Molecular Rotational and Vibrational Spectra

Derivation

Molecular Rotational and Vibrational Spectra

D-312 Home PU-306 Threads energy · matter · waves · light Depends on The Born-Oppenheimer Approximation, Electric-Dipole Selection Rules

Statement

For a diatomic molecule in a single electronic state, the rigid rotor gives rotational term values \(F(J)=\tilde{B}\,J(J+1)\); harmonic vibration gives \(G(v)=\tilde{\omega}_e\left(v+\tfrac12\right)\), corrected by the Morse potential to \(G(v)=\tilde{\omega}_e\left(v+\tfrac12\right)-\tilde{\omega}_e x_e\left(v+\tfrac12\right)^2\). Combining these with the electric-dipole selection rules \(\Delta v=+1,\ \Delta J=\pm1\) produces a rovibrational band split into a P branch (\(\Delta J=-1\)) and an R branch (\(\Delta J=+1\)) with an approximately \(2\tilde{B}\) line spacing and a missing line at the band origin.

Why it matters

Infrared and microwave spectra are read almost entirely through these two formulas: a single measured line spacing yields the moment of inertia and hence a bond length to picometre precision, while the anharmonic constant fixes the dissociation energy and the shape of the potential well. This is the primary experimental route to molecular geometry for gas-phase species.

The same term-value scheme underlies atmospheric radiative transfer, astrochemical line identification, and laser cooling of molecules. Because rotational and vibrational energy scales differ by \(\sim10^2\), the derivation also cleanly illustrates the hierarchy of separations the Born–Oppenheimer picture sets up.

Assumptions

Born–Oppenheimer separation holds.If dropped, nuclear motion couples to electronic state and the single effective potential \(V(r)\) is no longer well defined; terms cannot be written as \(G(v)+F(J)\).

Nuclear motion separates into radial (vibration) and angular (rotation) parts.If dropped, the \(J\)-dependent centrifugal term stays entangled with the vibrational problem and \(\tilde{B}\) and \(\tilde{\omega}_e\) are not independent constants.

Rigid-rotor limit: bond length fixed at \(r_e\) for rotation.If dropped, centrifugal stretching lowers levels by \(-\tilde{D}_J[J(J+1)]^2\) and the spacing is no longer uniform.

The potential is analytic near \(r_e\) and truncatable at low order (harmonic), then modelled by a Morse curve for anharmonicity.If dropped, higher derivatives of \(V\) contribute and the two-parameter \((\tilde{\omega}_e,\tilde{\omega}_e x_e)\) fit fails for high \(v\).

The electronic state is \(^1\Sigma\) (no electronic or vibrational angular momentum about the axis).If dropped, \(\Delta J=0\) becomes allowed and a Q branch appears at the band origin.

Derivation

\[ -\frac{\hbar^2}{2\mu}\nabla^2\psi(\mathbf{r}) + V(r)\,\psi(\mathbf{r}) = E\,\psi(\mathbf{r}), \qquad \mu=\frac{m_1 m_2}{m_1+m_2} \]

Born–Oppenheimer gives a single effective potential \(V(r)\); the two-body nuclear problem reduces to one particle of reduced mass \(\mu\) in a central field. A

\[ \psi(\mathbf{r}) = \frac{u(r)}{r}\,Y_{JM}(\theta,\phi), \qquad \hat{L}^2 Y_{JM}=\hbar^2 J(J+1)\,Y_{JM} \]

Central symmetry lets the wavefunction separate into a radial factor and a spherical harmonic, the eigenfunctions of \(\hat{L}^2\). A

\[ -\frac{\hbar^2}{2\mu}\frac{d^2u}{dr^2} + \left[ V(r) + \frac{\hbar^2 J(J+1)}{2\mu r^2} \right] u = E\,u \]

Substituting the separated form isolates a one-dimensional radial equation with an added centrifugal potential. B

\[ \text{Rigid rotor: } V(r)\to 0,\quad r\to r_e \;\Rightarrow\; E_J = \frac{\hbar^2}{2 I}J(J+1),\qquad I=\mu r_e^2 \]

Freezing \(r=r_e\) leaves only the centrifugal term, whose eigenvalue follows directly from \(\hat{L}^2\); the moment of inertia is \(I=\mu r_e^2\). B

\[ F(J)\equiv\frac{E_J}{hc} = \tilde{B}\,J(J+1), \qquad \tilde{B}=\frac{\hbar}{4\pi c\, I}=\frac{h}{8\pi^2 c\,\mu r_e^2} \]

Dividing energy by \(hc\) converts to spectroscopic term values (wavenumbers) and defines the rotational constant \(\tilde{B}\). A

\[ V(r) = V(r_e) + \tfrac12 k\,(r-r_e)^2 + \cdots,\qquad k=\left.\frac{d^2V}{dr^2}\right|_{r_e} \]

Taylor-expanding \(V\) about its minimum kills the linear term; the leading curvature term is the harmonic force constant \(k\). A

\[ -\frac{\hbar^2}{2\mu}\frac{d^2u}{dx^2}+\tfrac12 k x^2 u = E\,u,\quad x=r-r_e \;\Rightarrow\; E_v=\hbar\omega\!\left(v+\tfrac12\right),\ \ \omega=\sqrt{\frac{k}{\mu}} \]

Truncating at quadratic order (and dropping centrifugal coupling) is the quantum harmonic oscillator, whose spectrum is textbook. B

\[ G(v)=\frac{E_v}{hc}=\tilde{\omega}_e\!\left(v+\tfrac12\right),\qquad \tilde{\omega}_e=\frac{1}{2\pi c}\sqrt{\frac{k}{\mu}} \]

Converting to wavenumbers defines the harmonic vibrational constant \(\tilde{\omega}_e\). A

\[ V(r)=D_e\left[1-e^{-a(r-r_e)}\right]^2 \;\Rightarrow\; G(v)=\tilde{\omega}_e\!\left(v+\tfrac12\right)-\tilde{\omega}_e x_e\!\left(v+\tfrac12\right)^2 \]

The Morse potential is one of the few anharmonic wells with an exact closed-form spectrum; solving its radial equation yields a quadratic-in-\((v+\tfrac12)\) term. C

\[ \tilde{\omega}_e=\frac{a}{2\pi c}\sqrt{\frac{2D_e}{\mu}},\qquad \tilde{\omega}_e x_e=\frac{a^2\hbar}{4\pi c\,\mu}=\frac{\tilde{\omega}_e^{\,2}}{4\tilde{D}_e},\qquad \tilde{D}_e=\frac{D_e}{hc} \]

Matching the Morse eigenvalues to the term expansion identifies the physical constants; the anharmonicity is fixed by \(\tilde{\omega}_e\) and the well depth. C

\[ S(v,J)=G(v)+F_v(J),\qquad F_v(J)=\tilde{B}_v\,J(J+1),\quad \tilde{B}_v=\tilde{B}_e-\alpha_e\!\left(v+\tfrac12\right) \]

Adding rotational and vibrational terms; averaging \(\langle 1/r^2\rangle\) over an anharmonic level makes \(\tilde{B}_v\) shrink linearly with \(v\) (vibration–rotation coupling). B

\[ \tilde{\nu}=S(1,J')-S(0,J''),\qquad \Delta v=+1,\ \ \Delta J=\pm1 \]

The electric-dipole selection rules (assumed prior result) restrict the allowed transitions; \(\Delta J=0\) is forbidden for a \(^1\Sigma\) state. A

\[ \tilde{\nu}_R(J)=\tilde{\nu}_0+2\tilde{B}_1+(3\tilde{B}_1-\tilde{B}_0)J+(\tilde{B}_1-\tilde{B}_0)J^2,\quad J=0,1,2,\dots \]

Setting \(J'=J''+1\) (R branch) and collecting powers of \(J''\equiv J\); \(\tilde{\nu}_0=\tilde{\omega}_e-2\tilde{\omega}_e x_e\) is the \(0\to1\) band origin. B

\[ \tilde{\nu}_P(J)=\tilde{\nu}_0-(\tilde{B}_1+\tilde{B}_0)J+(\tilde{B}_1-\tilde{B}_0)J^2,\quad J=1,2,3,\dots \]

Setting \(J'=J''-1\) (P branch) with \(J''\equiv J\); no \(J=0\) term exists because \(J''=0\) has no \(\Delta J=-1\) partner. B

Result

\[ F(J)=\tilde{B}\,J(J+1),\qquad G(v)=\tilde{\omega}_e\!\left(v+\tfrac12\right)-\tilde{\omega}_e x_e\!\left(v+\tfrac12\right)^2 \]

\[ \tilde{\nu}_R(J)=\tilde{\nu}_0+2\tilde{B}(J+1),\qquad \tilde{\nu}_P(J)=\tilde{\nu}_0-2\tilde{B}J \quad (\tilde{B}_0\approx\tilde{B}_1\equiv\tilde{B}) \]

Reading. Rotational levels fan out quadratically in \(J\), so their spacing \(F(J+1)-F(J)=2\tilde{B}(J+1)\) grows linearly. Vibrational levels start at spacing \(\tilde{\omega}_e-2\tilde{\omega}_e x_e\) and converge toward dissociation. In the rovibrational band the R and P branches sit symmetrically about the origin \(\tilde{\nu}_0\), spaced by \(\approx 2\tilde{B}\), with a gap of \(4\tilde{B}\) at the centre where the forbidden \(\Delta J=0\) line would fall. Because \(\tilde{B}_1<\tilde{B}_0\), R lines slowly crowd together (forming a band head) while P lines spread apart.

Units check. \(\tilde{B}=\dfrac{h}{8\pi^2 c\,\mu r_e^2}\): \(\dfrac{\mathrm{J\,s}}{(\mathrm{cm\,s^{-1}})(\mathrm{kg})(\mathrm{m^2})}=\dfrac{\mathrm{J\,s}}{\mathrm{kg\,m^2\,cm\,s^{-1}}}=\dfrac{\mathrm{J\,s}}{\mathrm{J\,s\,cm}}=\mathrm{cm^{-1}}\) (using \(\mathrm{J}=\mathrm{kg\,m^2\,s^{-2}}\)). Likewise \(\tilde{\omega}_e x_e=\tilde{\omega}_e^2/4\tilde{D}_e\) has units \(\mathrm{cm^{-2}/cm^{-1}=cm^{-1}}\). All term values are wavenumbers, as required.

Limiting cases

Harmonic limit \(\tilde{\omega}_e x_e\to0\): equally spaced vibrational levels; overtones \(\Delta v=\pm2,\pm3\) vanish.
Pure rotation (\(\Delta v=0\)): microwave spectrum of lines at \(2\tilde{B}(J+1)\), spacing exactly \(2\tilde{B}\).
Rigid rotor \(\tilde{D}_J\to0\): centrifugal distortion negligible for low \(J\); levels strictly \(\tilde{B}J(J+1)\).
No vibration–rotation coupling \(\alpha_e\to0\): \(\tilde{B}_0=\tilde{B}_1\); P and R lines exactly evenly spaced, no band head.
High-\(J\) classical limit: \(F(J)\to\tilde{B}J^2\), matching the rigid classical rotor energy \(L^2/2I\).

Breaks when

High \(v\) near dissociation: the Morse quadratic underestimates the true term convergence; real potentials need cubic \(\tilde{\omega}_e y_e(v+\tfrac12)^3\) corrections or full RKR inversion, and the level count \(v_\mathrm{max}\) is only approximate.
High \(J\) (fast rotation): centrifugal stretching bends the bond, so \(F_v(J)=\tilde{B}_vJ(J+1)-\tilde{D}_J[J(J+1)]^2\); ignoring \(\tilde{D}_J\) overpredicts high-\(J\) line positions.
Open-shell or \(\Pi/\Delta\) electronic states: electronic angular momentum allows \(\Delta J=0\), adding a Q branch and lambda-doubling the derivation omits.
Breakdown of Born–Oppenheimer (light nuclei, near curve crossings): vibronic coupling mixes electronic states and the single \(V(r)\) picture fails.

Failure modes

Confusing \(\tilde{B}\)-in-energy with \(\tilde{B}\)-in-wavenumbers: writing \(E=\tilde{B}J(J+1)\) with \(\tilde{B}\) in cm\(^{-1}\) without the \(hc\) factor gives energies off by \(\sim10^{-23}\).
Using atomic masses instead of the reduced mass \(\mu\) in \(I=\mu r_e^2\); a very common two-fold or worse error in bond lengths.
Line spacing quoted as \(\tilde{B}\) instead of \(2\tilde{B}\): forgetting that adjacent lines differ by \(F(J+1)-F(J)=2\tilde{B}(J+1)\).
Expecting a line at the band centre: plotting a peak at \(\tilde{\nu}_0\) rather than the characteristic \(4\tilde{B}\) gap of a \(^1\Sigma\) band.
Sign slip in anharmonicity: adding \(+\tilde{\omega}_e x_e(v+\tfrac12)^2\), which makes levels diverge instead of converge.
Applying \(\Delta v=\pm1\) as if it forbade all overtones: it is exact only for a strictly harmonic dipole; anharmonicity makes weak overtones real.

Discussion

The clean factorization \(S(v,J)=G(v)+F(J)\) is a direct dividend of the Born–Oppenheimer hierarchy: electronic energies scale as \(\sim1\) eV, vibrational as \(\sim\sqrt{m_e/\mu}\), and rotational as \(\sim m_e/\mu\). With \(\mu\sim10^3\,m_e\) this predicts vibrational quanta \(\sim10^2\) times rotational quanta, exactly the \(\tilde{\omega}_e\sim3000\) cm\(^{-1}\) versus \(\tilde{B}\sim10\) cm\(^{-1}\) split seen in HCl. The spectrum is therefore a vibrational transition dressed with a fine comb of rotational structure.

The method of combination differences turns this structure into precision metrology. Since \(\tilde{\nu}_R(J-1)-\tilde{\nu}_P(J+1)=4\tilde{B}_0(J+\tfrac12)\) depends only on the lower vibrational state and \(\tilde{\nu}_R(J)-\tilde{\nu}_P(J)=4\tilde{B}_1(J+\tfrac12)\) only on the upper state, a linear fit of these differences against \((J+\tfrac12)\) extracts \(\tilde{B}_0\) and \(\tilde{B}_1\) separately, and hence the change in bond length upon vibrational excitation.

The Morse solution is special because its bound spectrum is exactly \(\tilde{\omega}_e(v+\tfrac12)-\tilde{\omega}_e x_e(v+\tfrac12)^2\) with no higher powers, letting the well depth be read off as \(\tilde{D}_e=\tilde{\omega}_e^2/4\tilde{\omega}_e x_e\). Real potentials deviate: the number of bound levels, \(v_\mathrm{max}\approx \tilde{\omega}_e/2\tilde{\omega}_e x_e-\tfrac12\), and the true \(\tilde{D}_0\) obtained by a Birge–Sponer extrapolation of observed level spacings usually fall below the Morse estimate, because the Morse tail is too steep. The residual is itself diagnostic of the potential's long-range \(-C_6/r^6\) behaviour.

Common misconceptions. The \(\Delta J=0\) gap is not an instrumental artefact — it is a genuine selection-rule consequence of angular-momentum conservation for a \(^1\Sigma\) state with no axial electronic angular momentum. And a rovibrational "band" is one vibrational transition, not many; the dozens of lines are all \(v=0\to1\), differing only in \(J\).

Worked examples

Bond length of H\(^{35}\)Cl from \(\tilde{B}\). Given \(\tilde{B}=10.59\ \mathrm{cm^{-1}}\).

\[ \mu=\frac{m_\mathrm{H}m_\mathrm{Cl}}{m_\mathrm{H}+m_\mathrm{Cl}}=\frac{1.008\times34.97}{35.98}\ \mathrm{u}=0.9722\ \mathrm{u}=1.6145\times10^{-27}\ \mathrm{kg} \]

\[ I=\frac{h}{8\pi^2 c\,\tilde{B}}=\frac{6.626\times10^{-34}}{8\pi^2\,(2.998\times10^{10}\,\mathrm{cm\,s^{-1}})(10.59\,\mathrm{cm^{-1}})}=2.643\times10^{-47}\ \mathrm{kg\,m^2} \]

\[ r_e=\sqrt{\frac{I}{\mu}}=\sqrt{\frac{2.643\times10^{-47}}{1.6145\times10^{-27}}}=\sqrt{1.637\times10^{-20}}\ \mathrm{m} \]

Reduced mass, then moment of inertia from \(\tilde{B}\), then \(r_e=\sqrt{I/\mu}\); \(c\) in cm s\(^{-1}\) to match \(\tilde{B}\) in cm\(^{-1}\). A

\[ r_e=1.279\times10^{-10}\ \mathrm{m}=127.9\ \mathrm{pm} \]

Reading. Agrees with the accepted HCl bond length (\(\approx127.5\) pm); a single spacing yields geometry to \(\sim1\%\).

Well depth and level count of HCl from anharmonicity. Given \(\tilde{\omega}_e=2990.9\ \mathrm{cm^{-1}}\), \(\tilde{\omega}_e x_e=52.8\ \mathrm{cm^{-1}}\).

\[ \tilde{D}_e=\frac{\tilde{\omega}_e^{\,2}}{4\,\tilde{\omega}_e x_e}=\frac{(2990.9)^2}{4(52.8)}\ \mathrm{cm^{-1}}=\frac{8.945\times10^{6}}{211.2}\ \mathrm{cm^{-1}}=4.236\times10^{4}\ \mathrm{cm^{-1}} \]

\[ D_e=hc\,\tilde{D}_e=(1.986\times10^{-23}\,\mathrm{J\,cm})(4.236\times10^{4}\,\mathrm{cm^{-1}})\,N_A=506.6\ \mathrm{kJ\,mol^{-1}} \]

\[ v_\mathrm{max}\approx\frac{\tilde{\omega}_e}{2\,\tilde{\omega}_e x_e}-\frac12=\frac{2990.9}{105.6}-0.5=27.8 \;\Rightarrow\; v_\mathrm{max}=27 \]

Morse relations give the well depth directly; the top bound level is where \(dG/dv=0\). Numbers only after the symbolic forms. B

\[ \tilde{D}_e\approx 42\,360\ \mathrm{cm^{-1}}\ (506\ \mathrm{kJ\,mol^{-1}}),\qquad \sim28\ \text{bound levels} \]

Reading. The Morse \(D_e\) overestimates the true \(\approx445\ \mathrm{kJ\,mol^{-1}}\) because the Morse tail is too steep; still the right order and a useful first estimate. Zero-point-corrected \(D_0=D_e-\tfrac12\tilde{\omega}_e+\tfrac14\tilde{\omega}_e x_e\approx40\,880\ \mathrm{cm^{-1}}\).

Problems

The pure-rotational spectrum of \(^{12}\mathrm{C}^{16}\mathrm{O}\) has \(\tilde{B}=1.931\ \mathrm{cm^{-1}}\). Find the bond length.
Solution
\(\mu=\dfrac{12.00\times15.995}{27.995}\,\mathrm{u}=6.856\,\mathrm{u}=1.1385\times10^{-26}\,\mathrm{kg}\). \(I=\dfrac{h}{8\pi^2 c\tilde{B}}=\dfrac{6.626\times10^{-34}}{8\pi^2(2.998\times10^{10})(1.931)}=1.449\times10^{-46}\,\mathrm{kg\,m^2}\). \(r_e=\sqrt{I/\mu}=\sqrt{1.449\times10^{-46}/1.1385\times10^{-26}}=\sqrt{1.273\times10^{-20}}=1.128\times10^{-10}\,\mathrm{m}=112.8\,\mathrm{pm}\), matching the accepted CO bond length.
For the HCl \(v=0\to1\) band with \(\tilde{\nu}_0=2886\ \mathrm{cm^{-1}}\) and \(\tilde{B}\approx10.4\ \mathrm{cm^{-1}}\) (treat \(\tilde{B}_0\approx\tilde{B}_1\)), give the first three R-branch and first three P-branch line positions.
Solution
Using \(\tilde{\nu}_R(J)=\tilde{\nu}_0+2\tilde{B}(J+1)\): \(R(0)=2886+20.8=2906.8\), \(R(1)=2886+41.6=2927.6\), \(R(2)=2886+62.4=2948.4\ \mathrm{cm^{-1}}\). Using \(\tilde{\nu}_P(J)=\tilde{\nu}_0-2\tilde{B}J\): \(P(1)=2886-20.8=2865.2\), \(P(2)=2886-41.6=2844.4\), \(P(3)=2886-62.4=2823.6\ \mathrm{cm^{-1}}\). Note the \(4\tilde{B}\approx41.6\ \mathrm{cm^{-1}}\) gap between \(R(0)\) and \(P(1)\) straddling the missing origin.
Estimate the most populated rotational level \(J_\mathrm{max}\) of HCl (\(\tilde{B}=10.59\ \mathrm{cm^{-1}}\)) at \(T=300\ \mathrm{K}\).
Solution
Maximising \((2J+1)e^{-\tilde{B}J(J+1)hc/k_BT}\) gives \(J_\mathrm{max}=\sqrt{\dfrac{k_BT}{2hc\tilde{B}}}-\dfrac12\). With \(k_BT/hc=(0.6950\,\mathrm{cm^{-1}\,K^{-1}})(300\,\mathrm{K})=208.5\ \mathrm{cm^{-1}}\): \(J_\mathrm{max}=\sqrt{208.5/(2\times10.59)}-0.5=\sqrt{9.84}-0.5=3.14-0.5=2.6\). So \(J_\mathrm{max}\approx3\); the R(2)/R(3) and P(3) lines are among the strongest.
In the HCl \(0\to1\) band the combination differences give \(\tilde{\nu}_R(J-1)-\tilde{\nu}_P(J+1)=4\tilde{B}_0(J+\tfrac12)\). If this difference is measured as \(188.0\ \mathrm{cm^{-1}}\) at \(J=4\), find \(\tilde{B}_0\) and the corresponding \(r_0\).
Solution
\(4\tilde{B}_0(4+\tfrac12)=188.0\Rightarrow \tilde{B}_0=\dfrac{188.0}{4\times4.5}=\dfrac{188.0}{18.0}=10.44\ \mathrm{cm^{-1}}\), the expected HCl lower-state constant. Then \(I=\dfrac{h}{8\pi^2 c\tilde{B}_0}=\dfrac{6.626\times10^{-34}}{8\pi^2(2.998\times10^{10})(10.44)}=2.681\times10^{-47}\,\mathrm{kg\,m^2}\), and \(r_0=\sqrt{I/\mu}=\sqrt{2.681\times10^{-47}/1.6145\times10^{-27}}=1.288\times10^{-10}\,\mathrm{m}=128.8\ \mathrm{pm}\), slightly longer than \(r_e\) because vibrational averaging over the anharmonic well lengthens the mean bond. The lesson is procedural: the lower-state constant, and hence \(r_0\), comes purely from \(R(J-1)-P(J+1)\), independent of the upper state.
A Morse oscillator has \(\tilde{\omega}_e=2170\ \mathrm{cm^{-1}}\) and \(\tilde{\omega}_e x_e=13.3\ \mathrm{cm^{-1}}\) (CO). Find (a) the fundamental \(1\leftarrow0\) and (b) the first overtone \(2\leftarrow0\) wavenumbers, and (c) how many bound levels the well supports.
Solution
\(G(v)=\tilde{\omega}_e(v+\tfrac12)-\tilde{\omega}_e x_e(v+\tfrac12)^2\). (a) Fundamental: \(G(1)-G(0)=\tilde{\omega}_e-2\tilde{\omega}_e x_e=2170-26.6=2143.4\ \mathrm{cm^{-1}}\). (b) Overtone: \(G(2)-G(0)=2\tilde{\omega}_e-6\tilde{\omega}_e x_e=4340-79.8=4260.2\ \mathrm{cm^{-1}}\) (slightly less than twice the fundamental, as expected). (c) \(v_\mathrm{max}\approx\tilde{\omega}_e/2\tilde{\omega}_e x_e-\tfrac12=2170/26.6-0.5=81.6-0.5\approx81\), so about 82 bound levels — the Morse estimate; the true CO well is deeper still.