Applications of group theory to vibrational spectroscopy and molecular orbital symmetry - One Line Questions

1. For a linear molecule, how many rotational degrees of freedom are there? — 2
2. For a molecule like methane (CH4) in the Td point group, how many IR active vibrational modes are there? — 3
3. How many vibrational degrees of freedom does a non-linear molecule with N atoms possess? — 3N - 6
4. What does the Mulliken symbol 'A' in a character table typically denote? — A non-degenerate representation symmetric with respect to the principal axis
5. What does the Mulliken symbol 'E' represent in a character table? — A doubly degenerate representation
6. If a vibrational mode is IR active, what is its corresponding symmetry representation in the character table? — A representation that transforms as x, y, or z
7. What is the symmetry of the totally symmetric representation in any point group? — A1
8. What is the symmetry of the z-axis rotation (Rz) in the C2v point group? — A2
9. For a molecule in the C3v point group, what is the symmetry of the translation along the z-axis? — A1
10. What are the symmetry labels for the translational motions (Tx, Ty, Tz) in the C2v point group? — A1, B2, B1
11. For a molecule in the D4h point group, what is the symmetry of the rotation about the z-axis (Rz)? — A2g
12. What is the symmetry of the vibrational modes that correspond to stretching of the C-H bonds in benzene (D6h point group)? — A1g + E2g
13. What is the symmetry of the reducible representation of the 3N displacement vectors for a molecule in Oh point group? — A1g + Eg + T1g + T2g + 2T1u + 2T2u
14. Which representation in a character table corresponds to the x, y, and z translations? — T1u
15. For a molecule in the D2h point group, which vibrational modes are typically IR active? — Au, B1u, B2u, B3u
16. What is the number of Raman active modes for a non-linear molecule in the Td point group? — Modes that transform as quadratic functions of coordinates
17. According to the mutual exclusion rule, for molecules with a center of symmetry, if a vibration is IR active, it is: — Raman inactive
18. What is the symmetry of a vibrational mode that results in a change of the molecular dipole moment along the z-axis? — A1
19. What is the symmetry of the x and y translations in the D2h point group? — B2u, B3u
20. In vibrational spectroscopy, a 'silent mode' is a vibrational mode that is: — Both IR and Raman inactive
21. How are the normal modes of vibration of a molecule classified in group theory? — By their symmetry species
22. Which of the following is a symmetry operation for a molecule in the C3v point group? — C3
23. Which symmetry point group does a linear molecule like CO2 belong to? — Dinfh
24. Which symmetry operation is defined as reflection through a plane of symmetry? — sigma
25. What is the point group for a square planar molecule like XeF4? — D4h
26. For a molecule to be Raman active, its vibrational modes must have a change in which property? — Polarizability
27. Which of the following symmetry operations is NOT present in the C2v point group? — sigma_d
28. Consider a C3h point group. Which symmetry operation is represented by a rotation by 120 degrees about the principal axis? — C3
29. In the context of molecular orbital symmetry, what does the label 'g' or 'u' signify for orbitals in centrosymmetric molecules? — Gerade (even) or Ungerade (odd) symmetry with respect to inversion
30. What is the condition for a vibrational mode to be IR active? — It must cause a change in dipole moment
31. The symmetry of molecular orbitals can be determined using group theory. What does the symmetry label of an MO represent? — Its spatial distribution and phase
32. What type of vibrational modes are observed in Raman spectroscopy for molecules with a center of symmetry? — Modes that are either IR or Raman active, but not both (mutual exclusion rule)
33. What is the primary application of group theory in vibrational spectroscopy? — Determining the selection rules for infrared and Raman spectroscopy
34. If a vibrational mode transforms as the representation 'T1u', it is expected to be active in which spectroscopy? — IR
35. Which symmetry operation involves a rotation followed by a reflection in the same plane perpendicular to the rotation axis? — Improper Rotation (Sn)
36. Which symmetry operation corresponds to the identity element (E) in group theory? — No change to the molecule
37. What is the character of the identity operation (E) for any irreducible representation? — The dimension of the representation
38. The number of vibrational modes that are Raman active can be determined by analyzing the symmetry of: — The polarizability tensor components
39. What information can be directly obtained from the character table of a molecule's point group? — The symmetry species of normal modes, translations, rotations, and quadratic functions
40. In Raman spectroscopy, what does the Raman shift correspond to? — The energy difference between vibrational levels
41. What is the symmetry of the normal modes of vibration of a molecule determined by? — The symmetry of the molecule
42. In molecular orbital theory, the symmetry of a molecular orbital must be compatible with the symmetry of: — Both the molecule's point group and the atomic orbitals
43. The number of distinct vibrational frequencies observed in the IR spectrum is limited by: — The number of IR active normal modes
44. In group theory, what does the character of a representation signify? — The trace of the transformation matrix for a symmetry operation
45. The symmetry of the molecular orbitals formed from atomic orbitals depends on: — The symmetry of the atomic orbitals and the geometry of the molecule
46. What is the total number of irreducible representations in the character table of a point group equal to? — The number of classes of symmetry operations
47. In molecular orbital theory, what symmetry requirement must be met for two atomic orbitals to form a molecular orbital? — They must have the same symmetry and appropriate overlap
48. What is the symmetry of the polarizability tensor components (e.g., alpha_xx, alpha_yy, alpha_zz) in group theory? — They transform as quadratic functions of coordinates
49. What is the primary role of group theory in predicting molecular properties from spectroscopy? — To determine the symmetry of electronic and vibrational states
50. Which spectroscopic technique is primarily used to study the vibrational modes of molecules? — Infrared (IR) Spectroscopy
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