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Comparing manybody approaches against the helium atom exact solution
by Jing Li, N. D. Drummond, Peter Schuck, Valerio Olevano
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Valerio Olevano 
Submission information  

Preprint Link:  https://arxiv.org/abs/1801.09977v3 (pdf) 
Date accepted:  20190321 
Date submitted:  20190304 01:00 
Submitted by:  Olevano, Valerio 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Computational 
Abstract
Over time, many different theories and approaches have been developed to tackle the manybody problem in quantum chemistry, condensedmatter 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 randomphase approximation (RPA) from a renormalized RPA (rRPA) in the framework of the selfconsistent RPA (SCRPA) originally developed in nuclear physics, and compare them with various other approaches like configuration interaction (CI), quantum Monte Carlo (QMC), timedependent densityfunctional theory (TDDFT), and the BetheSalpeter 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.
Author comments upon resubmission
We are all researchers, and researchers are exploring the world amazed like children, and like children we must do it by playing (a fact that research financing agencies, especially European, do not understand, unfortunately). And models are our preferred toys. We are absolutely convinced about the importance of studying models: only by playing with models we can grasp the full complexity of, e.g., manybody theory. However, also like children, at a certain point we must take awareness of the limit where the game is over and the reality starts: the confusion of the two planes is not at all good (for both researchers and children). So far as a model could be a beautiful toy to be reluctantly given up, we must be aware of its limits as portrait of the reality. I find "forced" the Editor's opinion that there is no "distinction" between models and real systems.
The Hubbard model, in its original version $\hat{H} = t \sum_{ij \sigma} \hat{c}^\dag_{i \sigma} \hat{c}_{j \sigma} + U \sum_i \hat{n}_{i \uparrow} \hat{n}_{i \downarrow}$, is a simplification of the reality thanks to a discretization of the space $r \in \mathrm{I\!R} \to i \in \mathrm{I\!I}$ and to a reduction of the range of the interaction, from the longestrange interaction in nature, the Coulombian $1/r$, to a shortrange interaction $U \hat{n}_{i \uparrow} \hat{n}_{i \downarrow}$ local to a site $i$. It is true that the site $i$ can be thought both as an effective discretized point in space $r_i$, but also as a more spreadinspace localized orbital $\phi_i(r)$. However along this way the interaction and the model become dependent on the choice.of the basis set. Suppose that we have two electrons in two different localized orbitals $\phi_i(r)$ and $\phi_j(r)$: they won't interact if we choose as basis set of the Hubbard model the $\{\phi_i\}$, but they would interact if we choose another basis set, say Wannier functions $w_k(r)$: the two wavefunctions can both have a contribution on some same Wannier function, $\langle w_k  \phi_i \rangle \ne 0 \quad \langle w_k  \phi_j \rangle \ne 0$ and so interact. In principle the physics must not depend on the chosen basisset, one should find the same result using different basisset in the calculation. But here it is not the case, we have a model presenting a basisset dependent physics.
Second and main point: the raison d'être of this model is the replacement of the manybody longdistance interaction by a local interaction (or an interaction local with respect to a given basis set $\phi_i$) in the hope to have a viable simplification of the manybody problem. Two electrons interact only if they are both in the same site $r_i$ or in the same state $\phi_i$, and do not interact at all if they are in two different sites or states $\phi_i$ and $\phi_j$. Even though the two sites are close or the distributions associated to the two orbitals $\phi_i , \phi_j$ are close in space or even overlap. Which is not the case in the real world. And it cannot become the case even by playing with the range/localization of the orbitals. One can extend the range of the interaction by making the model more complex and extending the interaction to nextneighbors sites/orbitals, but the longrange nature of the Coulomb interaction is such that a finiterank of nextneighbours cannot really capture it. Note that, along the way to be more and more realistic, a model can become more complex than the reality.
On the other hand, it is true that the Hubbard model proved to be a good representation of manybody systems with shortrange interaction, like for example cold atoms systems.
The Editor proposal for a Multiband Hubbard Monomer model to simulate the real helium atom is interesting indeed!
However, as above, how physical or unphysical this model turns out to be in particular question of: which interaction are you considering among the onsite multi bands? Are the electrons on different bands all interacting by the same strength U? Or are electrons interacting by U only if they are on the very same band with different spins? I also guess that the choice of the bands is not a marginal one. Probably, to simulate the helium atom, it would be more judicious to choose not really Bloch bands, and rather hydrogenic Z=2 singleparticle atomic orbitals Anyway, I encourage the Editor to develop such model really in the spirit of amazement discussed above: this is interesting per se. However at the end a comparison on the excitation spectrum between the model and the exact Hylleraas solution, is unavoidable if she/he wishes to validate the model as a faithfull representation of the reality. Notice that the model can safely be validated only against an exact solution, that is a solution like Hylleraas that coincides to any level of significant digits to the experiment. Indeed, if we only had the experiment, the model could have found an excitation spectrum with a level very close to, say, the experimental level at 19.8 eV that Hylleraas unambiguosly interpreted as $2^3\!S$ state. But the model could have interpreted that level very differently, I don't know, for example as a lower Hubbard band!
List of changes
Summary of Changes:
Page 1: "analytic" changed (twice) to "closedform" to correct an inaccuracy following Ref. 2 points 3 and 4.
Page 2: A paragraph introducing the Spherium model has been added together with the citation to a relevant reference, follwoing Ref. 2 point 5.
Page 2: The section introducing SCRPA and rRPA has been dropped since too cryptic, following Ref. 2 point 6.
Page 2, Added footnote [26] at page 21, to discuss the story of gaussian basisset adapted to DFT pure functionals or hybrids etc. and citations to the relevant references, following Ref. 1, point 5 and Ref. 2, point 2.
Page 3: Added all the Ref. 2 requested references for DFTLDA+TDLDA, point 7.
Page 5: Added mathematical details about the characteristics of the Hylleraaslike series, following Ref. 1 point 1 and Ref. 2 point 3 and 4.
Page 5: Added details about modifications to calculate L>0 angular momentum He solutions, following Ref. 2 point 8.
Page 5, Sec. III: Wiggly sentence dropped, follwoing Ref. 2 point 10.
Page 5, Sec. III.A Specification that lambda runs over both singlet and triplet states, follwoing Ref. 2 point 11.
Page 7, Formulas for the groundstate correlation energies and text: Specification that the sums over lambda runs only over the positive (resonant) energies, following Ref. 1 point 2.
Page 8, Sec III.C: "computing power" > "computing resources" (follwoing Ref. 2 point 15. And mentioning of other close to full CI methods, like CIPSI and CCSD, and relative citations, following Ref. 2 point 14.
Page 12:Sec IV.B: "exact DFT kinetic energy" > "exact KS kinetic energy" following Ref. 1, point 6.
Page 13, Sec. IV.C: Paragraph introduced to discuss the mitigation of gaussian basisset problems occurring in large molecules, following Ref. 1, point 5.
Page 15 and 16: Paragraph introduced to discuss the possibility of a scissor operator correction to the DFTLDA+TDLDA spectrum, follwoing Ref. 1, point 7.
Published as SciPost Phys. 6, 040 (2019)