## How to Calculate Interstitial Formation Energy for Aluminum

### Interstitial Formation Energy Calculation

The necessary tools to perform the basic calculations to investigate the interstital formation energy is shown in this page.

Figure 1. Consider one atom at the center-most part of the fcc Al system. Then add one more atom nearby that, coordinates of the new atom depends on whether octahedral (add (a/2, 0, 0)) or tetrahedral (add (a/4, a/4, a/4)) location is asked.

Atomic positions and a volume relaxation are performed with DFT calculations when the extra atom is inserted in one of the interstitial sites, preferably in the center of the simulation box. Similar to the vacancy formation calculations, this interstitial defect formation energy also needs a large supercell in order to accommodate more atoms, such that to keep the interaction of periodic defects minimized. To calculate the interstitial energy subtract energy per atom of Bulk Aluminum (∈) times the number of atoms (including the self interstitial) from the total energy of the structure containing the interstitial (Etot).

For a binary AB alloy, the number of calculations increases as one needs to calculate the interstitial formation energy for Element A in Matrix B and Element B in Matrix A; also for both interstitial positions: octahedral and tetrahedral positions.

Note: In order to get the octahedral position, consider any atom coordinates (x,y,z) that are part of the perfect crystal structure, then add (a/2, 0, 0) to it to get the coordinates for octahedral interstitial atom location. For a tetrahedral position add (a/4, a/4, a/4) to get the tetrahedral interstitial atom coordinates; "a" being the lattice parameter.

### Relaxation

For relaxation (geometric or ionic optimizations) always use ISMEAR = 1 and SIGMA = 0.1 or ISMEAR = 2 and SIGMA = 0.2. It is always recommended to do ionic relaxations at fixed volumes and plot the energy vs volume graph to determine the equilibrium volume. To get accurate energies obtain the structure after relaxation and run a static calculation (no relaxations NSW = 0) with ISMEAR = -5.

Proposed method to relax and get accurate energies

The system should be relaxed (ISIF=2) with ISMEAR = 1 (Methfessel-Paxton). At the end of the relaxation run VASP will generate new positions in the CONTCAR file. Copy the CONTCAR as POSCAR. Then run a static calculation (no relaxations, NSW = 0) with the tetrahedron method (ISMEAR = -5).

#### INCAR file for relaxation

```LWAVE = .FALSE.
LCHARG = .FALSE.
LREAL = Auto
ISMEAR = 1
ENCUT = 240.3
EDIFF = 1e-6
NSW=100
ISIF=2
IBRION=2
```

#### INCAR file for static calculation

```LWAVE = .FALSE.
LCHARG = .FALSE.
LREAL = Auto
ISMEAR = -5
ENCUT = 240.3
EDIFF = 1e-6
#NSW=100
ISIF=2
IBRION=2
```

### Running the calculation

To find the energy of the system using VASP

• Copy the following INCAR and KPOINTS file to a directory.
• Copy the POTCAR file to the same directory
• Copy the POSCAR file and insert atom (either octahedral or tetrahedral) and modify the total no. of atoms to create an interstitial.
• Then do
`ulimit -s unlimited`
and execute VASP by
`mpirun -np <no. of processors> <path of executable>`
• To submit to a computer cluster use the following method

#### Submitting job to cluster

Two files are needed to submit to the cluster. One is the pbs command script and the other is a job.sh shell script to invoke ulimit command on all allocated processors. They are presented below.Both files should be in your work directory with the rest of the VASP input files.

Pbs command script
```#PBS -N
#PBS -l nodes=4:ppn=4
#PBS -l walltime=48:00:00
#PBS -q q64p48h@raptor
#PBS -mea
#PBS -r n
#PBS -V
cd \$PBS_O_WORKDIR

mpiexec -np 16 ./job.sh
```

Job.sh script
```#!/bin/bash
ulimit -s unlimited
<Absolute path of VASP exe>
```

### Post-processing

After running the calculation use the equation to calculate interstitial energy. Etot is the Energy you get from VASP. This value can be extracted at the end of the calculation by executing
```tail -n1 OSZICAR|awk '{print \$5}'
```

### k-point convergence

The interstitial energy should be converged in terms of k-point grid. Therefore try different k-point grids. Remember that the ratio of k-points should be inversely proportional to the lengths of the lattice vectors (in this case the edges of the simulation box).

#### KPOINTS

```MK11x11x2	#header file
0
Monkhorst	#Style of Kpoints
3  3  3	#Numbers
0  0  0
```

#### POSCAR file with 3X3X3 orthogonal supercell containing 108 atoms

```Al fcc (111)
1.0
12.1499999999999986      0.0000000000000000      0.0000000000000000
0.0000000000000000     12.1499999999999986      0.0000000000000000
0.0000000000000000      0.0000000000000000     12.1499999999999986
108 <----Change to 109 after adding atom
Cartsian
0.0000000000000000     0.0000000000000000     0.0000000000000000
0.0000000000000000     2.0249999999999999     2.0249999999999999
2.0249999999999999     0.0000000000000000     2.0249999999999999
2.0249999999999999     2.0249999999999999     0.0000000000000000
0.0000000000000000     0.0000000000000000     4.0499999999999998
0.0000000000000000     2.0249999999999999     6.0749999999999993
2.0249999999999999     0.0000000000000000     6.0749999999999993
2.0249999999999999     2.0249999999999999     4.0499999999999998
0.0000000000000000     0.0000000000000000     8.0999999999999996
0.0000000000000000     2.0249999999999999    10.1250000000000000
2.0249999999999999     0.0000000000000000    10.1250000000000000
2.0249999999999999     2.0249999999999999     8.0999999999999996
0.0000000000000000     4.0499999999999998     0.0000000000000000
0.0000000000000000     6.0749999999999993     2.0249999999999999
2.0249999999999999     4.0499999999999998     2.0249999999999999
2.0249999999999999     6.0749999999999993     0.0000000000000000
0.0000000000000000     4.0499999999999998     4.0499999999999998
0.0000000000000000     6.0749999999999993     6.0749999999999993
2.0249999999999999     4.0499999999999998     6.0749999999999993
2.0249999999999999     6.0749999999999993     4.0499999999999998
0.0000000000000000     4.0499999999999998     8.0999999999999996
0.0000000000000000     6.0749999999999993    10.1250000000000000
2.0249999999999999     4.0499999999999998    10.1250000000000000
2.0249999999999999     6.0749999999999993     8.0999999999999996
0.0000000000000000     8.0999999999999996     0.0000000000000000
0.0000000000000000    10.1250000000000000     2.0249999999999999
2.0249999999999999     8.0999999999999996     2.0249999999999999
2.0249999999999999    10.1250000000000000     0.0000000000000000
0.0000000000000000     8.0999999999999996     4.0499999999999998
0.0000000000000000    10.1250000000000000     6.0749999999999993
2.0249999999999999     8.0999999999999996     6.0749999999999993
2.0249999999999999    10.1250000000000000     4.0499999999999998
0.0000000000000000     8.0999999999999996     8.0999999999999996
0.0000000000000000    10.1250000000000000    10.1250000000000000
2.0249999999999999     8.0999999999999996    10.1250000000000000
2.0249999999999999    10.1250000000000000     8.0999999999999996
4.0499999999999998     0.0000000000000000     0.0000000000000000
4.0499999999999998     2.0249999999999999     2.0249999999999999
6.0749999999999993     0.0000000000000000     2.0249999999999999
6.0749999999999993     2.0249999999999999     0.0000000000000000
4.0499999999999998     0.0000000000000000     4.0499999999999998
4.0499999999999998     2.0249999999999999     6.0749999999999993
6.0749999999999993     0.0000000000000000     6.0749999999999993
6.0749999999999993     2.0249999999999999     4.0499999999999998
4.0499999999999998     0.0000000000000000     8.0999999999999996
4.0499999999999998     2.0249999999999999    10.1250000000000000
6.0749999999999993     0.0000000000000000    10.1250000000000000
6.0749999999999993     2.0249999999999999     8.0999999999999996
4.0499999999999998     4.0499999999999998     0.0000000000000000
4.0499999999999998     6.0749999999999993     2.0249999999999999
6.0749999999999993     4.0499999999999998     2.0249999999999999
6.0749999999999993     6.0749999999999993     0.0000000000000000
4.0499999999999998     4.0499999999999998     4.0499999999999998
4.0499999999999998     6.0749999999999993     6.0749999999999993
6.0749999999999993     4.0499999999999998     6.0749999999999993 <------------------Insert interstitial atom
6.0749999999999993     6.0749999999999993     4.0499999999999998
4.0499999999999998     4.0499999999999998     8.0999999999999996
4.0499999999999998     6.0749999999999993    10.1250000000000000
6.0749999999999993     4.0499999999999998    10.1250000000000000
6.0749999999999993     6.0749999999999993     8.0999999999999996
4.0499999999999998     8.0999999999999996     0.0000000000000000
4.0499999999999998    10.1250000000000000     2.0249999999999999
6.0749999999999993     8.0999999999999996     2.0249999999999999
6.0749999999999993    10.1250000000000000     0.0000000000000000
4.0499999999999998     8.0999999999999996     4.0499999999999998
4.0499999999999998    10.1250000000000000     6.0749999999999993
6.0749999999999993     8.0999999999999996     6.0749999999999993
6.0749999999999993    10.1250000000000000     4.0499999999999998
4.0499999999999998     8.0999999999999996     8.0999999999999996
4.0499999999999998    10.1250000000000000    10.1250000000000000
6.0749999999999993     8.0999999999999996    10.1250000000000000
6.0749999999999993    10.1250000000000000     8.0999999999999996
8.0999999999999996     0.0000000000000000     0.0000000000000000
8.0999999999999996     2.0249999999999999     2.0249999999999999
10.1250000000000000     0.0000000000000000     2.0249999999999999
10.1250000000000000     2.0249999999999999     0.0000000000000000
8.0999999999999996     0.0000000000000000     4.0499999999999998
8.0999999999999996     2.0249999999999999     6.0749999999999993
10.1250000000000000     0.0000000000000000     6.0749999999999993
10.1250000000000000     2.0249999999999999     4.0499999999999998
8.0999999999999996     0.0000000000000000     8.0999999999999996
8.0999999999999996     2.0249999999999999    10.1250000000000000
10.1250000000000000     0.0000000000000000    10.1250000000000000
10.1250000000000000     2.0249999999999999     8.0999999999999996
8.0999999999999996     4.0499999999999998     0.0000000000000000
8.0999999999999996     6.0749999999999993     2.0249999999999999
10.1250000000000000     4.0499999999999998     2.0249999999999999
10.1250000000000000     6.0749999999999993     0.0000000000000000
8.0999999999999996     4.0499999999999998     4.0499999999999998
8.0999999999999996     6.0749999999999993     6.0749999999999993
10.1250000000000000     4.0499999999999998     6.0749999999999993
10.1250000000000000     6.0749999999999993     4.0499999999999998
8.0999999999999996     4.0499999999999998     8.0999999999999996
8.0999999999999996     6.0749999999999993    10.1250000000000000
10.1250000000000000     4.0499999999999998    10.1250000000000000
10.1250000000000000     6.0749999999999993     8.0999999999999996
8.0999999999999996     8.0999999999999996     0.0000000000000000
8.0999999999999996    10.1250000000000000     2.0249999999999999
10.1250000000000000     8.0999999999999996     2.0249999999999999
10.1250000000000000    10.1250000000000000     0.0000000000000000
8.0999999999999996     8.0999999999999996     4.0499999999999998
8.0999999999999996    10.1250000000000000     6.0749999999999993
10.1250000000000000     8.0999999999999996     6.0749999999999993
10.1250000000000000    10.1250000000000000     4.0499999999999998
8.0999999999999996     8.0999999999999996     8.0999999999999996
8.0999999999999996    10.1250000000000000    10.1250000000000000
10.1250000000000000     8.0999999999999996    10.1250000000000000
10.1250000000000000    10.1250000000000000     8.0999999999999996
```