Wave mechanics abounds in the evaluation of integrals, usually a complicated calculation. It is possible, however, to determine on symmetry grounds whether an integral of the type
J7A/C/B</T should be nonzero.
We first consider the simpler integral
where /A and /B might be two wave functions for a molecule. It turns out that such an integral must be zero unless the integrand is invariant under all operations of the symmetry group to which the molecule belongs. What this means speci
fically is that the product /A/B must form the basis for the totally symmetric irre
ducible representation (IR) of the group—the one for which all the traces for the various symmetry operations are unity.
The procedure for determining whether or not this requirement is met is as follows. First, the traces of the representation of a product fA X fB , called a direct product, are, for each symmetry operation R ,
X(R) = XA(R)XB(R). (17-17)
Consider, for example, a molecule in the C4 v point group. Table 17-11 gives the character table and the traces for the direct products of various IR's. Thus the direct product Αλ χ A2 is just A2, and likewise B± χ B2. The direct product E2 or Ε χ Ε does not correspond to any of the IR's, but on application of Eq. (17-16), one finds it to contain the IR's A1 + A2 + B± + B2. We conclude that an integral involving functions which are bases for the first three direct
prod-T A B L E 17-11. Direct Products in C4 V Symmetry
C4 v * Ε c2 2 C4 2 σν 2σ<ι
A1 1 1 1 1 1
A 1 1 1 - 1 - 1
Bt 1 1 - 1 1 - 1
B, 1 1 - 1 - 1 1
Ε 2 - 2 0 0 0
Α\ x A2 1 1 1 - 1 - 1
Βχ χ B2 1 1 1 - 1 - 1
χ Ε 2 - 2 0 0 0
Ex Ε 4 4 0 0 0
A1 Χ Ε X B2 2 - 2 0 0 0
ucts in the table must be zero since the direct products do not contain the totally symmetric IR, in this case Ax. An integral based on the direct product E2 would be nonzero, however, since it contains the Ax IR.
The principle may be extended to three functions. One simply takes the direct product of the representations for which fc and /B are bases, and then the direct product of the result with the representation for which fA is a basis. Again, unless the final result is or contains the totally symmetric IR of the point group, the integral must be zero. Alternatively stated, for the integral to be nonzero, the direct product of any two of the functions must have at least one I R in common with those spanned by the third function.
G E N E R A L R E F E R E N C E S
ADAMSON, A . W. (1969). "Understanding Physical Chemistry," 2nd ed., Chapter 21. Benjamin, N e w York.
COTTON, F. A . (1963). "Chemical Applications of Group Theory." Wiley (Interscience), N e w York.
COULSON, C. A . (1961). "Valence," 2nd ed. Oxford Univ. Press, London and N e w York.
FIGGIS, Β. N . (1966). "Introduction to Ligand Fields." Wiley (Interscience), N e w York.
C I T E D R E F E R E N C E S
ADAMSON, A . W. (1969). "Understanding Physical Chemistry," 2nd ed. Benjamin, N e w York.
COTTON, F. A . (1963). "Chemical Applications of Group Theory." Wiley, N e w York.
EXERCISES
17-1 Write out the series of symmetry operations associated with an S5 axis, that is, s h o w with explanation alternative designations wherever possible.
Ans. Sb\ 55 2 = C5 2, 56 8, S5* = Cb\ 55 5 = ah ,
«V
= C5 8, S§\ S6 8 = C6* ,V,
S | ° = E.17-2 Explain the symmetry elements present and the point group designation for the follow
ing: (a) A b o o k with blank pages; (b) a normally printed book; (c) a tennis ball, including the seam; (d) Siamese twins; (e) an ash tray in the shape of a round bowl and with four equally spaced grooves in the rim for holding cigarettes.
Ans. (a) C2 V ; (b) Cx ; (c) D2U ; (d) C2 V ; (e) C4 v.
17-3 Explain what symmetry elements are present and the point group designation for the following molecules: (a) Pyridine; (b) acetylene; (c) H2C = C = C H2; (d) C2He in the staggered configuration; (e) PC15 (a trigonal bipyramid); (f) A u C l4~ (a square planar ion); (g) ruthenocene (a pentagonal prism); (h) trans-Co(NHz)4Br2+ (ignore the H's);
(i) / r a w j - C o ( N H3)4C l B r+ (ignore the H's); (j) chloroform.
Ans. (a) C2 V ; (b) A x * ; (c) D2U ; (d) D3d ; (e) D3h; ( 0 ; (g) D6h ; (h) D4 h ; (i) C4 V ; (j) C3 V.
EXERCISES 755
17-4 Work out the multiplication table for the C2 V group, showing each multiplication in the manner of Fig. 17-9.
Ans. See Table 17-4.
17-5 Referring t o Exercise 17-4, s h o w that σ(χζ) and σ'(γζ) belong t o separate classes by carrying out the similarity transformations Χ~1σ(χζ)Χ and X~W{yz)X, where X is each symmetry operation.
Ans. X-xa{xz)X is o{xz) for X = Ε (X~x = E\ X = C2 (X-1 = C2) , X = σ(χζ) [Χ-1 = σ(χζ)], and X = o{yz) [X'1 = o{yz)]\
similarly X~W{yz)X is o{yz) for each case.
Thus a{xz) and o'iyz) d o n o t mix and therefore belong t o separate classes.
17-6 Complete the example in connection with Table 17-2, which showed that σν , oy\ and Oy"
belong to the same class.
17-7 Carry out the matrix multiplication
π ε
a-Ans.
[J jj].
17-8 Carry o u t the matrix multiplication
"1 2 0" '3 2 0_ 0 2 0 1 1 0 -0 0 3- -0 0
2-(Notice that the product matrix is blocked out in the same way as are those multiplied.)
17-9 S h o w what irreducible representations in C3 V are spanned by the reducible representation Ε 1C$ 3 σν
Γ 5 - 1 - 1
Ans. Γ = 2E
+
A%. 17-10 Find the traces of the reducible representation Γ5 σ generated by the set of five sigma bondsin PC15 (a triangular bipyramid), the irreducible representations spanned by Γδσ, and the types of hydrogen-like orbitals that might be used for bonding.
Ans. Γδσ& 2Αι + E' + A2 \ corresponding to s(pa !pw)pzdi i2 or s(da !2_t f2da ; t,)pi idz2.
17-11 Referring t o Exercise 17-10 and P C 15, find the traces for the reducible representation
Γ3 7 Γ ± of the set of pi bonds for the three chlorines in the trigonal plane, whose lobes lie
above and below the plane. Find also the irreducible representations spanned by Γ3 7 Γ ±
and the types of hydrogenlike orbitals that might be used for pi bonding.
Ans. rS7T± ~ A2" + E'\ corresponding
P R O B L E M S
17-1 Show what the point group designations are for (a) ferrocene (iron sandwiched between two cyclopentadienyl rings, which are staggered; see the accompanying diagram), (b) ethane in the eclipsed configuration, (c) C r ( N H3)5C l2 +, (d) c / s - C o ( N H3)4C l2 +, (e) C H2C I C H2C 1 in the staggered configuration, (f) naphthalene, (g) cyclohexane in the chair form. (List all symmetry elements; ignore H's in the ammine complexes.)
17-2 S h o w what the point group designations are for (a) Fe(CN)J", (b) HOC1 (a bent molecule), (c) ( H N B H )3 (a planar six-membered ring with alternating Ν and B), (d) C5H8, spiro-pentane (two mutually perpendicular triangles joined apically), (e) C O , (f) U 02F ; | ~ (a pentagonal bipyramid with apical oxygens), (g) diborane, B2He (two hydrogens are bridging and lie in a plane perpendicular t o that o f the other four hydrogens), ( h ) C H2C l B r . (List all symmetry elements.)
17-3 S h o w what the point group designations are for (a) a triangular antiprism, (b) trans-P t ( N H3)2C l2 (ignore the H's), (c) I F5 (four F's in a square plane and o n e apical-F), (d) D e w a r benzene. (List all the symmetry elements.)
17-4 Work out the multiplication table for the Dzh group. S h o w which are the various classes, that is, carry out the necessary similarity transformations.
17-5 Equation (17-10) gives a set of matrices which simplify to a set of two-dimensional ones and a set of one-dimensional ones. S h o w that the first set does in fact obey the C3 V multi
plication table.
17-6 Carry out the matrix multiplication
"1 3 T Ό 2 Γ 1 2 1 3 0 2 4 0 1 2 0 1
17-7 Determine what sets of orbitals should be appropriate for sigma bonding in a hypothetical Cr(NH3)i)+ (a pentagonal bipyramid).
17-8 A reducible representation in the Td point group has the traces: Ε = ?, C3 = 1, C2 = — 1, S4 = — 3 , and σα = 1. The trace under the symmetry operation Ε is missing. Explain what is the simplest choice for this missing number and what irreducible representations are spanned.
17-12 Calculate, in terms of Dq, the crystal field stabilization energy for (a) low-spin Fe(II), (b) high-spin Mn(II), and (c) M o ( I V ) , assuming octahedral geometry in each case.
Ans. (a) 2 4 , (b) 0, (c) 8.
SPECIAL TOPICS PROBLEMS 757
17-9 If an object has a C2 axis and a σν plane but n o other types of symmetry, s h o w that a second σν plane must also be present.
17-10 Verify Eq. (17-10) by carrying out the transformations of a point according to the sym
metry operations of Cs v.
17-11 Find the traces of the reducible representations in Z>4 h which are generated by carrying the set of (a) ρ orbitals and (b) d orbitals through the various symmetry operations. What irreducible representations are spanned in the t w o cases ?
17-12 S h o w what combinations of s, p, and d orbitals are suitable for sigma bonding of η atoms to a central one if (a) η = 2 and the molecule is angular, (b) η = 3 and the sym
metry is Z )3 h , (c) η = 4 and the η atoms lie at the corners of a square, η = 5 and the molecules have C4 v symmetry, (d) η = 6 and the η atoms lie at the corners of a triangular prism.
17-13 The following are the angular portions of 4f hydrogen-like orbitals. Determine the irre
ducible representations to which these orbitals belong in point groups 7 d , Oh , .
\m\ Function \m\ Function
3 (sin3 Θ) e±Zi<*> 1 (sin 0)(5 c o s2 θ - \)e^
2 (sin2 0)(cos θ) β±2ίΦ 0 § c o s3 θ - cos θ
17-14 Determine the hydrogen-like orbitals which are of the proper symmetry for pi bonding in a tetrahedral molecule.
17-15 List the symmetry operations in the group D6 and construct the group multiplication table.
17-16 Calculate, in terms o f Dq, the crystal field stabilization energy for transition metal c o m p l e x e s with o n e through nine d electrons, assuming octahedral geometry a n d (a) l o w spin; (b) high spin.
17-17 S h o w h o w the set o f five d orbitals s h o u l d split in a crystal field o f Τά symmetry and explain qualitatively what the energy ordering s h o u l d be.
17-18 A transition metal c o m p l e x , Μ Αβ, h a s Dza symmetry, with the ligands at the corners o f a triangular antiprism. T h e six Μ — A b o n d s are equivalent. F i n d the IR's spanned by the set o f six sigma b o n d s and infer the various hybrid orbital c o m b i n a t i o n s that w o u l d be possible.
SPECIAL TOPICS P R O B L E M S
17-1 F i n d what irreducible representations are spanned in C8 V by the direct product Ε χ E.
17-2 F i n d what irreducible representations are spanned in D3h by the direct products A2' x E' and Ef χ Ε'.
17-3 Evaluate the direct product o f Eg with T2g in Oh . 17-4 S h o w that in Oh , Eg χ (Alg χ T2fS) = (Eg χ Ale) χ Τ2β .