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Comparing many-body approaches against the helium atom exact solution
by Jing Li, N. D. Drummond, Peter Schuck, Valerio Olevano
This is not the current version.
|As Contributors:||Valerio Olevano|
|Arxiv Link:||https://arxiv.org/abs/1801.09977v2 (pdf)|
|Date submitted:||2019-01-15 01:00|
|Submitted by:||Olevano, Valerio|
|Submitted to:||SciPost Physics|
Over time, many different theories and approaches have been developed to tackle the many-body problem in quantum chemistry, condensed-matter physics, and nuclear physics. Here we use the helium atom, a real system rather than a model, and we use the exact solution of its Schr\"odinger equation as a benchmark for comparison between methods. We present new results beyond the random-phase approximation (RPA) from a renormalized RPA (r-RPA) in the framework of the self-consistent RPA (SCRPA) originally developed in nuclear physics, and compare them with various other approaches like configuration interaction (CI), quantum Monte Carlo (QMC), time-dependent density-functional theory (TDDFT), and the Bethe-Salpeter equation on top of the GW approximation. Most of the calculations are consistently done on the same footing, e.g. using the same basis set, in an effort for a most faithful comparison between methods.
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Submission & Refereeing History
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- Report 2 submitted on 2019-02-18 16:38 by Anonymous
- Report 1 submitted on 2019-02-13 14:55 by Anonymous
Reports on this Submission
Anonymous Report 2 on 2019-2-18 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1801.09977v2, delivered 2019-02-18, doi: 10.21468/SciPost.Report.835
- clear, pedagogical and self-contained
- of general interest
- scRPA and rPRA parts are genuinely new
- a bit verbose
- discussion in the introduction about the "exact" wave function of the He atom is very misleading.
- sometimes the claims are too strong
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.
See my points above.
Anonymous Report 1 on 2019-2-13 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1801.09977v2, delivered 2019-02-13, doi: 10.21468/SciPost.Report.823
1-clear and pedagogical
3-of broad interest
1-some of the conclusions valid for the specific case of Helium atom might not necessarily apply to larger systems
In this work the authors use the simple but realistic case of He atom to compare the quality of various methods and approximations used in condensed matter physics, nuclear physics, and quantum chemistry. The He atom has two electrons, hence electron correlation (which is in general hard to describe in all commonly used theories) is well displayed; moreover the system can be solved exactly, and hence be used as a benchmark.
The manuscript is well written, very pedagogical, and self-contained. I enjoyed very much reading it, since various abstract concepts, such as the correction to the Kohn-Sham kinetic energy contained in the exchange-correlation energy functional of DFT, becomes tangible by showing the numbers for the case of He.
I have only a couple of questions/comments:
1) If I understand well Eq. (3) is an ansatz. Maybe the authors should explicitly mention this.
2) Eq. (11) calculates both the resonant and antiresonant excitation energies. Does the sum over the excitation energies in the correlation energies (first equation of page 7, first column) run over both resonant and antiresonant energies? Maybe it would be useful to write it in the text.
3) The second equation of page 7 (first column) seems to imply that the correlation energy comes from the coupling between resonant and antiresonant energies. Is that correct? Is there a simple way to see this?
4) I found the renormalized RPA and SCRPA equations very interesting. The difference with the usual RPA/TDDFT/BSE equations is the presence of fractional occupation numbers. How do the authors define what is a particle or a hole, and hence the particle-hole basis set for the matrix equation (11) if occupation numbers are fractional? Do they fix them to the HF initial values of the occupation numbers?
5) The limits of some methods, such as CI, because of an incomplete basis set might be mitigated for larger systems. In medium-sized or large molecules a moderately large basis sets can be quite adequate because of the effect of basis set sharing, i.e. the fact that each atom profits from the basis functions on its many neighbors. Moreover some methods or approximations can be less dependent on basis set size (for example, in general wavefunction-based methods show a slower convergence with respect to the basis set size than DFT; or hybrid functional are more sensitive to the basis set size than pure functionals, see, e.g J. Chem. Phys. 121, 7632 (2004)). Finally the aug-cc-pVXZ family of basis sets have been developed for correlated wave function methods and are not optimum for other methods such as DFT (see, e.g., J.Phys.Chem.A 121, 6104 (2017); this hence plays a role into the comparison between CI and DFT/Green’s function-based methods results. It would be nice if the authors could comment on this in the paper.
6) In the second column of page 12 the authors mention “the exact DFT kinetic energy”, which has nothing to do with the exact kinetic energy. What do they mean for exact DFT kinetic energy is the exact KS kinetic energy, as they point out a few lines later. I think this can be confusing and it would be better to call it exact KS kinetic energy from the start. In principle the exact DFT kinetic energy as functional of the density is the exact kinetic energy.
7) The TDLDA calculations on top of an exact DFT KS spectrum seems similar in spirit (although not in the physics) to the rigid shift (scissor operator) that one often uses in solids. Is that correct? Maybe the authors could mention this in the manuscript.
In conclusions, I find this manuscript very interesting, the physics valid, and targeting a broad audience. I recommend publication as a regular article in SCIpost after minor corrections according to the comments above.
See my comments 1-7 of the report