Section 13-19
Section 13-19 Symmetry and Integration
13-18. In each of the following cases, resolve the given character set. If these were characters of integrands, would the integral vanish by symmetry? For example, C3v : 4 0 1 A1 ⊕ A2 ⊕ E No.
a) C4v:
5
−1
1
1
−3
b) D2h: 3 −1 −1 3 −1 3 3 −1 c) D3d: 8
2
0
0
0
0 13-19. Referring to the data in Problem 13-9, which integrals below must vanish by symmetry?
a) φ6xφ10 dv
b)
φ6yφ10 dv
c)
φ1xφ2 dv
d)
φ3xφ14 dv
e)
φ2zφ4 dv
13-20. Consider the possible electronic excitations of staggered ethane from its occu pied 1eg MO to the various empty MOs of symmetry a1g, a2u, eg, and eu. Which of these are symmetry allowed and how are they polarized?
13-21. Referring to the data in Problem 12-24, which transitions from MO 4 can be induced by y-polarized light? (Note: the molecular y axis is coincident with the symmetry z axis.)
13-22. The order of a group for a given object is equal to the number of equivalent locations for any nonspecial point in space about the object. Without referring to tables, predict the order of the group for a) a square (both faces identical).
b) a cube (all faces identical).
Multiple Choice Questions (Answer the following questions without referring to text or tables.)
1. Generate the symmetry operations for water, and choose the true statement from the set below.
a) H2O has two C2 axes.
b) H2O belongs to a group of order three.
c) H2O belongs to a group having exactly three classes.
d) H2O can have only nondegenerate MOs.
e) None of the above statements is true.
2. Which one of the following statements is true?
a) All reflections in a group must belong to the same class.
b) All linear molecules must have an inversion center.
c) An S6 axis has exactly two nonredundant operations.
Chapter 13 Group Theory d) Use of a basis set of hybrid orbitals built from a minimal basis set of STOs leads to a different variationally minimized energy than does use of the minimal basis set of STOs without hybridization.
e) None of the above is a true statement.
3. Which one of the following is not a symmetry element for eclipsed ethane?
a) i b) S3 c) C2 d) σv e) σh
4. Which of the following nondegenerate MOs is (are) possible for the CO2 molecule?
+
–
+
+
–
+
+
+
+
–
+
–
–
+
–
+
–
1
2
3
4
a) 1 only.
b) All four are possible.
c) 1, 2, and 3 only.
d) 2 and 3 only.
e) 1 and 3 only.
5. The presence of an S4 axis guarantees the presence of a) a point of inversion.
b) a σh reflection plane.
c) a C4 axis.
d) a C2 axis.
e) None of the above is correct.
6. A molecule having only the E symmetry operation can have a) only nondegenerate MOs.
b) MOs of any degeneracy.
c) an infinitely degenerate level.
d) only doubly-degenerate MOs.
e) None of the above is correct.
7. An MO that is symmetric or antisymmetric for every symmetry operation of a molecule
a) must be degenerate.
b) cannot be variationally altered.
c) must be the lowest- or highest-energy MO.
d) must be nondegenerate.
e) None of the above is correct.
Section 13-19 Symmetry and Integration8. The character for a reducible representation of C4 on a square planar molecule equals a) 0 b) 1 c) 2 d) −2 e) 4 9. Consider the planar molecule-ion NOCl+.
2
+
Cl
N
O
Cl
The number of reflection planes, σ , three-fold rotation axes, C3, two-fold rotation axes, C2, and inversion centers, i, that are symmetry elements for this moleculeion axis
σ
C3
C2
i
a.
1
0
1
0
a)
2
0
1
0
b)
2
0
1
1
c)
2
1
2
0
d)
None of the above is correct.
References [1] D. M. Bishop, Group Theory in Chemistry. Oxford University Press, London and New York, 1973.
[2] H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry. Wiley, New York, 1944.
13-18. In each of the following cases, resolve the given character set. If these were characters of integrands, would the integral vanish by symmetry? For example, C3v : 4 0 1 A1 ⊕ A2 ⊕ E No.
a) C4v:
5
−1
1
1
−3
b) D2h: 3 −1 −1 3 −1 3 3 −1 c) D3d: 8
2
0
0
0
0 13-19. Referring to the data in Problem 13-9, which integrals below must vanish by symmetry?
a) φ6xφ10 dv
b)
φ6yφ10 dv
c)
φ1xφ2 dv
d)
φ3xφ14 dv
e)
φ2zφ4 dv
13-20. Consider the possible electronic excitations of staggered ethane from its occu pied 1eg MO to the various empty MOs of symmetry a1g, a2u, eg, and eu. Which of these are symmetry allowed and how are they polarized?
13-21. Referring to the data in Problem 12-24, which transitions from MO 4 can be induced by y-polarized light? (Note: the molecular y axis is coincident with the symmetry z axis.)
13-22. The order of a group for a given object is equal to the number of equivalent locations for any nonspecial point in space about the object. Without referring to tables, predict the order of the group for a) a square (both faces identical).
b) a cube (all faces identical).
Multiple Choice Questions (Answer the following questions without referring to text or tables.)
1. Generate the symmetry operations for water, and choose the true statement from the set below.
a) H2O has two C2 axes.
b) H2O belongs to a group of order three.
c) H2O belongs to a group having exactly three classes.
d) H2O can have only nondegenerate MOs.
e) None of the above statements is true.
2. Which one of the following statements is true?
a) All reflections in a group must belong to the same class.
b) All linear molecules must have an inversion center.
c) An S6 axis has exactly two nonredundant operations.
Chapter 13 Group Theory d) Use of a basis set of hybrid orbitals built from a minimal basis set of STOs leads to a different variationally minimized energy than does use of the minimal basis set of STOs without hybridization.
e) None of the above is a true statement.
3. Which one of the following is not a symmetry element for eclipsed ethane?
a) i b) S3 c) C2 d) σv e) σh
4. Which of the following nondegenerate MOs is (are) possible for the CO2 molecule?
+
–
+
+
–
+
+
+
+
–
+
–
–
+
–
+
–
1
2
3
4
a) 1 only.
b) All four are possible.
c) 1, 2, and 3 only.
d) 2 and 3 only.
e) 1 and 3 only.
5. The presence of an S4 axis guarantees the presence of a) a point of inversion.
b) a σh reflection plane.
c) a C4 axis.
d) a C2 axis.
e) None of the above is correct.
6. A molecule having only the E symmetry operation can have a) only nondegenerate MOs.
b) MOs of any degeneracy.
c) an infinitely degenerate level.
d) only doubly-degenerate MOs.
e) None of the above is correct.
7. An MO that is symmetric or antisymmetric for every symmetry operation of a molecule
a) must be degenerate.
b) cannot be variationally altered.
c) must be the lowest- or highest-energy MO.
d) must be nondegenerate.
e) None of the above is correct.
Section 13-19 Symmetry and Integration8. The character for a reducible representation of C4 on a square planar molecule equals a) 0 b) 1 c) 2 d) −2 e) 4 9. Consider the planar molecule-ion NOCl+.
2
+
Cl
N
O
Cl
The number of reflection planes, σ , three-fold rotation axes, C3, two-fold rotation axes, C2, and inversion centers, i, that are symmetry elements for this moleculeion axis
σ
C3
C2
i
a.
1
0
1
0
a)
2
0
1
0
b)
2
0
1
1
c)
2
1
2
0
d)
None of the above is correct.
References [1] D. M. Bishop, Group Theory in Chemistry. Oxford University Press, London and New York, 1973.
[2] H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry. Wiley, New York, 1944.