Jurg Waser
Gates and Crellin Laboratories of Chemistry California Institute of Technology Posadena, 91 109
Hybrid Orbitals and Atomic Configuration
I
I n an important paper Kimball (1) investigated by group-theoretical methods the relationship between different hybrids of atomic orbitals (such as sp3 or dsp2) and the geometric configuration of the atoms bonded to the atom considered. While Kimball limited himself to s, p, and d orbitals, his approach was extended by Eisenstein (g) to include f orbitals. The material in the first paper, particularly, is discussed in several well-known texts on chemical bonding and group theory. However, it appears to this writer that nowhere is it sufficiently stressed (or perhaps even recognized) that some of the hybrids listed for a given configuration exhibit a higher symmetry than the configuration itself. The hybrid orbitals are therefore pointed not only towards the bonded atoms, but to an equal extent in opposite directions where there are no such atoms. This leads to less-than-optimal overlap between the hybrid orbitals and o orbitals of the neighboring atoms, and therefore to bonds of lower strength than might be attained with hybrids of lower symmetry. As an example consider the grouping XYa with tetrahedral arrangement of the atoms Y, for which spa and sd3 hybridization are both stated as being appropriate. Suitable orbitals for the first case are (3)
These four hybrids are pointed towards the alternate corners A , C, F, and H , respectively, of the cube shown in Figure 1. How does this come about in detail? The directional portion of the three p orbitals contributing to q~ is (Z
Figure 1. A cube. totrohedron.
+Y +
Z)/T
Alternate corners, such 0s A, C, F, ond
(5)
H, deflne
a
of which the magnitude is a maximum in the two opposite directions of corners A and G. However, the sign of expression (5) is positive towards A , and since s is independent of direction and positive, the contributions of s and p to ql reinforce each other towards A . The sign of expression (5) is negative towards corner G, and the contributions of s and p largely cancel in that direction. Analogous considerations apply to the orbitals 9%)q3,and q4. For sda hybridization appropriate orbitals can be obtained by replacing p,, p,, and p, in eqns. ( 1 ) through (4) by d,,, d,,, and d,,, respectively, leading, in the case of q, to a = I/z8 ' / & f dz. d w ) (6) The angle-dependent portion is now contributed by the d orbitals and has the form
+
(YZ
+ z z + zv)/r9
+
+ Y + zIs - r11/(2ra)
(7)
= [(z
The second form of this expression is similar to eqn. (5), and it is evident that eqn. (7) also reaches its maximum value in the directions of the opposite corners A and G. But now the sign of eqn. (7) is positive in either direction, so that the contributions of s and d to q5 add in the same way in eit,her direction. The "shape" of qs relative to corners A and G is identical. For the remaining sd3 hybrids the situation is similar: each of these hybrids points a t two diametrically opposite corners of the cube in exactly the same way. Thus, while sda hybrids are indeed in harmony with tetrahedral symmetry, they actually exhibit the symmetry of a cube. The nub of the matter is that s and d orbitals are centrosymmetric while p orbitals are antisymmetric with respect to inversion through a point (Fig. 2). Linear combinations of centrosymmetric functions, such as s and d orbitals, remain centrosymmetric, so that each sd3 hybrid is of necessity centrosymmetric. This is not the case for spa hybrids, because of the different symmetry properties of s and p orbitals relative to inversion through a point. In spite of the centrosymmetric nature of sd3 orbitals, the impression is often given that these hybrids play a major, if not an exclusive role in tetrahedral transitionmetal compounds. It is, however, noteworthy that one of the originators of the hybridization concept, Linus Pauling, did not suggest pure sd3 hybrids to describe tetrahedral bonding under circumstances in which d orbitals are expected to contribute. Rather, we find in his classical paper on the subject (4) the linear combination w = '/%s
+ (3/32)'/*(p.+ pv + p,) + (5/32)'ll(dus+ dzi Volume 48, Number 9, September 1971
d=u) (8)
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Figure 2. Schematic diogroms of r, p, and d orbitals. Note that r and d orbitolr are symmetric and p orbitals antisymmetric relative to inversion through the origin. In mothemotical terms, these orbital, ore: s = flrl; P. = (xlrlglrl; P, = ly/rlglrl; PS = Iz/rlglrl; d,, = lx~/r'lhlrl; 4- = (yr/r21h(r); d,, = Izx/r21hlrl; d,s-,* = [ ( x Z - y21/r31hlrl/21; d,, =
[(3z2- r2)/rqhlr)/2 4 3 , where the functions flr], girl, and hlrl depend
on the principd quantum number of the orbital and on the nature of the
otom considered.
Orbitals of different kinds are shown in different scaler.
From t,he point of view taken here substantial admixture of p to and to analogous orbitals plays the role of "breaking" the centric symmetry of pure sd3 hybrids. It has also been pointed out by Z. Z. Hugus, Jr. (6) that f orbitals may always he substituted for p orbitals in hybridization; f orbitals are antisymmetric in the same way that p orbitals are, so that sf3 hybrids have true tetrahedral symmetry. As a second example, we may consider a planar trigonal arrangement XY8. It is sometimes stated, or implied, that sp2 and sd2 are both appropriate hybrids for this geometry. However, if the axes x and y are chosen so as to lie in the plane of the X Y a grouping, the p orbitals involved in sp2 hybridization are p, and p,, while sd2 hybridization involves the orbitals d,, and d,,. It turns out that t.he symmetry of the sd2 hybrids is not just that of an equilateral triangle, but that of a regular hexagon, each of the three sd2 orbitals pointing
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in a symmetrical way towards diametrically opposite corners of the hexagon. The reason is, of course, that each of these hybrid orbitals is centrosymmetric. To break the centric symmetry of sd2 hybrids requires at least partial replacement of the d orbitals by orbitals that are antisymmetric relative to inversion through the center of the hexagon. Among further examples is the listing of d3sp hybridization for a trigonal bipyramidal configuration. These hybrids have a six-fold axis as main rotational symmetry rather than just a three-fold axis as required, and as is indeed shown by the more customary dsp3 hybrids. A related situation exists in regard to p3 hybridization, which is sometimes said to be appropriate for describing the bonding in trigonal pyramidal molecules such as p H 3 . However, even though the symmetry of p3 hybrids matches the symmetry of the p H 3 molecule, each of the three p orbitals is in fact antisymmetric with respect to inversion through the center of the P atom. Disregarding the plus or minus signs, the lobes of p3 hybrids have, in fact, octahedral symmetry, and to use them to describe the bonding in a trigonal X Y 3 configuration seems inappropriate. Use of pure p orbitals on atom X is detrimental to optimum overlap with a orbitals of the atoms Y , because the p orbitals of X point away from the atoms Y equally 8s much as they point towards them. But this time it is an admixture of one or more centrosymmetric orbitals, such, as s or d , that is required to provide hybrids truly pointed towards the atoms Y . Finally, as has been proposed notably by Rundle ( 6 ) ,p3 hybrids may be eminently suitable for bonds of order L/Z 07 less in octahedral configurations for electron-deficient compounds and in particular for interstitial compounds in which the environment of the iuterstitial atoms is often octahedral (7). In a similar way, sd3 hybrids would be suitable to describe bonds of order or less with a cubic configuration. In summary, the description of bonding by hybrids that point in unique directions, rather than b ~ i n gsimultaneously pointed in two diamet,rically opposite directions, requires the mixing of orbitals that are symmetric with orbitals that are antisymmetric with respect to inversion through a point. The first type, such as s, d , and g corresponds to an even value of the quantum number I, the second type, such as p and f to an odd value of!. I wish to thank Richard E. Marsh for valuable discussions. Literature Cited (1) K l l m ~ b b G. . E., J . Chcm. Phys., 8 , 188 (1940). (2) EIaENSTEIN, J. C.. J . Cham. Phus., 25, 142 (1956). a , "The Nature of the Chemical Bond" (3rd (3) Far example, P ~ a ~ m L., ed.), Cornell University Press. Ithaoa, N e v York, 1960. D. 118. (4) P m o r ~ o L.. . J . Amar. Chem. Soe.. 53, 1367 (1931). (5) Hoonr. Z. Z.. .In.. J . Amev. Chem. Soc.. 74, 1076 (1951). (6) Ruxolr;, R.E.,J . Amer, Chem. Soc., 69, 1327 (1947); J . Cham. Phys., 17, 671 (1949). (7) RUNDLE.R.E.. Ada Cryrtolloor., 1, I80 (1948).
