Introduction to Group Theory and Molecular Symmetry Representations

Introduction to Group Theory

• The term "group" in the context of a point group refers specifically to a collection of elements—in this case, symmetry operations—that fulfill established, well-defined criteria.

• A group is a formal mathematical structure.

• The branch of mathematics dedicated to the study of these structures is known as group theory.

Definition of a Group and Group Axioms

• A set of elements $\{A, B, C, \dots\}$ and a defined operation "$\circ$" that combines any two elements form a group if and only if the following four axioms are satisfied:

  1. Closure: The result of the operation "$\circ$" on any two elements (e.g., $A \circ B$) must also be an element of the same group. The group is described as "closed" under the operation.

  2. Identity Element: The set must contain an identity element, denoted as $E$. For any element $A$ in the group, applying the operation with identity must result in the element itself: $A \circ E = E \circ A = A$.

  3. Inverse Element: For every element $A$ in the set, there must exist an inverse element, denoted as $A^{-1}$, such that combining them yields the identity: $A \circ A^{-1} = A^{-1} \circ A = E$.

  4. Associativity: The operation "$\circ$" must be associative. For elements $A$, $B$, and $C$, the following must hold: $(A \circ B) \circ C = A \circ (B \circ C)$.

• Note on Commutativity: In group theory, the operation is generally not commutative ($A \circ B \neq B \circ A$). In the context of symmetry, the order in which operations are applied frequently changes the final result.

The Integers as a Group Example: $(\mathbb{Z}, +)$

• The set of integers ($\mathbb{Z}$) combined with the operation of addition ($+$) forms a group:

  • Closure: Adding two integers results in another integer (e.g., $1 + 2 = 3$, where $3 \in \mathbb{Z}$).

  • Identity: The integer $0$ acts as the identity, as adding it to any integer leaves it unchanged (e.g., $1 + 0 = 1$).

  • Inverse: For every integer $n$, there is an inverse $-n$ such that their sum is the identity (e.g., $1 + (-1) = 0$).

  • Associativity: Addition is associative, meaning $(1 + 2) + 3 = 1 + (2 + 3)$.

Algebra with Symmetry Operations

• The symmetry operations of an object form a point group.

• The group operation "$\circ$" represents the successive application of symmetry operations.

• Shorthand Notation: Similar to multiplication, the operator symbol is often omitted. For example, $C_2 \circ C_2$ is written as $C_2 C_2$ or $C_2^2$.

• Closure Constraint: The net effect of applying multiple operations successively must be equivalent to the effect of a single operation already within the group; otherwise, the group is not closed. For instance, $C_2^2 = E$.

Example Exercise: Symmetry Operations in $C_{2v}$

• The $C_{2v}$ group (e.g., in a water molecule) consists of four operations: $E$, $C_2$, $\sigma_v$, and $\sigma_v'$.

• Case 1: $C_2$ followed by $C_2$

  • Application of a $180^{\circ}$ rotation ($C_2$) followed by another $180^{\circ}$ rotation ($C_2$) returns the object to its original orientation.

  • Result: $C_2 C_2 = E$.

• Case 2: $C_2$ followed by $\sigma_v$

  • Rotating the molecule (e.g., water) by $C_2$ and then reflecting through a vertical plane ($\sigma_v$) results in a configuration equivalent to applying a different vertical plane reflection ($\sigma_v'$).

  • Result: $\sigma_v C_2 = \sigma_v'$.

The $C_{2v}$ Group Multiplication Table

• A multiplication table tabulates all possible combinations of the group's operations. The following represents the complete table for $C_{2v}$, where the top row is the first operation applied and the left column is the second operation applied:

Validating Group Axioms for $C_{2v}$

• Closure: No elements outside the set $\{E, C_2, \sigma_v, \sigma_v'\}$ appear in the multiplication table.

• Identity: $E$ is clearly present and behaves as the identity for all operations.

• Inverses: Every element has an inverse. In the case of $C_{2v}$, the elements happen to be self-inverses ($C_2 = C_2^{-1}$, $\sigma_v = \sigma_v^{-1}$, etc.), meaning the identity appears on the diagonal of the table. Note: Self-inversion is not a general property of all groups.

• Associativity: Operations follow the associative rule, such as $C_2 \circ (\sigma_v \circ \sigma_v') = (C_2 \circ \sigma_v) \circ \sigma_v'$.

Notation and Matrix Algebra Representation

• Right-to-Left Multiplication: Successive symmetry operations are written and read from right to left, similar to mathematical operators. For the phrase "$C_2$ followed by $\sigma_v$", the notation is $\sigma_v C_2$.

  • Logic: In the expression $\sigma_v C_2 X$, $C_2$ operates on $X$ first, and then $\sigma_v$ operates on the result.

• Matrix Algebra: Symmetry operations can be represented as matrices ($D$), and the object (coordinates) as a vector ($r_{orig}$).

  • The transformed vector is found via left-multiplication: $r_{trans} = D r_{orig}$.

  • Example Calculation: Rotating by $90^{\circ}$ counter-clockwise about the $z$-axis followed by a reflection in the $xz$-plane:

  • $\mathbf{D} = \sigma_v C_2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$.

Chemical Applications of Group Theory

• Group theory allows for the calculation and manipulation of molecular features based solely on symmetry.

• Vibrational Spectroscopy (IR): Symmetry dictates the number of observable stretch bands.

  • Example: For $[PdCl_2(NH_3)_2]$ complexes:

    • cis-isomer: Displays two IR bands for $Pd-L$ stretching vibrations.

    • trans-isomer: Displays only one IR band for $Pd-L$ stretching vibrations.

• Photoelectron Spectroscopy (PES): Used alongside MO (Molecular Orbital) diagrams to understand the energy required to remove electrons ($E_{ion} = h\nu - E_{kin}$).

Characters and Representations in Quantum Mechanics

• When symmetry operations act on molecular features like atoms or bonds, they can stay in place or move to an equivalent position.

• The Third Outcome: For "signed" features like orbitals/wavefunctions ($\psi$), an operation can keep the orbital in place but flip its sign ($+$, $-$).

• Wavefunction Refresher:

  • Orbitals have regions of positive and negative character separated by nodal planes (zero probability).

  • Born Interpretation: The probability of finding a particle is proportional to the square of the wavefunction ($\psi^2$).

  • The absolute sign of the wavefunction is not directly observable; only the relative sign matters.

• Character (\chi): This value identifies how a property is affected by a symmetry operation.

  • +1: Symmetric (orbital/feature is unchanged).

  • -1: Antisymmetric (orbital/feature changes sign).

  • 0: The orbital/feature moves to a different location.

Symmetry Properties of Orbitals in $H_2O$ ($C_{2v}$)

Oxygen is located at the invariant point of the water molecule, meaning its atomic orbitals do not move under $C_{2v}$ symmetry operations.

• s-orbital ($\phi_s$): Remains unchanged under all operations ($E, C_2, \sigma_v, \sigma_v'$).

  • Characters: $(1, 1, 1, 1)$. This spans the $a_1$ representation.

• $p_z$ orbital ($\phi_z$): Aligned with the principal axis; remains unchanged under all operations.

  • Characters: $(1, 1, 1, 1)$. This also spans the $a_1$ representation.

• $p_x$ orbital ($\phi_x$): Stays in place but changes sign under $C_2$ and $\sigma_v'$.

  • Characters: $(1, -1, 1, -1)$. This spans the $b_1$ representation.

• $p_y$ orbital ($\phi_y$): Stays in place but changes sign under $C_2$ and $\sigma_v$.

  • Characters: $(1, -1, -1, 1)$. This spans the $b_2$ representation.

• $d_{xy}$ orbital ($\phi_{xy}$): Stays in place but changes sign under $\sigma_v$ and $\sigma_v'$.

  • Characters: $(1, 1, -1, -1)$. This spans the $a_2$ representation.

The Character Table and Mulliken Symbols

• A character table lists all irreducible representations (or symmetry species) for a point group.

• Mulliken Symbols: Labels used for these representations.

  • a / b: Indicates symmetry (a) or antisymmetry (b) under rotation about the principal axis.

  • e / t: Designate 2-dimensional (e) and 3-dimensional (t) representations.

  • Subscripts 1 / 2: Refer to symmetry (1) or antisymmetry (2) under reflection in $\sigma_v$ or rotation about a perpendicular $C_2$.

  • Subscripts g / u: (German: gerade/ungerade) Refer to symmetry (g) or antisymmetry (u) under inversion ($i$).

  • Superscripts ' / '': Refer to symmetry (') or antisymmetry ('') under reflection through a horizontal plane ($\sigma_h$).

Character Table for $C_{2v}$: