E L S E V I E R inorganica Chimica Acta 252 (1996) 185-194

Geometrical and algebraical invariances and general angular dependences of sets of s-p hybrid orbitals. Their angular overlap model relevance

John M. Kennedy, Claus E. Sch~ffer Department of Chemistry. University of Copenhagen. Universitetsparken 5, DK-2100 Copenhagen O. Denmark

Received 21 March 1996; revised 19 June 1996

Abstract

An s-p hybrid orbital can be conceived as a linear combination of a scalar and a vector and can be written as

h°( O, qb) = h(sp"; 0,cp)-~ h()(, 0,q0 = s sin)(+ p~(0,~k) cos)(

where p = cot2x determines the shape of the hybrid and

p~(0,~b) =p,(pxlp~( O, ck) )~g + p~.(pylp~( O, fp) )~g + p.(p.lp~( O, dp) )~g

determines its direction. The coefficients to the unit vectors along the three Cartesian axes, written as p functions, are the angular overlaps between these unit -ectors and the unit vector along the direction of the hybrid. The situation of four mutually orthogonal s-p hybrids has been analyzed by using these concepts and it has been found that these hybrids invariably lie pairwis¢ in perpendicular planes. Moreover, if each hybrid h~ with shape parameter X~ is replaced by a vector with the direction of the hybrid and the length sin2x~, then their veclor sum is vanishing. The tour-hybrid problem has three degrees of freedom determining shapes and relative directions. Formulas are given for a general analysis of the situation, and particularly for the analysis ~ on the basis of three experimentally determined angles - - of the shape and direction of a single lone-pair, which is necessary in an angular overlap model (AOM) context. The forrnafism of the AOM is shown to function on the basis of hybrid orbitals, and this property of the AOM gives the clue to its handling of non-linearly ligating ligands.

Keywords: Hybrid orbitals; Angular overlap model; Analysis of lone pairs

1. Hybr id orbitals and figand fields

Hybridization models are well-defined mathematical mod- els based upon symme~, and orthogonality. However, in general, these models have left the scene and their symmetry background is now used in chemistry in a different way. Hybrids of s and p atomic orbitals make an exception. They are still being used in organic chemistry [ 1-3] and in model work where it is attempted to bridge [4,5] organic and inor- ganic views. This paper is a contribution to streamlining the s-p hybridization model for the treatment of non-linear liga- tion within ligand-field theory.

Ligand fields come closest t~, .nemistry when they have additive character, that is, when the ligand field is described as a sum of contributions from the individual ligands [6] or, even simpler, from the individual ligators [7,8]. These con- tributions are non-spherical in the sense that they describe the energy splitting of a set o f d orbitals [9], say. Except in certain special cases, for instance cases of trigonal ligation

0020-1693/96/$15.00 © 1996 Elsevier Science S.A. All t~ghts reserved PIISO020-1693 (96)05313-3

[ 8 ] such as when PF3 is the ligand, the ligators have generally been assumed to perturb a standard set of d orbitals to give a diagonal [7,8,10] energy matrix. This may be exemplified in the angular overlap model (AOM) by planarly [8] coordi- nating O i l 2.

The more symmetry-restricted situations that occur when C N - or monoatomic ligands are considered have been asso- ciated with the concept of linear ligafion [9,11 ]. In this case the ligand field has linear, conical symmetry - - point group C®v - - and allows a symmetry-based energy discrimination [9,11 ] within the d orbital set of functions, which attributes to the ~r function one energy, to the two ~r functions another energy, and to the two 8 functions yet another one. This case of linear ligation thus leaves two independent energy differ- ences as the quantities m semiempirical parameters- - to be determined by comparison of ligand-field data with ligand- field model requirements. When all the pans of the additive field have C®~ symmetry, an alternative parametrization may be chosen [ 9,10] which shows that in this restricted case the

186 J.M. Kennedy. C.E. Schiiffer / hzorganica Chimica Acta 252 (1996) 185-194

AOM is mathematically equivalent to the point charge or point dipole model. Thus the two sets, {e,,',e,/} and {12,14}, of two independent empirical parameters of the two models are related to each other [ 9,11-13 ] by a linear transformation.

In spite of this equivalence, the AOM parametrization has in the last 25 years been the preferred one in the chemical literature for additive ligand fields [ 14,15] and, of course, for molecular-orbital-oriented [16] discussions [ 17]. This is primarily because the parameters of the AOM appeal to the chemists' intuition about bonding. Secondarily, it is because the AOM is not restricted to treating the special situation of linear ligation, but on the contrary embodies sufficient flexi- bility to allow the treatment also of situations of non-linear ligation, which are chemically more general. This point, which has already been mentioned above in a historical per- spective, has been in focus in recent years in connection with the ligand-field treatment of bidentate, heteroaromatic ligands, which coordinate planarly [ 6,18-20].

No general solution to the problem of the treatment of non- linearly ligating coordinating atoms has been offered but a rather general idea [21] based upon the concept of misdi- rected orbitals [5] or misdirected valency [ 14] has been persistent in the literature. These misdirections were sup- posed to occur in the case of, for instance, the non-planarly coordinating ligand ethylenediamine, where the bonding within the non-planar ligand should leave the lone-pair orbi- tals, located on the nitrogen ligators, misdirected relative to the central ion, i.e. pointing in the general direction of the central ion but directed slightly away from it. This idea, which originally [21 ] was conceived in order to give a ligand-fieJd account of chirality based upon consequences of the puck- ering of a chelate ring, was more recently taken up again [ 5 ] in a hybridization discussion but never worked out in detail from a ligand-field point of view.

However, an idea that might be P~med the idea of yet unligated lone-pairs has, in the more symmetrical situation of/z-hydroxido and/~-alkoxido (/z-alkanolato) complexes [4], had a considerable success. Use of the AOM [ 10], combined with the idea of tour mutually orthogonal, conical s--p hybrid orb:,tals on the bridging ligator, has resulted in the so-called GHP model which has been able to account for the interrelation between geometry and antiferromagnetism, not only for a number of binuclear di-/L-hydroxido complexes [ 22-25 ] but also for similar complexes of nuclearity three and four [26,27]. The direction of the lone pair on the bridg- ing three-coordinate oxygen atom is of key importance here, and this is where the idea of general s-p hybridization comes in.

The present paper is concerned with a functional and geo- metrical analysis of lone-pair orbitals, which is the prereq- uisite for bringing these orbitals into the ligand-field world under the auspices of the AOM. In this paper the tools for bridging between hybridization and ligand-field models will be looked at mainly from the hybridization side. A useful geometrical invarianee of a real s-p hybrid set will be dem- onstrated and used as a frame for an overlap matrix which

relates the four s-p hybrids to an s orbital and a set of three p orbitals. Useful algebraical formulas emerge. The results are exemplified in order to throw light upon the hybridization- model starting point from where ligand-field modeling may begin.

Finally, s-p hybrid orbitals are used as basic perturbing orbitals of the AOM. Thereby, the problems of the pertur- bation from a non-linearly ligating ligand and from a skew lone-pair are in principle solved.

2. The geometry of a set of hybrids. A geometrical invariant for s -p hybrids

In casual language, an 'sp'-hybridized atom' really means an atom whose valencies take up p dimensions (p = l, 2, 3) in Euclidean space. This terminology is generally used in connection with boron, carbon and nitrogen. So, for example, to say that a nitrogen atom is sp3-hybridized is synonymous with saying either that it is four-coordinated, or that it is pyramidally three-coordinated and carries an additional lone- pair. In this paper the term an 'sp3-hybridized atom' means that one s orbital and three p orbitals are mixed to give a set of four equivalent hybrids directed along the corners of a regular tetrahedron.

In the same casual language, to say that an atom is 'sp 2- hybridized' is synonymous with saying that it has its valen- cies in the same pla~e, whereas in this paper it means that its set of hybrids is directed along the corners of an equilateral triangle. Similarly, our meaning of an 'sp-hybridized atom' is an atom that has a linear, centrosymmetric set of two hybrids. In the three examples all hybrids within a set are equivalent in the sense of being congruent. In addition, they have two-dimensional rotational symmetry m. In the rest of this paper, this symmetry will be conserved, but we shall be concerned with sets of hybrids that contain equivalent as well as with sets containing inequivalent members. The equiva- lence arises from the same mixes of s and p character. So, from here onwards, a set is described by the s-p composition of each of its members. The rest of this paper is mostly about sets of four inequivalent hybrids.

An orthonormal set of one real s orbital and three real p orbitals is transformed into an orthonormal set of four real hybrids by an orthogonal transformation. If a member of such

t In this paper we restrict ourselves to considering only pure o" hybrids tha~ are real, linear combinations of standard real s and p orbitals. This restriction, according to Eqs. (1) and (9), prevents us from associating interhybrid angles with valence angles when these are smaller than 90 °. Complex linear combinations of the real p orbitals [28.29] may overcome this limitation at the expense of no longer representing 'standing waves' and pure o- hybrids. For example [28], four mutually orthogonal, equivalent hybrids whose squared urbitals have maxima along the comers of a square in the XF plane can be built up out of the complete s-p function space by letting them contain an imaginary amount of Pz orbital. Here the shapes of the hybrid orbitals are hardly defined, but it can he stated that the squared hybrid orbitals do not have o" symmetry.

J.M. Kennedy, C E. Schiiffer / inorganica Chiraica Acre 252 (1996) 185-194 187

Table I Sets of hybrids embodying only equivalent members. Note that in describing the symmetry of each set, the presence of pure p orbitals in the s-p set of four orbitals is conventionally ignored. If the full set of four orbitals is considered, i.e. if the pure p orbitals are included, then the symmetries change: D3h becomes C3~; D®h becomes C~ (holohedr,.d symmetry D,u,)

Set 7 (°) a X (°) Symmetry

{sp3,sp3,sp~,sp ~ } 109471 I/2 30 T# {sp2,sp2,sp 2 } 120 (J3)/3 35.264 D3n {sp,sp} 180 (~2)/2 45 D~n

a set of hybrids is chosen so that it contains positive amounts of s and p~ orbitals, it has the expression

h z = a s + f l p z ( a > 0 , f l > 0 ) (1)

This hybrid's symmetry is cr (more specifically, or +(C~,.) about the Z axis), and the hybrid may be associated with the direction of its p orbital part. which is well-defined because a p orbital may be conceived a.~ a polar vector. If s and p~ are normalized to unity, and ifEq. (2) is valid

a2+ /32= 1 (2)

then h~ is also normalized to unity. A hybrid orbital is math- ematically an unusual construction in that it consists of parts that transform as tensors of different ranks. Thus for h~ under the condition of Eq. (2) the squared coefficient a 2 is the degree of scalar character (s character) and/32 is the degree of vector character (p character) of the particular s-p hybrid, which is denoted by sp p where p = (fl2/a~). a and fl make up a two-parameter characterization of the shape of a partic- ular hybrid, p makes up a one-parameter characterization. This may be replaced by one shape angle, X, which is another one-parameter characterization defined by either one of the two equations, (3).

a = s i n X; ( a / ~ ) = t a n X; ( 0 < X < (~ ' /2 ) ) (3)

which implies

a 2 = I / ( I +p) ; p=cot2x (4)

X may be called a parametrical angle as opposed to a geo- metrical angle.

For the special cases referred to above, the 'sF"-hybridized atom' for p= 1, 2, 3 comprises (p+ 1) sp" hybrids. These sets of equivalent hybrids lie in p-dimensional Euclidean space. It is only for these cases of p being an integer that it is possible to talk in simple terms about an 'sp"-hybridized atom', which implies that one is talking about equivalent hybrids of that atom. In each of the three cases, the interhybrid angles, 7p, are .known from symmetry. Application of Eq. (4) gives a and X. This information is gathered together in Table 1.

It may finally be noted that the situation a = 1, which is not included in Eqs. (3) and ( 4 ), represents the pure s orbital. In the present context this is more special than a pure p orbital because it fails to define a specific direction in real space. It

may be considered to define all directions in the sense that it has or symmetry in every direction.

Having considered the special cases of sets of only one type of hybrid, more gel~eral sets of four bybrids are now considered. A redistribution of the s and p cheracters amongst the members of the set of four equivalent sp 3 hybrids causes the set to descend in symmetry from tetrabedral, while each of the four members retain their symmetry C=~, which is the symmetry of a cone. We may call this symmetry conical symmetry to distinguish it verbally from the other linear sym- metry D=n which is then cylindrical symmetry. The most general sets of s-p hybrids lack symmetry and consist of four inequivalent hybrids. However, even though the symmetry of :he set of hybrids disappears, a geometrical invariant remains. It is thus shown here that the four s--p hybrids invar- iably lie pairwise in mutually perpendicular planes.

The expression for an stY" hybrid orbital, laP(0,~), in the direction (0,~b), is given in Eqs. (5) and (6)

h°(0,&) =h(sp°; 0 ,¢ , )=h(x; 0,~p)=as+13p,,(O,d#) (5)

p,,(0,~) = lp~ + mpy + npz

=Px sin 0cos O+py sin 0sin ~b+pz cos 0 (6)

where l, m and n are the direction cosines of the vector whose direction is defined by the polar coordinates (0 ,0) . Alterna- tively, the coefficients to the real standard orbitals, p~, py and Pz, are angular overlap integrals (angular overlaps) between these orbitals and the real p,, orbitals along the direction ( 0,q0, e.g.

m = (pylpo(0,~k)) = (po(0,~b) Ipy)

= (pyl p¢¢ 0,~b))=g = (pc(0 ,~) Ipy)~ (7)

It is seen that a general expression for an s--p hybrid orbital contains one degree of freedom for shape and two degrees of freedom for direction (cf. Eq. (5)) .

The orthogonality condition for two hybrids, hi and hi, directed along (Oi,~pi) and (0j,cpj), can then be expressed in Eq. (8) as a vanishing overlap integral between them

(hA hi) = aiai +/3~/3j(pil pj)

= a, aj + #,pj(p,-pj)

= , , % + ~ i~( l i l j + m3nj + n ~ j )

= aiaj + ~,/3j cos Yo = 8o (8)

where Be= 0 when i ~ j and 6#= 1 when i = j . The radial fuaction of the s orbital and that of the p orbital set, bath normalized to unity, have been integrated 2 out in Eqs. (7) and (8). The expression, (p~l pj), may therefore be conceived as the angular overlap integral between the two normalized p orbitals. It can altemativ=Iy be conceived as the scalar

2 The hybrids themselves are not f a e t ~ into a radial and an angular function unless it is assumed that the s and p orbitals have the same radial function. This assumption is ~ lane. Actually, the whole discus- sion in this paper concerns angular variations only.

188 J.M. Kennedy. CE. Schiiffer / lnorganica C~imica Acta 252 (1996) 185-194

product,(p~-pj), of two unit vectors. ~Ai is the angle between these two unit vectors, i.e. the interhybrid angle, which, to be precise, is the angle between the two symmetry axes of the two hybrid orbitals with tr symmetry, cos~/~ is accordingly the angular overlap integral between the (normalized) p orbi- tals contained in the expressions for the two hybrids. The last equality of Eq. (8) shows that the angle between the two s-p hybrids depends on the shape parameters of the two hybrids..'tAb is result is not new (for some history see Refs. [ 1 ] and [2] ) but can be re-expressed as

cos y~ = - a~ajlfl~fli= - t a n X~ tan )6 (9)

Every chemist knows that two sp 3 hybrids, h~ and h/ (X~ =)6 = 30°), have an interhybrid angle which is the tetra- hedrai valence angle, y(Td) = 109.471 °. Analogously, for a pair of sp 2 hybrids (Xi = )6 -- 35.264°), the interhybrid angle is 120 °.

It is much less well-known that the interhybrid angle of 180 °, which follows from Eq. (8) for a pair of equivalent sp hybrids, follows also for any other pair of hybrids whose sum of s characters is unity (p, = pj- t). Only for p~ =pj = 1, that is X~ =36 = 45°, is d~ere the well-known set of the two equiv- alent sp hybrids. The situation ofx~ ~)6 will be quantitatively elaborated in the next section.

It is useful in this paper to have a running, numerical example to bridge the different sections. Here comes the first part of this example.

We assume that v - have two hybrids. For reasons that will become clear in the tollowing chapter, we denote them by ht and ha. ht is an sp 2 hybrid (Xi - 35.264°) while ha is an sp a hybrid (Xa = 30°). Eq. (9) allows us to make the statement that the interhybrid angle between a pair of hybrids of exactly these shapes is invariably given by

cos Tla = - ( 1/6) I/2

~'la--- 114.095° (10)

As expected, this angle lies close to the middle of the interval between the 120 ° of two sp 2 hybrids and the 109.471 ° of the two sp a hybrids.

The discussion of the more general situation is now resumed. Two hybrids hi and hj define a plane. Using the vector character of the p orbitals, the vector product (p~ A p./) lies along the normal to this plane. Now consider four mutually orthogonal s -p hybrids: h~; hi; hk; hr. They can arbi- trarily be divided into two pairs. Consider the angle between the two normals. By combining the well-known vector identity

(Pa APh)" (Pc APd) ---- (Pa'Pc) {Pb'Pa) - (P~'Pa) {Pb'Pc)

(11)

with Eqs. (8) and (9), the two terms on the right-hand side of Eq. ( 1 !) cancel when Pa, Pb, Pc and Pd make up any combination of different Pl, P~, Pk and Pt. Therefore any two normals are invariably perpendicular to each other.

In conclusion, the symmetry axes of four real s-p hybrids lie pairwise in mutually perpendicular planes. Thus the s-p hydrids can be thought of as invariably lying pairwise in mutually perpendicular planes. This conclusion is at variance with that of Ref. [ 1 ] which claims the mutual perpendicu- larity to be a sub-case for s-p hybrid sets of C~ symmetry.

This simple, geometrical result makes up a strong, quali- tative restriction on a set of s-p hybrids. The restriction means a reduction in the number of degrees of freedom of the four directions in real space, defined by four s-p hybrids, which as we shall see is three, as compared with the number of degrees of freedom for four completely freely chosen direc- tions in space, which is five. This latter number may be counted up as four times two (polar coordinates referring to the four directions) minus three (Euler angles referring to the orientation of the set of four directions in the same coor- dinate system). The geometrical result for a general set of four s-p hybrids is conceptually useful in connection with a visualization of the set of hybrids and in the next section the result will be used to choose a particularly sparse quantitative description.

3. An orthogonal matrix relating four general s-p hybrids to the constituent real standard atomic orbitals

By using the geometrical restriction on the four s-p hybrids described in the previous section, it is always possible to place one pair, denoted belGw by ho and h,, in the ZX plane and the other pair, h i and h 3, in the YZ plane of a Cartesian coordinate system. This placement of the coordinate system will be used in the rest of this paper as a frame for relating the four s-p hybrids to real standard atomic orbitals. This relationship will be derived by considering the four hybrids as two pairs. Each pair is constructed by a two-step process.

The hybrid ofEq. ( 1 ) may under the condition of Eq. (2) be rewritten, together with its orthogonal companion, as follows

h z = s sin to+ p~ cos to

h_e=s cos to+P-e sin to ( 0 < to_< (¢r/2)) (12)

where, for h~, ta is equal to the parametric angle X defining the shape of h~. However, to is not equal to X for h _ r More- over, when the p functions, Px and p.,. have been brought actively into play (cf. Eq. (15) ), as they soon will be, to can no longer be considered to be a shape angle of an individual hybrid. Rather it may be conceived as a shape angle of the set of hybrids. If Px and py are brought in inactively, that means, just added to the basis, {he, h_~}, so that the new basis, {h~, h_ e, Px, Py}, is obtained, then the situation is the one in which the directions defined by the four basis orbitals are those of the octahedron deprived of the two of its cis comers that point at - X and - Y.

Before continuing the general discussion it is appropriate to make the following comments. Firstly, it should be noted

ZM. Kennedy. CE. Schiiffer / Inorganiea Chimica Acta 252 (1996) 185-194 189

that when p~ and py have not been brought actively into play, ca = 45 ° corresponds to the conventional hybridization situa- tion with two equivalent s -p hybrids of sp type. ca increasing above 45 °, which, of course, means transfer of s character into the hybrid h~ pointing toward the positive end of the Z axis, results in the transfer of p character into the orthogonal companion orbital h_~. Secondly, the unusual property of these two particular hybrids mentioned in the previous section is noted, i.e. they change shape concertedly without changing direction. Thirdly, the four orbitals h~, h_. , p~ and py form an orthonormal set and accordingly span the full s -p function space. Fourthly, an equality applies to the p parts of the hybrids of Eq. (12) because real p orbitals behave as vectors:

p~ = - p_~ (13)

while an inequality applies to the hybrids themselves

h ~ - h _ ~ (14)

even for ca = 45 ° because the amount of s orbital contents is in all hybrids, by definition, non-negative.

The building-up processes for hybrids, outlined above, will now be described, h~ and p~, on the one hand, and h_,. and p:., on the other hand, will be combined to form two mutually orthogonal sets of two orthonormal hybrids, one set having the symmetry axes of its individual hybrids in the ZX plane, the other in the YZ plane. Using a self-e'~planatory notation the four hybrids are

ho = h: sin to' + Px cos to'

h2=h~cos M - p ~ sin w' (O<_ca'<_ (~ ' / 2 ) )

h: = h_ ~ sin ca" + p.~ cos ca"

h3 = h _ . c~s ca"-py sin ca" (0 < ca"< (¢ r /2 ) ) (15)

Inserting the expressions (12) and (13) into the expressions (15) , ho, for example, becomes

ho = s sin ca' sin ca + p~ sin ca' cos ca + p~ cos a,' (16)

This result and the results for the other three members of the orthonormal set of s -p hybrids are given in Table 2. The subscripts z, x and y to the real standard p orbitals in Table 2 imply the Cartesian-coordinate description of the hybrids first referred to at the beginning of this chapter. Thus, the Z axis is special because it is the intersection line between the plane

Table 2

C ~

o I

Fig. 1. Disto:le,.i tetrahedron, defined by four s--p hybrid orbitals, inscribed in a cube. ~ set-up is that of Table 2, where the hybrids ho and he define the ZX plane, the hybrids h: and h s the YZ plane (these planes hatched). The directions from the common origin of the cube and the tetrahedron ",o the four comers of the distoRed tetrahedlon have been indicated by open circles. In the example of a bridging alkoxido ligand, oxygen sits at the orion, the metal ions M: and M3 on h: and h~, respectively, and C on !1,..

hoh 2 and the plane blh3 3. If the Zaxis is considered as apolar axis, the hybrids ho and h2 are in the northern hemisphere in the sense that their polar angles 0o and 02 are acute. !1o is on the ( + X ) side (longitude q0--O) and h2 on the ( - X ) side (longitude q,-- z,). The hybrids ht and h3 are in the southern hemisphere ( 0 : and 03 being obtuse) with h, on the ( + Y) side (longitude ~o= ~r/2) and h3 on the ( - Y) side (longitude ~p= 3¢r/2, cf. Eq. (17) ). Summing up, for the azimuthal or longitudinal angles, ~0~, of the hybrids, hi = h(x, ; 0~,q~,),

,p~ = i(w/2) (17)

If the hybrids are referred to a right-handed coordinate system, the order h,, h2 and h3 of the hybrids viewed along the direction of ho ,~==urs in clockwise order (Fig. I ). Per- mutation of the shapes of the memOers of any pair of the four s-p hybrids results in a transition to the mirror-image situa- tion, which is enantiomomhous in the case of C: symmetry.

The parametrical angles ca, to', and ~a" are al l in the closed interval [01 (¢d2) ]. As ca increases, s character f lows from the hybrids in the southern hemisphere to those in the northern hemisphere. The ratio of the total s character in the northern hemisphere to that in the southern hemisphere is given by tan~-ca. As ca' increases, s character f lows between the two hybrids in the ZX plane from the - X side to the + X side.

3 it is by choice that this intersectur is the unique one thai has become the Z axis. One of the other two intersectors might have been chosen instead. Thus the same geometrical situation may be described by three different sets of ~a type parameters (~a= a~oz (our choice), ~ : and a~o3, say) depending on which of the three intersectors be chosen as the Z axis of Table 2 and Fig. 1.

Interrelationship between the four conical s-p hybrid orbitals and their s and real standard p orbital constituents. This sparse table arises (i) from choosing one of the intersecting lines of mutually perpendicular planes of pairs of hybrids as unique (Z axis) and (ii) from letting the pair of hybrids, ho and he, dsfiae the ZX plane, the other pair, h: and h3, the YZ plane. See also lege~ld to Fig. 1

ho b: he h3

s sin to' s in o: s in oJ ~ c o s to c o s to' sin ca cos tat' COS ~, Pz sin to' cos to --sin of sin co cos oY cos to --cos 6/' sin o~ py 0 cos to" 0 - sin co" p~ cos ¢o' 0 -sin to' 0

190 J.M. Kennedy. C.E. Sch after / Inorganica Chimica Acre 252 (1996) 185-194

The ratio of the total s character of the hybrid on the + X side to that on the - X side is given by tan2to '. The variation of to" influences the s flow in the I,'Z plane analogously. This information is collected together in Eq. (18).

tan2to = (Cro+Ot~)l(ot~+ot3);

tan2to ' = o~/a~; tan2d ' = a~/a~ (18)

Thus, there is no polarization when to, to', and to" are all equal to 45 °. In th~.s case all four hybrids are equivalent and the set has Ta symmetry 4

It is important to realize that, for all changes in to, to' and d ' , every flow of s character is accompanied by a simulta- neous counterflow of p character: as to increases, p character flows down the Z axis; as to' and to" increase, p character flows down the X and Y axes, resgcctively.

The general set of four m~tually orthogonal s -p hybrids may be viewed as a two-dimensional rigid body with three degrees of internal freedom, placed in a body-fixed and body- defined coordinate frame. The dimensionality is only two in the sense that the spherical coordinate r is not used. An indi- vidual s -p hybrid in this view has one parameter that deter- mines its shape and it may be conceived as placed along its own symmetry axis to which the hybrid provides a positive direction. If one wants to specify the orientation of the set relative to for instance an external Cartesian coordinate frame, this requires three parameters [ 8], for example, Euler angles ( ~/,, 0, q~), while a single hybrid only requires a spec- ification of its direction, for instance by its two spherical polar coordinates (0, ~o).

Consider again the running example, the beginning of which is in the previous section in connection with Eq. (10). In this example there were two hybrids ht, which is an sp 2 hybrid, and ha, which is an sp ~ hybrid. When referred to Table 2, both these hybrids lie in the southern hemisphere. Eq. (19) shows that the s characters for all hybrids must add up to unity:

) ' a 2 = ! (19) i

Since a l 2 = ( l / 3 ) and t~32 = ( 1 / 4 ) , Eq. (18) can be used to determine

4 Nowadays there is a trend to refer to such polarizations as quantitative symmet~ reductions and to search for features which may be regarded as measures of the quantitative symmetry reduction ill an attempt to describe the geometry of something in a symmetry framework. We emphasize that what follows is just one such quantitative measure of the symmetry reduc- tion. This is defined on the basis of noting that each choice of Z axis gives its own well-defined value of an to parameter (cf. the generalization of Eq. ( 18 ) ) ~o that cos2tois the intersector's polarizer and sin2to the intersecwr's degree of symmetry relative to the tetrahedral situation. The product of the three intersectors" degrees of symmetry is a unique quantitative measure of the degree of symmetry of the hybrid set. This degree of symmetry has in the notation of Footnote 3 the expression sin2too#in2too2sin2too3. An alter- native quantitative measure of symmetry reduction can be taken from the degrees of symmetry in three mutually perpendicular directions. sin2tosin2to'sin2to", say. There has been a certain interest in the quantifica- tion of deviations from a given symmetry these days [30].

to=40.203 ° and d ' = 4 9 . 1 0 7 ° (20)

In this example, there is more s character in the hybrids of the southern hemisphere than in those of the northern hemi- sphere. The hybrids of the southern hemisphere are confined to the YZ plane: there is more s character in that on the + Y side than on the - Y side, i.e. more in hi than there is in h3.

This concludes the discussion of the interpretation of the parametrical angles to, to' and to".

Cyclic permutation of subscripts 1, 2 and 3 on the right- hand sides of Eq. (18) gives rise to two new triplets of parametric angles analogous to { to, to', to"}. Each new triplet corresponds to a Z axis being placed so as to coincide with another intersector of mutually perpendicular planes. The coordinate system for our present choice (Table 2) has been depicted in Fig. i.

In the next chapter it is shown how, from the choices already made in the running example, the geometry of the entire southern hemisphere is already determined.

4. Relationships between parametric angles and geometric angles

In the set-up of Table 2 the polar angles, 0~, of the four hybrids may be expressed in terms of the parametric angles as follows

cot 00 = cos to tan to'

cot 02 = cos to cot to'

cot 0~ = - sin to tan to"

cot 03 = - s i n tocot to" (21)

The four 0i may, in fact, be replaced by a more general notation which is independent of the coordinate system:

Oo = Y~.t3 ( , r - Or) = ~1.o2

02 = Y2.,3 ( z r - 03) = 3'3.02 (22)

where, for example, To.t3 is the hybrid-intersector angle that is the acute angle between the hybrid ho and the hybrid plane hth3, or, what is the same, the angle between the hybrid ho and the intersector of the two planes hoh 2 and hlh3, i.e. between the ho and the present Z axis. Application of Eqs. (5) , (6) , (18) and (19) to Table 2 reveals that a hybrid- intersector angle (of which there are 12) may be expressed by the following formula:

cos y , . j ,=[a2(af j + a 2 ) / [ 3 Z ~ ( l - a 2 - a ~ ) ] l / 2 (23)

The sum of both hybrid-intersector angles to a common plane is tile interhybrid angle in the plane perpendicular to the common plane:

~/~.ik + Yt.jk = "Fit (24)

There are four polar angles in our gecmetrical description but only three independent ones. A way of expressing this fact quantitatively can be seen directly from Eq. (2 ! ):

J.M. Kennedy. C E. Schiiffer / Inn rganica Chimica A cta 252 (1996) 185-194 191

cot 00 cot 02+cot 01cot 03

= cot 00 cot 02 + cot( ~r- 0: ) cot( z r - 03) = 1

cot `yi.s* cot 'Yl.)k + CO': ̀yj.il COt 'Yt¢.il = 1 (25)

The last line in Eq. (25) shows the result for each of the three lines of intersection of mutually perpendicular planes.

It is evident from Eqs. (23) and (24) that the intersector between two equivalent hybrids is the bisector of the inter- hybrid angle defined by the hybrids. When hi is equivalent to ht, and h s is equivalent to hk, Eq. (25) degenerates into Tor- kington's formula [31]. This applies to C2,, symmetry (as well as the only accessible supergroup symmetry, T a, when all four hybrids are equivalent).

A final very useful formula [ 1,2] provides the s characters of three hybrids given the three interhybrid angles defined by these three hybrids:

or2 = cos "Yij COS "yik/(COS `yij COS 'y/t-- COS ̀ yjk) (26)

Returning to the running example, it is known already that to=40.203 ° and to"=49.107 ° (Eq. (20)) and that at 2= 1/3 and t~32 = 1/4, whereupon Eq. (21) can be used to calculate the two hybrid-intersector angles into which `yt 3 of Eq. (10) is divided:

Yl.o2 = 7 r - 0~ = 53.301 °

3'3.o2 = 7r - 03 = 60.794 ° (27)

The geometry of the entire southern hemisphere of our exam- ple has now been revealed (including the placement of the Z axis (Table 2) ). In order to complete our example, so that it comprises a full set of four s-p hybrids, one more piece of parametric information is required. This information con- cerns the northern hemisphere, and we choose it by requiring h2 to be an sp 4 hybrid, i.e. a hybrid with 20% s character. The entire shape information of our running example (in connec- tion with Eqs. (10), (20) and (27)) is now available. It can be made explicit for the individual hybrids by use of Eqs. ( 19 ), (3) and ( 4 ), and for the hybrid set by use of Eq. (18). This information has been collected together in Eq. (28)

ho= h(spt47/t3~); ot~= 13/60; )(o--27.741 °

h2=h(sp4); a22= 1/5; ,'~'2 = 26.565 °

hi = h(sp2); at 2= 1/3; Xt =35.264°

h3 =h(sp3); a ] = 1/4; )(3=30 °

to=40.203°; to' =46.146°; to"=49.107 ° (28)

In Eq. (28) only three out of the 12 hybrid orbital shape parameters are independent. They are connected by Eqs. ( 1 ) - (4). The shape parameters can he used to calculate all geo- metrical parameters. From the shape parameters of our running example (Eq. (28)) , we calculate for instance the polar angle 02 (Eq. (21)) , and the interhybrid angles, `Yl2 and `Y23 (Eq. (9) ) , to be

02 = 53.729°;

`yt2 = 110.705°; "y23 = 106.7790 (29)

5. Establishment of the platform for ligand-field modeling

This section will illustrate the geometric empirical basis for applying the hybridization model, the results that this model can provide in terms of compositions (i.e. shapes) of hybrids and direction of lone-pairs, and finally, the particular analysis of these results that is the prerequisite for them to he treated further by the AOM. In addition, the hybridization part of the very successful ligand-field application, known as the GHP model [4], will he analyzed and generalized.

In the particular situation that is of interest in the ligand- field context, there is a ligating atom, ligator, bonded also to three neighboring atoms. The ligator, O, is placed at the origin of Fig. 1. The three valence angles at O are experimentally accessible by X-ray crystallography. The bridge to the hybridization model is built by identifying theseexperimental angles with three neighboring interhybrid angles, ̀ Y12, ̀Y23 and `Y3~, say. These three angles are associated with the face of the distorted tetrahedron defined by hI, h 2 and h3. We assume that each of the hybrids hj and ha forms a bond to its own metal ion, Mt and M3, say, and that h2 bonds to a hydrogen atom or an alkyl group, C (Fig. I ). In an AOM ligand-field context the a priori problem is concerned with an analysis of the lone-pair hybrid, ho, and the possible consequences of the analysis in the AOM formalism. The hybrid, ha, carries in this generalized example the potentially coordinating lone- pair, which was referred to in Section ! as the yet unligated lone-pair of the GHP model. The directions of the three hybrids, h~, h3 and h2, determine the direction of ho through the hybridization model.

At this stage we are able to resume our running example, which we left in connection with Eqs. (28) and (29). Con- sider the interhybrid angles `yi2 and 3'--3 of Eq. (29) and `YI3 of Eq. (10) to be identified with three valence angles that were assumed to have been determined experimentally. The information of Eqs. (28) and (29) can he generated from these three interhybnd angles alone. This is initiated by appli- cation of Eqs. (26) and (19), which give the ot values of the four hybrids. Then, Eq. (18) provides the to values (Table 2) and Eqs. (21) and (17) the spherical polar coordinates of each hybrid (cf. F_,q. (6)) .

We have thus demonstrated how the three experimental valence angles through the hybridization model can be used to generate not o;dy the remaining three valence angles and the 12 hybrid-in~ersector angles but also the compositions (i.e. shapes) ant', directions of all the four hybrids. We now return to our ger~'ralized example, which has no symmetry because the hybridized atom, O, functions as a figator bridge between two different metal ions.

192 J.M. Kennedy. CE. Schiiffer / Inorganica Chimica Acta 252 (1996) 185-194

The lone-pair hybrid, ho, is given in Eq. ( i 6). By inserting the data tabulated in Eq. (28) into Eq. (16), one obtains a first stage of this analysis directly

ho = ~/( 13/60)s + ~/(18.2/60) p~ + ~/(28.8/6t)) p~ (30)

Since real p orbitais transform as vectors, the direction of a real p orbital determines its tr and 7r characters, referred to any direction. In this case, the direction of p~. is perpendicular to the MnOM3 platte, i.e. the h~h3 plane, whereupon we are immediately able to say that the hybrid ho contai,'s (28.8/ 60) p~j_ characte~ with respect to both metal ions (Fig. 1) and the p part of 11o contains (28.8/47) p~± character and (18.2/47) p~ character. This is where the hybridization model and the AOM have been used together with success [4,22- 27 ] up till now. The p,~± part of ho provides in this successful model, the GHP model [4], the essential non-diagonal ele- ment of the AOM ligand field, which by connecting dt2g orbitals on two different chromium(III) ions provides the matrix elements that mix metal-metal electron transfer states (in the GHP case, states of CtaV®Cr n type) into the ground states (of Crm®Cr m type) and thereby, in a parametrical sense, allow quantitatively for the antiferromagnetic coupling between the two metal ions [ 32].

In order to reveal the basis for a more general application of the AOM, it is necessar~ to analyze the Pc orbital, which has its symmetry axis in the M~OM3 plane, in terms of its components p,~ and P~:I with respect to the two metal ions. For illustration of the analysis we now perform it with regard to the metal ion Ms.

In order to do this, the pC orbital must be rt, tated about the X axis by the angle, ~b = [ 2 ~ -/__. (p~,h I ) ] = [ ~" + Yl.o2 ]. The p,~ and P,~II vectorial contents of Pc lie in the F-Z plane and we are able to write

p :=p~ cos ~+P~u sin ~b

= - p,~ cos ~ti.o2 - - P,~u sin y~.o2 (31 )

By using Eqs. (28), (2), (23), (30) and (31), Eq. (32) follows (of. also Eqs. (5) and t6 ) )

ho = sx/(13/60) + [x/(47/60) ] [ - p,,~/(13/94)

- p,,u~/(23.4/94) + p,,± ~/(57.6/94) ]

= s~/(26/120) - p~.~/(13/120) - p~u~/(23.4/120)

+ p~a,J(57.6/120) (32)

This concludes the analysis. As we shall see in the next section the expression (32) contains ho in a form which is directly adapted to being used within the AOM.

In the scenario of the GHP model, which was embraced by our discussion, the hybridized atom is an otygen atom of a tt-hydroxido or a p,-alkoxido group bridging two metal ions. Here only one of the bonds from this ligating atom is of the more covalent kind with a hydrogen atom or a carbon atom of an alkyl group in the other end; the two other bonds are more polar in character involving metal ions in the other end.

The non-trivial part of the scenario is the existence of the lone-pair as discussed above.

In the original scenario, where the idea of the misdirected orbital was born [21], the ligator is connected to the rest of the ligand by three covalent bonds. Here it is the lone-pair, which is to form the coordinative, more polar bond to the central atom or ion. The idea of the misdirected lone-pair is substantiated by the combi:lation of the hybridization model and the AOM, which makes it clear that a loss of o" character must be accompanied by a gain of 1r character.

Independently of which of the two scenarios the experi- mental situation refers to, the analysis using the hybridization model remains the same.

In the next section it will be shown how the perturbation from a single hybrid orbital can be expressed within the for- malism of the AOM.

6. The angular overlap model based upon hybrid orbitals

The problem of collecting together the perturbation con- tributions from the individual ligands of a coordination sphere has been solved in the past [6,8,9]. Therefore, the problem that one has to face here is how to handle the AOM for describing the perturbation from a single ligand when the valence shell of this ligand is given in terms of a set of s-p hybrids.

The ligand F - provides an example of a linearly iigating ligand. Its valence shell consists of a 2s orbital and three 2p orbitals all of which are doubly occupied with electrons. This valence shell is equivalent to that given in terms of any com- plete set of s-p hybrid orbitals, e.g. a set of four inequivalent hybrid orbitals.

Since Table 2 has the Zaxis as its unique axis, the simplest discussion of the situation arises when the F - ligand is placed upon the Z axis. The result of using the AOM conventionally then is the parametric energy expressions

h(z 2 ) = e,, = e~o + ep~,

h ( y z ) =h(zx) =e~

h(xy) = h(x 2 - y2) = e~ (33)

where the inequalities e , ,>e~>e~ have invariably been found empirically to be valid and where only the energy differences, such as for instance

e,,' = e~, - es and e~' = e~,- e~ (34)

make up the observable quantities in a ligand-field context. Moreover, the parameters es,, and %,, cannot be separated by using experiments based upon mononuclear complexes.

The radial energy parameters, es,, and ep~r, are associated with the F - orbitals 2s and 2pc, respectively, while e~, is associated with the set of orbitals {Px Py}, which are seen from the perturbed central ion as pure ~" orbitals, e~ can from

J.M. Kennedy, CE. Schiiffer / lnorganica Chimica Acta 252 (1996) 185-194 193

this point of view be set equal to zero, which is only aquestion of the zero point for the energy. As it appears, one may in the model world split up the perturbation of F - into parts asso- ciated with its individual orbitals, and we shall be able to refer to the perturbation by or from these individual orbitals without losing any rigorousness.

The 25 operators of the AOM that act upon the set of central ion d orbitals [33] are mutually orthogonal. This is a suffi- cient condition for making it possible to consider each one independently of any other. It is, however, not a necessary condition for this independence, and we have assumed by Eq. (30) that one can actually write the perturbations from the 2s and 2p,~ orbitals as if they were independent. This means that in using the AOM we have to calculate all possible squares and cross products of angular overlaps, weight each of these symmetry/geometry quantities by thei~ appropriate radial parameter and then add the contributions from all the perturbing orbitals.

Let us, still keeping the F - ligand on the Z a:t~s, illustrate the procedure by first considering the perturbation from its 2s orbital through the representation of this orbital in the four general hybrids of Table 2. We know by symraetry that the AOM energy matrix only has one non-vanishing element, ~dz21As,r I dz2) = es~ and we see from the expression for the hybrid, h 0, of Eq. (16) that its angular coefficient, which is equal to its degree ofs character, is (sin m' sin m)2. Moreover, from the fact that the matrix constituting Table 2 is an orthog- onal matrix, it follows that the angular coefficients to the perturbation contributions of so" type from the four hybrids add up to unity as expected.

The contributions from the AOM operator ,~p,~ likewise have parts associated with the four hybrids and tl',ese partb add up to ep,, by Table 2.

The lr perturbations arise from the contents of p~ and p.~. in the hybrids. The p~ contents will perturb d~ while the py contents will perturb d~. It is seen by application of Table 2 that (d~lA~ld~,)= (dy~lA,ldy~) =e~.

The final check is, of course, that all the matrix elements that should vanish because of symmetry also do this through the algebra of the AOM. For example, the hybrid, h2, perturbs d~2 by a or perturbation and d~ by a ~r perturbation to produce a non-diagonal element (dz21Ao.~v (h2) Id~)~s = - s i n ca' cos w' cos ca, but this element is counterbalanced by the element (d~21A,~(ho) Id~)~g=cos ca' sin ¢o' cos ca. This kind of counterbalancing occurs generally because of the orthogon- ality property of the matrix contained in Table 2.

We see that the formalism of the AOM, which describes the pe,'turbation from the valence shell of F - , can isoconse- quentially be applied to a basis of the usual real s and p orbitals or, alternatively, to a basis of the four hybrid orbitals spanning the same function space. It should be stressed here, as already mentioned in connection with Eqs. (7) and (8), that no restrictive assumptions have been made regarding the radial functions of the 2s and 2p orbitals. The consequences of the radial functions are buried in the radial parameters e~,,, ep~, and ep~

As we saw above, linearly ligating ligands produce non- diagonal elements that counterbalance one another. However, a lone-pair hybrid orbital that desuoys the symmetry distinc- tion between the or and ¢r orbitals will in genera! produce non-vanishing non-diagor~al terms mixing or- and or- perturbations. These have independent radial parameters, eel, say, which, seen from the AOM, need not be simply related to e,, and e,, When 'deriving" these.AOM results from an extended Hfickel model, Rakitin et al. [34] found the tempting relation e,,,, = (e,,e~) ~/2 which may give problems when e,,' > 0 and e~' < 0 simultaneously, which certainly is possible within the AOM. This preblem may be connected with the zero point of energy for T:he AOM orbital energies.

Using the formalism describe(!l in this section, the pertur- bation from the lone-pair hybrids of Section 5 may be accounted for. However. in mo.~:t other cases of lone-pairs, additional assumptions are required, assumptions whose use- fulness will have to be demon:;trated by comparison with experiments. For example, in the case of the unidentate iigand, HO- , not only the direction but also the s--p compo- sition of the hybrid carrying the H atom must be known.

7. Conclusions

The s-p hybridization model may be conceived as defining an infinity of rigid bodies, each of which specifies four direc- tions in space in a restricted, geometrically well-definedway. Whereas the number of degrees of freedom for the non- restricted specification of four directions in space is five, that for the s-p hybridization model is only three. Visualizable geometrical (Fig. 1) and algebraical (Eqs. (23) and (25)) consequences of the restriction inherent in this model have been revealed, and the sparse Table 2 has been provided. This allows the most general set of mutually orthogonal s--p hybrids of conical symmetry, C~v, to be expressed as linear combinations of the real standard atomic s and p orbitals by using three parameters corresponding to the three degrees of freedom of the model (cf. Eq. (18)).

The following expression for an s orbital is, in effect, given in Table 2.

s = aoho + alht + ot2h2 + a3h3 (35)

On account of the scalar character of the s orbital and the vector character of the p orbitals, the p-part terms implicit in Eq. (35) must cancel, whereupon the linear dependence of thep parts of the hybrids of Eq. (35) is given by Eq. (36).

Oto/30p(ho) + al/31 p(hl) + ot2/]2p(h2) + a3/]3p(h3) = 0

(36)

wl:ere p(h,) is the normalized p function in h~ and can be thought of as a unit vector. (Note that there can only be three linearly independent vectors in real space. So the fact that each of the real hybrids is associated with a direction implies

194 J.M. Kennedy. CE. Schiiffer / Inorganica Chimica Acta 252 (1996) 185-194

that the vecte~ al p,~.rts o f the hybr ids mus t be linearly dependen t 5).

The hybridizat ion model , as d i scussed in this paper, and

the angula r over lap mode l have in c o m m o n the, the theoret-

ical part o f each mode l is founded upon the angular properties o f the a tomic orbitals. In addition, the or thogonal i ty require- men t on the hybr ids is used in the hybridizat ion mode l while the A G M cons is t s o f a fo rmal i sm that provides an a lways

symmet ry -cor rec t energy matr ix , const ructed on the basis o f

an input conta in ing geometr ica l information. The two mode l s were uni ted in this paper. W h e n the l igands

cons idered by use o f the A O M are linearly ligating, only wel l -known resul ts were obtained, but the analys is revealed how the cases o f non- l inear ly l igating l igands, at least in

principle, could be handled by introducing the hybridizat ion

mode l into the A O M .

A c k n o w l e d g e m e n t s

J .M.K. wi shes to thank The Royal Society ( in London)

for the award o f a European Science Exchange P rog ramme Fel lowship, The A n g l o - D a n i s h Society for a Hambros Bank Award and The Swedish Inst i tute for the Award o f a Swedish G o v e r n m e n t Scholarship .

References

[ I ] W.A. Bingel and W. Liittke, Angew. Chem., Int. Ed. Engl., 20 ( 1981 ) 899-911.

[2] B. Klahn, J. Mol. Struct. (Theochem.), 104 (1983) 49-77. [3] In extenso J. Mol. Struct. (Theochem.), 169 (1988) 1-574. [4] J. Glerup, D.J. Hodgson and E. Pedersen, Acta Chem. Scand., Ser. A.

37 (1983) 161-164.

s The full hybrids are, of course, linearly ind~peadent since they are mutually orthogonal. However, they are orthogcmal in their own four-dimen- sional function space. Only the p orbitals are special in that there is a one- to-one relationship between their mutual orthogonality in their own three-dimensional function space and their vectorial perpendicularity in real space.

[5] M. Brorson, RJ. Geue and AM. Sargeson, manuscript in preparation. [6l C.E. Sch[iffer ~ nd H. Yamatem, lnorg. Chem., 30 ( 1991 ) 2840-2853. [7] C.E. Schfiffer, Proc. R. Soc. London, Set. A, 297 (1967) 96-133. [8] C.E. Schliffer, Struct. Bonding (Berlin). 5 (1968) 6%95. [9] C.E. Schfiffer, Struct. Bonding (Berlin), 14 (1973) 69-110.

[ 10] C.E. Schfiffer and C.K. Jergensen, Mol. Phys., 9 (1965) 401-412. [ 11 ] C.E. Sch~iffer, in W.C. Price, S.S. Chissick and T. Ravensdale (eds.),

Wave Mechanics-- The First Fifty Years, Butterworths, London, 1973, Ch. 12, pp. 174-192.

[ 12] C.E. Sch~iffer and C.K. Jergensen, Mat. Fys. Medd. Dan. Vid. Selsk.. No. ]3 (1965) 1-20.

[ 13] H. Yamatera, Bull. Chem. Soc. Jpn., 31 (1958) 95-108. [ 14] M. Gerloch, Magnetism and Ligand-Field Analysis, Cambridge

University Press, UK, 1983. [ 15] A.B.P. Lever, Inorganic Electronic Spectroscopy, Elsevier,

Amsterdam, 2nd edn., 1984. [ 16l C.E. Sch~iffer, Pure Appl. Chen, 24 (1970) 361-391. [ 17] J.K. Burdett, MolecularShap~s, Wiley, New York, 1980. [ 18] A. Ceulemans, M. Dendooven and L.G. Vanquickenborne, lnorg.

Chem., 24 (1985) i 153-1158. [ 19] A. Cettlemans, M. Dendooven and L.G. Vanquickenborne, lnorg.

Chem.,24 (1985) 1159-1165. [20] A. Ceulemans and L.G. Vanquickenborne, Pure Appl. Chem., 62

(1990) 1081-1086. [21 ] A.D. Liehr, J. Phys. Chem., 68 (1964) 665-722. [22] H.R. Fischer, D.J. Hodgson and E. Pedersen, lnorg. Chem., 23 (1984)

4755-4758. [23] H.R. Fischer, D.J. Hodgson, K. Michelsen and E. Pedersen, lnorg.

Chim. Acta, 88 (1984) 143-150. [24] D.J. Hodgson, E. Pedersen, H. Toftlund and C. Weiss, lnorg. Chim.

Acta, 120 (1986) 177-184. [25] S. Larsen, K. Michelsen and E. Pelersen, Acta Chem. Scand., Ser. A,

40 (1986) 63-76. [26] P. Andersen, T. Damhus, E. Pedecs~a and A. Pedersen, Acta Chem.

Scand., Ser. A, 38 (1984) 359-376 [27] T. Damhus and E. Pedersen, Invrg Ch, m., 23 (1984) 695-699. [28] O. M~wtensson and Y. Ohm, T~,eor. Chim. Acta (Berlin), 9 (1967)

133-139. [29] P. Lindner and O. M~h'tensson, Ac~a Chem. Scand, 23 (1969) 429-

434. [ 30] H. gabrodsky, S. Peleg and D. Avnir, J. Am. Chem. Soc., 115 (1993)

8278-8289. [31l P. Torkington, J. Chem. Phys., 19 ( 1951 ) 528-533. [32] J. Glerup, Acta Chem. Stand, 26 (1972) 3775-3787. [33] C.E. Scbiiffer, Physica, l14A (1982) 28-49. [34] Yu.V. Rakitin, S.G. Khodasevich, V.V. Zclentsov and V.T.

Kalinnikov, Sov. J. Coord. Chem., 16 (1990) 640-645.