D. 5. Urch
Queen Mary College London, England
Pi-Bonding in Tetrahedral Molecules
Group theory ( 1 ) shows that in a tetrahedral ( T d )structure, XLa, orbitals may be classified as follows: Set No. 1
2 3 4 5 6
Orbitals Classifioation X, s orbital A, X, p orbitds Tx Ts E X, d orbit& Li, four s orbitals A, TP L,, four p orbitals, along X-L axes A, T% L,,eight other p orbitals E TI 2'2
+
++
+ +
Thus s (sets 1 and 4) and p (sets 2 and 5 ) orbitals of both X and L can interact to form the X-L o bonds which hold the tetrahedron together. Often X also has d orbitals which are of the correct symmetry to interact with ligand orbitals of set 6. The more electronegative the ligands, the less are these d orbitals shielded from the X nucleus. I n such a situation the d orbitals become more tightly bound; they become less diffuse and their energy becomes more comparable
Figure 1. Solid line. show redionr of the angular wove functions of dli2~U1 lleftl and d,* (right1 in the x r , xy, and yr ploner ar seen from the ligond atom, looking down the 18gmd 11)-central atom (XI .xi% Only those parts of each orbitol near to the ligond ore shown. Dotted liner representp orbitalran the ligond that mightoverlap with the dcrbilalr.
with that of a ligand p orbital. Two important factors for strong bonding (overlap and energy match) are therefore enhanced ( f ) . Craig, et al. (3) have shown that the d ( X ) - p ( L ) overlap integrals may be quite large for typical X - L bonds, when X is from the second row of the periodic table, and Moffit (4) has interpreted the bonding in S-O bonds in terms of sulfur's d orbitals forming d-p T bonds with oxygen. Indeed it may be regarded as established that a-bonding plays an important role in ensuring the stability of anions such as SiOa4-, Pola-, S O n 2and C D - . Cruicltshank (5) has correlated X-L bond lcngths in such anions with thc d-p a bond order and, on the other hand, a reduction in the efficiency of a-bonding in the third row of the periodic table has been suggested (6) as a contributory factor to the instability of Br04-. a-Bonding of the same formal nature has also been suggested as important in stabilizing tetrahedral molecules involving transition metals, e.g., Ni(CO)a. 502
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Journal of Chemical Education
Unfortunately, although important, and despite the simplicity of the theory, it is not very easy to visualize this three-dimensional T-bonding. The purpose of this note is to present diagrams which, it is hoped, will simplify the problem. The nature of the bonding, and especially the a-bonding, in the three groups, E, T2, and TIwill now be considered in turn. E Symmetry
The two orbitals on X which belone to this classification are d22.yland dSl. The view of trhe nearest lobes of these orbitals, as seen from a particular ligand, loolung down an L-X axis is shown in Figure 1. This diagram shows sections through the angular hydrogen-like wave functions in the xy, xz, and yz planes; i.e., the "walls" of the Cartesian octant in which the ligand is to be found. These functions must be "built out" into space to give an idea of the three-dimensional nature of the orbitals. Even so, when this has been done, the picture obtained is not that of the whole wave function, as the radial dependence has been ignored. The diagram does provide, however, a picture of the regions of space where the largest amplitudes may be expected and can therefore he used as a rough guide of overlap potential. The two p orbitals on the ligand atom which might be expected to play some part in bonding with the d orbitals are shown dotted. While some symmetry match is apparent it is not obvious that the overlap is particularly good. What can be done now is t o attempt t o redraw some of the orbitals in an equivalent way that will provide a more intuitively satisfying picture of the overlap.
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+
2. Sections through h I= 11d2 I d z * _ "2 d , 4 and (= 1 / 4 5 Ids* dd,n_ .'I1 are drown in d i d liner wing the same pro-
Figure
-
jection orin Figure 1. for best overlap.
Dotted liner again show ligand p orbitals oriented
Since the two d orbitals belong to a degenerated representation any linear combination of these orbitals may be used: Figure 2 shows J., and $1 where
and again p orbitals on L are shown dotted (rotated from the position in Figure 1 to achieve best overlap). Since the lobes of J., and $Zin the octant under consideration have very much the appearance of p orbitals, it is easier to see that the interaction with ligand p orbitals might well lead to good overlap and therefore strong bonding.
',.
Figure 3. Sections through those lobes of d,. 1-, d,, (- -1, and d,. (-1 thot ore nearest to the chosen ligond.
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.
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+
ignored since the L s orbital would then be very tightly bound. The T2orbitals from set 5 will interact with those from set 2, such interaction being conccntrated along the X-L axes where a bonds have, by definition, nodes. A useful simplificat is now to consider that the sets 3 and 6, which have rr potential have no interaction a t all with the n bonds and may be considered in isolation. I n Figures 3 and 4 one octant is again considered, looking down the X-T axis I n Figure 3 the relevant lobes of d,,, d, and d,, (the T2, d orbitals) are shown and in Figure 4 portions of the three molecular orbitals of type T 2 that can be generated from ligand p orbitals. It can be seen only one half of the 'tlobelet" of each of these three molecular orbitals overlaps well with the corresponding d orbital, the other half overlaps with the node in such orbitals. Much smaller overlap integrals (and therefore weaker bonding) will therefore be generated by T2 than by E type overlap.
Figure 4. Shows portions 1% in each core) of the three molecvlor orbitals of symmetry Ts to be found a t m y ligand derived from p AO'. of set 6 (see tablel. The three MO's ore drawn to match the d orbitols of Figure 3 with which they overlop.
Figure 5. One-fourth of each of the nonbonding TI molecular orbitals derived from the ligond p atomic orbitals of set 6. Each of the TI MO'r is ot right angles to one of the T2 orbitals (Fig. 51 and they hove been drawn accordingly.
7%Symmetry
Many groups of atomic orbitals (AO's) contribute to triply degenerate sets of molecular orbitals of this symmetry. The three p AO's of X comprise such a set in their own right but the others (from A 0 sets, 3, 4, 5, and 6) are all composite molecular orbitals (MO's) built from fractions of atomic orbitals. Thus each of the eight p ligand orbitals (set 6) contributes '/a to the E type MO's, 3/s to the T Iorbitals, and a/8 to the T , orbitals (for a more detailed discussion of the exact form of these MO's, see (6)). Although all eight p orbitals may be thought of as interacting with the E orbitals of X only two bonding and two antibonding molecular orbitals are formed. The remaining six nonbonding orbitals are distributed over the four ligands and it is these nonbonding molecular orbitals that are now divided into groups T I and T,. The other molecular orbitals of type T , are from set 4 which, if L is strongly electron attracting, may be
TI Symmetry
This family of three degenerate molecular orbitals will reside wholly on the ligands, taking no part in the bonding to X, since there are no orbitals of this symmetry on the X atom. One quarter of each orbital is shown in Figure 5 , using the same representation as before. Tl orbitals have their maximum amplitudes where T* orbitals have nodes. Literature Cited (1) MULLIKEN, R. S., P h g s Rev., 43, 279 (1933). (2) CODLSON, C. A., ''Valence," 2nd ed., Oxford University Press, London, 1952, p. 71. (3) CRAIG,D. P., MACCOLL, A,, NYHOLM, R. S., ORGEL, L. E., AND SUTTON, L.E., J . Chem. Soc., 1954,332. (4) MOFFIT, W., PTOC.ROY.Sac., AZOO, 409 (1950). (5) CRUICKGHANK, D. W. J., J. Chern.Soc., 1961,5486. (6) URCH,D. S., J . Inorg. Nucl. Chem., 25,771 (1963).
Volume 47, Number 9, September 1964
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