Journal of Molecular Modeling

, Volume 19, Issue 4, pp 1677–1683

Shape entropy’s response to molecular ionization

  • K. Pineda-Urbina
  • R. D. Guerrero
  • A. Reyes
  • Z. Gómez-Sandoval
  • R. Flores-Moreno
Original Paper

Abstract

In this work we define a shape entropy by calculating the Shannon’s entropy of the shape function. This shape entropy and its linear response to the change in the total number of electrons of the molecule are explored as descriptors of bonding properties. Calculations on selected molecular systems were performed. According to these, shape entropy properly describes electron delocalization while its linear response to ionization predicts changes in bonding patterns. The derivative of the shape entropy proposed turned out to be fully determined by the shape function and the Fukui function.

Keywords

Electron delocalization Fukui function Shannon’s entropy Shape function 

References

  1. 1.
    Gagliardi L, Roos BO (2007) Multiconfigurational quantum chemical methods for molecular systems containing actinides. Chem Soc Rev 36:893–903CrossRefGoogle Scholar
  2. 2.
    Roos BO, Borin AC, Gagliardi L (2007) The maximum multiplicity of the covalent chemical bond. Angew Chem Int Ed 46:1469–1472CrossRefGoogle Scholar
  3. 3.
    Bader R (1994) Atoms in Molecules: a quantum theory. Oxford Univ. Press, OxfordGoogle Scholar
  4. 4.
    Tognetti V, Joubert L, Cortona P, Adamo C (2009) Toward a combined DFT/QTAIM description of agostic bonds: the critical case of a Nb(III) complex. J Phys Chem A 113:12322–12327CrossRefGoogle Scholar
  5. 5.
    Nalewajski RF (2006) Information theory of molecular systems. Elsevier, CambridgeGoogle Scholar
  6. 6.
    Ihara S (1993) Information theory for continuous systems. World Scientific, SingaporeCrossRefGoogle Scholar
  7. 7.
    Hocker D, Li X, Iyengar SS (2011) Shannon entropy based time-dependent deterministic sampling for efficient on-the-fly quantum dynamics and electronic structure. J Chem Theory Comput 7:256–268CrossRefGoogle Scholar
  8. 8.
    Sagar RP, Hô M (2008) Shannon entropies of atomic basins and electron correlation effects. J Mex Chem Soc 52:60–66Google Scholar
  9. 9.
    Corzo HH, Laguna HG, Sagar RP (2012) Localization phenomena in a cyclic box. J Math Chem 50:233–248CrossRefGoogle Scholar
  10. 10.
    Noorizadeh S, Shakerzadeh E (2010) Shannon entropy as a new measure of aromaticity, Shannon aromaticity. Phys Chem Chem Phys 12:4742–4749CrossRefGoogle Scholar
  11. 11.
    Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27(379–423):623–656Google Scholar
  12. 12.
    Cover TM, Thomas JA (2006) Elements of information theory. Wiley, New YorkGoogle Scholar
  13. 13.
    El Saddik A, Orozco M, Asfaw Y, Shirmohammadi S, Adler A (2007) A novel biometric system for identification and verification of haptic users. IEEE Trans Instrum Meas 56:895–906CrossRefGoogle Scholar
  14. 14.
    Moddemeijer R (1989) On estimation of entropy and mutual information of continuous distributions. Signal Process 16:233–248CrossRefGoogle Scholar
  15. 15.
    Gadre SR, Sears SB, Chakravorty SJ, Bendale RD (1985) Some novel characteristics of atomic information entropies. Phys Rev A 32:2602–2606CrossRefGoogle Scholar
  16. 16.
    Hô MH, Sagar RP, Pérez-Jordá JM, Smith VH, Esquivel RO (1994) A numerical study of molecular information entropies. Chem Phys Lett 219:15–20CrossRefGoogle Scholar
  17. 17.
    Hô MH, Sagar RP, Smith VH, Esquivel RO (1994) Atomic information entropies beyond the Hartree-Fock limit. J Phys B Atomic Mol Opt Phys 27:5149–5157CrossRefGoogle Scholar
  18. 18.
    Hô MH, Sagar RP, Weaver DF, Smith VH (1995) An investigation of the dependence of Shannon-information entropies and distance measures on molecular-geometry. Int J Quantum Chem S29:109–115CrossRefGoogle Scholar
  19. 19.
    Esquivel RO, Rodríguez AL, Sagar RP, Hô MH, Smith VH (1996) Physical interpretation of information entropy: numerical evidence of the Collins conjecture. Phys Rev A 54:259–265CrossRefGoogle Scholar
  20. 20.
    Ramírez JC, Soriano C, Esquivel RO, Sagar RP, Hô MH, Smith VH (1997) Jaynes information entropy of small molecules: numerical evidence of the Collins conjecture. Phys Rev A 56:4477–4482CrossRefGoogle Scholar
  21. 21.
    Hô M, Smith VH, Weaver DF, Gatti C, Sagar RP, Esquivel RO (1998) Molecular similarity based on information entropies and distances. J Chem Phys 108:5469–5475CrossRefGoogle Scholar
  22. 22.
    Hô M, Weaver DF, Smith VH, Sagar RP, Esquivel RO, Yamamoto S (1998) An information entropic study of correlated densities of the water molecule. J Chem Phys 109:10620–10627CrossRefGoogle Scholar
  23. 23.
    Chattaraj PK, Chamorro E, Fuentealba P (1999) Chemical bonding and reactivity: a local thermodynamic viewpoint. Chem Phys Lett 314:114–121CrossRefGoogle Scholar
  24. 24.
    Nalewajski RF, Parr RG (2000) Information theory, atoms in molecules, and molecular similarity. Proc Natl Acad Sci 97:8879–8882CrossRefGoogle Scholar
  25. 25.
    Nalewajski RF (2002) Applications of the information theory to problems of molecular electronic structure and chemical reactivity. Int J Mol Sci 3:237–259CrossRefGoogle Scholar
  26. 26.
    Ayers PW (2000) Atoms in molecules, an axiomatic approach. I. Maximum transferability. J Chem Phys 113:10886–10898CrossRefGoogle Scholar
  27. 27.
    Parr RG, Ayers PW, Nalewajski RF (2005) What is an atom in a molecule? J Phys Chem A 109:3957–3959CrossRefGoogle Scholar
  28. 28.
    Ghiringhelli LM, Delle Site L, Mosna RA, Hamilton IP (2010) Information-theoretic approach to kinetic-energy functionals: the nearly uniform electron gas. J Math Chem 48:78–82CrossRefGoogle Scholar
  29. 29.
    Ghiringhelli LM, Hamilton IP, Delle Site L (2010) Interacting electrons, spin statistics, and information theory. J Chem Phys 132:014106CrossRefGoogle Scholar
  30. 30.
    Parr RG, Bartolotti LJ (1983) Some remarks on the density functional theory of few-electron systems. J Phys Chem 87:2810–2815CrossRefGoogle Scholar
  31. 31.
    Cedillo A (1994) A new representation for ground-states and its Legendre transforms. Int J Quantum Chem Symp 28:231–240CrossRefGoogle Scholar
  32. 32.
    Baekelandt BG, Cedillo A, Parr RG (1995) Reactivity indexes and fluctuation formulas in density-functional theory—isomorphic ensembles and a new measure of local hardness. J Chem Phys 103:8548–8556CrossRefGoogle Scholar
  33. 33.
    Ayers PW, De Proft F, Geerlings P (2007) A comparison of the utility of the shape function and electron density for predicting periodic properties: atomic ionization potentials. Phys Rev A 75:012508CrossRefGoogle Scholar
  34. 34.
    De Proft F, Ayers PW, Sen K, Geerlings P (2004) On the importance of the “density per particle” (shape function) in the density functional theory. J Chem Phys 120:9969–9973CrossRefGoogle Scholar
  35. 35.
    Ayers PW (2006) Information theory, the shape function, and the Hirshfeld atom. Theor Chem Accounts 115:370–378CrossRefGoogle Scholar
  36. 36.
    Ayers PW (2000) Density per particle as a descriptor of coulombic systems. Proc Natl Acad Sci 97:1959–1964CrossRefGoogle Scholar
  37. 37.
    Ayers PW, Cedillo A (2009) The shape function. In: Chattaraj PK (ed) Chemical reactivity theory: a density functional view. Taylor and Francis, Boca Raton, p 269Google Scholar
  38. 38.
    Parr RG, Yang WT (1984) Density-functional approach to the frontier-electron theory of chemical reactivity. J Am Chem Soc 106:4049CrossRefGoogle Scholar
  39. 39.
    Fuentealba P, Florez E, Tiznado W (2010) Topological analysis of the Fukui function. J Chem Theory Comput 6:1470–1478CrossRefGoogle Scholar
  40. 40.
    Cardenas C, Tiznado W, Ayers PW, Fuentealba P (2011) The Fukui potential and the capacity of charge and the global hardness of atoms. J Phys Chem A 115:2325–2331CrossRefGoogle Scholar
  41. 41.
    Geerlings P, De Proft F, Langenaeker W (2003) Conceptual density functional theory. Chem Rev 103:1793–1873CrossRefGoogle Scholar
  42. 42.
    Ayers PW (2008) The continuity of the energy and other molecular properties with respect to the number of electrons. J Math Chem 43:285–303CrossRefGoogle Scholar
  43. 43.
    Perdew JP, Parr RG, Levy M, Balduz JL Jr (1982) Density-functional theory for fractional particle number: derivative discontinuities of the energy. Phys Rev Lett 49:1691–1694CrossRefGoogle Scholar
  44. 44.
    Ayers PW, Levy M (2000) Perspective on “Density functional approach to the frontier-electron theory of chemical reactivity” by Parr RG, Yang W (1984). Theor Chem Account 103:353–360CrossRefGoogle Scholar
  45. 45.
    Ayers PW, Yang WT, Bartolotti LJ (2009) Fukui function. In: Chattaraj PK (ed) Chemical reactivity theory: A density functional view. CRC Press, Boca Raton, pp 255–267Google Scholar
  46. 46.
    Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140:A1133–A1138CrossRefGoogle Scholar
  47. 47.
    Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136:B864–B871CrossRefGoogle Scholar
  48. 48.
    Parr RG, Yang W (1989) Density functional theory of atoms and molecules. Oxford University Press, New YorkGoogle Scholar
  49. 49.
    Yang W, Parr RG, Pucci R (1984) Electron density, Kohn–Sham frontier orbitals, and Fukui functions. J Chem Phys 81:2862–2863CrossRefGoogle Scholar
  50. 50.
    Michalak A, De Proft F, Geerlings P, Nalewajski RF (1999) Fukui functions from the relaxed Kohn-Sham orbitals. J Phys Chem A 103:762–771CrossRefGoogle Scholar
  51. 51.
    Balawender R, Geerlings P (2005) DFT-based chemical reactivity indices in the Hartree Fock method. II. Hardness and Fukui function. J Chem Phys 123:124103CrossRefGoogle Scholar
  52. 52.
    Flores-Moreno R, Köster AM (2008) Auxiliary density perturbation theory. J Chem Phys 128:134105CrossRefGoogle Scholar
  53. 53.
    Flores-Moreno R, Melin J, Ortiz JV, Merino G (2008) Efficient calculation of analytic Fukui functions. J Chem Phys 129:224105CrossRefGoogle Scholar
  54. 54.
    Flores-Moreno R (2010) Symmetry conservation in Fukui functions. J Chem Theory Comput 6:48CrossRefGoogle Scholar
  55. 55.
    Yang W, Cohen AJ, De Proft F, Geerlings P (2012) Analytical evaluation of Fukui functions and real-space linear response function. J Chem Phys 136:144110CrossRefGoogle Scholar
  56. 56.
    Dirac PAM (1928) The quantum theory of the electron. Proc R Soc Lond A 117:610–624CrossRefGoogle Scholar
  57. 57.
    Vosko SH, Wilk L, Nusair M (1980) Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can J Phys 58:1200–1211CrossRefGoogle Scholar
  58. 58.
    Godbout N, Salahub DR, Andzelm J, Wimmer E (1992) Optimization of Gaussian-type basis sets for local spin density functional calculations. Part I: boron through neon, optimization technique and validation. Can J Phys 70:560–571Google Scholar
  59. 59.
    Andzelm J, Radzio E, Salahub DR (1985) Compact basis sets for LCAO-LSD calculations. Part I: method and bases for Sc to Zn. J Comput Chem 6:520–532CrossRefGoogle Scholar
  60. 60.
    Andzelm J, Russo N, Salahub DR (1987) Ground and excited states of group IVA diatomics from local-spin-density calculations: model potentials for Si, Ge, and Sn. J Chem Phys 87:6562CrossRefGoogle Scholar
  61. 61.
    Köster AM, Geudtner G, Calaminici P, Casida ME, Domínguez VD, Flores-Moreno R, Gamboa GU, Goursot A, Heine T, Ipatov A, Janetzko F, del Campo JM, Reveles JU, Vela A, Zuñiga-Gutierrez B, Salahub DR (2011) deMon2k, Version 3, The deMon developers, Cinvestav, México City. http://www.demon-software.com
  62. 62.
    Flores-Moreno R, Pineda-Urbina K, Gómez-Sandoval Z (2012) Sinapsis, Version XII-V, Sinapsis developers, Guadalajara. http://sinapsis.sourceforge.net

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • K. Pineda-Urbina
    • 1
  • R. D. Guerrero
    • 2
  • A. Reyes
    • 3
  • Z. Gómez-Sandoval
    • 1
  • R. Flores-Moreno
    • 4
  1. 1.Facultad de Ciencias QuímicasUniversidad de ColimaCoquimatlánMexico
  2. 2.Departamento de FísicaUniversidad Nacional de ColombiaBogota D.C.Colombia
  3. 3.Departamento de QuímicaUniversidad Nacional de ColombiaBogota D.C.Colombia
  4. 4.Departamento de QuímicaUniversidad de GuadalajaraGuadalajaraMexico

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