L e t t e r

Revised values of the bond valence parameters for [6]Sb(V)––Oand [3–11]Sb(III)––O

Stuart J. Mills*, I, Andrew G. Christy II, Emily C.-C. Chen III and Mati Raudsepp I

I Department of Earth and Ocean Sciences, University of British Columbia, 6339 Stores Rd, Vancouver BC, Canada, V6T 1Z4II Research School of Earth Sciences, The Australian National University, Canberra, ACT 0200, AustraliaIII Environmental Sciences, Department of Earth and Ocean Sciences, University of British Columbia, 6339 Stores Rd, Vancouver BC, Canada,

V6T 1Z4

Received January 16, 2009; accepted June 2, 2009

Bond valence / Sb(V)––O / Sb(III)––O /Lone-pair electrons / Polyhedral distortion

Abstract. Bond valence parameters r0 and b have beenre-determined for [3–11]Sb(III)––O and [6]Sb(V)––O, utilis-ing crystal structures of natural and inorganic compoundsfrom the Inorganic Crystal Structure Database. Bond va-lence parameters for Sb(III) were obtained from a best-fitr0–b curve for 242 independent SbOn polyhedra. For[6]Sb(V), a curve of best fitting r0–b pairs was determinedby fitting to 207 independent SbO6 octahedra; b was thendetermined by optimising bond valence sums on the oxy-gens of Sb2O5 and Sb2O4, given the limited, low qualitystructural data available for Sb(V) coordination numbersother than 6. Parameter values that minimised r.m.s. devia-tion from the ideal bond valence sums were r0 ¼ 1.925 �Aand b ¼ 0.455 �A for Sb(III) and r0 ¼ 1.904 �A, b ¼0.430 �A for [6]Sb(V). The increase in r0 for Sb(III) mayrepresent the repulsive effect of the lone-pair electrons,while the difference in b indicates higher polarisabilitywhen these electrons are present. Consideration of subsetsof data for differing coordination numbers demonstratesthat Sb(III) parameters are applicable to all SbOn coordina-tion numbers (CN ¼ 3–11). We also show that the appar-ent overbonding using the classical b value cannot be anartefact of unresolved site splitting. For Sb(V), indepen-dent determination of b allows bond lengths cautiously tobe estimated for CN 6¼ 6. This work confirms that the“universal” value b ¼ 0.37 �A is not adequate for heaviercations such as Sb.

Introduction

During recent studies on the mineralogy and crystallogra-phy of Sb- and Te-bearing minerals from the Black Pinemine, 14.5 km NW of Philipsburg, Granite County, Monta-

na, USA (Mills et al., 2008; Mills et al., 2009), we notedinconsistencies between the calculated bond valence sums(BVS) for Sb(V) octahedra, which were not in accord withour chemical data. The BVS systematically overestimatedthe bond valence sum on Sb(V). In order to better fit ourdata, we conducted systematic survey of Sb(V)––O bonds inminerals and synthetic inorganic compounds found in theInorganic Crystal Structure Database (Fachinformations-zentrum Karlsruhe, 2007) in order to calculate new bondvalence parameters which better fit the measured bondlengths for [6]Sb(V). To complement this dataset, we alsosurveyed data for Sb(III)––O, also found in minerals andsynthetic inorganic compounds. The Sb(III) cation has astereoactive lone pair, and occurs in a wide range of coordi-nation numbers (CN), so one issue that we address iswhether a single set of parameters is actually applicable toSb(III) in both low-CN and high-CN environments.

Bond valence sums and previous history

The bond valence model is a powerful and straightforwarddescription of acid–base bonding which is used to inter-pret and predict the bond lengths found in crystalline so-lids. An in depth discussion of the history, method and itsapplications can be found in Brown (2002). The bond va-lence model can be used to determine bond valences, sij,which are calculated from the bond lengths, rij, using thecorrelation function:

sij ¼ exp ½ðr0 � rijÞ=b� ð1Þwhere r0 and b are the empirical parameters which arechosen such that the sums of the bond valences aroundthe ions are the same as their formal valences (Brown,2002). Parameters r0 and b can be determined from well-defined and constrained crystal structures, particularlythose found in the more recent literature. Unlike r0, b isdifficult to fit robustly, and a “universal constant” valueb ¼ 0.37 �A is usually adopted (Brown & Altermatt, 1985).

LetterZ. Kristallogr. 224 (2009) 423–431 / DOI 10.1524/zkri.2009.1166 423# by Oldenbourg Wissenschaftsverlag, Munchen

* Correspondence author (e-mail: smills@eos.ubc.ca)

Several authors have tabulated r0 for large sets of cation–anion pairs (e.g. Brown & Wu, 1976; Brown & Altermatt,1985; Brese & O’Keefe, 1991; Brown, 2002).

Recently, however, several authors have noted thatb ¼ 0.37 �A does not adequately predict interatomic dis-tances around some heavier cations, such as those withstereoactive lone pairs and the quadrupolar U(VI) of uranyl.For these, independent values of b need to be determined,and since r0 and b are correlated, the corresponding r0 va-lues are quite different than those in the b ¼ 0.37 �A data-sets. Full r0-b fits have been undertaken for Pb(II)(r0 ¼ 1.963 �A, b ¼ 0.49 �A: Krivovichev & Brown, 2001),U(VI) (r0 ¼ 2.051 �A, b ¼ 0.519 �A: Burns et al., 1996) andTl(I) (r0 ¼ 1.927 �A, b ¼ 0.5 �A: Locock & Burns, 2004),whilst Sidey (2006) also redetermined the parameters forthe Bi(III)/Br(-I) pair and more recently the Sn(II), Sb(III),Te(IV) and I(V)/O2� ion pairs (Sidey, 2009). Sidey (2008)recently described apparent overbonding in a number of dif-ferent ion pairs and concluded that this overbonding is anartefact of poorly determined bond valence parameters.

Palenik et al. (2005) also noted problems with using theSb values listed by Brown & Altermatt (1985). Based on103 datasets for Sb(III) and 20 for Sb(V), they calculatednew BVS parameters. However, they assumed that b re-mained constant at 0.37, and only refined r0. Their newparameters were r0 ¼ 1.955 �A for Sb(III) and r0 ¼ 1.912 �Afor Sb(V). Sidey et al. (2008) and Sidey (2009) also notedsuch problems with the Sb bond-valence parameters. Sideyet al. (2008) determined the value of r0 = 1.908 �A for Sb(V)based on the crystal structure of Sb2O5 (Jensen, 1979) andb ¼ 0.409 �A was determined by adjusting from 0.37 �A togive BVS ¼ 5.0. For Sb(III), Sidey (2009) obtainedr0 ¼ 1.924 �A and b ¼ 0.47 �A by least-squares fitting thepower law curve of Brown & Wu (1976) to Eq. (1) above.These parameters are compared with other values below.

Derivation of new parameters

Several different methods have been employed to calculatenew bond valence parameters. Krivovichev & Brown (2001)and Krivovichev & Filatov (2001) chose parameters whichoptimised the sums for both the encapsulated anion (O2�)and the surrounding cations (see Eqs. (3) and (4) in Krivovi-chev & Brown, 2001). This method ensures that bond va-lence sums are optimised over a larger portion of the struc-ture than just an isolated cation–oxygen polyhedron. Wangand Liebau (2007, 2008) have pointed out that a weaknessof optimising bond valence sums on the central cation onlyis that charge imbalance may simply be transferred to theoxygens or beyond. However, simultaneous optimisation ofbond valence sums on more than one type of atom per struc-ture requires a larger number of parameters to be fitted si-multaneously to much larger and more heterogeneous data-sets, and may be fitting structural quirks that are driven bynext-nearest neighbours while losing generality. Similarly,the distinction made by Wang and Liebau (2007, 2008) be-tween stoichiometric and structural “bond valence sums”loses predictive power because it requires considerableknowledge of specific structures before it can be used; thisapproach has also been questioned by Sidey (2008).

Like the Tl cations of Locock and Burns (2004), theSb cations of our study do not exclusively occur in oxy-gen-centered metal polyhedra, simultaneous optimisationof Sb and O atoms was not routinely performed. Never-theless, we expect any resulting distortions to be small.Suppose Sb(III) is bonded strongly to three oxygens,which each have three other neighbours. If the error in ourestimated bond sum on Sb(III) takes the value of 0.3valence units (v.u.), then 0.1 v.u. on average is transferredto each oxygen, a 5% error in the expected bond valencesum on oxygen. The Brown & Altermatt (1985) parameterb ¼ 0.37 �A implies that the expected error in the otherbonds to oxygen would be 0.018 �A. It will be seen belowthat errors in the bond valence sum on Sb(III) are usuallymuch less than 0.3 �A, and propagated errors on otherbond sums and bond lengths would be expected to be cor-respondingly smaller.

Initially, we considered SbOn polyhedra in isolation,and refined the bond valence parameters to minimise thesum of the squared differences between the formal oxida-tion state and the experimental BVS using the equation:

F ¼P

iV �

Pj

sij

!2

ð2Þ

where F is the function to be minimised, V is the formaloxidation state of the central atom, and sij is the calculatedbond strength for the j th bond to the ith cation in the data-set. The root-mean-square deviation of BVS from theideal value, f, is given by f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiðF=nÞ

p. This iterative ap-

proach is equivalent to that described in Brown (2002)and used by Locock & Burns (2004) for the redetermina-tion of the Tl(I) bond valence parameters.

A strict set of criteria was used to obtain data whichwould minimise the errors associated with calculating thenew bond valence parameters. These were:

1) Sb is bonded only to oxygens.2) Sb with the appropriate valence is the only species

in the given site.3) Sb and its oxygens are on sites that are fully or-

dered and occupied.4) Sb––O bonds are within reasonable limits. For

Sb(V), these were between 1.4 and 2.4 �A. ForSb(III), which usually shows 3–4 short distancesand several much longer distances as a result of thestereoactive lone pair, the maximum distance wasextended to 3.5 �A, as per Locock and Burns (2004).

5) Data collected at low or high temperature or highpressures were excluded, so as to ensure that thefitted parameters were applicable to structures atambient conditions. This contrasts with the ap-proach of Palenik et al. (2005), who suggested thatlow-temperature data would provide more accuratebond valence data given the reduced effect of ther-mal perturbations. However, since our aim is to fitbond lengths measured near room temperature andpressure, we preference ambient database data.

6) Structures were only included in the final fits if theypassed a quality check. Initially, a criterion of maxi-mum R% value was used, even though it was foundthat this excluded some structures where preliminary

424 S. J. Mills, A. G. Christy, E. C.-C. Chen et al.Letter

bond valence calculations suggested that the SbOn

environment had been resolved well (e.g. manganos-tibite with R ¼ 11%; Moore, 1970), while includingothers where the BVS on Sb was unrealistic (e.g.Pb2Sb2O7 of Ivanov & Zavodnik, 1990, which gaveBVS ¼ 3.98 in trial fits even though R ¼ 3.3%).

7) It was decided to double-check quality of the SbOn

data using a two-pass BVS fit. After a preliminarybest fit was obtained for the full dataset, structureswere excluded if they gave BVS more than 0.5 dif-ferent from the ideal value, and the remaining datawere then fitted again.

Similar criteria to 1–4 and 6 were employed by Tyktoet al. (1999) and Locock & Burns (2004). We have addition-ally screened out non-ambient data or rejected additionalstructures after finding them to be outliers in preliminaryBVS fits. After this selection process, we were left with datafor 207 independent Sb(V)O6 polyhedra. Only five polyhe-dra were found in which Sb(V) adopted other coordinationnumbers: [4]Sb in Cs3SbO4 (Hirschle & Roehr, 2000), [5]Sbin Sb5O12(OH) � H2O (Jansen, 1978), Pb2Sb2O7 (Iva-nov & Zavodnik, 1990), Cs3Sb5O14 (Hirschle & Roehr,2001), and [8]Sb in Pt2Sb2O7 (Sleight, 1974). Unfortunately,all of these had to be rejected on the grounds of low reliabil-ity, despite their potential utility for fitting b. For Sb(III),data were retained for 242 independent polyhedra, spanninga wide range of coordination numbers.

Sb(III)––O parameters

The irregularity of Sb(III)On polyhedra gave us a widerange on bond strengths to work with, which was a greataid in refining both r0 and b, making it more straightfor-ward than for Sb(V)On. Because of repulsion by thestereoactive lone pair on Sb(III), coordination polyhedratypically have 3–4 short strong bonds, all on the side ofthe Sb atom opposite to the lone pair. Occasionally, thereare no significantly bonded oxygens on the same side asthe lone pair. Nearest neighbours on this side are then“non-bonded” species interacting via Van der Waalsforces, lone pair-lone pair dipole interactions, and so on.More usually, there are several bonded oxygens at dis-tances corresponding to small bond strengths that never-theless make up a significant fraction of the total BVS.Hyde & Andersson (1989) discussed both of these scenarios.They noted that a stereoactive lone pair can be regarded asa quasi-anion of similar non-bonded size to an oxygen butonly �1 �A from its parent cation, and that structures withstereoactive lone-pairs can often be derived from thosewithout such lone pairs if either:

i) The lone pair and a small number of oxygens forma relatively regular coordination polyhedron with alledges roughly similar in length, with the cation nearthe centre but displaced towards the lone pair, or

ii) A larger number of oxygens form a regular coordi-nation polyhedron with the lone pair at the centroidand the cation displaced towards a face of the sur-rounding O polyhedron.

Some but not all examples of case (i) can also be de-scribed using the case (ii) model. All examples of case (ii)

will have relatively long Sb––O bonds on the same side ofthe Sb as the lone pair.

Wang & Liebau (2007) discussed “overbonding” inasymmetrical coordination polyhedra, as measured usingthe bond valence parameters of Brown & Altermatt (1985),Brese & O’Keeffe (1991), and attempted to fit a correctionterm dependent on the degree of one-sidedness of the co-ordination polyhedron. This approach was critiqued bySidey (2008), who indicated that such a complication wasunnecessary if alternative valence parameters were used,either of the power law type (Brown & Wu, 1976) orlogarithmic r0–b form but with b no longer constrained tothe “universal” value of 0.37 �A.

In the current study, the cut-off distance for the longSb––O bonds was set at 3.5 �A, which was found necessaryto achieve an acceptable BVS for structures such as thatof chapmanite, Fe2Sb[SiO4]2(OH), where Sb is bonded tothree O at about 2 �A but another six at 3.4–3.45 �A(Zhukhlistov & Zvyagin, 1977). Coordination numbers forSb(III) ranged from 3 to 11, unevenly distributed withthree modes at CN ¼ 4, 7 and 9 and mean CN ¼ 6.07(Fig. 1). The diversity of Sb coordination environments isdemonstrated by Fig. 2, which shows the Sb––O bondlengths for all Sb atoms in this study with CN ¼ 9, ar-ranged in order of increasing fourth-shortest bond length.Note that “CN ¼ 9” includes many enviroments which canbe described as 3 þ 6 coordinated, some which are clearly4 þ 5, and a few which are neither.

For any chosen value of b, it was always possible tofind a value of r0 that minimized F and f for the dataset.These r0 values were estimated to the nearest 0.0005 �A.The optimal r0 correlated smoothly and negatively with b(Fig. 3a). The minimal f for each b, on the other hand,attained a well-defined minimum at b ¼ 0.455 (Fig. 3b).From this, we deduce r0 ¼ 1.925 �A and b ¼ 0.455 �A asthe best values applicable to the full Sb(III) dataset. Themean BVS for all Sb atoms was m ¼ 2.993 valence units(v.u.), and the standard deviation s ¼ 0.133 v.u.. This

Revised values of the bond valence parameters for [6]Sb(V)––O and [3–11]Sb(III)––O 425 Letter

Fig. 1. Histogram showing numbers of Sb(III) polyhedra of each co-ordination number considered in this study.

corresponds to an estimated error in length of bs/m ¼ 0.020 �A per Sb––O bond.

The Sb––O bonds in this dataset range in length from1.80 to 3.50 �A, and in strength from 1.30 to 0.033 v.u..Given that s varies by a factor of 40, it is reasonable toask whether the longer bonds are in fact of the same in-trinsic type as the shorter bonds, and whether it is appro-priate to model them using the same r0 and b values. Ifthis is not the case, then there will be a CN dependence ofthe best-fitting values, which would lessen the generalityand predictive power of the bond parameters. Neverthe-less, such a variation has been reported for U(VI) byBurns et al. (1997), where change in coordination of theU from 2 þ 4 to 2 þ 6 reduced r0 from 2.074 �A to2.042 �A while b increased from 0.510 to 0.519. It is pos-sible that this is because of the change in proportions ofthe very short, p-bonded uranyl bonds and the longer,weaker equatorial bonds, and may indicate that differentparameters should be used for these two bond types.

The CN sensitivity of our Sb(III) fits was tested bysplitting the data into groups based on coordination num-ber, and plotting b–r0 trajectories for each group. The re-sults are plotted in Fig. 4, which shows that the b–r0 lineshave very different gradients for different CN. In particu-lar, notice that the fitted r0 is almost constant for CN ¼ 3,given the very narrow range of bond lengths for that coor-

dination number, while r0 tends to decrease faster withincreasing b for larger CN. Nevertheless, the curves allintersect in the range b ¼ 0.42–0.46, and are tightlybunched at the global optimum b value. Therefore, thebest fit derived above is not sensitive to coordination num-ber over the whole range CN ¼ 3–11. The corresponding

426 S. J. Mills, A. G. Christy, E. C.-C. Chen et al.Letter

Fig. 2. Sb(III)––O bond lengths for CN ¼ 9, sorted in order of fourth-nearest oxygen distance. Note clear examples of 4 þ 5 coordination(left), 3 þ 6 coordination (right) and intermediate cases that areneither.

a�

b�

Fig. 3. (a) Best fitting r0 as a function of b for Sb(III) data. (b) Cor-responding values of f. Note distinct minimum.

f values are shown in Fig. 5. Since increase in b causesthe BVS to be less sensitive to variations in bond length,there is an innate tendency for these curves to trend down-wards with increasing b, with the result that no minimumis observed at all for small coordination numbers withsmall degrees of bond length variation. Very shallow mini-ma are observed for CN � 6, but these are near b ¼ 0.45for CN ¼ 8 and near b ¼ 0.55 for all the rest of thecurves, which suggests that they either do not shift sys-tematically with CN or are not accurately located. The fvalue is never smaller than 60% of that for the global va-lues, so use of CN-dependent parameters is not justifiedby a significant improvement in goodness-of-fit.

The distribution of bond valence sums obtained is com-pared with those using the bond valence parameters ofBrown & Altermatt (1985), Palenik et al. (2005) andSidey (2009) in Fig. 6 and Table 1. Note that the para-meters of Brown & Altermatt give a BVS distributionwith a mean that is slightly high (3.14) and a standarddeviation that is nearly 40% larger than that obtainedabove. The histogram of Fig. 6 shows that the BVS arealso significantly skewed towards high values for theseparameters. The parameters of Palenik et al. (2005) shiftthe mean BVS to be closely in accord with the ideal va-

lue, but because b is too low, the standard deviation is stilllarge, and some skew is still present. The new parametersobtained in this study give a distribution that is narrowerand more symmetrical than either of the earlier ones, withmore than 80% of BVS values in the range 2.85–3.15.They also perform better than the very similar parametersof Sidey (2009), which have a narrower spread due totheir larger b, but also have a mean that is slightly toohigh (Table 1). The close similarity between our para-meters and those of Sidey (2009) is reassuring that bothsets are robust, despite being derived differently. Bothmethods have their own strengths and weaknesses.

Sb(V)––O parameters

Obtaining bond valence parameters for Sb(V) to oxygen iscomplicated by the fact that the vast majority of coordina-tion polyhedra approximate regular SbO6 octahedra. Veryfew examples of CN 6¼ 6 have been refined. We found onlyfive examples, but the low quality made these unsuitable forfitting. In contrast, 207 examples of [6]Sb were used.

The [6]Sb(V) data allowed for calculation of the bestr0 for each of several values of b, as before. The trajec-

Revised values of the bond valence parameters for [6]Sb(V)––O and [3–11]Sb(III)––O 427 Letter

Fig. 4. Trajectories of r0 versus b for Sb(III) data subsets of differingCN. All curves intersect in the range b ¼ 0.42–0.46. Full dataset isin grey.

Fig. 5. Values of f corresponding to data of Fig. 4. Notice that thecurves for restricted ranges of bond length do not show distinct mini-ma. Full dataset is in grey.

Table 1. Statistics for various Sb(III) bond valence parameters (n ¼ 242).

r0/b BVS Reference

m s % 3 � 0.05 v.u. % 3 � 0.15 v.u. max min range

1.973/0.37 3.141 0.182 10.3 47.1 3.898 2.616 1.282 Brown & Altermatt (1985)

1.955/0.37 2.992 0.173 33.9 70.2 3.713 2.492 1.221 Palenik et al. (2005)

1.924/0.47 3.070 0.111 31.4 74.8 3.492 2.548 0.944 Sidey (2009)

1.925/0.455 2.993 0.133 34.3 80.2 3.489 2.520 0.969 this study

tory for [6]Sb(V) is compared with that for Sb(III) inFig. 7, where it can be seen that for [6]Sb(V), the varia-tion with b is slower and more linear. The two r0 trendsintersect near b ¼ 0.55 �A, but for smaller b, [6]Sb(V) hasthe smaller r0.

Unfortunately, because of the very narrow spread ofbond lengths in the [6]Sb(V) dataset, F and f did not exhi-bit a minimum but, instead, decreased monotonically withincreasing b. Therefore, b had to be estimated by anothermethod. One possibility was that employed by Sidey et al.(2008), who determined b by optimising bond valencesums on the oxygens of Sb2O5. A good quality refinementis available for the structure of this compound (Jansen,1978), which contains three symmetrically distinct oxygensites with a total of six distinct Sb(V)––O bond lengthswith approximate bond strengths of either 2=3 or 1. Giventhe small size of this dataset, we decided to optimise bondvalence sums on oxygen for this compound and also thetwo polymorphs of Sb2O4 (Orosel et al., 2005), whichhave another nine symmetrically distinct Sb(V)––O dis-tances to six distinct oxygen sites. The previously ob-tained parameters for Sb(III) were used for bonds to thatcation. Parameter pairs, {r0, b}, were taken from pointsalong the optimal trend for [6]Sb(V), and the root-mean-squared deviation, fO, for the oxygen bond valence sumswas calculated. A well-defined minimum (0.093 v.u.) in fOwas obtained at r0 = 1.904 �A and b ¼ 0.430 �A for Sb(V).Root-mean-squared deviations from ideality for the BVSon Sb(V) were 0.007 v.u. in Sb2O5, 0.074 v.u. in orthor-hombic Sb2O4 and 0.008 v.u. in monoclinic Sb2O4. Addi-tionally, root-mean-squared deviations for those on Sb(III)were 0.020 v.u., in orthorhombic Sb2O4 and 0.031 v.u. inmonoclinic Sb2O4. The small size of these deviations indi-cate that Sb(III) and Sb(V) are well ordered in the Sb2O4

polymorphs. The range of calculated Sb(V)––O bondstrengths was 0.62–1.02 v.u.. The mean and standard de-viation for BVS on Sb(V) were m ¼ 4.994 v.u. ands ¼ 0.195 v.u., respectively. The corresponding uncertaintyin bond length, bs/m was 0.018 �A per Sb––O bond. Note

428 S. J. Mills, A. G. Christy, E. C.-C. Chen et al.Letter

Fig. 6. Histograms of bond valencesums for Sb(III) data, using bond va-lence parameters of this study, Brown& Altermatt (1985), Palenik et al.(2005) and Sidey (2009).

Fig. 7. Trajectories of r0 versus b for [6]Sb(V) data, compared withcurve for Sb(III).

that r0 for Sb(III) is larger than for Sb(V) by 0.021 �A,which can be interpreted as a consequence of repulsion bythe additional non-bonding electrons in Sb(III), while thelarger b may represent increased polarisability when thelone pair is present.

As for Sb(III), these new BVS parameters are comparedwith those of Brown & Altermatt (1985), Palenik et al.(2005) and Sidey et al. (2008) in Fig. 8 and Table 2. TheBrown & Altermatt (1985) parameters applied to our struc-tural dataset give an average valence of Sb(V) ¼ 5.374 v.u.and s ¼ 0.224 v.u. while the range of valence sums is4.24–6.20 v.u.. Only 15.5% of the data fall within �0.15v.u. of the ideal BVS. Again, the data of Palenik et al.(2005) shift the mean to a more acceptable value, but thespread and skew of the data are still large. The Sidey et al.(2008) parameters were applied to our structural datasetand gave an average valence of Sb(V) ¼ 4.997 v.u. ands ¼ 0.205 v.u., which is also an improvement on the ear-lier parameters. The overall distribution for the parametersof Sidey et al. (2008) is similar to that for our values, butour standard deviation is slightly lower and thus more pre-cise. Our proportions of Sb(V)––O bonds both with BVS

within �0.05 v.u. and within � 0.15 v.u. of the ideal valueare higher (Table 2). This may be due to an underestima-tion of b by Sidey et al. (2008) using data from onlySb2O5. Nevertheless, the similarity in values and perfor-mance between their values and ours is reassuring thatboth are close to optimum. Note that optimisation of onlySb2O5 oxygens gave r0 ¼ 1.906 �A and b ¼ 0.419 with ourdata. Trying to determine b using the low-reliability bondlength data for CN 6¼ 6 suggested that b should be identi-cal to that for Sb(III)––O (0.455), and r0 slightly smaller at1.900 �A. Both of these alternative sets of parameters alsoperform slightly better than the Sidey et al. (2008) fit. Ifour preferred parameters are used to fit bond valence sumsfor Sb in the five cited structures with CN ¼ 4, 5 or 8, thenthe BVS obtained is 4.88 � 0.61 v.u., with a minimum andmaximum BVS of 4.23 and 5.64 v.u., respectively. Thestandard deviation corresponds to an estimated error inbond length of 0.054 �A. Since this is small, despite being aresultant of errors in the structures as well as uncertainty inour parameters, it seems that our parameters can reason-ably be applied to Sb(V)––O bond strengths very differentfrom those typical for CN ¼ 6.

Revised values of the bond valence parameters for [6]Sb(V)––O and [3–11]Sb(III)––O 429 Letter

Fig. 8. Histograms of bond valencesums for Sb(V) data, using bond va-lence parameters of Brown & Altermatt(1985), Palenik et al. (2005), Sideyet al. (2008) and this study.

Table 2. Statistics for various Sb(V) bond valence parameters (n ¼ 207 [6]Sb).

r0/b BVS Reference

m s % 5 � 0.05 v.u. % 5 � 0.15 v.u. max min range

1.942/0.37 5.376 0.244 6.3 15.5 6.203 4.636 1.567 Brown & Altermatt (1985)

1.912/0.37 4.957 0.225 19.3 50.7 5.719 4.275 1.444 Palenik et al. (2005)

1.908/0.409 4.997 0.205 17.9 53.1 5.684 4.373 1.311 Sidey et al. (2008)

1.904/0.430 4.994 0.195 18.8 57.0 5.644 4.400 1.244 this study

Polyhedral distortion and site splitting

The concave-upward form of the bond length–bond strengthcurve necessarily implies that any distortion away from uni-form bond lengths will result in an overall increase in size ofthe coordination polyhedron. This is the well-known “dis-tortion theorem” (cf. Brown, 2002). A quantitative bondlength–bond strength relationship allows quantification ofthe distortion–polyhedral size relationship. Consider a regu-lar octahedron of oxygens, located at �(R, 0, 0), �(0, R, 0),�(0, 0, R). If an Sb(III) cation were located at the centre ofthis O6 octahedron (Fig. 9a), our bonding parameters implythat the octahedron radius R must be 2.2405 �A, equivalent tothe Sb––O bond length for bond strength 0.5. However, sup-pose that the Sb cation is in fact displaced along the [1, 1, 1]direction by a distance d, so as to form 3 short and 3 longbonds to oxygen, but that the O6 octahedron itself remainsregular in shape (Fig. 9b). In order to maintain BVS ¼ 3 onthe cation, the octahedral radius R must expand. One canreadily calculate that the expansion is strongly nonlinear, butthat if d ¼ 1.0 �A, the Sb––O distances are 3 1.9648 �A and3 3.0530 �A, R ¼ 2.3645 �A, and hence the linear dimen-sions of the O6 polyhedron have increased by 5.53% and itsvolume by 16.9%. The pattern of three short bonds trans to

three long bonds is found frequently in Sb––S and Pb––Sbond lengths of sulphosalts, and results in trans pairs ofbond lengths lying along hyperbolic curves that are specificto each cation (Berlepsch et al., 2001).

A possible complication in fitting r0 for strongly dis-torted polyhedra is that the apparent position of the centralatom in a refinement may actually be an average of two ormore split sites. The resulting averaged bond length liealong chords to the bond-pair hyperbola for a cation (Ma-kovicky et al., 2006), and gives anomalously low bondvalence sums. This is known to be the case for some AsSn

polyhedra in the sulphosalt sartorite (Berlepsch et al.2003) and SbSn polyhedra in minerals such as dadsonite(Makovicky et al., 2006).

The volume increase of an Sb––O octahedron due topolyhedral distortion and the deviation from ideal bondlength–bond valence behaviour due to averaging of splitsites can both be illustrated and quantified with a simpleexample. Suppose the Sb atom of Fig. 9b is split over twosites [z, z, z] and [z, z, �z], so as to give an average posi-tion of [z, z, 0] (Fig. 9c). If the site splitting is not distin-guished (Fig. 9d), then this Sb is apparently (2 þ 2 þ 2)coordinated rather than (3 þ 3), and if the BVS ¼ 3 at the(3 þ 3) positions, then the BVS at the averaged positionwill be less, since it is closer to the centroid of the octahe-dron. For an Sb atom that is displaced from the octahedralcentroid towards a face centre by

ffiffiffiffi3p

z, Figure 10 showsboth the percentage increase in linear dimensions of theoctahedron, assuming that it stays regular in shape, andalso the percentage underbonding if the Sb position is mis-identified as the [z, z, 0] average of two split sites. Notethat the apparent underbonding of Sb reaches a maximumof about 3.3% for

ffiffiffiffi3p

z ¼ 0.7 �A, and then decreases as thenearest two oxygens become very close. If these split sitesare averaged rather than resolved, r0 will be overestimatedby the corresponding distance increment, which takes amaximum value of bln (1.033) ¼ 0.015 �A for this particu-

430 S. J. Mills, A. G. Christy, E. C.-C. Chen et al.Letter

a� b�

c� d�Fig. 9. (a) Sb surrounded symmetrically by an octahedron of six oxy-gens. (b) Displacement of Sb towards a face centre, giving (3 þ 3)coordination. (c) Site splitting between two (3 þ 3) coordinated posi-tions. (d) The apparent (2 þ 2 þ 2) coordinated position if the splitsites are unresolved.

Fig. 10. Plot against Sb displacement from centroid of percentage (i)increase in linear dimensions of O6 octahedron, (ii) apparent under-bonding of Sb at the unresolved split site of Fig. 9(d).

lar scenario. Hence, any error in r0 due to this effect islikely to be very small. Such site splitting can arise whenthe cation is somewhat too small for its polyhedron, whichis far more likely for relatively small As surrounded bylarge S than for larger Sb surrounded by smaller O. How-ever, site splitting can also be driven by coupling of thelone pair orientation to the occupancies and geometry ofneighbouring sites (Makovicky, pers. comm. 2009). Ineither case, unresolved split sites will not cause apparentoverbonding such as that which has prompted reinvestiga-tion of Sb––O bonding parameters, unless the cation is sofar displaced from the polyhedron centroid that the aver-age position is nearly 2-coordinate.

Conclusions

Our parameters and performance statistics confirm stronglythat b ¼ 0.37 �A is too small to provide a good fit for Sb––Obonds in either oxidation state. Apparent overbonding ofcations if this b value is used is unlikely to arise from aver-aging of split sites, which usually lead to apparent under-bonding and overestimation of r0. For cations such asSb(III) with stereoactive lone-pairs, the resulting asym-metric coordination environment allows determination ofboth r0 and b, provided that the fit includes the long, weakbonds that lie on the same side of the cation as the lone pair.The resulting parameters have been verified to be indepen-dent of coordination number, which enhances their predic-tive power. Cations such as Sb(V), which are almost exclu-sively restricted to one coordination number, allow a curveto be determined of r0 against b, but the appropriate locationon this curve can only be obtained by other means. Optimis-ing bond valence sums on the oxygens in a subset of chemi-cally simple but geometrically diverse structures appears tobe a successful strategy.

The new BVS parameters derived here were testedon the crystal structure of joelbruggerite, ideallyPb3Zn3(Sb5þ,Te6þ)As2O13(OH,O), where we reported‘overbonding’ of Sb(V) by �0.3 v.u. (Mills et al. 2009).Application of the new parameters eliminated the problemof ‘overbonding’ and lead to a lowered formal valence thatis in accord with the chemical data (Mills et al. 2009).

Acknowledgments. The Associate Editor, Emil Makovicky, and tworeferees provided helpful comments on the manuscript. NSERC Cana-da is thanked for a Discovery Grant to Mati Raudsepp.

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Revised values of the bond valence parameters for [6]Sb(V)––O and [3–11]Sb(III)––O 431 Letter