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Quantum Metrics for Continuous Shape Measures of Molecules

  • STRUCTURE OF MATTER AND QUANTUM CHEMISTRY
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Abstract

The field of continuous molecular shape and symmetry (dis)similarity quantifiers habitually called measures (specifically continuous shape measures (CShM) or continuous symmetry measures (CSM)) is obfuscated by the combinatorial numerical algorithms used in the field which restricts the applicability to the molecules containing up to twenty equivalent atoms. In the present paper we analyze this problem using various tools of classical probability theory as well as of one-particle and many-particle quantum mechanics. Applying these allows us to lift the combinatorial restriction and to identify the adequate renumbering of atoms (vertices) without considering all N! permutations of an N-vertex set so that in the end purely geometric molecular shape (dis)similarity quantifier can be defined. Developed methods can be easily implemented in the relevant computer code.

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Notes

  1. They are not measures in the strict mathematical sense of this word, like area, volume or probability, rather more precisely can be dubbed as quantifiers.

  2. Among N! permutations it is pretty easy to pick a wrong one since only one is correct.

  3. Positiveness of κ is important as is of \(\mathcal{R}\) being a proper rotation, otherwise a chiral shape could be superimposed with its enantiomer.

  4. This slightly modifies the suggestion of [5] where it has been proposed to superimpose centers of mass of the molecules to be compared. As far as it goes about the shapes formed by equivalent vertices this definition gives the same result as one of  [5].

  5. Size doesn’t matter. Irrespective to it, cube remains a cube.

  6. That of the functions integrable with their squares.

  7. An inverse of the trick known yet to Kirchhoff and Helmholz who used a limit of Gaussians to represent an identity impulse – the δ-“function”: \(\delta (x)\) = \({{\lim }_{{\sigma \to 0}}}\frac{1}{{\sqrt {2\pi \sigma } }}\exp \left( { - \frac{{{{x}^{2}}}}{{2{{\sigma }^{2}}}}} \right)\) to be used only under integral over x.

  8. L2 (\({{\mathbb{R}}^{3}}\)) scalar products for which we use the standard quantum theoretical bra-ket notation (see e.g., [5, 11–13]).

  9. Squared L2 (\({{\mathbb{R}}^{3}}\)) norms of \(\rho _{P}^{{(\sigma )}}\) and \(\rho _{T}^{{(\sigma )}}\) considered as its elements. It naturally applies only to distributions which are representable by the functions integrable with their squares.

  10. The direct consequence of the triangle inequality.

  11. (Pk − Tj)2 = (Pk − Tj')2; j ≠ j '.

  12. The difference is the sum of interference terms (known in quantum chemistry as “differential overlaps”) \(\chi _{{\mathbf{Q}}}^{{(\sigma )}}({\mathbf{R}})\chi _{{{\mathbf{Q}}'}}^{{(\sigma )}}({\mathbf{R}})\)—the price of the consistent quantum description, which any way rapidly decays with increase of the separation between Q and Q' and decrease of dispersion.

  13. Either of functions \({{\rho }_{Q}},\;\rho _{Q}^{{(\sigma )}},\;\psi _{P}^{{(\sigma )}}\) is that of one vector R as its argument which can eventually represent coordinates of only one particle, thus in the terminology of statistical physics and quantum mechanics those functions are one-particle densities or amplitudes.

  14. Like electrons where respective functions are known as Slater determinants.

  15. The outer or Grassmann product \({{ \wedge }^{N}}{{L}^{2}}({{\mathbb{R}}^{3}})\) of the L2(\({{\mathbb{R}}^{3}}\)) Hilbert spaces of functions of each variable Ri.

  16. One can understand by \({{ \oplus }_{m}}{{{\mathbf{Q}}}_{m}}\) the 3N components of N vectors Qm written one after another or N column vectors Qm arranged in a 3 × N matrix like in Eq. (15) of [4].

  17. “As small as possible” is not actually that small. Taking the separation between the nearest vertices in the pattern set for unity we estimate \(\sigma = \frac{1}{{2\sqrt 2 }}\), which assures the factor \(\frac{1}{{{{e}^{2}}}}\)—almost one tenth—for the Carbó index of two normal distributions centered in these vertices. Then in order to assure that for the “close” distance the quadratic approximation for the Gaussian filter is valid, it is enough to set its argument \( < {\kern 1pt} \frac{1}{9}\) that is the distance between the “close” vertices must be less than \(\frac{1}{{3\sqrt 2 }}\) which assures the precision of the quadratic approximation at the level of 0.7%—more than enough for whatever practical applications.

REFERENCES

  1. M. Pinsky and D. Avnir, Inorg. Chem. 37, 5575 (1998).

    Article  CAS  Google Scholar 

  2. D. Casanova, J. Cirera, M. Llunell, P. Alemany, D. Avnir, and S. Alvarez, J. Am. Chem. Soc. 126, 1755 (2004).

    Article  CAS  Google Scholar 

  3. H. Zabrodsky, S. Peleg, and D. Avnir, J. Am. Chem. Soc. 114, 7843 (1992);

    Article  CAS  Google Scholar 

  4. H. Zabrodsky and D. Avnir, J. Am. Chem. Soc. 117, 462 (1995).

    Article  CAS  Google Scholar 

  5. P. Alemany, D. Casanova, S. Alvarez, C. Dryzun, and D. Avnir, Rev. Comput. Chem. 30, 289 (2017).

    CAS  Google Scholar 

  6. C. Dryzun and D. Avnir, Phys. Chem. Chem. Phys. 11, 9653 (2009).

    Article  CAS  Google Scholar 

  7. M. Pinsky, C. Dryzun, D. Casanova, P. Alemany, and D. Avnir, J. Comput. Chem. 29, 2712 (2008). https://doi.org/10.1002/jcc.20990

    Article  CAS  PubMed  Google Scholar 

  8. G. G. Malenkov, J. Struct. Chem. 57, 793 (2016);

    Article  CAS  Google Scholar 

  9. A. L. Tchougreeff, AIP Conf. Proc. 1906, 030004 (2017). https://doi.org/10.1063/1.5012283; A. L. Tchougreeff, Theor. Chem. Acc. 137, 138 (2018). https://doi.org/10.1007/s00214-018-2322-0

  10. A. I. Kostrikin and Yu. I. Manin, Linear Algebra and Geometry (Nauka, Moscow, 1986; CRC, Boca Raton, 1989).

  11. L. Schwartz, Theorie des distributions (Hermann, Paris, 1951); A. Ya. Khelemskii, Lectures and Exercises on Functional Analysis (MTsNMO, Moscow, 2014; Am. Math. Soc., Philadelphia, 2006).

  12. R. Carbo and B. Calabuig, Int. J. Quantum Chem., 42, 1695 (1992);

    Article  CAS  Google Scholar 

  13. E. Besalu, R. Carbo, J. Mestres, and M. Solà, Top. Curr. Chem. 173, 31 (1995);

    Article  CAS  Google Scholar 

  14. P. E. Bowen-Jenkins, D. L. Cooper, and W. G. Richards, J. Phys. Chem. 89, 2195 (1985);

    Article  CAS  Google Scholar 

  15. R. Carbó and E. Besalú, Molecular Similarity and Reactivity: From Quantum Chemical to Phenomenological Approaches (Kluwer Academic, Amsterdam, 1995); R. Carbó and B. Calabuig, Int. J. Quantum Chem. 42, 1681 (1992);

    Article  Google Scholar 

  16. R. Carbó, B. Calabuig, and E. Besalú, Adv. Quantum Chem. 25, 255 (1994);

    Google Scholar 

  17. R. Carbó, L. Leyda, and M. Arnau, Int. J. Quantum Chem. 17, 1185 (1980).

    Article  Google Scholar 

  18. I. Mayer, Simple Theorems, Proofs, and Derivations in Quantum Chemistry (Kluwer Academic/Plenum, New York, 2003).

    Book  Google Scholar 

  19. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Nauka, Moscow, 1989; Pergamon, Oxford, 1977).

  20. L. Zülicke, Quantenchemie, Ein Lehrgang, Band 1: Grundlagen und allgemeine Methoden (Deutscher Verlag der Wissenschaften, Berlin, 1973).

  21. L. Piela, J. Kostrowicki, and H. A. Scheraga, J. Phys. Chem. 93 (8) (1989).

  22. G. Fubini, Atti Reale Ist. Veneto Sci., Lett. Art. 63, 502 (1904);

    Google Scholar 

  23. E. Study, Math. Ann. 60, 321 (1905).

    Article  Google Scholar 

  24. A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer, Berlin, 1933); B. N. Tutubalin, The Theory of Probabilities and Stochastic Processes. Fundamentals of the Mathematical Apparatus and Applied Aspects (Mosk. Gos. Univ., Moscow, 1992) [in Russian].

  25. D. C. Brody and L. P. Hughston, J. Geom. Phys. 38, 19 (2001);

    Article  Google Scholar 

  26. P. Facchi, R. Kulkarni, V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Ventriglia, Phys. Lett. A 374, 4801 (2010).

    Article  CAS  Google Scholar 

  27. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, Cambridge, 1992), Chap. 2.6.

    Google Scholar 

  28. F. R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1960).

    Google Scholar 

  29. S. Gerschgorin, Bull. Acad. Sci. URSS, Math. Natur., No. 6, 749 (1931).

  30. J. L. Littlewood, A Mathematician’s Miscellany (Methuen, London, 1953), p. 38.

    Google Scholar 

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ACKNOWLEDGMENTS

The author is thankful to Prof. Pere Alemany (Barcelona) who some years ago drew his attention to the problem of continuous shape and symmetry measures. Dr. Olga Chugreeva (Stuttgart) is acknowledged for valuable discussion.

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  1. A.N. Frumkin Institute of Physical Chemistry and Electrochemistry of RAS, Moscow, Russia

    A. L. Tchougréeff

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Correspondence to A. L. Tchougréeff.

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Dedicated to my dear teacher Dr. Alexander I. Shtern on occasion of his 80th birthday

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Tchougréeff, A.L. Quantum Metrics for Continuous Shape Measures of Molecules. Russ. J. Phys. Chem. 95, 1846–1856 (2021). https://doi.org/10.1134/S0036024421090272

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