JOURNAL OF CHEMICAL PHYSICS
VOLUME 119, NUMBER 13
1 OCTOBER 2003
Entrance channel localized states in ozone:
Possible application to helium nanodroplet isolation spectroscopy
Dmitri Babikova)
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
!Received 3 June 2003; accepted 8 July 2003"
Accurate calculations of the bound ro-vibrational states of ozone are performed in the region of high
vibrational excitation. Two unusual vibrational states localized in the far van der Waals region are
discovered. They can be considered as the ground vibrational states of even and odd symmetries
bound in the shallow van der Waals well. Properties of these states are presented and discussed,
which opens possibilities for experimental realization of the helium nanodroplet isolation
ro-vibrational spectroscopy of ozone. © 2003 American Institute of Physics.
#DOI: 10.1063/1.1604771$
I. INTRODUCTION
formed there and what is the structure of these states. Answering these questions theoretically will help in designing
the experimental set up.5
If the experiment is successful, the high resolution
HENDI spectra will provide new accurate information about
the rovibrational vdW states of the ozone molecule and will
help us to make a new round of improvements in the theoretical PES of ozone in the high energy and large O–O2
separation region, where theoretical ab initio calculations are
quite problematic. This will improve our capabilities in accurate modeling of many processes related to ozone chemistry, because the upper part of the PES in the regions of the
barrier and the vdW well is very important for low energy
scattering processes, such as following processes involved in
ozone formation:8
Helium nanodroplet isolation !HENDI" rovibrational
spectroscopy allows combining the tools of molecular beam
spectroscopy and the matrix isolation spectroscopy into a
powerful set of experimental techniques for studies of unusual molecular aggregates.1 Several amazing examples have
been demonstrated during the last years. Among those are
observation of quartet 1 4 A 2! states of the Na3 trimer,2 formation of cyclic water hexamer,3 and the discovery of self assembly of long chains of polar molecules.4 It is encouraging
that the chemically neutral environment of He nanodroplets
induces only tiny shifts in the ro-vibrational spectra, so that
experimental data can be directly compared with theoretical
results obtained for gas phase molecules. It is even more
surprising, that the quantum symmetry rotational selection
rules are preserved in the spectra of molecules isolated in the
helium nanodroplets. Furthermore, HENDI allows studying
very unusual nonequilibrium states of molecules that could
not be created in any other way.
It has been thought5 that HENDI might help to form and
study the highly excited large amplitude vibrational states of
ozone molecules (O3 ) localized in the shallow van der Waals
!vdW" well. Figure 1 shows a one dimensional cut of the
potential energy surface !PES" of ozone along the minimum
energy path for dissociation/recombination, as one O atom
leaves the molecule: O3 →O2 !O. In the ground rovibrational state, at the bottom of deep covalent well, the ozone
molecule has the shape of an isosceles triangle with R 1
"R 2 %2.4 a.u. and & %117°. It dissociates by losing one end
!terminal" O atom, which passes above the barrier and the
vdW well as it leaves !see Fig. 1". It has been shown only
recently that the top of the barrier, that separates the covalent
and the vdW wells, is below the dissociation limit. Accordingly, a global PES, accurate in the region of the barrier and
the vdW well, has been constructed.6,7 It is, however, still a
question whether localized vibrational states can be formed
in the shallow vdW well, how many of such states can be
O2 !O⇒O3* ,
!Here M is any atmospheric atom or molecule that stabilizes
the metastable O*
3 state to stable ozone O3 .) Motivated by
these advantages we have undertaken calculations of upper
ro-vibrational states in ozone. Our results are encouraging
and we report them in this paper.
II. THEORETICAL METHOD
As discussed in our earlier papers,6,7 a very sophisticated
potential energy surface !PES" for ozone was used for this
work. It is based on accurate ab initio calculations9,10 and
includes a correction in the barrier region to make it agree
with even more accurate ab initio calculations performed
along the minimum energy path.11,12 It is the most accurate
PES currently available for the ground electronic state of
ozone. It has the full symmetry of the system and goes
smoothly to all the correct dissociation limits. It exhibits a
small barrier along the dissociation path, but the top of that
barrier is below the dissociation limit !see Fig. 1". Several
slices of the PES have been presented in Ref. 6 as relevant to
formation of ozone metastable states; here we show several
slices relevant to formation of the bound states, particularly
those in the vdW well region. Three two-dimensional slices
of the PES are shown in Fig. 2 as stereographic projections
a"
Electronic mail: [email protected]
0021-9606/2003/119(13)/6554/6/$20.00
O*
3 !M⇒O3 .
6554
© 2003 American Institute of Physics
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J. Chem. Phys., Vol. 119, No. 13, 1 October 2003
Nanodroplet isolation spectroscopy
6555
FIG. 1. One-dimensional cut of the potential energy surface for ozone along
the minimum energy path for dissociation/recombination (O3 ↔O2 !O).
The horizontal axis gives the distance between the most distant end O atom
and the central O atom. The barrier between the deep covalent well and the
shallow van der Waals well is below the dissociation limit.
using symmetrized hyperspherical coordinates.13 The slices
correspond to fixed values of the hyperradius ' !the overall
size of the triatomic molecule"; free to vary are the two hyperangles ( and ) !shape of the triatomic". At ' "4 a.u. one
sees three deep covalent minima of the ozone molecule !Fig.
2a"; they correspond to the three possible permutations of
nuclei in the equilibrium isosceles configuration of
O3 : OO!O" , O!OO" and OO" O! . At ' "6 a.u. !Fig. 2b" one
sees six shallow vdW minima of the ozone molecule. Each
pair of those can be reached by stretching one of the two
bonds !see Fig. 1" to form either O–O! O" or OO! – O" ; two
such configurations for each of three possible permutations
give six vdW minima. A slice in the more distant vdW region
at ' "8 a.u. !Fig. 2c" is also shown; at this value of ' the
PES already exhibits behavior of the asymptotic channels
!compare to Fig. 3c, Ref. 6".
In this work we perform full quantum calculations of the
bound states of ozone in all six dimensions of the problem
(3•(N#1)"6 for N"3 atoms" using an exact Hamiltonian.
No approximations are made. A coupled channel !CC" approach using APH hyperspherical coordinates and a hybrid
FBR/DVR14 is employed. Propagation of the coupled channel equations is performed using a Numerov method.15 The
parallel computer code of Kendrick, applied previously to
calculate bound states in HO2 !Ref. 16" and Na3 !Refs. 17,
18" molecules, was used. Convergence issues have been discussed in Ref. 6.
Oxygen isotopes 16O are spinless bosons, so that the total wavefunction of the O3 molecule must be symmetric with
respect to exchange of any two 16O atoms. The ground electronic state of O3 in the vdW region, 1A! , is asymptotically
connected to O( 3 P)!O2 ( 3 * #
g ) products and is antisymmetric.19 Thus, the nuclear motion wavefunction must
FIG. 2. Two-dimensional slices of the potential energy surface in hyperspherical coordinates. The fixed value of the hyperradius ' is given for each
frame; two hyperangles are varied. The O2 !O dissociation limit is taken as
the energy zero. !a" ' "4 a.u., contours are given between #1.1 and 0.1 eV
with a step of 0.1 eV. The three deep covalent minima of O3 are clearly seen.
!b" ' "6 a.u., contours are given at 0.0, 0.05 and 0.1 eV. The six shallow
van der Waals minima of O3 are clearly seen. !c" ' "8 a.u., contours are the
same as in !b". The asymptotic channels character of the surface is observed.
also be antisymmetric, so that the product of electronic and
nuclear wavefunctions is symmetric. This fact imposes symmetry restrictions on rovibrational wavefunctions, similar to
those discussed in the literature.20 For example, for a nonrotating ozone molecule (J"0 ! , the plus sign labels even
inversion parity states", the rotational component of the
nuclear motion wavefunction is symmetric, so that only antisymmetric !odd" vibrational states are allowed. However,
for J"1 ! the rotational component of the nuclear wavefunction is antisymmetric, so that only symmetric !even" vibra-
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6556
J. Chem. Phys., Vol. 119, No. 13, 1 October 2003
Dmitri Babikov
FIG. 3. !a" Surface function energies for several low-lying channels of odd
symmetry and J"0 ! . The barrier, the deep covalent region, and the shallow
vdW well are reflected by the surface functions. See text for details. !b"
Probability as a function of hyperradius for three upper J"0 ! vibrationals
states. The state n"106 is localized in the far van der Waals region.
FIG. 4. !a" Surface function energies for several low-lying channels of even
symmetry and J"1 ! . The barrier, the deep covalent region, and the shallow
vdW well are reflected by the surface functions. See text for details. !b"
Probability as a function of hyperradius for three upper J"1 ! vibrationals
states. The state n"138 is localized in the far van der Waals region.
tional states are allowed. For J"1 # and J$1 both even and
odd vibrational states are present.19 As a consequence, odd
vibrational bands contain the odd states of J"0 ! , while
even vibrational bands contain even states of J"1 ! . To obtain a spectrum with both even and odd vibrational states, it
is necessary to perform calculations for at least J"0 ! and
J"1 ! . Everyone who wants to do calculations of the spectra
for O3 and their assignment should take these considerations
into account. We note that previous bound state
calculations21 for O3 were performed for both even and odd
vibrational states of J"0 and this is not quite correct.
Convergence studies show that as many as 60 channels
of proper symmetry are necessary. Figures 3a and 4a plot the
surface function energies of several low-lying channels for
the J"0 ! and J"1 ! calculations, respectively. In the
asymptotic region ( ' $14 a.u.), the surface function energies
for both J"0 ! and J"1 ! correlate to odd rotational states
of O2 ( 3 * #
g ) diatomics, as restricted by symmetry selection
rules:22 v "0, j"1,3,5,7,... The energy of the lowest surface
function reflects the property of the PES barrier: it shows a
barrier at ' +5.27 a.u., and its top (E+0.089 25 eV) is
slightly below the dissociation threshold (E+0.097 75 eV).
#Energy required for dissociation O3 →O2 !O is the energy
of the ground ro-vibrational state ( v "0,j"1) of O2 .] Both
the deep covalent region (3.75 a.u.% ' %5 a.u.) and the shallow vdW well are clearly seen, with the vdW minimum at
' +6.17 a.u. and E+0.073 00 eV. Since we are searching for
the states bound in the vdW well, we will focus our attention
on the high-lying vibrational energy levels with E
$0.073 eV.
$0.073 eV) are included. States are labeled by the total
number of nodes n (n"0 is the ground vibrational state".
Values of n$100 in Table I clearly correspond to many
quanta of vibrational excitation. Analysis of the corresponding wavefunctions shows that all states at energies below the
barrier top (E%0.089 25 eV) are localized entirely in the covalent region. This is, probably, because the vdW well is
III. RESULTS
Table I gives calculated energies of vibrational states for
J"0 ! and J"1 ! . Only states with energies of interest (E
TABLE I. Calculated energies of vibrational states for J"0 ! and J"1 ! in
the energy range E$0.073 eV. States are labeled by the total number of
nodes in the wavefunction.
E !eV"
n
J"0
!
98
99
100
101
102
103
104
105
!vdW"106
107
0.074 758 6
0.076 775 0
0.083 305 5
0.085 432 7
0.087 329 7
0.091 838 1
0.092 277 0
0.092 784 1
0.095 217 2
0.096 380 3
J"1 !
129
130
131
132
133
134
135
136
137
!vdW"138
139
0.075 244 1
0.076 186 1
0.077 408 5
0.081 910 0
0.082 832 8
0.089 064 0
0.091 129 1
0.091 816 0
0.093 218 1
0.095 850 6
0.096 320 2
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J. Chem. Phys., Vol. 119, No. 13, 1 October 2003
Nanodroplet isolation spectroscopy
6557
FIG. 5. Two-dimensional slices in hyperspherical coordinates of the wavefunctions for the three upper J"0 ! vibrational states depicted in Fig. 3a:
n"105, 106, 107. The value of the hyperradius ' is fixed !see text for
details"; two hyperangles are varied. The structure of the van der Waals state
n"106 is very different from the structures of the other two states associated with the covalent well !compare to Fig. 2". The odd character of vibrational states is reflected by the nodal lines at ) "0°, 60°, 120°, etc.
FIG. 6. Same as Fig. 5 but for the three upper J"1 ! vibrational states
depicted in Fig. 4a: n"137, 138, 139. The structure of the van der Waals
state n"138 is very different from the structures of the other two states
associated with the covalent well !compare to Fig. 2". The even character of
vibrational states is reflected by the non-zero probability at ) "0°, 60°,
120°, etc.
very shallow and due to the large local ZPE cannot accommodate any states ‘‘truly trapped’’ in the vdW well. However, we found that all states at energies above the barrier top
(E$0.089 25 eV) exhibit at least some non-negligible probability in the vdW region. Figures 3b and 4b plot the probability density along the hyperradius coordinate ' !integrated
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6558
J. Chem. Phys., Vol. 119, No. 13, 1 October 2003
over hyperangles" for three upper states with J"0 ! and J
"1 ! , respectively. It is seen very clearly that some of states
are still located mostly in the covalent well region at '
%5.27 a.u., and only a portion of the probability density is
located outside the barrier in the close vdW region at
5.27 a.u.% ' %6.2 a.u. This behavior is typical for almost all
states at energies above the barrier top. There is, however
one exception in each case. State n"106 for the J"0 ! case
and state n"138 for the J"1 ! case are located essentially
entirely in the far vdW region at 7.6 a.u.% ' %8.7 a.u.! Both
of these states exhibit a very smooth Gaussian-like shape
without any visible nodes. Thus, they appear to be ground
vibrational states in the vdW well. This point is further emphasized in Figs. 5 and 6, where two-dimensional slices are
plotted of the wavefunctions shown in Figs. 3b and 4b, respectively. Slices of the two vdW states are made in the far
vdW region at ' "8 a.u. !compare to Fig. 2c", while slices of
other states are made in the covalent well region at '
"4 a.u. !compare to Fig. 2a". First of all, Figs. 5 and 6 emphasize the quantum symmetry selection rules, showing that
all the wavefunctions for J"0 ! exhibit nodal lines at )
"0°, 60°, 120°, etc. !i.e., they are odd under the exchange
of any two O atoms", while the wavefunctions for J"1 !
exhibit nonzero probability at ) "0°, 60°, 120°, etc. !i.e.,
they are even under the exchange of any two O atoms". Furthermore, Figs. 5 and 6 demonstrate that the two vdW states,
(n"106,J"0 ! ) and (n"138,J"1 ! ), have indeed no
nodes corresponding to vibrational excitation, while covalent
well states at energies just below and above the energies of
vdW states exhibit rich nodal structure corresponding to
many quanta of vibrational excitation. It is worth mentioning
that the wide spread of the wavefunction for the vdW states
suggests that the ozone molecules in these states are able to
perform large amplitude bending motion.
Clearly, states (n"106,J"0 ! ) and (n"138,J"1 ! ) can
be considered as ground vibrational states of odd and even
symmetries in the vdW well. For the lower energy state of
these two, the (n"106,J"0 ! ) state, the maximum probability point of the wavefunction corresponds to ' +8.061 a.u.,
( +52.48°, ) +&27.91°. It is approximately shown by an
arrow in Fig. 7, where three-dimensional views for the three
upper J"0 ! vibrational states are presented. !Only one of
three equivalent parts is shown for each wavefunction: for
the n"106 vdW state the part on the right side of the frame
in Fig. 5 is shown; for covalent states n"105 and n"107 the
part on the left side of the frame in Fig. 5 is shown." In
bond-angle coordinates the maximum probability point of
the wavefunction corresponds to R 1 +2.288 a.u., R 2
+6.234 a.u., & +148.75°. In addition, Fig. 7 clearly exposes
the nodal planes of all odd J"0 ! vibrational states. Also, the
complex nodal structure of two covalent well vibrational
states (n"105 and n"107) is clearly seen, and this is very
different from the simple structure of (n"106) vdW state.
For the higher energy vdW state, the (n"138,J"1 ! ) state,
the maximum probability point of the wavefunction corresponds to ' +8.049 a.u., ( +46.00°, ) +&24.95°, or R 1
+2.292 a.u., R 2 +6.366 a.u., & +135.08°.
Finally, we want to demonstrate that though these simple
looking states can be practically considered as ground vibra-
Dmitri Babikov
FIG. 7. Three-dimensional view in hyperspherical coordinates of the wavefunctions for the three upper J"0 ! vibrationals states depicted in Fig. 3a
and Fig. 5: n"105, 106, 107. Two arrows show an approximate position of
the maximum probability points in the wavefunction for van der Waals state
n"106. See text for analysis.
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J. Chem. Phys., Vol. 119, No. 13, 1 October 2003
Nanodroplet isolation spectroscopy
6559
In order to facilitate experimental detection of these
states it would also be useful to calculate vertical electronic
excitation energies and oscillator strengths, starting at the
configuration with maximum probability found in this work.
Also it would be potentially useful to calculate vibrational
spectra of electronically excited ozone molecules. This, however, requires knowledge of at least a part of the PES for
excited electronic state !or states" and it is not yet developed.
This research is in our plans for the future.
FIG. 8. Log-scale plot of the probability as a function of hyperradius for two
van der Waals states: (n"106,J"0 ! ) and (n"138,J"1 ! ). The probability
is localized in the far van der Waals region. A tiny tail extends into the
covalent region and exhibits significant structure !nodes".
tional states in the vdW well, they are not such states strictly
speaking. This is seen, of course, from the energy of these
states, which is higher than the barrier energy. Thus, the
wavefunction can extend also to the region of covalent well
without tunneling. This is indeed the case and is seen from
the log plot of the low probability part of the wavefunction
as a function of the hyperradius, presented in Fig. 8 for both
(n"106,J"0 ! ) and (n"138,J"1 ! ) states. The wavefunctions for the vdW states have significant structure !nodes"
throughout the wide range of hyperradius (4 a.u.% '
%7 a.u.). However, the probability is quite small in this region. Only in the very narrow far vdW region (7.6 a.u.% '
%8.7 a.u.) is the probability appreciable.
IV. CONCLUSIONS
We have performed calculations of high-lying vibrational states of the ozone molecule on an accurate PES using
an exact Hamiltonian and taking into account all quantum
symmetry constraints. Two states are identified as the ground
vibrational states of odd and even symmetries bound in the
shallow van der Waals well: (n"106,J"0 ! ) and (n
"138,J"1 ! ), respectively. These states are located at energies 20.4 cm#1 and 15.3 cm#1 , respectively, below the dissociation threshold and are localized in the far van der Waals
region. The shape of the ozone molecule in these states is
very different from its equilibrium isosceles shape in the
deep covalent minimum; it represents a bent triatomic with
one significantly elongated side: R 1 +2.288 a.u., R 2
+6.234 a.u.,
& +148.75° and R 1 +2.292 a.u., R 2
+6.366 a.u., & +135.08°, for (n"106,J"0 ! ) and (n
"138,J"1 ! ) states, respectively. Wavefunctions of these
states contain no nodal structure in the shape coordinates
!except the nodes of odd state imposed by the symmetry" and
their shapes suggest that ozone molecules in these states can
easily perform large amplitude bending motion. Information
about these states should help in the practical realization of
HENDI rovibrational spectroscopy of ozone molecules stabilized in the far van der Waals region.
ACKNOWLEDGMENTS
This work was performed under the auspices of the U.S.
Department of Energy !under Contract No. W-7405-ENG36" and used resources of the National Energy Research Scientific Computing Center, which is supported by the Office
of Science of the U.S. Department of Energy under contract
No. DE-AC03-76SF00098. The author acknowledges Professor Kevin Lehmann at Princeton for proposing the subject
of this paper. Russell Pack, Bob Walker, and Brian Kendrick
are gratefully acknowledged for their help in preparation of
the manuscript.
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