Download Photodissociation Dynamics R. Schinke

CPA11-
Photodissociation Dynamics
~
Vex A
t>0
ABCx
Max-Planck-Institut für Strömungsforschung, Göttingen, Germany
−
E = <ΦexΗexΦex>
Φex (t)
σtot
Eavai.
hω
hω'
α
Introduction
Computational Methods
Direct and Indirect Photodissociation
Product State Distributions
Emission Spectroscopy of Dissociating Molecules
Nonadiabatic Effects in Dissociating Molecules
Unimolecular Dissociation in the Ground Electronic
State
Real-time Dynamics
Outlook
Related Articles
References
A + BC(α)
P(α)
Ef
D0
ω-ω'
ψgr,i
,,,,,
,,,
8
9
10
11
potential energy
~
Vgr X
1
2
3
4
5
6
7
E
,,,,,,
,,,,
,
t=0
Reinhard Schinke
Ei
ABC
(R)
σif
dissociation coordinate R
Abbreviations
BO D Born Oppenheimer; FC D Franck Condon; PES D
potential energy surface; TS D transition state.
Figure 1 Schematic representation of the photodissociation of a
molecule ABC into products A C BC.˛/. gr,i is the initial wave
Q and ex .t/ is the wave
function in the ground electronic state X
packet in the excited electronic state with average energy E. tot
is the energy-dependent total dissociation cross section, P.˛/ is the
.R/
is
probability for filling quantum state ˛ of the products, and if
the Raman cross section for transition i ! f in the lower state via
resonant excitation of the upper electronic state
1 INTRODUCTION
1.1 What is Photodissociation?
Photodissociation is the breaking of one or several bonds
in a molecule through the absorption of light. Depending on
the number of photons required to fragment the molecule, one
distinguishes between single- and multiphoton dissociation. In
the case of dissociation with a single photon, usually in the
optical or the ultraviolet region, the molecule is promoted from
Q to an excited electronic state, for
the ground electronic state X
Q
example A, as schematically illustrated in Figure 1. In multiphoton dissociation, several or many photons, mostly in the
infrared region, excite the molecule until the energy exceeds
the dissociation energy of the weakest bond and thus enables
the molecule to break apart (see Multiphoton Excitation). In
the present article we will exclusively focus on dissociation
with single photons. An example is the photodissociation of
isocyanic acid (equation 1) with two spin-allowed and one
spin-forbidden, chemically different product channels,
(1)
In what follows Vgr and Vex will denote the potential
in the ground and the excited electronic state, respectively.
Depending on the shape of Vex along the dissociation bond
the molecule will fall apart immediately, on a time-scale
smaller than a typical internal vibrational period, or after
some particular lifetime, which can range from several to
thousands of internal vibrational periods. Immediate (or direct)
dissociation occurs if Vex .R/ is purely repulsive from the
Franck Condon (FC) point, where the molecule starts its
motion in the upper electronic state, until the fragments are
irreversibly formed. Indirect (or delayed) dissociation, on the
other hand, requires that the molecule is trapped for some time,
either by a potential barrier or some subtle dynamical effect,
before sufficient energy is accumulated in the dissociation
mode, enabling the bond to break.1
1.2 Goals and Observables
The principal goal of photodissociation studies is to obtain
the clearest picture of the molecular dynamics in the excited
electronic state as the molecule leaves the FC region, traverses
the ‘transition state’ (i.e., the barrier, if there is any), and
finally reaches the asymptotic channel(s), where the fragments
are formed. Key questions are: What is the lifetime of the
molecule in the upper state and how does this lifetime depend
on the excitation energy? Which bonds break and what is the
branching ratio for the possible chemical channels? If more
than one bond breaks, does this occur in a concerted or a
sequential process? Is the dissociation governed by one and
only one electronic state or does the fragmentation take place
on several potential energy surfaces? How is the total available
energy Eavai. D Ephoton Do (i.e., the photon energy minus the
dissociation energy) distributed among the translational and
internal degrees of freedom of the fragments? What are the
probabilities for filling the various electronic, vibrational, and
CPA11-
2 PHOTODISSOCIATION DYNAMICS
rotational states of the fragments after the bond is broken?
All of these questions are ultimately determined by the shape
of the potential energy surface of the particular excited state,
and therefore Vex is the cornerstone for understanding the
photodissociation of polyatomic molecules.
The last twenty years have witnessed tremendous advances
in the experimental investigation of photodissociation
processes.2 4 This became possible through the development
of powerful laser systems in the UV, allowing efficient
excitation of molecules even in a molecular beam, and the
development of new detection schemes in order to probe
the different quantum states of the products. Today a typical
experiment uses three laser systems:5,6 a first laser prepares
the molecule in a particular vibrational rotational level in the
ground electronic state, the second pumps the molecule to the
excited state, and with the third laser one probes the fragments.
Information about the dissociation process is inferred from
basically three observables (see Figure 1):
1. The absorption spectrum, or total absorption cross section
tot .Ephoton / measures the probability with which the molecule can absorb light with a particular energy Ephoton D
hv or h̄ω. The energy dependence of tot , especially the
occurence of progressions of sharp structures with more
or less constant energy spacings, reveals information about
the shape of the potential near the FC region, where the
molecule is still bound.
2. The Raman spectrum, i.e., the dispersed spectrum of
light that the dissociating molecule emits to a lower-lying
electronic state on its way from the FC region to the
fragments, if.R/ , provides knowledge about a wider region
of the potential including the region of the transition state.
3. The probabilities P.˛/ with which the energetically accessible quantum states ˛ of the fragments are populated
essentially reflect the motion of the molecule from the
transition state to the asymptote(s), i.e., the forces acting
between the different fragments in the exit channel (the
so-called final state interaction).
Although consideration of all these quantities for as many
excitation frequencies as possible gives a detailed overview of
the whole dissociation process, its actual time-dependence can
only be inferred in an indirect way; in a ‘traditional’ experiment the pulse length of the excitation laser exceeds the typical
time-scale for molecular motion by orders of magnitude so
that it is inherently impossible to resolve the temporal behavior of the state of the molecule. Only when lasers with pulse
lengths in the sub-picosecond region became available did it
become possible to study directly, by means of pump probe
experiments, the evolution of the molecule from the point of
excitation, through the transition state, into products.7 9
Photodissociation is an ideal field for studying molecular motion in excited electronic states. The interplay between
sophisticated experiments on one hand and state-of-the-art calculations on the other has significantly advanced the general
understanding of processes in molecules. However, besides
this more academic interest, photodissociation is of vital
importance for other aspects of physical chemistry, too, for
example the complex chemistry in the atmosphere (see Photochemistry). Many basic chain reactions, such as the ozone
cycle, are initiated by the absorption of light.10 13 In many
cases atomic or molecular radicals like O, Cl, NO, or OH
are produced in a dissociation process, and in turn undergo
secondary reactions with other species and so on. Understanding such fundamental processes, which are crucial for the
future of the earth, in full detail, is certainly rewarding and
computational chemistry plays a major role in this task.
1.3 Potential Surfaces
The potential curves shown in Figure 1 represent either a
diatomic molecule with only one bond coordinate or a cut
through a multidimensional potential energy surface14,15 of a
polyatomic molecule such as HNCO (see Potential Energy
(Hyper)Surface (PES)). In general, a molecule with N (> 2)
atoms has 3N 6 internal degrees of freedom Qi and the PES
is a function of all of them, i.e., V D V.Q1 , Q2 , . . ./. Figure 2
depicts a two-dimensional cut through the six-dimensional PES
of HNCO(AQ 1 A00 ) as functions of the two bond distances RHN
and RNC . The two exit channels correspond to the two chemically different products H C NCO and NH C CO that can be
formed. Although the other four degrees of freedom, RCO , ˛, ˇ,
and , are fixed in this picture, the potential is not independent
of them and a complete dynamics calculation should in principle include all of them. Figure 3 depicts excited-state potential
energy surfaces for several other molecules. Below we will
use these examples for illustrating the sensitivity of observables like the absorption spectrum with respect to topographic
details. The dissociation dynamics is exclusively governed by
the structure of the PES in the upper electronic state. In the
classical mechanics approach of photodissociation, studying
the fragmentation dynamics of a polyatomic molecule essentially amounts to placing ‘billiard balls’ on these PESs, near
the FC point, and following their trajectories towards one or
the other exit channel.
Potential surfaces are the key quantities for understanding molecular processes. Due to the existence of wells,
barriers, avoided crossings, conical intersections etc., rather
complicated topographies, which can hardly be guessed or
Figure 2 Two-dimensional representation of the potential energy
surface for the AQ 1 A00 state of HNCO as a function of the NC and the
HN bond distances RNC and RHN , respectively. The other coordinates
are frozen
CPA11-
PHOTODISSOCIATION DYNAMICS
3
H
R
γ
C
r
O
Figure 4 Definition of Jacobi coordinates R, r, and for the
fragmentation of HCO into H and CO
calculations and this situation will not essentially change in
the foreseeable future.
1.4 Coordinates
Before we describe the various theoretical approaches, it
is worthwhile to define the coordinates commonly used in
dynamics calculations. The most convenient coordinates for
describing the breakup of the parent molecule into two or
several fragments (half-collision) are the Jacobi coordinates
routinely employed in scattering theory (see Reactive Scattering of Polyatomic Molecules and State to State Reactive
Scattering). For a triatomic molecule like HCO, which fragments into H and CO, the Jacobi coordinates are defined as
follows (Figure 4): R is the vector pointing from the H atom to
the center-of-mass of CO, r is the vector pointing from atom C
to atom O, and the angle is defined by cos D R Ð r/.Rr/.
For a tetratomic molecule like HNCO, which splits up into
NH and CO, for example, R is the vector joining the centersof-mass of NH and CO, respectively. The advantage of Jacobi
coordinates is that the expression for the kinetic energy is
particularly simple. Coordinates which are usually employed
for calculating the bound states of a molecule (i.e., normal
mode coordinates) are not appropriate as they do not correctly
describe the rupture of a bond.
2 COMPUTATIONAL METHODS
2.1 Light Matter Interaction
The evolution of the molecular system, with Hamiltonian
Hmol , in a time-dependent external electric field Efield .t/ is
governed by the time-dependent Schrödinger equation
ih̄
Figure 3 Two-dimensional potential energy surfaces for excited
states of several XNO systems. r is the NO vibrational bond distance
and R is the distance from X to the center-of-mass of NO. The dots
mark the Franck Condon point, i.e., the equilibrium configuration
in the ground electronic state. Reprinted, with permission of the
American Chemical Society, from Ref. 45
parametrized in a simple manner, are common. Potential surfaces must be calculated by ab initio methods16 18 point by
point, scanning at least that part of the PES that is sampled during the breakup. Typically about 5 10 grid points are
required for each coordinate. Finally, all the calculated points
on the PES have to be fitted to an analytical function which
then is employed in the dynamics calculations. The lack of
high-quality complete PESs is still the bottleneck for realistic
∂
O mol C dO Ð Efield .t/]F.t/
F.t/ D [H
∂t
.2/
where the function F.t/ describes the time-dependence of the
entire molecular system (including electrons and nuclei). The
field matter interaction is represented by dO Ð Efield .t/ where dO
is the electric dipole operator. Equation (2) must be solved
subject to the initial condition that the molecule is in a particular vibrational rotational level in the ground electronic state
before it is exposed to the field. When the field is switched on
it continuously promotes molecules (or, to be precise, probability distribution) from the ground to an excited electronic
state.1,19 Except for comparably simple cases the solution of
the time-dependent Schrödinger equation is impossible unless
special assumptions and approximations are introduced.
In almost all photodissociation experiments one uses light
sources with pulse lengths of the order of nanoseconds or
longer, so that the electric field Efield .t/ to a very good
approximation can be represented by an infinitely long pulse
CPA11-
4 PHOTODISSOCIATION DYNAMICS
with frequency ω, i.e.
Efield .t/ D Eo cos ωt
.3/
where Eo is the time-independent field strength. Provided that,
in addition to the infinitely long duration, Eo is so small
that the rate of probability transfer from the ground to the
excited state is very small, the absorption process can be
treated using first-order perturbation theory concerning the
light matter interaction.1,19 Under these circumstances, the
photon populates only states in the excited manifold, which
are resonant with the photon energy, i.e., for which
Ef
Ei D h̄ω
.4/
where Ei and Ef are the energies of the initial and the
final state, respectively. Using first-order perturbation theory
for the field matter interaction does not mean, however, that
the motions of the molecule in the two electronic states are
approximated; on the contrary, the corresponding equations of
motion are solved without further assumptions. The case of
excitation with short and/or strong laser pulses will be briefly
discussed at the end of this article.
2.2 Photodissociation Cross Sections
Under the assumption of long laser pulses and weak
light matter interaction, formally quite simple expressions for
the various photodissociation cross sections can be derived
(Fermi’s Golden Rule).1,19,20 For consistency we will use the
same notation as in Ref. 1.
The partial photodissociation cross section for absorbing a
photon with frequency ω and producing the fragments in a
particular quantum state ˛ is given by1,21,22
.ω, ˛/ D CEphoton jt.Ef , ˛/j2
.5/
where C D /.h̄εo c/ is a constant, the factor D .2h̄/ 1
stems from the normalization of the continuum wave functions,
Ephoton is the photon energy, and
D
E
.˛/ .e/ t.Ef , ˛/ D ex,f
ex,gr gr,i
.6/
is the partial photodissociation amplitude, a complex function.
.e/
ex,gr is the component of the electric transition dipole function
for the ground and the excited electronic states in the direction
of the electric field vector Efield . The ‘quantum number’ ˛
comprises a whole set of quantum numbers necessary for
specifying the different fragments including the particular
chemical channels.
.˛/
The two stationary wave functions gr,i and ex,f
are solutions of the time-independent nuclear Schrödinger equations in
the ground and in the excited electronic states,
O gr
.H
Ei /gr,i D 0
.7/
O ex
.H
.˛/
Ef /ex,f
.8/
D0
where Ei is the energy of the ground state and Ef D Ei C
Ephoton is the resonant energy in the excited manifold. The
wave functions in the ground electronic state, describing the
state of the parent molecule before the absorption of the photon, are bound-state wave functions and as such are localized
around the equilibrium position; they exist only for specific
energies Ei (see Rovibronic Energy Level Calculations for
Molecules). In contrast, the wave functions in the excited state
are continuum wave functions, i.e., they extend to infinite interfragment separations (R ! 1) and behave like plane waves
in the various product channels ˛ (see Reactive Scattering of
Polyatomic Molecules). Being continuum wave functions they
exist for any energy above the lowest dissociation threshold.
The situation is even more complicated: every possible combination of quantum numbers ˛ defines, through the boundary
conditions in the fragment channels, a unique wave function
.˛/
distinguished by the upper index (˛).
ex,f
The total photodissociation cross section,
tot .ω/ D
X
.ω, ˛/
.9/
˛
is the quantity that is measured in ‘traditional’ spectroscopy
(absorption spectrum). The partial dissociation cross sections
can be measured in a modern experiment, in which the fragments and their different states are probed by e.g., spectroscopic means. For obvious reasons, they contain much more
information about the dissociation dynamics than the total
cross section: while the total cross section reveals clues about
the molecular motion only near the point of excitation, the
.ω, ˛/ reflect the entire history of the dissociating molecule,
from the FC region all the way to the products. In many
examples one considers product state distributions, which are
defined for a fixed frequency by
P.˛/ D
.ω, ˛/
tot .ω/
.10/
2.3 Time-independent Approach
The most direct, although numerically probably the hardest way of calculating photodissociation cross sections is to
solve the two nuclear Schrödinger equations (7) and (8) for
energies Ei and Ef D Ei C h̄ω, respectively, and to directly
compute the overlap elements in equation (6) by numerical
integration over all nuclear coordinates. The calculation of
the bound-state wave function in the ground state is usually
not problematic and several efficient methods are available
(see Rovibronic Energy Level Calculations for Molecules).23
Because of the nonvanishing boundary conditions at infinite
fragment separations, the calculation of the set of continuum
.˛/
is much more involved. A rather genwave functions ex,f
eral numerical procedure based on direct integration of the
close coupling equations in an expansion of suitable basis
functions has been described in chapter 3 of Ref. 1. By plotting the two wave functions in coordinate space one can get
a qualitative understanding of the energy dependence of the
photodissociation spectrum or the final state distributions. Particularly illustrative is the time-independent method based on
the flux redistribution during the fragmentation process, developed by Alexander and co-workers.24,25 Since the mid-1980s,
several other time-independent methods have been developed,
which are more efficient than the close coupling approach; like
bound-state calculations they use expansions in finite basis sets
in all coordinates, including the dissociation coordinate (see,
for example, Dobbyn et al.26 ).
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PHOTODISSOCIATION DYNAMICS
2.4 Time-dependent Approach
Instead of solving the time-independent Schrödinger equation (8) for the nuclear motion in the excited electronic state,
one can work in the time-domain and solve the time-dependent
Schrödinger equation
∂
ih̄
∂t
O ex ex .t/ D 0
H
.11/
for the time-dependent wave packet ex .t/. At first glance,
it appears unreasonable to introduce an extra ‘coordinate’,
the time t, in addition to the nuclear coordinates. However,
the time-dependent Schrödinger equation is an initial value
problem, i.e., the wave packet at time t D 0 is specified, and
as such it is, under certain conditions, easier to solve than the
time-independent analogue (equation 8), which is a boundary
value problem (see Wave Packets). If we demand that1,27 29
ex .0/ D .e/
ex,gr gr,i
.12/
it is easy to show that the total absorption cross section is
given by
Z
C1
dt eiEf t/h̄ S.t/
tot .ω/ D CEphoton
.13/
1
The overlap of the evolving wave packet at time t with the
initial function at t D 0,
S.t/ D hex .0/jex .t/i
.14/
is called the autocorrelation function. In other words, the
energy-dependent absorption spectrum is simply the Fourier
transform of the time-dependent autocorrelation function.
Equation (13) establishes the connection between the
energy- or frequency-domain on one hand and the time-domain
on the other. S.t/ is the key quantity as it reflects in a rather
direct way the dynamics of the system in the upper state, for
example, how fast the initial wave packet leaves its place
of birth, whether it ever comes back to its starting position
or not, and if so, how often does it recur? Because of the
intimate relation between the absorption spectrum and the
autocorrelation function, it is possible, with only elementary
knowledge of the properties of the Fourier transformation, to
explain structures in the spectrum by the time-evolution of the
wave packet and vice versa. Examples will be briefly discussed
below.
It is essential to underline that the time-dependence considered here must not be mixed up with the time-dependence of
the actual photoabsorption process in which the electric field
Efield .t/ steadily pumps the molecule from the ground electronic state to the excited state. The formalism described above
corresponds to a picture in which an infinitely short light pulse
instantaneously promotes the molecule to the excited-state
PES. Because the initial wave packet is not an eigenfunction
of the upper-state Hamiltonian, it immediately starts to move
according to the equations of motion on the PES Vex .
Partial photodissociation cross sections can be obtained by
propagating the wave packet to long times, until ex .t/ moves
freely in the exit channels, and projecting it on the asymptotic
states, i.e.1,30
.ω, ˛/ D CEphoton
D
E 2
m
lim eik˛ R ϕ˛ ex .t/ h̄ka t!1
.15/
5
where m is the reduced mass for this particular (chemical)
exit channel and the k˛ [2m.Ef ε˛ ] 1/2 are the corresponding
wavenumbers. The ϕ˛ are the wave functions for the products
and the ε˛ are the corresponding energies.
The advantage of the time-dependent approach is that
information for all energies can be extracted from a single
wave packet. This is possible, because the wave packet ex .t/
is nothing other than a superposition of all stationary wave
.˛/ and therefore contains everything one needs to
functions ex
calculate the cross sections.1 The stationary wave functions
are calculated for a specific energy Ef and contain the entire
‘history’ of the fragmentation process. On the contrary, the
wave packet is a function of time but it contains the stationary
states for all energies. This is most intriguingly expressed by
the relation between the total stationary wave function and the
wave packet,
Z
tot .E/ D
dt eiEf t/h̄ ex .t/
.16/
i.e., the total wave function for a given energy (i.e., a linear
.˛/
for a particular energy) is just the
superposition of all ex,f
Fourier transform of the wave packet and vice versa.1
The essential work to be done is the propagation of
the wave packet on the excited-state PES. Many efficient
propagation methods have been derived and applied to numerous examples.31 33 The propagation in time is usually done
by expansion of the time-evolution operator
Û.t/ D e
O ex t/h̄
iH
.17/
in terms of appropriate functions (for example Chebychev
polynomials34 ). The central operation in all propagation methods is the application of the Hamiltonian to the wave packet,
O ex ex .t/. This is facilitated by discretizing ex .t/ on
i.e., H
a multidimensional grid in coordinate space which allows an
efficient evaluation of the kinetic energy terms.35 Because the
computer time rapidly grows with each additional degree of
freedom, exact propagation schemes are restricted to small
systems with three or at most four atoms. Therefore, several
approximate propagation schemes have been introduced all of
which rely, in some way or another, on a separation of the
multidimensional wave packet (see Time-dependent Multiconfiguration Hartree Method). An advantage of time-dependent
methods is that quantum mechanics can be mixed with classical mechanics, i.e., while some degrees of freedom are treated
quantum mechanically, others are approximated by classical
trajectories (see Mixed Quantum-classical Methods).
Figure 5 illustrates the motion of a wave packet in a twodimensional coordinate space; shown are snapshots of the
evolving packet for several times. The example represents the
photodissociation of FNO in the S1 state.36 The initial wave
packet, a two-dimensional Gaussian-type function, is prepared
at the FC point, i.e., the equilibrium in the ground electronic
state. In the very first moments the wave packet follows the
steepest descent and slides down the slope towards the tiny
potential barrier, just like a classical ‘billiard ball’. Near the
barrier it splits into two parts; one portion enters the exit
channel and slides down the steep potential trough leading to
fragments F and NO in a time much shorter than an internal
vibrational period. The other part is trapped in the inner region
of the potential ridge, performs one full oscillation, and recurs
to its starting position. Here, a new round begins, another
CPA11-
6 PHOTODISSOCIATION DYNAMICS
Figure 5 Snapshots of the wave packet in the S1 excited electronic state of FNO for various times, ranging from 0 (a) to 36.3 fs (f). Reprinted,
with permission of the American Institute of Physics, from Ref. 36
portion of the wave packet leads to dissociation whereas the
remaining wave packet performs a second oscillation. This
continues until the entire wave packet has finally escaped
from the inner region and travels freely in the F C NO exit
channel. The consequences for the autocorrelation function
and the absorption cross section will be discussed in the next
section.
2.5 Classical Trajectory Approach
The general understanding of molecular dynamics rests
mainly upon classical mechanics; this holds true for full
bimolecular collisions (see Classical Trajectory Simulations
of Molecular Collisions) as well as half-collisions, i.e., the
dissociation of a parent molecule into different products.
The classical picture of photodissociation closely resembles
the time-dependent picture: the electronic transition from the
ground to the excited electronic state is assumed to take
place instantaneously so that the internal coordinates (Qi ) and
corresponding momenta (Pi ) of the parent molecule remain
unchanged during the excitation step (vertical transition).1
After the molecule is promoted to the PES of the upper state
it starts to move subject to the classical equations of motion
(Hamilton’s equations)
d
∂Hcl
d
ex
and Pi D
Qi D
dt
∂Pi
dt
∂Hcl
ex
∂Qi
.18/
where Hcl
ex .Qi , Pi / is the classical analogue of the quantum
mechanical Hamilton operator. The space-time curves Q.t/
and P.t/, i.e., the trajectories, are followed until the fragments
are completely separated.
Like the time-dependent wave packet approach, the classical approach is an initial value problem. However, while it
suffices to propagate a single wave packet from which all relevant quantities can be extracted, many individual trajectories
have to be calculated before meaningful cross sections can
be obtained. Initial conditions Q.t D 0/ and P.t D 0/ are randomly chosen from the .2 ð N/-dimensional phase space, if
the number of degrees of freedom is N (Monte Carlo sampling technique; see Integrating the Classical Equations of
Motion). In order to mimic the quantum mechanical motion
of the parent molecule before excitation, the initial coordinates
and momenta have to be weighted by a distribution function
which resembles the quantum mechanical distribution function
in the ground state, jgr,i j2 , as closely as possible. The Wigner
distribution function is usually taken.1 For an ensemble of N
uncoupled harmonic oscillators, all in the lowest vibrational
state i D 0, the Wigner function is given by
PW .Q, P/ /
N
Y
e
2˛i Qi2 /h̄
e
O
Pi2 /.2˛i ,h̄/
.19/
i
where the ˛i determine the widths of the Gaussians and are
related to the corresponding frequencies of the oscillators.
CPA11-
PHOTODISSOCIATION DYNAMICS
The classical trajectory approach is very general. While
the quantum mechanical approaches are restricted to small
systems with no more than, at most, 3 6 internal degrees of
freedom, the trajectory technique can be applied to much larger
systems without great difficulty. Direct comparisons with exact
quantal calculations have shown that classical calculations
are usually reliable when the dissociation is fast and direct
and if the released energy is large.37 39 Quantum mechanical
effects like interferences and resonances inherently cannot be
described in a classical way. If such effects are important,
semiclassical methods that take into account the quantum
mechanical superposition principle have to be employed.40,41
The great assets of the trajectory approach are its simplicity
and its interpretative power. In many cases running only a
small batch of trajectories has been found sufficient to reveal
the overall dissociation dynamics.
3 DIRECT AND INDIRECT PHOTODISSOCIATION
Photodissociation can be broadly classified as direct dissociation and indirect or delayed fragmentation. Whether a
particular system belongs to the one or the other category
depends ultimately on the topology of the PES. Although
both types can be describe in the time-independent as well as
the time-dependent approach, in the following we will mainly
employ time-dependent arguments for summarizing the main
features.27,28
3.1 Direct Photodissociation
After the wave packet is launched on the upper-state PES,
it immediately leaves the FC region and its center moves in
the direction of the steepest descent. As a result of this initial
motion the overlap with the wave packet at time t D 0 rapidly
decays to zero as illustrated in Figure 6(a). If the PES is purely
repulsive, the wave packet, not even a small part of it, will
never come back to its origin with the consequence that the
autocorrelation function remains zero for all times. According
to equation (13), the total dissociation cross section is proportional to the Fourier transform of S.t/. The Fourier transform42
of a structureless Gaussian-type function in the time-domain
∆E0
(d)
(c)
S(t)
∆E
T
time
T0 E0 D 8h̄ ln 2 ³ h
.20/
which is known as the fourth Heisenberg uncertainty relation
(see Uncertainty Principle (Heisenberg Principle)).
The temporal width T0 is determined by the speed with
which the wave packet moves out of the FC position and therefore by the slope of the PES near the FC point: the steeper the
potential, the more rapidly the wave packet leaves its place of
birth, the broader is the spectrum and vice versa. The maximum of the spectrum is determined by the vertical excitation
energy, i.e., the difference between the energy of the initial
state and the (potential) energy in the upper state evaluated at
the ground-state equilibrium. In the time-dependent view the
general shape of the absorption spectrum can be explained by
the reflection principle.1
3.2 Indirect Photodissociation
If the PES is not purely repulsive, i.e., if direct dissociation
is (partly) prohibited by a potential barrier in the exit channel
or some other dynamical effect, some part of the wave packet
is trapped for a longer time and performs an oscillatory motion
as observed for FNO(S1 ) in Figure 5. Whenever a part of
the wave packet recurs to its starting position, the overlap
with ex (0) increases and the autocorrelation function shows a
series of recurrences as schematically illustrated in Figure 6(c).
The recurrences reflect the vibrational motion of the molecule
in the inner region of the PES, before it finally escapes through
the transition state (point of no return) and dissociates. If T is
the period of the oscillation in the time-domain, the Fourier
transformation of S.t/ leads to structures in the spectrum,
which are separated by
E D
2h̄
T
.21/
The number of recurrences reflects the damping of the wave
packet in the inner part of the PES and therefore the lifetime .
The longer the series of recurrences, the longer is the lifetime,
and the narrower are the absorption features. The full widths
at half the maximum of the individual absorption peaks are
related to the lifetime by
D
h̄
D h̄k
.22/
with k D 1/ being the dissociation rate. In the time-independent view the sharp structures are termed resonances.43,44
Mathematically they are described as Lorentzian functions1
σtot (E)
S(t)
∆T0
2
(b)
with width T0 is a smooth Gaussian-type function in the
energy-domain with width E0 as sketched in Figure 6(b).
The two widths T0 and E0 are inversely related to each
other by
σtot (E)
(a)
7
energy
Figure 6 Illustration of the relation between the time-dependence
of the autocorrelation function jS.t/j (left-hand side) and the energy
dependence of the absorption cross section tot .E/ (right-hand side).
(a) and (b): direct dissociation; (c) and (d): indirect dissociation
.E/ /
.E
./2/
Eres /2 C ./2/2
.23/
where Eres is the position of the resonance. The linewidth 
reflects how strongly the excited parent molecule, moving in
the bound region of the upper-state PES, is coupled to the
fragment channel. Note that each resonance has a distinct
linewidth. The dependence of  on the particular resonance
excited can be smooth and simple, oscillatory or much more
complicated.44 In any case, the energy- (or state-) dependence
CPA11-
8 PHOTODISSOCIATION DYNAMICS
of  reveals a great deal of information on the dissociation
dynamics.
The breadth of the envelope of the absorption spectrum in
Figure 6(d) is, as in the case of direct dissociation, related to
the width of the first peak of the autocorrelation function. It
basically reflects the number of vibrational states in the upper
manifold that have appreciable FC overlap with the initial-state
wave function gr,i and are therefore excited.
FNO(S2)
2.0
3.0
4.0
5.0
6.0
7.0
8.0
ClNO(S1)
3.3 Transition from Direct to Indirect Dissociation
0.6
absorption spectrum
The lifetime of the molecule in the excited state, with
respect to dissociation, is governed by the shape of the potential, especially along the dissociation coordinate. Very fast and
very slow dissociation are just the extreme cases with plenty of
possibilities in between. The transition from direct to indirect
dissociation is smooth and gradual as illustrated in Figures 3,
7, and 8.45 Shown in Figure 3 are calculated PESs for several molecules of the type XNO which dissociate into X and
NO (X D F, Cl, and CH3 O). The systems are ordered from
0.8
1.0
1.2
1.4
nx = 0
kx
nx = 2
0.2
0.4
0.6
0.8
2
1.0
3
FNO (S2)
1.2
1.4
1.2
1.4
FNO(S1)
4
nx = 0
20
1.8
ClNO(T1)
0 12
1
40
1.6
nx = 5
1.6
1.8
2.0
2.2
2.4
CH3ONO(S1)
0
ClNO (S1)
40
1.4
20
ClNO (T1)
 S(t)
TNO
2.0 2.2
E [eV]
2.4
2.6
Tbend
20
0
40
FNO (S1)
TNO
20
0
CH3ONO (S1)
60
40
20
0
1.8
Figure 8 Absorption cross sections calculated from the autocorrelation functions displayed in Figure 7. nŁ and k Ł denote the NO stretching and the XNO bending quantum numbers, respectively. Reprinted,
with permission of the American Chemical Society, from Ref. 45
0
40
1.6
0
40
80
120
160
200
t[fs]
Figure 7 Autocorrelation functions for the systems displayed in
Figure 3. TNO is the NO stretching period and Tbend is the period
of bending motion. Reprinted, with permission of the American
Chemical Society, from Ref. 45
top to bottom in terms of their steepness along the dissociation coordinate R (i.e., the distance from the fragment X to
the center-of-mass of NO). The corresponding autocorrelation
functions jS.t/j and absorption spectra tot .E/ are depicted in
Figures 7 and 8, respectively. At first glance, the potentials
look rather similar. However, the small but unique differences
have pronounced effects on the overall dissociation dynamics.
The PES for FNO(S2 ) is very repulsive with the consequence that the F NO bond breaks immediately, on a timescale shorter than an internal vibrational period. The autocorrelation function is very narrow and structureless leading to
a broad and featureless spectrum. The ClNO(S1 ) potential is
considerably less repulsive so that the bond breaking requires
a longer time than for FNO(S2 ). A tiny recurrence is seen
which produces very broad diffuse structures in the absorption
spectrum. The slope ∂Vex /∂R for ClNO(T1 ) is almost zero near
the FC point so that the Cl NO bond breaks very slowly. This
allows the system to perform, during the breakup, several NO
vibrations and in addition one bending oscillation manifested
by the recurrences with periods TNO and Tbend . The fingerprints of these recurrences are pronounced structures in the
absorption spectrum. The dissociation of FNO(S1 ) has been
discussed before. Since a tiny barrier hinders immediate dissociation the overall lifetime is longer than for ClNO(T1 ). The
CPA11-
PHOTODISSOCIATION DYNAMICS
remarkable feature in the dissociation of FNO in the S1 state is
the coexistence of fast and slow dissociation leading to interesting asymmetric line shapes in the spectrum.36,46 Finally, the
barrier which blocks the exit channel is about two times higher
for the last system, CH3 ONO(S1 ), which leads to a significant
increase of the lifetime. The long series of quite regular recurrences implies a sequence of narrow, well separated absorption
lines in the spectrum.
These examples intriguingly illustrate several points: (i)
The ‘smooth’ transition from fast to slow dissociation, (ii) the
sensitive dependence of the molecular dynamics on the PES,
and, last but not least, (iii) the high accuracy that is required for
realistic dynamics studies on calculated potential surfaces. In
the language of scattering theory the structures in the energydependent spectra are called resonances and in the context
of spectroscopy they are termed diffuse vibrational structures.
Irrespective of what we call them, they reveal interesting
information about the dissociation dynamics, especially how
the inner region of the PES is coupled to the exit channel.
4 PRODUCT STATE DISTRIBUTIONS
After the bond is broken, the total available energy released
is distributed among all degrees of freedom and the fragments
are produced in distinct internal states, e.g., electronic, vibrational, or rotational states. The distribution with which the
possible states are populated reveals further details about the
dissociation dynamics, especially about the forces acting in
the exit channel, beyond the transition state (TS).1 By transition state we mean that point (or line) on the PES where
the bond irreversibly starts to break. In the case of a direct
process on a repulsive potential the TS is just the FC point,
to which the molecule is promoted by the absorption; if the
dissociation is indirect, the TS is commonly taken as the barrier that separates the inner from the outer region. Vibrational
state distributions39 are mainly governed by the dependence of
the PES on the fragment coordinate r whereas rotational state
distributions primarily reflect the dependence of Vex on the
orientation angle .47 49 Classical trajectories are very useful
for understanding the general principles which are responsible for filling the various vibrational and rotational levels. The population of the individual electronic states of the
atoms or molecules is governed by the strength of nonadiabatic (i.e., non-Born Oppenheimer) coupling between different electronic states (see below).
Because of energy constraints very often only a few vibrational states can be populated. In contrast, the energy content of
rotational levels is much smaller and it is often found that many
rotational levels are populated, which makes analysis much
more interesting. For this reason we will restrict the discussion to rotational distributions; vibrational excitation follows
very similar principles and can be described in analogy.1
4.1 Franck Condon Mapping
If Vex depends only weakly on the orientation angle , the
torque ∂Vex /∂ is small and so is the degree of rotational
excitation, which in the classical approach is described by
dj.t/
D
dt
∂Vex .R, r, /
∂
.24/
9
here, j.t/ is the classical analogue of the rotational quantum
number quantum j D 0, 1, . . .. The degree of rotational excitation of the product, which is already present in the parent
molecule, remains more or less unchanged during the fragmentation process. Under such conditions the final rotational state
distribution P.j/ is approximately given by a projection of the
initial (bending) wave function ϕbend ./ on the rotational wave
functions of the free diatomic rotor, which are the spherical
harmonics Yj,0 ., /, i.e.50
D
E 2
P.j/ / ϕbend ./Yj,0 ., / .25/
With the help of further approximations (semiclassical limit of
the rotor wave functions, harmonic bending wave functions)
one can derive a very simple analytical expression for the
projection integral,
2
h
i .k/
jh̄
P.j/ / sin2 je C . 1/k
ϕbend
4
mω
Q bend
.26/
where e is the equilibrium angle at the TS, m
Q is the moment
.k/
is the harof inertia, ωbend is the bending frequency, and ϕbend
monic wave function for the kth bending level with argument
jh̄/mω
Q bend .
Equation (26) predicts that in the limit of very weak exit
channel interaction the rotational probability is proportional to
the angular momentum space distribution of the bending wave
function of the parent molecule. Thus, if the dissociation starts
from the ground bending level, k D 0, the resulting rotational
distribution is a unimodel distribution with maximum near the
lowest rotational state. On the other hand, if the dissociation
begins in an excited bending state, k ½ 1, P.j/ will have a
bi-, tri-, or multimodal structure, just like the corresponding
wave function in the coordinate space. These distributions are
superimposed by ‘fast’ oscillations caused by the sinusoidal
term in equation (26). In the case of vanishing exit channel
coupling the final rotational state distribution is a more or less
direct mapping of the initial wave function. A representative
example is the photodissociation of H2 O in the first absorption
band;51 both the multimodel structure, if one starts from
excited bending states, and the ‘fast’ oscillations have been
experimentally verified.50,52
4.2 Rotational Reflection Principle
The situation is quite different if the -dependence of the
potential is substantial beyond the TS. In that case the rotor
experiences a large torque and is excited (or de-excited) on its
way to the asymptote. The resulting rotational state distribution
is usually ball-shaped with a peak at large values of j. The
maximum jmax reflects more or less directly the strength of the
rotational coupling.1 An example following the dissociation
of ClNO into Cl and NO is shown on the right-hand side of
Figure 9.
In the case of strong exit channel coupling classical
mechanics gives the simplest interpretation. The central quantity is the rotational excitation function J.0 /, which is nothing
other than the final rotational angular momentum of the rotor,
j.t D 1/, as a function of the initial angle 0 . It has to
be determined by calculating several classical trajectories for
different values of 0 . The corresponding function for the fragmentation of ClNO, shown on the left-hand side of Figure 9,
CPA11-
10 PHOTODISSOCIATION DYNAMICS
50
50
40
30
30
20
20
j
J(γ0) 
IM
40
W (γ0)
10
0
100
10
0
120
γ0 [deg]
140
0.5
P ( j)
Figure 9 Illustration of the rotational reflection principle for the
photodissociation ClNO ! Cl C NO. J.0 / is the rotational excitation function as a function of the initial angle 0 and W.0 / is the
weighting function. The dashed curves correspond to a simple kinematic model (impulsive model, IM). Reprinted, with permission of
the American Institute of Physics, from J. Chem. Phys., 1990, 93,
1098
is, in the relevant angular range, a monotonic function of the
initial angle. The classical approximation for the final state
distribution is then given by1
dJ P.j/ ³ W[0 .j/] d0 While the molecule breaks apart in the excited electronic
state it can spontaneously emit a photon and thereby make
a transition back to the ground state (see Figure 1). This
process is known as radiative recombination or Raman scattering (i.e., the molecule absorbs one photon with frequency
ω and emits another one with a different frequency ω0 ). Since
the ‘lifetime’ in the excited state, typically 10 13 10 12 seconds, is very short compared to the radiative lifetime of about
10 9 10 8 seconds, the probability for this process is exceedingly small. Nevertheless, the emitted photons can be counted
and dispersed.54,55 The emission spectrum contains basically
two types of information: (i) The energies of the vibrational
levels in the ground electronic state, which are related to
the emitted frequencies by Ei C h̄ω D Ef C h̄ω0 if Ei and Ef
denote the energies of the initial and the final state, respectively. (ii) The intensities of the emission lines, which reflect
the dissociation dynamics in the excited electronic state as well
as the vibrational dynamics in the ground electronic state. The
frequencies can be used to determine the energies of highly
excited vibrational states, close to the dissociation threshold,
which may not be accessible otherwise. The intensities contain
further clues about the bond-breaking mechanism and therefore
are relevant for photodissociation studies.
5.1 Time-independent View
1
.27/
0 D0 .j/
where the particular angle 0 .j/ is determined by the equation
J.0 / D j D 0, 1, . . .
5 EMISSION SPECTROSCOPY OF DISSOCIATING
MOLECULES
Raman scattering is a two-photon process and is described
by second-order perturbation theory. The cross section for a
transition from initial state jgr,i i with energy Ei to a final
state jgr,f i with energy Ef through absorption of a photon
with frequency ω is given by the Kramers Heisenberg Dirac
formula56
.28/
The weighting function W.0 / is also shown in Figure 9; it
basically reflects the angular dependence of the ground-state
wave function, jϕbend .0 /j2 . Since the excitation function is
monotonic, equation (28) has exactly one solution for any
given value of j so that there is an unique relation between j on
one hand and 0 on the other. The final rotational state distribution P.j/ is then a direct reflection of the initial angular distribution function mediated by the rotational excitation function
J.0 /, hence the name rotational reflection principle.37 This
kind of dynamical mapping is completely different from the
FC mapping of the initial wave function in the case of very
weak final state interaction.
Rotational state distributions like the one in Figure 9 have
been measured for many systems. If the initial wave function
has nodes along the angular coordinate, these nodes will be
reflected as undulations in the state distribution. If the decay
proceeds through a transition state, instead of the bending wave
function of the parent molecule it is the wave function at the
TS that is reflected.53 The rotational reflection principle not
only provides a very simple explanation of final rotational
state distributions, it also shows that complicated ‘brute force’
calculations are not always necessary; sometimes simple mechanistic pictures can provide simple insight as well as quantitative agreement with full quantum mechanical calculations.37
.R/
if
.ω/ / ωω03 jtres C tnonres j2
.29/
where the resonant term is defined by
Z
tres D lim
!0
dE
X
.Ei C h̄ω
E C i/
1
˛
D
ED
E
.˛/
.˛/ ð gr,i gr,ex ex,E
ex,E
ex,gr gr,f
.30/
.˛/
The ex,E
are the continuum wave functions in the excited
state for a particular energy E and a distinct final channel ˛.
The nonresonant term is much smaller than the resonant one
and therefore it is usually neglected.
The integration over the (continuous) energy E and the
summation over all channels ˛ are necessary in order to
take into account the excitation of all states in the excited
manifold (completeness relation). It is this integration over all
energies in the excited state and the summation over all product
channels that makes the Kramers Heisenberg Dirac formula
very difficult to apply. In order to compute the emission
spectrum, one first has to determine continuum wave functions
and their overlap with all bound-state wavefunctions for all
possible energies and subsequently to integrate over E. While
this is rather straightforward for a system with one degree of
freedom, it becomes very tedious for a triatomic molecule.
In addition, the time-independent expression does not offer
simple interpretations.
CPA11-
PHOTODISSOCIATION DYNAMICS
5.2 Time-dependent View
As for the absorption cross section, it is possible to derive
an alternative time-dependent expression. The time-dependent
formulation of Raman scattering is computationally simpler
and provides a better means of extracting dynamics information from the emission spectrum.27,28 Its derivation from
the time-independent expression is rather simple and basically
exploits the equality1,57
1
Ei C h̄ω
E C i
D
i
h̄
Z
1
dtei.Ei Ch̄ω
ECi/t/h̄
.31/
o
The Raman cross section is then given by
Z
1
D
E 2
.R/
if
.ω/ / dtei.Ei Ch̄ω/t/h̄ f .0/i .t/ o
.32/
,
where f .0/ D gr,ex gr,f and i .t/ is the wave packet that
is propagated in the excited state with the initial condition
i .0/ D gr,ex gr,i . Thus, what one has to do is: (i) place
the initial wave function, multiplied by the transition dipole
moment, on the upper-state PES, (ii) propagate this wave
packet as has been described above, (iii) calculate the timedependent overlap with the bound-state wave functions in the
ground electronic state (cross correlation functions), and (iv)
Fourier transform this overlap to yield the emission spectrum. This procedure is very similar to the time-dependent
computation of absorption cross sections and it is numerically much simpler than the time-independent approach. More
importantly, it allows one to interpret the emission spectrum
in terms of the evolution of the excited-state wave packet,
which at least for short times has a simple behavior.
An example of an emission spectrum from a dissociating molecule is depicted in Figure 10. Shown is the intensity
distribution for water excited in the first absorption band,
Intensity
160.0 nm
171.4 nm
1
2
3
4
5
6
N
Figure 10 Low resolution Raman spectrum for H2 O excited in the
first continuum for two wavelengths. N is the total number of OH
stretching quanta. The open bars represent experimental results and
the shaded and black bars are the results of two different calculations.
Reprinted, with permission of the American Institute of Physics, from
Ref. 58
11
Q 1 A1 , as a function of the total number of OH stretchAQ 1 B1 X
ing quanta in the electronic ground state of water.51,58 When
water is excited to the upper state it first performs a symmetric
stretch motion in which both OH bond distances are simultaneously elongated. This initial motion leads to the excitation
Q and
of highly excited vibrational stretching states in H2 O(X)
hence to the quite long progression in the emission spectrum.
Two different calculations, using the same upper-state PES
but different transition dipole moment functions, quantitatively
reproduce the measured spectrum. In conclusion, emission
spectra reflect, in a certain way, the early motion of the wave
packet in the excited state and therefore they provide information about the dissociation dynamics, which supplements the
information gained from the absorption spectrum and the final
state distributions.
6 NONADIABATIC EFFECTS IN DISSOCIATING
MOLECULES
Up to this point we exclusively discussed processes taking
place on a single PES without coupling to other electronic
states, i.e., processes for which the Born Oppenheimer (BO)
approximation, the workhorse of molecular dynamics, is valid.
This is more the exception rather than the rule. Many bondbreaking processes, if not most, evolve on several potential
energy surfaces and violate the BO approximation. This is not
unexpected if one recalls that with increasing excitation energy
the density of electronic states generally increases as well,
i.e., the separation between different PESs diminishes and as a
consequence transitions between different states become more
and more probable. Such transitions generally occur in the
vicinity of avoided crossings and conical intersections where
the mixing between different states is, by definition, largest
(see Curve Crossing Methods).
The theoretical description of nonadiabatic transitions
is quite difficult. For a correct description one needs
those elements which are usually neglected in the BO
approximation,59,60
+
*
+
∂ 2 .a/
.a/ ∂ .a/
.a/

and


0
0
k k ∂Q k 6Dk
∂Q2 k 6Dk
*
.33/
where the .a/
k are electronic wave functions, calculated in
an adiabatic way for fixed nuclear coordinate Q (see Nonadiabatic Derivative Couplings). These coupling elements are
usually ignored in dynamical treatments with the argument that
the motion of the electrons is much faster than the nuclear
motion so that the two can be adiabatically decoupled, i.e.,
one first solves the Schrödinger equation for the electrons with
the nuclear coordinates being fixed, which yields the potential
energy surfaces, and subsequently one solves the equations of
motion for the nuclei on these potential energy surfaces. However, when two potential surfaces are close to each other, this
approximation might break down and decoupling electronic
and nuclear motion become unjustified.
As a consequence one has to take into account simultaneous
motion on several potential energy surfaces and the coupling
elements between them. This is an extraordinary task as the
calculation of the kinetic coupling elements in equation (33)
is extremely difficult when several nuclear degrees of freedom
are active. Rather than working in the adiabatic representation
CPA11-
12 PHOTODISSOCIATION DYNAMICS
(index a) one makes a uniform transformation to a diabatic
representation (index d) in which the above matrix elements
are as small as possible,1,61
1.d/
!
2.d/
cos sin !
D
.a/
1
D
sin !
.34/
.a/
2
cos where the .d/
k are the new, diabatic electronic wave functions
and .Q/ is the coordinate-dependent mixing angle. The price
to be paid is that the electronic Hamilton operator in the new
representation is not diagonal. However, the coupling due to
the nondiagonal elements of the electronic Hamilton operator
is easier to take into account than the kinetic energy coupling
terms and therefore the diabatic representation is in practice
the method of choice.
In order to adequately describe the coupled motion on two
excited-state PESs we have to consider two nuclear wave
packets .d/
exk .t/, k D 1 and 2, if we prefer the time-dependent
view. The time-evolution of the molecular system is then
described by two time-dependent Schrödinger equations,
∂
ih̄
∂t
.d/
ex
1
.d/
ex
2
!
"
D
T11
0
0
T22
!
C
.d/
V11
.d/
V12
.d/
V21
.d/
V22
!#
.d/
ex
1
.d/
ex
2
!
.35/
where the coupling is represented by the nondiagonal potential
.d/
elements V.d/
12 D V21 and T11 and T22 represent the kinetic
energy operators in the two diabatic states. The coupled
Schrödinger equations have to be solved subject to the initial
conditions
.d/
ex
.t D 0/ D gr,exk gr,i
k
.36/
with gr,exk being the transition dipole function between the
ground state and the kth excited electronic state.
An instructive example for dissociation dynamics on two
excited states is the dissociation of H2 S in the first absorption band.62 In the diabatic picture, the electromagnetic field
1.0
0.8
PA(t)
P(t)
0.6
0.4
PB(t)
0.2
0
0
20
40
t[fs]
Figure 11 Time-evolution of the population of the diabatic states
B.1 B1 / and A.1 A1 / in H2 S, PB .t/ and PA .t/, respectively. The full
line shows the modulus of the autocorrelation function. Reprinted,
with permission of the American Institute of Physics, from Ref. 62
promotes the molecule essentially to a binding state, which
by itself can not dissociate. However, this binding state is
strongly coupled to a state with repulsive PES which allows the
molecule to decay. This process is called electronic predissociation in contrast to the vibrational predissociation discussed
in Section 3. Figure 11 shows how the populations of the two
diabatic states depend on time. Merely 20 femtoseconds are
required to reshuffle the population from the binding to the
dissociative state, i.e., damping due to nonadiabatic coupling
to the dissociative state is very efficient. The tiny recurrence
after roughly 30 femtoseconds leads to very diffuse vibrational
structures in the absorption spectrum similar to the ones seen
in Figure 8. Other examples of nonadiabatic transitions during
fragmentation, which have been theoretically studied, include
the Renner Teller effect in HCO.63,64 In this case, the coupling between the two relevant electronic states is caused by
coupling of electronic and nuclear angular momenta.
7 UNIMOLECULAR DISSOCIATION IN THE
GROUND ELECTRONIC STATE
In the preceding sections we discussed cases in which the
molecule is first electronically excited to an upper-state PES
before it dissociates, in the second step, on this PES. Dissociation can, of course, also take place directly on the ground-state
PES, without ‘detour’ via an excited state. This process is
known as unimolecular dissociation and plays an important
role in combustion processes or atmospherical chemistry65,66
(see Quantum Theory of Chemical Reaction Rates and Unimolecular Reaction Dynamics). Experimentally, unimolecular dissociations can be directly measured either by overtone
excitation67 (OH stretching vibration, for example) or by stimulated emission pumping from an excited electronic state.68 On
the theoretical side, all the methods described in the previous
sections can be used to study unimolecular processes.
Ground-state potentials of chemically bound molecules, like
H2 O, HNO, or NO2 have deep wells supporting hundreds or
even thousands of bound states below the first dissociation
threshold. Depending on the shape of the PES, the manifold
of true bound states does not abruptly terminate at the fragmentation threshold but extends into the continuum, where the
true bound states become ‘quasi-bound’ states or resonances
(in the language of scattering theory). Unlike real bound states,
resonances have a finite lifetime and therefore finite widths
 D h̄/ in an energy resolved spectrum. The resonances
above the threshold of the ground-state PES are, of course,
equivalent to the resonances shown in Figure 8 for the photodissociation of the XNO systems. However, in those cases
the potential wells are very shallow and therefore the series
of resonances are comparably short. As a consequence of the
substantially higher density of states for ground-state potentials
there are many more resonances, which makes their investigation very interesting from a dynamical point of view.44,69
Resonances are expected to occur whenever the PES has a
well. Storage of energy in internal modes ‘perpendicular’ to
the dissociation mode delays the dissociation and thereby traps
the system for some particular time.
There are three quantities which specify a resonance: (i)
the position Eres , (ii) the width , and (iii) the final state
distribution P.˛/ of the products resulting from the decay
of the resonance. In principle, all these three observables
CPA11-
PHOTODISSOCIATION DYNAMICS
depend uniquely on the particular resonance state excited.
The question then is, whether the dependence is ‘regular’ and
‘predictable’ in the sense that resonances belonging to a certain
progression show similar behavior or whether the decay is
‘statistical’ and ‘unpredictable’. The answer to this question
depends on the overall dynamics of the system, whether it is
regular or chaotic, and therefore on quantities like the density
of states (i.e., the number of states per unit energy interval)
and the coupling between the different degrees of freedom.
Systems with deep potential wells and consequently a
high density of states are a real challenge for exact quantum
mechanical theories. Advances in numerical approaches and
computer technology have made possible exact calculations
for realistic molecular systems only recently. In the following
we briefly describe one particular system, HCO, for which the
results of exact dynamics calculations using an accurate PES70
can be compared with state-of-the-art experimental data71 at
an unprecedented level of sophistication. Because of lack of
space the discussion must be very short but is intended to
4
This work
Tobiason et al.
Γ[cm−1]
3
(0,v2,0)
2
13
stimulate interest in other examples. Traditional descriptions of
fragmentation on ground-state potentials use mainly statistical
approximations66 (see Statistical Adiabatic Channel Model).
Q has a relatively shallow potential well that supHCO(X)
ports merely 15 bound states. However, a small barrier of
about 0.1 eV hinders the direct dissociation into H and CO.
This barrier together with very weak coupling between the
CO and the H CO vibrational modes allows one to temporarily store a large amount of internal energy, much more than
is necessary for cleaving the H CO bond, in the molecule.
The coupling between these internal modes and the dissociation degree of freedom means that slowly enough energy is
transferred into the dissociation mode; this allows the molecule
to break apart. The lifetime depends on the coupling strength
and can be as much as 50 picoseconds despite the fact that
the total energy exceeds the dissociation energy by more than
1 eV. The consequence is a long progression of narrow resonance states high above the dissociation threshold. The widths
of these resonances exhibit large fluctuations of up to three
orders of magnitude, even at the highest energies considered,
which manifest a distinct state-sensitivity of the dissociation
dynamics.72
Because of the weak intermode coupling, many of the resonances can be uniquely assigned to three quantum numbers,
1 (H CO stretch), 2 (CO stretch), and 3 (HCO bending),
and therefore it is meaningful to investigate how the resonance
width depends on these three quantum numbers. In Figure 13
1
0
20
(0,v2,1)
Γ[cm−1]
15
10
5
0
80
(0,v2,2)
Γ[cm−1]
60
40
20
0
1
2
3
4
5
6
ν2
7
8
9
10
11
12
Figure 12 Resonance widths  in the unimolecular dissociation
of HCO for three progressions. 1 , 2 , and 3 are the number of
quanta in the HC stretching, the CO stretching, and the HCO bending
mode, respectively. Comparison between experimental and calculated
results. Reprinted, with permission of the American Institute of
Physics, from Ref. 70
Figure 13 Upper part: dynamics of the vibrational wave packet in
the electronic A state of Na2 for delay times between two and three
picoseconds. Lower part: experimental and theoretical pump-probe
ionization signal as a function of delay time between the two laser
pulses. Reprinted, with permission of Elsevier Science B.V., from
Ref. 79
CPA11-
14 PHOTODISSOCIATION DYNAMICS
we depict resonance widths for three progressions and compare
the predictions of theory with experimental values. First of all,
the agreement between theory and experiment is, on the average, very good. Second, one sees a distinct mode-specificity:
the widths for the pure CO stretching progression (0, 2 , 0) are
the smallest, i.e., because of the weak coupling to the dissociation mode they have the longest lifetimes; adding one or two
quanta of the bending degree of freedom significantly increases
(decreases) the widths (lifetimes). Furthermore, adding one
quantum of the H CO stretch (not shown in the figure) significantly accelerates the fragmentation. HCO is a prototype for
a regular and mode-specific system.
8 REAL-TIME DYNAMICS
In the preceding sections we outlined how one can theoretically describe photodissociation processes and infer information about the bond-breaking mechanism and ultimately
about the multidimensional PES when light sources with weak
intensities and long durations are used as in all traditional
experiments. Typical time-scales for fragmentation are 10 12
seconds or even shorter whereas pulse lengths of conventional
lasers are in the range of 10 9 seconds. Therefore, it is inherently impossible to analyze directly the time-dependence of
the dissociating molecule. A ‘revolution’ in molecular dynamics set in at the end of the 1980s, when new laser systems with
pulse lengths in the range of 10 13 (100 femtoseconds) or so
became available. With light sources that short it is possible
to ‘see’ directly molecular vibrations in the laboratory.7 9,73
A typical pump probe experiment uses two lasers, both
with pulse durations in the sub-picosecond regime. The first
(pump) laser excites the system from the ground to an upper
electronic state. Since the laser has a small temporal width t,
it excites a wave packet, i.e., a coherent superposition of all
stationary (time-independent) states, in the excited manifold
rather than a single stationary level as in an experiment with
an infinitely long light pulse; the energetic width of the wave
packet is given by Et ³ h. This wave packet is not an
eigenstate of the upper PES and therefore it starts to move
as described in Section 2.4. The motion of this wave packet
is then probed with the second (probe) laser, which is fired
with a well defined delay time after the first pulse. The probe
laser may promote the system to a third electronic state or
into the ionic continuum. The absorption spectrum or the
ionization yield as a function of the delay time and the probe
frequency then provide information about the time evolution
of the molecule on the upper potential energy surface.
Theoretically74 77 the pump and probe process is described
by three coupled Schrödinger equations for the three nuclear
wave packets, one evolving in the ground state, a second
one moving on the first excited PES, and the third wave
packet representing the motion in the second excited state (the
extension to more states, if required, is straightforward, at least
formally),
   H
gr
gr
∂ 
ih̄
ex1  D  Wex1 ,gr
∂t
ex2
0
Wgr,ex1
Hex1
Wex2 ,ex1
  
gr
Wex1 ,ex2   ex1  .37/
respectively. The coupling element
Wgr,ex1 D Wex1 ,gr D Epump .t/.e/
gr,ex1 cos.ωpump t/
.38/
couples the ground state to the first excited state and likewise
Wex1 ,ex2 D Wex2 ,ex1 D Eprobe .t/.e/
ex1 ,ex2 cos.ωprobe t/
.39/
couples the two excited states. gr,ex1 , and ex1 ,ex2 are the
corresponding transition dipole functions.
Equation (37) is very general and is applicable for arbitrary
electric fields, independent of the field strengths. The pulses
Epump .t/ and Eprobe .t/ can have any shape and peak amplitude.
Numerical methods for solving the set of coupled equations
are described in the literature.78 In all the preceding Sections
we considered merely the lowest two states, gr and ex1 ,
in the limit of the electric field being infinitely long and
the coupling being very weak. Under such conditions the
equations can be solved in first-order perturbation theory
yielding equation (5) for the absorption cross section. If these
conditions are not fulfilled the above coupled equations must
be solved numerically.
Figure 13 shows an example of a pump probe experiment.79 The upper part depicts the vibrational motion of Na2
in an excited electronic state as a function of time. The
oscillations of the wave packet between the inner and the outer
turning points are reflected in the ionization signal as a function
of the delay time in the lower panel of Figure 13. In this way it
is possible to make the internal motion of molecules ‘visible’
in real time. The pump probe signals are quite reminiscent of
the autocorrelation functions depicted in Figure 7.
The coupled equations (37) can also be used to study multiphoton excitation and multiphoton dissociation of molecules
in strong laser fields, both in the ultraviolet and the infrared
regions.80 82 If the field is sufficiently strong, in addition to
excitation from the ground state to the excited state the reverse
process is also possible. In other words, the whole molecule
is shaken up.
Finally, coherent control83,84 of the motion of a molecular
system with short and/or intense lasers has become a hot
topic in the 1990s (see Control of Microworld Chemical and
Physical Processes). The general question is very simple: can
one find a particular pulse or sequence of pulses that forces
the molecule to do what it refuses to do without the applied
light field? Several control schemes have been suggested and
applied to simple systems, mostly diatomics. The idea of
coherent control is very appealing; some day it might become
possible to open chemical reaction pathways, which are either
energetically forbidden or just too inefficient without external
field. However, a lot more investigations, both experimental
and theoretical, are required before this goal is achieved. The
internal motion of a realistic, multimode system is, except for
few simple cases, very complicated and to unravel this motion
is formidable in itself.
9 OUTLOOK
0
Hex2
ex2
where Hgr and Hexi .i D 1, 2/ are the unperturbed Hamilton
operators in the ground and the two excited electronic states,
Photodissociation is a fantastic field for studying the internal dynamics of polyatomic molecules. Nowadays it is possible
to investigate particular questions on an unprecedented level
of detail. Especially the close interplay between sophisticated
experiments and state-of-the-art theories has tremendously
CPA11-
PHOTODISSOCIATION DYNAMICS
advanced our understanding of this important elementary
reaction.
In the last decade or so a number of efficient numerical methods for solving the quantum mechanical equations of
motion have been developed. On the other hand, the calculation of accurate potential energy surfaces required for realistic
dynamics studies has also made significant progress. Both
developments have led to a number of very fine applications,
mostly for triatomic molecules, which can be considered as
prototypes for more complex systems.
For the future I see basically three routes for extending
the theoretical applications. 1. Non-adiabatic effects. Most
systems studied today are based on the Born Oppenheimer
approximation, that is, the dynamics evolves on a single potential energy surface. There are many examples for which this
is not the case, and very often these are the more important
cases as far as real applications are concerned (OClO and O3 ,
to name only two). The bottlenecks for calculations including
two or several coupled electronic states are the potential energy
surfaces and, even more importantly, the coupling elements
between the different states. Work in this direction is regarded
as very important. 2. Systems with chemically different product channels. Almost all examples that I know of have either
only one exit channel or two equivalent product channels like,
e.g., H2 O. Chemically as well as dynamically more interesting,
however, are systems like HNCO, which have two different
channels, H C NCO and HN C CO. Such molecules are for
obvious reasons much more difficult to treat than systems
with a single exit channel. Even the calculation of realistic
potential energy surfaces is much more involved, not to mention the propagation of quantum mechanical wave packets.
3. Applications to large systems. Eventually it may become
feasible to study even a system like HNCO, with ‘only’ six
internal degrees of freedom, in full complexity. However, that
is certainly impossible for still larger systems with more than
four atoms, organic molecules, for example. If one wants to
describe photochemical systems in organic chemistry, it is
absolutely essential to devise theoretical methods that can handle many degrees of freedom. The combination of classical
mechanics for the less important bonds and quantum mechanical wave packets for the most crucial degrees of freedom
seems to be most promising (see Mixed Quantum-classical
Methods). However, the determination of a reasonable multidimensional potential energy surface will still be the main
obstacle.
Theoretical chemistry has been very fruitful in the past
for our understanding the breakup of comparatively simple
molecules. I am convinced that theoretical chemistry will be
likewise successful in explaining more complex molecules.
10 RELATED ARTICLES
Spectroscopy: Computational Methods; Classical Trajectory Simulations of Molecular Collisions; Control of
Microworld Chemical and Physical Processes; Integrating
the Classical Equations of Motion; Rovibronic Energy Level
Calculations for Molecules; Mixed Quantum-classical Methods; Nonadiabatic Derivative Couplings; Classical Dynamics of Nonequilibrium Processes in Fluids; Photochemistry;
Quantum Theory of Chemical Reaction Rates; Reactive Scattering of Polyatomic Molecules; State to State Reactive
15
Scattering; Statistical Adiabatic Channel Model; Multiphoton Excitation; Time-dependent Multiconfiguration Hartree
Method; Transition State Theory; Unimolecular Reaction
Dynamics; Curve Crossing Methods; Vibronic Dynamics in
Polyatomic Molecules; Wave Packets.
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