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 lines when approaching Balmer jump

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

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

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

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. 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: +1 -1 + 

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 / c the vibrational frequency in [cm1]
Equidistant levels

16. Anharmonic oscillator
Better model for molecular vibrations, accounting for Coulomb 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 inertia L 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. Vibrating rotator Simultaneous rotation and vibration
Energy: sum of anharmonic oscillator plus non-rigid rotator
Total energy (incl. electronic energy) [cm1]:
Electronic
energy
Vibrational
Rotational

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

22. Observed molecular spectra: optical
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

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. Rotational structure of electronic bands
(0,0) band head
(1,1) band head
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 head forming in R branch
Band shaded to the red
If 
band head in R branch,
band shaded to the red
band head in P branch,
band shaded to the blue
Example: AlH molecule, theory (top) and observations (bottom)

28. 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
d3  a3 system
B2  X2 system
A2  X2

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
Scissoring
Twisting