D. De Fazio, T. V. Tscherbul 2, S. Cavalli 3, and V. Aquilanti 3

Transcription

1 D. De Fazio, T. V. Tscherbul, S. Cavalli 3, and V. Aquilanti 3 1 Istituto di Struttura della Materia C.N.R., Roma, Italy Department of Chemistry, University of Toronto, M5S 3H6, Canada 3 Dipartimento di Chimica dell' Università, 0613 Perugia, Italy

2 TIME-INDEPENDENT SCHRÖDINGER EQUATION ELECTRONICALLY ADIABATIC POTENTIAL ENERGY SURFACE µ + V Ψ = EΨ THE CALCULATIONS ARE PERFORMED USING A STANDARD COUPLED-CHANNEL TECHNIQUE AND THE HYPER-QUANTIZATION ALGORITHM DEVELOPED IN OUR LABORATORY For details see: J. Chem. Phys. 109 (1998) 379; J. Chem. Phys. 109 (1998) 3805; Phys. Chem. Chem. Phys. 1 (1999) 1091; Phys. Chem. Chem. Phys. 4 (00) 401

3 THE COUPLED CHANNEL EQUATIONS ρ + µ ( E ε ) f ( ρ) W f ( ρ) = 0 n nn' n' nn' nn' ARE THEN PROPAGATED USING A LOG DERIVATIVE METHOD AND SOLVED SUBJECT TO SCATTERING BOUNDARY CONDITIONS f nn' 1 ( ρ) nn' n nn' ρ ik n ( δ exp[ ik ρ] S exp[ ik ρ] ) TO YIELD A SCATTERING MATRIX S AT FIXED VALUES OF TOTAL ENERGY E AND TOTAL ANGULAR MOMENTUM J n

4 Quantum scattering calculations provide the full S-matrix involving all open initial and final channels Coherent partial wave sum σ vjω, v' j' Ω' Incoherent partial wave sum σ vjω, v' j' Ω' ( E, ϑ) = ( E) = π k 1 4k vj vj J J (J + 1) d (J + 1) S J ΩΩ' J vjω, v' j' Ω' ( π ϑ) S J vjω, vjω ( E) The presence of many partial waves in a full collision experiment make reactive resonances elusive to direct observation

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6 Q ( E) = i S J J + ds de J From the scattering matrix Smith (1960) defined the collision lifetime matrix The diagonal elements Q nn are the weighted lifetime of the collision originating in the initial state n at the energy E and total angular momentum J The eigenvalues of Q provide the lifetime of the intermediate metastable states, i.e. resonances

7 From the scattering matrix Smith defined the collision time-delay matrix Q ( E) = i S J J + ds de J Q vj,v j is the average delay time of the collision in the initial state vj q ( E) = T Q J T + q α is the average delay time of the collision in the compound state α provides resonance position and lifetime p ( J, v, j ) = t l vjl Eigenvectors provide the probability for the decay into a channel v, j F.T. Smith, Phys. Rev. 118(1960)349

8 At an isolated resonance one eigenvalue is much larger than the others and shows a Lorentian like profile q max Γa = E a is the resonance positionand Γ a Γa ( E Ea ) + is the width 4

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10 Resonance position as a function of total angular momentum Slopes of the approximately straight lines B A =0.19 mev B B =0.115 mev E r E J = 0 r + B r J ( J + 1) B C =0.077 mev B D =0.086 mev

11 E r E J = 0 r + B r J ( J + 1)

12 Resonance width as a function of total angular momentum A J=0 Vibrational predissociation D C Tunneling through the centrifugal barrier τ A = 131 fs τ B = 334 fs τ C = 4539 fs B τ D = 117 fs

13 The eigenvectors analysis: the decay probabilities C D 1 D HD(v=0, j=0) P Q HD(v=0, j=1) 0.3 P Q J

14 POTENTIAL ENERGY PROFILE F+H (v=0) 0.68 v =3 V = E, ev V =0.067 Barrier height V = V =0 Reaction coordinate HF (v ) + H

15 Van der Waals F+HD T-conf. Entrance Channel The Long-range Interaction Van der Waals D + HF Collinear conf. Exit Channel V (kcal/mol) FXZ PES PES-III V (kcal/mol) PES-III FXZ PES SWMHS PES -0.3 SW PES r F--HD (bohr) r FH--D (bohr) r H F H r D F D

16 How does a resonance affect the ICS and DCS? ΔJ ICS Γ = B(J + 1) τ rot τ Number of partial waves contributing to the resonance structure NARROW τ >> τ rot Δ J <<1 J-selected resonance peaks BROAD τ <<τ rot ΔJ >>1 Resonance structure quenched by overlapping J DCS Forward-backward symmetric peaking Ridge structure superposition of J-shifted resonances W.H. Miller and J. Zhang, J. Phys. Chem. 95(1991)1

17 3 FXZ PES PES-V FXZ PES Integral Cross Section (10 - nm ) 1 F + HD -->HF + D 0 4 PES-V 3 FXZ PES 1 F + HD --> DF + H Collision Energy (mev)

18 K. Liu et al (00) Vibrational Branching Ratio v =3 v = PES III FXZ PES v =1 v = Collision Energy (mev) PES III FXZ PES

19 Integral Cross Section (10 - nm ) F + HD (v=0, j=0) --> HF(v ) + D v = v = 1 v = 3 v = 0 F + HD (v=0, j=1) --> HF(v ) + D v = v = 1 v = 3 v = Collision Energy (mev) Integral Cross Section (10 - nm ) j = 0 State-to-state ics j = 1 j = F + HD(v=0,j=0) --> HF(v =3,j ) + D Total j = 3 j = Collision Energy (mev)

20 Integral Cross Section (10 - nm ) F + HD(v=0,j=0) --> HF(v'=3,j') + D Collision Energy (mev)

21 Assignment of quantum numbers at peaks B,C,D,F E C D B Is peak A a compound state?

22 q max, ps 1,5 1 0,5 6 J = 0 J = J = 1 J = 3 10 F 8 q max, ps 6 4 F 3 1 F 1 F Collision energy, mev Collison energy, mev

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24 ZERO-POINT EFFECTS COLLINEAR SADDLE ENTRANCE CHANNEL EXIT CHANNEL

25 Resonances Wavefunctions

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27 F + HD (v=0,j=1) The agreement with experiments is seen to be qualitatively good. Fully quantitative agreement between QM calculations and experiment in all the reaction attribute must wait further refinements of the PES!

28 Temperature (Kelvin) Rate constant (cm 3 molecule -1 sec -1 ) F + HD --> HF + D F + HD --> HF + D FXZ PES PES V T c = 178 K Bell '35 Bell '59 Arrhenius Bell '35 Arrhenius d-arrhenius T c = 119 K d-arrhenius Bell '59 F + HD --> DF + H F + HD --> DF + H FXZ PES PES V Arrhenius Bell '35 Arrhenius Bell '59 T c = 105 K Bell '35 Bell '59 T c = 84 K d-arrhenius d-arrhenius /T (Kelvin -1 )

29 48 46 (a) (b) E F + H (v=0, j=, l= ) Energy (mev) I II III II III D I R F - H (Å) P J = 0

30 F+HD-->HF (DF)+D(H) SW PES 10 ics (A ) Wigner law ics = *(E coll ) -1/ Wigner law ics = *(E coll ) -1/ e-06 1e Collision Energy mev

31 F+HD -->HF(DF) + H(D) SW PES 10 HF k * (cgs) 1 DF Wigner limit = Wigner limit = T (Kelvin)