Determination of the Number and
1.1.Hsleh Monash University clayton, Victoria 3168 Australia Andrew
II Symmetry of the M-C-0
Deformation Modes
A direct g w p theoretical m e w
Groun theorv has been extensivelv used in the determinationof [he number and symmetry oithe 4COJ fundamentals in metal carbonvls (1-4). The s i m ~ l wDrocedure t ( 4 ) emplop the internal bond displacement coord/nates: these, in metal carhonvls. are the corres~ondinnchanges in the C-0 bond distances; Arco, which are to beregarded as localized bond vectors. The number, nR, of such C-0 bond vectors not shifted by the symmetry operation R is first determined. The number and symmetry of the u(C0) normal modes can then be obtained using the reduction formula (1)
i;1g
ai = x(R)xi(R) (1) where h is the order of the point group and xi(R)is the character of the (irreducible) representation of the ith symmetry species with respect to the symmetry operation R. In this particular case, x(R), the character of the (reducible) representation for the system is given by x(R) = nn (2) The number and symmetry species of the u(MC) and 6(CMC) modes in metal carbonyls can be similarly determined using the appropriate bond (M-C) or angle (LCMC) displacement coordinates. By contrast, since the M-C-0 groups in metal carhonyls are almost always, crystallographically linear, a similar attempt to derive the number as well as the symmetry of the 6(MCO) vibrations will be singularly unsuccessful. Thus, they are often obtained indirectly from the svmmetrv snecies of the (3N - 6 ) normal modes of the N-atom m e t i c&bonyl by subtracting those arising from the vlCO). uIMCI. ~ ( C M Cand I if anv. 4M-M'J and other vihrations: t his tedious procedure a i d the lack of any well documented, straightforward method prompt this report of a more direct group theoretical approach. group. I t is First, consider an isolated linear M-C-0 trivial by inspection or otherwise that there is only one 6(MCO) mode which in this case is doubly degenerate (II in the C ,, point group). The 6(MCO) vibrational coordinates can be resolved into two mutually perpendicular components, say, in the xz and yz planes for a MCO group in the z direction.
From the transformation matrix, we have t(E) = 2. C.: Anticlockwise rotation about the z axis hy an angle of 0 = 2nln moves them to new position cos R -sin 6 = s ~ Rn cos R For clockwise rotation cos R sm 8 [-sin R eos8 Whence, r(C,) = 2 cos 8.
'II:[ [;:I
Improper Rotatlons a: Reflection
in xz plane
or in the yz plane both afford €(a)= 0. i: Inversion gives 4 ) = -2 from the transformation
S,: This is essentially the same as C, since after rotation by an anele of 0 = 2 d n . the reflection is taken through . the xy plane. Thus, the character for both the proper and the improper rotations is r(R) = 2 cos 0 (3) This expression provides us with the character of the transformation matrix when the symmetry operation is performed on a single M-C-0 moiety. For a metal carbonyl containing several linear M-C-0 groups, we have x(R) = n d R ) (4) where nR is now the number of M-C-0 groups not shifted by the symmetry operation R. Combining eqns. (1) and (4), we obtain 1
ai = iE
Symbolizing these vibrational displacements by 6,, and a,, we consider next the effect of the geometrical transformation by the various symmetry operations on 6,, and a,,. From each of the transformation matrices associated with the corresponding symmetry operations, the character of the operations, r(R), which is the trace of the matrix, can be deduced. The effect of each of the five symmetry operations is now considered in turn. Proper Rotatlons
E:The identity operation may be represented as follows
~RL(R)x'(R) (5) The use of this formula is illustrated by the following examples. Example 1. The solid state structure (5)of the compound C1Mn(C0)5 has an idealized Cq. symmetry.
The following table can be constructed in conjunction with the Cp. character table.
Application of the reduction formula (eqn. (5)) gives I'A(MCO)
= A1
+ A2 + E l + B 2 + 3E
Example 2. F e ( c 0 ) ~has a trigonal bipyramidal (D3d structure in the gas phase ( 6 ) OC OC-Fe
I 4" m I'
Whence
CO
r a c ~ c o ) = A1
+As
+ 2E + 3 T 1 t 3 T 2
It is of interest to note that unlike other deformation modes of non-linear moieties in a molecule, e.g., 6(CMC) modes in metal carbonyls, the problem of redundancy does not arise in the case of the deformation vibrations of linear M-C-0 groups in metal carbonyls. The reduction formula (5) may also find its application in other compounds containing a number of linear groups of the same kind, e.g. cyanides and isocyanides of metals. Using the Dab character table together with eqn. (5). we obtain Polynuclear metal carbonyls can be similarly treated. Although X-ray crystallography may detect disorder or distortion in the molecule, a fairly high symmetry of the complex may be assumed. This is partly justified by the fact that the vihrational spectra of metal carbonyls are often taken in solution. Examule 3. Thecluster Ird(C0h2consists of a tetrahedron of iridium metal atoms each carrying three terminal carhonyl aroups (Fia. 4). Its solid statestructure (7) thus hasanideal-
The author wishes to express his appreciation to Professor B. 0. West for his interest in this work, Dr. A. D. E. Pullin for constructive criticism, and Dr. J. E. Kent for helpful discus. sion. Literature CHed (1) Cotton,F.A.,"Chemieal ApplieationsofGmupThwry." 2nd Ed.. Wiley-Inumience, New York. 1971. (2) Hall, h H.. "Group Theory and Symmetry in Chemiatry," MeGraw-Hill. New York. 1969. (3) Davidson. G., "lntmduetory GmupTheorylor Chsmiatr."Elsevier. Inndun, 1371. (4) Oarensherg. M. Y..and Damnsheberg, D. J..J. CHEM. EDUC..47.33 (19701, (5) Bryan. R.F.,andGreene.P.T.. J. Chem. 5'or.A. 1559 (19711. ( 6 ) Rndorff, W., and Hofmenn, U.Z.Physh. Chem. (Leip%ig),28B,351 (1985). (7) Wei. C. H..Wilkas. G.R., and Dahl. L. F.. J Amer Cham. Sm.. 83.4792 (1967).
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Volume 54, Number 7, July 1977 1 421
