Molecular vibrations and spectroscopic selection rules via symmetry

Intuition Beginner

A molecule is not a rigid sculpture. Its atoms jiggle around their resting positions, and the patterns of that jiggling are its normal modes of vibration. Each mode has a definite frequency, and when infrared light or a laser hits the molecule, only certain modes light up in the spectrum. The puzzling fact chemists noticed early is that you can predict which modes appear, and in which kind of experiment, before doing any hard calculation at all. The secret is the molecule's symmetry.

Think of a water molecule. It has a left-right mirror symmetry: swap the two hydrogens and you get back the same molecule. Any vibration must respect that symmetry in one of a few allowed ways. It can keep the mirror image identical, or it can flip sign under the mirror. Sorting the jiggling patterns by how they behave under each symmetry move is exactly what group theory does for us, and that sorting is what tells us the spectrum.

So the plan is simple to state. List the symmetry moves of the molecule. Watch how the cloud of all possible atomic displacements transforms under each move. Bookkeeping then splits that cloud into a handful of symmetry types, and each type carries a label that says whether a mode is visible to infrared light, visible to a Raman laser, both, or neither.

Visual Beginner

The picture shows the bent water molecule and the three ways it can vibrate. In the symmetric stretch both bonds lengthen together; in the bend the bond angle opens and closes; in the antisymmetric stretch one bond lengthens while the other shortens. Two of these keep the mirror symmetry, and one flips sign under it. The labels in the side column ($A_1$, $A_1$, $B_2$) are the symmetry names, and the checkmarks record that every one of water's modes shows up in both the infrared and the Raman experiment.

Worked example Beginner

Take water and count its motions. Water has $3$ atoms, and each atom can move in $3$ independent directions, so there are $3\times 3=9$ ways the atoms can be displaced. Of those $9$, three describe the whole molecule sliding through space (translation) and three describe the whole molecule tumbling (rotation). That leaves $9-6=3$ genuine vibrations. So water has exactly $3$ normal modes, matching the three pictures above.

Now count the symmetry moves of water. There are four: doing nothing (call it $E$); rotating by $180$ degrees about the axis through the oxygen ($C_2$); and reflecting through each of the two mirror planes. These four moves form water's symmetry group, named $C_{2v}$.

For each move we ask a single number: how many atoms stay put? Under "do nothing", all $3$ atoms stay. Under the $180$-degree turn, only the oxygen stays (the two hydrogens swap), so $1$. Under the mirror that contains all three atoms, all $3$ stay. Under the other mirror, only the oxygen stays, so $1$.

What this tells us: those four counts ($3,1,3,1$), combined with a fixed weight for each kind of move, are the raw data that the next tier turns into the exact list of symmetry labels for the three modes — no differential equations required.

Check your understanding Beginner

Formal definition Intermediate+

Let a molecule have $N$ atoms with equilibrium positions ${\mathbf{r}}_1,\dots ,{\mathbf{r}}_N$ in ${\mathbb{R}}^3$, invariant as a set under a finite point group $G\le \mathrm{O}(3)$. Attach to each atom its small Cartesian displacement ${\mathbf{u}}_a\in {\mathbb{R}}^3$, and collect these into the $3N$-dimensional configuration space $V_{\mathrm{tot}}={⨁}_{a=1}^N{\mathbb{R}}^3$. A symmetry operation $R\in G$ permutes the atoms (sending atom $a$ to atom ${\sigma }_R(a)$) and rotates each displacement vector by the orthogonal matrix of $R$. The resulting action on $V_{\mathrm{tot}}$ is the total (Cartesian displacement) representation ${\Gamma }_{\mathrm{tot}}:G\to \mathrm{GL}(V_{\mathrm{tot}})$.

The character of ${\Gamma }_{\mathrm{tot}}$ has a geometric closed form. Only atoms fixed by $R$ (those with ${\sigma }_R(a)=a$) contribute to the trace, and each such atom contributes the trace of the $3\times 3$ orthogonal matrix of $R$ acting on its local displacement. For a proper rotation by angle $\theta$ that trace is $1+2\cos \theta$; for an improper operation (rotation-reflection by $\theta$, including reflections at $\theta =0$ and inversion at $\theta =\pi$) it is $-1+2\cos \theta$. Writing $N_{\mathrm{unmoved}}(R)$ for the number of atoms fixed by $R$,

$${\chi }_{\mathrm{tot}}(R)=N_{\mathrm{unmoved}}(R)(\pm 1+2\cos {\theta }_R),$$

with $+$ for proper and $-$ for improper $R$. Because characters are class functions, this need only be evaluated once per conjugacy class. The reduction formula (character orthogonality, 07.01.04) then gives the multiplicity of each irreducible representation ${\Gamma }_i$ of $G$:

$$a_i=\frac{1}{|G|}\sum _{R\in G}\overline{{\chi }_i(R)}{\chi }_{\mathrm{tot}}(R)=\frac{1}{|G|}\sum _{\text{classes }c}|c|\overline{{\chi }_i(c)}{\chi }_{\mathrm{tot}}(c),$$

so that ${\Gamma }_{\mathrm{tot}}={⨁}_i{a}_i{\Gamma }_i$. The vibrational representation is obtained by subtracting the parts carried by rigid translation and rigid rotation:

$${\Gamma }_{\mathrm{vib}}={\Gamma }_{\mathrm{tot}}\ominus {\Gamma }_{\mathrm{trans}}\ominus {\Gamma }_{\mathrm{rot}},$$

where ${\Gamma }_{\mathrm{trans}}$ transforms as the coordinate triple $(x,y,z)$ and ${\Gamma }_{\mathrm{rot}}$ as the axial triple $(R_x,R_y,R_z)$; both are read straight off the right-hand columns of the character table.

A non-example to keep in view: ${\Gamma }_{\mathrm{tot}}$ is not, in general, irreducible, and its decomposition is not the eigenvalue decomposition of the dynamical matrix. Symmetry fixes which irreducible species occur and with what multiplicity; it does not fix the frequencies. Two modes of the same symmetry species can have any frequencies the force field dictates, and degeneracies forced by a two- or three-dimensional irrep ($E$, $T$) are exact, whereas accidental near-degeneracies between different species carry no protection.

Key theorem with proof Intermediate+

The selection rule is a statement about when a transition-moment integral can be nonzero. A radiative transition between vibrational states ${\psi }_i$ and ${\psi }_f$ has amplitude proportional to $⟨{\psi }_f\mid \hat{O}\mid {\psi }_i⟩$, where $\hat{O}$ is the relevant operator: the electric dipole $\hat{\mathbf{\mu }}$, transforming as $(x,y,z)$, for one-photon infrared absorption (the matrix element entering the Fermi golden rule, 12.07.02); the polarizability $\hat{\alpha }$, transforming as the quadratic forms $(x^2,y^2,z^2,xy,xz,yz)$, for Raman scattering.

Theorem (symmetry selection rule). Let $G$ be the molecular point group and let ${\psi }_i,\hat{O},{\psi }_f$ transform under irreducible representations ${\Gamma }_i$, ${\Gamma }_O$, ${\Gamma }_f$ respectively. Then $⟨{\psi }_f\mid \hat{O}\mid {\psi }_i⟩=0$ unless the direct-product representation ${\Gamma }_f\otimes {\Gamma }_O\otimes {\Gamma }_i$ contains the totally symmetric representation ${\Gamma }_1$ (equivalently, unless ${\Gamma }_O\otimes {\Gamma }_i$ contains ${\Gamma }_f$).

Proof. The integral $⟨{\psi }_f\mid \hat{O}\mid {\psi }_i⟩$ is a number, fixed under every $R\in G$ because $G$ is a symmetry of the molecular Hamiltonian and of the integration measure. The integrand transforms under the representation ${\Gamma }_f^{\ast }\otimes {\Gamma }_O\otimes {\Gamma }_i$; for real orthogonal point-group representations ${\Gamma }_f^{\ast }={\Gamma }_f$. Averaging the integrand over the group projects onto its totally symmetric part, and only that part survives integration: applying the projection operator $P_1=\frac{1}{|G|}{\sum }_R\Gamma (R)$ for ${\Gamma }_1$, the value of the integral equals the value of its symmetric projection. The multiplicity of ${\Gamma }_1$ inside ${\Gamma }_f\otimes {\Gamma }_O\otimes {\Gamma }_i$ is, by the reduction formula and ${\chi }_{{\Gamma }_1}\equiv 1$,

$$a_1=\frac{1}{|G|}\sum _R{\chi }_f(R){\chi }_O(R){\chi }_i(R),$$

using ${\chi }_{\Gamma \otimes {\Gamma }^{\prime }}={\chi }_{\Gamma }{\chi }_{{\Gamma }^{\prime }}$ for direct products (07.01.06). If $a_1=0$ the integrand has no symmetric component and the integral vanishes. Folding two factors via ${\chi }_f{\chi }_O={\chi }_{{\Gamma }_f\otimes {\Gamma }_O}$ and second orthogonality, $a_1\ne 0$ is equivalent to ${\Gamma }_f$ appearing in ${\Gamma }_O\otimes {\Gamma }_i$. $■$

For a fundamental transition out of the (totally symmetric) vibrational ground state, ${\Gamma }_i={\Gamma }_1$, so ${\Gamma }_O\otimes {\Gamma }_i={\Gamma }_O$ and the rule collapses to a one-line test: a mode is infrared-active iff its symmetry species appears among $(x,y,z)$, and Raman-active iff it appears among the quadratic forms. This is read directly from the rightmost columns of the character table.

Bridge. The foundational reason a spectrum can be predicted from a four-by-four table is that integration is a projection onto the symmetric subspace, and this is exactly the same orthogonality machinery that produced the reduction multiplicities $a_i$ a moment ago — putting these together, the same sum $\frac{1}{|G|}{\sum }_R\overline{{\chi }_i}\chi$ both decomposes the displacement cloud and tests the dipole integral. This builds toward the worked ${\mathrm{H}}_2\mathrm{O}$ and $\mathrm{C}{\mathrm{O}}_2$ reductions below, and the totally-symmetric-component criterion appears again in the crystal-field subduction of atomic terms; the device that identifies vanishing integrals with absent symmetric components generalises far past vibrations, and is dual to the Wigner-Eckart factorisation of angular-momentum matrix elements. The central insight is that selection rules are a representation-theoretic accounting identity, not a dynamical accident; the bridge is the projection operator $P_1$.

Exercises Intermediate+

Advanced results Master

The character formula ${\chi }_{\mathrm{tot}}(R)=N_{\mathrm{unmoved}}(R)(\pm 1+2\cos {\theta }_R)$ deserves a structural reading. The displacement representation factors as a tensor product of two representations: the permutation representation ${\Gamma }_{\mathrm{perm}}$ on the set of atoms (character $N_{\mathrm{unmoved}}(R)$, the count of fixed points) and the vector representation ${\Gamma }_{\mathrm{vec}}$ of $G\le \mathrm{O}(3)$ on ${\mathbb{R}}^3$ (character $\pm 1+2\cos {\theta }_R$). Thus ${\Gamma }_{\mathrm{tot}}={\Gamma }_{\mathrm{perm}}\otimes {\Gamma }_{\mathrm{vec}}$, and the multiplicativity of characters under direct products recovers the geometric formula at once. The atomic permutation representation is itself the representation of $G$ on functions on the orbit-set of nuclei, so when several symmetry-equivalent atoms exist (the two H of water, the three H of ammonia, the six C of benzene) the orbit structure controls $N_{\mathrm{unmoved}}$.

Symmetry coordinates make the reduction operational. The projection operator for irrep ${\Gamma }_i$,

$$P^{(i)}=\frac{\dim {\Gamma }_i}{|G|}\sum _{R\in G}\overline{{\chi }_i(R)}{\Gamma }_{\mathrm{tot}}(R),$$

applied to any Cartesian displacement produces a vector lying in the ${\Gamma }_i$-isotypic subspace. The image vectors are symmetry-adapted linear combinations (SALCs): block-diagonalising the $3N\times 3N$ dynamical (Hessian) matrix into one block per irreducible species, of size $a_i^{\mathrm{vib}}\times a_i^{\mathrm{vib}}$. This is the rigorous content of the Wilson FG-matrix method: the secular determinant $\det (\mathbf{GF}-\lambda 1)=0$ for $3N-6$ coupled coordinates decouples into independent symmetry blocks, each diagonalised separately, with degeneracies $2$ or $3$ enforced exactly by the dimension of $E$ or $T$ irreps. Symmetry never sets the frequencies, but it determines which coordinates can mix, and therefore the qualitative shape of every mode and the exact pattern of degeneracies.

Overtones and combination bands extend the rule beyond fundamentals. A fundamental excites one quantum of one mode and tests ${\Gamma }_O\otimes {\Gamma }_{\mathrm{vib}}$ against ${\Gamma }_1$; an overtone of a non-degenerate mode places two quanta in the same coordinate, transforming as the symmetric square ${\mathrm{Sym}}^2({\Gamma }_{\mathrm{vib}})$, while a combination band of two distinct modes transforms as their direct product. For a doubly degenerate mode the first overtone decomposes via the symmetric-square character ${\chi }_{{\mathrm{Sym}}^2}(R)=\frac{1}{2}[\chi (R{)}^2+\chi (R^2)]$, which can contain the totally symmetric representation even when the fundamental does not — so symmetry-forbidden fundamentals routinely acquire allowed overtones. This same symmetric-versus-antisymmetric-square bookkeeping is what controls Fermi resonance and the activity of binary combinations.

Synthesis. The foundational reason a single character table predicts an entire vibrational spectrum is that the projection operator $P_1$ identifies "the integral is nonzero" with "the integrand contains ${\Gamma }_1$", and this is exactly the reduction-formula accounting that decomposes ${\Gamma }_{\mathrm{tot}}$: putting these together, decomposition and selection are one computation run twice. The factorisation ${\Gamma }_{\mathrm{tot}}={\Gamma }_{\mathrm{perm}}\otimes {\Gamma }_{\mathrm{vec}}$ generalises the geometric character formula to any orbit structure, the symmetric-square device generalises the rule to overtones, and the SALC projection is dual to the FG-matrix block-diagonalisation — three faces of one orthogonality relation. The central insight is that the whole apparatus, fundamentals through overtones through degeneracy patterns, is the great orthogonality theorem applied to the molecule's own geometry; this pattern recurs whenever a finite symmetry acts on a physical state space, and identifies spectroscopy with the harmonic analysis of a finite group.

Full proof set Master

Proposition (geometric character of the displacement representation). Let $G\le \mathrm{O}(3)$ act on the $3N$-dimensional Cartesian displacement space $V_{\mathrm{tot}}$ of an $N$-atom molecule. For $R\in G$ realised as a proper rotation by angle $\theta$ or an improper rotation-reflection by angle $\theta$,

$${\chi }_{\mathrm{tot}}(R)=N_{\mathrm{unmoved}}(R)(\pm 1+2\cos \theta ),$$

the sign being $+$ for proper and $-$ for improper $R$.

Proof. Order the basis of $V_{\mathrm{tot}}$ by atom, $({\mathbf{u}}_1,\dots ,{\mathbf{u}}_N)$, each ${\mathbf{u}}_a\in {\mathbb{R}}^3$. In this basis ${\Gamma }_{\mathrm{tot}}(R)$ is a block matrix: the block in row ${\sigma }_R(a)$, column $a$ is the $3\times 3$ orthogonal matrix $D(R)$ representing $R$ on ${\mathbb{R}}^3$, and all other blocks vanish, because $R$ sends the displacement of atom $a$ to a rotated displacement of atom ${\sigma }_R(a)$. The trace receives a contribution only from diagonal blocks, i.e. from atoms with ${\sigma }_R(a)=a$. Each such fixed atom contributes $\mathrm{tr}D(R)$. Hence ${\chi }_{\mathrm{tot}}(R)=N_{\mathrm{unmoved}}(R)\mathrm{tr}D(R)$.

It remains to compute $\mathrm{tr}D(R)$. Conjugating into a standard frame, a proper rotation by $\theta$ about a chosen axis has matrix $$ D(R) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \ \sin\theta & \cos\theta & 0 \ 0 & 0 & 1 \end{pmatrix},\qquad \operatorname{tr} D(R) = 1 + 2\cos\theta. $$ An improper operation $S_{\theta }$ is such a rotation composed with reflection in the plane normal to the axis, multiplying the $(3,3)$ entry by $-1$, giving $\mathrm{tr}D(S_{\theta })=-1+2\cos \theta$. The trace is basis-independent, so these values hold in the molecular frame. Special cases: reflection $\sigma$ is $S_{\theta }$ at $\theta =0$, trace $-1+2=1$; inversion $i$ is $S_{\theta }$ at $\theta =\pi$, trace $-1-2=-3$; the identity $E$ is a rotation at $\theta =0$, trace $3$. Substituting back yields the stated formula. $■$

Proposition (translations and rotations as fixed character columns). Under any point group $G\le \mathrm{O}(3)$, the rigid translations of the molecule span a representation with the same character as the vector representation ${\Gamma }_{\mathrm{vec}}$ (character $\pm 1+2\cos {\theta }_R$), and the rigid rotations span the axial-vector (pseudovector) representation with character $+1+2\cos {\theta }_R-2{\delta }_{\mathrm{improper}}$ — equivalently ${\Gamma }_{\mathrm{vec}}$ twisted by the determinant.

Proof. A uniform translation displaces every atom by the same vector $\mathbf{t}\in {\mathbb{R}}^3$; under $R$ this vector transforms by $D(R)$, so the three translational coordinates carry exactly ${\Gamma }_{\mathrm{vec}}$, character $\mathrm{tr}D(R)=\pm 1+2\cos \theta$. An infinitesimal rigid rotation is generated by an axial vector $\mathbf{\omega }$ (angular velocity); under an orthogonal transformation $R$ an axial vector transforms by $\det (R)D(R)$, because the cross product defining $\mathbf{\omega }$ picks up the determinant. Its character is $\det (R)\mathrm{tr}D(R)=(\pm 1)(\pm 1+2\cos \theta )$, which equals $+1+2\cos \theta$ for proper $R$ ($\det =+1$) and $-(-1+2\cos \theta )=1-2\cos \theta$ for improper $R$ ($\det =-1$). These are precisely the characters tabulated against $(x,y,z)$ and $(R_x,R_y,R_z)$ in any character table, justifying their subtraction in ${\Gamma }_{\mathrm{vib}}={\Gamma }_{\mathrm{tot}}\ominus {\Gamma }_{\mathrm{trans}}\ominus {\Gamma }_{\mathrm{rot}}$. $■$

These two propositions together make the reduction algorithmic: compute ${\chi }_{\mathrm{tot}}$ from the first, subtract the two known characters from the second, apply the reduction formula, and read activity off the coordinate and quadratic columns.

Connections Master

The mechanical normal-mode theory of 05.00.11 supplies the dynamical content this unit organises by symmetry: the eigenvalue problem $\det (\mathbf{V}-{\omega }^2\mathbf{T})=0$ for small oscillations is exactly the secular equation that the projection operators block-diagonalise here, so symmetry pre-sorts the eigenvectors before any diagonalisation. This unit is the representation-theoretic refinement of that purely mechanical account.

The chemistry-side treatment of point groups and character tables in 16.02.01 is the prerequisite vocabulary — Schoenflies symbols, Mulliken irrep labels, the assignment of a molecule to its group — while the present unit supplies the math-side derivation of the $3N$ reduction and the selection rule that 16.02.01 states. The dependency that previously ran only chemistry-to-chemistry is here anchored back into representation theory through 07.01.04 and 07.01.06.

The Fermi golden rule of 12.07.02 is where the transition-moment integral $⟨{\psi }_f|\hat{O}|{\psi }_i⟩$ acquires its physical meaning as a transition rate; the symmetry selection rule proved here is the statement of when that rate is forced to vanish, and the same totally-symmetric-component criterion reappears in the Wigner-Eckart factorisation of 12.15.02 for rotational and electronic matrix elements. The character-orthogonality engine of 07.01.04 is the common foundation of all three.

Historical & philosophical context Master

The use of group representations to predict spectra began with Wigner's 1931 monograph ^{[Wigner 1931]}, which established that symmetry operations act on quantum states by (anti)unitary operators and that selection rules are statements about when a matrix element is forced to vanish by symmetry. Bethe's 1929 paper on the splitting of atomic terms in crystals ^{[Bethe 1929]} supplied the companion technique of subducing rotation-group representations to a finite point group, the electronic analogue of the vibrational reduction. The systematic application to molecular vibrations was codified by Wilson, Decius and Cross in Molecular Vibrations ^{[Wilson 1955]}, whose FG-matrix method and symmetry-coordinate construction remain the working spectroscopist's standard. Sternberg's Group Theory and Physics ^{[Sternberg 1994]} places the same material inside a unified representation-theoretic narrative spanning finite, compact, and non-compact groups, framing the periodic table, vibrational spectra, and particle classification as instances of one idea. The pedagogical line from Wigner through Tinkham ^{[Tinkham 1964]} to Cotton ^{[Cotton 1990]} and Bishop transmitted the character-table technique from physicists to chemists, where it became routine laboratory practice for assigning infrared and Raman bands.

Bibliography Master

@book{Sternberg1994,
  author    = {Sternberg, Shlomo},
  title     = {Group Theory and Physics},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {1994},
  isbn      = {0-521-24870-1}
}

@book{WilsonDeciusCross1955,
  author    = {Wilson, E. Bright and Decius, J. C. and Cross, Paul C.},
  title     = {Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra},
  publisher = {McGraw-Hill},
  address   = {New York},
  year      = {1955},
  note      = {Dover reprint, 1980}
}

@book{Wigner1931,
  author    = {Wigner, Eugene P.},
  title     = {Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren},
  publisher = {Vieweg},
  address   = {Braunschweig},
  year      = {1931},
  note      = {English translation: Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959}
}

@article{Bethe1929,
  author  = {Bethe, Hans},
  title   = {Termaufspaltung in Kristallen},
  journal = {Annalen der Physik},
  volume  = {395},
  number  = {2},
  pages   = {133--208},
  year    = {1929},
  doi     = {10.1002/andp.19293950202}
}

@book{Tinkham1964,
  author    = {Tinkham, Michael},
  title     = {Group Theory and Quantum Mechanics},
  publisher = {McGraw-Hill},
  address   = {New York},
  year      = {1964},
  note      = {Dover reprint, 2003}
}

@book{Cotton1990,
  author    = {Cotton, F. Albert},
  title     = {Chemical Applications of Group Theory},
  edition   = {3rd},
  publisher = {Wiley},
  address   = {New York},
  year      = {1990},
  isbn      = {0-471-51094-7}
}

@book{Bishop1993,
  author    = {Bishop, David M.},
  title     = {Group Theory and Chemistry},
  publisher = {Dover},
  address   = {New York},
  year      = {1993},
  isbn      = {0-486-67355-3}
}