Content. 1. Overview of molecular spectra 2. Rotational spectra 3. Vibrational spectra 4. Electronic spectra

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1 Content 1. Overview of molecular spectra 2. Rotational spectra 3. Vibrational spectra 4. Electronic spectra Molecular orbital theory Electronic quantum numbers Vibrational structure of electronic transitions Rotational structure of electronic transitions Transition strengths 5. Hund s cases (angular momenta coupling schemes) 6. Applications

2 Molecular orbital theory LCAO-method (Linear Combination of Atomic Orbitals) Bonding Antibonding Separation Separation

3 Molecular orbital theory Potential energy curves With minimum Bonding Without minimum Antibonding

4 Molecular orbital theory: g/u symmetry Parity of the electronic wavefunction Even ( gerade in German) subscript g, e.g. σ g orbital Odd( ungerade in German) subscript u, e.g. σ u orbital Only for linear molecules with inversion center (homonuclear diatomic molecules, or e.g. CO 2 )

5 Molecular orbital theory σ-orbitals: rotational symmetry about internuclear axis

6 Molecular orbital theory π-orbitals: antisymmetric under 180 rotation about internuclear axis

7 Molecular orbital theory Fill orbitals with electrons Start at lowest energy Pauli principle H 2

8 Molecular orbital theory Heteronuclearmolecules Unequal contribution from different atomic orbitals

9 Molecular orbital theory Polyatomic molecules Still LCAO Hybrid orbitals more ideal as starting point BeH 2

10 Molecular orbital theory Most relevant for spectroscopy: potential energy cures Every curve corresponds to one electronic state Next: Only: diatomic molecules Quantum numbers characterizing electronic states Notation: term symbol Initial assumption: Hund s case (a) (later more general)

11 Orbital angular momentum Λ Precession of Labout internuclear axis Axial symmetry of electric field in diatomic molecule Constant component Λ Possible values M = L, L 1, L 2,, L Good quantum number: Λ = L ML Values Λ = 0, 1, 2, 3, Designation Σ Π Φ Λ

12 Orbital angular momentum Λ If Λ> 0 (Π,, states): electronic state double-degenerate (reversing L leaves energy unchanged) If Λ= 0 (Σstates): Non-degenerate, but two energetically distinct states exist Σ + symmetric under reflection at plane through internuclear axis Σ antisymmetric under reflection at this plane Don t confuse with g/u symmetry (parity of electronic + wavefunction, homonuclear!) e.g. Σ, Σ, Σ g g u Λ Σ +, Σ

13 Spin Spin unaffected by electric field Λ= 0 (Σstates): Spin fixed in space (in absence of rotation or magnetic field) Λ> 0 (Π,, states): Internal B-field along internuclear axis (caused by electrons) Spin precesses about internuclear M s denoted by Σ Good quantum number: Σ = S, S 1, S 2,, S (positive and negative values!) multiplicity 2S + 1 Σundefined if Λ= 0 (Σstates)

14 Total electronic angular momentum Ω Ω=Λ+ Σ Λand Σparallel vectors simple algebraic addition Quantum number of total electronic angular momentum Ω = Λ + Σ 2S+ 1 Λ Λ+Σ Term symbol: e.g. 2 Π 1/2 or 3 0 or 4 Π 1/2

15 Spin-orbit interaction multiplets If Λ> 0 (Π,, states): (2S+ 1) different values of Λ+ Σ different energies Te = T0 + ζ ΛΣ Multiplet splitting: equidistant (in contrast to atoms) Always (2S+ 1) different states, even if S> Λ

16 Notation for electronic states Electronic states labeled with letters X: ground state A, B, C, : excited states with same multiplicity as ground state Sorted by increasing energy a, b, c, : different multiplicity than ground state increasing energy

17 Notation for electronic states Examples of band systems d 3 Π a 3 Πsystem A 2 Π X 2 Σsystem B 2 Σ X 2 Σsystem

18 Vibrational structure of electronic transitions Electronic-vibrational transition band Rotational substructure of bands neglected at first No selection rules on υ υ= 0, ±1, ±2, υ υ Labeling of bands: ( υ, υ ) e.g. (0,0) band (1,0),(0,2) bands Frank-Condon principle: Probabilities / relative intensities of bands Appearance of spectra Electronic transition (UV/optical) Vibrational transition (infrared) Rotational transition (sub-mm)

19 Vibrational structure of electronic transitions Band progression: series of bands with same υ or υ e.g. (0,0) band, (1,0) band, (2,0) band Almost equidistant (0,0), (1,1), (2,2), bands Almost same frequency

20 Frank-Condon principle Born-Oppenheimer approximation Electronic movement rapid compared to nuclear motion Nuclear separation constant during electronic transition Frank-Condon principle: Large transition probability if maxima of wavefunctions at same separation R Nuclei spend most time at turnaround points of vibration (except vibrational ground state: at center) Energy Excited state Groundstate

21 Frank-Condon principle Transitions most probable to/from intersection points (energy of level with potential energy curve) 3 types of intensity patterns in band progression Depends on change of mean internuclear distance

22 Frank-Condon principle Example of 2 nd type: CO A 1 Π X 1 Σsystem band progression with υ = 1 Example of 3 rd type: I 2 vapor band progression with υ = 0

23 Rotational structure of electronic transitions Substructure? Formation of band head? (0,0) band head (1,1) band head

24 Rotational structure of electronic transitions Substructure of individual band Consider electronic transition Energy difference of electronic transition: = T + G + F = + F J F J ɶ e 0 ν 0 constant in specific band ( ) ( ) ν ν Fvariable many lines within a band Λ, Σ, υ, J Λ, Σ, υ, J F possibly large because electronic states different (in contrast to infrared rotational-vibrational spectra)

25 Branches of a band Band structured in branches Series of lines for different J values Electric dipole selection rules In general J = J J = 0, ± 1 Sometimes J = J J = ±1 If at least one Λ 0 J = 0, ± 1 three branches two branches If Λ= 0 in both states J= 0 forbidden (e.g. 1 Σ 1 Σ) or very weak

26 Branches of a band Definition of branches: R branch: J = J J = + 1 Q branch: J = J J = 0 P branch: J = J J = 1 Wavenumbers υ υ υ ( ) Fυ ( J) ( ) υ ( ) ( ) F ( J) R branch: ɶ ν = ν + F J + 1 Q branch: ɶ ν = ν + F J F J 0 P branch: ɶ ν = ν + F J υ

27 Remember: Energies (cm 1 ) of branches: ɶ ν ɶ ν R 0 Q 0 Branches of a band 2 ( ) = ( + 1) ( + 1) 2 ( + 1) ( ) ( ) ( B B ) J ( B B υ υ υ υ ) J ( ) ( ) 2 = ν + 2B + 3B B J + B B J υ υ υ υ υ = ν + + ɶ νp = ν 0 B + B J + B B J Rewrite: 2 2 D B F J B J J D J J B J J υ υ υ υ υ υ υ υ ( ) ( ) 2 R and P branch: ɶ ν = ν 0 + B υ + B υ m + B υ B υ m with m= J+ 1 (R branch), m= J(P branch) 2 Q branch: ɶ ν = ν 0 + ( B υ B υ ) m + ( B υ B υ ) m with m= J

28 Branches of a band Rewrite (Fortratparabolae): R and P branch: ɶ ν = ν 0 + B + B m + B B m ( ) ( ) 2 υ υ υ υ with m= J+ 1 (R branch), m= J(P branch) Q branch: ɶ ν = ν 0 + B υ B υ m + B υ B υ m with m= J ( ) ( ) 2 If B υ < B υ band head in R branch, band shaded to the red If B υ > B υ band head in P branch, band shaded to the blue

29 Band heads and shading Example: AlH molecule, theory (top) and observations (bottom) Band shaded to the red Band head forming in R branch Bandhead in Q branch

30 Transition strengths Definition analogous to atomic transitions 2 ul lu D = Dmm mm mm S S S υ Jm υ J m = = = Separation of electronic, vibrational, and rotational transition probabilities m m D 2 2 mm = De q υ υ SJJ where D e 2 q υ υ S J J electronic transition probability vibrational transition probability (Frank-Condon factor) rotational transition probability (Hönl-London factor) 2

31 Hönl-London factors Hönl-Londonfactorcontrolsintensity variation of lines within rotational branches Simple expressions in Hund s case (a)

32 Intensity variation in rotational branches MgH 2 Π 2 Σsystem in solar spectrum Berdyugina et al. 2002

33 Content 1. Overview of molecular spectra 2. Rotational spectra 3. Vibrational spectra 4. Electronic spectra 5. Hund s cases (angular momenta coupling schemes) Hund s case (a) Hund s case (b) Intermediate Hund s case (a-b) 6. Applications

34 Interaction between different types of motion Simultaneous electronic, vibrational, and rotational motion interactions Anharmonicpotential energy: Interaction electronic vibrational motion Vibrating rotor: Interaction vibrational rotational motion Hund s cases: Interaction electronic rotational motion Different coupling schemes of angular momenta (orbital angular momentum, spin, rotation) Hund s cases (a), (b), (c), (d), and (e)

35 Hund s case (a) Land Sstrongly coupled to internuclear axis Λand Σdefined Weak interaction with rotation Total electronic angular momentum Ω = Λ + Σ Ω = Λ + Σ Total angular momentum J = Ω + R J = Ω, Ω + 1, Ω + 2,...

36 Hund s case (a) Strong spin-orbit interaction multiplet splitting > rotational splitting J Ω

37 Hund s case (b) Lstrongly coupled to internuclear axis Λdefined Snot coupled to internuclear axis Σand Ωundefined Λ= 0 and S 0 Or weak spin-orbit coupling Total angular momentum apart from spin N = Λ + R N = Λ, Λ + 1, Λ + 2,... Total angular momentum J = N + S J = N + S, N + S 1, N + S 2,..., N S rotational levels (given J) split into (2S+ 1) states

38 Hund s case (b) Weak spin-orbit interaction multipletsplitting < rotational splitting N 2 Σstate 3 Σstate J N J

39 Hund s cases (c) and (d) Hund s case (c) Hund s case (d)

40 Intermediate Hund s cases Hund s cases are special, limiting cases Σstates (Λ= 0): strictly Hund s case (b) In general: intermediate case Most frequent: intermediate Hund s case (a-b)

41 Transition Hund s case (a) to (b) Spin uncoupling J Often: Case (a) for small rotation Case (b) for fast rotation J N

42 Selection rules electronic transitions Rigorous rules: J = 0, ± 1 (but J = 0 / J = 0) g u (while g / g, u / u) Rules valid for Hund s cases (a) and (b): Λ = 0, ± 1 S = Σ Σ, Σ Σ (while Σ / Σ )