The Rule of Mutual Exclusion

Infrared absorption and Raman scattering probe vibrations through two physically different mechanisms, covered separately when we discussed group frequencies and the Raman scattering process. Infrared absorption requires that the molecular dipole moment $\mu$ change as the nuclei move along a normal coordinate $Q$. Raman scattering requires that the polarizability $\alpha$, the ease with which an applied electric field induces a dipole, change along $Q$. Both are derivatives evaluated at the equilibrium geometry: for IR, the requirement is a nonzero $\partial\mu/\partial Q$; for Raman, a nonzero $\partial\alpha/\partial Q$. In a molecule that possesses a center of inversion, every atom at position $\mathbf{r}$ has a symmetry-equivalent atom at $-\mathbf{r}$. This single structural feature has a dramatic consequence for these two derivatives. The dipole moment components ($x$, $y$, $z$) are vectors: under inversion, a vector reverses sign. The polarizability components (such as $x^2$, $y^2$, $xy$) are quadratic in the coordinates: under inversion, a product of two sign changes cancels, leaving the component unchanged. This is exactly the split between ungerade (odd, "u") and gerade (even, "g") behavior familiar from atomic orbital labels like $\sigma_g$ and $\sigma_u$. A normal mode is IR-active only if it has the same symmetry as $x$, $y$, or $z$ — necessarily ungerade. A normal mode is Raman-active only if it has the same symmetry as one of the quadratic functions — necessarily gerade. Since a mode belongs to exactly one irreducible representation of the point group, and gerade and ungerade representations are disjoint sets, a mode cannot be both. That is the entire rule, compressed into one sentence of group theory.

The second useful picture is the actual normal modes of carbon dioxide, since they are the textbook proof of the rule and worth seeing explicitly, alongside the acetylene-type stretching pair that shows the same split in a different centrosymmetric skeleton.

where: $\hat\mu$ transforms as $x,y,z$ (ungerade), and the integral is the IR transition dipole.

where: $\hat\alpha$ transforms as quadratic functions $x^2, y^2, z^2, xy, xz, yz$ (gerade).

where: in any point group containing $i$, every irreducible representation is either purely gerade or purely ungerade.

where: this empty intersection is the rule of mutual exclusion itself.

Example 1 — Carbon dioxide, confirming the rule from first principles. CO2 is linear, symmetric, point group $D_{\infty h}$, with the carbon atom sitting exactly at the center of inversion. It has $3N - 5 = 3(3) - 5 = 4$ vibrational degrees of freedom (linear molecule, $N=3$), consisting of one symmetric stretch, one asymmetric stretch, and one doubly degenerate bend. Symmetry labels from the $D_{\infty h}$ character table: symmetric stretch transforms as $\Sigma_g^+$ (gerade), asymmetric stretch as $\Sigma_u^+$ (ungerade), bend as $\Pi_u$ (ungerade, doubly degenerate, accounting for the "4" vibrations as $1+1+2$). Apply the rule mode by mode:

• Symmetric stretch, $\Sigma_g^+$: gerade, so it can be Raman-active and must be IR-silent. Experimentally it appears as a strong, polarized Raman line at 1388 cm$^{-1}$, and is absent from the IR spectrum.
• Asymmetric stretch, $\Sigma_u^+$: ungerade, so it can be IR-active and must be Raman-silent. Experimentally, a strong IR band appears at 2349 cm$^{-1}$ (this is the famous CO2 IR absorption responsible for its role as a greenhouse gas), with no counterpart Raman line.
• Bend, $\Pi_u$: ungerade, IR-active at 667 cm$^{-1}$, Raman-silent.

Check: three distinct fundamental frequencies observed (1388, 2349, 667 cm$^{-1}$), split cleanly between the two techniques with zero overlap, exactly as the rule predicts. Sanity check on degrees of freedom: $1(\Sigma_g^+) + 1(\Sigma_u^+) + 2(\Pi_u) = 4$ vibrational modes, matching $3N-5$.

Example 2 — Distinguishing cis- and trans-1,2-dichloroethylene by IR/Raman overlap. Consider the two structural isomers of 1,2-dichloroethylene, ClHC=CHCl. The task: predict which isomer shows coincident IR and Raman bands, and which does not, then use this to assign an unlabeled spectrum pair. trans-1,2-dichloroethylene has the two Cl atoms and two H atoms arranged so the molecule has a center of inversion at the midpoint of the C=C bond (point group $C_{2h}$, which contains $i$). All $3N-6 = 3(6)-6 = 12$ vibrational modes distribute among the irreducible representations $A_g$, $B_g$ (gerade, Raman-active) and $A_u$, $B_u$ (ungerade, IR-active) of $C_{2h}$. Because $C_{2h}$ contains $i$, the rule of mutual exclusion applies rigorously: every $A_g/B_g$ mode is Raman-only, every $A_u/B_u$ mode is IR-only, and no band should appear at the same frequency in both spectra. cis-1,2-dichloroethylene has the two Cl atoms on the same side; its point group is $C_{2v}$, which does not contain an inversion center. In $C_{2v}$ the relevant representations ($A_1$, $A_2$, $B_1$, $B_2$) are not split into disjoint gerade/ungerade sets, and most modes ($A_1$, $B_1$, $B_2$) are simultaneously IR- and Raman-allowed. The rule of mutual exclusion simply does not apply here, so coincidences are expected and routinely observed. Numeric check using real assignments: the C=C stretch of trans-dichloroethylene is observed by Raman near 1580 cm$^{-1}$ and is IR-inactive (no matching IR band at that frequency, as required for an $A_g$ mode in $C_{2h}$). In contrast, the C=C stretch of the cis isomer appears at a similar frequency (near 1590 cm$^{-1}$) in both IR and Raman spectra, consistent with $C_{2v}$ symmetry allowing dual activity. Diagnostic conclusion: if a laboratory records IR and Raman spectra of an unknown dichloroethylene sample and finds that no vibrational frequency appears in both spectra, the compound is the trans (centrosymmetric) isomer. If several bands coincide, it is the cis isomer. This single comparison settles a structural assignment that would otherwise require full crystallographic data.

The systematic use of symmetry arguments to predict vibrational spectra grew out of the marriage between group theory and molecular spectroscopy in the 1920s and 1930s, following the foundational quantum-mechanical treatments of molecular vibrations. Eugene Wigner's application of group representation theory to quantum mechanics supplied the mathematical selection-rule machinery, while spectroscopists such as Gerhard Herzberg compiled and organized the vast body of experimental infrared and Raman data on small molecules like CO2 and C2H2 that made the rule an everyday working tool rather than an abstract theorem. Herzberg's monographs on molecular spectra and molecular structure, published across the mid-twentieth century, remain the reference point for exactly this kind of gerade/ungerade bookkeeping.

Ask the tutor: How does the Laporte selection rule for electronic transitions extend this same gerade/ungerade logic to UV-Vis spectra of centrosymmetric chromophores?