Proc. Nat. Acad. Sci. USA Vol. 72, No. 11, pp. 4200-4202, November 1975 Chemistry Valence-bond theory of compounds of transition metals (valence bonds/metal carbonyls/spd bond orbitals/bond angles) LINUS PAULING Linus Pauling Institute of Science and Medicine, 2700 Sand Hill Road, Menlo Park, California 94025 Contributed by Linus Pauling, August 14, 1975 ABSTRACT An equation relating the strength (bond- fying assumption that the bond orbitals have cylindrical forming power) of an spd hybrid bon orbital to the angles it symmetry about the bond axis. A treatment without this as- makes with other bond orbitals is formulated and applied in the discussion of the structures of transition-metal carbonyls sumption has been carried out by McClure (6). and other substances by the valence-bond method. The rath- The two equivalent and mutually orthogonal spd orbitals er simple theory gives results that agree well with those ob- with maximum strength in two directions making the angle tained by the complicated and laborious calculation of sets a with one another can be found by use of Lagrange's meth- of orthogonal hybrid bond orbitals with maximum strength. od of undetermined multipliers (4). Their strength is During the last two decades there has been intense activity S(a) = (3 - 6x + 7.5x2)1/2 in the field of the synthesis of compounds of the transition metals, such as those containing carbonyl and cyclopentadi- + (15 + 6x - 7..5x2)"/2 [1] enyl groups. Hundreds of these compounds have been made, and the structures of scores of them have been determined in which x = cos2(a/2). Values of S(a) are shown in Fig. 1. by the x-ray diffraction method. The theoretical treatment The best bond orbitals lie at a equal to 73.15' or 133.620. of the substances has been carried out by the molecular-or- The ranges 65°-85° and 120°-145° involve only a small de- bital method, and has yielded only qualitative results. For crease in S, and we may conclude that these ranges are fa- example, the short Re-Re distance 222 pm reported for the vored by atoms of the transition metals (7). ion Re2CI82- by Kuznetzov and Koz'min (1) was interpreted For an orbital i at angles aij with other orbitals j the value by Cotton as showing that the rhenium atoms are connected of Si is assumed to be by a quadruple bond, and Cotton formulated a simple mo- Si = 3 - Z[3 - S(a11)] [2] lecular-orbital theory that provided an explanation of the eclipsed configuration of the two ReCL4 groups, as follows J. (2): "The d.a2 y2orbital on each metal atom was assumed to This assumed additivity is seen to have moderate reliability be employed mainly in Re-Cl bonding and the remaining from the approximation of the points in Fig. 2 to the line four d orbitals on each metal atom were used to form the corresponding to Eq. 2. For example, each of the six octahe- Re-Re bond. Overlap of the d52 orbitals give rise to a a dral d2sp3 orbitals has four others at 900 and one at 180°, bond; overlap of corresponding pairs of the dX2 and dy4 or- giving 3 - S = 0.0852, which is 12% greater than the actual bitals leads to formation of a pair of r bonds. Finally, over- value (4), 0.0760. The point H4 is for Hultgren's best set of lap of the dxy orbitals gives rise to a a bond. The eclipsed four cylindrical orbitals with equal strength, with a = configuration is a consequence of the a component of the 73.820 and 136.10° and 3 - S = 0.008 by both direct calcu- bond since that is the only component that is angle-depen- lation and Eq. 2. The points 1 to 9 are for the set of nine best dent." Neither this nor any other molecular-orbital treat- spd orbitals reported by McClure, and R1, R2, R3 are for ment (3) provides any significant discussion of bond angles eight orbitals, as reported by Racah (see next section). and bond lengths or of the relative stability of alternative structures. Sets of eight or nine spd bond orbitals I have now found a simple relation between the strength Racah (8) pointed out that sets of eight equivalent spd orbit- (the bond-forming power) of a hybrid spd bond orbital and als could be formed. They are directed toward the corners of the angles that it makes with other similar orbitals that per- a square antiprism, point group 8 2m. The best set (R1 in mits the prediction to be made of bond angles, bond lengths, Fig. 2) has 3 - S = 0.0114, in good agreement with the the relative stability of alternative structures, and other value 0.0108 from Eq. 2. The angle with the 8 axis is 60.900 properties of compounds of the transition metals. and the bond angles are 72.340 (8), 76.32' (8), 121.80° (4), and 140.930 (8), all within the ranges corresponding to close Relation between bond strength and bond angles for to the maximum bond strength. Racah also determined the spd bond orbitals best orbitals for the tetragonal dodecahedron, which Hoard The bond strength S of an orbital has been defined (4) as the (9) had found in his x-ray study of K4Mo(CN)8-2H20. The value in the bond direction of the angular part of the wave best set of orbitals (with the sum of S a maximum) consists of function, with the angular wave functions normalized to 4wr. four with S = 2.9954, angle 34.550 with the 4 axis, and four It is a reasonably good measure of the energy of the bond with S = 2.9676, angle 72.78° with this axis. The bond an- that can be formed by the orbital, with corrections needed, gles are 69.100 (2), 72.670 (4), 75.80° (8), 95.030 (4), 132.72' of course; for promotion of electrons, partial ionic character, (2), 141.77° (4), and 145.560 (2). The values of 3 - S given deviation from electroneutrality, strain energy of bent by Eq. 2, 0.0048 and 0.0377, agree with those given by bonds, and other factors. The values of S for some spd bond Racah, 0.0046 and 0.0324 (R2 and R3 in Fig. 2). The angles orbitals were discussed long ago (4). Hultgren (5) then dis- of the Mo-CN bonds in the crystal are within one degree of cussed the problem thoroughly, with, however, the simpli- the values given by Racah's calculation. 4200 Downloaded by guest on October 1, 2021 Chemistry: Pauling Proc. Nat. Acad. Sci. USA 72 (1975) 4201 0.08 St 2.97 - TWO spd HYBR I D ORB ITA LS 2.96 p 2.95 - 400 600 800 1000 1200 1400 1600 1800 ct, ANGLE BETWEEN BOND DIRECTIONS FIG. 1. The strength S in two directions at angle a of two best 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 orthogonal spd orbitals as a function of a. FIG. 2. Comparison of values of 3 - S calculated by evaluation of the best sets of orthogonal orbitals (vertical) and by use of Eq. 2 The value of Eqs. 1 and 2 can be illustrated by their appli- (horizontal). Point 9 is between R2 and H4. cation to the square antiprism. The value of S is found by a few minutes of calculation to be a maximum (at 2.9789) for gives 2.9816 for McClure's set. This tetragonal structure is angle 61.40 with the 8 axis, only 0.50 from the value found accordingly only slightly less stable than McClure's and con- by Racah. Similarly, for the tetragonal dodecahedron this stitutes an alternative stable arrangement for nine spd simple calculation makes SI + S2 a maximum for angles bonds. 340 and 72.0°, 10 from Racah's values. SI and S2 have the values 2.9944 and 2.9644, respectively, close to those found Molecules containing quadruple bonds by Racah. The bond angles are 68.0° (2), 74.00 (4), 75.20 (8), I have pointed out (7) that the orientation of eight or nine 95.50 (4), 133.40 (4), 142.00 (4), and 144.00 (2). The angle bonds toward the corners of a square antiprism or a square 95.5° is the most seriously strained one. It is between pairs of antiprism with a pyramid permits the formation of a bonds in the oblate tetrahedron, and its contribution to S2, quadruple bond and the calculation of ratio of the -0.0276, explains its low value. quadruple-bond length to the single-bond length. These The nine best spd orbitals found by McClure are oriented structures-also account for the observed eclipsed configura- as shown in Fig. 3. They have values of S between 2.9539 tion of the other bonds and the values of the bond angles, and 2.9936. Of the 36 bond angles, 21 lie between 66.490 observed to be 1040 in Re2CJ82- and about the same in other and 92.570, average 74.040 (only 0.890 from the best bond complexes. The calculated value for this angle is 113'; a angle), and 15 lie between 112.920 and 140.490, average smaller value is expected, however, because it would relieve 133.350 (0.200 from the other best bond angle). The individ- the strain in the four bent bonds of the covalent bond and in ual values of 3 - S (Fig.