Robert G. A. R. Maclagan University of Canterbury Christchurch 1. New Zealand
Symmetry, Ionic Structures and d Orbitals in SF6
The use of d orbitals in describing the bonding of molecules like SF6 was introduced by Pauling (I).While the beauty of the hvhrid orbitals has continued to be admired bv. aenera.. t i o n of chemists, the practical importance of d orhitali hay heeu seriouslv challeneed. It was discoverrd thdt thed orbitals in some states of atoms like S could be so diffuse that they could not reasonably be expected to participate to a significant extent in bonding. This dilemma was resolved by work in groups led by Coulson (2),Craig (3)and Cruickshank ( 4 ) .The situation in 1969 was reviewed by Coulson (5).Since then the speed and efficiency of quantum chemistry computer programs have improved to the extent that ab-iniriocalculations on mulecules as complex as SF6 are now pussi ble. I'opulatim analyses of the wavefunctions obtained should allow us to investigate the participation of d orbitals in the bonding of such molecules from a new . oers~ective. However. care must . be taken in interpreting such population analyses since the amount of d orbital oarticination to be expected depends upon which ionic structuies are'important. he degree bf d orbital participation required by symmetry varies from one ionic structure to another. Apart from d orbitals required by symmetry considerations for certain structures, there are always likely to be contributions to the d orbital population because of polarization effects. However, as Coulson comments, these do not have any significant chemical meaning when one talks about the role of d electrons in bonding. ~
~
d Orbitals In SFe
What d orhital participation is required by symmetry for various structures of SF6? For the covalent structure, it is reasonable to assume that the hybridization is determined by the pairing of six equivalent octahedral sulfur hybrid orbitals with fluorine orbitals to form a bonds. The procedure by which group theory is applied to determining which atomic orhitals can contribute to the hybrid orhitals is described in detail by Cotton (6).T h e representation, ' l for which the six hybrid orbitals form a basis reduces as:
r, = A,, + E, + TI, ,and p orbitalson the S atom are nut ol E, iymmetr).. d,,and urb~talsmust he included if we wish to describe the d.2. bonding including a covalent structure. If the orbitals used to construct the hybrid orbitals are those with the lowest energy for a given symmetry, then the 3s,3p,, 3p,, 3p,, 3d,2 and 3dx2-y2 orbitals are used. For S+ in a S+(F& structure, the five hybrid orbitals forming a bonds form a representation of the symmetry group C4,. (See Fig. l(ii)). Overall Oh symmetry is obtained by "resonance" between these structures. The wavefunction has the form: X ?
*
=
1 L4,-+ m-, + my + ,6 + 4, + L l 6
where &, for example, is the wavefuuction where the F- is on the positive z-axis. The axial hybrid orbital is of A1 symmetry, while the four equatorial bonds form a basis for a representation which reduces as:
r, = A , + R, + E s and p orhitals are not of BI symmetry, so adX2-y2orbital (or 428 1 Journal of Chemical Education
Hybrldiratlonas a Function of Structure Type-S
in SFe
Hybridization Charge on S
(OrlhogonalOrbitals) (Averaged over all Symmetry
resonance forms)
S: p: d
rather a combination of dr2~y2and d,r orbitals when other orientations are included) must he included if the S+ (Fdstructure is to be included. Two structures are possible for S2+(Fg)? one with two Fions trans to each other (Fig. l(iii)) and the other with them cis to each other (Fig. l(iv)). For the three structures in the trans case, the four bonding hybrid orbitals form a basis for a representation of the Dph point group which reduces as: r,=A,,+B,,+E,
Again we require d orbitals since s or p orhitals do not form a basis for theB1, irreducible representation. For two F- ions along the z axis, the B I , orbital is a d,z-,,$ orhital. Orienting them along the x or y axis leads to contributions from the d,a orbital as well. For the twelve structures in the cis case, the hybrid orhitals form a basis for a representation of the Czu group which reduces as: Here for the first time we have a case where d orbitals are not required by symmetry. A set of orthogonal orbitals can be constructed from 3s, 3p,, 3p, and 3p, orbitals. 3d orbitals are not required by symmetry for either the 8 structures of Ca, symmetry or the~twelvestructures of Cn, symmetry for S"+ (Fs)". For the Czu structure we would expect the average configuration to be spI2l"pY2/:'p,2/:3 rather than p,p,p, because of a lower valence state promotion energy. The actual configuration will be somewhere in between these two extreme possibilities. In all the above structures. the maximum nossible number of covalent bonds is formed. AS the recent valence-bond calculations on methane by Raimondi. Camoion, . . and Kardus (7) show, contributions from two other types of structure
cannot he ignored. These are (i) Structures where a hybrid orhital is s&coupled with the orhitals of a fluorineitom other than the one to which it points. If the hybrid orhitals are still required to he orthogonal; d orhitals arestill required for S in SF6, and S+ in S+ (F& hut not for S2+ in S2+ (F#. (ii) Structures like SF4 (F+)(F-), where the configuration of the S atom is between s2p3d and sp% For S+Fa(F-)(F+)(F-), the structure where there is an ion on each Cartesian axis does not require d orhitals, hut the other two possihle structures have a confieuration for Sf between s202d and m3d. If we canapproximate the wavefunction for SF^ by the wavefunctions for S2+ ( F d - cis structures and structures where the S atom has an even higher positive charge, then d orbitals are not needed in the h a w set. I t we c x ~ l u d ~ orbitals from our basis set, we are automatically excluding the contributions from the covalent, S+Ffi- and S2+ (F# trans structures with the maximum number of covalent bonds. The question of the importance of 3d orhitals in describing the electronic structure of molecules thus depends upon the importance of structure where the central atom has a small or zero positive charge. Molecular Orbital Calculations on SFs Unfortunately, a classical ah initio valence-bond calculation on SF6 has not yet been performed. However a generalized valence-bond (GVB) and a numher of molecular orbital (MO) calculations have been reoorted recentlv. The first ~uhlished ah initio molecular orbi& calculation df SF^, by ~endazzoli, Palmieri, Cadioli, and Pincelli (a),used a basis set with only s and p orhitals. A gross atomic population (charge) of 1.01 was found for the S atom. That the charge on the S atom is less than the +2 limit predicted by the symmetry arguments for structures with the maximum possible numher of covalent bonds, is due to the use of structures like SF4 (F+)(F-). The fact that we can partially expand mathematically a fluorine orhital function in terms of the sulfur orhital functions could also make a minor contribution. (The overlap integral S(~SF,~P,)has a value of about 0.32.) The energy is significantly improved by the addition of d orhitals to the basis set. The calculation with the lowest enerev to date. that hv Hav (9)gives a charge on the sulfur atom of%l.91. he popdtion;
of the sulfur orhitals are s1.055 ~2.0" d0.g52. On the basis of this population we would expect the most important set of structures to he the S2+ (Fd2- trans set (Fig. l(iii)). The trans structure for S2+ (Fe)2- also has a lower F- - F- repulsion. Other molecular orhital calculations give a range of values from +0.39 to +2.10 for the charge of the sulfur atom. As calculationswith better basis sets are performed, a final "best" value of at least +1.0 and perhaps as high as or higher than +2.0 for the charge on the S atom in SFGseems likely. Conclusions Pauling's sp3d2 hyhrids for SF6 are introduced in most elementary valency courses. The evidence from ah initio calculations suggests that the covalent structure for SF6 is not as important as structures in which the S atom carries a positive charee. Thus we should not teach so3d2 hvhrids as de& 6e introduced scrihinp th;. ~ o n t i ~ r a t i uf o nS in SF^. ~ h may as one of the simolest forms of hvbrid orhitals containing d orhitals. I t should be emphasized that d orhitals are still;eauired hv svmmetrv for some ionic structures. Their use dhould not be confused with the inclusion of polarization functions e.g. 2p functions on the H atoms or 3d functions on the C atom;n C H ~ . Our discussion of the d orhital problem concentrated on S in SFs. A similar prohlem exists with other hypervalent molecules. For P+ in PF5, a d orbital is required to describe the structure where the F- is in an eauatorial nosition hut not in an axial position. d orhitals are not required by symmetry for S+ in SFa. Like SFE.thev are reauired for the covalent struc~ SF^. tures of both P F &d Literature Clted I11 Psu1ing.L.. J.Amer Chem Soc.. 53.1367 11931). I21 Codson. C. A.andGiantureo.F.A.,bCham. Soc.. A. 161Sl19681. (31 Craig, 0. P.,and Thirunsmachandran,T..J. Chrm. Phyr.. 45.3366 (19661; Craig, D. P..andZadi,C.. J. Chsm Phyr.. 37.ML.6M 119621. 14) C r u i c k h k , D.W. J.. Web.tpr,B. C.,snd Mayera.0. F.J Chem. Phys., 40.3733 (1%): Cruiekrhank,D. W. J., Webiter,B.C.andSpinnler.M.A.,lnfrrn. J Quantumchem.. , e ".x,,Qm, .",**"s."".,.
151 Coulmn,C.A.,NaLure, 221,1106
11969).
161 Cation. F. A,, "Chemical AppliesliunnofGroupThwrv:2nd Ed., Wiley.Ltencienco. New York, 1971. 17) Raimondi, M..Csmpian, W.,and Karp1ua.M.. M d e c Phys, 34,1483 (19771. IS1 Bendazzali. G. L.. Palmiori. P.. Csdioli, B., and Pincelli. U.. Mulec. Phyr.. 19. 865 119701.
191 Hay, P. J., J. Amer
Ch~m.S
