Microwave spectrum, large-amplitude motions, and ab initio calculations for N2O5

Article abstract of DOI:10.1063/1.472586

The rotational spectrum of dinitrogen pentoxide (N₂O₅) has been investigated between 8 to 25 GHz at a rotational temperature of ～2.5 K using a pulsed-molecular-beam Fourier-transform microwave spectrometer. Two weak b-dipole spectra are observed for two internal-rotor states of the molecule, with each spectrum poorly characterized by an asymmetric-rotor Hamiltonian. The observation of only 6-type transitions is consistent with the earlier electron-diffraction results of McClelland et al. [J. Am. Chem. Soc. 105, 3789 (1983)] which give a C₂ symmetry molecule with the b inertial axis coincident with the C₂ axis. Analysis of the ¹⁴N nuclear hyperfine structure demonstrates that the two nitrogen nuclei occupy either structurally equivalent positions or are interchanging inequivalent structural positions via tunneling or internal rotation at a rate larger than ～1 MHz. For the two internal rotor states, rotational levels with K_a + K_c even have I_N=0, 2, while levels with K_a + K_c odd have I_N= 1, where I_N is the resultant nitrogen nuclear spin. This observation establishes that the equilibrium configuration of the molecule has a twofold axis of symmetry. Guided by ab initio and dynamical calculations which show a planar configuration is energetically unfavorable, we assign the spectrum to the symmetric and antisymmetric tunneling states of a C₂ symmetry N₂O₅ with internal rotation tunneling of the two NO₂ groups via a geared rotation about their respective C₂ axes. Because of the Bose-Einstein statistics of the spin-zero oxygen nuclei, which require that the rotational-vibrational-tunneling wave functions be symmetric for interchange of the O nuclei, only four of the ten vibrational-rotational-tunneling states of the molecule have nonzero statistical weights. Model dynamical calculations suggest that the internal-rotation potential is significantly more isotropic than implied by the electron-diffraction analysis.

Full text of DOI:10.1063/1.472586

Microwave spectrum, largeamplitude motions, and ab initio calculations for N2O5
J.U. Grabow, A. M. Andrews, G. T. Fraser, K. K. Irikura, R. D. Suenram, F. J. Lovas, W. J. Lafferty, and J. L.
Domenech
Citation: The Journal of Chemical Physics 105, 7249 (1996); doi: 10.1063/1.472586
View online: http://dx.doi.org/10.1063/1.472586
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/105/17?ver=pdfcov
Published by the AIP Publishing
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Microwave spectrum, large-amplitude motions, and ab initio calculations
for N O5
2
a),b),c)
a),d)
a)
J.-U. Grabow,
A. M. Andrews,
G. T. Fraser,
e)
a)
a)
K. K. Irikura, a)R. D. Suenram, F. J. Lovas, and
W. J. Lafferty
National Institute of Standards and Technology, Gaithersburg, Maryland 20899
J. L. Domenech
Instituto de Estructura de la Materia, CSIC, Serano 119, 28006 Madrid, Spain
͑
Received 21 March 1996; accepted 22 July 1996͒
The rotational spectrum of dinitrogen pentoxide ͑N O ͒ has been investigated between 8 to 25 GHz
2
5
at a rotational temperature of ϳ2.5 K using a pulsed-molecular-beam Fourier-transform microwave
spectrometer. Two weak b-dipole spectra are observed for two internal-rotor states of the molecule,
with each spectrum poorly characterized by an asymmetric-rotor Hamiltonian. The observation of
only b-type transitions is consistent with the earlier electron-diffraction results of McClelland et al.
͓
J. Am. Chem. Soc. 105, 3789 ͑1983͔͒ which give a C symmetry molecule with the b inertial axis
2
14
coincident with the C axis. Analysis of the N nuclear hyperfine structure demonstrates that the
2
two nitrogen nuclei occupy either structurally equivalent positions or are interchanging inequivalent
structural positions via tunneling or internal rotation at a rate larger than ϳ1 MHz. For the two
internal rotor states, rotational levels with K ϩK even have I ϭ0, 2, while levels with K ϩK
a
c
N
a
c
odd have I ϭ1, where I is the resultant nitrogen nuclear spin. This observation establishes that the
N
N
equilibrium configuration of the molecule has a twofold axis of symmetry. Guided by ab initio and
dynamical calculations which show a planar configuration is energetically unfavorable, we assign
the spectrum to the symmetric and antisymmetric tunneling states of a C symmetry N O with
2
2
5
internal rotation tunneling of the two NO groups via a geared rotation about their respective C2
2
axes. Because of the Bose–Einstein statistics of the spin-zero oxygen nuclei, which require that the
rotational–vibrational–tunneling wave functions be symmetric for interchange of the O nuclei, only
four of the ten vibrational-rotational-tunneling states of the molecule have nonzero statistical
weights. Model dynamical calculations suggest that the internal-rotation potential is significantly
more isotropic than implied by the electron-diffraction analysis. © 1996 American Institute of
Physics. ͓S0021-9606͑96͒02740-7͔
I. INTRODUCTION
ported for this species, which could provide the necessary
spectroscopic constants for simulating the atmospheric spec-
trum as a function of temperature.
The highly reactive dinitrogen pentoxide ͑N O ͒ mol-
ecule plays an important role in the chemistry of the upper
2
5
A starting point in any high-resolution spectroscopic
study of N O is the electron-diffraction results of McClel-
atmosphere, functioning as a nighttime reservoir for NO and
2
2
5
NO , both of which are implicated in the catalytic destruc-
3
7
land et al. Their experiments show that N O has a nonpla-
2
5
tion of stratospheric ozone. Direct evidence for the presence
1
nar geometry with two equivalent NO groups joined by an
2
of N O in the stratosphere has been obtained by Toon et al.
2
5
oxygen atom to form a C symmetry configuration with a
2
from infrared spectra recorded at sunrise by the Atmospheric
Trace Molecule Experiment aboard Spacelab 3. Their results
have generated renewed interest in the spectroscopy of
strongly bent O N–O–NO angle of 111.8͑6͒° ͑see Fig. 1͒.
2
2
To adequately model the electron-diffraction patterns, Mc-
7
2
–6
Clelland et al. invoked large-amplitude torsions of the NO2
N O , motivating efforts to measure precise spectroscopic
2
5
groups about their C axes. They modeled the torsional po-
parameters for modeling the band profiles used in extracting
the atmospheric concentration of N O . Despite all this ef-
2
tential using atom–atom Lennard-Jones repulsions between
2
5
the oxygen atoms on opposing NO groups, supplemented by
fort, no rotationally resolved infrared spectrum has been re-
2
cosine potentials in the internal rotation angles to mimic any
electronic effects propagated through the N–O–N backbone.
a͒
Optical Technology Division.
NIST Guest Researcher.
7
b͒
The structural results of McClelland et al. are qualita-
c͒
Present address: Institut f u¨ r Physikalische Chemie, Christian-Albrechts-
tively supported by Hartree–Fock calculations using small
Universit a¨ t, D-24098 Kiel, Germany.
8
4
–31 G basis sets and by density-functional calculations
d͒
Permanent address: Science and Technology Division, Institute for De-
fense Analysis, 1801 N. Beauregard Street, Alexandria, VA 22311.
using Gaussian-type orbitals and triple-zeta-plus-polarization
e͒
9
9
Physical and Chemical Properties Division.
basis sets. The more recent density-functional calculations
J. Chem. Phys. 105 (17), 1 November 1996
0021-9606/96/105(17)/7249/14/$10.00
© 1996 American Institute of Physics
7249
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7250
Grabow et al.: Microwave spectrum of N O5
2
groups about their C axes. The present results suggest that
2
5
the microwave spectrum of Colmont et al. arises either from
an excited state of N O or from impurities.
2
5
II. EXPERIMENT
A. Measurements
Nitrogen pentoxide was synthesized by dehydrating dry
nitric acid with phosphorous pentoxide, following the proce-
dure of Caesar and Goldfrank12 and others. The reaction
product was distilled, with the vapor collected in a stream of
13
FIG. 1. Electron-diffraction geometry for N O and the coordinate system
2
5
oxygen and ozone to insure complete oxidation of any NO2
used to discuss the structural analysis. The electron-diffraction geometry of
McClelland et al. ͑Ref. 7͒ has ␣ϭ␣ ϭ␣ ϭ30° rϭ1.183͑2͒ Å, Rϭ1.492͑4͒
1
2
byproduct to N O . Care was taken to avoid introduction of
2
5
Å, ␥ϭ111.8͑16͒°, and ␤ϭ133.2͑6͒°.
water vapor into the system since N O is rapidly converted
2
5
to nitric acid in the presence of water. The purity of the
resulting product was periodically tested using Fourier-
transform infrared spectroscopy.
give a nearly planar configuration, with out of plane dihedral
angles of ϳ22°, compared to the electron-diffraction angles
of ϳ30°. In addition, the density functional calculations pre-
The Balle–Flygare pulsed-nozzle Fourier-transform
spectrometer used in the present investigation has been de-
7
14
scribed previously.15 To increase the sensitivity and resolu-
tion, we follow Grabow and Stahl,16 and direct the nozzle
pulse down the cavity axis. In this configuration each of the
transitions appears as a doublet, separated by 50–75 kHz
symmetrically about the line center. In a frequency-domain
picture these doublets occur because of the opposite Doppler
shifts of the two electromagnetic waves emitted parallel and
antiparallel to the direction of the molecular-beam propaga-
tion. In a time-domain picture, the doublets arise from the
modulation of the electromagnetic field due to the motion of
the macroscopic dipole moment of the inhomogeneously po-
larized molecular beam through the TEMplq mode pattern of
the resonator. The moving macroscopic dipole moment
causes a modulation of the electromagnetic field equivalent
to the splitting observed in the frequency domain.
dict a small electric-dipole moment for the molecule of 0.1–
0
.3 D, significantly less than the dielectric-constant measured
10
value of 1.4 D for N O in carbon tetrachloride solvent.
2
5
5
Recently, Colmont, guided by the electron-diffraction
results, has reported the observation of the microwave spec-
trum of N O using source frequency modulation with a
2
5
waveguide sample cell. A near rigid-rotor series of lines
were assigned to N O5 based on a fit to a Watson11
2
asymmetric-top Hamiltonian to a root-mean-square deviation
of ϳ1.1 MHz, or about ten times the measurement uncer-
tainty. Efforts were made to remove possible nitric acid im-
purities arising from the reaction of N O with water, by
2
5
comparing the observed spectrum with a HNO spectrum
3
obtained under similar conditions. Surprisingly, no direct
evidence was obtained for tunneling splittings arising from
large-amplitude motions, or for 14N nuclear quadrupole hy-
perfine structure, presumably because of the high-J data
sampled ͑Jу15͒.
In the present study we have examined the microwave
spectrum of N O between 8–25 GHz at a rotational tem-
2
5
perature of ϳ2.5 K using the high sensitivity and high reso-
lution ͑ϳ2 kHz͒ of a pulsed-molecular-beam Fourier-
transform microwave spectrometer. Definitive rotational
assignments of the transitions are made by modeling the
J-dependent nuclear quadrupole hyperfine patterns. The
spectrum reveals two essentially equal-intensity asymmetric-
rotor-like series, each poorly fit to a Watson Hamiltonian.
The hyperfine patterns show that the two nitrogen nuclei
sample equivalent chemical environments on a time scale
faster than ϳ1 ␮s.
The observed nuclear-spin statistical weights require that
the equilibrium nuclear configuration of N O has a twofold
2
5
axis of symmetry, in agreement with the electron-diffraction
results. Model dynamical calculations and new high-level ab
initio results described here, lead us to assign the two states
to two tunneling components of a C symmetry N O . The
observed splittings are consistent with a tunneling pathway
corresponding to a geared internal rotation of the two NO2
2
2
5
FIG. 2. Low-resolution survey spectrum for N O showing the three J com-
2
5
Ϫ
ponents of the K ϭ2Ϫ1, Q branch for the A1 state. The spectrum was
a
acquired in ϳ45 min.
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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Grabow et al.: Microwave spectrum of N O5
7251
2
FIG. 3. Low-resolution survey spectrum for N O showing the 313–202 and
2
5
ϩ
322
HNO3 .
–313 transitions for the A1 state and the 651–642 impurity transition for
A molecular beam of N O was formed by flowing, at a
2
5
pressure of ϳ115 kPa, a mixture of ϳ20% by volume He in
Ne over N O solid at Ϫ20 to 0 °C through a 1 mm diameter
2
5
17
pulsed-nozzle valve using a glass and Teflon gas manifold.
Low-resolution ͑0.2–0.5 MHz͒ broadband survey spectra
were recorded between 8.82–9.02, 9.814–11.423, 11.500–
1
7.488, 17.665–19.225, and 21.055–22.378 GHz as de-
18
scribed by Andresen et al. Sample low-resolution survey
spectra are shown in Figs. 2 and 3, and a sample high-
resolution spectrum is shown in Fig. 4. The main contami-
nant lines in the spectra are due to nitric acid originating
from the hydration reaction of the N O with residual water
FIG. 5. Geometries of four conformations of N O5 optimized at the
2
QCISD/6-311G* level ͑active core͒ under the symmetry constraints indi-
cated.
2
5
vapor in the gas-handling system.
B. Ab initio calculations
Four conformations were considered computationally.
Structures 1, 2, 3, and 4 correspond approximately to the
͑
͑
␣ ,␣ ͒ torsional coordinate values ͑␣,␣͒, ͑0,90°͒ or ͑90°,0͒,
90°,90°͒, and ͑0,0͒, respectively. These conformations be-
1 2
long to the point groups C , C , C , and C , respectively
2
s
2v
2v
͑see Fig. 5͒. Geometries for all four conformations were fully
optimized within the appropriate symmetry restrictions,
without constraining the O–NO moieties to be planar. Be-
2
cause of the point-group restrictions, however, the optimized
geometries do not necessarily represent minima on the cor-
responding potential energy surfaces. Geometries, relative
energies, and dipole moments were determined at the
HF/6-311G*, MP2/6-311G* ͑active core͒, QCISD/6-311G*
͑
active core͒, SVWN/6-311G*, BLYP/6-311G*, and
Becke3LYP/6-311G*. The last three are density functional
methods and were included to aid comparison with a prior
9
density functional study of N O . SVWN is local, employ-
2
5
19
ing Slater’s exchange functional ͑␣ϭ2/3͒ and the Vosko–
Wilk–Nusair correlation functional. BLYP includes the
ϩ
FIG. 4. High-resolution spectrum of the 221–212 transition for the A state
20
1
1
4
of N O . The hyperfine pattern arises from the two Iϭ1 N quadrupolar
2
5
nonlocal ͑gradient͒ corrections to the exchange and correla-
nuclei giving a resultant nitrogen nuclear spin of 1. The arrows point out the
‘Doppler doublets.’’
tion functionals due to Becke21 and to Lee, Yang, and Parr,
22
‘
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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7252
Grabow et al.: Microwave spectrum of N O5
2
a,b
TABLE I. Transition line centers and effective nuclear quadrupole coupling constants for N O in MHz.
2
5
ϩ
Ϫ
JЈ ϪJKЉ
K
a c
␯ ͑A ͒
eQqeЈff
eQqeЉff
␯ ͑A ͒
eQqeЈff
eQqeЉff
K
K
0
1
0
1
a
c
111
212
313
414
515
616
404
505
615
716
717
220
321
422
523
624
725
826
927
–000
–101
–202
–303
–404
–505
–313
–414
–523
–624
–625
–211
–312
–413
–514
–615
–716
–817
–918
9 171.278 78͑92͒
12 658.155 06͑41͒
16 002.971 40͑91͒
19 220.199 94͑97͒
22 331.870 18͑84͒
25 367.586 6͑18͒
10 381.773 83͑66͒
14 604.645 ͑5͒
•••
0.1900͑23͒
Ϫ0.4349͑19͒
Ϫ0.5978͑39͒
Ϫ0.6703͑49͒
Ϫ0.6933͑37͒
⌬ϭ
•••
•••
•••
Ϫ0.4402͑22͒
Ϫ0.6031͑18͒
Ϫ0.6563͑64͒
•••
•••
Ϫ0.6406͑16͒
Ϫ0.6468͑31͒
Ϫ0.6404͑54͒
Ϫ0.6619͑51͒
Ϫ0.031͑10͒
Ϫ0.594͑17͒
•••
11 190.263 22͑46͒
14 736.326 57͑42͒
18 218.538 2͑13͒
21 636.5͑2͒
Ϫ0.6063͑18͒
Ϫ0.6140͑15͒
Ϫ0.6352͑68͒
•••
•••
•••
•••
Ϫ0.663͑17͒
•••
11 186.355 11͑28͒
15 069.303 0͑23͒
10 961.705 22͑22͒
14 382.2͑2͒
Ϫ0.6369͑11͒
Ϫ0.665͑12͒
Ϫ0.509͑18͒
•••
Ϫ0.6008͑12͒
Ϫ0.649͑13͒
Ϫ0.426͑18͒
•••
•••
•••
•••
•••
•••
•••
•••
•••
11 563.1͑1͒
•••
•••
14 957.879 84͑62͒
14 466.829 1͑10͒
13 876.9524 7͑45͒
13 235.854 6͑10͒
12 597.807 8͑12͒
12 021.017 78͑97͒
11 563.1͑1͒
0.6383͑20͒
Ϫ0.2089͑18͒
Ϫ0.3896͑54͒
Ϫ0.4317͑16͒
ϩ0.12504͑50͒
ϩ0.0768͑42͒
ϩ0.0655͑54͒
•••
12 791.831 04͑91͒
12 648.242 40͑96͒
12 489.265 08͑33͒
12 339.551 90͑10͒
12 225.642 95͑62͒
12 173.439 88͑62͒
12 206.5͑1͒
0.6302͑29͒
0.0c
Ϫ0.2441͑13͒
⌬ϭ
Ϫ0.4136͑23͒
⌬ϭ
Ϫ0.1841͑29͒
Ϫ0.3542͑50͒
Ϫ0.4171͑11͒
ϩ0.0996͑50͒
Ϫ0.4667͑25͒
ϩ0.0330͑31͒
•••
c
0.0
Ϫ0.2265͑18͒
⌬ϭ
⌬ϭ
⌬ϭ
•••
•••
•••
⌬ϭ
•••
•••
•••
•••
•••
12 345.879 58͑57͒
12 611.56͑3͒
ϩ0.0387͑29͒
•••
1
028–1019
•••
1
129–111,10
•••
•••
•••
13 024.21͑1͒
•••
•••
221
322
423
524
625
726
827
–212
–313
–414
–515
–616
–717
–818
15 752.271 51͑96͒
16 013.863 03͑79͒
16 369.835 15͑82͒
16 826.370 74͑58͒
17 390.775 28͑50͒
18 070.759 09͑54͒
•••
0.6357͑45͒
0.0
Ϫ0.2478͑46͒
Ϫ0.3765͑28͒
Ϫ0.4373͑45͒
Ϫ0.6057͑28͒
Ϫ0.6685͑46͒
Ϫ0.7004͑22͒
ϩ0.2693͑28͒
ϩ0.2363͑19͒
•••
13 357.684 49͑67͒
13 757.239 65͑61͒
14 288.619 06͑76͒
14 950.469 81͑64͒
15 740.621 975͑85͒
16 655.643 0͑12͒
17 690.478 16͑15͒
0.6292͑31͒
0.0c
Ϫ0.2517͑41͒
Ϫ0.3939͑30͒
⌬ϭ
Ϫ0.4410͑31͒
Ϫ0.5911͑20͒
Ϫ0.6563
Ϫ0.7037͑28͒
ϩ0.25564͑47͒
Ϫ0.7377͑62͒
ϩ0.20289͑86͒
c
c
⌬ϭ
⌬ϭ
•••
Ϫ0.5178͑57͒
⌬ϭ
aExperimental uncertainties are one standard deviation as determined by a least squares fit of each rotational transition to a line center and one or two
quadrupole coupling constants.
b
⌬
ϭeQqЈ ϪeQqЉ .
eff eff
c
Constrained in the fit.
respectively. Becke3LYP ͑also called B3LYP͒ is similar, but
also includes the HF density in a parameterized way.23 It is
closely related to the three-parameter functional that Becke
optimized for the thermochemistry of small molecules.24 Vi-
brational analysis ͑stationary point characterization͒ was
done only at the HF and MP2 levels. Relative energies of the
four conformations were also computed at the
CCSD͑T͒/6-311G*, frozen-core CCSD͑T͒/6-311G*, and
frozen-core CCSD͑T͒/6-311ϩG* levels, in all these cases
using the QCISD/6-311G* geometries.
GAUSSIAN 92/DFT,27 GAUSSIAN 94,28 and ACES II29,30 program
packages.
31
III. RESULTS
A. Microwave spectrum
1. Rotational analysis
The observed transitions for N O corrected for hyper-
2
5
fine splittings are listed in Table I. A complete set of the
measured transitions which include the frequencies of the
individual hyperfine components is available from PAPS.32
Two sets of essentially equal intensity b-type asymmetric-
rotor-like transitions are observed, assigned to two internal-
The electric-field gradient was computed at the nitrogen
nuclei at the QCISD//QCISD/6-311G* level. To calibrate the
accuracy of these results, the analogous calculation was done
ϩ
Ϫ
for nitric acid ͑HNO ͒, for which experimental results are
rotor/tunneling states of the molecule, denoted A1 and A1 in
the table. Since collisional relaxation is allowed between the
two tunneling states, the nearly equal intensity of the two
sets of transitions indicates that the energy difference be-
tween the two states must be small compared to the
molecular-beam temperature of ϳ2.5 Kϵ52 GHz. The large
number of other internal-rotor states possible for a system
with two tops are not observed, either because they are not
populated due to the cold molecular-beam temperature, or
because they cannot exist due to the Bose–Einstein statistics
of the spin-zero oxygen nuclei being interchanged by the
3
2
5
available. Further test calculations on HNO , using lower
3
levels of theory and both larger and smaller basis sets,
showed that the orientation of the principal axes of the field-
gradient tensor is insensitive to these computational details,
as has been found previously.26
The d-type polarization functions were used in spherical
form ͑5D͒ for all calculations except the QCISD geometry
optimizations and electric-field gradients, in which case soft-
ware restrictions required the Cartesian form ͑6D͒. All
electrons were active in correlated calculations unless
otherwise indicated. Computations were done using the
internal rotation. Transitions are observed for the K ϭ2–1
a
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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Grabow et al.: Microwave spectrum of N O5
7253
2
and 1–0 subbands of the two states, for J levels up to 11. No
evidence was found for transitions between the two states,
presumably because these transitions are either too weak or
forbidden. We note that the ‘‘permanent’’ b-type dipole mo-
ment component itself must be small, less than ϳ0.5 D, as
estimated by the amount of power needed to create a ␲/2
polarizing pulse.
HϭH ϩH ͑I ,J͒ϩH ͑I ,J͒,
͑1͒
rt
Q
1
Q
2
where
H ͑I ,J͒ϭ
͑eQqJ͒i
I ͑2I Ϫ1͒J͑2JϪ1͒
Q
i
2
i
i
2
3
2
2
2
i
ϫ͓3͑I –J͒ ϩ ͑I –J͒ϪI J ͔.
͑2͒
i
i
Attempts to fit the rotational series of either of the
internal-rotor/tunneling states to a Watson11 asymmetric-
rotor Hamiltonian proved unsuccessful. Because the hyper-
fine analysis discussed below definitively establishes the J
Here, H is the rotation–internal-rotation Hamiltonian, I is
rt
i
the nuclear-spin angular momentum of the ith nitrogen
nucleus, and (eQq ) is the coupling constant of the ith ni-
J
i
trogen nucleus and is a characteristic of a particular rotation–
internal-rotation state. Because the two nitrogen nuclei
are equivalent, only the average of the (eQq ) ,
and K quantum numbers of the transitions, it is improbable
a
that misassigned transitions are responsible for our inability
to fit the transitions to an asymmetric-rotor Hamiltonian.
More likely, the rotational progressions are strongly affected
J
i
(
eQqJ)aveϭ[(eQq ) ϩ(eQq ) ]/2, is determined for a par-
J 1 J 2
ticular rotation-internal-rotation state. The eigenvalues of H
by the internal rotation of the NO groups. In the high
2
are evaluated in the energy representation of H , with the
rt
internal-rotation-barrier limit, the rotational transitions of the
two observed internal states for the same set of rotational
quantum numbers will converge to the same frequency.
However, we observe large splittings of the rotational tran-
quadrupole interaction treated by first-order perturbation
theory in this basis. In the initial analysis of the hyperfine
interactions, no explicit knowledge of H is required; each
rt
observed transition is fit to a hyperfine-free line center and
two average quadrupole coupling constants, (eQqJ)ave , one
for the upper state and the other for the lower state. The
sitions. For instance, the 2 –1 transitions for the two states
12
01
are separated by ϳ1.5 GHz, while the 2 –2 transitions are
21
12
separated by ϳ2.4 GHz.
If we apply a rigid asymmetric rotor analysis and assume
that the 2 –1 transition occurs at Aϩ3C and the 2 –2
(
eQqJ)ave were scaled by (2Jϩ3)/J for JÞ0 to remove the
J dependence from the constants. The resulting effective av-
erage coupling constants for each state, eQqeff are listed in
Table I next to the hypothetical hyperfine-free line centers.
The eQqeff for the 1 , 1 , 1 , 2 , 2 , and 2 states
are related to the components of the quadrupole coupling
tensor along the principal inertial axes, eQqaa , eQqbb , and
eQqcc , averaged over the two nuclear environments. For the
1
2
01
21
12
transition occurs at 3(A–C), we find that the A and C rota-
tional constants are approximately 7103 and 1852 MHz for
01
11
10
12
11
21
ϩ
Ϫ
the A1 state and 6137 and 1684 MHz for the A1 state. The
effect of nonrigidity is seen by noting that these A and C
ϩ
values for the A1 state predict the 1 –0 transition at
11
00
ϳ8955 MHz ͑i.e., AϩC͒ compared to the observed value of
171 MHz. The apparent B rotational constants for the two
states are also significantly different. The Kϭ0 combination
differences from the K ϭ1Ϫ0 subband yield an approximate
101
, 1 , 1 , 2 , 2 , and 2 states the eQqeff are equal to
11 10 12 11 21
9
ϪeQqaa , ϪeQqbb , ϪeQqcc , eQqcc , eQqbb , and eQqaa
,
respectively, independent of the asymmetry parameter, ␬.
This approximation breaks down if the two internal-rotor/
tunneling states are interacting with each other or with other
states through Coriolis interactions, since, for example, a
state which correlates with a 101 state may now be an admix-
ture of 101 and 110 character due to a b-type Coriolis inter-
action operating between the two tunneling states. From an
analysis of the hyperfine structure of transitions involving
the 1 , 1 , 1 , 2 , 2 , and 2 states we obtain
a
ϩ
(
BϩC)/2 value for the A1 state of 1894 MHz, correspond-
ing to a B value of ϳ1936 MHz, and an approximate (B
Ϫ
ϩC)/2 value for the A1 state of 1855 MHz, corresponding
to a B value of ϳ2026 MHz.
2. Hyperfine analysis
01
11
10
12
11
21
The Iϭ1, 14N quadrupole hyperfine structure allows de-
eQq ϭ0.6291͑54͒ MHz, eQq ϭϪ0.1948͑59͒ MHz, and
aa bb
finitive assignments of the J and K quantum numbers to the
eQq ϭϪ0.4343͑80͒ MHz. For an unperturbed vibrational
a
cc
transitions and provides information about the geometry of
the molecule. The observed quadrupole hyperfine structure is
characteristic of two equivalent quadrupolar nuclei, demon-
strating that N O has two different nuclear-spin modifica-
or internal-rotor/tunneling state, eQq ϩeQq ϩeQq ϭ0,
aa
bb
cc
for eQqii determined in this manner. Equivalently, the
eQqeff for all the K K components of a given J sum to zero
a
c
within a tunneling or vibrational state. Both of these condi-
2
5
tions. For both internal-rotor/tunneling states, rotational
tions are violated by the coupling constants listed in Tables I.
ϩ
states with K ϩK even have a resultant nitrogen nuclear
For Jϭ2 of the A1 state, the eQq ’s sum to Ϫ0.0178͑65͒
a
c
eff
Ϫ
spin of I ϭ0,2, while rotational states with K ϩK odd
MHz, while for the A1 state they sum to 0.0316͑58͒ MHz. If
N
a
c
have I ϭ1. This fact implies that in the high-barrier limit
when the two tunneling states become degenerate, states with
we add these sums for the two internal-rotor states, we obtain
a value of 0.0138͑87͒ MHz, which is effectively zero for an
uncertainty of two standard deviations. This result suggests
that the two internal-rotor states are interacting via Coriolis
interactions, redistributing the quadrupole coupling constant
components between them.
N
K ϩK even and K ϩK odd will have different nuclear
a
c
a
c
spins. The presence of two different spin modifications in the
high-barrier limit requires that the equilibrium configuration
of the molecule has a twofold axis of symmetry.
For each transition, the hyperfine patterns were fit to the
In addition to guiding the rotational assignment, the 14N
quadrupole hyperfine structure provides precise structural in-
frequencies calculated from the energy-level expression33
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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7254
Grabow et al.: Microwave spectrum of N O5
2
formation on N O . To obtain this information, we first as-
2
5
sume that the electric-field-gradient tensors at the two nitro-
gen nuclei are similar to that found in HNO . Ab initio
3
calculations presented in the next section support this as-
sumption. Making this approximation allows us to calculate
the components of the electric field gradient, and thus of the
quadrupole coupling constant, along the principal inertial
axis system of N O by projecting the quadrupole tensor of
2
5
the individual NO groups onto the inertial frame. These pro-
2
jections are principally a function of the NO2 torsional
angles, ␣ and ␣ , and the NON valence angle, ␥, defined in
1
2
Fig. 1. Because experimentally we observe that the two ni-
trogen nuclear environments are dynamically equivalent, we
average the projections of the two NO groups before com-
2
paring with experiment. The angles ␣ and ␣ are referenced
1
2
to the planar configuration, where ␣ ϭ0 and ␣ ϭ0. ␣ ϭ␣
1
2
1
2
corresponds to a C2 symmetry structure, ␣ ϭ180°Ϫ␣ ,
1
2
␣1
ϭ0, ␣ ϭ90°, and ␣ ϭ90°, ␣ ϭ0 correspond to C sym-
2
1
2
s
FIG. 6. Plot of calculated eQqaa , eQqbb , and eQqcc as a function of
torsion angle, ␣ϭ␣ ϭ␣ for three values for the NON angle, ␥, using the
metry structures, and ␣ ϭ␣ ϭ90° corresponds to a C sym-
1
2
2v
1
2
model discussed in the text. The experimental values for the coupling con-
stants are represented by the three horizontal lines in the figure. The two
vertical lines show the values of ␣ where the electric-dipole selection rules
change from b type to c type. The best agreement between the calculated
and observed values occurs for ␣ϭ41° and ␥ 120.5°.
metry structure. We assume that the two NO groups are
3
planar and equivalent and fix the NO bond lengths and NO2
7
angles at the electron-diffraction values: rϭ1.183͑2͒ Å,
Rϭ1.492͑4͒ Å, and ␤ϭ133.2͑6͒°. Essentially identical re-
sults are obtained if we use the ab initio values for r, R, and
␤
, presented below.
The principal axes of the quadrupole coupling tensors of
the two nitrogen nuclei are approximated by placing an
x,y,z) axis systems on each of the two NO groups with the
ab initio calculations discussed below, we examine the pre-
dicted quadrupole coupling constants for three types of struc-
tures: a C configuration with ␣ϭ␣ ϭ␣ , a C structure
(
2
z-axes fixed normal to the NO planes and the x axes fixed
2
2
1
2
2v
along the NO bonds which form the NON backbone. For
with ␣ϭ␣ ϭ␣ ϭ0 or 90°, and a C structure with ␣ ϭ0,
␣ ϭ90°, or equivalently, ␣ ϭ90°, ␣ ϭ0. The results for pos-
2 1 2
1 2 s 1
25
HNO , Albinus et al. have found this to be an accurate
3
representation of the orientation of the principal-axis system;
sible C and C configurations are shown in Fig. 6, where
2 2v
the x axis of the principal-axis system of HNO is rotated by
we plot calculated values for the quadrupole coupling con-
stants, eQqaa , eQqbb , and eQqcc , as a function of ␣ for
three values of the N–O–N angle, ␥ϭ110.0°, 120.5°, and
130.0°. Recall for the electron-diffraction results
␥ϭ111.8͑6͒° and for the ab initio calculations ␥ϭ103°–
123°, depending on level of theory and values of ␣ and ␣ .
3
only 1.1° from the unique NO bond axis. Our ab initio cal-
culations discussed later support the assumption that the
principal-axis system of the electric-field gradient tensor of
N O has one component along the NO bond of the NON
2
5
backbone and distorts minimally with changes in the internal
rotation angle. Thus for N O we approximate the two quad-
1
2
The solid horizontal lines give the three experimental values
2
5
rupole coupling tensors by eQq ϭϪ0.070 MHz,
for the coupling constants ͓eQq ϭ0.6291͑54͒ MHZ,
zz
aa
eQq ϭϩ1.103 MHz, and eQq ϭϪ1.033 MHz, the prin-
cipal axis system components of HNO3.
Guided in part by the electron diffraction results and the
eQq ϭϪ0.1948͑59͒ MHz, and eQq ϭϪ0.4343͑80͒
MHz͔, taken by averaging the relevant results in Table I. The
two solid vertical lines in the figure show the values of ␣
xx
yy
bb
cc
25
TABLE II. Estimated quadrupole coupling constants for various geometries of N O in MHz.
2
5
␣1
␣2
ЄNON
eQqaa
0.4003
eQqbb
eQqcc
0°
0°
110°
Ϫ0.3303
Ϫ0.4990
Ϫ0.6515
0.0424
Ϫ0.0989
Ϫ0.2275
Ϫ1.0330
Ϫ1.0330
Ϫ1.0330
Ϫ0.0700
Ϫ0.0700
Ϫ0.0700
Ϫ0.5515
Ϫ0.5515
Ϫ0.5515
0.3159
1
1
20°
30°
0.5690
0.7215
0.5091
0.6504
0.7790
0.7171
0.8097
0.8935
0°
90°
90°
110°
1
1
20°
30°
90°
110°
1
1
20°
30°
0.2233
0.1395
Experiment
0.6291͑54͒
Ϫ0.1948͑59͒
Ϫ0.4343͑80͒
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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Grabow et al.: Microwave spectrum of N O5
7255
2
TABLE III. Computed and experimental geometries and dipole moments ͑␮͒ for conformations 1–4 of N O ͑C , C , C2v, and C2v symmetry, respectively͒
2
5
2
s
with all electrons active and the 6-311G* basis set. Terminal dihedral angles ͑N–O–N–O͒ are denoted by the symbol ␶. Other parameters are defined in Fig.
1
. For conformation 1, the average angles ␣ ϭ␣ ϭ␣ϵ͑␶ ϩ180°ϩ␶ ͒/2.
1 2 1 2
B3LYP
QCISD
Parameter
1
2
3
4
1
2
3
4
Expt.a
1.498
Ra ͑Å͒
1.568
1.493
1
.513
1.531
1.525
1.185
1.470
1.485
1.485
Rb ͑Å͒
r1 ͑Å͒
1.460
1.182
1.447
1.188
1
1
.188
1.192
1.188
1.187
1.188
r4 ͑Å͒
r2 ͑Å͒
1.198
1.182
1.190
1.196
1.188
1.187
1.188
1.191
1.188b
.199
1.186
r3 ͑Å͒
␤a ͑°͒
134.9
133.5
1
33.8
115.3
0.7
133.3
107.3
91.2
133.5
132.9
132.6
132.9
133.2
␤b ͑°͒
132.2
110.1
91.5
132.5
109.1
91.5
␥
␶1
͑°͒
͑°͒
120.9
0.0
111.6
38.6
105.9
91.1
120.6
0.0
111.8
3
ϳ30b
␶4
␶2
͑°͒
͑°͒
0.0
Ϫ91.5
0.0
Ϫ91.5
Ϫ151.3
Ϫ91.2
180.0
0.0
Ϫ145.0
Ϫ91.1
180.0
0.0
ϳϪ150b
ϳ30
␶3
␣1
͑°͒
͑°͒
180.0
90.0
180.0
90.0
29.7
90.0
36.8
90.0
␣2
͑°͒
0.0
0.78
0.0
0.43
c
␮
͑D͒
0.03
0.03
0.25
0.24
0.21
0.17
1.39
a
Geometry ͑conformation 1͒ from Ref. 7 and dipole moment ͑in CCl solution͒ from Ref. 10.
4
b
Local C2v symmetry was assumed in Ref. 7.
c
Ϫ30
1
Dϭ3.3357ϫ10
C m.
͑
47Ϯ1° and 133Ϯ1°͒ where the b and c inertial axes switch
listed; the results from the HF, SVWN, BLYP, and MP2
calculations may be obtained directly from the authors. Fig-
ure 5 shows the optimized geometries for structures 1–4 at
our highest level of theory ͑QCISD͒. The deviation from
for the N–O–N angles considered. The b and c axis switch-
ing arises due to the definition of rotational constants which
requires that AϾBϾC, and leads to a flipping of the selec-
tion rules from b-type ͑␮ Þ0͒ to c type ͑␮ Þ0͒. As seen in
the figure, the calculated quadrupole coupling constants are a
sensitive function of ␣. The best agreement with experiment
is obtained for ␣ϭ41° and ␥ϭ120.5° with 1␴ uncertainties
of less than 1° on each angle. These choices of angles predict
b-type selection rules, as observed experimentally.
b
c
planarity of the O–NO groups, defined as the supplement of
2
the dihedral angle between terminal oxygen atoms, is 3.6°,
3
.0° ͑for the out-of-plane group͒, 2.2°, and 0° ͑by symmetry͒
for structures 1–4, respectively. The computed relative ener-
gies, without any corrections for vibrational zero-point en-
ergy, are collected in Table IV. The numbers of imaginary
vibrational frequencies obtained for structures ͑1,2,3,4͒ are
͑1,0,2,1͒ and ͑0,0,2,2͒ at the HF/6-311G* and MP2/6-311G*
Only the C configuration discussed above is consistent
2
with the observed quadrupole coupling constants. This is
seen in Fig. 6 where we give calculated eQq’s for the two
possible C2v configurations ͑␣ϭ0 or ␣ϭ90°͒ and in Table II
levels, respectively. For structure 1 ͑C symmetry͒, which is
2
7
where we compare these configurations with the C configu-
the one determined experimentally, the computed vibra-
s
ration ͑␣ ϭ0, ␣ ϭ90° or ␣ ϭ90°, ␣ ϭ0͒. The experimental
1
2
1
2
tional frequencies and infrared intensities are listed in Table
V.
Electric field gradients at the nitrogen nuclei were char-
acterized by the angle formed between the chemically unique
N–O bond and the nearest principal axis of the electric-field
results are clearly inconsistent with a C2v or C geometry for
s
the molecule. In particular, for the C configuration the cal-
s
culated value for eQqcc is Ϫ0.5515 MHz, independent of ␥,
and differs significantly from the observed value of
Ϫ0.4343͑80͒ MHz.
gradient tensor. For HNO , the computed angle is 0.3°, in-
3
dicating that the N–O bond lies along a principal axis of the
H
B. Ab initio calculations
electric-field gradient tensor. This value compares favorably
7
with the experimental angle of 1.1°. For N O , the com-
The computed geometries and dipole moments for struc-
tures 1–4 are summarized in Table III. For conciseness, only
the results from the B3LYP and QCISD calculations are
2
5
puted angles are 7.2°, 1.3°, 2.4°, and 4.2° for structures 1–4,
respectively.
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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7256
Grabow et al.: Microwave spectrum of N O5
2
TABLE IV. Computed relative energies in cmϪ1 for conformations 1–4.
The 6-311G* basis set was used and all electrons were active in correlated
calculations, except as noted.
TABLE V. Computed vibrational frequencies in cmϪ1 and infrared intensi-
ties in km/mol for conformation 1 ͑C2 symmetry͒ of N O . The HF and
2
5
MP2 frequencies have been scaled by 0.90 and 0.95, respectively.
Relative energy of conformation
HF/6-311G* MP2/6-311G*
Calculation
1
2
3
4
Symm.
Freq.
Int.
Freq.
Int.
HF
MP2
SVWN
BLYP
B3LYP
QCISD
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Ϫ28.6
Ϫ82.4
265.5
288.5
120.4
Ϫ8.2
Ϫ35.6
Ϫ29.2
154.5
2751.4
2242.7
3360.5
2630.3
2484.7
2312.4
2188.9
2172.9
2400.7
1508.2
583.5
174.0
81.7
380.1
1065.4
868.4
863.8
1176.0
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
1801
1451
974
879
712
461
247
81
1757
1344
893
787
767
528
7i
681.8
82.3
17.0
12.7
7.4
0.5
2.4
0.8
1005.6
381.4
709.7
90.6
17.4
22.1
0.8
1879
1303
822
767
631
347
213
64
1861
1224
719
690
519
226
18
182.8
48.5
45.0
1.1
2.3
3.3
a
CCSD͑T͒
val-CCSD͑T͒
val-CCSD͑T͒
0.3
0.3
a,b
a,b,c
178.8
289.5
217.3
30.0
233.2
574.6
13.8
a
Using the QCISD/6-311G* geometry.
Frozen-core; only valence electrons correlated.
Using the 6-311ϩG* basis set.
b
c
C. Internal-rotation theory
tunneling N O with a C minimum, such as found in the
2
5
2
In the present section we present a dynamical model
which allows the calculation of the rotational-tunneling spec-
trum of N O for a given structure and torsional potential.
electron diffraction and hyperfine analysis, each rotation–
vibration state is eightfold degenerate, corresponding to the
eight configurations pictured in Fig. 7. Tunneling between
the eight minima lifts the eightfold degeneracy. Starting from
a single reference configuration, four of the minima are
2
5
We start with a group theoretical discussion of N O using
2
5
34
the molecular-symmetry group of Longuet–Higgins.
A
model Hamiltonian is developed which treats all the internal
sampled by a geared rotation of the two NO groups about
2
modes as rigid except for the two NO torsions. Diagonal-
their C axes as pictured in Fig. 8. An antigeared inversion
2
2
ization of the associated Hamiltonian matrix is facilitated by
using symmeterized basis functions to factor the matrix. As
shown below, the resulting calculations give insight into the
nature of the two internal-rotor/tunneling states observed for
N O .
of each of these configurations through either a planar or
nonplanar C2v transition state, also shown in Fig. 8, allows
access to the other four minima. The vibrational wave func-
tions for the eight configurations generates an eight-
dimensional reducible representation of G16 as
2
5
ϩ
Ϫ
ϩ
Ϫ
ϩ
Ϫ
Allowing internal-rotation of the two NO groups, the
A1 ꢀ A1 ꢀ B2 ꢀ B2 ꢀ E ꢀ E . Thus tunneling between all
the eight configurations of Fig. 7 will split the Jϭ0 vibra-
2
most general molecular symmetry group for N O is G .
2
5
16
ϩ
Ϫ
ϩ
Ϫ
This group has been previously applied to help interpret the
tional state into six tunneling sublevels of A , A , B , B ,
E , and E symmetry.
1 1 2 2
35
ϩ
Ϫ
large-amplitude motions observed in the water dimer. The
character table for G16 is shown in Table VI,35 where the
permutation-inversion operations use the numbering of the
nuclei as given in Fig. 1. For a noninternally rotating or
The qualitative effects of tunneling between the eight
minimum can be gleaned by constructing a tunneling Hamil-
tonian matrix for a Jϭ0 N O5
2
TABLE VI. Character table for the G16 molecular symmetry group used for N O .
2
5
͑
12͒
(ab)͑14͒͑23͒
(ab)͑13͒͑24͒
(ab)͑1324͒
(ab)͑1423͒
͑12͒*
͑34͒*
(ab)͑14͒͑23͒*
(ab)͑13͒͑24͒*
(ab)͑1324͒*
(ab)͑1423͒*
E
͑12͒͑34͒
͑34͒
E*
͑12͒͑34͒*
ϩ
A1
1
1
1
1
2
1
1
1
1
2
1
1
1
1
Ϫ2
1
1
1
1
Ϫ2
1
Ϫ1
1
Ϫ1
0
1
Ϫ1
1
1
Ϫ1
Ϫ1
1
0
1
Ϫ1
Ϫ1
1
1
1
Ϫ1
Ϫ1
0
1
1
Ϫ1
Ϫ1
0
1
1
1
1
2
Ϫ1
Ϫ1
Ϫ1
Ϫ1
Ϫ2
1
1
1
1
Ϫ1
1
Ϫ1
0
Ϫ1
1
Ϫ1
1
1
Ϫ1
Ϫ1
1
0
Ϫ1
1
1
Ϫ1
0
1
1
Ϫ1
Ϫ1
0
Ϫ1
Ϫ1
1
ϩ
A2
ϩ
B1
ϩ
B2
1
ϩ
Ϫ
E
Ϫ2
Ϫ1
Ϫ1
Ϫ1
Ϫ1
2
A1
Ϫ
A2
Ϫ
B1
Ϫ
B2
Ϫ1
0
1
0
Ϫ
E
0
0
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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Grabow et al.: Microwave spectrum of N O5
7257
2
␾1
␾2
␾3
␾4
hg
␾5
hn
␾6
hi
hn
hi
␾7
hp
␾8
hi
hp
hi
␾1
␾2
␾3
␾4
␾5
␾6
␾7
␾8
E0
Ϫ
͉
hg
͉
͉
͉
0
Ϫ
͉
͉
͉
͉
Ϫ
͉
͉
Ϫ
͉
͉
Ϫ
͉
͉
Ϫ
͉
͉
Ϫ
͉
hg
͉
E0
Ϫ
͉
hg
͉
0
Ϫ
͉
hi
͉
Ϫ
Ϫ
Ϫ
Ϫ
͉
͉
Ϫ
͉
hi
͉
Ϫ
Ϫ
Ϫ
Ϫ
͉
͉
0
Ϫ
͉
hg
E0
Ϫ
͉
hg
E0
hi
hp
Ϫ
͉
hp
͉
͉
͉
Ϫ
͉
hn
͉
͉
͉
Ϫ
Ϫ
͉
͉
hg
hn
͉
͉
0
Ϫ
͉
͉
hg
hp
hi
hn
hi
͉
͉
Ϫ
͉
hi
͉
͉
hp
hg
͉
͉
Ϫ
͉
hi
͉
͉
hn
hg
͉
͉
,
͑3͒
Ϫ
Ϫ
͉
hi
Ϫ
Ϫ
͉
E0
͉
0
͉
͉
Ϫ
Ϫ
͉
hi
hp
hi
͉
͉
hn
͉
Ϫ
Ϫ
͉
͉
Ϫ
Ϫ
Ϫ
͉
͉
Ϫ
Ϫ
͉
hg
͉
͉
E0
hg
Ϫ
͉
hg
͉
͉
0
΄
΅
͉
͉
Ϫ
Ϫ
͉
hi
͉
͉
͉
͉
hi
͉
0
Ϫ
͉
͉
E0
hg
Ϫ
hg
͉
Ϫ
͉
͉
͉
hp
͉
Ϫ
͉
͉
͉
hn
͉
͉
hg
0
Ϫ
͉
E0
where ␾ –␾ are the vibrational wave functions for the eight different configurations of Fig. 7,
element describing the coupling between nearest neighbor minima for the geared internal rotation,
matrix element for the antigeared inversion through a planar transitions state,
antigeared inversion through a nonplanar C2␯ transition state, and
rotation of one of the NO groups by 180°. The coefficients of these matrix elements are negative to insure that the totally
symmetric state is lowest in energy. The eigenvalues and eigenvectors resulting from diagonalization of the matrix are
͉
hg
͉
is a tunneling matrix
is the corresponding
is the tunneling matrix element for the
1
8
͉
hp
͉
͉
hn
͉
͉
hi
͉
is a tunneling matrix element for the independent
2
ϩ
A1 ͑␾ ϩ␾ ϩ␾ ϩ␾ ϩ␾ ϩ␾ ϩ␾ ϩ␾ ͒/
ͱ
8
ͱ
8
ͱ
8
ͱ
8
E Ϫ2
͉
͉
͉
͉
hg
hg
hg
hg
͉
͉
͉
͉
Ϫ
ϩ
Ϫ
ϩ
͉
͉
͉
͉
hn
hn
hn
hn
͉
͉
͉
͉
Ϫ
ϩ
Ϫ
ϩ
͉
͉
͉
͉
h
p͉Ϫ2
͉
͉
͉
͉
hi
hi
hi
hi
͉
͉
͉
͉
1
2
3
4
5
6
7
8
0
Ϫ
A1 ͑␾ Ϫ␾ ϩ␾ Ϫ␾ Ϫ␾ ϩ␾ Ϫ␾ ϩ␾ ͒/
E ϩ2
0
hp
hp
hp
͉
͉
͉
Ϫ2
ϩ2
ϩ2
1
2
3
4
5
6
7
8
ϩ
B2 ͑␾ Ϫ␾ ϩ␾ Ϫ␾ ϩ␾ Ϫ␾ ϩ␾ Ϫ␾ ͒/
E ϩ2
0
1
2
3
4
5
6
7
8
Ϫ
B2 ͑␾ ϩ␾ ϩ␾ ϩ␾ Ϫ␾ Ϫ␾ Ϫ␾ Ϫ␾ ͒/
E Ϫ2
0
1
2
3
4
5
6
7
8
,
͑4͒
ϩ
E
͑␾ Ϫ␾ ϩ␾ Ϫ␾ ͒/
ͱ
ͱ
ͱ
ͱ
4
E Ϫ
͉
͉
͉
͉
hn
hn
hn
hn
͉
͉
͉
͉
ϩ
ϩ
Ϫ
Ϫ
͉
͉
͉
͉
hp
hp
hp
hp
͉
͉
͉
͉
1
3
7
5
0
͑
␾ Ϫ␾ ϩ␾ Ϫ␾ ͒/
2
4
4
4
E0Ϫ
4
8
6
Ϫ
E
͑␾ Ϫ␾ Ϫ␾ ϩ␾ ͒/
E ϩ
0
1
3
7
5
͑
␾ Ϫ␾ Ϫ␾ ϩ␾ ͒/
2
E0ϩ
4
8
6
where E is the zero of energy for the degenerate nontunnel-
observed states while still having the molecular dynamics
described by the G16 molecular symmetry group requires 17
oxygen substitution ͑Iϭ5/2͒ on all the oxygens of the NO2
groups. The cost of such an experiment is prohibitive in our
case.
0
ing vibrational states. Figure 9 shows the predicted energy
level diagram for the 000 state under the assumption that the
tunneling matrix elements have the size order
O
2
͉ ͉Ͼ͉hn͉Ͼ͉hp͉Ͼ2͉hi͉.
hg
1
4
16
For the dominant isotopomer ͑ N O ͒ the total wave
The group theory allows us to construct a symmeterized
basis set for discussing the dynamics of N O . Neglecting
2
5
ϩ
Ϫ
function must be of A1 or A1 symmetry in G . Since the
nuclear-spin functions transform as 6A ꢀ 3B , only rovi-
bronic states of A , A , B , and B symmetry are allowed.
The A states have a statistical weight of 6 ͑corresponding to
16
2
5
ϩ
1
ϩ
the small amplitude coordinates, an appropriate coordinate
system for describing the rotation–internal rotation dynamics
1
ϩ
1
Ϫ
1
ϩ
1
Ϫ
1
Ϯ
1
36
of N O consists of the three Euler angles ͑␪,␾,␹͒ relating
2
5
Ϯ
a resultant N nuclear spin of I ϭ0,2͒ and the B1 states have
the body-fixed and space-fixed Cartesian coordinate systems
N
a statistical weight of 3 ͑resultant N nuclear spin of I ϭ1͒.
and the two internal-rotation angles, ␣ and ␣ ͑see Fig. 1͒.
N
1
2
Ϫ
The electric dipole moment operator transforms as A , giv-
We fix the right-handed body-fixed axis system so that the y
axis is normal to the N –O–N plane, the x axis bisects the
1
Ϯ
1
ϯ
1
Ϯ
2
ϯ
2
Ϯ
1
ϯ
1
Ϯ
2
ϯ
2
Ϯ
ϯ
ing A ↔A , A ↔A , B ↔B , B ↔B , and E ↔E
a
b
selection rules on the rovibronic species.
The large number of vanishing states in N O is a con-
N –O–N angle with the positive x direction pointing from
a b
the N’s to the O, and the z axis is parallel to a line connect-
ing N to N , with the positive z direction pointing from N
2
5
sequence of the spin-zero oxygen nuclei, which allow only
rovibronic species which are symmetric with respect to in-
terchange of the equivalent oxygen nuclei. In the case of
b
a
b
to Na .
In Table VII we show the transformation properties of
the Euler and internal-rotation angles under the effects of the
operations of the molecular-symmetry group. Also given are
1
5
N O ͑which was not studied in the present investigation͒
2
5
ϩ
Ϫ
the total wave function must be of B1 or B symmetry.
Again, the allowed rovibronic states have A , A , B , and
B1 symmetry, however, for spin-1/2 N the A1 states have
a statistical weight of 1 ͑I ϭ0͒ and the B1 states have a
1
ϩ
1
Ϫ
1
ϩ
1
37
the equivalent rotations of Bunker for each operation. From
Ϫ
15
Ϯ
the transformation properties of the Euler angles, or similarly
from the equivalent rotations, we can determine the symme-
try properties of the rotational wave functions specifying the
Ϯ
N
statistical weight of 3 ͑I ϭ1͒. To increase the number of
N
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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7258
Grabow et al.: Microwave spectrum of N O5
2
FIG. 7. The eight configurations responsible for the eightfold structural
degeneracy of a nontunneling/noninternally rotating N O . The vibrational
2
5
FIG. 9. Qualitative energy level diagram for Jϭ0 N O showing the lifting
2 5
wave functions for the eight configurations are denoted ␾ through ␾ . The
1
8
of the eightfold structural degeneracy of the rigid molecule through internal-
rotation tunneling of the oxygen nuclei. The h’s are tunneling matrix ele-
ments and are defined in the text.
even-numbered oxygen nuclei are labeled with asterisks for clarity.
relative orientation of the (x,y,z) body-fixed axis system to
r
11
the space-fixed axis system. For a I representation the
37
ϩ
Wang symmetric-rotor functions have the symmetries A ,
B , A , and B for the ͉K Kc͘ϭ͉ee , ͉eo , ͉oo , and ͉oe
͘ ͘ ͘ ͘
a
states, respectively, where e and o refer to the limiting K or
1
Ϫ
1
Ϫ
1
ϩ
1
a
K quantum number being even or odd. Similarly, using the
c
transformation properties of the internal-rotation angles we
can use projection operators to construct symmeterized
internal-rotation wave functions from an unsymmeterized
basis set consisting of products of sin͑n␣ ͒ or cos͑n␣ ͒ with
1
1
sin͑m␣ ͒ or cos͑m␣ ͒ where n,m are integers. These sym-
2
2
meterized internal rotor functions are listed in Table VIII.
Products of the symmeterized internal-rotor functions and
the Wang symmetric rotor functions are used as basis func-
TABLE VII. Equivalent rotations and effects of symmetry operations on
internal rotation and Euler angles for N O .
2
5
0
0
0
0
␲
E
␣1
␣2
␣2ϩ␲
␣2
␣2ϩ␲
␣1
␣1ϩ␲
␣1
␪
␾
␾
␾
␾
␹
␹
␹
␹
R
R
R
R
͑
͑
͑
͑
(
(
(
12͒͑34͒
12͒
34͒
ab͒͑14͒͑23͒
ab)͑13͒͑24͒
ab)͑1423͒
ab)͑1324͒
␣1ϩ␲
␣1ϩ␲
␣1
␪
␪
␪
␣2
␲Ϫ␪ ␾ϩ␲ 2␲Ϫ␹ Rx
␲
␣2ϩ␲
␣2ϩ␲
␣2
␲Ϫ␪ ␾ϩ␲ 2␲Ϫ␹ Rx
␲
␲Ϫ␪ ␾ϩ␲ 2␲Ϫ␹ Rx
␲
␣1ϩ␲
␲Ϫ␪ ␾ϩ␲ 2␲Ϫ␹ Rx
␲
E*
2␲Ϫ␣1 2␲Ϫ␣2 ␲Ϫ␪ ␾ϩ␲ ␲Ϫ␹
␲Ϫ␣1
␲Ϫ␣1
Ry
␲
͑
͑
͑
(
(
12͒͑34͒*
12͒*
34͒*
␲Ϫ␣2
2␲Ϫ␣2 ␲Ϫ␪ ␾ϩ␲ ␲Ϫ␹
␲Ϫ␪ ␾ϩ␲ ␲Ϫ␹
␲Ϫ␪ ␾ϩ␲ ␲Ϫ␹
Ry
␲
Ry
␲
2␲Ϫ␣1 ␲Ϫ␣2
Ry
␲
ab)͑14͒͑23͒* 2␲Ϫ␣2 2␲Ϫ␣1
ab)͑13͒͑24͒* ␲Ϫ␣2
␪
␪
␪
␪
␾
␾
␾
␾
␹ϩ␲
␹ϩ␲
␹ϩ␲
␹ϩ␲
Rz
␲
␲Ϫ␣1
␲Ϫ␣1
2␲Ϫ␣1
Rz
␲
FIG. 8. Possible tunneling pathways for oxygen interchange in N O . The
(ab)͑1324͒*
(ab)͑1423͒*
2␲Ϫ␣2
␲Ϫ␣2
Rz
2
5
␲
bridging O is furthest away from the reader. The even-numbered oxygen
nuclei are labeled with an asterisks for clarity.
Rz
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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Grabow et al.: Microwave spectrum of N O5
7259
2
a,b
TABLE VIII. Symmetrized internal-rotor basis functions for N O .
2
5
ϩ
A1
cos͑n␣ ͒cos͑m␣ ͒ϩcos͑n␣ ͒cos͑m␣ ͒
nуmу0
nуmу2
nϾmу0
nуmу2
nϾmу0
nϾmу2
nϾmу0
nуmу2
nϾmу1
nϾmу1
nϾmу1
nуmу1
nуmу1
nуmу1
nϾmу1
nуmу1
nу1,mу0
nу1,mу2
nу1,mу0
nу1,mу2
nу1,mу0
nу1,mу2
nу1,mу0
nу1,mу2
n even, m even
n even, m even
n even, m even
n even, m even
n even, m even
n even, m even
n even, m even
n even, m even
n odd, m odd
n odd, m odd
n odd, m odd
n odd, m odd
n odd, m odd
n odd, m odd
n odd, m odd
n odd, m odd
n odd, m even
n odd, m even
n odd, m even
n odd, m even
n odd, m even
n odd, m even
n odd, m even
n odd, m even
1
2
2
1
sin͑n␣ ͒sin͑m␣ ͒ϩsin͑n␣ ͒sin͑m␣ ͒
1
2
2
1
Ϫ
A1
sin͑n␣ ͒cos͑m␣ ͒ϩsin͑n␣ ͒cos͑m␣ ͒
1 2 2 1
cos͑n␣ ͒sin͑m␣ ͒ϩcos͑n␣ ͒sin͑m␣ ͒
1
2
2
1
ϩ
B1
cos͑n␣ ͒cos͑m␣ ͒Ϫcos͑n␣ ͒cos͑m␣ ͒
1 2 2 1
sin͑n␣ ͒sin͑m␣ ͒Ϫsin͑n␣ ͒sin͑m␣ ͒
1
2
2
1
Ϫ
B1
sin͑n␣ ͒cos͑m␣ ͒Ϫsin͑n␣ ͒cos͑m␣ ͒
1 2 2 1
cos͑n␣ ͒sin͑m␣ ͒Ϫcos͑n␣ ͒sin͑m␣ ͒
1
2
2
1
ϩ
A2
cos͑n␣ ͒cos͑m␣ ͒Ϫcos͑n␣ ͒cos͑m␣ ͒
1 2 2 1
sin͑n␣ ͒sin͑m␣ ͒Ϫsin͑n␣ ͒sin͑m␣ ͒
1
2
2
1
Ϫ
A2
sin͑n␣ ͒cos͑m␣ ͒Ϫsin͑n␣ ͒cos͑m␣ ͒
1 2 2 1
cos͑n␣ ͒sin͑m␣ ͒Ϫcos͑n␣ ͒sin͑m␣ ͒
1
2
2
1
ϩ
B2
cos͑n␣ ͒cos͑m␣ ͒ϩcos͑n␣ ͒cos͑m␣ ͒
1 2 2 1
sin͑n␣ ͒sin͑m␣ ͒ϩsin͑n␣ ͒sin͑m␣ ͒
1
2
2
1
Ϫ
B2
sin͑n␣ ͒cos͑m␣ ͒ϩsin͑n␣ ͒cos͑m␣ ͒
1 2 2 1
cos͑n␣ ͒sin͑m␣ ͒ϩcos͑n␣ ͒sin͑m␣ ͒
1
2
2
1
ϩ
E1
cos͑n␣ ͒cos͑m␣ ͒ϩcos͑n␣ ͒cos͑m␣ ͒
1 2 2 1
sin͑n␣ ͒sin͑m␣ ͒ϩsin͑n␣ ͒sin͑m␣ ͒
1
2
2
1
ϩ
E2
cos͑n␣ ͒cos͑m␣ ͒Ϫcos͑n␣ ͒cos͑m␣ ͒
1 2 2 1
sin͑n␣ ͒sin͑m␣ ͒Ϫsin͑n␣ ͒sin͑m␣ ͒
1
2
2
1
Ϫ
E1
sin͑n␣ ͒cos͑m␣ ͒ϩsin͑n␣ ͒cos͑m␣ ͒
1 2 2 1
cos͑n␣ ͒sin͑m␣ ͒ϩcos͑n␣ ͒sin͑m␣ ͒
1
2
2
1
Ϫ
E2
sin͑n␣ ͒cos͑m␣ ͒Ϫsin͑n␣ ͒cos͑m␣ ͒
1 2 2 1
cos͑n␣ ͒sin͑m␣ ͒Ϫcos͑n␣ ͒sin͑m␣ ͒
1
2
2
1
a
Functions are not normalized.
b
The subscripts on the E species are not symmetry labels but refer to the two sets of functions which make up
the degenerate E species.
ϩ
tions for diagonalizing the rotation–internal rotation Hamil-
tonian of N O .
2
Tϭp ␮p,
͑7͒
2
5
To construct the Hamiltonian operator for N O we first
express the space-fixed Cartesian coordinates of each atom in
2
5
ϩ
where the elements of p ϭ(P ,P ,P ,P ,P ) are the
x y z ϩ Ϫ
terms of ͑␪,␾,␹,␣ ,␣ ͒ and the fixed bond angles and bond
generalized momenta associated with the angular velocities
of ␻ , and ␮ is the inverse inertia tensor, the elements of
1
2
ϩ
lengths ͑R,r,␥,␤͒ to obtain a kinetic energy expression in
terms of the Euler and internal-rotation angles and their ve-
locities. After ignoring terms associated with translation of
the center of mass we obtain the following expression for the
classical kinetic energy:
which are functions of ␣ and ␣ .
1
2
3
7,38
Using standard procedures
for converting the classi-
cal Hamiltonian to the quantum mechanical operator we ob-
tain the following rotation–internal-rotation Hamiltonian for
N O :
2
2
2
2
5
2
TϭI ␻ ϩI ␻ ϩI ␻ ϩ2I ␻ ␻ ϩ2I ␻ ␻
x y z
xx yy zz xy x y xz x z
2
Ϫ
2
ϩ
ϩ2I ␻ ␻ ϩI ͑ ␣˙ ϩ ␣˙ ͒ϩ&I ␭ ␻ ␣˙
Ϫ
yz
y
z
0
0
z
z
1
2
Ϫ&I ␭ ␻ ␣˙
,
ϩ
͑5͒
Hϭ
Pi␮ijPjϩVeff͑␣1 ,␣2͒ϩV͑␣1 ,␣2͒,
͑8͒
0
x
x
͚
where Iij (i, jϭx,y,z) are the components of the total iner-
tial tensor of the molecule and are functions of ␣ and ␣ , ␻
1
2
i
(
iϭx,y,z) are the angular velocities about the i’th axis, I is
where the summation runs over the five momenta, Veff is a
small effective potential arising from the transformation of
the kinetic-energy operator to quantum mechanical form,39
and V is the internal-rotation potential. Since Veff is expected
to be negligible compared to V we will ignore its contribu-
tions to H in the present discussion. The symmetries of the
various operators of H are summarized in Table IX.
0
the moment of inertia of an NO group about its C axes,
2
2
␭xϭcos͑␥/2͒ and ␭ ϭsin͑␥/2͒ are the direction cosines be-
z
tween the NO top C axes and the x and z axes, respec-
2
2
tively, ␣˙ ϭ( ␣˙ ϩ ␣˙ )/& and ␣˙ ϭ( ␣˙ Ϫ ␣˙ )/&. Because
ϩ
1
2
Ϫ
1
2
of our choice of the inertial frame, ␣˙ ϩ couples only with ␻x
and ␣˙ Ϫ couples only with ␻z .
In the spirit of Hougen, Bunker, Johns36 and others, Eq.
The Hamiltonian matrix of H is set up in the symmeter-
ized rotation–internal-rotation basis. The evaluation of the
͑5͒ can be rewritten as
matrix elements containing the ␮ components or V is sim-
ϩ
ij
2
Tϭ␻ I␻,
͑6͒
plified by expanding each of these operators in the basis
functions of the correct symmetry. In our initial calculation
of the rotation–tunneling levels we follow McClelland et al.7
and choose a trial potential of the form
where I is a 5 dimensional generalized inertia tensor and
ϩ
␻
ϭ(␻ ,␻ ,␻ , ␣˙ , ␣˙ ) is a row vector and ␻ is its trans-
x y z ϩ Ϫ
pose. Expressing T in terms of momenta we have
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7260
Grabow et al.: Microwave spectrum of N O5
2
TABLE IX. Symmetries of various operators in G16
.
any direct measure of the Jϭ0 tunneling splitting and be-
cause the vanishing statistical weights for many of the states
prevents their observation. Alternatively, we have used the
ab initio calculations to model the general features of the
tunneling potential for N O . The ab initio calculations indi-
Angular momentum operators
P
A1
P
B1
P
B1
P
A1
P
Ϫ
B1
x
y
z
ϩ
Ϫ
ϩ
Ϫ
Ϫ
Ϫ
2
5
␮ij matrix elements
cate that the configurations where ␣ ϭ␣ ϭ0 and
1
2
iگj
x
x
A1
y
z
ϩ
Ϫ
ϩ
Ϫ
ϩ
ϩ
ϩ
␣ ϭ␣ ϭ90° correspond to high energy maxima on the
B1
B1
A1
B1
B1
A1
B1
1 2
ϩ
Ϫ
Ϫ
Ϫ
y
A1
A1
A1
internal-rotation potential. The lowest energy extrema on the
ϩ
ϩ
ϩ
z
ϩ
Ϫ
A1
A1
potential energy surface correspond to ␣ ϭ0 and ␣ ϭ90° or
1
2
ϩ
ϩ
B1
␣1
ϭ90° and ␣ ϭ0 and the C configuration observed experi-
ϩ
2 2
A1
mentally. The ab initio calculations suggest that the most
reasonable tunneling motion for interchanging the bonding
roles of the two NO groups is the geared pathway of Fig. 8.
2
A two dimensional potential energy surface which qualita-
tively represents these features is given by
V0
V͑␣ ␣ ͒ϭ
͑2Ϫcos 2␣ Ϫcos 2␣ ͒
1 2
1
2
2
V͑␣ ,␣ ͒ϭϪV
cos͓2͑␣ Ϫ␣ ͔͒
2 1
1
2
geared
2
n
␦͓1Ϫsin ͑␣ ϩ␣ ͔͒
2 1
ϩU0͚
͑r /r ͒ ,
͑9͒
ϩVantigearede
0
ij
ϩVlinear͑cos 2␣ ϩcos 2␣ ͒,
͑10͒
1
2
where the first term allows for electronic contributions to the
potential due to coupling of the two NO groups through the
NON frame and the second term mimics the van der Waals
where the Vgeared term characterizes the barrier at ␣ ϭ0 and
2
1
␣ ϭ90° ͑or ␣ ϭ90° and ␣ ϭ0͒, the V
term forces the
2
1
2
antigeared
repulsions of the O atoms between the two NO groups, with
NO groups to move preferentially in a geared motion, and
2
2
rij being the separation of the ith and jth O nuclei of the two
Vlinear is necessary to allow the potential minimum to move
away from an ␣ ϭ␣ ϭ45° configuration.
NO units. The electron-diffraction potential of McClelland
2
1
2
7
Ϫ1
Ϫ1
et al. has V ϭ665͑84͒ cm , U ϭ35 cm , r ϭ2.8 Å, and
The qualitative features of the spectrum are only repro-
0
0
0
Ϫ1
nϭ12.
duced by having very small barriers ͑Ͻϳ10 cm ͒ hindering
Initial dynamical calculations were undertaken on the
electron-diffraction potential to compare predictions from
this surface with the experimental results. Because of the
anisotropy of this potential, the basis-set expansion requires
internal rotor states with n and m as large as 24 for the basis
functions shown in Table VIII, leading to diagonalization of
matrices as large as 481ϫ481 for Jϭ1. The results from the
calculations show that the electron-diffraction potential is
much too anisotropic and favors a tunneling pathway which
is inconsistent with experiment. The predicted tunneling
the geared internal rotation of the two NO groups. For ex-
2
ample, if we use values of V
, Vantigeared , ␦, and Vlinear of
geared
Ϫ1
Ϫ1
0, 120 cm , 2, and Ϫ40 cm , which gives a barrier of ϳ7
Ϫ1
cm to geared internal rotation, we calculate large splittings
of the transitions ͑2.0 GHz for the two 1 –0 lines, 1.3
11
00
GHz for the 2 –1 lines, and 3.7 GHz for the 2 –1 lines͒
20
11
21
10
for the two tunneling states, of the same order of magnitude
as the experimental observations. In addition, this potential
has a minimum at ␣ ϭ␣ ϭ41°, as expected from the quad-
1
2
rupole hyperfine analysis, and no other low lying tunneling
states are predicted to be observable experimentally, in
agreement with the detection of only one pair of tunneling
states.
Ϫ
ϩ
splittings for the A1 and A1 are less than 100 kHz and es-
sentially identical rotational constants are calculated for the
two states ͑Aϭ7019 MHz, Bϭ2032 MHz, Cϭ1779 MHz͒.
ϩ
Ϫ
The zero-statistical-weight, Jϭ0, B2 and B2 states are also
ϩ
degenerate with each other, as are the zero-weight, E and
IV. DISCUSSION
Ϫ
ϩ
Ϫ
ϩ
E
states. The B and B2 pair are separated from the A
2
1
Ϫ
ϩ
Ϫ
and A1 pair by 8 MHz, with the E and E states falling in
The microwave spectrum of N O is consistent with the
2
5
Ϫ
ϩ
7
between. Experimentally, we observe that the A1 and A1
transitions are split in some cases by more than 2 GHz, sug-
gesting that the tunneling splittings are more on the order of
GHz than kHz. The dominant electron-diffraction tunneling
pathway, splits the A1 and B2 species symmetrically from
the E species, and mimics the independent internal rotation
earlier electron-diffraction results which determined a C2
symmetry structure for the molecule. The present microwave
and ab initio calculations further demonstrate that N O has
2
5
a large-amplitude motion corresponding to the geared rota-
Ϯ
Ϯ
tion of the two NO groups about their twofold axes. The
2
Ϯ
lack of any direct measurement of the tunneling splitting
associated with this motion makes it difficult to precisely
determine the barrier, however, our model simulations sug-
of one of the two NO groups by 180° about its C axis,
2
2
corresponding to the tunneling matrix element h of Eqs. ͑3͒
i
Ϫ1
and ͑4͒.
gest that the barrier is on the order of 10 cm , significantly
Ϫ1 7
We have attempted to vary the potential parameters of
the electron-diffraction potential to obtain results which are
more consistent with experiment. This process turns out to
be difficult because the spectroscopic data is only weakly
sensitive to the potential barrier heights due to the lack of
smaller than the electron-diffraction value of ϳ200 cm . A
possible source of error in the electron-diffraction analysis
may be due to the lack of correction for background nitric
acid impurity, which was found to be a dominant impurity in
the present experiments.
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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Grabow et al.: Microwave spectrum of N O5
7261
2
The small magnitude of the barrier in N O makes it
the discrepancy with the solution measurement. The sample
2
5
essential to understand the large-amplitude potential and its
effect on the rotational and vibrational states of the molecule
may have been contaminated with HNO , the product of the
3
42
facile hydrolysis of N O . Since ␮͑HNO ͒ϭ2.17Ϯ0.02 D,
2
5
3
to accurately model the thermodynamic properties of N O5
such contamination would lead to a high measured value.
Alternatively, N O may adopt a different conformation in
2
in atmospheric chemistry applications. For molecules where
the internal rotor is a symmetric top, precise internal-rotation
barriers have been determined without direct measurement of
the tunneling splitting. However, for such molecules there is
no dependence of the overall rotational constants on the in-
ternal rotation angles, making it easier to model the Coriolis
contributions to the rotational constants from which the bar-
2
5
solution than in the gas phase. Only a small differential sol-
vation energy ͑Շ2 kJ/mol͒ is required to favor structure 2
over 1. Since conformation 2 is predicted to have a larger
dipole moment than structure 1 ͑Table III͒, solvation effects
may explain the high solution value for ␮.
rier is inferred. In the case of N O , the rotational constants
2
5
ACKNOWLEDGMENTS
vary greatly with internal rotation angles due to the large
masses of the oxygen nuclei. Because of the significant de-
pendence of the rotational constants on the internal-rotation
The authors would like to thank the Upper-Atmosphere
Research Program of NASA and NATO ͑Grant No.
CRG931359͒ for partial support of this work. J.-U.G. ac-
knowledges postdoctoral fellowship support from the Deut-
sche Forschungsgemeinschaft.
angles, more general internal-axis models ͑IAM͒ such as ap-
plied by Coudert and Hougen40 to ͑H O͒ would need to
2
2
include the internal-rotation dependence of the rotational
constants explicitly.
1
Additional information on the structure and tunneling
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The geared pathway is also affected since now all four
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does not affect the qualitative features of any of the tunnel-
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5
6
7
equivalent spin-zero oxygens of the NO groups.
8
9
2
The structure inferred from the electron-diffraction
7
͑
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1
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1
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2
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19
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9
,41
from experiment.
However, the density-functional results
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2
2
0
1
41
the experimental assumption of multiple degeneracies is
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22
2
3
2
2
4
5
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26
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ϭ0.24 D, and values from 0.14 to 0.31 D were obtained in
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9
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4
1
0
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J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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On: Tue, 23 Dec 2014 12:30:42

7262
Grabow et al.: Microwave spectrum of N O5
2
2
8
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identified is necessarily the best available for the purpose.
bury, New York 11797-2999. Fax: 516-576-2223, e-mail: paps@aip.org.
The price is $1.50 for each microfiche ͑98 pages͒ or $5.00 for photocopies
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Airmail additional. Make checks payable to the American Institute of
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2
Veff
has
the
form
Veffϭ͚ ͓(Ϫ5/32g )␮ (P g)(P g)ϩ(1/8g)
ij
ij
i
j
(
P ␮ )(P g)ϩ(1/8g)␮ [P (P g)]͔, where the i and j summation is over
i ij j ij i j
the two tops ͑i, jϭ4,5͒, g is a determinant of the ␮ inverse tensor matrix,
and the P operations are confined to be within the first surrounding pa-
i
renthesis or brackets.
32
40
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J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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