S. F. A. Kettle The University Sheffield, England
Ligand Group Orbitals of Octahedral Complexes
I n recent years it has become appreciated that it is a very crude approximation to regard the interaction between cation and ligands in a transition metal complex as purely ionic. Recent articles in THIS JOURNAL (1,2) have discussed this point a t length so we shall not elaborate further here. A more realistic picture which allows for the effects of covalency within a complex is provided by ligand field theory. A necessary first step in this approach is to form the linear combinations of ligand orbitals appropriate to the symmetry of the complex molecule. These linear combinations are grouped into sets and for each of these there is usually a set of orbitals of the cation which have a similar symmetry and degeneracy. The problem which we shall discuss is that of finding the correct linear combination of ligand orbitals for an octahedral complex. A table of these is usually given in articles on ligand field theory but the combinations are derived, a t best, by inspection. The a,, ligand group orbital1 (1/V%) (A B C D E F)-where we have labeled the six u ligand orbitals A-F as shown in Figure I-is quite reasonable, in that the six ligand orbitals, which we know to be equivalent, contribute equally. An e, combination, (l/dz)(2A - B - C - D E 2F), appears unreasonable, however, in that if it is permitted why are not also a host of other combinations in which the six ligand orbitals are nonequivalent? This problem is best resolved by a systematic derivation of the correct combinations, after which invented combinations can be discussed. The basis of the derivation is group theory. There are now many texts which include, or are devoted to, an exposition of group theory for the chemist (.5-8).2 In contrast to the complicated algebra needed to derive the important relationships, the actual arithmetical operations involved in the application of group theory are usually extremely simple. I n this article we shall use group theory, and take the algebra for granted. The reader who wishes to investigate the mathematical basis of the steps involved is referred to the above texts. We must, however, introduce the terminology of group theory.
+ + +
+ +
An octahedron (a figure with eight triangular faces, twelve edges and six corners) shows considerable symmetry. I t has, for example, three distinct fourfold axes, passing through opposite pairs of corners. The corner A (which in Figure 1 represents a ligand) canby the operation of rotation about the symmetry axis through EF-be converted into B, C, or D (Fig. 2). The operations which convert A into B and D are fourfold rotations (i.e., *90° = *360a/4) but the rotation converting A into C (180" = 360/2) is really about a twofold axis. That is, we have coincident fourfold and twofold axes, with two distinct fourfold rotation operations (denoted 2CJ and a single twofold rotation operation (denoted CJ. It may be helpful to regard the symbol C as a simplification of C, denoting a rotation. Each of the other two fourfold axes, through AC and BD, similarly have two distinct fourfold rotation operations-a total of 6C4. Since AC, BD, and EF are also twofold axes there are a total of 3C* operations. E A
- +
The symbolism used here is that generally adopted (see footnote 7 in lef. ( 8 ) and p. 72 of (5)). In brief, a implies an orbital which is desoribed by a one-dimensional representation (vide infm); e, a two-dimensional; and t, a three-dimensional. Note added in pmof: Since this article was written, two further books have appeared: JAFP*, H. H., AND ORCHIN,M, "Symmetry in Chemistry," (particularlyChap. 5); and MURRELL, J . N., KETTLE, S.F.A., AND TEDDER, J . M., "Valence Theory" (Chaps. 8 and 13). Both are published by John Wiley & Sons, Inc., New York, 1965.
Figure 1.
Figure
2.
Perpendicular to each pair of opposite faces of the octahedron and passing through the midpoint of them is a threefold axis. Since there are four such pairs of faces there are also four threefold axes. Figure 3 shows one such aspect; the reader is looking directly down the axis perpendicular to the midpoint of face BCF of Figure 1. I n Figure 3, A can be rotated about one of these axes to give either D or E; i.e., there are two threefold rotation operations about this, and therefore about each C3 axis, giving a total of 8C3operations. Passing through each pair of opposite edges of the octahedron are twofold axes quite distinct from those found earlier (the octahedron in Fig. 4 is drawn slightly off-axis). Since there are six pairs of edges there are six twofold axes, denoted 6C,', the prime distinguishing these twofold axes from the others. As is always a
Figure 3.
Figure 4.
Volume 43, Number 1 , iunuory 1966
/ 21
the case for twofold rotation axes, the number of axes is the same as the number of distinct operations. There remains one symmetry operation which must be introduced: the apparently trivial operation of leaving the octahedron alone. This opcration, denoted by the lett,er E (or sometimes I),is called the "identity operation." The operations nhich have been discussed above are a complete list of t,he group of operations possessed by a molecule of symmetry O ( 0 for Octahedral). It is rather unfortunate that no n~olcculcis known of symmetry 0, for the rest of this article will he spent discussing the bonding in a molecule of this symmetry; our results, however, are readily modified to apply to the group Oh. Octahedral molecules and complexes belong to this later group which has twice as many symmetry opcrat,ions as the group 0. I n particular, the group Oh includes the operation of inversion in a center of symmetry. By the term "the group 0" we really mean "the complete gronp of operations fiC4,.7G,8C3,G G ' , and E." The label O is n shorthand for the set of symmetry operations. For each group there exists a table called a character table. The character table of the group 0 is shown in Table 1. Table 1.
Character Table for the Octahedral Group
Irreducible represeninl,ions
-41 A,
E
TI
TZ
0
Statements (c) and (e) are mathematical requirements and are true for any character table. We now proceed to show that the behavior of sets of atomic orbitals on a cation at the center of the octahedron under the operations of the gronp can be described by the irreducible representations in Table 1. A closely related example has been discussed by Manch and Fernclius (9),but we shall need to elaborate the technique that they use. Other related discussions are given in references (3-8). If an orbital is converted into itself by a particular operation of the group this is represented by the number +l. If it is converted into minus itself (i.e., the phases of all the lobes changed) then this is denoted by -1. If an orbital is converted into a completely differcnt orbital, this is represented by the number 0. Fractions also occur but we shall defer a discussion of these until later. Bccause an s orbital is spherically symmetrical, all of the operations of the gronp convert it into an orbital which is indistinguishable from the original one. That is E
6Cr
3C!
8C3
6C2'
E
1 1
1
1 1
1 1 -1
1 -1
1
-1
n 1
-1
2 -1
-1
0 0
o
-1
1
The charact,er table of the group Oh is four times larger; it is for this reason that we shall work with the group O and later modify various labels to those which we would have obtained had we used the larger group. We note several things about t,he character table: (a) A complete list of the symmetry operations which convert the octahedron into itself appears across the tap of the table. ( h ) Thwn the left side artre various letters which we shall tae as labels. Associated with each label is a row of numbers, with one number for each type of symmetr," operation. The rows of numbers are each called "irreducihle representation" (of the group) and the letters are just a shorthand way of indicat,ingwhich set of numhers (which irreducible representation) we are talking about. The numbers themselves are called "charzcters"-hence the name "character table." (e) The number of irreducible representations (5) equals the number of distinct sets of symmetry operations (4). l d ) The capital letter E appears both along the top and down the left side. This is unfortunate for it means two different things (the identity operation and a label). It will always he clear from the context which is referred to. In general, E (identity operation) is seldom encomtered, slthongh tthis is not true for this article. ( e ) If the numbers in the E (identity operat,ion) column are 1 4 9 9 = 24) the total equals squared and added ( 1 3 8 6 = 24) the n ~ ~ m h eofr symmetry operations (1 6 (4).
+ + + + + + + +
22 / Journal o f Chemical Fducotian
3C9
8Ca
1
1
1
6C%' 1
It is most important to note that the result is not
E 2 3 3
6C4
s orbital on the central cation
6C4 6
3Cz 3
8Cs 8
6Cn' 6
But the first (correct) set of numbers is exactly those which are denoted A , in the character table (Table 1). Hence we can say that the s orbital "transforms as Al," or that the orbital is an a, orbital (lower case letters are used to denote orbitals) since the A, irreducible representation completely describes its transformation properties. We now consider in detail three p orbitals on the central cation. Under the identity operation E they are, of course, unchanged. It follows that each contributes 1 to the character, giving E
P., P", P*
3
The effect of the six Ca operations is more complex. We consider a representative example of these operations; the reader can satisfy himself that the same result holds for the other five (a model is a great help for this). Figure 5, in which the p orbitals are distinguished by their differcnt outlining, shows the effect of a Ca operation before it is applied, in the middle of the operation, and finally when it is completed. This Ca opcration converts p, into p, and p, into p,, while p, remains itself. Because the p, orbital is converted by the rotation into a completely diierent orbital, it contributes 0 to the character. Similarly p, contributes 0 hut p, contributes 1 as i t remains unchanged.
Figure 6.
p, but p,, which is perpendicular to the twofold axis, becomes -p,. The character is 0 0 - 1 = -1.
We have
-+
6C4
1
P.,P",P,
6Cz'
Other C4 axes give different individual transformations but the same final anewer (1). The 3Cz axes are coincident with the C4 iust discussed. The effect of operating with them on the p orbitals is shown in Figure 6. Here p, becomes -p, and contributes -1, p, becomes -p, and also contributes -1, while p, remains unchanged and contributes 1. We have 3C2
pz,p.,p,
-1
(=-1-l+l)
The 3Ca rotations are readily dealt with: as usual we consider one representative operation. The behavior of the orbitals is illustrated in Figure 7, where it is seen that they cyclically permute: P.
+
(=0+0+1)
- - P"
P=
P*
All change their identity so each contributes 0 t o the character. That is
P=,P,, Ps
-1
To summarize: E P,,P,,P,
3
6C4 1
3Cz -1
8C3 0
6Cs' -1
This is the same as the T, irreducible representation in Table 2, so we may say that the three p orbitals transform (together) as TIand give them (together) the orbital symmetry label tl. The three orbitals together form a complete set. That is, the operations of the group acting on any one member of the sct convert it into the various members of the set (or into mixtures of them). To put it another way, we could not have considered p, and p, and ignored p, because some of the symmetry operations of the 0 group convert p, and p, into p,. That is, p,, p,, and p, form a complete set but p, and p, do not.
SC, ,.P Pm Pa
0
Last, we must consider the 6CZf operations. The operation is illustrated in Figure 8. Here p, -t p, and p,
I n a precisely similar way the orbitals d,,, d,,, and rl,, form a complete set which transforms as T2and d,.-,. and d,. form a complete set which transforms as E, so that the sets of orbitals are labeled as t~ and e respectively. The effect of representative symmetry operations on the members of the tz set is shown in Figures 9-11 and summarized below (Table 3). I n
Figure 8.
Table 3 dz.
&* d,
Character
E
C4
d="
-d%" -d,
4.
dB, l + l + l = 3
&s
-l+O+O=-1
cz &"
-dvs -d,
1-1-1=-1
cz
C*'
dvz
4
[hu
0 + 0 + 0 = 0 Volume 43, Number
d,
dvz
1 + 0 + 0 = 1
I , Januory 1966 / 23
Figures 9-11 the different d orbitals are characterized by the way they are outlined, a consistent scheme being followed. Only a few lobes of the orhitals are shown, to keep the diagrams as simple as posBible.
7
Figure 9.
sentation is obtained when we consider the transformation of o orbitals (i.e., pointing toward the center of the octahedron) on the six ligands a t the corners of an octahedron under the operations of the group 0 (Fig. 14).
Ligmd n orbitals
Figure 1 1 .
Figures 12 and 13 similarly show the transformation of the d,%-,> and d,. orbitals. (See Table 4.) The behavior of these orhitals under the C3 rotation operations is more complex-they are mixed-and is discussed in an appendix. The character for this operation is - 1.
Figure 13.
The labels of the irreducible representations of the full octahedral group On are simply derived from those of the group 0. As noted earlier, the operations of the Oh group include that of inversion in a center of symmetry. For every irreducible representation in the group 0 there are two in the group Oh; one is centrosymmetric (g) and the other centro-antisymmetric (u). So (for example) corresponding to the irreducible representation T 2 in 0, there are Tz, and T2, in On. We can easily decide which since we know whether or not the atomic orhitals are centro-symmetric. We conclude that an s orbital on the central cation transforms as al, in Oh, the p orbitals as t,, and the two sets of d orhitals as t,, and e, respectively. So far we have only encountered irreducible representations. An example of a educible repre-
4A
Journal of Chemical Education
E 6
6C4 2
3C2 2
8Cs 0
6C2' 0
+ +
+ 4B + 4C + 4 0 + 4E + 4F
Normalizing (i.e., rewriting the orbital so that the sum of the squares of coefficients equals I), the at
Table 4
/
14.
This reducible representation equals Al E TI (A,, E, T I , if we had considered the transforms, tions under the operations of the group Oh). It follows from this that we may take linear combinations of the ligand o orbitals to give combinations which are quite independent and carry the labels a,,, e,, and t,,. These combinations are those encountered in ligand field theory; their form will now be derived. We give each ligand orbital the label A-F as shown in Figure 1. We then concentrate on one of these orbitals and consider in more detail its transformations under the operation of the group. In Table 2 are listed the 24 operations of the group 0, and beneath each is the ligand o orbital into which A is transformed by the particular operation. Within each set of operations, 8CJ for example, the order in which the operations are considered is unimportant; what matters is that all are included. We can now obtain the ligand group orhitals. The procedure is to multiply the orbitals in Table 2 by the appropriate set of characters. To obtain the al orbitals one uses the A, characters, to obtain the e orbitals, the E characters, and so on. The orbital in Table 2 listed beneath the identity operation is multiplied by the identity operation character; the six orbitals listed beneath the Ca operations are each multiplied by the C, character, and so on. After this multiplication has been carried out for all of the orbitals, the products are added together. The sum is either the required ligand group orbital or is simply related to it. Consider the at group orbital. Multiplying each of the orhitals by 1 (the value of each of the a] characters) and adding the products together gives
+ +
24
Figure
Under the E operation a11 six o orbitals remain unchanged giving a character of 6. The 6C4 (and 3C2) operations leave the two orbitals lying along the rota, tion axis unchanged but interchanges the others. For each of these operations we have a character of 2. Both the 8C3 and the 6C2' operations interchange all of the orbitals giving a character of 0. To summarize:
Figure 10,
Figure 12.
=
orbital (algin Oh) is obtained If we try to form an A2 orbital we find that the products cancel out, giving no contribution from any orbital. This confirms our earlier conclusion that only al, e, and tl group orbitals exist. Turning to the e orbitals, the sum obtained after multiplication is 4A
+ 4C - 2B - 2D - 2E - 2F
which after normalizing is Had we considered, in Table 2, the transformation of
R or E rather than that of A, we would have obtained from B:
+ 4D - 2A - 2E - 2C - 2F + 4F - 2A - 2B - 2C - 2 0
48
from E : 4E
Physically, this new flu set is oriented along different coordinate axes to the set given in the table, the latter being that appropriate to the usual choice of coordinate axes for an octahedron. What of the combination A B-C D -E F? This is equal to $,(2) $,,(l),and because i t mixes orbitals of different symmetry species it is not acceptable. The rule is this: if some invented combination can be decomposed into a linear combination of orbitals of one of the symmetry species of Table 5, it is acceptable (although new partners will also have to be found). If the combination mixes orbitals of different species, it is not acceptable. For simplicity we have used integral mixing coefficients. I n general fractions will occur. We have now determined the symmetry species of the atomic orbitals on the metal cation and the symmetry and form of the ligand a group orbitals.
+ +
+
Neither of these is suitable the second E function for neither is linearly independent of +,(I). If we sum these two we obtain -4A - 4C
+ 2 8 + 2 0 + 2E + 2F
which is - 1times the sum obtained when we considered the transformation of A. The difference between the sums derived from B and E is which on normalizing gives the second E orbital $0)
=
'/dB
,,,
+ D - E - F)
\
The TIfunctions are readily obtained. The transformation of A yields 4 A - 4C which, normalized, gives
tl"
Metal orbitals
. . ., , r
!
Liqand orbitals
+(I) = ( l / d a ( ~ - C)
The transformation of B and E gives end +r,(3) = ( l / d % E - F )
respectively. This completes the list of ligand a group orhitals; they are collected in Table 5. What of other combinations, for example, A BC-D -E F? This is simply +,, (1) $,,(2) +,,(R), and since it is composed solely of tl orbitals it is perfectly acceptable as an alternative tl orbital. The other two tl orhitals would have to be different from those given in Table 3; acceptable combinations are + r , (1) - +U (2) (= A - B - C D )and + I , (1) +z, (2) 2$t, (3) (= A B -C-D 2E f 2F).
+
+
+
+ +
+
+
+
Table 5
-SymmetryIn the In the group
0
group Oh
-
Form (See Fig. 1)
If we use the rule that only orbitals of the same symmetry species have non-zero overlap integrals, the familiar schematic diagram shown in Figure 15 is obtained. Figures 16-19 show the interacting pairs of orbitals. The molecular orbital theory of transition metal complexes has been recently described in articles in THIS JOURNAL by Liehr (10) and by Gray ( I ) , and the reader is referred to these for a fuller discussion. The sets of orbitals given in Table 5 are not isoenergetic. The differences in energy between the sets
Figure 16.
-
Figure 17.
Volume 43, Number 1, January 1966
/
25
are usually neglected, but we have included them in Figure 15. Where interactions between the atoms forming the octahedron cannot be neglected-for example in the Ta6C1122+ and Mo6CL4+cations, where there are octahedra of metal atoms-these energies are most important. We conclude this article by showing how such energy differencesmay be calculated in the approximation usually used-that in which nearest-neighbor interactions are the only ones considered. The time-independent Schrodinger equation may be written XJ., = E.J.,, which tells us that EX, when it operates on the eigenfunction J.,,, produces the function J.. again, multiplied by the energy of that function E,. As is well known, pre-multiplying either side of the equation by J., and integrating over all space gives an expression for En:
where we have assumed that J., is real. If it is also 3C& dr. If we now assume normalized, then E, = f J.. that this equation can be applied to the orbitals in Table 5 their energies can be calculated. As an example we will consider the case of J.,(2): '/%(B D - E - F). Substituting this into the energy equation we obtain
+
E&)
=
114
f( B + D - E
- F)X(B
+ D - E - F)dr
The righthand side of this equation can be expanded to give four integrals of the type f BXBdr, eight of the type f BEE&, and four of the type f B3CDdr. These integrals are distinguished by being respectively based on one orbital (B), on orbitals on adjacent atoms (B and E ) , and on atoms trans to each other (B and D). The nearest-neighbor approximation means that we shall neglect this last set. The first (coulomb) integral is given the symbol a and the second (resonance) integral the symbol p. Since the labels we have attached to the orbitals are purely arbitrary f BxB& = f DxDds, etc., = a; and f B3CEdr = f B x F ~ T ,etc., = 0. Hence the energy of +,(I) is or - 26. It may readily be shown that J..(2) also has this energy, which indeed i t must since the two orhitals transform as a pair. Similarly, the tl. set
Figure 18.
Figure 19.
26 / Journal of Chemical Education
+
all have energy a and the a, orbital a 48. It is always found, in the nearest neighbor approximation, that the energy of the totally symmetric a combination is or (number of nearest neighbors) p.
+
Appendix: The Behavior of d,~,, and d,. under a C, Rotation Operation In Fig. 20 we ahow the effect of a Cs rotation on d,x,z and d.2. For simplicity we show only those lobes of the orbitals which point toward the observer. The operation haa the effectof These are not new converting d,l.,r into d,c.r and d,. into d.2. orbitals hut linear combinations of d . ..I and d.2. We must, then, is contained in redefine our problem as "how much of d.~.,l d,.,r and how much of d.2 is contained in d,~?" We can do this most readily by asking "how much of zs - yz is contained in yP - zB and 9 in z2?" but first we must get our labels correct. Although zP - yB correctly describes the nodal behavior of d,*.,l (positive lobes along the z axis, negative along the y), zPdoes not give the "eollt~r"associated with the d3. orbital. The correct combinationis (l/%6)(2z2 - z 2 - yn)(strictly, oneshould orbital; not the d..). The factor talk about the do,r.,r.,xl of I/& iis introduced because it makes the sum of squares of is, two. Similarly, "coefficients" the same as for 2' - y-that d,, is really d p ~ ~ * . s ) . We write yP - z a = C,(zP- yP)
(l/d3T(zz'-
y k 9)=
+ ( c p / d 5 )(222 - - y2)
CS(ZP
2%
-Y)+ ( ~ ' l d 3 )(2z1 - z2 - ye)
I t is readily shown that C, = C4 -- -1/2 and C2 = -Ca = 2 The coefficientsCIand C4tell us how much of d.z.* is contained in d,,.,i and how much of d , ~in d,, respectively, so the character given by these two orbitals under the Cs opera(-'/*) = -1. tion is (-]/*)
+
f
* f
Figure 20.
Literature Cited (1) GRAY,H. B., J. CEEM.Enuc. 4 1 , 2 (1964). F. A., J. CHEM.EDUC.41,466 (1964). (2) COTTON, , A., "Chemical Aspects of Group Theory," (3) C O T ~ NF. Interscience Publishers, New York, 1963. (4) EYRINQ,H., WALER~,J., AND KIMBALL, G. E., "Quantum Chemistry," John Wiley and Sons, Inc., New York, 1944, Chap. 10. (5) ROBERTS,D., "Notes on Molecular Orbital Calculation," W. A. Benjamin, Inc., New York, 1962, Chap. 4. (6) STREITWEISER, A., "Molecular Orbital Theory for Organic Chemists," John Wiley and Sons, Inc., New York, 1961, Chao. 3. (7) B A R R ~ W , G. H., "Molecular Spectroscopy," McGrawHill Book Co., New York, 1962, Chap. 8. (8) PHILLIPS,L. F., "Basic Quantum Chemistry," John Wiley and Sons, New York, 1965, Chap. 4. (9) MANCH. W.. ANDFERNELIUS. W. C., J. CHEM.EDUC.38.192 (1961'). (10) LIEHR,A. D., J. CHEM.EDUC.39, 135 (1962).
