14.12.03 · genchem-pchem / spectroscopy

Vibrational spectroscopy: harmonic oscillator selection rules, anharmonicity, and IR-active modes

shipped3 tiersLean: nonepending prereqs

Anchor (Master): Herzberg — Infrared and Raman Spectra (1945)

Intuition Beginner

Chemical bonds act like springs. When a bond stretches and compresses, it vibrates at a frequency that depends on two things: how stiff the bond is (the force constant) and how heavy the atoms are (the reduced mass). A stiff, light bond like H-F vibrates fast — its stretching frequency is about 4000 inverse centimetres. A softer, heavier bond like C-Cl vibrates slowly — about 700 inverse centimetres. Infrared light carries exactly the right energies to set bonds vibrating, so when IR light passes through a sample, the frequencies that are absorbed map directly onto the bond-stretching and bond-bending motions present in the molecule.

This is why IR spectroscopy is the chemist's workhorse for identifying functional groups. The C=O stretch always absorbs near 1700 inverse centimetres, whether the carbonyl is in an aldehyde, a ketone, an ester, or a carboxylic acid. The O-H stretch always appears near 3400 inverse centimetres, broadened by hydrogen bonding. A C-H stretch near 3000 inverse centimetres signals the presence of hydrogen atoms bonded to carbon. By reading the positions, shapes, and intensities of the absorption bands, a chemist identifies which bonds and functional groups are present — often within minutes and with a sample of less than a milligram.

Not every vibration absorbs IR light. The selection rule is simple: a vibration is IR-active only if it changes the molecular dipole moment. A homonuclear diatomic like H $_{2}$ or N $_{2}$ has no dipole moment and stretching the bond does not create one, so these molecules are IR-silent. A heteronuclear diatomic like HCl has a dipole that changes as the bond stretches, so HCl absorbs IR strongly. For polyatomics, the criterion is applied mode by mode: each of the $3 N - 6$ normal modes of a molecule with $N$ atoms either changes the dipole moment (IR-active) or does not (IR-silent).

Visual Beginner

A diatomic molecule vibrating along its bond axis traces out a potential-energy curve. The harmonic approximation treats this curve as a parabola, with equally spaced vibrational energy levels like the rungs of a ladder. The spacing $ℏ ω$ between rungs is the vibrational frequency, set by the force constant $k$ and reduced mass $μ$ : $ω = k / μ$ .

The harmonic model predicts that only the $v = 0 \to 1$ transition (the fundamental) is allowed. Real molecules are anharmonic — the potential is not a perfect parabola — and this allows weak overtone transitions at roughly $2 ω$ , $3 ω$ , and so on. The fundamental is always the strongest absorption. The anharmonicity also means the energy levels are not exactly equally spaced: the spacing decreases slightly with increasing $v$ , converging to the dissociation limit.

Worked example Beginner

The C-O stretching frequency in carbon monoxide is observed at 2143 inverse centimetres. The reduced mass of CO is $1.139 \times 1 0^{- 26}$ kg. Calculate the force constant of the C-O bond and classify the bond order.

Step 1. The vibrational frequency in wavenumber units is $\tilde{ν} = (1/2 π c) k / μ$ . Rearranging for $k$ :

$k = (2 π c \tilde{ν})^{2} μ = (2 π \times 2.998 \times 1 0^{10} \times 2143)^{2} \times 1.139 \times 1 0^{- 26} = 1902$ N/m.

Step 2. Classify by bond order. Single bonds (C-C, C-O) have force constants around 500 N/m. Double bonds (C=C, C=O) around 1000 N/m. Triple bonds (C $\equiv$ C, C $\equiv$ O) around 1500-2000 N/m. The value of 1902 N/m confirms that CO has a triple-bond character, consistent with its Lewis structure $\equiv$ O: with a formal bond order of 3.

Step 3. The force constant is a measure of bond stiffness. A larger $k$ means a stronger, shorter bond that vibrates at higher frequency. This is why triple bonds absorb at higher wavenumbers than double bonds, and double bonds higher than single bonds — the basis of the group-frequency approach to IR spectroscopy.

Check your understanding Beginner

Exercise (easy, true-false).

The IR absorption frequency of the O-H stretch in ethanol is the same as in water.

Hint

The vibrational frequency depends on the force constant (bond strength) and the reduced mass. Consider how hydrogen bonding and the molecular environment affect the O-H bond.

Answer

False. While both O-H stretches appear in the same general region (3200-3600 cm $^{- 1}$ ), the exact position depends on the molecular environment. Free (non-hydrogen-bonded) O-H stretches appear near 3600 cm $^{- 1}$ as sharp peaks; hydrogen-bonded O-H stretches are red-shifted to 3200-3500 cm $^{- 1}$ and broadened. Water in liquid phase shows a broad hydrogen-bonded O-H stretch, while dilute ethanol in a nonpolar solvent shows a sharper peak closer to 3600 cm $^{- 1}$ . The position and breadth of the O-H band are diagnostic of hydrogen bonding.

Formal definition Intermediate+

The vibrational motion of a diatomic molecule is modelled by the quantum harmonic oscillator. The Hamiltonian for a diatomic with reduced mass $μ$ and force constant $k$ is

\hat{H}_{vib} = \frac{p ^ ^{2}}{2 μ} + \frac{1}{2} k \overset{x}{^}^{2},

where $x = r - r_{e}$ is the displacement from equilibrium. The energy eigenvalues are

E_{v} = ℏ ω (v + \frac{1}{2}), v = 0, 1, 2, \dots, ω = \frac{k}{μ} .

The zero-point energy $E_{0} = ℏ ω /2$ means the molecule never stops vibrating. In wavenumber units the level spacing is $\tilde{ν} = ω / (2 π c) = (1/2 π c) k / μ$ .

Definition (harmonic-oscillator selection rule). The electric-dipole selection rule for the harmonic oscillator is $Δ v = \pm 1$ . This follows from the matrix element $⟨ v^{'} ∣ \overset{x}{^} ∣ v ⟩ = ℏ/ (2 μ ω) [v + 1 δ_{v^{'}, v + 1} + v δ_{v^{'}, v - 1}]$ , which is nonzero only for $v^{'} = v \pm 1$ . The transition is IR-active only if the dipole moment changes during the vibration, i.e., $(d μ / d x) ∣_{x = 0} \neq = 0$ .

Definition (Morse potential and anharmonicity). The Morse potential $V (r) = D_{e} [1 - e^{- a (r - r_{e})}]^{2}$ provides a more realistic model where the potential is asymmetric: steeper than harmonic at short range and flattening toward dissociation at long range. The vibrational energy levels are

E_{v} = ℏ ω_{e} (v + \frac{1}{2}) - ℏ ω_{e} x_{e} (v + \frac{1}{2})^{2},

where $ω_{e} = a 2 D_{e} / μ$ and $x_{e} = ℏ ω_{e} / (4 D_{e})$ is the anharmonicity constant. The negative quadratic term causes the level spacing to decrease with increasing $v$ , and the number of bound states is finite: $v_{max} = ⌊ 1/ (2 x_{e}) - 1/2 ⌋$ .

Definition (IR activity for polyatomic molecules). A molecule with $N$ atoms has $3 N - 6$ normal modes ( $3 N - 5$ for linear molecules). Each normal mode $Q_{k}$ is IR-active if the dipole moment changes along that coordinate: $\partial μ / \partial Q_{k} \neq = 0$ . Group theory determines activity from symmetry: a mode is IR-active if it transforms as the same irreducible representation as one of the translation components ( $x$ , $y$ , or $z$ ) of the molecular point group.

Counterexamples to common slips

Fundamental versus overtone intensity. The $Δ v = \pm 1$ selection rule is exact only for the harmonic oscillator. Anharmonicity relaxes this: overtones ( $Δ v = \pm 2, \pm 3, \dots$ ) are observed but are typically 10 to 100 times weaker than the fundamental. Treating the observed fundamental as "the only band" misses the overtone and combination bands that are diagnostic in gas-phase and matrix-isolation IR spectroscopy.
Number of IR-active modes. A molecule with $3 N - 6$ normal modes does not necessarily show $3 N - 6$ IR peaks. Modes may be degenerate (e.g., the two bending modes of CO $_{2}$ are degenerate at 667 cm $^{- 1}$ ), IR-inactive by symmetry, or too weak to observe. The number of observed IR bands is typically less than $3 N - 6$ .
Force constant versus bond energy. A large force constant means a stiff bond, not necessarily a strong one. The C-H force constant (about 500 N/m) is similar to the C-C force constant, but the C-H bond energy (413 kJ/mol) is much larger than the C-C bond energy (348 kJ/mol). Force constants and bond dissociation energies probe different aspects of the potential-energy surface.

Key result Intermediate+

Theorem (normal-mode decomposition and spectroscopic activity). Let a molecule with $N$ atoms have a potential energy $V$ expanded about the equilibrium geometry as $V = V_{0} + \frac{1}{2} \sum_{ij} F_{ij} ξ_{i} ξ_{j}$ , where $ξ_{i}$ are mass-weighted displacement coordinates and $F_{ij}$ is the mass-weighted Hessian. Diagonalising $F$ gives eigenvalues $λ_{k} = ω_{k}^{2}$ and eigenvectors $L_{k}$ (the normal modes). The translational and rotational eigenvalues are zero; the remaining $3 N - 6$ (or $3 N - 5$ ) positive eigenvalues give the normal-mode frequencies. A mode $Q_{k}$ is IR-active if and only if the transition dipole moment $⟨ 1_{k} ∣ \hat{μ} ∣ 0_{k} ⟩$ is nonzero, which is equivalent to $\partial μ / \partial Q_{k} \neq = 0$ .

Proof. The molecular potential energy surface near equilibrium is a quadratic form in the $3 N$ nuclear displacement coordinates. Removing the six (or five) zero-frequency translational and rotational degrees of freedom by projecting onto the vibrational subspace leaves a $(3 N - 6) \times (3 N - 6)$ symmetric positive-definite matrix. The spectral theorem guarantees a real orthogonal diagonalisation $F = L Λ L^{T}$ , where $Λ = diag (λ_{1}, \dots, λ_{3 N - 6})$ with $λ_{k} > 0$ . The normal-mode coordinates are $Q_{k} = \sum_{i} L_{ik} ξ_{i}$ , and the Hamiltonian separates into $3 N - 6$ independent harmonic oscillators: $H = \sum_{k} [\overset{p}{^}_{k}^{2} /2 + ω_{k}^{2} Q_{k}^{2} /2]$ . The dipole operator expanded to first order in the normal coordinates is $μ = μ_{0} + \sum_{k} (\partial μ / \partial Q_{k}) Q_{k} + \dots$ . The first term ( $μ_{0}$ , the permanent dipole) does not couple different vibrational states. The second term gives transition dipole moments $⟨ 1_{k} ∣ Q_{k} ∣ 0_{k} ⟩ (\partial μ / \partial Q_{k})$ , which are nonzero if and only if $\partial μ / \partial Q_{k} \neq = 0$ . This is the IR-activity criterion. $□$

Bridge. This result connects the purely mathematical diagonalisation of the Hessian matrix to the observable IR spectrum. The normal-mode analysis predicts how many IR-active bands a molecule should have and at what frequencies. The connection to group theory (14.02.01) is that the irreducible representations of the molecular point group determine which normal modes have nonzero $\partial μ / \partial Q_{k}$ , providing a rapid symmetry-based prediction of the IR spectrum without explicit matrix diagonalisation.

Worked example at intermediate level

Determine the number of IR-active normal modes for formaldehyde (H $_{2}$ CO, point group $C_{2 v}$ ) and predict the symmetry species of each mode.

Formaldehyde has 4 atoms, so $3 N - 6 = 6$ normal modes. The reducible representation of the $3 N$ displacements decomposes under $C_{2 v}$ as $Γ_{3 N} = 4 A_{1} + 1 A_{2} + 3 B_{1} + 4 B_{2}$ . Subtracting translations ( $A_{1} + B_{1} + B_{2}$ for $z, x, y$ ) and rotations ( $A_{2} + B_{1} + B_{2}$ for $R_{z}, R_{y}, R_{x}$ ) gives the vibrational modes:

Γ_{vib} = 3 A_{1} + 1 A_{2} + 1 B_{1} + 1 B_{2} .

In $C_{2 v}$ , the translation components $x$ , $y$ , $z$ transform as $B_{1}$ , $B_{2}$ , $A_{1}$ respectively. All four symmetry species of the vibrations ( $A_{1}$ , $A_{2}$ , $B_{1}$ , $B_{2}$ ) match at least one translation component except $A_{2}$ . Therefore: five IR-active modes ( $3 A_{1} + 1 B_{1} + 1 B_{2}$ ) and one IR-inactive mode ( $A_{2}$ ). The $A_{2}$ mode (a twisting motion) is Raman-active but IR-silent. Experimentally, formaldehyde shows five strong IR absorption bands, consistent with this prediction.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

The vibrational frequency of HCl is 2886 cm $^{- 1}$ and that of DCl is 2091 cm $^{- 1}$ . Show that this ratio is approximately $μ_{DCl} / μ_{HCl}$ , confirming that the force constant is unchanged by isotopic substitution.

Hint

$ω \propto k / μ$ . If $k$ is the same, the frequency ratio is $μ_{1} / μ_{2}$ .

Answer

$μ_{HCl} = (1 \times 35.45) /36.45 = 0.9725$ amu. $μ_{DCl} = (2 \times 35.45) /37.45 = 1.892$ amu. Predicted ratio: $μ_{DCl} / μ_{HCl} = 1.892/0.9725 = 1.945 = 1.395$ .

Observed ratio: $2886/2091 = 1.380$ .

The agreement is within 1 percent. The small discrepancy arises because the observed frequencies include anharmonic corrections, and the anharmonicity constant $x_{e}$ differs slightly between HCl and DCl (the zero-point amplitude is different, so the effective $x_{e}$ changes). The harmonic prediction is recovered by using the harmonic frequencies $\tilde{ω}_{e}$ instead of the observed fundamentals.

Exercise 5 (medium, symbolic).

CO $_{2}$ (linear, $D_{\infty h}$ ) has four normal modes. Determine which modes are IR-active and which are Raman-active, and verify the mutual-exclusion principle.

Hint

The four modes are: symmetric stretch ( $σ_{g}^{+}$ ), antisymmetric stretch ( $σ_{u}^{+}$ ), and two degenerate bends ( $π_{u}$ ). IR-active modes must transform as the translation components ( $x, y$ or $π_{u}$ ; $z$ or $σ_{u}^{+}$ ). Raman-active modes must transform as components of the polarisability tensor ( $x^{2}, y^{2}, z^{2}$ etc.).

Answer

The four normal modes and their symmetries:

Symmetric stretch $ν_{1}$ : $σ_{g}^{+}$ (both C=O bonds stretch in phase). The dipole moment remains zero throughout: IR-inactive. The polarisability changes: Raman-active.
Antisymmetric stretch $ν_{3}$ : $σ_{u}^{+}$ (one C=O stretches while the other compresses). The dipole moment oscillates: IR-active. The polarisability change averages out: Raman-inactive.
Two degenerate bending modes $ν_{2}$ : $π_{u}$ (the molecule bends in either of two perpendicular planes). The dipole moment oscillates perpendicular to the molecular axis: IR-active. Raman-inactive.

The mutual-exclusion principle is satisfied: $σ_{g}^{+}$ is Raman-only; $σ_{u}^{+}$ and $π_{u}$ are IR-only. No mode is both IR and Raman active. This is why CO $_{2}$ has no dipole moment but still shows IR absorption bands — the antisymmetric stretch and bends generate a dynamic dipole even though the static dipole is zero.

Exercise 6 (hard, symbolic).

The intensity ratio of the first overtone ( $0 \to 2$ ) to the fundamental ( $0 \to 1$ ) in a Morse oscillator can be estimated by expanding the dipole moment as $μ (x) = μ_{0} + μ_{1} x + μ_{2} x^{2} + \dots$ and evaluating the matrix elements in the harmonic basis. Show that $I_{02} / I_{01} \approx (μ_{2} / μ_{1})^{2} \times (ℏ/ (2 μ ω))$ and estimate this ratio for HCl given that $μ_{2} / μ_{1} \approx 1$ angstrom $^{- 1}$ .

Hint

In the harmonic basis, $⟨ 2∣ x^{2} ∣0 ⟩ = (ℏ/ (2 μ ω)) (1 + 2 \times 0 + 1) = 2 ℏ/ (2 μ ω)$ and $⟨ 1∣ x ∣0 ⟩ = ℏ/ (2 μ ω)$ . Compute the ratio of squared matrix elements.

Answer

The transition dipole for $0 \to 1$ : $M_{01} = μ_{1} ⟨ 1∣ x ∣0 ⟩ = μ_{1} ℏ/ (2 μ ω)$ .

The transition dipole for $0 \to 2$ : $M_{02} = μ_{2} ⟨ 2∣ x^{2} ∣0 ⟩ = μ_{2} 2 ℏ/ (2 μ ω)$ (using the harmonic oscillator matrix element $⟨ 2∣ x^{2} ∣0 ⟩ = 2 ℏ/ (2 μ ω)$ ).

The intensity ratio (proportional to $∣ M ∣^{2}$ ):

$I_{02} / I_{01} = ∣ M_{02} / M_{01} ∣^{2} = \frac{2 μ _{2}^{2} ( ℏ/ ( 2 μ ω ) ) ^{2}}{μ _{1}^{2} ℏ/ ( 2 μ ω )} = 2 (μ_{2} / μ_{1})^{2} ℏ/ (2 μ ω) = (μ_{2} / μ_{1})^{2} ℏ/ (μ ω)$ .

For HCl: $ℏ/ (μ ω) = ℏ/ (μ \times 2 π c \times 2886)$ . With $μ = 1.627 \times 1 0^{- 27}$ kg and $ω = 2 π \times 2.998 \times 1 0^{10} \times 2886 = 5.44 \times 1 0^{14}$ rad/s: $ℏ/ (μ ω) = 1.055 \times 1 0^{- 34} / (1.627 \times 1 0^{- 27} \times 5.44 \times 1 0^{14}) = 1.19 \times 1 0^{- 22}$ m $^{2}$ .

With $μ_{2} / μ_{1} \approx 1 0^{10}$ m $^{- 1}$ (1 angstrom $^{- 1}$ ): $I_{02} / I_{01} \approx (1 0^{10})^{2} \times 1.19 \times 1 0^{- 22} = 1.19 \times 1 0^{- 2}$ . The overtone is about 1 percent of the fundamental intensity, consistent with the experimental observation of overtone bands at 1-2 percent of the fundamental in HCl.

The Morse potential, dissociation, and vibrational thermodynamics Master

The harmonic oscillator is the starting point for understanding molecular vibrations, but real bonds are anharmonic. The Morse potential $V (r) = D_{e} [1 - e^{- a (r - r_{e})}]^{2}$ captures the essential physics: the potential is approximately quadratic near equilibrium, but it flattens toward the dissociation energy $D_{e}$ as the bond is stretched, and it rises steeply for compressed bond lengths. The Schrodinger equation for the Morse potential can be solved exactly, yielding the energy levels

E_{v} = h c \tilde{ω}_{e} (v + 1/2) - h c \tilde{ω}_{e} x_{e} (v + 1/2)^{2},

where $\tilde{ω}_{e} = (a /2 π c) 2 D_{e} / μ$ is the harmonic frequency and $x_{e} = h c \tilde{ω}_{e} / (4 D_{e})$ is the anharmonicity. The parameter $a = \tilde{ω}_{e} 2 π^{2} c μ / (D_{e} h)$ sets the width of the potential well. The Morse model has three experimentally determined parameters ( $\tilde{ω}_{e}$ , $x_{e}$ , and $r_{e}$ ) compared with two for the harmonic oscillator ( $\tilde{ω}$ and $r_{e}$ ), and the extra parameter quantifies the departure from harmonicity.

The Morse potential predicts three observable phenomena absent from the harmonic model. First, the decreasing level spacing produces a convergence of overtone frequencies: the $0 \to 1$ fundamental is at $\tilde{ω}_{e} (1 - 2 x_{e})$ , the $0 \to 2$ first overtone is at $2 \tilde{ω}_{e} (1 - 3 x_{e})$ , the $0 \to 3$ second overtone is at $3 \tilde{ω}_{e} (1 - 4 x_{e})$ , and so on. The overtones are not exact integer multiples of the fundamental — each successive overtone is slightly shifted to lower frequency by the cumulative anharmonic correction. Measuring the positions of several overtones determines both $\tilde{ω}_{e}$ and $x_{e}$ simultaneously.

Second, the dissociation energy can be extracted from the spectroscopic constants. Setting $E_{v_{max}} = D_{e}$ and solving gives $D_{e} = \tilde{ω}_{e}^{2} / (4 \tilde{ω}_{e} x_{e}) = \tilde{ω}_{e} / (4 x_{e})$ in wavenumber units. For H $_{2}$ , $\tilde{ω}_{e} = 4401$ cm $^{- 1}$ and $x_{e} = 0.0276$ , giving $D_{e} = 4401/ (4 \times 0.0276) = 39, 862$ cm $^{- 1}$ = 4.95 eV, in excellent agreement with the directly measured dissociation energy of 4.48 eV (the small difference arises from the difference between $D_{e}$ and $D_{0}$ ). The Birge-Sponer extrapolation — plotting $Δ E_{v}$ against $v$ and extrapolating the straight line to $Δ E = 0$ — is the standard graphical method for extracting $D_{e}$ from vibrational spectra.

Third, the finite number of bound states means that at sufficiently high excitation the molecule dissociates. This connects vibrational spectroscopy to chemical kinetics: photodissociation thresholds correspond to reaching $v_{max}$ , and the dissociation products carry the excess energy as translational kinetic energy. Predissociation — where the vibrational level lies above a dissociation asymptote but the molecule is temporarily trapped behind a small barrier — produces line broadening in the vibrational spectrum, providing a spectroscopic window onto dissociation dynamics.

Group frequencies and the fingerprint region Master

The normal-mode analysis shows that each molecule has a unique set of vibrational frequencies. In practice, however, certain vibrational frequencies are remarkably transferable across different molecules: the C=O stretch appears near 1700 cm $^{- 1}$ whether the carbonyl is in formaldehyde, acetone, acetic acid, or a peptide bond. This transferability arises because the C=O stretching mode is largely localised to the C=O group — the normal-mode displacement pattern concentrates most of the motion on the carbonyl carbon and oxygen, with only small participation from the rest of the molecule.

This localisation is the basis of group-frequency analysis. The IR spectrum is divided into two regions. The functional-group region (4000-1500 cm $^{- 1}$ ) contains stretching vibrations that are diagnostic of specific bonds: O-H and N-H stretches (3200-3600 cm $^{- 1}$ ), C-H stretches (2800-3100 cm $^{- 1}$ ), C $\equiv$ N and C $\equiv$ C stretches (2100-2260 cm $^{- 1}$ ), C=O stretches (1650-1800 cm $^{- 1}$ ), C=C stretches (1500-1680 cm $^{- 1}$ ). The fingerprint region (below 1500 cm $^{- 1}$ ) contains C-C, C-O, C-N, and C-X single-bond stretches, bending modes, and coupled motions that are specific to the entire molecular skeleton. Two molecules with the same functional groups but different connectivity show the same bands in the functional-group region but different patterns in the fingerprint region.

The positions of group frequencies shift systematically with the electronic environment. A carbonyl adjacent to an electron-withdrawing group is strengthened (the C=O bond has more double-bond character), shifting the stretch to higher wavenumber: acid chlorides (1810 cm $^{- 1}$ ), esters (1735 cm $^{- 1}$ ), ketones (1715 cm $^{- 1}$ ), amides (1680 cm $^{- 1}$ ). Conjugation with a C=C double bond weakens the C=O by delocalising the pi electrons, shifting the stretch to lower wavenumber: conjugated ketones absorb near 1685 cm $^{- 1}$ . Hydrogen bonding to the carbonyl oxygen further weakens the bond, producing additional red shifts and peak broadening. These systematic trends, first compiled by Colthup ^{[Colthup1950]}, form the basis of modern IR spectral interpretation.

Raman spectroscopy: the complementary technique Master

Raman spectroscopy probes molecular vibrations through inelastic light scattering rather than absorption. When monochromatic light of frequency $ν_{0}$ illuminates a sample, most scattered photons have the same frequency (Rayleigh scattering), but a small fraction ( $1 0^{- 6}$ to $1 0^{- 8}$ ) are shifted in frequency by amounts equal to molecular vibrational frequencies. Stokes-shifted photons (frequency $ν_{0} - ν_{vib}$ ) correspond to vibrational excitation; anti-Stokes photons (frequency $ν_{0} + ν_{vib}$ ) correspond to vibrational de-excitation. The Stokes lines dominate at room temperature because most molecules are in the vibrational ground state.

The selection rule for Raman activity is different from IR activity: a vibration is Raman-active if the molecular polarisability changes during the vibration ( $\partial α / \partial Q \neq = 0$ ). For centrosymmetric molecules, the mutual-exclusion principle applies: modes that are IR-active are Raman-inactive and vice versa. Combining IR and Raman spectroscopy of a centrosymmetric molecule recovers the complete vibrational spectrum. Even for non-centrosymmetric molecules, the two techniques have different sensitivities: symmetric stretches (which strongly modulate the polarisability) are Raman-strong but often IR-weak, while asymmetric stretches (which strongly modulate the dipole moment) are IR-strong but Raman-weak.

The Raman effect was predicted theoretically by Smekal in 1923 and discovered experimentally by Raman and Krishnan in 1928 ^[Raman1928], earning Raman the 1930 Nobel Prize in Physics. Modern Raman spectroscopy uses laser excitation and CCD detection to record spectra with high signal-to-noise. Resonance Raman spectroscopy — where the laser frequency matches an electronic transition — enhances Raman scattering by factors of $1 0^{3}$ to $1 0^{6}$ for modes coupled to the chromophore, providing selective enhancement of specific vibrational modes. Surface-enhanced Raman scattering (SERS) on rough metal surfaces amplifies signals by $1 0^{6}$ to $1 0^{14}$ , enabling detection of single molecules. Coherent anti-Stokes Raman scattering (CARS) microscopy provides label-free vibrational imaging of biological tissue with subcellular resolution.

Connections Master

Rotational spectroscopy 14.12.02. Each vibrational transition carries a rotational fine structure: the P-branch ( $Δ J = - 1$ ) and R-branch ( $Δ J = + 1$ ) bracket the vibrational band origin, and the Q-branch ( $Δ J = 0$ ) appears when the electronic or vibrational angular momentum permits. The rotational constants $B_{0}$ and $B_{1}$ of the ground and excited vibrational states are extracted from the P/R branch line positions, connecting the vibrational and rotational analyses.
Electronic spectroscopy 14.12.04. Electronic transitions produce vibronic bands whose spacing equals the vibrational frequency of the excited electronic state. The Franck-Condon principle governs which vibrational levels of the excited state are populated, and the resulting intensity pattern encodes the displacement between ground and excited-state potential-energy surfaces. Understanding vibrational spectroscopy is a prerequisite for interpreting the vibronic structure of UV-Vis spectra.
Statistical mechanics 14.07.02. The vibrational partition function $q_{vib} = 1/ [1 - exp (- h ν / k_{B} T)]$ per mode determines the vibrational contributions to heat capacity, entropy, and Gibbs free energy. The vibrational frequencies measured by IR spectroscopy feed directly into thermodynamic calculations via the partition function. At low temperature, $q_{vib} \to 1$ (all molecules in $v = 0$ ); at high temperature, $q_{vib} \to k_{B} T / (h ν)$ (classical limit).
Molecular orbital theory 14.05.02. The force constants of bonds are predicted by MO theory: the bond order determines the electron density in the bonding orbital, which sets the restoring force against displacement. The correlation between bond order and vibrational frequency (single < double < triple) is a direct spectroscopic consequence of the MO picture.
Organic functional groups 15.02.01. The IR group-frequency tables used for structure determination map directly onto the functional-group classification of organic chemistry. The carbonyl stretch, the O-H stretch, the C-H stretch — each is a spectroscopic signature of a specific functional group from the organic chemistry taxonomy.

Historical notes Master

The study of molecular vibrations through infrared spectroscopy has its origins in the discovery of infrared radiation itself. William Herschel detected infrared light in 1800 by observing that a thermometer placed beyond the red end of a solar spectrum registered heating. The first IR absorption spectra of molecules were recorded in the 1880s by Abney and Festing, who observed that specific organic functional groups absorbed at characteristic wavelengths. Their work established the qualitative connection between molecular structure and IR absorption that underpins modern spectral interpretation.

The theoretical framework developed over the following decades. The quantum harmonic oscillator was solved by Schrodinger in 1926, providing the energy-level formula $E_{v} = ℏ ω (v + 1/2)$ and the selection rule $Δ v = \pm 1$ . The Morse potential was introduced by Philip Morse in 1929 ^[Morse1929] as a model for diatomic vibrations that could be solved exactly while capturing anharmonicity. Herzberg's monumental three-volume work Molecular Spectra and Molecular Structure (1939-1966) ^{[Herzberg1945]} systematised the entire field of molecular spectroscopy, including the vibrational spectra of hundreds of molecules, and earned him the 1971 Nobel Prize in Chemistry.

The development of IR instrumentation transformed the technique from a specialist research tool into a routine analytical method. The first commercial double-beam IR spectrometer (Perkin-Elmer Model 21, 1944) made IR spectroscopy accessible to industrial and academic chemistry laboratories. The introduction of Fourier-transform IR (FT-IR) spectrometers in the 1970s — replacing the slow mechanical scanning of dispersive instruments with rapid interferometric measurement — improved sensitivity by orders of magnitude (the Jacquinot and Fellgett advantages) and reduced scan times from hours to seconds.

Raman spectroscopy had a slower development trajectory because the effect is inherently weak. The invention of the laser in 1960 provided the intense monochromatic source that Raman spectroscopy required, and by the 1970s laser Raman spectroscopy was routine. The development of notch filters (to reject the intense Rayleigh line), holographic gratings, and CCD detectors in the 1980s and 1990s brought Raman spectroscopy to its current level of sensitivity and convenience.

Colthup's 1950 correlation tables ^{[Colthup1950]} — which mapped functional groups to characteristic IR frequency ranges — transformed IR spectroscopy from a research technique into a universal analytical method for organic chemists. The tables have been refined and extended many times since, but the basic framework remains: each functional group has a diagnostic IR signature, and reading the spectrum is reading the molecular structure.

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}

@article{Colthup1950,
  author = {Colthup, N. B.},
  title = {Spectra-Structure Correlations in the Infrared Region},
  journal = {Journal of the Optical Society of America},
  volume = {40},
  year = {1950},
  pages = {397--400},
}

Prerequisites

14.12.01 pending
14.12.02 pending

Used in

14.12.04

Tier anchors

beginner: Atkins, Ch. 11.3-11.4
intermediate: Banwell & McCash, Ch. 3; Herzberg — Molecular Spectra II (1945), Ch. 2
master: Herzberg — Infrared and Raman Spectra (1945)

References

Atkins, P. & de Paula, J. — Physical Chemistry, 12e (Oxford UP, 2022) · Ch. 11.3-11.4 Vibrational spectroscopy
Banwell, C. N. & McCash, E. M. — Fundamentals of Molecular Spectroscopy, 4e (McGraw-Hill, 1994) · Ch. 3 Vibrational spectroscopy
Herzberg, G. — Molecular Spectra and Molecular Structure II: Infrared and Raman Spectra of Polyatomic Molecules (Van Nostrand, 1945) · Ch. 2 Vibrational and rotational spectra of diatomic molecules
Pavia, D. L., Lampman, G. M., Kriz, G. S. & Vyvyan, J. R. — Introduction to Spectroscopy, 5e (Cengage, 2015) · Ch. 2 Infrared spectroscopy
Morse1929 · Morse, P. M., Phys. Rev. 34 (1929), 57-64 — diatomic molecules by the Morse potential
Herzberg1945 · Herzberg, G. — Infrared and Raman Spectra of Polyatomic Molecules (Van Nostrand, 1945); Nobel Prize 1971
Raman1928 · Raman, C. V. & Krishnan, K. S., Indian J. Phys. 2 (1928), 387-398 — discovery of Raman scattering
Colthup1950 · Colthup, N. B., J. Opt. Soc. Am. 40 (1950), 397-400 — correlation tables for infrared spectra

Estimated time

beginner: 12m
intermediate: 30m
master: 45m