AY216 1

16 & 17: MolecularSpectroscopy

James R. Graham

UC, Berkeley

AY216 2

Reading

• Tielens, Chs. 2– Overview only

• Herzberg, “Molecular Spectra & MolecularStructure” Vol. 1, Ch. 5(1950)

• Herzberg, “The Spectra & Structure of SimpleFree Radicals”, (Dover 1971), esp. Parts I & II

• Townes & Schawlow, “MicrowaveSpectroscopy” (Dover 1975)

• Rybicki & Lightman Ch. 10• Shu, “Physics of Astrophysics: I” Chs. 28-30• Dopita & Sutherland, Ch. 1

AY216 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– H2 1-0 S(1) at 2.12 µm and CO 1-0 at 4.6 µm

• Rotational transitions are less energetic and occur at mid-IRthrough mm wavelengths

– H2 0-0 S(0) 28.2 µm & 0-0 S(1) 17.04 µm– CO J=1-0 at 2.7 mm

• UV and optical transitions of molecules tend to include electronictransitions

AY216 4

A Little History• Discovery of Interstellar Molecules

– First optical detections (absorption)• CH X 2Π –A 2Δ 4300.3 Å (Dunham et al. 1937)• CN X 2Σ+–B 2Σ+ 3876.8 Å (Swings & Rosenfeld 1937)• CH+ X 1Σ+–A 1Π 4232.5 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)• NH3 1.3 cm & H2O 1.4 cm (Cheung et al. 1968, 1969)• H2CO 6.2 cm (Snyder at al. 1969)• CO 2.6 mm (Wilson Jefferts & Penzias(1970)

– Rocket UV• H2 X 1Σg

+–A 1Σu+ 1108Å; X 1Σg

+–B 1Π u 1008 Å (Carruthers

1970)– Over 140 interstellar molecules known

• wwwusr.obspm.fr/departement/demirm/list-mol.html

AY216 5

ISO Mid- & Far-IR Spectroscopy

• Bright filaments (left) trace dust heated by HD 147889 (B2V) located off the image (vanDishoeck AARA 2004 42 119)

• The dense cloud containing IRAS 16293-2422 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 16293-2422 (left bottom);Elias 29 (right bottom)

ISOCAM 7 & 15 µmimage of the ρ Ophmolecular cloud

AY216 6

ISO Mid- & Far-IR Spectroscopy

• Orion Peak 1 shock, showing a rich forest ofH2 rotation and rotation-vibration lines

AY216 7

Electronic Structure of Diatomic Molecules

• Compared to an atom an additional electrostaticinteraction 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 moleculeψ = c1χ1 + c2χ2

with normalization constants c

– For a homonuclear moleculeψ = 1/√2 (χ1 ± χ2 )

AY216 8

The Pauli Exclusion Principle

• No two e- ’s in a molecule can haveidentical quantum numbers

• Suppose that e- ’s 1 & 2 are in statesa & b respectively– The wavefunction for two electrons is

ψ = ψ1(a) ψ2(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 whichψ= ψ1(a) ψ2(b) ψ - ψ1(b) ψ2(a)

• Particles of half-integer spin must haveantisymmetric wavefunctions

ψ1(a)

ψ2(b)

AY216 9

Bonding & Antibonding• ψ ~ (χ1 ± χ2 )

– Charge density is ∝ ψ2

• ψ ~ (χ1 + χ2 )– Charge density is high

in the overlap region• ψ ~ (χ1 - χ2 )

– Charge density has aminimum at the mid-point between the twonuclei

• (χ1 + χ2 ) is a bondingorbital

• (χ1 - χ2 ) anti-bondingorbital– σg1s is a lower energy

state than σu1s

σg1s

σu*1s

ψ = N(χ1 + χ2 )

ψ = N(χ1 - χ2 )

g and u refer to symmetry of thewave function with respect toinversion through a point at thecenter of the molecule– g: gerade (even)– u: ungerade (odd)

AY216 10

Building up Orbitals: H2 & He2

• Molecular electronwavefunction isassembled bypopulating the singleelectron states– Each σ orbital can

accommodate twoelectrons

– H2 has bond order 1– He2 has bond order 0

AY216 11

σ and π Orbitals

• σ orbitals– Cylindrical

symmetry aboutinternuclear axis

– Nondegenerate

• π orbitals– Lack cylindrical

symmetry–onenodal planethrough the z-axis

– Doublydegenerate

AY216 12

Building up Diatomic Molecules

• Hund’s rule applies and the term with thehighest multiplicity is the ground state– The ground state of O2 is a triplet # # # # # #

2

πg*2p

0,1242222O2

0242222N2

042222C2

0,122222B2

Sσg2pπu2pσu*2sσg2sσu*1sσg1s

AY216 13

B2 (σg1s)2(σu*1s)2(σg2s)2(σu*2s)1(πu2p)2 (σg2p)1

• σu*2s, πu2p, & σg2p are very close together– The spin paring energy to put two electrons in σu*2s requires

more energy so that S = 2 not 1 is the ground state

AY216 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 angularmomentum, L

– Each electron spin couples to give the total spinangular momentum, S

• Coupling between L and S can be neglectedfor 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 goodquantum number: Λ = Σ, Π, Δ, … (0, 1, 2, …)

AY216 15

Summary of Symmetry Labels

• When individual molecular orbitals haverotational symmetry about the about theinternuclear 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)

AY216 16

Ground Electronic States

• Electronic statesare designated2S+1Λg/u

+/-

• H2+

– 1e- in σg1s ∴ Λ = 0– One e- ∴ S = 1/2– σg1s wavefunction is

symmetric onreflection throughany plane containingthe internuclear axisso it is labeled “+”

2Σg+

• H2– Two e- in σg1s ∴Λ = 0– Two e- ∴ S = 0

1Σg+

L

€

Λ = 0,1,2...L

S

€

Σ = (S,S −1,S − 2...− S)

€

Λh

€

Σh

€

Ωh = Λ + Σ( )h

Hund’s case (a)

AY216 17

Ground Electronic States of H2

• Two e- in σg1s ∴ Λ = 0– σ wavefunctions have

cylindrical symmetry: + x + = +– g x g = g– Two e- ∴ S = 0

1Σg+

– Bond energy = 4.476 eV

• One e- in σg1s & one e- inσu1s ∴ Λ = 0– g x u = u– Two e- ∴ S = 0 or 1

3Σu+ or 1Σu

+

1s 1s

σg

σu1Σg

+

1s 1s

σg

σu3Σu

+

1s 1s

σg

σu1Σu

+

AY216 18

H2 Potential Curves

1s 1s

σg

σu1Σg

+

1s 1s

σg

σu3Σu

+

1s 1s

σg

σu1Σu

+

?

AY216 19

Ground Electronic States of H2

• Two e- in σg1s ∴ Λ = 0– σ wavefunctions have

cylindrical symmetry: + x + = +– g x g = g– Two e- ∴ S = 0

1Σg+

– Bond energy = 4.476 eV

• One e- in σg1s & one e- inσu1s ∴ Λ = 0– g x u = u– Two e- ∴ S = 0 or 1

3Σu+

1s 1s

σg

σu1Σg

+

1s 1s

σg

σu3Σu

+

1s 1s

σg

σu1Σu

+

AY216 20

Excited Electronic States of H2

• Additional states arise from2s and 2p electrons– One e- in σg1s & one e- in πu2p∴ Λ = 0,1• + x - = -• g x u = u• S = 1

3Πu-

– One e- in σg1s & one e- in σg2s∴ Λ = 0• g x g = g• S = 0

1Σg+

1sσg

σu1s

2s2sσu

σg

1Σg+

1s

σg

σu1s

2s2s

2p 2p

σu

σg

πg

πu3Πu

-

σg

σu

AY216 21

Hybrid Orbitals

• Forms of bonding combining s and p atomicwavefunctions 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 oforganic chemistry

– sp (acetylene: H-C≡C-H)– sp2 (ethylene: H2C=CH2; or benzene: C6H6)– sp3 (methane: CH4)

• Stable O and N molecules that form with H– H2O and H2O2

– NH3 and N2H4

AY216 22

Electronic Transitions• The matrix elements are of the form, e.g., <ψ1|µ|ψ2>,

where µ is the operator for the dipole moment– Molecular term designations describe the symmetry of ψ– To find the selection rules, find combinations ψ1, ψ2 , 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 H2 X 1Σg+ state is 4.476 eV

– Naïvely expect that FUV radiation λ < 2769 Å will dissociateinterstellar H2

– The transition 1Σg+ → 3Σu

+ is forbidden and therefore H2 isprotected

AY216 23

Photodissociation of H2

• 1Σg+ → 3Σu

+ is notdipole permitted– Photodissociation of

H2 in the ISM mustproceed via highenergy photons

– H2 cannot form viaradiativeassociation of twoHI atoms in 1s

Σ

AY216 24

H2 1Σg

+ - 1Σu+ Transition(s)

• FUSE spectra of ISM H2 in absorption of the1Σg

+ – 1Σu+ transition

– Lyman band of lines and not a single transition

Sembach 2001 AJ 121 992

AY216 25

Vibrational & Rotational Bands

• Electronicstates are splitinto vibrational(v) androtational (J)sublevels

• For anabsorption line– P: ΔJ = -1– R: ΔJ = +1

• For transitionswhere Ω = 0 inboth states, theQ-branch, ΔJ =0 is absent

A

X v”

J”+1J”J”-1

v’

J’+1J’J’-1

P(J) R(J)

AY216 26

Interstellar Molecular R & P Branches

• Interstellar C3– X 1Σg+ – A 1Πu

• When multipletransitionsbetweenelectronic statesare observed lineratios give theexcitationconditions– The synthetic

spectrum for Trot =80 K at R = 105

• Famous exampleis CN, CH+ & CH

AY216 27

O2

• For O2 the only electrons that contributeto the angular momentum are the twoπg*2p electrons– Two πg

*2p electrons yield 3Σg-, 1Σg

+, & 1Δg

– The state of of highest multiplicity is theground state: 3Σg

-,

• Sound familiar?– Similar to figuring out the terms which arise

from two p electrons

AY216 28

Heteronuclear Molecules• Molecules like CN & CO have sufficiently similar

nuclear charges that they can be treated usinghomonuclear 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 arenow slightly different

• CO is isoelectronic with N2– 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: 1Σ+

– First excited state (σ2p)1 (π*2p)1

• The two unpaired electrons yield singlet and triplet states• 3Π and 1Π

AY216 29

Rotational & Vibrational Structure

• Molecular transitions can be categorized asrotational, vibrational and electronic– Typically, the energies are very different

• Erot ~ 10-3 – 10-2 eV: rotational energy of the molecule• Evib ~ 10-2 – 10-1 eV: KE & PE of the nuclei associated with

vibration about their equilibrium positions• Eel ~ 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

ψtot ≈ ψnucψel– Essentially the motions of the heavy nuclei are

much faster than that of the light electrons

AY216 30

The Born-Oppenheimer Approximation

• A further approximation involves thefactorization of ψnuc

ψnuc = ψvib ψrotso that

ψtot = ψelψvib ψrot• This factorization justifies writing the total

energy of a moleculeEtot = Eel + Evib + Erot

• The electronic part is characterized by apotential curve, with a minimum at theequilibrium radius, re, if the molecule is stable

AY216 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 ~ me/mn ~ 10-4

– Additional effects are magnetic interactions between theirvarious orbital & spin angular momenta

• To the first approximation molecular dynamicsreduces to

1. Rigid body motion2. Normal modes of oscillation (3N-6)

• (1) describes to the rotational motion of moleculeswith wavelengths in the far-IR/mm bands; and (2)vibrational motions observable at near-IRwavelengths

AY216 32

Harmonic Oscillator

• The vibration of a diatomic molecule can betreated as the stretching and compression of aspring (the molecular bond)– Approximated as an harmonic oscillator the

potential is

V(x) = kx2/2

andEv=hν(v + 1/2)

where

ν = (1/2%)(k/µ)1/2

and µ is the reduced mass

AY216 33

Anharmonic Oscillators

• The potential energy curve for realmolecules is not parabolic– Not harmonic oscillators

• More generallyG(v) = ωe(v+1/2) - xe(v+1/2)2 + ye(v+1/2)3 +…

0.01013.292169.7512C16O 1Σ+

0.8121.334401.211H2 1Σ+

g

ye

(cm-1)

xe

(cm-1)

ωe

(cm-1)

http://physics.nist.gov/PhysRefData/MolSpec/Diatomic/index.html

AY216 34

H2 & CO

• Because it is a light molecule 1H2 has muchhigher fundamental vibrational frequency than12C16O– The vibrational levels of 1H2 lie higher up the

vibrational potential and effects of anharmonicity arelarger

– Δv=1 for H2 is at 2.40 µm; CO at 4.66 µm

5341.5

10257.4

G(2)

(cm-1)

1081.6

2170.4

G(0)

(cm-1)

4259.9

8087.0

G(2)-G(0)

(cm-1)

2143.23224.812C16O

4161.26331.61H2

G(1)-G(0)

(cm-1)

G(1)

(cm-1)

AY216 35

Vibrational Levels of H2

109876543210

v=1-02.4 µm

Fernandes et al. 1997 MNRAS 290 216

• CO has a permanentdipole moment– Stong vibrational bands

• H2 does not!

AY216 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 groundthroughout much of mm and sub-mm.

• Very dry conditions are needed at high frequencies, e.g., the Atacamadesert. The record is CO 9-8 1.087 THz, detected from 5525 m in N. Chile(Marrone et al. 2004)

• Rotational motion of molecules is determined by moments ofinertia & associated angular momentum– Classically, any object has three orthogonal principal moments of

inertia (symmetric inertia tensor) and simple expressions for therotational energy and the angular momentum

– Customary to classify the rotational characteristics of moleculesbased on the values of the principle moments of inertia

AY216 37

An Elementary Example: CO

• For a rigidly rotating diatomic or linearmolecule the rotation energy levels are

• Conventionally, the rotational constantBe = h/8%2cI is quoted in cm-1 or Hz– For CO Be = 1.9225167 cm-1 or ~ 2.77 K

€

Erot = 12 Iω

2; l2 = I2ω 2 = J J +1( )h2

= 12 J J +1( )h2 I

= BeJ J +1( ) hc where I = miri2

i∑

AY216 38

An Elementary Example: CO

• Electric dipole transitionshave ΔJ = ± 1– For a rigid rotator the

frequencies are integermultiples of the J=1-0transition

– Exact frequencies are notprecise multiples becauseCO is not perfectly rigid€

ν J +1,J = Erot J +1( ) h − Erot J( ) h

= 2Be J +1( )J=0

J=1

J=2

J=3

115.271 GHz

230.538 GHz

345.795 GHz

5.53 K

16.60 K

33.19 K

vv0 2v0 3v0 4v0

AY216 39

Electronic, Vibrational & RotationalEnergy Levels of CO

• Comparisonof energiesof theelectronic(singlet),vibrationalandrotationalstates of CO

AY216 40

Real Molecules

• In real molecules the bond is stretchedby rotation and the rotational energystates are approximated by fittingformulae

– Effective B decreases with increasing J,correcting for centrifugal distortion

• For CO B = 57,635.9683 MHz, D = 0.1835055 MHz, & H =1.725x10-7 MHz

€

EJ h = BJ J +1( ) −DJ 2 J +1( )2 + HJ 3 J +1( )3 −K

AY216 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 rotationaltransitions– H2 0-0 S(0) is at 10.7 THz– The corresponding transitions of light hydrides are

at THz frequencies• CH+ (M.W. = 13) B = 417.62 GHz

– Many of these high frequency transitions areblocked by terrestrial H2O and O2• Ground-based observation of species abundant in the

atmosphere is challenging

AY216 42

Rotational Spectra• ISO/LWS spectrum

of CRL 618 (C-richPPN)– Continuum-

subtracted spectra &model spectrum

– CO, 13CO, HCN,H2O, and OH areindicated by arrows

– HNC J=22-21(150.627 µm) toJ=17-16 (194.759µm) indicated byvertical lines

Herpin et al. 2000 A

pJ 530 L129

AY216 43

Rotational Lines in Orion• The 607-725

GHz (450 µm)spectrum ofthe starforming Orioncloud

• Dominated byrotationaltransitions ofCO, CS, SO,SiO, HCN,HCO+, H2CO,SO2 & CH3OH

• More than 103 transitions, many unidentified• Strongest transition is CO J=6-5 however, the integrated SO2 and

CH3OH dominate the cooling in this region

CO J=6-5

— ⊕

H2O

— ⊕

O2

Schilke et al. 2000 A

pJS 132 281

AY216 44

Rotational Lines in Orion• The 607-725

GHz (450 µm)spectrum ofthe starforming Orioncloud

• Dominated byrotationaltransitions ofCO, CS, SO,SiO, HCN,HCO+, H2CO,SO2 & CH3OH

• More than 103 transitions, many unidentified• Strongest transition is CO J=6-5 however, the integrated SO2 and

CH3OH dominate the cooling in this region

Schilke et al. 2000 A

pJS 132 281

AY216 45

Sub-mm Atmospheric Transmission

• Mauna Kea, 1mm H2O

— ⊕

H2O

— ⊕

O2

AY216 46

H2 Symmetry & Selection Rules• The angular momentum eigenfunctions for J are

the Legendre polynomials– ψ(J=0) = 1, ψ(J=1) = cosθ, ψ(J=2) = (3cos2θ-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 Jwith statistical weight 3

• Orthohydrogen

• Dipole permitted rotational transitions have ΔJ = 1– These transitions do not occur in H2

– Observed rotational spectra are electric quadrupoleΔJ = 2, e.g., 0-0 S(0)

AY216 47

Linear Molecules & Symmetric Rotors

• The principal moments of inertia are designatedIa, Ib, and Ic– Conventially Ia ≤ Ib ≤ Ic

• A molecule which is linear or has a rotationalsymmetry axis a symmetric top– Either Ic = Ib > Ia or Ic > Ib = Ia

• Linear molecules, .eg., CO, have a small Ia about the axisof the molecule so they are prolate symmetric rotor

• Other molecules, e.g., benzene, have the largest momentof inertia about the symmetry axis and are oblatesymmetric rotors

• Molecules which are spherically symmetric, e.g, methanehave three equal moments of inertia and are sphericalrotors

– Molecules Ic ≠ Ib ≠ Ia are asymmetric rotors

AY216 48

Symmetric Rotors• Classically, the energy of rotation, E, is

• Consider a symmetric rotor Ix = Iy = IbSince angular momentum is P2 = Px

2 + Py2 + Pz

2

€

E = 12 Ixωx

2 + 12 Iyωy

2 + 12 Izωz

2

=Px2

2Ix+Py2

2Iy+Pz2

2Iz

€

E =P 2

2Ib+ Pz

2 12Ic

−12Ib

Ic

IbIa

AY216 49

Symmetric Rotors

• The total angular momentum, P, of the rotatingmolecule is quantized as is the z component

J = 0, 1, 2… , K = 0, 1, 2, …• Hence the energy is

€

P 2 = J(J +1)h2, and Pz2 = K 2h2

€

E =h2

2IbJ(J +1) +

h2

2Ic−

h2

2Ib

K 2, or

E = BJ(J +1) + (C − B)K 2

AY216 50

Linear Molecules

• For linear molecules like H2, CO, or HCN– IC = IB >> IA

• IA is the moment of inertia about the internuclearaxis

– Recover the elementary expression for a rigidrotator

E = B(J+1)J

AY216 51

Symmetric RotorsE = B(J+1)J + (C-B)K2

• For K = 0 the energy levels are those of alinear 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 levelsstarting 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

AY216 52

Oblate & Prolate Symmetric-Top Molecules

Oblate

Prolate

AY216 53

Selection Rules

• By symmetry there can be no dipolemoment perpendicular to the axis of aaxis of a symmetric top– No torque along that axis due to E fields

associated with radiation• ∆K = 0

– The dipole moment lies along the molecularaxis• This axis preceses around the total angular

momentum with frequency P/2$Ib thus ∆J = ±1

– Levels with J = K are metastable

AY216 54

NH3• NH3 was the first

polyatomic moleculedetected in the ISM(Cheung et al. 1968)

• Oblate symmetric top withpyramidal symmetry– Any molecule with 3-fold

(or greater) rotationalsymmetry is a symmetrictop

• The ground state 10-00transition, i.e., J=1-0, K=0-0 occurs at 572.49815GHz– Only observable from

airborne telescopes(SOFIA) or space

Ho &

Tow

nes 1983 AA

RA

21 239

10-00

AY216 55

NH3• An inversion transition

occurs when the N tunnelsthrough the plane of the threeH atoms– In contrast to most non-

planar molecules, thepotential barrier is weak andtunneling occurs rapidly

– The corresponding frequencyfalls in the microwave range

– Each of the inversiondoublets splits due to theelectrostatic interactionbetween the electricquadrupole moment of the Nnucleus and the electrons

• Weaker magnetic hyperfineinteractions associated withthe H nuclei yield a total of18 hyperfine transitions

Ho &

Townes 1983 A

ARA

21 239

AY216 56

NH3

Rydbeck et al. 1977 ApJ 215 L35

AY216 57

Asymmetric Rotators

• Three different principal moments of inertiaIa ≠ Ib ≠ Ic

• Only J and E are conserved– Total angular momentum J remains a good

quantum number

• J states are labeled by two approximatequantum numbers:– Projection of J on two molecular axes: K– & K+– Notation: JK–K+– K– & K+ only become good quantum numbers

(conserved projections of angular momentum) in thelimit of prolate & oblate symmetric tops

AY216 58

H2CO (Formaldehyde)

* in frequency units A = h/8%2IA; B = h/8%2IB; C = h/8%2IC

Mangum

& W

ootten 1993 A

pJS 89 123

• H2CO is nearlysymmetric prolate top– A = 281,970.672 MHz*– B = 38,836.046 MHz– C = 34,002.203 MHz

• Small asymmetry aboutthe C-O axis causes adeviation from pureprolate symmetry– Splits the degeneracy

of energy levels with K> 0

AY216 59

H2CO (Formaldehyde)

• The 6 cm 4830 MHztransition of H2CO wasdetected in the radio in1969– Always seen in

absorption, even indark clouds,suggesting that itabsorbs the 3Kbackground

303

202

101

000

111

110

212

211

72,838 MHz

145,603 MHz

Para

Ortho

4830 MHz

14,488 MHz

150,488140,838 MHz

AY216 60

Rotational Levels of Ortho-H2O

• A = 835,839.10 MHz• B = 435,347.353 MHz• C = 278,139.826 MHz

AY216 61

Nuclear Spin

018O

5/217O

016O

1/215N

114N

1/213C

012C

1/2H

Spin INucleus