Molecular Line Spectra

The permitted rotation rates and resulting line frequencies are determined by the quantization of angular momentum. The quantization rule for the permitted electron orbital radii $$a_{\rm n} = { n \hbar \over m_{\rm e} v}$$ in the Bohr atom quantizes the orbital angular momentum $L = m_{\rm e} a_{\rm n} v$ in multiples of $\hbar \equiv h / (2 \pi)$: $$\bbox[border:3px blue solid,7pt]{L = n \hbar}\rlap{\quad \rm {(7D1)}}$$ The rule that angular momentum is an integer multiple of $\hbar$ is universal and applies to the angular momentum of a rotating molecule as well.

Consider a diatomic molecule whose two atoms have masses $m_{\rm A}$ and $m_{\rm B}$ and whose centers are separated by the equilibrium distance $r_{\rm e}$. The individual atomic distances $r_{\rm A}$ and $r_{\rm B}$ from the center of mass must obey $$r_{\rm e} = r_{\rm A} + r_{\rm B} \qquad {\rm and} \qquad r_{\rm A} m_{\rm A} = r_{\rm B} m_{\rm B}$$

In the inertial center-of-mass frame, $$L = I \omega~,$$ where $I$ is the moment of inertia and $\omega$ is the angular velocity of the rotation. Nearly all of the mass is in the two compact (much smaller than $r_{\rm e}$) nuclei, so $I = (m_{\rm A} r_{\rm A}^2 + m_{\rm B} r_{\rm B}^2)$ and $L = (m_{\rm A} r_{\rm A}^2 + m_{\rm B} r_{\rm B}^2) \omega$.  It is convenient to rewrite this as $$L = \biggl( {m_{\rm A} m_{\rm B} \over m_{\rm A} + m_{\rm B}} \biggr) r_{\rm e}^2 \omega$$ or $$\bbox[border:3px blue solid,7pt]{L = m r_{\rm e}^2 \omega}\rlap{\quad \rm {(7D2)}}$$ where $$\bbox[border:3px blue solid,7pt]{m \equiv \biggl( {m_{\rm A} m_{\rm B} \over m_{\rm A} + m_{\rm B}} \biggr)}\rlap{\quad \rm {(7D3)}} $$ is the reduced mass of the molecule.

The rotational kinetic energy associated with this angular momentum is $$E_{\rm rot} = {I \omega^2 \over 2} = {L^2 \over 2 I}$$ The quantization of angular momentum to integer multiples of $\hbar$ implies that rotational energy is also quantized. The corresponding energy eigenvalues of the Schroedinger equation are:
$$\bbox[border:3px blue solid,7pt]{E_{\rm rot} = { J (J + 1) \hbar^2 \over 2 I}~, \qquad J = {\rm 0,~1,~2,} \dots }\rlap{\quad \rm {(7D4)}} $$

Note the inverse relation between permitted rotational energies and the moment of inertia $I$. If the upper-level rotational energy is much higher than $kT$, few molecules will be collisionally excited to that level and the line emission from molecules in that level will be very weak. For example, the minimum rotational energy of the small and light H$_2$ molecule is equivalent to a temperature $T = E_{\rm min} / k \approx 500$ K, which is much higher than the actual temperature of most interstellar H$_2$. Only relatively massive molecules are likely to be detectable radio emitters in very cold (tens of K) molecular clouds. 

Quantization of rotational energy implies that changes in rotational energy are also quantized. The energy change of permitted transitions is also governed by the quantum-mechanical selection rule $$\bbox[border:3px blue solid,7pt]{\Delta J = \pm 1}\rlap{\quad \rm {(7D5)}}$$ Going from $J$ to $J -1$ releases energy $$ \Delta E_{\rm rot} = [ J (J + 1) - (J - 1)J] { {\hbar}^2 \over 2 I} = {\hbar^2 J \over I}~. $$ The frequency of the photon emitted during this rotational transition is $$\nu = {\Delta E_{\rm rot} \over h} = {\hbar J \over 2 \pi I}~, \qquad J = {\rm 1,~2,} ...~,$$ where $J$ is the angular-momentum quantum number corresponding to the upper energy level. In terms of the molecular reduced mass $m$ and equilibrium nuclear separation $r_{\rm e}$, $$\bbox[border:3px blue solid,7pt]{\nu = { h J \over 4 \pi^2 m r_{\rm e}^2 }~, \qquad J = {\rm 1,~2,} \dots }\rlap{\quad \rm {(7D6)}}$$

Thus a plot of the radio spectrum of a particular molecular species in an interstellar cloud will look like a ladder whose steps are all harmonics of the fundamental frequency that is determined solely by the moment of inertia $I = m r_{\rm e}^2$ of that species. The relative intensities of lines in the ladder depend on the temperature of the cloud.  Since $\nu \propto m^{-1} r_{\rm e}^{-2}$, the lowest frequency of line emission depends on the mass and size of the molecule.  Large, heavy molecules in cold clouds may be seen at centimeter wavelengths, but smaller and lighter molecules emit only at millimeter wavelengths.

Example: The laboratory spectrum of the $^{12}$C$^{16}$O carbon-monoxide molecule shows that the fundamental $J = 1 \rightarrow 0$ transition emits a photon at $\nu = 115.271208$ GHz. (See the on-line spectral-line catalog called Splatalogue for accurate frequencies of radio spectral lines.) The distance $r_{\rm e}$ between the C and O nuclei can be estimated from
$$r_{\rm e} = {1 \over 2 \pi} \biggl( { h J \over m \nu} \biggr)^{1/2}$$ where the reduced mass is
$$m = {m_{\rm C} m_{\rm O} \over m_{\rm C} + m_{\rm O}} \approx m_{\rm H} \biggl( {12 \cdot 16 \over 12 + 16} \biggr) \approx 1.67 \times 10^{-24} {\rm ~g~} \times 6.86 \approx 1.15 \times 10^{-23} {\rm ~g}~.$$ Thus the equilibrium distance between the C and O nuclei is
$$ r_{\rm e} = {1 \over 2 \pi} \biggl( { 6.63 \times 10^{-27} {\rm ~erg~s} \times 1 \over 115.271208 \times 10^9 {\rm ~Hz~} \times 1.15 \times 10^{-23} {\rm ~g}} \biggr)^{1/2} \approx 1.13 \times 10^{-8} {\rm ~cm}$$

The centrifugal forces acting on the nuclei increase with $J$, so the bond will stretch and $r_{\rm e}$ will increase slightly with $J$. Spectral lines emitted by more rapidly rotating $^{12}$C$^{16}$O molecules will have frequencies slightly lower than the harmonics $2 \nu_{1-0}$, $3 \nu_{1-0}$, $...$ of the $J = 1 \rightarrow 0$ line: $2 \nu_{1-0} = 230.542416$  GHz and the actual $J = 2 \rightarrow 1$ frequency is $\nu_{2-1} = 230.538000$ GHz. Chemists use these line frequencies to determine $r_{\rm e}$, and the difference between $2 \nu_{1-0}$ and $\nu_{2-1}$ is a measure of the "stiffness" of the carbon-oxygen chemical bond. Since the actual frequency is only slightly less than the harmonic frequency, the stiffness is quite high and it is clear that the fundamental vibrational frequency of the CO molecule will be much higher than the fundamental rotational frequency. Thus the vibrational line spectrum of CO begins at infrared wavelengths.

Equation 7D6 can be used to calculate frequencies for rare isotopic molecules (e.g., $^{13}$C$^{16}$O) that might be more difficult to measure in the lab.
$${m(^{13}{\rm C}^{16}{\rm O}) \over m(^{12}{\rm C}^{16}{\rm O})} = { 13 \cdot 16 / (13 + 16) \over 12 \cdot 16 / (12 + 16)} \approx 1.0460$$ so we expect
$${ \nu_{1-0} (^{13}{\rm C}^{16}{\rm O}) \over \nu_{1-0} (^{12}{\rm C}^{16}{\rm O}) } \approx \biggl[{m(^{13}{\rm C}^{16}{\rm O}) \over m(^{12}{\rm C}^{16}{\rm O})} \biggr]^{-1}$$ $$ \nu_{1-0} (^{13}{\rm C}^{16}{\rm O}) \approx 115.271208 / 1.0460 \approx 110.204 {\rm ~GHz}$$
The actual $^{13}$C$^{16}$O $J = 1 \rightarrow 2$ frequency is 110.201354 GHz.

This simple analysis shows that polar diatomic molecules emit a harmonic series of radio spectral lines at millimeter wavelengths. Bigger and heavier linear polyatomic molecules have ladders of lines starting at somewhat lower frequencies.

Nonlinear molecules such as the symmetric-top ammonia (NH$_3$) with two distinct rotational axes have more complex spectra consisting of many parallel ladders.

Energy levels of ammonia (NH$_3$) in the lowest vibrational state (Wilson, T. L. et al. 1993, A&A, 276, L29).  On the abscissa, $K$ is the quantum number corresponding to the $z$-component of the angular momentum.  Transitions between the two spin states of the nitrogen atom cause the line splitting shown and yield emission at frequencies near 24 GHz. NH$_3$ is a very useful thermometer for molecular clouds (Ho, P. T. P., & Townes, C. H. 1983, ARA&A, 21, 239).

Molecular Excitation

Molecules are excited into $E_{\rm rot} > 0$ states by ambient radiation and by collisions in a gas. The minimum gas temperature $T_{\rm min}$ needed for significant collisional excitation is roughly
$$T_{\rm min} \sim { E_{\rm rot} \over k}$$ From equations 7D4 and 7D6,
$$E_{\rm rot} = {J (J + 1) \hbar^2 \over 2 I} \qquad {\rm and} \qquad \nu = {h J \over 4 \pi^2 I}$$ so
$$T_{\rm min} \sim { J (J+1) h^2 \over 2 \cdot 4 \pi^2 I k} = { h J \over 4 \pi^2 I} { h (J + 1) \over 2 k}$$ Thus a minimum gas kinetic temperature
$$\bbox[border:3px blue solid,7pt]{T_{\rm min} \approx { \nu h (J + 1) \over 2 k}}\rlap{\quad \rm {(7D7)}}$$ is required to excite the $J \rightarrow J -1$ transition at frequency $\nu$.

Example: What is the minimum gas temperature needed for significant excitation of the $^{12}$C$^{16}$O $J = 2-1$ line at $\nu \approx 230.5$ GHz?

$$T_{\rm min} \approx { \nu h (J + 1) \over 2 k} \approx { 230.5 \times 10^9 {\rm ~Hz~} \cdot 6.63 \times 10^{-27} {\rm ~erg~s~} (2 + 1) \over 2 \times 1.3 8 \times 10^{-16} {\rm ~erg~K}^{-1} } \approx 16.6 {\rm ~K}$$

The $T_{\rm min} = E_{\rm u} / {\rm K}$ values for many molecular lines may be found in the on-line spectral-line catalog Splatalogue.  If $T_{\rm min} \gg 2.7$ K, then radiative excitation by the cosmic microwave background is ineffective. 

Molecular Line Strengths

Larmor's formula for a time-varying dipole can be applied to estimate the average power radiated by a rotating polar molecule. The electric dipole moment $\vec{d}$ of any charge distribution $\rho(\vec{x})$ is defined as the integral
$$\vec{d} \equiv \int \vec{x} \rho(v) dv~,$$ over the volume $v$ containing the charges. In the case of two point charges $+q$ and $-q$ separated by $r _{\rm e}$ on the $x$-axis,
$$ \vert d \vert = q r_{\rm e}~.$$

For neutral molecules, $q < \vert e \vert$. For a molecule rotating with angular velocity $\omega$, the projection of the dipole moment perpendicular to the line of sight varies with time as $q r_{\rm e} \exp( - i \omega t)$. Recall that
$$ r_{\rm A} m_{\rm A } = r_{\rm B} m_{\rm B} $$ so
$$ \dot{v}_{\rm A}= \ddot{r}_{\rm A} = \omega^2 r _{\rm A} \qquad {\rm and} \qquad \dot{v}_{\rm B} = \ddot{r}_{\rm B} = \omega^2 r _{\rm B}$$ From our derivation of the Larmor formula, we found for each charge $$E_\bot = { q \dot{v} \sin \theta \over r c^2}~,$$ where $r$ is the distance from the source. We add the contributions of both charges to get the total $E_\bot$:
$$ E_\bot = { q ( \omega^2 r_{\rm A} + \omega^2 r_{\rm B} ) \sin \theta \over r c^2} \exp ( - i \omega t)$$ Thus the instantaneous power emitted is
$$P = {2 q^2 \over 3 c^3} \omega^4 \vert r_{\rm e} \exp( -i \omega t) \vert^2~,$$ where $r_{\rm e} = r_{\rm A} + r_{\rm B}$. The time-averaged power is $$ \langle P \rangle = {2 q^2 \over 3 c^3} (2 \pi \nu)^4 {r_{\rm e}^2 \over 2}~.$$
$$\langle P \rangle = { 64 \pi^4 \over 3 c^3} \nu^4 \biggl( { q r_{\rm e} \over 2} \biggr)^2$$
$$\langle P \rangle = { 64 \pi^4 \over 3 c^3} \nu^4 \vert \mu \vert^2~,$$ where
$$\vert \mu \vert^2 \equiv \biggl( {q r_{\rm e} \over 2} \biggr)^2$$ defines the mean electric dipole moment $\mu$. For a radiative transition characterized by upper and lower levels U and L, the spontaneous emission coefficient is
$$A_{\rm UL} = { \vert P \vert \over h \nu_{\rm UL}} $$
$$\bbox[border:3px blue solid,7pt]{A_{\rm UL} = {64 \pi^4 \over 3 h c^3} \nu_{\rm UL}^3 \vert \mu_{\rm UL} \vert^2}\rlap{\quad \rm {(7D8)}}$$ Notice that the spontaneous emission coefficient is proportional to frequency cubed, so spectral lines are more prominent at higher radio frequencies.

Many important interstellar molecules (e.g., H$_2$, O$_2$, etc.) are symmetric and hence have no permanent electric dipole moments and aren't radio line sources. Those with permanent dipole moments $\mu$ (e.g., CO molecules have $\mu \sim 10^{-19}$ statcoul cm $= 0.1 $ Debye, where 1 Debye $\equiv 10^{-18}$ statcoul cm) have
$$\bbox[border:3px blue solid,7pt]{\vert \mu_{{\rm J}\rightarrow{\rm J}-1} \vert^2 = {\mu^2 J \over 2 J + 1 }}\rlap{\quad \rm {(7D9)}}$$ (This equation reflects the complex angular wavefunctions involved, and we won't derive it here.)

Example: Estimate the spontaneous emission coefficient $A_{10}$ for the CO $ J = 1 \rightarrow 0$ line, given the CO dipole moment $\mu \approx 0.11$ Debye.

$$\vert \mu_{1\rightarrow0} \vert^2 \approx {(0.11 \times 10^{-18} {\rm ~statcoul~cm})^2 \cdot 1 \over (2+1)} \approx 4.0 \times 10^{-39} {\rm ~statcoul }^2 {\rm ~cm}^2$$ $$A_{10} = {64 \pi^4 \over 3 h c^3} \nu_{1-0}^3 \vert \mu_{1-0 } \vert^2$$ $$A_{10} \approx { 64 \pi^4 \cdot (115 \times 10^9 {\rm ~Hz})^2 \cdot 4.0 \times 10^{-39} {\rm ~statcoul}^2 {\rm ~cm}^2 \over 3 \cdot 6.63 \times 10^{-27} {\rm ~erg~s} \cdot (3 \times 10^{10} {\rm ~cm~s}^{-1})^2 }$$ $$A_{10} \approx 7.1 \times 10^{-8} {\rm ~s}^{-1} \sim 2.3 {\rm ~yr}^{-1}$$ This is close to the value listed by the Splatalogue,  $A_{10} \approx 7.202 \times 10^{-8}$ s$^{-1}$.

The typical time $A_{10}^{-1} \approx 10^7$ s to emit one photon spontaneously is much longer than the average time between molecular collisions in an interstellar molecular cloud with density $10^2$ to $10^3$ H$_2$ molecules cm$^{-3}$ (H$_2$ is by far the most abundant molecule), so the CO can approach LTE with the excitation temperature of the $J= 1\rightarrow0$ line being nearly equal to the kinetic temperature $T$ of the molecular cloud.  For any molecular transition, there is a critical density $$\bbox[border:3px blue solid,7pt]{n^* \approx {A_{\rm UL} \over \sigma v}}\rlap{\quad \rm {(7D10)}}$$ at which the radiating molecule suffers collisions at the rate $n$(H$_2$)$\sigma v$ equals $A$.  Typical collision cross sections are $\sigma \sim 10^{-15}$ cm$^2$.  Since the most abundant molecule is H$_2$, the average velocity is $v \approx (3 k T / m)^{1/2} \sim 10^5$ cm s$^{-1}$ for $T \sim 100$ K. Transitions with high emission coefficients [e.g., the HCN (hydrogen cyanide) $J = 1 \rightarrow 0$ line at $\nu \approx 88.63$ GHz has $A \approx 2.0 \times 10^{-5}$ s$^{-1}$] are collisionally excited only at very high densities ($n^* \approx 10^5$ cm$^{-3}$ for HCN $J = 1 \rightarrow 0$).  They are valuable for pinpointing the very dense gas directly associated with the formation of individual stars, unlike the CO $J = 1 \rightarrow 0$ transition which is more widespread.

The discovery of ammonia (NH$_3$) in the direction of the galactic center by Cheung et al. (1968, Phys Rev Lett, 21, 1701) immediately led to the realization that the interstellar medium must contain regions much denser than previously expected because the critical density needed to excite the NH$_3$ line is $n^* \approx 10^3$ cm$^{-3}$.

Some Astronomical Applications of Molecular Lines

Formation of interstellar glycoaldehyde, a sugar consisting of two carbon atoms, two oxygen atoms, and four hydrogen atoms.  This pre-biotic molecule is a building block for DNA and RNA. The GBT detected its line emission from Sagittarius B2 near the galactic center.  Image credit

Red- and blue-shifted CO outflows from a young stellar object in the dark cloud L1157. Image credit

The protostar NGC 1333 IRS 4A (gray spot) has ejected two supersonic jets whose Doppler velocities are indicated by the false colors (blue = approaching, red = receding) in this VLA image of a 43 GHz SiO (silicon monoxide) line.  SiO is an excellent tracer of shocked molecular gas. Image credit

CO $J = 1 \rightarrow 0$ emission from a shell ejected by the evolved carbon star TT Cyg, imaged by the Plateau de Bure interferometer (Olofsson et al. 2000, A&A, 353, 583).

The CO $J = 1 \rightarrow 0$ contours superimposed on a gray-scale optical image of NGC 5194 trace the spiral arms and outline regions in which stars are forming (Regan, M. W. et al. 2001, ApJ, 561 218).  Strong CO emission is ubiquitous in star-forming galaxies, and one of the ALMA science goals is to detect star formation in normal galaxies like our own out to redshifts $z \sim 3$.

The circumnuclear water maser in NGC 4258. The central group of spectral lines is at the systemic recession velocity of about 500 km s$^{-1}$.  The redshifted lines at about 1300 km s$^{-1}$ and the blueshifted lines at about $-400$ km s$^{-1}$ reflect motions of the gas orbiting the central black hole. Image credit

Keplerian orbits of the circumnuclear water masers in NGC 4258. They have been used to measure both the mass $M = 39 \pm 1 \times 10^6 ~M_\odot$ of the central black hole and its distance $D = 7.2 \pm 0.5$ Mpc.  Image credit

This VLA image of the 346 GHz CO $J=3\rightarrow 2$ emission from the z = 6.42 quasar J1148+5251 redshifted to 46.61 GHz (Walter et al. 2004, ApJ, 615, L17) constrains the co-evolution of stars and supermassive black holes when the universe was $< 10^9$ years old.