Molecular Spectra and Molecular Structure - Lec 37

Page 1 of 9 pages Lecture #37: Laser Double Resonance Studies of Electronic Spectroscopy and Vibrational State Mixing in Highly Vibrationally Excited C2H2 Today: Double Resonance Technique for preparing and probing highly vibrationally excited molecules Background information on acetylene: geometry, normal modes, selection rules, principal axes ! 1 !+ Zero-Order vibrational state mixing in X g Vibrational Overtone-Laser Induced Fluorescence (LIF) Double Resonance v′, J′, K′a  1A A u v=0 ~42,000 Detect Florescence λ2 ~300nm ~ ~ LIF (Probe) ~ ~ 3ν3, J,  = 0 ~10,000  ∑+ X g 1 λ1 ~1000nm ~1µm 0 - 2 photons - Double resonance - Electronic F-C may differ for 3ν3 vs. v = 0 - Pulsed (~10ns) lasers Vibrational Angular Momentum (about bond axis) Vibrational Overtone Excitation (PUMP) v=0 A closer look at Overtone Excitation: ! µ is dipole moment operator ! Transition strength ! "v# µ "v## % ! * ! "v# µ "v## = & "v#µ"v## d' $% "’s are true molecular wavefunctions 2 Page 2 of 9 pages Important approximations: ! Simplify µ : ! µ = µ a ê a + µ b ê b + µ c ê c Expand µa… with respect to Normal mode displacement coordinates (q ) $ #2 µ a ' $ #µ a ' 1 µ a ( q1 …q 3N!6 ) = µ a,e + " & ) q jq k +… ) qk + " & 2 j,k % #q j#q k (e % #q k (e k !###"### $ !####"####$ 1. Linear Dipole Moment Approximation due to electrical anharmonicity Simplify ψv. Let's consider a single normal coordinate for now. 2. !v = " a v #!v wavefunctions for normal coordinate of interest (v = 0, 1, 2, …) v SHO Neglect mechanical (potential) anharmonicity ψv ≈ φv (harmonic oscillator approximation) RESULTS: A) SHO (Simple Harmonic Oscillator) wavefunctions and Linear Dipole Moment !v" µ !v"" # $ v" !q k $ 0 % 0!if and only if!v = 1 ⇒ no OVERTONE transitions (∆v > 1) allowed in the harmonic oscillator, linear dipole moment limit. B) SHO wavefunctions and Electronic Anharmonicity 1 1 !v" µ !v"" # $ v !q k +! q 2k +! q k3 +… $ o 2 6 terms like ! v q 2k ! 0 ≠ 0 for ∆v = 2. C) Mechanical (potential) anharmonicity !v" µ !v"" # % a v$ v ~ q k % a v$ v v v (X-H Systems) (C, N, O, …) V = 1/2kr2 (SHO) V(r) V(r) terms like a1!1 q k a 0 ! 0 " 0 (if ψ2 contains φ1 character, then ∆v > 1 allowed) even for linear dipole moment function * * * * r In practice, need highly anharmonic bands (X-H stretches) σ's (absorption cross section) still small for ∆v = 3 (v = 0 → 3ν3) in C2H2, σ = 4 × 10–23 cm2. σ decreases ≈ 10× for each unit increase in ∆v small σ → low fractional excitation: 10mJ, 10ns pulse → f ≈ 10–4 can excite single rovibrational eigenstates in small polyatomic molecules Page 3 of 9 pages C2H2 P  A v = 0 → 3v3 R || transition ⇒ no Q branch, tune λ1 to transition to select v,J for λ2 λ2 3v3 scan λ1 v=0 1.037 µm λ1 LIF probe step: Electronic Transition - use Condon approximation to get vibrational part of transition probability. Pv!v" ! #v!|v"$2 |Mif|2 F-C factor - need to evaluate these v! v!! = $ # "# %*v! %v!!d& ≡ 0 if integrand is an odd function Group Theory helps us identify these cases. Non-zero v! v!! magnitudes given by overlap integrals ∆ in equilibrium coordinate ∆v > 0 no ∆ in equilibrium coordinate ∆v = 0 (propensity rule) !!X !: C2H2 A ∆vC–C > 0 ∆vtrans > 0 ∆vC–H ≈ 0 ∆vcis ≈ 0 !ν ← X ! 3ν For our spectrum A 6 3 (νcis) (3νC–H) So ∆vCC = 0 ∆vtrans = 0 How do we see anything? ! A Re CH Re CC θe CCH ∆vC–H = –3 ∆vcis = +1 1.097Å 1.375Å 122.48° low vibrational levels (< 1000 cm–1) ! X Re CC θe CCH 1.058Å 1.208Å 180° vibrational excited levels (= 9600 cm–1) C-H stretch trans-bend C-C-stretch  X Page 4 of 9 pages Franck-Condon factors important Previous studies - progressions in vtrans and vC-C Our work: transitions from (1 1 1 20 00) brighter than those from (0 0 3 00 00) trans C-C ! -state is really not a pure C-H stretching state. Maybe 3ν3 in the X Rotational level dependence of LIF signal (mixing of modes?) Consider Intramolecular Coupling model (vibrational state mixing) coupling mechanisms ! + vCH “Bright” vC-C Vtrans “Dark” Zero-Order States (e.g. normal modes) Coriolis, Fermi, DarlingDennison, Centrifugal molecular eigenstates Overtone Absorption Spectrum “Filter” for “Bright” state character (can only see CH stretch character) At high levels of vibrational excitation, “3ν3” is probably an admixture of zero-order states, not just a pure C-H stretch. Previous work: 3!3 is in anharmonic resonance with v1 + v2 + v3 + 2v4 (1 1 1 2° 0°) C-C trans This anharmonic mixing affects all rotational levels ≈ equally, does not account for J-dependence in LIF signals. Page 5 of 9 pages How about weak ∆J = 0 interactions between nearly degenerate rotational levels? Search for mostly dark eigenstate using LIF. Hold λ2(UV) fixed, scan λ1(IR), monitor UV fluorescence. P(21) R(19) λ2 monitor 3νCH, J LIF scan λ IR absorption λ1 3ν3, P(21) enormous expansion of horizontal axis λ2 = 2993.2203 nm λ2 = 2993.2119 nm Detect LIF signal at a λ1(IR) where there is “no” IR absorption λ1 photon E passes P/R branch check Page 6 of 9 pages What is the J level of the “Dark” eigenstate? Use LIF probe spectrum as a fingerprint. Pump P(23) of “Bright” state Pump P(21) of “Dark” state 2-State model: |1〉 and |2〉 are eigenstates state 1 = c1s s + c1! ! |s〉 “Bright” zero-order state |〉 “Dark” zero-order 2 = cs2 s + c 2! ! te |s〉 and |〉 as product wave functions of zero-order local (C-H stretch) and normal mode basis functions |s〉 = |v〉 |0〉 v ~ C-H stretch |〉 = |0〉 |n〉 n ~ collectively C-C stretch, bends |vg〉 = |0〉 |0〉 |vg〉 = ground vibrational level (v = 0) Rewrite |1〉 and |2〉 in terms of basis functions: 1 = c1s s + c1! ! = c1s v 0 + c1! 0 n 2 = cs2 v 0 + c 2! 0 n Now calculate v! µ v v!! . µv operates only on the |v〉 basis vectors in bond-dipole approximation. 1 µ v vg = c1s v µ v 0 0 0 + c1! 0 µ v 0 n 0 2 µ v vg = cs2 v µ v 0 0 0 + c 2! 0 µ v 0 n 0 orthonormality → 1 µ v vg = c1s v µ v 0 2 µ v vg = cs2 v µ v 0 ∴ overtone transition probabilities given by amount of zero-order “s” character in the ( eigenfunction Pv!v!! " csi absorption scan. 2 ) if c 1 s ! cs2 , only one of the 2 eigenstates will appear in overtone Now consider 2-photon signal in 2nd (LIF) step, νC-C and νtrans are “bright”.  character has become bright! Page 7 of 9 pages electronic dipole moment operator matrix element P1 ! D c v µ v 0 #$c1s v" v n" 0 + c1! v" 0 n" n %& 1 fg s 2 Franck-Condon Factors 2-step transition probability ! LIF signal detected for double resonance scheme through state |1' ! -state vibrational levels with v′ = 0 (v = 0 in A ! ) since Our probe transitions terminate in low-lying A CH 2 Franck-Condon says ∆vC-H = 0, v! v " 0 and P1 ! D fg c1s c1! v µ v 0 v" 0 n" n . Similarly, 2 P2 ! D fg cs2 c 2! v µ v 0 v" 0 n" n . c1s c1! = cs2 c 2! due to normalization ⇒ P1 = P2. For a 2-state model, the normalization requirements → c1s c1! = cs2 c 2! ⇒ P1 = P2 ∴ 2-laser signal is independent of intermediate state! The trade-off between excitation and detection efficiency. v! " A |1" excite many, detect few |1" |s" (C–H) " X |!" |2" (C–C) trans cis |2" excite few, detect most v = 0,J We detect other “dark” eigenstates. We know v″, J″, and a transition energy for all of these predominantly dark eigenstates. ∴ we can calculate term values. Page 8 of 9 pages TA! ,v = !v + Bv J(J +1) Linear polyatomic, rigid rotor limit. Let “D” subscript refer to “dark” eigenstate. T3!v = ! 3!3 + B3!3 J(J +1) – TD = vD + BD J(J+1) (!T# 3! 3 " TD ) = ( ! 3! 3 " ! D ) + ( B3! 3 " BD ) (J)(J ! #"+1) # $ "#$ energy gap between transitions known from rotational combination difference in v = 0. ∴ make reduced term value plot ∆T vs. J(J + 1) m = ∆B, intercept = ∆ν ➁ ① +2 ③ ∆T = 0, ~3ν3 0 –2 J(J + 1) 3!3 ! " # v0 = 9639.86 9632.86 9629.98 9629.34 Bv = 1.15839 1.181 1.182 1.182 Bv ! B3!3 What is vibrational character of ①, ➁ and ③? Large B ⇒ Bending levels. Are ①, ➁ and ③ perturbers of 3ν3? Observed perturbations of 3ν3 consistent with expected location (see direction of arrows) of extra states in every instance. How many states are interacting? ~3 states/cm–1 in this region, ~1.5/cm–1 have proper symmetry to interact with 3ν3. In ±2 cm–1 range we observe 4 vibrational states. ∴ nearly all possible nearby states are interacting (albeit weakly) with 3ν3. What are the coupling matrix elements for these states? We can't get intensity information, ∴ base mixing coefficients and coupling matrix elements on energy shifts measured in high-resolution studies on 3ν3. Page 9 of 9 pages effective Hamiltonian matrix: Wij are coupling elements " 0 W12 % $ ' Wij = W*ji #W21 ! & δ = zero-order energy difference (assume zero-order energy difference is the same as eigenvalue splitting since energy shifts observed  splittings) DIAGONALIZE ⇒ En = ! 2 + 4W12 W21 ! ± 2 2 The |energy shift| due to the perturbation for one of the states is " 2 + 4W12 W21 " ! 2 2 #$2 ! E n + " 2 %& ' " 2 ! E n = E n ! E on = ! w12 w 21 = 2 Eg. for 2 points in state ①: ! En J 17 18 T3!3 " TD # $ –1 0.0017 cm –0.0006 cm–1 –0.20 0.60 w12 w 21 0.0185 0.0190 similarlity suggests that this is a perturbation Others .01 → .06 cm–1 for W12 W21 due to Coriolis, Fermi? Consistent with expected coupling matrix elements observed in other polyatomics.