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A heteronuclear diatomic molecule consisting of carbon and sulfur. Its permanent electric dipole moment permits an electric-dipole rotational spectrum. The linear rigid rotor model relates its rotational constant to its reduced mass and bond length.

Real chemical bonds can stretch as molecular rotation increases. The resulting larger moment of inertia lowers the energy eigenvalues relative to a rigid rotor. Within one vibrational quantum state, a leading effective correction isBoth $B_v$ and $D_v$ are in frequency units here; $v$ labels the vibrational quantum state. For positive $D_v$, the adjacent-line spacing becomesso the departures grow with $J$. Molecular vibration also changes the effective rotational constant: even the lowest vibrational quantum state does not have motionless atomic nuclei.^[ref]

The quartic correction is a limited expansion, not a model to extrapolate to arbitrary $J$. It is small relative to the rigid contribution only while $(D_v/B_v)J(J+1)\ll 1$; eventually higher terms, changed internal motion, or chemical bond breaking matter. In the synthetic plot, the largest upper-state label is $J=5$, giving a stipulated correction ratio of $0.03$. No universal maximum valid $J$ follows without a particular molecule and precision target.

The quantum numbers also have limits of meaning. Here $J$ describes spin-free rotation. In a molecule with electronic spin, rotation alone is commonly labelled $N$, while $J$ includes electronic angular momentum; including nuclear spin introduces additional total-angular momentum labels and resolved splittings. An applied electric field can mix different field-free $J$ states through the Stark effect. For identical atomic nuclei, the Pauli exclusion principle can restrict combinations of rotational and nuclear-spin states. The simple $2J+1$ degeneracy and every-integer-$J$ ladder therefore describe the stated idealization, not all observed sublevels.^[ref]

The nuclear separation at the minimum of an electronic potential-energy curve. It differs from effective separations inferred from vibrationally averaged rotational constants.

For a diatomic molecule, let $m_1,m_2$ be the two masses and $r$ their fixed separation. The distances from the centre of mass are $r_1=m_{2}r/(m_1+m_2)$ and $r_2=m_{1}r/(m_1+m_2)$. A perpendicular axis through the centre of mass therefore has moment of inertiaThe reduced mass $\mu$ here is not the electric dipole moment, for which we use $\mathbf{d}$. For a longer linear molecule, replace $\mu r^2$ by ${\sum }_i{m}_i{\rho }_i^2$, where ${\rho }_i$ is each distance along the axis from the centre of mass. In the point-mass idealization, the axial moment of inertia is zero; turning about that axis changes no nuclear positions, so no independent axial rotational degree of freedom is included.

For the field-free linear rigid rotor, assume a fixed internal quantum state with no electronic angular momentum or electronic spin, and omit couplings to nuclear spin. The rotational Hamiltonian is ${\hat{H}}_{\mathrm{rot}}={\hat{\mathbf{L}}}^2/(2I)$. Its orientation wavefunctions are spherical harmonics $Y_{JM}$, and its angular momentum eigenstates satisfywhere $J=0,1,2,\dots$ and $M=-J,-J+1,\dots ,J$. Each $E_J$ belongs to $2J+1$ angular momentum eigenstates distinguished by the magnetic quantum number $M$, an instance of degenerate energy levels. The Planck constant is $h=2\pi \hslash$; $B$ is the frequency-valued rotational constant. Notice that the energy eigenvalues scale as $J(J+1)$, not $J^2$.

Subtracting consecutive energy eigenvalues givesThus increasing level gaps produce a regularly spaced series of lines at $2B,4B,6B,\dots$, with spacing $2B$. Allowed positions do not guarantee that every line is visible: electric-dipole selection rules, occupation of the initial quantum state, and instrument response also matter.

Recovering molecular geometry from rotational constants is an inverse problem. In the synthetic assignment, $B=5\times 10^9\mathrm{Hz}$ gives $I=h/(8{\pi }^{2}B)\simeq 1.6784\times 10^{-45}\mathrm{kg}{\mathrm{m}}^2$. For a diatomic molecule with known masses, rotational constant explains how to infer an effective bond length. For a longer linear molecule, one $I$ cannot determine every bond length: several different arrangements can have the same moment of inertia.

For a fixed three-atom linear model with positions $0,a,a+b$ and positive masses $m_1,m_2,m_3$, subtracting the centre of mass givesThis is one equation for two unknown bond lengths. Even fixing the order of the chemical atoms leaves a family of solutions. Additional isotope substitutions change known masses and supply independent equations, provided the effective molecular geometry can be treated as shared. For nonlinear molecules, three principal moments of inertia give frequency-valued constants $A=h/(8{\pi }^2{I}_a)$, $B=h/(8{\pi }^2{I}_b)$, and $C=h/(8{\pi }^2{I}_c)$; a single evenly spaced comb is generally insufficient.

Vibrational averaging, unresolved hyperfine structure, internal motion, and centrifugal distortion limit structural inference. A ground-vibrational fitted rotational constant is not automatically the equilibrium constant. Ordinary field-free rotational spectroscopy also gives identical spectra for opposite enantiomers under the usual parity-symmetric molecular Hamiltonian; assigning composition and conformation does not by itself determine handedness. Molecular identification from a rich fingerprint can therefore succeed even when a unique full molecular geometry cannot be reconstructed.

Molecular vibration changes internal separations and bond angles, unlike the fixed internal molecular geometry of a rigid rotor. Its coupling to molecular rotation makes measured rotational constants depend on the vibrational quantum state.

A diatomic molecule has one internal stretching coordinate: its two atomic nuclei approach and separate about an equilibrium bond length. The diatomic bond stretch turns this motion into a particularly useful model. The local potential energy curvature supplies the restoring force, while the reduced mass accounts for both moving partners. Thus the same chemical bond can vibrate more slowly after an isotope substitution even when its electronic structure is nearly unchanged.

This separation makes infrared spectroscopy informative: transition frequencies constrain chemical bond stiffness, and isotope scaling of diatomic vibration tests the mass dependence. The University of Florida HCl and DCl FTIR experiment illustrates the measurement. Detectability also requires vibrational infrared activity; having vibrational energy eigenvalues alone does not guarantee an infrared absorption line.

An oblate symmetric top has two equal smaller principal moments of inertia and a larger moment of inertia about its symmetry axis. A planar trigonal molecule, such as boron trifluoride, is an example. Its axial rotational constant is smaller than its transverse rotational constant.

Using the stipulated integer isotope masses givesThe rotational constant in wavenumber units isHere $c$ must be expressed in $\mathrm{cm}{\mathrm{s}}^{-1}$ when $I$ is in SI units. Equivalently calculate $B=h/(8{\pi }^{2}I)$ in $\mathrm{Hz}$ first, then divide by $c$ in $\mathrm{cm}{\mathrm{s}}^{-1}$.

The calculated rotational constant gives ${\theta }_{\mathrm{rot}}=1.18721\mathrm{K}$. At room temperature, taken as $300\mathrm{K}$, the continuous maximum is $J_{\ast }=10.7404$; comparing the adjacent integer populations gives $J_{\max }=11$. Taking $298.15\mathrm{K}$ instead gives $J_{\ast }=10.7057$ and the same integer answer.

At $8.7\mathrm{K}$, $J_{\ast }=1.41417$. In particular,and $N_1>N_0$, so $J_{\max }=1$. The integer comparison avoids treating a noninteger angular momentum quantum number as a physical state.

Use rovibrational spectroscopy combination differences so that the band origin cancels. With the assignments above, $R(2)$ and $P(4)$ end at the same upper level $J^{\prime }=3$, givingLikewise $R(3)$ and $P(3)$ start at the same lower level $J=3$, soOnly the printed wavenumbers are needed:Thus $B^{\prime }-B^{\prime \prime }\simeq 0.00429{\mathrm{cm}}^{-1}$. The rotational constants differ only slightly, so the average moment of inertia changes little on vibrational excitation. The upper value is slightly larger, corresponding to a slightly smaller effective moment of inertia. This sign is possible for a bending excitation; the intuition that excitation must always lengthen a diatomic bond is not a general rule for a polyatomic bend.

The rounded printed values are internally consistent: fitting all six positions to the same elementary model gives approximately ${\tilde{\nu }}_{\ast }=711.9945{\mathrm{cm}}^{-1}$, $B^{\prime \prime }=1.47797{\mathrm{cm}}^{-1}$ and $B^{\prime }=1.48203{\mathrm{cm}}^{-1}$, with residuals below $0.002{\mathrm{cm}}^{-1}$. The displayed combination-difference values have more digits than their physical accuracy; the meaningful scale is about $1.48{\mathrm{cm}}^{-1}$.

Let $J$ label the lower rotational state and write the elementary band model asHere $B^{\prime \prime },B^{\prime }$ are lower and upper rotational constants, in inverse centimetres; a constant term associated with upper-state vibrational angular momentum can be included in ${\tilde{\nu }}_{\ast }$. The P branch has $J^{\prime }=J-1$, and the R branch has $J^{\prime }=J+1$.

In this model the six printed labels are assigned asThe low-frequency series steps away from the central origin towards increasing lower $J$. The first two high-frequency R positions are approximately $714.96$ and $717.93{\mathrm{cm}}^{-1}$, so the labels at $720.91$ and $723.90$ correspond to $R(2)$ and $R(3)$; they are not the first two rotational transitions.

There is a physical limitation to taking this schematic model literally. A strict $l=0\to |l|=1$ bending band has $J^{\prime }\ge 1$, so a genuine $P(1)$ transition to $J^{\prime }=0$ is forbidden. The printed $709.04$ feature has the formal $P(1)$ position of the elementary model. The numerical analysis below follows the printed line sequence; it does not infer a physically allowed $J^{\prime }=0$ bending level from that feature.

The expanded feature resolves the Q branch: each line has $\Delta J=0$ but starts from a different lower rotational level. In the simple rigid rotor band model,Because the upper rotational constant is larger, the lines extend towards higher wavenumber. Their adjacent spacings areso they crowd together at the low-frequency edge and become increasingly separated. A strictly resolved bending-band analysis may distinguish the upper parity component sampled by Q lines from that sampled by P/R lines; it is not necessary to impose identical high-resolution constants on every branch in this sketch.

The intensity envelope reflects rotational state populations and the rotational transition strength. For this perpendicular Q branch, the usual rotational line-strength factor gives the approximate dependenceAt small $J$, the increasing magnetic degeneracy makes the line intensity rise; at large $J$, Boltzmann suppression dominates. The maximum is therefore at an intermediate $J$. This is a population envelope across distinct transitions, rather than one intrinsically broad vibrational line.

The two extra progressions are rotational transitions within thermally populated $v=1$ and $v=2$ vibrational levels. Isotopic purity rules out attributing them to different isotope masses. Their rotational constants areThe Boltzmann distribution makes each higher vibrational manifold less populated, so its rotational spectrum is weaker. A Morse oscillator explores larger separations as it is excited: the bond's vibrationally averaged moment of inertia increases and the rotational spacing decreases. More precisely, $B_v$ involves an average of the inverse square separation; it is not simply the inverse square of the mean separation.

The observed decrement is constant to the given precision:This fits the leading vibration-rotation interaction law $B_v=B_e-{\alpha }_e(v+\frac{1}{2})$.

A useful vibration-rotation expression, with energies measured from the potential minimum, isHere $v=0,1,\dots$ is the vibrational quantum number, $J=0,1,\dots$ the rotational angular-momentum quantum number, ${\tilde{\nu }}_e$ the harmonic spectroscopic wavenumber, $x_e$ a dimensionless anharmonic coefficient, and $B_v$ the vibrational-state-dependent rotational constant. $B_e$ is its equilibrium value and ${\alpha }_e$ describes vibration-rotation coupling. All wavenumber constants use the same inverse-length units; $h$ is the Planck constant and $c$ the speed of light.

The elementary harmonic rigid rotor result follows by setting $x_e={\alpha }_e=0$, with ${\tilde{\nu }}_e=(2\pi c{)}^{-1}\sqrt{k/\mu }$ and $B_e=h/(8{\pi }^{2}cI)$, where $I=\mu r_e^2$. The molecule's electronic energy would add a common constant.

Read the stronger members of the inset doublets at about $2945$, $2926$ and $2906\mathrm{c}{\mathrm{m}}^{-1}$. Their spacings are roughly $19$–$20\mathrm{c}{\mathrm{m}}^{-1}$, so the common-$B$ estimate isUsing the harmonic wavenumber from the preceding calculation and labelling lines by the lower-state $J$,These are graph-based estimates. Finite reading precision and different upper/lower rotational constants limit their accuracy to a few cm−1; extra decimal places would not be justified. The R1 result refers to $J=1\to 2$, while R0 is the line nearest the high-wavenumber side of the band gap.

$I_{C_4}\simeq 2.43\times 10^{-45}\mathrm{kg}{\mathrm{m}}^2=146.6m_{\mathrm{u}}{\mathring{\mathrm{A}}}^{,2}$. The parallel-band progression gives $2\tilde{B}=0.42{\mathrm{cm}}^{-1}$, so $\tilde{B}=0.210{\mathrm{cm}}^{-1}$. The perpendicular Q-feature spacing givesFor an oblate symmetric top, $I_{C_4}>I_{\perp }$, hence $\tilde{C}<\tilde{B}$ and $\tilde{C}=0.115{\mathrm{cm}}^{-1}$. The rotational constant relation then givesThe estimated constants neglect centrifugal distortion and the change of rotational constants in the excited vibration.

For conventional CISD, the leading matrix-vector contraction scales as $O(o^2{v}^4)$ in occupied and virtual orbital dimensions. If both grow proportionally to a system-size parameter $N$, the familiar estimate is $O(N^6)$ work per iteration; the iteration count and integral-storage strategy add implementation-dependent costs.

Calculated potential energy surfaces can predict equilibrium geometries and harmonic force constants, permitting comparison with diffraction-derived structures, rotational constants and vibrational spectra. Calculating several roots or different charge states can also estimate excitation or ionization energies. Such predictions depend on basis and truncation quality; truncated CI is not generally size extensive and is not automatically reliable for strongly multiconfigurational bond breaking. The quoted polynomial scaling is for CISD, not full configuration interaction, whose determinant space grows combinatorially.

The subbranch spacing gives $\tilde{A}-\tilde{B}=4.9\mathrm{c}{\mathrm{m}}^{-1}$, hence $\tilde{A}\simeq 5.15\mathrm{c}{\mathrm{m}}^{-1}$. Rotation about the C–I symmetry axis has momentC and I lie on this axis, so only the three hydrogens contribute to $I_a$. If $\theta$ is the I–C–H angle, their perpendicular distances are $r_{\mathrm{CH}}\sin \theta$. Taking integer hydrogen mass givesThe molecular arrangement selects the obtuse solution, so $\theta \simeq 108^{\circ }$. The supplementary acute angle describes the opposite axial direction, not the bond angle to iodine. This is a rigid-geometry estimate based on an effective rotational constant and the stipulated bond length, rather than a precision equilibrium-structure measurement.

A regularly repeated set of sharp features is characteristic of the unresolved Q subbranches of a perpendicular rovibrational band. The transition dipole is perpendicular to the symmetry axis, giving $\Delta K=\pm 1$; for the Q branch $\Delta J=0$. Holding the rotational constants fixed between vibrational states cancels the $BJ(J+1)$ term, leavingMany J transitions therefore coincide within each K-changing Q subbranch in this approximation. The different K subbranches are separated byThe P and R branches have $\Delta J=-1$ and $+1$, respectively. Their extra J-dependent terms spread many transitions across a broad background at modest resolution. A parallel band, with $\Delta K=0$, would not generate this repeated Q-subbranch spacing under the stated assumptions.

Express the rotational constant as a spectroscopic wavenumber $\tilde{B}=h/(8{\pi }^{2}cI)$. The rotational absorption selection rule $\Delta J=+1$ givesSuccessive lines therefore have spacing $2\tilde{B}$, regardless of the starting $J$. Both observed spacings are $20.88\mathrm{c}{\mathrm{m}}^{-1}$, giving $\tilde{B}=10.44\mathrm{c}{\mathrm{m}}^{-1}$. The three lines are consequently the $5\to 6$, $6\to 7$ and $7\to 8$ transitions within the stated precision; the first observed line is not the $0\to 1$ line.

Using the given isotopic masses, the reduced mass isWith $\tilde{B}=1044{\mathrm{m}}^{-1}$ and $c$ in $\mathrm{m}{\mathrm{s}}^{-1}$, the bond length follows fromThe small common $0.01\mathrm{c}{\mathrm{m}}^{-1}$ offset relative to integer multiples of the spacing is immaterial at the precision of this model. A real ground-vibrational-state rotational constant determines an effective vibrationally averaged bond length; it need not equal the exact equilibrium separation.