Madelung Constants BY PROF. DR. R. HOPPE INSTITUT FUR ANORGANISCHE U N D ANALYTISCHE CHEMIE DER UNIVERSITAT GIESSEN (GERMANY) Dedicated to Prof. Wilhelni Klernrn on the occasion of his 70th birthday Madelung constants are simple numbers which depend on the type of structure investigated. They are needed for the calculation (using the Born-Haber cycle) of lattice energies and erithalpies of formation of ionic compounds. Each Madelung constant is the sum of partial Madelung constants which represent the contributions of the individual ions to the total lattice energy. The partial Madelung constants depend on the ionic charge and, clearly though not stringently, on the coordination number. On the other hand, each Madelung constant can be represented by a sum of Madelung constants for related simple primitive A B structures. Surprisingly, these Madelung constants are numerically interrelated in a simple manner, and are related to the partial Madelung constants of interstitial sites, - Madelung constants of parameter-dependent structures (e.g. of the rutile or anatase type) and their variations with the structure-determining quantities are of particular interest. Madelung constants also yield information in the case of complex compounds and - surprisingly of non-metal compounds (e.g. XeF2, XeF4, XeFyXeF4). For many problems both inside and outside the realm of inorganic chemistry, it is necessary or desirable to obtain reasonably reliable information about the free energy of formation of unknown compounds or of compounds that have not yet been subjected to thermochemical investigation. A very recent example of such a problem is that of the thermodynamic stability of xenon fluorides [ I , l a l . Entropies can nearly always be found with a high degree of accuracy from thermochemical data for analogous compounds. The calculation of enthalpies of formation, on the other hand, is difficult even today. An important aid to the calculation of enthalpies of formation, and one that is often used because of its fundamental simplicity, is the Born-Haber cycle. It can be seen from Fig. 1 that the two simple steps I and 11, i.e. the formation of single atoms of the gaseous substances and the formation, from these atoms, of isolated gaseous ions, are followed by a third, more complicated step, in which all the ions combine to form the crystal structure Even for compounds which can be considered as consisting largely of ions, it is difficult to calculate the enthalpy of formation AHo corresponding to step IV (Fig. 1) with sufficient accuracy. There are several reasons for this. In the first place (and this is a fundamental weakness of the Born-Haber cycle), AH" is a relatively small difference of values that are often very large (ionization potential Isp, electron affinity E,, enthalpy of sublimation AH:ubl, dissociation energy Do, lattice energy U,), so that small individual errors can lead to a considerable absolute error. Secondly, the calculation of the lattice energy associated with step I11 [ I ] R. Hoppe, H . Mottorich, K . - M . Rodder, and W. Dihne, 2. anorg. allg. Chern. 324, 214 (1963); N . Burrleft, Endeavour 23, 3 (1964). [ l a ] R. Hoppe, Fortschr. chern. Forsch. 5, 213 (1965). Angew. Cliem. ititerrrat. Edit. Val. 5 (1966) / N o . I (cf. Fig. 1) is not simple. This lattice energy can be broken down into three terms as shown in Equation (I), U, = + [MF x f(zi) x NL x e 2 ] / r ~ ~Eric + Ecorr (I) where NL is the Avogadro number, e is the unit charge, rKAis the shortest distance between cations and anions, and f(zi) is a function of the ionic charge, which is often written out in full for binary compounds [f(zi) = z1 x 221, but is better included in the Madelung constant M F in the case of more complicated compounds. (For the sake of simplicity, it is always assumed that T = 0 OK.) +6ClG,, Fig. 1. Born-Haber cycle f o r NaCl (c) and K?[PtCI,] (b). The index f denotes the solid state. 95 In order to calculate the first term in Equation (l), i.e. the “Madelung component of the lattice energy”, we must know the Madelung constant. This is determined by the charge on the ions and by their positions in the unit cell; in the remainder of this paper it will for binary compounds always be based on the shortest distance between cations and anions. The calculation of the Madelung constants is laborious without the aid of modern computers; consequently, Madelung factors have so far been calculated only for about 50 types of structures (most of which are very simple, often parameter-free structures) (cf. list in 131). The term E,c, which takes into account the Born repulsion between the electron clouds that is due to non-Coulombic forces, must be present to explain the finite value of the equilibrium distances rKA. Thus if we were to consider only the Madelung component of the lattice energy, this would reach its “optimum” value at rKA = o! Further correction takes into account the mutual polarization of the ions, the zero-point energy, lattice-energy components due to additional bonding forces (e.g. bonds within complex ions), etc. (cf. [2931); these correction terms are combined in the in Equation (1). As can be seen from the review term E,, by Waddington [31, satisfactory accuracy in the calculation of lattice energies by means of Equation (1) can at present be achieved only for relatively few inorganic compounds, e.g. the alkali metal halides and alkaline earth metal chalcogenides. Impressive calculations have recently also led to the lattice energies of the higher alkali metal oxides “+I. tions[**J of the same compound, so that differences in the enthalpies of formation are essentially due to differences in the Madelung components of the lattice energies. A similar argument applies to the variation of c/a or of the oxygen coordinates in isotypic compounds of the same structural type, such as NaSc02, NaY02, NaInOz, and NaTlO2 151. 11. Partial Madelung Constants It follows implicitly from Ewald’s method [61 for the calculation of Madelung constants that the Madelung constant, and hence the Madelung component of the lattice energy, can be expressed as a sum of terms which are due to the different ions in the unit cell [71. We shall call these terms partial Madelung constants (PMF) 18991. If in a compound A,Bn all the A particles occupy equivalent positions and if the same holds for all the B particles, Equation (2a) is valid: MF(AmBn) = m x PMF(A) + n x PMF(B) (2a) For example, for the NaCl type [Oi - Frn3m, Z = 4, with Na+ in 4(a) and C1- in 4(b)], we haveEquation (3), MF(NaC1) = PMF(Na+) f PMF(C1-) (3) and similarly, for the CaFz type [Oz - Fm3m, Z = 4, with Ca2+ in 4(a) and F in S(c)], we have Equation (4). A. General +2x MF(CaF2) = PMF(Ca*+) I. The Madelung Constant and the Lattice Energy Equation (1) can be used for the calculation of enthalpies of formation only if sufficient Madelung constants are known and if the uncertainty associated with the terms En, and (in particular) E,,,, can be minimized. This makes the calculation of more Madelung constants desirable. The knowledge of Madelung constants is even more important in cases where the Madelung component of the lattice energy can be used alone, e.g. for investigating which modification of a compound (e.g. Ti02) is thermodynamically stable under normal conditions, or why a parameter-dependent structure [*] (e.g. the rutile type) favors certain parameters (e.g. c/a w 1/2112 and xo= 0.3), instead of other “adjacent”va1ues. It can often be assumed in such cases that the terms ECn and Ecorr are practically the same for “adjacent” modifica[2] M . Born: Atomtheorie des festen Zustandes. Teubner, Leipzig, 1923; Handbuch d. Physik, Springer, Berlin 1927, Vol. 24, p. 420. M. Born and A . Lande, S.-B. preuB. Akad. Wiss., physik.math. kl. 45, 1048 (1918); M. Born and A . Lande, Verh. dtsch. physik. Ges. 20, 210 (1918); F. Haber, ibid. 21, 750 (1919); M . F. C. Ladd and W. H. Lee, Trans. Faraday SOC.54, 34 (1958); E. Madelung, Physik. Z . 19, 524 (1918). [3] T. C. Waddington: Advances in Inorganic Chemistry and Radiochemistry. Academic Press, New York 1959, Vol. 1, p. 157. [4] L . A . D’Orario and R . H . Wood, J . physic. Chem. 69, 2550 (1965); R . H . Woodand L . A . D’Orario, ibid. 69,2558 (1965); 69, PMF(F-) (4) If, on the other hand, the particles A occupy j different types of positions (multiplicity zj), while the particles B occupy r different types of positions (multiplicity z,), the general equation is (2b), where the factors mj = zj/Z r j MF(AmBn) = x m j x PMF(Aj) j=l + x n r x PMF(Br) (2b) r=l and n, = z,/Z ensure that the individual PMF values are correctly weighted ( Z = number of formula units per unit cell). Thus for Mn2O3 [Tl - la3, Z 16, with MnI in 8(a), MnII in 24(d), and 0 in 48(e)], we obtain MF(Mn203)= 1/2 PMF(MnI)+ 312 PMF(Mnn)+ 3 PMF(0) (5) In contrast to the Madelung constants themselves, which offer only an overall picture of the energy relationships of a structural type, the partial Madelung constants provide information about the potential energies of the individual ions. For the calculation of partial Madelung constants, see [7-91. 2562 (1965). [*I In a parameter-dependent structure, the dimensions of the unjt cell (e.g. the ratio cia for tetragonal or hexagonal crystals) or the coordinates of the particles in the unit cell can be altered [ * *] “Adjacent” modifications differonly slightly in the structuredetermining parameters. [5] R . Hoppe, Bull. SOC.chim. France 1965, 1115; Chem. Zvesti 19, 172 (1 965). [6] P. P. Ewald, Ann. Physik 64, 253 (1921). [7] R . Hoppe, Dissertation, Universitat Miinster, 1954; 2. anorg. allg. Chem. 283, 196 (1956). [8] R . Hoppe, Habilitation Thesis, Universitxt Miinster, 1958. [9] H. G. v . Schnering, Habilitation Thesis, Universitat Miinster, without affecting the symmetry. 1963. 96 Angew. Chem. internat. Edit. / Vol. 5 (1966) 1 No. I 111. Partial Madelung Constants as Structural Arguments [9,101 The partial Madelung constants of the various ions can also be used as structural arguments: Using modern Fourier methods, very accurate information about the position of the center of gravity of the particles in a crystal can be obtained by X-ray analysis of single crystals, often with an average error of only h0.005 A. It is only in special cases, however, that electron density distributions can be found from precision measurements and used as a basis for conclusions regarding the type of chemical bonding [10al. It is therefore still useful to employ concepts such as ionic radius, sum of the radii, ratio of the radii, coordination number, etc., in the interpretation of crystal structures (which are described only geometrically by the positions of the particles). Of course, there is a remarkable discrepancy between these concepts, which are often only vaguely defined, and the accuracy of modern methods of structural analysis. Thus in addition to attempts to obtain improved definitions of these fundamental crystal-chemical quantities (cf. e.g. [ill), every possibility of obtaining further information from the structural geometry should be examined. Such a possibility seems to be offered by the partial Madelung constants: A three-dimensional description of the potentials and potential energies of the individual particles corresponding to the three-dimensional network of the centers of gravity is obtained by calculation of the partial Madelung constants. This discontinuous “grid” of partial Madelung constants may then be made as “dense” as required by the calculation of partial Madelung constants for interstitial sites. It has been found that even in the case of compounds with considerable non-ionic character, a better understanding of the reasons for the occurrence of certain structures is obtained by comparison of the partial Madelung constants with those of chemically related compounds (e.g. BrF3, XeF4, and XeF2) [llal. It has been known for some time that the Born-Haber cycle can also be applied in the case of compounds that do not consist of ions. Thus Kapustinskii[121 demonstrated that the lattice energies of non-ionic compounds can often be calculated with reasonable accuracy with the aid of the effective Madelung constants (cf. Section A.IX), which were introduced by Klemm [131. (In favorable cases, the lattice energies of covalent compounds are found with an accuracy of about +10 kcal/mole.) [lo] R. Hoppe and H. G. v. Sclinering, unpublished work (1962-1964). [IOa] Cf. in particular E. W o f i l e t a/.,Z . Elektrochem., Ber. Bunsenges. physik. Chem. 63, 891 (1959); Z . physik. Chem. N.F. 3, 273, 296 (1955); 4, 36, 65 (1955); 10, 98 (1957). [I 11 R. Hoppe, 18. IUPAC-Congress, Montreal (Abstracts of Scientif. Papers, page 73). [ I la ] Cf. for example [la]. [I21 K . F. Kapustinskii, Z . physik. Chem. B 22, 257 (1933); Zh. fiz. Khim. 5 , 59 (1943); Acta physicochim. USSR 18, 370 (1943); Quart. Rev. chem. SOC.(London) 10, 283 (1956); A. F. Kapustimkii and K. B. Yatsimirskii, Zh. ohshch. Khim. 19, 2191 (1949); 26, 941 (1956). [I31 W. Klemm, Z . physik. Chem. B 12, I (1931). Angc~w.Cliem. internat. Edit. 1 Vol. 5 (1966) / No. 1 IV. Madelung Constants of AB Types with Single Primitive Unit Cells [8,10,13al Simple relationships exist between Madelung constants of some compounds [7,141, for example: MF(CaF2) = MF(CsCI) +2 x MF(ZnS) (6) I n spite of reports to t h e contrary[ls], there is a n infinite number of such relationships for a given structure, since t he Madelung constant represents t h e s u m of all attractive and repulsive interactions of t h e ions in t h e crystal a n d can be broken d o w n i n a n infinite number of ways into partial sums. The Madelung constants of AB compounds with simple primitive unit cells (e.g. the CsCl type) are particularly important, since the Madelung constant of any structural type can be expressed as a sum of Madelung constants of single primitive (not necessarily isometric) AB types [*I. This follows from the fact that any structure can be derived by superposition of suitable, not necessarily isometric single primitive AB types. The Madelung constants of single primitive AB types are therefore “building units” for all other Madelung constants; they will be denoted in the remainder of this article by “MF. For t h e sake of simplicity, we shall discuss only Madelung constants of single primitive AB types with cubic unit cells. However, similar relationships exist for non-cubic single primitive AB types. For AB types with single primitive cubic unit cells, we shall always use the arrangement A+ in O,O,O and Bin x,y,z, the corresponding Madelung constant being denoted by oMF(x,y,z)k; for example MF(CsC1) “MF(1/2,1/2,1/2)k When based on the shortest distance A-B, these Madelung constants all lie between 1 (the “Madelung constant” of the gaseous molecule AB) and MF(CsC1) = 1.76267. . . (cf. Table 1). Table 1. “MF(x, y, z)k for special values of x, y, z [8]. ‘12 XI4 0 114 0 0 l/4 0 0 1,7626747730. 1,607425.. . 1.594367 1.412511... 1.388259 1.370681 1.141774... 1.087781... 1.035808... The variations of “MF(x,O,O), “MF(x,l/2,0), ‘MF (x,1/2,1/2) and OMF(x,x,x) in the range 0 x 112 are shown in Figure2. It can be seen that the variations follow roughly the same pattern, and are all characterized by the rapid fall-off from the maximum. The < < [l3a] R . Hoppe and H . G. Y . Schnering, unpublished work (1965). [I41 F. Bertaut, J. Physique Radium 13, 499 (1952); C. R. hebd. Seances Acad. Sci. 239, 234 (1954); P. Naor, 2. Kristallogr., Kristallgeometr., Kristallphysik, Kristallchem. 110, 2 (1958). [I51 C. C. Benson, Canad. J. Physics 34, 888 (1956); G. C. Eenson and F. van Zeggeran, J. chem. Physics 26, 1083 (1957). [ * ] The dimensions of the unit cells of these AB types need not be the same as those of the structural type, thz Madelung constant of which is to he expressed as a sum. 97 Madelung constants of single primitive A+B- types with non-cubic unit cells are also smaller than 1.76267.. ., but some are smaller than 1.0, and they may even be negative (cf. Section X). 1.1 1.i 1.E / 1.5 m +z 2 1.4 VI. Relationships between the Madelung Constants of Single Primitive AB Types “71 The Madelung constants of more complex structural types can all be expressed (cf. Section IV) as the sum of the Madelung constants of single primitive cells of the same size. It was assumed, therefore, that the Madelung constants of primitive AB types are not mutually related (“independent” Madelung constants) [*]; however, this assumption is wrong 1171: Any single primitive AB structure, irrespective of the size and symmetry of its unit cell, can be described by a non-primitive pseudo-cell obtained by multiplication of one, two, or all three of the principal axes; this naturally does not change the Madelung constant, which can therefore be expressed as the sum of the Madelung constants of other single primitive AB types. If the nonprimitive pseudo-cell is based on m times the a-axis, n times the b-axis, and p times the c-axis, we have m-1 0-1 D-1 1.3 1.2 In this equation, the primes in C’, r’, s’, and t’ indicate that the relationship r’ = s’ = t’ = 0 must not be valid. The factors arSt and ar’sy are simple numbers depending o n the geometry of the structure, and ensuring that the values of ‘MF are correctly weighted and normalized in the summation. 1.1 1.0 0.2 I I 0.4 0.6 For example, if we describe the CsCl structure by eight times the primitive cell, we obtain X- Fig. 2. Madelung constants (“MF) of cubic AB types with single primitive unit cells [arrangement: A+ in (O,O,O) and B- in (x,y,z)J. V. Partial Madelung Constants of Single Primitive AB Types Single primitive AB types, irrespective of size and/or symmetry of their unit cell have commutative sub-lattices [71. The partial Madelung constants of A+ and B- are then numerically equal, each being half of the total Madelung constant. Where the ions are monovalent, P M F ( x , ~ , z )is~ between 0.5 and 0.88 (cf. Table 1). Moreover, some non-primitive AB structures, such as the rock salt, sphalerite and wurtzite types, also have commutative sub-lattices. In these cases the partial Madelung constants of the A+ and B- particles are again numerically equal, each again being half of the total Madelung constant. An important example of a simple AB type with non-commutative sub-lattices is the NiAs type, in which the Madelung constant, as calculated by Zemannr161, is not a simple function of the partial Madelung constants of A+ and B-, which differ in their numerical values [*I. The partial Madelung constants of polyvalent ions in single primitive AB types are z; times as great as those of monovalent ions in the same lattice type (zi = number of unit charges on the ion in question). These unexpected and surprising relationships between the Madelung constants of single primitive AB types, which exist in every case irrespective of size and symmetry of the unit cell, indicate that we are still far from a real understanding of the Madelung constants. They also suggest that other numerical relationships may exist between the Madelung constants of single primitive AB types with isometric unit cells. VII. Relationships between the Partial Madelung Constants of Primitive AB Types and of Interstitial Sites “71 If we consider an infinitely large crystal of a primitive AB structure, and assume that only one interstitial site is occupied by an additional ion, the partial Madelung constant of this ion can be expressed as a sum of partial Madelung constants of single primitive AB types, and can then provide information about the energy of interstitial sites; defects can also be taken into account in this calculation by the use of additional terms. For example, the partial Madelung constant of a cation X+ occupying the position 1/2,1/2,0 in one unit cell only of a CsC1type crystal (A+ in O,O,O and B- in 1 / 2 , 1 / 2 , * / ~ > is: PMF(X+) = [16] J. Zemann, Acta crystallogr. 11, 55 (1958). [*] They were recently calculated as functions of c/a; see [13a]. 98 P M F ( ~ ,0,0) 1 1 122 x PMF(2, z,0) = 0.24329466 (9) [I71 R. Hoppe, unpublished work. Angew. Chem. internat. Edit. 1 Vol. 5 (1966) 1 No. I VIII. Reduced Partial Madelung Constants In isotypic compounds of the same formula type, but with different valences(e.g.Cu+Cl-, Znz+Sz-, Ga3+As3-), the partial Madelung constants of the ‘‘cations” are related as the squares of the valences zp, if the valence factor f(zi) fromEquation (1) is included in the Madelung constant. Similar relationships exist for more complex structures. In addition to the partial Madelung constants PMF(Am’) and PMF(Bn- ), it is therefore convenient to define reduced partial Madelung constants [lo, 181 as the partial Madelung constants divided by z,: i.e. *PMF(Am+) E PMF(AmC’ and m2 *PMF(Bn-) E PMF(Bn-) ___ n2 These quantities enable us to compare the potentials in compounds of different types and with different ionic charges. The reduced partial Madelung constants are strongly affected by the peculiarities of the crystal structures, and (unfortunately) are by no means constant. A number of typical values are given in Table 2, and Figure 3 shows how the reduced partial Madelung constants of cations vary with the coordination number and the ionic charge. The reduced partial Madelung constants as found from Figure 3 are maximum values, and the values found for real structures are usually lower. For the “coordination number” in such cases, see Section 1X. 1X. The Madelung Constant as a Function of StructureDetermining Factors such as Composition and Coordination Number Many attempts have been made to estimate Madelung constants. Klemm [131, in his simplified expression (10) for the lattice energy, introduced an “effective” Madelung constant a’. He showed that a’ is practically constant for chemically similar compounds of analogous composition, but depends strongly on the formula type and on the bonds present. (The terms Enc and ECor,of Equation (1) are here included in a‘). Kapustinskii 1121 then showed that the lattice energies of many compounds can be calculated from Equation (1 l), often with good accuracy. (zA and zB are the ionic charges of A and B, and Z is the number of atoms per formula unit.) In this casethe Madelung constant is considered as independent of the coordination numbers of the particles A and B in the binary compound AmBn, since Equation (11) is based on Equation (12): MF(AmBn) = z ~ z x~ Zxx const. 2 ~ ZAX ZB x Z XMF(NaC1) ~ 2 (12) Table 2. Reduced partial Madelung constants *PMF of the cations in typical binary crystal structures. Ion Corn p ound *PMF(Am+) Na+ CaZ+ Bi3+ Sc’f Sn4+ A4 NaCl CaF2 BiFs ScF3 SnF4 AB4 2 A in: 1/4,1/4,114 0.874.. . 0.819.. . 0.797.. . 0.685.. . 0.657.. . 0.701.. . - + 3/4.3/4, ’ 1 4 Nbs+ I It is surprising to find that (contrary to the results of calculations, which give a different Madelung constant for each structural type) MF(NaC1) may be used for all possible AB structures. For other binary structural types, a simple calculation with the aid of Equation (12) gives effective Madelung constants that should depend only on the composition, i.e. on the formula type, and not on structural pecularities! The explanation for this is as follows: In binary compounds, we have: ZA 0.551.. . NbCls x ZB x Z SE EZ?. (13) Thus, according to Kapustinskii [Equations (1 1) and (la]: MF(AmBn) = 2 2 2 x MF(NaC1) 2 = zzt x PMF(NaLaCI). (14) \Kz=~ 0 11\19131 1 -3 2 m 1, 5 Fig. 3. Maximum values of reduced partial Madelung constants [13al of cations as a function of charge (m) and coordination number (Kz). It can be seen that, although Kapustinskii himself failed to notice it, this expression implicitly represents any Madelung constant as the sum of partial Madelung constants 1101. Kapustinskii also assumed that all compounds can be regarded as “NaCl derivatives” [in Equation (1 1) the shortest interionic distance rKA is assumed for the coordination number 61, a view that has also been expressed by other authors (cf. e.g. 1191). To understand this apparently bold simplification, it must be remembered that a deviation from the coordination number 6 [I81 D . H . Templeton, J . chern. Physics 21, 2097 (1953); 23, 1826 (1955). Angew. Chem. internnt. Edit. 1 Val. 5 (1966) No. 1 [I91 T. C . Waddington, Trans. Faraday SOC. 56, 305 (1960). 99 in a real structure rather frequently leads to lower coordination numbers, and nearly always results in lower partial Madelung constants, and hence in smaller Madelung components of the lattice energy. However, such a deviation can occur only if additional (e.g. covalent) bonds ensure that the decrease in the Madelung component of the lattice energy is overcompensated in some other way. If the overcompensation is comparatively slight, Kupustinskii’s equation (14) is applicable, even for compounds that cannot be considered as consisting predominantly of ions. For coordination numbers above 6 , the Madelung component of the lattice energy again changes only slightly, as can be seen from a comparison of the Madelung constants of the NaCl and CsCl types. Moreover, the shortest distance dKA, too, depends on the coordination number and always changes in the same direction as the Madelung constant. Thus alterations of the Madelung constant are in part “internally” compensated. MF=1.370681 d~-ildA.o=2 MF= 1386276 d,-,ldp.g = 2 n n n n If one does not wish to calculate lattice energies from Equation (1 l), but to estimate Madelung constants from Equation (12), it is more convenient in the case of binary compounds [71 to use Equation (15) in which curate: p + ($/P)Z = = MF(NaC1) is replaced by the more ac(16) 1.81 In Equation (16), p is the “average” coordination number of A a n d B [i.e. 1/3(8+2x4)=5.33 for the CaF2 type]. Templetun 1181later also attempted to represent Madelung constants of binary compounds as functions of the coordination number; he used the empirical equation (17) M F = 1.89 - l/ph (17) in which Ph is the weighted harmonic mean of the coordination numbers of A and B. The fact that all the straight lines in Fig. 3 meet at 1.8912 is probably closeIy connected with Equation (17). The approximations used by Huppe [Eq. (15)] and by Templeton give practically the same results, as can be seen from the examples in Table 3. Table 3. Madelung constants MF of the CsCI, NaCI, and ZnS types. _ _ _ ~ 1 Hoppe [7] CsCl NaCl Zn+S- I Templeton 1181 I I 1.761 1.727 1.638 I I 1.765 1,723 1.640 Exact value 1.7626.. . 1.7475.. . 1.6380.. lei Fig. 4. Simplz AB structures with the same “simple” coordination number, and the corresponding Madelung constants (sizes of the particles chosen arbitrarily). which are the “nearest” neighbors. To illustrate this, Figure 4 shows some simple AB structures the coordination numbers of which, when counted on the basis of the simple definition, are the same. The Madelung constants1171 of these structures, on the other hand, show the influence of the arrangements of the A and B sub-lattices. From Figure 5 it can be seen how the Madelung constant is affected by the nature of the A and B sub-lattices (as a function of the coordination number) (cf. also Table 4). . However, there is no strict relationship between the Madelung constant and the coordination number! Such a functional relationship does not exist even in the case of the Madelung constants “MF of single primitive AB structures. The reason is the simplicity of the concept of the coordination number, i.e. the fact that only the nearest neighbors (with opposite charges) are counted for compounds considered as consisting of ions, whereas the partial Madelung constants take into account the interactions of one ion with all the others, however far away and whathever the sign of their charge. Furthermore, in the case of more complex structures (e.g. crystal structures of the NaSi type [201), it is impossible to be completely objective in deciding Table 4. Madelung constants of simple cubic AB types. ordination number. C.N. 100 co- cubic cubic body-centered face-centered ,ub lattices start in ~ 2 A+: O,O,O B-: ‘/2,0,0 M F = 1.37068.. . o,o,0 o,o, 0 ‘/4,1/4,1/4 ‘14. l / 4 , 0 MF MF = 1.41504. = 1.40221.. . A + : O,O,O B-: ’ l z , 118. ‘Is M F = 1.56968.. . 3 A+: O,O,O B-: l / z , l / z 1 0 MF = 1.59436 MF = 1.63804.. . A + : O,O,O B-: 1 / z , O , O MF = 1.74755.. . 8 [20] W. Klemm, J. Witte, and H. G. v. Sclznering, Z . anorg. allg. Chem. 327, 260 (1964). cubic primitive - C.N. = A + : 0.0.0 B-: ’ 1 2 , ‘Iz, ‘/z MF = 1.76267.. . Angew. Chem. internut. Edit. 1 Vol. 5 (1966) 1 No. I I. Madelung Constants of AB Types [ 10.171 Homogeneous affine transformations lead to changes in the lengths of the axes and in the angles formed by them. For the sake of simplicity, the following examples are confined to special transformations in which the angle remains unchanged, and in which the changes in the principal axes can be expressed by one figure (the axial ratio c/a). 10’ ’ 2 I I 6 i Kz I 1. C h a n g e s i n t h e M a d e l u n g C o n s t a n t s on T e t r a g o n a l C o m p r e s s i o n o r E l o n g a t i o n of a C u b i c U n i t Cell[sl L 8 Fig. 5. Madelung constants and “simple” coordination numbers of simple cubic A B types with commutative sub-lattices. Primitive face-centered (A), and body-centered (0) sub-lattices. K z = coordination number. (m), The estimation of Madelung constants or the deduction of “effective coordination numbers” from calculated partial Madelung constants will always raise difficulties, since the precisely defined Madelung constant cannot be readily related to the rather vague though useful concept of the “coordination number”. Estimations of this nature are best carried out with the aid of Equation (18), in which *PMFi is the reduced partial Madelung constant and zi is the charge; pi corresponds to mj or n, of Equation (2b), and the summation is carried out as in Equation (2b). The * P M F i are found from comparable structures, or from graphs as in Figure 7. The following general rule is valid here, as for any other structure: Any affine transformation of a structural type with a finite Madelung constant gives a new structure which also has a finite Madelung constant if the shortest distance dA-B remains smaller than the shortest distances dA-A and dB-B. On the other hand, when the ratio dA-A/dA-B or dBPB/dApBapproaches zero, the Madelung constant approaches --m. It is assumed here that the Madelung constant is always based on the shortest distance dA-B. Table 5 contains some typical examples of Madelung constants of AB structures with single primitive unit cells derived from cubic unit cells by tetragonal compression or elongation. A characteristic feature is that Table 5. Changes in the Madelung constants M F of cubic AB structures on tetragonal affine transformation of the unit cell [81. A+ in: O,O,O;B- in: X,Y,Z @I. B. Special Part The Madelung constant of a structural type is of interest for the discussion of enthalpies of formation. However, it acquires a more general importance only when considered in relation to other Madelung constants. An “observed” crystal structure is only one of many, since it is “surrounded” by a large number of other structures (e.g. thermodynamically less stable modifications) which are fundamentally possible, but which do not occur at thermodynamic equilibrium or have not so far been found. It is particularly interesting in this connection to find how the Madelung constant of a structural type changes: (a) whengreater or smaller changes in the size and/or in the symmetry of the unit cell occur, while the coordinates of the particles remain constant in relation to the new axes, i.e. when the unit cell undergoes homogeneous affine transformations, (b) when the coordinates of the particles change in parameter-dependent structural types, while the size of the unit cell remains unaltered, or (c) when the changes described under (a) and (b) occur simultaneously. In the following sections, therefore, we shall discuss, not only individual Madelung constants, but mainly their dependence on the variations mentioned. Augew. Chem. internot. Edit. Vol. 5 (1966) No . I 1/2.0,0 1.386.. . --m 1/2,1/2,0 --% ‘/2.1/2,1/2 --%. O,0,lj2 ~/2.0,~/2 --co 1/4,1/4.0 -cc l/4,0,0 -a 1.386.. . 1.127.. . 1.095 . . . 1.158 . . . 1.244.. . 0 .9 3 4 .. . 0 .9 7 1 .. . 1.770.. . 1.370.. . 1.594 . . . 1.762 . . . 1.594.. . 1.087.. . 1.035.. . 0.774.. . 1.378.. . 1.615 . . . 0.911 . . . 0 .8 4 8 .. . -a 1.615 . . . --% --5 the Madelung constant of the CsCl type falls off rapidly towards --m on compression or elongation. That of the NaCl structure, on the other hand, remains finite, as can be seen from Table 6. Table 6. The Madelung constant of the NaCl type o n tetragonal affine transformation of the unit cell [8]. c/a 0 1 m I MF 1.3562.. . 1.7475.. . 1.615.. . 2. C h a n g e s i n t h e M a d e l u n g C o n s t a n t s o n H e x a g o n a l C o m p r e s s i o n o r E l o n g a t i o n of a C u b i c U n i t Cell[lol The CsCl and NaCl structures, which are fundamentally different in their cubic arrangements, are found to be 101 affinely transformed in the hexagonal arrangement (cf. Table 7). Table 7. NaCl and CsCl types as affine transformations of the same structure. The letters A, a, B, b, C, c denote the positions of the particles with the coordinates given. The cation sites are denoted by capital letters and the anion sites by small letters. Table 8. Schematic representation of the ZnS structure. (For explanation ofthe symbols, see Table 7.) 1 oooooooooo A '1l12 ++++i+++++ c 8/12 ++++++++++ b 7/12 A (and a) : 0.0,z B (and b): 1lz,2/3,z C (and c): 2/3, 1 1 3 , ~ c/a = : NaC1-type c/a = 1/3/2 : CsCI-type oooooo A +++++ib 0 0 0 0 0 0 c I/a ++++++ a n ++++++ c 000000 0000000 o00 C n 4/~2 0 0 0 0 0 0 0 0 0 0 3iI2 ++++++++++a oooooooooo A 0 A ( a n d a ) : O,O,z B(and b): 1/3,2/3,z C (and c): 2/3, lI3,z c/a = 1/65 : Zinc blende type Table 9. Schematic representation of the ZnO structure. (For explanation of the symbols, see Table 7.) oooooo A As we pass from the CsCl type (c/a = 1/6'/2)to theNaCl type (cla = vc),the simple coordinationnumber changes continuously from 8 to 6. If there were a direct relationship between the simple coordination number and the Madelung constant, the latter should change in the same direction as the coordination number, i.e. a(MF)/a(c/a)+ 0 between the values given for c/a. However, it can be seen from Figure 6 that this is not the case. The two Madelung constants MF(CsC1) and MF(NaC1) are, in fact, separated by a minimum. 1 718 4i8 oooooooooo A A(anda): O,O,z B (and b): 1/3,2/3,z +++++++ ++t-b oooooooooo B 3i8 ++++++++++ a 0 oooooooooo A c/a = 1813:ideal ZnO type approaching 1 for c/a + 0, land --co for c/a + -a, (cf. Figure 7). The Madelung constant of the wurzite type shows a similar dependence on c/a, although (as can be seen from Table 9) this type consists of hexagonally close-packed sequences of A and B particles. t Y \ \ 10 rn cla - Fig. 6. Dependence of the Madelung constants of the CsCl and NaCl types on the ratio c/a in the hexagonal arrangement[lO]. The change in the Madelung constants is also affected by the fact that during hexagonal elongation along [OOl], the shortest distance dA-B in the c s c l type increases rapidly for two (or six on analogous compression) of the original eight equally distant neighbors. The steep fall in MF(CsC1) for both larger and smaller values of c/a is directly related to this effect. Conversely, the relatively small decrease in MF(NaC1) on both sidesof cia = V a i s due to the fact that the six nearest neighbors d o not split u p into non-equivalent groups. The characteristic changes in the Madelung constants of these two binary structural types are related to the fact that, although the structures of many ternary compounds (double oxides, double fluorides) are variants of the NaCl type (e.g. the important cc-NaFeO2 type), no corresponding variants of the CsCl type have so far been found. The sphalerite type, in which the A and B sub-lattices are arranged somewhat differently (Table 8) also belongs to this group. Here the Madelung constant MF(ZnS) decreases monotonically on both sides of c/a = 1 / g ' / 3 , 102 0 m 1.0 2.0 3.0 40 cla- Fig. 7. Dependence of the Madelung constant of the sphalerite type on the ratio c/a in the hexagonal arrangement. An example of an AB structure with non-commutative sub-lattices is the NiAs type (cf. Table lo), the Madelung constant of which (Fig. 8) also passes through a maximurn[161, and approaches --a, for c/a + 0 and for cia -+ 00. 4 Maximum l.*O 1 601 20 140 160 cia - 180 2 00 Fig. 8. Dependence of the Madelung constant of the NiAs type (noncommutative sub-lattices) o n the ratio cia [161. Ordinate: Madelung constant. Angew. Clietn. infernat. Edit. 1 Vol. 5 (1966) / No. I we start with a body-centered arrangement of the anions with regularly occupied quasi-octahedral holes (cf. Figure 10) the significance of the ratio c/a = 1/2'/2 becomes immediately obvious: F o r this ratio the cubic sub-lattice (as indicated in Figure 10) and the favorable arrangement of the cations (with the charges +2 or +4) remain unchanged. Table 10. Schematic representation of the NiAs structure. ( N i : c ; As: A, B; for explanation OF the symbols, see Table 7). 1 oaoooooooo A A : 0.0,~ 4-++ ! c oooooooooO B +f+i i--I-+++i c ooooooooooA B: 1/2,z/3,Z C: 2/3,'/3,2 + I- 3/4 '/I '14 0 i--i: + T 11. Madelung Constants of Complicated Structural Types Recently calculated Madelung constants of some important structural types are listed in Table 11 (see below). n n n 0 0z = o 1 1. T h e M a d e l u n g C o n s t a n t of the R u t i l e Structure In the rutile type AB2, each cation A has only six equally distant neighbors B if the axial ratio c/a and the parameter x of the anion are related as Bollnow [211 has calculated MF(rutile) for a number of values of cia and x that satisfy Equation (19). A recent detailed publication [221 suffers from the disadvantage that MF(ruti1e) is based, not on the shortest A-B distance (or some other simple difference), but on an undefined average of the distances to the six neighbors 4+ 2!), which are generally not all equal; the nature of this average cannot be deduced from the values given, but it is obviously not a simple mean. Moreover, since the difference between the Madelung constant and the Madelung component of the lattice energy is overlooked, some of the conclusions drawn are incorrect. 559 056 m 583 0 60 607 629 0 6L cla - 652 674 Figure 9 shows MF(ruti1e) as a function of cia and x [lo]. The following points can be observed: I . The Madelung constant has a maximum for c/a :-= 1/2'/2 and x - = 5/16 [in accordance with Equation (19)]. It became known only recently that this maximum is due to the structural geometry of the rutile type [171. Thus if Anyew. Chem. internot. Edit. 1 C'ol. 5 11966) / N o . 1 Fig. 10. Structural geometry of the rutile type. T h e lengths and directions of the arrows show the displacement of the anions t o the true positions (sizes of the particles chosen arbitrarily). 2. No known compound with a rutile structure corresponds t o this ideal case. This is at least partly due t o the fact that the "grid" of Madelung factors (Figure 9) is calculated for point-shaped ions and does not take into account finite and even fixed ionic radii. In typical rutile-type compounds AB2, such as difluorides (e.g. ZnFz) and dioxides (e.g. SnOn), the ratio of the distances dA-B (1.9 to 2.1 8,) and dB-B (2.6 to 2.7 8,) is expected to be between 0.7 and 0.8. From the maximum of the Madelung constant at cia = 1212 and x = 5/16, it follows 697 720 0 68 Fig. 9. Two-dimensional diagram of a three-dimensional representation of the Madelung constants of rutile as a function of xanion and cla in accordance with Equation (19). The figures beside the curves are 10,. [MF(rutile)-4] [21] 0. F. Bo//~oM', Z . Physik 33, 741 (1925). [22] W. H . Buur, Acta crystallogr. 14, 209 (1961). 11491101 that dA-B/dB-B = 0.83 [by comparison, for cia = 0.6 and 0.295, we find dA-B/dB-B = 0.721; the deviations towards cia < 1212 are therefore understandable to some extent. 3. Even with a knowledge of the Madelung constants, it is not possible t o discuss the question of why the parameters of a given rutile-type compound (e.g. ZnF2) have the observed values (c/a = 0.666, x = 0.303: cf. [239 instead of other values close to these. X = Thus if these parameters are slightly changed, it can be assumed that the shortest distance dA-B [2.015 8, ( 2 x ) for ZnFz] will remain constant as is required by the concept of [23] W. H . Buur, Acta crystallogr. 9, 515 (1956); I / , 488 (1958). 103 fixed ionic radii, or that the molar cell volume [20.87 cm3 for ZnFz] will remain unchanged, as is to be expected if the volume increments of the various ions are constant. Examples are known which show that neither of these two concepts (which are mutually exclusive, cf. 1241) is strictly correct. This means that in the Madelung component of the lattice energy [Equation (l)] of “adjacent” rutile-type compounds, the numerator can be given accurately, but not the denominator (the shortest distance dA-B). This difficulty is encountered with all parameter-dependent structures. 4. Finally it should be borne in mind that abnormally short distances dBWBare found in dioxides with rutile structures, such as SnO2 and Ti02 (the shortest 0-0 distance in Ti02 is 2.53 A). The only other cases in which such short distances are found are complex anions, e.g. SO:-, PO:-, or where strong hydrogen bonding occurs. These short distances are due to the strong polarization of the 0 2 - particles by the highly charged cations (Ti4f etc.); they show that the picture of a compound consisting of rigid ions must be modified for these dioxides. 2. T h e M a d e l u n g C o n s t a n t of t h e A n a t a s e Structure The anatase structure also depends on two parameters, namely the ratio c/a of the tetragonal unit cells (cf. Figure 11) and the parameter zo of the anions. For c/a = 2.0 and zo = 1/4, we have a variant of the NaCl type in which half of the cation sub-lattice has been removed n systematically. Figure 12 shows the variation of the Madelung constant MF(anatase), and Figure 13 shows the variation of the partial Madelung constants for two limiting cases: in the first case, c/a varies while zo (=1/4) remains constant [six originally equally distant B neighbors of each A particle become unequally distant (4 + 2 ) as c/a increases], and in the second case zo changes with c/a in accordance with Equation (20) z*, = + a*/2c2 (20) (in this case, each A particle remains surrounded by six equally distant B particles). It is interesting to note that in compounds of the “filled-up’’ anatase type (or-LiFeO2 type), that have been studied so far, zo = zO*, i.e. for any given c/a the arrangement of the 0 particles is such as to give the optimum Madelung constant [5,24a1. 3. M a d e l u n g C o n s t a n t s of C o m p l e x S t r u c t u r e s It can be seen from Figure l b that in the case of coordination compounds, the Born-Haber cycle can be applied in various ways; the term “complexing energy” is also used with various meanings (it can be defined by steps IIIa or IIIb’ in Figure 1b [251). In stating the Madelung constants of complex structures, therefore, it is particularly important to specify the reference distance. The values in Table 11 are based on the shortest central ion-ligand distance in each case. n Table 11. Madelung constants of other structural types. Compound Type Type Si02 High cristobalite Low cristobalite Rutile type (ZnFz parameter) individual type individual type individual type Li3Bi type individual type individual type individual type individual type individual type individual type AZfB; AZ+B; Az+B, ZnFz NbC14 SnF4 SnI4 BiF3 FeCI3 T~IJ MoCli NbOI? NbOCI? MoOClz Ref. 4.4534. . 4.428 4.7656.. 191 [91 [lo, 13al 12.650.. A4+Bi 14.084.. A4+Bi 12.439.. A4+Bi 9.573.. A3+B, 8.261. . A3+B;7.738.. A3+B, 7.673.. A3+B, A4 +Bz-CT 13.542.. A 4 i B 2 - c ~ 14.311.. 14.099.. A4+B2-C; 191 [lo, 13al 1101 171 [91 [91 191 191 individual type Fig. 11. Unit cell of the anatase type (sizes of the particles chosen arbitrarily). MF individual type individual type PI [91 10.093.. 7.867.. 7.508.. 16.199.. 11,154.. individual type individual type individual type individual type individual type individual type individual type individual type 16.18,. . 11.553.. 14.591.. 11.791.. 15.873 11.792.. 15.981.. 11.031. . Madelung Constants of Molecular Lattices 20 lrlL9112) 22 24 26 cla 28 30 Fig. 12. Madelung constant M F of the anatase type as a function of cla and zo [13al. [24] W’. Biltr: Raurnchernie der festen Stoffe. W. de Gruyter, Leipzig 1934. 104 Compounds such as BrF5, SiF4, SbC13, and XeF2 form solids with structures in which the building units are individual molecules. Typical covalent bonds seem to [24a] R . Hoppe, B. Schepers, H . J. Rohrbortr, and E. Vielhnber, Z . anorg. allg. Chem. 339, 130 (1965). [25] R . Hoppe, Z . anorg. allg. Chem. 291, 4 (1957). Atrgew. Chem. internat. Edit. Vol. 5 (1966) No. 1 z0.z; 090 B- I distinctly polar, in agreement with spectroscopic findings (cf. [la]). This explains the apparently large enthalpy of sublimation (12.3 kcal/mole). I. Maximum 080t 2. T h e M a d e l u n g C o n s t a n t of t h e XeF2.XeF4 Structure o“20 22 26 2L c/a 28 30 Fig. 13. Partial Madelung constants P M F of the anatase type as functions of c/a and zo [13al. be present here. With the exception of Klemnz’s effective Madelung constants 1131, little was previously known about the Madelung constants of such structures. It is of interest, however, how these Madelung constants are related to those of typically ionic compounds. The Madelung constants of XeF2, XeF2.XeF4, and XeF4 are therefore briefly discussed in the following sections. 1 . T h e M a d e l u n g C o n s t a n t of t h e XeF2 S t r u c t u r e [la] Linear F-Xe-F molecules are present in the tetragonal unit cell of xenon difluoride (cf. Figure 14). The PMF values and the Madelung constant of this structure are given in Table 12. PMF(Xe2+) for the coordination number 2 fits readily into the pattern of PMF(A2f) values for higher coordination numbers, as does the Madelung constant MF(XeF2) itself [*I. Thus XeF2 is I I lk - @ XeD,@Fin<<XeF2y> Xe”,OFinc<XeF,.> Fig. 15. Crystal structure of XeF2.XeF4 (a = 6.64 A, b = 7.33 A, c = 6.40 A, B = 92 “ 4 0 ) (sizes of the particles chosen arbitrarily). Table 12. Comparison of Madelung constants and partial Madelung constants of XeF2, XeFrXeFd, and XeF4 [la]. I I I I I I 1 -1 I 1.932 9.4138 Reference distance Xe-F[A] P M F (Xe*+) P M F (Xe4+) P M F (FI-) P M F (FII-) P M F (FIII-) MF 0.60535 0.6749 0.86372 0.8 6209 16.200 0.63250 Si-F 9.03938 PMF(Si4’) 8.8262 0.86372 0.9157 PMF(F-) 0.8233 1 3.7992 I 1.909 =* -= 0.8184 -~ -- 3.87550 12.6972 12.4910 12.4889 3. T h e M a d e l u n g C o n s t a n t of t h e XeF4 Structure The monoclinic unit cell of XeF4 contains discrete, practically square XeF4 groups (cf. Figure 16); the PMF values and the Madelung constant are given in Table 13. Comparison with the data for XeF2.XeF4 Table 13. Madelung constants and partial Madelung constants of a number of tetrahalides with respect to the shortest distance A-B=I [la, 9, 101. I Fig. 14. Tetragonal unit cell of XeFz (sizes of the particles chosen arbitrarily). [*] For details, see [la]. Atigew. Clienr. internat. Edit. Vol. 5 (1966) 1 No. I PMF(A4+) PMF (FI-) PMF(FII-) P M F (FIII-) _ MF [Xe[4lF41 I I &“bI6lC141 I 1 1 1 1 I 9.4138 0.8233 I 0.8184 _ &ISn[6lF41 12.6972 ~ 10.5153 1.0364 0.7480 14.0841 I 8.9515 1.1102 ;:;;I: 12.6505 1 [Si[4lF41 8.8262 C.9157 12.4889 105 0F I shows that the PMF(F-) values and the Madelung constant are of the expected magnitude. Comparison with the data for SiF4 shows that the PMF(F-) values for XeF4 are smaller than those for SiF4. The reason is the electrostatically unfavorable planar shape of the XeF4 unit. The smaller value of PMF(Si4+) as compared with PMF(Xe4+) indicates the more intimate entanglement of the XeF4 groups. From the heats of sublimation (SiF4: 6.15 kcal/mole, XeF4: 15.3 kcal/mole) the Madelung constant of the XeF4 type would be expected to be larger than that of the SiF4 type. Table 13 shows that this is indeed the case. However, in considerations of this kind one has to be satisfied with modest accuracy. ..xeN Fig. 16. Crystal structure of XeF4 (a = 5.05 A, b = 5.92 A, c = 5.77 k, , fi = 99.6 ”). For the sake of clarity, only some of the F atoms are shown (sizes of the particles chosen arbitrarily). Received: October 5th, 1965 [A 491/275 IE] German version: Angew. Chem. 78, 52 (1966) Translated by Express Translation Service, London Structure Formation and Molecular Mobility in Water and in Aqueous Solutions BY PROF. DR. E. WICKE INSTITUT FOR PHYSIKALISCHE CHEMIE DER UNIVERSITAT MtfNSTER (GERMANY) Dedicated to Prof. W. Klemm on the occasion of his 70th birthday The study of the structure of water and of aqueous solutions has recently received new impetus from the efforts at commercial desalineation of sea water and from developments in molecular biology. The current view that, apart from single molecules, water contains only one type of structural element, namely ,,flickering ”network structures with tetrahedrally hydrogen-bonded water molecules (two-states model) is proving inadequate in the interpretation of new experimental data and in the calculation of fhermodynamic functions. After a critical discussion of the basis of this model and of the concept of hydrogen bonds, a second kind of structural element, i.e. a third state, is suggested: small aggregates of molecules containing mainly non-tetrahedral hydrogen bonds as well as some tetrahedral ones, and packed more densely than allowed by the lattice-like structure. These aggregates - dimers to hexamers - can be regarded as the primary products of disruption of the network structures, and displace the latter as structural components in water with increasing temperature or concentration of solutes. This “combined” model allows a consistent interpretation of the properties of water and of the various effects of dissolved substances. Introduction Intermolecular structures have long been held responsible for the anomalies in the physicochemical properties of water, and are also used to explain its behavior as a solvent. The most important method up to a few years ago of detecting these structures was based on the radial distribution function of the intermolecular distances as found from X-ray diffraction photographs of liquid water. The development of microwave and in particular NMR spectroscopy now enables us to take a closer look at the interactions between neighboring H20 molecules and between dissolved particles and H20 molecules. Information about the mobility of H20 molecules, particularly with regard to rotary motion (reorientation), 106 provides a better picture of the “structure”, as well as a more sensitive means of determining the effect of solute particles on the short-range order of the H20 molecules. In the following pages the development of current views will be discussed systematically and critically. It appears that we have been too ready to assume that the liquid structure of water is due to an intermolecular linkage of the same type as in ordinary ice (this will be referred to below as linkage by “tetrahedral” hydrogen bonds). Thus in addition to the larger structural units, i.e. the fluctuating networks or clusters, in which linkage is in fact mainly due to tetrahedral H bonds, too little attention has been paid to the possibility of a second, smaller type of aggregate, in which the molecules are held together by more dipolar non-tetrahedral hydrogen bonds. Atigew. Cheiir. intermit. 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