Table of Contents

1. What is the QHA_2D program ?
2. What is the quasi-harmonic approximation ?
3. Power of the quasi-harmonic approximation
4. Why is QHA_2D useful ?
5. Files needed for running QHA_2D
6. How to run QHA_2D
7. Test
8. How to cite
9. Contributing
10. References

What is the QHA_2D program ?

QHA_2D is a program for computational chemistry and physics that performs the quasi-harmonic approximation reading the frequencies at each volume calculated with CRYSTAL.

• Extracts all the frequencies within all the k points in the supercell for a given volume.

• Calculates the pressure at finite temperature as a function of volume, as well as the entropy, Helmholtz and Gibbs free energy.

• Produces tables summarizing the results for all the volumes analyzed.

• For a pair of solid state phases, if produces a PDF document with plots of the following type:

• Outputs the pressure-temperature phase diagram for the thermodynamic phase stability of both solid phases:

• The underlying criteria for producing this phase boundary is by evaluating where two Gibbs free energy intersect:

The program was developed as part of David Carrasco de Busturia PhD project at Prof. Nicholas Harrison's Computational Materials Science Group, Imperial College London. The program was used to investigate the phase diagram and phase transitions mechanisms on the calcium carbonate system.

What is the quasi-harmonic approximation ?

Since the birth of quantum chemistry, almost every calculation was performed in the athermal limit (0K) and no pressure effects were considered (0Pa). One of the most exciting challenges in an ab intio calculation is to obtain information of the system at a finite temperature and pressure. This allow us to obtain a more realistic picture of the system in the everyday world, where temperature and pressure cannot be neglected and are indeed the driving force for many transformations in nature.

One of the most famous techniques for taking into account the effect of the temperature in the computed properties of molecules and crystals is ab intio molecular dynamics [1-3], in which the Schrodinger equation is solved at each MD time step within the Born-Oppenheimer approximation. Unfortunetely, this is a very computationally expensive technique. Therefore, there is a huge interest in developing accurate and reliable models for the inclusion of a combined effect of pressure and temperature in the standard first-principles quantum chemical methods.

The sole effect of pressure can be easily taken into account if the structure is fully relaxed through a volume constraint geometry optimization process. If this process is repeated over a series of equidistant volumes ranging from compression to expansion a energy-volume or pressure-volume relation equation of state is obtained, which describes the behaviour of a solid under compression and expansion in the athermal limit. On the contrary, the inclusion of temperature is not so straightforward to implement. Following the harmonic approximation (HA) formalism, the lattice dynamics of crystal vibrations (i.e. phonons) are calculated at each volume.

The HA indeed presents serious limitations that arise from the fact that only constant volume quantities can be calculated, so that thermal expansion cannot be explained. Simpler and less computationally expensive, but still effective tecnhiques can be used based on the so called quasi-harmonic approximation (QHA), which corrects the HA deficiencies by maintaining the same harmonic expression but introducing an explicit dependence of vibration phonon frequencies on volume. For a more detailed explanation, please check the pdf in this repository.

Power of the quasi-harmonic approximation

By implementing a quasi-harmonic approximation framework we are able to calculate the phase diagram of a solid substance (like the one shown in the above figure) and the thermodynamic stability of different polymorphs, without the need of running computationally expensive ab initio molecular dynamics calculations. In addition, this leads to the exploration of phase transitions at a fnite temperature and pressure.

Why is QHA_2D useful ?

The actual version of CRYSTAL17 does perform an atomated quasi-harmonic approximation calculation for a given set of volumes. However, the QHA built-in module in CRYSTAL presents some deficiencies:

• If the phase transition is driven by a soft phonon mode, i.e. a phonon mode that becomes negative at a certain volume constraint, the QHA module in CRYSTAL will give some problems.

• Unfortunalety, the optimization at a constant volume is performed within the supercell scheme. If the supercell is big, (as it should be in order to ensure convergence the entropy), this might leads to an optimization porcess which is without doubt, more difficult in the supercell scheme: the cell is bigger, there are more atoms, and this can lead to convergence problems, or flase minima.

• Not relevant (unwanted) supercell phase transitions as a consequence of the optimization being performed in the supercell.

• Ideally, CRYSTAL should perform the optimization in the the primitive cell prior to making the supercell for the phonons calculation at a finite k point, but this is not so trivial to implement in the main code, according to the developers. Hopefully, this will be taken into account in future versions of the code. But for the moment, the QHA_2D code presented in this repository is an easy and effective solution for evaluating thermodynamic properties of crystals at a finite temperature and pressure (the real world).

Files needed for running QHA_2D

• Say you want to compute the pressure-temperature phase diagram of two sold phases I and II. QHA_2D requires the frequency calculation outputs at each volume, for each of the two phases. These frequencies calculations can be either in the Gamma point or at finite k points.

• The name of all these frequency outputs have to end as *.out

• Please ensure that you are using a sufficient big supercell for the entropy to be converged with the number of k points (mention the other code to sort out the supercell expansion matrix).

How to run QHA

• Get the code: git clone https://github.com/DavidCdeB/QHA_2D
• Give permissions to all the scripts: chmod u+x *.sh *.py
• Create the Files_Outputs folder inside the QHA_2D folder that has just been cloned: cd ./QHA_2D && mkdir Files_Outputs
• Create the folders that will contain the constant-volume frequency outputs for each phase: mkdir Calcite_I && mkdir Calcite II
• Copy all the frequencies outputs for each volume, for each phase, to the folders Calcite_I and Calcite_II. For example, Calcite_I folder will contain the frequency output for each j-th volume for the Calcite I phase.
• Remember that name of all these frequency outputs have to end as *.out
• The file system at this point looks like the following:

The following is a summarized flow chart on the structure of codes:

• Run ./boundary_1_node.sh

Prerequisites

To run, QHA requires Python with certain packages:

• Python 2.7 or higher. Packages: numpy, scipy, re, os, glob, itertools, subprocess, sys (All of these come with a default Anacaonda installation).

• Standard bash version in your system.

Test

Under the TEST folder, you will find all the programs needed, together with a Files_Outputs folder with the frequency outputs of two phases: calcite I and calcite II. If you run the program, you will obtain the main.pdf with all the plots needed.

How to cite

Please cite the following reference when using this code:

D. Carrasco-Busturia, "The temperature - pressure phase diagram of the calcite I - calcite II phase transition: A first-principles investigation", Journal of Physics and Chemistry of Solids, vol. 154, p. 110 045, 2021. DOI: https://doi.org/10.1016/j.jpcs.2021.110045.

Here the bibtex:

@article{CARRASCOBUSTURIA2021110045,                                                                        
title = {The temperature - pressure phase diagram of the calcite {I} - calcite {II} phase transition: A first-principles investigation},
journal = {Journal of Physics and Chemistry of Solids},
volume = {154},
pages = {110045},
year = {2021},
issn = {0022-3697},
doi = {https://doi.org/10.1016/j.jpcs.2021.110045},
url = {https://www.sciencedirect.com/science/article/pii/S0022369721001116},
author = {David Carrasco-Busturia}
}

Contributing

QHA is free software released under the Gnu Public Licence version 3. All contributions to improve this code are more than welcome.

• Have a look at GitHub's "How to contribute".

• If you are familiar with git: fork this repository and submit a pull request.

• If you are not familiar with git:

  • If something should be improved, open an issue here on GitHub
  • If you think a new feature would be interesting, open an issue
  • If you need a particular feature for your project contact me directly.

References

[1] R. Car, M. Parrinello, Phys. Rev. Lett. 1985, 55, 2471

[2] F. Buda, R. Car, M. Parrinello, Phys. Rev. B 1990, 41, 1680

[3] F. D. Vila, V. E. Lindahl, J. J. Rehr, Phys. Rev. B 2012, 85, 024303