3.3 Applications of Group Theory in Inorganic Chemistry

Symmetry and Point Groups

Fundamental Symmetry Concepts

Symmetry elements are geometric features of a molecule (an axis, a plane, a point) that correspond to operations you can perform and leave the molecule looking identical. Mastering these is the foundation for everything else in group theory.

The core symmetry elements are:

• Proper rotation axis ($C_n$): Rotation by $360°/n$ about an axis. A $C_3$ axis means a 120° rotation maps the molecule onto itself.
• Mirror plane ($\sigma$): Reflection through a plane. These come in three flavors: $\sigma_h$ (perpendicular to the principal axis), $\sigma_v$ (containing the principal axis), and $\sigma_d$ (bisecting $C_2$ axes perpendicular to the principal axis).
• Inversion center ($i$): Every atom at position $(x, y, z)$ maps to $(-x, -y, -z)$. Octahedral complexes have one; tetrahedral complexes do not.
• Improper rotation axis ($S_n$): A $C_n$ rotation followed by reflection through a perpendicular plane. Note that $S_1 = \sigma$ and $S_2 = i$.

Point groups classify molecules by collecting all their symmetry elements into a single label. You assign a point group by working through a systematic flowchart: check for special groups (linear, cubic), identify the highest-order rotation axis, then look for perpendicular $C_2$ axes, mirror planes, and so on. Common examples: $\text{H}_2\text{O}$ is $C_{2v}$, $\text{BF}_3$ is $D_{3h}$, and $[\text{Co(NH}_3)_6]^{3+}$ is $O_h$.

Character tables are the reference sheets for each point group. Each row is an irreducible representation (symmetry species like $A_1$, $T_{2g}$, etc.), and each column is a class of symmetry operations. The numbers in the table (characters) tell you how basis functions transform under each operation. The rightmost columns list which linear/rotational functions and quadratic functions belong to each representation, which is directly useful for spectroscopic selection rules.

Symmetry-Adapted Linear Combinations

When you build molecular orbitals, you don't just throw atomic orbitals together randomly. Symmetry-adapted linear combinations (SALCs) are specific combinations of atomic orbitals that each transform as a single irreducible representation of the molecular point group. Only orbitals of the same symmetry species can mix, which dramatically simplifies MO construction.

To generate SALCs:

1. Choose a set of equivalent atomic orbitals (e.g., the six $\sigma$-bonding orbitals of an octahedral complex).
2. Determine the reducible representation by tracking how many orbitals are unmoved by each symmetry operation (that count is the character).
3. Decompose the reducible representation into irreducible representations using the reduction formula.
4. Apply the projection operator for each irreducible representation to one of the basis orbitals. This produces the explicit linear combination.
5. Normalize each SALC so its coefficients squared sum to one.

The resulting SALCs then combine with metal orbitals of matching symmetry to form bonding and antibonding molecular orbitals.

Representations in Group Theory

A reducible representation describes how an entire set of orbitals or displacement vectors transforms under the symmetry operations of a group. You build it by applying each operation and recording the trace (sum of diagonal elements) of the transformation matrix.

An irreducible representation cannot be broken down further. These are the fundamental building blocks listed as rows in the character table. Every reducible representation is a unique sum of irreducible representations.

To decompose a reducible representation, use the reduction formula:

$n_i = \frac{1}{h} \sum_R N_R \cdot \chi(R) \cdot \chi_i(R)$

where $n_i$ is the number of times irreducible representation $i$ appears, $h$ is the order of the group, $N_R$ is the number of operations in class $R$, $\chi(R)$ is the character of the reducible representation, and $\chi_i(R)$ is the character of the irreducible representation. This formula is one of the most-used tools in applied group theory, so get comfortable with it.

Molecular Orbital Theory and Spectroscopy

Principles of Molecular Orbital Theory

Molecular orbital (MO) theory describes bonding by combining atomic orbitals across the entire molecule into delocalized molecular orbitals. Each combination produces a bonding MO (lower energy, constructive overlap), an antibonding MO (higher energy, destructive overlap), and sometimes non-bonding MOs (no net interaction).

The LCAO (linear combination of atomic orbitals) approach is the standard method: molecular orbitals are written as weighted sums of atomic orbitals. The key symmetry constraint is that only atomic orbitals belonging to the same irreducible representation can combine. This is where SALCs pay off.

For inorganic complexes, the important orbital interactions are:

• $\sigma$-bonding: Ligand lone pairs directed at the metal overlap with metal $s$, $p$, and $d$ orbitals of appropriate symmetry. In an $O_h$ complex, the six $\sigma$-SALCs span $A_{1g} + E_g + T_{1u}$, matching the metal $s$, $d_{z^2}/d_{x^2-y^2}$, and $p_x/p_y/p_z$ orbitals respectively.
• $\pi$-bonding: Ligand $\pi$-donor or $\pi$-acceptor orbitals interact with metal $t_{2g}$ orbitals ($d_{xy}$, $d_{xz}$, $d_{yz}$) in octahedral geometry. $\pi$-donors raise the $t_{2g}$ set, shrinking $\Delta_o$; $\pi$-acceptors lower it, increasing $\Delta_o$.

MO energy level diagrams for octahedral and tetrahedral complexes show how these interactions produce the familiar $t_{2g}$/$e_g$ (or $e$/$t_2$) splitting patterns. Bond order is calculated as $\frac{1}{2}(\text{bonding electrons} - \text{antibonding electrons})$.

Vibrational Spectroscopy and Selection Rules

Vibrational spectroscopy (IR and Raman) probes the normal modes of a molecule, and group theory tells you exactly how many modes exist and which ones are observable.

A nonlinear molecule with $N$ atoms has $3N - 6$ vibrational modes ($3N - 5$ for linear molecules). To find the symmetry species of these modes:

1. Construct the reducible representation for all $3N$ Cartesian displacement vectors by counting unmoved atoms under each operation and multiplying by the per-atom character contribution.
2. Subtract the translational ($T_x, T_y, T_z$) and rotational ($R_x, R_y, R_z$) representations, which you read directly from the character table.
3. Decompose the remaining reducible representation into irreducible representations. Each one corresponds to a vibrational mode of that symmetry.

Selection rules then determine which modes show up in each type of spectrum:

• IR-active: The vibrational mode must belong to the same symmetry species as $x$, $y$, or $z$ (i.e., it must involve a change in dipole moment).
• Raman-active: The mode must belong to the same symmetry species as a quadratic function ($x^2$, $xy$, etc.), meaning it involves a change in polarizability.

In centrosymmetric molecules (those with an inversion center), the mutual exclusion rule applies: no mode can be both IR- and Raman-active. This is a useful diagnostic for determining whether a molecule has an inversion center.

Ligand Field Theory and Electronic Spectroscopy

Ligand field theory (LFT) explains the electronic structure, color, and magnetism of transition metal complexes by considering how ligand electron density perturbs the energies of metal d-orbitals.

The central quantity is the crystal field splitting parameter: $\Delta_o$ for octahedral and $\Delta_t$ for tetrahedral complexes (with $\Delta_t \approx \frac{4}{9}\Delta_o$ for the same metal-ligand pair). The magnitude of $\Delta$ depends on both the metal ion and the ligands, ranked by the spectrochemical series:

When $\Delta$ is large relative to the electron pairing energy, electrons fill lower orbitals first, giving a low-spin configuration. When $\Delta$ is small, electrons spread across orbitals to maximize spin, giving a high-spin configuration. This distinction only matters for $d^4$ through $d^7$ octahedral complexes.

Electronic (UV-Vis) spectroscopy probes d-d transitions between these split levels. Two key selection rules govern their intensity:

• Laporte rule: Transitions between states of the same parity are forbidden ($g \rightarrow g$ or $u \rightarrow u$). In octahedral complexes, all d-d transitions are formally Laporte-forbidden since d-orbitals are all gerade. They gain weak intensity through vibronic coupling (molecular vibrations temporarily break the inversion symmetry).
• Spin selection rule: Transitions must conserve spin multiplicity ($\Delta S = 0$). Spin-forbidden transitions are even weaker than Laporte-forbidden ones, which is why $[\text{Mn(H}_2\text{O)}_6]^{2+}$ (high-spin $d^5$, all transitions spin-forbidden) is nearly colorless.

Tetrahedral complexes lack an inversion center, so the Laporte rule doesn't apply, and their d-d transitions tend to be more intense than octahedral ones.

Structure and Reactivity

Crystallography and Solid-State Structures

X-ray crystallography determines the three-dimensional arrangement of atoms in a crystal by measuring how X-rays diffract off the periodic electron density. The unit cell is the smallest repeating box that, when translated in three dimensions, generates the entire crystal lattice.

Symmetry in the solid state is described by space groups, which combine point group operations (rotations, reflections, inversions) with translational operations (screw axes, glide planes). There are exactly 230 unique space groups. The site symmetry that a molecule occupies in a crystal can differ from its free-molecule point group, which affects solid-state spectroscopic properties.

Common inorganic crystal structure types to know:

• Close-packed structures: Cubic close-packed (ccp/fcc) and hexagonal close-packed (hcp) represent the most efficient packing of equal spheres (74% packing efficiency). Many metals adopt these structures.
• Ionic structures: NaCl (rock salt, 6:6 coordination), CsCl (8:8 coordination), zinc blende and wurtzite (both 4:4, tetrahedral coordination). The adopted structure depends on the radius ratio of cation to anion.
• Perovskite ($\text{ABX}_3$): Features corner-sharing $\text{BX}_6$ octahedra with A cations in the cavities. Widely important in ferroelectrics, superconductors, and catalysis.

Stereochemistry and Isomerism

The three-dimensional arrangement of ligands around a metal center determines many physical and chemical properties of coordination compounds. Isomers share the same formula but differ in structure or spatial arrangement.

Structural isomers differ in connectivity:

• Linkage isomers: Same ligand binds through different donor atoms (e.g., $\text{NO}_2^-$ binding through N vs. O).
• Ionization isomers: Exchange between a coordinated ligand and a counterion.
• Coordination isomers: Different distribution of ligands between two metal centers.

Stereoisomers have the same connectivity but different spatial arrangements:

• Geometric (cis/trans): In square planar $[\text{Pt(NH}_3)_2\text{Cl}_2]$, the cis isomer is the anticancer drug cisplatin while the trans isomer is inactive. In octahedral complexes, fac/mer isomerism arises with three identical ligands.
• Optical isomers (enantiomers): Non-superimposable mirror images. Tris-chelate complexes like $[\text{Co(en)}_3]^{3+}$ exist as $\Delta$ and $\Lambda$ enantiomers.

Point group analysis is the definitive way to identify chirality: a molecule is chiral if and only if it lacks all improper rotation axes ($S_n$), which includes lacking both $\sigma$ planes and an inversion center $i$.

Reaction Mechanisms and Symmetry

Symmetry considerations constrain which reaction pathways are accessible by determining whether orbital interactions along a proposed pathway are favorable.

The Woodward-Hoffmann rules apply orbital symmetry conservation to pericyclic reactions. The approach involves:

1. Identify the symmetry element preserved throughout the reaction (often a mirror plane or $C_2$ axis).
2. Classify the frontier molecular orbitals of reactants and products as symmetric (S) or antisymmetric (A) with respect to that element.
3. Construct a correlation diagram connecting reactant and product orbitals of the same symmetry.
4. If the ground-state occupied orbitals of the reactant correlate smoothly with ground-state orbitals of the product, the reaction is thermally allowed. If they correlate with excited-state orbitals, the reaction is thermally forbidden but photochemically allowed.

In inorganic chemistry, symmetry also governs:

• Ligand substitution: Associative mechanisms (common for square planar $d^8$ complexes) proceed through a trigonal bipyramidal transition state, while dissociative mechanisms (common for octahedral complexes) go through a square pyramidal intermediate. The symmetry of the transition state determines which orbitals stabilize or destabilize the pathway.
• Electron transfer: Inner-sphere and outer-sphere mechanisms have different symmetry requirements. Orbital symmetry matching between donor and acceptor facilitates electron transfer.
• Organometallic reactions: Oxidative addition and reductive elimination require frontier orbital symmetry compatibility between the metal complex and the incoming/departing substrate. For example, concerted oxidative addition of $\text{H}_2$ to a $d^8$ complex is symmetry-allowed when the filled metal $d$ orbital and empty $\sigma^*$ of $\text{H}_2$ have matching symmetry.