Electronic structure of liquid xenon in the context of light dark matter direct detection

1—Clearly written.
2—Well organized structure.
3—Clear motivation of the work
4—Complete theoretical overview of dark matter induced ionization in xenon detectors.
5—Good comparisons to experiment to justify DFT approximations
6—interesting dark matter induced xenon ionization rate results

1—The writing is overly verbose in places
2—Detailed electronic structure analysis is not consistently strong throughout
3—The discussion of the dark matter induced ionization rates in Sec IV C is underdeveloped compared to the rest of the manuscript.

The manuscript titled “Electronic structure of liquid xenon in the context of light dark matter direct detection” by R. Catena et al. perform an in-depth study of the electronic structure and electronic density of both liquid and solid xenon, and estimate the dark matter scatting rate with an eye towards the XENON10 and XENON1T experiments. Various properties such at the 4d levels were found to be sensitive to the treatment of exchange-correlation effects. Moreover, a marked change in the 5p band width between solid/liquid phases influences the dark matter scatting rate. Overall, I find this work quite comprehensive, readable --though verbose at times--, and provide a good baseline for future studies in this field. Before I can accept this article, I have a few questions and comments/suggestions for the authors below.

(A) When the authors say (first paragraph of Sec. III) “…use of a plane-wave basis set for the expansion of the Kohn-Sham states is particularly convenient for ensuring convergence.” What is meant? Convergence as in accuracy? Number of SCF steps needed? Each basis/implementation has their own pros/cons, so it would be good to be clear about the methods used and why so that non-experts in electronic structure can clearly get the point.

(B) Since the Hubbard U is an empirical fix, I am curious as to the performance of metaGGA (e.g. SCAN, HSE, mBJ, …) functionals and/or MBPT corrections (e.g. GW) on the xenon 4d states? Furthermore, references to previous work in this section are quite slim. Is there truly very little work in this area? Are there no other theoretical calculations to compare with?

(C) For the lattice constants, yes LDA over binds and PBE under binds so it makes sense the vdW correction helps to fix PBE, but is it for the right reason? Would the vdW correction be minimal for a functional that captures a good lattice parameter out-of-the-box? Or is this a general limitation of semi-local functionals, making their description of the subtle long-range interactions between xenon atoms out of reach? Since I see this work setting the baseline in this area, I think it would be good to fully tease out the key problem areas for electronic structure modeling to tackle in this field.

(D) Can the authors give some reference(s) to justify the appropriateness of the Lennard-Jones potential for xenon? There are a number of pairwise, many-body potentials, and specialized models in the field, so some reasoning as to why Lennard-Jones would be good to call out. Moreover, are the parameters used similar to those that would be obtained with DFT?

(E) In Fig. 5, can the authors offer some insights into why the peak in liquid phase theoretical RDF at 15 [a.u.] appears to be shifted to higher larger distances? Also, is there a physical reason why the peak in the experimental RDF at $\sim$13 [a.u.] is missing in the liquid phase theoretical results?

(F) It will be good to present the calculated dielectric function to facilitate more direct comparison with the electronic structure results.

(G) Since a number of different functional+vdw+Hubbard corrections were discussed in Sec. III, it is unclear what was actually used for the data in Sec. IV. Please call this out in text and clearly specify this in the relevant figure captions.

(H) Since Sec. III E and F feel auxiliary to the flow of the text, I suggest moving Sec. III E and F to an appendix.

(I) When comparing/analyzing the electronic densities, have the authors compared against an all-electron code, such as Wien2k? In such codes there is no pseudopotential problem and they could provide a good benchmark to ensure the quality of the planewave calculations. This will be important to certify the important high momentum regimes of the density. Also, currently Fig. 9 does not add much to the discussion, but adding comparisons to other pseudopotentials and/or all-electron codes would enrich the results.

(J) When modeling the final states (page 15 bottom left-hand column), this discussion appears to be similar to those that occurred in the 1970s surrounding the modeling of ARPES spectra, i.e. pros/cons of the 1 step and 3 step model. Could this field learn from these previous discussions? Additionally, in this connection, calculated ARPES spectra is highly sensitive to the surface potential. Can the authors speak to this issue in the context of xenon detectors a bit more and how it might effect the utilization of the theoretical predictions such as those in Fig. 14 for XENON10 and XENON1T experiments?

(K) Why is there a kink in the blue lines of Fig. 14 (a)?

(L) The text in a number of places throughput the manuscript is heavy, I suggest editing to help with the flow.

In addition to addressing questions (A)-(L) in the Report section, please fix the minor issues below:

1) In the abstract, can the authors be specific as to the numerical range of ‘low’ when they state “…5p levels induced by the liquid phase is relevant only at low dark matter masses…”? Similarly, when they say “For most of the probable mass range the energies…”, can ‘probable’ be defined. Adding the mass ranges explicitly will help the reader at a glance determine the relevance of this work to their own interstates without digging in the text.

2) Footnote 3, change ‘consistently’ to ‘consistent’

3) In figure 5, it will be good to specify the temperature at which the experimental data was collected.

4) First paragraph in the left-hand column of Page 10, it says “We obtain an fcc structure and the sharp Dirac-delta-like peaks in the RDF shown with purple solid lines in Fig. 5.” I believe the authors mean as shown in blue solid lines.

5) Since there are many types of ‘material response functions’ in condensed matter, is there a better name for W? For instance, when modeling dark matter scatting in solids authors have used similar terms when discussing the dielectric response, which is different from the meaning here.

6) I find it strange to indicate vdw-DF, vdw-DF2, and rVV10 on the horizontal axis of Fig. 3 (a) but not plot anything. Since QE does not have this combination of treatments implemented, I suggest just stating it in the text (as was done) and leave them out of the figure.

Ask for major revision

1- The authors clearly stated the problem which they wanted to address. Dark matter (DM) detection experiments are designed by considering the range of the masses of assumed DM particles. For heavy candidates, above GeV, recoil of the atomic nucleus is relevant. However, for light mass DM candidate particles, ie., lighter than 1 GeV, DM - electron interactions are important. In regard to the existing detectors containing liquid xenon, a formalism is necessary to compute scattering rates for the electrons of the liquid xenon. The introduction is very good at explaining both the problem and the approach taken by the authors to investigate it.

2- The authors obtained an expression for the dark matter (DM) - electron interaction rate which can be used either for an isolated atom or for the liquid state. This provides a convenient way to compare the effect of different targets on the interaction rate.

3- In the case of an isolated atom, Hartree-Fock type approaches can be sufficient. However, for the description of the electronic structure in condensed phases, the preferred state-of-the-art methods are based on density functional theory (DFT). Hence, a comparison of the atom versus liquid state requires also a comparison of different electronic structure methodologies. Furthermore, the DFT implementations are not unique, there are different exchange-correlation functional parametrizations, different ways of incorporating van der Waals interactions, and the issues of DFT+U methods and the treatment of spin-orbit coupling .. all these imply a wide spectrum of computational tools available for such a study. The authors provide a comparison of many such methods and justify their choice. They also provide a comparison of standard Roothan-Hartree-Fock method and the chosen DFT method.

The manuscript reports the results of a theoretical study on if the electronic structure of liquid xenon matters significantly in the description of the dark matter - target material interactions. Overall, many aspects of the problem were investigated carefully. The introduced approximations were explained, and made plausible. Such detailed discussions increase the reliability of the obtained results. The text is clear, easy to follow. The manuscript satisfies all the six general acceptance criteria.

1- The implementation of the DFT+U method requires the removal of the effects already included through the exchange-correlation functional in the DFT part. This, so-called double count correction, can be done in various ways, fully-localized limit, around the mean field, etc. This information is not provided here. Please provide it.

2- Page 8, first column: "LDA PZ (lower panel)" --> LDA PZ (left panel)

3- Page 8, second column: In the last paragraph of the subsection III.C.1, I think it will be better if the theoretical lattice parameter obtained by the preferred functional is explicitly stated. In the current text, this information (11.42 au) is given in Fig. 5 (figure and caption). The reason is that in this section the lattice parameter plays the decisive role, but at the end the reader does not know the best value. This information comes after two pages.

4- Page 10, first column: "this density density matches": remove one of the density words.

Publish (surpasses expectations and criteria for this Journal; among top 10%)