Journal of Molecular Structure 795 (2006) 4–8
www.elsevier.com/locate/molstruc
The microwave and submillimeterwave spectrum of 13C1-methyl formate
in its ground torsional state (H13COOCH3)
F. Willaert a, H. Møllendal b, E. Alekseev a,1, M. Carvajal a,2, I. Kleiner c, J. Demaison a,*
a
Laboratoire de Physique des Lasers, Atomes, et Molécules, UMR CNRS 8523, Université de Lille I, Bat. P5, F-59655 Villeneuve d’Ascq Cédex, France
b
Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, NO-0315 Oslo, Norway
c
Laboratoire Interuniversitaire des Systèmes Atmosphériques, UMR CNRS 7583, 61 av. Charles de Gaulle, F-94010 Créteil Cédex, France
Received 19 May 2005; accepted 14 February 2006
Available online 30 March 2006
This paper is dedicated to Professor Gisbert Winnewisser.
Abstract
The rotational spectrum of 13C1 methyl formate (H13COOCH3) has been observed in the frequency range 7–610 GHz. Two hundred and ninety
one transitions up to JZ58 (KmaxZ24) were assigned to the A-species of the ground torsional state. They could be fitted to a standard Watsonian
involving 19 parameters (up to one decic centrifugal distortion constant). About 260 E-transitions were also assigned. A global analysis of all
these transitions using the internal axis method gave a satisfactory fit permitting to determine the three internal rotation parameters (IaZ
3.132(4) mÅ2, V3Z4912(6) J/mol and :(i,a)Z52.30(5)8, which is the angle between the principal axis a and the internal rotation axis i) and
allowing us to make an accurate prediction of the rotational spectrum.
q 2006 Elsevier B.V. All rights reserved.
Keywords: Internal rotation; Microwave spectroscopy; Methyl formate
1. Introduction
Methyl formate, HCOOCH3, is abundant in numerous
interstellar clouds [1,2] although its mechanism of formation is
not yet understood [3]. Furthermore, its rotational spectrum is
dense and intense, which has led to the identification of a large
number of interstellar spectral lines. This is remarkable
because the rotational spectrum of methyl formate is
complicated by the internal rotation of the methyl group,
which splits each rotational line into a doublet (characterized
by the symmetry labels A and E [4]). As the molecule is light
and as the barrier to internal rotation is not high, these splittings
are large and the transition frequencies are extremely difficult
to calculate with accuracy. For this reason, the identification of
interstellar methyl formate was made possible only thanks to an
* Corresponding author. Tel. C33 3 20 43 44 90; fax: CC33 3 20 33 70 20.
E-mail address: jean.demaison@univ-lille1.fr (J. Demaison).
1
Present address: Institute of Radio Astronomy of NASU, Chervonopraporna 4, 61002 Kharkov, Ukraine.
2
Present address: Departamento Fisica Aplicada, Facultad de Ciencias
Experimentales, Universidad de Huelva, Campus de el Carmen, 21071 Huelva,
Spain.
0022-2860/$ - see front matter q 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.molstruc.2006.02.013
intense laboratory work, which has led to the assignment of
more than 3000 lines up to 608 GHz in the vibrational ground
state [5–13]. Quite recently, the rotational spectrum of the first
excited torsional state was observed in the frequency range
7–200 GHz [13]. The monosubstituted species DCOOCH3 was
also studied in great detail [14]. On the other hand, for the 13C
isotopologs, only approximate values of the rotational
constants are known [5], which is not enough to permit the
identification of these species in interstellar space. To fill this
gap, we have undertaken to measure the rotational spectrum of
H13COOCH 3 from the microwave up to the
submillimeterwave.
The current view of the formation of interstellar methyl
formate is that it derives from precursor molecules methanol
(CH3OH) and formaldehyde (H2CO). This reaction has been
recently studied and the conclusion is that it cannot produce
enough methyl formate to explain its observed abundance.
Another, more plausible reaction could be the radiative
association reaction between the methyl cation and formic
acid (HCOOH) although it does not yet produce enough methyl
formate [3]. Anyhow, as the 13C isotopologs of methanol [2],
formaldehyde [15–17] and formic acid [18,19] have been
identified in interstellar clouds, the presence of H13COOH is
also likely.
F. Willaert et al. / Journal of Molecular Structure 795 (2006) 4–8
Finally, methyl formate is a prototypical molecule for
studying interactions among small-amplitude vibrations, the
large-amplitude internal rotation, and overall rotation.
2. Experimental details
An enriched sample of H13COOCH3 (99% 13C) was
purchased from Aldrich, Milwaukee.
The microwave spectrum was studied between 7 and
62 GHz with the Oslo Stark spectrometer, which is described
briefly in Ref. [20]. The accuracy of the measurements was
generally better than 100 kHz.
The submillimeterwave measurements were performed in
Lille with a source-modulated spectrometer using phasestabilized backwardwave oscillators working in the frequency
range 300–700 GHz. The sources were Russian ISTOK
backwardwave oscillators. They were phase-locked on the
emission of a harmonic of a 2–20 GHz Hewlett–Packard (HP
83711A) synthetizer whose frequency was first multiplied by
six and amplified by a Millitech active frequency multiplier.
The intermediate frequency beat near 320 MHz was compared
with the 32nd harmonic of the 10 MHz signal from a second
HP synthetizer (HP 3325B). The mixer was a new planar
Schottky diode optimized for the range 500–650 GHz and
provided by the University of Virginia. It was placed in a
parabolic structure. The large step frequency tuning of the
source was obtained by changing the frequency of the 2–
20 GHz synthetizer. The small step frequency scan is provided
by means of 10 MHz (9.75–10.25 MHz) synthesizer. The
typical aquisitition time was 0.1 s per point. A sine-wave
frequency modulation of 5 kHz (modulation depth: 400 kHz)
was applied to the 10 MHz and the signal was demodulated at
twice this frequency. The circular absorption cell, of 6 cm in
diameter and 110 cm long, was in stainless steel. The
absorption was detected with a liquid helium cooled InSb
bolometer (QMC). The measurements were made at room
temperature and the pressure in the absorption cell was about
10–30 m Torr. Computer processing was used to improve the
signal/noise ratio and to measure the frequency of the lines
whose accuracy is better than 50 kHz for most lines.
3. Analysis of the A-type transitions
The assignment of the spectrum was rather easy because the
rotational constants of Curl [5] provided a first prediction of
approximate frequencies. The A-species transitions were first
fitted to a standard Watsonian (A-reduction, representation Ir)
up to decic terms [21]. For the A-species, the internal rotation
can indeed be completely absorbed into contributions to the
rotational constants and to the centrifugal distortion constants
[22,23]. Obviously, as the internal rotation effects are large, the
convergence of the Watsonian is slow. Nevertheless, this
permitted an easy identification of many A-lines and,
furthermore, it was possible to obtain a good fit: 291 lines
were fitted using only 19 free parameters. The standard
deviation of the fit was only 77 kHz and the least well
determined parameter is the octic constant LJJK, whose value is
5
Table 1
Molecular parameters for the A-species of H13COOCH3 in the A-reduction
(Ir representation)
Parameter
Unit
A
B
C
DJ
DJK
DK
dJ
dK
FJ
FJK
FKJ
fJ
fJK
fK
LJJK
LJK
[KJ
[K
PKJ
MHz
MHz
MHz
kHz
kHz
kHz
kHz
kHz
Hz
Hz
Hz
Hz
Hz
Hz
mHz
mHz
mHz
mHz
mHz
Value
19802.29381(644)
6865.44182(139)
5262.50261(127)
6.02231(113)
K17.0181(142)
80.735(107)
1.904361(267)
7.3075(171)
0.012679(246)
1.4770(266)
4.7263(576)
0.0065802(736)
0.90604(381)
10.581(263)
K0.10607(541)
K2.6549(456)
K1.3722(578)
K10.929(255)
0.0008102(281)
19.6 times larger than its corresponding standard deviation, see
Table 1. However, the frequency consistency between the
measured lines and their predicted values is not a sufficient
criterion to warrant a correct assignment because the spectrum
is dense and because, in order to obtain a good fit, it is
necessary to free many parameters. To check the assignments,
the relative intensities of the lines was useful as well as the
width of the internal rotation doublets (see below), at least
when the two components could be assigned. A careful analysis
of the residuals was also performed [24] in order to avoid
possible misassignments. Particularly, the ‘jackknifed’ residual
(residual divided by its standard error calculated after
elimination of the corresponding frequency from the fit) was
systematically calculated. It is extremely sensitive to errors
even when the residuals are small [25]. The leverages (diagonal
~ where X is the
~ K1 X
terms of the ‘hat’ matrix HZ XðXXÞ
Jacobian matrix [25]) were also calculated in order to eliminate
influential data (no parameter is determined by either a single
or a very small number of lines). The determined parameters
are accurate enough to permit prediction of all strong A-lines in
the frequency range 7–700 GHz. The frequencies of the
measured A-lines are given in Table S1 of the Supplementary
material.
4. Internal rotation analysis
The next step is to assign the E-type lines. For this goal, an
approximate value for the A–E splittings was obtained using
the internal rotation parameters of the parent species [8,13].
The assigned A- and E-transitions were introduced into a
global fit. To obtain molecular parameters from the observed
frequencies, a least-squares program written in Lille and based
on the internal axis method was used [26]. The torsional
Hamiltonian Hir is first set-up in its own internal axes system
(rho representation) and diagonalized numerically for each K
6
F. Willaert et al. / Journal of Molecular Structure 795 (2006) 4–8
value and the eigenvectors are stored to calculate the torsional
integrals. The rotational Hamiltonian Hrr is set-up in the rho
representation (using the exact torsional integrals and using
basis functions exp(3kCs) with k ranging from K8 to C8) and
added to the diagonal matrix of Hir. The quartic and sextic
centrifugal distortion terms are defined in the principal axis
system (representation Ir) and added to the Hamiltonian. The
elements off-diagonal in nt (torsional quantum number) are
removed by a Van Vleck transformation. The eigenvalues of
the 2JC1 by 2JC1 Hamiltonian matrix are then calculated
directly by matrix diagonalization. Basis functions exp(3kCs)
with k ranging from K8 to C8 and a Van Vleck transformation
up to ntZ3 were sufficient to obtain stable results within a few
kilohertz.
When enough lines were assigned, a rather large subset of
the 480 transitions (284 A-lines and 196 E-lines) was subjected
to a global least-squares fit in order to determine approximate
rotational and internal rotation parameters. In order to obtain a
reasonable fit, a few high-J or high-K lines had to be eliminated
from the fit. This is not surprising because, as for the parent
species, H13COOCH3 is a flexible molecule with two lowfrequency vibrations (besides the methyl internal rotation at
about 130 cmK1): the COC bending at about 318 cmK1 and the
C–O torsion at about 332 cmK1. Furthermore, the coupling
term Dab is large (K4830 MHz), thus a J-dependence of this
term is expected. The results of the fit are given in the first
column of Table 2 (under the heading: global fit). Although the
standard deviation of the fit, sZ344 kHz is larger than the
experimental uncertainty (100 kHz or better), it is still
reasonable and all the parameters are well determined. As
expected, the standard deviation of the A-lines, sZ292 kHz is
smaller than that of the E-lines, which is sZ493 kHz. It is
worth noting that only 12 parameters were free during the fit. It
is also important to note that the used model is rather crude
because it does not take into account the interactions between
small-amplitude vibrations and internal rotation. Therefore, the
residuals do not follow the normal law and the standard
deviations of the parameters are overoptimistic. For instance, if
we only fit half the lines (retaining those of lowest-J), the
barrier drops to 1164 cal/mol, i.e. a change seven times greater
than its standard deviation. To obtain a reliable confidence
interval, it seems prudent to use a range of about 20 standard
deviations. With this precaution, the accuracy is 28 cal/mol for
V3, 18 for :(i,a), and 0.08 mÅ2 for Ia, which is still
satisfactory. Finally, it is also worth noting that, taking into
account the terms off-diagonal in nt, is not extremely important
for the quality of the fit, but has a significant effect on the value
of the parameters, particularly the centrifugal distortion
constants DJK, DK, and dK. For instance, if the Van Vleck
transformation is limited to ntZ1, the values of these
parameters are (in kilohertz): DJKZK20.748(7); DKZ
75.16(16) and dKZK3.491(9).
To try to predict more accurately the E-lines, they were
separately fitted using the same model. The frequencies of the
fitted E-lines are reported in Table S2 of the Supplementary
material. The fit is indeed better albeit with more free
parameters, which are reported in the last column of Table 2.
It is extremely satisfactory to observe that the derived
parameters are close to those obtained by the global fit.
Finally, as a check, only the A-lines were fitted, again with the
same model. In this case, in order to obtain a well-conditioned
fit, it was found necessary to fix the internal rotation parameters
V3, :(i,a), and Ia to the values of the global fit. The parameters
are also reported in Table 2 (under the heading fit of the
A-transitions). They are also close to the parameters obtained
with the global fit. This consistency of the parameters indicates
that our analysis is likely to be satisfactory.
Table 2
Molecular parameters for H13COOCH3
Parameter
Unit
Global fit
Fit of A-transitions
Fit of E-transitions
A
B
C
V3
:(i,a)
Ia
DJ
DJK
DK
dJ
dK
FJK
FKJ
fJ
fJK
fK
Nc
sd
MHz
MHz
MHz
cal/mol
Degree
mÅ2
kHz
kHz
kHz
kHz
kHz
Hz
Hz
Hz
Hz
Hz
19799.451(17)
6865.1534(16)
5261.4087(25)
1174.1(14)a
52.299(52)
3.1316(44)
5.89779(37)
K26.4351(67)
81.60(16)
1.83897(14)
9.008(18)
K0.0510(17)
19799.462(11)
6865.1490(12)
5261.3955(15)
1174.0894b
52.29905b
3.13159b
5.89672(31)
K26.4791(56)
83.02(18)
1.83944(11)
9.184(23)
K0.0613(15)
19799.372(17)
6865.1415(18)
5261.3800(48)
1175.7(26)
52.114(98)
3.1596(66)
5.89583(56)
K26.640(11)
79.26(29)
1.84594(687)
9.210(49)
0.0760(87)
K0.989(41)
0.00139(16)
0.1036(82)
1.66(17)
241
256
a
b
c
d
kHz
0.0250(48)
480
381
287
241
V3Z4912(6) J/mol. Derived internal rotation parameters: rZ0.082232, FZ173998 MHz, DabZK4830.3 MHz.
Fixed at the value of the global fit (previous column).
Number of fitted transitions.
Standard deviation of the fit.
F. Willaert et al. / Journal of Molecular Structure 795 (2006) 4–8
Table 3
Internal rotation contributions to the rotational constants of H13COOCH3
(MHz)a
DA
DB
DC
Secondorder
Forthorder
Denominator
Total
Exp.
3.602
0.725
0.000
0.003
0.003
K0.004
K0.719
K0.394
1.113
2.89
0.33
1.11
2.84
0.29
1.09
a
See Eq. (1). Calculated with FZ174000 MHz, sZ31.466, raZ0.074989,
rbZ0.033642.
5. Discussion
It is interesting to compare the parameters of Table 1 with
those of Table 2. For the rotational constants, the relations
between the two sets of parameters are the sum of three terms:
second-order, fourth-order, and denominator corrections [26]:
DA Z AA KA ðglobalÞ
ð2Þ
ð4Þ
Z Fr2a W0A
K2Fr2a r2b W0A
C r2b ðCKAÞW0ðdÞ
(1a)
DB Z BA KB ðglobalÞ
ð2Þ
ð4Þ
Z Fr2b W0A
K2Fr2a r2b W0A
C r2a ðCKBÞW0ðdÞ
(1b)
DC Z C A KC ðglobalÞ
ð4Þ
Z 0 C 3Fr2a r2b W0A
C ½r2a ðBKCÞ C r2b ðAKCÞW0ðdÞ
(1c)
In these formulae, we have adopted the standard notation of
the principal axis method [4,26]. The calculated values are
compared to the experimental ones in Table 3. It is seen that, in
order to obtain a good agreement, it is necessary to take into
account the denominator correction.
For the centrifugal distortion constants, it is possible to
proceed to a similar analysis but, as for the rotational constants,
the formulae, which do not take into account the denominator
correction [22,23], give poor results. The problem is that the
analytical equations of the higher-order denominator corrections are rather cumbersome [26] and it was not attempted to
use them. What can be said is that the two sets of quartic
centrifugal distortion constants are close and that the small
differences observed for DJK, DK, and dK can be explained by
Table 4
Experimental analysis calculated quartic centrifugal distortion constants (kHz)
of methyl formate
H13COOCH3
HCOOCH3
DJ
DJK
DK
dJ
dK
a
b
c
Exp.a
Calc.b
Exp.c
Calc.b
6.02
K27.86
85.14
1.87
9.81
6.14
K22.47
84.12
1.90
2.97
5.90
K26.44
81.60
1.84
9.01
5.94
K22.63
83.78
1.82
3.07
Ref. [8].
B3LYP/cc-pVTZ, see text and footnote a of Table 3.
This work, see Table 2.
7
the higher-order denominator correction, which is basically
quite similar to the Van Vleck transformation used in the IAM
method (see discussion in Section 4). It is also interesting to
note that the quartic centrifugal distortion constants are quite
close to those of the parent species, see Table 4. To check that
the values of these constants are correct, we have calculated
them for both species (parent and 13C1) using an ab initio
harmonic force field. The Kohn–Sham density functional
theory [27] using Becke’s three-parameter hybrid exchange
functional [28] and the Lee–Yang–Parr correlation functional
[29], together denoted as B3LYP, was used with
the correlation-consistent polarized triple zeta basis sets
cc-pVTZ [30]. All calculations were performed with GAUSSIAN
03 [31]. The results are reported in Table 4. It is observed that
the calculated constants are close to their experimental
counterparts, except for dK, as usual. This confirms that our
analysis allowed us to determine meaningful constants.
It is tempting to compare the experimental value of the
internal rotation parameter :(i,a) with the angle between
the C–O bond (assumed to be the internal rotation axis) and the
a-principal axis. However, the existing experimental structures
(either from microwave [5] or from electron diffraction [32])
are not accurate. Particularly, they assume that the methyl
group is symmetric, which is a very poor assumption (ab initio
calculations indicate that the out-of-plane C–H bond is 0.003 Å
longer than the in-plane C–H bond [33,34]). Furthermore, Curl
[5] found a methyl tilt (angle between the internal rotation axis
and the C–O bond) of about 58 for HCOOCH3. The ab initio
structures [33,34] confirm that the :(C–O,a) angle is about 58
larger than the internal rotation parameter :(i,a). However,
when the 12C1 atom is substituted by a 13C atom, there is a
rotation of the principal axis system of about 0.878. This is
reasonable agreement with the change of the :(i,a) angle,
which is: 1.18(11)8 [8].
Finally, it may be noted that our value for the barrier to
internal rotation is in reasonable agreement with that found for
the parent species, V3Z4772(12) J/mol [8].
Acknowledgements
E.A. and M.C. thank the CNRS for financial support (project
CERC3). We are indebted to the Laboratoire Européen Associé
de Spectroscopie Moléculaire for financial support.
Supplementary data
Supplementary data associated with this article can be
found, in the online version, at doi:10.1016/j.molstruc.2006.02.
013.
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