Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://cryst.geol.msu.ru/courses/aspmk/article02.pdf Äàòà èçìåíåíèÿ: Tue Feb 16 16:13:28 2016 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:24:33 2016 Êîäèðîâêà:

M. HJORTH, R. NORRESTAM AND H. JOHANSEN than in the experimental investigation, and they are located much closer to the central atom (0.35 versus 0.82 ,~). This is again due to the thermal smearing of the experimental densities, and similar differences have been seen in many other studies. Qualitatively the two investigations do, however, compare reasonably well as far as bonds, lone pairs and the nonbonding d orbital are concerned. The dense positive peak found close to the central atom in the experiment is not matched by the theoretical results; it is probably caused by experimental errors and series-termination effects. There is a peak in the direction of the axial water ligand, but it is questionable whether this can account for the experimental peak. The negative region on the vanadyl V--O bond close to vanadium is, on the other hand, well reproduced.
References

7

ABRAHAMS,S. C. (1974). Acta Cryst. B30, 261-268. BREITENSTEIN, M., DANNOHL, H., MEYER, H., SCHWEIG, A. & ZITTLAU, W. (1982). Electron Distribution and the Chemical Bond, edited by P. COPPENS(~. M. B. HALL, pp. 255-281. New York: Plenum Press. HALL, M. B. (1986). Chem. Scr. 26, 389-394. HANSEN, N. K. (~ COPPENS, P. (1978). Acta Cryst. A34, 909-921.

HERMANSSON,K. (1985). Acta Cryst. B41, 161-169. International Tables for X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press. (Present distributor Kluwer Academic Publishers, Dordrecht.) JOHANSEN, H. (1976). Acta Cryst. A32, 353-355. JOHANSEN, H. (1986). MOLPLOT. Contour plotting program. Technical Univ. of Denmark. JOHANSEN, H. & TANAKA, K. (1985). Chem. Phys. Lett. 116, 155-159. KURKI-SUONXO,K. (1977). Isr. J. Chem. 16, 115-123. LUNDGREN, J.-O. (1985). Crystallographic Computer Programs. Report UUIC-B13-4-06D. Institute of Chemistry, Univ. of Uppsala, Sweden. MCCANDLISH, L. E., STOUT, G. H. & ANDREWS, L. C. (1975). Acta Cryst. A31, 245-249. MASLEN, E. N., RIDOUT,S.C. & WATSON, K. J. (1988). Acta Cryst. B44, 96-101. MASLEN, E. N., WATSON, K. J. & MOORE, F. H. (1988). Acta Cryst. B44, 102-107. MULLIKEN, R. S. (1955). J. Chem. Phys. 23, 1833-1840. NORRESTAM,R. (1976). CADABS. Program for CAD-4 absorption correction. Technical Univ. of Denmark. REES, B. (1978). Acta Cryst. A34, 254-256. TACHEZ, M. & THEOBALD,F. (1980a). Acta Cryst. B36, 249-254. TACHEZ, M. & TH~BALD, F. (1980b). Acta Cryst. B36, 17571761. TACHEZ, M., TI-~OBALD,F., WATSON,K. J. & MERCIER,R. (1979). Acta Cryst. B35, 1545-1550. THEOBALD, F. & GALY, J. (1973). Acta Cryst. B29, 2732-2736. TORIUMI, K. & SAITO, Y. (1983). Adv. Inorg. Chem. Radiochem. 27, 27-81.

Acta Cryst. (1990). B46, 7-23

Classification of Structures with Anionic Tetrahedron Complexes using Valence-Eleetron Criteria
BY E. PARa3-m AND B. CrtABOT

Laboratoire de Cristallographie aux Rayons X, Universitd de Gendve, 24 quai Ernest Ansermet, CH- 1211 Gen6ve 4, Switzerland
(Received 25 May 1989; accepted 4 July 1989)

Abstract

A classification is proposed for the structures of CmC'm,A, compounds which contain anionic tetrahedron complexes formed with all the C' and A atoms, the C' atoms centering the tetrahedra, and with the C atoms outside the complex. The classification is based on the observation that compounds with these structures can be considered as general valence compounds, which permits the use of the generalized (8-N) rule. The parameters considered are: (a) VECA, the partial valence-electron concentration with respect to the anion A, calculated from the position of the elements in the Periodic Table; (b) AA(n/m') or C'C" (this depends on the VECA value) which correspond to the average number of electrons 0108-7681/90/010007-17503.00

per tetrahedron available for bonds between anions or between central atoms (and/or for electron lone pairs on the central atom), respectively; and finally (c) C'AC' which expresses the average number of C'--A--C" links originating from a tetrahedron. Using simple rules it is possible to calculate the most probable value for C'AC" without knowing the structure. Based on these parameters, classification codes are proposed, written as AA(n/m')VECA/C, AC,, °8/C'AC' or c'C'VECA/C'AC' depending on the VECA value. Each of these codes expresses an average of the heteronuclear and homonuclear bonds in the anionic tetrahedron complex. Base tetrahedra can be defined for particular sets of VECA, C'AC" and AA(n/m') or C'C" values. All tetrahedron complexes are interpreted as a linkage of one or more © 1990 International Union of Crystallography


8

CLASSIFICATION OF ANIONIC TETRAHEDRON COMPLEXES

also known under the name of the Zintl-KlemmBusmann (ZKB) concept (Sch~ifer, Eisenmann & Mfiller, 1973; Klemm & Busmann, 1963). The generalized (8-N) rule was originally only applied to semiconductors; however, Zintl has demonstrated Introduction that the valence concept can also be used with Anionic tetrahedron complexes are found in many certain intermetallic compounds.* iono-covalent compounds such as silicates, phosIn a formal approach, we consider these comphates and their analogues, but also in semi-metallic pounds as 'ionic compounds' with the C atoms 'Zintl phases'. By grouping the elements according to contributing all their valence electrons to the comtheir functions in the structure, the general formula plex made up of the A and C' atoms. This electron of these compounds can be expressed as CmC'm,An transfer allows the A atoms (but also the C' atoms by with electronegativity values of the corresponding electron sharing and/or electron lone pairs, if any) to elements increasing from left to right. The obtain completely filled octets. Thus, the model aselementary unit of an anionic tetrahedron complex sumes heteropolar bonding between the C atoms and consists of a C' (central atom)-centered ,4 4 (anions) the anionic complex, and covalent two-electron tetrahedron. This may be linked in different ways bonds within the complex. In this sense, these comwith one or more other tetrahedron (a) by sharing pounds agree with the ZKB concept established for anion corners and/or edges thus forming C'--A--C" binary compounds such as NaT1 or NaSi, which links, (b) by interpenetration with a second tetrahe- states that the non-noble atoms transfer their elecdron and replacement of a C'--A bond by a C'--C' trons to the more noble component, the 'anion bond or (c) by the formation of .4--.4 bonds between former'. With the resulting outer-electron configuradifferent tetrahedra. The C' atom may also have an tion, an 'anion partial structure' is constructed electron lone pair in place of a tetrahedral anion (whose atomic arrangement corresponds to the neighbour, the corresponding defect tetrahedron element having the same number of valence elecbeing called a ¢, tetrahedron. The C atoms (cations) trons). In the case of the CmC'm,An compounds condo not participate in the tetrahedral anion complex. sidered here, a binary anionic tetrahedron complex is There is no general method available for classify- formed with, now, heteronuclear C'--A bonds and ing these kinds of structures. Liebau (1985) has also, if necessary, with A--A or C'--C' homonuclear developed an elaborate classification system for sili- bonds. In the strict definition of the ZKB concept, cates and phosphates; however, in these compounds the C' atoms should also be considered as anions. neither A--A nor C'--C" bonds nor electron lone However, to conserve the notation used with the pairs occur on the central atoms. Further, to com- generalized (8-N) rule, we will treat the C' atoms as mence the classification one has to know the struc- 'cations ex-officio'and only the A atoms as anions. ture first. We thought it worthwhile to develop a Thus, we will speak below about polycationic valence primary classification system to (a) relate the com- compounds (with covalent C'--A and C'--C' bonds position of the compound to the valence electrons of and/or lone pairs on C'), polyanionic valence comthe participating atoms and (b) evaluate the average pounds (with C'--A and A--A bonds) and normal number of C'--A, A--A or C'--C" bonds (and/or valence compounds (with C'--A bonds only). electron lone pairs on C') and the number of According to the definition given above for the C'--A--C" links originating from a tetrahedron anionic tetrahedron complex, all the anions A share which characterize the anionic tetrahedron complex. covalent two-electron bonds with the C' atoms and As will be seen, in many cases it will be possible to all the valence electrons of the compound are predict important bonding features of the anionic assumed to be transferred to the C'm,An tetrahedron tetrahedron complex and to classify the structure complex. However, the total electron transfer is only with only the knowledge of the compound formula a model. For many multicomponent compounds it and the valence electrons of the participating might be argued that the A atoms also share covalent elements. bonds with the C cations and even that C-centered .4 4 tetrahedra are formed. In this latter case one could alternatively include the C atoms in the tetraThe model behind the classification scheme The proposed classification requires the multicomponent compounds to satisfy the generalized (8-N) rule, also called the Mooser-Pearson rule (Hulliger & Mooser, 1963; Pearson, 1964; Hulliger, 1968; Kjekshus & Rakke, 1974), which as far as the subdivision of polyanionic compounds is concerned is *According to the definition given by Sch/ifer (1985a.b) the term 'Zintl phases' is applied today to semimetallic or even metallic compounds where the underlying ionic and covalent bonds play such an important role that chemically based valence rules, normally reserved for semi- or nonconducting compounds, can be used to account for the stoichiometry and the observed structural features.

kinds of these base tetrahedra. The classification is applied to some 290 structure types with anionic or neutral tetrahedron complexes.


E. PARTHI~ AND B. CHABOT hedron complex and describe the complete structure as a tetrahedral structure. The possibility of describing a structure in this way does not interfere with the possibility of also using the iono-covalent model. Another limiting case occurs with compounds of composition C'm,A, which have no C cations. Here, the number of valence electrons provided by the central atoms C' is just sufficient to complete the octets of the A atoms. These C'm,A, compounds also have tetrahedral structures for which valenceelectron rules have been given before (Parth6, 1972). From the generalized (8-N) rule we shall calculate the average number of electrons per A or (7' atom which are used for the formation of A~A or C'--C' bonds (and/or lone pairs on the C' atoms). We shall also calculate from the stoichiometry of the compound a parameter labelled C'AC', which expresses the average number of C'--A--C' links originating from one tetrahedron. This parameter corresponds in the case of normal-valence compounds to the tetrahedron-sharing coefficient TT (Parth6 & Engel, 1986; Engel, 1986). As we shall see, the values for C'AC" cannot be calculated with certainty because this parameter does not depend on the composition alone. However, from simple empirical rules, it is possible to obtain the correct C'AC' value of a given structure type with a probability of 95%. This makes C'AC" another interesting parameter to be considered for the proposed classification.

9

generalized (8-N) rule, as reformulated by Parth+ (1972, 1973), can be written in the modified form:
VEGA = 8 + (m'/n)C'C" - AA

(2)

where C'C" is the average number of electrons per central atom for C'--C" bonds and/or for lone pairs on the C' atoms. If there are only C'--C' bonds, the C'C' value corresponds to the average number of C'~C" bonds per C' atom. AA is the average number of electrons per anion for A--A bonds. The A A value corresponds to the average number of A--A bonds per A atom. It can be concluded from observations that A--A and C'--C" bonds (or LP on C') do not coexist in a structure except for very rare cases. Consequently, the VEGA value allows the differentation between three kinds of valence compounds: (a) Normal-valence compounds for which C'C'= AA = 0 and thus:
VEC A = 8.

The great majority of the compounds with anionic tetrahedron complexes are normal-valence compounds (silicates in particular) where there are neither A--A bonds, nor C'--C" bonds or lone pairs associated with the central atoms. (b) Polyanionic valence compounds with AA > O, C'C" = 0 and then:
VEGA < 8 ~ AA = 8- VEGA.

(3)

The generalized (8-N) rule applied to ternary compounds with binary anionic tetrahedron complexes
For an application of the generalized (8-N) rule to a compound CmC'm,A, with an anionic tetrahedron complex of composition C',,,A, we shall proceed as described below. First, following a formulation proposed by Parth6 (1973), we shall express the electronic quantities by the partial valence-electron concentration with respect to the anion, VECA, defined as:
VECA = (mec + m'ec, + neA)/n

The polyanionic valence compounds are characterized by homonuclear A~A bonds between anions belonging to different tetrahedra. Since, in a compound of composition CmC'm,A,, the average number of anions per tetrahedron of the anionic tetrahedron complex is n/m', the average number of A--A bonds per tetrahedron is AA(n/m'). This quantity applies to the bonds between different tetrahedra. (c) Polycationic valence compounds with C'C'> O, AA =0 and thus:
VEGA > 8 ---, C'C" = (n/m')( VEGA -

8).

(4)

for CmC'm,A, (1)

where ec, ec, and eA are the numbers of valence electrons of the neutral atoms, and m, m' and n are composition parameters. The value of VEGA can be easily calculated from the composition of a given compound and from the positions of its component elements in the Periodic Table (for exceptions and the contribution of the transition elements see Table 2). Next, concerning the structural features, we make the assumption that, if there are electrons left which are not needed by the anions, they are retained by the central atoms C' and are used either for C'--C" bonds or for lone pairs (LP). In this case the

The polycationic valence compounds are characterized by homonuclear C'--C" bonds between the central atoms and/or lone pairs (non-bonding orbitals) attached to the central atoms. Since each tetrahedron of the anionic tetrahedron complex has one C' central atom, the C'C" value corresponds to the number of electrons which, on the average, are available for C'--C' bonds and/or lone pairs per tetrahedron. We retain here the conventional term 'polycationic' valence compound although some of the elements corresponding to the central atoms, such as P, S, or Se, are normally not considered as cations. Equation (4) allows the total number of valence electrons which rest with a central atom to be


10

CLASSIFICATION OF ANIONIC TETRAHEDRON COMPLEXES case of the isolated tetrahedron) tetrahedra linked by A--A bonds or by C'--C" bonds and/or C'--A--C' links. The tetrahedra can only be corner-linked, not edge-linked (this would correspond to a cycle of bonds). A derivative formula of the general tetrahedral structure equation allows the calculation of the average number of atoms in the molecular tetrahedron grouping, labelled N'A/M, according to:
N'A/M =

obtained. For our purposes, it is necessary to indicate how many of these are used for a C'--C' bond (one electron per central atom) or for an electron lone pair (two electrons per central atom). We introduce a parameter, called x, which relates the average number of C'--C" bonds (Nc,-c,) to the number of lone pairs (N/,e) per central atom as follows:
x = Nc,-c,/(Nc,-c, + 2NLp) = Nc,-c/C'C'.

(5)

2/(VEC'-6)

for VEC" > 6.

(8)

An analysis of the 109 structure types of polycationic valence compounds listed in Table 2 allows the formulation of a simple empirical rule to predict when C'--C" bonds or lone pairs are expected: (a) when C'C "= 1 then x = 1: this means that all C' atoms participate in C'--C" bonds (93% of 42 structure types); (b) when C'C '= 2 then x = 0: this means that all C' atoms have a lone pair (98% of 44 structure types); (c) when C'C" = 3 then x = ~: this means that all C' atoms have one lone pair and one C'--C" bond (5 of 5 structure types). We have found too few polycationic valence compounds with C'C' values different from 1, 2 or 3 to formulate an empirical rule concerning their most probable value of x. The general tetrahedrai structure equation applied to

For example, in the case of KSO4 with VEC' = 31/5 it is found that N'A/M = 10, which corresponds to the observed number of atoms in the anionic tetrahedron complex. More examples can be found in Fig. 5. The C'AC' parameter for different kinds of valence

compounds
The C'AC' parameter, which denotes the average number of C'--A--C" links per tetrahedron, depends on the number of C'--A bonds of a central atom and on the ratio n/m'. The number of C'--A bonds of a central atom, denoted Nc,--A, is four in the case of normal or polyanionic valence compounds but less than four in the case of polycationic valence compounds. It should be noted that a C'--C' bond replaces two anions (one each on two tetrahedra), but an electron lone pair only one. It follows that:
Nc,--A = 4 -- [(1 + x)C'C']/2

(9)

ternary compounds with binary anionic tetrahedron complexes
Ternary compounds with a binary anionic tetrahedron complex may be treated as compounds with a tetrahedral anion partial structure. The general tetrahedral structure equation as given by Parth6 (1972, 1973) can be applied to the charged anionic tetrahedron complex, assuming that the cations C have transferred all their valence electrons to the complex. Using a total valence-electron concentration value referred to the charged complex given by:
VEC" =(mec + m"ec, + neA)/(m" + n)

where x is the fraction of the number of valence electrons used for C'--C" bonds over the total number of valence electrons which rest with the C' atoms to form C'--C" bonds and/or lone pairs as defined by (5). All possible C'AC" versus n/m' equations can be derived by combining the following three equations:
NC'--A = No + Nl + N2 + N3

(10) (11) (12)

n/m' = No + ~NI + ~N2 + aN3 C'AC" = N~ + 2Nz + 3N3

I

(6)

one can write the general tetrahedral structure equation as:
VEC" = 4 + N'NB 0

(7)

where N'NSO is the number of non-bonding orbitals per atom of the charged anionic tetrahedron complex. For example, an isolated charged tetrahedron has 12 non-bonding orbitals, that means N'NB o = 12/5 and consequently, VEC' = 32/5. A special case occurs when VEC'> 6 and when the anionic tetrahedron complex consists of one or more non-cyclic molecular tetrahedron groupings with a finite number of linked tetrahedron. By tetrahedron grouping we understand (besides the limiting

where No is the mean number of 'unshared' anions per tetrahedron (they do not participate in C'--A--C' links to other tetrahedra), N~ is the mean number of shared anions per tetrahedron which participate in one C'--A--C" link to one other tetrahedron, N2 is the mean number of shared anions per tetrahedron which participate in two C'--A--C' links to two other tetrahedra, and N3 is the mean number of shared anions per tetrahedron which participate in three C'--A--C' links to three other tetrahedra. To derive an equation for C'AC" as a function of n/m' it is necessary to specify first which of the No to N3 values will be zero for the particular case of interest and then to eliminate the remaining non-zero values of N by combining the three equations.


E. PARTHt~ AND B. CHABOT In the common case (98% of 293 structure types) there is an equipartition of the C'--A--C' links over the tetrahedron corners occupied by the A anions. Equipartition means that there is either none or only a difference of one in the number of C'--A--C" links which pass through the different anion corners of a tetrahedron; this corresponds to a minimization of C'AC' for a given n/m" value. An equivalent definition of the equipartition (Zoltai, 1960) is that the difference between the smallest and the largest number of tetrahedra participating in the sharing of a tetrahedral corner in a structure cannot be more than one. For the general case of equipartition the different anions of one tetrahedron participate only in, say, k or k-1 C'--A--C" links to other tetrahedra. Thus, in (10), (11) and (12), only the parameters Nk and Nk-~ have values different from zero. This leads to:
C'AC" = k[2Nc,_A - (k + 1)(n/m')],

11

with Nc,-A/(k - 1) _> n/m" > Nc,-A/(k + 1). For the evaluation of k, the inequality can be rewritten as:
k - 1 < [Nc,_A/(n/m')] <--k + 1. Normal-valence compounds (VECA = 8 and Nc,- A = 4)

(13)

In the case of normal-valence compounds the parameter C'AC" is identical to the tetrahedronsharing coefficient TT, discussed by Parth6 & Engel (1986) and Engel (1986). (a) To calculate C'AC" for normal-valence compounds with equipartition of the C'--A--C" links (167 of 172 types) we use (13) with NC,--A = 4. The resulting equation becomes similar to the one derived by Engel (1986; see equation III.19). The three particular solutions for k = 1, 2 and 3, already given by Parth~ & Engel (1986), are: (i) For unshared anions or anions participating in one C'--A--C' link to another tetrahedron (k = I):
C'AC" = 2[4 - (n/m')]

with (1/k)Nc'-A >- n/m" >_ [ 1/(k + ger k, which corresponds to the are two to the larger number) of which the different anions of a pate, can be obtained from

1)]Nc,_A. The intenumber (or if there C'--A--C" links in tetrahedron partici-

for 4 _> n/m" _ 2. (15a)

(ii) For anions participating either in one or two C'--A--C" links to other tetrahedra (k = 2):
C'AC' = 6[~ - (n/m')]

for 2 > n/m" >_ ~. (15b)

k < [Nc,_A/(n/m')] <- k + 1.

(iii) For anions participating either in two or three C'--A--C" links to other tetrahedra (k = 3): C' AC' = 1212 - (n/m')] for 4/3 _ n/m" --- 1. (15c) The n/m" ranges of these three equations, together with symbolic drawings indicating the different numbers of C'--A--C" links which may pass through a tetrahedron corner corresponding to different k values, are given on the bottom line of Fig. 1 (Nc,--A = 4 which corresponds to C'C "= 0). Only (15a) is shown in the diagram. (b) For the rare cases (5 of 172 types) where there is no equipartition of the C'--A--C" links one uses (14) with Nc,--A replaced by 4. Two solutions are of interest: (i) for anions either unshared or participating in two C'--A--C" links to other tetrahedra (k = 2):
C'AC" = 3[4 - (n/m')]

Equation (13) is a modified form of Zoltai"s expression for the sharing coefficient (Zolta'i, 1960) which is also extended here to include compounds with C'--C" bonds and/or lone pairs. The three solutions of (13) for k = 1, 2, or 3 are: (a) For unshared anions or anions participating in one C'--A--C" link to another tetrahedron (k = 1):
C'AC' = 2[Nc,_A - (n/m')] for Nc,__ A >- n/m" >_ ~Nc,--A

(13a) two

and thus 0 <_ C'A C" <_ Nc,_,~. (b) For anions participating in one or C'--A--C" links to other tetrahedra (k = 2):
C'AC" = 6[~Nc,--A - (n/m')] 2
for ~Nc,_ A >- n/m" >- INc,_ A 1

(13b)

and thus Nc,--A <- C'AC" <_ 2Nc,_ A. (c) For anions participating in two or three C'--A--C" links to other tetrahedra (k = 3):
C'AC" = 12[~Uc,_A - (n/m')] 1 t 1 for ~Nc,__ A >- n/m >_ ;~Nc,--A

for 4 _ n/m" -> 4; (16a)

(ii) for anions participating either in one or in three C'--A--C' links to other tetrahedra (k = 3): C'AC' = 811 - (n/m')] for 2 >_ n/m" _ 1. (16b)
(VECA<8 and

(13c)

and thus 2Nc,--A <-- C'AC' <_ 3Nc,--A. Of the different imaginable possibilities for nonequipartition, we want to consider only the case where the anions participate in k or k- 2 links to other tetrahedra. The resulting equation obtained from (10), (11) and (12), assuming that Ark and Nk-2 are non-zero, is:
C'AC "= (2k. - 1)Nc,--A -- (k 2 - 1)(n/m'),

Polyanionic NC,--A = 4)

valence

compounds

(14)

The calculation of C'AC" is not affected by the presence of extra A--A bonds between the tetrahedra. Equations (15a) and (15b) can be applied here without any change. Structures of polyanionic valence compounds with C'--A--C" links corresponding to (15c), (16a) or (16b) are not known so far.


12

CLASSIFICATION OF ANIONIC TETRAHEDRON COMPLEXES

Polycationic valence compounds Uc,--A = 4 -- [(1 + x)C'C']/2)

(VECA > 8 and

For the general case of an equipartition of the C'--A--C" links (99% of 109 structure types), C'AC' is calculated from (13) using the NC,--A value obtained from (9). Thus, for the polycationic valence compounds, different n/m' ranges apply for (13), depending on the C'C' and x values. They can be read from Fig. 1 which shows a diagram for C'AC" as a function of [(1 +x)C'C']/2 and n/m'. Thick inclined lines separate the three regions where, from right to left, (13a), (13b) and (13c) are valid. The ordinate corresponds to the number of C'--A bonds lost by the central atom of a C'A, tetrahedron when C'--C" bonds are formed and/or when there are lone pairs on C'. The solutions of (13a) for Nc,--A = 4, 3 and 2, that is [(1 + x)C'C']/2 = 0, I and 2, are shown in the diagram. Each C'AC" equation used in this work was obtained from the combination of (10), (11) and (12). All of them contain the parameter N~, the subscript of which may take integral values between 0 and 3. When y = 3, according to the definition given above, the anion participates in four two-electron A--C" bonds. This case corresponds to the maximum possible number of A--C" covalent bonds which allows

the anion to complete its octet shell by electron sharing. Compounds with anionic tetrahedron complexes are known, however, where the anions may participate in more than four bonds to central atoms and which, thus, do not correspond to our model of valence compounds. It is still possible, in such a case, to derive C'A C" equations as a function of n/m' from (10), (11) and (12) extended by N: terms with y > 3. For example, in metallic KCu4S3 (Brown, Zubieta, Vella, Wrobleski, Watt, Hatfield & Day, 1980), there are CuS4 tetrahedra which are linked in such a way that two anions participate in three C'--A--C' links and two in seven C'--A--C" links. Thus, with 4 -- N3 + N7 from (10), n/m' = ~N3 + ~N7 from (11) and C'AC' = 3N3 + 7Nv from (12), one finds C'AC' = 4111 8(n/m')]. For KCu4S3 the calculated C'AC" is 20, which is the correct value. A final remark can be added that (10), (11) and (12) can be easily modified to obtain a relation between the average number of C'--A--C' links and the n/m" ratio assuming that the anion complex is built up of polyhedra different from tetrahedra [this possibility has already been noted by Engel, (1986)]. The digit 4 in (9) has to be replaced by a value which corresponds to the number of corners of the new polyhedron and all parameters have to be related to the new kind of polyhedron.

C'AC'= 2 (2-n/m') 2. . . . . .

\

\cc' ' :2(,:
.\

I

'

c

'

'

'

x:O)

t
OJ

-c)
-,lm

.

.

.

.

.

.

.

.

p-

,

......

',

\\ \'~1 X

X

~'

C'AC'=2(3-n/m'}

~.

,...,..,_, .........

(x :O)

,
,
I

.",, h\ %,,, .
~,~_.-o-~

%.,,:
'

%:, :
;
~

%,
N

~\k=,.3~

\ 4/3

<.-o-c

k:2\

\\

\\

,~ 2

X

k:l

,N

~

~. r~/rl3' ~4

0

I

:3

C'C'=0

Fig. 1. C'AC" diagram as a function of [(1 + x)C'C']/2 and n/m" for the case of equipartition of the C'--A--C' links. Thick inclined lines separate the three domains which differ in the way the anions are shared and for which, from right to left, the equations (13a), (13b) and (13c) are valid. Anion sharing is represented by a symbolic drawing in each of these domains. A small black circle corresponds to a central atom, the blank symbols represent the anions. A complete blank circle corresponds to an anion which belongs to only one tetrahedron (no C'--A--C" link), a half circle to an anion which is shared with one other tetrahedron (it participates in one C'--A--C" link), a triangle to an anion shared with two other tetrahedra (two C'--A--C' links) and a square corresponds to an anion shared with three other tetrahedra (three C'--A--C" links). Within the range of a C'AC" - n/m' equation, the number of C'--A--C' links of a tetrahedron varies from one limiting case where all the anions have links (as shown by the blank symbol on the right) to the other limiting case (represented by the blank symbol on the left of the black circle). The number of anion neighbours of a central atom is Nc.-A = 4 - [(1 + x)C'C']/2. The composition of a single tetrahedron depends, however, on the number of C'--A--C' links in which its anions participate (half circles, triangles and squares count, respectively, for one half, one third and one quarter of an anion.)


E. PARTH]~ AND B. CHABOT

13

Prediction of the most probable base tetrahedron(a) from the chemical formula of the compound
We consider as base tetrahedra the isolated C'A4 tetrahedron but also the tetrahedra which share anions, those in which each anion may extend one bond to an anion of another tetrahedron and finally, the tetrahedra which have a C'--C" bond or a lone pair instead of a C'--A bond. In Fig. 2 graphs are shown of the 32 possible base tetrahedra subject to the restriction that a central atom has no less than two anion neighbours and that an anion is either unshared or shared with only one other tetrahedron. Note that the planar graph presentation suggests a certain order of the ligands

which is however irrelevant, all four ligands of a central atom in a tetrahedron being geometrically equivalent with respect to each other. The C'AC" value of each base tetrahedron in Fig. 2 can be calculated by counting the half-circles, the AA(n/m') value by counting the number of indicated A--A bonds per tetrahedron and the C'C" value by adding twice the number of non-bonding orbitals to the number of C'--C' bonds. Inserting the values for AA or C'C" and n/m" into (2) one can calculate a VECA value for each base tetrahedron. This value should correspond to the VECA value of a valence compound with an anionic tetrahedron complex built up only with this particular base tetrahedron. The base tetrahedra in Fig. 2 are arranged according to their

AA
n/m'

(n/re')
2 9

~.A=C'C': 0

CiC '

41

+
47/0

5

1 (x=z)

2 (x:Ol

?_ (x--i)

3'(x:I/3)

4 (x:O)

+
I 7.75/0
o 8/0

37.25/0

z7.s/0

72
3 7.143/1
2 7.429/1

I 7.714/1

o 8/1

5:1
2 7. 333/2

o5I 7. 667/2
0 8/2

o-~

9 6

I 8. 333/0

28. 667/0

2-,

52

17.6/3

08/3

I 8.4/1

28.8/1

5' &

2:1
08/4

c46
2 9/2 2 9/0

'8.5/2

39.5/0

4 i0/0

±

5:2
I 8. 667/3 29. 333/3 2

o

9.333/1

3 I0/i

4 I0. 667/1

@
I:I
2 lO/2

D-~I
41.2/2

311/2

Fig. 2. Graphs of the 32 possible base tetrahedra where each central atom has at least two anion neighbours and where each anion is either unshared or shared with only one other tetrahedron (0 _< C'AC' <- Nc, A, k = 1). The base tetrahedra are arranged according to their n/m', AA(n/m') or C'C' values and in the last case also their x value. Assuming an equipartition of the C'.--A--C' links and of the A--A links over the corners of the tetrahedron, each base tetrahedron is unambiguously identified by means of the "4A("/"')VECA/C'AC code (when VECA < 8), the °8/C'AC" code (when VEC,4 = 8) or the cC'VECA/C'AC code (when VECA > 8). The value of AA(n/m') corresponds to the average number of A--A bonds per tetrahedron and C'C' to the average number of C'--C" bonds and/or the number of electrons used for lone pairs on the central atom per tetrahedron. The A--A bonds and C'--C' bonds are indicated by short heavy lines extending from the blank symbol and the filled circle, respectively. A lone pair is shown as a heavy bar on a filled circle.


14

CLASSIFICATION OF ANIONIC TETRAHEDRON COMPLEXES In Fig. 3 base tetrahedra are shown where one or more anions participate on more than one C'--A--C' link. The list does not include all the possibilities; however, all the base tetrahedra found in the crystal structures in Table 2 are presented. In the three base tetrahedra at the bottom of Fig. 3 there is no equipartition of the C'--A--C' links. To distinguish these exceptional base tetrahedra their codes are surrounded by square brackets. An observed anionic tetrahedron complex either corresponds directly to a base tetrahedron or can be interpreted as a linkage of different base tetrahedra. Inversely, using the equations given above, it is possible to express the expected structural features of a compound in terms of the most probable base tetrahedron(a): one first calculates the AA or C'C' value and next, the most probable C'AC' value

n/m' and their AA(n/m') or C'C" values and in the
latter case also the x value. To classify the base tetrahedra we use the parameters AA(n/m') or C'C', FECA and C'AC" written in the form AA~"/"')VECA/ C'AC', °8/C'AC" or c'C'FECA/C'AC'. When FECA < 8, then the superscript corresponds to the AA(n/m') value. For FECA = 8 the superscript is 0. When FECA > 8, then the superscript expresses the C'C" value. In the case of AA(n/m')VECA/C'AC', the AA parameter is simply the difference from 8 of the VECA value. In the case of c'C'VECA/C'AC', however, the relation between C'C" and VECA also depends, according to (4), on the n/m' ratio. The proposed codes allow unambiguous characterization of the base tetrahedra provided there is an equipartition of the C'--A--C' links and A--A bonds over the comers of the tetrahedron.

Base tetrahedra with equipartition of C'--A'--C' links n/m' VECA < 8 3/2 ,~ V 1/27.667/7

n/in' VECA = 8

+
08/5 4/3

11/6

+
08/6 7/6 18"857/5

5/3

+
1/1

3/2

+
08/8 3/4 19"333/9

4/3

+
08/9 x=l

5/4

1/1 []

08/7

°8/12

n/m'

18"75/4 VECA > 8

19/6

+
29"5/4 29'714/5 210/6 Base tetrahedra with non-equipartition of C'--A--C' links n/m' 2/1 3/2 5/3

210.667/9

lEFt!

x=0

[°8/6]

[°8/81

[29.2/4] x=0

Fig. 3. Graphs of the base tetrahedra which occur in the structures listed in Table 2 and where one or more anions participate on more than one C'--A--C" link (k > I). For the meaning of the filled and blank symbols, see the legend to Fig. 1. Note that, in the base tetrahedron on the top, a shared anion (half circle) has a bond to another anion. Thus there is one half of an A--A bond for 3/2 anions and the ,4A value is consequently l/3 and AA(n/m') = 1/2.


E. PARTHI~ AND B. CHABOT (assuming that there is an equipartition of the C'--A--C' links and that, for C'C'> O, the partitioning of the electrons between C'--C' bonds and lone pairs on C' follows the rule given above). In this way the most probable AA("/")VECA/C'AC', °8/C'AC" or cC'VECA/C'AC' code for the compound can be established. Depending on the code, two cases have to be distinguished to find the most probable base tetrahedron(a): (a) The most probable code of the compound is identical with the code of a base tetrahedron (260 of 293 types). Here, one expects the tetrahedron complex to be built up with only this single base tetrahedron (this is verified in 242 of 260 types). Examples are to be found in Fig. 4 where the predicted most probable single base tetrahedra of five compounds with composition CC'A3 can be compared with the observed tetrahedron complexes. The VECA values vary from 7.333 to 8-667 and, consequently, the predicted base tetrahedra are all different. In the case of BaTeS3, the predicted base tetrahedron (28"667/0) is identical to the observed isolated O-tetrahedron. For the four other compounds the anionic tetrahedron complex can be obtained by linking the corresponding single base tetrahedron with itself. In 18 of 260 types of this category the tetrahedron complex is built up from more than one kind of base tetrahedron. As an example one may study the schematic drawings of the observed tetrahedron complexes found in particular (alkali)-sulfur(selenium)oxygen compounds shown in Fig. 5. The predicted most probable single-base tetrahedra are not shown

15

here; however, using the observed n/m' ratios and the listed codes of the anionic tetrahedron complexes, the drawings of the base tetrahedra can be easily found in Fig. 2. It can be noted that each observed tetrahedron complex is built up with one kind of base tetrahedron in all compounds except for Se205 (most probable code ~8.4/1) and K2S20 5 (28.8/1). TO construct the observed tetrahedron complex two different kinds of base tetrahedra are needed for both compounds (°8/2 plus 29/2 for Se205 and ~8.333/0 plus 39-5/0 for K2S205). The reason why two different base tetrahedra are necessary in these two S or Se compounds seems to lie in the splitting of the oxidation states of the pairs of chalcogen atoms into 4 and 6 instead of 5. (b) The most probable code of the compound is not identical with the code of a base tetrahedron (33 of 293 types). Normally, whenever the C'AC" and/or the C'C" and/or the AA(n/m') values of a compound code are not integers, the tetrahedron complex is built up from more than one single base tetrahedron (one exception can be found in Fig. 3). As an example, one may consider the compound Zn2P3S9for which the most probable calculated code is 1/38.111/1.333, assuming x = 1. No base tetrahedron with this code is possible. Restricting the search to base tetrahedra with the same n/m" value, the two most probable ones with codes close to the code of the compound are °8/2 and 18.333/0. There must be twice as many of the first kind in order to obtain the correct C'C' and C'AC' values. The predictions are in full agreement with the observed anionic tetrahedron complex in Zn2P3S9: one finds one isolated con-

AISiP 3 VEC·-- 7.335 AA =213 C'AC' = 2

CsSiTe 3 VECA =7.667 AA= 1/3 C'AC' = 2

AgPS 3 VECA =8.0 AA =C'C' =0 C'A C' = 2

HgPS 3 VECA= 8.333 C'C' = I C'AC'=O

BoTeS 3 VECA=8.667 C'C'= 2 C'AC'= O

z 7. 333/2

~7.667/2

°8/2

~8.333/0

z 8.667/0

Fig. 4. Graphs of the predicted most probable single-base tetrahedra (upper drawings) and of the observed tetrahedron linkages of the anion complexes (lower drawings) in five compounds with composition CC'A3 where VECAvaries from 7.33 to 8.667. The classification code is given for each anionic tetrahedron complex.


16

CLASSIFICATION OF ANIONIC TETRAHEDRON COMPLEXES (i) For a normal-valence compound one expects two base tetrahedra, of the type °8/C'AC', with their n/m" values closest to the n/m' value of the compound. Their proportions are chosen so that the average n/m' value is equal to the n/m' value of the compound. (ii) For polycationic compounds the following cases are considered: --If the n/m" ratio of the compound is the same as that of listed base tetrahedra, two base tetrahedra with different C'C" values are selected from that group. --If base tetrahedra exist with the same C'C' value as for the compou4d, two with different n/m' values are selected. --For more complicated cases there is no simple rule. However, the pairs of C'C" and n/m' values are expected to be not too different between tetrahedra. Application of (17) permits the verification that the chosen base tetrahedra and their proportions

densed double tetrahedron, as in HgPS3, together with two isolated edge-linked double tetrahedra, as in AgPS3 (see Fig. 4). The average code of a structure built up of different base tetrahedra and the overall composition of the anionic complex can be obtained from the following equations:

AA(n/m') = ( AA(n/m') )or C'C" = (C'C')
VEC A = (VECA(n/m') ) / (n/m')

C'AC'=(C'AC')

(17)

where n/m'= (n/m') and where the angled brackets indicate that the enclosed data are the average values of the base tetrahedra involved. Note also that x =

(xc'c')/(c'c').

To find the most probable base tetrahedra of a compound whose code is not identical with a code of a base tetrahedron, it is helpful to use the following guidelines:

KS04
Persulfate

K2SO4
Sulfate

K2S20 7
Pyrosulfate

SO5
Trioxide

KSO5
_13ithionate

~7.75/0 VEC'=6.2 N'A/M =10

08/0 VEC'=6.4
N'

°8/I VEC'= 6.222 N 'A/M = 9
:5

osbestos modificotion 08/2 VEC'=6

J8 555/0 VEC':6.25 NA/M=8
i

AIM

Se205
Pentoxide

No2 S05
Sulfite

K 2S205
Pyrosulfite

SeO2
Dioxide

NoSO2
Dithionite

oto
not found t8.4/I VEC'= 6 Se205 18.4/2 28.667/0 VEC'= 6.5 N' =4 AIM MgTe205 28.8/I K2S205 28.8/0 VEC' =6.2 86 N' =7 AIM 2 9/2 VEC'= 6 39.5/0 VEC'= 6.:555 N' =6 AIM

Fig. 5. Schematic presentation of the observed tetrahedron complexes in different (alkali-)sulfur(selenium)-oxygen compounds. Filled circles: S or Se; blank or half-circles: O. The classification code and its VEC' (equation 6) and N'A,M (equation 8) values are given for each anionic tetrahedron complex. With the exception of Se205 and K2S205 each observed tetrahedron complex can be constructed with a base tetrahedron which can be predicted from the calculated VECA, AA or C'C" and most probable C'AC' values. In the case of Se205 and K2SzO5 two drawings are given, those on the left corresponding to the expected tetrahedron complexes, and those on the right to the ones observed with the compounds. They are constructed from two different base tetrahedra (for their codes see text). The single base tetrahedron which is expected for K2S205 is found with MgTe2Os.


E. PARTHI~ AND B. CHABOT

17

Table 1. The 55 classification codes AA(n/m')VECA/C'AC', °8/C'AC' and c'C'VECA/C'AC" of CmC'm'A,, compounds arranged according to their n/m', AA(n/m') and C'C" values
A code within square brackets indicates that there is no equipartition of the C'--A--C' links over the apices of the tetrahedron. The number within parentheses below each code indicates how many different structure types with this code are listed in Table 2.

Polyanionic valence compounds AA(n/m') values
n/m'
4 15/4 11/3 7/2 10/3 13/4 16/5 3 14/5 11/4 8/3 5/2 z7"6/3 (l) 27.33/2 (I) '7'67/2 (3) '7.71/1 (i) 4 47/0 (I) 2 27.5/0 (l) 1 '7.75/0 (3) 1/2

Normalvalence
compounds °8/0 (54) °8/0"5 (1) 08/0"67 (2) °8/1 (11) °8/1"33 (2) °8/1"5 (1) °8/1"6 (1) 08/2 (31) °8/2"4 (I) °8/2"5 (I) °8/3 (ll) °8/3'33 (2) °8/4 (31) ~8/61 (4) o8/5"5 (3) I/3 !/2 2/3

Polycationic valence compounds C'C' values
1 4/3 3/2 2 5/2 3 7/2

t'38"11/1"33 (1)

'8-33/0 (20)

28"67/0 (12)

4/38.5/0 (1) ~8.4/1 (4) '8.4/2 (i) ,/28.22/3 (2)
18.5/2 (4) '8.5/3 (1)

28'8/0 (1) 28"8/1 (5)

7/3 9/4 2

29/2 (16) ~a8.86/2.5 (2) 3:28.86/3 (I)

39.5/0 (2)

7/4

5/3 13/8 3/2 ,'27.67/7 (1) °8/7 (7) ~8/81 (I) °8/8 (3)

'8-6/3-33 (1) '8.67/3 (3)

29"2/2"67 (1) 29.23/2.75 (2) 29.33/3 (4) [29.33/4] (1) 5ai0/2 (2) 5al0/2.5 (2)

4/3 5/4

1 3/4

°8/12 (5)

938-67/8

'9/6

210/6

311/2

(I)

(8)

(2)

(5)
7:212.67/1.5 (4)

agree with the most probable code of the compound. There are cases, however, where different combinations of different base tetrahedra lead to the same code for the compound. Classification of structures with anionic tetrahedron complexes The structures with anionic tetrahedron complexes are characterized by their VECA, AA(n/m') or C'C' and C'AC" values. We have proposed writing these

parameters in the coded forms AA(n/m')VECA/C'AC', °8/C'A C' or c'c"VECA/C'A C'. The advantage of using these parameters as codes for the compounds is that they combine observable structural features with the number of available valence electrons. We have seen above how the most probable code and the most probable base tetrahedron(a) of a compound can be derived from its chemical formula. Inversely, one can state what the value of the valence-electron concentration of a compound would have to be in order that a structure


18

CLASSIFICATION OF ANIONIC TETRAHEDRON COMPLEXES

Table 2. Classification codes of structure types with anionic tetrahedron complexes and the codes of their base

tetrahedra
Codes enclosed by square brackets correspond to tetrahedron complexes where the equipartition of the C'--A--C' links is not respected. The formal charge is indicated as a superscript for each element where the number of valence electrons cannot be predicted from its position in the Periodic Table. Type formulae which are preceeded by a hyphen correspond to an uncharged tetrahedron complex where there are no cations C present. References were taken mainly from Strukturberichte (SB) or from Structure Reports (SR). Note that the number of atoms in the Pearson code of rhombohedral structures applies to the triple hexagonal cell; in addition, in the case of hydrates and ammonium cations, the H atoms have been ignored for the atom count in the unit cell. I. Polyanionicvalencecompounds: VECa < 8, AA > 0 and C'C" = 0 Structure types are arranged in groups of same VECa and AA(n/m') values. These groups (separated by dashed lines) are ordered according to increasing VECa and then according to decreasing AA(n/m') values. In each group, the structures are ordered first according to increasing m/m', next according to the number of atoms in the unit cell (see Pearson code) and then according to increasing space-group number. SpaceType Pearson group Code(s) of base "~A(~/"'~VEC,dC'AC'n/m" formula code number tetrahedron(a) ~7/0 4 Cd~GeAs4 oP28 (62) 47/0* ............................................. 27.333/2 3 A1SiP~ oP20 (62) 27.333/2 ............................................. 27"5/0 4 KzGeTe4 mP28 (14) -'7.5/0 ............................................. ~7.6/3 5/2 Na~AI2Sb5 mP28 (11) ~7.6/3 ............................................. ~7.667/2 3 CsSiTe~ mS40 (9) ~7.667/2 ~7'667/2 3 Ca~Ga2As6 0/'26 ~7.667/2 3 CasAl2Bi6 oP26 ............................................. "27.667/7 3/2 NazAI2Sb~ mP56 ............................................. t7-714/1 7/2 Cs~Sn_,Te+ mS52 ............................................. ~7,75/0 4 KSO4 aP12 ~7.75/0 4 (NH4) ~+SO~ mP24 ~7.75/0 4 Na~SiSe+ raP64 (55) (55) (14) (15) (2) (14) (14) t7.667/2 t7.667/2 v27"667/7 t7"714/1 ~7.75/0 ~7.75/0 ~7.75/0 Classification code Classification code
o8/C'AC" n/m"

Ref. SR40A,15 SR45A,9 SR51A,54 SR51A,6 SR52A,26 SR42A,27 SR51A,4 SR51A,6 SR52A,86 SB3,138 SR41A,427 SR52A,89

°8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °810 o8/0 °8/0 °8/0 08/0 °8/0 °8/0 08/0 °8/0 08/0 08/0 08/0 °8/0

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

SpacePearson group Code(s) of base Type formula code number tetrahedron(a) Ref. TI~*SbS4 aPI6 (1) °8/0 SR48A,15 Na2ZnSiO4 mPI6 (7) °8/0 JLBP69 Cu~AsS4 oPl6 (31) °8/0 SR35A,19 Cu.,CdGeS4 oPl6 (31) °8/0 SR34A,47 Cu3SbS4 tll6 (121) °8/0 SR21,349 Cu2Fe2*SnS4 t/l 6 ( 121) °8/0 SR44A,56 (NH4)~ SbS, + cll 6 (217) °8/0 SR41 A,421 Cu2SrSnS~ hP24 (144) °8/0 SR42A,77 Na2MgSiO4 raP32 (7) °8/0 SR48A,337 TI~*Pb: *GeS4 mP32 (14) °8/0 SR46A,82 Li2Co +SiO4/3 2 oP32 (33) °8/0 SR45A,364 K3AsS4 oP32 (33) 08/0 SR40A,22 (NI-L)~* AsS4 oP32 (62) °8/0 SR42A,302 Cs~TaSe4 oP32 (62) °8/0 YRI88 K~PS4.H20 oP36 (18) °8/0 SR45A,275 Na4SnS4 tP18 TI~*GeS4 mS36 TI~* SnS4 raP36 Na4SnTe4 oP36 Na4GeSe4.14H20 mS46 Ba,SiAs4 cP72 Na~SnS4.14H,O mS96-4 NasFe 3*$4 Na~ZnO4 NasSnSb4 oP80 hP22 cFl04 (114) (9) (14) (19) (5) (218) (15) (61 ) (186) (227) (14) (15) (2) (5) (15) (14) (14) (46) (I 1) (15) (5) (9) (I 5) (I 5) (40) (15) (2) (13) (14) (14) (33) (12) (14) (62) (71) (2) (169) (14) (15) (2) (44) (12) (14) (36) (62) 08/0 °8/0 °8/0 °8/0 °8/0 08/0 °8/0 °8/0 °8/0 °8/0 °8/0 + °8'1 °8/0 + (08/I )2 SR41A,29 SR44A,66 SR51 A,86 SR50A,62 SR42A,301 SR48A, I 7 SR39A,259 SR53A,57 SR40A,214 EK88 TMS87 TKR82

* For the polyanionic base tetrahedra shown in Fig. 2 an equipartition is assumed of the A--A bonds over the four anions of a tetrahedron. This is not the case in CdzGeAs4 where one As has two homonuclear bonds, two As one and the fourth As no bonds. II. Normal-valence compounds: VECA = 8, AA = 0 and C'C' = 0 Structure types are arranged in groups of same n/m' values. These groups (separated by dashed lines) are ordered according to decreasing n/m' values. In each group, the ordering of the structures is the same as in I. Classification code
o8/C'A C' n/m'

°8/0.5 15/4 Na2Ba6Si40~s mP54 ............................................. °8/0"667 I 1/3 AgTP3SI i rnS224 -56 08/0-667 11/3 Agl0Si3SH aP50-2 ............................................. °8/1 7/2 HgaP2S7 mS22 °8/I 7/2 K2S20, mS44 °8/I 08/I °8/I °8/1 °8/I °8/1 °8/1 °8/I °8/I °8/I.333 7/2 7/2 7/2 7/2 7/2 7/2 7/2 7/2 7/2 Ba3Sn2S7 CajSi~O,
mP48

08/0 + (~8:1)2 SR42A,124 08 '1 °8'1 °8/I °8/I °8/0 + (08:1)2 + °8:2 08' I °8/I °8/1 °8q 08 I °8 1 (08:1)5 + °8/3 08'1 + o8,'2 SR44A,80 SR24,378 SR37A,28 SR44A,314 SR37A,336 EJS84 SR43A,92 SR46A,64 DJIR82 SR39A,95 EHS86 MBB83

Ca3Si.~O7
BaaGa2S7 Ag4P2S7 Cu4Ni2~Si2S7 Cu~ 2*Si2S7 NarSn2S7 NarGe2Se~

m P48 0/96 raP26
mS52

°8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

SpacePearson group Code(s) of base Type formula code number tetrahedron(a) Ref. - SiF, cllO (217) 08/0 SR18,353 AIPS4 oP12 (16) 08/0 SR24,405 BPS~ LT o112 (23) 08/0 SR28,153 InPS4 tll 2 (82) °8/0 SR44A,71 Cr3* PS, mS24 (5) 08/0 SR43A,48 GAPS4 mP24 (14) °8/0 SR39A,61 LiAICI4 raP24 (14) °8/0 PSR82 NaA1CI~ oP24 (19) 08/0 PSR82 ZrSiO4 t/24 ( 141) °8/0 SR22,314 Bi~" PS4 o/96 (73) °8/0 SR41A,35 PrPS4 t196 (142) °8/0 SR51A,77 La4Ge~St,
hR114

mS28
mS56 mS60 mS60

10/3 NaBa3Nd~SirO2o oSI32

(161) (164) (14) (4) (I 1) (121) (14) (14) (14) (14) (62) (62) (33) (146) (64)

°8/0 08/0 °8/0 °8/0 08/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °8/0 °810 °8/0

SR41A,63 SR38A,135 SR52A,249 SR45A,69 SR38A,100 SR19,337 SR41A,29 SR37A,25 SR39A,80 SR39A,80 SR22,447 SR28,245 SR37A,26 F87 SR51A,17

°8/I.333 10/3 NasP3OtoLT mS'72 ............................................. o8/I.5 2 13/4 Pb3*P~Ot3 aP40 ............................................. °8/1.6 16/5 Na~Mg2P5Ot6 mP52 ............................................. °8/2 3 -AIBr~ mPl6 °8/2 3 -SO3 mPl6 °8/2 3 - SO3 oP48 °8/2
08/2 08:'2

(°8'1)2 + 08/2 SR22,402 AD87

(°8/I), + (°8/2)~SR45A,316 °8/2
°8/2

°8:2
08/2 °8/2

SRI0,104 SR18,367 SR8,148

Pd]+ P2S8 hPl3 Ba3As2Ss.7H20 raP80 Eu22"GeS4 Sr2GeS4 Cu2HgI4/3 BazSnS4 a Pb~ÂGeS4 Pb~*SiS4 Pb~ÂSiSe, K2SO4/3 MgFe-" SiO4 Â Ba2SnS~/3 LiGaSiO4 a Ba~Ga~S8
raP14 rnP14 tl14

mP28 mP28 mP28 mP28 0/'28 oP28
oP56

°8/2 °8,/2 °8,'2 °8,;2 °8, 2 °8,'2 0S 2 "8/2 · ~2 °8:2 °8, 2

3 3 3 3 3 3 3 3 3 3 3 3 3 3

AgPS.a Pb2* GeS3 TaCuS3 TI~* PS3 Eu2" GeS3 lnGaS3 LaGaS3 Ba~Ga2S6 TI~+GeS~ Cu2GeSe~ TI~*SnS~ Na2GeS3 Li,SiO~ K2Agl ~

mS20 raP20 oP20 0/20
aP30 hP30 raP60 mS44 aPl2 o112

SR44A,89 SR40A,72 °8/2 SR29,46 08;2 SR50A,55 °8;2 SR44A,59 °8/2 GAKM87 08 1 + 08/2 + 98,'3 JJD82 °8/2 °8'2 °8,'2 °8/2 08'2 08/2 °8 '2 EJS84 SR44A,66 SR37A,73 SR51A,85 SR38A,100 SR48A,335 SR 16,204

hRI26 oS120

mS24 raP24 oS24 oP24


E. PARTHI~ AND B. CHABOT
Table 2
Classification code
°8/C'AC' n/m'

19

(cont.)
Classification code
°8/C'AC" n/m'

°8/2 °8/2 °8/2 °8/2 °8/2 °8/2 °8/2 °8/2

3 3 3 3 3 3

SpacePearson group Code(s) of base Type formula code number tetrahedron(a) Ref. K2CuCI3 oV24 (62) °8/2 SP..12,188 Eu25ÂCuS3 oP24 (62) o8/2 SR53A,46 Na2GeS3.7H20 aP26 (2) 08/2 SR38A,291 K2SnS3.2H20 oP40 (62) °8/2 SR41A,278 Rb2TiO3 oS48 (64) °8/2 SR40A,185 CasSn2As6 oP26 (55) °8/2 SR52A, 13 (I 2) (14) (14) (62) (14) (62) (64) (14) (14) °8/2 08/2
°8/2

3 Rb3InS3 mS28 3 Na3Fe3+S~ raP28 °8/2 3 Cs3GaSe3 raP28 °8/2 3 Ca3AIAs3 oP28 °8/2 3 Sr3GaSb3 raP56 08/2 3 Ba3GaSb3 oP56 °8/2 3 Ba3AISb3 oS56 °8/2 3 NasGeP3 raP36 ............................................. °8/2"4 14/5 NdPsO~4 raP80 ............................................. 08/2.5 11/4 CaP4Ot~ raP64 ......... "- °8i3.... ;/2-" "-'P2S5 "'" aP28"" °8/3 5/2 - P205 hR84 °8/3 5/2 BaGe2S5
cF128

°8/2 °8/2 o8/2 °8/2 08/2

SR46A,87 SR45A,89 SR50A, I 8 CSS82 CSS87 SR52A,6 CSS82 SR52A,76

(°8/2)3 (°8/3)2 SR40A,257 +

(14) °8/2 + 08/3 SR40A,256 ........... (2) "" o'8/3 SR30A,353 (161) °8/3 SR29,360 (203) (33) (15) (62) (63) (92) (9) (2) (2) (15) (12) (152) (72) (122) (7) (14) (164) (125) (87) (ll7) (140) (140) (66) °8/3 °8/3 o8/3 °8/3 °8/3 °8/3 °8/3 °8/2 + °8/4 °8/3 SR39A,22 SR51A,55 SR42A,92 ES82b SR37A,93 SR33A.363 SR42A,300 SR51A,27 SR46A,87

08/4 °8/4 °8/4 °8/4 °8/4 °8/4 [°8/6] °8/4 °8/4 [08/6] °8/4 °8/4 o8/4 °8/4 08/4 °8/4 °8/4 08/4 08/4 08/4 °8/4 °8/4

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

SpacePearson group Code(s) of base Type formula code number tetrahedron(a) Ref. CaAI2Se4 oS36-8 (66) °8/4 SR45A,5 BaAI2Se4 tP30-2 (126) °8/4 SR48A,5 BaGa2S4 cP84 (205) °8/4 EJS82b Baln2Se4 oF224 (70) °8/4 SR46A,25 NaNdGa4S8 oF224 (70) °8/4 IGGF88 SrGa2Se4 05"756 (6) °8/4 SR50A,36 ScCuS2 hP4 (156) [°8/6] SR37A,75 BaZnO2 hPl 2 (152) °8/4 SR24,254 TP * Fe3+Se2 rnSl6 (12) °8/4 SR45A,88 LaCuS2 mPi 6 (14) [08/6] SR48A,45 KFe3+S2 mSl6 (15) °8/4 SR10,123 ErAgSe 2 oPl6 (19) °8/4 SR43A,58 NaFe 3Â02 fl oPl 6 (33) 08/4 SR 18,422 SrZnO~ oPl6 (62) °8/4 SR26,396 CsFe3+ $2 oi16 (71) 08/4 SR46A,46 CuFe 3* $2 tll6 (122) °8/4 SR39A,51 NalnTe2 tll6 (140) °8/4 SR39A,100 Ti t ÂGaSe2 rnS64 (9) °8/4 SR44A,64 Sr31n~P4 oP18 (58) °8/4 SR53A,38 Ca3AI2As4 K2SiP2 K2ZnO2 mS36 o/20 o/20 (15) (72) (72) (14) (31) (31) (15) (9) (9) (36) (169) (62) (15) ( l l) (62) (i 76) (159) (72) (186) (216) (164) (139) (62) o8/4 08/4 °8/4 [°8/6] °8/5 + °8/6 °8/5 + °8/6 °8/5 + °8/6 °8/7 °8/7 °8/7 °8/7 08/7 °8/7 [08/8] 08/7 08/8 08/8 °8/8 °8/12 °8/12 °8/12 °8/12 °8/12 SR48A,3 SR51A,77 SR33A,325 SR53A,57 SR50A,4 SR31A,169 SR40A,302 SR42A,86 SR50A,35 SR33A,259 HF55 SR40A,56 SR50A,48 SR52A,3 CS82 SR26,292 SR26,292 SR48A,38 SBl,78 SBI,76 SR32A,5 SR44A,18 SR40A,31

°8/3 5/2 Na2Ge2Se5 oP36 °8/3 5/2 TI~+Ge2S5 mS'72 °8/3 5/2 K2Si2Te5 oP80-8 o8/3 5/2 Na2Ge2S5 oS72 08/3 5/2 Pb~+B2Ss tP72 °8/3 5/2 Cs,Ge4S~o.3H20 mS92-8 °8/3 5/2 Cs4Ga2Se5 aP22 °8/3 5/2 Rb4ln2S5 aP22 ............................................. 08/3-333 7/3 BaTF~33*S~ mSl08 °8/3.333 7/3 Cs~Ga3Se7 mS60 ............................................. °8/4 2 - SiO2 quartz hP9 °8/4 2 - SiS2 oil 2 08/4 °8/4 o8/4 [08/6] °8/4 08/4 08/4 o8/4 o8/4 08/4 2 2 2 2 2 2 2 2 2 2 - GeS~ II HP - GeS2 LT - GeS2 HT Fe2 +Ga~S, LT BaAI:Te4 II BaF~ +S~ fl SrF~+Sa SrAl~Te~ BaF~+S~ a SrAI2Se~
tIl 2 raP36 raP48

(08/3)2 + °8/4 SR37A,17 °8/2 + (08/4)2 SR51A,27 °8/4 °8/4 08/4 °8/4 08/4 [08/6] °8/4 08/4 °8/4 08/4 08/4 o8/4 DL78 SB3,37 SR30A,55 SR42A,91 SR41A,72 SR46A,79 EJS82a SR45A,32 SR45A,32 SR39A,100 SR45A,32 SR45A,5

[°8/61 2 Ti~+AgTe2 raP24 ............................................. °8/5.5 7/4 BaAI4S7 oP24 °8/5.5 7/4 SrB407 oP24 °8/5.5 7/4 CaAl407 mS48 ............................................. °8/7 3/2 - Ga2S3 a mS20 °8/7 3/2 - Ga2Se3fl mS20 °8/7 3/2 - B203 II HP oS20 °8/7 3/2 - AI2S3a hP30 °8/7 3/2 CuF~5+S3LT oP24 08/7 3/2 Na~Mn~+S~
mS56

[08/8] 3/2 K3AI2As3 mPI 6 08/7 3/2 Ca~AI2Ge~ oP32 ............................................. 08/8 4/3 - Si3N~fl hPl4 08/8 4/3 - Si3N4a hP28 °8/8 4/3 Cs~Zn3S~ o/36 ............................................. °8/12 l - ZnS wurtzite hP4 °8/12 °8/12 °8/12 °8/12 1 l 1 1 -ZnS zinc blende ck'8 CaAl2Si~ hP5 BaZn2P~ tllO BaCu2S~ oP20

hP7
tIl 6-2 tPl6-2 tIi6-2 tPl4

ti16-2 oS28

III. Polycationic valence compounds: VECA > 8, AA = 0 and C'C' > O. Structure types are arranged in groups with the same VECA and C'C' values. These groups (separated with dashed lines) are ordered first according to increasing VECA values and then according to increasing C'C" values. In each group, the ordering of the structures is the same as in I. Classification code
cc VECA/C'A C" n/m" x

Type formula ~'38.111/I.33 3 1 Zn2P3S9 ............................................................................................. tr28-222/3 9/4 0 - P409 t'28.222/3 9/4 0 - P4S9II ............................................................................................. 2138-667/8 I 1 Na2Ga3Sb3 ............................................................................................. t8-333/0 3 - CCI3 18.333/0 ~8-333/0 18.333/0 ~8-333/0 18.333/0 ~8.333/0 18-333/0 18.333/0 ~8-333/0 18.333/0 18.333/0 ~8-333/0 18.33370 3 3 3 3 3 3 3 3 3 3 3 3 3 TiP2S6 In2P3S9 HgPS3 Sn2+PS3 LiGaBr3 Fe2+PS3 Pb 2+PSe3 CsSO3 Fe 2+PSe3 Cr3+SiTe3 KSO3 HgPSe3 CuCr3+P2S6

Pearson code
mS56

Spacegroup number (12) (167) (206) (62) (62) (43) (14) (2) (7) (1 l) (12) (14) (186) (146) (148) (I 50) (15) (15)

Code(s) of base tetrahedron(a) (08/2)2 + 18.333/0 (08/3)3 + 29"33313 (°8/3)3 + 29-333/3 o8/9 + 9/6 + i9.333/9 J8.333/0 18"333/0 18-333/0 18-333/0 18.333/0 18.333/0 ~8"333/0 ~8.333/0 18-333/0 18.333/0 t8-333/0 18.333/0 t8.333/0 J8.333/0

Ref. SR44A,90 SR32A,252 SR34A,I 18 SR53A,14 SR23,520 SR46A,109 SR44A,71 SR44A ,79 SR40A,94 SR53A,103 SR39A,76 SR54A,65 SR46A,364 SR39A,76 M88 SR20,336 SR44A,79 CLDR82

hRI56 c/208
oP32 oP32

oF'/2
mP56 aPl 0

raP20 raP20
mS20

raP20 hP20
hR30 hR30 hP30 mS40 mS52-12


20

CLASSIFICATION OF ANIONIC TETRAHEDRON COMPLEXES Table 2

(cont.)
Spacegroup number (163) (14) (14) (12) (14) (15) (33) (2) (2) (14) (14) (2) (4) (14) (15) (15) (59) (163) ( 11 ) Code(s) of base tetrahedron(a) t8.333/0 ~8.333/0 ~.333/0 ~8.333/0 t8.333/0 t8.333/0 ~8.4/1 t8.4/I '8.4/1 t8"333/0 + (18"4/I)2 + 18"5/2 °8/2 + 29/2 t8.4/1 + )8-667/3 ~8.4/1 + '8.667/3 )8.4/I + ~8.667/3 ~8"5/2 °8/3 + 29"333/3 °8/4 + ~8.667/3 + 29.333/3 t8.667/3 08/4 + 210/2 )8'667/3 ~9/6 ~9/6 )9/6 ~9/6 ~9/6 a9/6 ~9/6 ~9/6 (~8"333/0)2 + 29/0 o8/3 + .aI I/2 o8/3 + 311/2 °8/3 + (29"333/3)3 28.667/0 28.667/0 28.667/0 28-667/0 28"667/0 28.667/0 28-667/0 28.667/0 28.667/0 28.667/0 28.667/0 28-667/0 '8.333/0 + ~9.5/0 28.8/I 28"667/0 28-667/0 28'667/0 28.8, + 29/2 + 29/2 + 29/2 I Ref. SR53A,58 SR53A,57 TMKR82 SR44A,94 SR45A,75 SR45A,75 SR5 IA, I 7 SR52A,49 SR50A,41 SR50A,41 SR46A,230 SR50A,39 SR53A,34 ES82a ES82a SR34A,240 KE88 SR42A,125 SR51 A, 18 SR42A,85 WIKN82 SR18,176 SR17,166 SR42A,86 SR17,167 SR.24,128 SR45A,73 SR41A,101 SR34A,321 SR30A,353 SR30A,354 SR48A,193 SR30A,285 SR42A,43 SR41 A,426 NA83 SR32A,229 ZE88 SR20,30 ZE88 SR53A,48 SR40A, 13 SR42A,30 SR33A,37 SR21,371 SR41 A,360 CSS85 SR48A, I I SRSOA,9 CSS85 SB5,4 ENBS82 SKHT87 SR38A,9 SH88 SR45A,275 SR44A,27 SR39A,26 SR46A, I 0' SR29,21 SR23,42 SR48A, I 1 SH88

Classification code CCVECA/C'AC" t8.333/0 18'333/0 ~8-333/0 ~8-333/0 ~8-333/0 ~8-333/0 x8.4/1

n/m' 3

x 1

Type formula Mn] ÂSi2Te6 TI~ Â PSe3 Ag~PS~ K3SiTe~ K~SnTe~ K~GeTe~ Ba2Ge2Tes

Pearson code hP22
raP48 mP72

3 3 3 3 3 5/2

1 1 l I I 1

mS28 raP28 mS56
oP36

t8.4/1 5/2 1 Na4Ge2Ses II ~8.4/1 5/2 1 Na4Ge2Tes I1 t8"4/1 5/2 1 Na4Ge2Tes I ............................................................................................. ~8.4/2 5/2 0 - Se2Os ............................................................................................. ~8-5/2 2 1 LiGeTe2 )8"5/2 2 I Ba3Sn2P4 ~8-5/2 2 I Ca3Si2As, ~8.5/2 2 1 Sr~Si2As4 ............................................................................................. t8-5/3 2 0 - PO2 ............................................................................................. ~8-6/3.333 5/3 I/3 K~Sn~Ass ............................................................................................. 8'667/3 3/2 1 - Si2Te3 18.667/3 3/2 I Ba3Si4P6

aP22 aP22 mP88 raP28 aP24
mP36 raP36 mS36 mS48 oP56 hP40-20

raP26

)8.667/3 3/2 1 Na2Ga2Se3 hR18-4 (166) ............................................................................................. ~9/6 1 - lnSe HP raP8 (10) ~9/6 1 - InS 0/:'8 (58) )9/6 1 -GaSe 2H hP8 (187) t9/6 1 - GaS 2H hP8 (194) ~9/6 1 -GaSe 3R hRI2 (160) ~9/6 I - GaSe 4H hPl 6 (I 86) ~9/6 I - GaTe mS24 (12) 19/6 1 - SiP mS48 (36) ............................................................................................. 4/38"5/0 8/3 1 NasP3Os.14H20 aP60 (2) .......................... .- .................................................................. ~'28"857/2-5 7/4 I/3 - P4S7 a raP44 (14) ~a8'857/2-5 7/4 1/3 - P4S7 fl oP44 (60) ............................................................................................. 3a8"857/3 7/4 0 - P407 raP44 (13) ............................................................................................. 28"667/0 3 0 - AsBr~ oPl 6 (19) 28-667/0 28"667/0 28"667/0 28'667/0 28"667/0 28"667/0 28-667/0 28"667/0 28'667/0 28"667/0 28"667/0 3 3 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 BaTeS3 Na2SO3 AgHgAsS3 AgPb 2Â AsS3 K2TeSe3 CuPb 2 ÂAsS3 Na2TeSe3 KHgSbS3 A~Mn 2 ÂSb2S6 Na3AsS3 Ag~AsS3 oP20
hP12 mS24 raP24 raP24 oP24 mS48 mS48

(62) (147) (9) (14) (14) (31) (15) (15) (14) (198) (I 5) (11) (60) (4) (14) (29) (15) (135) (14) (33) (41) ( 1) (2) (14) (14) (14) (9) (14) (14) (61)

raP26 cP28 mS56
mPI8 oP32

28-8/0 5/2 1/2 K25205 ............................................................................................. 28"8/1 5/2 0 MgTe2Os 28"8/1 5/2 0 Ba2As2Se5 28-8/1 5/2 0 Ca2Sb2S5 28"8/1 5/2 0 Ba2As2Ss 28.8/1 5/2 0 Sr2Sb2Ss. 15H20 ............................................................................................. 29/2 2 0 - SeO2 29/2 29/2 29/2 29/2 29/2 29/2 29/2 29/2 29/2 29/2 29/2 29/2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 T1 ~+ HgAs3S6 AgPb 2 +Sb3S6 TI~ Â Hg3(AssSb2)S2o KAsSe2 (NH4) t ÂSbS2 NaAsS2 BaSnS2 CsSbS2 AgSbS2 T1 t ÂAsS2 CsSbSe2 CsAsSe2

raP36 raP36 oP72 mS96 tP24
mP44 oP44 oS148 aPl 6 aPI6 mPl6 mP16 mP 16 mS32 mP32 raP32 oP32

29/2 29/2 29/2 (-'8-8/I)2 + 29/2 + (29"333 '3)2 29/2 29/2 29/2 29/2 29/2 29/2 29/2 29/2 29/2


E. PARTHI~. AND B. CHABOT

21

Table 2 (cont.)
Classification code cc VECA/C'A C" L9/2 29/2 29/2 29"2/2-667
. . . . . . . . . . . . . . . . .

n/m' 2 2 2
5/3
. . . . . . . . .

x 0 0 0
0
. . . . . . . . .

Type formula RbAsSe2 NaAsSez AgAsS2 TIj ÂSb3S5
. . . . . . . . . . . . . . .

Pearson code mS64 oP64 mS96

Spacegroup number (15) (61) (I 5) (14)
. . . . . . . . . . . . . . . . . . .

Code(s) of base tetrahedron(a) 29/2 29/2 29/2 29/2 + (29"333/3)2
. . . . . . . . . . . . . . . . . .

Ref. SH88 SR45A,27 SR29,25 GNE82 SR52A,248 SR52A, I 2
. . . . . . . .

raP36
. . . . . .

29-231/2'75 29'231/2.75
. . . . . . . . . . . . . . . . .

13/8 13/8
. . . . . . . . .

0 0
. . . . . . . . .

Rb2AssS~3.H20 Cs2AssSt3
. . . . . . . . . . . . . . . .

raP96 oP184
. . . . . . . . . . .

(14) (60)
. . . . . . . . . . . . .

29/2 + (29"333/3)3 29/2 + C'9"333/3)3
. . . . . . . . . .

29.333/3 29.333/3 29.333/3 [29.333/4] 29-333/3
. . . . . . . . . . . . . . . . .

3/2 3/2 3/2 3/2 3/2
. . . . . . . . .

0 0 0 0 0
. . . . . . . . . . .

- As2031 - As203 II - As2S3 -Sb2S3 - As203
. . . . . . . . . . . . . .

raP20 mF20 raP20 oP20 cb80
. . . . . . . . . . .

(14) (14) (14) (62) (227)
. . . . . . . . . . . . .

29.333/3 29.333/3 29-333/3 29.5/4 + [29.2/4] 29.333/3
. . . . . . . . . . . . . . . . . .

SR44A,183 SR41A,213 SR38A,30 SR42A,22 SR44A,342 SR45A,76 KE88

210/6 210/6
. . . . . . . . . . . . . . . . .

1 I
. . . . . . . . .

0 0
. . . . . . . . . . . .

- GeS KSnAs
. . . . . . . . . . . . . . -P4S5

o/>8 hP6
. . . . . . . . . .

(62) (186)
. . . . . . . . . . . . . . . .

210/6 210/6
. . . . . . . . . . . . . . .

~'z10/2 Sal0/2
. . . . . . . . . . . . . . . . .

5/4 5/4
. . . . . . . .

2/5 2/5
. . . . . . . . . . . .

ot - P4Se5
. . . . . . . . . . . . .

mPI8 oP36
. . . . . . . . . . .

(4) (33)
. . . . . . . .

18.5/2 + 29.333/3 + 31 I/2 + '16/1" ~8.5/2 + 29.333/3 + 31I/2 + 416/1*
. . . . . . . . . . . . . . . . . . . . . . .

SR30A,353 SR37A,121 SR41A,101 SR39A,18

.

5a10/2-5 5'210/2'5
. . . . . . . . . . . . . . . . .

5/4 5/4
. . . . . . . .

1/5 1/5
. . . . . . . . . . . .

- P4S513
-- As455 . . . . . . . . . . . . . .

mPl8 mPI 8
. . . . . . . . . . .

(11) (I 1)
. . . . . . . . . . . . .

29'333/3 + 311/2 29-333/3 + 311/2
. . . . . . . . . . . . . . . . . .

39.5/0 39.5/0
. . . . . . . . . . . . . . . . .

2 2
. . . . . . . .

1/3 I/3
. . . . . . . . . . . . .

- PI2 NaSO2
. . . . . . . . . . . . .

aP6 rnPl6
. . . . . . . . . . .

(2) (13)
. . . . . . . . . . . . . . . .

39-5/0 39.5/0
. . . . . . . . . . . . . . .

SR20,254 SR20,331 SRI6,269 SR42A,31 SR28,55 SR39A,19 SR31A,82
. . . . .

311/2 31I/2 31I/2 311/2 31I/2
. . . . . . . . . . . . . . . . .

I 1 I 1
1
. . . . . . . .

I/3 1/3 I/3 1/3
I/3
. . . . . . . . . . .

- AsS realgar - AsS II - SN - AsS fl - SeN
. . . . . . . . . . . . . . .

rnP32 raP32 rnP32 rnS32 mS32
. . . . . . . . . . .

(14) (14) (14) (15) (15)
. . . . . . . . . .

31I/2 29-333/3 + (311/2)2 + 416/1" 311/2 31I/2 31I/2
. . . . . . . . . . . . . . . .

~'212.67/1.5 7a12'67/1.5 7/212-67/1-5 7,212.67/1.5

3/4 3/4 3/4 3/4

3/7 3/7 3/7 3/7

- As4S3/3 - As4S3a - P4S3 - P4Se3

oP28 oP28 oP56 oPl 12

(62) (62) (62) (62)

29-333/3 29.333/3 29.333/3 29-333/3

+ + + +

(416/I)3" (416/1)3" (416/1)3" (416/1)3*

SR39A, 17 SR38A,29 SR50A,55 SR23.402

References: AD87, AVERBUCH-POuCHOT, M. T. & DURIF, A. (1987). Acta Cryst. C43, 631-632; CLDR82, COLOMBET, P., LEBLANC, A., DANOT, M. & ROUXEL, J. (1982). J. Solid State Chem. 41, 174-184; CS82, CORDIER, G. & SCHAFER, H. (1982). Z. Anorg. Allg. Chem. 490, 136-140; CSS82, CORDIER, G., SAVELSBERG, G. & SCH.~'ER, H. (1982). Z. Naturforsch. Teil B, 37, 975-980; CSS85, CORDtER, G., SCHT, FER, H. & SCHWlDETZKY, C. (1985). Rev. Chim. Min~r. 22, 722-727; CSS87, CORDIER, G., SCILg, FER, H. d~, STELTER, M. (1987). Z. Naturforsch. Tell B, 42, 1268-1272; DJLR82, DOGGUV, M., JAULMES, S., LARUELLE, P. ~: RIVET, J. (1982). Acta Cryst. B38, 2014--2016; DL78, DONNAY, J, D. & LE PAGE, Y. (1978). Acta Cryst. B34, 584--594; EHS86, EISENMANN, B., HANSA, J. & SCI-LZ, FER, H. (1986). Rev. Chim. Mindr. 23, 8-13; EJS82a, EISENMANN,B., JAKOWSKI, M. & SCH.~FER, H. (1982). Rev. Chim. Minc;r. 19, 263-273; EJS82b, EISENMANN,B., JAKOWSKi, M. & SCH~ER, H. (1982). Mater. Res. Bull. 17, 1169-1175; EJS84, EISEN~NN, B., JAKOWSKI, M. & SCr~,~R, H. (1984). Rev. Chim. Min~r. 21, 12-20; EK88, EISENMANN,B. & KLEIN, J. (1988). Z. Naturforsch. Teil B, 43, 69-71; ENBS82, ENGEL, P., NOWACKI, W., BALIC-ZUNIC, Z. · SCAVNICAR,S. (1982). Z. Kristallogr. 161, 159-166; ES82a, EISENMANN,B. ~; SCH)~FER, H. (1982). Z. Anorg. ,41lg. Chem. 484, 142-152; ES82b, EISENMANN,B. & SCHXFER, H. (1982). Z. Anorg. Allg. Chem. 491, 67-72; F87, FLEET, M. E. (1987). Z. Kristallogr. 180, 63-75; GAKM87, GUSEINOV, G. G., AMIRASLANOV, I. R., KULIEV, A. S. & MAMEDOV, KH. S. (1987). Inorg. Mater. 23, 766-768; GNE82, GosToJIc, M., NOWACKI, W. & ENGEL, P. (1982). Z. Kristallogr. 159, 217-224; HF55, HAHN, H. & FRANK, G. (1955). Z. Anorg. Allg. Chem. 278, 333-339; IGGF88, IBANEZ, R., GRAVEREAU, P., GARCIA, A. & FOUASSIER,C. (1988). J. Solid State Chem. 73, 252-258; JJD82, JULIEN-POuzOL, M., JAULMES, S. & DAGRON, C. (1982). Acta Cryst. B38, 1566-1568; JLBP69, JOUBERT-BETTAN, C. A., LACHENAL, R., BERTAUT, E. F. & PARTHI~, E. (1969). J. Solid State Chem. 1, 1-5; KE88, KLEIN, J. & EISENMANN, B. (1988). Mater. Res. Bull. 23, 587-594; M88, MARSH, R. E. (1988). J. Solid State Chem. 77, 190-191; MBB83, MALINOVSKII,YU. A., BATURIN, S. V. & BONDAREVA,O. S. (1983). Soy. Phys. Dokl. 28, 809-812; NA83, NAKAI, I. & APPLEMAN, D. E. (1983). Am. Mineral. 68, 235-244; PSR82, PERENTHALER, E., SCHULZ, H. & RABENAU,A. (1982). Z. Anorg. Allg. Chem. 491,259-265; SH88, SHELDRICK,W. S. t~ HAEUSLER, H.-J. (1988). Z. Anorg. dllg. Chem. 561, 139-148; SKHT87, SAWADA, H., KAWADA, 1., HELLNER, E. & TOKONAMI, M. (1987). Z. Kristallogr. 181), 141-150; TKR82, TOFFOLI, P., KHODADAD, P. & RODIER, N. (1982). Acta Cryst. B38, 2374-2378; TMKR82, TOFFOLI, P., MICHELET, A., KHODADAD, P. & RODIER, N. (1982). Acta Cryst. B38, 706-710; TMS87, TAMAZYAN, R. A., MALINOVSKII,YU. A. t~ S1ROTA, M. I. (1987). Soy. Phys. Crystallogr. 32, 519-522; WIKN82, WATANAaE, Y., IWASAKI, H., KURODA, N. & NISHINA, Y. (1982). J. Solid State Chem. 43, 140-150; YRI88, YUN, H., RANDALL, C. R. & IaERS, J. A. (1988). J. Solid State Chem. B76, 109-114; ZE88, ZAGLER, R. & EISENMANN,I. (1988). Z. Kristallogr. 183, 193-200. * In the 416/1 base tetrahedron the central atom has only one shared anion neighbour (n/m" -- 1/2) and this base tetrahedron is therefore not shown in Fig. 2. According to x = 1/2 the central atom has two C'--C' bonds and one non-bonding orbital.

with a particular anionic tetrahedron complex might be formed. An important characteristic of a classification based on these codes is the absence of a relation with the overall compositions of the compounds. For structures to have the same code one of the conditions is that the anionic tetrahedron complexes must

have the same n/m" and VECA values. For example, the structures of such apparently unrelated compounds as InzP3S9, LiGaBr3, KSO3 or K3SiTe3 have the code 18.333/0. On the other hand, the compounds CasGa2As6 and CasSn2As6 can be considered for the consistent assignment to different codes. These compounds have the same overall formula, the


22

CLASSIFICATION OF ANIONIC TETRAHEDRON COMPLEXES compound AgPS3, shown in the middle part of Fig. 4. The predicted most probable base tetrahedron can be edge-linked to result in the observed molecular double tetrahedron, but it could also be two-cornerlinked to give an infinite tetrahedron chain or a ring. These differences in linkage cannot be predicted from simple calculations. They seem to depend on subtle differences in the environment. A good demonstration for this is offered by the °8/2 compounds Na2GeS3 and NazGeS3.7H20. They have the same classification code as AgPS3. Na2GeS3 has infinite chains of two-corner-linked GeS4 tetrahedra, but the hydrated compound Na2GeS3.7H20 has molecular edge-linked double tetrahedra. The water molecules occupy interstitial sites and do not contribute to the valence-electron pool but influence the kind of linkage of the GeS4 tetrahedra. Any subclassification to characterize the different kinds of linkages of a given base tetrahedron(a) which may occur with the different structures belonging to one (main) category will be based entirely on experimentally determined structure features and is thus beyond the scope of this work. It can be added that Liebau (1985) has already proposed two parameters: linkedness and connectedness which may be used advantageously for a subclassification of structures with not too complicated tetrahedron linkages.
Limits to the use of the classification

same space group, the same Wyckoff sequence, quite similar unit-cell dimensions and also infinite chains of two-corner-linked tetrahedra (C'AC'= 2). From these considerations one might presume that the structures are isotypic; however, they are not. The first compound (code 17.667/2) is a polyanionic valence compound, while the second (code °8/2) is a normal-valence compound without As--As bonds between neighbouring tetrahedra. For the 290 structure types studied, one can distinguish 55 different classification codes. They are listed in Table 1 as a function of the n/m' ratios of the compounds. Among them, only three are surrounded by square brackets which indicate that there is no equipartition. The reader should note that the list is certainly not complete. Particularly for silicates, which are all normal-valence compounds with equipartition, more classification codes of the type °8/C'AC' with 2 < C'AC' < 4 exist. The classification of the structure types, as proposed in Table 2, is performed in such a way that the codes of the compounds are grouped according to the most probable C'AC" values and not necessarily according to the observed ones. This allows a given structure type to be found more easily in the list even if there is no equipartition of the C'--A--C" links. Not included or only exceptionally included are structure types which are found mostly with oxygenor halogen-containing compounds. For each type is given: the classification code, the n/m" ratio, the type formula name, the Pearson code (Pearson, 1967), the space-group number, the codes and numbers of the base tetrahedra involved and a literature reference. For polycationic compounds the x values are also listed. We have considered as the C' atom the element positioned immediately to the left of the anion. There are isolated cases, such as T1l Hg3(As8Sb2)S20, where there are two kinds of equivalent central atoms. The n/m' value here is assumed to be 2 and the resulting code is 29/2. There is on average a good agreement between the most probable base tetrahedra predicted from the composition and the observed base tetrahedra. If this is not the case one has to remember that the calculated parameters are average values. Whenever more complicated base tetrahedron combinations are found in a structure the average structural features will correspond to the calculated parameter values used in the code of the compound. The proposed classification characterizes the base tetrahedron(a) used for the construction of the tetrahedron complex but not the details of linkages. The C'AC" parameter denotes only the average number of C'--A--C' links per tetrahedron; however, it is left open whether the tetrahedra are corner- or edgelinked. As an example, we may consider the °8/2

The limits to the use of the classification are as follows: (a) An anionic tetrahedron complex has to be formed and all the C' and A atoms must participate on the complex. (b) The valence-electron contribution of the elements is clear in most cases, but there are exceptions: we always have TI ~Â, Pb 2Â, but sometimes Eu 2Â. In the case of transition elements the valence is calculated from the observed structure of the tetrahedron complex. (c) From a certain point on, the Zintl concept does not hold and the predicted structural features of the (metallic) compounds do not agree with observation. For example, for BaTGa4Sb9 (Cordier, Schaefer & Stelter, 1986) one calculates the code 1/47.889/3.5, but two relatively long Sb--Sb bonds are observed for nine anions instead of one (short covalent) bond. (d) The occurrence of C'--C" bonds and/or electron lone pairs on the C' atoms can be predicted only for C'C "= 1, 2 or 3 with 95% probability. (e) Non-equipartition of the C'--A--C' links is unlikely but has been observed exceptionally in structures with codes [°8/6], [°8/8] and [29.333/4]. Inspite of certain limitations in the applicability of this classification scheme for structures with anionic


E. PARTHE AND B. CHABOT tetrahedron complexes, we believe that this approach is a useful one because: (a) It permits, based on the generalized (8-N) rule, a systematization of the structures. (b) It allows the recognition of certain systematic trends such as the equipartition of the C'--A--C' links, the competition of C'--C' bonds with lone pairs, and so on. (c) It will aide the synthesis of new compounds with particular structural features. We would like to thank Dr Karin Cenzual and Dr Nora Engel for very useful comments. We acknowledge the help of Mrs Birgitta Kiinzler and Mrs Christine Boffi with the preparation of the drawings and the text. This study was supported by the Swiss National Science Foundation under contract 2-5.537.

23

References

BROWN, D. B., ZUBIETA, A., VELLA, P. A., WROBELSKI,J. T., WATT, T., HAT~ELD, W. E. & DAY, P. (1980). lnorg. Chem. 19, 1945-1950. CoROrER, G., SCHAEFER, H. & STEELER, M. (1986). Z. Anorg. Chem. 534, 137-142.

ENGEL, N. (1986). Contribution d l'Etude des Structures d Groupements Tdtraddriques Anioniques. Etudes" Structurale Thdorique du Systdme CaO-AI203-SiO2-H20. Thesis, Univ. of Genrve, Switzerland. Present address: Musrum d'Histoire Naturelle, Geneva. HULL1GER, F. (1968). Struct. Bonding (Berlin), 4, 83-229. HULLIGER, F. & MOOSER, E. (1963). J. Phys. Chem. Solids 24, 283-295. KJEKSHUS, A. & RAKKE, T. (1974). Struct. Bonding (Berlin), 19, 45-83. KLEMM, W. & BUSMANN, E. (1963). Z. Anorg. Allg. Chem. 319, 297-311. LIEBAU,F. (1985). Structural Chemistry of Silicates, ch. 4.2, Berlin: Springer. PART~, E. (1972). Cristallochimie des Structures Tdtraddriques. New York: Gordon & Breach. PARTHE, E. (1973). Acta Cryst. B29, 2808-2815. PARTH~, E. & ENGEL, N. (1986). Acta Cryst. B42, 538-544. PEARSON, W. B. (1964). Acta Cryst. 17, 1-15. PEARSON, W. B. (1967). In Handbook of Lattice Spacings and Structures of Metals, Vol. 2. New York: Pergamon. SCH.MTER,n. (1985a). Annu. Rev. Mater. Sci. 15, 1-41. SCH.~-ER, H. (1985b). J. Solid State Chem. 57, 97-111. SCH)~'ER, H., EISENMANN, B. & MOLLER, W. (1973). Angew. Chem. Int. Ed. Engl. 12, 694--712. Structure Reports (1940-1986). Vols. 8-53. Published for the International Union of Crystallography. Strukturberichte (1913-1939). Vols. 1-7. Leipzig: Akademishe Verlagsgesellschaft. ZOLTAI,T. (1960). Am. Mineral. 45, 960-973.

Acta Cryst. (1990). B46, 23-27

Electron Density Distribution around Hydrogen Atoms in Linear Molecules
BY S. IKUTA

General Education Department, Tokyo Metropolitan University, Yakumo, Meguro-ku, Tokyo 152, Japan
AND M. ISHIKAWA, M. KATADA AND n. SANO

Faculty of Science, Tokyo Metropolitan University, Fukasawa, Setagayaku, Tokyo 158, Japan
(Received 22 June 1989; accepted 6 September 1989)

Abstract

Electron densities around H atoms in linear HX molecules (X--H, F, C1, OLi, ONa, CN, CP, Bell, Li and Na) were calculated in two directions relating to the major and minor radii. A Hartree-Fock level of theory with the 6-311G(2d, p) or MC-311G(2d, p) basis sets was applied. Van der Waals radii of the H atoms in the isolated molecules were estimated at the position where the electron density is 0-005 a.u. (0.0337 e A-3). The anisotropy of both the electron density distributions and the van der Waals radii of the H atoms was clearly confirmed in all the molecules. The radii and the degree of anisotropy are linearly related to both the Mulliken atomic charges on the H atom and the electronegativity of the 0108-7681/90/010023-05503.00

substituent X. The minor radius of the H atom depends strongly on the substituent X and increases as the electronegativity of X decreases. The predicted radii provide us with useful hints for considering the size of the H atom in molecules in the crystal. 1. Introduction The van der Waals radius of an atom in molecules in the major direction (sideways-on radius, rs), which is perpendicular to the chemical bond, is usually larger than that in the minor direction (head-on radius, rh) parallel to the chemical bond. This is called 'polar flattening' and has been pointed out quantitatively by Nyburg & Faerman (1985). They examined many © 1990 International Union of Crystallography