Correlation Tables

Why This Matters

Correlation tables are the roadmap connecting symmetry concepts across representation theory—they show you how irreducible representations transform when you move between groups, combine systems, or analyze complex structures. You're being tested on your ability to trace these relationships: group-subgroup correlations, direct product decompositions, reducible-to-irreducible breakdowns, and the physical consequences of symmetry in spectroscopy and crystallography. Mastering correlation tables means understanding not just what representations exist, but how they relate to each other.

Think of correlation tables as translation dictionaries between different symmetry languages. When a molecule's symmetry is lowered, when two independent systems combine, or when you need to decompose a complicated representation into simpler pieces, correlation tables tell you exactly what happens. Don't just memorize table entries—know why representations split, combine, or remain unchanged, and what physical properties (vibrational activity, orbital degeneracy, spectroscopic selection rules) follow from these relationships.

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Foundational Tables: Building Blocks of Representation Analysis

These tables establish the core vocabulary of representation theory. Character tables encode everything about a group's irreducible representations in a compact format, while irreducible representation tables provide the dimensional foundation for all subsequent analysis.

Character Tables

• Characters are traces of representation matrices—they capture essential information about how group elements transform basis functions without requiring full matrix details
• Each row corresponds to an irreducible representation, with columns showing characters for each conjugacy class of symmetry operations
• Orthogonality relations between rows and columns enable decomposition of any representation and determination of symmetry-adapted bases

Irreducible Representation Tables

• List all irreducible representations with their dimensions—for finite groups, dimensions squared must sum to the group order $|G| = \sum_i d_i^2$
• Serve as the atomic building blocks from which all other representations are constructed through direct sums
• Dimension patterns reveal group structure—abelian groups have only one-dimensional irreps, while non-abelian groups must have higher-dimensional ones

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Group Relationship Tables: Navigating Between Symmetry Levels

When symmetry changes—through restriction to subgroups or extension via direct products—these tables track how representations transform. The key principle is that irreducible representations of a larger group may become reducible when restricted to a smaller group, and representations of component groups combine systematically in direct products.

Correlation Tables for Subgroups

• Track how irreps of a group $G$ decompose when restricted to a subgroup $H \subset G$—an irrep of $G$ may split into multiple irreps of $H$
• Branching rules specify exactly which subgroup irreps appear and with what multiplicities
• Essential for symmetry lowering analysis—when a perturbation reduces molecular symmetry, these tables predict how degenerate levels split

Correlation Tables for Direct Products

• Construct representations of $G_1 \times G_2$ from irreps of component groups—if $\Gamma_1$ is an irrep of $G_1$ and $\Gamma_2$ of $G_2$, then $\Gamma_1 \otimes \Gamma_2$ is an irrep of the product
• Dimensions multiply: $\dim(\Gamma_1 \otimes \Gamma_2) = \dim(\Gamma_1) \times \dim(\Gamma_2)$
• Critical for systems with independent symmetries—spin-orbit coupling, vibronic states, and multi-electron configurations all require direct product analysis

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Structural Symmetry Tables: Points, Space, and Operations

These tables connect abstract group theory to concrete physical structures. Point groups describe finite molecular symmetry, space groups add translational periodicity for crystals, and symmetry operation tables map how specific transformations act on physical quantities.

Correlation Tables for Point Groups

• Catalog symmetry operations that leave at least one point fixed—rotations $C_n$, reflections $\sigma$, improper rotations $S_n$, and inversion $i$
• Molecular shape determines point group assignment, which then dictates allowed spectroscopic transitions and orbital degeneracies
• Schoenflies notation ($C_{2v}$, $D_{3h}$, $T_d$) labels point groups by their characteristic operations and is standard in chemistry applications

Correlation Tables for Space Groups

• Extend point groups by including translational symmetry—230 distinct space groups classify all possible three-dimensional crystal symmetries
• Combine point group operations with Bravais lattice translations to describe the full symmetry of periodic structures
• International (Hermann-Mauguin) notation standard in crystallography; understanding point group substructure is key to interpreting space group symbols

Correlation Tables for Symmetry Operations

• Map how each symmetry operation transforms basis functions—orbitals, vibrations, or other physical quantities
• Matrix representatives show explicit transformation rules; characters summarize the essential information
• Selection rules emerge directly from how transition operators transform under symmetry operations

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Physical Application Tables: From Symmetry to Observables

These tables connect representation theory directly to measurable physical properties. The power of symmetry analysis lies in predicting which transitions are allowed, how modes transform, and when representations can be simplified.

Correlation Tables for Molecular Vibrations

• Classify vibrational modes by their symmetry species—each normal mode transforms as a specific irrep of the molecular point group
• Selection rules follow from symmetry: IR-active modes transform as $x$, $y$, or $z$ (translation); Raman-active modes transform as quadratic functions ($x^2$, $xy$, etc.)
• Mutual exclusion rule for centrosymmetric molecules—no mode can be both IR and Raman active, a direct consequence of inversion symmetry

Correlation Tables for Reducible Representations

• Decomposition formula: $n_i = \frac{1}{|G|} \sum_R \chi(R) \chi_i^*(R)$ gives the multiplicity of each irrep $\Gamma_i$ in a reducible representation
• Physical interpretation: a reducible representation describes a composite system; decomposition reveals its independent symmetry-adapted components
• Projection operators use character table data to construct symmetry-adapted linear combinations from arbitrary basis functions

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Quick Reference Table

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Self-Check Questions

1. When a molecule's symmetry is lowered from $T_d$ to $C_{3v}$ by a perturbation, which type of correlation table would you use to determine how a triply degenerate $T_2$ representation splits?

2. Compare and contrast how character tables and irreducible representation tables are used in decomposing a reducible representation—what information does each provide?

3. A molecule has inversion symmetry. Using correlation tables for molecular vibrations, explain why you would never observe the same vibrational mode in both IR and Raman spectra.

4. If you need to analyze spin-orbit coupling in an atom, you must combine spatial and spin symmetry. Which type of correlation table applies, and how do the dimensions of the resulting representations relate to those of the component representations?

5. Given a reducible representation with characters $\chi(E) = 6$, $\chi(C_3) = 0$, $\chi(\sigma_v) = 2$ in point group $C_{3v}$, use the decomposition formula to identify which irreps appear—what physical property might this representation describe?