Peer review of: Anharmonic quantum nuclear densities from full dimensional vibrational eigenfunctions with app (2020)

Reviewers' comments: 
 
Reviewer #1 (Remarks to the Author): 
 
In this interesting and well-written manuscript, the authors describe a rigorous procedure based on 
semiclassical dynamics to visualize nuclear probability densities. They use the procedure to illustrate clearly 
the difference between the harmonic and anharmonic vibrational wavefunctions. As their main result, the 
authors show that the anharmonic wavefunction with a single excitation in the O-H stretch mode, unlike its 
harmonic counterpart, involves multiple atoms, even those that are far away from the OH functional group. 
In particular, the method overcomes the major drawbacks of classical analysis (which ignores ZPE) and 
quantum analysis within the harmonic picture (which ignores anharmonicities). It is remarkable that in Fig. 
2b, the density is captured even in the classically forbidden region. Although the effects of anharmonicity 
have been studied in many ways in the last few decades, the simple picture provided by the present method 
is most certainly worth attention and I support its publication. 

I have several questions and suggestions: 
1) Even in the harmonic analysis, the normal mode dominated by the OH stretch is not perfectly isolated 
from the motion of other atoms (because it is a normal, and not local mode, after all). In Fig. 4, it seems 
perfectly isolated (as far as my eyes can distinguish). Could the authors list numerically the relative 
displacements of all 11 atoms in the OH stretch normal mode within harmonic approximation, and within the 
anharmonic model (to fix the magnitude, consider the classical turning point at the ZPE) ? E.g., in a table in 
the supplementary information. 
2) How is the “OH stretch” defined in the anharmonic model? To be complete, the authors should provide a 
precise definition that they used even for the harmonic model. 
3) The authors present nuclear density differences in Figs. 2, 3, and 4 (between harmonic and anharmonic in 
the first two figures, between the ground and excited state in Fig. 4), but the actual nuclear densities are not 
shown. Could the authors show these densities analogously to Figs. 2b, 3a, 4a, and 4b.? It would be nice to 
see how important the change in the density is compared to the magnitude of the density. 
4) In electronic structure theory, when comparing ground and excited states, one can either plot the 
difference density (such as those shown in Fig. 4) or the transition density. Could this method be used to 
compute and visualize the transition density between the vibrational states? 
5) Since the list of references is already quite large, the authors could mention other works that visualize 
nuclear densities beyond the harmonic approximation. 
 
Otherwise, I have several minor comments: 
1. Are the densities plotted using a readily available software for visualizing, e.g., molecular orbitals or 
electronic densities? If yes, please mention it in Methods or in the SI. Otherwise, would it be possible, in 
general, to prepare the file containing nuclear density to be compatible with such softwares (e.g. GaussView, 
VMD, Avogadro, Molden, …)? 
2. On page 5, the sentence “Therefore, differently from classical normal mode displacements,…” is obscure. 
For me, quantum and classical normal mode coordinates are the same coordinates, Depending on whether 
one includes rotations and translations, either 3N or 3N-6 (3N-5) in total. Could the authors expand on what 
difference they have in mind? 
3. On page 8, in Fig. 1, I am not sure if the interruption of the x-axis scale is needed. It seems to me that 
only ~10% of the energy range is removed. If more space is saved, the authors should clearly indicate the 
range to the left of the interruption. (The problem is aggravated because the tick 20,000 cm^-1 is right 
under the interruption sign so it is not clear if it corresponds to the scale on the left or on the right.) 
4. On page 9, the sentence “The value of ZPE energy … is well above …which is a huge amount of energy in 
comparison with the harmonic vibrational energy level spacing.” is stated at the beginning of the page, but 
its significance is only explained in the first sentence of the next section (still on page 9) mentioning that the 
ZPE is overlooked in classical approaches. This is a great point but should be clarified immediately. 
5. The title of Fig. 2 “…ZPE nuclear densities” should perhaps be changed to something clearer, e.g., in 
analogy to Fig. 1 to “…for the ground state nuclear densities” 
6. In Fig. 3 the legend “Anharmonic increase” should be just “Anharmonic”, right? The increase is only 
shown in panel c. 
  
 
Reviewer #2 (Remarks to the Author): 
 
The manuscript by Aieta et al. discusses a computational investigation on geometry and vibrational modes of 
protonated Glycine obtained through a quantum description of the nuclei that include anharmonic effects. 
The average geometry of the molecule is found to be slightly elongated with respect to classical results, and 
the vibrational OH stretching acquires (due to anharmonicity) components on other normal mode as well. 
 
The computational achievement (based on a clever scheme developed previously by M. Ceotto's group) is 
certainly remarkable. What concerns me most is the value of the new insights obtained from such 
challenging calculations. It is stated that the presented results contradict the "group frequency hypothesis". 
I do not think this is the case: here it is shown that if anharmonicity is taken into account, the OH stretching 
has some degree of mixing with other modes (mostly NH). But this is barely surprising, one of the expected 
effects of anharmonicity is to mix modes that are independent at the harmonic level (and mixing modes that 
are the closest in energy is the most natural occurrence). Already at textbook level the concept of group 
frequency is understood to be a useful but oversimplified picture of reality, the present work confirms this 
view. 
 
In terms of quantify the extent of the effect, pictures such as fig. 3a may be misleading: they give the 
impression that OH stretching and other modes are mixed with comparable weights, but these are only the 
anharmonic nuclear densities on top of the harmonic ones. If the total (harmonic+anharmonic) density 
would be plotted, I expect a less impressive mixing, with the isosurfaces on NH rather smaller than those on 
OH. 
 
Concerning the ground state geometry modifications, again the results are quantitatively non trivial, but 
qualitatively expected: the asymmetry in bonding potentials, such as in Morse's, leads to change in the 
average bond length, even at the level of vibrational ground state. 
 
There are also some specific issues: 
1. Computational procedures such as those in ref. 59 for glycine have been used to provide accurate 
molecular geometries. It is unclear to me how the anharmonic corrections found here should be integrated 
in such schemes: are they additional corrections to be accounted for in such protocols? 
 
2. From the supporting material, the procedure seems to provide also the coefficients of mixing of the 
various normal modes into the vibrational eigenfunction: could these be used to quantify the mixing of the 
OH stretching with the other modes? 
 
3. Fig. 3b: The harmonic density distribution of the first vibrational excited state of the OH stretching has no 
node. Why is that so? 
 
4. Fig. 2m: the pictorial representation using arrows of the anharmonic density plotted in fig. 2b is certainly 
useful. However I cannot understand why arrows on H4,H5 and H6 are missing, since there is anharmonic 
density also on these atoms. 
 
In summary: I find the theoretical and computational achievement remarkable, the idea of representing the 
geometry and vibrational modes by real space nuclear densities effective and the new quantitative insights 
useful. Overall, all these points might warrant the broad interest expected for a Nature Communications 
paper. On the other hand, I think that the article in its present form is making too strong and unjustified 
claims, for the reasons I discussed above. 
 
 
 
Reviewer #3 (Remarks to the Author): 
 
The paper reminds me a set of old studies on X-H stretchings in polyatomic molecules, which could be 
described starting from either normal-modes or local-modes. The former are harmonic oscillators delocalized 
on the whole molecule, perturbed and coupled by quartic anharmonic interactions. The latter are anharmonic 
oscillators localized on individual X-H bonds with harmonic interbond coupling. It was discovered that two 
descriptions are completely equivalent and that their two Hamiltonians yields identical eigenvalues if their 
parameters obey proper relations (called x-K relations). 
 
Exactly as in the present paper, it was noticed that when the energy is increased it tends to concentrates on 
individual bonds. I don't remember anymore the literature on this field, but it can be found by searching for 
"local modes and x-K relations". It could be useful to the authors. 
 
Despite the fact that it is a nice and well written paper, I don't think this work is suitable for NatureComm. 
After citing the aforementioned old works, this paper can be submitted to a more specialized journal.