Molecular Symmetry: Introduction and Application

Molecular Symmetry: An Introduction

Molecular symmetry is a powerful concept in chemistry that helps us understand and predict a molecule’s properties, from its spectroscopic behavior to its polarity and chirality. It provides a systematic way to classify molecules based on their geometric characteristics. Far from being a purely theoretical exercise, understanding molecular symmetry offers profound insights into how molecules interact with light, their reactivity, and even their biological activity.

What is Molecular Symmetry?

At its core, molecular symmetry describes the spatial arrangement of a molecule’s atoms. If certain operations, such as rotation or reflection, leave the molecule looking identical to its original state, then the molecule possesses elements of symmetry. These “elements” are geometric entities like axes, planes, or points, while the “operations” are the actual transformations performed.

Think of it like this: if you rotate a square by 90 degrees around its center, it looks exactly the same. The center of the square is an axis of symmetry, and the 90-degree rotation is a symmetry operation. Molecular symmetry applies this same logic to three-dimensional molecules.

Key Symmetry Elements and Operations

There are five fundamental types of symmetry elements and their corresponding operations:

  1. Identity Element (I or E):
    • Operation: Do nothing.
    • Description: Every molecule possesses this element. It might seem trivial, but it’s crucial for the mathematical framework of group theory. It leaves the molecule completely unchanged, serving as the basis for all point groups.
  2. n-Fold Axis of Symmetry (Cn​):
    • Operation: Rotation by 360∘/n around the axis. This operation is performed n times to return to the original orientation.
    • Description: If a molecule looks identical after rotation by a specific angle (360∘/n), it has a Cn​ axis. For example, a water molecule (H2​O) has a C2​ axis, meaning it looks the same after a 180∘ rotation. Benzene (C6​H6​) has a C6​ axis as its highest order (principal) axis, along with multiple C2​ axes. Linear molecules like HCN (heteronuclear) or O2​ (homonuclear) have an infinite-fold axis (C∞​), as rotation by any angle around the internuclear axis leaves them unchanged.
  3. Plane of Symmetry (σ):
    • Operation: Reflection through the plane.
    • Description: If reflecting all atoms through a plane results in an indistinguishable configuration, the molecule has a plane of symmetry.
      • Vertical Plane (σv​): This plane contains the highest-order Cn​ axis. In water (H2​O), the plane containing all three atoms is a σv​ plane, and there’s another σv​ plane perpendicular to it, bisecting the H−O−H angle. Ammonia (NH3​) has three σv​ planes.
      • Horizontal Plane (σh​): This plane is perpendicular to the highest-order Cn​ axis. Boron trifluoride (BF3​), a planar molecule, has a σh​ plane that coincides with the molecular plane.
      • Dihedral Plane (σd​): A vertical plane that bisects the angle between two C2​ axes perpendicular to the principal Cn​ axis. These planes are often found in molecules with alternating layers or staggered conformations, such as allene (C3​H4​) or staggered ethane.
  4. Centre of Inversion (i):
    • Operation: Inversion through the center point. Every atom is moved through the center to an equal distance on the opposite side.
    • Description: If reflecting every atom through a central point to an equal distance on the opposite side yields an identical configuration, the molecule possesses a center of inversion. Examples include s-trans-buta-1,3-diene, benzene, and carbon dioxide. If a molecule has a center of inversion, any atom at position (x, y, z) must have an identical atom at (-x, -y, -z).
  5. n-Fold Rotation-Reflection Axis (Sn​):
    • Operation: Rotation by 360∘/n followed by reflection through a plane perpendicular to the axis.
    • Description: This is a compound operation. For instance, methane (CH4​) has S4​ axes, and allene has an S4​ axis. Importantly, if a molecule possesses an Sn​ axis, it may or may not have a separate Cn​ axis or a σh​ plane as standalone elements. For example, S1​ is equivalent to a plane of symmetry (σ), and S2​ is equivalent to a center of inversion (i).

Point Groups: Classifying Molecules by Symmetry

The complete collection of symmetry elements that a molecule possesses defines its “point group.” Point groups are a mathematical classification system based on all the symmetry operations that can be performed on a molecule while leaving at least one point (the center of mass) fixed. Molecules are classified into specific point groups, which simplifies the prediction of their properties. Some common point groups include:

  • Low Symmetry Groups:
    • C1​: Contains only the identity element (I or E). Molecules in this group are asymmetric (e.g., CHFClBr).
    • Cs​: Contains the identity and one plane of symmetry (σ). Example: H2​CCl2​ (dichloromethane with C and Cl atoms in a plane).
    • Ci​: Contains the identity and one center of inversion (i). Example: s-trans-1,2-dichloroethylene.
  • High Symmetry Groups:
    • Cn​: Contains only a Cn​ axis (and I). Example: Hydrogen peroxide (H2​O2​) in its non-planar conformation.
    • Cnv​: Contains a Cn​ axis and n vertical planes (σv​). Example: Water (C2v​), Ammonia (C3v​). These are very common for pyramidal and bent molecules.
    • Dn​: Contains a Cn​ axis and n C2​ axes perpendicular to Cn​. Example: Tris(ethylenediamine)cobalt(III) complex.
    • Cnh​: Contains a Cn​ axis and a horizontal plane (σh​). Example: s-trans-glyoxal (C2h​).
    • Dnh​: Contains a Cn​ axis, n C2​ axes perpendicular to Cn​, and a σh​ plane. Example: Benzene (D6h​), Boron trifluoride (D3h​), and linear molecules like CO2​ (D∞h​). These are typical for planar and highly symmetrical molecules.
    • Dnd​: Contains a Cn​ axis, an S2n​ axis, and n dihedral planes (σd​). Example: Staggered ethane (D3d​).
    • Td​: Tetrahedral symmetry. Example: Methane (CH4​), Carbon tetrachloride (CCl4​). These molecules have four C3​ axes, three C2​ axes, six σd​ planes, and three S4​ axes.
    • Oh​: Octahedral symmetry. Example: Sulfur hexafluoride (SF6​), PtCl62−​. Characterized by four C3​ axes, three C4​ axes, an inversion center, and multiple planes of symmetry.
    • Ih​: Icosahedral symmetry. Example: Buckminsterfullerene (C60​). This is a very high symmetry group with sixty C5​ axes and other elements.
    • Kh​: Spherical symmetry. Atoms belong to this point group, as they can be rotated by any angle around any axis passing through their center and still appear identical.

Molecular Chirality and Dipole Moments

Molecular symmetry plays a crucial role in determining two very important molecular properties: chirality and the presence of a permanent dipole moment.

Chirality

A molecule is chiral if it is non-superimposable on its mirror image. This means that its mirror image cannot be rotated in any way to perfectly align with the original molecule. These mirror image forms are called enantiomers, and they are optically active, meaning they can rotate plane-polarized light. Chirality is fundamental in biochemistry, as many biological systems are enantioselective (e.g., enzymes often only interact with one enantiomer of a drug).

Symmetry Rule for Chirality: A molecule is chiral if and only if it does not possess any Sn​ symmetry element. Since a plane of symmetry (σ) is equivalent to an S1​ operation and a center of inversion (i) is equivalent to an S2​ operation, this rule implies that chiral molecules cannot have a plane of symmetry (σ) or a center of inversion (i).

For example, CHFClBr (bromochlorofluoromethane) has no symmetry elements other than the identity (C1​ point group), so it is chiral. Hydrogen peroxide (H2​O2​), with only a C2​ axis (C2​ point group), is also chiral, though its enantiomers are difficult to separate due to rapid interconversion at room temperature.

Permanent Dipole Moment

A permanent dipole moment is a measure of the overall charge asymmetry in a molecule. It arises from the unequal sharing of electrons in polar covalent bonds and the molecule’s overall geometry. It’s a vector quantity, indicating both magnitude and direction, usually measured in Debye (D). The presence of a permanent dipole moment is critical for a molecule’s interaction with electric fields, its solubility properties (e.g., “like dissolves like”), and its rotational spectroscopy.

Symmetry Rule for Permanent Dipole Moment: A molecule possesses a permanent dipole moment if and only if none of its symmetry operations can reverse or cancel the direction of the dipole moment vector. More formally, the dipole moment vector must be totally symmetric (i.e., its components along the Cartesian axes must transform as the totally symmetric irreducible representation of the molecule’s point group). This essentially means that for a molecule to have a dipole moment, there must be at least one axis along which a net separation of charge can exist without being eliminated by a symmetry operation.

Practically, this means a molecule has a permanent dipole moment if:

  • It belongs to point groups C1​, Cs​, Cn​, or Cnv​. In these groups, a unique axis or plane often allows for a net dipole.
  • Molecules in point groups like Dnh​, Td​, Oh​, or Ih​ (which include i or σh​ elements that would reverse or cancel any potential dipole) do not have a permanent dipole moment. For instance, a σh​ perpendicular to the principal axis would eliminate any dipole along that axis. A center of inversion (i) would mean that for every charge +q at (x,y,z), there’s an equal charge -q at (-x,-y,-z), leading to cancellation.

For example:

  • Water (H2​O, C2v​): Has a significant dipole moment along its C2​ axis (often designated as the z-axis), as the Tz​ (translation along z) transforms as the totally symmetric A1​ representation. The bond dipoles of the O−H bonds sum to a net molecular dipole.
  • Benzene (C6​H6​, D6h​): Has no permanent dipole moment because its high symmetry (planar, hexagonal) cancels any potential charge asymmetry. All individual bond dipoles cancel out due to symmetry.
  • Carbon dioxide (CO2​, D∞h​): Although it has highly polar C=O bonds, the linear and symmetrical arrangement cancels the individual bond dipoles, resulting in a zero net dipole moment.
  • Ammonia (NH3​, C3v​): Possesses a permanent dipole moment along its C3​ axis due to the lone pair of electrons on nitrogen and the pyramidal geometry.

Applications in Spectroscopy

Molecular symmetry is indispensable for understanding spectroscopic selection rules. These rules dictate which transitions (e.g., electronic, vibrational, rotational) are “allowed” or “forbidden” when a molecule interacts with electromagnetic radiation. An “allowed” transition means it has a non-zero probability of occurring and will be observed in a spectrum, whereas a “forbidden” transition has zero probability and will not be observed.

Vibrational Spectroscopy

In infrared (IR) and Raman spectroscopy, molecular vibrations are studied. Each normal mode of vibration in a molecule belongs to a specific symmetry species of the molecule’s point group. These normal modes are independent ways in which a molecule can vibrate.

  • Infrared (IR) Activity: A vibrational mode is IR active if it causes a change in the molecule’s permanent dipole moment during the vibration. This change must occur along one of the Cartesian axes (x, y, or z). This occurs if the vibrational mode’s symmetry species is the same as the symmetry species of a translation (Tx​, Ty​, or Tz​) of the molecule, which transform as irreducible representations of the point group.
  • Raman Activity: A vibrational mode is Raman active if it causes a change in the molecule’s polarizability tensor during the vibration. Polarizability describes how easily the electron cloud of a molecule can be distorted by an electric field. The polarizability components (e.g., αxx​, αyy​, αxy​, etc.) belong to specific symmetry species (transforming as quadratic functions like x2, xy, etc., or rotations), and if a vibrational mode’s symmetry matches any of these, it’s Raman active.

Complementarity Rule: For molecules with a center of inversion (i), a fundamental vibration cannot be both IR and Raman active. If a mode is IR active, it is Raman inactive, and vice-versa. This is known as the mutually exclusive rule or complementarity rule and is a powerful tool for structure determination. For example, in CO2​ (D∞h​, has i), the symmetric stretch is Raman active but IR inactive, while the asymmetric stretch and bend are IR active but Raman inactive.

Example: Water (H2​O, C2v​ Point Group) Water has three normal modes of vibration:

  • Symmetric stretch (ν1​, A1​ symmetry): Changes dipole moment along z-axis. Both IR and Raman active.
  • Bend (ν2​, A1​ symmetry): Changes dipole moment along z-axis. Both IR and Raman active.
  • Asymmetric stretch (ν3​, B2​ symmetry): Changes dipole moment along y-axis. Both IR and Raman active. Since water does not have a center of inversion, all three modes can be both IR and Raman active, which is observed experimentally.

This shows how symmetry arguments predict the activity of different vibrations, which is crucial for interpreting complex spectra and gaining insights into molecular structure and bonding.

Electronic Spectroscopy

Electronic transitions involve the promotion of an electron from one molecular orbital to another upon absorption of light (typically in the UV-Vis region). The selection rules for these transitions are also governed by symmetry, often more rigorously than vibrational transitions. The transition is allowed if the direct product of the symmetry species of the initial electronic state (Γinitial​), the transition moment operator (Γdipole​ – which transforms as Tx​, Ty​, or Tz​), and the final electronic state (Γfinal​) contains the totally symmetric representation (A1​ or A in most point groups). Mathematically: Γinitial​×Γdipole​×Γfinal​⊃A1​ (or A). If the initial state is totally symmetric (as is often the ground state), then the rule simplifies to: Γdipole​×Γfinal​⊃A1​. This means the final electronic state must have the same symmetry as one of the Cartesian coordinates (x, y, or z).

This rigorous application of symmetry helps spectroscopists to assign observed bands in UV-Vis spectra to specific electronic transitions and to understand the electronic structure of molecules.

Conclusion

Molecular symmetry is a cornerstone of modern chemistry. It provides an elegant and rigorous framework for classifying molecules and predicting their fundamental properties. From determining whether a molecule is chiral, predicting the presence of a permanent dipole moment, to understanding its interactions with electromagnetic radiation through selection rules for various spectroscopies (IR, Raman, Electronic), the principles of molecular symmetry are invaluable for chemists, physicists, and materials scientists alike. Mastering these concepts unlocks a deeper understanding of the molecular world around us, enabling rational design of new materials and interpretation of complex experimental data.

Molecular Symmetry: Multiple Choice Questions

Instructions: Choose the best answer for each question. Explanations are provided after each question.

1. Which of the following is NOT considered a fundamental symmetry element? a) Cn​ axis of symmetry b) Plane of symmetry (σ) c) Identity element (I or E) d) Dipole moment e) Centre of inversion (i)

Explanation: A dipole moment is a molecular property, not a symmetry element. The fundamental symmetry elements are axes of rotation, planes of reflection, centers of inversion, rotation-reflection axes, and the identity element.

2. What does an n-fold axis of symmetry (Cn​) imply about a molecule? a) Reflection through a plane b) Rotation by 360∘/n c) Inversion through a central point d) No change in appearance e) Rotation by 180∘ only

Explanation: A Cn​ axis signifies that a molecule returns to an indistinguishable configuration after a rotation of 360∘/n around that axis.

3. Which molecule possesses a C2​ axis of symmetry? a) Benzene (C6​H6​) b) Water (H2​O) c) Ammonia (NH3​) d) Methane (CH4​) e) Hydrogen cyanide (HCN)

Explanation: Water (H2​O) has a C2​ axis passing through the oxygen atom and bisecting the H−O−H angle. Benzene has C6​, Ammonia C3​, Methane has multiple C3​ and C2​ but not a single C2​ as its highest, and HCN has C∞​.

4. A horizontal plane of symmetry (σh​) is defined as being perpendicular to what? a) A C2​ axis b) Any plane of symmetry c) The highest-order Cn​ axis d) The molecular plane e) A σv​ plane

Explanation: A σh​ plane is always perpendicular to the highest-order rotational axis (Cn​) in the molecule.

5. Which molecule is an example of one with a center of inversion (i)? a) H2​O b) NH3​ c) s-trans-buta-1,3-diene d) CH4​ e) CHFClBr

Explanation: s-trans-buta-1,3-diene has a center of inversion, meaning reflection of every atom through the center yields an identical configuration.

6. The operation of rotation by 360∘/n followed by reflection through a plane perpendicular to the axis describes which symmetry element? a) Cn​ b) σ c) i d) Sn​ e) I

Explanation: This is the definition of an n-fold rotation-reflection axis (Sn​).

7. Which point group contains only a Cn​ axis (and the identity element)? a) Cnv​ b) Dn​ c) Cn​ d) Cnh​ e) Dnh​

Explanation: The Cn​ point group is defined by having only a Cn​ axis and the identity element.

8. Ammonia (NH3​) belongs to which point group? a) C2v​ b) C3v​ c) D3h​ d) Td​ e) Cs​

Explanation: Ammonia has a C3​ axis and three σv​ planes, characteristic of the C3v​ point group.

9. What is the symmetry rule for a molecule to be chiral? a) It must have at least one σ plane. b) It must have a center of inversion (i). c) It must not have any Sn​ symmetry element. d) It must have a Cn​ axis where n>1. e) It must have at least two C2​ axes.

Explanation: A molecule is chiral if it lacks any Sn​ symmetry element, which includes σ (S1​) and i (S2​).

10. Which of the following molecules is typically chiral? a) H2​O b) CO2​ c) CH4​ d) CHFClBr e) BF3​

Explanation: CHFClBr (bromochlorofluoromethane) has no symmetry elements other than identity, making it non-superimposable on its mirror image, hence chiral.

11. A molecule has a permanent dipole moment if its point group is one of the following, EXCEPT: a) C1​ b) Cs​ c) Cn​ d) Cnv​ e) Dnh​

Explanation: Molecules belonging to Dnh​ point groups (e.g., benzene, BF3​) possess high symmetry (including σh​ or i) that cancels out any potential dipole moments.

12. Why does carbon dioxide (CO2​) have a zero net dipole moment despite having polar C=O bonds? a) It has a plane of symmetry. b) It is a linear molecule with a symmetrical arrangement. c) It lacks a C2​ axis. d) Its vibrational modes are inactive in IR. e) Its electronic transitions are forbidden.

Explanation: The linear and symmetrical arrangement of the two polar C=O bonds in CO2​ causes their individual dipole moments to cancel each other out, resulting in a zero net dipole moment.

13. In infrared (IR) spectroscopy, a vibrational mode is active if it causes a change in what molecular property? a) Nuclear spin b) Electron configuration c) Permanent dipole moment d) Molecular mass e) Bond length only

Explanation: For a vibrational mode to be IR active, it must result in a change in the molecule’s net dipole moment during the vibration.

14. What determines whether a vibrational mode is Raman active? a) Change in electron spin b) Change in bond angle c) Change in the molecule’s polarizability tensor d) Presence of a Cn​ axis e) Only if it is also IR active

Explanation: A vibrational mode is Raman active if it causes a change in the molecule’s polarizability during the vibration.

15. Which of water’s (H2​O) vibrational modes is NOT both IR and Raman active? a) Symmetric stretch (ν1​) b) Bend (ν2​) c) Asymmetric stretch (ν3​) d) All three are both IR and Raman active e) None of the above

Explanation: All three normal modes of vibration for water (ν1​ symmetric stretch, ν2​ bend, and ν3​ asymmetric stretch) are both IR and Raman active due to their symmetry properties and the changes they induce in dipole moment and polarizability.

16. What does the “identity element” (I or E) represent in molecular symmetry? a) A rotation of 360∘ b) A reflection through any plane c) Doing nothing to the molecule d) Inversion through the center e) The principal axis of rotation

Explanation: The identity element represents an operation of “doing nothing” to the molecule, which leaves it unchanged. It is a necessary element in group theory.

17. If a molecule has a C4​ axis, how many degrees of rotation (360∘/n) will leave it indistinguishable? a) 90∘ b) 180∘ c) 45∘ d) 360∘ e) 72∘

Explanation: A Cn​ axis indicates rotation by 360∘/n. For C4​, it’s 360∘/4=90∘.

18. Which type of plane of symmetry contains the highest-order Cn​ axis? a) Horizontal plane (σh​) b) Dihedral plane (σd​) c) Vertical plane (σv​) d) Any plane of symmetry e) Mirror plane

Explanation: A vertical plane of symmetry (σv​) is defined as containing the highest-order Cn​ axis.

19. What is the common symbol used for the S1​ operation? a) i b) E c) σ d) C1​ e) S2​

Explanation: The S1​ operation (rotation by 360∘ followed by reflection through a perpendicular plane) is equivalent to a simple reflection, so it’s denoted by σ.

20. Buckminsterfullerene (C60​) belongs to which highly symmetrical point group? a) Td​ b) Oh​ c) D6h​ d) Ih​ e) Kh​

Explanation: Buckminsterfullerene (C60​) has icosahedral symmetry and belongs to the Ih​ point group.

21. A molecule is achiral if it possesses which of the following symmetry elements? a) A Cn​ axis b) Only the identity element c) A plane of symmetry (σ) d) A Dn​ point group e) Only σd​ planes

Explanation: A plane of symmetry (σ) is an S1​ element. Any molecule with an Sn​ element (including σ or i) is achiral.

22. How is the direction of a permanent dipole moment vector related to symmetry operations? a) It must be reversed by at least one operation. b) It must be perpendicular to all symmetry axes. c) It must be totally symmetric to all operations of the point group. d) It must be parallel to all planes of symmetry. e) It is unrelated to symmetry operations.

Explanation: The dipole moment vector cannot change its direction under any symmetry operation of the molecule, meaning its components must transform as the totally symmetric irreducible representation.

23. Which of the following point groups implies a molecule will NOT have a permanent dipole moment? a) C1​ b) Cs​ c) C2v​ d) D2h​ e) C3​

Explanation: Molecules in the D2h​ point group (e.g., ethylene, naphthalene) possess symmetry elements (like a center of inversion or multiple perpendicular mirror planes) that ensure a zero net dipole moment.

24. What does the “g” or “u” subscript indicate in molecular orbital (MO) labels for homonuclear diatomic molecules? a) Electron spin b) Degeneracy c) Symmetry with respect to inversion d) Number of nodal planes e) Bonding or antibonding character

Explanation: “g” (gerade) indicates symmetry with respect to inversion through the center of the molecule, while “u” (ungerade) indicates antisymmetry.

25. In the context of vibrational selection rules, what does “IR active” mean? a) The vibration changes the molecule’s mass. b) The vibration changes the molecule’s magnetic moment. c) The vibration causes a change in the molecule’s permanent dipole moment. d) The vibration is observable only with laser light. e) The vibration is non-degenerate.

Explanation: An IR active vibration results in a net change in the molecule’s dipole moment.

26. If a molecule’s vibrational mode has the same symmetry species as a translation (Tx​, Ty​, or Tz​), what can be concluded about its IR activity? a) It is forbidden. b) It is always Raman active but not IR active. c) It is IR active. d) It is only active in the far-infrared region. e) It requires a strong magnetic field to be observed.

Explanation: This is the direct symmetry rule for infrared activity: if a vibrational mode’s symmetry matches a translational mode’s symmetry, it is IR active.

27. What mathematical concept is used to describe the symmetry classification of properties like electronic and vibrational wave functions? a) Calculus b) Algebra c) Group theory (specifically, character tables) d) Geometry e) Statistics

Explanation: Character tables, derived from group theory, are used to classify the symmetry properties of molecular properties such as wave functions.

28. Which of the following is NOT a characteristic of a degenerate point group? a) Contains a Cn​ axis with n>2. b) May have degenerate properties (e.g., wave functions with identical energies). c) All characters in its character table are either +1 or -1. d) Examples include C3v​ and D3h​. e) The number of symmetry species equals the number of classes.

Explanation: Degenerate point groups have symmetry species (like E or T) whose characters can be other than +1 or -1, indicating multi-dimensional representations of the symmetry operations. Non-degenerate point groups have only +1 or -1 characters.

29. The “direct product” of two symmetry species is obtained by what operation? a) Summing their characters b) Dividing their characters c) Multiplying their characters under each symmetry element d) Subtracting their characters e) Taking the square root of their characters

Explanation: The direct product of two symmetry species is obtained by multiplying their characters under each corresponding symmetry operation.

30. What does the “multiplex or Fellgett advantage” in Fourier Transform (FT) spectroscopy refer to? a) The ability to use multiple detectors simultaneously. b) Recording all frequencies in the spectrum simultaneously. c) The improved resolution compared to dispersive instruments. d) The ease of sample preparation. e) The reduced need for high-power sources.

Explanation: The multiplex advantage means that all frequencies (or wavenumbers) in the spectrum are recorded simultaneously, leading to a much shorter acquisition time for a comparable spectrum.

31. Which of the following molecules has an infinite-fold axis of symmetry (C∞​)? a) H2​O b) NH3​ c) CH4​ d) HCN e) BF3​

Explanation: Linear molecules like HCN possess an infinite-fold axis of symmetry along their internuclear axis, as rotation by any angle leaves them indistinguishable.

32. What is the relationship between S2​ and a center of inversion (i)? a) S2​ is equivalent to i. b) S2​ is the inverse of i. c) S2​ is always perpendicular to i. d) S2​ and i are unrelated symmetry elements. e) S2​ is a generating element for i.

Explanation: The S2​ operation (rotation by 180∘ followed by reflection through a perpendicular plane) is equivalent to inversion through a center point, so S2​=i.

33. If a molecule belongs to the C1​ point group, what is true about its symmetry elements? a) It has a C2​ axis and a plane of symmetry. b) It has only the identity element. c) It has a center of inversion. d) It has a C3​ axis. e) It has multiple planes of symmetry.

Explanation: The C1​ point group is the lowest symmetry group, containing only the identity element. Such molecules are completely asymmetric.

34. In the C2v​ point group, if the C2​ axis is taken as the z-axis and the molecular plane as the yz-plane, what is the effect of exchanging the x and y axis labels on the symmetry species of a vibration? a) It will have no effect. b) A1​ and A2​ species will swap. c) B1​ and B2​ species will swap. d) All species will become totally symmetric. e) All species will become non-totally symmetric.

Explanation: In C2v​, the subscripts 1 and 2 often relate to symmetry or antisymmetry with respect to specific planes (e.g., σv​(xz) vs. σv​(yz)). Swapping x and y axes would swap these definitions, thus swapping B1​ and B2​ species.

35. What is the primary purpose of a “character table” in molecular symmetry? a) To list all possible bond lengths in a molecule. b) To show the arrangement of atoms in a crystal lattice. c) To classify molecular properties (like vibrations or orbitals) according to their symmetry. d) To determine the molecule’s mass and density. e) To predict the reactivity of a molecule.

Explanation: Character tables are fundamental tools that classify the behavior of molecular properties under each symmetry operation of a given point group.

36. For an electronic transition to be “allowed” by symmetry, what condition regarding the direct product of initial state, transition moment operator, and final state symmetry species must be met? a) It must be anti-symmetric. b) It must contain the totally symmetric representation. c) It must be a degenerate representation. d) It must be a direct product of two identical species. e) It must result in a zero value.

Explanation: A transition is electronically allowed if the direct product of the initial electronic state’s symmetry, the transition moment operator’s symmetry, and the final electronic state’s symmetry contains the totally symmetric irreducible representation.

37. The Kh​ point group is associated with the symmetry of what entity? a) A regular octahedron b) A linear molecule c) A sphere (e.g., an atom) d) A regular tetrahedron e) A planar molecule

Explanation: The Kh​ point group describes spherical symmetry, which is characteristic of atoms.

38. What is the relationship between the C1​ operation and the identity element I? a) C1​ is the inverse of I. b) C1​ is a generating element for I. c) C1​ is equivalent to I. d) They are unrelated. e) C1​ is a subset of I.

Explanation: A C1​ operation is a 360∘ rotation, which leaves the molecule unchanged, making it equivalent to the identity element I.

39. If a molecule undergoes an IR active vibration, what is observed? a) A change in its center of mass. b) Absorption of electromagnetic radiation at a specific frequency. c) Emission of visible light. d) A shift in its nuclear magnetic resonance signal. e) A change in its overall shape, but no interaction with light.

Explanation: IR active vibrations absorb electromagnetic radiation in the infrared region, corresponding to a change in the molecule’s dipole moment.

40. Why is molecular symmetry considered a “powerful concept” in chemistry? a) It allows for the synthesis of new elements. b) It solely focuses on the kinetic energy of molecules. c) It helps classify molecules and predict their properties like spectroscopy, polarity, and chirality. d) It is only applicable to diatomic molecules. e) It simplifies the process of calculating bond energies.

Explanation: Molecular symmetry provides a systematic framework to understand and predict a wide range of molecular properties, making it a powerful tool in chemical analysis and prediction.

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