# Random displacements#

Phonopy supports generating two types of random displacements.

• Random directions with constant displacement distance

• Random sampling of harmonic oscillator probability densities of phonon modes

In both cases, phonopy generates dataset of displacements of supercells in phonopy_disp.yaml, supercells with displacements, and a perfect supercell. The number of the supercells with displacements are specified by RANDOM_DISPLACEMENTS tag (or --rd option) in either case.

Forces on atoms in supercells with displacements are calculated by a force calculator and force constants are computed by a force constants calculator. As the force constants calculator, currently only ALM is supported.

## Random directions with constant displacement distance#

### Example#

An example to generate the random direction diplacements with distance of 0.03 Angstrom in example/NaCl-rd directory is as shown below,

 % phonopy --rd 10 --dim 2 2 2 --pa auto --amplitude 0.03 -c POSCAR-unitcell
_
_ __ | |__   ___  _ __   ___   _ __  _   _
| '_ \| '_ \ / _ \| '_ \ / _ \ | '_ \| | | |
| |_) | | | | (_) | | | | (_) || |_) | |_| |
| .__/|_| |_|\___/|_| |_|\___(_) .__/ \__, |
|_|                            |_|    |___/
2.20.0

Compiled with OpenMP support (max 10 threads).
Python version 3.11.4
Spglib version 2.1.0

Crystal structure was read from "POSCAR-unitcell".
Unit of length: angstrom
Displacements creation mode
Number of supercells with random displacements: 10
Displacement distance: 0.03
Settings:
Supercell: [2 2 2]
Spacegroup: Fm-3m (225)
Use -v option to watch primitive cell, unit cell, and supercell structures.

"phonopy_disp.yaml" and supercells have been created.

Summary of calculation was written in "phonopy_disp.yaml".
_
___ _ __   __| |
/ _ \ '_ \ / _ |
|  __/ | | | (_| |
\___|_| |_|\__,_|


In the generated phonopy_disp.yaml, dataset of displacements is stored as follows:

% cat phonopy_disp.yaml
...
displacements:
- #    1
- displacement: # 1
[  -0.0201336051051884,  0.0137526506253330,  0.0174786311319239 ]
- displacement: # 2
[  -0.0176810186962932,  0.0037245664471804, -0.0239480517504424 ]
- displacement: # 3
[  -0.0028626950563326, -0.0032145292137935,  0.0296895904139501 ]
- displacement: # 4
[   0.0251664434224757,  0.0149681266628547,  0.0065272742908559 ]
- displacement: # 5
[  -0.0136393731379345, -0.0226372187886819, -0.0141959087739228 ]
- displacement: # 6
[   0.0299122348895043, -0.0003839832933863, -0.0022606991718315 ]
- displacement: # 7
[  -0.0265153637733336, -0.0030921513486749, -0.0136884653633881 ]
- displacement: # 8
[   0.0284712562630672, -0.0049105574746481, -0.0080792321473599 ]
...


where the displacements are given in the Cartesian coordinates. The supercell with displacements in the VASP interface is found as

% cat POSCAR-001
generated by phonopy
1.0
11.3806029523513423 0.0000000000000000 0.0000000000000000
0.0000000000000000 11.3806029523513423 0.0000000000000000
0.0000000000000000 0.0000000000000000 11.3806029523513423
Na Cl
32 32
Direct
0.9982308841465181 0.0012084289982625 0.0015358264588532
0.4984463899873917 0.0003272732088778 0.9978957132718971
0.9997484584016929 0.4997175431541499 0.0026087888786082
0.5022113453503161 0.5013152314271505 0.0005735438023965
0.9988015245593718 0.9980108945999205 0.4987526224371979
0.5026283523829750 0.9999662598462494 0.4998013550616521
0.9976701266282333 0.4997282963511142 0.4987972108841070
0.5025017353107100 0.4995685151748806 0.4992900875128334
...


It is easy to verify consistency between them, e.g.,

np.dot(ph.displacements[0], np.linalg.inv(ph.supercell.cell)) + ph.supercell.scaled_positions


Once forces on atoms in these supercells with displacements, FORCE_SETS can be generated by the usual manner. For the NaCl-rd example,

% phonopy -f force-calcs/disp-{001..010}/vasprun.xml


where phonopy_disp.yaml has to be located in the current directory where the above command is executed. The FORCE_SETS file is written in the Type 2 format. To compute force constants for the type-2 displacements-forces dataset, an external force constants calculator is necessary, e.g., ALM (ttadano/ALM). With installed ALM, force constants can be calculated by

% phonopy --alm --writefc


With this command, the force constants are written in FORCE_CONSTANTS.

## Random sampling of harmonic oscillator probability densities of phonon modes#

Random displacements corresponding to a specific temperature are generated with the combination of the RANDOM_DISPLACEMENTS (--rd) and RANDOM_DISPLACEMENT_TEMPERATURE tags. This is based on the harmonic phonon approximation. Since each phonon follows the probability density function of the harmonic oscillator, the random displacements are generated by randomly sampling the probability density functions of phonons at the commensurate points. More details are found in the paper “Implementation strategies in phonopy and phono3py” (see Citation of phonopy).

### Example#

To generate the random displacements in this approach, phonon information is required. At the first step, phonons at the commensurate points are calculated. Then, the random displacements are generated in the next step. If FORCE_CONSTANTS file exists in the current directory, as an example, this is performed by

% phonopy --readfc --rd 1000 --temperature 300


where 1000 supercells with displacements at 300K are generated.

### Visualization of displacement distance distribution#

Distribution of displacement distances may be visualized and checked by a python script like below.

import matplotlib.pyplot as plt
import numpy as np
from phonopy.interface.phonopy_yaml import PhonopyYaml
phyml = PhonopyYaml()
plt.hist(np.linalg.norm(phyml.dataset['displacements'].reshape(-1, 3), axis=1), 100)
plt.show()


### Another example#

In the below, a step-by-step example is presented.

1. Compute supercell force sets in the usual manner with small displacements. Generate FORCE_SETS in the current directory.

2. Create new directory and copy FORCE_SETS and input crystal structure, or phonopy_disp.yaml to the new directory, then change to the new directory.

3. Using FORCE_SETS, generate supercells with random displacements at finite temperature using this tag. phonopy_disp.yaml is generated at this step. Therefore, if phonopy_disp.yaml already exists in this directory, it is overwritten. The required number of supercells depends on your system and also the purpose. It can be 10, or can be 1000.

4. Using those supercells with random displacements, calculate supercell forces with some force calculator with VASP, QE, etc.

5. Generate FORCE_SETS in this directory, which overwrites previous FORCE_SETS.

6. Run the force constants calculation, e.g., with ALM. ALM does the fitting force constants using the dataset of displacements and forces stored in FORCE_SETS. Run also phonon calculation. This is done for example,

phonopy -v --alm --mesh 10 10 10 -p


or

phonopy -v --alm --writefc
phonopy --readfc --mesh 10 10 10 -p
`

Force constants calculated as above contain a part of temperature effect from atomic displacements. To even better include the temperature effect, this calculation has to be done self-consistently, i.e., it is required to repeat the loop over the steps 2-5 until the phonon structure is converged. The solution corresponds to a stochastic self-consistent harmonic approximation (SSCHA) approach.

It is also possible to perform the same calculation as above using the phonopy API.