CHM2311 Section 3: Molecular Symmetry & Group Theory

• CHM2311 Section 3 covers symmetry and group theory, focusing on molecular symmetry.

Introduction to Molecular Symmetry

• Molecular symmetry relates to the inherent beauty of an object based on its shape.
• Beauty can be found in rotation, inversion, and combined rotation and reflection (improper rotation).

Types Of Symmetry

• 2-fold rotation: An object looks the same after a 180-degree rotation
• 3-fold rotation: Requires a rotation of 120 degrees
• 4-fold rotation: Requires a rotation of 90 degrees
• Inversion: Requires beauty by inversion (going through an inversion center)
• Beauty by combined rotation and reflection involves improper rotation.

Structures, Shapes, Symmetry & Group Theory

• Bonding features dictate the molecular shape
• Symmetry is aesthetic, precise, and demonstrated by geometry/mathematics rules, with a distinct connection to shape/symmetry and bonding.
• Knowing a molecule's symmetry properties aids in describing orbitals used in bonding, predicting infrared spectra/optical activity, and interpreting electronic spectra.
• Symmetry involves an inherent feel for the symmetry of objects.

Symmetry and Group Theory

• Symmetry classifies molecules and helps derive selection rules for spectroscopic transitions.
• Group Theory systematically classifies molecules by symmetry.

Symmetry operations and elements

• Symmetry operation: An action leaving the object unchanged after transformation.
• Symmetry element: A geometrical figure (point, line, or plane) where the operation occurs.
• Point group: A group of operations leaving at least one point in a molecule unchanged.
• Space group: Grouping operations for extended 3D entities.

Molecular Symmetry: The Next Step Includes

• Defining and understanding symmetry, its elements, and operations
• Classifying shapes by identifying point groups
• Understanding the symmetry of electronic wave functions, including using character tables and representations.
• Molecular orbitals are formed from the combination of atomic orbitals.
• σg (bonding) is formed from [1SA + 1SB]
• σu (antibonding) is formed from [1SA - 1SB]

Defining Symmetry

• A symmetry operation is a reorientation of a body such that the initial and final orientations are indistinguishable.
• A symmetry element is the geometric entity (point, line, plane) with respect to which a symmetry operation is performed

Symmetry Elements and Operations

• There are five symmetry operations: identity, rotation, reflection, inversion, and rotation-reflection.
• Identity operation (E): Leaves the object unchanged, every object possesses this element.
• Rotation operation (Cn): Rotates the object through 360/n degrees about an axis of rotation.
• Reflection operation (σ): Reflects the object through a mirror plane.
• Inversion operation (i): Moves each point through the object's center to an opposite, equidistant position.
• Rotation-reflection operation (Sn): Rotates the object by 360/n degrees, followed by reflection through a plane perpendicular to the rotation axis.

Symmetry Elements and Operations Explained

• Molecules can be described in terms of symmetry, including axes, planes, and points.
• Rotation, reflection, and inversion are symmetry operations that result in equivalent positions of atoms.
• Rotation Operation: Object is rotated through 360/n degrees on an axis of rotation which are objects that can have more than one roation axis!
• When a symmetry operation yields the same result as a simpler operation, it is written as the simpler operation
• If an object has multiple rotation axes, the axis with the largest n is the highest order rotation axis or the principle axis.

Principal Axis

The principal axis is the highest order rotation.

• Reflection Operation: The object is reflected through a mirror plane and there are three types of reflection operations:
• σh mirror planes are perpendicular to the principal axis of rotation.
• σv mirror planes that include the principal axis of rotation.
• σd represents a mirror plane that bisects two C₂ axes.
• The slide presents the reflection of water as having two σv mirror planes.

Differentiating σv and σd Mirror Planes

• Both σv and σd are collinear with the principle rotation axis; however, it is only possible to have a σd plane in molecules where there are
• The principle axis Cn.
• Improper rotation axis Sn is 4-fold or larger i.e. n >=4
• In a molecule with both σv and σd planes, σv bisect as many atoms as possible while σd bisect bonds.

Inversion Operation

• Moves each point through the object's center to a position opposite and equidistant (mathematically: inversion of (x, y, z) to (−x, – y, – z)).

Example

• Staggered conformation of Ethane (C2H6) (YES Inversion centre!)

Example

• Methane (CH4) (NO Inversion centre!)

Improper Rotation

• This consists of rotation by 360/n degrees, then reflecting through a plane perpendicular to the rotation axis.
• S4 illustrated in methane as an example.
• Neither the the rotation of the σh are symmetry operations for methane but combined as an improper rotation it is.

In General: Improper Rotation

• Two successive Sn operations equal a Cn/2 operation (S² = Cn/2)
• S₁ equals a reflection (σ), call it σ. It equals i call is i

Summary of Symmetry Elements and Operations

• Identity (E): Leave the molecule alone.
• Proper Axis (Cn): Rotate the molecule by 360°/n degrees around the axis.
• Horizontal Plane (σh): Reflect the molecule through the plane perpendicular to the principle axis.
• Vertical Plane (σv): Reflect the molecule through a plane containing the principle axis.
• Dihedral Plane (σd): Reflect the molecule through a plane containing the principle axis and bisects 2 Cn' axes.
• Improper Axis (Sn): Rotate the molecule by 360°/n degrees around the improper axis and then reflect the molecule through the plane perpendicular to the improper axis.
• Inversion center (i): Invert the molecule through the inversion center.

Symmetry Elements in a Cube

• Contains three 4-fold axes

Conventions Regarding Coordinate Systems

• Use a right-handed system; if one rotational axis exists, it's the z-axis (vertical).
• For several axes, the highest order is the z-axis; if several highest order axes exist, the one through most atoms is the z-axis.
• If a molecule is planar and the z-axis lies in this plane, the x-axis is chosen to be normal to the plane.

Assigning Point Groups

• Molecular symmetry is defined by the group of symmetry elements it contains which are the set of symmetry elements together on the molecule's point group together.

Assigning Point Groups with High Symmetry

• Determine whether the molecule belongs to one of the special cases of high symmetry:
• If it is not then find the rotation axis with the highest order Cn e.g the highest order Cn axis for the molecule
• Determine whether the molecule belongs to one of the special cases of high symmetry:
• Is the molecule linear? If yes, then have Coy or Doh point group.
• Does the molecule contain more than one rotational axis where n > 2? If yes, then have Td, Oh or In point group.
• If there is no rotation axis (i.e., n = 1), it belongs to one of the low symmetry point groups (C₁, Cs, or C₁)

Groups of High Symmetry

Groups of high symmetry: have more than one Cn where n > 2 OR They have a C„ where n = ∞

High Symmetry Point Groups

Group Ta: These molecules have tetrahedral geometry and shape and all terminal atoms C„: 4x C3

Assigning Point Groups: Cont

4. If this has C₂' axes perpendicular to the C„ axis:
   • YES: D set of point groups; continue to step 5
   • NO: C or S set of point groups; continue to step 5
5. They they have mirror plane perpendicular-
   • YES: classified as C„h or D„h
   • NO: continue to step 6
6. Are there mirror plane that contain C„ axis? * -YES: classified as C„v or D„d -NO: C set of point groups
7. Is there SN axis collinear: -YES: then classified -NO: Classified C „

Characteristics

Common Point Group Information

• Cav: very common point group; C„ axis + no planes containing Cn
• Special case: H∞N C∞y and H2 axis , Cn and S, if N axis is present
• With Cnh axis and c perpendicular Cn,Sn and Lf n is present _ They all have E

Group Description for Symmetry Point Groups

• Dooh : linear with i
• Td : tetrahedral
• Og : octahedral
• Kli : spherical
• „must be ever otherwise it becomes the C„h *

Point Groups

• At least one point in the object must be fixed under the symmetry operations.

1. Main symbol

• C_Highest symmetry axis is a proper axis of rotation
• D Highest symmetry element IS a proper axis with L fold axis perpendicular to principal axis. T, O, Special systems for highly dynamic structures! tetra and Octa

S_ highest symmetry is a improper axis!
3) Alpha Indicative for axis plane 4) SUBSCRIPT

• Indicates prescience off center synoetry

Assigning point groups

M symmetry labels the S, it contains is set and colled molecules

Irreducible Representation

. The point represents some effect! To find coordinate

Qualitative consequences *

• Polarity* ** M could have have dipole in polar or plane .
• Chirality* ** M that cannot be super immpsoed are all Dissymetric ( Optically active) also has molecule images

Matrix Representations of Point Groups

• Every symmetry operation corresponds to a transformation matrix that shows coordinates. Matrix representation for a point group provides symmetry within the point!
• A more general representation shows a way to look at an object with the C2v point and its following systems (with x y z)
• Identity operation (C2v EX _.)* The „ „ and z and are unchanged by the identity operation
• Fold rotation C. At Cthe Zcordinates Given by x=x1, y=x(1) and z=xx/ This states the TRANSFORMATION Matriz!!!__ _reflection: with the axis of rotation

Character

__ character symbolises the transformation, by symertry__ . __It is define as the some of the numbers on the diagonals on transformation matrix* * For Group: It is with transformation matrices.

• Irreducible Presentation:
• Now can right to express the rotation for each transformation on coordinates, this representation effects each coordinate This help define the transformation and the elements and operation of it!

Summary of Representation

• We have lockers at three kids represents on this. Then matrices presents on each side!

• Character Tables The body of The trace gives us more into about their symmetries operations! It helps find functions for a.

What Do the Charts tell us?

• They have no effect on what direction, in this it has no effect by charecto . Also as they help not some facts on it . NOTE that symmetry is very important! All are related.

Symmetry

• Symmetric: The rotation does have the same character the same number of same! Also help relate between and make more defined!!

Assignment

The exercise is useful . All properties

• There is 3 in this! Helpful Transformations of Oxygen , the molecule must be placed in the right order following
• This all means it is easy to work with.

Representations of Point Groups

Point group representations show how the coordinates of an object are affected by the symmetry operations; matrix, irreducible, and reducible representations! Then from to that.

Point properties

The graph of a group, all it makes must be the main number or the symbol! It is what needs to be satisfied.

• Also the same for other
• In the character table (i.e.,symmetric and what is needed)
• The sum it makes All parts must be very exact!