Skip to main content
SpringerLink
Log in

Particle coordinates and discrete molecular description: a geometric point of view on a twofold dimensionality environment

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

In the present study a specific kind of widespread mathematical objects: descriptor matrices are defined and studied. These are matrices connected with several problems concerning many fields of interest in theoretical chemistry, classical and quantum mechanics, or quantitative structure-properties relations. The twofold dimensionality structure of descriptor matrices is analyzed and the properties of descriptor matrices are also disclosed with respect origin shifts and rotations. A tensor to study schematically the three dimensional nature of many particle structures, the characteristic form tensor is defined. The construction of similarity matrices from descriptor matrices and the connection with quantum similarity are finally discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In fact, there must be also considered present an alternative parallel description in dual space, involving the transposes of all vectors and matrices which are employed in this work. The twofold dimensions appearing in the main discussion will be therefore reversed looking at this dual case. However, such issue will not be discussed here, in order to keep the formalism as simple as possible.

  2. Here the Dirac’s notation is used for column vectors, which will be written using a ket symbol: \(\left| \mathbf{a} \right\rangle \), while row vectors will be noted by a bra symbol \(\left\langle \mathbf{a} \right| \). Both symbols correspond to the transpose one from the other. As the field of reference along this study will be the real field, then there is no conjugation involved.

  3. The symbol \({*}\) involving two vectors of a given vector space: \(\forall \left| \mathbf{a} \right\rangle ,\left| \mathbf{b} \right\rangle \in \text{ V }_D :\left| \mathbf{c} \right\rangle =\left| \mathbf{a} \right\rangle {*}\left| \mathbf{b} \right\rangle \rightarrow \forall I:c_{I} =a_{I} b_{I} \rightarrow \left| \mathbf{c} \right\rangle \in \text{ V }_{D} \) denotes an inward product, acting on two vectors and yielding another vector of the same vector space. The symbol involving a vector: \(\alpha =\left\langle {\left| \mathbf{a} \right\rangle } \right\rangle =\sum _I {a_{I}} \) corresponds to the complete sum of its elements. Therefore, the expression: \(\left\langle {\left| \mathbf{a} \right\rangle {*}\left| \mathbf{b} \right\rangle } \right\rangle =\sum _I {a_{I} b_{I} } \equiv \left\langle \mathbf{a} | \mathbf{b} \right\rangle \) corresponds to the scalar product of the two vectors. In the previous definitions have been used column vectors but the same definitions hold for row vectors, matrices or hypermatrices. If the involved vectors are functions: \(\left| f \right\rangle \equiv f\left( \mathbf{r} \right) \), the inward products \(\left| f \right\rangle {*}\left| g \right\rangle \equiv f\left( \mathbf{r} \right) g\left( \mathbf{r} \right) \) are coincident with products of functions, and the complete sum of a function becomes an integral over the definition domain:\(\left\langle {f\left( \mathbf{r} \right) } \right\rangle =\int _D {f\left( \mathbf{r} \right) d\mathbf{r}.} \) A scalar product of two functions can be written in this notation as: \(\left\langle {\left| f \right\rangle {*}\left| g \right\rangle } \right\rangle =\int _{D} {f\left( \mathbf{r} \right) g\left( \mathbf{r} \right) d\mathbf{r}\equiv \left\langle f | g \right\rangle }.\)

  4. Provided that the symbol \(\left[ \!\left[ \mathbf{A} \right] \!\right] \), applied over a matrix with functions as elements: \(\mathbf{A}\left( \mathbf{r} \right) =\left\{ {A_{IJ} \left( \mathbf{r} \right) } \right\} \), might be considered that yields the matrix of the integrals of the function elements: \(\mathbf{G}=\left[ \!\left[ \mathbf{A} \right] \!\right] =\left\{ {G_{IJ} =\int _D {A_{IJ} \left( \mathbf{r} \right) d\mathbf{r}} } \right\} .\)

References

  1. R. Carbó-Dorca, A. Gallegos, Á.J. Sánchez, Notes on quantitative structure-properties relationships (QSPR) (1): a discussion on a QSPR dimensionality paradox (QSPR DP) and its quantum resolution. J. Comput. Chem. 30, 1146–1159 (2008)

    Article  Google Scholar 

  2. R. Carbó-Dorca, Notes on quantitative structure-properties relationships (QSPR) (3): density functions origin shift as a source of quantum QSPR (QQSPR) algorithms in molecular spaces. J. Comput. Chem. (2012). doi:10.1002/jcc.23198.

  3. R. Carbó-Dorca, E. Besalú, Centroid origin shift of quantum object sets and molecular point clouds: description and element comparisons. J. Math. Chem. 50, 1161–1178 (2012)

    Article  Google Scholar 

  4. R. Carbó-Dorca, Quantum similarity, in Concepts and Methods in Modern Theoretical Chemistry, ed. by S.K. Ghosh, P.K. Chattaraj, vol. 1 (Taylor & Francis, London, 2013)

  5. R. Carbó-Dorca, Mathematical aspects of the LCAO MO first order density function (5): centroid shifting of MO ShF basis set, properties and applications. J. Math. Chem. (2012). doi:10.1007/s10910-012-0083-x

  6. R. Carbó-Dorca, A naïve geometrical perspective of Fukui functions: definition of Fukui function skew symmetric matrices described on density function sets. J. Math. Chem. (2012). doi:10.1007/s10910-012-0120-9

  7. R. Carbó-Dorca, E. Besalú, Shells, point cloud huts, generalized scalar products, cosines and similarity tensor representations in vector semispaces. J. Math. Chem. 50, 210–219 (2012)

    Article  Google Scholar 

  8. R. Carbó-Dorca, About the concept of chemical space: a concerned reflection on some trends of modern scientific thought within theoretical chemical lore. J. Math. Chem. (2012). doi:10.1007/s10910-012-0091-x

  9. R. Carbó-Dorca, Enfolded conformational spaces: definition of the chemical quantum mechanical multiverse under Born–Oppenheimer approximation. IQC technical report TR-2012-12. J. Math. Chem. (submitted)

  10. P. Bultinck, H. De Winter, W. Langenaeker, J.P. Tollenaere (eds.), Molecular Similarity and QSAR in Computational Medicinal Chemistry for Drug Discovery (Marcel Dekker Inc., New York, 2004)

    Google Scholar 

  11. J.P. Doucet, A. Panaye, Three Dimensional QSAR (CRC Press, Boca Raton, FL, 2010)

    Google Scholar 

  12. Y.C. Martin, Quantitative Drug Design: A Critical Introduction (CRC Press, Boca Raton, FL, 2010)

    Book  Google Scholar 

  13. J.C. Dearden, M.T.D. Cronin, K.L.E. Kaiser, SAR QSAR Environ. Res. 20, 241 (2009)

    Article  CAS  Google Scholar 

  14. M.T.D. Cronin, T.W.J. Schultz, Mol. Struct. (Theochem) 622, 39 (2003)

    Article  CAS  Google Scholar 

  15. W. Karcher, J. Devilliers (eds.), Practical Applications of QSAR in Environmental Chemistry and Toxicology (Kluwer, Dordrecht, 1990)

    Google Scholar 

  16. H. Kubinyi (ed.), 3D QSAR in Drug Design (ESCOM, Leiden, 1993)

    Google Scholar 

  17. C. Hansch, C. Leo, Exploring QSAR. ACS Professional Reference Book (ACS, Washington DC, 1995)

  18. F. Sanz, J. Giraldo, F. Manault (eds.), QSAR and Molecular Modelling (Prous Science, Barcelona, 1995)

    Google Scholar 

  19. N.C. Cohen (ed.), Molecular Modelling in Drug Design (Academic Press, San Diego, CA, 1996)

    Google Scholar 

  20. J. Devilliers (ed.), Comparative QSAR (CRC Press, Boca Raton, FL, 1998)

    Google Scholar 

  21. R. Carbó-Dorca, D. Robert, L. Amat, X. Gironés, E. Besalú, Molecular quantum similarity in QSAR and drug design, in Lecture Notes in Chemistry, vol. 73 (Springer, Berlin, 2000).

  22. N.C. Firth, N. Brown, J. Blagg, Plane of best fit: a novel method to characterize the 3D of molecules. J. Chem. Inf. Mod. 52, 2516–2525 (2012)

    Article  CAS  Google Scholar 

  23. W.H.B. Sauer, M.K. Schwarz, Molecular shape diversity of combinatorial libraries: a prerequisite for broad bioactivity. J. Chem. Inf. Comput. Sci. 43, 987–1003 (2003)

    Article  CAS  Google Scholar 

  24. F. Lovering, J. Bikker, C. Humblet, Escape from flatland: increasing saturation as an approach to improving clinical success. J. Med. Chem. 52, 6752–6756 (2009)

    Article  CAS  Google Scholar 

  25. A.Y. Meyer, Molecular mechanics and molecular shape. III. Surface area and cross-sectional areas of organic molecules. J. Comput. Chem. 7, 144–152 (1986)

    Article  CAS  Google Scholar 

  26. R. Todeschini, V. Consonni, Molecular Descriptors for Chemoinformatics (Wiley-VCH, Germany, 2009)

    Book  Google Scholar 

  27. R. Carbó, B. Calabuig, Molsimil-88: molecular similarity calculations using a CNDO approximation. Comput. Phys. Commun. 55, 117 (1989)

    Article  Google Scholar 

  28. R. Carbó-Dorca, E. Besalú, L.D. Mercado, Communications on quantum similarity (3): a geometric-quantum similarity molecular superposition (GQSMS) algorithm. J. Comp. Chem. 32, 582–599 (2011)

    Article  Google Scholar 

  29. R. Carbó-Dorca, E. Besalú, A general survey of molecular quantum similarity Huzinaga symposium. Fukuoka. J. Molec. Struct. Theochem. 451, 11–23 (1998)

    Article  Google Scholar 

  30. R. Carbó-Dorca, L. Amat, E. Besalú, X. Gironès, D. Robert, Quantum Molecular Similarity: Theory and Applications to the Evaluation of Molecular Properties, Biological Activity and Toxicity. Mathematical and Computational Chemistry: Fundamentals of Molecuar Similarity (Kluwer Academic/Plenum Publishers, Dordrecht, 2001)

    Google Scholar 

  31. P. Bultinck, X. Gironés, R: Carbó-Dorca. Molecular quantum similarity: theory and applications. in, Reviews in Computational Chemistry, vol. 21, ed. by K.B. Lipkowitz, R. Larter and T. Cundari (Wiley, Hoboken, 2005), pp. 127–207

  32. R. Carbó, B. Calabuig, Molecular quantum similarity measures and N-dimensional representation of quantum objects. I. Theoretical foundations. Int. J. Quantum Chem. 42, 1681 (1992)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Química Computacional, Universitat de Girona, 17071 , Gerona, Catalonia, Spain

    Ramon Carbó-Dorca

Authors
  1. Ramon Carbó-Dorca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ramon Carbó-Dorca.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carbó-Dorca, R. Particle coordinates and discrete molecular description: a geometric point of view on a twofold dimensionality environment. J Math Chem 51, 1569–1583 (2013). https://doi.org/10.1007/s10910-013-0165-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-013-0165-4

Keywords

Search

Navigation