Correlation effects on the interelectronic distributions of localized electron pairs

Regular Article


It has been shown previously that a wealth of chemical insight may be drawn from studying the interelectronic distribution function for localized electron pairs within a molecular system (Hennessey et al. in Phys Chem Chem Phys 16(46):25548–25556, 2014); however, these accounts have been limited due to the relative inaccuracy of the underlying Hartree–Fock electronic structure model that has been employed. In the current work, we study how the interelectronic distribution function of single electron pairs, represented by localized molecular orbitals, changes between a Hartree–Fock and density functional electronic structure model. We find that localized electron pairs expand, relative to a Hartree–Fock model, in a small but significant way when modelled with density functional theory. More specifically, we generally find that compact electron pairs (i.e. those pairs having narrow distribution functions near small interelectronic distances) result in a smaller change due to correlation than electron pairs more broadly distributed. This counterintuitive effect is attributed to the fact that compact electron pairs are generally held close together by a relatively rigid confining potential (i.e. attraction from local nuclei and repulsion from neighbouring electrons). However, in cases where a series of species all have nearly identical nuclear structure in the vicinity of the LMO of interest, more compact electron pairs did in fact experience the greatest change in interelectronic separation and electron repulsion upon the incorporation of a correlated model. Also, if one allows the geometry of the molecular species to relax, bonds associated with LMOs having compact interelectronic distributions tend to elongate the most with a correlated model (relative to Hartree–Fock) and the resultant deformation of the interelectronic distribution functions then show the most significant changes due to correlation. The results presented herein demonstrate the utility of Kohn–Sham density functional theory for the description of localized chemical space within the scope of the localized pair model.


Density functional theory Electron correlation effects Interelectronic distribution functions Intracules Electron pairs Localized pair model Localized orbitals Chemical bonding 


  1. 1.
    Zielinski ZAM, Pearson JK (2013) The localized pair model of electronic structure analysis. Comput Theor Chem 1003:79–90CrossRefGoogle Scholar
  2. 2.
    Hennessey DC, Sheppard BJH, Mackenzie DECK, Pearson JK (2014) Predicting bond strength from a single Hartree–Fock ground state using the localized pair model. Phys Chem Chem Phys 16(46):25548–25556CrossRefGoogle Scholar
  3. 3.
    Bader RFW (1994) Atoms in molecules: a quantum theory. Oxford University Press, OxfordGoogle Scholar
  4. 4.
    Matta CF, Boyd RJ (2007) The quantum theory of atoms in molecules: from solid state to DNA and drug design. Wiley, HobokenCrossRefGoogle Scholar
  5. 5.
    Coulson CA, Neilson AH (1961) Electron correlation in the ground state of helium. Proc Phys Soc 78(5):831–837CrossRefGoogle Scholar
  6. 6.
    Boyd RJ, Coulson CA (1973) Coulomb hole in some excited states of helium. J Phys B At Mol Phys 6(5):782–793CrossRefGoogle Scholar
  7. 7.
    Ugalde JM, Boyd RJ (1985) Angular aspects of exchange correlation and the fermi hole. Int J Quantum Chem 27(4):439–449CrossRefGoogle Scholar
  8. 8.
    Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136(3B):B864–B871CrossRefGoogle Scholar
  9. 9.
    Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140(4A):A1133–A1138CrossRefGoogle Scholar
  10. 10.
    Stowasser R, Hoffmann R (1999) What do the Kohn–Sham orbitals and eigenvalues mean? J Am Chem Soc 121:3414–3420CrossRefGoogle Scholar
  11. 11.
    Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, Koseki S, Matsunaga N, Nguyen KA, Su S, Windus TL, Dupuis M, Montgomery JA (1993) General atomic and molecular electronic structure system. J Comput Chem 14(11):1347–1363CrossRefGoogle Scholar
  12. 12.
    Edmiston C, Ruedenberg K (1963) Localized atomic and molecular orbitals. Rev Mod Phys 35(3):457–464CrossRefGoogle Scholar
  13. 13.
    Bode BM, Gordon MS (1998) MacMolPlt: a graphical user interface for GAMESS. J Mol Graph Model 16(3):133–138CrossRefGoogle Scholar
  14. 14.
    Wolfram Research, Inc., Mathematica, Version 7.0, Champaign, IL (2016)Google Scholar
  15. 15.
    Proud AJ, Walker MP, Pearson JK (2013) The analysis of polarization effects on the interelectronic separations in the atoms and molecules of the G1 test set. Int J Quantum Chem 113(1):76–82CrossRefGoogle Scholar
  16. 16.
    Hollett JW, Gill PMW (2011) Intracule functional models. V. Recurrence relations for two-electron integrals in position and momentum space. Phys Chem Chem Phys 13(7):2972–2978CrossRefGoogle Scholar
  17. 17.
    Schlegel HB (1982) An efficient algorithm for calculating abinitio energy gradients using s, p Cartesian Gaussians. J Chem Phys 77(7):3676–3681CrossRefGoogle Scholar
  18. 18.
    Obara S, Saika A (1986) Efficient recursive computation of molecular integrals over Cartesian Gaussian functions. J Chem Phys 84(7):3963–3974CrossRefGoogle Scholar
  19. 19.
    Obara S, Saika A (1988) General recurrence formulas for molecular integrals over Cartesian Gaussian functions. J Chem Phys 89:1540–1559CrossRefGoogle Scholar
  20. 20.
    Ahlrichs R (2006) A simple algebraic derivation of the Obara–Saika scheme for general two-electron interaction potentials. Phys Chem Chem Phys 8(26):3072–3077CrossRefGoogle Scholar
  21. 21.
    Pearson JK, Crittenden DL, Gill PMW (2009) Intracule functional models. IV. Basis set effects. J Chem Phys 130(16):164,110CrossRefGoogle Scholar
  22. 22.
    Perdew JP, Schmidt K (2001) Jacob’s ladder of density functional approximations for the exchange-correlation energy. AIP Conf Proc 577(1):1–20CrossRefGoogle Scholar
  23. 23.
    Becke AD (1988) Density-functional exchange-energy approximation with correct asymptotic behavior. Phys Rev A 38(6):3098–3100CrossRefGoogle Scholar
  24. 24.
    Lee C, Yang W, Parr RG (1988) Development of the Colle–Salvetti correlation-energy formula into a functional of the electron density. Phys Rev B Condens Matter 37(2):785–789CrossRefGoogle Scholar
  25. 25.
    Mienlich B, Savin A, Stoll H, Preuss H (1989) Results obtained with the correlation energy density functionals of becke and Lee, Yang and Parr. Chem Phys Lett 157:200–206CrossRefGoogle Scholar
  26. 26.
    Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77(18):3865–3868CrossRefGoogle Scholar
  27. 27.
    Perdew JP, Burke K, Ernzerhof M (1997) Generalized gradient approximation made simple [Phys. Rev. Lett. 77, 3865 (1996)]. Phys Rev Lett 78(7):1396–1396CrossRefGoogle Scholar
  28. 28.
    Zhao Y, Truhlar DG (2006) A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions. J Chem Phys 125(19):194101CrossRefGoogle Scholar
  29. 29.
    Tao J, Perdew JP, Staroverov VN, Scuseria GE (2003) Climbing the density functional ladder: nonempirical meta-generalized gradient approximation designed for molecules and solids. Phys Rev Lett 91(14):146401CrossRefGoogle Scholar
  30. 30.
    Becke AD (1993) Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98(7):5648CrossRefGoogle Scholar
  31. 31.
    Stephens PJ, Devlin FJ, Chabalowski CF, Frisch MJ (1994) Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields. J Phys Chem 98(45):11623–11627CrossRefGoogle Scholar
  32. 32.
    Zhao Y, Truhlar DG (2008) The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals. Theor Chem Acc 120(1–3):215–241CrossRefGoogle Scholar
  33. 33.
    Pearson JK, Gill PMW, Ugalde JM, Boyd RJ (2010) Can correlation bring electrons closer together? Mol Phys 107(8–12):1089–1093Google Scholar
  34. 34.
    Per MC, Russo SP, Snook IK (2009) Anisotropic intracule densities and electron correlation in H2: a quantum Monte Carlo study. J Chem Phys 130(13):134103CrossRefGoogle Scholar
  35. 35.
    Hollett JW, McKemmish LK, Gill PMW (2011) The nature of electron correlation in a dissociating bond. J Chem Phys 134(22):224103CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of ChemistryUniversity of Prince Edward IslandCharlottetownCanada

Personalised recommendations