Quantum Crystallography in the Last Decade: Developments and Outlooks

Round 1

Reviewer 1 Report

This is a nice review of recent developments in quantum crystallography emphasizing wave function and multipole-based methods. I think it will be an important reference for those wishing to understand these methods and their performance.

My only criticism is on p. 3, line 117 where the authors suggest that HAR stands separate from wave function and multipole methods as not being concerned with electronic structure. Consider, for example, a study of urea where HAR is used to obtain a charge density derived from the DFT which differs substantially from a multipole model, even with differences in net atom charges (https://journals.iucr.org/m/issues/2016/04/00/fc5014/).

In addition other quantum crystallographic approaches are sometimes used to refine atom positions, e.g., hydrogen atoms in the multipole model. 

This criticism should be addressed by eliminating the classification of the methods in the manuscript according to their relevance to electronic structure. A note advising the reader of the focus of the review on the wave function and multipole methods is appropriate, and this does not need further justification.

Author Response

We thank this reviewer,

See the attached document for a detailed reply

Author Response File: Author Response.docx

Reviewer 2 Report

I find no significant problems with this review on quantum crystallography. Its content is relevant, the presentation is balanced, thus I recommend publication. Being a review article, in my view it is not only the scientific content, but also the presentation that decides on its value. Thus, below I point out, apart from minor errors, also passages in the text that can be improved.

"crystals ... inherently apply a Fourier transformation when illuminated" (of themselves) -- I would say it is rather physics in the guise of Maxwell's equations or Huygens' principle that performs the Fourier transformation. Indeed, this is the case for any matter (not necessarily crystalline).

In the range of lines 69-76 the text seems to be not yet in a final shape, probably you would like to reformulate here.

"Three are the main applications..." I am not entirely sure whether you intended here to use poetical language (citing perhaps the line "three are the fathers" (of Israel) from the Who knows One) or whether this is unintentionally malformed scientific prose -- if the former, I would suggest dropping the "main", it weakens the sentence.

The section "Fitting the wavefunction" should have number 2, not 1.

In the paragraph starting at line 140, you make the point that the wavefunction is not uniquely determined by the scattering measurement. However, here the step that the experiment determines the one-electron density is missing, you just point out that for a given electron density many wavefunctions can be found. In section "3. Fitting the density" you show that in detail. So I would suggest that you at least mention the point here. However, perhaps it would be even better to pull out the first part of section 3 (about what the experiment is actually sensitive to) and put it before "Fitting the wavefunction", as a pedagogical introduction to the basics of both approaches.

"heart" instead of "hearth"

What do you want to imply with "perturbative" in the "perturbative restraint"? To me, it is just a general restraint, while "perturbative" makes me think of perturbation theory, which is something different. Perhaps you want to drop it, or be more explicit if there is some meaning that has escaped me.

What is the role of Delta in Eq. (1)? It does not depend on Psi, so if you minimize the functional with respect to Psi, you could as well have left it out. So Eq. (1) is just an example of the general principle of regularization (it would probably not hurt to mention this concept, this being a review article), and lambda would normally be chosen so that chi^2 is not too far above N_r-N_p (but see me next comment), that is, a given quantile of the chi^2-distribution.

And what you call chi^2 is actually rather called the reduced chi^2: your definition should give about 1 for a good fit, while the conventional chi^2 does not have your normalization 1/(N_r-N_p) -- to me, chi^2 is what you take as independent variable when you evaluate the chi^2-distribution, which is unambiguously defined.

This is not restricted to quantum crystallography, but also very valid in conventional crystallography, and often not enough appreciated: I would like to see sigma_h^2 depend on the modelling, and not being called "experimental error", as it implies that it is an "error" that is a function purely of the experiment. No, it is an uncertainty, an error would rather be the deviation between model and actual measurement. And it is not fixed by the experiment: if we consider counting statistics, the dominant contribution to this uncertainty in scattering experiments (other systematic errors are much harder to handle and in any case do not fit into your Eq. (2)), the uncertainty in the sense of variance is equal to the parameter of the Poisson random variable, which is your (eta*F_h^calc)^2, and not F_h^exp^2.

And why do you write Eq. (2) on the basis of the amplitudes and not the intensities? I would find the intensities to be much more natural, they are directly measured, they are non-negative (and specifically real), and their measurement follows a nice statistical distribution (Poisson).

You make the point about constrained or restrained. If the latter is just your proposal and not yet adopted by the community (I see no references here), I would rather propose to call it "regularized" instead of coining yet another term -- as I said, "regularization" is a classical field of applied mathematics, and what you have here is just an example of it. If you don't know, just start with "Regularization (mathematics)" on the English wikipedia.

You say "a molecular wavefunction is needed ... for the calculation of polarizabilities". I would say, you even need the excitations, that is, not one wavefunction (presumably of the groundstate), but all eigenfunctions of the Hamiltonian. Am I missing something here?

"solid-state" instead of "solid-sate"

Starting in line 197, you talk about resolution. You seem to use this for the number of used reflections -- of course, this gives the real-space resolution. But in conventional crystallography, "resolution" unambiguously means something different: it is how well you can resolve the distinct peaks in reciprocal space, connected to real-space correlation lengths. Please be unambiguous here.

"the goal was to ... quantify" instead of "... quantifying"

"the electron density, which is best defined as the one-electron density" --- what else should it be?

"a wavefunction is necessary to obtain..." -- I do not get what this sentence wants to say.

"that the scattered radiation is causally related ..." again, I find this sentence either clumsy and obvious, or it has a meaning that escapes me.

"subject to the electric magnetization" -- should this say electronic magnetization? And why is it even necessary, do you really want to oppose this to nuclear magnetization?

The nuclear cross section is not necessarily larger than the magnetic cross section. Given the orders of magnitude on which the scales of these quantities could be varying if the fundamental physical constants would be slightly different, I find it noteworthy how comparable they actually are.

Line 489: "enhance" means "make larger", but in the positive sense of "make better", for instance "enhance understanding". If you want to imply that the problems grow, rather use something like "exacerbate"

Line 492: "up to very high resolutions" -- I repeat, to me, this is definitely non-standard usage of resolution.

"joint refinement" instead of "join"

You talk about liquid nitrogen temperature, and that the thermal displacements would become even smaller at liquid helium temperature. Now among the elemental systems, Cs has the lowest Debye temperature at 40 K apart from the noble gases, with the majority of systems being around room temperature, and the arguably most important molecular crystal (water ice) has a Debye temperature on the order of 300 K. So I have problems with the present presentation: the lower you go in temperature, the smaller the displacements will be, but you should not off-hand imply that 10 K gives necessarily significantly smaller displacements than 100 K. And finally, you should be more specific here: assuming a very low Debye temperature and thus classical behaviour, a factor of three between room temperature and 100 K corresponds to a square root of three in the thermal displacements. If pressure would give the same reduction at constant temperature, this would imply an increase of the force constant by a factor of three, and thus an increase of the frequency by a square root of three. In the general case, this does not happen at 10 GPa, much less a stiffening of the frequencies by a square root of 30 if you go to 100 GPa. You should phrase this here as a discussion of a specific system and give the reference.

Author Response

We thank this reviewer,

please find attached a detailed response

Author Response File: Author Response.docx