Valences vector

Further, A is used to generate the valence vector, v [11]. This vector indicates the vertex degree for each atom in G ... 

The valence vector sum can also be treated as a field since it can be calculated for an atom placed at any point in the structure, not just the known site of an atom. Mathematically it represents the slope of the valence map, U r), at the position r ... 

Bickmore et al. [23] have adopted an alternative approach to the description of stereoactivity by showing that the valence vector sum (Eq. 16) can be used as a measure of the stereoactivity of the lone pair, noting the correlation between vector valence sum and the valence of the shortest primary M-O bond, which as shown above is equal to the bonding strength of the catirni. 

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. 

Model 3 is SrO rich at the surface, consisting of an SrOs termination with 1/3 of the oxygen sites vacant. The bond valence sums are somewhat close to the atomic valences, leading to an Sll of 0.21, indicating the possibility of problems. The surface Sr has coordination sphere with one half completely empty, and thus it fails the valence vector sum test and is less stable than indicated by the Sn. 

Model 5 surface consists of an SrOy termination with a half-filled Ti layer on top. All bond valence sums are within 0.30 of the atomic valence, leading to an SII of 0.17. However, it also fails the valence vector sum test. 

F. 5 Compariscm of surface Iree energy [24] and SII Sot SrTiOs (111) siuface models. Differences between SII and surface free energy can be accounted for whai considering the valence vector sums. Figure adapted with permission from [15]... 

For models 3 through 8 (Fig. 4), the SII fall in the order model 5 < model < model 8 < model 3 < model 6 < model 4. The first five of these are all close together (0.17 < SII < 0.22) while model 4 is significantly higher (Sll = 0.29). Except for model 8, all of these have at least one metal atom at the surface with an incomplete coordination sphere for models 3 and 4 an Sr atom, for model 7 a Ti atom, and for models 5 and 6 an Sr atom and half a Ti atom per (1x1) unit cell. These do not satisfy the valence vector sum mle (see Sect. 6.2 in Chap. 2 of this volume by Brown [16]) and are less stable than indicated by the SII (Fig. 5). Of the five models with similar SIIs, model 8 has the most reasonable bonding. This agrees with the DFT calculations [24]. 

The Sr faceted model has terminal Sr atoms with good bond valence sums [15], The bond valence sum, however, does not reveal the fuU instability of the terminal Sr atom. The majority of the coordination sphere for the apical Sr atom is empty. While the valence vector sum has not been calculated, it has been qualitatively noted that this leads to a less stable structure than indicated by the SII [22]. Bond valence theory therefore predicts that this and related structures are not stable, and they have not been observed experimentally. 

While a surface structure with (2 x 2) periodicity has been observed, Ciston and coworkers concluded that the observed diffraction intensities do not match with an octapolar structure [20]. Instead, the diffraction data indicates a (2 x 2)-a type structure. In this structure, the atoms in the terminal layer can occupy any or all of three different possible sites (Fig. 12) and the different occupations are virtually indistinguishable crystallographically. Assuming that Mg and O can only have oxidation states of 2 and no adsorbates are present, two of the three sites must be occupied in order to maintain valence neutrality. Like the octapolar structures, the (2 x 2)-a- can have either a magnesium bulk termination with the surface sites filled by oxygen ((2 x 2)-a-0), or an oxygen bulk terminatimi with the surface sites filled by magnesium ((2 x 2)-a-Mg). The (2 x 2)-a-Mg structure considered by Ciston has all three sites occupied by Mg atoms, would not be valence neutral, and does not appear to exist. A (2 x 2)-a-Mg structure with two sites occupied would be valence neutral, but the terminal Mg atoms would not satisfy the valence vector sum rule. 

One set of stmctures where such an analysis would be beneficial are the Sr adatom models proposed for several SrTi03 (n x n) reconstmctions [60-62]. These models match experimental STM images quite weU. However, having an Sr adatom means that the stmctore must somewhere be reduced by 2 electrons per Sr adatom. A bond valence analysis would quickly reveal the location and effects of this reduction. This could then be compared with experimental data, such as XPS, to see if the reduction in the model structure matches with some experimental observation, as was done by Becarra-Toledo and coworkers for the SrTiOs trilines [11]. Including other considerations from bond valence theory, such as the valence vector sum rule. 

Valence vector A vector with the magnitude of the bond valence directed from the... 

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