Valence-Bond Concepts in Coordination Chemistry and the Nature of Metal-Metal Bonds Linus Pauling and Zelek S. Herman Linus Pauling Institute of Science and Medicine, 440 Page Mill Road, Palo Alto, CA 94306

In a recent discussion of some aspects of coordination chemistry, including its history ( I ), it is pointed out that the first successful application of bonding theory to coordination compounds was made by use of the valence-bond method. This method (2) was formulated on the basis of the principles of quantum mechanics, and its application to coordination compounds provided a rational explanation of the existence of tetrahedral, octahedral, and square planar complexes and of their magnetic properties. The author of the article, how- ever, then states thatthe valence-bond theory has been sup- planted by other theories: the crystal-field theory, the li- gand-field theory, and especially the molecular-orbital theory. We, on the other hand, believe that the valence-bond theory still has much value.

Among the most interesting coordination compounds are the carhonvl comoounds of the transition metals. which have been known for about a century. Some of them (Cr(CO)6, Fe(CO)s, Ni(C0)4, etc.) contain one metal atom, whereas others (MndCO)m, CoACO)s, etc.) contain two or more metal atoms. Durina recent years thousands of comoounds of transition metals containing carhonyl groups or ocher groups with somewhat similar properties have been synthesized and many hundreds have had their structures determined by the X-ray diffraction method ( 3 , 4 ) . Interest in the nature of the bonds in these molecules is indicated by another recent paper in this Journal (5). In this paper, which has the title "Orbital Concepts and the Metal-Metal Bond," the author uses the molecular-orbital theory in discussing the ground state and excited states of some metal carhonvls involvine metal-metal honds. Our discussion of the ground state of th&e molecules with use of the valence-bond theorv is eiven helow. -

Each of the two methods of discussing coordination com- pounds has its own advantages and disadvantages. The mo- lecular-orhital method in its many approximate formulations seems to us to be the more useful one for discussina the excited states of molecules and the properties, such aH color, that depend upon the excited states, whereas the valence-bond rn&hod is.sirnplt,r to understand and is more powerful in its i~pplici~tion 10 thrdisrussion of thr normal state ofthe mole- cules. In the following paragraphs we discuss the valence-bond method and apply it to some coordination compounds of metals, especially those involving metal-metal honds. We mention here that the idea that transition metal atoms can form as many as nine covalent honds allows for the application of valence-bond theory to compounds of the transition metals in a more effective way than has been possible heretofore.

Hybrid Orbitals and the Tetrahedral Carbon Atom The first success of the valence-bond method (2) was its

straightforward exolanation of the eauivalence of the four sing6 honds formed by a carbon atom and of their arrange- ment in space, directed toward the corners of a regular tet- rahedron. I t had been recognized that in the formation of covalent bonds two electrons, with opposed spins, are in- volved, each occupying an orhital on one or the other of the two honded atoms. This was in accord with G. N. Lewis's idea that a covalent bond consists of a pair of electrons shared between the two honded atoms (6). The problem with carbon

Table 1. The Spherical Harmonics, Normallzed to 4r, for Angular Momentum Quantum Number t 5 2. A right-handed

coordinate system is employed, with 8 relorring to the polar angle (0 5 8 5 2s) and 6 to the azimuthal angle (0 5 9 5 r).

e = o S = 1 e = 1 p, = & sin0 cos6

p, = 6 sin0 sin6 p, = \/ij c o d

e = 2 ds = &ii(3cos20 - I) d, = fi sin0 c o d C O S ~

d, = fi sin0 cas0 sin6 ds-9 = fi sin20 c o d $ d, = f i s i n 2 0 sin26

was that the four orhitals in the L-shell of the carbon atom usually were described as nonequivalent. There are one 2s orhital and three 2p orbitals in this shell, so that one might expect one of the honds to be different from the other three. An important simplifying assumption was made in the de- velopment of the theory of hyhrid orbitals (2): that the radial parts of the wave functions for the orhitals are closely enough similar for the s orhital and the p orhitals to permit them to be taken as identical. Attention could then he focused on the angular parts of the orhitals, as given in Table 1. The as- sumption was then made that thebest hond orbital, formed as a linear combination of the s andp orhital functions, is the one with the greatest concentration in the hond direction (2). This best hyhrid bond orhital involves one quarter of the s function and three-quarters of a p function. I t was also found that a second, third, and fourth orhital equivalent to the first one could he constructed. and i t came out diredlv from the theory that the hond dirkctions of these four belt orhitals would he toward the corners of a regular tetrahedron: that is. the hond angle between the best sp" honds is 109.47~, the tetrahedral angle. The way in which these simple calculations are made is discussed in the book "The Nature of the Chem- ical Bond" (7).

A diagram showing the value of the best hyhrid sp3 bond orhital as a function of the polar angle a is given in Figure 1. The function has a nodal cone a t the angle 109.41° with the hond direction. I t is the fact that this angle a t which the function has the value 0 is just equal to the tetrahedral angle that permits the four equivalent best bond orhitals to he formed in the tetrahedral positions. In general, two or more best hybrid hond orhitals can he formed provided that the angles between them in pairs are all nodal angles for the best hond orbital function.

The tetrahedral arrangement of the four single honds formed by a carbon atom is largely responsible for the struc- ture and properties of organic compounds. Hence, the simple theory outlined above may he described as of fundamental sienificance to organic chemistrv. In the same wav. a similar treatment of thebyhrid orhitais formed hy the i ; p , and d orhitals mav he described as fundamental to the structural chemistry of many other elements, especially the transition metals, as is discussed in the following paragraphs.

582 Journal of Chemical Education

Hybrld spd Bond Orbitals A system of structural chemistry for compounds of the

transition metals has been slowly developing over the last half century. An important part of this system has been the rec- ognition that the elements chromium, manganese, iron, and cobalt and their congeners are often able to form nine single bonds. The factors that determine whether these transition metals are enneacovalent (have covalence 9) or have a smaller covalence are discussed below. The first person to point out that transition metals may have a large value for their cov- alence was Irving Langmuir, in 1921 (8). He made use of the principle that the distribution of electrons in any stable commund must be such that even atom has an electric charee close to zero. He pointed out t h i t the compound nickel t'e- tracarbonvl, Ni(COh, can be assimed a structure in which the nickel atom is neutral hy having this atom possess one un- shared pair of electrons and form a double hmd with enrh of the four rnrhonyl groups. This leads to covalenre 8 for nickel. I t would involve the use of the five 3d orbitali, the one 4s or. I~ital, and the thrre 4p orbitals of the nickel atom, with one orbital (3dj occupied by the unshared pair and tht, other eight. as hvbrid hond orhiu~li. involved in the ibrmatiun of the four double bonds. The existence of double bonds in nickel car- bonyl was verified by Brockway and Cross in 1935 through their determination of the structure of the molecule by the electron-diffraction method; the nickel-carbon hond length turned out to be that for a double hond, rather than that for a sinale bond (9). A recent determination (10) lists the . . nickei-carbon length as 1.838 f 0.002 A.

The formation of eight or nine covalent bonds by a transi- tion-metal atom raises the question of the nature of the hyhrid bond orbitals that can be formed from the s, p, and d orbitals by comhination of the angular wave functions given in Table 1. The rules of quantum mechanics require that each of the hybrid orbitals formed by linear comhination he orthogonal to each of the others; that is, that the integral of theproduct of the two functions over the surface of asphere bezero. In addition, the individual functions are normalized, with the integral of the square of the function over the surface of the sphere equal to 4n. The best spd orbital in the z direction is obtained as the sum of the functions s , p,, and d,* with suit- able coefficients. To obtain the maximum value of the strength of the function, which is its magnitude along the hond axis, the coefficients of the three functions must he taken propor- tional to the values of the functions in this direction. For nnrmnlizntion thr sum of'thr squares of the three cuefticients is ea~tnl to 1. ?'he hest .sud iunrtion has the form shown in Figure 1, with nodal cones a t the twoangles 13.15' and 133.62O (11).

These angles, 73.15O and 133.6Z0, for the best spd hyhrid bond orbitals are analogous to the tetrahedral anrle, 109.47O. for the carbon atom. he best single spd bonds are formed at these angles, and good bonds can be formed when the bond angles do not deviate very much from one or the other of these two values.

I t is easy to set up the equations for a pair of mutually or- thogonal equivalent hyhrid spd hond orbitals with the largest streneth in two directions at the anele a from one another. The v - discussion of this question and related questions is given in a number of recent oauers (11-22). The eauation for the strength S of two eq&lent'spd hihrid orbitals a t angle a with one another is eiven bvean. (1). and the function isshown - . . . . . in Figure 2.

S(a ) = (3 - 6r + 7.5r2)ln + (1.5 + 6x - 7.5x2)'" (1)

withx = cos2(a/2). The maximum strength has the value 3. The quantity 3 -

S is the defect in the strength associated with the hond angle a. Because the solution of the problem of finding the hest set of several orbitals with the maximum strength in the direc- tions toward the corners of an assumed polyhedron may be

Figure 1. (a) The best sp hybrid wbitel with axis at 4 = 0'; nodes occur at the tetrahedral angle (109.47'). (b)The best spdhybrid orbital with axis at 4 = 0'. nodes occur at 73.15' and 133.62'.

2 . g 5 t I 1 , , , , 1 40' 60' 80' 100' 120" 140' 160" 180'

ru. A N G L E BETWEEN BOND DlRECTiONS

3.00

2.99

2.98

S t 2.97

2.96

Figure 2. The strength S i n two directions at angle a of the two best onhogonal spdorbitals as a function of N.

-

-

- - TWO rpd H Y B R I D ORBITALS

-

rather difficult. an a~proximation has been formulated (13). which may be r&ed ihe pair-d~frct-additivity approximil;ion. Thp assumption is made that a defect for an orbital at vnriow values of the angle a with the other orbitals in the set of n orbitals is equal to the sum of then - 1 defects associated with the individual values of the bond angle a. A thorough test of this postulate has been made, and it has been found that the error is in general very small (23).

The problem of finding the best ways of arranging nine hybrid spd hond orbitals is that of finding the nine directions in space that come closest tomaking the angles equal to 13.15' or 133.62". Two good arrangements have been found (13). They are shown in Figure 3. One of them is the trigonal prism with three eauatorial cam: that is. with three bonds directed . . out toward the centers of the nearly square prismatic faces (24). The other is the tetraeonal anti~rism with one nolar can These are the most likelfcoordinaiion polyhedrak~~ecte 'd for a transition metal forming single bonds with nine li- gands.

An atom of cobalt, for example, has nine outer electrons, and also has the nine spd orbitals available for the formation of covalent bonds. I t accordingly might form the hydride CoH9. We can predict from knowledge of atomic radii that the

Volume 61 Number 7 July 1964 583

Table 2. Sfngle-Bond Radii (A) lor Transltlon Metals with Covalence 9 a

Cr Mn Fe Co Ni 1.26 1.25 1.24 1.23 1.22

Ma Tc Ru Rh Pd 1.39 1.38 1.37 1.36 1.35

W Re 0s t Pt 1.40 1.39 1.38 1.37 1.38

Single-born radii for H, C, N. and 0 are 0.30 A, 0.772 A, 0.74 A, arn 074A. respec- tively.

hvdroeen atoms would lie about 1.53 A from the cobalt atom. eitherat the corners of the trigonal prism with three equatorid caps or a t the corners of the tetraeonal antiprism with one

cap. An isoelectronic complex ion, theknneabydrido- rhenate anion [ReHsI2-, is known, and its structure, in the potassium salt, has been determined (25). The hydrogen atoms lie a t the corners of the trigonal prism with three equatorial caps, as predicted from the hybrid-orbital calcu- lation, and at the distance 1.68 A, nearly equal to the Re-H single bond length of 1.69 A (see Table 2). Moreover, the ob- served value of the polar angle determining the axial ratio of the prism, 45'. agrees with the calculated angle, 45O.

The hydrogen atom is small enough that nine of these atoms can he arranged around a central transition-metal atom without undue crowding. The fluorine atom is also small enough to permit such a complex to form. So far as we are aware, however, no such enneafluoride complex has been re- ported.

The best sets of eight spd hybrid bond orbitals are those corresponding to the tetragonal antiprism and the Hoard polyhedron (the coordination polyhedron observed for the octacyanomolybdenum ions). These two arrangements are shown in Figure 4. The Hoard polyhedron (26) is a dodeca- hedron with eiaht vertices and twelve trianeular faces. It is the figure outlined by the vertices of two iGerpenetrating te- tragoual bispbenoids oriented in the same way as the positive and negative tetrahedra of a cube. The structure has a fourfold axis of rotary inversion with two mutually perpendicular twofold axes and two diagonal planes of symmetry.

Multiple Bonds: The Two Theories Compared The carhonyl group can be attached to a central atom by

either a double bond or a single bond. In nickel tetracarbonyl, as mentioned above, the nickel atom forms double bonds with each of the four attached carhonyl groups. In thevalence-bond description each double bond can be described as consisting of two bent sinele bonds. Thus. for nickel tetracarbonvl the four single bonhs defining two' opposite sides of the ipper sauare of the tetraeonal antiprism mav be involved in double hinds to two of thecarbonyicarhon atoms (Ni=C=O:), and similarlv the sinele bonds cor res~ondin~ to the lower sauare form the double bonds to the bther two carhonyl carbon atoms, thereby giving a tetrahedral arrangement of the four carhonyl groups about the nickel atom. In iron pentacarbonyl, Fe(CO)s, one carbonyl group is attached to the iron atom by a single bond involving a pair of electrons donated by the carbon monoxide molecule, Fe--C=O:+. This transfers one electron to the iron atom, thus increasing the number of its valence electrons from 8 to 9 and permitting it to he en- neacovalent and to form double bonds with the other four carhonyl groups. We assume that the single bond and the four double bonds resonate amone the five iron-carbon nositions. . so t h ~ the five bonds are essentially equivalent (Fig. 51, each having XUq double-hond chararter. Using 1.96 h and 1.76 A as the F e C single and double hond lengths, resp~.crively (12). wt, rnlrulate an Fe+: bond lrnrrh of I.&) A in Fe(COk. which may be compared to the average observed value oj1.82 A (27).

584 Journal of Chemical Education

Figure 3. Polyhedra formed by directions of maximum values (bond directions) of me ben set of nine spdmbiiis: (a) higonal prism wim mree equatarial caps: (b) telragonal amiprism with one polar cap.

Figure 4. Polyhedra f o M by directions of maximum values (bond directions) of the best set of eight spdorbitals (a) square antiprism; (b) tetragonal dodeca- hedron (Hoard polyhedron).

Figure 5. h structure of F W O b (data from Ref. (27))

In this molecule and in many other molecules, when the different bonds do not conform to the symmetry of the mol- ecule it is necessary to consider resonance among the several structures in which the bonds have been redistributed. Thus, for the valence-bond treatment the order is hybridization first, resonance second; that is, we first form the best set of hybrid bond orbitals for each atom and assign the valence- hond structures, and then, when necessarv. combine the resonating structures to getthe best resultant structure.

In the molecular-orbital treatment the order is reversed: resonance first, then hybridization. The explicit consider- ation of resonance in the molecular-orbital treatment leads immediately to the assignment of energy values for excited states as well as for the normal state. This procedure, reso- nance first, requires, however, that the geometrical structure of the complex be assumed, rather than derived, as it is in the application of the valence-bond theory. Furthermore, in order to attain the correct description of the ground state of many molecules in the molecular-orbital treatment i t is necessary

to resort to the laborious process of configuration interaction (resonance). This is especially true for molecules containing metal-metal bonds (28-32). Indeed, according to Trogler (5), "It is logical to suspect the suitability of simple molecular orbital calculations for metal-metal bonded compounds."

Electron Transfer to Achieve Maxlmum Covalency The simplest way for an atom such as iron to become en-

neacovalent is through the transfer of an electron to it from an electropositive atom, as in the compound K+[HFe(CO)&, in which the iron atom has received an electron from the po- tassium atom, permitting i t to form a single,bond with the hydrogen atom and a double hond (Fe=C=O:) with each of the four carbonyl groups. Another way is through the forma- tion of a single bond with the carbon atom of a carbonyl group, giving the structure F e - - - C e + , as mentioned above in the discussion of Fe(CO)s. We may point out that the assumptions of enneacovalence and electron transfer lead immediately to the formula Fe(CO)& for iron carbonyl. Similar considerations lead to the formulas Cr(CO),j, (OC)sMn-Mn(CO)s, and (OC)4Co-Xo(C0)4 for the simplest carbonyl compounds of the other iron-group elements.

Bridaina Carbonvl Group - - Yet a third structure exists by which a transition mr-1 can

hind a carbonvl moup in wmpound.i haring at least two metal atoms, name$, by having a bridging carbonyl group sharing electrons with two metals. In principle this could occur with or without being accompanied by the formation of a single metal-metal bond

dt\M . / M Y 'c' 'c'

Only the second structure (or similar structures with multiple metal-metal bonds) has, however, been observed.

Discussions of some molecules by the valence-bond method are given in the following sections.

Chromium Hexacarbonyl

For Cr(CO)6 we assign a structure with three carbonyls attached by single honds and three by double bonds, the en- neacovalent chromium atom thus having the formal charge -3. This would he ruled out by the electroneutrality principle (charges -1 to 1 permitted) except for the fact that the C I - C hond has a large amount of ionic character. With 2/9 = 22.2% of ionic character for each of the nine bonds the resultant charge on the chromium atom would be reduced to -1, which is acceptable. In fact, this value for the electronegativity dif- ference for carbon and chromium corresponds to the value 1.5 for the electronegativity of chromium, which may well be a better value than the usually assigned 1.6 (7).

The six carbonyl groups in Cr(CO),j are equivalent by virtue of resonance among the 20 structures representing the dif- ferent assignments of the three single bonds, and each chro- mium-carbon hond has 50% double-bond character. In fact the crystal is found to consist of octahedral molecules with a CI-C hond length of 1.913 A (33), between the C I - C sin- gle-bond length and douhle-bond length values (12).

Dicobalt Octacarbonyl The cohalt atom has nine valence electrons and nine sp3d5

bond orbitals and accordingly can form nine covalent bonds. I t can use two electrons and two orbitals to form a double bond (two bent single honds) with a carbonyl group, Co=C=O: Hence, it could attach four carbonyl groups by double bonds and form one more bond, with another cobalt atom, to give the

molecule C02(C0)& with structure (0C)aCo-Co(C0)a. Two other reasonable structures are predicted by valence-hond considerations. With one of these there is a cobalt-cobalt bond and two briding carhonyl groups, with each cobalt atom also bonded to three carbonyl groups by double bonds. The favored bond angles between single bonds require that the two bridging carhonyl groups lie on one side of the dicohalt axis. The third structure involves a cobalt-cobalt bond, four bridging carhonyl groups, and two carhonyl groups attached hv double honds to each cohalt atom. This structure has not hken reported. The three possible structures are shown in Figure 6.

The structure of crystalline Co2(CO)s has heen determined (34) by X-ray examination; i t is shown in Figure 7. This low- temperature form is the isomer with a cobalt-cobalt bond and two hriding carbonyl groups. The latter lie on one side of the dicohalt axis, as predicted. When a solution of Coz(C0)s is heated under CO pressure, three tautomeric forms occur (35-38). One tautomer has been assigned the structure of the

Flgve 7. The aystal sbuctve of me low-tamparatwe teutomer of COACO)~ (data from Ref. (34).

Volume 61 Number 7 July 1984 585

crystalline form and the second that of the structure with no bridging carhonyl groups. As yet the structure of the third tautomer is unknown although matrix isolation spectra of Coz(C0)s do not provide evidence of absorptions ascribable to hridging CO griups in the third tantom& (39).

DI-Iron Enneacarbonyl The valence-bond theory predicts two reasonable structures

for di-iron enneacarhonyl, Fez(C0)9, the first polynuclear carhonyl to have been discovered (40). In each structure each iron atom, to become enneacovalent, attaches one carbonyl group by a single covalent hond, thus gaining an electron (Fe--C=O:+). With its other 8 valences it forms double honds to nonbridging carhonyls, single bonds to hridging carhonvls, and a sinale Fe-Fe hond. In the first structure there is one bridging car<cmyl and earh iron atom hdds four non- bridaina rarbonvls hv bonds with 75"; of double-hond rhar- acte; his struc&e has not been reported. A second structure has three bridging carbonyls with each iron atom holding three nonbridging carhonyls. An X-ray study of this structure was reported in 1939 (41). and recently has been repeated with modern refinements (42). The dimensions of the molecule. given in Figure 8, agrke closely with those predicted by the valence-bond theory. The third structure, witb five bridging carhonyl groups, may be unstable because of steric hin- drance.

More Complex Molecules Whereas in the molecular-orbital treatment nearly every

molecule requires a separate consideration of enerav levels of one-electron o rb i t ak the valence-bond treatmentis simple and general. For example, we may consider the metal tricar- bonyl radicals M(C0)z. With Co, Rh, and Ir no electron transfer is needed to achieve enneacovalence, and hence three duul~le bmds are formed to the rsrhonyl gnmps, lraving three valences on the metal atom. An example is 1(0('),,(:1~11(1- CIC'U(COJ:III, in which therr arp ConC tetrahedra, with earh

i Fe

o c @ o

Figure 8. The crystal structure of Fe2(CO)~ (data from Ref. (47)).

586 Journal of Chemical Education

cobalt atom forming single bonds with two other cobalt atoms and with the carbon atom.

In the Fe, Ru, and 0 s tricarbonyl groups one CO is held by a single bond and two by douhle honds, leaving the metal atom with four other valences. An example is H ~ C I F ~ ( C O ) ~ , in which the iron atom forms four single bonds with the four carbon atoms of the cyclobutane ring.

In 0410s(CO)& the four osmium atoms lie a t four nonad- jacent corners of a cube, witb the four oxygen atoms near the other four corners and with the Os-0 honds alone the edees of theruhe ( 4 1 1 asshown in F~gurr S.'l'huseach<&en at&, : 0 < , hns lost an electron in order co he trirovnlent. and each osmium atom has picked up an electron from one of these oxygen atoms rather than from a carhonyl group. In this molecule all three carbonyls of the Os(C0)a group are held by douhle bonds.

Bond Angles An example of the predictive power of the valence-bond

theory is provided by Figure 10, showing the bond angle for tricarhonyl groups as a function of the bond number n (equal to 1 for single honds and 2 for double bonds). The value 73.15"

F w r e 9 me slrunne ot 0, WCOhi. (mla from Ref (43)) Ths molec.le has a lelragonal ax s colncdent u th the I - a x s and tne lwa lndcponoenl Or-0s dostances are 3 253 A awl 3 190 A rsspectrve y

Fioure 10. The curve of bond anaie versus bond number far tricerbonvl orouos ~ ~~ . ~ - - - 01 transmn meta s is oer u r n from the Smpe valencs.bon0 theory ol nyorld spdbond orbmls. wilhout any use of emp r c a inlormalion Tne exper menta valuer lclrr: eslareaveragez for Co (Table 31. Fe (Table 4). Ir ( n lr,Co&01,,1, Ru (in two molecules) and 0s (in two molecules) ( 18).

Table 3. Values of O C - 0 - S O Bond Angles In the Cobalt carhon atom. The idea that transition metals can form as . Tricarbonyl Group many as nine covalent bonds permits valence-bond theory to

he extended to comnounds of the transition metals in a more Average anale.

effective way than has been possible before.

Acknowledgment

This work was supported in part by a grant from the Na- tional Science Foundation. We thank Joan Engels for assis- tance in drawing the figures.

. . .. R3C--CGC-CR3 99.8 CeF& 100.5 R3CCH3 100.5 R~CO)BHZNIC~H& 103.0

Average 101.0

for three single honds is the smaller of the two values for best spd hybrids, and the value 101.85" for three double honds results from using the equatorial and end set of orhitals of the trigonal prism with equatorial caps. For n = 1.5 the value 90- corresponds to resonance of three single and three double honds, as in Cr(CO)& with carhonyl groups a t the corners of a regular octahedron. The line corresponds to the equation

Bond angle = 13.15' + 38.1% - 1) - 1O0(n - 112 (2)

Thus, for cobalt tricarbonyl groups this simple theory gives 101.85" as the expected value for the OC=Co=CO bond angle. Experimental values, given in Table 3, average 101.OO, with mean deviation 1.1' (18). For the iron tricarhonyl group the value n = 1.67 (average for one single-bonded and two douhle-bonded carhonyls) leads to 94.5" for the bond angle, not far from the average 95.6O (mean deviation l.2O) for 45 compounds containing this group (18).

Conclusion

The valence-bond approach has been used in discussing other aspects of the problem of the structure of molecules involving transition metals and groups containing atoms of carhon, sulfur, phosphorus, arsenic, and other atoms with medium electronegativity (19), including atomic radii (12). hond lengths (14,16,19), hond angles (15,18-20), the hond- forming power of hybrid orbitals (11, 13), the nature of the quadruple hond and other multiple honds (12,141, the sta- bility of octahedral complexes (20), and the structure of pyrite and related minerals (22). Much of the discussion is rather simple, involving use of the angles 73.15' and 133.62" in the same way as that of the tetrahedral angle 109.41° for the

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