Vibrational Transitions and the Harmonic Oscillator

Every commercial FTIR (Fourier-transform infrared) spectrometer used in synthetic organic chemistry labs is, at its core, a machine for measuring exactly the transition this article derives: which bonds have $(d\mu/dx)_0\neq0$, and at what wavenumber their $v=0\to1$ jump occurs. The carbonyl stretch near $1700\text{–}1750\ \text{cm}^{-1}$, the O–H stretch near $3200\text{–}3550\ \text{cm}^{-1}$, and the characteristic fingerprint region below $1500\ \text{cm}^{-1}$ are all direct, routine applications of the fundamental-frequency formula $\tilde\nu = (2\pi c)^{-1}\sqrt{k/\mu}$, used daily to confirm functional groups in pharmaceutical quality control, polymer characterization, and reaction monitoring. Atmospheric science depends critically on the changing-dipole requirement: greenhouse gases ($\text{CO}_2$, $\text{H}_2\text{O}$, $\text{CH}_4$, $\text{N}_2\text{O}$) are potent infrared absorbers precisely because their vibrational normal modes modulate a dipole, while the overwhelmingly abundant $\text{N}_2$ and $\text{O}_2$ that make up 99% of the atmosphere are completely transparent in the infrared for the symmetry reason worked out above — a genuinely load-bearing fact for climate science, not a textbook curiosity. Isotope-edited infrared spectroscopy — replacing $^{12}\text{C}$ with $^{13}\text{C}$, or $^1\text{H}$ with $^2\text{H}$, in a specific bond — is a standard tool in mechanistic biochemistry and structural biology, used to assign vibrational bands in proteins and to track reaction intermediates in time-resolved infrared studies of enzymes, all resting on the $\tilde\nu\propto\mu^{-1/2}$ mass-dependence derived here. Surface and materials science use the same selection rule in reflection-absorption infrared spectroscopy (RAIRS) to determine molecular orientation on catalytic surfaces, since only dipole components perpendicular to a metal surface survive the relevant surface selection rule — a direct descendant of the transition-dipole-moment argument developed above. And in astrochemistry, infrared telescopes (JWST prominently among them) identify molecules in interstellar clouds and exoplanet atmospheres by matching observed fundamental and overtone bands to laboratory-measured $\tilde\nu_{\text{fund}}$ values computed from exactly this theory.

Ask the tutor: How does adding anharmonicity to the potential change both the energy-level spacing and the strictness of the $\Delta v=\pm1$ selection rule?

Picture a diatomic molecule as two point masses, $m_1$ and $m_2$, joined by a spring of force constant $k$. Quantum mechanically, once the center-of-mass motion is separated out, the relative coordinate $x = r - r_e$ (displacement from equilibrium bond length) obeys the one-dimensional Schrödinger equation for a particle of reduced mass $\mu = m_1 m_2/(m_1+m_2)$ moving in the potential $V(x) = \tfrac12 kx^2$. This is exactly the harmonic oscillator eigenstates problem solved in every quantum chemistry course, and its solutions are Hermite-polynomial wavefunctions $\psi_v(x)$ with energies

The ladder of levels is perfectly evenly spaced — the hallmark of the harmonic approximation. A photon can in principle carry away or supply any energy gap $E_{v'}-E_v$, but whether the molecule actually absorbs it is an entirely separate question, governed by how the oscillating electric field of light couples to the molecule's charge distribution. That coupling is captured by the transition dipole moment,

where $\hat{\boldsymbol\mu}(x)$ is the molecule's instantaneous dipole moment as a function of bond displacement. Fermi's golden rule tells us the absorption rate is proportional to $|\boldsymbol\mu_{v'v}|^2$; if this integral vanishes, the transition is forbidden regardless of how much photon energy is available. This is the same logic used throughout Selection Rules from Symmetry: a transition needs a nonzero matrix element of the relevant operator, and symmetry (or, here, the algebra of Hermite polynomials) can force that integral to be exactly zero. The key physical fact is that $\hat\mu(x)$ is not constant. As the bond stretches or compresses, the electron cloud redistributes, and the dipole moment changes — even for a homonuclear-looking local picture, any heteronuclear bond has a dipole that depends on $x$. Taylor-expanding around the equilibrium geometry,

turns the transition dipole integral into a sum of integrals of $\psi_{v'}^* x^n \psi_v$. The constant term $\mu_0$ contributes only when $v'=v$ (orthonormality of the $\psi_v$), which is irrelevant to absorption. The linear term, weighted by $(d\mu/dx)_0$, is the one that matters for ordinary infrared spectroscopy, and it is exactly solvable using the ladder-operator representation of $x$.

Notice how the figure encodes two facts at once: the mechanical fact that all rungs of the harmonic ladder are separated by the same $\hbar\omega$, and the electromagnetic fact that only the rung-to-adjacent-rung jump is switched on by a linearly-varying dipole. The overtone at $2\hbar\omega$ is drawn in gray precisely because, in the strict harmonic-oscillator/linear-dipole model, its transition dipole integral is identically zero.

Homonuclear diatomics such as $\text{N}_2$ and $\text{O}_2$ have $\mu(x)\equiv 0$ by symmetry at every bond length, so every derivative $(d\mu/dx)_0$, $(d^2\mu/dx^2)_0,\dots$ vanishes identically. No amount of vibrational energy spacing helps: the transition dipole integral is zero term by term, and these molecules are rigorously infrared-inactive in their fundamental stretch (which is why atmospheric $\text{N}_2$ and $\text{O}_2$ do not contribute to the greenhouse effect, while $\text{CO}_2$ and $\text{H}_2\text{O}$, with vibrations that do modulate a dipole, do).

1. The harmonic oscillator energy ladder.

where: $v=0,1,2,\dots$ is the vibrational quantum number, $k$ is the bond force constant (curvature of the potential at the minimum), and $\mu$ is the reduced mass of the two vibrating nuclei. Reading: consecutive levels are separated by the same energy $\hbar\omega$ regardless of $v$ — a direct consequence of the parabolic potential, and the reason a single frequency $\omega$ characterizes the whole ladder.

2. The transition dipole moment and its Taylor expansion.

where: $\langle v'|v\rangle = \delta_{v'v}$ by orthonormality, $\langle v'|x|v\rangle$ is the position matrix element between oscillator states, and higher terms bring in $x^2, x^3,\dots$ matrix elements. Reading: infrared intensity for the fundamental comes almost entirely from the first derivative term; a molecule with $(d\mu/dx)_0 = 0$ has a forbidden fundamental even if $\mu_0\neq0$.

3. The position matrix element in the harmonic oscillator basis.

where: the prefactor $\sqrt{\hbar/2\mu\omega}$ sets the natural length scale of the oscillator, and the Kronecker deltas show the matrix element is nonzero only when $v' = v\pm1$. Reading: this single equation is the $\Delta v=\pm1$ selection rule — it is a statement about the algebra of raising/lowering operators, not an approximation.

4. The fundamental vibrational frequency observed in IR.

where: $\tilde\nu_{\text{fund}}$ (units $\text{cm}^{-1}$) is the wavenumber of the $v=0\to1$ absorption line, $c$ is the speed of light, and $k,\mu$ are as above. Reading: this is the number chemists actually measure and tabulate; it packages the pure mechanics of the ladder ($\omega=\sqrt{k/\mu}$) together with the fact that only the adjacent-level jump is dipole-allowed.

Example 1 — Foundational: the fundamental of gas-phase $^{1}\text{H}^{35}\text{Cl}$. HCl has an experimentally known fundamental absorption at $\tilde\nu_{\text{fund}} = 2886\ \text{cm}^{-1}$. Use this to extract the force constant, and confirm self-consistently that this is a physically reasonable single bond. Reduced mass, using $m_{\text H} = 1.008\ \text{u}$ and $m_{\text{Cl}} = 34.97\ \text{u}$ (chlorine-35):

Convert to kilograms using $1\ \text{u} = 1.6605\times10^{-27}\ \text{kg}$:

Rearranging $\tilde\nu_{\text{fund}} = \dfrac{1}{2\pi c}\sqrt{k/\mu}$ for $k$:

Compute $2\pi c\,\tilde\nu_{\text{fund}}$ with $c = 2.998\times10^{10}\ \text{cm/s}$:

Then

This matches the textbook-quoted HCl force constant of roughly $480$–$516\ \text{N/m}$ (values vary slightly depending on whether anharmonic corrections are included), consistent with a moderately strong single bond — stiffer than a typical C–C single bond ($\sim 450\ \text{N/m}$) but far softer than a C≡C triple bond ($\sim 1600\ \text{N/m}$). As a limiting-case check: if HCl were instead treated (incorrectly) as a bond between two infinitely heavy atoms, $\mu\to\infty$ would force $\tilde\nu_{\text{fund}}\to0$ for the same $k$ — reduced mass, not just force constant, is essential to matching the observed frequency, which is exactly why deuterated DCl absorbs near $2091\ \text{cm}^{-1}$ rather than $2886\ \text{cm}^{-1}$ for essentially the same bond stiffness.

Example 2 — Applied: predicting whether $\text{CO}_2$'s bending mode is IR active, and estimating its wavenumber contribution to atmospheric absorption. Linear $\text{CO}_2$ (symmetric, $D_{\infty h}$) has a permanent dipole moment of zero at equilibrium. Naively one might guess it is IR-silent like $\text{N}_2$. But $\text{CO}_2$ has three vibrational modes: symmetric stretch ($\Sigma_g^+$, IR-inactive, $\mu_0=0$ and $d\mu/dx=0$ by symmetry since both C=O bonds stretch oppositely and dipole changes cancel), asymmetric stretch ($\Sigma_u^+$), and doubly-degenerate bend ($\Pi_u$). For the bending mode, displacing the two oxygens perpendicular to the O=C=O axis (in opposite directions) breaks the linear symmetry and induces a net transient dipole perpendicular to the molecular axis, even though the equilibrium dipole is zero. Formally, $(d\mu/dx)_0 \neq 0$ for this normal coordinate because the symmetry species of the bending mode ($\Pi_u$) matches the symmetry species of the dipole operator components ($x,y$, which transform as $\Pi_u$ in $D_{\infty h}$) — precisely the group-theoretic test from Selection Rules from Symmetry applied here to a specific normal mode rather than the raw bond-length coordinate. Because the derivation above shows the mechanism (linear dipole term connects $v=0\to1$) is symmetry-independent once $(d\mu/dQ)_0\neq0$ for the correct normal coordinate $Q$, the same $\Delta v=\pm1$ rule applies mode-by-mode. Experimentally the bend appears at $\tilde\nu \approx 667\ \text{cm}^{-1}$ and the asymmetric stretch at $\tilde\nu \approx 2349\ \text{cm}^{-1}$ — both strong, both central to $\text{CO}_2$'s role as a greenhouse gas, while the symmetric stretch near $1333\ \text{cm}^{-1}$ is formally IR-forbidden (it does appear weakly via Fermi resonance with the bend overtone, a genuinely anharmonic effect outside the scope of the pure harmonic-oscillator model).

Example 3 — Exam/competition level: comparing predicted intensities of the fundamental versus the first overtone in $\text{CO}$, given a nonzero $d^2\mu/dx^2$. Suppose for carbon monoxide the dipole moment function is modeled (illustrative, non-round numbers chosen to avoid masking errors) as

(These illustrative slopes are chosen only to demonstrate the relative size of harmonic vs. first-overtone transition dipoles; real CO parameters are similar in order of magnitude but the anharmonic mechanical mixing dominates in practice.) With $k = 1902\ \text{N/m}$ and $\mu_{\text{CO}} = \dfrac{(12.00)(15.995)}{12.00+15.995}\ \text{u} = 6.860\ \text{u} = 1.139\times10^{-26}\ \text{kg}$, compute the oscillator length scale:

Fundamental ($v=0\to1$). Only the linear term contributes (Step 3 above shows $\langle1|x^2|0\rangle=0$ by parity/selection rules for $x^2$, which connects only $\Delta v = 0,\pm2$):

First overtone ($v=0\to2$). The linear term gives zero here (it only connects $\Delta v=\pm1$); the surviving contribution comes from the quadratic term, whose matrix element for a harmonic oscillator is $\langle2|x^2|0\rangle = \sqrt{2}\,\ell_0^2$:

Compute $\ell_0^2 = 1.132\times10^{-24}\ \text{m}^2$, so

Intensity ratio. Since absorption intensity scales as $|\mu_{v'v}|^2$,

The overtone is predicted roughly $10^4$ times weaker than the fundamental — consistent with real infrared overtones being faint, easily missed shoulders rather than prominent bands. As a limiting-case check: if the dipole function were purely linear ($b=0$), $\mu_{20}$ would vanish exactly and the overtone would be strictly, perfectly forbidden — recovering the ideal harmonic-oscillator selection rule derived above. Real molecules show weak but nonzero overtones precisely because $b\neq0$ and because the mechanical potential itself is not perfectly parabolic (true anharmonicity, mixing higher $v$ states into what we call "$v=2$").

The harmonic oscillator model of molecular vibration, and the selection rules built on it, emerged from the convergence of several research programs in the 1920s–1930s rather than a single discoverer, but the physicist most closely associated with turning quantum mechanics into a working spectroscopic tool for vibration-rotation spectra is Robert S. Mulliken, alongside the foundational quantum treatment of molecular vibrations by Erwin Schrödinger's equation itself and its early application to diatomics by spectroscopists such as Ralph de Laer Kronig and Friedrich Hund. The person most worth profiling here, though, is the American physical chemist and spectroscopist who did the most to connect quantum-mechanical vibrational theory to precision infrared measurement: Richard M. Badger, whose empirical force-constant relations bridged theory and experiment for exactly this class of problem, alongside the deeper theoretical grounding laid by Schrödinger's 1926 wave mechanics applied immediately to the harmonic oscillator by Werner Heisenberg and Paul Dirac's operator methods — the ladder-operator technique used in the derivation above is due to Dirac's 1930 formulation of quantum mechanics.