Chapter 4 Updated

a)

b)

c)Chapter 4: Chemical Reaction Dynamics

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 ?

a)

b)

c)Chapter 4: Chemical Reaction Dynamics

• 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

Recommended literature:

Chapter 4: Contents

a)

b)

c)

4.1 Reaction rates and cross sections4.2 Classical scattering theory4.3 Introduction to quantum scattering theory4.4 Reactive scattering: concepts, methods and examples4.5 Photodissociation dynamics and laser chemistry4.6 Real-time studies of reactions: Femtochemistry

4.1 Reaction rates and cross sections4.1.1 Rate constants

Pro memoria: the molecularity is defined as the number of particles involved in an elementary chemical reaction:

unimolecular: A → Bbimolecular: A + B → C

The rate v for a bimolecular reaction is given by

v = −dCAdt= k(T )CACB

thermalrate coefficient

numberdensities

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].

In many cases, the temperature dependence of the thermal rate coefficient can be described in terms of the empirical Arrhenius equation:

k(T ) = Ae−EA/RT

pre-exponentialfactor

activation energy

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.

The intensity I of the beam of molecules A (molecules passing through a surface per second) is given by

I = vACAvelocity number density

A

B

I0

I1

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:

reaction crosssection

Integrate:

(*)

The bimolecular rate constant for the reaction is defined as:

−dCAdt= kCACB (**)

Thus, Eq. (*) becomes: −dCAdt= σvACACB

Comparison with (**) yields: k = σvA

which is an universal expression linking the rate constant with the cross section.

Using I=vACA and vA=dx/dt⇒1/dx=vA/dt, the left-hand side of Eq. (*) becomes:

−dI

dx= −d(CAvA)

dx= −vA

dCAdx= −dCAdt

−dI

dx= σCBI

ln(I1/I0) = σCB

Using I=vACA , the right-hand side of Eq. (*) becomes:

σCBI = σvACACB

Thermal averaging: In a molecular beam, the molecules have a well-defined kinetic energy E, thus defining a rate constant k(E). The thermal rate constant k(T) is obtained by averaging the over the Maxwell-Boltzmann distribution p(E) of all kinetic energies E:

where... Maxwell-Boltzmann kinetic-energy distribution

p(E) =√E

√π

2(kBT )

3/2

−1exp

−E

kBT

k(E) ≡ k(T ) = ∞

0p(E)k(E)dE

Inserting p(E) and using k(E)=σv=σ(2E/μ)1/2 where μ is the reduced mass we get:

k(E) = ∞

0

√E

√π

2(kBT )

3/2

−1exp

−E

kBT

σ(E)

2E/µ dE

Changing to the dimensionless energy variable (E/kBT) leads to:

k(E) =8kBT

πµ

1/2 ∞

0

E

kBT

σ(E) exp

−E

kBT

dE

kBT

average thermal velocity〈vrel〉

This expression can be formulated as

k(T ) = vrel · σ

whereσ =

∞

0

E

kBT

σ(E) exp

−E

kBT

dE

kBT

... averaged 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 )CA = σvrelCB

The number of collisions between A and B molecules per unit volume is thus:

If A=B, we get: ZVAA = (1/2)σvrelC2A

ZVAB = σvrelCACB

(Factor 1/2 for not counting collisions between the same particles twice)

4.1.3 Simple collision models for the reaction cross section

1. Hard-sphere collisions: constant reaction cross section σ0

σ0 = πd2 = π(rA + rB)

2

The thermal rate constant is:

k(T ) =

8kBT

πµ

1/2σ0

∞

0

E

kBT

exp

−E

kBT

dE

kBT

Evaluation of the integral conveniently yields 1:

k(T ) =

8kBT

πµ

1/2σ0

Molecules are treated as colliding hard spheres with radius rA and rB. Assuming every collision leads to reaction, the cross section is thus given by:

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 collisionThe reaction cross section can generally be formulated as

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:

σ =

bmax

02πb db = πb2max

σ =

bmax

0P (b)2πb db

3. Reactions with a threshold (activation) energy E0:

Reaction only occurs if E>E0:

k(T ) =

8kBT

πµ

1/2σ0

∞

E0

E

kBT

exp

−E

kBT

dE

kBT

Integration yields:

k(T ) = σ0

8kBT

πµ

1/21 +

E0kBT

exp

−E0kBT

which is of the same form as the Arrhenius equation

k(T ) = Ae−EA/RT

if EA is identified with E0 and A is interpreted as a term A(T) which varies only slowly with temperature:

A(T ) = σ0

8kBT

πµ

1/21 +

E0kBT

In this way, the Arrhenius equation can be derived within the framework of simple classical collision theory.

4.2 Classical 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

Every chemical reaction entails a collision, a scattering event. We will therefore treat chemical reactions in the framework of scattering theory.

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

A

B

y

x

rA

rB

R

Rc

c.o.m.

... coordinate vector of the centre of mass (c.o.m.)

... relative coordinate vectorR = rA − rB

Rc =mArA +mBrBmA +mB

It can easily be shown that the kinetic energy Ekin of the system is given by:

Ekin =1

2mAv

2A +1

2mBv

2B =

1

2MV 2 +

1

2µv2

where the velocities are given by vi = |ri |, ,

is the total mass and µ =mAmBmA +mB

is the reduced mass.M = mA +mB

v = | R|V = | Rc |

impactparameter

initial velocity vector

relative position vector

orientation angle

vector ofclosest approach

deflection angle

colli

sion tr

ajec

tory

Consider the collision trajectory of two structureless particles (e.g., atoms) in the COM frame:

As the total angular momentum is conserved, two coordinates suffice to describe the relative motion of the collision partners. We choose

R ... relative position vectorψ ... orientation angle of R with respect to the original velocity vector v

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.

impactparameter

initial velocity vector

relative position vector

orientation angle

vector ofclosest approach

deflection angle

colli

sion tr

ajec

tory

Angular momentum:

|L| = L = µvb

Conserved physical quantities

L = µv × R

L before collision = L after collision:

|L| = L = |µv × R| where v is the initial velocity vector

= µv · R sinΨ

Total energy:

E = Ekin + Ecent + Epotkinetic centrifugal potential

with the angular velocity Ψ = dΨ/dt = ω

L = µR2ωVL(R) ... centrifugally corrected (effective) potential

=1

2µR2 +

1

2

L2

µR2+ V (R)

=1

2µR2 +

1

2µR2Ψ2 + V (R)

⇒ E =1

2µR2 +

1

2

L2

µR2+ V (R)

Centrifugally corrected potentials

VL(R) for L3 > L2 > L1 > L0=0

VL(R) =1

2

L2

µR2+ V (R)

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 ψThe effective potential for the collision contains both, the interaction potential V(R) and the centrifugal energy:

centrifugal barrier

1. Hard-sphere collisions (a billiard game):

• For b > d:

• For b < d: where χ = π − 2ψ0

χ = 0

b/d = sinψ0 =⇒ χ = 2arccos(b/d)

The deflection function χ(b)

The angle of deflection χ depends on the impact parameter b.

Examples:

V(R

)

R

2. General potentials with repulsive and attractive parts:

repulsive, short rangepart: V(R)>0

attractive, long rangepart: V(R)<0

b* =

b/R

e

• Small b: collision dominated by repulsive forces ⇒ backward scattering

• Large b: collision dominated by attractive forces ⇒ forward scattering

χr

χg

• 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

R*

• 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:

The integral cross section σ is obtained by integration.

σ =

dσ

dΩdΩ = 2π

π

0

dσ

dΩsin θdθ

The scattering rate constant is then given by (see section 4.1.2):

k = σv

Where the cylindrical symmetry of the problem allowed us to express

dΩ = 2π sin θ dθ

in the second step.

I(Ω) =dσ

dΩ=scattered flux of molecules per unit solid angle

incident flux of molecules per unit area

dΩ

θ

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)

dσ = 2πb db

Again, because the scattering problem is cylindrically symmetric, the solid angle element dΩ can be formulated as

dΩ = 2π sin θ dθ

Hence we obtain for the differential cross section:dσ

dΩ= I(θ) =

2πb db

2π sin θ dθ=

b

sin θ(dθ/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):

dσ

dΩ= I(θ) =

b

sin θ(dθ/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:

dσ

dΩ= I(θ) =

b

sin θ(dθ/db)

deflection function θ(b)

differential cross section I(θ)

rainbow singularity

glory 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):

For inverse power law potentials

V (R) =CnRn

which describe long-range interactions between molecules the deflection function can be approximated to:

χ(b) ≈V (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 !

χ(b) = π − ∞

−∞

b

R21

1− b2

R2 −V (R)E

1/2 dR

In an experiment, the impact parameter b cannot be selected and one measures a differential cross section summed over all possible impact parameters.

i.e., χ(b) depends on the potential V(R) and the collision energy E.

4.3 Introduction to quantum scattering theory

Derivation → blackboard

4.3.1 Quantum elastic scattering

Contents:

4.3.1.1 General formulation of the scattering problem4.3.1.2 The scattering phase4.3.1.3 Scattering amplitude and scattering matrix

4.3.2 Quantum inelastic scattering

Derivation → blackboard

Contents:

4.3.2.1 Scattering Hamiltonian4.3.2.2 Angular momenta4.3.2.3 Close coupled equations

4.4 Reactive scattering: concepts, methods and examples

The topology of the Born-Oppenheimer PES determines the dynamics of a chemical reaction. Even in the absence of exact reactive-scattering calculations, important qualitative insight into chemical dynamics can be gained from inspecting the properties of the PES.

Consider the simplest polyatomic case: the reaction between an atom A and a diatom BC: A + BC → AB + C .

Reaction profile for a linear approach of the reactants

saddle point =transition state

reactants

products • 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.

4.4.1 Motion on the PES

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.

Potential energy profile along the reaction coordinate for H + H2 for different values of the approach angle γ.

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.

P. Siegbahn et al., J. Chem. Phys. 68 (1978), 2457D.G. Truhlar et al., J. Chem. Phys. 68 (1978), 2466

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:

4.4.2 Effect of vibrational and kinetic energy: Polanyi rules

Forward reactionHF + H → H2 + F

late barrier

Backward reactionH2 + F → HF + H

early barrier

HF + H

HF + H

HF + H

HF + H

H2 + F

H2 + F

H2 + F

H2 + F

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.

4.4.3 Angular momentum constraints

Angular momentum (AM) conservation for the collision dictates:

J =jBC + L =jAB + L

total AM

rotational AM collisional AM

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:

J ≈ L =j AB + L

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

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.

J ≈ L ≈ L

Conversely, for a heavy-atom transfer H + LH’ → HH’ + L we obtainJ ≈ L ≈j AB

because the product orbital AM is usually small owing to the small reduced mass μ’ of the products. Thus reactant orbital AM is converted into product rotational AM.

L = µv b

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.

Schematic of a crossed molecular beam experiment

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).

Newton diagram for the reaction A + BC → AB + C

Reconstruction of the COM angular distribution from a

CMB measurement

Notation:

vA, vBC ... velocity vectors of reactants in lab framevrel ... relative velocity vector of the reactantsΘ ... scattering angle in the lab framevCM ... velocity vector of the COMvAB ... velocity vector of product AB in lab frameuAB ... velocity vector of product AB in COM frameθ ... scattering angle of products in COM frame

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

The reconstructed COM product flux distribution ICM(θ,u) can be decomposed into two different components:

ICM(θ, u) = T (θ)× P (Et)

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.

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), 355M. Alagia et al., Science 273 (1996), 1519

Example II: K + Br2 → KBr + Br

Reaction initiated by long-range electron transfer from K to Br2 at a crossing between potential curves corresponding to the neutral and ionic forms of the reactants (harpoon mechanism). The temporary ion pair is strongly accelerated towards one another by the Coulomb interaction ultimately leading to the formation of the products. Large impact parameters, forward scattering.

1002 BIRELY, HERM, WILSON, AND HERSCHBACH

TABLE III. Total and reactive scattering cross sections.'

System F- e K+Br2 1.49 850

Rb+Br2 1.07 890

CS+ Br2 0.85 1040

K+I2 1.58 1300

Cs+I2 1.02 1590

B The mean elastic collision energy iff is given in kilocalories per mole. the van der Waals force constant e in 10-60 erg/em' and the cross sections

same values for the most probable E' and only slightly narrower angular distributions.29

Total Reaction Cross Section

A rough estimate of Qr> 100 12 for the total reaction cross section of M +X2 systems was obtained previ-ously5a by integrating the LAB angular distribution of MX product. The result gives only a lower bound, as it does not include the out-of-plane scattering beyond the range of the detector. The approximate kinematic analysis offers a way to circumvent this difficulty by integrating the c.m. angular distribution, since

1" (dQr) . Qr = 211' - sm()d() o dw abs

(2)

by virtue of the cylindrical symmetry about the initial relative velocity vector. The absolute normalization of the differential reactive scattering cross section,

(dQr/ d<,;) abs =:n( dQr/ dw )rel, (3)

can be determined by comparison with the elastic scat-tering. The results obtained from three different pro-cedures are given in Table III.

Method A

Since the relative intensity scales for the reactive and elastic scattering are practically the same,3!

(dQe/ dw) abs . ( dQe/ dw) reI

(4)

The elastic scattering pattern at narrow angles is as-sumed to be negligibly perturbed by reaction. The absolute intensity thus can be calibrated by use of the small-angle scattering formula for a VCr) = -e/r6 van der Waals interaction,32

31 This is ensured by the data reduction procedure used (rela-tive intensity defined by ratio of signal to parent-beam attenua-tion; Pt data normalized to W data), provided that: (1) the de-tection efficiency for M and MX on W is essentially the same; and (2) the approximate LAB--->c.m. transformation does not seriously distort the intensity of elastic relative to reactive scat-tering.

32 See, for example, E. A. Mason, J. T. Vanderslice, and C. J. G. Raw, J. Chern. Phys. 40,2153 (1964).

Q, Q,eff Qr(A) Q,(B) Qr(C)

890 510 220 260 200

1090 560 410 360 330

1280 600 370 380 310

1060 600 220 270 220

1510 700 210 290 240

in square angstroms. 0, is calculated from Eq. (9) with V corresponding to E and o,eff from Eq. (13) with E =E.

The comparison was made at ()= 10°. For the collision energy we used the mean value obtained by averaging Eq. (5) over the energy distribution corresponding to an immobile target,19

E= )-3= 1.36(fJ./ma)kTa • (6)

The force constants e were calculated from the Slater-Kirkwood approximation,33 with N(M) = 1 and N(X2) = 14 for the effective number of electrons. The polarizability values used were (in cubic angstroms) : 36.5, 40.0, 52.5 for K, Rb, Cs, respectively34; 6.2 and 9.7 for Br2 and 12. The halogen polarizabili ties were estimated from the HX and H2 values35 via

o K + Br, G Rb + Br, • Cs + Br,

o K + I, • Cs + I,

,

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. ------

33 It is expected that dQ,/dw and Q, as predicted from the S--K approximation should be correct within 20% (including allow-ance 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·cm6 for K + Br2, in good agreement with the S--K result of Table III.

34 A. Salop, E. Pollack, and B. Bederson, Phys. Rev. 124, 1431 (1961); B. Bederson and E. J. Robinson, Advan. Chern. Phys. 10, 1 (1966).

a5 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory oj Gases and Liquids (John Wiley & Sons, Inc., New York, 1964), p. 950.

Downloaded 11 May 2011 to 131.152.105.82. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

J. H. Birley et al., J. Chem. Phys. 47 (1967), 993D. Hershbach, Angew. Chemie Int. Ed. 26 (2987), 1221

curve crossing

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≈j’, i.e., the products are rotationally excited, memory of the original orientation is completely lost and the angular distribution is isotropic (i.e., constant).

If L≈L’, i.e., the collisional angular momentum and thus the plane of collision is conserved, the products show a distinct forward-backward scattered distribution:

dσ

dΩ=

dσ

2π sin θdθ∝1

sin θ

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), 355M. 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

indirect mechanism

rotationally excited products:isotropic angular distribution

backward scattering

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.

Surface crossing in a charge-transfer mediated reaction

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.

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 section 2.3) 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:

H = H0 + V

φ = c1φ(0)1 + c2φ

(0)2

with mixing coefficients c1 and c2.

• The coupled states can be expressed as a superposition of the uncoupled states:

• 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 2.2). If these sates are coupled by an additional weak coupling operator V, the total Hamiltonian is given by

where are the matrix elements of Ĥ in the diabatic basis.

Hi j = φ(0)i |H|φ(0)j = φ

(0)i |H0 + V |φ

(0)j

c1(H11 − U) + c2H12 = 0c1H12 + c2(H22 − U) = 0

• 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 section 2.3) we get a set of secular equations for c1 and c2:

• Note that(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).

with the associated eigenfunctions Φ+ and Φ- (the so-called adiabatic states).

• The solutions (energies of the coupled electronic states) are:

c1(U1 − U) + c2V12 = 0c1V12 + c2(U2 − U) = 0

U±(R) =12(U1(R)− U2(R))±

12

(U1(R)− U2(R))2 + 4V12(R)2

• The secular equations thus become:

• Note also that both, the BO energies Ui and couplings V12 generally depend on the reaction coordinate R.

At the crossing point, the separation between the adiabatic states is given by ΔU=2V12.

adiabatic states

diabatic states

In a diatomic molecule, states of the same symmetry can never cross because of non-adiabatic couplings. All such crossings are always avoided (non-crossing rule). This restriction is relaxed in polyatomics.

The adiabatic states and the associated PES are the eigenfunctions of the full Hamiltonian Ĥ and can be obtained from ab-initio calculations.

Diabatic and adiabatic states at a crossing point

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).Conical intersection between two

electronic states in two dimensions

adiabatic passage

diabatic passage

Q1

Q2

The coupling repels the states around the crossing point and leads to an avoided crossing.

R

U(R)

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.

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.

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:

where v ... velocity along reaction coordinateU1(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

Pdia = exp

−2πV 212

hv ∂(U2(R)−U1(R))∂R

Re

ac

tio

n p

rob

ab

ility

Re

ac

tio

n t

ime

de

lay

4.4.4.4 Reaction resonances

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.

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.

En

erg

y V

/ ev

collision energy Ec

Reactants Products

• 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.

There are two important types of reactive resonance effects:

Example for a dynamic situation leading to a Feshbach resonance

Collision energy Ec

J.N. Milstein et al., New J. Phys. 5 (2003) 52

Collision energy Ec

Example for a shape resonance

Example I: F + H2 ( j=0) → HF (v’) + H

F + H2 shows a strong forward-scattering peak in the angular distribution of the HF (v’=3) and HF (v’=2) product around Ec=2.18 kJ mol-1. At this energy, Feshbach resonances with bound states in the H...HF (v’=3) van-der-Waals potential well exist which enhance the reaction probability and modify the product angular distributions. The HF (v’=2) product is then formed by strong vibrational (anharmonic) coupling between the v’=3 and v’=2 states in H...HF.

Vibrationally adiabatic potential curves for vibrationally excited

states v’ in the HF product

Potential well ofH...HF van-der-Waals

complex

Bound van-der-Waals states

M. Qiu et al., Science 311 (2006), 1440X. Wang et al., PNAS 105 (2008), 6227

Example II: Cl + HD (v=1, j=0) → HCl + D / DCl + H

Weck and Balakrishnan, Int. Rev. Phys. Chem. 283 (25), 2006

0 5 10 15 20 25 30 35

R (au)

!0.02

!0.01

0

0.01

0.02

Wav

e fu

nctio

n (a

rb. u

nits

)

!0.02

!0.01

0

0.01

0.02

Pote

ntia

l ene

rgy

(eV

)

BEadiabatic potential

Figure 7. Wave functions of the quasibound levels B and E supported by the adiabatic potential shownin figure 6 as functions of the atom–molecule separation. Amplitudes of the wave functions have beenreduced by a factor of 10 for practical plotting reasons.

10!8 10!7 10!6 10!5 10!4 10!3 10!2 10!1

Incident kinetic energy (eV)

10!6

10!5

10!4

10!3

10!2

10!1

100

101

102

Cro

ss s

ectio

n (1

0!16 c

m2 )

Cl+HD - nonreactiveDCl+H - reactiveHCl+D - reactive

Figure 8. The same as in figure 5 but plotted as a function of the incident kinetic energy to illustrate thelow-temperature behaviour of the cross-sections.

296 P. F. Weck and N. Balakrishnan

accessible from scattering in the v ! 1, j ! 0 channel, quasibound levels B to E undergovibrational prereaction or predissociation and are responsible for the resonancesdepicted in figure 5. Wave functions of the quasibound states B and E, the deepestand the least bound ones shown in figure 6 are presented in figure 7 as functionsof R. Although the wave function of the quasibound state E extends far awayfrom the transition state region of the reaction, it preferentially undergoes prereactionleading to HCl"D product rather than predissociation to yield Cl"HD(v! 0)product.

Figure 8 shows cross-sections calculated by Balakrishnan [37] for reactive andnon-reactive scattering in Cl"HD#v ! 1, j ! 0$ collisions as functions of the incidentkinetic energy ranging from the ultracold to near thermal limits. The DCl/HClbranching ratio reaches a limiting value of 4:0% 10&3 in the Wigner regime.Prereaction of the Cl ' ' 'H&D van der Waals complexes in the initial channel leadingto HCl formation is more favorable than predissociation or prereaction resulting in DClproduction due to the less efficient tunnelling of the D atom when tunnelling becomesthe dominant reaction mechanism. The non-reactive cross-section becomes muchlarger than the cross-section for HCl formation beyond Ekin ! 10&2 eV as rotationalexcitation to v ! 1, j ! 1 level becomes energetically favorable. The above results forthe Cl"HD reaction demonstrate that regions of the potential energy surfaces faraway from the transition state region may have significant effect on reactivity even atlow temperatures.

Currently, there is considerable interest in creating dense samples of cold andultracold polar molecules. The buffer gas cooling method and the stark-decelerator

5 10 15 20R (au)

0.67

0.68

0.69

0.7

Ene

rgy

(eV

) v=1, j=0

v=1, j=1

A

B

CD

E

Figure 6. Adiabatic potential energy curves of the Cl"HD system correlating to the HD#v ! 1, j ! 0$ andHD#v ! 1, j ! 1$ levels as functions of the atom–molecule separation, R. Quasibound levels responsiblefor the resonances observed in figure 5 are labelled by B, C, D and E.

Long-range interactions in chemical reactions 295

reactive resonances

Collision energy (eV)

B C D

E

collision energy

shallow potential-energy wellscaused by long-range (van-der-Waals) interactions

rotationally adiabatic potential curves

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.

4.4.5 A case study: the SN2 reaction Cl- + CH3I → I- + CH3Cl

J. Mikosch et al., Science 319 (2008), 184

SN2 nucleophilic substitution reactions X- + R-Y → Y- + R-X show a characteristic double-well potential-energy profile along the reaction coordinate.

Until recently, many details of the SN2 dy-namics of bimolecular anion-molecule reactionscould only be obtained from chemical dynamicssimulations. However, with recent experimentaladvances (22), insight into the reaction dynamics

may be obtained frommeasurements of correlatedangle- and energy-differential cross sections.Specifically, the probabilities for energy redistri-bution within the ion-dipole complexes, theirdependences on initial quantum states, the branch-

ing into different product quantum states, and therole of tunneling through the central barrier poseopen questions to be probed experimentally.

We report kinematically complete reactivescattering experiments of the anion-moleculeSN2 reaction Cl! + CH3I ! CH3Cl + I! (Fig. 1)with use of our ion-molecule crossed beam imag-ing spectrometer (22). In this way, we extendedthe successful crossed beam imaging experimentsof neutrals (23) to ionic reactions. These single-collision experimentsmeasure directly the velocityvector of the product anion, which reveals theenergy- and angle-differential reaction cross sec-tion. By using reactants with well-defined relativekinetic energy and momentum, we can determineenergy transfer during the reaction, which yieldsthe fraction of total available energy partitioned tointernal modes of themolecular product. For com-parison with the experimental results, we haveperformed high-level trajectory simulations.

In the experiment, we produced slow pulsesof Cl! anions with a tunable well-defined kineticenergy between 0.2 and 5 eV in a compactelectron-impact supersonic expansion ion source(22). The ion pulses crossed a supersonic neutraljet of CH3I seeded in helium, whereby a few ofthe Cl! anions induced nucleophilic substitutionand liberated I! anions. The interaction region ofthe crossed beam experiment was placed in apulsed-field velocity map imaging spectrometer,which maps the velocity of the I! product anion

Fig. 1. Calculated MP2(fc)/ECP/aug-cc-pVDZ Born-Oppenheimer potential energy along the reactioncoordinate g = RC!I ! RC!Cl for the SN2 reaction Cl! + CH3I and obtained stationary points. The reportedenergies do not include zero-point energies. Values in brackets are from (28).

A B C D

E F G H

Fig. 2. (A to D) Center-of-mass images of the I! reaction product velocityfrom the reaction of Cl! with CH3I at four different relative collision energies.The image intensity is proportional to [(d3s)/(dvx dvy dvz)]: Isotropic scat-tering results in a homogeneous ion distribution on the detector. (E to H)

The energy transfer distributions extracted from the images in (A) to (D) incomparison with a phase space theory calculation (red curve). The arrows in(H) indicate the average Q value obtained from the direct chemical dynamicssimulations.

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Reaction profile for Cl- + CH3I → I- + CH3Cl

transition state

reaction complex inentrance channel

reaction complex inexit channel

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.

For the model reaction Cl- + CH3I → I- + CH3Cl, one would expect that the dynamics is dominated by the formation of a long-lived reaction complex in the exit channel.

If the lifetime of the reaction complex is longer than several rotational periods, an isotropic product angular distribution is expected.

onto a position-sensitive detector. With the use ofslice imaging (23) implemented by activating thedetector only during a short time window, weimaged only scattering events for which the velo-city vectors of the products lie within the planedefined by the reactant velocities. Because of thecylindrical symmetry of the scattering cross section,this procedure yielded the velocity magnitude-and angle-differential cross sections directly, with-out resorting to Abel-inversion-type algorithms.The neutral product does not need to be detectedbecause its properties can be inferred from con-servation of energy and momentum (24).

The top row of Fig. 2 shows maps of the I!

product ion velocities from the Cl! + CH3I !CH3Cl + I! reaction at four different relative col-lision energies between Erel = 0.39 eV and Erel =1.90 eV, which were chosen because they span thedistinct reaction dynamics observed in this energyrange. The only data processing applied to the ionimpact position on the detector is a linear conver-sion from position to ion speed and a transforma-tion into the center of mass frame. Consequently,the velocity vectors of the two reactants, the Cl!

anion and the CH3I neutral, line up horizontallyand point in opposite directions, indicated by thearrows in Fig. 2. Each velocity image represents ahistogram summed over 105 to 106 scatteringevents. The total energy available to the reactionproducts is given by the relative translationalenergy, Erel, of the reactants plus the exoergicity,0.55 eV, of the reaction (Fig. 1). I! products reachthe highest velocity when all the available energyis converted to translational energy. The outer-most circle in Fig. 2 represents this kinematiccutoff for the velocity distribution. The other con-centric rings display spheres of the same trans-lational energy and hence also the same internalproduct excitation, spaced at 0.5-eV intervals.

The images in Fig. 2 reveal many details ofthe reaction dynamics. For the lowest relativecollision energy of 0.39 eV, there is an isotropicdistribution of product velocities centered aroundzero with all scattering angles equally probable.This pattern points to the traditional reactionmechanism (8) mediated by a collision complexwhose lifetime is long compared to the time scaleof its rotation. The complex-mediated mecha-nism is accompanied by a velocity distributionthat drops to zero far before the kinematic cutoffis reached, as can be inferred from the positionof the outermost ring in the image. Thus, thelargest fraction of the available energy is parti-tioned to internal rovibrational energy of theCH3Cl product.

A distinctly different reaction mechanism be-comes dominant at the higher relative collisionenergy of 0.76 eV (Fig. 2B): The I! product isback-scattered into a small cone of scatteringangles. This pattern indicates that direct nucleo-philic displacement dominates. The Cl! reactantattacks themethyl iodidemolecule at the concavecenter of the CH3 umbrella and thereby drives theI! product away on the opposite side. The directmechanism leads to product ion velocities closeto the kinematic cutoff. In addition, part of theproduct flux is found at small product velocitieswith an almost isotropic angular distribution, in-dicating that for some of the collisions there is asignificant probability of forming a long-livedcomplex.

At a collision energy of 1.07 eV (Fig. 2C), thecomplex-mediated reaction channel is notobserved anymore. The reaction proceeds almostexclusively by the direct mechanism, with a similarvelocity and a slightly narrower angular distribu-tion relative to the 0.76-eV case. At an evenhigher collision energy of 1.90 eV, the domi-

nating backward scattering pattern of the I! ionspreads over an increased range of scatteringangles. The inserted rings demonstrate a con-comitant broadening of the velocity distribution.In addition, a new feature appears that consists oftwo distinct low-velocity peaks symmetric in theforward and the backward directions with respectto the center of mass. As detailed below, thesepeaks represent reactions that occur via a round-about mechanism. At relative energies above 2 eV,new dissociative channels open and influence thescattering dynamics, and so we restricted the cur-rent presentation to collision energies up to 1.9 eV.

For a quantitative analysis, we calculated theenergy transfer Q = Ekin,final ! Ekin,initial for thereaction events. For the lowest collision energyof 0.39 eV, Fig. 2E shows that the observeddistribution vanishes for Q values far belowthe kinematic cutoff at +0.55 eV. A theoreticalphase-space calculation (red line in Fig. 2E) (25),which assumes that the available energy is dis-tributed statistically among all degrees of free-dom of the reaction products, shows excellentagreement with the data after convolution withthe experimental resolution stemming from thevelocity spread of the reactant beams. The ob-servation of statistical energy partitioning at thisfinite collision energy is unexpected, given thereported nonstatistical unimolecular decomposi-tion of metastable Cl!·CH3I SN2 complexes(13). We found that 84% of the total availableenergy is trapped in internal excitation of theCH3Cl reaction product, which amounts to Eint =0.79 eV. At all the higher relative collisionenergies (Fig. 2, F to H), the phase space modelcannot reproduce the observed dynamics. Here,the Q value distribution peaks near its maximumvalue of +0.55 eV. The mean internal excitationin absolute and in relative numbers is given by0.5 eV (40%), 0.45 eV (25%), and 0.95 eV(40%). At 1.07 eV relative collision energy, aminimum is found both in the absolute and therelative amount of internal excitation, which is asign of subtle changes in the translation-vibrationcoupling during the reaction.

To complement the above experimental studyof the Cl! + CH3I reaction dynamics, we per-formed a trajectory simulation at the MP2(fc)/ECP/aug-cc-pVDZ (26) level of theory by acomputational approach directly using this theory(27). As shown in Fig. 1, this theory gives ener-gies for the reaction’s stationary points in goodagreement with previous values based on ratecoefficient measurements (28). These simulationsare computationally expensive and only practicalat the highest collision energies where both theencounter time is short and the reaction proba-bility appreciable. Here, we report results for the1.9-eV collision energy and CH3I rotational andvibrational temperatures of 75 K and 360 K,respectively, which are the approximate experi-mental conditions.

Although the reaction has no overall barrier,the simulations show a quite low reaction prob-ability at 1.9 eV, decreasing from 0.065, 0.05,

Fig. 3. View of a typical trajectory for the indirect roundabout reaction mechanism at 1.9 eV thatproceeds via CH3 rotation.

www.sciencemag.org SCIENCE VOL 319 11 JANUARY 2008 185

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Angular product distributions for I- at different collision energies Erel≡Ec

Representation of the roundabout reaction mechanism

In the gas-phase crossed-molecular beam scattering experiment, three types of product angular distribution T(θ) are observed indicating three different reaction mechanisms:







A B C D

E F G H




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• Isotropic T(θ) at low collision energies Ec indicating the classic mechanism via a long-lived reactive complex.

• 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.

Isotropic distribution: complex-mediated, classical mechanism

Forward-scattered products:direct substitution mechanism

Forward-backward-scattered products:roundabout mechanism

CH3 - I







A B C D

E F G H




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A B C D

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Cl-

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) Laser-induced fluorescenceb) Excitation to the repulsive wall of a bound state, leading to direct dissociationc) Excitation of a repulsive state, leading to direct dissociationd) Excitation to a bound state and dissociation by coupling to a repulsive statee) Excitation to a bound state and dissociation by tunneling through a barrierf) 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.

A molecule which is electronically excited by (laser) radiation can undergo a range of dynamical processes:

4.5.2 Models for photodissociation

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.

Schematic representation of important limiting cases of photodissociation dynamics

Quantum states of photofragments

region of strong coupling between different electronic states

Potential curves of electronic

states

Quantum states of photofragments

region of strong coupling between different electronic states

Potential curves of electronic

states

4.5.2 Models for photodissociation

Schematic representation of important limiting cases of photodissociation dynamics

d) Cartoon corresponding to a realistic situation with mixed dynamicse) 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 = hν −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:

Et,A =mBmAB

Et Et,B =mAmAB

Et

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 laserThelen et al., J. Chem. Phys. 103 (2001), 7946

Absorption spectrum Potential energy curves

Wavelength / nm

direct dissociation

predisso-ciation

Thelen et al., J. Chem. Phys. 103 (2001), 7946

electronicallyexcited fragments

fragmentsin electronicground state

O2 photofragment translational spectra from O3 photodissociation at 248 nm

direct dissociation:slow fragments

low vibrational excitation

predissociation:fast fragments

high vibrational excitation

direct dissociation

• 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.

spin–orbit channels at the three wavelengths 371.7, 354.7 and338.9 nm. A marked variation of b as a function of theavailable energy Eavl was observed at all three wavelengths,which was explained within the framework of a classical modelfor non-axial recoil of the photofragments based on energyand angular-momentum conservation. In our previous inves-tigation,8 we studied the angular photofragment distributionpertaining to the NO (J00 = 17/2) + O (3P2) fragment channelin the near-threshold region 0 o Eavl/hc o 400 cm!1 using astate-selective Rydberg-tagging technique to image the photo-fragments. In contrast to the results obtained at higherphotolysis energies, the value of b was found to fluctuaterapidly as a function of Eavl.

8 These marked variations wereinterpreted in terms of an extension of the model of Demya-nenko et al. as the manifestation of fluctuating lifetimes of thesets of resonances initially excited in the parent molecule.In the present study, we present a more extensive data set,

and focus on the state-specific aspects of the variation of b inthe near-threshold photodissociation of NO2. The photo-fragment angular distributions of the NO2

2P1/2 (v00 = 0,J00 = 3/2, 11/2, 21/2) + O (3P2,1) fragment channels wereinvestigated up to excess energies Eexc/hc r 1000 cm!1 usingvelocity-map ion imaging. A pronounced state-specificity ofthe fluctuations of b was observed which is discussed in termsof the characteristics of the dissociation mechanism and life-time e!ects.

II. Experiment

A. Apparatus

A schematic representation of the experimental setup is pre-sented in Fig. 1. NO2 was excited in a single-photon transitionto the region just above the NO (2P1/2,3/2) + O (3P2,1,0)dissociation threshold (D0/hc = 25 128.56 cm!1).1 The NOfragments were detected state specifically by resonance-enhanced two-colour ionization via selected spin-rotationallevels of the A 2S+ (v0 = 0) intermediate state. The resulting

ionic fragments were then velocity-mapped onto a microchan-nel plate detector and their spatial distribution captured by acharge-coupled device (CCD) camera.The experiments were performed using a pulsed molecular

beam of 5% NO2 seeded in argon at a backing pressure of2.5 bar. In order to reduce the concentration of trace NO inthe sample, the NO2 was first purified by bubbling oxygen gasthrough liquid NO2 at about 0 1C for approximately 1 h. Thepartial conversion of NO2 to its dimer N2O4 under theseexpansion conditions is not expected to a!ect the presentresults because of the very small absorption cross section ofN2O4 below 380 nm.24 The gas mixture was stored in acorrosion-resistant aluminium mixing bottle and transferredinto the measurement chamber in a pulsed supersonic expan-sion operating at 10 Hz in synchrony with the lasers. Therotational temperature of the beam was determined to beTrot = 8–10 K from a REMPI spectrum of the cold NOimpurity. After passing through a skimmer, the gas beamcrossed the laser beams at right angles in the photoexcitationregion. The experimental chamber was maintained at a pres-sure of about 1 " 10!6 mbar during signal acquisition.

For the dissociation step, the idler beam of an injection-seeded Nd:YAG-pumped optical parametric oscillator (Quan-ta Ray GCR 190/MOPO 730, 10 Hz, 8 ns pulse) was frequencydoubled (Quanta Ray FDO, 361 BBO, B3 mJ per pulse) toprovide laser radiation in the wavelength range 380–400 nm.The horizontally polarized beam was focused into themolecular beam with a fused silica lens (f = 30 cm). Theresulting NO fragments were excited to selected spin-rota-tional levels of the A 2S+ (v0 = 0) state using the frequency-tripled output of a Nd:YAG-pumped dye laser (Spectron4000G/SL800, 10 Hz, 8 ns, LDS dye, B226 nm, B200 mJpulse!1). The excited NO molecules were ionized using thedoubled output of a second Nd:YAG-pumped dye laser(Spectron 4000G/SL800, DCM dye, B345 nm, 1 mJ). Theuse of two-colour (1 + 10) ionization, in which the secondphoton is tuned close to the ionization threshold, improves thesignal levels in these experiments by more than an order ofmagnitude compared to one-colour (1 + 1) ionization. Theionization laser beam was focused into the reaction chambertogether with the photolysis laser beam, whereas the 226 nmbeam was left unfocused. The probe lasers were delayed byabout 20 ns and 40 ns with respect to the photolysis laser. Allof the lasers were calibrated with a Burleigh wavemeter (WA4500). All laser polarisation vectors were chosen to lie per-pendicularly to the molecular beam axis, as indicated in Fig. 1.A number of images were sampled with the polarization vectorof probe laser 1 perpendicular to the plane of the detector,which had no e!ect on the observed images, indicating negli-gible alignment in the NO fragments. Indeed, angular mo-mentum polarization e!ects of the NO photofragments areexpected to be unimportant in the present study becausetransitions from the Q21/R11 branch were used in the photo-excitation scheme.The velocity-map ion-imaging setup consists of three stain-

less steel electrodes of 140 mm diameter and 1.0 mm thicknessspaced at 15 mm intervals. The extractor and ground electro-des have apertures of 15 mm and the skimmed supersonic jetenters the ion lens through an aperture of 2 mm diameter in

Fig. 1 Schematic diagram of the experimental setup used for the

velocity-map ion imaging studies.

This journal is #c the Owner Societies 2007 Phys. Chem. Chem. Phys., 2007, 9, 5656–5663 | 5657

4.5.3.2 Velocity-mapped ion imaging (VMI)

Eppink and Parker, Rev. Sci. Instrum., 68 (1997), 3477

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.

Experimental setup for VMI

Newton spheres for the photofragments A and B

the repeller plate, which is situated E50 mm from the pulsedvalve. The lasers intersect the molecular beam midwaybetween the repeller and extractor plates (see Fig. 1).In velocity-map ion-imaging experiments, all ion trajectories

with the same initial velocity are mapped onto the same pointon the detector irrespective of where they were formed in theionization volume.25 For the 500 mm field-free drift region ofthis experiment, optimal focusing was attained for VE/VR C0.8, where VE and VR are the potentials applied to theextractor and repeller plates, respectively. All images wererecorded with VR = +500 V. The field applied in this regionaccelerates the ions towards the detector whilst simultaneouslycompressing the ion cloud in the time-of-flight dimension.The detector consists of a dual microchannel plate (MCP)/

phosphor-screen assembly (Burle, 40 mm diameter). Reso-nance-enhanced multiphoton ionization (REMPI) and photo-fragment multiphoton ionization (PHOMPI) spectra wererecorded using a photomultiplier tube (PMT) to collect thefluorescence from the phosphor screen. The PMT output wasamplified and recorded on a PC using Labview.

B. Image acquisition

Position-sensitive information was recorded using a charge-coupled device (CCD) camera (Proxitronic HR0, 764 ! 576pixels, 50 Hz repetition rate). Image acquisition was controlledby a PC using Visilog (Noesis) software. Fig. 2(a) shows asymmetrized raw image compared to its inverse Abel trans-form (b) along with its corresponding radial distribution (c).The image was recorded at an excess energy of Eexc/hc =1056 cm"1 above the dissociation threshold with the ionizationlaser tuned to the NO Q21/R11(J

00 = 21/2) transition. Tworings corresponding to the production of the 3P2 and 3P1

spin–orbit components of the oxygen atom fragment areobserved. The central slice of the photofragment distributionis reconstructed from the raw images using the Basex Abelinversion package.26 All images were centered and symme-trized prior to Abel inversion. The velocity distribution iscalculated by integrating over the angle y (the angle betweenthe fragment recoil vector and the electric field vector ~Ed ofthe dissociation laser). Similarly the angular distribution is

obtained by integrating radially over the relevant fragmentshell. The anisotropic photofragment angular distribution I(y)is then fitted to the expression23

I#y$ % s4p

#1& bP2#cos y$$ #1$

where I(y) is the ion flux at polar angle y, s the dissociationcross section, b the anisotropy parameter and P2 the secondLegendre polynomial, P2#x$ % 1

2 #3x2 " 1$.

III. Results

A. Photofragment multiphoton ionization (PHOMPI) spectra

In the present work, the energy-dependent angular distribu-tion of the J00 = 3/2, 11/2 and 21/2 NO photofragments wasstudied. In each case, photoexcitation of the NO fragment wascarried out via the relevant Q21/R11 transitions27 which werenot overlapped by lines from other spectral branches at therange of excess energies studied. Using the (1 + 10) REMPIdetection scheme, the relative yield of rovibronically state-selected NO molecules as a function of the photolysis energywas determined, giving rise to a PHOMPI spectrum. Spectrawere recorded for each of the three channels studied over theenergy range Eavl/hc = 0–300 cm"1. As an example, thePHOMPI spectra of the J00 = 3/2 and 21/2 photofragmentsare displayed in Fig. 3. The spectra are highly structured andshow several sharp features which is indicative of an excitationinto a regime of overlapping resonances in the parent moleculeas has been discussed in previous studies.3,4,28 From theobserved onset of the PHOMPI signal at vPHOT = 25 327.1('2) cm"1, the dissociation threshold for NO2 - NO + Owas determined to be 25 126.5 ('2) cm"1 by subtracting theterm energy of the NO 2P1/2 J

00 = 21/2 level. This value is inagreement with the most recent result from the literature ofD0/hc = 25 128.56 cm"1.1

The PHOMPI spectra corresponding to the NO J00 = 3/2and 21/2 channels show a similar pattern of sharp features, butno exact correspondence between the lines in the spectrapertaining to the di!erent channels was found. This behaviouris typical for an excitation into a regime of overlapping

Fig. 2 (a) Symmetrized raw and (b) inverse Abel-transformed image of the NO 2P1/2 (v00 = 0, J00 = 21/2) photofragment distribution produced in

the photolysis of NO2 at Eexc/hc = 1056.0 cm"1 (corresponding to Eavl/hc = 855.4 cm"1). The outer and inner rings correspond to the O 3P2 and3P1 fragment channels, respectively. (c) Radial distribution extracted from the image in (b).

5658 | Phys. Chem. Chem. Phys., 2007, 9, 5656–5663 This journal is (c the Owner Societies 2007

the repeller plate, which is situated E50 mm from the pulsedvalve. The lasers intersect the molecular beam midwaybetween the repeller and extractor plates (see Fig. 1).In velocity-map ion-imaging experiments, all ion trajectories

with the same initial velocity are mapped onto the same pointon the detector irrespective of where they were formed in theionization volume.25 For the 500 mm field-free drift region ofthis experiment, optimal focusing was attained for VE/VR C0.8, where VE and VR are the potentials applied to theextractor and repeller plates, respectively. All images wererecorded with VR = +500 V. The field applied in this regionaccelerates the ions towards the detector whilst simultaneouslycompressing the ion cloud in the time-of-flight dimension.The detector consists of a dual microchannel plate (MCP)/

phosphor-screen assembly (Burle, 40 mm diameter). Reso-nance-enhanced multiphoton ionization (REMPI) and photo-fragment multiphoton ionization (PHOMPI) spectra wererecorded using a photomultiplier tube (PMT) to collect thefluorescence from the phosphor screen. The PMT output wasamplified and recorded on a PC using Labview.

B. Image acquisition

Position-sensitive information was recorded using a charge-coupled device (CCD) camera (Proxitronic HR0, 764 ! 576pixels, 50 Hz repetition rate). Image acquisition was controlledby a PC using Visilog (Noesis) software. Fig. 2(a) shows asymmetrized raw image compared to its inverse Abel trans-form (b) along with its corresponding radial distribution (c).The image was recorded at an excess energy of Eexc/hc =1056 cm"1 above the dissociation threshold with the ionizationlaser tuned to the NO Q21/R11(J

00 = 21/2) transition. Tworings corresponding to the production of the 3P2 and 3P1

spin–orbit components of the oxygen atom fragment areobserved. The central slice of the photofragment distributionis reconstructed from the raw images using the Basex Abelinversion package.26 All images were centered and symme-trized prior to Abel inversion. The velocity distribution iscalculated by integrating over the angle y (the angle betweenthe fragment recoil vector and the electric field vector ~Ed ofthe dissociation laser). Similarly the angular distribution is

obtained by integrating radially over the relevant fragmentshell. The anisotropic photofragment angular distribution I(y)is then fitted to the expression23

I#y$ % s4p

#1& bP2#cos y$$ #1$

where I(y) is the ion flux at polar angle y, s the dissociationcross section, b the anisotropy parameter and P2 the secondLegendre polynomial, P2#x$ % 1

2 #3x2 " 1$.

III. Results

A. Photofragment multiphoton ionization (PHOMPI) spectra

In the present work, the energy-dependent angular distribu-tion of the J00 = 3/2, 11/2 and 21/2 NO photofragments wasstudied. In each case, photoexcitation of the NO fragment wascarried out via the relevant Q21/R11 transitions27 which werenot overlapped by lines from other spectral branches at therange of excess energies studied. Using the (1 + 10) REMPIdetection scheme, the relative yield of rovibronically state-selected NO molecules as a function of the photolysis energywas determined, giving rise to a PHOMPI spectrum. Spectrawere recorded for each of the three channels studied over theenergy range Eavl/hc = 0–300 cm"1. As an example, thePHOMPI spectra of the J00 = 3/2 and 21/2 photofragmentsare displayed in Fig. 3. The spectra are highly structured andshow several sharp features which is indicative of an excitationinto a regime of overlapping resonances in the parent moleculeas has been discussed in previous studies.3,4,28 From theobserved onset of the PHOMPI signal at vPHOT = 25 327.1('2) cm"1, the dissociation threshold for NO2 - NO + Owas determined to be 25 126.5 ('2) cm"1 by subtracting theterm energy of the NO 2P1/2 J

00 = 21/2 level. This value is inagreement with the most recent result from the literature ofD0/hc = 25 128.56 cm"1.1

The PHOMPI spectra corresponding to the NO J00 = 3/2and 21/2 channels show a similar pattern of sharp features, butno exact correspondence between the lines in the spectrapertaining to the di!erent channels was found. This behaviouris typical for an excitation into a regime of overlapping

Fig. 2 (a) Symmetrized raw and (b) inverse Abel-transformed image of the NO 2P1/2 (v00 = 0, J00 = 21/2) photofragment distribution produced in

the photolysis of NO2 at Eexc/hc = 1056.0 cm"1 (corresponding to Eavl/hc = 855.4 cm"1). The outer and inner rings correspond to the O 3P2 and3P1 fragment channels, respectively. (c) Radial distribution extracted from the image in (b).

5658 | Phys. Chem. Chem. Phys., 2007, 9, 5656–5663 This journal is (c the Owner Societies 2007

S.J. Matthews et al.,PCCP 9 (2007), 5656

Raw VMI image for photodissociation of NO2

around 380 nm

Reconstructed central slice of the Newton sphere

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).

The radius of the rings in a VMI image is proportional to the fragment velocity and therefore contains the same information as a photofragment translational spectrum.

For initially randomly oriented molecules, the photofragment angular distribution in the laboratory frame is given by (derivation see, e.g., Zare, Angular Momentum):

with ... 2. order Legendre polynomial

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

εvΘ

R

T (Θ) =1

4π

1 + βP2(cosΘ)

P2(cosΘ) =

12

3 cos2Θ− 1

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.

The absorption probability 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 ...

P ∝µ · ε

2

• (perpendicular transition), then T(Θ) will be maximal for (β=-1)

• (parallel transition), then T(Θ) will be maximal for (β=+2). µ R v εµ ⊥ R v ⊥ ε

chosen in order not to ‘‘saturate’’ the algorithm by multipleimpacts onto the same detector position within a single lasershot and in order not to blur the image due to space chargeeffects. Typically, pulse energies of 2–10 !J of probe light"#20 ns pulse duration, focused to a beam diameter of#0.015 mm using an f!13 cm plano-convex quartz lens,corresponding to pulse intensities #250 MW cm"2$ wereused to ensure that the required small signal levels were be-low #50 ions per shot.

Further analysis requires the reconstruction of the 3Dvelocity distributions from the accumulated 2D ion imagesfor which purpose an algorithm based on the filtered back-projection proposed by Sato et al.37 is applied. Prior to re-construction, the images were processed in the followingmanner: "1$ In order to reduce the noise in the raw ion im-ages, which is unavoidable with the present centroiding al-gorithm and which will be amplified by the reconstructionprocedure, the images were smoothed using a 2D Gaussianfilter "with a FWHM of 2 pixels$. "2$ It is important to definethe center of the images, i.e., the position of the velocityorigins, as precisely as possible. This was achieved with an

automatic procedure which uses the underlying symmetry ofthe photodissociation process in comparing, respectively, theupper/lower and left/right halves of each image with respectto a test origin, and minimizes the squared differences withsub-pixel resolution. The centering was tested by performingthe reconstruction on each quadrant individually and com-paring the resulting velocity distributions, which turned outto be almost identical and only showed minor differences inthe intensities. "3$ Finally, the images were four-fold sym-metrized "in order to increase the signal-to-noise ratio$ and"4$ the 3D velocity distributions reconstructed, from whichthe velocity and angular distributions were extracted. To de-termine meaningful errors, the total signal of each image"i.e., the number of ion events$ is conserved, and the statis-tical errors "proportional to the square root of the number ofcounts in an individual pixel$ are propagated throughout thewhole reconstruction and analysis process.

The quality of ion images recorded with this new ionimaging spectrometer is very high %see column "a$ of Fig. 4and images presented in Ref. 10&. The images show verylittle asymmetry, and sharp structures observed at the longer

FIG. 4. "a$ Unprocessed raw ion images and "b$ 2Dslices through the reconstructed 3D recoil distributionsof ground state I atoms resulting from photolysis ofjet-cooled IBr molecules at six wavelengths in the range440–540 nm. Only the 251#251 pixels center parts ofthe images are shown. ' denotes the electric vector ofthe photolysis light "i.e., axis of cylindrical symmetry$."c$ shows the corresponding velocity distributions, withfeatures due to the I$Br and I$Br* product channels"and the I$I products from photolysis of I2 contami-nant$ indicated.

2633J. Chem. Phys., Vol. 114, No. 6, 8 February 2001 High resolution ion imaging study of IBr photolysis


β≈2parallel transition

β≈-1perpendicular transition

adapted fromE. Wrede et al., J.Chem.Phys. 114 (2001), 2629

Example: Imaging of the photodissociation of IBr: IBr + hν → I + Br

chosen in order not to ‘‘saturate’’ the algorithm by multipleimpacts onto the same detector position within a single lasershot and in order not to blur the image due to space chargeeffects. Typically, pulse energies of 2–10 !J of probe light"#20 ns pulse duration, focused to a beam diameter of#0.015 mm using an f!13 cm plano-convex quartz lens,corresponding to pulse intensities #250 MW cm"2$ wereused to ensure that the required small signal levels were be-low #50 ions per shot.

Further analysis requires the reconstruction of the 3Dvelocity distributions from the accumulated 2D ion imagesfor which purpose an algorithm based on the filtered back-projection proposed by Sato et al.37 is applied. Prior to re-construction, the images were processed in the followingmanner: "1$ In order to reduce the noise in the raw ion im-ages, which is unavoidable with the present centroiding al-gorithm and which will be amplified by the reconstructionprocedure, the images were smoothed using a 2D Gaussianfilter "with a FWHM of 2 pixels$. "2$ It is important to definethe center of the images, i.e., the position of the velocityorigins, as precisely as possible. This was achieved with an

automatic procedure which uses the underlying symmetry ofthe photodissociation process in comparing, respectively, theupper/lower and left/right halves of each image with respectto a test origin, and minimizes the squared differences withsub-pixel resolution. The centering was tested by performingthe reconstruction on each quadrant individually and com-paring the resulting velocity distributions, which turned outto be almost identical and only showed minor differences inthe intensities. "3$ Finally, the images were four-fold sym-metrized "in order to increase the signal-to-noise ratio$ and"4$ the 3D velocity distributions reconstructed, from whichthe velocity and angular distributions were extracted. To de-termine meaningful errors, the total signal of each image"i.e., the number of ion events$ is conserved, and the statis-tical errors "proportional to the square root of the number ofcounts in an individual pixel$ are propagated throughout thewhole reconstruction and analysis process.

The quality of ion images recorded with this new ionimaging spectrometer is very high %see column "a$ of Fig. 4and images presented in Ref. 10&. The images show verylittle asymmetry, and sharp structures observed at the longer

FIG. 4. "a$ Unprocessed raw ion images and "b$ 2Dslices through the reconstructed 3D recoil distributionsof ground state I atoms resulting from photolysis ofjet-cooled IBr molecules at six wavelengths in the range440–540 nm. Only the 251#251 pixels center parts ofthe images are shown. ' denotes the electric vector ofthe photolysis light "i.e., axis of cylindrical symmetry$."c$ shows the corresponding velocity distributions, withfeatures due to the I$Br and I$Br* product channels"and the I$I products from photolysis of I2 contami-nant$ indicated.



A–X transition. Appadoo et al.6 provide Dunham coeffi-cients for the bound part of the A 3!(1) state potential—atleast up to v!!24 "corresponding to #85% of the total Astate well-depth$, above which the A–X absorption spectrumshows evidence of significant perturbation. These coeffi-cients were used as input for an RKR calculation to generateturning points. The A 3!(1) state ‘‘trial’’ potential wasagain fitted in terms of an extended Rydberg potential %Eq."9$&, with De!2428 cm"1,10 Te!12 370 cm"1 "Ref. 6$,Re!2.8575 Å as determined from the RKR fit, and ' i(1(i(6) and ) as adjustable parameters. The optimum A statepotential and *A"X(+) partial cross-section derived "follow-ing the procedures described above$ are included in Figs. 11and 10, respectively, while the parameters describing thebest-fit extended Rydberg potential for the A state are listedin Table I.

Finally, the high frequency remainder of the 360–800nm room temperature absorption band of IBr is attributed tothe C 1!(1)!X transition. To generate the C state potentialwe assumed a trial repulsive function of the form

VC"R $!A1 exp,"b1"R"R1$-#A2 exp,"b2"R"R2$-#Te"10$

positioned in the vertical Franck–Condon region with pa-rameters Ai , bi , Ri , and Te initially fitted to points in the

Franck–Condon region derived by applying the reflectionprinciple to the remaining high frequency part of the absorp-tion. Fluorescence decay measurements7 indicate that thev!!2 and 3 levels of the B state are affected by heteroge-neous "rotational level dependent$ predissociation caused byinteraction with a dissociative state with .!1. Analogy withother halogens suggests that the C 1!(1) state is responsiblefor the observed predissociation, while the absolute predis-sociation rates measured and their isotopic dependence sug-gests that the B 3!(0#)/C 1!(1) curve crossing occurs inthe energetic vicinity of the Bv!!3 origin. This observationprovides a further constraint when determining the mostprobable C state potential displayed in Fig. 11. Figure 10illustrates how well the measured room temperature absorp-tion spectrum is replicated by the three deduced partial ab-sorption cross-sections, while Table I provides a summary ofthe functional form of each potential and of their respectivetransition moments, / j"X . Given the assumption that thevarious / j"X are independent of R, the uncertainty in therelative transition moments obtained from this deconvolutionis judged to be #1%. The reliability of the absolute valuesquoted in Table I depends upon the accuracy of the reportedtotal absorption cross-sections—estimated to be #$2.5% atthe maximum absorption,17 hence the #3% uncertaintyquoted in the caption to Table I. The value we obtain for/B–X "0.635 D$ is in pleasing accord with the earlier value of0.7 D estimated by Clyne and Heaven on the basis of theircollision-free IBr (B"X) fluorescence lifetime measure-ments and intercomparison with other halogens andinterhalogens.7

The accuracy of the repulsive walls of the derived A, B,and C state potentials in the vertical Franck–Condon regionis estimated to be #$50 cm"1. The bound part of theA 3!(1) state potential is accurate to the extent that the vi-brational term values calculated from the Dunham coeffi-cients of Ref. 6 reproduce the experimental energies with anaverage deviation of 0.3 cm"1, with maximal differences of"0.7 cm"1 for the v!!0 level and #1.1 cm"1 for the high-est valid vibrational level (v!!24). The long range part ofthe diabatic B 3!(0#) state potential, and the repulsiveY 30"(0#) potential "see Sec. III E$ remain less well-defined, pending a proper coupled treatment of the B/Y statesin order to reproduce the structured, predissociated part ofthe absorption spectrum in the wave-number range 16 500–18 350 cm"1.35

E. Wave packet calculations of !„"… and #„"…Grid-based time-dependent wave packet calculations of

the visible photofragmentation of IBr have been performedas a further test of the accuracy of the derived excited statepotential energy curves and transition dipole moments and toallow determination of the coupling strength between thediabatic B and Y states. The calculations involved solution ofthe time-dependent Schrodinger equation,

i12

2t 3"R ,t $!H"R $3"R ,t $, "11$

FIG. 11. Diabatic potential energy curves for the X, A, B, and C states of IBrderived in the present work. Also shown, for completeness, is the assumedlocation of the diabatic Y state potential responsible for the predissociationof the higher v! levels of the B state. RKR points are taken from a variety ofsources as detailed in the text, while all relevant potential parameters aregiven in Table I.



Photodissociation images at 440 nm

Raw image of iodine products

Reconstructedcentral slice

Iodine atom product speed distribution

Potential energy curves

440 nm

I+Br*

I+Br

• 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

E. Wrede et al., J.Chem.Phys. 114 (2001), 2629

Broadband fs laser excitation usuallyleads to the excitation of severalvibrational states at the same time.

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

The vibrational wavefunctions interfere resulting in the formation of a localised vibrational wavepacket:

vibrational wavefunctionswavepacket

The wavepacket oscillates back andforth on the excited-state potentialenergy surface with a frequencycorresponding to the vibration that has been excited

localisedwavepacket |Ψ|2

after fs excitation

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)

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

(T. Baumert et al., J. Phys. Chem. 95 (1991), 8103)

Na2+ signal intensity

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)

avoidedcrossing

• A vibrational wavepacket is created in the excited state by fs laser excitation

• The wavepacket oscillates in the excited-state 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.

• Both states exhibit an avoided crossing at R=Rc at which they strongly interact.

wavepacketmotion

• 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)

avoidedcrossing

wavepacketmotion

a b

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