Abstract
The resonance bond number n, as defined in this paper, is designed to describe the strength of an XO bond as a function of the kinds of atoms present and which atoms are bonded. The calculation of n is made on a fragment extracted from the crystal encompassing the XO bond. If this fragment consists of only the X atom and its coordinating O atoms, then n is numerically equal to the Pauling bond strength, s. In this study a graph-theoretic algorithm is developed permitting the calculation of n using fragments including up to 50 atoms. This algorithm was used to calculate n for all of the bonds in ten silicate crystals. Since bond strength is be inversely related to bond length, we examined the relationship between these two variables and found that n can be used to explain over 70 percent of the variation of XO bond lengths from their average values in the crystals.
A fit of the parameter n/r, where r is the row number in the periodic table of the metal atom X, to the observed bond lengths in these crystals yielded the equation R(XO)=1.39(n/r)−0.22 which explains over 95.5 percent of the variation of bond lengths in the crystals. The fact that the same formula with s replacing n was found in an earlier study to be a good estimator of average bond lengths in crystals shows that n relates to individual variations in bond lengths in crystals in the same way that s relates to average bond lengths in crystals.
Using minimum energy SiO, AlO and MgO bond lengths and harmonic force constant data calculated for these bonds in hydroxyacid molecules, theoretical equations similar to those used by Pauling to explain bond length variations in hydrocarbons are derived. Bond lengths calculated with these equations for the 10 crystals shows that 95 percent of the variation of the observed bond lengths in these crystals can be explained in terms of n by this purely theoretical model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baur W (1970) Bond length variation and distorted coordination polyhedra in inorganic crystals. Trans Am Crystallogr Assoc 6:125–155
Bent HA (1968) Tangent-sphere models of molecules: VI. Ion-packing models of covalent compounds. J Chem Educ 45:768–778
Bondy JA, Murty USR (1976) Graph Theory with Applications. North-Holland, New York, p 264
Brown ID, Shannon RD (1973) Empirical bond-strength-bondlength curves for oxides. Acta Crystallogr A29:266–282
Cruickshank DWJ (1962) X-ray results on aromatic hydrocarbons. Tetrahedron 17:155–161
Geisinger KL, Gibbs GV, Navrotsky A (1985) A molecular orbital study of bond length and angle variations in framework structures. Phys Chem Minerals 11:266–283
Gibbs GV, D'Arco P, Boisen Jr. MB (1987a) Molecular mimicry of bond length and angle variations in germanate and thiogermanate crystals: A comparison with variations calculated for C-, Si-, Sn-containing oxide and sulfide molecules. J Phys Chem 91:5347–5354
Gibbs GV, Finger LW, Boisen Jr. MB (1987b) Molecular mimicry of the bond length-bond strength variations in oxide crystals. Phys Chem Minerals 14:327–331
Gibbs GV, Hamil MM, Louisnathan SJ, Bartell LS, Yow H (1972) Correlations between SiO bond length, SiOSi angle and bond overlap populations calculated using extended Huckel molecular orbital theory. Am Mineral 57:1578–1612
Gibbs GV, Meagher EP, Newton MD, Swanson DA (1981) A comparison of experimental and theoretical bond length and angle variations in minerals, inorganic solids and molecules. In: O'Keeffe M and Navrotsky A (eds) Structure and bonding in crystals, Vol. 1, Academic Press, New York, 195–225
Hehre WJ, Random L, Schleyer PR, Pople JA (1986) Ab initio molecular orbital theory. John Wiley and Sons Inc., New York, p 548
Lager GA, Gibbs GV (1973) Effect of variations in OPO and POP angles on PO bond overlap populations for some selected ortho-and pyrophosphates. Am Mineral 58:756–764
Pauling L (1929) The principles determining the structure of complex ionic crystals. J Am Chem Soc 51:1010–1026
Pauling L (1952) Interatomic distances and bond character in the oxygen acids and related substances. J Phys Chem 56:361–365
Pauling L (1960) The nature of the chemical bond. 3rd edition, Cornell University Press, Ithaca, New York, p 644
Pauling L, Brockway LO (1937) Carbon-carbon bond distances. The electron diffraction investigation of ethane, propane, isobutane, neopentane, cyclopropane, cyclopentane, cyclohexane, allene, ethylene, isobutene, tetramethylethylene, mesitylene, and hexamethylbenzene. J Am Chem Soc 59:1223–1236
Pauling L, Brockway LO, Beach JY (1935) The dependence of interatomic distance on single bond-double bond resonance. J Am Chem Soc 57:2705–2709
Shannon RD (1976) Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallogr A32:751–767
Shannon RD, Prewitt CT (1969) Effective ionic radii in oxides and fluorides. Acta Crystallogr B25:925–946
Zoltai T, Stout JH (1984) Mineralogy concepts and principles. Burgess Publishing Co., Minneapolis, Minnesota, p 505
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Boisen, M.B., Gibbs, G.V. & Zhang, Z.G. Resonance bond numbers: A graph-theoretic study of bond length variations in silicate crystals. Phys Chem Minerals 15, 409–415 (1988). https://doi.org/10.1007/BF00311046
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00311046