Fragmentation of Na3+ clusters following He impact: theoretical analysis of fragmentation mechanisms

Article abstract of DOI:10.1063/1.481561

The fragmentation of an Na₃⁺ cluster following a collision with a fast He atom was theoretically studied. Trajectory results indicate that very different mechanisms contribute to the fragmentation process even in the simplest pathway when the cluster always stays in the ground electronic state and fragments adiabatically. The results are explained in terms of the binary, diatom, sequential, and complex mechanisms.

Full text of DOI:10.1063/1.481561

Fragmentation of Na 3 + clusters following He impact: Theoretical analysis of
fragmentation mechanisms
D. Babikov, E. Gislason, M. Sizun, F. Aguillon, and V. Sidis
Citation: The Journal of Chemical Physics 112, 9417 (2000); doi: 10.1063/1.481561
View online: http://dx.doi.org/10.1063/1.481561
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 112, NUMBER 21
1 JUNE 2000
Fragmentation of Na¿3 clusters following He impact: Theoretical analysis
of fragmentation mechanisms
D. Babikov and E. Gislasona)
Department of Chemistry, University of Illinois at Chicago, SES (M/C 111), 845 West Taylor Street,
Chicago, Illinois 60607-7061
M. Sizun, F. Aguillon, and V. Sidis
´
ˆ
´
Laboratoire des Collisions Atomiques et Moleculaires, Bat. 351, Unite Mixte de Recherche C8625,
´
Universite Paris-Sud XI, 91405 Orsay Cedex, France
͑Received 28 January 2000; accepted 2 March 2000͒
The process of Naϩ3 cluster ion fragmentation following excitation by a fast He atom is studied using
a theoretical procedure developed earlier. The collision with He leaves Naϩ3 in any of three
electronic states, and each of these can fragment into three product channels ͑Naϩ2 ϩNa, Na2ϩNaϩ
and NaϩNaϩNaϩ͒. Attention is focused on understanding both the dynamics of the Naϩ3 –He
interaction and the post-collisional fragmentation of the excited cluster. Four simple fragmentation
mechanisms are proposed to describe the major features of the process. Contributions of these
mechanisms to different fragmentation pathways are determined and their dependence on the initial
internal energy of the cluster is studied. Fragmentation intensity maps are calculated and good
agreement with experiment is obtained. © 2000 American Institute of Physics.
͓S0021-9606͑00͒01620-2͔
I. INTRODUCTION
trajectory surface hopping ͑TSH͒ approximation. Special at-
tention is paid to the conical intersection of the two excited
electronic states and to the avoided crossing in the product
region between the ground and first excited state. The earlier
theoretical work focused on calculating total cross sections4
and their dependence on the initial cluster internal energy.5
In addition, doubly differential cross sections of intensity
plotted against the Naϩ3 –He scattering angle and the kinetic
energy of the products were calculated4,6 and compared with
the experimental results for process ͑1͒.
In the present work we seek to determine the various
fragmentation mechanisms that lead to the product channels
in ͑1͒. For a proper understanding of a mechanism one needs
to know both how the He interacts with the various Na nuclei
during the Naϩ3 –He encounter and also how the Na nuclei
interact with each other as the Naϩ3 fragments after the He
has departed. Most of this work is done for fragmentation to
channel A on the ground electronic state; this corresponds to
the common case of electronically adiabatic dissociation on
the ground state of a polyatomic molecule. We also apply
our methodology to fragmentation processes that involve an
excited electronic state, because we wish to know if this
gives rise to additional important mechanisms. Finally, we
examine how the initial internal energy of the Naϩ3 cluster
affects the relative importance of the various mechanisms.
The fragmentation of the sodium cluster ion Naϩ3 follow-
ing collision with a He atom at 263 eV center-of-mass en-
ergy has recently been investigated experimentally.1–3 The
following processes were observed:
Naϩ3 ϩHe→Naϩ2 ϩNaϩHe ͑channel A)
→Na2ϩNaϩϩHe ͑channel B)
→NaϩNaϩNaϩϩHe ͑channel C).
͑1͒
The experimental results show evidence for two qualitatively
different fragmentation mechanisms. The first involves an
electronic excitation process, which brings the cluster to a
dissociative state that rapidly fragments. The second in-
volves an impulsive process whereby sufficient momentum
is transferred from the He atom to the atoms in the cluster to
induce fragmentation on the ground electronic state. How-
ever, a more detailed examination of the experimental data
suggests that several simple mechanisms are involved in the
fragmentation process.
We have studied process ͑1͒ theoretically.4–6 The calcu-
lations make use of the fact that the Naϩ3 –He collision is a
fast process (tcollϳ10Ϫ15 s) in comparison to the cluster vi-
brations (tvibϳ10Ϫ13 s). This allows us to study the Naϩ3 –He
encounter separately from the fragmentation of Naϩ3 , and, in
addition, the collision is treated by assuming that the Na
nuclei are fixed. In the encounter the He passes near or
through the Naϩ3 . The fragmentation of the Naϩ3 on the three
electronic states after the He departs is studied using the
II. THEORY
The theoretical procedure is described in detail
elsewhere6 and is briefly summarized here. The three lowest
singlet adiabatic potential-energy surfaces ͑PESs͒ of Naϩ3 are
computed from Kuntz’s DIM ͑diatoms in molecules͒ treat-
ment of this system.7,8 We identify the ground state as state
1, and the first and second excited states as states 2 and 3,
a͒
Author to whom correspondence should be addressed. Electronic mail:
gislason@uic.edu.
0021-9606/2000/112(21)/9417/10/$17.00
9417
© 2000 American Institute of Physics
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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000
Babikov et al.
potential energy seen by the He is calculated using the semi-
classical energy conserving trajectory ͑SCECT͒ method.10 In
most cases the He interacts strongly with only one Na atom,
but multiple hits do occur. After the He departs, the Naϩ3 is
left with population in the three electronic states, and addi-
tional momentum has been transferred to each Na nucleus.
Clusters with insufficient energy to fragment are not consid-
ered further. The final step of the calculation is a TSH study
of the fragmentation of Naϩ3 with proper treatment of the
conical intersection and avoided crossing.11,12
In the experimental work,1–3 the Naϩ3 clusters are prob-
ably produced as the result of thermal fragmentation of
larger clusters, which were previously ionized by an electron
beam. After such a process Naϩ3 clusters in the ground elec-
tronic state have negligibly low ͑thermal͒ rotational energy,
but significant amounts of vibrational excitation ͑typically up
to the dissociation limit of Naϩ3 ͒. Consequently, we have
performed calculations for three values of the initial vibra-
tional energy Evibϭ0.02, 0.5, and 1 eV. The first value cor-
responds to the zero-point-energy of the cluster, and the last
is just 0.14 eV below the dissociation limit to channel A.6 A
total of 50 000 collisions with randomly chosen initial con-
ditions was studied. The Naϩ3 –He impact parameter and the
orientation of the cluster plane were chosen randomly. The
calculation was stopped when two Na–Na distances ex-
ceeded 18.5 a.u. At that point the product channel and prod-
uct properties were determined. In a few cases the excited
cluster had not dissociated after a time of 400 000 a.u. These
events were not considered further.
To better understand the fragmentation mechanisms we
have also done a series of calculations for a fixed initial
shape of Naϩ3 . This is an elongated T-shaped cluster when
the distance r between two nuclei is 6.4 a.u. and the distance
R from the third atom to the center-of-mass of the other two
is 11.5 a.u. This configuration corresponds to a classical
turning point of Naϩ3 with 1.0 eV of vibrational energy. We
have seen earlier6 that this is a typical configuration for ex-
citation to state 2 for a hot Naϩ3 cluster. One property moni-
tored during these calculations was the location of the He
when it passes through the plane of Naϩ3 . These calculations
give a useful qualitative picture of the excitation process in
the Naϩ3 –He collision. To distinguish the two types of atoms
in this bound cluster ion, we shall refer to the two close
atoms as the ‘‘dimer’’ and the distant atom as the ‘‘mono-
mer.’’ Clearly the two atoms of the dimer are strongly
bound, whereas the monomer is only weakly bound to the
dimer. We also monitored which atom is ejected from the
cluster when it fragments into an atom and a diatomic.
Any process induced by the Naϩ3 –He collision can be
assigned to a particular pathway by specifying the final prod-
uct channel ͑A, B, or C͒ and the electronic state of Naϩ3 ͑1, 2,
or 3͒ just after the collision with He.4–6 Thus, collision and
dissociation into two and three fragments, respectively, on
the ground state correspond to pathways A1 and C1. Simi-
larly, excitation of state 2 or 3 by He, followed by adiabatic
fragmentation corresponds to pathways B2, C2, or C3. Fi-
nally, the ‘‘off-diagonal’’ elements ͑A2, A3, B1, and B3͒
correspond to pathways with nonadiabatic transitions during
the fragmentation process. Specification of these pathways
FIG. 1. Contour plots of the three PESs for T-shaped Naϩ3 as a function of
Jacobi coordinates ͑r,R͒. Zero energy corresponds to completely separated
atoms. Contour lines are given between Ϫ2 and 2 eV with interval of 0.25
eV. The PES of state 1 ͑upper view͒ exhibits a well of Ϫ2.12 eV in equi-
lateral geometry. For large R, it displays a valley leading to the Naϩ2 ϩNa
products ͑Ϫ0.98 eV͒. The PES of state 2 ͑intermediate view͒ at large R has
a valley, leading to the Na2ϩNaϩ products ͑Ϫ0.75 eV͒. State 3 ͑lower view͒
leads to the three-body fragmentation NaϩNaϩNaϩ. States 1 and 2 have an
avoided crossing in the product region ͑labeled A.C.͒. States 2 and 3 exhibit
a conical intersection ͑labeled C.I.͒ in equilateral geometry.
respectively. States 1 and 2 have an avoided crossing in the
product region and states 2 and 3 exhibit a conical intersec-
tion in the equilateral configuration ͑see Fig. 1͒. The cou-
plings between electronic states of Naϩ3 in the presence of He
are calculated in the DIM formalism, but they also include
Na–He–Na three-center-interaction terms.6 The Na3ϩ is pre-
pared with zero rotational energy and at a fixed classical
vibrational energy with random phases. Next the Naϩ3 –He
collision at a relative energy of 263 eV is treated using the
classical path approximation9 for the He motion with the Na
nuclei frozen ͑the sudden approximation͒. The instantaneous
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Fragmentation of Naϩ3 clusters
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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000
Figures 2͑b͒ and 2͑c͒ show two important examples of
possible fragmentation processes that give products at par-
ticular ⌽ values. Figure 2͑b͒ shows the collision where He
collides strongly with only one atom of the cluster and ejects
it from the cluster. The other two atoms are passive specta-
tors and continue in the forward direction. In that case the
collision and fragmentation planes coincide and ⌽ϭ0° or
⌽ϭ360°. We shall see that these ‘‘binary’’ collisions are
very important in the fragmentation process. By comparison
Fig. 2͑c͒ shows the case where the collision with He ejects
two sodium atoms from the cluster. In that case the third Na
atom is a passive spectator and continues in the forward di-
rection. Again the collision plane coincides with the disso-
ciation plane, but now ⌽ϭ180° for this ‘‘diatom’’ process.
Our definition of ⌽ coincides with the definition of the ex-
perimental group1–3 for product channel A, but for channel B
we reverse their definitions for ⌽ϭ0 and ⌽ϭ180°. This
difference does not cause any confusion in this paper be-
cause no experimental ⌽-distributions were obtained for
channel B.
FIG. 2. Definition of ⌽ angle in Naϩ3 fragmentation after the collision with
a He atom. The figure shows the Naϩ3 –He center-of-mass reference frame.
Initial velocity vectors for all species are dashed and the final velocity vec-
tors are solid. Vector C locates the center-of-mass of Naϩ3 after the collision.
Vector D connects the diatomic center-of-mass and the atomic fragment. ͑a͒
Definition of ⌽ angle ͑general situation͒. See text for further details. The
fragmentation plane is shaded. ͑b͒ Example of the direct binary collision:
⌽ϭ0 or 360°; ͑c͒ Example of the ‘‘diatom’’ process: ⌽ϭ180°.
It is also possible to define ⌽ for the case of three body
fragmentation. The relative velocities between each pair of
Na nuclei are determined, and the smallest relative velocity
identifies the ͑pseudo͒ diatomic fragment. The third Na
nucleus is identified as the ejected atom. The definition of ⌽
is then made as before.
Another important observable of the fragmentation pro-
cess is ⑀, the sum of the kinetic energies of the two or three
fragments in the Naϩ3 -center-of-mass reference frame. This
energy, as well as the angle ⌽, are experimental observables.
Intensity contour maps as a function of the energy spectrum
of the dissociation fragments and the ⌽ angle have been
obtained experimentally,1–3 and we will compare our results
with those data.
During the collision with He each of the Na nuclei re-
ceives a momentum transfer due to the interaction with He. It
is useful to denote the three Na nuclei by X, Y, and Z, and
identify the Na atom that receives the largest momentum
transfer as X. Thus, we say that He ‘‘hits’’ atom X. ͑Of
course, all three Na nuclei gain momentum from He, but it is
usually the case that one receives a much larger impulse, so
it is reasonable to say that He hits X.͒ If the cluster then
fragments into an atom and a diatomic, there are two possi-
bilities. In one case XϩYZ is produced, and in the other X is
combined with Y or Z. We assume for definiteness in that
case that the products are XYϩZ. When the product is XY it
is useful to define another angle ␺. We identify the vector
momentum transferred to atom X as p, where p in general is
not confined to the plane of the Naϩ3 . Then ␺ is defined as the
angle between p and the vector drawn from X to the product
atom Z ͑see below͒.
implies that there is at most one hop between surfaces during
the fragmentation process. ͑Pathway A3, however, requires a
minimum of two transitions as the system hops from state 3
to state 2 and then from state 2 to state 1.͒ In fact, multiple
hops do occur but they are rare.
An important characteristic of process ͑1͒ is the time
scale of the fragmentation. We denote the fragmentation time
as Tfr , and we define it as the time when two Na–Na dis-
tances become greater than 18.5 a.u. The zero of time is
taken as the instant when the He departs the Naϩ3 fragment,
and we begin the TSH calculation. Atomic units are used and
for comparison typical vibrational periods for the normal
modes of Naϩ3 are 10 000 a.u.
One of the most important observables in the two-body
fragmentation process is the angle between the collision
plane and dissociation plane, denoted ⌽. This angle is rou-
tinely used to interpret experimental data.1–3 The definitions
of both planes and the angle ⌽ are given in Fig. 2͑a͒. The
collision plane is defined as containing the initial and final
velocity vectors of the He atom. The fragmentation plane
contains the final velocity vector of the He atom and the
vector connecting the center-of-mass of the diatomic frag-
ment with the atomic sodium fragment ͑vector D in Fig. 2͒.
These two planes cross each other along the line containing
the final velocity vector of He. It is obvious that the center-
of-mass of the cluster ͑vector C in Fig. 2͒ also moves along
this line after the collision. ⌽ϭ0° is defined as the case
where the collision plane and the fragmentation plane coin-
cide and the velocity of the center-of-mass of the diatomic is
closer to the initial velocity vector of the He atom. This
situation is shown in Fig. 2͑b͒.
III. ANALYSIS OF MECHANISMS FOR PATHWAY A1
We have calculated total cross sections for each path-
way, and the results for Eintϭ1.0 eV are summarized in the
fragmentation matrix shown in Table I. These cross sections
have been discussed in details in earlier papers.5,6 Here we
note that the cross section for producing state 1 is about four
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9420
J. Chem. Phys., Vol. 112, No. 21, 1 June 2000
Babikov et al.
TABLE I. Fragmentation matrix.a
cess takes place adiabatically on the ground electronic state.
A number of results are immediately apparent. First, there is
a high density of points ͑trajectories͒ at small times near ⌽
ϭ0 or 360°. We shall see that most of these events are due
to a binary collision where He hits one Na atom hard and
ejects it from the Naϩ3 cluster ͓see Fig. 2͑b͔͒. Second, there
are many points at large times, much larger than the vibra-
tional periods of Naϩ3 , and we shall see that these come from
long-lived complexes. Finally, there is a secondary peak at
small times near ⌽ϭ180°. This corresponds to the diatom
process described in the previous section ͓see Fig. 2͑c͔͒. We
conclude that a variety of mechanisms make significant con-
tributions to the fragmentation of Naϩ3 .
To determine and separate the various mechanisms we
consider first the time dependence of all trajectories for path-
way A1 with Eintϭ1.0 eV. Figure 4 gives a semilog plot of
intensity versus Tfr . At long time we find that the plot is
approximately linear, indicative of long-lived complexes that
dissociate statistically. We have fit this data as shown in the
figure, and we estimate that all collisions with
TfrϾ30 000 a.u. correspond to long-lived complexes. This is
confirmed by analysis of typical trajectories, which show that
in general these trajectories correspond to rovibrational exci-
tation just above the dissociation limit along with a complete
sharing of energy among all degrees of freedom. We there-
fore, define a ‘‘complex’’ collision to be any trajectory with
Tfr greater than 30 000 a.u. Also shown in Fig. 4 are the time
Electronic state after
He–Naϩ3 collision
Product
channel
Total values
͑observables͒
1
2
3
A:
B:
C:
8.05
0.65
0.84
9.54
1.28
0.94
0.30
2.52
0.00
0.00
0.19
0.19
9.33
1.59
1.33
AϩBϩC:
12.25
aThe columns labeled 1, 2, and 3 indicate the electronic states immediately
after the Naϩ3 –He collision, and the rows labeled A, B, and C denote the
three possible product channels ͓see Eq. 1͔. Each numerical entry is a total
cross section in Å2. Results are calculated for Eintϭ1 eV. The final column
gives the cross section for each row summed over states 1, 2, and 3; this is
the total product cross section for each channel. Similarly, the final row
gives the cross sections for each column summed over channels A, B, and
C; these are the total cross sections for populating states 1, 2, and 3 in the
collision with He.
times larger than that for state 2, which in turn is more than
ten times greater than for state 3. Pathway A1 is the domi-
nant process, but most other pathways have significant cross
sections. Only A3 and B3 are unimportant.
Figure 3 shows a distribution of fragmentation products
plotted against the fragmentation time Tfr and the angle ⌽ for
the nine pathways for Eintϭ1.0 eV. We consider first the A1
pathway, which has the largest cross section ͑see Table I͒. In
addition, it is the simplest to analyze because the entire pro-
FIG. 3. Distribution of fragmentation
events plotted against the fragmenta-
tion time Tfr and the angle ⌽ for all
nine pathways. The labeling corre-
sponds to the elements of the fragmen-
tation matrix. Results were obtained
for Evibϭ1 eV.
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Fragmentation of Naϩ3 clusters
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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000
FIG. 5. Distribution of fragmentation events corresponding to different frag-
mentation mechanisms plotted against the fragmentation time Tfr and the
angle ⌽ for Evibϭ1 eV and pathway A1. See text for further details.
FIG. 4. Semilog plot of fragmentation intensity vs time Tfr for some frag-
mentation pathways. For the pathway A1 ͑top͒ a linear fit is done and the
two dashed lines meet at T ϭ30 000 a.u.
Ј
behaviors for pathways B2, C1, and C3. It is clear that there
are no complex trajectories in these cases ͑see also Fig. 3͒.
After we subtract the complex trajectories we next con-
sider, from the remaining trajectories, the binary collisions.
As discussed earlier we identify the three Na nuclei as X, Y,
and Z, and the nuclei that receives the largest impulse from
He is denoted X, the ‘‘hit’’ atom. We define ‘‘binary’’ col-
lisions as those trajectories where the hit atom X is also the
ejected atom, so that the diatomic product is YZ. In most
cases we expect the products to appear near ⌽ϭ0 or 360°.
The long-lived complex and binary trajectories can be sepa-
rated from the others, and the results are shown in Fig. 5. As
expected, the ⌽-distribution of trajectories is uniform for the
complex mechanism.
The trajectories that remain show intensity peaks at ⌽
ϭ0, 180°, and 360° ͑see Fig. 5͒. They correspond to cases
where the hit atom X emerges in the diatomic product XY. As
discussed earlier there are two types of ideal collisions that
lead to peaks at ⌽ϭ180° and also at ⌽ϭ0 or 360°. The first
is the case shown in Fig. 2͑b͒, where the struck atom X reacts
with atom Y, while atom Z remains a passive spectator and
continues in the forward direction. In that case ⌽ϭ180°,
and we call that a ‘‘diatom’’ reaction. The angle ␺, shown in
Fig. 6, should be greater than 90° for this case to be certain
that there is no XZ collision. The second ideal collision oc-
curs when the struck atom X hits atom Z head on. This cor-
responds to ␺ϭ0. Since they have the same mass, Z is
ejected with all of the energy, X slows to the same velocity
as Y, and the XY molecule appears in the forward direction.
In that case ⌽ϭ0 or 360°. We call this the ‘‘sequential’’
mechanism. Of course, real collisions will not be so simple,
and there will be a spread in the ␺ and ⌽ angles. Figure 6
shows a two dimensional plot of all remaining trajectories
against ␺ and ⌽. The angle ␺ϭ90° separates trajectories
into two discrete groups.
We conclude that the angle ␺ is a useful way to separate
diatom and sequential trajectories. Therefore, we define the
‘‘diatom’’ mechanism to be all trajectories where the product
is XY and ␺Ͼ90°. In addition, we define the ‘‘sequential’’
mechanism to be all trajectories producing XY with ␺
Ͻ90°. The distribution of points for these two mechanisms
plotted against Tfr and ⌽ are also shown in Fig. 5. We have
now separated all possible trajectories into four mechanisms.
The separation scheme is summarized in Table II. We
call the binary and diatom mechanisms ‘‘direct’’ because the
hit Na atom X does not interact with one or both of the YZ
pair ͑at least in the ideal case͒. By comparison, we call the
sequential and complex mechanisms ‘‘indirect’’ because
atom X must interact strongly with both Y and Z. Table II
also gives the fraction of all trajectories that fall into the four
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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000
Babikov et al.
nuclei. In fact, for the fixed T-shaped cluster this is the only
important mechanism when the distant atom ͑the monomer͒
is hit. The calculations show that long-lived complexes are
caused by softer hits on one of the Na atoms; in this case, the
typical Na–He impact parameter is about 2 a.u. The sequen-
tial mechanism occurs when the He hits on the outside of the
cluster and knocks one of the Na atoms directly toward a
second, knocking that second atom out of the cluster. For the
fixed T-shaped cluster this always involves knocking one of
the atoms in the dimer towards the monomer and knocking
the monomer out of the cluster. Finally, the diatom mecha-
nism only occurs when one Na atom is far from the other
two. The He passes through the inside of the cluster ion and
hits one atom of the dimer. This atom recoils out of the
cluster and pulls the second atom of the dimer along with it.
This process occurs readily because both atoms in the dimer
are only weakly bound to the monomer.
Table II also gives the contributions of the four basic
fragmentation mechanisms to pathway A1 at the other two
initial internal energies ͑Eintϭ0.02 and 0.5 eV͒ of the cluster.
͑Absolute values of corresponding total cross sections were
published elsewhere.5,6͒ The direct binary mechanism domi-
nates in all cases, especially when the cluster is cold ͑94% at
Eintϭ0.02 eV͒. We can see that the contribution of the com-
plex mechanism is very small when the cluster is vibra-
tionally cold, and it increases more then 10 times when the
cluster is initially hot (Eintϭ1.0 eV). Finally, the contribu-
tion of the sequential mechanism does not depend on internal
cluster energy, while the diatom mechanism disappears com-
pletely when the cluster is cold.
FIG. 6. Distribution of fragmentation events corresponding to the sequential
and diatom mechanisms of fragmentation plotted against the angles ␺ and ⌽
for Evibϭ1 eV and pathway A1. The top shows the definition of the angle ␺.
In the bottom part the vertical dashed line at ␺ϭ90° illustrates the last step
of the separation procedure.
categories for Evibϭ1.0 eV. It is not surprising that 63% of
all reactions occur by the binary mechanism, because this
was already apparent in Fig. 3. The binary mechanism has
long been known to be very important in fragmentation
processes.13 It is somewhat more surprising to find that
nearly one-third ͑29%͒ of all reactions come from long-lived
complexes. Finally the diatom and sequential mechanisms
make relatively small and equal contributions.
To better understand these results we have made a de-
tailed examination of trajectories for the hot cluster, both for
the fixed T-shaped Naϩ3 cluster ͑see Sec. II͒ and also for the
general case. The analysis shows that the binary mechanism
occurs whenever the He atom makes a direct hit ͑a Na–He
impact parameter less than 2 a.u.͒ on any of the three Na
We believe we can explain all of these results. The Na3ϩ
cluster is fairly ‘‘large;’’ the bond length at the equilateral
minimum is 6.78 a.u. By comparison, the He is small, and it
must pass within 2 a.u. of an Na nucleus to give it a large
impulse. Simple geometry guarantees that only rarely does a
He atom collide strongly with two Na nuclei. Thus, the only
way to impart considerable impulsive energy to the Naϩ3 ion
is for the He to hit one nucleus very hard. In most cases that
Na is ejected, giving rise to the binary mechanism. If the
Naϩ3 is initially excited, a softer hit is sufficient to excite the
cluster above the fragmentation limit. In addition, all of the
vibrational modes of Naϩ3 are already excited, so it is much
easier for the hit atom to share energy with the other two and
TABLE II. Scheme for separation of mechanisms.a
Fragmentation
Separation steps and criteria
Contributions ͑%͒ to A1
mechanisms
Step I
Step II
Step III
ZPE
0.5 eV
1 eV
Direct
Binary
Diatom
Sequential
Complex
T ϽT
94
0
4
82
1
4
63
4
4
Ј
Ј
Ј
Ј
XϩYZ
XYϩZ
XYϩZ
fr
T ϽT
␺Ͼ90°
␺Ͻ90°
fr
Indirect
T ϽT
fr
T ϾT
2
13
29
fr
aSee text for the definitions of the fragmentation mechanisms and more complete description of the separation
criteria. Step I separates trajectories corresponding to the complex mechanism of fragmentation. Tfr is the
fragmentation time and T ϭ30 000 a.u. Step II separates the binary mechanism from the rest of trajectories. X
Ј
is the Na atom in the cluster XYZ that is hit by the He atom. Step III assigns the rest of the trajectories to either
the diatom or the sequential mechanism of fragmentation. The angle ␺ is shown in Fig. 6. The contributions of
these mechanisms to pathway A1 are shown for three values of the initial internal energy of the Naϩ3 cluster.
ZPE, the zero-point energy of Naϩ3 , is 0.02 eV.
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Fragmentation of Naϩ3 clusters
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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000
TABLE III. Fragmentation mechanism contributions to other pathways.a
give rise to the complex mechanism. The sequential mecha-
nism can occur whenever the He knocks one Na atom di-
rectly at another, so this mechanism should persist even
when the cluster is cold. Finally, we have seen that the dia-
tom mechanism can only occur when one Na atom in the
cluster is weakly bound to the other two, so this mechanism
vanishes for the cold cluster.
Two additional conclusions were reached from this
analysis. First, whenever the diatom mechanism occurs, the
He atom has passed through the interior of the Naϩ3 cluster.
Similarly, if the sequential mechanism has occurred, the He
has hit the Naϩ3 on the outside of the cluster.
Contribution ͑%͒ to different fragmentation pathways
Fragmentation
mechanisms Eint
B1
C1
A2
B2
C2
C3
Binary
a
b
c
a
b
c
a
b
c
a
b
c
39
83
78
0
0
3
32
2
2
29
15
17
64
73
78
7
6
5
29
21
17
0
0
0
85
51
28
15
36
34
0
9
28
0
91
68
43
9
26
35
0
6
22
0
79
73
75
17
22
15
5
6
10
0
73
77
63
23
18
25
4
5
12
0
Diatom
Sequential
Complex
4
10
0
0
0
0
0
0
IV. OTHERS PATHWAYS OF FRAGMENTATION
aResults for Eintϭ0.02, 0.5, and 1.0 eV are identified by (a), (b), and (c),
The previous section describes how impulsive energy
transfer from He to one Na nucleus followed possibly by
additional Na–Na collisions determines the fragmentation
mechanisms for pathway A1. When the cluster is hot the
major conclusions are that the binary mechanism dominates,
that complex formation is also important, and that both the
diatom and sequential mechanisms, which involve secondary
Na–Na collisions, make small but significant contributions.
The fragmentation intensity for A1 plotted against ⌽ and Tfr
in Figs. 3 and 5 foretold most of these conclusions. Peaks
due to the binary mechanism are seen at short time near ⌽
ϭ0 and 360°. By comparison, complex trajectories appear at
large time with a uniform ⌽ distribution. Finally, the diatom
mechanism gives a broad peak at relatively short time near
⌽ϭ180°.
We wish to apply the mechanisms developed for path-
way A1 to other pathways to see what light they can shed on
those fragmentation processes. There are two quite different
types of pathways to consider. First, there are those ͑B1 and
C1͒ that originate in the ground electronic state. ͑Recall that
we can apply the mechanisms developed for two body dis-
sociation to three body fragmentation such as C1 by follow-
ing the procedure described in the theory section.͒ These are
discussed below, and we shall see that the four mechanisms
are very useful for understanding these pathways.
The other type of pathway is one that originates in an
excited electronic state. In that case repulsive energy release
along one or more of the Naϩ3 coordinates also contributes to
and may dominate the fragmentation process. The mecha-
nisms developed in the previous section and summarized in
Table II uniquely separate all possible fragmentation trajec-
tories into four groups. Consequently, the procedure can be
applied to dissociation in an excited electronic state. How-
ever, it is not clear what can be learned in this case.
Experimentalists1–3 show that a certain fraction of collisions
with He gives fragmentation in excited electronic states with
very little impulsion given to the three Na nuclei of the clus-
ter ion. In this case the fragmentation is determined by the
repulsive PESs of the excited states, and it is irrelevant
which of three Na nuclei receives the strongest hit from the
He atom. We call this limiting case the pure ‘‘electronic’’
mechanism. An application of our separation procedure sum-
marized in Table II shows that in this limit there will be no
complex formation and the percentages for the binary, dia-
respectively.
tom and sequential mechanisms will all be equal. In addition,
the fragmentation intensity plotted against ⌽ and Tfr in the
fashion of Fig. 3 will show a vertical band restricted to small
fragmentation times and with a uniform ⌽ distribution. We
shall see that the results for pathways B2 and A2 are close to
this limit when the cluster is hot. Nevertheless, we shall also
find for these pathways as well as for C2 and C3 that certain
of the mechanisms developed for A1 continue to play a role
in fragmentations that originate in excited electronic states.
The distributions of fragmentation products for different
pathways are shown in Fig. 3 for Eintϭ1.0 eV. Several
mechanistic conclusions can be reached from these data.
First, we see that the binary mechanism dominates in most
pathways ͑the exception is A2͒. Second, complex formation
is possible only in those pathways that involve the stable
ground electronic state ͑A1, B1, and A2͒. Third, complete
fragmentation ͑C1, C2, and C3͒ is always a fast process and
is very fast in state 3. Fourth, pathways A2 and B2 exhibit a
slight concentration of trajectories near ⌽ϭ90° and 270°.
We now consider pathway B1 in detail. This requires a
hop from the ground to the first excited electronic state. An
examination of Fig. 3 shows that the binary and complex
mechanisms are quite important, and the diatom mechanism
is relatively less important than for A1. The various percent-
ages for all four mechanisms at three values of Eint are sum-
marized in Table III. Overall, the relative importance of each
mechanism is similar to that for A1 for Eintϭ0.5 and 1.0 eV
͑see Table II͒. Examination of trajectories for the fixed
T-shaped Naϩ3 ͑see Sec. III͒ shows that for a hot cluster B1
comes predominantly from He hitting one of the atoms in the
dimer, knocking it out of the cluster, and leaving behind a
diatomic product with large vibrational energy. It is remark-
able, however, to see that the fraction of binary collisions
drops to 39% for cold Naϩ3 ; in fact, we see in Table III that
the binary mechanism is only slightly more effective in this
case than sequential or complex. To understand this result
we recall that state 1 dissociates adiabatically to Naϩ2 ϩNa
͑channel A͒. To produce channel B ͑Na2ϩNaϩ͒ the interme-
diate diatomic product Na2ϩ must be produced with at least
0.27 eV vibrational energy, so it can reach the avoided cross-
ing to form Na2. This is very unlikely if the Naϩ3 is originally
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9424
J. Chem. Phys., Vol. 112, No. 21, 1 June 2000
Babikov et al.
cold.5,6 In a ‘‘pure’’ binary collision, the ejected Na atom
would not interact in any way with the remaining Na2 dimer
as it left the cluster. In that case the Na2 dimer would have
zero vibrational energy, so the fraction of binary collisions
giving pathway B1 would be zero for cold clusters. The
reality is that the departing Naϩ ion does interact with the
Na2 dimer to some extent, giving to it some vibrational en-
ergy, and a small fraction of binary collisions will lead to
pathway B1 even for a cold cluster. Nevertheless, the binary
mechanism is not an efficient way to give vibrational energy
to the diatomic product when the cluster is cold. By compari-
son, the sequential mechanism is an efficient way to put vi-
brational energy into the diatomic product.
Pathway C1 results from three-body fragmentation in
the ground electronic state. Figures 3 and 4 show that that
fragmentation is fast ͑there are no complex trajectories͒ and
is dominated by binary collisions with some evidence that
the diatom mechanism contributes. Table III shows the se-
quential mechanism is also quite important. An examination
of the calculations for the fixed T-shaped cluster shows that
for both binary and sequential mechanisms pathway C1 pri-
marily occurs when the He atom hits one of the atoms in the
dimer species, thereby breaking the shortest ͑and strongest͒
bond in the Naϩ3 cluster.
We next consider the pathways that originate in state 2.
In the T-shaped configuration this excited state’s PES is
bound along the dimer bond r but repulsive along the dimer-
monomer coordinate R.5,6 Pathway B2 is produced when
state 2 dissociates adiabatically to Na2ϩNaϩ. Examination
of Fig. 3 shows that there is a quite uniform band of intensity
as a function of ⌽ except near ⌽ϭ0 and 360° where binary
collisions accumulate and near ⌽ϭ90° and 270°. There are
some trajectories with fragmentation times larger than 30 000
a.u., but Fig. 4 shows that these do not arise from complex
collisions. Table III shows that the percentages for the other
three mechanisms are nearly the same when Eintϭ1.0 eV.
This coupled with the behavior in Fig. 3 suggests that the
system is nearly in the limit where the fragmentation is de-
termined solely by the repulsion in the excited state. This
occurs partly because state 2 is produced primarily when the
He passes between two Na nuclei, and no nucleus receives a
large impulse. Nevertheless, binary collisions are clearly fa-
vored over the diatom and sequential mechanisms, so the
impulse given to Naϩ3 by He still plays a role in the fragmen-
tation. An examination of the calculations for the fixed
T-shaped Naϩ3 for pathway B2 shows that the monomer is
ejected in 96% of the trajectories. This is not surprising,
given that the PES is bonding for the dimer but repulsive
along the dimer–monomer coordinate. The calculations also
show that state 2 is excited most often when the He passes
through the interior of the Naϩ3 cluster; this explains why the
diatom mechanism is more important than the sequential.
Finally, the calculations explain the intensity increases in
Fig. 3 near ⌽ϭ90° and 270°. These trajectories come from
collisions where He passes through the axis of the dimer just
on the outside of one of the dimer atoms and then the mono-
mer atom is ejected. In this case the collision plane contains
the two Na nuclei in the dimer, whereas the fragmentation
plane contains the monomer atom and is perpendicular to the
plane of the Naϩ3 ion. These two planes are perpendicular to
each other, and this corresponds to ⌽ϭ90° and 270°. Of
course, this is the description of an ideal collision, and in
general the two planes will not be always perpendicular, so
there will be a range of ⌽ values near 90° and 270°.
We now consider the other pathways that arise from
state 2. Almost all intermediate Na2 molecules produced in
state 2 have enough vibrational energy to reach the avoided
crossing ͑more then 0.04 eV͒ and make the transition to the
ground electronic state and form the pathway A2.5,6 Thus it
is not surprising that the results in Fig. 3 and the percentage
of the various mechanisms are so similar for pathways B2
and A2. However, it is interesting that a small fraction of the
fragmentation in pathway A2 for hot Naϩ3 clusters involve
the complex mechanism. Presumably this occurs because the
transition to the ground state at the avoided crossing puts
additional vibrational energy into the Naϩ3 cluster, increasing
the chance of forming a long lived intermediate complex.
The results for three body fragmentation in pathway C2 are
also similar to those for B2. Inspection of the calculations for
the fixed T-shaped Naϩ3 shows, however, that in this case the
He atom always hits one of the nuclei in the dimer hard,
thereby breaking the shortest ͑and strongest͒ bond in the
cluster.
Excitation of state 3 is very unlikely. For this state the
PES is strongly repulsive along all three Na–Na pairs, so C3
is the most important pathway. Figures 3 and 4 show that
fragmentation is very fast. The mechanistic results, shown in
Table III, are similar to those for pathways B2. Thus, the
binary mechanism dominates, and the diatom mechanism is
FIG. 7. Theoretical intensity contour maps plotted
against the angle ⌽ and the relative kinetic energy of
fragments ⑀ for pathways A1 and A2 obtained for
E
vibϭ1 eV.
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Fragmentation of Naϩ3 clusters
9425
J. Chem. Phys., Vol. 112, No. 21, 1 June 2000
tributes 56%, IM2 adds 34%, and EM gives 10%. We can
roughly associate their IM1 with our binary collisions in
pathway A1, IM2 with diatom, sequential and complex col-
lisions in A1; and EM with all trajectories in A2. Our theo-
retical numbers to compare with theirs are 54%, 32%, and
14%. Again the comparison is very good.
VI. SUMMARY
This article reports a theoretical study of the fragmenta-
tion of an Naϩ3 cluster ion following a collision with a fast
He atom. Trajectory results are presented in the form of cor-
relations between the time of fragmentation Tfr and the angle
⌽ between the planes of collision and fragmentation. Such a
representation makes clear that very different mechanisms
contribute to the fragmentation process even in the simplest
pathway when the cluster always stays in the ground elec-
tronic state and fragments adiabatically. The results are ex-
plained in terms of four simple fragmentation mechanisms,
representing possible scenarios of the Naϩ3 –He collision and
motion of the Na nuclei during the ensuing fragmentation of
the Naϩ3 cluster. These are the binary, diatom, sequential, and
complex mechanisms. A way to separate these fragmentation
mechanisms is demonstrated, and their contributions are cal-
culated.
FIG. 8. A comparison of theoretical and experimental intensity contour
maps plotted against ⌽ and ⑀ for product channel A. The theoretical results
were obtained for Evibϭ1 eV by summing the results for A1 and A2 shown
in Fig. 7.
The dependence of the four fragmentation mechanisms
on the initial internal energy of the Naϩ3 cluster is also stud-
ied. We found that complex formation as well as the diatom
mechanism of fragmentation are not efficient when the clus-
ter is cold. This result gives an interesting prediction for
future experimental studies of vibrationally cold clusters.
Finally we have applied our methodology to the analysis
of other fragmentation pathways that involve an electronic
excitation of the Naϩ3 cluster by the He atom and/or elec-
tronic transitions in the excited Naϩ3 while it fragments. We
found that our four mechanisms of fragmentation are still
useful in this case. They provide deep insight into the frag-
mentation process.
more important than sequential. For the fixed T-shaped clus-
ter we find that excitation only occurs when the He hits near
one of the atoms in the dimer. In all cases the atom ejected
with the fastest velocity comes from the dimer.
V. COMPARISON WITH EXPERIMENT
Figure 7 shows theoretical intensity contour maps plot-
ted against the angle ⌽ and the relative kinetic energy of
fragments ⑀ for pathways A1 and A2 for Eintϭ1.0 eV. For
A1 the peaks near ⌽ϭ0 and 360° are due to binary colli-
sions that can lead to large values of ⑀. The other peak near
⌽ϭ180° is due to collisions which follow the diatom
mechanism. The results for A2 do not show the large ⑀ val-
ues seen for A1, because excitation of state 2 does not re-
quire a hard collision of He with an Na nucleus. The peaks
near ⌽ϭ90° and 270° appear prominently in the Figure.
Figure 8 shows a comparison of theoretical and
experimental1–3 intensity contour maps plotted against ⌽ and
⑀ for product channel A ͑doubly differential experimental
data for other pathways are not available͒. The theoretical
results were obtained by adding together the results for path-
ways A1 and A2 shown in Fig. 7. ͑The cross section for A3
is negligible.͒ Overall, the agreement is very good. It re-
quires the superposition of the impulsive and electronic pro-
cesses, A1 and A2, respectively, to give the proper shape to
the contour map. The experimentalists interpreted their re-
sults by introducing three different mechanisms for cluster
fragmentation.2 These are IM1, one step impulsive mecha-
nism; IM2, two step impulsive; and EM, electronic mecha-
nism. For product channel A they estimated that IM1 con-
ACKNOWLEDGMENTS
We would like to thank J. A. Fayeton, M. Barat, and Y.
J. Picard at LCAM ͑Orsay͒ for providing their experimental
data and for helpful discussions.
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