MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.61 Physical...

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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.61 Physical Chemistry I Fall, 2017 Professor Robert W. Field Lecture 30 and part 31: Electronic Spectroscopy. Franck-Condon. Highlights • “Electronic Structure” and “Dynamics” atom → diatomic molecule → small polyatomic → large molecules gas phase → condensed phase molecule breaks  radiative decay → non–radiative decay� R @ dephasing • Kinds of spectra: rotation, vibration, electronic • Some ways to record spectra • Selection Rules • Franck–Condon Principle ΔR = 0, ΔP = 0 quantitative and qualitative F–C factors diatomic molecule (1 mode) → polyatomic molecule (3N − 6 modes) FC “dark” modes, FC “bright” modes • Absorption and Emission Spectra: gas vs. condensed phase • Non–radiative decay: “Dynamics” 1. molecule breaks 2. 3–level system → quantum beats 3. vibrational wavepacket: Ehrenfest hxit , hpit diagonal anharmonicity → wavepacket spreads intermode anharmonicity → new modes light up 5.61 Lecture 30/31 Fall, 2017 Page 2 4. Electronic Curve Crossings → new stuff from other electronic state 5. Bixon-Jortner Theory → rapid loss of ability to radiate 6. Photochemistry: photon-induced reactions, unimolecular and bimolecular 7. F¨orster: Donor → Acceptor processes hR−3 i A molecule is a collection of e− and nuclei. No analytic solution for ≥ 3 bodies * Born–Oppenheimer Approximation: clamp the nuclei (no vibration or rotation) and solve for the electronic energy levels at a grid of nuclear geometries: Ei (R). connect the dots → potential energy surfaces PES Vi (R) Understand the relative energies and shapes of the PESs in terms of LCAO–MOs: configurations → Electronic States MO’s: bonding, nonbonding, antibonding, structure-demanding (e.g. σ vs. π bonds, spn hydridization) * Once we have a PES, we solve nuclear Schr¨odingier Equation for the vibrational levels. The 3N − 6 normal mode frequencies “encode” the geometric structures (molecular shape and force constants). Next we solve semi–rigid rotor for rotational levels and vibration dependence of rotational constants. More encoding of PES in observable energy levels. Ψevv = ψel χvib φrot From diatomic to polyatomic molecules: there is a lot of information about the spectrum and structures of a polyatomic molecule. How do we organize it? atom → diatomic molecule → small polyatomic molecule → big stuff gas phase (hHit =constant) → condensed phase–energy transfer to surroundings Radiative decay (population decays, molecule remains intact) molecule breaks �  Nonradiative decay � @ @ molecule loses ability to radiate, could change identity (isomerization) R - assemble picture as complexity increases 5.61 Lecture 30/31 Fall, 2017 Page 3 Kinds of Spectrum (everything except NMR is based on electric dipole transition moments: typically 103 times larger than magnetic dipole transition moments). rotation microwave µ 6= 0 [HCl] d~ µ vibration infrared 6= 0 [CO2 ] dQi electronic ultraviolet µe1 ,e2 6= 0 transition moment (not zero, even for H2 ) Kinds of things a diatomic molecule can do: Molecular cation .......... .. ... ... ... .. . ... . .... ... .... .... .... .... .... ... . . . .... ....... .... ......... ...... ............................ AB+ + e− A* + B ......................................................... ... ............................... ... .................. ... .............. ... ........... . . . . . . . . . . .... . ......... .... ........ .... ........ .... ....... .... ...... . . . . . . .... ..... ..... ..... ..... ...... ..... ..... .... ... ...... . . . ...... ... ...... ... ... ...... .. ... ....... ... ....... ... ... ........ . . ... . ........ .. .... ........ ... ......... ... ......... ..... ... ........... ... . . . . . . . . ... .......... . . ........... ... .... ............ ... .... ............. ... ... ............. . . . .... ................ ... . . .... ................. . . .... .................... ..... . . . . . ............ .. ....... ............ ................... .................................. ........... ............ .......... . . . . . . . . ...... . . . . . ...... ...... ...... ...... ..... . . . . .... .... .... ... . . . . ... ... ... ... ... ... ... .... ... . . . ... ... ... .... ... .... ... .... . . . ... .. ... .... .... .... .... .... .... ..... . . . . .... . ...... ...... ....... ........ .................................... Repulsive state AB Less bound excited state. Predissociated by A+B repulsive state A+B AB Bound Ground State (“X” or S0 ) Kinds of Spectroscopy C  C  H    H  light source ... ...... ... .... .... .... ... .. ... ... .. .. .. .. .. ... ... ... .. ... .. .. .... ... .. ... .... ... ... ... ... .. . . .. . . .. . ... ... . .. .. .. .. .. .. .. ... .. . . ... . ... .... ... .. ... ... ...... . lens @ @ @ � � � gas cell detector 5.61 Lecture 30/31 Fall, 2017 Page 4 Methodology • absorption • induced fluorescence (using laser) • multi–photon ionization, resonance enhanced [ions are easier to collect and detect than photons, why?] • more sophisticated schemes: 2-color, stepwise vs. coherent (STIRAP = Stimulated Raman Adiabatic Passage) Rules that govern what we can see: ~ is a one–e− operator µ ~ = µ X e~ri electron i ψel = Slater determinant: product of 1e− spin–orbitals Δso = 1 and α 6↔ β for electric dipole: parity+ ↔ − for magnetic dipole: parity+ 6↔ − total e− spin ΔS = 0 Other point group symmetry elements: important for small molecules and metal–centered inorganic complexes [Group Theory] Electronic Transitions Many vibrational bands Vibrational Band System → a lot of information about the change in shape of PES for excited electronic state (0 ) vs. ground electronic state (00 ). Universal notation: Upper state first, single 0 for upper state, double 00 for lower state. 5.61 Lecture 30/31 Fall, 2017 Page 5 Franck–Condon Principle governs relative intensities of the vibrational bands in an electronic transition The Franck-Condon principle is based on sudden promotion of one e− , so fast that nuclei respond only after the e− excitation. ΔR = 0 “vertical transitions” ΔP = 0 no change in momentum. How is momentum encoded in eikt ? Transition occurs at value of R where spatial oscillation frequency (momentum) is the same in the upper and the lower electronicvibrational states. “Stationary Phase Approximation” Mostly favors turning point to turning point transitions. Z FC Factor = qv0 ,v00 ≡ 2 ∞ χv0 (R) χv00 (R) dR 0 This is the square of an overlap integral between initial and final vibrational states. If you know V 0 (R) and V 00 (R) you can then compute accurate FC factors. There is no simple formula. If you do not know V 0 and V 00 , you can use the vibrational intensity distribution to qualitatively describe ΔV = V 0 (R) − V 00 (R) ≡ ΔV (R). ... ... ... ... ... ... ... ... ... ... ............................................... . . . . . . . . . . . . . . . . . . ........... ............. . . . ... ... . . . . . . . . . . ......... ....... ... ... . . . . . ........ . . . . ... ... ...... ...... . . . . . . . . . . . . .. ... ... .... ........ ... ... ..... ....... ... ... .... ....... ... ... ... ...... .. ... ...... ... . . . . . . ... .. ... ... ... .... .... .... .... ... .... .... .... .... ... .... ... .... ..... . . . . . . . . ....... ....... .. .......................... ........... .................. ............ - ... ... ... ... ... ......... ............................... ... .............. ... ........... ... ......... . . . . . . . ... .... ...... ... ...... ... ..... ... .... ... ... . . .... . .... .... .... .... ... .... ..... ...... . . . . ....... .. ....... .......... ...........................  FC Allowed Region .... .... ................ ................ ............. ............. ........... ........... .......... .......... . . . . . . . . . . . . . . . . ... ... ........ ........ ....... ....... ...... ...... ...... ...... .... .... . . . . .. .. .... .... .... .... ... ... ... ... .. .. .. . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... . . ... ... ... ... ... ... ... ... ... ... .. .. .... ... ... ... . . . .... . . . . .... . .... ... .... .... ... .... ..... .... .... .... ....... ... ..... ........ ............. ................... ............................ ... v=0 ........ ............... ............ ........... ......... . . . . . . . .. ........ ....... ...... ...... .... . . .... .... ... ... ... ... .. . ... ... ... ... ... ... ... .. ... . ... ... ... ... ... .. .... ... . . . .... . .... .... ...... .... ....... ... .................................. FC allowed region from v 00 = 0 is over the range of R from turning point to turning point in the V 0 potential. 5.61 Lecture 30/31 Fall, 2017 Page 6 3 cases • Upper state less bound and shifted to larger Re : long vibrational “progression” because inner wall is steep. • Upper state same shape as lower state: only Δv = 0 transitions. No progression. Hot bands 0 − 0, 1 − 1, 2 − 2: “sequence” structure. • Upper state more bound and shifted to shorter Re : short progression because outer wall of V 0 potential is not steep. How do we determine absolute vibrational quantum numbers? Isotope shifts. Recall that ω = [k/µ]1/2 where µ is a reduced mass, which depends on isotopologue. Band shapes: depend on rotational constants B 0 > B 00 B 0 = B 00 .... .... .... .... .... .... .... .... .... ..... ...... ...... ...... ...... ....... ....... ......... ........... ............... ..... - ν “Blue degraded” “head” in P branch usually ω 0 > ω 00 More bound than ground state B 0 < B 00 ............................... .................................... ....... .......... ....... ...... ...... ...... ...... ...... .... .... .... .... . . . . . . .... ... .... ... . .... . . . ........ . .. .... . - ν No bandhead (like IR spectrum) ω 0 ≈ ω 00 . .... .... .... .... . . . .. .... .... .... .... . . . . .... ...... ....... ....... ........ . . . . . . . ..... ......... ............ ................ ....................................... - ν Red degraded head in R branch usually ω 0 < ω 00 Less bound than ground state What about FC factors for polyatomic molecules? 3N − 6 Vibrational nodes. Benzene has 12 atoms, 30 vibrational modes! Horrible? Not so bad because many vibrational modes are identical in the upper and lower electronic states. *When ω 0 = ω 00 , those are called “FC dark” modes because they follow the Δv = 0 propensity rule. 5.61 Lecture 30/31 Fall, 2017 Page 7 *The few modes where ω 0 = 6 ω 00 are called “FC bright” modes because several Δv 6= 0 transitions for that mode are observable. These are the vibrational modes that “generate” the change in geometry between the excited and ground electronic states. For example, acetylene is linear and with a C ≡ C bond in the electronic ground state and the first excited singlet state is trans–bent with a C = C bond. The CC stretch and trans–bending normal modes are FC bright because they generate the change in geometry. The vibrational intensity distribution (which modes are FC bright and the v 0 distribution of qv0 ,v00 =0 ) gives qualitative information about ΔV (R). Stimulated Emission Pumping (“Pump and Dump”) ... ... ... ... ... ... ... ... .................. ............................... ... ............... ... ............ ... .......... ....... . ... . . . . . ... .... ...... .... .... .... .... .... ... . .... . . .. ..... ..... ....... ....... ......... ............................... 6 PUMP DUMP ............ .............. ............ ........... ....... ......... ? ........ ....... ...... ...... ..... ... . . . ... ... .... ... ... ... .. ... . ... ... ... ... ... .. ... .. . . ... ... ... .. ... ... .... .... .... . . . . .... .... ...... .... ....... ................................... gives information about highly excited vibrational levels in the electronic ground state. I invented SEP! (Both the experiment and the terminology.) **Gas Phase vs. Condensed Phase Electronic Spectra Absorption vs. Emission Spectra. 5.61 Lecture 30/31 Fall, 2017 Page 8 ... ........................................... ... ................. ... ............ .......... ... ........ . . . . . ... . ..... ... ...... ... ... ... .. ... .. . . ... . ... ... .... .... .... .... .... .... . . .... . ..... ...... ........ ...... .................................. GAS PHASE ......... ............... ............ ........... ......... . . . . . . . . .... ....... ....... ...... ...... ... . . .. .... .... ... ... ... .. . . ... ... ... ... ... ... ... .. ... . ... ... ... .. ... ... .... .... . . .... . . .... .... ...... ... ...... .... ............. ................... ... Absorption spectrum is simpler than the emission spectrum because the FC region for absorption from v 00 = 0 is narrow in energy range and especially because in emission from an excited state, the inner and outer turning points on the upper potential curve are far apart. In condensed phase, the vibrational population in the excited state is rapidly relaxed (by collisions with solvent or lattice) to v 0 = 0. Relaxation is always much faster than spontaneous radiative decay. ... . ... ... ... ... ... .. . ... ... ... ... ... ..... .. ... ..... .. . ... ........ . . . ... .... ... .... .... .... .... ... ..... . . . .. ...... ......... ....... ........................... R ... .. .. ... .. .. .. ... .. . . .. .. ... .. ... ... ... ... .. ... . .. ... ... ... ... .. ... .. . . ... ... ... ... ... ... ... .. ... . . ... ... ... ... .... .... .... ........ ........... ............ So in condensed phase, the absorption spectrum is from v 00 = 0 and the emission spectrum is from v 0 = 0. Get the classic picture of absorption vs. emission spectrum. 5.61 Lecture 30/31 Fall, 2017 Page 9 ...................................... ....................................... ...... ...... ....... ........ ..... ..... ...... ..... .... .... .... .... ... .... ...... .... . . . .... .... .... ... . . . . .. . emission red - absorption blue [Herzberg insisted: “red on the right.”] This gas phase↔condensed phase difference is due to hHit = constant (conservation of energy) for gas and hHit rapidly decaying due to intermolecular interactions in condensed phase. Collision-Free Dynamics ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... ...... ....... ........ ......................... - ... .. .. ... ... ... ... .. ... . ... ... ... ... ... ... ... .. ... . . .... ... .... .... .... .... .... ... . . ..... . .... ........ .......................... direct dissociation .. ... ... ... ... ... ... ... ... ... ... .. ... ... .. ... . ... . .... ... .... ... ... .... ... ... .... .. .... .... .. . .... . .... .... .... ...... .... ....... ..... ........ ....... ................................. ...... .... ..... ...... ...... .. .. .. ... ... ... ... ... .. ... . ... ... ... ... ... ... ... .. . ... ... ... ... ... ... ... .. ... . ... ... ... .... ... ... .... ... . . .... . ......... ........... ........... predissociation via a repulsive state ...... ......... ............ .... .... .... ... ... .. . ... ... .. ... ... .. . .... . ... ...... .. . ... ........ .. ... ........... . ....................... .. ... . ............. ... .. . . ... . . ... . . . .... . ... .... . . ......... ......... ........... - ... .. ... ... ... ... ... .. . ... ... ... ... ... ... ... ... .. . ... ... ... ... ... .. ... .. . . ... ... ... ... ... ... .... .... .... . . . . ...... .. .......................... predissociation by tunneling through a barrier on a single potential surface Now let’s look beyond the collision–free (no energy removed) non–radiative decay mechanisms as shown in the above figure. We will see now that there are other decay mechanisms that do NOT involve breaking the molecule! “Dynamics” 5.61 Lecture 30/31 Fall, 2017 Page 10 Dynamics Begin with simplest case • ground state 1 • dark excited state 2 |2i(0) bright excited state 3 |3i(0) |3i |3i(0) 3 |2i(0) 2 ... ... .. . ... .... V23  − 3(0) S=0 − S=1 A A |2i 2(0) − 6 − µ13 |1i(0) 1 + ? + 1(0) µ12 = 0 perhaps because |2i(0) is a triplet state S=0 ⎛ (0) H ⎞ 0 0 E1 ⎜ ⎟ (0) V23 ⎠ ⎝ 0 E2 (0) 0 V23 E3 µ ⎞ 0 0 µ13 ⎝ 0 0 0 ⎠ µ13 0 0 ⎛ + ↔ − selection rule parity is conserved Notice the complementary structures of H and µ. Short pulse to excite from |1i(0) = |1i h > (E3 − E2 ) 2πτ τ is pulse duration 5.61 Lecture 30/31 Fall, 2017 Page 11 Uncertainty principle broadened pulse covers the frequencies of both E2 ← E1 and E3 ← E1 transitions. |Ψ(t = 0)i = |3i(0) = (1 − α2 )1/2 |3i + α |2i due to mixing of zero-order states by V23 6 6 6 (spin-orbit operator) not an eigenstate eigenstates (This Ψ(t) is a normalized state.) α= (0) V23 (0) E2 (0) − E3 (0) What is α in terms of E2 , E3 , and V23 ? You should already know how to derive this!   |Ψ(t)i ∝ µ13 (1 − β 2 )1/2 |3i e−iE3 t/} + β |2i e−iE2 t/} These are eigenstates weighted by the bright state |3i(0) character in each. What is β? What are the values of E2 , E3 ? by perturbation theory! Again, you already know how to derive these Get Quantum Beats! I(t) ∝ µ213 [1 + 2β(1 − β 2 )1/2 cos ω23 t] ω23 = (E2 − E3 )/ } If β = 2−1/2 ............................ ..... ...... .... .... .... ... ... ... . . ... ... ... . .... . . ... . .... .. ... . ... . ... .. . ... .. . ... . . . ... . . . .... .... .... .... ... ... ..... .... ....... .......................... I(t)..................................... cosinusoidal 100% modulation - t Get maximum modulation depth for β = 2−1/2 . |3i and |2i are 50 : 50 mixed: |3i(0) ± |2i(0) This modulation is superimposed on an exponential decay (not shown). The phase of the modulation is what you expect when you prepare and detect the same “bright state”. 5.61 Lecture 30/31 Fall, 2017 Page 12 What would I(t) look like if the 3 ← 1 transition is “bright” for excitation but the 2 → 1 transition is bright for detection? Now we are ready to consider a “wavepacket” built from several vibrational levels of the same electronic state. Again we use a short excitation pulse out of electronic state 1 to create a coherent superposition state in one bright vibrational level of electronic state 2. ⎛ ⎞ 0 0 0 ... E1 0 ⎜ 0 E2 0 0 0 . . .⎟ ⎜ ⎟ ⎟ + }ω 0 0 . . . 0 0 E H=⎜ 2 ⎜ ⎟ ⎝0 0 0 E2 + 2}ω 0 . . .⎠ 0 0 0 0 E2 + 3}ω . . . The state with E = E2 + n}ω is denoted |2, ni. The transition moment h1, 0|µ|2, ni ≡ µ10,2n . ⎞ ⎛ 0 µ10,20 µ10,21 µ10,22 µ10,23 . . . ⎜µ10,20 0 0 0 0 . . .⎟ ⎜ ⎟ ⎟ 0 0 0 0 . . . µ µ=⎜ 10,21 ⎜ ⎟ ⎝µ10,22 0 0 0 0 . . .⎠ µ10,23 0 0 0 0 ... This gives a vibrational wavepacket ... ... ... ... ... ............................. ... .............. ... ........... ... ......... . . . . . . . ... ...... ........................ ... ........ ............ ... ....... ..... ... ........................ .......... .. ... . . . .. .... .... .... .... .... .... .... ..... ...... . . . . ... ....... ............. .................... ......... 2  6 6 ..... ................ ............. .......... ......... . . . . . . . ........ ....... ...... ..... ..... . . . .... .... ... ... ... ... .. . ... ... ... ... ... .. ... .. . . ... ... ... ... .... ... ... .... .... . . . .... .... .... .... .... ...... ....... ..................................... *Many vibrational levels of electronic state 2 are “coherently” excited. *Wavepacket starts at inner turning point of electronic state 2. 1 v 00 = 0 *hxit and hpit follow Newton’s Laws If upper potential curve is harmonic but ω 0 6= ω 00 , wavepacket remains localized and oscillates at ω 0 = [k/µ]1/2 . The phases of the eigenstates in the wavepacket are such that the ability of the Ψ(t) to fluoresce to the |1, 0i state oscillates at ω. 5.61 Lecture 30/31 Fall, 2017 Page 13 Now add some anharmonicity in the electronic state 2 potential curve. • Wavepacket continues to oscillate where hxit and hpit follow Newton’s Laws, but the wavepacket spreads out (dephases). • The total population decay rate from state 2 is unaffected, but the ability to fluor...
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  • Fall '17
  • Atom, Excited state, electronic states, Δv

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