A partition function for atoms and bonds in molecules

  • Ibrahim E. AwadEmail author
  • Raymond A. PoirierEmail author
Regular Article


A new weight function is proposed to work with the atoms and bonds in molecules theory. The molecular radial electron density (\(\rho _{\mathrm{rad}}{({\mathbf {r}} )}\)) and bond electron density (\(\rho ^{A-B}{({\mathbf {r}} )}\)) are visually illustrated using the proposed weight. The molecular properties including the total number of electrons, the electron–nuclear potential energy, and Coulomb potential energy are calculated numerically using the proposed weight. The computed molecular properties using the proposed weight are compared to those obtained using the Becke weight and as well with molecular properties calculated analytically. Our findings show that the proposed weight gives better bonding-region representations for both \(\rho _{\mathrm{rad}}{({\mathbf {r}} )}\) and \(\rho ^{A-B}{({\mathbf {r}} )}\) than those obtained using the Becke weight. In addition, the proposed weight performs equally well or better than the Becke weight at numerical integration of molecular properties.


Weight function Numerical integration atoms and molecule Molecular radial density Bond densities 



We gratefully acknowledge the support of the Natural Sciences and Engineering Council of Canada (NSERC) and the Atlantic Excellence Network (ACENET) and Compute Canada for the computer time.

Supplementary material

214_2019_2468_MOESM1_ESM.pdf (12.8 mb)
Supplementary material 1 (pdf 13073 KB)


  1. 1.
    Mayer I (2013) Relation between the Hilbert space and “fuzzy atoms” analyses. Chem. Phys. Lett. 585:198–200CrossRefGoogle Scholar
  2. 2.
    Mulliken RS (1935) Electronic structures of molecules XI. Electroaffinity, molecular orbitals and dipole moments. J. Chem. Phys. 3:573–585CrossRefGoogle Scholar
  3. 3.
    Löwdin PO (1953) Approximate formulas for many center integrals in the theory of molecules and crystals. J. Chem. Phys. 21:374–375CrossRefGoogle Scholar
  4. 4.
    Clark AE, Davidson ER (2003) Population analyses that utilize projection operators. Int. J. Quantum Chem. 93:384–394CrossRefGoogle Scholar
  5. 5.
    Salvador P, Ramos-Cordoba E (2013) Communication: an approximation to Bader’s topological atom. J. Chem. Phys. 139:071103CrossRefGoogle Scholar
  6. 6.
    Awad IE, Poirier RA (2019) Atoms and bonds in molecules: molecular radial energy densities. Comput. Theor. Chem. 1153:44–53CrossRefGoogle Scholar
  7. 7.
    Blanco MA, Martín Pendás A, Francisco E (2005) Interacting quantum atoms: a correlated energy decomposition scheme based on the quantum theory of atoms in molecules. J. Chem. Theory Comput. 1:1096–1109CrossRefGoogle Scholar
  8. 8.
    Fradera X, Austen MA, Bader RFW (1999) The Lewis model and beyond. J. Phys. Chem. A 103:304–314CrossRefGoogle Scholar
  9. 9.
    Bader RFW, Nguyen-Dang T-T, Tal Y (1981) A topological theory of molecular structure. Rep. Prog. Phys. 44:893CrossRefGoogle Scholar
  10. 10.
    Bader RFW (1994) Atoms in Molecules: A Quantum Theory. Wiley, ChichesterGoogle Scholar
  11. 11.
    Bader RFW (2005) Monatshefte fur Chemie 136:819–854CrossRefGoogle Scholar
  12. 12.
    Hirshfeld FL (1977) Bonded-atom fragments for describing molecular charge densities. Theor. Chim. Acta 44:129–138CrossRefGoogle Scholar
  13. 13.
    Bultinck P, Alsenoy CV, Ayers PW, Carbó-Dorca R (2007) Critical analysis and extension of the hirshfeld atoms in molecules. J. Chem. Phys. 126:144111CrossRefGoogle Scholar
  14. 14.
    Manz TA, Sholl DS (2010) Chemically meaningful atomic charges that reproduce the electrostatic potential in periodic and nonperiodic materials. J. Chem. Theory Comput. 6:2455–2468 PMID: 26613499CrossRefGoogle Scholar
  15. 15.
    Verstraelen T, Ayers PW, Van Speybroeck V, Waroquier M (2013) Hirshfeld-E partitioning: AIM charges with an improved trade-off between robustness and accurate electrostatics. J. Chem. Theory Comput. 9:2221–2225 PMID: 26583716CrossRefGoogle Scholar
  16. 16.
    Manz TA, Sholl DS (2012) Improved atoms-in-molecule charge partitioning functional for simultaneously reproducing the electrostatic potential and chemical states in periodic and nonperiodic materials. J. Chem. Theory Comput. 8:2844–2867 PMID: 26592125CrossRefGoogle Scholar
  17. 17.
    Vanpoucke DEP, Bultinck P, Van Driessche I (2013) Extending Hirshfeld-I to bulk and periodic materials. J. Comput. Chem. 34:405–417CrossRefGoogle Scholar
  18. 18.
    Becke AD (1988) A multicenter numerical integration scheme for polyatomic molecules. J. Chem. Phys. 88:2547–2553CrossRefGoogle Scholar
  19. 19.
    Besaw JE, Warburton PL, Poirier RA (2015) Atoms and bonds in molecules: topology and properties. Theor. Chem. Acc. 134:1–15CrossRefGoogle Scholar
  20. 20.
    Warburton PL, Poirier RA, Nippard D (2011) Atoms and bonds in molecules from radial densities. J. Phys. Chem. A 115:852–867 PMID: 21189033CrossRefGoogle Scholar
  21. 21.
    Poirier RA, Awad IE, Hollett JW, Warburton PL MUNgauss, Memorial University, Chemistry Department, St. John’s, NL, A1B 3X7. With contributions from Alrawashdeh A, Becker JP, Besaw J, Bungay SD, Colonna F, El-Sherbiny A, Gosse T, Keefe D, Kelly A, Nippard D, Pye CC, Reid D, Saputantri K, Shaw M, Staveley M, Stueker O, Wang Y, and Xidos JGoogle Scholar
  22. 22.
    Gill PMW, Johnson BG, Pople JA (1993) A standard grid for density functional calculations. Chem. Phys. Lett. 209:506–512CrossRefGoogle Scholar
  23. 23.
    Wolfram Research, Inc., (2007) Mathematica 11.2.
  24. 24.
    El-Sherbiny A, Poirier RA (2009) Comprehensive study of some well-known molecular numerical integration methods. Can. J. Chem. 87:1313–1321CrossRefGoogle Scholar
  25. 25.
    Gill PMW, Chien S-H (2003) Radial quadrature for multiexponential integrands. J. Comput. Chem. 24:732–740CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of ChemistryMemorial UniversitySt. John’sCanada

Personalised recommendations