Chao-Yang Hsu and Milton Orchin University of C~ncinnati Cincinnati, Ohio 45221
Ligand Group Orbitals and Normal Molecular Vibrations Symmetry simplifications
The generation of ligand group orbitals (Igo's) of molecular complexes and a determination of the symmetry. species to which they belong are essential prerequisites for the of molecular orbital enerm diagrams. ~ -construction -. Where feasible the appropriate lgo's are mos&onveniently arrived at bv ins~ection:the individual ligand orhitals are combined b; additions and suhtractions t o match any of the s, p, or d orbitals of the central atom of the complex. Thus, for example, in a D4h complex in the coordinate system of Figure 1, the (normalized) combination which matches the d,=-,. atomic orbital of the central atom is, by inspection, l , ( a l - oz + (r3 - 04) and this lgo belongs, of course, to the same symmetry species, bl,, as the matching atomic orbital. However, the inspection method has obvious limitations. Thus, for example, in an Oh complex whose ligands possess occupied p a orbitals, there are Igo's which belong to species tl, and tzu but there are no atomic orbitals on the central atom which belong to these species, as reference to the character table for On readily shows. Also in some geometries, the matching is not very obvious. There are available a number of procedures (1, 2) which ~ e r m i tsymmetry simplifications of the problem but we believe that none are suite as simple or require less mastery of group theoretical methods than the procedure we o u t h e herein, although access to and understanding of character tables are still required. ~~~~~~
~
Figure 2. Orientations of ligand orbitals of square planar MLn: (a) n orbitals; (b) in-plane orbitals: (c) out-of-plane ry orbitals.
*,
ligand (Fig. 2a). I t is therefore obvious that the z-coordinates (hence the four M-L stretches) transform identically to the a orhitals (both belong to AI, BI, E,). The x coordinate on each ligand (Fig. 1) lies in the horizontal plane (xy) and these four coordinates are equivalent to the disposition of the four in-plane p a orbitals of the ligands, a*, (Fig. 2b). Hence the set of four a x Igo's and the four in-plane vibrational modes transform as the same symmetry species (Az, Bz, + E,). Reference to the Dph chara d e r tahle shows however that the rotation, R,, belongs to Az,; hence there are three genuine in-plane vibrations (BZ, E,). The y-coordinate on each ligand corresponds to the r, orbital, (Fig. 2c); hence the set of ir, lgo's and the out-of-plane vibrational modes transform as the same symmetry species (Az. + Bzu E,). The D4h character tahle shows that R,, R, transform as E, and hence there are only two genuine out-of-plane vihrations (AzU + BzY). It should thus be clear that by separately combining the x-, y-, and z-coordinates on each of the n ligands we get the 3n = 12 normal vihrations (three of which will be rotations in a nonlinear molecule) as well as the 3n = 12 Igo's. In our example of the Dqh case, the combinations of x-, y-, and z-coordinates leading to the symmetry species of the in-plane, out-of-plane, and stretching vibrations are identical to the Igo's (and their symmetry species) derived from the a,, a , , and o ligand orbitals, respectively.
+
+
+
+
+
Symmetry Simplifications for Determining LGO's
Figure 1. Coordinafe sysfem of square planar MLI
In this article we also demonstrate the close connection, indeed the 1:l correspondence, between the lgo's of a complex and its normal vihrational modes. The standard procedure for determining the symmetry species of the 3n normal vihrations of, for example, a ML4 complex of D4h symmetry (Fig. 1) begins by transforming the 3 X 5 = 15 internal coordinates under each of the symmetry operations of Dlh and continues in a relatively straightforward manner (3). Although rarely employed for the purpose, this general procedure can distinguish between stretching modes, in-plane, and out-of-plane modes1 by separately combining coordinates of the same sub-set, e.g., all z-, or all x-, or all y-coordinates of Figure 1. Comhining coordinates in this way leads to a direct correspondence of the normal vibrations with the Igo's of the complex. The plus z-coordinate on each ligand of ML4 (Fig. 1) points toward the M atom just as does the a orbital on each
We now proceed to describe a method for deriving the Igo's of complexes of various geometries. Having obtained these, we will then demonstrate their correspondence to the various normal vibrations. Step I. Place the complex in an appropriate coordinate system* and classify the ligand orhitals into sets based on their "obvious" point group symmetry. In D o (Fig. 1) this means that for the o orbitals we use Dm. for the *, orbitals, C4,, and for the n, orhitals, Cm. Step 2. (a) Descent in symmetry. Further divide the sets of orbitals into smaller sets in point groups that are subgroups ( 4 ) of those chosen in Step (1) until no mare than two ligand orhitals are in a single set. Thus the four ir orbitals in D m become two sets in Dm. namely ( c , , ~ and ) (c2,0r),based on the Cz axes, (Fig. 1)which characterize Dm.
'The in-plane and out-of-plane vibrations can also he ascertained hv transforming the in-plane angles and the out-of-plane angles; see ~ e f( .6 ) . ?By convention, the complex is placed in a right-hand coordinate system, but each ligand is always placed in a left-hand coordinate system; see Ref. ( 3 ) . Volume 51, Number 11. November 1974
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(h) Make linear comhinations (addition and subtraction) of the orbitals in the same set. Thus as shown in Tahle 1, for the two sets from Step 2(a) we get the linear combinations: ol + 0 3 , ox - 0.3; 02 + or, sz - a,. These combinations must be orthogon-
(e) The Igo's, $, and $,, made by combining sets must he orthogonal
S@,@,dr = 0; and each lgo must be normalized
.~
n1 --(31. ,. (c) Assign
the comhinations to their appropriate symmetry species in the lower symmetry. In our example this can he done by recognizing that sx - os matches the p, orbital on the central atom and rm - cn the p, orbital3 and hence reference to the Dzn character table shows that the comhinations belong to br. and bm, respectively. Step 3. (a) Ascent in symmetry. Correlate the combinations in the lower symmetry with species in the higher symmetry. If two combinations in the lower symmetry belong to the same species and can be transformed into each other by an operation in the higher point group, make linear combinations. Thus from Table 1 we see that CI + q and oz or belong to a# in Dl*and can he transformed into each other by C4 in Dl*.Hence the comhinations (ex + $3) + (-2 + (r*) are necessary in the ascent in symmetry. (h) Assign the comhinations obtained in Step 3(a) to their appropriate symmetry species in the original point group. Thus the combination [($I + 04 - (cl c4)] matches the d - yz orbital on the central atom and belongs to species bl,. Similarly the combinations - os, and oz - G, correspond to x and y and in DG, belong jointly to e.. If the comhination (wave function) does not correspond to a function in the point group, then treat the lgo as a comhination of ligand axes with arrows pointing in the positive direction (vectors) and transform these axes under the symmetry operation of the point group and make the assignment. Thus from Table 1, the comhination (yl yr) - (y2+ y ~ does ) not correspond to any function in DM.However by transforming this comhination of axes (Fig. 1) under the symmetry operations of Dm we find that this lgo transforms as bz,.
Ir~(@,)= l ~1d ~ see Ref. (3). Construction of Normal Modes of Vibration
+
In a ML, complex there are 3n lgo's and 3(n 1) - 6 = - 3 normal vibrations (for nonlinear complexes). T h e three excess lgo's correspond t o the three rotations. The stretching modes are obtained from the a lgo's and the bending modes from the a lgo's. The sign of each component ligand orbital in the group orbital may he used to determine the direction of vibration of t h a t atom. The positive end of the a orbital is taken a s the direction of motion. In order to determine the motion of the central atom we consider t h e motion of each ligand a s a vector and recall t h a t a genuine vibration cannot involve motion of the entire molecule. Accordingly the direction of motion of the central atom is opposite to the vector sum of the ligands.
3n
+
+
The matching in this example is relatively simple but in other geometries, e.g., Td. it is not SO immediately obvious. It should he pointed out that p orhitals correspond to translations, and d orbitals to polarizabilities. The matching can also be done by reference to the symmetry species of rotations.
+
Table 1. M 4 in Ddh(Coordinate System Fig. 1; Vibrational Modes, Figure 3) "Obrious"
Table 2. MLBin Oh (Coordinate System Figure 4) "Obvious" 1.0.
symm. orbital set
-ubgmup------
rrt
comb.
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wmm. (funct.)
bv(Rx) b*"(~)
-
D‘hCombination
Yl ya
Oh
wmm.
+- xrx*
(funet.)
1.g.o.
funet.
mmm.
MLs in
Oh
The basic strategy here is to descend to the D4h subgroup and then classify the orbitals on ligands five and six in this point group (Fig. 4 and Table 2). Step (3) requires comment. From the s orbitals we have four combinations belonging to e, in D4h We see from Table 2 that two combinations correspond to translation in the x-direction (or p,) and two to the y-direction (or p,). The two e d x ) are combined to give the normalized linear combinations (1/2)[(xz - x4) - (xg -YE.)] and (1/2)[(xz - xr) + (x5 - ye)] which correspond t o one of the component tl, and one of the component tz, wave functions in Uh. respectively. One of the combinations of y's in D4h leads to the second t l , component in O h and the third tl, correlates with z of az, symmetry in D4h. ML5 in
D3h
Derivation of LGO's and Normal Modes of Vibration of Complexes in Various Geometries
I t is important to remember that in Dah,(Fig. 5) the apical ligands do not transform to the equatorial ones by any operation in the group and hence these are a separate set; this set is carried over directly to D3h. The equatorial set is however, descended to lower symmetry, (Table 3). The Igo's and the corresponding normal vibrations are shown in Figure 6.
MLI in D 4 h
ML. in Td
In Table 1 we summarize Steps (1-3) for the 12 Igo's and Figure 3 shows the motion of the normal modes conesponding to the symmetry species of the Igo's.
This is a rather difficult problem and can be handled satisfactorily in either of two ways. If we place ML4 in the coordinate system of Figure 7, a descent to Dzd symmetry is convenient. The assignment of symmetry species (Table 4) is best made by matching combinations with the sym-
Figure 3. Normal vibrations and the corresponding ligand group orbitals of square planar MLn.
Figure4 Coordinate system of octahedral MLe. Figwe 6. Normal vibrations and the corresponding iigand group orbitals of trigonal bipyramidal MLr.
Figure 5. Coordinate system of trigonal bipyramidal MLs.
Figwe 7. Coordinate system of tetrahedral MLI, subgroup.
emphasizing D z ~
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Table 3.
M k in Dlh(Coordinate System Figure 5: Vibrational Modes, Figure 6) Subgroop--
1.0.
"Obvious" S y m m e t r y o r b i t a l eet
set
Table 4. M L in 1.0.
"Obvious" Symm. orbital s e t
1.0.
"Obvious" Sym. orbital sst
-"bemu~ e t
..
.
comb.
symm. (funet.)
comb.
-
lJ2d
s m . (fund.)
combination
function
symm.
-
Ta1.g.o.
funot.
symm.
Td (Coordinate System Figure 8)
-
Ca-
,
aymm. (funct.1
Dah
1.g.o.
Td(Coordinate System Figure 7)
.
Table 5. M L in M u b g r o u p - . set
Symm. (Function)
mmb.
combination
symm. (funet.1
Ta 1.8.0.
fund.
sym.
metry species of orbitals (or translations) and rotations. The descent in symmetry can also be made through Csu and for this purpose the coordinate system shown in Figure 8 can be employed. The appropriate procedure is then outlined in Table 5. ML2in D,h
Although this otherwise cumbersome problem has been simplified (5) it is handled even more simply by recognizing that only one set of two o, two s,, and two sy lgo's exist (Table 6 and Fig. 9). The vibrational modes corresponding to the lgo's are shown in Figure 10. Literature Cited (11 Phelan. N.F.,andO~ehm,M., J . CHEM.EDUC., 43.571 (1966). (2) Katt1a.S.F.A.. J.CHEM.EDUC., 43,652(1966). (31 Orchin. M.. and Jsffi. H. H... "Svmme~v. . . John Wilev L . .. Orbitals. and Soectra." ~ans;Ine.,New York, 1972. 141 Jam. H. H.. and Orchin. M.. "Symmetrv in Chemistni." John Wilev & Sons. hc.. ~~~~~
Figure 8 . Coordinate s y s t e m of tetrahedral subgroup.
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MLI emphasizing C3,
~~~~~. ~
-~
(51 ~ t ~ ~ m e n , O . P . , a n d L i p p i n c o E.R.. t t , J.CHEM.EDUC., 49,341. (1972).
(61 Thomsa. C.A..J.CHEM.EDUC.. 51.91 (1974).
~~~.
Table 6. M b in D,h (Coordinate System Fig. 9; Vibrational Modes. Figure 10)
_CC_
% =I*? Figure9. Coordinate systemof M 4 in D,h
1;
4- '5
nu X1-
X2
nu YI+ y2
Figure 10. Normal vibrations and the corresponding ligand group orbitals of a MLP. 0.h complex.
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