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Anharmonicity is departure from the motion of a harmonic oscillator, often caused by nonquadratic terms in a potential energy. It can make the oscillation frequency depend on amplitude and generate higher harmonics. A quartic oscillator has one such anharmonic potential energy.

Quantizing the stretching coordinate gives a quantum harmonic oscillator with reduced mass $\mu$:Here $v$ is the vibrational quantum number and $\hslash =h/(2\pi )$ the reduced Planck constant. Consecutive energy eigenvalues have gap $\hslash \omega$; the ground state has zero-point energy $\hslash \omega /2$ above the well minimum. The offset $V_0$ cancels from every transition energy.

A short proof uses creation and annihilation operators: with $\ell =\sqrt{\hslash /(\mu \omega )}$, write $q=\ell (a+a^{†})/\sqrt{2}$. The Quantum Hamiltonian becomes $\hat{H}-V_0=\hslash \omega (a^{†}a+1/2)$. Nonnegativity of $a^{†}a$ gives the lowest level; repeated application of $a^{†}$ raises its energy by $\hslash \omega$ each time. This also gives $⟨q{⟩}_v=0$ andThe ground state therefore has a nonzero spread of bond lengths, even though its mean stretching displacement vanishes. A stationary energy eigenstate has a time-independent position probability density; the classical oscillation curve is not a trajectory of an atomic nucleus in that state.

The parabolic model formally extends $q$ to the whole real line, whereas an actual bond length satisfies $r>0$. It is useful for low levels concentrated near $r_e$, with typical stretching displacements much smaller than $r_e$. It fails near breakup of the chemical bond and when neglected anharmonicity becomes substantial.

Keep the same bond force constant $k$ and approximate the nuclear masses of ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, and ${}^{35}\mathrm{Cl}$ by $1u$, $2u$, and $35u$, respectively. Here $u$ is the unified atomic mass unit, and $\mathrm{D}$ denotes deuterium. These are chosen mass number approximations, not measured atomic masses or natural-abundance averages. With the same chlorine isotope in both molecules,This is the isotope scaling of diatomic vibration prediction. Doubling the light partner's mass nearly doubles the reduced mass, lowering the frequency by about $28.3\%$. The factor differs from $1/\sqrt{2}$ because the chlorine partner also moves.

For the same small initial stretch $A$ and zero velocity, both models start with the same internal mechanical energy $k{A}^2/2$, but the DCl oscillation period is longer by $\sqrt{72/37}\approx 1.394972$. In the centre of mass frame, $u_{\mathrm{H}}=-35q/36$ and $u_{\mathrm{Cl}}=q/36$ for HCl; $u_{\mathrm{D}}=-35q/37$ and $u_{\mathrm{Cl}}=2q/37$ for DCl. These ratios explain which partner moves and are separate from the change in frequency.

The common-$k$ assumption belongs to the leading Born-Oppenheimer approximation: fixed nuclear positions and electric charges determine the electronic potential energy, while nuclear masses govern motion on that curve. Quantitative comparison requires actual atomic masses, departures from that approximation, and separation of anharmonicity and molecular rotation from the measured band. Even with common $k$, the quantum harmonic oscillator predicts a different zero-point energy for each isotope.^[ref]

An actual infrared spectroscopy route is to compare gaseous hydrogen chloride containing ordinary hydrogen with its deuterium counterpart. A documented teaching experiment uses a stopcock-equipped gas cell with potassium bromide windows and a Nicolet Avatar instrument for Fourier transform spectroscopy. A nitrogen-purged flask and drying tube supply the mixture; the sample also contains sulfur dioxide. An empty sample-chamber background precedes the filled-cell scan. The absorbance trace and peak-picked spectroscopic wavenumbers are the readout. The analysis assigns and indexes ${}^1{\mathrm{H}}^{35}\mathrm{Cl}$ and ${}^2{\mathrm{H}}^{35}\mathrm{Cl}$ transitions separately and propagates transition uncertainties. The instructions distinguish resolvable chlorine isotope peaks from adequate precision for the intended parameter analysis. ^[ref]

A separate published teaching dataset reports a Nicolet Nexus 670 instrument, a $10\mathrm{cm}$ gas cell at $20\mathrm{Torr}$, and chlorine isotope peaks resolved to baseline. These apparatus details describe that dataset, not the preceding experiment. ^[ref]

The inference connects several models. Fit correctly assigned transitions to separate molecular rotation from molecular vibration, then compare the inferred vibrational scales using isotope scaling of diatomic vibration. Agreement supports the approximate separation between a nearly unchanged chemical bond stiffness and a changed nuclear mass. It does not make a real molecule an exact classical spring: anharmonicity shifts the fundamental transition away from the local harmonic oscillator scale, and coupling to molecular rotation shifts individual lines.

For an independent test, calculate the reduced masses from the specified isotopes and compare the inferred values $k=\mu (2\pi c{\tilde{\nu }}_{\mathrm{harm}}{)}^2$. Agreement within uncertainty tests the common-curvature assumption; choosing separate $k$ values just to fit each frequency would not test it. The documented background addresses instrument and ambient contributions, but does not by itself subtract window absorption, identify contaminating species, or calibrate the spectroscopic wavenumber scale. Those require separate controls. Finite spectral resolution, blended peaks, incorrect isotope assignments, and departure from the harmonic approximation near a potential minimum limit the comparison. No numerical measured line positions are assumed here.

A transition starting from a thermally populated vibrationally excited state. Its intensity depends on the lower-state population through the Boltzmann distribution. Anharmonicity and coupling between normal modes can shift it away from the corresponding vibrational fundamental.

A vibrational fundamental is the transition $v=0\to 1$. A first vibrational overtone is $v=0\to 2$. A hot band starts from a thermally populated vibrationally excited level, such as $v=1\to 2$.

For a strictly harmonic potential and a dipole linear in displacement, the allowed transitions have $\Delta v=\pm 1$. A fundamental and a $1\to 2$ hot band then share the same frequency, while the overtone has zero intensity. Anharmonicity separates the adjacent gaps and can give overtone intensity; nonlinear displacement dependence of the dipole can also contribute. A hot band is defined by its lower state, not merely by being weak.

Anharmonicity places the $1\to 2$ hot band atonly $3.7\mathrm{c}{\mathrm{m}}^{-1}$ below the fundamental. It can be detected with adequate spectral resolution and signal-to-noise; it is not guaranteed to appear as a separate peak in an unresolved band contour. The predicted ten-percent population-based intensity is appreciable, but overlapping rotational branches and linewidth can hide it beneath the stronger fundamental. No instrumental resolution or linewidth is supplied, so an unconditional assertion of visibility or invisibility would go beyond the information given.

The first vibrational overtone is $v=0\to 2$, with $\Delta v=2$. It lies near twice the fundamental wavenumber, generally slightly below it because of anharmonicity. The linear-dipole harmonic model forbids it; anharmonic wavefunctions or higher terms in the dipole expansion give a weak intensity. Thus it is generally weaker than the corresponding fundamental.

A hot band starts from a thermally populated vibrationally excited state, for example $v=1\to 2$. It can satisfy $\Delta v=1$ and need not have a small transition moment, but its lower-state population is reduced by the Boltzmann distribution:It is therefore usually weak at low temperature and grows relative to the fundamental on heating. Anharmonicity shifts its frequency away from the $0\to 1$ transition; in a polyatomic molecule, excitation in another mode can also produce a hot-band sequence.

There are three stretching fundamentals, although four stretching-region features are listed. The low feature at $1746\mathrm{c}{\mathrm{m}}^{-1}$ is predominantly C=O stretch ($A_1$). The two features at $2783$ and $2843\mathrm{c}{\mathrm{m}}^{-1}$ are the symmetric and antisymmetric C–H stretches; their order is not determined by this mode count alone.

The $3482\mathrm{c}{\mathrm{m}}^{-1}$ feature is consistent with the first C=O vibrational overtone, close to $2(1746)=3492\mathrm{c}{\mathrm{m}}^{-1}$ and shifted downward by anharmonicity. Its symmetry is $A_1\otimes A_1=A_1$, so symmetry permits the weak anharmonic transition. It is not a fourth stretching fundamental. The combination-band evidence in the next part distinguishes the two C–H assignments.

The strong band is the $v=0\to 1$ vibrational fundamental, and the weaker band near twice its wavenumber is the $v=0\to 2$ first vibrational overtone. At room temperature these high-frequency oscillators have overwhelmingly ground-state vibrational populations. In the harmonic-potential, linear-dipole limit the infrared selection rule is $\Delta v=\pm 1$; anharmonicity and nonlinear dipole dependence make the overtone weakly allowed. A $1\to 2$ hot band would lie near, and slightly below, the fundamental wavenumber, rather than near twice it.

The right spring changes length by $u_{n+1}-u_n$ and pulls atom $n$ to the right with force $C(u_{n+1}-u_n)$. The left spring contributes $-C(u_n-u_{n-1})$. Adding these restoring forces and applying Newton's second law givesUniform displacement makes both spring extensions vanish, as translational invariance requires. Search for a normal mode $u_n=u_0{e}^{i(kna-\omega t)}$. Substitution yieldsOne representative first Brillouin zone is $-\pi /a\le k\le \pi /a$. This is one positive-frequency acoustic branch: near zero wavevector, $\omega \simeq a\sqrt{C/m}|k|$, so the sound speed is $v_s=a\sqrt{C/m}$. Opposite signs of $k$ label opposite propagation directions, not a second polarization branch.

The model assumes small displacements, harmonic nearest-neighbor forces, one scalar displacement per atom and uniform equilibrium spacing. It excludes damping, anharmonic expansion and transverse polarizations. In a three-dimensional monatomic crystal there would normally be three acoustic polarizations; that counting cannot be imported into this explicitly one-dimensional scalar model.

Let $a$ be a stable equilibrium separation: $V^{\prime }(a)=0$, $K=V^{\prime \prime }(a)>0$. With bond extension $\delta =x-a$, the Taylor expansion isThe harmonic approximation applies when the typical thermal and zero-point bond fluctuations satisfy $|\delta |\ll a$ and the higher terms are negligible, for example $|V^{\prime \prime \prime }(a)\delta |\ll 3K$ and $|V^{\prime \prime \prime \prime }(a){\delta }^2|\ll 12K$. The relevant small quantity is the relative displacement of neighbouring atoms. Small bond excursions about a stable minimum give an amplitude-independent spring constant $K=V^{\prime \prime }(a)$. Large thermal excursions, bond breaking or appreciable anharmonicity invalidate the approximation.

The Debye model replaces the crystal by a harmonic elastic continuum with acoustic dispersion $\omega =v_{s}k$. It counts three polarizations and introduces a cutoff so that the total number of vibrational modes is $3N$, where $N$ is the atom count. Periodic-box counting givesFor an isotropic average speed, $3/v_D^3=1/v_L^3+2/v_T^3$. With cutoff frequency ${\omega }_D$ fixed by the mode count,This density of states is quadratic because a three-dimensional shell has volume proportional to $k^{2}dk$ and the acoustic dispersion is linear. The cutoff is a mode-counting device; it is not a claim that every actual branch stays linear to the zone boundary.

Each phonon mode has mean thermal energy $\hslash \omega /(e^{\beta \hslash \omega }-1)$, using Bose-Einstein statistics with zero phonon chemical potential. The fixed harmonic zero-point contribution has zero temperature derivative. With $x=\hslash \omega /(k_{B}T)$,and differentiating at fixed volume givesFor $T\ll {\Theta }_D$, extend the upper limit to infinity. Expanding $1/(e^x-1)={\sum }_{j\ge 1}{e}^{-jx}$ gives ${\int }_0^{\infty }{x}^3/(e^x-1)dx=6\sum j^{-4}={\pi }^4/15$. integration by parts makes the heat-capacity integral $4{\pi }^4/15$, soOnly modes with $\hslash \omega \lesssim k_{B}T$ are appreciably excited; their number scales as $T^3$, explaining the power law physically. At high temperature every mode contributes $k_B$, giving the Dulong-Petit limit $C_V\to 3N{k}_B$.

The model assumes the harmonic approximation, a macroscopic three-dimensional solid and acoustic continuum behaviour for the thermally populated modes. Anisotropic sound speeds change the low-temperature coefficient rather than its cubic power. Real optical branches and nonlinear acoustic dispersion limit the full-temperature approximation; electronic excitations can add a linear term in a metal, and anharmonicity produces thermal expansion and distinctions between $C_P$ and $C_V$. At temperatures below the lowest finite-sample mode, the bulk continuum law also fails. Thus the $T^3$ result is the low-temperature lattice contribution, not a universal total heat capacity for every solid.

A quartic oscillator adds an anharmonic term to the quadratic potential, for example $H=(p^2+q^2)/2+ϵq^4$.

The quartic term is a specific anharmonicity of the oscillator potential.

For a bond free of applied tension, use a local potential energy $V(a+x)=V(a)+\kappa x^2/2+\gamma x^3/6+\cdots$ with $\kappa >0$. Classical equipartition gives $⟨x^2{⟩}_0=k_{B}T/\kappa$; vanishing mean force then yields $⟨x⟩=-\gamma k_{B}T/(2{\kappa }^2)$ to first order in anharmonicity. Thus the linear thermal expansion coefficient is $\alpha =-\gamma k_B/(2a{\kappa }^2)$. For a Morse potential, $\gamma =-3\kappa /\xi$, giving positive expansion $3k_B/(2a\kappa \xi )$. This is a local bound-state approximation; the isolated cubic truncation is not a globally stable equilibrium potential.

A vibrational overtone excites more than one vibrational quantum: $v=0\to 2$ is the first overtone, and $0\to 3$ the second. Such transitions vanish in the combined harmonic-potential and linear-dipole approximation. Anharmonicity changes wavefunctions and level spacing; nonlinear terms in the electric dipole moment can also supply otherwise missing transition matrix elements.^[ref]

For an original illustrative level model, setThen $E_1-E_0=\hslash \omega (1-2ϵ)$ and $E_2-E_0=2\hslash \omega (1-3ϵ)$. With chosen $ϵ=0.02$, the fundamental gap is $0.96\hslash \omega$ and the first-overtone gap $1.88\hslash \omega$, below twice the fundamental, $1.92\hslash \omega$. This low-level illustration does not determine transition intensity or justify extrapolation to arbitrarily large $v$. It shows why even a well-resolved fundamental band origin need not equal the harmonic curvature scale ${\tilde{\nu }}_{\mathrm{harm}}$.