SciPost Submission Page

The seniority quantum number in Tensor Network States

by K. Gunst, D. Van Neck, P.A. Limacher and S. De Baerdemacker

This is not the current version.

Submission summary

As Contributors: Klaas Gunst
Preprint link: scipost_202008_00004v1
Code repository:
Date submitted: 2020-08-04 15:20
Submitted by: Gunst, Klaas
Submitted to: SciPost Chemistry
Academic field: Chemistry
  • Quantum Physics
  • Theoretical and Computational Chemistry
Approaches: Theoretical, Computational


We employ tensor network methods for the study of the seniority quantum number -- defined as the number of unpaired electrons in a many-body wave function -- in molecular systems. Seniority-zero methods recently emerged as promising candidates to treat strong static correlations in molecular systems, but are prone to deficiencies related to dynamical correlation and dispersion. We systematically resolve these deficiencies by increasing the allowed seniority number using tensor network methods. In particular, we investigate the number of unpaired electrons needed to correctly describe the binding of the neon and nitrogen dimer and the $D_{6h}$ symmetry of benzene.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2020-11-25 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202008_00004v1, delivered 2020-11-24, doi: 10.21468/SciPost.Report.2231


This manuscript descibes the optimization of seniority-number
restricted wavefunctions using a tensor network
representation. Overall, the work is very nice and well organized. I
only have minor comments and suggestions.

Requested changes

1. As the authors say, one critical aspect in DOCI calculations is the
choice of the right orbital basis. The authors should comment on the
prospect of doing orbital optimization within their tensor-network

2. The authors should emphasize more the low-entanglement structure of
the DOCI wavefunction. The important consequence of this is that the
tensor network can be optimized rather efficiently.

The authors should comment on the entanglement structure once higher
seniorities are involved. In particular, if one has to reach
relatively high seniorities or if the entanglement structure increases
considerably, then it may not be readily apparent why one should use
an MPS with restricted seniorities vs the full MPS in DMRG.

3. A clarification regarding, e.g., Fig 2, would be appreciated. It
seems to me that the weights plotted correspond to a splitting of the
|Psi> = c0 |v=0> + c2 |v=2> + ...
Yet, this is not described in any Eqn and I assume it can lead to

4. While I'm not completely familiar with Ref 44, I cannot agree with
the statement

"Size consistency of the AP1roG wave function, or by extension any
seniority-zero method, is guaranteed when the spin orbitals are
optimized such that the energy is in a variational minimum."

In fact, the optimal DOCI orbitals for N2 are delocalized and
therefore the solution is not size consistent.

  • validity: high
  • significance: high
  • originality: top
  • clarity: top
  • formatting: excellent
  • grammar: excellent

Author:  Klaas Gunst  on 2020-12-15

(in reply to Report 2 on 2020-11-25)
answer to question

We would like to thank the referee for his/her thorough reading and the comments made in the report. We reply to the comments in the report point-by-point in the following:

  1. We have commented upon the usage of DOCI-TNS in combination with orbital optimization in the conclusions.

  2. We have further commented on the entanglement present in the seniority-restricted wave functions at the end of section 2.

  3. We have elaborated on this in section 2.2.

  4. While the orbital-optimized DOCI wave function is indeed not separable for dinitrogen, this is due to the fact that DOCI is unable to describe (at least without introducing ghost orbitals) a single nitrogen atom as it has an odd amount of electrons. When DOCI is able to describe the constituents of the system, it is size-consistent.

As we agree that it is at least debatable if DOCI is size consistent, we have left out the given sentence in the manuscript. This does not impinge the general message of the paragraph.

Anonymous Report 1 on 2020-10-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202008_00004v1, delivered 2020-10-30, doi: 10.21468/SciPost.Report.2130


Gunst, Van Neck, Limacher, and De Baerdemacker study the convergence of the electronic structure of many-electron systems with seniority number, using a tensor network representation of the wave function. The analysis is clear and the conclusions reflect similar findings in earlier studies. To my knowledge, this work is the first to present a seniority-based analysis involving tensor networks. I particularly like the analysis of the toy model for N2 dissociation, which provides a clear illustration of the orbital dependence of the relevance of different seniority sectors. Overall, I think this is a nice contribution, and I have only minor comments and questions.

Requested changes

1. Page 2: “For example, the complete active space self consistent field method (CASSCF) and density matrix renormalization group (DMRG) are capable of capturing strong correlations…” DMRG applied within the full space should also account for dynamic correlation, so this statement should be modified to acknowledge that the authors are referring to DMRG in an active space.

2. Page 2: “Interestingly, the antisymmetric product of one-reference orbital geminals (AP1roG), also known as pair-coupled cluster doubles (pCCD), appears to provide a reliable approximation to the DOCI ground state solution…” I would change “solution” to “energy” since pCCD does not necessarily provide a good description of other properties. For example, right pCCD wave functions can be good approximations to DOCI ones, whereas the left pCCD wave function can be significantly different (JCP 142, 214116 (2015)), meaning that the respective density matrices could differ.

3. Page 5: it would be nice if the notation in Eq. 4 was clarified.

4. Page 6: FCI is used for the first time, but the acronym was not defined.

5. Page 6: “For example, DOCI calculations with 162 electron pairs and 261 spatial orbitals have been executed in a few minutes on a common laptop.” This assertion requires a citation. Also, a clarification would be appreciated: did this calculation involve an orbital optimization?

6. Page 7: Table 1: It should be clear that the resource requirements do not include the orbital optimization step often performed in DOCI.

7. Page 9: “When including up to seniority four the energies are close to FCI around the binding distance…” No FCI results are presented for N2, so this statement should be modified to suggest simply that the energy has converged with respect to seniority number by omega = 4. Otherwise, FCI results should be included or a citation provided.

8. Page 12: “When including progressively higher seniorities, the stable configuration moves closer to the expected D6h symmetric benzene.” How is the stable configuration determined? Is a full geometry optimization performed, or do the authors just scan the angle theta? If they are just scanning an angle, I wonder how the remaining geometric paramaters change. Either way, some additional information would be useful for anyone trying to reproduce this study.

9. Section 3.3: This is a somewhat silly request, but I would prefer that the authors remove the language about “bonding” in favor of some other phrase (e.g., “is very weakly bonding” -> “is very weakly bound” on page 12, “to be bonding” -> “to be bound” on page 13, etc.) because the favorable interaction in neon dimer can hardly be thought of as a bond.

10. Page 13: “The over-corrected dissociation underestimates the dissociation energy a bit with respect to the FCI BSSE-corrected calculations…” FCI results are not presented. Do the authors mean to say tensor network calculations with full seniority?

11. Page 14: “This suggests the breaking of at least one electron pair at each Ne atom is needed, inducing polarization effects in each atom which give rise to the dispersion energy.” This is a reasonable conclusion to draw regarding the importance of seniority four sector, except that the basis used is a delocalized one, correct? Can the physical picture of broken pairs at atomic centers be applied here?

12. Lastly, it seems to me that the seniority-based expansion is not necessarily good for reducing the overall complexity of a calculation, if one wants quantitative accuracy. Seniority eight is required for exact restoration of symmetry in benzene, and seniority of at least four is required to capture dispersion. This leads me to some questions. First, what is the prospect for generally useful seniority-based expansions of the wave function? Is the best route to stop at seniority zero and then apply some correction (e.g., AP1roG + ERPA)? Second, if a better route is to perform some high-seniority-number (8?) calculation in a tensor network framework, I wonder how the complexity of intermediate seniority calculations compares to the DOCI or full seniority limits in tensor network calculations. Can the authors comment?

  • validity: high
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: perfect

Author:  Klaas Gunst  on 2020-12-15

(in reply to Report 1 on 2020-10-30)
answer to question

We would like to thank the referee for his/her thorough reading and the comments made in the report. We have addressed the points raised in the resubmission. We would like to further answer on two questions raised by the referee:
8) We have used the experimental value as given in the NIST database for the distances between the center of mass and the different atoms. We keep these fixed and indeed only variate the angle. We have added this to the manuscript. As to give some insight in the change of the remaining geometric parameters, we have done a more exhaustive scan of the geometries in a restricted active space. We have performed orbital-optimized DOCI calculations in an active space spanned by the 2pz-orbitals of each C atom in the same sto6g basis-set. This very restricted active space of 6 orbitals and 6 electrons exhibits the same symmetry-breaking as the calculations presented in the paper with the exception that seniority 6 restores the symmetry in this case. While these calculations in the restricted active space are not conclusive, they do suggest that the symmetry-breaking does not vanish when optimizing the geometry while distorting. We have attached these results (benzene_min_act.pdf) for the ground state energies in the different seniority subspaces in this small active space. A step-size of 0.05 degrees and 0.005Å for the center-of-mass to carbon distance was used. The carbon-hydrogen distances were also varied and the best distance was used for each point shown in the contour plots.
11) The DOCI-optimized orbital for the Neon dimer are delocalized on one of the two atoms for larger separations. We have clarified this in the manuscript.



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