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
J / A / C / B A -should be nonzero.
We first consider the simpler integral
where fA and fB 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 fAfB 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 C4v point group. Table 17-11 gives the character table and the traces for the direct products of various IR's. Thus the direct product A± X A2 is just A2, and likewise Βλχ 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 + Bx + B2. We conclude that an integral involving functions which are bases for the first three direct
prod-TABLE 1 7 - 1 1 . Direct Products in C4V Symmetry
C4 V: Ε C2 2C4 2σν 2ad
Ax 1 1 1 1 1
A2 1 1 1 - 1 - 1
Bx 1 1 - 1 1 - 1
B2 1 1 - 1 - 1 1
Ε 2 - 2 0 0 0
A1 X A2 1 1 1 - 1 - 1
Bxx B2 1 1 1 - 1 - 1
B1 χ Ε 2 -2 0 0 0
Ex Ε 4 4 0 0 0
Α χ Ε χ 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 A±. 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 fB 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 I R 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.
GENERAL REFERENCES
ADAMSON, A. W. (1969). "Understanding Physical Chemistry," 2nd ed., Chapter 21. Benjamin, New York.
COTTON, F. A. (1963). "Chemical Applications of Group Theory." Wiley (Interscience), New York.
COULSON, C. A. (1961). "Valence," 2nd ed. Oxford Univ. Press, London and New York.
FIGGIS, Β . N. (1966). "Introduction to Ligand Fields." Wiley (Interscience), New York.
CITED REFERENCES
ADAMSON, A. W. (1969). "Understanding Physical Chemistry," 2nd ed. Benjamin, New York.
COTTON, F. A. (1963). "Chemical Applications of Group Theory." Wiley, New York.
E X E R C I S E S
1 7 - 1 Write out the series of symmetry operations associated with an S5 axis, that is, show with explanation alternative designations wherever possible.
Ans. S6\ S5* = C52, S*63, .S54 = C54, S55 = oh , 556 = Ch\ 557, 558 = C.,3, 559, Sl° = E.
1 7 - 2 Explain the symmetry elements present and the point group designation for the follow
ing: (a) A book 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) C2V ; (b) Cx ; (c) D2d ; (d) C2v ; (e) C4 v.
1 7 - 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) C2H6 in the staggered configuration; (e) PC15 (a trigonal bipyramid); (f) AuCl4~ (a square planar ion); (g) ruthenocene (a pentagonal prism); (h) /riw.s-Co(NH3)4Br2+ (ignore the H's);
(i) /rarts-Co(NH3)4ClBr+ (ignore the H's); (j) chloroform.
Ans. (a) C2V ; (b) A »h ; (c) D2d ; (d) Ad ; (e) Ah ;
(f) Ah ; (g) Ah ; (h) / )4h ; (0 c4v ; (j) c3 v.
EXERCISES 755
17-4 Work out the multiplication table for the C2V group, showing each multiplication in the manner of Fig. 17-9.
Arts. See Table 17-4.
17-5 Referring to Exercise 17-4, show that σ(χζ) and o\yz) belong to separate classes by carrying out the similarity transformations X~xa{xz)X and X~1a/(yz)X, where X is each symmetry operation.
Ans. Χ~1σ(χζ)Χ is σ(χζ) for X = E{X~X = E\ X = C2 (X'1 = C2), X = σ(χζ) [X-1 = σ(χζ)1 and X = o'(yz) [X~x = o'(yz)];
similarly X~1o(yz)X is o\yz) for each case.
Thus σ(χζ) and o\yz) do not mix and therefore belong to separate classes.
17-6 Complete the example in connection with Table 17-2, which showed that σν , σν', and σν"
belong to the same class.
17-7 Carry out the matrix multiplication
L L] [ 2 3] '
S
' [1 LÎ]
Ans.
17-8 Carry out the matrix multiplication 1 2 0
(Notice that the product matrix is blocked out in the same way as are those multiplied.)
17-9 Show what irreducible representations in C3V are spanned by the reducible representation Ε 2 C3 3σν
Γ 5 - 1 - 1
Ans. Γ = 2Ε
+
Α2. 1 7 - 1 0 Find the traces of the reducible representation Γ5σ generated by the set of five sigma bondsin PCI5 (a triangular bipyramid), the irreducible representations spanned by Γ5σ, and the types of hydrogen-like orbitals that might be used for bonding.
Ans. Γ5 σ^ 2Αι + Ε' + A2\ corresponding to s(pa.p1/)pzd22 or
s(d
a;2_
î/2d
a;î/)p
ed
z2.
17-11 Referring to 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 Γ3π±
and the types of hydrogenlike orbitals that might be used for pi bonding.
Ans. Γ31 • A2 + E\ corresponding to p2
PROBLEMS
1 7 - 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) Cr(NH3)5Cl2 +, (d) c/5-Co(NH3)4Cl2+, (e) CH2C1CH2C1 in the staggered configuration, (f) naphthalene, (g) cyclohexane in the chair form. (List all symmetry elements; ignore H's in the ammine complexes.)
1 7 - 2 Show what the point group designations are for (a) Fe(CN)4.-, (b) HOC1 (a bent molecule), (c) (HNBH)3 (a planar six-membered ring with alternating Ν and B), (d) C5H8, spiro-pentane (two mutually perpendicular triangles joined apically), (e) CO, (f) U 02F g ~ (a pentagonal bipyramid with apical oxygens), (g) diborane, B2H6 (two hydrogens are bridging and lie in a plane perpendicular to that of the other four hydrogens), (h)CH2C!Br.
(List all symmetry elements.)
1 7 - 3 Show what the point group designations are for (a) a triangular antiprism, (b) trans-Pt(NH3)2Cl2 (ignore the H's), (c) I F5 (four F's in a square plane and one apical-F), (d) Dewar benzene. (List all the symmetry elements.)
1 7 - 4 Work out the multiplication table for the Z>3h group. Show which are the various classes, that is, carry out the necessary similarity transformations.
1 7 - 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. Show that the first set does in fact obey the C3V multi
plication table.
1 7 - 6 Carry out the matrix multiplication
"1 3 2" "0 2 r 1 2 1 3 0 2 4 0 1 2 0 1
1 7 - 7 Determine what sets of orbitals should be appropriate for sigma bonding in a hypothetical Cr(NH3)*+ (a pentagonal bipyramid).
1 7 - 8 A reducible representation in the Td point group has the traces: Ε = ?, C3 = 1, C2 = — 1,
£ 4 = —3, and σά = 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.
1 7 - 1 2 Calculate, in terms of Dq, the crystal field stabilization energy for (a) low-spin Fe(II), (b) high-spin Mn(II), and (c) Mo(IV), assuming octahedral geometry in each case.
Ans. (a) 24, (b) 0, (c) 8.
SPECIAL TOPICS PROBLEMS 757
1 7 - 9 If an object has a C2 axis and a σν plane but no other types of symmetry, show that a second σν plane must also be present.
1 7 - 1 0 Verify Eq. (17-10) by carrying out the transformations of a point according to the sym
metry operations of C3 V.
1 7 - 1 1 Find the traces of the reducible representations in Z)4h 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 two cases ?
1 7 - 1 2 Show 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 )3h , (c) η = 4 and the η atoms lie at the corners of a square, η = 5 and the molecules have C4V symmetry, (d) η = 6 and the η atoms lie at the corners of a triangular prism.
1 7 - 1 3 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 , Dih . 1 m 1 Function 1 Ml 1 Function
3 (sin3 0) é?±3î* 1 (sin 0)(5 cos2 θ - \)e±i(f>
2 (sin2 0)(cos Θ) e±2i* 0 f cos3 θ — cos θ
1 7 - 1 4 Determine the hydrogen-like orbitals which are of the proper symmetry for pi bonding in a tetrahedral molecule.
1 7 - 1 5 List the symmetry operations in the group D6 and construct the group multiplication table.
1 7 - 1 6 Calculate, in terms of Dq, the crystal field stabilization energy for transition metal complexes with one through nine d electrons, assuming octahedral geometry and (a) low spin; (b) high spin.
1 7 - 1 7 Show how the set of five d orbitals should split in a crystal field of Γά symmetry and explain qualitatively what the energy ordering should be.
1 7 - 1 8 A transition metal complex, M A6, has D3d symmetry, with the ligands at the corners of a triangular antiprism. The six M—A bonds are equivalent. Find the IR's spanned by the set of six sigma bonds and infer the various hybrid orbital combinations that would be possible.
SPECIAL TOPICS P R O B L E M S
1 7 - 1 Find what irreducible representations are spanned in C3V by the direct product E x E.
1 7 - 2 Find what irreducible representations are spanned in D3h by the direct products A 2' x E' and Ε' χ Ε'.
1 7 - 3 Evaluate the direct product of Eg with T2e in Oh . 1 7 - 4 Show that in Oh,Ee x (Alg x T2g) = (Ee x Alg) x T2g .