Vibrational Modes and IR/Raman Selection Rules

Symmetry-predicted IR/Raman activity remains a front-line structural tool across chemistry and materials science. In organometallic synthesis, counting carbonyl-stretch bands (exactly as done for fac/*mer-$[M(CO)_3X_3]$ above) is still the fastest way to confirm isomeric purity of a metal carbonyl compound in solution, long before a crystal structure is available; the same logic extends to nitrosyl, cyanide, and dinitrogen complexes. In <strong>solid-state and materials chemistry</strong>, Raman spectroscopy of centrosymmetric perovskites and metal oxides exploits mutual exclusion to detect subtle symmetry-lowering phase transitions (e.g., a cubic-to-tetragonal transition in $\text{BaTiO}_3$ breaks inversion symmetry and turns on new IR bands that were Raman-only in the cubic phase). In <strong>coordination chemistry</strong>, IR/Raman selection rules distinguish octahedral trans isomers (centrosymmetric, $D_{4h}$, obeying mutual exclusion) from cis isomers (lower symmetry, $C_{2v}$, with IR/Raman coincidences), a technique widely used for platinum and palladium anticancer drug candidates related to cisplatin, where confirming the cis geometry is pharmacologically critical. In <strong>analytical and industrial infrared spectroscopy</strong>, the symmetry-derived mode count underlies quantitative interpretation of fingerprint-region bands for quality control of polymers, pharmaceuticals, and gas-phase pollutants (e.g., CO₂ and SO₂ atmospheric monitoring rely directly on the linear-vs-bent vibrational analysis worked out above). Finally, in <strong>computational chemistry</strong>, every electronic-structure package that computes a vibrational frequency calculation performs, internally, exactly this symmetry decomposition to label computed normal modes with Mulliken symbols and flag IR/Raman activity automatically, connecting this century-old group-theoretical method directly to modern ab initio* workflows.

:::quiz[The fac isomer of [M(CO)₃X₃] in C₃ᵥ gives how many IR-active carbonyl stretching bands?]

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Ask the tutor: How would the vibrational analysis and IR/Raman predictions change for square-planar trans-$[PtCl_2(NH_3)_2]$ versus its cis isomer?

A molecule with $N$ atoms has $3N$ Cartesian coordinates describing the instantaneous position of every nucleus. Of these, $3N$ degrees of freedom split into: 3 translations of the center of mass, 3 rotations about the center of mass (2 for a linear molecule, since rotation about the molecular axis does not correspond to a physical motion of point masses on that axis), and the remainder are genuine internal vibrations,

The trick of group theory is that $\Gamma_{3N}$ can be built and reduced without ever writing down a force constant. Each symmetry operation of the point group either moves every atom (contributing zero to the character, since a displaced atom's local coordinate vector does not map back onto itself) or leaves some atoms fixed in place. Only unmoved atoms contribute, and each contributes a fixed "per-atom character" that depends only on the type of operation (rotation angle, whether it is proper or improper). Multiplying (unmoved atoms) × (per-atom contribution) for every class gives the full character of $\Gamma_{3N}$, ready for reduction by the standard formula. Once $\Gamma_{vib}$ is decomposed into irreducible representations, the character table itself supplies the spectroscopic answer. The electric dipole moment operator $\hat\mu$ has three Cartesian components, and the intensity of an IR transition is governed by the transition dipole integral

which is nonzero only if the direct product of the vibrational mode's symmetry with $x$, $y$, or $z$ contains the totally symmetric representation. Since the ground vibrational state is always totally symmetric, this reduces to requiring that the normal mode itself transform as one of $x,y,z$. Analogously, Raman scattering intensity depends on the polarizability derivative, and the polarizability $\alpha$ is a symmetric tensor whose six independent components transform as the quadratic functions $x^2, y^2, z^2, xy, xz, yz$ (or symmetry-adapted combinations like $x^2-y^2$). A mode is Raman-active iff it shares an irreducible representation with one of these quadratics.

The reducible representation of the full set of Cartesian displacement vectors has, for each symmetry operation $R$, the character

where: $N_{unmoved}(R)$ is the number of atoms left in place by operation $R$ (an atom contributes 0 if it moves to a different site); the bracket is $+1+2\cos\theta$ for a proper rotation $C_n$ (or $E$, where $\theta=0$, giving $+3$); and the bracket is $-1+2\cos\theta$ for an improper rotation $S_n$ (including $\sigma$, where $\theta=0$ gives $+1$, and $i$, where $\theta=\pi$, gives $-3$... more precisely $i$ is $S_2$ with $\theta = \pi$, giving $-1 + 2\cos\pi = -3$, so each unmoved atom under $i$ contributes $-3$; a lone atom sitting exactly at the inversion center is the only one that can be "unmoved" by $i$). Reading: this formula converts a geometric fact (which atoms stay put) into an algebraic one (a character), for every class of the point group, without ever diagonalizing a force-constant matrix. Once $\chi_{3N}(R)$ is tabulated for every class, the number of times irreducible representation $i$ appears is given by the reduction formula,

where: $h$ is the order of the group; $g(R)$ is the number of operations in the class of $R$; $\chi_i(R)$ is the character of irreducible representation $i$ for that class. This is exactly the standard Reduction Formula machinery, applied here to $\Gamma_{3N}$ rather than to some abstract basis. Reading: each irreducible representation's multiplicity in the full 3N-dimensional space is an overlap integral (in the discrete, finite-group sense) between the reducible character vector and each row of the character table. Finally, the vibrational representation is obtained by subtraction,

where: $\Gamma_{trans}$ is identified directly from the character table as the span of $x, y, z$; $\Gamma_{rot}$ is the span of $R_x, R_y, R_z$ (only 2 components survive for a linear molecule, since rotation about the internuclear axis is not a rotation of point masses and carries zero moment of inertia). Reading: this subtraction removes exactly six (or five, if linear) representations corresponding to rigid-body motion, leaving only the internal, bond-stretching/bending degrees of freedom that a spectrometer can detect.

Example 1 — Foundational: water, H₂O ($C_{2v}$), full numeric check. This simply instantiates the symbolic derivation above with real numbers. $N=3$ atoms, $C_{2v}$ point group, $h=4$. Unmoved atoms: $E\to3$, $C_2\to1$ (only O), $\sigma_v(xz,\text{molecular plane})\to3$, $\sigma_v'(yz)\to1$ (only O). Characters:

Reduction (as carried out step-by-step above) gives $\Gamma_{3N}=3A_1+A_2+3B_1+2B_2$, and after removing $\Gamma_{trans}=A_1+B_1+B_2$ and $\Gamma_{rot}=A_2+B_1+B_2$,

i.e. $3N-6 = 3$ vibrations: symmetric stretch ($A_1$, ~3657 cm⁻¹), bend ($A_1$, ~1595 cm⁻¹), asymmetric stretch ($B_1$, ~3756 cm⁻¹). Since $A_1$ carries $z$ and $x^2,y^2,z^2$, and $B_1$ carries $x$ and $xz$, all three fundamentals are both IR- and Raman-active — consistent with water's experimental spectrum showing three distinct bands in both techniques, and consistent with $C_{2v}$ having no inversion center (so mutual exclusion does not apply).

Example 2 — Applied: linear XeF₂ vs bent ClO₂⁻ — using symmetry to settle a geometry question. Suppose an unknown triatomic $AB_2$ species shows exactly two IR-active fundamentals and the mutual exclusion rule holds strictly (no coincidences between IR and Raman bands). Is the molecule linear centrosymmetric ($D_{\infty h}$) or bent ($C_{2v}$)? For a bent $AB_2$ in $C_{2v}$, Example 1 shows all three fundamentals are IR-active (and also Raman-active) — that would give three IR bands and complete IR/Raman coincidence, contradicting the premise. For a linear, centrosymmetric $AB_2$ (point group $D_{\infty h}$, e.g. XeF₂ or CO₂), redo the unmoved-atom count. Relevant classes: $E$, $C_\infty^\phi$ (all rotations about the axis), $\sigma_v$ (infinite set containing the axis), $i$, $S_\infty^\phi$, $C_2'$ (perpendicular $C_2$ axes). For the linear case it is faster to use the standard result: $\Gamma_{3N}$ for a linear triatomic reduces to

with $\Gamma_{trans} = \Sigma_u^+ + \Pi_u$ (the $z$-stretch and the doubly-degenerate $(x,y)$ translations) and $\Gamma_{rot} = \Pi_g$ (rotations about the two axes perpendicular to the molecular axis; rotation about the molecular axis is not a coordinate at all for point atoms on a line). Subtracting,

giving $3N-5 = 4$ vibrations (the $\Pi_u$ bend is doubly degenerate, so it counts as one distinct frequency but 2 vibrational degrees of freedom: $1+1+2=4$). Since $D_{\infty h}$ is centrosymmetric, $\Sigma_g^+$ (the symmetric stretch) is gerade — Raman-active, IR-silent — while $\Sigma_u^+$ (antisymmetric stretch) and $\Pi_u$ (bend) are ungerade — IR-active, Raman-silent. That is exactly two distinct IR-active frequencies ($\Sigma_u^+$ and $\Pi_u$) with complete mutual exclusion from the one Raman-active band ($\Sigma_g^+$). The premise therefore identifies the molecule as linear and centrosymmetric, exactly the classic argument historically used to confirm that CO₂ is linear (not bent) from its combined IR/Raman data.

Example 3 — Exam/competition level: distinguishing fac- and mer-[M(CO)₃X₃] by carbonyl-stretch counting. Octahedral tricarbonyl complexes $[M(CO)_3X_3]$ exist as two geometric isomers: fac (three CO ligands mutually cis, occupying one triangular face) in point group $C_{3v}$, and mer (three CO ligands in a meridional, T-shaped arrangement, two trans pairs) in point group $C_{2v}$. Predict the number of IR-active C≡O stretching bands for each isomer, treating only the three C≡O stretches as a local basis (a standard simplification — the CO stretches are much higher in frequency and well separated from all other modes, so they can be treated as an independent local-mode representation rather than doing the full $3N$ analysis). fac isomer, $C_{3v}$. The three CO groups are related by the $C_3$ axis passing through the unique threefold vertex. Build the reducible representation $\Gamma_{CO}$ using the three C≡O bond vectors as basis functions: under $E$, all 3 are unmoved, contributing $\chi=3$; under $C_3$ (and $C_3^2$), no CO bond maps onto itself (all three cycle), contributing $\chi=0$; under $\sigma_v$ (three vertical mirror planes, each containing one CO group and bisecting the other two), exactly 1 CO bond lies in the plane and is unmoved, contributing $\chi=1$. So

Using the $C_{3v}$ character table ($h=6$; $A_1: 1,1,1$; $A_2: 1,1,-1$; $E: 2,-1,0$):

So $\Gamma_{CO} = A_1 + E$ (dimension check: $1+2=3$ ✓). In $C_{3v}$, $A_1$ transforms as $z$ (IR-active) and $E$ transforms as $(x,y)$ (IR-active, doubly degenerate). Both bands are IR-active: fac gives two IR carbonyl bands (one non-degenerate, one doubly degenerate, observed as a single peak position but full intensity of two modes). mer isomer, $C_{2v}$. Now the three CO ligands sit at two mutually trans positions plus one position trans to an X ligand; the $C_2$ axis passes through the unique CO and the trans X. Under $E$: all 3 CO bonds unmoved, $\chi=3$. Under $C_2$: the unique CO (on the axis) is unmoved; the other two COs are trans to each other and swapped by the $C_2$ rotation (through the equatorial plane) — so only 1 is unmoved, $\chi=1$. Under $\sigma_v$ (the plane containing the unique CO and both trans COs — since in mer geometry there is a mirror plane containing all three CO ligands): all 3 unmoved, $\chi=3$. Under $\sigma_v'$ (perpendicular plane, containing only the unique CO and bisecting the other two): 1 unmoved, $\chi=1$. So

Reducing with the $C_{2v}$ table ($h=4$):

So $\Gamma_{CO} = 2A_1 + B_1$ (dimension $2+1=3$ ✓). Both $A_1$ (transforms as $z$) and $B_1$ (transforms as $x$) are IR-active, giving three distinct IR carbonyl bands for mer (two $A_1$ + one $B_1$, all non-degenerate, all at different frequencies since force constants along different local symmetry coordinates differ). The clean, testable prediction — two CO stretching bands for fac, three for mer — is precisely the criterion used routinely in synthetic inorganic chemistry (e.g., for $[Mo(CO)_3(NCMe)_3]$-type precursors or Re(I) tricarbonyl photophysics complexes) to assign geometric isomers directly from a solution IR spectrum, without needing a crystal structure.

The systematic use of group theory to classify molecular vibrations grew out of the broader introduction of representation theory into quantum chemistry in the 1930s, most closely associated with E. Bright Wilson Jr., whose 1934–1955 work (culminating in the classic textbook Molecular Vibrations, co-authored with Decius and Cross, 1955) established the systematic $GF$-matrix method and the symmetry-based classification of normal modes still taught today. Wilson's insight was to separate the purely kinematic/symmetry question — how many modes of each symmetry type exist, and which are IR/Raman active — from the dynamical question of their actual frequencies, which requires force constants (the $F$ matrix) and atomic masses (the $G$ matrix). This article covers exactly the symmetry half of that separation; the frequencies themselves are a separate, numerical problem.