In the present manuscript, Li et al. report an exhaustive study of the helium atom using electronic structure methods ranging from RPA-type methods to more conventional methods such as QMC, CI and TD-DFT.
I am to admit that, at first sight, I was a bit scared by the length of the paper, but I have learnt to like throughout this pleasant reading.
In my humble opinion, I still believe that the present manuscript is too long but every single author has his/her own style, and you have to accept this I guess.
Although the He atom has been studied to death in the last century, I believe that the present manuscript is still very interesting as it studies new, unconventional methods.
I have, however, a long list of comments that the authors should consider.
- abstract: the helium atom is indeed a real system but, in chemistry at least, this is what ressemble the most a model.
- abstract: studying methods on equal footing is not a trivial task.
The authors have chosen to use the same basis set throughout their study.
This is one way of doing it but I don't think this is the best.
My reason for saying this is that each method has its own sensitivity to basis set incompleteness. For example, density-based methods are known to converge faster wrt the size of the one-electron basis set than CI or CC methods. Therefore, a 6-31+G* B3LYP calculation might be much more converged (in a basis set sense) than a CCSD/aug-cc-pVTZ.
My belief is that one should always work in a near complete basis to assess faithful quantum chemistry methods.
The authors should at least comment on this.
- abstract and Introduction: sadly, we don't have the exact solution for the He atom.
The best we have is a very accurate numerical solution.
The wave function of the He is non-analytic and cannot be obtained in closed form.
- I strongly disagree with the authors on the "exact" solution of the He atom.
There is no mathematical guarantee that an expansion in terms of Hylleraas coordinates does converge.
As shown by many authors, one must introduce (non-analytic) logarithmic terms to ensure that the series converge to the right limit.
An expansion in terms of Hylleraas coordinates yields an analytic function.
As correctly mentioned by the authors, the exact wave function of He is non-analytic, therefore one cannot get the exact He wave function with Hylleraas coordinates.
Nakatsuji has shown that one must insert non-analytic terms to ensure a proper convergence.
Actually, Bartlett has shown that one cannot find a solution using Hylleraas variables within the Frobenius method as one hits a contradiction very quickly (https://journals.aps.org/pra/pdf/10.1103/PhysRevA.30.1506).
One could also use the Fock expansion but, as shown by Morgan and coworkers, the radius of convergence of such an expansion is zero (https://link.springer.com/content/pdf/10.1007/BF00526420.pdf).
Therefore, there is no guarantee that the numerical calculation converges to the right value.
- The authors mentioned that the present study is much better than conventional models but they do not cite any.
Also, as mentioned above, the helium atom is indeed a real system but, in chemistry at least, this is what ressemble the most a model.
It would be interesting to mention the pros and cons of each model (helium, Hubbard, spherium, etc) as, for example, in J. Chem. Theory Comput. 14, 3071 (2018).
- Page 2: "In short, it consists of self-consistently re-injecting into the parameters of the RPA equations the correlations from the eigensolutions out of the RPA equations themselves. An obvious place where in the standard RPA equations correlations are disregarded is in single-particle occupation numbers."
It would be nice to make that statement clearer. Thankfully, this is the short version...
- Page 3: "Although the DFT- LDA + TDLDA helium-atom excitation spectrum has been discussed several times in the literature, numerical results have never been published to our knowledge."
Additional references would be welcome.
- Page 4: Expansion (3) is only suitable for S states (singlet and triplet).
However, it is not suitable for higher angular momentum (P, D, etc) where one needs to use a different ansatz.
- Page 5: Is it really appropriate to report Eq. (4) as this type of expansions is not used in the present manuscript?
Moreover, the addition of logarithmic terms in the wave function has been already mentioned earlier in the manuscript.
- Page 5: The second sentence of Sec. III needs to be modified.
- Page 5 and 6: for each formula involving summation of RPA quantities, it would be very useful to know if the summation runs over singlet and/or triplet excitations (and why).
- Last equation in left row of page 5: typo with two equal signs.
- Equations (12) and (13): very interesting part. I always wanted to know more about rRPA and scRPA, and it is very well explained here. However, it would be nice to have the exact definition of the quantity \chi.
It must be related to the eigenvectors (X+Y) of the RPA problem but it's better to be sure that's indeed the case.
- Page 8 Sec. C: These days, selected CI methods (like CIPSI) can be used to get near-FCI energies for ground and excited states of molecules with several heavy atoms.
See, for example, J. Chem. Phys. 147, 034101 (2017) or J. Chem. Theory Comput. 14, 4360 (2018) or 10.1021/acs.jctc.8b01205
One can even do DMC on top of it in order to complete the basis set
See, for example, J. Chem. Phys. 149, 034108 (2018).
I believe that it should be mentioned as the present claim isn't fair.
- Page 8: I am very surprised that the authors cannot perform a proper FCI calculation on He with d-aug-cc-pV5Z.
It's only a CISD with 2 electrons and 115 orbitals. I'm pretty sure it can be performed with GAMESS in a few minutes.
- Page 8 Sec. D: The problem with RPA/GW methods is that the combination of things that can be done is just infinite, and one gets lost very quickly. Could the author provide a clear diagram summarising which methods they use to get which quantity?
GW is already using dRPA to get the neutral excitations used to construct the self-energy.
- Page 18: The depletion/repletion of the occupation number is small because it is a single-reference system (the HF wave function is a very good approximation).
In metallic systems, as mentioned by the authors, the situation is drastically different as they are usually strongly mutli-reference if one uses a localized basis set.