Of Molecules and Media

Gert van der Zwan

June 19, 2014

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Molecules Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 3 / 42 Positions of nuclei Electron density Peak positions Intensity profiles ● ● ● ● Quantum Chemists calculate: Spectroscopists see:

What is a Molecule? Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 4 / 42 ) ν ( ) 1 − dνǫ 7 10 band in cm Z ( 1 9 ν − − Energy difference Oscillator strength 10 cm ? ⇒ ⇒ 1 ) = × − = 319 . in nm ( λ =4 in L mol ) f ν ( Peak position Peak intensity Peak Width ǫ ● ● ●

What is a Molecule? II Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 5 / 42 il- f × ν D. e ¯h 5 2 e ∼ πm the charge of the electron. This 3 4 e is a dimensionless quantity, de- = f 2 µ is the mass and e m where The oscillator strength The transition dipolestrength moment by the is relation related to the oscillator fined as the ratiothetical) of absorption the strength of absorption alating strength single electron. to harmonically Oscillator osc strengths the are (hypo- between 0 and 1. dipole moment is in Cm,(D). a At more 300 common nm unit an istransition oscillator the dipole Debye strength moment of of 1 corresponds to a

Units Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 6 / 42 i ) } ~ R { , } ~r i { ( vibrations ; V state | + j 2 j M state P 2 E N = ∈ i X i + e 2 i state m electronic density p | 2 H| e = ∈ i i X = state H | Hamiltonian: Hamiltonian, states, dipole moments with Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 7 / 42 , 2840. 35 , (2011), New J. Chem. , et al

Quantum Chemistry E. Ferrari Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 8 / 42 , 714. 92 , 2011, Dyes and Pigments , et al. value between -2 and +2 kcal/mol. Taking into account G ∆

Quantum Chemistry II “The problem is that ifa you have a real equilibriumthe this real means accuracy of thethe quantum results chemistry are of meaningless 1–2 in kcal/mol, most cases. ” L. Antonov Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 9 / 42

Quantum Chemistry III Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 10 / 42 i i j i j ~ R j q state state | | N ˆ ~µ ˆ ∈ ~µ X j | | i i + i ~r state state e h h ∈ i X = = e i ij − ~µ ~µ = ˆ ~µ State dipole moment: Transition dipole moment: ● ●

Hamiltonian, states, dipole moments II Dipole Operator: Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 11 / 42 a L 1 x b b CD1 L L CB a 1 L NE1 CG G reflect magnitude. not CE2 CD2 y CZ2 CE3 CZ3 CH2

Tryptophan (indole) Red: transition dipole moments; Green:moments. state Arrow dipole length does Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 12 / 42
′ ~e iϕ Ae + ~e ~r · ~ k E ~ i · + ˆ ~µ iωt − − e = 0 E int H ) = t ( light ~ E Linear and non–linear spectroscopies: Stark Spectroscopy: constant electric fieldthe in light addition field. to “Quantum relaxation” (Redfield theory): randomly fluctuating electric fields. Polarization fields due to media; reaction field; QM/MM. ● ● ● ●

Interaction Interaction Hamiltonian with external fields: Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 13 / 42 k E th a B k | i ih i | i λ E states X ∈ i = 0 H

q

-q

+ Interaction of a classical, oscillating, electric field, wi Quantum molecule Interaction with Light: Absorption quantum molecule. Light Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems i i 14 / 42 | 0 on h ~µ = i i ; e , Ch. 6 | i ˆ ~µ ) | t ( ;0 g , ρ h ) t ( = ~ E · ge ~µ ˆ : homogeneous (lifetime) ~µ γ − 0 2 H γ h ¯h + i ∗ ge 2 − ~µ ) ge eg = ν ) γ~µ t − ( ∂t ν ( ∂ρ e X Principles of Nonlinear Optical Spectroscopy 1 ¯h : Franck–Condon factor; 2 i| ) = i ν | ( 0 0 A Quantum Liouville Equation Solve to first order in the ‘perturbation’ gives the absorpti S. Mukamel, spectrum of a single molecule: |h Interaction with Light: Absorption II broadening. Still to do: average overbroadening. orientations and inhomogeneous Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 15 / 42 Benzene has 30Why do vibrations. wewhy see are so the bands few, so and sharp?

Vibrations Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 16 / 42 2 ) s ~ E · ~µ + 4( 2 ) s ~ E · ) 0 ~µ − 1 ~µ ( − hν ( q = ′ hν Orients the molecules. Changes the absorption wavelength: Changes the transition dipole moment. Changes the width. ● ● ● ●

Interaction with a static field A static electric field: Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 17 / 42 y. , 1583. 68 1 − 8cm , 1995, . 16 . 1 ≈ − ~ E · Biophys. J. cm ~µ 20000 ≈ hν properties based on simultaneous fitting of absorption, b L 1 and a L Changes are small: A 1D dipole in a 1 MV/cm field gives 1 D.W. Pierce and S.G. Boxer, fluorescence excitation anisotropy, and Stark spectroscop Stark Spectroscopy compare to Molecules Units Hamiltonian Indole Interaction Absorption Vibrations Stark Molecules ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Media Exercises and Problems 18 / 42

Media Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 19 / 42 ˚ A. q to q Dipole moments go from − Water has a dipolement of mo- 1.86 D. 1 D isan electron displacement over 0.2 of ● ● ●

Dipoles and Dielectrics External fields align the dipoles: Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems c 20 / 42 i H β − ~ E ˆ ~µe 1) h − r ǫ ~ Ω Tr ( d 0 ǫ Z = 1 Q ~ P = ~ P is exceedingly complex. r ǫ Polarization is the average dipole moment of the system: To lowest order in the external field: Dielectric Constant The relation between molecular dipoleconstant moment and dielectri Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 21 / 42 T B µE k 3 ≈ θ ~ E 1) θ cos − cos ~ E r βµE ǫ T ( θe 2 0 βµE B ǫ µ k θe 3 cos 2 = TV B θ = ~ E cos k Nµ i 0 d T ǫ ~µ cos 2 1 h 3 B 1 µ d − k R 1 3 1 − 1 = R V N − = = r i ǫ i θ ~µ h cos V N h = Dilute gas of non–polarizable molecules. so that ~ P Polarization: So that Average orientation in the external field: Dielectric Constant II First attempt: Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 22 / 42 ext ~ E +2 3 r ǫ 2 TV B = k Nµ 0 ~ ǫ P 9 0 ǫ 1 3 = 1 + − +2 ext r r ~ ǫ ǫ E = int ~ E Lorentz Cavity Field: Internal Field Debye: Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 23 / 42 = 1nm 2 ~µ , r ·  2 r ~r~r =1D 3 µ − 1  for · 1 ~µ 1 3 − r 0 1 πǫ 4 035cm . = dd =5 V 3 r 0 2 µ πǫ 4

Dipole–dipole interaction Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 24 / 42 ~µ 1) out − +1 out ~ E r r ~ D ǫ ǫ · × 2 2( ρ ~n ~n 3 a =0 = = = 0 1 ~ in in ~ E D πǫ ~ ~ E 4 D · ~ ∇· × = ~ ∇ × ~n ext ~n R ~ E ~ E r +1 ǫ Maxwell equations: Reaction field: r 3 ǫ 2 = int ~ E

Onsager’s Reaction Field Internal field due to Onsager: Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 25 / 42 1 ant − cm 1450 , quite large. R ~ ≈ − E · 2 R TV ~µ U B − k Nµ 0 ǫ = = R U 1) − r ǫ r ǫ + 1)( r ǫ (2 V/m, and the energy is 9 10 × 4 . 1 For a 6 D dipolefield in is a cavity with 0.5 nm radius in water the Reaction fields can bedipole in quite its strong, own and reaction field, the energy of a Onsager’s result for the relation between dielectric const Dielectric Constant IV and molecular dipole moment: Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 26 / 42 , 216. 101 , (2002), J. Mol. Liq.

Dielectric Constant, V Results of calculations (MD/MC): B. Guillot, Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 27 / 42 , emission s nν + 0 ν . 2 i| n | 0 |h 2 µ . s nν − 0 ν Transition energies: Absorption Oscillator strengths: ● ●

Vibrational Stokes Shift Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 28 / 42 g ~µ · g ~µ µ ∆ 0 0 A A ≡ − g 0 ~µ hν = + 1) 1) r ǫ R,g − ~ (2 E r 3 ǫ · a Absorption 0 ~µ 2( ∆ πǫ 4 − e = U R,g = ~ E abs hν First approximation: Reaction field: Solvent Dependent Stokes Shift Absorption frequency: Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 29 / 42 e ~µ · ~µ ∆ 0 A 2 − | 0 ~µ ∆ hν | = 1) − +1 em ǫ ǫ 2 hν 2( h 3 a 0 1 πǫ 4 dipole moment: Relaxed Emission = ν ∆ (without polarizability) Reaction field in equilibrium with the new (excited state)

Solvent Dependent Stokes Shift II Lippert–Mataga Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 30 / 42 , 1 − sensitive to ⇒ D: 6 | ≈ ~µ ∆ | , with a L 1 Emission is from For tryptophan shifts range from 1400–5400 cm depending on polarity of the environment. Tryptophan environment. Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 31 / 42 , , are 1 106 − , (2002), Intramolecular e.g. J. Phys. Chem. A Excited State Proton Transfer (ESIPT). Larger Stokes shifts, sometimes more than 10000 cm Usual cause: excited state reactions, H. Joshi, C. Gooijer, G. van der Zwan, 11422. ESIPT also reported: Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 32 / 42

ESIPT II Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems e- 33 / 42 w situa- s themselves , 998. because the sur- 69 D D iωτ iωτ − − , (1978), r 1 ǫ ) = ω ( dielectric friction r ǫ J. Chem. Phys Debye: : rotational correlation time of the solvent dipole, Debye r D laxation time. the theory of dielectric relaxation is rather complicated. tion. Since this friction also applies to the solvent dipole rounding solvent dipoles need to adjust themselves to the ne A rotating dipole experiences J.B. Hubbard and P. Wolynes, Dielectric Friction τ Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 34 / 42 (1) L 0 iωτ A − 1 = ) ··· ω ( = ~µ ) 1) ω ( − A + 1) ) ω ω ( ( ) = r r ǫ ǫ ω ( 2 2( R ) 3 ~ E τ a 0 1 − : t πǫ ( 4 D A +1 τ =0 t r 3 0 t ǫ ) = Z 2 ω at ( ~µ = A ~µ L ) = τ t ( R ~ E Longitudinal relaxation time: Onsager’s solution is still valid: Turn on Dielectric Friction II with Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 35 / 42 are : solvent : solvent L 0 τ 0 A → , 6353. ζ → L 52 ζ L τ iωτ 0 ⇒ A − 1 , (1970), 0 = →∞ ⇒ → 0 A cannot keep up. ω ω needs to be dragged along. − ● ● ) iω ω ( J. Chem. Phys. A ) = friction on a rotating dipole ω ( ζ Nee and Zwanzig: T.W. Nee and R.W. Zwanzig, Dielectric Friction III Dielectric friction plays a rolekinetics, in electron, isomerization and reaction proton transfer.moved Whenever or charges dipoles rotated, there is dielectric friction. Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems ] 36 / 42 ) ω ( ζ , 4181. 89 . ) g t ~µ ( (0) ζ ζ − e , (1985), = ~µ ) ) = ∞ ∞ ( ( ~µ ∆ em em ν ν − − ) J. Phys. Chem. t ( (0) em em ν ν ) = t ∆( difference dipole moment is the time–dependent friction [Fourier transform of ) t ( ζ G. van der Zwan and J.T. Hynes, Dynamical Stokes Shift on the Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 37 / 42 at different times 029 ns). For the first time . 6 . , 8179. = 0 = 0 P 101 x t ∆ (a) FluorescenceC153 in a spectra hexane/EtOHat mixture of step raw data aredots. shown The by solid the sult lines are of the smoothing re- Steady–state the raw spectrum data. inane hex- (dashed line)EtOH (dot-dashed and line); (b) in cen- ter pure of thenoexponential band fit (dots) (solid and line);width mo- (c) of thedata band; taken inset with higher shows olution. time res- ( , (1997), J. Phys. Chem. A , et al.

Dynamical Stokes Shift II F. Cichos Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 38 / 42 Kamlet–Taft parameters. viz. Molecules and Media are both collections of dipoles. Molecules must be treated quantumhave mechanically: state we and transition dipole moments. Media are treated classically: weand have polarizability dipole (ignored moments in thisirrelevant). lecture, but not Polarization is not the onlythat relevant influences property spectroscopy: of protons media canimportant also role, play an Interaction between molecules and mediacomplex is topic. a For very those whoto want Finland to in know august: more: www.jyu.fi/summerschool. come ● ● ● ● ●

Remarks and Conclusions Dielectrics Internal Field Interaction Reaction Field Stokes Shift ESIPT Dielectric Friction Dynamics conclusions Molecules Media ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Exercises and Problems 39 / 42

Exercises and Problems Exercises Problems Literature Molecules Media Exercises and Problems ❖ ❖ ❖ 40 / 42 xane to for water from the cal of slide 29? lvent sion electric djusts be for water? Do these c T below which the system has a macroscopic c T temperature permanent dipole moment. What would so–called electrets exist? problem given in 1. about 50 times) than the solvent dipoles rotate. constant in these slides taking the polarizability of the so molecules into account. including polarizability. Remember that polarizabilityinstantaneously a to a new situation. do experiments in, as in the Cichos reference? 1. Show that Debye’s expression (on slide 21) leads to a criti 2. Show that Onsager’s solution, on slide 24, does not suffer 4. Derive Eq. (1). Explain how the field can relax much faster ( 3. In which direction does the polarity increase in the figure 5. For a rainy afternoon: rederive all expressions for the di 6. And, while you’re at it, derive the Lippert–Mataga expres 7. Why is it a good idea to use a dilute solution of ethanol in he

Exercises Exercises Problems Literature Molecules Media Exercises and Problems ❖ ❖ ❖ 41 / 42 er st eld of ger literature, re the urbation is zation and t equation for a (see exercise 1) is c T magnets who makes this temperature(since go Onsager) away. electrets In were the discovered, mean so time where is Onsa called the Curie temperature. Why is there no Onsager in the fi wrong? quantum system in equilibrium withsolutions its of reaction this field. non–linear Explo equation. reaction field. Nevertheless in Starkmade spectroscopy on the the pert quantum level,make whereas the for molecule the a reaction classical fieldpolarizability system we (dipolar with fir spring). dipole Derive moment a and self–consisten magnetization is treated. In magnetization 2. The Stark field is most of the time much weaker than the Onsag 1. What are the essential differences between the way polari 3. Explain the behavior of the width in Cichos’ experiment. As far as I know these topics have not yet been addressed in the Problems so they might make a good subject for a report. Exercises Problems Literature Molecules Media Exercises and Problems ❖ ❖ ❖ 42 / 42 , Oxide, at Lies Marker, nt grating Chem. Phys. ¨ am, A. Tortschanoff, A. Cannizzo, , 1549–1557 45 , 1002–1013. 8 , (2012), , (2014), , 47–52. 422 Behind the High Sensitivity of the Thioflavin-T Fluorescent Acc. Chem. Res. and M. Chergui, Ultraviolet transient absorption, transie and photon echo studies of(2013), aqueous tryptophan, ACS NANO Excitation Wavelength Dependent Fluorescence of Graphene 2. N. Amdurski, Y. Erez, and D. Huppert, Molecular Rotors: Wh 3. A. Ajdarzadeh, C. Consani, O. Br 1. S.K. Cushing, M. Li, F. Huang, and N. Wu, Origin of Strong

Literature Exercises Problems Literature Molecules Media Exercises and Problems ❖ ❖ ❖