16 & 17: Molecular Spectroscopy

Transcription

1 16 & 17: Molecular Spectroscopy James R. Graham UC, Berkeley 1

2 Reading Tielens, Chs. 2 Overview only Herzberg, Molecular Spectra & Molecular Structure Vol. 1, Ch. 5(1950) Herzberg, The Spectra & Structure of Simple Free Radicals, (Dover 1971), esp. Parts I & II Townes & Schawlow, Microwave Spectroscopy (Dover 1975) Rybicki & Lightman Ch. 10 Shu, Physics of Astrophysics: I Chs Dopita & Sutherland, Ch. 1 2

3 Molecular Emission Atoms do not produce many lines at IR or radio wavelengths Molecules produce many such lines Vibrational & rotational transitions Only occur with two or more nuclei The energies of such transitions cover a wide range General rule of thumb Vibrational transitions occur at near to mid-ir H S(1) at 2.12 µm and CO 1-0 at 4.6 µm Rotational transitions are less energetic and occur at mid-ir through mm wavelengths H S(0) 28.2 µm & 0-0 S(1) µm CO J=1-0 at 2.7 mm UV and optical transitions of molecules tend to include electronic transitions 3

4 A Little History Discovery of Interstellar Molecules First optical detections (absorption) CH X 2 Π A 2 Δ Å (Dunham et al. 1937) CN X 2 Σ + B 2 Σ Å (Swings & Rosenfeld 1937) CH + X 1 Σ + A 1 Π Å (McKellar 1940) Prediction of microwave emission by Townes, Shklovsky et al. in the 1950s OH 18 cm (Weaver & Williams 1964; Weinreb et al. 1963) NH cm & H 2 O 1.4 cm (Cheung et al. 1968, 1969) H 2 CO 6.2 cm (Snyder at al. 1969) CO 2.6 mm (Wilson Jefferts & Penzias(1970) Rocket UV H 2 X 1 Σ g+ A 1 Σ u Å; X 1 Σ g+ B 1 Π u 1008 Å (Carruthers 1970) Over 140 interstellar molecules known wwwusr.obspm.fr/departement/demirm/list-mol.html 4

5 ISO Mid- & Far-IR Spectroscopy ISOCAM 7 & 15 µm image of the ρ Oph molecular cloud Bright filaments (left) trace dust heated by HD (B2V) located off the image (van Dishoeck AARA ) The dense cloud containing IRAS is to the E Dark patches are very dense cores that are optically thick even at 15 µm Bright point sources are mostly low-mass YSOs & most of the extended 7µm emission is PAHs Spectra of objects in ρ Oph: WL 6 (left top); ρ Oph W (right top); IRAS (left bottom); Elias 29 (right bottom) 5

6 ISO Mid- & Far-IR Spectroscopy Orion Peak 1 shock, showing a rich forest of H 2 rotation and rotation-vibration lines 6

7 Electronic Structure of Diatomic Molecules Compared to an atom an additional electrostatic interaction is introduced in molecules Repulsion between +ve charged nuclei Quantitative treatment is complex Qualitative ordering of energy levels is achieved using molecular orbital theory A molecular orbital,ψ, is constructed as a linear combination of atomic wave functions χ For any diatomic molecule ψ = c 1 χ 1 + c 2 χ 2 with normalization constants c For a homonuclear molecule ψ = 1/ 2 (χ 1 ± χ 2 ) 7

8 The Pauli Exclusion Principle No two e - s in a molecule can have identical quantum numbers Suppose that e - ψ s 1 & 2 are in states 1 (a) a & b respectively ψ 2 (b) The wavefunction for two electrons is ψ = ψ 1 (a) ψ 2 (b) This is unacceptable because the e - s are identical & indistinguishable ψ must be a linear combination of the two possibilities so it is impossible to tell which is which ψ= ψ 1 (a) ψ 2 (b) ψ - ψ 1 (b) ψ 2 (a) Particles of half-integer spin must have antisymmetric wavefunctions 8

9 Bonding & Antibonding ψ ~ (χ 1 ± χ 2 ) Charge density is ψ 2 ψ ~ (χ 1 + χ 2 ) Charge density is high in the overlap region ψ ~ (χ 1 - χ 2 ) Charge density has a minimum at the midpoint between the two nuclei (χ 1 + χ 2 ) is a bonding orbital (χ 1 - χ 2 ) anti-bonding orbital σ g 1s is a lower energy state than σ u 1s ψ = N(χ 1 + χ 2 ) ψ = N(χ 1 - χ 2 ) g and u refer to symmetry of the wave function with respect to inversion through a point at the center of the molecule g: gerade (even) u: ungerade (odd) σ g 1s σ u* 1s 9

10 Building up Orbitals: H 2 & He 2 Molecular electron wavefunction is assembled by populating the single electron states Each σ orbital can accommodate two electrons H 2 has bond order 1 He 2 has bond order 0 10

11 σ and π Orbitals σ orbitals Cylindrical symmetry about internuclear axis Nondegenerate π orbitals Lack cylindrical symmetry one nodal plane through the z-axis Doubly degenerate 11

12 Building up Diatomic Molecules σ g 1s σ u *1s σ g 2s σ u *2s π u 2p σ g 2p π g *2p S B ,1 C N O ,1 Hund s rule applies and the term with the highest multiplicity is the ground state The ground state of O 2 is a triplet###### 12

13 B 2 (σ g 1s) 2 (σ u *1s) 2 (σ g 2s) 2 (σ u *2s) 1 (π u 2p) 2 (σ g 2p) 1 σ u *2s, π u 2p, & σ g 2p are very close together The spin paring energy to put two electrons in σ u *2s requires more energy so that S = 2 not 1 is the ground state 13

14 Ground Electronic States Russell-Sanders coupling applies For each e -, the coupling of its own orbital angular momentum & spin is neglected Each e - with orbital angular momentum σ, π, δ, (0, 1, 2, ) couples to give total orbital angular momentum, L Each electron spin couples to give the total spin angular momentum, S Coupling between L and S can be neglected for small nuclear charges can be neglected The coupling of L to the internuclear axis is strong L precesses about the internuclear axis The projection of L onto this axis, Λ is a good quantum number: Λ = Σ, Π, Δ, (0, 1, 2, ) 14

15 Summary of Symmetry Labels When individual molecular orbitals have rotational symmetry about the about the internuclear axis σ: cylindrical symmetry π: one nodal plane &c. When the total wavefunction is symmetric on reflection through any plane containing the internuclear axis so it is labeled + with respect to inversion through a point at the center of the molecule (only for homonuclear case) g: gerade (even) u: ungerade (odd) 15

16 Ground Electronic States Electronic states are designated 2S+1 Λ g/u +/- Hund s case (a) L S H 2 + 1e - in σ g 1s Λ = 0 One e - S = 1/2 σ g 1s wavefunction is symmetric on reflection through any plane containing the internuclear axis so it is labeled + 2 Σ g + Λ = 0,1,2...L Σ = (S,S 1,S 2... S) Λh Σh Ωh = ( Λ + Σ)h H 2 Two e - in σ g 1s Λ = 0 Two e - S = 0 1 Σ g + 16

17 Ground Electronic States of H 2 Two e - in σ g 1s Λ = 0 σ wavefunctions have cylindrical symmetry: + x + = + g x g = g Two e - S = 0 1 Σ g + 1s 1 Σ g + σ u σ g 1s Bond energy = ev 3 Σ u + σ u 1s 1s One e - in σ g 1s & one e - in σ u 1s Λ = 0 g x u = u Two e - S = 0 or 1 σ g 1s 1 Σ u + σ u σ g 1s 3 Σ u + or 1 Σ u + 17

18 H 2 Potential Curves 1s 1 Σ u +? σ u 1s σ g 1s 3 Σ u + σ u 1s σ g 1s 1 Σ g + σ u 1s σ g 18

19 Ground Electronic States of H 2 Two e - in σ g 1s Λ = 0 σ wavefunctions have cylindrical symmetry: + x + = + g x g = g Two e - S = 0 1 Σ g + 1s 1 Σ g + σ u σ g 1s Bond energy = ev 1s 3 Σ u + σ u 1s One e - in σ g 1s & one e - in σ u 1s Λ = 0 g x u = u Two e - S = 0 or 1 3 Σ u + σ g 1s 1 Σ u + σ u σ g 1s 19

20 Excited Electronic States of H 2 Additional states arise from 2s and 2p electrons One e - in σ g 1s & one e - in π u 2p Λ = 0,1 + x - = - g x u = u S = 1 3 Π u - 2p 2s 1s σ u π 3 u Π - 2p u σ g π g σ u σ g σ u σ g 2s 1s One e - in σ g 1s & one e - in σ g 2s Λ = 0 g x g = g S = 0 1 Σ g + 2s 1s σ u σ g σ u σ g 2s 1s 1 Σ g + 20

21 Hybrid Orbitals Forms of bonding combining s and p atomic wavefunctions are possible Best know example of hybrid orbitals is in C The four valence electrons (2s & 2p), which can combine in multiple ways explains the richness of organic chemistry sp (acetylene: H-C C-H) sp 2 (ethylene: H 2 C=CH 2 ; or benzene: C 6 H 6 ) sp 3 (methane: CH 4 ) Stable O and N molecules that form with H H 2 O and H 2 O 2 NH 3 and N 2 H 4 21

22 Electronic Transitions The matrix elements are of the form, e.g., <ψ 1 µ ψ 2 >, where µ is the operator for the dipole moment Molecular term designations describe the symmetry of ψ To find the selection rules, find combinations ψ 1, ψ 2, and the operator that are symmetric. The selection rules for electric dipole transitions are ΔΛ = 0 or ±1 (polarization // or to the internuclear axis) Σ + Σ - g g, u u And when S is not coupled to L (Hund s case a) ΔS = 0 Binding energy is of the H 2 X 1 Σ g + state is ev Naïvely expect that FUV radiation λ < 2769 Å will dissociate interstellar H 2 The transition 1 Σ g + 3 Σ u + is forbidden and therefore H 2 is protected 22

23 Photodissociation of H 2 1 Σ g + 3 Σ u + is not dipole permitted Photodissociation of H 2 in the ISM must proceed via high energy photons H 2 cannot form via radiative association of two HI atoms in 1s Σ 23

24 H 2 1 Σ g Σ u + Transition(s) Sembach 2001 AJ FUSE spectra of ISM H 2 in absorption of the 1 Σ g + 1 Σ u + transition Lyman band of lines and not a single transition 24

25 Vibrational & Rotational Bands Electronic states are split into vibrational (v) and rotational (J) sublevels For an absorption line P: ΔJ = -1 R: ΔJ = +1 For transitions where Ω = 0 in both states, the Q-branch, ΔJ = 0 is absent A v X v P(J) R(J) J +1 J J -1 J +1 J J -1 25

26 Interstellar Molecular R & P Branches Interstellar C 3 X 1 A 1 u When multiple transitions between electronic states are observed line ratios give the excitation conditions The synthetic spectrum for T rot = 80 K at R = 10 5 Famous example is CN, CH + & CH 26

27 O 2 For O 2 the only electrons that contribute to the angular momentum are the two π g *2p electrons Two π g* 2p electrons yield 3 Σ g-, 1 Σ g+, & 1 Δ g The state of of highest multiplicity is the ground state: 3 Σ g-, Sound familiar? Similar to figuring out the terms which arise from two p electrons 27

28 Heteronuclear Molecules Molecules like CN & CO have sufficiently similar nuclear charges that they can be treated using homonuclear techniques The g/u symmetry with respect to inversion through the center of the molecule is lost The energies of the two 1s, the two 2s, &c. atomic orbitals are now slightly different CO is isoelectronic with N 2 Ground state (σ1s) 2 (σ*1s) 2 (σ2s) 2 (σ*2s) 2 (π2p) 4 (σ2p) 2 All occupied molecular orbitals are filled Λ=0, S=0: 1 Σ + First excited state (σ2p) 1 (π*2p) 1 The two unpaired electrons yield singlet and triplet states 3 Π and 1 Π 28

29 Rotational & Vibrational Structure Molecular transitions can be categorized as rotational, vibrational and electronic Typically, the energies are very different E rot ~ ev: rotational energy of the molecule E vib ~ ev: KE & PE of the nuclei associated with vibration about their equilibrium positions E el ~ 1 10 ev: electrostatic energy The Born-Oppenheimer approximation Due to the very different energies of electronic and nuclear the interactions can be ignored Assume that the wave functions separable ψ tot ψ nuc ψ el Essentially the motions of the heavy nuclei are much faster than that of the light electrons 29

30 The Born-Oppenheimer Approximation A further approximation involves the factorization of ψ nuc ψ nuc = ψ vib ψ rot so that ψ tot = ψ el ψ vib ψ rot This factorization justifies writing the total energy of a molecule E tot = E el + E vib + E rot The electronic part is characterized by a potential curve, with a minimum at the equilibrium radius, r e, if the molecule is stable 30

31 Molecular Dynamics In the Born-Oppenheimer picture, the nuclei vibrate & rotate about their equilibrium separation Neglect of coupling between nuclear and electronic motion leads to errors in the electronic energy levels ~ m e /m n ~ 10-4 Additional effects are magnetic interactions between their various orbital & spin angular momenta To the first approximation molecular dynamics reduces to 1. Rigid body motion 2. Normal modes of oscillation (3N-6) (1) describes to the rotational motion of molecules with wavelengths in the far-ir/mm bands; and (2) vibrational motions observable at near-ir wavelengths 31

32 Harmonic Oscillator The vibration of a diatomic molecule can be treated as the stretching and compression of a spring (the molecular bond) Approximated as an harmonic oscillator the potential is and where V(x) = kx 2 /2 E v =hν(v + 1/2) ν = (1/2%)(k/µ) 1/2 and µ is the reduced mass 32

33 Anharmonic Oscillators The potential energy curve for real molecules is not parabolic Not harmonic oscillators More generally G(v) = ω e (v+1/2) - x e (v+1/2) 2 + y e (v+1/2) 3 + ω e (cm -1 ) x e (cm -1 ) y e (cm -1 ) 1 H 1 2 Σ + g C16O 1Σ

34 H 2 & CO Because it is a light molecule 1 H 2 has much higher fundamental vibrational frequency than 12 C 16 O The vibrational levels of 1 H 2 lie higher up the vibrational potential and effects of anharmonicity are larger Δv=1 for H 2 is at 2.40 µm; CO at 4.66 µm G(0) G(1) G(2) G(1)-G(0) G(2)-G(0) (cm -1 ) (cm -1 ) (cm -1 ) (cm -1 ) (cm -1 ) 1 H C 16 O

35 Vibrational Levels of H 2 CO has a permanent dipole moment Stong vibrational bands H 2 does not! Fernandes et al MNRAS v= µm 35

36 Rotational Lines Under cold ISM conditions rotational transitions and some atomic fine structure transitions carry most of the radiation The first detected rotational transitions were cm maser lines of OH Most commonly observed lines are microwave lines of CO Rotational transitions of CO have been detected from the ground throughout much of mm and sub-mm. Very dry conditions are needed at high frequencies, e.g., the Atacama desert. The record is CO THz, detected from 5525 m in N. Chile (Marrone et al. 2004) Rotational motion of molecules is determined by moments of inertia & associated angular momentum Classically, any object has three orthogonal principal moments of inertia (symmetric inertia tensor) and simple expressions for the rotational energy and the angular momentum Customary to classify the rotational characteristics of molecules based on the values of the principle moments of inertia 36

37 An Elementary Example: CO For a rigidly rotating diatomic or linear molecule the rotation energy levels are E rot = 1 Iω 2 ; l 2 = I 2 ω 2 = J J +1 2 ( )h 2 = 1 J J +1 2 ( )h 2 I = B e J( J +1) hc where I = 2 m i r i i Conventionally, the rotational constant B e = h/8% 2 ci is quoted in cm -1 or Hz For CO B e = cm -1 or ~ 2.77 K 37

38 An Elementary Example: CO Electric dipole transitions have ΔJ = ± 1 For a rigid rotator the frequencies are integer multiples of the J=1-0 transition ν J +1,J = E rot J +1 ( ) h E rot ( J) h ( ) = 2B e J +1 Exact frequencies are not precise multiples because CO is not perfectly rigid J=3 J=2 J=1 J= GHz GHz GHz v 0 2v 0 3v 0 4v K K 5.53 K v 38

39 Electronic, Vibrational & Rotational Energy Levels of CO Comparison of energies of the electronic (singlet), vibrational and rotational states of CO 39

40 Real Molecules In real molecules the bond is stretched by rotation and the rotational energy states are approximated by fitting formulae E J h = BJ J +1 ( ) DJ 2 J +1 ( ) 2 + HJ 3 J +1 ( ) 3 K Effective B decreases with increasing J, correcting for centrifugal distortion For CO B = 57, MHz, D = MHz, & H =1.725x10-7 MHz 40

41 Rotational Spectra CO molecular weight is 28 Large moment of inertia J=1-0 rotational transition at 115 GHz Relatively low frequency The submm and far-ir is rich with rotational transitions H S(0) is at 10.7 THz The corresponding transitions of light hydrides are at THz frequencies CH + (M.W. = 13) B = GHz Many of these high frequency transitions are blocked by terrestrial H 2 O and O 2 Ground-based observation of species abundant in the atmosphere is challenging 41

42 Rotational Spectra ISO/LWS spectrum of CRL 618 (C-rich PPN) Continuumsubtracted spectra & model spectrum CO, 13 CO, HCN, H 2 O, and OH are indicated by arrows HNC J=22-21 ( µm) to J=17-16 ( µm) indicated by vertical lines Herpin et al ApJ 530 L129 42

43 Rotational Lines in Orion The GHz (450 µm) spectrum of the star forming Orion cloud Dominated by rotational transitions of CO, CS, SO, SiO, HCN, HCO +, H 2 CO, SO 2 & CH 3 OH H 2 O CO J=6-5 O 2 Schilke et al ApJS More than 10 3 transitions, many unidentified Strongest transition is CO J=6-5 however, the integrated SO 2 and CH 3 OH dominate the cooling in this region 43

44 Rotational Lines in Orion The GHz (450 µm) spectrum of the star forming Orion cloud Dominated by rotational transitions of CO, CS, SO, SiO, HCN, HCO +, H 2 CO, SO 2 & CH 3 OH Schilke et al ApJS More than 10 3 transitions, many unidentified Strongest transition is CO J=6-5 however, the integrated SO 2 and CH 3 OH dominate the cooling in this region 44

45 Sub-mm Atmospheric Transmission Mauna Kea, 1mm H 2 O H 2 O O 2 45

46 H 2 Symmetry & Selection Rules The angular momentum eigenfunctions for J are the Legendre polynomials ψ(j=0) = 1, ψ(j=1) = cosθ, ψ(j=2) = (3cos 2 θ-1)/2, &c. Even J levels are symmetric & odd J are antisymmetric The proton has spin 1/2 Antisymmetric nuclear spins ( ) combine with even J with statistical weight 1 Parahydrogen Symmetric nuclear spins ( ) combine with odd J with statistical weight 3 Orthohydrogen Dipole permitted rotational transitions have ΔJ = 1 These transitions do not occur in H 2 Observed rotational spectra are electric quadrupole ΔJ = 2, e.g., 0-0 S(0) 46

47 Linear Molecules & Symmetric Rotors The principal moments of inertia are designated I a, I b, and I c Conventially I a I b I c A molecule which is linear or has a rotational symmetry axis a symmetric top Either I c = I b > I a or I c > I b = I a Linear molecules,.eg., CO, have a small I a about the axis of the molecule so they are prolate symmetric rotor Other molecules, e.g., benzene, have the largest moment of inertia about the symmetry axis and are oblate symmetric rotors Molecules which are spherically symmetric, e.g, methane have three equal moments of inertia and are spherical rotors Molecules I c I b I a are asymmetric rotors 47

48 Symmetric Rotors Classically, the energy of rotation, E, is E = 1 2 I xω x I yω y I zω z 2 = P 2 x + P y + P z 2I x 2I y 2I z 2 2 Consider a symmetric rotor I x = I y = I b Since angular momentum is P 2 = P x 2 + P y 2 + P z 2 E = P 2 2I b + P z 1 2I c 2I b 2 1 I c I b I a 48

49 Symmetric Rotors The total angular momentum, P, of the rotating molecule is quantized as is the z component P 2 = J(J +1)h 2, and P z 2 = K 2 h 2 J = 0, 1, 2, K = 0, 1, 2, Hence the energy is E = h2 J(J +1) + h2 h2 K 2, or 2I b 2I c 2I b E = BJ(J +1) + (C B)K 2 49

50 Linear Molecules For linear molecules like H 2, CO, or HCN I C = I B >> I A I A is the moment of inertia about the internuclear axis Recover the elementary expression for a rigid rotator E = B(J+1)J 50

51 Symmetric Rotors E = B(J+1)J + (C-B)K 2 For K = 0 the energy levels are those of a linear molecule K is a projection of J, with 2J+1 different values K J thus K = -J, -J+1, J-1, J Energy depends on K so there are J+1 distinct levels starting at K = J For given K there is an infinite number of J levels For a prolate top (cigar) C > B At given J energy levels increase with K For an oblate top (pancake) C < B At given J energy levels decrease with K 51

52 Oblate & Prolate Symmetric-Top Molecules Prolate Oblate 52

53 Selection Rules By symmetry there can be no dipole moment perpendicular to the axis of a axis of a symmetric top No torque along that axis due to E fields associated with radiation K = 0 The dipole moment lies along the molecular axis This axis preceses around the total angular momentum with frequency P/2$I b thus J = ±1 Levels with J = K are metastable 53

54 NH 3 NH 3 was the first polyatomic molecule detected in the ISM (Cheung et al. 1968) Oblate symmetric top with pyramidal symmetry Any molecule with 3-fold (or greater) rotational symmetry is a symmetric top The ground state transition, i.e., J=1-0, K=0-0 occurs at GHz Only observable from airborne telescopes (SOFIA) or space Ho & Townes 1983 AARA

55 NH 3 An inversion transition occurs when the N tunnels through the plane of the three H atoms In contrast to most nonplanar molecules, the potential barrier is weak and tunneling occurs rapidly The corresponding frequency falls in the microwave range Each of the inversion doublets splits due to the electrostatic interaction between the electric quadrupole moment of the N nucleus and the electrons Weaker magnetic hyperfine interactions associated with the H nuclei yield a total of 18 hyperfine transitions Ho & Townes 1983 AARA

56 NH 3 Rydbeck et al ApJ 215 L35 56

57 Asymmetric Rotators Three different principal moments of inertia I a I b I c Only J and E are conserved Total angular momentum J remains a good quantum number J states are labeled by two approximate quantum numbers: Projection of J on two molecular axes: K & K + Notation: J K K + K & K + only become good quantum numbers (conserved projections of angular momentum) in the limit of prolate & oblate symmetric tops 57

58 H 2 CO (Formaldehyde) H 2 CO is nearly symmetric prolate top A = 281, MHz* B = 38, MHz C = 34, MHz Small asymmetry about the C-O axis causes a deviation from pure prolate symmetry Splits the degeneracy of energy levels with K > 0 Mangum & Wootten 1993 ApJS * in frequency units A = h/8% 2 I A ; B = h/8% 2 I B ; C = h/8% 2 I C 58

59 H 2 CO (Formaldehyde) The 6 cm 4830 MHz transition of H 2 CO was detected in the radio in 1969 Always seen in absorption, even in dark clouds, suggesting that it absorbs the 3K background ,603 MHz ,488 MHz , ,838 MHz MHz ,838 MHz Ortho Para 59

60 Rotational Levels of Ortho-H 2 O A = 835, MHz B = 435, MHz C = 278, MHz 60

61 Nuclear Spin Nucleus H 12 C 13 C 14 N 15 N 16 O 17 O 18 O Spin I 1/2 0 1/2 1 1/2 0 5/2 0 61