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Chapter 4: Chemical Reaction Dynamics a) Chemical reaction dynamics is concerned with unraveling the mechanism of chemical reactions on a quantum mechanical level. Some key questions: How does the BO-PES influence a chemical reaction ? What are the driving forces behind a chemical process ? How does the kinetic energy and the internal quantum state of the reactants (electronic, vibrational, rotational) influence the chemical reactivity ? Which reaction product channels are available and how is energy partitioned between them ? What are the physical constraints on a chemical reaction, i.e., are there chemical “selection rules” ? What is the role of angular momentum ? b) Chapter 4: Chemical Reaction Dynamics a) Recommended literature: • M. Brouard, C. Vallance (eds.), Tutorials in Molecular Reaction Dynamics, RSC Publishing 2010 • R.D. Levine, Molecular Reaction Dynamics, Cambridge University Press 2005 • M. Brouard, Reaction Dynamics, Oxford Chemistry Primers, Oxford University Press 1998 • H.H. Telle, A.G. Urena, R.J. Donovan, Laser Chemistry, Wiley 2007 • B.J. Whitaker, Imaging in Molecular Dynamics, Cambridge University Press 2003 • M.S. Child, Molecular Collision Theory, Dover 1996 • J.Z. Zhang, Theory and Application of Quantum Molecular Dynamics, World Scientific 1999 b) a) Chapter 4: Contents 4.1 Reaction rates and cross sections 4.2 Classical scattering theory 4.3 Introduction to quantum scattering theory 4.4 Reactive scattering: concepts, methods and examples 4.5 Photodissociation dynamics and laser chemistry 4.6 Real-time studies of reactions: Femtochemistry b) 4.1 Reaction rates and cross sections 4.1.1 Rate constants Pro memoria: the molecularity is defined as the number of particles involved in an elementary chemical reaction: unimolecular: bimolecular: A→B A+B→C The rate v for a bimolecular reaction is given by v= dCA = k(T )CA CB dt thermal rate coefficient number densities If [C]=[molecules cm-3], then the dimension of the rate constant k(T) is [k]=[cm3 molec.-1 s-1] or simply [k]=[cm3 s-1]. 4.1.2 Reaction cross sections Consider an experiment in which a beam of molecules A with intensity I0 enters a chamber filled with a gas of molecules B. A reacts with B, and after passing a distance l through the chamber, the intensity is reduced from I0 to I1 because of reactive collisions. A I0 B I1 The intensity I of the beam of molecules A (molecules passing through a surface per second) is given by I = vA CA velocity number density If we assume that the B molecules are much slower than the molecular beam of A molecules (vB=0), the attenuation of the intensity of the beam can be cast into a Lambert-Beer-type form of expression: dI (*) = CB I dx reaction cross section ln(I1 /I0 ) = CB ⇥ Integrate: The bimolecular rate constant for the reaction is defined as: dCA = kCA CB (**) dt Using I=vACA and vA=dx/dt 1/dx=vA/dt, the left-hand side of Eq. (*) becomes: dI d(CA vA ) dCA dCA = = vA = dx dx dx dt Using I=vACA , the right-hand side of Eq. (*) becomes: CB I = vA CA CB Thus, Eq. (*) becomes: Comparison with (**) yields: dCA = vA CA CB dt k = vA which is an universal expression linking the rate constant with the cross section. 4.1.3 Simple collision models for the reaction cross section The classical collision density ZA (defined as the number of collision per second) of a molecule A with molecules B is given by: ZA = k(T )CAB = h ihvrel iCB The number of collisions between A and B molecules per unit volume is thus: V ZAB = h ihvrel iCA CB If A=B, we get: V ZAA = (1/2)h ihvrel iCA2 (Factor 1/2 for not counting collisions between the same particles twice) 1. Hard-sphere collisions: constant reaction cross section σ0 Molecules are treated as colliding hard spheres with radius rA and rB. Assuming every collision leads to reaction, the cross section is 2 2 thus given by: = ⇡d = ⇡(r + r ) 0 A B 2. The impact parameter b: The impact parameter b is defined as the distance of closest approach of the reactants in the absence of an interaction potential: • b≈0: head-on collision • b>>0: glancing collision The reaction cross section can generally be formulated as = Z bmax P (b)2⇡b db 0 Where P(b) is the probability for reaction at collision at a given value of b (the opacity function). b, P(b) and σ are usually dependent on the collision energy. If P(b)=1, we recover the hard-sphere collision model: Z bmax 2 = 2⇡b db = ⇡bmax 0 4.2 Classical scattering theory Every chemical reaction entails a collision, a scattering event. We will therefore treat chemical reactions in the framework of scattering theory. Molecules are quantum systems - so why use classical models ? • The essential physical concepts are much easier to understand in a classical picture • Classical scattering models are still used for even rather small molecules (>3 atoms !) for which a quantum treatment is prohibitively expensive Types of scattering events: • Elastic scattering: total kinetic energy and the internal state of the collision partners are conserved • Inelastic scattering: total kinetic energy and internal state of the reaction partners change, the chemical structure is conserved • Reactive scattering: kinetic energy, internal state and chemical structure change 4.2.1 Kinematics of molecular collisions: the centre-of-mass system For collisions between molecules, the relevant kinematics are defined by their motion relative to one another, and not by their absolute motion in the laboratory coordinate frame. On thus transforms the system into the centre-of-mass coordinate frame defined by R c.o.m. rA Rc mA~rA + mB ~rB ~ Rc = mA + mB ... coordinate vector of the centre of mass (c.o.m.) ~ = ~rA R ... relative coordinate vector ~rB y A B rB x It can easily be shown that the kinetic energy Ekin of the system is given by: 1 1 1 1 2 2 2 2 Ekin = mA vA + mB vB = MV + µv 2 2 2 2 ~˙ ~˙ c | , v = |R| where the velocities are given by vi = |~r˙i |,V = |R mA mB M = mA + mB is the total mass and µ = is the reduced mass. mA + mB 4.2.2 Elastic scattering Elastic collisions, i.e., collisions in which the kinetic energy is conserved, are the simplest form of scattering events. We will discuss classical elastic collisions to introduce the basic concepts of scattering theory. R ... relative position vector ψ ... orientation angle of R with respect to the original velocity vector v or ct je tra on si lli co As the total angular momentum is conserved, two coordinates suffice to describe the relative motion of the collision partners. We choose y Consider the collision trajectory of two structureless particles (e.g., atoms) in the COM frame: initial velocity vector deflection angle vector of closest approach impact parameter relative position vector orientation angle Conserved physical quantities ~ = µ~ ~ v ⇥R Angular momentum: L or ct je tra on si lli co ~ sin = µ~ v ·R y L before collision = L after collision: ~ = L = |µ~ ~ where v is the initial velocity vector |L| v ⇥ R| initial velocity vector ~ = L = µv b |L| deflection angle vector of closest approach Total energy: E = Ekin + Ecent + Epot kinetic centrifugal potential impact parameter relative position vector orientation angle 1 1 2 = µṘ + µR2 ˙ 2 + V (R) with the angular velocity ˙ = d /dt = ! 2 2 2 1 1 L = µṘ2 + + V (R) 2 2 2 µR L = µR2 ! 2 1 1 L ) E = µṘ2 + + V (R) 2 2 2 µR VL(R) ... centrifugally corrected (effective) potential Centrifugally corrected potentials Centrifugal energy = energy taken up in the rotation of the position vector R Collisional angular momentum L = angular momentum associated with the rotation of R about ψ The effective potential for the collision contains both, the interaction potential V(R) and the centrifugal energy: 1 L2 VL (R) = + V (R) 2 2 µR centrifugal barrier Centrifugally corrected potentials VL(R) for L3 > L2 > L1 > L0=0 The deflection function χ(b) The angle of deflection χ depends on the impact parameter b. Examples: 1. Hard-sphere collisions (a billiard game): • • For b > d: =0 For b < d: =⇡ 2 0 where b/d = sin 0 =) = 2 arccos(b/d) 2. General potentials with repulsive and attractive parts: repulsive, short range part: V(R)>0 R V(R) b* = b/Re χg R* attractive, long range part: V(R)<0 χr • • • • Small b: collision dominated by repulsive forces backward scattering Large b: collision dominated by attractive forces forward scattering Rainbow angle χr: maximum negative deflection angle at impact parameter br≈Re where the potential is most attractive Glory angle χg: deflection angle at impact parameter bg≈R* where attractive and repulsive forces cancel • Experimentally, it is not possible to distinguish between positive and negative deflection angles χ because of the cylindrical symmetry of the collision process. One can only measure the absolute value of the deflection angle θ=|χ|. Experimental observables in molecular-collision experiments The intensity of scattered molecules I(Ω), i.e., the flux of molecules scattered into the solid angle Ω, defines the differential cross section dσ/dΩ: dΩ d scattered flux of molecules per unit solid angle I(⌦) = = d⌦ incident flux of molecules per unit area θ The integral cross section σ is obtained by integration. = Z d d⌦ = 2⇡ d⌦ Z 0 ⇡ d sin ✓d✓ d⌦ Where the cylindrical symmetry of the problem allowed us to express d⌦ = 2⇡ sin ✓ d✓ in the second step. The scattering rate constant is then given by (see section 4.1.2): k= v Calculating the differential cross section from the deflection function θ(b) If we assume that the opacity function is unity, P(b)=1, the differential cross section can be expressed as (see sec. 4.1.3) d = 2⇡b db Again, because the scattering problem is cylindrically symmetric, the solid angle element dΩ can be formulated as d⌦ = 2⇡ sin ✓ d✓ Hence we obtain for the differential cross section: d 2⇡b db b = I(✓) = = d⌦ 2⇡ sin ✓ d✓ sin ✓(d✓/db) If more than one value of b contribute to the same scattering angle θ, we have to sum over all contributions and arrive at the following dependence of the differential cross section on the deflection function θ(b): X b d = I(✓) = d⌦ sin ✓(d✓/db) Singularities in the differential cross section (dσ/dΩ=∞): • • Glory (θ=0) singularity: sin θ = 0 Rainbow singularity: (dθ/db) = 0 (maximum of the function θ(b) ) Illustration: deflection function θ(b) X b d = I(✓) = d⌦ sin ✓(d✓/db) differential cross section I(θ) glory singularity rainbow singularity Calculating the deflection function θ(b) from the potential V(R) It can be shown (see, e.g., R.D. Levine, Molecular Reaction Dynamics): (b) = ⇡ Z 1 1 b ⇣ R2 1 1 b2 R2 V (R) E ⌘1/2 dR i.e., χ(b) depends on the potential V(R) and the collision energy E. For inverse power law potentials Cn V (R) = n R which describe long-range interactions between molecules the deflection function can be approximated to: V (b) (b) ⇡ E in the limit of large impact parameters b (momentum approximation). Hence, in this limit the deflection function is a direct measure of the potential ! In an experiment, the impact parameter b cannot be selected and one measures a differential cross section summed over all possible impact parameters. 4.3 Introduction to quantum scattering theory 4.3.1 Quantum elastic scattering Contents: 4.3.1.1 General formulation of the scattering problem 4.3.1.2 The scattering phase 4.3.1.3 Scattering amplitude and scattering matrix Derivation → blackboard 4.3.2 Quantum inelastic scattering Contents: 4.3.2.1 Scattering Hamiltonian 4.3.2.2 Angular momenta 4.3.2.3 Close coupled equations Derivation → blackboard 4.4 Reactive scattering: concepts, methods and examples 4.4.1 Motion on the PES The topology of the Born-Oppenheimer PES determines the dynamics of a chemical reaction. Even in the absence of exact QM reactive-scattering calculations, important insight into chemical dynamics can be gained from analysing classical collision trajectories on the BO-PES. Consider the simplest polyatomic case: the reaction between an atom A and a diatom BC: A + BC → AB + C . products Reaction profile for a linear approach of the reactants • The path of minimum energy from the reactants to the products of the PES is termed reaction path or reaction coordinate. • The energy barrier (saddle point on the surface) separating reactant and product “valleys” is termed transition state. saddle point = transition state reactants If the total energy in the reactants (the sum of collisional energy Ec, vibrational energy Ev, rotational energy Er, and electronic energy Ee if applicable) is higher than the barrier height, the reaction can proceed in principle. The available energy Eavl after the collision is distributed among the products. For an A + BC reaction, the barrier height in general changes for different approach angles. If more energy is stored in the reactants, the barrier can also be crossed for approach angels differing from the optimal value. Thus, the cone of acceptance of the reaction can be increased. Potential energy profile along the reaction coordinate for H + H2 for different values of the approach angle γ. P. Siegbahn et al., J. Chem. Phys. 68 (1978), 2457 D.G. Truhlar et al., J. Chem. Phys. 68 (1978), 2466 4.4.2 Effect of vibrational and kinetic energy: Polanyi rules For asymmetric reactions, the transition state is usually located closer to either the reactant or the products (early or late barrier). From an inspection of the favourable reaction trajectories, it can be seen that: For an early barrier, translational excitation (high kinetic energies) of the reactants promotes the reaction and leads to vibrationally excited products. Vibrational excitation hinders the reaction. For a late barrier, vibrational excitation promotes the reaction and leads to products with a high kinetic energy. Translational excitation of the reactants hinders the reaction. Forward reaction HF + H → H2 + F late barrier H2 + F Backward reaction H2 + F → HF + H early barrier HF + H HF + H H2 + F H2 + F HF + H HF + H H2 + F 4.4.3 Angular momentum constraints Angular momentum (AM) conservation for the collision dictates: rotational AM collisional AM total AM 0 J = jBC + L = jAB + L0 before collision after collision If the reactants are internally cold (e.g., from supersonic cooling in a molecular beam), then the initial rotational AM can be neglected: 0 J ⇡ L = jAB + L0 In addition, for reactions involving the transfer of a light atom L from a heavy atom H’ to another heavy atom H (H + LH’ → HL + H’), we get J ⇡ L ⇡ L0 because the large rotational energy spacing of HL suppresses rotational excitation of the product so that orbital AM is conserved. This is called the kinematic effect. Conversely, for a heavy-atom transfer H + LH’ → HH’ + L we obtain 0 J ⇡ L ⇡ jAB because the product orbital AM L0 = µ0 v 0 b0 is usually small owing to the small reduced mass μ’ of the products. Thus reactant orbital AM is converted into product rotational AM. 4.4.4 Reaction mechanisms from angular scattering The angular distribution of scattering products reflecting the differential scattering cross section can be measured in crossed molecular beam experiments. The angular distribution of the scattering products is measured with a moveable detector in the laboratory frame. The distribution of scattering angles θ and product velocities uAB in the centre-of-mass (COM) frame can be inferred from a Newton diagram (velocity diagram). Schematic of a crossed molecular beam experiment Notation: vA, vBC ... velocity vectors of reactants in lab frame vrel ... relative velocity vector of the reactants Θ ... scattering angle in the lab frame vCM ... velocity vector of the COM vAB ... velocity vector of product AB in lab frame uAB ... velocity vector of product AB in COM frame θ ... scattering angle of products in COM frame Newton diagram for the reaction A + BC → AB + C Reconstruction of the COM angular distribution from a CMB measurement The reconstructed COM product flux distribution ICM(θ,u) can be decomposed into two different components: ICM ( , u) = T ( ) ⇥ P (Et0 ) product angular distribution product translational energy distribution (kinetic energy release) The COM product flux distributions are usually represented in a polar plot. The contour lines indicate the product flux scattered into a certain angle θ with a given velocity u (or kinetic energy Et’). Example: Product flux distribution for the HCl product in the reaction H2 + Cl → HCl + H. The reaction mechanism manifests itself directly in the angular distribution of the reaction products. Two important types of mechanisms can be distinguished: • • Direct mechanisms entail a direct scattering event Indirect (or complex-forming) mechanisms entail the formation of an intermediary reaction complex 4.4.4.1 Direct reactions Two important limiting cases: • Stripping reactions: dominated by long-range interactions between the reaction partners. Occur at large impact parameters, lead to forward scattering, i.e., the product angular distribution peaks at θ=0°. (For A + BC, “forward” is defined with respect to the direction of the incoming atom A.) • Rebound reactions: dominated by short-range interactions. Occur at small impact parameters, lead to backward scattering, i.e., the product angular distribution peaks at θ=180°. Example I: Cl + H2 → HCl + H Classical reaction showing rebound dynamics with a highly constrained linear transition state. The small cone of acceptance leads to small impact parameters and backward scattering. P. Casavecchia, Rep. Prog. Phys. 63 (2000), 355 M. Alagia et al., Science 273 (1996), 1519 sis offers a way to circumvent this difficulty by rating the c.m. angular distribution, since " 1 (dQr) . sm()d() (2) o dw abs II: K + Br2 → KBr + Br rtue ofExample the cylindrical symmetry about the initial ve velocity vector. The absolute normalization of Reaction long-range ifferential reactiveinitiated scatteringby cross section, electron transfer from K to Br2 at a (3) (dQr/ d<,;) abs =:n( dQr/ dw )rel, crossing between potential curves e determined by comparison with the elastic corresponding to the neutral andscatg. Theionic resultsforms obtained fromreactants three different proof the (harpoon es are given in Table III. mechanism). The temporary ion pair Method A is strongly accelerated towards one the Coulomb ce theanother relative by intensity scales forinteraction the reactive ultimately to the formation of lastic scattering areleading practically the same,3! the products. Large (dQe/ dw) abs impact forward. scattering. (4) parameters, ( dQe/ dw) reI Qr = 211' elastic scattering pattern at narrow angles is asd to be negligibly perturbed by reaction. The ute intensity thus can be calibrated by use of the -angle scattering formula for a VCr) = -e/r 6 van Waals interaction,32 his is ensured by the data reduction procedure used (relaH. Birley et al., J. Chem. Phys. 47 (1967), 993 by ratio of signal to parent-beam attenuatensity J. defined D. Hershbach, Angew. Chemie Int. Ed. 26 (1987), 1221 Pt data normalized to W data), provided that: (1) the de- 9.7 for Br2 and 12. The halogen polarizabili ties were estimated from the HX and H2 values 35 via o K + Br, G Rb + Br, • Cs + Br, , K + I, • Cs + I, o 00 L-L--'--'---'---L-L....J---"----,-L.L--'---'---'---"----'-I..--J O· 30· 60· 900 1200 1500 1800 CM SCATTERING ANGLE e FIG. 16. Comparison of approximate C.m. angular distributions of reactive scattering. The curves (--) are calculated from the Legendre polynomial expansions given in Table II. ------ It is expected that dQ,/dw and Q, as predicted from the S--K approximation shouldcurve be crossing correct within 20% (including allowance for the uncertainty in the polarizabilities). This is indicated by extensive data on relative cross sections (see Ref. 24) and recent absolute measurements for several reference systems; see E. W. Rothe and R. H. Neynaber, J. Chern. Phys. 42, 3306 (1965); ibid. 43, 4177 (1965); and H. G. Bennewitz and H. D. Dohmann, Z. Phygik 182, 524 (1965). Small angle scattering measurements of Ref. 27 (h) give <:l=870XlO--60 erg·cm 6 for K + Br2, in good agreement with the S--K result of Table III. 33 4.4.4.2 Indirect reactions Indirect reactions proceed via the formation of a long-lived reaction complex (corresponding to a reaction intermediate, i.e., a minimum on the PES along the reaction path) which lives longer than several rotational periods. During this time, the collision partners lose part or all of the memory of their original orientation (see also section 4.4.3): If L≈L’ (e.g., in a light-atom transfer), i.e., the collisional angular momentum and thus the plane of collision is conserved, the products show a distinct forwardbackward scattered distribution: d⇤ d⇤ 1 = / d 2⇥ sin d sin If L≈j’ (e.g., in a heavy-atom transfer), i.e., the products are rotationally excited, memory of the original orientation is completely lost and the angular distribution is isotropic (i.e., constant). L≈J=L’ L≈J=j’ Example I: OH + CO → CO2 + H The reaction of CO + OH (a major channel for the production of CO2 in combustion processes) proceeds via the formation of an intermediate HOCO product. The angular distribution shows prominent forward-backward scattering peaks indicating the indirect mechanism with a propensity for the conservation of collisional AM. P. Casavecchia, Rep. Prog. Phys. 63 (2000), 355 M. Alagia et al., J. Chem. Phys. 98 (1993), 8341 Example II: angular product distribution and reaction paths: O(1D) + H2 → OH + H This reaction can either proceed through an indirect insertion mechanism of the O atom into the H-H bond forming an intermediary water molecule which breaks apart or by a direct abstraction mechanism via an excited electronic state. Depending on the collision energy, both pathways can be open and can be distinguished by their different angular product distributions. direct mechanism backward scattering rotationally excited products: isotropic angular distribution indirect mechanism 4.4.4.3 Dynamics at curve crossings: adiabatic and diabatic states Charge-transfer mediated reactions such as the harpooning reaction in K + Br2 are classical examples of reactions dominated by the crossing of two potential energy surfaces. In fact, many chemical processes are dominated by such non-adiabatic dynamics when the system crosses from one PES to another. Such processes involve a breakdown of the Born-Oppenheimer approximation. Surface crossing in a charge-transfer mediated reaction The crossing from one PES to another necessitates coupling terms in the molecular Hamiltonian which are usually neglected in the BO approximation, e.g., • • the adiabatic correction terms Ĉn (see chapter 1.3) which couple states of the same symmetry and the same multiplicity spin-orbit interaction which couples states with different multiplicities (see problem sheet 3) Although usually small, such couplings become important when two electronic states come close in energy, i.e., at crossing points. Mathematical description: • Let Φ1(0) and Φ2(0) be electronic states in the BO approximation (so-called diabatic states), i.e., solutions of a BO-Hamiltonian Ĥ0 (see section 1.2). If these states are coupled by an additional weak coupling operator V, the total Hamiltonian is given by Ĥ = Ĥ0 + V̂ • The coupled states can be expressed as a superposition of the uncoupled states: = c1 (0) 1 + c2 (0) 2 with mixing coefficients c1 and c2. • By inserting into the nuclear Schrödinger equation ĤΦ=EΦ, multiplying from the left bey either Φ1(0) or Φ2(0) and integrating over the nuclear coordinates (see chapter 1.3) we get a set of secular equations for c1 and c2: c1 (H11 U) + c2 H12 = 0 c1 H12 + c2 (H22 where Hij = (0) i |Ĥ| diabatic basis. (0) j ⇥ = (0) i |Ĥ0 + V̂ | U) = 0 (0) j ⇥ are the matrix elements of Ĥ in the • Note that for the matrix elements Hij = i |Ĥ| j ⇥ = i |Ĥ0 + V̂ | j ⇥ : (i) Hii≡Ui(R) (i=1,2), the BO energies of Φ1(0) and Φ2(0) (ii) H12≡V12(R) because Φ1(0) and Φ2(0) are orthonormal eigenstates of Ĥ0 and V mixes Φ1(0) and Φ2(0). • Note also that both, the BO energies Ui and couplings V12 generally depend on the reaction coordinate R. • The secular equations thus become: (0) c1 (U1 (0) (0) U) + c2 V12 = 0 c1 V12 + c2 (U2 • (0) U) = 0 The solutions (energies of the coupled electronic states) are: p 1 1 U± (R) = 2 (U1 (R) U2 (R)) ± 2 (U1 (R) U2 (R))2 + 4V12 (R)2 with the associated eigenfunctions Φ+ and Φ- (the so-called adiabatic states). The coupling repels the states around the crossing point and leads to an avoided crossing. U(R) adiabatic states diabatic states At the crossing point, the separation between the adiabatic states is given by ΔU=2V12. The adiabatic states and the associated PES are the eigenfunctions of the full Hamiltonian Ĥ and can be obtained from ab-initio calculations. R Diabatic and adiabatic states at a crossing point In a diatomic molecule, states of the same symmetry can never cross because of nonadiabatic couplings. All such crossings are always avoided (non-crossing rule). This restriction is relaxed in polyatomics. The crossing of two states is referred to as a conical intersection. The term originates from the shape of the two potential energy surfaces in the crossing region in two dimensions (2D cut through the PES along two internal coords Q1 and Q2). Q1 Q2 diabatic passage adiabatic passage Conical intersection between two electronic states in two dimensions Conical intersections dominate the dynamics of many chemical processes involving excited electronic states (see several examples in this chapter). Moreover, in many cases energy barriers on an adiabatic PES are caused by avoided crossings. Landau-Zener theory: When a crossing is traversed in the course of a reaction, the system can stay on the same adiabatic surface (adiabatic passage) or cross to the other adiabatic surface (i.e., stay on the same diabatic surface, diabatic passage). The probability Pad for diabatic passage (i.e., crossing from one adiabatic surface to the other) can be calculated using the semiclassical Landau-Zener equation: ( ) 2 2 V12 Pdia = exp hv (U2 (R)RU1 (R)) where v ... velocity along reaction coordinate U1(R), U2(R) ... BO-PES associated with the diabatic states Φ1(0) and Φ2(0) The probability for adiabatic passage Pdia is then Pad = 1 Pdia Thus, the probability for diabatic passage is high if the coupling V12 is weak and the velocity and the difference of the potential gradients are large. Thus, the probability for adiabatic passage is high if the coupling V12 is strong and the velocity and the difference of the potential gradients are small. 4.4.4.4 Reaction resonances Energy V / ev Reactants Products Reaction time delay Reaction resonances can modulate the reaction cross section by several orders of magnitude in a small energy interval. They can therefore have drastic effects on the dynamics of a reaction. collision energy Ec Reaction probability Reaction resonances are a distinctly quantum mechanical phenomenon which lead to strong fluctuations in the reaction cross section and the collision time. They appear when the collision energy is in resonance with a suitable bound state of the system thus enhancing the reaction probability. There are two important types of reactive resonance effects: • • Feshbach resonances: the bound state is an excited state of the system (e.g., rotationally, vibrationally or electronically excited) Shape (or orbiting) resonances: the bound state is located behind a centrifugal barrier Obviously, the occurrence of resonances strongly depends on the collision energy, collisional angular momentum and quantum state of the reactants. Collision energy Ec J.N. Milstein et al., New J. Phys. 5 (2003) 52 Example for a dynamic situation leading to a Feshbach resonance Collision energy Ec Example for a shape resonance (here ≣L ... collisional angular momentum) ognized the metastable argon in another, either as neat gasesthat or seeded of p He ground He state. metastable helium atom second ground beam, and atom or molecule atomWhen and theacolliding atom or At atom state.any When a metastable helium atom of the second beam, and any atomoforthe molecule formed bymolecule. the colliding Department of Chemical Physics, Weizmann Institute of acollides noble carrier gas. By changing the takes compoene sho withdistances another atom or molecule with an that canbeam be entrained intoanother the supersonic beaminwith the repulsion over, collides with atom Rehovot, or molecule an short that can be entrained into the supersonic can beterm trapped by the poten Science, Israel. whe ionization energy lower as than 19.8 atemperchargecould be used inionization principle. By varying the19.8 relative of the gas mixtures well aseV, the argo whereas at intermediate distances there is energy lower than eV, asition chargecould be used in principle. By varying the relative emerges from thea shalcontribution *To whom correspondence should be addressed. E-mail: low van der Waals well. The complex part of the transfer takes place whereby an electron velocities between the two beams,velocities we continulow transfer process takes place whereby an electron between theprocess two beams, we continuand 120 wecase of pea potential (21).K), In the el [email protected] ature of the valve (between 355 potential G(R)/2 is related to the ionization probfrom theenergies neutral species jumps into the vacancy ously tune collisional energies fromously 350 Ktune downcollisional pot from the neutral species jumps into the vacancy from 350 K downchanged the relative mean velocitydistance. between the duc ability at a given internuclear Because the 1sbound orbital for of helium, coinciding withofthe to 10 mK. The lower bound for collision abil the 1s orbital of helium, coinciding with the the to 10 energy mK. Theoflower collision energybeams 1000 m/s probability down to zero. Atrapidly zero theover charge transfer decays is unexpected. One would naïvely expect the ejection of an electron from the 2s orbital to thefrom the1 ejection of an electronit isfrom themodeled 2s orbital the Fig. unexpected. One would naïvely expect therelative velocity between the beams, the by residual date with separation, usually atosingle (Eq. 1) lowest collision temperature we canis achieve to be continuum with continuum (Eq. 1) lowest collision temperature we can achieve to be exponential is much limited by the temperature of the hottest beam energy stemsterm. fromElectronic the finitemotion velocity dis- tion perf * þ M → He þ M þ þ e− collision He ð1Þ exp compared with nuclear motion, and the poten among the two, in the range of 0.1limited to 1 K. by Thethe temperature of the hottest beam þ tribution offaster the beams within colproc He*supersonic þM → He þ Mbe þ e− the ð1Þ PI rea autoionization process can viewed as a verreduction in energy occurs becauseamong of longitudiPI has been studied in detail at higher enfast the two, in the range of 0.1 to 1 K. The lision volume. equ He (3S tical process within the Born Oppenheimer apnal momentum compression in phase space durergies, with many interesting results summarized reduction in energy occurs because of longitudiPI has been studied in detail at higher en- aut We measured the interesting time-of-flight for helium both droge part proximation. The resulting ion signal and neutral ing free propagation. Our nozzle-opening duration incompression a review article by Siska (40).durPI reactions tica nal momentum in phase space ergies, with many results summarized interaction is described by a neutral-ion potential is much shorter compared with the propagation to have different entrance and exit channelsreactant (Fig. 1). beams nel in as well as the product. The metafrom proo ing freephase propagation. Our nozzle-opening duration between in a review by channel. Siska (40). PI reactions part surfacearticle in the exit the detector time, and the initial spherical In the entrance channel, the interaction wasentrance measured by channels using a (Fig. microsolu inte much shortermetastable comparedhelium with and the propagation tostable havebeam different 1).is solid A schematic ofand theexit experimental system space distribution deforms assumingis a cigar shape. another atom or molecule channel plate, whereas the ground state beam sym detector anddescribed the initial spherical phase the entrance channel, thepulsed interaction between shown in Fig. 2A. The metastable helium assurf In our study of low-energy PI the reactions, we time, can be by using a complex opticalInpoau was measured by using an ionizing quadrupole trea supersonic beam was created by using an Even- (dash merged a supersonic beam of metastable helium tential. The real part of the interatomic potential space distribution deforms assuming a cigar shape. metastable helium and another atom or molecule Lavie valve (41) cooled down tothe 55 K. Immewith a supersonic beam containing either argonstudy contains the appropriate Waals mass spectrometer (QMS). Toameasure product ion sho In our of low-energy PI long-range reactions,van weder can be described by using complex optical po- botto diately after the valve there is a dielectric barrier or molecular hydrogen. The excited 3S state interaction, the leading term of which ion, scales as resen we turned offpart the ofionizing elementpotential of the sum sup merged a supersonic beam of metastable helium tential. The real the interatomic of helium has an energy of 19.8 eV above the R−6, where R is the distance between the metastable discharge (42), which was used to excite the exit-c Lav with a supersonic beam containing either argonQMS contains the appropriate van der Waals in order to observe the ions in the wav ground-state heliumlong-range to the 23Sformed level. The beam poten or molecular hydrogen. The excited 3S statechemi-ionization interaction, the leading term of which scales as Wem/sfirst had a meancollisions. velocity of ~770 with divided a standard diat −6 disc of helium has an energy of 19.8 eV above thethe Rion, signal where Rby is the distance between metastable deviation of 15 m/s, corresponding a temperthe product of the the areato of both face ature of 50 mK in the moving frame of reference. gro neutral beams and then normalized by using the cula The beam then passed through a 4-mm-diameter had data fromskimmer earlier located high-collision-energy experi10 cm from our valve orifice high dev ments (43,and 44). Thus, we are able to present our the www.sciencemag.org SCIENCE VO entered a 20-cm-long magnetic quadrupole, atur which had a 10° curve with a curvature radius of results for hydrogen (Fig. 3A) and for argon (Fig. the * The cm. We createdrate a quadrupole 3B) on the114.7 absolute reaction scale. magnetic cau field by passing a current pulse through 1-mm skim At higher collision energies, 1 meVas with diameter copper wires arrangedabove in quadratures, and (11.5 K) and 20 inmeV K) quadrature for the Arconsisted and H2of inve shown Fig. (230 2B; each whi nine wires in a our three-by-three pattern. At the peak this systems, respectively, results are in very good 114 of 1000experimental A, the transverse quadrupole trap ram agreement current with earlier measurements. depth was about 2.7 K and 3 mm by 3 mm in fiel A classicalsize. treatment of the collisional process is pos diam Only low-field–seeking Zeeman sublevels sufficient to explain theinmain results above sho were confined two dimensions duringthis the star transit through thefalls quadrupole guide. As such, rece nin energy. reaction rate at lower velocities Fig. 2. (A) Schematic of the experimental system within the vacuum chambers, showing the two source The the metastable helium beam leaving the magnetic supersonic valves followed by two skimmers, the curved magnetic quadrupole guide with its assembly, and curr because the inner classical turning-point position the the QMS entrance at the end. The blue beam is magnetically guided, whereas the red beam is unaffected. quadrupole was 100% spin-polarized in a single dep scales withquantum energy.state Forwith lower thetheclasThe merged volume is in purple. (B) A front view of the quadrupole guide. the energies, projection of total the Example I: Shape resonances in the Penning ionization of He* with H2 A.B. Henson et al., Science 338 (2012), 234 • Penning ionization: Ionization by energy transfer from an excited species (important chemical process in high-energy environments like flames and plasmas) • The prototypical Penning ionization process of He (= He (1s2s) 3S) with H2 was recently studied using a new mergedmolecular-beam method using a magnetic guide for one beam (see also chapter 5) • Fig. 3. (A) Reaction rate measurements for (3S) He* and H2 PI are shown in black with error bars. The lowest collision energy achieved is 0.75 with standard deviation of 0.07 meV, corresponding to 8.7 T 0.8 mK temperature. Blue dash-dot line is the reaction rate calculated by using the most recent potential from (44). Red solid line is the calculated reaction rate by using the TangToennies potential with parameters that give the best Pronounced modulation in the reaction rate were observed as a function of the collision energy indicating the presence of numerous shape resonances size1 of 1wer Å tran Fig. 2. (A) Schematic of the experimental system within the vacuum chambers, showing the two source pote A B the supersonic valves followed by two skimmers, the curved magnetic quadrupole guide with its assembly, and suri Expt. qua the QMS entrance =5 at the end. The blue beam is magnetically guided, whereas the red beam is unaffected. corr The merged volume is in purple. (B) A front view of the quadrupole guide. qua mor =6 inve hyd Fig. 3. (A) Reaction rate A B sim measurements for (3S) He* and and H2 PI are shown in black with error bars. The ab i Theory lowest collision energy the achieved is 0.75 with stanalso dard deviation of 0.07 meV, ord corresponding to 8.7 T −0.01 −0.02 −0.01 0 5 10 20 15 R (au) 25 30 −0.02 35 Example II: Cl + HD (v=1, j=0) → HCl + D / DCl + H Wave functions of the quasibound levels B and E supported by the adiabatic potential shown as functions of the atom–molecule separation. Amplitudes of the wave functions have been a factor of 10 for practical plotting reasons. The Cl + HD (v=1,j=0) reaction is predicted to have pronounced rotational Feshbach resonances caused by bound states of the van-der-Waals complex Cl...HD of the reactants. Long-range interactions in chemical reactions 102 rotationally adiabatic potential curves 0.7 Cl+HD - nonreactive DCl+H - reactive HCl+D - reactive 101 E D 100 C 0.69 10−1 Energy (eV) Cross section (10−16 cm2) v=1, j=1 10−2 10−3 B v=1, j=0 collision energy A 0.68 10−4 reactive resonances 10−5 10−6 10−8 10−7 10−4 10−3 10−6 10−5 Incident kinetic energy Collision energy (eV)(eV) E 0.67 BC D 10−2 10−1 5 10 15 R (au) 20 The same as in figure 5 but plotted as a function of the incident kinetic energy to illustrate the Figure 6.shallow Adiabaticpotential-energy potential energy curveswells of the Cl þ HD system correlating to the HDðv ¼ 1, ature behaviour of the cross-sections. HDðvby ¼ 1,long-range j ¼ 1Þ levels as(van-der-Waals) functions of the atom–molecule separation, R. Quasibound levels re caused interactions for the resonances observed in figure 5 are labelled by B, C, D and E. Weck and Balakrishnan, Int. Rev. Phys. Chem. 283 (25), 2006 REPORTS Until recently, many of the S 2 dy- - may be obtained from measurements of correlated - +details 4.4.5 A case study: the SN2 reaction Cl CH →I + CH Cl 3I reactions namics of bimolecular anion-molecule angleand 3 energy-differential cross sections. N could only be obtained from chemical dynamics simulations. However, with recent experimental J. Mikosch et al., Science 319(22), (2008), advances insight184 into the reaction dynamics SN2 nucleophilic substitution reactions X- + R-Y → Y- + R-X show a characteristic double-well potential-energy profile along the reaction coordinate. According to the conventional picture, the reaction proceeds via a back-side attack on the R-Y bond leading to an inversion of the molecular configuration. Specifically, the probabilities for energy redistribution within the ion-dipole complexes, their dependences on initial quantum states, the branch- transition state reaction complex in exit channel reaction complex in entrance channel For the model reaction Cl- + CH3I → I- + CH3Cl, one would expect that the Reaction profile for Cl- + CH3Ipotential → I- +energy CH3along Cl the reaction 1. Calculated MP2(fc)/ECP/aug-cc-pVDZ Born-Oppenheimer dynamics is dominated by the formation Fig. for the S 2 reaction Cl + CH I and obtained stationary points. The reported coordinate g = R − R of a long-lived reaction complex in the energies do not include zero-point energies. Values in brackets are from (28). exit channel. C−I C−Cl N − 3 If the lifetime of the reaction complex is longer than several rotational periods, an isotropic product angular distribution is expected. A B Forward-scattered products: direct substitution mechanism Isotropic distribution: complexmediated, classical mechanism 184 A advances (22), insight into the r pulsed-field velocity servation map imaging spectrometer, of energy and momentum (24). which maps the velocityThe of top therow I − product anionmaps of the I − of Fig. 2 shows B C of the outermost ring in the image. Thus, the largest fraction of the available energy is partiproduct ion velocities from the Cl− + CH3I → tioned to internal rovibrational energy of the CH3Cl + I − reaction at four different relative col- CH3Cl product. lision energies between Erel = 0.39 eV and Erel = A distinctly different reaction mechanism be1.90 eV, which were chosen because they span the comes dominant at the higher relative collision distinct reaction dynamics observed in this energy energy of 0.76 eV (Fig. 2B): The I − product is Forward-backward-scattered products: range. The only data processing applied to the ion back-scattered into a small cone of scattering roundabout impact position on the detectormechanism is a linear conver- angles. This pattern indicates that direct nucleosion from position to ion speed and a transforma- philic displacement dominates. The Cl− reactant tion into the center of mass frame. Consequently, attacks the methyl iodide molecule at the concave the velocity vectors of the two reactants, the Cl− center of the CH3 umbrella and thereby drives the − anion and the CH3I neutral, D line up horizontally I product away on the opposite side. The direct and point in opposite directions, indicated by the mechanism leads to product ion velocities close arrows in Fig. 2. Each velocity image represents a to the kinematic cutoff. In addition, part of the histogram summed over 105 to 106 scattering product flux is found at small product velocities events. The total energy available to the reaction with an almost isotropic angular distribution, inproducts is given by the relative translational dicating that for some of the collisions there is a energy, Erel, of the reactants plus the exoergicity, significant probability of forming a long-lived 0.55 eV, of the reaction (Fig. 1). I − products reach Cl- complex. CH3 - I At a collision energy of 1.07 eV (Fig. 2C), the the highest velocity when all the available energy is converted to translational energy. The outer- complex-mediated reaction channel is not most circle in Fig. 2 represents this kinematic observed any more. The reaction proceeds almost cutoff for the velocity distribution. The other con- exclusively by the direct mechanism, with a similar centric rings display spheres of the same trans- velocity and a slightly narrower angular distribulational energy and hence also the same internal tion relative to the 0.76-eV case. At an even higher collision energy of 1.90 eV, the domiproductrel excitation, c spacedHat 0.5-eV intervals. Downloaded from www.sciencem coordinate g = RC−I − RC−Cl for the SN2 reaction Cl− + CH3I and obtained stationary points. The reported energies do not include zero-point energies. Values in brackets are from (28). E G energies E ≡E Angular product distributions forF I- at different collision In the gas-phase crossed-molecular beam scattering experiment, three types of product angular distribution T(θ) are observed indicating three different reaction mechanisms: Fig. 2. (A to D) Center-of-mass images of the I− reaction product velocity from the reaction of Cl− with CH3I at four different relative collision energies. The image intensity is proportional to [(d3s)/(dvx dvy dvz)]: Isotropic scattering results in a homogeneous ion distribution on the detector. (E to H) • • • en rea of dis the ph wh trib do agr the ve ser fin rep tio (13 en CH 0.7 en can the val in 0.5 (40 mi rel sig co of for EC co (27 gie ag coe are at en bil 1.9 vib res me The energy transfer distributions extracted from the images in (A) to (D) in comparison with a phase spacectheory calculation (red curve). The arrows in (H) indicate the average Q value obtained from the direct chemical dynamics simulations. Isotropic T(θ) at low collision energies E indicating the classic mechanism via a longlived reactive complex. 11 JANUARY 2008 VOL 319 SCIENCE www.sciencemag.org Forward-scattered scattered I- (w.r.t. to incoming Cl-) indicating a fast, direct nucleophilic displacement of the I-. Additional forward-backward-scattered I-products at highest Ec indicate a new indirect “roundabout” reaction mechanism. ren Fig. 3. View of a typical trajectory for the indirect roundabout reaction mechanism at 1.9 eV that proceeds via CH3 rotation. Representation of the roundabout reaction mechanism www.sciencemag.org SCIENCE VOL 319 the ab 11 JANUAR Fig. 1. Calculated MP2(fc)/ECP Fig. 1. Calculated MP2(fc)/ECP/aug-cc 4.5 Photodissociation dynamics and laser chemistry Chemical processes of molecules excited by light are of relevance for a range of environments and applications, e.g.,: • • • • Photochemistry: study and control of chemical reactions by radiation Atmospheric chemistry Interstellar chemistry Radiation damage to biological molecules In general, the following properties are of relevance for the photodissociation dynamics of molecules: • • • • • • The dissociation energy of the molecule D0 The symmetries of the involved electronic states The absorption cross sections for photoexcitation Timescales for the dissociation event Product yields if more than one dissociation channel is open Angular distributions of the photofragments 4.5.1 Dynamics of electronically excited states A molecule which is electronically excited by (laser) radiation can undergo a range of dynamical processes: a) Laser-induced fluorescence b) Excitation to the repulsive wall of a bound state, leading to direct dissociation c) Excitation of a repulsive state, leading to direct dissociation d) Excitation to a bound state and dissociation by coupling to a repulsive state e) Excitation to a bound state and dissociation by tunneling through a barrier f) Excitation to a bound state and dissociation by internal conversion to the dissociation continuum of the ground state Processes d)-f) are referred to as predissociation. 4.5.2 Models for photodissociation Schematic representation of important limiting cases of photodissociation dynamics Potential curves of electronic states Quantum states of region of strong coupling between photofragments different electronic states a) Adiabatic model: the molecule follows a single potential energy curve during fragmentation. Applicable if the recoupling region is traversed very slowly. b) Sudden (diabatic) model: the dissociative molecular states are directly mapped onto the fragment states. Fragment state distributions are determined by the overlap of the molecular with the fragment wavefunctions and symmetry/angular momentum constraints. Applicable if the recoupling region is traversed very fast. c) Statistical model: all accessible fragment states are equally populated. Applicable in the limit of very strong coupling between electronic states. 4.5.2 Models for photodissociation Schematic representation of important limiting cases of photodissociation dynamics Potential curves of electronic states Quantum states of region of strong coupling between photofragments different electronic states d) Cartoon corresponding to a realistic situation with mixed dynamics e) Transition state model: dynamics is dominated by a single transition state. In the statistical limit in which the energy is distributed over all accessible molecular states, this situation can be described by transition state theory (see, e.g., lecture PC IV). Often a good representation of the photodissociation dynamics in polyatomics. How can we determine experimentally which situation applies ? 4.5.3 Experimental methods 4.5.3.1 Photofragment translational spectroscopy PTS is an important method to unravel the energetics and product state distribution in a photodissociation event AB + hν → A + B. The total kinetic energy release Et’ (or KER) of the photofragments A and B is given by (neglecting internal and kinetic energy of AB which are usually small in comparison): Et 0 = h D0 Eint,A Eint,B where Eint,A and Eint,B are the internal energies of the fragments A and B. Their kinetic energies are given by momentum conservation: mB mA Et,A = Et 0 Et,B = Et 0 mAB mAB Thus, the total kinetic energy release can be calculated by measuring the velocity of only one of the fragments. In general, the lighter fragment carries away most of the kinetic energy. =Et’ Example: photodissociation of ozone O3 in the Hartley bands: O3 + hν → O2 + O • • • • Process relevant for shielding the earth’s surface from cosmic UV radiation Energy released causes stratospheric temperature inversion Complicated process with different competing reaction channels Experiment: study photodissociation at 248 nm using an excimer laser Thelen et al., J. Chem. Phys. 103 (2001), 7946 Absorption spectrum Potential energy curves direct dissociation electronically excited fragments predissociation fragments in electronic ground state Wavelength / nm Thelen et al., J. Chem. Phys. 103 (2001), 7946 O2 photofragment translational spectra from O3 photodissociation at 248 nm • • The resolved peaks at low Et’ in the spectrum correspond to O2 (1Δ) photofragments in well-defined vibrational states v produced by dissociation on the 1 1B2 surface. The broad peak at high Et’ corresponds to unresolved, highly excited vibrational states in the O2 (3Σ) photofragment by predissociation on the 2 1A1 surface. direct dissociation: slow fragments low vibrational excitation direct dissociation predissociation: fast fragments high vibrational excitation fragments were detected state specifically by resonanceenhanced two-colour ionization via selected spin-rotational levels of the A 2S+ (v 0 = 0) intermediate state. The resulting 4.5.3.2 Velocity-mapped ion imaging (VMI) VMI has become a standard method for measuring both, the KER and the photofragment angular distributions at the same time. After photodissociation, one of the fragment species is ionized by REMPI. The expanding Newton sphere (the 3D velocity distribution) of the fragments is then accelerated by electric fields and crushed onto an ion detector. By a specially designed electrostatic lens system, all molecules with the same velocity vector are mapped onto the same spot on the detector. Eppink and Parker, Rev. Sci. Instrum., 68 (1997), 3477 Experimental setup for VMI Fig. 1 Schematic diagram of the experimental setup used for the velocity-map ion imaging studies. This journal is # c the Owner Societies 2007 Newton spheres for the photofragments A and B ~d of the fragment recoil vector and the electric field vector E the dissociation laser). Similarly the angular distribution is pertaining t is typical f The central slice of the Newton sphere can be reconstructed mathematically from the raw image, e.g., by an inverse Abel transformation, or experimentally by only switching on the detector when the central slice arrives (slice imaging). Raw VMI image for Reconstructed central Fig.photodissociation 2 (a) Symmetrized (b)of inverse Abel-transforme of raw NO2and slice the Newton sphere "1 S.J. Matthews et al., around the photolysis of380 NOnm cm (correspond 2 at Eexc/hc = 1056.0 PCCP 9 (2007), 5656 3 P1 fragment channels, respectively. (c) Radial distribution e The radius of the rings in a VMI image is proportional to the fragment velocity and therefore contains the same information a photofragment 5658 | as Phys. Chem. Chem. Phys.,translational 2007, 9, 5656–5663 spectrum. Fig. 2 (a) Symmetrized raw and (b) inverse Abel-transformed image of the NO 2P1/2 (v00 For initially randomly photofragment angular "1 the oriented photolysis ofmolecules, NO2 at Eexc/hcthe = 1056.0 cm"1 (corresponding to Edistribution avl/hc = 855.4 cm 3 in the laboratory frame is given by (derivation e.g.,distribution Zare, Angular P1 fragment channels, respectively. see, (c) Radial extractedMomentum): from the image in ⇤ 1 ⇥ P2 (cos ) = 12 3 cos2 1 ... 2. order Legendre polynomial 1 + P2 (cos ) with T( ) = 5658 | Phys. Chem. Chem. Phys., 2007, 9, 5656–5663 4⇥ β ... anisotropy parameter (-1≤β≤+2) Θ ... angle between the velocity vector v of the photofragments and the polarization vector ε of the photodissociation laser ε Θ R v 2 The absorption probability P µ ⇥ · ⇥ will show a maximum for molecules with the transition dipole moment μ oriented parallel to ε. In a diatomic molecule, v is always parallel to the bond vector R . Thus if ... J. Chem. Phys., Vol. 114, No. 6, 8 February 2001 • µ k R (parallel transition), then T(Θ) will be maximal for ⇥v k ⇥ (β=+2). • µ ? R (perpendicular transition), then T(Θ) will be maximal for ⇥v ? ⇥ (β=-1) β≈-1 perpendicular transition β≈2 parallel transition adapted from E. Wrede et al., J.Chem.Phys. 114 (2001), 2629 If β=0, then T(Θ) is isotropic. In this case the dissociation is slower than several rotational periods and the information about the original molecular orientation is lost. Thus, the value of β contains information about the symmetry of the excited state (which determines whether the transition is parallel or perpendicular, see section 2.2) as well as about the timescales of the dissociation process. Example: Imaging of the photodissociation of IBr: IBr + hν → I + Br E. Wrede et al., J.Chem.Phys. 114 (2001), 2629 • • • • • IBr: Hund’s case a: notation of states: 2S+1|Λ|(|Ω|) Parallel transition: Δ Ω=0, perpendicular transition: Δ Ω=±1 Photodissociation at 440 nm shows two velocity components corresponding to the formation of I+Br and I+Br* I+Br: β≈-1: indicates perpendicular transition dissociation via the A, and C states I+Br*: β≈2: indicates parallel transition dissociation via the B state J. Chem. Phys., Vol. 114, No. 6, 8 February 2001 Potential energy curves J. Chem. Phys., Vol. 114, No. 6, 8 February 2001 440 nm High resolution ion imaging study of IBr photolysis I+ B r* I+ B r Photodissociation images at 440 nm Raw image of iodine products Reconstructed Iodine atom product speed distribution central slice 4.6 Real-time studies of reactions: femtochemistry Bond-breaking processes happen on the timescale of molecular vibrations (femtoseconds, 10-15 s) real-time studies require the generation of ultrafast laser pulses Broadband fs laser excitation usually leads to the excitation of several vibrational states at the same time. The vibrational wavefunctions interfere resulting in the formation of a localised vibrational wavepacket: wavepacket vibrational wavefunctions The wavepacket oscillates back and forth on the excited-state potential energy surface with a frequency corresponding to the vibration that has been excited fs pump-probe experiments: a vibrational wavepaket is generated by a first fs laser pulse (the pump), the time evolution of the wavepacket is studied with a second fs pulse after a variable delay (the probe) localised wavepacket |Ψ|2 after fs excitation Example I: real-time observation of molecular vibrations • Step 1: create a vibrational wavepacket consisting of the v=11-15 states in the first excited electronic state of Na2 using a 50 fs laser pulse • Step 2: study the motion of the wavepacket by a probe pulse triggered after a variable time delay Na2+ signal intensity (T. Baumert et al., J. Phys. Chem. 95 (1991), 8103) Example II: transition state dynamics in NaI Zewail and co-workers, Annu. Rev. Phys. Chem. 41 (1990), 15 • Consider two lowest electronic states of NaI with potential energy curves V0(R) and V1(R) • Both states exhibit an avoided crossing at R=Rc at which they strongly interact. • A vibrational wavepacket is created in the excited state by fs laser excitation • The wavepacket oscillates in the excitedstate potential well. Every time it approaches the avoided crossing, part of the population crosses to the ground-state adiabatic potential curve on which the molecule dissociates. wavepacket motion avoided crossing • Experiment: probe wavepacket motion with a second fs laser pulse - at the inner turning point of the excited-state potential (trace b) - at large internuclear distances on the ground-state surface (trace a) wavepacket motion b a avoided crossing