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Localization-Delocalization Matrices and Electron Density-Weighted Adjacency/Connectivity Matrices: A Bridge Between the Quantum Theory of Atoms in Molecules and Chemical Graph Theory

  • Chérif F. MattaEmail author
  • Ismat Sumar
  • Ronald CookEmail author
  • Paul W. AyersEmail author
Chapter
Part of the Challenges and Advances in Computational Chemistry and Physics book series (COCH, volume 22)

Abstract

Chemical graph theory (CGT) starts by defining matrices that represent the molecular graph then proceed to extract numbering-independent matrix invariants to be used as molecular descriptors in empirical quantitative structure to activity (or property) relationships (QSAR/QSPR). Two proposed matrix representations of molecular structure are presented in this chapter as alternatives to simple connectivity molecular graphs. Firstly, it is proposed to use a more “nuanced” connectivity matrix by weighing the “ones” entered in a CGT molecular graph matrix by the bond critical point electron densities associated with each bond path to yield what we term the “electron density-weighted adjacency/connectivity matrices (EDWAM/EDWCM)”. In a second approach, it is proposed to use the localization and delocalization indices of the quantum theory of atoms in molecules (QTAIM) to construct a richer representation of the molecular graph, a “fuzzy” graph, whereby an edge exists between any two atoms (measured by the delocalization index between them) whether they share a bond path or not. Such a fuzzy graph is represented by what we term “electron localization-delocalization matrix (LDM)”. We show that the LDM representations of a series of molecules provide a powerful tool for robust QSAR/QSPR modeling.

Keywords

Quantitative Structure Activity Relationship Molecular Graph Bond Critical Point Quantitative Structure Property Relationship Bond Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The authors thank Dr. Todd A. Keith, Professor Lou Massa, Dr. Nenad Trinajstić, Dr. Sonja Nikolić, and Mr. Matthew J. Timm for helpful discussions. Financial support of this work was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC), Canada Foundation for Innovation (CFI), Saint Mary’s University, McMaster University, and Mount Saint Vincent University.

References

  1. 1.
    Dmitriev IS (1981) Molecules without chemical bonds (English Translation). Mir Publishers, MoscowGoogle Scholar
  2. 2.
    Balaban AT (1976) Chemical applications of graph theory. Academic Press, New YorkGoogle Scholar
  3. 3.
    Balaban AT (1985) Applications of graph theory in chemistry. J Chem Inf Comput Sci 25:334–343CrossRefGoogle Scholar
  4. 4.
    Hall LH, Kier LB (1976) Molecular connectivity in chemistry and drug research. Academic Press, BostonGoogle Scholar
  5. 5.
    Bonchev D, Rouvray DH (1991) Chemical graph theory: introduction and fundamentals. OPA, AmsterdamGoogle Scholar
  6. 6.
    Balasubramanian K (1994) Integration of graph theory and quantum chemistry for structure-activity relationship. SAR & QSAR Eviron Res 2:59–77CrossRefGoogle Scholar
  7. 7.
    Diudea MD, Gutman I, Lorentz J (1999) Molecular topology. Nova Science Publishers Inc, Hauppauge NYGoogle Scholar
  8. 8.
    Janezić D, Milicević A, Nikolić S, Trinajstić N (2007) Graph theoretical matrices in chemistry. In: Mathematical chemistry monographs, vol 3. University of Kragujevac, KragujevacGoogle Scholar
  9. 9.
    Todeschini R, Consonni V (2009) Molecular descriptors for chemoinformatics, 2nd Edn, vols. I and II). Wiley-VCH Weinheim, WeinheimGoogle Scholar
  10. 10.
    Bader RFW (1990) Atoms in molecules: a quantum theory. Oxford University Press, OxfordGoogle Scholar
  11. 11.
    Popelier PLA (2000) Atoms in molecules: an introduction. Prentice Hall, LondonCrossRefGoogle Scholar
  12. 12.
    Matta CF, Boyd RJ (eds) (2007) The quantum theory of atoms in molecules: from solid state to DNA and drug design. Wiley-VCH, WeinheimGoogle Scholar
  13. 13.
    Bader RFW (1998) A bond path: a universal indicator of bonded interactions. J Phys Chem A 102:7314–7323CrossRefGoogle Scholar
  14. 14.
    Bader RFW (2009) Bond paths are not chemical bonds. J Phys Chem A 113:10391–10396CrossRefGoogle Scholar
  15. 15.
    Martín-Pendás A, Francisco E, Blanco MA, Gatti C (2007) Bond paths as privileged exchange channels. Chem Eur J 13:9362–9371Google Scholar
  16. 16.
    Runtz GR, Bader RFW, Messer RR (1977) Definition of bond paths and bond directions in terms of the molecular charge distribution. Can J Chem 55:3040–3045CrossRefGoogle Scholar
  17. 17.
    Keith TA, Bader RFW, Aray Y (1996) Structural homeomorphism between the electron density and the virial field. Int J Quantum Chem 57:183–198CrossRefGoogle Scholar
  18. 18.
    Keith TA (2015) AIMAll/AIMStudio. http://aim.tkgristmill.com/
  19. 19.
    Fradera X, Austen MA, Bader RFW (1999) The Lewis model and beyond. J Phys Chem A 103:304–314CrossRefGoogle Scholar
  20. 20.
    Matta CF (2014) Modeling biophysical and biological properties from the characteristics of the molecular electron density, electron localization and delocalization matrices, and the electrostatic potential. J Comput Chem 35:1165–1198CrossRefGoogle Scholar
  21. 21.
    Sumar I, Ayers PW, Matta CF (2014) Electron localization and delocalization matrices in the prediction of pK a’s and UV-wavelengths of maximum absorbance of p-benzoic acids and the definition of super-atoms in molecules. Chem Phys Lett 612:190–197CrossRefGoogle Scholar
  22. 22.
    Timm MJ, Matta CF, Massa L, Huang L (2014) The localization-delocalization matrix and the electron density-weighted connectivity matrix of a finite graphene flake reconstructed from kernel fragments. J Phys Chem A 118:11304–11316CrossRefGoogle Scholar
  23. 23.
    Matta CF (2014) Localization-delocalization matrices and electron density-weighted adjacency matrices: new electronic fingerprinting tools for medicinal computational chemistry. Future Med Chem 6:1475–1479CrossRefGoogle Scholar
  24. 24.
    Dittrich B, Matta CF (2014) Contributions of charge-density research to medicinal chemistry. Int U Cryst J (IUCrJ) 1:457–469CrossRefGoogle Scholar
  25. 25.
    Sumar I, Cook R, Ayers PW, Matta CF (2015) AIMLDM: a program to generate and analyze electron localization–delocalization matrices (LDMs). Comput Theor Chem 1070:55–67CrossRefGoogle Scholar
  26. 26.
    White D, Wilson RC (2008) Parts-based generative models for graphs. In: 19th International Conference on Pattern Recognition (ICPR 2008), pp 1–4Google Scholar
  27. 27.
    Pye CC, Poirier RA (1998) Graphical approach for defining natural internal coordinates. J Comput Chem 19:504–511CrossRefGoogle Scholar
  28. 28.
    Pye CC (1997) Applications of optimization to quantum chemistry, PhD Thesis. Memorial University of Newfoundland, Saint John’s (NF), CanadaGoogle Scholar
  29. 29.
    Müller AMK (1984) Explicit approximate relation between reduced two- and one-particle density matrices. Phys Lett A 105:446–452CrossRefGoogle Scholar
  30. 30.
    Massa L (2014) Personal communicationGoogle Scholar
  31. 31.
    Picard RR, Cook RD (1984) Cross-validation of regression models. J Am Stat Assoc 79:575–583CrossRefGoogle Scholar
  32. 32.
    Lide DR (2007–2008) CRC handbook of chemistry and physics, 88th Edn. CRC PressGoogle Scholar
  33. 33.
    Lide DR (2006) CRC handbook of chemistry and physics, 87th Edn. CRC PressGoogle Scholar
  34. 34.
    Jover J, Bosque R, Sales J (2008) QSPR prediction of pK a for benzoic acids in different solvents. QSAR Combin Sci 27:563–581CrossRefGoogle Scholar
  35. 35.
    Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev B 136:864–871CrossRefGoogle Scholar
  36. 36.
    Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford University Press, OxfordGoogle Scholar
  37. 37.
    Guo H-B, He F, Gu B, Liang L, Smith JC (2012) Time-dependent density functional theory assessment of UV absorption of benzoic acid derivatives. J Phys Chem A 116:11870–11879CrossRefGoogle Scholar
  38. 38.
    Pavia DL, Lampman GM, Kriz GS, Vyvyan JR (2009) Introduction to spectroscopy, 4th edn. Brooks/Cole Cengage Learning, Belmont, CA, USAGoogle Scholar
  39. 39.
    Kamath BV, Mehta JD, Bafna SL (1975) Ultraviolet absorption spectra: some substituted benzoic acids. J Appl Chem Biotechnol 25:743–751CrossRefGoogle Scholar
  40. 40.
    Krygowski TM, Szatylowicz H, Stasyuk OA, Dominikowska J, Palusiak M (2014) Aromaticity from the viewpoint of molecular geometry: application to planar systems. Chem Rev 114:6383–6422CrossRefGoogle Scholar
  41. 41.
    Krygowski TM, Stepien BT (2005) σ- and π-electron delocalization: focus on substituent effects. Chem Rev 105:3482–3512CrossRefGoogle Scholar
  42. 42.
    Krygowski TM, Cyranski MK (2001) Structural aspect of aromaticity. Chem Rev 101:1385–1419CrossRefGoogle Scholar
  43. 43.
    Kruszewski J, Krygowski TM (1972) Definition of aromaticity basing on the harmonic oscillator model. Tetrahedron Lett 3839–3842Google Scholar
  44. 44.
    Chen Z, Wannere CS, Corminboeuf C, Puchta R, Schleyer PvR (2005) Nucleus-independent chemical shifts (NICS) as an aromaticity criterion. Chem Rev 105:3842–3888CrossRefGoogle Scholar
  45. 45.
    Schleyer PvR, Manoharan M, Wang Z-X, Kiran B, Jiao H, Puchta R, Hommes NJRVE (2001) Dissected nucleus-independent chemical shift analysis of π-aromaticity and antiaromaticity. Org Lett 3:2465–2468Google Scholar
  46. 46.
    Schleyer PvR, Maerker C, Dransfeld A, Jiao H, Hommes NJRVE (1996) Nucleus-Independent chemical shifts: a simple and efficient aromaticity probe. J Am Chem Soc 118:6317–6318Google Scholar
  47. 47.
    Memory JD, Wilson NK (1982) NMR of aromatic compounds. Wiley, New YorkGoogle Scholar
  48. 48.
    Mitchell RH (2001) Measuring aromaticity by NMR. Chem Rev 101:1301–1315CrossRefGoogle Scholar
  49. 49.
    Gomes JANF, Mallion RB (2001) Aromaticity and ring currents. Chem Rev 101:1349–1383CrossRefGoogle Scholar
  50. 50.
    Keith TA, Bader RFW (1993) Topological analysis of magnetically induced molecular current distributions. J Chem Phys 99:3669–3682CrossRefGoogle Scholar
  51. 51.
    Keith TA, Bader RFW (1996) Use of electron charge and current distributions in the determination of atomic contributions to magnetic properties. Int J Quantum Chem 60:373–379CrossRefGoogle Scholar
  52. 52.
    Badger GM (1969) Aromatic character and aromaticity. Cambridge University Press, CambridgeGoogle Scholar
  53. 53.
    Slayden SW, Liebman JF (2001) The energetics of aromatic hydrocarbons: an experimental thermochemical perspective. Chem Rev 101:1541–1566CrossRefGoogle Scholar
  54. 54.
    Cyranski MK (2005) Energetic aspects of cyclic & σ-electron delocalization: Evaluation of the methods of estimating aromatic stabilization energies. Chem Rev 105:3773–3811CrossRefGoogle Scholar
  55. 55.
    Sivaramakrishnan R, Tranter RS, Brezinsky K (2005) Ring conserved isodesmic reactions: a new method for estimating the heats of formation of aromatics and PAHs. J Phys Chem A 109:1621–1628CrossRefGoogle Scholar
  56. 56.
    Poater J, Fradera X, Duran M, Solà M (2003) The delocalization index as an electronic aromaticity criterion: application to a series of planar polycyclic aromatic hydrocarbons. Chem Eur J 9:400–406CrossRefGoogle Scholar
  57. 57.
    Poater J, Fradera X, Duran M, Solà M (2003) An insight into local aromaticities of polycyclic aromatic hydrocarbons and fullerenes. Chem Eur J 9:1113–1122CrossRefGoogle Scholar
  58. 58.
    Matta CF, Hernández-Trujillo J (2005) Bonding in polycyclic aromatic hydrocarbons in terms of the the electron density and of electron delocalization. J Phys Chem A 107:7496–7504 (Correction: J Phys Chem A (2005) 109:10798)Google Scholar
  59. 59.
    Matito E, Duran M, Solà M (2005) The aromatic fluctuation index (FLU): a new aromaticity index based on electron delocalization. J Chem Phys 122:014109CrossRefGoogle Scholar
  60. 60.
    Poater J, Duran M, Solà M, Silvi B (2005) Theoretical evaluation of electron delocalization in aromatic molecules by means of atoms in molecules (AIM) and electron localization function (ELF) topological approaches. Chem Rev 105:3911–3947CrossRefGoogle Scholar
  61. 61.
    Portella G, Poater J, Bofill JM, Alemany P, Solà M (2005) Local aromaticity of [n]acenes, [n]phenacenes, and [n]helicenes (n = 1–9). J Org Chem 70:2509–2521CrossRefGoogle Scholar
  62. 62.
    Matito E, Poater J, Solà M (2007) Aromaticity analyses by means of the quantum theory of atoms in molecules. In: Matta CF (ed) The quantum theory of atoms in molecules: from solid state to DNA and drug design. Wiley-VCH, Weinheim, pp 399–423CrossRefGoogle Scholar
  63. 63.
    Firme CI, Galembeck SE, Antunes OAC, Esteves PM (2007) Density, degeneracy, delocalization-based index of aromaticity (D3BIA). J Braz Chem Soc 18:1397–1404CrossRefGoogle Scholar
  64. 64.
    Howard ST, Krygowski TM (1997) Benzenoid hydrocarbon aromaticity in terms of charge density descriptors. Can J Chem 75:1174–1181CrossRefGoogle Scholar
  65. 65.
    Suresh CH, Gadre SR (1999) Clar’s aromatic sextet theory revisited via molecular electrostatic potential topography. J Org Chem 64:2505–2512CrossRefGoogle Scholar
  66. 66.
    Cyranski MK, Stepien BT, Krygowski TM (2000) Global and local aromaticities of linear and angular polyacenes. Tetrahedron 56:9663–9667CrossRefGoogle Scholar
  67. 67.
    Palusiak M, Krygowski TM (2007) Application of AIM parameters at ring critical points for estimation of π-electron delocalization in six-membered aromatic and quasi-aromatic rings. Chem Eur J 13:7996–8006CrossRefGoogle Scholar
  68. 68.
    Mandado M, Gonzalez Moa MJ, Mosquera RA (2008) Aromaticity: exploring basic chemical concepts with the quantum theory of atoms in molecules. Nova Science Publishers, Inc., New YorkGoogle Scholar
  69. 69.
    Ebrahimi AA, Ghiasi R, Foroutan-Nejad C (2010) Topological characteristics of the ring critical points and the aromaticity of groups IIIA to VIA hetero-benzenes. J Mol Struct (THEOCHEM) 941:47–52CrossRefGoogle Scholar
  70. 70.
    Nigam S, Majumder C (2011) Aromaticity: from benzene to atomic clusters. In: Aromaticity and metal clusters. Chattaraj PK (Ed.), CRC Press, New YorkGoogle Scholar
  71. 71.
    Krygowski TM, Ciesielski A, Bird CW, Kotschy A (1995) Aromatic character of the benzene ring present in various topological environments in benzenoid hydrocarbons. Nonequivalence of indices of aromaticity. J Chem Inf Comput Sci 35:203–210CrossRefGoogle Scholar
  72. 72.
    Sumar I, Cook R, Ayers PW, Matta CF (2016) Aromaticity of rings-in-molecules (RIMs) from electron localization-delocalization matrices (LDMs). Phys Scripta 91:013001 (pp 13)Google Scholar
  73. 73.
    Cook R, Sumar I, Ayers PW, Matta CF (2015) Aromaticity of rings-in-molecules (RIMs) from electron localization-delocalization matrices (LDMs) from self-organized maps. In preparationGoogle Scholar
  74. 74.
    Mager PP (1984) Multidimensional pharmacochemistry: design of safer drugs. Academic Press Inc, LondonGoogle Scholar
  75. 75.
    Kier LB, Hall LH, Frazer JW (1991) An index of electrotopological state for atoms in molecules. J Math Chem 7:229–241CrossRefGoogle Scholar
  76. 76.
    Attwood TK, Parry-Smith DJ (1999) Introduction to bioinformatics. Prentice Hall, LondonGoogle Scholar
  77. 77.
    Doucet J-P, Weber J (1996) Computer-aided molecular design: theory and applications. Academic Press, ltd, LondonGoogle Scholar
  78. 78.
    Carbó R, Leyda L, Arnau M (1980) How similar is a molecule to another? an electron density measure of similarity between two molecular structures. Int J Quantum Chem 17:1185–1189CrossRefGoogle Scholar
  79. 79.
    Carbó-Dorca R, Mezey PGE (1996) Advances in molecular similarity, vol 1. Jai Press Inc, LondonGoogle Scholar
  80. 80.
    Carbó-Dorca R, Mezey PGE (1998) Advances in molecular similarity, vol 2. Jai Press Inc, LondonGoogle Scholar
  81. 81.
    Carbó-Dorca R, Robert D, Amat L, Gironés X,  Besalú E (2000) Molecular quantum similarity in QSAR and drug design. Springer, BerlinGoogle Scholar
  82. 82.
    Bonaccorsi R, Scrocco E, Tomasi J (1970) Molecular SCF calculations for the ground state of some three-membered ring molecules: (CH2)3, (CH2)2NH, (CH2)2NH2 +, (CH2)2O, (CH2)2S, (CH)2CH2, and N2CH2. J Chem Phys 52:5270–5284CrossRefGoogle Scholar
  83. 83.
    Petrongolo C, Tomasi J (1975) The use of electrostatic molecular potential in quantum pharmacology. 1. Ab initio results. Int J Quantum Chem Quantum Biol Symp 2:181–190Google Scholar
  84. 84.
    Bonaccorsi R, Scrocco E, Tomasi J (1976) Group contributions to electrostatic molecular potential. J Am Chem Soc 98:4049–4054CrossRefGoogle Scholar
  85. 85.
    Tomasi J (1981) Use of the electrostatic potential as a guide to understanding molecular properties. In: Truhlar DG, Politzer P (eds) Chemical applications of atomic and molecular electrostatic potentials. reactivity, structure, scattering, and energetics of organic, inorganic, and biological systems. Plenum Press, New YorkGoogle Scholar
  86. 86.
    Tomasi J, Cappelli C, Mennucci B, Cammi R (2010) From molecular electrostatic potentials to solvations models and ending with biomolecular photophysical processes. In: Matta CF (ed) Quantum biochemistry: electronic structure and biological activity, vol 1. Wiley-VCH, Weinheim, pp 131–170CrossRefGoogle Scholar
  87. 87.
    Truhlar DG, Politzer P (eds) (1981) Chemical applications of atomic and molecular electrostatic potentials. reactivity, structure, scattering, and energetics of organic, inorganic, and biological systems. Plenum Press, New YorkGoogle Scholar
  88. 88.
    Murray JS, Politzer P (1998) Electrostatic potentials: chemical applications. In: Schleyer PvR (ed) Encyclopedia of computational chemistry. Wiley, Chichester, UKGoogle Scholar
  89. 89.
    Politzer P, Murray JS (2007) Molecular electrostatic potentials and chemical reactivity. Reviews in computational chemistry. Wiley, Hoboken, NJGoogle Scholar
  90. 90.
    Gadre SR, Kulkarni SA, Suresh CH, Shrivastava IH (1995) Basis set dependence of molecular electrostatic potential topography: a case study of substituted benzenes. Chem Phys Lett 239:273–281CrossRefGoogle Scholar
  91. 91.
    Suresh CH, Gadre SR (1998) Novel electrostatic approach to substituent constants: doubly substituted benzenes. J Am Chem Soc 120:7049–7055CrossRefGoogle Scholar
  92. 92.
    Gadre SR (1999) Topography of atomic and molecular scalar fields. Computational chemistry: reviews of current trends. World Scientific, Singapore, pp 1–53Google Scholar
  93. 93.
    Gadre SR, Shirsat RN (2000) Electrostatics of atoms and molecules. Universities Press, HyderabadGoogle Scholar
  94. 94.
    Roy D, Balanarayan P, Gadre SR (2008) An appraisal of Poincaré-Hopf relation and application to topography of molecular electrostatic potentials. J Chem Phys 129:174103CrossRefGoogle Scholar
  95. 95.
    Cook R (2015) (in preparation)Google Scholar
  96. 96.
    Babić-Samardzija K, Lupu C, Hackerman N, Barron AR, Luttge A (2005) Inhibitive properties and surface morphology of a group of heterocyclic diazoles as inhibitors for acidic iron corrosion. Langmuir 21:12187–12196Google Scholar
  97. 97.
    Popelier PLA (2010) Developing Quantum Topological Molecular Similarity (QTMS). In: Matta CF (ed) Quantum biochemistry: electronic structure and biological activity. Wiley-VCH, Weinheim, pp 669–691Google Scholar
  98. 98.
    Harding AP, Wedge DC, Popelier PLA (2009) pK a prediction from “quantum chemical topology” descriptors. J Chem Inf Mod 49:1914–1924CrossRefGoogle Scholar
  99. 99.
    Popelier PLA, Chaudry UA, Smith PJ (2002) Quantum topological molecular similarity. Part 5. Further development with an application to the toxicity of polychlorinated dibenzo-p-dioxins (PCDDs). J Chem Soc Perkin Trans 2:1231–1237CrossRefGoogle Scholar
  100. 100.
    O’Brien SE, Popelier PLA (2002) Quantum topological molecular similarity. Part 4. A QSAR study of cell growth inhibitory properties of substituted (E)-1-phenylbut-1-en-3-ones. J Chem Soc Perkin Trans 2:478–483CrossRefGoogle Scholar
  101. 101.
    O’Brien SE, Popelier PLA (2001) Quantum molecular similarity. 3. QTMS descriptors. J Chem Inf Comput Sci 41:764–775CrossRefGoogle Scholar
  102. 102.
    O’Brien SE, Popelier PLA (1999) Quantum molecular similarity. Part 2: the relation between properties in BCP space and bond length. Can J Chem 77:28–36CrossRefGoogle Scholar
  103. 103.
    Popelier PLA (1999) Quantum molecular similarity. 1. BCP space. J Phys Chem A 103:2883–2890CrossRefGoogle Scholar
  104. 104.
    Cook R,. Sumar I, Ayers PW, Matta CF (2016) Electron localization-delocalization matrices (LDMs): theory and applications. SpringerGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Chemistry and PhysicsMount Saint Vincent UniversityHalifaxCanada
  2. 2.Department of ChemistryDalhousie UniversityHalifaxCanada
  3. 3.Department of ChemistrySaint Mary’s UniversityHalifaxCanada
  4. 4.Department of ChemistryMcMaster UniversityHamiltonCanada
  5. 5.TDA Research, IncWheat RidgeUSA

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