Repulsion, electron-pair unequal

A stable atomic coordination will also have electrons filling space in all directions. This phenomena is well known and has been considered at least since the introduction of valence shell electron pair repulsion theory [21]. In bond valence theory, this has been formalized as the valence vector sum mle In a stable coordination sphere the sum of the bond valence vectors around an ion is zero [22] (See Sect. 6.2 in Chap. 2 of this volume by Brown [16]). For metal cations at oxide surfaces, this requires bonds to O anions equally in all directions. As lone electron pairs can also fulfill this requirement for O anions, they can be stable with a nonzero valence vector sum, i.e. with bonds distributed unequally through space (See Sect. 7.1 in Chap. 2 of this volume by Brown [16]). Quantitative work on surfaces has so far been limited to bond valence sums. Since bond valence sums ignore these geometric concerns, metal terminated stmctures will be less stable than indicated by a bond valence sums analysis. This is seen in the differences between the Sll and the DFT calculated energies for cation terminated surfaces. Valence vector sums should be as useful for surface stmctures as for bulk stmctures. So far, however, the valence vector sum mle has been applied only qualitatively to surface stmctures, and so we will limit our discussion in this chapter to applying it qualitatively. 

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