Antiferrornagnetism in Transiti on Metal Complexes Part 7.'-Critique of the Heisenberg Model and a Re-examination of the System of Three Copper(r1) ions in a Linear Array BY RICHARD F. A. KETTLE"W. JOTHAM,-/-SIDNEY AND JOHNA. MARKS$ School of Chemical Sciences, The University of East Anglia, Norwich NK4 7TJ Methods which have been used to modify the basic Heisenberg-Dirac-Van Vleck treatment of magnetically concentrated systems are discussed. The electronic states arising from the interactions between three collinear copper@) ions are investigated by a variety of methods using orthogonalised atomic or molecular orbital basis sets to examine the explicit form of the exchange integrals involved in the treatment of such systems. The complementary roles of chemical bonding and electron- repulsion effects are illustrated and it is concluded that, in this system, a low-lying doublet state, unrecognised in the spin-exchange treatment, may be of magnetic importance.One aspect of the increasing availability of variable-temperature magnetic suscep- tibility balances in recent years has been the identification of many compounds of the first transition series with sub-normal magnetic moments, which are generally recog- nised to consist of infinite or discrete antiferromagnetically coupled polymetallic units, and a handful of similar systems with an indisputable ferromagnetic coupling.2 This new information has led some workers either to seek more sophisticated models to describe their systems or, alternatively, to question the extant models on essentially empirical grounds.The models available detail spin hamiltonia and assume an effective coupling of the dipolar type as given in eqn (1) and (2). From these, pro- perties of interest, such as the temperature dependence of the magnetic susceptibility, may be calculated. In eqn (1) 2 = %o+A?e+Za (1) X0is the term representing the uncoupled system, Zeis the dipolar coupling term and #'a represents additional coupling terms which may be introduced in an extended model. The dipolar term may be expanded as a summation over metal pairs where Jij is the effective exchange integral and Siz,Six and Si,are the z, x and y components of the spin on the ith metallic centre. k is an essentially arbitrary constant which some workers assign as -2 to emphasise that there is, at least, a formal parallel to true exchange interactions (but which we shall take as -1 precisely to avoid this analogy).The value of y is unity in the isotropic Heisenberg model, the one most frequently considered. This model is a simple and quite good approximation to most real systems and therefore provides a convenient reference against which comparisons withother models are normally made. So, in the example of particular interest in this paper, the case t Department of Science, Kesteven College of Education, Stoke Rochford, Lincolnshire NG33 5EJ. Department of Chemistry, University Pilau Penang, Penang, Malaysia. 125 126 ANTIFERROMAGNETISM IN TRANSITION METAL COMPLEXES of three Cu" ions in a linear array (with negligible coupling between the terniinal atoms), the model predicts a spin quartet of energy Eo-+J and doublet states of energy ,Yo and Eo+ J.From these energies the equation for the molar susceptibility is found to be that given by (3), Ng2P2 1.5+1.5 exp (J/kT)+15exp (1.5JIkT)+ Na& = -3kT 2+2 exp (J/kT)+4exp (1.5JlkT) (3) in which all symbols have their usual meaning. Five main methods have been used to modify the basic Heisenberg model in the interpretation of the bulk susceptibility of some magnetically concentrated compounds. In the first of these, workers have allowed quantities other than the exchange integrals to be treated as variables to improve the agreement between observed and calculated susceptibilities, a procedure difficult to justify for those quantities which can be determined or estimated from independent experimental evidence.However, in infinite systems, the introduction of a temperature-correcting factor (T+0) is in the spirit of the Weiss molecular field approximation. 9 A second, closely related, method involves the introduction of temperature- dependent variables. For example, Lines has shown that when large unquenched orbital angular momentum is present, as in some Co2+clusters, the g factor may be replaced by a temperature and exchange-energy dependent factor. The Ising model, in which y in eqn (2) is given a zero value, has been used widely in the interpretation of infinite coupled systems such as linear metal chains, because it can be used to generate analytical equations for the properties of interest, whereas the Heisenberg model can only be used analytically with additional assumptions.6 However, for some cases it appears theoretically preferable to use an empirical equa- tion obtained from the Heisenberg m0de1.~ The fourth approach involves the introduction of other terms into the spin- coupling hamiltonian (1).For example, Sage has given the results of a high tempera- ture expansion of the magnetic susceptibility which includes second and third order terms in the dipolar coupling.8 Lines, Ginsberg and Martin have investigated the role of antisymmetric bilinear exchange terms of the type (4), Xa= C k'[Si.K, . Sj] (4)i,j where K, is an antisymmetric exchange tensor, in addition to the dipolar coupling terms in the Hamilt~nian.~ Evidence for this coupling, which is only significant when the local symmetry of the metal ion is low and the orbital angular momentum is not fully quenched, has been intensively sought.lo It now appears that this antisymmetric coupling can account for the very strange magnetic susceptibility data of the Cu40X6Y4 tetramers such as Cu,0Cl,(OPPh3)4.9~ l1 The final modification abandons the explicit use of eqn (1) and attempts to assess those electronic states which may be thermally populated and their spin characteris- tics.This approach admits the possibility that states arising from a variety of single- metal-ion configurations may be considered, although the extra states which are relevant at temperatures below 500 K are few in number.Using this approach we have shown that there is aprima facie case for the inclusion of a low-lying singlet state arising from partially filled d,2 orbitals for Cu" dimers, such as copper(I1) acetate, in addition to the familiar singlet-triplet pattern given by the dipolar coupling rneth0d.l Because of the relative crudity of the theoretical model, evidence for the existence of such a low-lying singlet state must ultimately be obtained from experimental data. We have therefore reinterpreted all available susceptibility data on dimeric copper@) R. W. JOTHAXI, S. F. A. KETTLE AND J. A. MARKS species using our model. In these investigations g and Nu were taken from experi- mental data so that the same number of adjustable parameters were used as in the conventional singlet-triplet model.In general this reinvestigation supported the postulate that a low-lying singlet state may exist in dimeric copper(1r) species.1* l3 No model which has been reported to date makes more than parametric allowance for the existence of the ligands which, almost invariably, bridge the interacting metal ions. This is an unsatisfactory situation in view of the commonly held view that it is the " through ligand " interactions which play a dominant role in the coupling between magnetic centres. Unfortunately, it is only marginally within the capability of current computers to include the ligands explicitly in an ab initio calculation and, even then, the accuracy obtainable falls far short of that needed to comment in any detail on the magnetic problem.Accurate calculations involving the metal ions only are much more feasible but the numerical accuracy with which the necessary integrals can be evaluated is quite illusory because of the neglect of ligands. In this situation we have preferred to tackle the problem algebraically and to use order-of-magnitude values for integrals. This approach permits an answer to questions of feasibility, of whether it is conceivable that a certain situation can occur. Whether or not it does has, of course, to be determined experimentally. We have been motivated in this approach by the occasional complete failure of the Heisenberg-Dirac-Van Vleck model when applied to some weakly-coupled transition metal ion systems.This may be indicative of a more general weakness in the theory than is immediately apparent, and which could to some extent be concealed by its adjustable parameters, and suggests that a general search for additional low-lying levels may be rewarding. In the present paper we report the results of such a theoretical investigation on a system of three copper(I1) ions in a linear array on a fourfold axis; the molecular symmetry considered is D4,,.This system, whilst having the advantage of being theoretically tractable by our methods, has the disadvantage that relatively few species are known to which the resultant theory may be applied. The level of sophistication is similar to that used by us in our work on dimeric copper(I1) systems.This means that we attempt to identify the major contribution to the state energies, arguing that when these are identical for two states there is a prima facie case that the two states may be of comparable energy. The only states of direct relevance to magnetic measurements are those within a few hundred wavenumbers of the ground state and so our study is aimed at detecting states which have the same major energy determining terms as the ground state. However, an accuracy of a few hundred cm-' is well beyond the limits of the present calculation so that they may be regarded only as indicative. The geometry which we consider permits two arrangements of ligands, is shown in fig.1. The energy level pattern given by the Heisenberg-Dirac-Van Vleck model for the case of three copper(I1) ions in a linear array is given in fig. 2. The corresponding equation for the temperature variation of the magnetic susceptibility is given by (3). In the limit J = 0, xh reduces to the sum of the susceptibilities of three independent copper(I1) ions, but when it is large xh has the form appropriate to a single copper ion. This latter case corresponds, physically, to the spin on the central atom being aligned anti-parallel to the other two. Full details of the present calculations are given in an Appendix and as a deposited document. As for the case of binuclear copper(I1) species, it is clear from our analysis that the ground and any low-lying states are derived from hole configurations involving the dz2 and d,zLY2 orbitals on the three copper atoms and that all major energy terms important for our purpose arise from within these configurations. In both the eclipsed and the staggered configurations (fig.l), the set of three ]z') orbitals transforms as 2Alg+A2uin D4h 128 ANTIFERROMAGNETISM IN TRANSITION METAL COMPLEXES symmetry. In the eclipsed codguration the set of three Ix2-y2) orbitals transforms as 2B1,+B2,,,but in the staggered configuration this set transforms as B1,+B2,+BZu, so that the two cases must be considered separately. The energy level diagrams which we obtained for the case of (a)staggered and (b)eclipsed terminal and central d,~-~2 orbitals are given in fig.3 and 4 respectively and are to be compared with fig. 2. The problem of the involvement of 42 orbitals cannot be unambiguously resolved by the present calculation. In particular, it depends on the relative magnitudes of core and repulsion integrals involving dz2 orbitals as compared with the corresponding integrals involving dX2-,,2 orbitals. However, a similar situation held in the case of binuclear copper(I1) species and, in view of our conclusions for that case which point towards the involvement of such levels, we believe that there is a case for admitting the possibility of a low-lying state in linear trinuclear copper(I1) systems. In the D4,,geometry which we have considered it is clear that any such low-lying state will be 2A2u(for both staggered and eclipsed geometries, as shown in fig.3 and 4). De-noting the energy of this doublet as relative to the ground state, the expression for the temperature variation of the susceptibility which results is Ng2/I2 1.5 + 1.5 exp(J/kT) + 15 exp(l.5J/kT) + 1.5 exp( -A/kT)+Na. (5)xfrl == -2 +2 exp(J/kT) +4 exp(l.SJ/kT) +2 exp( -A/kT)3kT 2 Z 2 t + (a)eclipsed (b) staggered @ Cu(I1) ion ligand site FIG.1.-The idealised geometry of three linear, six-coordinate, copper(@ ions together with the atomic and molecular axes used in this work. -0 (a)trinuclear Cu(I1) complex (6)trinuclear Cu(W complex (equilateral triangle) (linear) FIG.2.-The energy level patterns for three antiferromagnetically coupled copper(I1) ions predicted by the spin-exchange hamiltonian.R. W. JOTHAM, S. F. A. KETTLE AND J. A. MARKS The system of three co-linear copper ions has the advantage that it is one of the very few amenable to our appr~ach.~ It suffers from the disadvantage that there are, presently, few systems which may provide a test of the relative applicability of the susceptibility eqn (5) and (3). The series of compounds which most nearly correspond to the model discussed in this paper is that reported by Gruber et al. and reinvestigated by Figgis and Martin.14* l5 These workers prepared a series of five trinuclear copper complexes in which a central copper ion was further complexed by two copper-Schiffs base groups. In these complexes the central copper atom is linked to each of the terminal copper atoms by a bridging ligand, and very little coupling between the terminal copper atoms would be expected.The bulk suscepti- bility data for these compounds are described rather poorly by eqn (3) and the best- fit values reported for gyromagnetic ratio vary widely in the range 2.10-2.20. (In I I I I I I I I \ \ \ \ \ \ \ \ f t==A2, a I I 31 x2- y2> staggered eclipsed FIG.3.-The low-lying energy levels predicted by this work for three co-linear copper(@ ions. The degeneracy of the lower set of levels in the staggered configuration is to be contrasted with the non- degeneracy of the eclipsed configuration and the predictions of the spin-exchange hamiltonian (fig.2). this context the widely scattered g-values calculated from e.s.r. data seem improb- able.") However, no general improvement results from the application of eqn (5) (see table 1). Either more accurate experimental data, with careful avoidance of paramagnetic or other impurities, are needed for these compounds, or, alternatively, we conclude that they indicate that additional low-lying excited states are not impor- tant in the description of these compounds. A compound of a similar type is Cu3C1,(2-picoline N-oxide),*2H20, which has an almost constant magnetic moment of 1.03 B.M. per copper atom over the range 77-273 K,16*l7 and is presumed to have a linear structure similar to that of Cu3CI,-(CH3Cv2. Conversely, the room-temperature moment of the linear trimer Cu3Cl,(adenine H),4H20 is as high as 1.86 B.M.per copper atom.19 The series of anhydrous N-2-(2-hydroxyethylthio)henylarenesulphonamidatocopper(11) chelates 11-5 130 ANT IFERR0MAG N E TI S M IN TR A N SIT I0 N MET A L C 0MP L E X E S TABLE1.-MAGNETIC PARAMETERS FOR SYSTEMSOF THREE COPPER(II) IONS published data this work no. of 109~~l10901 1094 refer-%IT -J/ m3 m3 -J/ A! m3 compounda ence model * data 9 cm-1 mol-1 mol-1 modeld cm-1 cm-1 mol-1 14[CU(ES)I~CU(C~O~)~.~H~~ L 10 2.10 80 2.27 1.50 L 89.2 -1.98. LE 86.5 279.9 1.oo T 68.1 -1.42 TE 67.1 362.6 1.03 15 L 22 2.09 86 2.27 2.00° Lf 77.9 -13.4 LEf 57.1 11.3 8.92 Tf 60.8 -11.5 TEf 48.4 9.6 7.90 Cu(HPNS)I3 20 T 14 2.10 344 2.27 4.22 L 316.7 -1.23 LE 289.9 197.1 1.09 T 257.8 -1.13 TE 254.1 442.8 1.12 CU(HPTS)I~ 20 T 13 2.15 344 2.27 3.54O L 323.4 -1.890 LE 321.6 709.2 1.89.T 265.9 -2.028 TE 265.9 Q) 2.020 ~,~-PS)J~CU(C~O~)~*~H~O[CU( 14 L 10 2.15 360 2.21 0.47 L 351.7 -0.50 LE 336.6 356.9 0.45 T 286.4 -0.40 TE 286.4 00 0.40 [Cu(EHA)] ~CU(CIO 4) 2'2H20 14 L 11 2.13 400 2.27 0.42 L 413.1 -0.64 LE 413.1 co 0.64 T 341.3 -0.78 TE 341.3 00 0.78 15 L 18 2.04 380 2.21 5.2Se Lf 748.4 -22.08 LEf 748.4 to 22.0 T f 748.4 -22.0 TE f 748.4 co 22.0e [Cu( 1,3 -PH A)] ~CU(CIO 4) 2.3 H20 14 L 9 2.20 480 2.27 0.99 L 432.1 -2.53' LE 360.6 71.2 2.47 T 352.3 -2.48 TE 309.0 61.0 2.45 15 L 10 2.10 472 2.27 2.30e Lf 840.4 -3.25 LEf 840.4 165.9 3.25 Tf 840.4 -3.25 TEf 840.4 165.9 3.25" 14 L 14 2.12 460 2.27 0.39 L 507.2 -0.89 LE 507.2 to 0.89 T 426.2 -0.97 TE 426.2 00 0.97 15 L 12 2.17 474 2.27 2.08e Lf 685.6 -1.10 LEf 685.6 co 1.10 Tf 607.9 -1.14 TE f 607.9 Q) 1.14 16,17 L 5 2.05 00 2.27 3.62= L 00 -7.46 0 LE 00 03 7.46 T 00 -7.46 TE 03 00 7.46 I21 T 16 2.00 00 2.09 0.70 L 00 4.52 LE Q) to 4.52 T 00 -4.52 8 TE 00 to 4.52 e a ES = NN'-ethylenebis(salicylaldimine), EHA = NN'-ethylenebis(0-hydroxyacetophenimine), HITS = N-[2-(2-hydro-xyethylthio)phenyl]-4-toluenesulphonamidato-, HPNS = [2-(2-hydroxyethylthio)phenyl]-2-naphthalenesulphonamidat0-.2-PNO = 2-picoline N-oxide, 1,2-PS = NN'-1,2-propylenebis(salicylaldimine), 1,3-PHA = NiV'-1,3-propylenebis(o, hydroxyacetophenimine), PA0 = deprotonated pyridine-2-carbaldehyde oxime.0 = ( 2. {x(calc.)-$expt.)}2/n . i=l i >' g = 2.130, Noc = 2.83 x 10-9 m3 mol-1 = 225X 10-6 c.g.s. L = linear, LE = linear plus an additional excited state, T = equilatoral triangle, TE = equilatoral triangle and additional excited state. o 3 1.89 X 10-9 m3 mol-1 = 150X 10-6 c.g.s. : this error is such that it seems probable that either the data is unreliable or the model considered is inappropriate. f g = 2.160 used. R. W. JOTHAM, S. F. A. KETTLE AND J. A. MARKS are thought to be triangular systems,20 but we have investigated them utilising both linear and triangular models (in each case with and without the inclusion of additional thermally populated excited states). The trimeric complex Cu,L,(OH) S04*2H20 (where L = deprotonated pyridine-2-carbaldehyde oxime) is known to involve an equilateral triangle of copper atoms from X-ray data on the crystals.21 The magnetic moment is recorded as 1.00 B.M.(mol Cu)-' over a wide temperature range. It is interesting to note that, in the absence of structural data, it appears to be very difficult to distinguish magnetically even between the linear and triangular models for any of these compounds. CONCLUSION Although it is improbable that in the near future it will be possible to carry out detailed calculations on systems such as those considered here, but with explicit inclusion of the ligands, such that the results may be used to discuss magnetic data, it is, nonetheless, of interest to consider the relationship between such an approach and that presented here.Two situations may arise. First, the pattern of low-lying energy levels may be isomorphous with those considered here. This will be the situation if, as generally assumed, the low-lying levels are dominantly transition-metal in parentage. Each integral in our approach would then become a sum of integrals, the additional integrals being ligand or ligand-metal in origin, modulated by products of mixing coefficients. The most that could be anticipated is that the sum of these integrals would be of the same order of magnitudes as those evaluated on our metal-orbital-only basis (hence our philosophy of regarding order-of-magnitudes as of more importance than exact numerical values).Quite new energy terms would arise from extended configuration interaction but we would hope to have included the major contributions on our approach. Secondly, quite new low-lying levels might appear, largely ligand in parentage. If this were the case it is possible that none of the extant models would be applicable. For instance, it might well be essential that spin-orbit coupling be included explicitly. The indication is that future theoretical advances in this field may well depend on the development of simple methods by which more explicit recognition can be given to the presence of the ligands in polynuclear complexes. APPENDIX Although the case of three co-linear copper ions is one of the few which can be treated algebraically, it only becomes so by transformations which bring all large terms to diagonalpositions. The residual off-diagonal terms are then incorporated by perturbation theory.The basis sets which were chosen for each of the cases discussed are given in the appro- priate place in the text. The transformation properties of symmetry adapted combinations, &-A3 are 21 A, 1-3 1Z2> (a13 (QlJ (a2u) Ixz-y2) eclipsed (b1J (big)' (b2J (Al) Ix2-y2) staggered @2g) @Ig) (bJ. The transformation properties of the ten product functions, with S, > 0 for each basis are given in table 1. In each case, application of the projection operator technique gave rise to one quartet and eight doublet states as given in table 2. * The tables referred to in this appendix have been deposited with the N.L.C.132 ANTIFERROMAGNETISM IN TRANSITION METAL COMPLEXES The principal component terms of all of the energy matrices used in this paper are defined in table 3, in which lower case letters indicate Ix2-yz). The rounded and angular brackets have their normal meaning,2 and results obtained by applying the Mulliken approximation 23 for the smaller, but not negligible, terms are also given in table 3. Complete energy matrices can now be written out in block diagonal form, but it is con- venient to carry out a partial diagonalisation in some of the submatrices so that the larger terms are already diagonalised. We give the necessary eigenvectors and the partially diagonalised matrices for the cases of staggered and eclipsed Ix2-y2) orbital bases (QMO) in tables 4 and 5 respectively. The case of 1z2) orbitals is isomorphous to the latter case, and the appropriate matrix may be obtained directly from table 5 by introducing upper case subscripts and the appropriate symmetry labels.Final energy levels were determined by the normal perturbation technique. Terms in higher powers of S than 2 are omitted from the tables. For comparison with the results obtained by the use of the OAO basis it is con- venient to retain the core terms in G,, in table 5. Although this term may be regarded as zero in the OM0 basis, the corresponding term derived from the OAObasis has a significant value. The OAO basis used as an alternative method for the discussion of the eclipsed Ix2-y2) orbitals was constructed in the following way.24 The method consists of orthogonalizing the atomic orbital basis 4 through the matrix transformation where is the new orthogonal basis, and 9is the overlap matrix constructed with the overlap integrals Sii= 1, Sij=S,,,i=jf1,Sij=0,i=j+n,n>2.We have assumed that negligible overlap occurs between non-adjacent atomsz5 The S matrix has the form 9is non-singular and therefore the transformation 9-4 may be easily constructed by first diagonalizing 9to give 9'.If P is an orthogonal transform such that 9'= P9P-1 (A4) then 9-s = p-1p-+p. (A5) Alternatively, one may use a general relation developed for S matrices of dimension N with the form 26 Sii= 1, Si,ifl= S,Si,ift= 0,r > 1, for i = 1,2... N sin(iqn/N + 1)sin(jqn/N + 1)2c (A611J(Csp+)..= N+l -4=1 1+2S~os(qn/N+l)~ ' ldq<N. The resulting orthogonalized basis has the form R. W. JOTHAM, S. F. A. KETTLE AND J. A. MARKS (1+JZS)'] b + [(1+JZS)' +-(1-JZS)' +2(1-2S2)']c) where we omit the suffix from Subas it is the only non-zero overlap integral involved. (XI, XZ, x3) transform isomorphously as (a,b, c) in D4,,symmetry, so that we may replace (6) by A; = &(l+ $x2 -k x3)-A; = $(xi-J2~2+~3) (A8) 1A; = -Jz(x1-x3). The evaluation of the energy matrix for the eclipsed Ix2-y2) configuration is identical to the previous approach as the two bases are both orthogonal. However, a great simplifica- tion occurs when the core and repulsion integrals are evaluated in terms of their components in the QAO basis because of the orthogonality condition (xilxi) = aij.Only eight core and repulsion integrals arise with the OAO basis : J2= (x1x1 IIx1x1) = (x3x3 Ilx3x3) J$ = (XlXl lIx2;c2) J; = (x2x211x2x2) J: = (xi% k33/3) G: = (Xlllxl) = (x3lix3) G$ = (x2 11x2) GZ = (x1Ilxd GZ = (x1llx3). To obtain the Blg and B2umatrices for the QAO basis from those given in table 5, wz merely omit all terms in Sand S2 and replace each other term by its starred counterpart and, following the same diagonalization procedure, the eigenvalues are obtained by substituting the explicit values of the Lowdin integrals which are given in (A10)Jz= Jaa(l+3s')$-*S2Jab JZ = $S2(Jou Jbb) fJab( 1 YS')f $S2Jac JG = Jbb(1 f 5s2)+3S2J0h JZ = Jot( 1 +%S2) $s2Jab GZ = Ga+$S2(3G,+ Gb)+ SG,, (A10) GZ = Gb+iS2(Ga+ 3Gb)-I-2SGab GZ = Gub(l 2S2)+ @(G,+ Gb) G: = $S2(3G, + Gb) sGub-Part VI, J.P. Fishwick, R. W. Jotham, S. F. A. Kettle and J. A. Marks, J.C.S. Dalton, 1974, 125. (a)M. Kato, M. G. Jonassen and J. C. Fanning, Chem. Rev., 1964, 64, 99; (b)G. F. Mokoszka and G. Gordon, Transition Metal Chem., 1969, 5, 181; (c! E. Sinn, Co-ord. Chem. Rev., 1970, 5, 313 ; (d)P. W. Ball, Co-ord. Chem. Rev., 1969, 4, 361; (e)R. L. Martin in New Pathways in Inorganic Chemistry, ed. E. A. V. Ebsworth, A. G. Mad-dock and A. G. Sharpe (Cambridge, 1968). R. W. Jotham and S. F. A. Kettle, J. Chem. SOC.A, 1969, 2821. S. Koide and T.Oguchi, Ado. Chem. Phys., 1963, 5, 189. 134 ANTIFERROMAGNETISM IN TRANSITION METAL COMPLEXES M. E. Lines, J. Chem. Phys., 1971,55,2977. ti M. E. Fisher, J. Math. Phys., 1963, 4, 124. R. W. Jotham, Chem. Comm., 1973, 178. M. L. Sage, Inorg. Chem., 1971, 10,44. M. E. Lines, A. P. Ginsberg and R. L. Martin, Phys. Rev. Letters, 1972, 28, 684. lo P. Erdos, J. Phys. Chem. Solids, 1966, 27, 1705. J. A. Barnes, G. W. Inman, Jr. and W. E. Hatfield, Inorg. Chem., 1971, 10, 1725. R. W. Jotham and S. F. A. Kettle, Inorg. Chem., 1970, 9, 1390. l3 R. W. Jotham, S. F. A. Kettle and J. A. Marks, J.C.S. Dalton, 1972, 428, 1133. S. J. Gruber, C. M. Harris and E. Sinn, J. Chem. Phys., 1968, 49, 2183. l5 B. N. Figgis and D. J. Martin, J.C.S. Dalton, 1972, 2174. I6 E. Sinn, Inorg. Nuclear Chem. Letters, 1969, 5, 193.' H. Miyoshi, H. Ohya-Nishiguchi and Y.Deguchi, Bull. Chem. SOC.Japati, 1972, 682. l8 R. D. Willett and R. E. Rundle, J. Chem. Phys., 1964, 40, 838. P. de Meester and A. C. Skapski, J.C.S. Dalton, 1972,2400. 2o S. Emori, M. Inoue, M. Kishita, M. Kubo, S. Mizukami and M. Kono, horg~Chem., 1968, 7, 2419. 21 R. Beckett, R. Cotton, B. F. Hoskins, R. L. Martin and D. G. Vince, Austral. J. Chem., 1969, 22, 2527.'* P. W. Anderson, Phys. Rev., 1959, 115, 2. 23 J. B. Goodenough, Magnetismand the Chemical Bond (Interscience, New York, 1963). 24 P. 0.Lowdin, J. Chem. Phys., 1950, 18, 365. 25 R. S. Mulliken, J. Chim. phys., 1949, 46,497, 675. 26 G. W. Wheland, J. Amer. Chem. SOC.,1941, 63, 2035.