Molecular spectroscopy Multispectral imaging (FAFF 020, FYST29) fall 2017

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1 Molecular spectroscopy Multispectral imaging (FAFF 00, FYST9) fall 017 Lecture prepared by Joakim Bood

2 Molecular structure Electronic structure Rotational structure Vibrational structure Molecular transitions Outline Rotational transitions (pure rotational) Rovibrational transitions (rotation + vibration) Rovibronic transitions (rotation + vibration + electronic) Population distributions

3 log(compexity) log(size) Molecules Molecules are formed by binding atoms together in a way such that the total energy becomes lower than the sum of the energies of the individual atoms.

4 Example of a molecular spectrum M. Jonsson, J. Borggren, M. Aldén, J. Bood, Appl. Phys. B 10, (015)

5 Multispectral imaging of species distributions in turbulent flames Selectivity from excitation and emission wavelengths

6 Absorption Major types of molecular spectroscopy I 0 I dz I z, ν = I 0 e σ ν Nz (Beer-Lambert z law) Emission (e.g. fluorescence) Scattering Spectrograph & detector Laser Lens

7 Res. fluorescence 1 10 ev Fluorescence Liquid and solid phase broad energy bands Free atoms Free molecules Condensed matter Abs. Emission Energy band Electronic states Vibrational states Rotational states 0.1 ev ev Energy band Human tissue contains a lot of water, lipids and proteins broad energy bands instead of the sharp energy levels of free atoms and molecules.

8 Light absorption in gases, liquids, and solids Gas absorption Gives rise to sharp spectral lines. Most light is transmitted. P. Lundin, Doctoral Thesis, LRAP-488, Lund University, 014 Absorption in liquids and solids Gives rise to broad spectral features. The transmitted light from wine is red, while blue, green, and yellow light is absorbed.

9 Intensity Fluorescence in gases, liquids, and solids Gases Liquids and solids Wavelength

10 The electromagnetic spectrum Nuclear state changes Inner electron transitions Outer electron transitions Vibrational state changes Rotational state changes Spin state changes, NMR

11 Spectral quantities and units Quantity Formula SI unit Comment Frequency n = c/l Hz or s -1 Wavelength l = c/n m often nm or Å Wavenumber ν = 1/λ m -1 Sometimes s instead of ν (often cm -1 ) Energy E = hn = hc/l J = hcν The formulas above are only strictly valid for n = 1 In spectroscopy, and in particular Raman spectroscopy, it is very common to work with wavenumbers in cm -1. It is convenient as it is directly proportional to energy.

12 Energy Molecular energy structure Electronic energy Energy level diagram Vibrational energy Electronic levels Vibrational levels Rotational levels Rotational energy Internuclear distance

13 Quantum mechanics - Molecules The Schrödinger equation: n m r V r n r En n r H n r En n Born-Oppenheimer approximation Vibrational motion is much faster than rotational motion E tot tot E el el vib E vib rot E rot E tot = T e + G(v) + F(J)

14 Electronic structure Svanberg: 3.1

15 Potential energy curves (PECs) E 1 (R), E (r), and E 4 (R) are bound (stable) states E (R) is an unbound (unstable/repulsive) state B-O approx. E molecule = E e- + E nuclei It also means molecule = e- nuclei Going beyond the approximation of clamped nuclei, the PECs describe the potentials in which the nuclei can vibrate (more about this later). R eq B-O approximation: m p 000 m e The electrons are moving much faster than the nuclei The electronic states are at any moment essentially the same as if the nuclei were fixed. Thus, in the B-O approximation R is fixed.

16 The electronic orbital angular momentum Electron orbital angular momentum, L, of molecules is quantized M L 3 z Only the component along the internuclear axis, L z, is a constant of motion quantum numbers: M L =L, L-1,.., -L Quantum number introduced: = M L, =0,1,,..,L =0,1,, =0 means a -state =1 means a P-state = means a D-state L Svanberg: 3.1 and Fig. 3.

17 Electronic spin Molecules also have spin angular momentum. S precesses around the internuclear axis S can have S+1 different projections,, on the internuclear axis S+1 = multiplicity of the state Total angular momentum: = + Projections on the z-axis: S z = Σ ħ = S,S-1,,-S Svanberg: 3.1 L z = Λ ħ Ω = Λ + Σ = 0, 1,,, P, D,

18 Term symbols for diatomic molecules For an atom: S+1 L J For a diatomic: S+1 Λ Ω Λ: Projection of orbital angular momentum onto the internuclear axis. Magnitude: Λ = Λ ħ Symbols: 0 1 Symbol P D Atoms: L 0 1 Symbol S P D S: Σ: Ω: Total electronic spin (sum of spins in unfilled shells). Magnitude: S = S ħ Projection of S onto the internuclear axis (only defined when 0) Magnitude: Σ = Σ ħ Allowed values: = S,S-1,,-S (S+1 values) Sum of projections along the internuclear axis of electron spin and orbital angular momentum. Ω = Λ + Σ = +,+-1,,- (S+1 values for S)

19 Rotation and vibration of molecules Svanberg:

20 Energy Molecular energy structure Electronic energy Energy level diagram Vibrational energy Electronic levels Vibrational levels Rotational levels Rotational energy Internuclear distance

21 Quantum mechanics - Molecules The Schrödinger equation: n m r V r n r En n r H n r En n Born-Oppenheimer approximation Vibrational motion is much faster than rotational motion E tot tot E el el vib E vib rot E rot

22 Molecular rotation Classical picture Moment of inertia: I qq q mixi x i m i i q Reduced mass (diatomic molecule): 1 1 m A 1 m B m m A mb m A B I R where R is the equilibrium bond length (internuclear distance) of the molecule Svanberg: 3.

23 Molecular rotation Quantum mechanical picture T Classical mechanics: 1 qq q q I q J I q qq where J qq = I qq q Quantum mechanics: (J q is an operator of the angular H J I x xx J I y yy J I z zz momentum) There is no potential energy associated with pure rotational motion

24 Four types of rigid rotors Linear rotors - One moment of inertia is zero (e.g. CO, HCl) Symmetric rotors - Two equal moments of inertia, one different (e.g. NH 3 ) Spherical rotors - Three equal moments of inertia (e.g. CH 4 ) Asymmetric rotors - Three different moments of inertia (e.g. H O)

25 Diatomic molecules - Basics A diatomic molecule has a total wave function consisting of an electronic wave function, and a nuclear wave function. The nuclear wave function can be separated into a rotational wave function and a vibrational wave function. In the Born-Oppenheimer approximation these are initially treated separately: E tot =E el + E vib + E rot Time scales Electronic interaction: s Vibrations: s Rotations: s

26 x J I H eigenvalues: J = J x +J y +J z J Diatomic molecules Eigenfunctions: E J, M J J ( J 1) I ( J = 0,1,,. M J = J, J-1,., -J ) The rotational constant, B: (Spherical harmonics, known from the atomic structure) J+1 possible M J values for each rotational level J+1-fold degenerate. B Joule = ħ I B cm 1 = B = ħ 4πcI Svanberg: 3. F J = BJ J + 1 (cm -1 )

27 Energy Rotational energy levels for a diatomic molecule Solutions to the Schrödinger equation: E J F(J) J h 8 E J hc I J( J h 8 ci 1) J ( J 1) I = μr e m1m m m 1 BJ ( J 1) [ cm 1 ] m 1 m C r 1 r R e 0B F(4) J=0, 1,,. 1B 6B B 0 F(3) F() F(1) F(0) The energy separation increases with increasing J Svanberg: 3.

28 Energy levels for non-rigid rotator F(J) J BJ ( J 1) DJ 1 ( J 1) [ cm ] J Rigid rotator Non-rigid rotator D centrifugal distortion constant F(18) = 683,40 cm -1 for N without D-correction F(18) = 68,7 cm -1 for N with D-correction Svanberg: Fig 3.6

29 Solutions to Schrödinger equation for harmonic oscillator: Vibrational energy levels G v = ω e v + 1 (cm -1 ) G(0) = 0.5 e, G(1) = 1.5 e, G()=.5 e A better description of the potential is given by the Morse function: E Deq. [ 1 exp{ a( req. r)}] where a is a constant for a particular molecule. Energy corrections can now be introduced. G v = ω e v + 1 ω ex e v + 1 (cm -1 ) Svanberg: 3.3

30 Interaction between rotation and vibration The vibrational and rotational energies can not be treated quite independent. A molecule can vibrate times during a rotation. The rotational constant B v in a vibrational state v, can be expressed as B v B e ( v 1/ ) e [ cm 1 ] The centrifugal distortion constant D v in a vibrational state v, can be expressed as D v D e ( v 1/ ) e [ cm 1 ]

31 Molecular constants T e e e x e B e e D e From Huber and Herzberg: Constants of diatomic molecules Molecular constants are available at:

32 Molecular transitions Svanberg: 4..

33 Energy Energy Energy Molecular transitions Rotational transitions Rovibrational transitions Rovibronic transitions Internuclear distance Internuclear distance Internuclear distance microwaves infrared (IR) vis ultraviolet (UV)

34 Energy Pure rotational transitions A heteronuclear diatomic molecule possesses a permanent dipole moment and therefore emits radiation when the rotation is changed. Selection rule: DJ = 1 J 9 Rigid molecule: F( J ) BJ ( J 1) 8 7 Transitions at the following wavenumbers: n ( J 1 J ) B( J 1) J = 0,1,, Elastic molecule: F ( J ) BJ ( J 1) DJ ( J 1) 0 0 n Transitions: n ( J 1 J ) B( J 1) 4D( J 1) 3 J = 0,1,, Svanberg: 4.., Fig. 4.10

35 Energy Rovibrational transitions (IR transitions) A vibration is IR-active only if it involves a change in the dipole moment. For diatomic molecules, therefore only heteronuclear molecules are IR-active. Thus, rovibrational transitions are not possible in homonuclear diatomics (since they are symmetric). Dv = 1 are by far strongest DJ = 1 (same as for pure rot. trans.) Internuclear distance

36 Rovibrational transitions Total energy: Upper state energy: E v, J = G v + F(J) E = G v + B J J + 1 Lower state energy: E = G v + B J J + 1 Expression for P-branch transitions: P J = G v G v B + B J + (B B )J ν v v = G v G v (band origin) P J = ν v v B + B J + (B B )J R J = ν v v + B + B J (B B ) J + 1 Both branches in a single formula: ν = ν v v + am + bm m = J for P-branch transitions Svanberg: page m = J + 1 for R-branch transitions where a = B + B and b = B B

37 The Fortrat parabola Band origin Band head ν = ν v v + am + bm m= -1,-,-3, m= 1,,3, for P-branch for R-branch n where a = B + B and b = B B B > B Plotting the result of the expression in a diagram with m vs ν results in a so-called Fortrat daigram. n Band head When B > B there is a bandhead in the R-branch and the band is said to be degraded towards red because the parabola has its vertex towards higher frequencies (like in this example) Band origin When B < B there is a bandhead in the P-branch and the band is degraded towards violet. Svanberg: Fig. 4.13

38 Energy Rovibronic transitions Selection rules (diatomics): Electronic: (major) Vibrational: D = 0,1 DS = 0 D = 0 D = 0 No strict rule, dependent on Franck-Condon overlap (more about this later) Internuclear distance Rotational: DJ = -1, 0, or +1 (P, Q, R branch) (not J = 0 to J = 0)

39 Rovibronic transitions Total energy: Upper state energy: E T e, v, J = T e + G v + F(J) E = T e + G v + B J J + 1 Lower state energy: E = T e + G v + B J J + 1 P J = T e T e + G v G v B + B J + (B B )J ν v v (band origin) P J = ν v v B + B J + (B B )J R J = ν v v + B + B J (B B ) J + 1 Q J = ν v v + B B J + (B B )J Q-branch is possible for electronic trans. P and R branches: ν = ν 00 + am + bm Q branch: ν = ν 00 + bm + bm where a = B + B and b = B B Fortrat diagrams m = J m = J + 1 m = J for P-branch transitions for R-branch transitions for Q-branch transitions

40 Absorption spectrum of OH Bandhead in the R-branch B >B Q R P

41 Energy Probability distributions for vibrational states It is more probable to find a molecule at the end positions than at the equilibrium position for excited vibrational states! R e Internuclear distance (R)

42 Strength of rovibronic transitions The strength of a vibrational transition depends on the overlap between the vibrational wave functions in the two states (Franck-Condon Factor, FCF). Franck-Condon principle: An electronic transition takes place so rapidly that a vibrating molecule does not change its internuclear distance appreciable during the transition. This means that the transitions can be represented by vertical arrows. a) FCF maximum for Dv = 0 transitions b) FCF maximum for Dv > 0 transitions Svanberg: page 60

43 Vibrational band structure There is essentially no selection rule for vibrational states when undergoing electronic transitions. Svanberg: page 60

44 Population distributions T 300 K T > 1000 K

45 Fractional population Vibrational population distributions The population distribution may generally be written N N j j g j g e / kt j e j / kt j ( j is the energy of a state j) Population distribution of N v=0 v=1 v= v=3 The vibrational population distribution can be written: N N v e (1 e vhc / kt hc / kt e e ) Temperature (K) N v : Population in state v N: Total population Svanberg: page 59

46 Rotational population distributions The rotational pop. distribution can be written: N J 1 BJ ( J 1) hc / kt N Q rot (J 1) e Q rot kt hcb Population / arbitrary units 10 T=300 9 T=400 8 T=000 K T=100 K f Where is max. population? ( J) (J f ( J ) J J max 1) e 0 BJ( J 1) hc/ kt kt Bhc 1 Rotational quantum number, J Svanberg: page 59

47 Data bases and simulation tools HITRAN/HITEMP: PGOPHER: PGOPHER, a program for rotational, vibrational and electronic spectra. 43

48 The End