Rotations of Molecules

Molecular rotation is the quantized reorientation of a molecule’s mass distribution in space. In a semirigid molecule, rotational spacings are usually much smaller than vibrational and electronic spacings, so a rotor Hamiltonian is often the first effective theory of nuclear motion on a potential-energy surface.

Rotational constants measure inverse moments of inertia. Turning those constants into bond lengths requires a model of vibrational averaging and, for polyatomic molecules, enough isotopic or structural information to constrain the geometry.

Required background. Molecular Hamiltonian and Born–Oppenheimer separation define the nuclear problem. Rigid rotor and rotational spectra derive the angular eigenstates and ideal line ladder used here.

From Molecular Motion to a Rotor

Separate translation first

For nuclei with laboratory positions $\mathbf R_A$ and masses $M_A$, define the nuclear center of mass

$$\begin{aligned} \mathbf R_{\mathrm{cm}} &= \frac{1}{M_{\mathrm{tot}}} \sum_A M_A\mathbf R_A,\\ M_{\mathrm{tot}} &= \sum_A M_A. \end{aligned}$$

In the leading Born–Oppenheimer nuclear model, coordinates relative to this nuclear center of mass separate a free translational term from internal motion. Exact separation of the whole molecule uses the center of mass of nuclei and electrons and retains finite-electron-mass recoil corrections; see Molecular Hamiltonian.

On one Born–Oppenheimer surface, an internal nuclear Hamiltonian has the schematic organization

$$\hat H_{\mathrm{nuc}} = \hat T_{\mathrm{vib}} +\hat T_{\mathrm{rot}} +\hat T_{\mathrm{rv}} +U(\mathbf q),$$

where $\mathbf q$ denotes internal shape coordinates and $\hat T_{\mathrm{rv}}$ contains rotation–vibration coupling. The decomposition requires a molecule-fixed frame; an Eckart frame is the standard local choice near an equilibrium geometry.

Rovibrational coupling develops the joint term values, P/Q/R branches, and Coriolis effects obtained when these couplings are retained.

Freeze shape, not quantum mechanics

For a nonlinear molecule in a nondegenerate electronic state with no electronic angular momentum, let $\mathbf I$ be the nuclear inertia tensor at a reference geometry. Neglecting spin, vibration, and rotation–vibration coupling gives

$$\hat H_{\mathrm{rot}} = \frac{1}{2} \hat{\mathbf J}^{\mathsf T} \mathbf I^{-1} \hat{\mathbf J}.$$

In principal axes $a,b,c$,

$$\hat H_{\mathrm{rot}} = \frac{\hat J_a^2}{2I_a} +\frac{\hat J_b^2}{2I_b} +\frac{\hat J_c^2}{2I_c}.$$

For a linear molecule $\mathbf I$ has a zero eigenvalue: this inverse is taken only on the two perpendicular rotational directions, not on the molecular axis.

The molecule is not literally a perfectly rigid classical object. “Rigid rotor” means that shape coordinates have been projected onto a reference vibrational state and their residual effects are initially neglected. Rotational constants measured in a vibrational state already contain vibrational averaging.

The approximation is controlled by scale separation. A useful diagnostic is

$$\epsilon_{\mathrm{rv}} \sim \frac{E_{\mathrm{rot}}}{E_{\mathrm{vib}}}.$$

Small $\epsilon_{\mathrm{rv}}$ supports a rotor expansion, but high rotational excitation, soft vibrations, internal rotation, near-degenerate vibronic states, or dissociation can invalidate a low-order treatment.

Diatomic Rigid Rotor

Internal radial Hamiltonian

For two nuclei, use the relative vector

$$\mathbf R=\mathbf R_2-\mathbf R_1, \qquad \mu=\frac{M_1M_2}{M_1+M_2}.$$

For a nondegenerate $^1\Sigma$ electronic state, neglecting nuclear spin and electronic recoil corrections, the leading internal nuclear Hamiltonian is

$$\hat H_{\mathrm{int}} ={} -\frac{\hbar^2}{2\mu} \frac{1}{R^2} \frac{\partial}{\partial R} R^2 \frac{\partial}{\partial R} + \frac{\hat J^2}{2\mu R^2} +U(R).$$

The first term changes bond length, the second reorients the internuclear axis, and $U(R)$ is the chosen adiabatic potential. Freezing $R$ at a reference value $R_\star$ gives

$$I_\star=\mu R_\star^2, \qquad \hat H_{\mathrm{rot}} = \frac{\hat J^2}{2I_\star}.$$

The corresponding spin-free rotational states are

$$\begin{gathered} |J,M\rangle, \qquad J=0,1,2,\ldots,\\ M=-J,\ldots,J, \end{gathered}$$

with energies

$$E_J = B_EJ(J+1), \qquad B_E=\frac{\hbar^2}{2I_\star}.$$

In zero external field, the $M$ degeneracy is $2J+1$. This degeneracy concerns orientation relative to a laboratory axis. Nuclear-spin statistical weights and fine or hyperfine structure are additional factors, not part of the scalar rigid-rotor result.

Two rotational degrees of freedom

For point nuclei on a line, rotation about the internuclear axis does not change the nuclear configuration. The linear rotor therefore has two orientational degrees of freedom. Its axial principal moment is zero in the ideal nuclear model, while the two perpendicular moments are equal:

$$I_a=0, \qquad I_b=I_c=\mu R_\star^2.$$

Writing an “axial rotational constant” proportional to $1/I_a$ is not a physical extra rotational ladder. Electronic orbital angular momentum, spin, bending angular momentum, and finite-size corrections can add structure in real molecules, but they require an enlarged Hamiltonian.

When the rotational label is N

Here $J$ describes rotation when electronic spin and electronic angular momentum vanish. The standard more general convention is

$$\mathbf N=\mathbf R_{\mathrm{rot}}+\mathbf L_{\mathrm{el}}, \qquad \mathbf J=\mathbf N+\mathbf S, \qquad \mathbf F=\mathbf J+\mathbf I_{\mathrm{nuc}}.$$

$\mathbf R_{\mathrm{rot}}$ is nuclear end-over-end rotation; $\mathbf N$ excludes electron spin but includes electronic orbital angular momentum. Adding $\mathbf L_{\mathrm{el}}$ again to $\mathbf N+\mathbf S$ would double count it. Different coupling limits determine which intermediate quantum numbers remain useful.

Identical nuclei

For a homonuclear diatomic, exchanging the nuclei reverses the directed molecular axis. The total molecular wavefunction must have the exchange symmetry required by nuclear statistics. Rotational states of alternating $J$ parity can therefore pair with different nuclear-spin species or be absent. This is why ortho and para forms cannot be handled by multiplying every level by one universal spin degeneracy.

Rotational Constants

Three common unit conventions

The same inertia is encoded in several constants:

Convention	Definition
Energy	$B_E=\hbar^2/(2I)$
Frequency	$B_\nu=B_E/h=h/(8\pi^2I)$
Wavenumber	$\widetilde B=B_E/(hc)=h/(8\pi^2cI)$

Accordingly,

$$\begin{aligned} E_J &=B_EJ(J+1),\\ \frac{E_J}{h} &=B_\nu J(J+1),\\ \frac{E_J}{hc} &=\widetilde B J(J+1). \end{aligned}$$

Spectroscopic papers often write all three constants simply as $B$. The units determine the meaning. In this page, $B_E$, $B_\nu$, and $\widetilde B$ are kept distinct whenever ambiguity matters.

The wavenumber $\widetilde\nu$ is a frequency divided by $c$:

$$\widetilde\nu=\frac{\nu}{c},$$

whereas the angular wavevector magnitude in vacuum is $k=2\pi\widetilde\nu$. If $\widetilde\nu$ is reported in $\mathrm{cm}^{-1}$, $c$ must be expressed in $\mathrm{cm\,s^{-1}}$ when converting to hertz.

Equilibrium and vibrational-state constants

For a diatomic equilibrium distance $R_e$,

$$I_e=\mu R_e^2, \qquad \widetilde B_e = \frac{h}{8\pi^2c\mu R_e^2}.$$

The experimentally fitted ground-vibrational constant $\widetilde B_0$ is not generally $\widetilde B_e$. Rotation samples a vibrational probability distribution, so approximately

$$B_v \propto \left\langle v\left| \frac{1}{R^2} \right|v\right\rangle.$$

For a diatomic, the standard low-order expansion is

$$\widetilde B_v = \widetilde B_e -\alpha_e\left(v+\frac{1}{2}\right) +\cdots.$$

For a polyatomic molecule with mode degeneracies $d_i$,

$$B_{\mathbf v} = B_e -\sum_i \alpha_i \left(v_i+\frac{d_i}{2}\right) +\cdots.$$

Thus a distance obtained directly from $B_0$,

$$R_0 = \sqrt{\frac{h}{8\pi^2c\mu\widetilde B_0}},$$

is an effective ground-state distance, not automatically the minimum $R_e$ of the potential. Precision work must state whether a reported structure is $r_e$, $r_0$, a substitution structure $r_s$, or another defined effective structure.

Moment of Inertia

Inertia tensor and principal axes

At a fixed geometry, with nuclear coordinates

$$\boldsymbol\rho_A = \mathbf R_A-\mathbf R_{\mathrm{cm}},$$

the nuclear inertia tensor is

$$I_{\alpha\beta} = \sum_A M_A \left[ \rho_A^2\delta_{\alpha\beta} -\rho_{A,\alpha}\rho_{A,\beta} \right].$$

Diagonalizing this real symmetric tensor defines principal axes and principal moments. The conventional ordering is

$$I_a\le I_b\le I_c,$$

so the frequency rotational constants obey

$$\begin{gathered} A\ge B\ge C,\\ A=\frac{h}{8\pi^2I_a}, \qquad B=\frac{h}{8\pi^2I_b},\\ C=\frac{h}{8\pi^2I_c}. \end{gathered}$$

Translation of every nucleus by the same vector does not change the center-of-mass inertia tensor. Rotation of the coordinate axes changes its matrix entries but not its three eigenvalues.

For a planar rigid geometry, choosing $c$ perpendicular to the plane gives the perpendicular-axis relation

$$I_c=I_a+I_b.$$

The ground-state inertial defect

$$\Delta_0 = I_c-I_a-I_b$$

need not vanish, even for a planar equilibrium structure. Zero-point out-of-plane motion, electronic contributions, and nonrigid effects contribute to $\Delta_0$.

Isotopic substitution

Changing an isotope changes masses while leaving the electronic potential nearly unchanged at the Born–Oppenheimer level. The resulting shift

$$A,B,C \longrightarrow A',B',C'$$

provides new constraints on atomic coordinates. For a diatomic with an isotope-independent reference distance,

$$\frac{B'}{B} \approx \frac{\mu}{\mu'}.$$

For a polyatomic molecule, substitution also shifts the center of mass and can rotate the principal axes. Kraitchman substitution coordinates use changes in principal moments to estimate the absolute coordinates of the substituted atom. They are powerful, but they do not recover coordinate signs by themselves and become unstable for coordinates close to a principal plane or axis.

Three cautions are essential:

1. three rotational constants provide only three independent inertia constraints for one isotopologue;
2. ground-state moments contain vibrational averaging;
3. a highly precise fit does not by itself make the inverse structural problem unique.

Multiple isotopologues, chemical constraints, ab initio vibrational corrections, and uncertainty propagation are normally combined for a defensible equilibrium structure.

Rotational Selection Rules

The transition operator matters

The leading interaction with a microwave electric field is

$$\hat H_{\mathrm{int}}(t) = -\hat{\boldsymbol\mu} \mathbin{\cdot} \mathbf E(t).$$

A transition requires a nonzero matrix element

$$\langle f| \hat{\boldsymbol\mu}\mathbin{\cdot}\boldsymbol\epsilon |i\rangle.$$

Energy-level spacings alone do not determine whether a line appears. The permanent body-fixed dipole, rotational wavefunctions, polarization, parity, exchange symmetry, and any coupled spins all enter the line strength.

For a polar linear rotor in a nondegenerate closed-shell electronic and vibrational state, electric-dipole pure rotational transitions obey

$$\Delta J=\pm1, \qquad \Delta M=0,\pm1.$$

Absorption normally follows $J\rightarrow J+1$. The ideal frequency is

$$\nu_{J\rightarrow J+1} = 2B_\nu(J+1).$$

A homonuclear diatomic has no permanent electric dipole in a field-free isolated state, so its ordinary electric-dipole pure rotational spectrum is absent. The rotational levels still exist. Rotational Raman scattering, weak multipole or magnetic mechanisms, collision-induced absorption, and rovibrational transitions probe different operators and have different rules.

Symmetric-top and asymmetric-top rules

For a symmetric top, $K$ is the projection of rotational angular momentum on the symmetry axis. A dipole component parallel to that axis gives

$$\Delta K=0,$$

while a perpendicular component gives

$$\Delta K=\pm1.$$

The angular rule is $\Delta J=0,\pm1$, with $J=0\not\leftrightarrow J=0$. For a parallel permanent dipole, $\Delta J=0,\Delta K=0$ connects degenerate rigid-top levels and gives no nonzero-frequency absorption line; the ordinary ladder has $\Delta J=\pm1$. If a perpendicular component is allowed, changing $K$ can instead give a nonzero-frequency $\Delta J=0$ transition. Molecular symmetry determines which dipole components exist.

An asymmetric top has no exact $K$, but its levels are conventionally labelled $J_{K_a,K_c}$ by correlation with prolate and oblate symmetric-top limits. If a dipole component lies along principal axis $a$, $b$, or $c$, the corresponding parity rules are:

Transition type	Changes of limiting labels
$a$ type	$\Delta K_a$ even, $\Delta K_c$ odd
$b$ type	$\Delta K_a$ odd, $\Delta K_c$ odd
$c$ type	$\Delta K_a$ odd, $\Delta K_c$ even

Here “even” includes zero and “odd” includes positive or negative odd integers. The general angular condition is

$$\Delta J=0,\pm1, \qquad 0\not\leftrightarrow0.$$

Mixing can lend intensity to rules based on approximate quantum numbers; it cannot remove a zero protected by an exact symmetry that the Hamiltonian and transition operator still obey. Nuclear-spin symmetry can remove or reweight levels, and spin coupling can replace $J$ by a chain of $N,J,F,\ldots$ labels. Selection rules must therefore be attached to a stated Hamiltonian, operator, and coupling scheme.

Centrifugal Distortion

Why the rigid ladder bends

At larger $J$, the rotational kinetic energy favors a larger average bond length and therefore a larger moment of inertia. In a simple diatomic model, write

$$R=R_e+x, \qquad U(R) \approx U(R_e)+\frac{1}{2}kx^2.$$

For fixed

$$Q=J(J+1),$$

the effective radial energy is approximately

$$E_{\mathrm{eff}}(x;J) = \frac{1}{2}kx^2 + \frac{\hbar^2Q} {2\mu(R_e+x)^2}.$$

Minimization to leading order gives

$$x_J \approx \frac{\hbar^2Q} {\mu kR_e^3}.$$

Substitution back into the energy produces

$$E_J \approx B_EQ-D_EQ^2,$$

where

$$D_E = \frac{\hbar^4} {2\mu^2kR_e^6} >0.$$

If $k=\mu\omega_{\mathrm{vib}}^2$, then

$$D_E \approx \frac{4B_E^3} {(\hbar\omega_{\mathrm{vib}})^2}.$$

In spectroscopic wavenumber units this becomes the familiar scale estimate

$$\widetilde D_e \approx \frac{4\widetilde B_e^3} {\widetilde\omega_e^2}.$$

This derivation explains the sign and scale, but fitted distortion constants are effective Hamiltonian parameters. In polyatomic molecules they summarize couplings to many vibrational coordinates, and their numerical values depend on the Hamiltonian reduction and convention.

Effective term values

For a linear molecule in a fixed nondegenerate vibrational state,

$$\begin{gathered} F_v(J) = B_vQ-D_vQ^2+H_vQ^3+\cdots,\\ Q=J(J+1), \end{gathered}$$

where all constants and $F_v$ use the same frequency or wavenumber units.

Writing $x=J+1$, the adjacent absorption line is

$$\nu_{J\rightarrow J+1} ={} 2B_vx -4D_vx^3 + 2H_vx^3(3x^2+1) +\cdots.$$

The positive $D_v$ term lowers high-$J$ lines relative to the rigid ladder. Because the correction grows as $(J+1)^3$, a fit that works for low $J$ can extrapolate poorly far beyond the measured range.

Evaluated HCN example

The NIST evaluated triatomic tables report the following ground-vibrational constants:

Constant	HCN	DCN
$B_0$ ($\mathrm{MHz}$)	$44\,315.9757$	$36\,207.4627$
$D_0$ ($\mathrm{kHz}$)	$87.24$	$57.83$

For HCN, retaining $B_0$ and $D_0$ gives:

$J\rightarrow J+1$	$\nu$ ($\mathrm{GHz}$)	Rigid excess ($\mathrm{MHz}$)
$0\rightarrow1$	$88.631602\,\mathrm{GHz}$	$0.349\,\mathrm{MHz}$
$1\rightarrow2$	$177.261111\,\mathrm{GHz}$	$2.792\,\mathrm{MHz}$
$2\rightarrow3$	$265.886432\,\mathrm{GHz}$	$9.422\,\mathrm{MHz}$

The increasing residual is the centrifugal-distortion signature. The nitrogen nucleus also has an electric quadrupole moment, so an observed $J=1\rightarrow0$ feature resolves into several hyperfine components under sufficient resolution. A rotational centroid, a hyperfine component, and the rigid-rotor prediction are three different frequencies.

The isotope shift from HCN to DCN is much larger than the distortion correction because deuteration changes the entire center-of-mass inertia. It is not legitimate to interpret that shift as a change in bond strength without a separate nuclear-motion analysis.

Symmetric and Asymmetric Tops

General rigid-top Hamiltonian

In frequency units, the principal-axis Hamiltonian is

$$\frac{\hat H_{\mathrm{rot}}}{h} = A\frac{\hat J_a^2}{\hbar^2} +B\frac{\hat J_b^2}{\hbar^2} +C\frac{\hat J_c^2}{\hbar^2}.$$

The relations among $A,B,C$ classify the ideal rotor.

Rigid-rotor classes are statements about principal moments of inertia. Because each rotational constant is inversely proportional to its moment, the inequalities reverse between $I_a,I_b,I_c$ and $A,B,C$. Molecular examples can carry inversion, internal-rotation, spin, or vibronic structure beyond the shapes shown.

Linear and spherical tops

A linear rotor has

$$I_a=0, \qquad I_b=I_c,$$

and only the two perpendicular rotations produce the scalar ladder $E_J\propto J(J+1)$.

A spherical top has

$$I_a=I_b=I_c, \qquad A=B=C.$$

Its rotational Hamiltonian is also proportional to $\hat J^2$, but the configuration space is the full orientation of a three-dimensional body rather than only a molecular axis. Methane and sulfur hexafluoride are standard high-symmetry examples, although their ordinary pure rotational electric-dipole spectra vanish in the rigid equilibrium limit because they have no permanent dipole.

Symmetric tops

A prolate symmetric top has

$$I_a<I_b=I_c, \qquad A>B=C,$$

while an oblate symmetric top has

$$I_a=I_b<I_c, \qquad A=B>C.$$

Let $B_\parallel$ be the constant for the symmetry axis and $B_\perp$ the common perpendicular constant. Then

$$\frac{E_{J,K}}{h} = B_\perp J(J+1) + (B_\parallel-B_\perp)K^2,$$

with

$$J=0,1,2,\ldots, \qquad K=-J,\ldots,J.$$

For a prolate top, $B_\parallel=A$; for an oblate top, $B_\parallel=C$. In a field-free rigid symmetric top the energy depends on $K^2$, so $K$ and $-K$ are degenerate before additional interactions are included.

Asymmetric tops

Most polyatomic molecules satisfy

$$I_a<I_b<I_c, \qquad A>B>C.$$

Because no body-axis projection commutes with the full asymmetric-top Hamiltonian, there is no exact $K$ quantum number and no universal closed-form energy formula. For each $J$, one constructs and diagonalizes a finite matrix in a symmetric-top basis.

For $A>C$, Ray’s asymmetry parameter is

$$\kappa = \frac{2B-A-C}{A-C}, \qquad -1\le\kappa\le1.$$

The spherical top $A=B=C$ makes this ratio undefined. The prolate symmetric-top limit is $\kappa=-1$ and the oblate limit is $\kappa=+1$. The conventional labels

$$J_{K_a,K_c}$$

record correlation with those two limits. They are useful labels, not simultaneous exact projections of $\mathbf J$ on two body axes.

For semirigid asymmetric tops, precision work usually adds quartic and then sextic centrifugal-distortion operators. Watson’s reduced Hamiltonians remove parameter indeterminacies, but different reductions and axis representations produce different numerical parameter sets. Constants should therefore be compared only after matching conventions.

Microwave Spectroscopy Connection

Microwave line frequencies constrain differences of effective rotational energies. Rotational spectroscopy explains line assignment, fits, and evaluated catalogs. The HCN example above illustrates why the rigid frequency, a distortion-corrected rotational centroid, and a hyperfine component must be distinguished.

Positions, strengths, and populations

Line positions come from energy differences:

$$h\nu_{fi}=E_f-E_i.$$

Integrated strengths also depend on the transition moment and lower-state population. For an ideal linear rotor in thermal equilibrium,

$$N_J \propto g_{\mathrm{ns}}(J) (2J+1) \exp\left[ -\frac{E_J}{k_BT} \right],$$

where $g_{\mathrm{ns}}(J)$ is the nuclear-spin statistical weight. Ignoring that factor and treating $J$ as continuous, the most populated region is near

$$J_{\mathrm{peak}} \approx \sqrt{ \frac{k_BT} {2hc\widetilde B} } -\frac{1}{2}.$$

This continuous estimate is useful when several $J$ levels are populated. A negative value is replaced by the boundary $J=0$; at low temperature compare the discrete populations directly. The estimate concerns level population, not the intensity of an individual line. Dipole components, Hönl–London factors, stimulated emission, partition functions, instrumental response, optical depth, collisions, and nonequilibrium populations can all matter.

What different patterns constrain

Spectroscopic evidence	Primary constraint
Line spacings and assignments	$A,B,C$ and distortion constants
Isotopologue shifts	mass distribution and structural coordinates
Stark shifts	body-fixed dipole components
Hyperfine splittings	nuclear quadrupole, spin-rotation, and spin-spin couplings
Relative intensities	populations, dipole components, and line-strength factors
Residual trends	missing distortion, coupling, perturbation, or misassignment

A rotational spectrum is often a distinctive molecular fingerprint because different isomers and isotopologues have different inertia tensors. Enantiomers in an isolated achiral environment, however, have the same ordinary rotational constants. Additional chiral discrimination requires a chiral interaction or phase-sensitive multi-axis protocol.

A low-order distortion expansion is useful only over a limited range of rotation. Localized level perturbations, tunneling splittings, and large-amplitude motion require the relevant coupling or internal coordinate, not merely more powers of $J(J+1)$. Constants from different Watson reductions or axis conventions should be transformed before comparison.

Exercises

1. Inertia and an effective bond length

A closed-shell diatomic has $B_0=57.64\,\mathrm{GHz}$ and reduced mass $\mu=6.856\,\mathrm u$. Estimate $I_0$ and $R_0$. Use $1\,\mathrm u=1.66054\times10^{-27}\,\mathrm{kg}$.

Solution

In frequency units,

$$I_0 = \frac{h}{8\pi^2B_0}.$$

With $B_0=5.764\times10^{10}\,\mathrm{s^{-1}}$,

$$I_0 \approx 1.46\times10^{-46}\,\mathrm{kg\,m^2}.$$

The reduced mass is

$$\begin{aligned} \mu &= 6.856(1.66054\times10^{-27}),\\ &\approx 1.139\times10^{-26}\,\mathrm{kg}. \end{aligned}$$

Therefore

$$\begin{aligned} R_0 &= \sqrt{\frac{I_0}{\mu}}\\ &\approx 1.13\times10^{-10}\,\mathrm m\\ &= 1.13\,\text{\AA}. \end{aligned}$$

Because the input is $B_0$, this is an effective ground-state distance $R_0$, not automatically $R_e$.

2. Diatomic isotope scaling

Approximate the masses of $^{12}\mathrm C$, $^{13}\mathrm C$, and $^{16}\mathrm O$ by their mass numbers. If the bond length is unchanged, estimate $B(^{13}\mathrm C{}^{16}\mathrm O)/B(^{12}\mathrm C{}^{16}\mathrm O)$.

Solution

The reduced masses are

$$\mu_{12,16} = \frac{12(16)}{12+16} = \frac{48}{7}\,\mathrm u$$

and

$$\mu_{13,16} = \frac{13(16)}{13+16} = \frac{208}{29}\,\mathrm u.$$

Since $B\propto1/\mu$ at fixed distance,

$$\begin{aligned} \frac{B_{13,16}}{B_{12,16}} &= \frac{\mu_{12,16}}{\mu_{13,16}}\\ &\approx 0.956. \end{aligned}$$

The heavier isotopologue has the smaller rotational constant and more closely spaced lines.

3. Distorted HCN line

Using $B_0=44\,315.9757\,\mathrm{MHz}$ and $D_0=0.08724\,\mathrm{MHz}$, calculate the HCN $J=2\rightarrow3$ centroid frequency through quartic distortion.

Solution

For $J\rightarrow J+1$,

$$\nu = 2B_0(J+1) -4D_0(J+1)^3.$$

At $J=2$,

$$\begin{aligned} \nu &= 6B_0-108D_0\\ &= 265\,895.8542\,\mathrm{MHz} -9.42192\,\mathrm{MHz}\\ &= 265\,886.4323\,\mathrm{MHz}. \end{aligned}$$

This is a rotational centroid within the truncated model. Nitrogen quadrupole coupling splits an actual high-resolution feature into hyperfine components.

4. Rotation and electron spin

In a $^2\Sigma$ state, take $L_{\mathrm{el}}=0$ and $S=1/2$. Which $J$ values arise from $N=0$ and $N=1$? Neglect spin–rotation coupling and verify the number of states in each manifold.

Solution

For $N=0$, only $J=1/2$ occurs, with two magnetic substates. For $N=1$, angular-momentum addition gives $J=1/2,3/2$, with $2+4=6$ substates. These agree with $(2N+1)(2S+1)$. In this limit $N$ is the nuclear rotor quantum number and the common energy is $B_EN(N+1)$; a spin–rotation interaction splits the two $N=1$ components.

5. Classify a rotor

A molecule has principal moments in the ratio

$$I_a:I_b:I_c=3:12:12.$$

Classify the rotor and find the ratio $A:B:C$.

Solution

Because

$$I_a<I_b=I_c,$$

the molecule is a prolate symmetric top. Rotational constants are inversely proportional to moments:

$$A:B:C = \frac{1}{3}:\frac{1}{12}:\frac{1}{12} = 4:1:1.$$

6. Identify an asymmetric-top transition type

Classify each transition by its $a$-, $b$-, or $c$-type parity rule:

$$2_{12}\rightarrow1_{01}, \qquad 2_{02}\rightarrow1_{01}.$$

Solution

For

$$2_{12}\rightarrow1_{01},$$

both changes are odd:

$$\Delta K_a=-1, \qquad \Delta K_c=-1.$$

It is therefore $b$ type.

For

$$2_{02}\rightarrow1_{01},$$

the changes are

$$\Delta K_a=0, \qquad \Delta K_c=-1.$$

Even $\Delta K_a$ and odd $\Delta K_c$ identify an $a$-type transition.

7. Thermal rotational population

For a linear rotor with $\widetilde B=2.00\,\mathrm{cm}^{-1}$ at $T=100\,\mathrm K$, estimate the most populated $J$, ignoring nuclear-spin weights. Use $k_B/(hc)=0.6950\,\mathrm{cm^{-1}\,K^{-1}}$.

Solution

The continuous estimate is

$$J_{\mathrm{peak}} \approx \sqrt{ \frac{k_BT} {2hc\widetilde B} } -\frac{1}{2}.$$

Since

$$\frac{k_BT}{hc} = 69.50\,\mathrm{cm}^{-1},$$

we find

$$\begin{aligned} J_{\mathrm{peak}} &\approx \sqrt{\frac{69.50}{4.00}} -\frac{1}{2}\\ &\approx 3.67. \end{aligned}$$

The nearest integer is $J=4$. Exact discrete populations should be compared at $J=3$ and $J=4$, and observed line intensity still requires a transition-strength factor.

8. Precision versus structural meaning

An asymmetric-top fit determines $A_0,B_0,C_0$ to relative precision $10^{-8}$. Explain why this does not establish every equilibrium bond length and angle to relative precision $10^{-8}$.

Solution

The fitted constants determine three ground-state principal moments with high precision. A polyatomic geometry has more than three internal coordinates, so one isotopologue does not make the inverse problem unique. Moreover, $A_0,B_0,C_0$ include zero-point vibrational averaging, while an equilibrium structure refers to the potential minimum.

Recovering $r_e$ requires additional information such as multiple isotopologues, structural constraints, and calculated rotation–vibration corrections. Frequency precision, effective-Hamiltonian precision, and structural accuracy are distinct quantities.

References

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