Journal of Molecular Spectroscopy 244 (2007) 160–169
www.elsevier.com/locate/jms
Rotational spectrum and structure of asymmetric
dinitrogen trioxide, N2O3
J. Demaison
a
a,*
, M. Herman a, J. Liévin a, L. Margulès b, H. Møllendal
c
Laboratoire de Chimie quantique et Photophysique, CP160/09, Université libre de Bruxelles (U.L.B.), ave. F.D. Roosevelt, 50, B-1050 Brussels, Belgium
b
Laboratoire de Physique des Lasers, Atomes, et Molécules, UMR CNRS 8523, Université de Lille I, F-59655 Villeneuve d’Ascq Cédex, France
c
Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, NO-0315 Oslo, Norway
Received 11 April 2007; in revised form 7 June 2007
Available online 19 June 2007
Abstract
The rotational spectra of the ground vibrational state and the m9 = 1 torsional state have been reinvestigated and accurate spectroscopic constants have been determined. The torsional frequency, m9 = 70(15) cm1, has been determined by relative intensity measurements. The assignment of the infrared spectrum has been slightly revised and an accurate harmonic force field has been calculated.
The equilibrium structure has been determined using different, complementary methods: experimental, semi-experimental and ab initio,
leading to r(NN) = 1.870(2) Å, in particular.
2007 Elsevier Inc. All rights reserved.
Keywords: Microwave; Ab initio; Structure; Force field; N2O3
1. Introduction
Due to chemical equilibrium, NO2 and N2O4 always
coexist. Adding NO in the gas phase leads to the formation
of N2O3. It may exist in two isomeric forms, sym and asym
but only the asymmetric species has been detected in gas
phase. This heterodimer is more stable than a van der
Waals complex but its binding energy is much lower than
that of a typical covalent bond. For this reason, the study
of the chemical bonding of N2O3 is quite interesting. Particularly, as in N2O4 [1], the NAN bond length is substantially longer than in N2H4 but has not yet been determined
accurately.
The microwave spectrum of N2O3 was first measured by
Kuczkowski [2]. He also obtained the rotational constants
for the 15N isotopologues and determined an approximate
effective value (r0) for the NAN bond length. These
*
Corresponding author. Permanent address: Laboratoire de Physique
des Lasers, Atomes, et Molécules, Université de Lille I, F-59655
Villeneuve d’Ascq Cédex, France. Fax: +33 3 20 33 70 20.
E-mail address: jean.demaison@univ-lille1.fr (J. Demaison).
0022-2852/$ - see front matter 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jms.2007.06.003
measurements were extended by Brittain et al. [3] who measured the microwave spectra of six isotopologues and
derived a complete substitution (rs) structure. The spectra
of the four lowest vibrational states were analyzed and
approximate vibrational frequencies were obtained from
relative intensity measurements. The electric dipole
moment was determined for the ground vibrational state.
The nuclear quadrupole coupling constants of the nitrogen
atoms were also measured by conventional Stark spectroscopy [4] and using a pulsed-beam Fourier transform microwave spectrometer [5,6].
The infrared spectrum of N2O3 was studied repeatedly,
starting with the work of D’Or and Tarte [7]. The work
prior to 1991 is summarized in Ref. [8]. The more recent
investigations can be reviewed as follows. Some controversy remained for a long time in the literature concerning
the bands near 260 and 160 cm1 either assigned to the
NAN stretching or NO rocking modes. The analysis of
the Raman spectrum in a NO matrix and a force field calculation definitely indicated that the band at 266 cm1 is
the NAN stretching mode [9]. Another problem in the literature concerned m8, the NO2 out-of-plane wag, assigned
J. Demaison et al. / Journal of Molecular Spectroscopy 244 (2007) 160–169
from gas phase infrared spectra to the band at 337 cm1
[10]. The Raman investigation of another band, at
627 cm1, gave unambiguous evidence for a fundamental
excitation, therefore attributed to m8. However, this modified assignment disagrees with ab initio predictions [1].
Concerning m1 (NO stretching), the fundamental and first
three overtone bands were rotationally analyzed from spectra recorded under high resolution with a Fourier transform spectrometer [11]. The accuracy of the ground state
parameters was improved in the latter reference by fitting
rotational transitions together with ground state combination differences (GSCDs). More recently, m1 was recorded
with a Fourier transform spectrometer under jet-cooled
experimental conditions which considerably simplifies the
rotational structure [12]. The m3 band at 1304 cm1 was
also analyzed using a slit-jet tunable-diode-laser spectrometer [13].
A valence-bond study of the origin of the long, weak
NAN bond was recently published [14].The properties of
N2O3 (and other nitrogen oxides) have been calculated by
the density functional method (DFT) [1]. A good agreement was found with the experimental results except for
the torsion, m9 and for m5 and m8 identified around 410
and 630 cm1, respectively. It was suggested to reverse
the assignment, namely the band situated at 630 cm1
should be associated with the NO2 in-plane rock and that
at 410 cm1 with the NO2 out-of-plane wag, in agreement
with the earlier infrared assignment.
Despite this body of work, there are still some problems
to solve. First, the structure is an empirical rs structure
which might be rather inaccurate. Then, the torsional frequency was determined from relative intensity measurements in the microwave range. Its accuracy seems to be
rather poor. Furthermore, its value does not agree with
the ab initio predictions [1]. Finally, there remains some
controversy concerning the assignment of the vibrational
spectrum. These various problems will be addressed in
the present paper.
The paper is organized as follows. Section 2 describes the
techniques used for the measurement of the rotational spectrum. Section 3 describes the computational methods used
in this study. Section 4 is dedicated to the determination
of accurate rotational constants and centrifugal distortion
constants for the ground and torsional states. The torsional
frequency is also obtained using two different methods (relative intensity measurements and vibrational dependence of
the inertial defect). In Section 5, the structure is determined
by combining high level ab initio calculations and the available experimental rotational constants. Section 6 discusses
the harmonic force field and the assignment of the vibrational spectrum, and Section 7 focuses on the NN bond
length. A brief conclusion is provided in Section 8.
2. Experimental details
14
The rotational spectra of the main isotopologue
N216O3 were measured by mixing NO and NO2 at about
161
Table 1
Summary of the fitted transitions for the ground vibrational state of N2O3
N#a
142
5
16
5
73
110
a
b
c
d
e
J 00max
20
2
4
4
11
33
K 00max
10
2
4
1
9
30
(kHz)b
Spectral range (GHz) r
c
GSCDs
20–24
13–40
6–16
40–80
78–255
10400
200
200
50
200
50
Refs.
[12]
[1]
[2]
[4]
This workd
This worke
Number of transitions.
Mean accuracy.
Infrared ground state combination differences.
Oslo spectrometer.
Lille spectrometer.
equal pressure in a cell cooled between 40 and 60 C.
The total pressure was about 8 Pa.
The microwave spectrum was studied between 40 and
80 GHz with the Oslo Stark spectrometer, which is
described in Ref. [15]. The lines are rather weak and some
of them are rather broad and skew as they are presumably
perturbed by nuclear quadrupole coupling. The accuracy
of the measurements is therefore estimated to be no better
than 200 kHz.
The millimeter wave measurements were performed in
Lille with a source-modulated spectrometer using phasestabilized backward wave oscillators working in the
frequency range 78–255 GHz [16]. The accuracy of the
measurements is about 50 kHz for most lines. A summary
of the measurements for the ground state is given in
Table 1. One hundred and sixty-nine new lines were
measured for the torsional state m9 = 1 with 2 6 J00 6 34
and 0 6 K 00a 6 28. A complete list of frequencies is given
as supplementary material (Table S1 for the ground state
and Table S2 for the excited torsional state).
3. Methods of computation
Most correlated-level ab initio electronic structure computations of the present study have been carried out at two
levels: second-order Møller–Plesset perturbation theory
(MP2) [17] and coupled cluster (CC) theory with single
and double excitation [18] augmented by a perturbational
estimate of the effects of connected triple excitations
[CCSD(T)] [19]. The Kohn–Sham density functional theory
[20] was also used in this study. Both the B3LYP (Becke
3-parameter [21] Lee–Yang–Parr [22]) and B3PW91 (Becke
3-parameter [21] Perdew–Wang-1991 [23]) exchange-correlation functionals were considered.
We used correlation-consistent polarized n-tuple zeta
basis sets cc-pVnZ [24] with n 2 {D, T, Q}, that are
abbreviated as VnZ in the text. To account for the electronegative character of the atoms, the augmented VnZ
(aug-cc-pVnZ, AVnZ in short) basis sets [25] were also
employed.
The core–core and core–valence correlation effects on
the computed equilibrium geometries [26], were estimated
162
J. Demaison et al. / Journal of Molecular Spectroscopy 244 (2007) 160–169
thanks to the correlation-consistent polarized weighted
core–valence quadruple zeta (cc-pwCVQZ) [27,28] basis
sets. For first-row atoms, it is sufficient to use the MP2
method to estimate this correction [29]. The frozen core
approximation (hereafter denoted as fc), i.e., keeping the
1s orbitals of N, and O doubly occupied during correlated-level calculations, was used extensively. Some geometry optimizations were also carried out by correlating all
electrons (hereafter denoted as ae).
The CCSD(T) calculations were performed with MOLPRO [30–33] electronic structure program packages, while
most other calculations utilized the GAUSSIAN03 (g03)
program [34]. Most calculations were performed on the
HP-XC4000 cluster of the ULB/VUB computing center.
a negative inertial defect is expected for a planar species
[37]. In the present case, however, the inertial defect is positive because of the dominant contribution of the low-frequency in-plane mode m7 = 160 cm1, as further discussed
later and presented in Table 3. The planarity of the molecule is confirmed by the so-called planarity defect [35]
Ds ¼ scccc s2 Cs1
¼ 0:0644ð56Þ kHz
AþB
ð1Þ
which presents the expected trends for a planar molecule
[38].
4.2. Torsional state m9 = 1
The assignment of the new data was not difficult because
a reliable starting set of constants was available from the
literature [13]. A Watson’s Hamiltonian using the A-reduction in the Ir representation was used [35]. The earlier
microwave measurements as well as the available infrared
GSCDs were included in the procedure. Table 1 summarizes the fitted data. The main difficulty was to find appropriate weights because the measurements are of very
different origin and, furthermore, some lines are broadened
by unresolved quadrupole hyperfine structure. To solve
this problem, a robust regression, the iteratively reweighted
least squares (IRLS) method [36], was used. The principle
of this method is to estimate the weights from the residuals
of a previous iteration. The resulting rotational parameters
are listed and compared to the literature values [13] in
Table 2. The improvement of the accuracy is significant.
One can notice that the value of the inertial defect is
small and positive, D0 = 0.135 uÅ2. Actually, when an
out-of-plane vibration has a frequency which is much lower
than any of the in-plane vibrations, as is the case for N2O3,
The assignment of the new data was made starting from
the approximate rotational constants of Brittain et al. [3]
arising from the observation of eight rotational transitions, only. The resulting rotational constants are given
in Table 2.
As pointed out in Section 1, the torsional frequency
could not be directly measured by infrared spectroscopy.
It was indirectly determined to be m9 = 63 cm1 from the
origins of the m5 + m9 combination and m5 fundamental
bands [9]. The value predicted from microwave intensity
measurements, m9 = 124(25) cm1 is in striking disagreement with the IR determination [3]. Furthermore, all ab initio values are around 150 cm1 [1]. We have performed
relative intensity measurements, using the method
described by Esbitt and Wilson [39]. We used a-type
R-branch transitions with J quantum numbers ranging
from 6 to 10. The highest Ka-members of these series of
lines were selected because presenting smaller nuclear
quadrupole splittings. For many other lines, the nuclear
quadrupole coupling of the three nitrogen nuclei with the
overall rotation indeed leads to distortion from the normal
Lorentzian shape, therefore lowering the quality of the
results. The selected lines also present the advantage of
being modulated at comparatively low Stark voltages,
which makes it easier to determine the base line accurately.
Table 2
Rotational and centrifugal distortion constants of N2O3
Table 3
Experimental and calculated (B3LYP/AVTZ) inertial defects Dm =
Ic Ib Ia of N2O3 (uÅ2)
4. Analysis of the rotational spectra
4.1. Ground state
Ground state
a
A
B
C
Dj
Djk
Dk
dj
dk
Ujk
Ukj
a
b
c
MHz
MHz
MHz
kHz
kHz
kHz
kHz
kHz
Hz
Hz
b
Previous
This work
Previous
This work
12454.02(18)
4226.57(2)
3152.99(2)
3.3(2)
7.0(10)
12454.153(25)
4226.6053(10)
3152.9591(10)
3.46221(43)
5.2221(26)
18.15(24)
0.91219(23)
10.0119(94)
0.0420(13)
0.3832(18)
12329.35
4210.80
3156.70
12329.567(33)
4210.9413(14)
3156.7170(13)
3.55303(51)
4.9890(34)
9.80(29)
0.93417(30)
7.524(11)
0.0739(15)
0.4966(26)
c
0.86(14)
c
c
c
Ref. [13].
Ref. [3].
Not determined.
c
c
c
c
c
c
c
m(cm1)a
m
m9 = 1
1
2
3
4
5
6
7
8
9
0
1832
1652
1305
773
627
241
160
414
63
—
a
b
c
Calc.
Exp.
Harm.b
Coriolisc
Total
0.007
0.003
0.015
0.040
0.089
0.118
0.177
0.000
0.000
0.007
0.016
0.055
0.075
0.130
0.098
0.043
0.777
0.234
0.923
0.016
0.12
0.08
0.22
0.30
0.31
0.20
1.08
0.11
0.80
0.13
Ref. [8].
Harmonic contribution to the inertial defect.
Coriolis contribution to the inertial defect.
Refs.
0.12
12
0.15
13
0.17
1.18
3
3
0.91
0.14
This work
This work
J. Demaison et al. / Journal of Molecular Spectroscopy 244 (2007) 160–169
Another feature of the selected transitions is that they actually consist of two coalescing Ka-lines.
The MW cell was packed with dry ice during the measurements and the temperature was therefore assumed to
be 195 K. The intensities of lines with identical rotational
quantum numbers were compared in the ground and m9
states. The a-component of the electric dipole moment
and the line width were assumed to be identical for the
two sets of transitions. Four separate measurements were
made and yielded values for m9 ranging between 78 and
68 cm1. The average value is 70 cm1, with an accuracy
estimated to be ±15 cm1.
We also estimated the m9 frequency from the changes in
the inertial defect upon vibrational excitation, using the
formula derived by Hanyu et al. [40]
Dðms ¼ 1Þ Dðm ¼ 0Þ ¼ h 1
2p2 c m9
constants, respectively. For most polyatomic molecules,
the accuracy of the r0 structure calculated from the experimental I0’s is rather poor. A more reasonable approximation, closer to the equilibrium structure re, is provided by
the so-called rs structure. One way to calculate this structure is to assume that the corrections eg’s are isotopologue-invariant. They can then be eliminated using the
Kraitchman’s equations. This specific procedure is valid
whenever isotopic differences of inertial moments dominate
the accuracy [42]. Another way to achieve rs structures is to
determine the quantities eg together with the structural
parameters in a fit of the inertial moments [43]. However,
the rs structure, though expected to be much better than
the r0 one, is sometimes worse [44]. For this reason, more
sophisticated assumptions have been devised, by Watson
et al. among others, defining the so-called mass-dependent
ð1Þ
structures, rm [45]. A further refinement, leading to the rm
structure assumes that e varies as I according to:
qffiffiffiffiffi
eg ¼ cg I m
ð4Þ
g
ð2Þ
The result is m9 = 64 cm1 in good agreement with the present relative intensity measurements and with the literature
IR value. It should actually be remembered that the Hanyu
formula underestimates the value of the related vibrational
frequency [41], thus further strengthening the present result
confirming the lower vibrational frequency value.
where cg is a parameter to be determined. When there is no
small Cartesian coordinate value and no hydrogen atom in
the molecule, and provided the system of normal equations
is well conditioned [44,45], it often gives results close to the
equilibrium structure.
The different experimental structures, r0, rs and rð1Þ
m , are
given in Table 5. Unfortunately, the condition number,
j = 682 (for rs), is rather large indicating that the system
is not well conditioned. However, at each step of improveð1Þ
ment of the model ðr0 ! rs ! rm
Þ, the standard deviation
5. Structure
5.1. Empirical structures
The experimental rotational constants used in this work
are given in Table 4. In order to avoid a possible bias due
to measurements of different origins, all rotational constants are taken from Ref. [3]. Furthermore, it was checked
that, in the present case, the use of the rotational constants
of Table 2 does not modify the results. We will first briefly
review the methods used. The experimental data permit to
determine the ground state moment of inertia I0, which are
different from the equilibrium ones, Ie, with
I 0g ¼ I eg ðmi ; d 2ij Þ þ eg ðF kl ; F klm Þ
163
Table 5
Empirical structures of N2O3 from ground state rotational constants
(distances in Å and angles in degrees)
NN
NO
NOcis
NOtrans
NNOcis
NNOtrans
ONN
ð3Þ
Here g = a,b,c refers to one of the principal inertial axes, mi
is the mass of atom i, dij the interatomic distance between
atoms i and j, and Fkl and Fklm quadratic and cubic force
ð1Þ
r0
rs
rm
1.889(9)
1.139(4)
1.188(4)
1.217(8)
111.3(6)
116.3(4)
105.2(2)
1.871(4)
1.144(5)
1.192(4)
1.218(4)
112.4(3)
116.6(4)
105.0(3)
1.868(4)
1.137(5)
1.191(4)
1.213(4)
112.2(2)
116.8(4)
105.1(3)
Table 4
Ground statea and equilibrium rotational constants (MHz) for N2O3
ONNO2
O15NNO2
ON15NO2
ONN18OcO
ONNO18Ot
18
ONNO2
O15N15NO2
a
b
c
A0
B0
C0
D0b
Ae
Be
Ce
Dec
12453.59
12294.22
12454.74
11630.73
12150.24
12443.91
12294.86
4226.65
4186.48
4213.18
4202.02
4059.46
4015.72
4172.67
3152.91
3120.34
3145.47
3083.96
3040.34
3033.40
3112.71
0.139
0.139
0.140
0.151
0.136
0.142
0.138
12531.19
12367.91
12531.41
11698.84
12231.90
12521.20
12367.74
4274.15
4233.71
4259.99
4249.65
4102.96
4060.42
4219.20
3186.91
3153.82
3179.02
3117.03
3072.33
3065.88
3145.76
0.009
0.011
0.010
0.013
0.003
0.013
0.011
Ref. [3].
Ground state inertial defect (uÅ2).
Equilibrium inertial defect (uÅ2).
164
J. Demaison et al. / Journal of Molecular Spectroscopy 244 (2007) 160–169
of the fit decreases, indicating that its quality improves. It is
ð1Þ
worth noting that the rs and rm
structures are compatible.
These structures will be compared and discussed at the end
of Section 5.3.
5.2. Ab initio structure
The ab initio structure was first calculated with the
AVTZ basis set using different level of theory: HF, MP2,
CCSD and CCSD(T). The B3LYP and B3PW91 methods
were also used. The results are given in Table 6. The large
dispersion of the results is not surprising because the value
of the T1 diagnostic at 0.022 is larger than the cut-off value
of 0.020. It thus indicates that the single-reference
CCSD(T) method is no more suitable for properly describing electron correlation effects [46]. At the HF level, the
bond lengths are too small, while they are too large at
the MP2 level. They are intermediate at the CCSD level.
The CCSD(T) values are close to, but smaller than the
MP2 values. The CCSD(T) structure is probably the closest
one to the equilibrium structure. However, the AVTZ basis
set is not yet converged as can be seen in Table 6 where the
VTZ, VQZ and AVQZ results are also listed. Adding the
core correlation correction (0.002 Å for the NO bond
lengths and 0.004 Å for the NN bond) and the effect of
basis set enlargement AVTZ fi AVQZ (calculated at the
MP2 level) gives the structure listed in Table 7. It should
be close to the equilibrium one although it is likely that
the NAN bond length is too short. It is also interesting
to note that the ab initio NOc and NOt bond lengths are
almost identical whereas the experimental values are significantly different as can be seen in Table 5. This might be
explained by the inability of the standard AVTZ basis set
to describe long range interactions. To check this point,
we have used the doubly and triply augmented VTZ basis
sets [47]. However, their use do not result into significant
asymmetry for the two NO bond lengths. The B3LYP
and B3PW91 structures are almost identical, except for
the NN bond length. They will be discussed at the end of
Section 5.3.
The ab initio NAN bond length might be too short
because of the basis set superposition error (BSSE), which
is a consequence of the incompleteness of the basis set. It
causes the molecular interactions to be artifactually too
attractive. Thus, the calculated intermolecular distance
NAN will be too small when the complex geometry is optimized. The conventional way to correct for BSSE is to use
the counterpoise (CP) method [48,49] as implemented in
g03. A calculation at the MP2/VQZ level of theory shows
that the NAN bond length is increased by 0.010 Å when
the counterpoise correction is taken into account.
Although this increase is significant, the NAN bond still
remains much too short.
It is, however, possible to explain the long NAN bond
length by performing a multiconfigurational calculation
at the complete active space multiconfiguration SCF
(CASSCF) level [50,51] using the AVTZ basis set. It
appears that a configuration mixing occurs involving a
double excitation from the NAN in-plane molecular
Table 7
Equilibrium structures of N2O3 (distances in Å and angles in degrees)
ð1Þ
NN
NO
NOc
NOt
NNOc
NNOt
ONN
ONOe
rm
B3LYPa
Ab initiob
c
rSE
e
1.868(2)
1.137(2)
1.191(2)
1.213(2)
112.2(2)
116.8(3)
105.1(2)
131.0
1.869
1.131
1.199
1.199
112.1
116.6
106.3
131.3
1.825
1.140
1.201
1.201
112.1
116.2
105.8
131.7
1.870(2)
1.139(2)
1.199(2)
1.206(2)
111.5(2)
117.4(2)
104.9(1)
131.1
ON + NO2d
1.1408
1.1946
1.1946
133.51
a
AVTZ basis set.
See Table 6: CCSD(T)/AVTZ + MP2/AVQZ MP2/AVTZ + MP2(ae)/wCVQZMP2/wCVQZ.
c
Semi-experimental equilibrium structure, see text.
d
Structure of the constituting species, Ref. [70] for NO and Ref. [71] for
NO2.
e
Value derived from ONO = 360 NNOc NNOt.
b
Table 6
Ab initio structures of N2O3 (distances in Å and angles in degrees)
Basis
Method
NAN
NAO
NAOc
NAOt
NNOc
NNOt
ONO
ONN
AVTZ
HF
MP2
B3LYP
B3PW91
CCSD
CCSD(T)
Range
1.6018
1.8865
1.8693
1.8381
1.7331
1.8389
0.2847
1.1165
1.1529
1.1308
1.1307
1.1406
1.1458
0.0364
1.1671
1.2084
1.1992
1.1946
1.1951
1.2060
0.0413
1.1692
1.2084
1.1989
1.1944
1.1966
1.2063
0.0392
116.69
108.54
112.11
112.31
114.07
111.88
8.15
112.45
119.42
116.62
116.20
114.33
116.44
6.97
130.87
132.04
131.27
131.49
131.60
131.68
1.18
109.46
103.40
106.25
106.37
107.18
105.56
6.06
VTZ
VQZ
AVQZ
wCVQZ
wCVQZ
reb
MP2
MP2
MP2
MP2
MP2 (ae)
1.8785
1.8729
1.8770
1.8729
1.8687
1.8251
1.1542
1.1497
1.1493
1.1494
1.1472
1.1400
1.2075
1.2047
1.2054
1.2045
1.2023
1.2007
1.2069
1.2046
1.2054
1.2044
1.2023
1.2012
108.59
108.74
108.72
108.73
108.77
112.10
119.13
119.18
119.30
119.19
119.10
116.23
132.29
132.08
131.98
132.08
132.13
131.67
103.39
103.55
103.59
103.55
103.61
105.80
a
b
T1 = 0.0223.
CCSD(T)/AVTZ + MP2/AVQZ MP2/AVTZ + MP2(ae)/wCVQZMP2/wCVQZ.
J. Demaison et al. / Journal of Molecular Spectroscopy 244 (2007) 160–169
orbital to the corresponding antibonding orbital with
respective mixing coefficients of 0.976 and 0.215. Going
from SCF to CASSCF corresponds to a lengthening of
the NAN bond from 1.603 Å to 1.767 Å. Correlating the
whole set of valence electrons using the multiconfiguration-reference configuration interaction (MRCI) method
[52,53] enhances the antibonding character and gives a
bond length of 1.801 Å. The MRCI calculation confirms
that the NAN bond length calculated at the CCSD(T) level
is too short.
5.3. Semi-experimental equilibrium structure
A more reliable way to estimate the equilibrium structure is to use a set of experimental ground state rotational
constants corrected by ab initio computed vibration-rotation interaction constants (a) [54]. The resulting rotational
constants can be used to determine the corresponding semiexperimental equilibrium structures, rSE
e . Theoretically, one
can even initially correct the experimental rotational constants for the magnetic effect [55]. We have estimated the
size of this correction by calculating the unknown values
of the related g constants using the Gaussian03 program
package, at the B3LYP/AVTZ level of theory. The results
are: gaa = 0.298; gbb = 0.088; and gcc = 0.026. The
corrected values of the rotational constants are given by
the relation [56]
Bacorr ¼
Baexp
1 þ Mmp gaa
ð5Þ
in which gaa is expressed in units of the nuclear magneton,
m is the electron mass, Mp the proton mass and a = a,b,c.
This correction turns out to be negligible, the largest deviation being 2 MHz for A.
To calculate the semi-equilibrium structure in N2O3, one
needs to calculate the vibration–rotation interaction constants. This requires computing the force field up to cubic
terms. As the proper description of electron correlation is
important for N2O3, we tried different levels of theory
(MP2, B3LYP, B3PW91 and CCSD(T) with the basis sets
VTZ and AVTZ).
The anharmonic normal coordinate force fields were
determined for all isotopologues whose ground state rotational constants are known. The molecular geometry was
first calculated. Then, the associated harmonic force field
was evaluated analytically in Cartesian coordinates. The
cubic (/ijk) and semidiagonal quartic (/ijkk) normal coordinates force constants were determined with the use of a
finite difference procedure involving displacements [57]
along reduced normal coordinates (step size Dq = 0.03)
and the calculation of analytic second derivatives at these
displaced geometries. The evaluation of anharmonic spectroscopic constants was based on second-order rovibrational perturbation theory [58].
The resulting force field is expected to rely on the quality
of the structure and of the harmonic force field. We
165
Table 8
Experimental and calculated quartic centrifugal distortion constants of
N2O3 (kHz)
Dj
Djk
Dk
dj
dk
a
Exp.
B3LYP/AVTZ
B3LYP/AVTZa
3.4621(4)
5.222(3)
18.2(2)
0.9122(2)
10.012(9)
2.450
8.038
11.756
0.640
8.785
3.304
5.212
15.452
0.858
8.881
After scaling, see text.
checked that the closest structure to re is from B3LYP/
AVTZ. It is indeed relatively close to the experimental
ð1Þ
rm
structure, see Table 5, as will be later confirmed. Furthermore, as already discussed by Stirling et al. [1],
B3LYP/AVTZ reproduces well the experimental fundamental vibrational frequencies. In addition, the quartic
centrifugal distortion constants and the inertial defects calculated with this harmonic force field are also in good
agreement with the experimental values, see Tables 3 and
8. Also, the B3LYP/AVTZ force field is the one providing
the smallest semi-experimental equilibrium inertial defect
(which should ideally be zero), see Table 3. The same conclusion was achieved for HONO which also presents both a
large T1 diagnostic and a very long NAO bond [59].
The semi-experimental structure is given in Table 7, in
which it is compared to other structures. It is rather close
ð1Þ
to the empirical rm
and B3LYP/AVTZ structures. With
the exception of the NAN bond length, it is also in good
agreement with the CCSD(T) structure. The same comparison occurred in HONO in which the CCSD(T) method was
found accurate except for the long NAO bond length [59].
The main discrepancy between the present semi-experimental and B3LYP/AVTZ structures lies with the NOc and the
NOt bond lengths. Experimentally, it was found that
r(NOt) > r(NOc) while no significant difference results from
the ab initio calculations. However, whereas the asymmetry
is quite large for the rs structure, Drs = rs(NOt) rs
(NOc) = 0.026(2) Å, it is much reduced for the semi-experimental structure, Dre = 0.0069(14) Å. Nevertheless, as
further discussed in the next section, this difference seems
to be genuine if one assumes that the standard deviation of
the fit is a reliable indicator of the uncertainty.
6. Harmonic force field
The experimental harmonic force field of N2O3 has
already been determined by Nour et al. [9]. However, as
noted in the introduction, the assignment of the NO2
out-of-plane wag is in disagreement with ab initio results.
Another problem arises from the assumption by Nour
et al. that the stretching force constants for the NOc and
NOt bonds are identical. Although they are expected to
be close, they should differ in the same respect as the two
bond lengths, a longer bond being associated with a smaller
force constant.
166
J. Demaison et al. / Journal of Molecular Spectroscopy 244 (2007) 160–169
Selecting the value 627 cm1 for the NO2 in-plane rock
(m5) and 414 cm1 for the NO2 out-of-plane wag (m8) gives
scaling factors k5 = 0.84(2) and k8 = 0.85(4). These show
the expected order of magnitude [62], comforting the
revised assignment based on the ab initio calculations
[10]. The scaled force field reproduces well the experimental
harmonic frequencies as well as the quartic centrifugal distortion constants, see Tables 8 and 11. However, it is not
accurate enough to point out the difference between the
two stretching force constants for the NOc and NOt bonds.
It has to be noted that the range of the scaling factors is
much larger than usual [62]. It demonstrates the difficulty
to calculate an accurate theoretical force field for N2O3.
To improve the force field, we determined the diagonal
force constants from the experimental data, fixing most
of the non-diagonal constants at the scaled ab initio values.
The result, given at the bottom of Table 10, does not reveal
any significant difference between the cis and trans NO
bond lengths.
The harmonic force field was first calculated in Cartesian coordinates at the B3LYP/AVTZ level of theory (as
in Section 5.3). It was then transformed into a representation based on a set of nonredundant internal coordinates
which are defined in Table 9. Then, this theoretical force
field was scaled in the usual way [60] using nine different
scaling factors. These were refined by fitting them with
the program ASYM40 [61] to the observed data described
ðthÞ
below. Practically, the theoretical force constants F ij are
scaled by the factors ki to give the scaled force constants
ðscÞ
F ij
pffiffiffiffiffiffiffiffi ðthÞ
ðscÞ
F ij ¼ k i k j F ij
ð6Þ
We used the same experimental data as Nour et al. [9],
except that the fundamental frequencies mi were transformed into harmonic frequencies xi using the B3LYP/
AVTZ anharmonic force field of Section 5.3. Furthermore,
the five quartic centrifugal distortion constants of Table 2
were also used. The results are given in Table 10.
7. Discussion of the NN bond length
Table 9
Symmetry coordinates of N2O3
An interesting result of this work is the determination of
NAN bond length. It appears to be quite long
re(NAN) = 1.870 Å, as expected. It is much longer than
in N2, re(N„N) = 1.098 Å [63] and in N2H2,
re(N@N) = 1.247 Å [64]. It is probably more informative
to compare it to other single bonds. From this point of
view, the prototype of the NAN single bond is found in
hydrazine, N2H4. Although it has been extensively studied
by microwave spectroscopy [65] and by electron diffraction
[66], there is no equilibrium structure available, the highest
level of ab initio calculation being MP2/6-311+G(3df, 2p)
Symmetry coordinate
a0
S1 = Dr(ON1)
S2 = Dr(N1N2)
S3 = Dr(N2Ot)
S4 = Dr(N2Oc)
S5 = D\(ON1N2)
S6 = 61/2[2D\(N1N2Oc) D\(OcN2Ot) D\((N1N2Ot)]
S7 = 21/2[D\(OcN2Ot) D\((N1N2Ot)]
a00
S8 = Dc(N1N2OcOt)a
S9 = Ds(ON1N2Oc)
a
Out-of-plane bending.
Table 10
Harmonic force constants and scaling factors for N2O3
i
kia
Theoretical (unscaled) harmonic force fieldb
1
2
3
4
5
6
7
8
9
0.9153(13)
0.7157(67)
0.8639(38)
1.0911(63)
0.837(23)
1.195(17)
0.8662(66)
0.8465(44)
0.2624(86)
16.593
0.721
0.634
0.752
0.328
0.077
0.420
0.357
0.005
Experimental harmonic force constants for N2O3
1
15.263(12)
2
0.583
3
0.564
4
0.752
5
0.287
6
0.080
7
0.374
8
0.30187(82)
9
0.00246
a
b
1.029
0.036
0.012
0.180
0.164
0.090
11.075
1.698
0.170
0.348
0.333
10.934
0.232
0.096
0.494
1.337
0.320
0.182
0.796
0.300
0.993
10.73(19)
1.341(11)
0.145
0.354
0.288
10.76(19)
0.222
0.110
0.480
1.460(26)
0.089(14)
0.155
0.647(14)
0.305
0.9134(29)
0.067
0.7345(37)
0.0281
0.0110
0.140
0.152
0.071
0.01762(31)
Scaling factor (uncertainty in parentheses).
The units correspond to energies in aJ, bond lengths in Å and angles in radians.
J. Demaison et al. / Journal of Molecular Spectroscopy 244 (2007) 160–169
Table 11
Harmonic frequencies xi (cm1) of N2O3
i
mi(exp)a
xi(exp)b
xi(calc)c
Exp. calc.
1
2
3
4
5
6
7
8
9
1829.5
1610.0
1304.3
773.0
615.5
241.0
205.0
414.0
63.0
1853.2
1690.1
1329.3
784.7
621.6
248.6
212.7
424.4
65.8
1853.4
1689.9
1329.0
784.2
621.7
247.4
216.0
424.4
65.8
0.2
0.2
0.3
0.5
0.1
1.2
3.3
0.0
0.0
a
Estimated gas phase values, Refs. [8,9].
b
Estimated harmonic frequency: xi ¼ mi ðexpÞ þ ½xi mi ðB3LYP=
AVTZÞ.
c
Calculated with the experimental force constants of Table 10.
Table 12
Ab initio CCSD(T) structure of hydrazine, N2H4 (distances in Å and
angles in degrees)
VTZ
NAN
NAHoc
NAHi
NNHo
NNHi
HoNNHo
HoNNHi
1.4448
1.0120
1.0154
106.208
110.789
154.723
89.919
VQZ
1.4382
1.0105
1.0137
106.711
111.189
153.915
89.639
wCVQZ
aea
fca
1.4340
1.0092
1.0123
106.943
111.374
152.561
90.464
1.4377
1.0106
1.0138
106.716
111.198
153.794
89.726
V5Z
reb
1.4361
1.0103
1.0134
106.953
111.400
152.934
90.021
1.4324
1.0089
1.0119
107.180
111.576
151.701
90.759
a
ae = all electrons correlated, fc = frozen core approximation.
V5Z + wCVQZ(ae) wCVQZ(fc). Derived parameter: \(HNH) =
107.588 degrees.
c
The hydrogen atoms which take the inner and outer positions are
denoted as i and o, respectively.
b
Table 13
YZ bond lengths (Å) and \(XYZ) bond angles (degrees) in XYZ2
molecules
X
Y
Z YZc
YZt
XYZc
XYZt
Refs.
ON
HO
HO
HO
HN@
HN@
H2NCH@
HOCH@
H2N
N
N
B
B
C
C
C
C
N
O
O
H
F
H
F
H
H
H
1.206
1.192
1.188
1.313
1.086
1.305
1.077
1.077
1.009
111.5
115.6
120.4
122.3
124.2
128.2
121.8
121.9
111.6
117.4
114.1
116.8
119.4
118.8
123.4
120.1
119.6
107.2
This work
[72]
[73]
[74]
[75]
[76]
[77]
[72]
This work
1.199
1.208
1.194
1.323
1.090
1.320
1.081
1.082
1.012
[67]. For this reason, we have calculated the structure of
N2H4 in the present work using the CCSD(T) method
and basis sets up to quintuple zeta. The results are reported
in Table 12. The T1 diagnostic being only 0.0073 at the
CCSD(T)/wCVQZ(ae) level of theory, the derived structure is expected to be reliable. In N2H4, re(NAN) =
1.432 Å is therefore much shorter than in N2O3. Another
interesting molecule is N2O2, the C2v dimer of NO for
ð1Þ
which rm
¼ 2:263ð1Þ Å [68]. The harmonic force field of
this dimer has also been determined [69]. It gives for the
167
Table 14
Structure of a few XNO and XNO2 molecules ((distances in Å and angles
in degrees)
XNO
NO
HNO
FNO
ClNO
BrNO
HONO
NCNO
ONNO
a
b
XNO2
r(NO)
Refs.
1.151
1.209
1.132
1.136
1.133
1.169
1.205
1.152
70
59
59
59
79
[59]
NO2
FNO2
ClNO2
BrNO2
O2NNO2
a
68
\(ONO)
r(NOc)
133.5
135.7
131.9
131.0
134.8
1.195
1.177
1.191
1.196
1.191
HONO2 130.5
ONNO2 131.1
1.208
1.199
r(NOt)
Refs.
[71]
[59]
b
[78]
[80]
1.192
1.206
[72]
b
ð1Þ
rm structure using the rotational constants of Ref. [81].
This work.
NAN stretching force constant, fNN = 0.193(14) aJÅ2.
This value is smaller than for N2O3, in agreement with
the fact that the NN bond length is longer in N2O2.
Another important point is the asymmetry of the NO2
group. The \(NNOt) angle at 117.4(2) degrees is much larger than the \(NNOc) angle whose value is only 111.5(2)
degrees. Generally, in XYZ2 molecules, when the two
\(XYZ) bond angles are significantly different, the longest
bond length YZ corresponds to the largest \(XYZ) bond
angle, see Table 13. This result confirms the likely asymmetry of the two r(NO) bond lengths, the r(NOt) bond being
longer as expected. However, it seems difficult to determine
an accurate value of the difference between the two lengths,
the most reliable value being probably the semi-experimental one, Dre = 0.0069(14) Å.
The NO bond length is given for a few molecules in
Table 14. It varies from 1.132 Å in FNO to 1.208 Å in
HNO3 (NOc).
8. Conclusions
The rotational spectra of the ground vibrational state
and of the m9 = 1 torsional state of N2O3 have been reinvestigated in the present work and accurate spectroscopic constants have been determined. The torsional frequency,
m9 = 70(15) cm1, has been determined by relative intensity
measurements, confirming the lowest of the two values proposed in the literature, as suggested from previous IR
results. The assignment of the infrared spectrum has been
slightly revised and an accurate harmonic force field has
been determined. The equilibrium structure has been determined using different, complementary methods and the
specificity of N2O3 as a complex has been confirmed by
the determination of the NN bond length, re(NN) =
1.870(2) Å.
Acknowledgments
This work was sponsored, in Brussels, by the Fonds
National de la Recherche Scientifique (FNRS, contracts
FRFC and IISN) and the «Action de Recherches Concertées
168
J. Demaison et al. / Journal of Molecular Spectroscopy 244 (2007) 160–169
de la Communauté française de Belgique». It was also performed within the ‘‘LEA HiRes ’’ and the EU project
QUASAAR (MRTN-CT-2004-512202). The authors are
indebted to ULB for providing an invited international
chair to Dr. Demaison (2006).
[29]
Appendix A. Supplementary data
[30]
Supplementary data for this article are available on
ScienceDirect (www.sciencedirect.com) and as part of the
Ohio State University Molecular Spectroscopy Archives
(http://msa.lib.ohio-state.edu/jmsa_hp.htm).
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