ISM Lecture 12

1. ISM Lecture 12 Molecular Clouds I: Molecular spectroscopy, H2

2. 12.1 Molecular structure Ryb & Lightman Chap. 11 Lectures H. Linnartz • All information about molecule is contained in Schrödinger equation (R positions of nuclei, x positions of electrons): • Hamilton operator:

3. Born-Oppenheimer approximation (1927) • Mass of nuclei >> mass of electrons  nuclei move slowly compared with electrons  • Separate wave function into electronic and nuclear part, and determine motion of electrons first with nuclei held fixed Electronic potential energy surface

4. 12.2 Electronic energies Example: H2+ ion • Solution of electronic Schrödinger equation for H2+ ion leads to two low-lying potential energy curves as functions of R • The lower state is bound, whereas the upper state has no minimum and is thus unbound • De = dissociation energy of H2+ into H + H+ • Re = equilibrium internuclear distance • Electronic potential curves for isotopic species H2, HD, D2are exactly identical

5. Potential curves for H2+ ion R H-----H H + H+ De Re

6. Notation electronic states • Electronic states are classified / labeled according to symmetry • Atoms: spherical symmetry S, P, D, … for L = 0, 1, 2, … • Diatomics: cylindrical symmetry S, P, D, … for L = 0, 1, 2, … where L = orbital angular momentum along the internuclear axis • Polyatomics: finite point group symmetry e.g., H2O: C2V symmetry A1, B1, B2

7. Notation molecular states • As for atoms, the total spin S of the molecule is indicated by the multiplicity 2S+1 as a superscript: 1S+, 3P, … • Often the ground electronic state is denoted with the letter X, excited electronic states of the same multiplicity with A, B, C, D, … in order of increasing energy. Excited electronic states with different spin multiplicity with a, b, c, d, … • Example H2: X1Sg+, B1Su+, C1Pu, a3Sg+, b3Su+

8. 12.3 Nuclear motion • Born-Oppenheimer: where it is assumed that • Diatomic molecule in center-of-mass system: • J = nuclear angular momentum operator • m= reduced mass of system • Assume Radial part Angular part Angular Radial

9. Nuclear motion (cont’d) Rotational equation Vibrational equation

10. a. Vibration • Take vibrational equation and assume that Eel(R) is bound. Take J = 0, and expand Eel(R) around minimum • Harmonic oscillator equation, solution: v=0, 1, 2, …

11. Vibration (cont’d) • Vibrational levels are equidistant, but depend on m H2, HD, D2, … have different vibrational spectra • Note that even for v = 0, Evib 0 (zero-point vibrational energy): Evib = ½ we • In reality, potential is anharmonic  levels are not exactly equidistant

12. Vibrational levels Harmonic vs. anharmonic vibrational levels

13. b. Rotation • If nuclei fixed at Re rigid rotator equation • Moment of inertia: • Rotational constant: • DE between adjacent J levels increases with J, depends on m: J=0, 1, 2, …

14. 12.4 Molecular transitions (summary) • The nuclear and electronic motions in molecules are nearly decoupled (Born-Oppenheimer approximation) • Energy difference two electronic states typically a few eV => VIS and UV wavelengths • Energy difference two vibrational states typically 0.1-0.3 eV => 500-3000 cm-1 => IR wavelengths • Energy difference two rotational states typically 0.001 eV => few cm-1 => (sub)millimeter wavelengths

15. Summary energy levels

16. Energy levels of CO

17. 12.5 Examples a. Rotational Spectra • CO J = 1 – 0 n = 115 GHz  l = 2.6 mm J = 2 – 1 n = 230 GHz  l = 1.3 mm J = 3 – 2 n = 345 GHz  l = 0.87 mm • Typically l a few mm for J = 1 – 0 in heavy diatomics (CS, SiO, SO, …) • Hydrides have much higher rotational frequencies (near l = 400 mm) because m is much smaller

18. Selection rules • Molecule must have permanent dipole moment  no strong rotational spectra observed for H2, C2, O2, CH4, C2H2, … • ΔJ = 1  only transitions between adjacent levels • For symmetric molecules like H2, only quadrupole transitions occur with ΔJ = 2, e.g. • J = 2  0 λ= 28 μm • J = 3  1 λ= 17 μm • J = 4  2 λ= 12 μm

19. CO vs H2 rotational levels Note much wider spacing for H2 compared with CO

20. Symmetric-top rotators • Non-linear molecules can rotate about three axes • If there is a three-fold (or higher) symmetry axis, two of the principal moments of inertia must be identical  “symmetric top” • Examples: NH3, CH3CN, CH3CCH • Rotational levels have two quantum numbers: J, and projection of J on symmetry axis, K (K  J) • Selection rules: DJ =  1, DK = 0  • Levels with J = K are metastable

21. Rotational energy levels of symmetric-top molecules • Right: oblate top • Left: prolate top

22. Energy levels of NH3 • Inversion doubling of rotational transitions • Frequency about 23 GHz • Used to build first masers

23. Asymmetric rotors • Three different principal moments of inertia  • Complicated spectra • Three quantum numbers: • Total angular momentum J • Projection of J on two molecular axes, K–, K+ • Notation: JK–K+ • Example: H2O (Herschel Space Observatory!)

24. Rotational levels of ortho-H2O

25. b. Vibrational spectra • CO: v = 1 – 0 band λ = 4.67 μm  2140 cm–1 v = 2 – 0 band λ = 2.35 μm  4250 cm–1 • H2: v = 1 – 0 band λ = 2.40 μm  4150 cm–1 • Various chemical groups in molecules have very characteristic vibrational frequencies, e.g. C – H stretch, C  C stretch, CH2 angle bending etc.

26. Characteristic group frequencies

27. Vibration-rotation spectra • Associated with each v-level is a stack of rotational levels  band v – v is composed of a number of lines vJ – vJ • Selection rules: • No restrictions on Δv • ΔJ = 0, 1, but J = 0  0 forbidden • ΔJ = +1: R branch • ΔJ = 0: Q branch • ΔJ = –1: P branch

28. HCN vibration-rotation spectrum Q P R P R P R

29. L Doubling • Spectra are more complicated if there is a non-zero electronic angular momentum • Example: OH (L = 1) • Electrons orbit with net angular momentum parallel or anti-parallel to electric dipole moment  • Splitting of each rotational level characterized by the two different sign choices for L

30. Rotational energy levels of OH • Two branches due to spin splitting • Each rotational level is split due to L doubling and hyperfine interaction • Known maser transitions are marked

31. Transitions of some astrophysically important molecules

32. 12.6 H2 molecule • The H2 nuclei each have nuclear spin I = ½  can be combined to form a triplet and a singlet nuclear function • Triplet: Ψns is symmetric • Singlet: Ψns is antisymmetric • Total wave function Ψ = Ψel Ψvib Ψrot Ψnsmust be antisymmetric with respect to exchange of any two particles (Fermi-Dirac statistics)

33. a. Ortho- and Para-H2 • Ground electronic state of H2 X1Sg+: symmetric • Ground vibrational state v = 0: symmetric  Ψrot Ψns must be antisymmetric • This leads to two distinct forms of H2: • Para-H2: Ψns antisymmetric, J even (J = 0, 2, 4,…) • Ortho-H2: Ψns symmetric, J odd (J = 1, 3, 5, …)

34. Nuclear spin statistics of 1H2 Total statistical weight = gN(2J+1)

35. Ortho- and Para-H2 (cont’d) • Para-H2 and ortho-H2 cannot be interchanged by normal inelastic collisions, e.g., H2 (J=0) + H  H2 (J=1) + H • Only reactive collisions in which H nuclei are interchanged can change para-H2 into ortho-H2, e.g. H2 (J=0) + H+ H+ + H2 (J=1)  H–H +  H+   H+ +  H–H

36. b. H2 vibration-rotation spectra Vibration-rotation spectra Near-IR Pure rotation spectra Mid-IR

37. c. Electronic spectra • Electric dipole allowed transitions: DS = 0, DL = 0, 1 • Lyman system: B1Su+ – X1Sg+ • Werner system: C1Pu – X1Sg+ • For example B–X (10,0) R(1) line: • Upper level B1Su+ v = 10 J = 2 • Lower level X1Sg+ v = 0 J = 1

38. H2 potential curves

39. 12.7 H2 Photodissociation Tielens Chap. 8.7 • H2 is most abundant molecule in interstellar clouds • In order to compute H2 abundance, we need to understand its formation and destruction processes • In diffuse interstellar clouds (i.e. clouds with AV 1 mag), H2 is destroyed by photo-dissociation via a complicated process

40. H2 photodissociation (cont’d) • In general, photo-dissociation is a quite simple process: absorption into repulsive excited state leads to dissociation: direct photo-dissociation • See H2+ potential curves for example • For H2, however, there are no allowed transitions to repulsive states from the ground state at energies less than 13.6 eV  no direct photodissociation possible • However, photodissociation of H2 can occur via an indirect, 2-step process: spontaneous radiative dissociation

41. H2 photodissociation • 90% of absorptions into B and C states are followed by emission back into bound vibrational levels of the X state • 10% of the absorptions are followed by emission into the unbound vibrational continuum, leading to dissociation (as indicated in the figure)

42. Photodissociation rate • In general, the photodissociation rate of a molecule in the interstellar radiation field I is where spd is the photodissociation cross section in cm2 • For H2, the absorption through all of lines into the B and C states needs to be summed, including the probability h~0.1 that absorption leads to dissociation (use relation between s and f)

43. Interstellar radiation field Note linear scale; uncertanties ~50%

44. Attenuation of radiation in clouds • Inside an interstellar cloud, I(l) will be diminished by several effects • Continuum attenuation by dust grains: calculation depends on scattering properties of grains such as albedo and scattering phase function • For 912 Å < l < 1200 Å, it results in Intensity inside cloud Intensity at edge

45. Line attenuation of H2 • tline~1 for N~1017 cm-2

46. Self-shielding of H2 • Typical diffuse cloud of 1 pc, n 100 cm–3  N  3  1020 cm–2  • H2 lines become very optically thick at small depth into the cloud; this is called self-shielding • H2 molecules at edge of cloud absorb all available photons at certain wavelengths, so that molecules lying deeper in the cloud “see” virtually no photons at all, and are not dissociated • Since H2 protects itself from dissociation, the transition from a cloud that is mostly atomic to mostly molecular is very sharp. This is confirmed by observations

47. H2 photodissociation with depth

48. Observed H2 column densities

49. 12.8 H2 formation • Define • In steady state, H2 formation and destruction rates must be equal: • For diffuse clouds with f << 1, kpd must have the unattenuated interstellar value • Observations imply R~3x10-17 cm3 s-1

50. H2 formation mechanisms • This value of R can only be produced by reactions on the surfaces of grains • For H2, gas phase reactions are too slow by at least 8 orders of magnitude! • Two mechanisms for H2 formation on grains: • Diffusive mechanism • Eley-Rideal mechanism (direct mechanism) • Surface can be silicates, carbonaceous, ice, …