Anharmonicity and Rovibrational Structure

The rigid rotor and harmonic oscillator give equally spaced vibrational levels and a rotational ladder of constant line spacing. A measured band departs from both: the vibrational overtones crowd together, the rotational lines are not quite evenly spaced, and the two branches turn back on themselves at high rotational quantum number. Each departure is a small correction to the leading model, and each carries a physical constant. This lesson computes those corrections and reads the molecular parameters they encode. Throughout, energies are quoted as term values $G(v)$ and $F(J)$ in the spectroscopist's units of wavenumber (inverse centimeters), where a term value $T$ and an energy $E$ are related by $E=hcT$.

The Morse potential and anharmonic levels

The parabolic approximation fails away from the bottom of the well. A real bond softens as it stretches and dissociates at large separation, so the potential climbs steeply on the inner (repulsive) side and levels off to a finite dissociation asymptote on the outer side. The Morse potential captures both features in a form that is still exactly solvable:

where $D_e$ is the well depth measured from the minimum, $r_0$ the equilibrium separation, and $a$ sets the curvature. Expanding about $r_0$ recovers a harmonic term $U\approx D_e{a}^2(r-r_0{)}^2$, so the small-oscillation frequency is

with $\mu$ the reduced mass. The Schrödinger equation for this potential has the closed-form eigenvalues

exact with only two terms and no higher powers. The anharmonicity constant${\omega }_e{x}_e$ is positive, so each successive level sits slightly below where the harmonic ladder would place it.

The spacing between adjacent levels is the first difference of the term values,

a straight line that decreases with $v$. The ladder terminates when the spacing reaches zero, at the largest bound quantum number

Beyond $v_{\max }$ the molecule is unbound. Summing the spacings up to that point, or equivalently taking the area under the $\Delta G$ line, gives the dissociation energy measured from the ground vibrational level,

The linearity of $\Delta G$ is the basis of the Birge-Sponer extrapolation: plot the measured level spacings against $v$, fit a straight line, and read off both ${\omega }_e$ (the intercept) and ${\omega }_e{x}_e$ (half the slope). The total area under the line, out to where it crosses zero, is $D_0$. Real potentials fall below the Morse form near dissociation, so the linear extrapolation slightly overestimates $D_0$, but it fixes the constants from just the first few overtone bands.

Worked example — dissociation energy of H₂ from its constants. For the hydrogen molecule ${\omega }_e=4401{\text{cm}}^{-1}$ and ${\omega }_e{x}_e=121{\text{cm}}^{-1}$. The largest bound level is $v_{\max }=4401/(2\cdot 121)-1\approx 17$, and the depth is $D_e={\omega }_e^2/(4{\omega }_e{x}_e)=4401^2/(484)\approx 4.00\times 10^4{\text{cm}}^{-1}$, or $4.96\text{eV}$ using $1\text{eV}=8065{\text{cm}}^{-1}$. Subtracting the zero-point term $G(0)=2170{\text{cm}}^{-1}$ gives $D_0\approx 3.78\times 10^4{\text{cm}}^{-1}=4.69\text{eV}$, within a few percent of the measured $4.48\text{eV}$; the small excess is the Morse overestimate near dissociation.

Centrifugal distortion of the rotor

A rotating molecule is not perfectly rigid. As it spins, the bond stretches under the centrifugal load until the restoring force balances it, so the moment of inertia grows with the rotational quantum number and the levels fall below the rigid-rotor prediction. The radial motion sees an effective potential that adds the rotational barrier to the vibrational well,

whose minimum shifts outward from $r_0$ as $J$ increases.

Perturbing the rigid rotor to first order in the stretch gives the rotational term with a quartic correction,

where $D_J$ is the centrifugal distortion constant (positive, so it lowers the levels). The Kratzer relation ties $D_J$ to the vibrational frequency,

so a stiffer bond (larger ${\omega }_e$) distorts less. For a light hydride $D_J$ is of order $10^{-3}{\text{cm}}^{-1}$ against a rotational constant $B$ of order $10{\text{cm}}^{-1}$, a part in $10^4$ per unit of $J(J+1)$ — negligible at low $J$, but growing as $J^4$ it becomes visible by $J\sim 20$, where it pulls the high-$J$ lines closer together.

Vibration-rotation coupling

The rotational constant is not a single number but depends on the vibrational state, because a higher vibrational level has a larger mean-square bond length and therefore a larger moment of inertia. Averaging $1/r^2$ over the anharmonic vibrational wavefunction gives a linear dependence,

where $B_e$ is the constant at the potential minimum and ${\alpha }_e>0$ the vibration-rotation coupling constant, typically a percent of $B_e$. The constant governing an observed band is not $B_e$ but the pair $B_0$ and $B_1$ for the two vibrational levels involved.

This coupling reshapes the P and R branches. In the fundamental $v:0\to 1$ band with $\Delta J=\pm 1$, the line positions relative to the band origin ${\nu }_0$ are

Because $B_1<B_0$ the quadratic coefficient $B_1-B_0$ is negative. In the R branch it opposes the linear growth, so the lines spread more slowly, stop, and turn back at a band head; in the P branch it reinforces the spreading, so the P lines fan out ever wider. The single combination-difference ${\nu }_R(J-1)-{\nu }_P(J+1)=4B_v(J+\frac{1}{2})$ isolates one vibrational level's rotational constant from the band alone, so $B_0$ and $B_1$, and thus ${\alpha }_e$, are extracted without knowing ${\nu }_0$.

The isotope effect

Replacing an atom by a heavier isotope changes the reduced mass but not the electronic potential $U(r)$, since the potential is set by the electrons. Every constant that depends on $\mu$ therefore shifts by a predictable ratio. Writing $\rho =\sqrt{\mu /{\mu }^{\prime }}$ for the two isotopologues,

• vibrational frequency ${\omega }_e\propto {\mu }^{-1/2}$, so ${\omega }_e^{\prime }=\rho {\omega }_e$;
• anharmonicity ${\omega }_e{x}_e\propto {\mu }^{-1}$, so ${\omega }_e^{\prime }{x}_e^{\prime }={\rho }^2{\omega }_e{x}_e$;
• rotational constant $B_e\propto {\mu }^{-1}$, so $B_e^{\prime }={\rho }^2{B}_e$;
• coupling ${\alpha }_e\propto {\mu }^{-3/2}$, so ${\alpha }_e^{\prime }={\rho }^3{\alpha }_e$.

The heavier isotopologue has the lower frequencies and the tighter rotational spacing. For hydrogen versus deuterium the effect is large: the reduced mass of ${\text{H}}_2$ is $0.504\text{u}$ and of ${\text{D}}_2$ is $1.007\text{u}$, so $\rho =\sqrt{0.504/1.007}=0.707$. The deuterium vibrational frequency is $0.707$ of hydrogen's, and its rotational lines are spaced at half the interval.

The isotope shift of the zero-point energy has a chemical consequence: because $G(0)\propto {\omega }_e\propto {\mu }^{-1/2}$, a C-D bond sits deeper in its well than a C-H bond by the difference in zero-point energy, which is why deuterated bonds break more slowly. Spectroscopically, resolving both isotopologues in a natural sample fixes the atomic mass ratio from the band positions alone.

Thermal population and the band envelope

The intensity of a rotational line is proportional to the population of its lower level, which is the degeneracy $2J+1$ times the Boltzmann factor,

The degeneracy raises the low levels and the exponential suppresses the high ones, so the population peaks at an intermediate $J$. Setting $\mathbf{d}{N}_J/\mathbf{d}J=0$ gives the most populated level,

which for a typical light molecule at room temperature is near $J\approx 7$-$10$. The band intensity therefore rises from the origin, reaches a maximum a few lines out in each branch, and falls off again, tracing a characteristic double-humped envelope over the P and R branches.

The envelope is a thermometer. Because $J_{\max }$ grows as $\sqrt{T}$, the separation of the two intensity peaks widens with temperature in a known way, so fitting the contour of an unresolved band returns the gas temperature. In astrophysical and combustion spectra, where individual lines may blur together, the band shape alone measures both the rotational constant and the temperature of the emitting gas.

With the anharmonic, centrifugally distorted, temperature-broadened band in hand, the next lesson turns to the transitions the infrared dipole rule forbids, reached instead by scattering and by electronic excitation.