Download Chapter 4: Chemical Reaction Dynamics

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