Low-Cost Vibrational Free Energies in Solid Solutions with Machine Learning Force Fields

The rational design of alloys and solid solutions relies on accurate computational predictions of phase diagrams. The cluster expansion method has proven to be a valuable tool for studying disordered crystals. However, the effects of vibrational entropy are commonly neglected due to the computational cost. Here, we devise a method for including the vibrational free energy in cluster expansions with a low computational cost by fitting a machine learning force field (MLFF) to the relaxation trajectories available from cluster expansion construction. We demonstrate our method for two (pseudo)binary systems, Na1–xKxCl and Ag1–xPdx, for which accurate phonon dispersions and vibrational free energies are derived from the MLFF. For both systems, the inclusion of vibrational effects results in significantly better agreement with miscibility gaps in experimental phase diagrams. This methodology can allow routine inclusion of vibrational effects in calculated phase diagrams and thus more accurate predictions of properties and stability for mixtures of materials.

I recommend the paper to be published a�er major revision regarding the following points.
1.I wonder if the authors could discuss the possible extension of this approach beyond mixing energies in mixed systems, for example could one employ a this approach to defect forma�on energies, surface interface energies etc?In these cases one typically also relaxes many structures to obtain their minimum energy in order to e.g.calculate defect forma�on energies.Do the authors think it would be possible to train MLFFs on these relaxa�on trajectories and "for free" get the defect vibra�onal forma�on free energy, or would this not be feasible?
2. In total quite a number of training structures is used for training the final MLFF.Do the authors have some arguments why using the relaxa�on trajectories is beter than e.g.spending the same amount of computa�onal resources to train a MLFF using a conven�onal approach?
3. I think would be useful and instruc�ve for readers if the authors could include figures of the total and vibra�onal mixing free energy as a func�on of concentra�on for some select temperatures.This would give some context on how small changes in the vibra�onal free energy would for example shi� a phase-boundary by say 100K.Addi�onally, this would help understand how small (or large) the vibra�onal free energy errors are shown in Fig 1d,e,f) and Fig 3d,e,f).Furthermore, could the authors calculate the phase diagram obtained from the MLFF-200 model?Again, I think this would help the reader understand how small energy and entropy errors propagate into complex thermodynamic proper�es such the phase diagram.
4. The harmonic approxima�on can lead to significant errors for strongly anharmonic materials and/or at high temperatures.Therefore it would be interes�ng to see, for maybe just for one or a few configura�ons, how accurate the harmonic approxima�on is at the relevant temperatures compared to the full anharmonic free energy for the system studied.This would help further validate the approach in the manuscript.
5. The authors report "Generally, we observe that the mixed phases are significantly stabilised due to vibra�onal free energy.", is there some explana�on for this?The mixing vibra�onal free energy of a configura�on can o�en be an�-correlated with its mixing energy, i.e. strong s�ff bonds yields a low energy but also high frequencies and thus a low vibra�onal entropy (as observed in previous work where cluster expansion have been combined with vibra�onal free energies).Is this something the authors could explore further to provide an explana�on for their observa�on that the inclusion of vibra�onal effects stabilizes mixed phases? 1 6.I did not understand the authors discussion on the error cancella�on in the pure phases, can the authors expand and clarify this?For cluster expansions in general the error may be larger for the end members due to the large bias in the training structures towards concentra�ons around 50% and almost no data points close to x=0 or x=1.
7. What is the mo�va�on for using the constraints when construc�ng the CEs?And what are the consequences (in terms of mixing energies and thermodynamic proper�es) of applying such constraints?For example maybe showing the phase diagram with and without the constraints would help mo�vate the usage of constraints.Addi�onally, I think the authors should specify how these constraints are imposed as there are many different ways one can do this.9. Showing the parity plots of e.g.energy and forces for the MLFF would be good for the reader.It is for example hard to evaluate if an error of 4.8meV/Å is small or not, specially since most of the training structures likely have forces close to zero?
10.It would be helpful with some addi�onal details regarding the relaxa�on process.Why are the structures ratled and then relaxed, and why then symmetrized and then relaxed again?And how is the symmetriza�on done?
11.It would be useful some more informa�on regarding the phonopy calcula�ons, e.g.how large was the q-point mesh used?Do the authors not include LO-TO spli�ng in the calcula�ons (is this not relevant for the free energy)?
12. The term "end-members" was introduced without any explana�on.
13.A sugges�on is to also cite sklearn and possibly sources therein for the implementa�on of ARDR.
Author's Response to Peer Review Comments: Dear Prof. Editor, We thank you for considering our manuscript for publica�on in the Journal of Physical Chemistry Leters.Below, we will give a point-by-point response to the reviewers' comments.The original reviewer comments are marked in blue, our responses in black, and changes to the manuscript highlighted in yellow.
We have also submited two versions of the manuscript, one with changes highlighted in yellow, and one without highligh�ng.
In addi�on, we have taken into account the style changes requested by the editor.
Yours Sincerely, Kasper Tolborg and Aron Walsh

Reviewer 1
The key advancement presented in this work is the inclusion of vibra�onal energy within the cluster expansion methodology for studying alloys and solid solu�ons.This �es nicely into the recent trend where modern machine learning and regression advancements have added new interest to cluster expansions, and the authors' work is a very valuable addi�on to this narra�ve.
This study has the poten�al for delivering significant improvements in the precision of models for alloys and solid solu�ons for a small computa�onal overhead rela�ve to previous methods, which is undoubtedly of great relevance to the community of physical and materials chemistry.
I propose a few points for the authors to consider before resubmission, which I believe do not cons�tute major barriers to publica�on.
We thank the reviewer for their posi�ve comments, and we will respond to their points below.
Firstly, in the first paragraph it is implied that this approach could allow us a beter understanding of high-entropy and many-component systems.Yet, these systems pose inherent challenges within "standard" cluster expansion due to their composi�onal and configura�onal complexity [1,2,3].It appears that the surrogate model applied may face similar or more significant scalability issues, and I think this casual associa�on is slightly misleading.I would prefer the link not implied, or that it was qualified.
We thank the reviewer for poin�ng this out.We did not intend to imply that our method would be easily applicable to high-entropy compounds for which the standard cluster expansion is struggling.We have slightly reworded the first paragraph, so we refer to high entropy systems only as a mo�va�on for studying alloys and solid solu�ons, whereas the first principles calcula�on of phase diagrams now refers to alloys and solid solu�ons to not imply that our method is targeted high entropy systems.Furthermore, we have expanded upon our discussion in the final paragraph on the future work and applicability to more complex systems sta�ng now (where the three references are the ones suggested by the reviewer): "For these, conven�onal cluster expansion modelling is already faced with challenges due to the many composi�onal degrees of freedom [1,2,3], and we expect the present methodology to face similar challenges." Regarding the AgPd phase diagram, the inclusion of vibra�onal energy introduces an interes�ng qualita�ve discrepancy when compared to experimental findings, with the emergence of a plateau absent in empirical data.As this is a proof-of-concept work, I don't see a need for immediate resolu�on in this manuscript.However, an acknowledgment of this discrepancy, discussing poten�al contributors such as model selec�on or training data, would be beneficial for the reader We thank the referee for poin�ng this out and we are happy to address it.The plateau was merely a result of a too coarse grid on the temperature axis for the interpola�on used when iden�fying the phase boundaries from the data arising from the Monte Carlo simula�ons.We have updated the figures both for AgPd and (Na/K)Cl with a finer temperature spacing, which leads to a disappearance of the plateau.
Lastly, for enhancing the manuscript, I suggest including a reference to itera�ve structure selec�on based on predicted uncertain�es, which could further op�mise the training data genera�on and efficacy of the modelling process [4].
We thank the reviewer for sugges�ng this reference.We have added it in the final sec�on as another example (besides Ref. 36 and 37 in the original manuscript) of an efficient method for selec�ng training data for the cluster expansion.
Finally, I'd just like to reiterate the fact I think this is an exci�ng development in the treatment of solid solu�ons for the materials chemistry community, and I am excited to see how the authors and the community go on to build on these developments.
We again thank the referee for this posi�ve and encouraging comment, and we share the excitement for future developments in the field.
The paper "Low-Cost Vibra�onal Free Energies in Solid Solu�ons with Machine Learning Force Fields" concerns a new computa�onal approach for modeling vibra�onal free energies in alloys and mixed systems.The inclusion of vibra�onal free energies when modeling alloys can be very important for many systems, as have previously been shown and is shown by the authors.Therefore, the development and improving methods to carry out such modeling more efficiently is important and relevant for computa�onal materials science in general.

What is the immediate significance of this advance?
The methodology outline in the manuscript will allow researchers to more accurately model alloys and mixed systems by taking into account vibra�onal free energy in a computa�onal efficient way.The convergence tes�ng and detailed analysis of the accuracy of the models will be helpful for future studies on this topic.
We thank the reviewer for their posi�ve comments and technical sugges�ons.We will address those point by point below.

Technical sugges�ons
I recommend the paper to be published a�er major revision regarding the following points.

1.
I wonder if the authors could discuss the possible extension of this approach beyond mixing energies in mixed systems, for example could one employ a this approach to defect forma�on energies, surface interface energies etc?In these cases one typically also relaxes many structures to obtain their minimum energy in order to e.g.calculate defect forma�on energies.Do the authors think it would be possible to train MLFFs on these relaxa�on trajectories and "for free" get the defect vibra�onal forma�on free energy, or would this not be feasible?This is a very good sugges�on, and we indeed hope that a similar methodology would be feasible for defects, surfaces and interfaces for which the computa�onal workflow with many relaxa�ons of similar structures is quite similar to the workflow here.We have added a paragraph at the end of the manuscript to highlight these poten�al applica�ons, although we note that they will require extensive tes�ng before widely employed: "Finally, a similar methodology of training a force field on relaxa�on trajectories to calculate vibra�onal free energies could poten�ally be used when modelling defects, surfaces and interfaces in crystals.Also for these systems, several relaxa�ons of similar structures are performed with DFT, and vibra�onal free energies have been shown to be important for quan�ta�ve proper�es in several cases (Mosquera-Lois et al., 2023;Kempisty et al., 2019)"

2.
In total quite a number of training structures is used for training the final MLFF.Do the authors have some arguments why using the relaxa�on trajectories is beter than e.g.spending the same amount of computa�onal resources to train a MLFF using a conven�onal approach?
We agree that quite a lot of training structures are used for the final MLFF, and that o�en fewer training structures are used for construc�ng cluster expansions of (pseudo)binary systems.However, we note that all relaxa�ons are performed on small supercells, meaning that all DFT calcula�ons are rela�vely fast.Using a more conven�onal approach for MLFFs of training on molecular dynamics trajectories typically includes simula�ons in larger supercells, suffering from the poor size scaling of DFT.Furthermore, using the enumerated symmetryunique small supercells ensures that all relevant local environments are properly sampled.As we note in the final paragraphs, future studies should indeed inves�gate if beter selec�on of training data -both for the cluster expansion and trajectories for the MLFF -is possible.
As noted in the response to comment 3, we have now also included a phase diagram calculated from the MLFF-200 force field, which also results in good agreement with experimental observa�ons.Thus, it is not unlikely that smaller data sets can be used in future with this methodology.
3. I think would be useful and instruc�ve for readers if the authors could include figures of the total and vibra�onal mixing free energy as a func�on of concentra�on for some select temperatures.This would give some context on how small changes in the vibra�onal free energy would for example shi� a phaseboundary by say 100K.Addi�onally, this would help understand how small (or large) the vibra�onal free energy errors are shown in Fig 1d,e,f) and Fig 3d,e,f).Furthermore, could the authors calculate the phase diagram obtained from the MLFF-200 model?Again, I think this would help the reader understand how small energy and entropy errors propagate into complex thermodynamic proper�es such the phase diagram.
The mixing (internal) energies as a func�on of concentra�on are given in Fig. S9a and S18a (figure numbers in the revised manuscript), and Fig. S9b+c and S18b+c gives the mixing energy including vibra�onal contribu�ons at 800 K for both systems.From these, it can, for example, be seen that the difference between Fig. S18a and S18c (~10 meV/f.u.) is enough to shi� the miscibility gap by ~150 K in AgPd.
To test the effects of the accuracy of the force field and vibra�onal free energy, we have calculated the phase diagram with vibra�onal contribu�ons using the MLFF-200 force field for the (Na/K)Cl system.The cluster expansion reconstruc�on and the calculated phase diagram are shown in Fig. S10 and S12.We note that MLFF-200 gives a poor fit to pure NaCl (Fig. S6), and thus the phase diagram using only MLFF free energies is very poor.However, using the DFT vibra�onal free energies for the end-members again gives good agreement with experiment.Furthermore, we have calculated the phase diagram using a cluster expansion, which is not constrained to reproduce the (free) energy of the end-members exactly as suggested in comment 7.The results are shown in Fig. S11 and S13 and furthermore highlights how differences in the cluster expansion fi�ng can lead to differences in the resul�ng phase diagram.
We have added the following paragraph to discuss these effects: "To inves�gate the effect of accuracy of the force field and of constraining the energy of the endmembers, we provide two addi�onal test: (i) we use the force field only trained on the first 200 relaxa�on trajectories (denoted MLFF-200 in Fig. 1); and (ii) we follow the same procedure as above, but do not enforce the cluster expansion to reproduce the (free) energies of the pure phases exactly.The cluster expansion fit and calculated phase diagram of (i) are shown in Fig. S10 and S12.We note that the vibra�onal free energy of NaCl is poorly reproduced with MLFF-200 (Fig. S6), and thus only relying on the MLFF gives a poor phase diagram.However, using the vibra�onal free energy from DFT for the endmembers results in a phase diagram in good agreement with experiment also for this force field, which is trained on significantly less data.Similarly, the cluster expansion fit and calculated phase diagram of (ii) are shown in Fig. S11 and S13.We note that this leads to large devia�ons in the (free) energy of the endmembers between cluster expansion and reference energies, since the large amount of data for mixed phases bias the cluster expansion towards these.Generally, this leads to smaller mixing (free) energies (Fig. S11) and results in a miscibility gap at lower temperatures (Fig. S13) compared to the constrained model shown in Fig. 2.

These different tests highlight the precision of the phase diagram reconstruc�on resul�ng
from the different model choices, both in terms of conven�onal cluster expansion choices such as whether to constrain the model to end-members, and in terms of accuracy of the vibra�onal free energy.In all cases, inclusion of vibra�onal entropy significantly improves the agreement with experimental results." 4. The harmonic approxima�on can lead to significant errors for strongly anharmonic materials and/or at high temperatures.Therefore it would be interes�ng to see, for maybe just for one or a few configura�ons, how accurate the harmonic approxima�on is at the relevant temperatures compared to the full anharmonic free energy for the system studied.This would help further validate the approach in the manuscript.
We agree with the reviewer that anharmonic effects are o�en important at high temperatures.However, we consider a treatment of anharmonic effects beyond the scope of the current proof-of-concept paper for the following reasons: (i) Comparison of anharmonic and harmonic absolute free energies are not possible as the compared phases are then not treated on an equal foo�ng.Thus, many configura�ons would need to be calculated at the same anharmonic level of theory to gain insight into whether anharmonicity would have an effect on mixing energies, or if the effects are similar for all systems. (ii) Inclusion of anharmonic effects comes with a significant computa�onal overhead, even if the calcula�on of force constants in rela�vely fast, especially when 100s or 1000s of structures need to be evaluated.This means that our methodology is only feasible within the harmonic approxima�on for which the computa�onal overhead of the phonon calcula�ons (a�er the force constants are determined) is rela�vely light.

(iii)
Calcula�on of anharmonic effects would require benchmarking of the accuracy of higher order force constants (or derived effec�ve harmonic phonon dispersions and free energies) against DFT to understand if the calculated free energies are meaningful.This would add a significant computa�onal cost to the present study as well as other studies using the method.
In total, inves�ga�ng anharmonic effects is beyond the present scope, but we acknowledge that they may have an effect.We have thus added a paragraph towards the end of the paper describing that the present methodology is only feasible within the harmonic approxima�on, which we, however, believe is already a significant step forward compared to neglec�ng vibra�onal contribu�ons completely.The paragraph reads: "We note that the present methodology is developed for including vibra�onal effects within the harmonic approxima�on.Anharmonic effects have been shown to contribute to the stability of complex alloys (Grabowski et al., 2019), but its inclusion in a cluster expansion would significantly increase the computa�onal cost and require further benchmarking."5.The authors report "Generally, we observe that the mixed phases are significantly stabilised due to vibra�onal free energy.", is there some explana�on for this?The mixing vibra�onal free energy of a configura�on can o�en be an�-correlated with its mixing energy, i.e. strong s�ff bonds yields a low energy but also high frequencies and thus a low vibra�onal entropy (as observed in previous work where cluster expansion have been combined with vibra�onal free energies).Is this something the authors could explore further to provide an explana�on for their observa�on that the inclusion of vibra�onal effects stabilizes mixed phases?
The comment cited by reviewer refers to he (Na/K)Cl system for which vibra�onal contribu�ons stabilize the mixed phases.During the revision, we realized that we had also men�oned that the inclusion of vibra�onal free energy stabilizes the mixed phases of AgPd.As one can see from Fig. S10, the mixed phases at around x=0.5 are in fact slightly destabilized at 800 K, which is the reason why the mixed phases around x=0.8 experience a rela�ve stabiliza�on by vibra�onal entropy and are therefore predicted to be miscible at lower temperatures.We have corrected this observa�on in the main text, so it now reads: "Compared to the ´Na1-xKxCl case, a smaller effect of vibra�onal entropy and a different kind of stabilisa�on is observed.In Ag1-xPdx, the structures around x~0.5 are slightly destabilised rela�ve to the end-members at elevated temperatures, which results in a rela�ve stabilisa�on of mixed phases around x~0.8 as will be clear from the phase diagram." Regarding the idea that mixed phases are o�en stabilized by vibra�onal entropy, the reviewer is completely right that there is o�en a tendency that higher energy configura�on have larger vibra�onal entropy, since weaker bonds is the origin of both.However, as shown by Manzoor et al. ( 2018) (Ref.10 in both original and revised manuscript), there are also several examples in which the opposite is true.The present study did not intend to deeply inves�gate this rela�on and its origin, and we do not believe that studying the rela�ons deeply in the two systems inves�gated here would add significantly to our general understanding of this phenomenon, since several counterexamples of effects in the opposite direc�on exist.We agree that the subject merits further study in a wider range of systems.
6.I did not understand the authors discussion on the error cancella�on in the pure phases, can the authors expand and clarify this?For cluster expansions in general the error may be larger for the end members due to the large bias in the training structures towards concentra�ons around 50% and almost no data points close to x=0 or x=1.
The lower degree of error cancella�on in the pure phases as men�oned in the last paragraph on page 2 refers to their vibra�onal free energies, and not the cluster expansion.We simply mean that when rela�vely few phonon branches are present, the vibra�onal free energy, which is calculated as an integral over these, becomes much more prone to small devia�ons in their frequencies.For systems with many phonon branches, one can expect some cancella�on of errors in the vibra�onal free energy, if some phonon frequencies are predicted a bit too high and others a bit too low, which will o�en be the case.We have edited the paragraph to clarify this, so it now reads: "The error in the vibra�onal free energy for the end-members, especially NaCl is significantly larger than for the mixed systems, Fig. S6.We atribute this to a poor cancella�on of errors for systems with few atoms, and thus few phonon branches, despite the seemingly similar agreement in terms of phonon dispersion and DOS.Since the vibra�onal free energy is calculated from a sum over the phonon frequencies, one can expect mixed systems with more phonon branches to have both overes�mated and underes�mated frequencies resul�ng in a cancella�on of errors, which is less likely for the pure phases with few atoms." 7. What is the mo�va�on for using the constraints when construc�ng the CEs?And what are the consequences (in terms of mixing energies and thermodynamic proper�es) of applying such constraints?For example maybe showing the phase diagram with and without the constraints would help mo�vate the usage of constraints.Addi�onally, I think the authors should specify how these constraints are imposed as there are many different ways one can do this.
Using constraints for pure phases is rela�vely standard prac�ce when construc�ng cluster expansions, since we are interested in reproducing the mixing (free) energy, which is the relevant property to be sampled in the subsequent Monte Carlo simula�on.If no constraints are applied, the pure phases will not have zero mixing energy, which they ideally should.Since we consider the reference (DFT) energies the "ground truth" energies, we believe it is reasonable to constrain our cluster expansion model to reproduce these.However, to show the effect of not constraining to the end-members, we have constructed a cluster expansion model and calculated a phase diagram for (Na/K)Cl without constraints.The results are shown in Fig. S11 and S13, and discussed in the paragraph highlighted in the response to comment 3. Furthermore, we have added the following note on how the constraints are implemented: "The cluster expansion is fited to mixing (free) energies, and constrained to reproduce the (free) energies of the end-members exactly with the get_mixing_energy_constraints module of ICET" We thank the reviewer for poin�ng this out, and we have included structural drawings of all the test cases and noted their composi�ons in Fig. S8 and S17 in the Suppor�ng Informa�on.To clarify, they are not random structures, but selected structures from the training set on which the cluster expansion is constructed.This should be clear now, when the structural drawings are included.9. Showing the parity plots of e.g.energy and forces for the MLFF would be good for the reader.It is for example hard to evaluate if an error of 4.8meV/Å is small or not, specially since most of the training structures likely have forces close to zero?
We thank the reviewer for this sugges�on.We have added parity plots for the forces on the test set between MLFF and DFT for two selected MLFFs in Fig. S5 and S15.We believe that such an out-of-sample error is more meaningful than the error on the training set.Since the energies are weighted very low in the fi�ng procedure because they are not used for calcula�ng the relevant vibra�onal proper�es, we do not benchmark our force field against DFT energies.
10.It would be helpful with some addi�onal details regarding the relaxa�on process.Why are the structures ratled and then relaxed, and why then symmetrized and then relaxed again?And how is the symmetriza�on done?
The structures are ratled, since star�ng from the subs�tuted structures on the parent la�ce resulted in a much lower quality MLFF in our ini�al tests.Star�ng from ratled structures does require in more relaxa�on steps (about 50 % more steps here), but it is o�en used to avoid ending in structures that are local saddle point on the poten�al energy surface, most strikingly for Jahn-Teller ac�ve species.The symmetriza�on is performed a�er the ini�al ionic convergence, since we o�en end up in structures very close to a higher symmetry, but the numerical devia�ons from this symmetry may lead to failure of the DFT code, e.g., if direct and reciprocal la�ces are determined to be of different symmetry as a result of the numerical imprecision.
We have added a few sentences on this, including the size of ratling and men�oned that the symmetriza�on is performed with spglib: "For each system, all symmetry unique structures with up to 8 metal atoms in the unit cell are included, giving 631 structures for both systems.All relaxa�ons are started from slightly ratled structures using the ratle func�on of the Atomic Simula�on Environment with a standard devia�on of 0.05 Å (Larsen et al., 2017).Performing the relaxa�ons without ratling results in too poor sampling of the poten�al energy surface for training the subsequent MLFF.While the ratling does increase the number of steps in the relaxa�on trajectories, it also has the benefit of breaking the ideal symmetries to search for lower energy configura�ons close to the ideal symmetry.The structures are re-symmetrised using spglib (Togo et al., 2018) a�er convergence of the first ionic relaxa�on, since small devia�on from ideal symmetries a�er the first relaxa�on resulted in poor numerical stability.A�er this, relaxa�on is restarted as the plane wave basis changes with the size and shape of the simula�on cell." 11.It would be useful some more informa�on regarding the phonopy calcula�ons, e.g.how large was the q-point mesh used?Do the authors not include LO-TO spli�ng in the calcula�ons (is this not relevant for the free energy)?
We have added a few sentences with more informa�on on phonopy parameters.We do not include LO-TO spli�ng, since determining these would require significant addi�onal calcula�ons for all training data.It is true that it may affect the vibra�onal free energy, but we expect that the effect is similar on all structures, since the formal charges are the same for K and Na in (Na/K)Cl, and in AgPd, LO-TO spli�ng does not appear as the system is metallic.We have added the following note on this: "A 20x20x20 and 40x40x40 q-mesh are used for integrals over the phonon dispersions to calculate DOS and vibra�onal free energies for Na1-xKxCl and Ag1-xPdx, respec�vely.No LO-TO spli�ng is included in the models.Ag1-xPdx is metallic and thus have no contribu�on of LOTO spli�ng.For Na1-xKxCl, LO-TO spli�ng is relevant, but it requires access to Born effec�ve charges and dielectric constants, which would need to be calculated for each structure, adding a large computa�onal cost.Since the formal ionic charges of Na and K are the same, we expect the effect of LO-TO spli�ng to be similar on all structures, and thus we neglect it in the present study." 12. The term "end-members" was introduced without any explana�on.
We have writen that by end-members we mean the pure phases at first instance to make it clear that this is what we mean.We thank the referee for no�ng this.
13.A sugges�on is to also cite sklearn and possibly sources therein for the implementa�on of ARDR.
We thank the referee for sugges�ng this and have added a reference to scikit-learn as the implementa�on of ARDR used by ICET.

8.
Can the authors spell out more precisely what structure the phonon dispersions are calculated for in Fig 1a,b,c) and Fig3a,b,c), e.g.what are he concentra�ons, are the structures completely random or ordered etc?

8.
Fig 1a,b,c) and Fig3a,b,c), e.g.what are he concentra�ons, are the structures completely random or ordered etc?