

from the fifth step to for instance `si4el_001`

, `si4el_011`

and `si4el_111`

.
By plotting the *Doppler spectra* in the (001) direction calculated with 4 and
12 valence electrons, you should obtain a figure like this:

The dataset with 4 valence electrons is **not complete enough** to describe the
positron **wavefunction** around the nucleus. This is reflected in the
unphysically high probability at high momenta in the spectrum.

Further explanation of the influence of the PAW dataset on the *Doppler spectra*
can be found in [Wiktor2015]. In case you need to generate
your own dataset for momentum distribution calculations,
you can follow the tutorial on generating PAW datasets (PAW2).