LONG RANGE COULOMB FORCES AND LOCALISED BONDS

C. Preiser¹, I. D. Brown², M. Kunz³, J. Lösel⁴, A. Skowron⁵ , M. Trömel¹  

1 Institut für Anorganische Chemie, Johann Wolfgang Goethe-Universität, Marie Curie-Str. 11,
    D-60439 Frankfurt am Main, Germany
    e-mail: cp@solid.anorg.chemie.uni-frankfurt.de
² Brockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario, Canada L8S 4M1
³ Labor fuer Kristallographie, ETH Zentrum, CH-8092 Zürich, Switzerland
⁴ SmithKline Beecham Pharmaceuticals, New Frontiers Science Park, Third Avenue, Harlow, Essex CM19 5AW, UK
⁵ Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4M1

Keywords: bond valence, electric flux, electric field, point charge, coordination, ionic bond, covalent bond, Madelung energy, Coulomb field, localised bond, long range force, electrostatic model.

The bond valence model has been proved to be a very successful model in describing and modelling inorganic solids [1]. The basic assumptions of this model are

• that many inorganic solids can be described by localised bonds between neighbouring atoms carrying charges of opposite sign,
• that each bond can be assigned a bond valence s in a way that the sum of the bond valences for all the bonds one atom forms equals its atomic valence V (i. e. its formal oxidation number).

         (CN = coordination number)

The bond valence s could be shown to be a function of the bond length R, dependent on two empirically determined parameters R₁, and b [2].

On the other hand it has been known from calculations of lattice energies (Madelung energies) that the contribution of atoms to the Coulomb energy which are much farther away than the nearest neighbours cannot be neglected, because the Coulomb energy, being proportional to the inverse of the interatomic distance R, drops to zero very slowly with distance.

In this investigation an electrostatic model was suggested where the atoms were represented by point charges. The electric field caused by all the charges in the crystal was calculated as the gradient of the electrostatic potential which was determined by the Ewald method [3]. The field could be represented by field lines connecting atoms which were carrying charges of opposite sign.

The following results were obtained:

• Although the Coulomb force is a long range force the lines of field terminated at atoms which were very close to the atom where the lines started.
• An unambiguous coordination number was derived (Atoms which were connected by lines were considered to be coordinated and bonded to each other in this model.) [4].
• Taking the collection of all lines joining two particular atoms a partitioning of space into bonding regions was obtained, since every point in space has to lie on a field line.
• The electric fluxes belonging to these bonding regions between two particular atoms could be shown to be approximately proportional to the bond valences (derived using equation (2)) in cases where an almost spherical charge distribution around the atoms could be assumed.
• In cases where the atoms were far away from being spherically symmetrical (for example lone pair compounds) a way could be shown how a physically meaningful introduction of multipoles would lead to a better agreement between fluxes and bond valences.
• In this model it is unnecessary to make the ambigious distinction between "more ionic" and "more covalent" bonds.

Conclusion: The electrostatic model which is using the properties of the Coulomb field of a point charge distribution provides a physical derivation of the empirical concept of localised bonds which allows a qualitative and quantitative description of many inorganic crystals.
 
The figure shows the calculated field of rutile (TiO₂) in the crystallographic (110)-plane {(x,y,z); x+y=1}.The light lines represent the field lines, the heavy lines show the boundaries between the bond regions.

[1] Brown, I.D.: Acta Cryst. B48 (1992) 553-572.
[2] Brown, I.D., Altermatt, D. Acta Cryst. B41 (1985) 244-247.
[3] Ewald, P.P.: Ann. Phys 64 (1921) 253-287.
[4] Preiser, C., Trömel, M., Brown, I. D., Kunz, M., Lösel, J.:Z. Kristallogr. Suppl. 12 (1997) 64.