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How constraint programming can help chemists to generate Benzenoid structures and assess the local Aromaticity of Benzenoids

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Abstract

Benzenoids are a subfamily of hydrocarbons (molecules that are only made of hydrogen and carbon atoms) whose carbon atoms form hexagons. These molecules are widely studied in theoretical chemistry and have a lot of concrete applications. Then, there is a lot of problems relative to this subject, like the enumeration of all its Kekulé structures (i.e. all valid configurations of double bonds). In this article, we focus our attention on two issues: the generation of benzenoid structures and the assessment of the local aromaticity. On the one hand, generating benzenoids that have certain structural and/or chemical properties (e.g. having a given number of hexagons or a particular structure from a graph viewpoint) is an interesting and important problem. It constitutes a preliminary step for studying their chemical properties. In this paper, we show that modeling this problem in Choco Solver and just letting its search engine generate the solutions is a fast enough and very flexible approach. It can allow to generate many different kinds of benzenoids with predefined structural properties by posting new constraints, saving the efforts of developing bespoke algorithmic methods for each kind of benzenoids. On the other hand, we want to assess the local aromaticity of a given benzenoid. This is a central issue in theoretical chemistry since aromaticity cannot be measured. Nowadays, computing aromaticity requires quantum chemistry calculations that are too expensive to be used on medium to large-sized molecules. In this article, we describe how constraint programming can be useful in order to assess the aromaticity of benzenoids. Moreover, we show that our method is much faster than the reference one, namely NICS.

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Notes

  1. A radical structure arises when a system has an odd number of electrons. All electrons but one pair form bonds. The lonely electron is called a radical.

  2. Remember that a graph \((V^{\prime },E^{\prime })\) is a sub-graph of a graph (V,E) if \(V^{\prime }\subseteq V\) and \(E^{\prime } = E \cap (V\times V).\)

  3. For instance, there exist 6,510 benzenoid structures for 9 hexagons, among which only five are coronoids (see Tables 3 and 5 for more examples).

  4. Table constraints list explicitly the allowed (or disallowed) combinations of values that a specific set of variables can take [40].

  5. BenzAI is an open source software for chemists that includes the work presented in this article (generation of benzenoid structures and estimation of the aromaticity) in a user-friendly graphical interface. More information and source code can be found at https://benzai-team.github.io/BenzAI/.

  6. If |V1|≠|V2|, it is trivial to show that the benzenoid has no Kekulé structure.

  7. In [13], this step is achieved by enumerating all the Kekulé structures thanks to the CSP model P1. This choice was justified by the long computation time of the determinant as well as by the fact that \(B-B[\mathcal {C}]\) was not a benzenoid. The extension of Rispoli’s result and the use of a more efficient library for the computation of the determinant allow us to make a different choice here. Note that we compared experimentally (not reported here) the determinant calculation, the solution counting for the model P1 or one of Mann and Thiel [47], and not surprisingly, the first method is the most efficient.

  8. Available at https://benzai-team.github.io/BenzAI/.

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Acknowledgements

The authors would like to thank Mohamed Sami Cherif and the anonymous reviewers for their useful comments.

Funding

This work has been funded by the Agence Nationale de la Recherche project ANR-16-CE40-0028.

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Authors and Affiliations

  1. Aix Marseille Univ, CNRS, Centrale Marseille, ISM2, Marseille, France

    Yannick Carissan & Denis Hagebaum-Reignier

  2. Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France

    Nicolas Prcovic, Cyril Terrioux & Adrien Varet

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  1. Yannick Carissan
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Appendix Detailed results for the subset \({\mathscr{B}}_{1}\)

Appendix Detailed results for the subset \({\mathscr{B}}_{1}\)

Figures 29-34 describe the benzenoid structures considered in the subset \({\mathscr{B}}_{1}\). These structures have various sizes or shapes. They may admit or not some symmetries. Moreover, in these figures, we also specify the values computed by CRECP and NICS in blue and red respectively. For sake of readability, we only provide them for a single hexagon per symmetry class. Indeed, all the hexagons of a symmetry class have the same value whatever the considered method.

Fig. 29
figure 29

Results on the set of 48 molecules of the subset \({\mathscr{B}}_{1}\)

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Fig. 30
figure 30

Results on the set of 48 molecules of the subset \({\mathscr{B}}_{1}\) (Fig. 29 continued)

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Fig. 31
figure 31

Results on the set of 48 molecules of the subset \({\mathscr{B}}_{1}\) (Fig. 30 continued)

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Fig. 32
figure 32

Results on the set of 48 molecules of the subset \({\mathscr{B}}_{1}\) (Fig. 31 continued)

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Fig. 33
figure 33

Results on the set of 48 molecules of the subset \({\mathscr{B}}_{1}\) (Fig. 32 continued)

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Fig. 34
figure 34

Results on the set of 48 molecules of the subset \({\mathscr{B}}_{1}\) (Fig. 33 continued)

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Carissan, Y., Hagebaum-Reignier, D., Prcovic, N. et al. How constraint programming can help chemists to generate Benzenoid structures and assess the local Aromaticity of Benzenoids. Constraints 27, 192–248 (2022). https://doi.org/10.1007/s10601-022-09328-x

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  • DOI: https://doi.org/10.1007/s10601-022-09328-x

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