Molecular Spectroscopy

1. Molecular Spectroscopy Svetlana Berdyugina Kiepenheuer Institut für Sonnenphysik, Freiburg, Germany Hot Molecules in Exoplanets and Inner Disks

2. Outline • Observed spectra: band structure • Electronic energy:angular momentum, spin, quantum numbers, term notation • Vibrational energy:(an)harmonic oscillator • Rotational energy:models of rotators (dumbbell…) • Combining different contributions: • Vibrating rotator • Structure of electronic band systems 6. Multi-atomic molecules: Vibrations

3. Introduction: Detection of molecules • Optical: 0.3 – 1 m • electronic transitions • Infrared : 1 – 200 m • vibrational transitions • Submm – Radio • rotational transitions

4. Observed spectra: atoms • Balmer series • Infinite number of lineswhen approaching Balmer jump

5. Electronic band systems: electronic states fixed per system Band: Vibrational states fixed Observed molecular spectra: optical

6. Band: • individual lines • band head • rotational structure of electronic term Observed molecular spectra: optical

7. Rotation-vibration bands: strong fundamental band other bands at about twice, three times,… the frequency Detailed structure: rotational structure  equally spaced lines Observed molecular spectra: infrared Schematic distributionof bands

8. Electronic energy • Diatomic molecules: • Join two atoms with and  total and total  electric field (axial symmetry about internuclear axis) • Coupling of angular momenta depends on involved energies • Most common coupling case: • and couple (individually) strongly to internuclear axis treat and separately

9. Orbital angular momentum • Precession of about internuclear axis • Constant component • Good quantum number: Values: electronic terms:

10. +1 -1 -1 +1 +  Orbital angular momentum • If   0: term double degenerated (approximately): reversing does not change energy • If  = 0 (-terms): non-degenerate,but two (energetically) distinct molecular terms exist:

11. Spin • Spin unaffected by electric field • Spin precesses about internuclear axis (B-field caused by electrons) • Good quantum number:spin component in the internuclear axis direction •  = S, S-1, …S (positive and negative!) • Multiplicity: 2S+1 • Exception: if  = 0 (-states)  spin projection  not defined

12. Total electronic angular momentum  • and parallel simple algebraic addition • Quantum number of total electronic angular momentum:

13. Multiplet • If   0: (2S+1) different values of (+) electronic term splits into multiplet • Term notation:

14. Vibrational Energy: Harmonic oscillator • Simplest model for vibrations • System can be represented by single harmonic oscillator with reduced mass  • Equation of motion: • Frequency:

15. Harmonic oscillator in QM • Energy levels: • Vibrational quantum number: • Vibrational energy term [cm1]with  = osc/ cthe vibrational frequency in [cm1] • Equidistant levels

16. Anharmonic oscillator • Better model for molecular vibrations, accounting forCoulomb barrier for small r, dissociation for large Ev • Higher order corrections to term values: • Vibrational constants: • Energy levels not equidistant • All  allowed in transitions(“higher harmonics”)

17. Rotational Energy: Rigid rotator Classical mechanics: • Energy: with I moment of inertiaL angular momentum QM: • Energy levels:where J = 0, 1, 2, 3, … angular momentum quantum number • Rotational energy term [cm1]: • Rotational constant:

18. Rigid rotator: spectrum • Selection rule: J = 1 • Transition energies[cm1]: • Equidistant lines with separation 2B

19. Non-rigid rotator • Internuclear distance r not fixed • Centrifugal force increasing r with faster rotation • Correction to energy levels:rotational constants D << B • Transition energies[cm1]:  line separation increases with increasing J • Consistent with observed sub-mm observations pure rotational transitions

20. Electronic energy Vibrational energy Rotational energy Vibrating rotator • Simultaneous rotation and vibration • Energy: sum of anharmonic oscillator plus non-rigid rotator • Total energy (incl. electronic energy) [cm1]:

21. Electronic energy Vibrational energy Rotational energy Vibrating rotator • Every electronic state has vibrational rotational structure Rotational structure withinevery vibrational level

22. For given band system: electronic states fixed Every band corresponds to fixed vibrational level in upper and lower electronic states. Sub-structure due to rotational levels Observed molecular spectra: optical

23. Most important selection rules of electronic transitions • Total angular momentum: • If   0 for at least one state of transition: J = 0, +1, -1 => Q, R, P branches • If  = 0 for both states: J = 1 (no Q branch) •  = 0, 1 • S = 0 •  = 0, 1, 2, 3, … • Note: J is total angular momentum J = , +1, +2, … • Note: J = 0 was not allowed for (non-)rigid rotator.It appears when including moment of inertia about internuclear axis

24. Vibrational structure of band systems Ingredients: •  = 0, 1, 2, 3, … • Almost equdistant vibrational states (cf. G() formula) Resulting vibrational structure (for fixed electronic transition): • Progressions (0,0) band, (1,0) band,(2,0) band, … almost equidistant • (0,0), (1,1), (2,2), …bands almost same frequency strongly overlapping (“fine” structure in shown spectrum)

25. (0,0) band head (1,1) band head Rotational structure of electronic bands • substructure? • How does band-head form?

26. Rotational structure of electronic bands • J = 0, 1  three branches • Transition energy [cm1] for rotational branches:R branch (J = +1):Q branch (J = 0):P branch (J = 1):where J = J'J", one prime upper level, two primes lower level • Remember:  F(J) and (F'F")are parabolas energy distribution within a branch is parabola (with increasing J)!

27. Rotational structure of electronic bands Band headforming in R branch Band shadedto the red If  band head in R branch, band shaded to the red If  band head in P branch, band shaded to the blue Example: AlH molecule, theory (top) and observations (bottom)

28. d3  a3 system A2  X2 B2  X2 system Notation Electronic transitions (band systems): • Upper level always written first, then lower level,e.g. 1  1 • To distinguish electronic states of same type in same molecule:use letters X, A, B,…, a, b, … Bands (fixed vibrational levels): • (up, low) band,e.g. (0,0) band, (1,0) band • Example: (0,0) band of d3  a3 system

29. Summary • Total energy = electronic (Ee) + vibrational (Ev) + rotational (Er) • Ee >> Ev >> Er • Vibrational-rotational structure of electronic states • Simple model: vibrating rotator  explains features of spectrum

30. Multi-atomic molecules: Vibrations Stretching: a change in the length of a bond Bending: a change in the angle between two bonds Rocking: a change in angle between a group of atoms Wagging: a change in angle between the plane of a group of atoms Twisting: a change in the angle between the planes of two groups of atoms Out-of-plane: a change in the angle between any one of the bond and the plane defined by the remaining atoms

31. Multi-atomic molecules: CH2 Symmetric stretching Asymmetric stretching Wagging Rocking Twisting Scissoring