Journal of Molecular Spectroscopy 254 (2009) 55–68
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Journal of Molecular Spectroscopy
journal homepage: www.elsevier.com/locate/jms
The microwave spectrum of the mono deuterated species of methyl
formate HCOOCH2 D
L. Margulès a, L.H. Coudert b,*, H. Møllendal c, J.-C. Guillemin d, T.R. Huet a, R. Janečkovà a
a
Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523, Bât. P5, Université de Lille I, 59655 Villeneuve d’Ascq Cedex, France
LISA, UMR 7583 CNRS et Universités Paris 12 et Paris 7, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France
c
Center for Computational and Theoretical Chemistry, Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, 0315 Oslo, Norway
d
Sciences Chimiques de Rennes, UMR 6226 CNRS/École Nationale Supérieure de Chimie de Rennes, 35700 Rennes, France
b
a r t i c l e
i n f o
Article history:
Received 17 October 2008
In revised form 16 December 2008
Available online 10 January 2009
Keywords:
Internal rotation
Partially deuterated species
Tunneling
Microwave
Mono deuterated
Methyl formate
IAM approach
Methyl group
a b s t r a c t
The microwave spectrum of mono deuterated methyl formate ðHCOOCH2 DÞ was measured in three
regions by Fourier transform microwave spectroscopy, conventional Stark modulation techniques, and
submillimeter wave spectroscopy. Effects due to the large amplitude torsional motion of the partially
deuterated CH2 D methyl group were evidenced and lead to two sets of transitions. Both sets display a
rigid-rotator behavior. In one of them, a small tunneling splitting ranging from 0 to 20 MHz was
observed. This splitting was accounted for using a phenomenological IAM-like treatment accounting
for its rotational dependence. This model was used to analyze line frequencies of the 1388 measured
microwave lines. The unitless standard deviation of the fit is 1.5 and the spectroscopic parameters
obtained are consistent with the structure of the molecule and the nature of the large amplitude tunneling motion.
Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction
The normal species of methyl formate ðHCOOCH3 Þ is a non-rigid
molecule displaying internal rotation of its methyl group. Because
of its astrophysical relevance [1,2], its microwave spectrum has
already been thoroughly investigated [3–12] leading to an accurate
determination of the hindering potential for the internal rotation.
Except for the few transitions measured by Curl [3], much less spectroscopic information is available for the mono deuterated species
HCOOCH2 D, with a partially deuterated CH2 D methyl group. Just like
the normal species, the mono deuterated species is expected to
undergo internal rotation of its methyl group. However, due to the
deuterium atom, the hindering potential should no longer display
threefold symmetry, but should still have three minima. As in the
case of the CH2 DOH isotopic species of methanol [13], the minimum
corresponding to the C S -symmetry configuration should be below
the two other energetically equivalent minima.
Experimental and theoretical investigations of the microwave
spectrum of HCOOCH2 D were undertaken in this paper in order
to study the tunneling pattern due to the internal rotation as well
as its rotational dependence. The microwave spectrum of mono
* Corresponding author. Fax: +33 1 45 17 15 64.
E-mail addresses: coudert@lisa.univ-paris12.fr, coudert.laurent@wanadoo.fr
(L.H. Coudert).
0022-2852/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.jms.2008.12.007
deuterated methyl formate has been recorded using Fourier transform microwave spectroscopy, conventional Stark modulation
techniques, and submillimeter wave spectroscopy. The tunnelingrotational energy levels were calculated using the IAM-like
approach developed in Refs. [14,15], which allows us to take into
account rotation–torsion coupling effects. This model was used
to perform an analysis of the microwave data.
The paper has four remaining sections. Section 2 is the experimental section. The rotation–torsion energy level calculation is
performed in Section 3. The results of the analysis of the microwave data are presented in Section 4. Section 5 is the discussion.
2. Experimental
The sample used for the measurements were synthesized as follows: formic acid and sulfuric acid (reagent grade, 95–98%) were
purchased from Aldrich, deuteromethyl alcohol was purchased from
Cambridge Isotope Laboratories, Inc. Formic acid (2 g, 43 mmol) and
deuteromethyl alcohol (1.02 g, 31 mmol) were introduced in a one
necked cell equipped with a stirring bar and a stopcock. The solution
was cooled around 80 °C and sulfuric acid (0.3 g, 3.0 mmol) was
added. The mixture was then cooled in a liquid nitrogen bath and
evacuated in vacuum. The stopcock was closed and the solution
was heated up to 40 °C and stirred overnight at this temperature.
The cell was then adapted to a vacuum line equipped with two traps
56
L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
Fig. 1. The traces obtained with the Fourier transform microwave spectrometer for the 211
202 rotational transition of mono deuterated methyl formate. The quadrupole
hyperfine structure associated with the D atom can clearly be seen for the D-in plane configuration (a) and for the + and tunneling components of the D-out of plane
configurations in (b) and (c), respectively. Each hyperfine component is Doppler split and is labeled with the F quantum number. The analysis of these hyerperfine patterns is
presented in Section 4.2.
and the solution was distilled. High boiling compounds were
trapped in the first trap immersed in a bath cooled at 70 °C. Mono
deuterated methyl formate (1.8 g, 30 mmol) was condensed in the
second trap immersed in a liquid nitrogen bath (196 °C). The yield
of the reaction was 97%.
Three sets of measurements were carried out and the recorded
transitions involve the two configurations of mono deuterated
methyl formate described in Section 3: the D-in plane and the
D-out of plane configurations. As stated in this section, transitions
arising from the former configuration follow a rigid-rotator pattern
while those arising from the latters display a tunneling splitting.
The first set of measurements was performed using the new
molecular beam Fourier transform microwave spectrometer in Lille
covering the 2–20 GHz spectral range. Methylformate vapors at a
pressure of 20 mbar were mixed with neon carrier gas at a backing
pressure of 1.5 bar. The mixture was introduced into a Fabry–Perot
cavity at a repetition rate of 1.5 Hz. Molecules were polarized within
the supersonic expansion by a 2 ls pulse and the free induction
decay signal was detected and digitized at a repetition rate of
120 MHz. After transformation in the time domain signal, molecular
lines were observed as Doppler doublets, with a signal point every
0.92 kHz, resulting from the average of about 100 coadded signals.
The transition frequency was measured as an average of the two
Doppler components and for most of the lines the uncertainty of
the measurements is estimated to be less than 2 kHz.
The hyperfine structure of 25 transitions rotation-tunneling
transitions was recorded in the 4.6–18 GHz range. A typical
202 rotational transiexample of signals, associated with the 211
tions, is displayed in Fig. 1. The quadrupole hyperfine structure
associated with the D atom (nuclear spin value of 1) is clearly observed. In the case of the D-in plane configuration, weak additional
features are observed. They are most probably due to the spin–spin
interaction between the two equivalent hydrogen atoms, and were
not further considered.
The second set of measurements was carried out in the 7–
80 GHz region using the Stark-modulated spectrometer of the University of Olso. Details of the construction and operation of this
spectrometer have been given elsewhere [16,17]. The spectrum
was taken at room temperature, or at roughly 20 °C, at a pressure
of approximately 10 Pa, employing a Stark field strength of about
1100 V/cm. The frequency of 332 individual transitions was measured with an estimated accuracy of 0.1 MHz.
The third set of measurements was performed in Lille from 140 to
660 GHz using as sources Russian Istok backward wave oscillators
(BWO). They were phase locked on a harmonic from an HP synthesizer. Up to 250 GHz the signal from the synthesizer was directly
mixed on a russian planar Schottky diode with a part of the signal
from the BWO. From 350 to 660 GHz an active sextupler from millitech (75–100 GHz) and a Schottky planar diode placed in a parabolic
structure (from Virginia Diodes Inc.) optimized in this range were
used. The detector is an InSb liquid He-cooled bolometer from
QMC. In order to improve the sensitivity of the spectrometer, the
sources were frequency modulated at 5 kHz. The absorption cell
was a stainless steel tube (6 cm diameter, 110 cm long), the pressure
L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
Fig. 2. The trace of the exactly overlapped 591;58
581;57 and 592;58
582;57
rotational transitions arising from the D-out of plane configurations of mono
deuterated methyl formate. Two tunneling components can be seen at 621009.849
and 621011.303 MHz and are identified by the and + signs, respectively.
57
Fig. 4. The initial configuration chosen in Section 3.1. Circles indicate atom
positions in the x0 y0 z0 axis system. The three hydrogen or deuterium atoms of the
methyl group are labeled 1, 2, and 3. The other atoms are labeled with letters. The
carbon atom of the methyl group is at the origin of the x0 y0 z0 axis system. The z0 axis
is the axis of internal rotation and is parallel to the bond between the methyl group
carbon atom and the oxygen atom. Hydrogen atom number 3 is hidden as it lies
below the x0 z0 plane.
After dealing with the coordinate system and the internal rotation
hindering potential, the IAM-like formalism developed for multidimensional tunneling in Refs. [14,15] is applied to account for the
rotational dependence of the tunneling pattern. This allows us to
set-up the Hamiltonian matrix and to compute tunneling-rotational levels.
3.1. Coordinate system and potential energy function
used during measurements was 2 Pa (20 lbar). The accuracy for isolated lines is better than 30 kHz and 1031 transitions could be
recorded. Figs. 2 and 3 show the traces recorded for two different
rotational transitions pertaining to the configuration of methyl formate displaying a tunneling splitting. Its rotational dependence is
clearly seen in Figs. 2 and 3 as the frenquency difference and the
ordering of the two tunneling components are different.
The coordinates needed for the present investigation are a the
angle of internal rotation of the methyl group and v; h; / the usual
Eulerian angles. The molecule-fixed xyz axis system is attached to
the molecule using the principal axis system and the Ir representation. Laboratory-fixed coordinates of the atoms are obtained using
the same scheme as in Section 3 of Ref. [18] for the principal axis
method. In agreement with this reference, an initial configuration
is first chosen and atom positions are written in an x0 y0 z0 axis system. This initial configuration is illustrated in Fig. 4 which also
defines the atom numbering. As shown by this figure, the C 3 axis
of symmetry of the methyl group coincides with the z0 axis. The
carbon atom of the methyl group is located at the origin of the
x0 y0 z0 axis system. The remaining atoms, that is, the atoms belonging to the frame, lie in the x0 z0 plane. The three atoms of the methyl
group are numbered 1, 2, and 3. In the case of the normal species of
methyl formate, atoms 1, 2, and 3 are hydrogen atoms. In the case
of the mono deuterated species, atom 1 is the deuterium atom
while atoms 2 and 3 are hydrogen atoms.
Laboratory-fixed coordinates of the atoms are obtained using Eq.
(7) of Ref. [18]. The rotation matrix S1
i ða; 0; 0Þ occurring in this equation is defined in the same way as in this reference. The rotation matrix T 1
PAM and the vector A, also occurring in this equation, are defined
as follows. In the case of the normal species, the same definition as in
Ref. [18] is used and they are also independent on the internal angle
of rotation a. In the case of the mono deuterated species, the two previous quantities are a-dependent and they will be written:
3. Theory
T 1
and AðaÞ:
PAM ðaÞ
In this section, the model used to calculate the rotational-tunneling sublevels of mono deuterated methyl formate is described.
The rotation matrix T 1
PAM ðaÞ is parameterized with a-dependent
Eulerian-type angles v1 ; h1 ; /1 as:
Fig. 3. The trace of the 5111;40
5011;39 rotational transition arising from the D-out of
plane configurations of mono deuterated methyl formate. Two tunneling components
can be seen at 645584.852 and 645588.311 MHz and are identified by the + and signs, respectively. Due to the rotational dependence of the tunneling splitting, the
ordering of the tunneling components is the opposite as in Fig. 2.
ð1Þ
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L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
Fig. 5. The three angles of Eq. (2), v1 , h1 , and /1 , are plotted in degrees as a function
of the angle of internal rotation a, also in degrees. For a ¼ 0 and 180°, it can be seen
that v1 and /1 are both zero.
1
T 1
PAM ðaÞ ¼ S ðv1 ; h1 ; /1 Þ;
ð2Þ
where Sðv1 ; h1 ; /1 Þ is the direction cosine matrix defined in Appendix I of Ref. [19].
Using the structure given in Table I of Ref. [20] for the cis conformer allows us to retrieve geometrical parameters for the initial
configuration and to obtain values for the Eulerian-type angles in
Eq. (2). In order to have a methyl group with a C 3 axis of symmetry,
as explained in the beginning of the present section, some of the
geometrical parameters of Ref. [20] were slightly altered. More
precisely, the following values were taken for the parameters
describing the methyl group: \OCX1 ¼ \OCH2 ¼ \OCH3 ¼
110:3 ; CX1 ¼ CH2 ¼ CH3 ¼ 1:0851 Å, and \HOCH1 ¼ \HOCH2 ¼
120 , where X is H for the normal species and D for the mono deuterated one. For the normal species, this choice leads to
v1 ¼ /1 ¼ 0 and h1 ¼ 58:10 . In this case the rotation matrix T 1
PAM
reduces to a 58.10° rotation about the y0 axis. In the case of the
mono deuterated species, the angles v1 ; h1 , and /1 are plotted in
Fig. 5 as a function of a. This figure emphasizes that we no longer
have 2p=3 periodicity. As required by symmetry, for a ¼ 0 and p,
0
the rotation matrix T 1
PAM ðaÞ is a rotation about the y axis through
57.25° and 56.89°, respectively.
The permutation-inversion symmetry group to be used for the
mono deuterated species of methyl formate is isomorphic to the C S
point group and contains two operations, the identity E and the permutation-inversion operation ð23Þ . The symmetry labels of the C S
point group will, therefore, be used in the present investigation. In
order to evaluate the symmetry transformations of the angle of
internal rotation and of the Eulerian angles under ð23Þ , Table VIII
of Ref. [18] can be used because the coordinate system used in this
reference is the same as the one used in the present investigation.
The hindering potential of the normal species of methyl formate
was first obtained by Ogata et al. [11] using the results of a global
analysis of the microwave data of this molecule. As it is customary
for molecules displaying hindered rotation of a methyl group, the
potential energy function VðaÞ is expanded with a Fourier series
compatible with the threefold axis of symmetry:
VðaÞ ¼ V 3 ð1 cos 3aÞ=2 þ V 6 ð1 cos 6aÞ=2 þ V 9 ð1 cos 9aÞ=2;
ð3Þ
where V 3 ; V 6 , and V 9 are three constants. The values reported for
V 3 ; V 6 , and V 9 by Ogata et al. [11] are 379.439(2), 29.017(5), and
3:095ð3Þ cm1 , respectively.
The effective potential in Eq. (3) contains contributions from the
zero-point energy of the 17 small amplitude vibrational modes of
the molecule. In the case of the normal species, these effects are
consistent with the threefold symmetry of the molecule. In the
Fig. 6. The effective potential energy function of the mono deuterated species of
methyl formate HCOOCH2 D is plotted as a function of the angle of internal rotation
a in degrees.
case of the mono deuterated species, these effects lead to an effective potential energy function which no longer displays threefold
symmetry but only C S . As in the case of the normal species, the
potential energy function is expanded with the help of a Fourier
series compatible with the lower C S symmetry:
VðaÞ ¼ V 1 ð1 cos aÞ=2 þ V 2 ð1 cos 2aÞ=2 þ V 3 ð1 cos 3aÞ=2;
ð4Þ
where V 1 ; V 2 , and V 3 are three constants. Values for these constants
are not know yet. It can nonetheless be assumed that the dominant
term in the potential energy function of Eq. (4) is the term in cos 3a,
as in the normal species. This leads to a potential energy function
still displaying three minima. With the help of ab initio calculations
and taking into account zero-point energy corrections, it was possible to evaluate the relative energies of these minima. The minima
for a 120 and 240° were found to be about 10 cm1 above the
a ¼ 0 minimum. This yields the following approximate values for
V 1 ; V 2 , and V 3 : 13.35, 0, and 379.439 cm1, respectively. The resulting hindering potential is plotted in Fig. 6. With the present values
of V 1 ; V 2 , and V 3 , the lowest minima of the PEF is located at a ¼ 0;
the two higher energy equivalent local minima are at a ¼ 119:8
and a ¼ 360 119:8 ¼ 240:2 . The C S -symmetry configuration
corresponding to the a ¼ 0 minimum will be referred to as the
D-in plane configuration. The two configurations corresponding to
the minima at a ¼ 119:8 and 240.2° will be referred to as the
D-out of plane configurations.
3.2. The IAM approach
Applying the IAM formalism developed in Refs. [14,15], requires
first obtaining the tunneling path(s) connecting the various nonsuperimposable configurations of the molecule. It is then possible,
using Eq. (5) of Ref. [15], to determine the contribution of each
path to the rotational dependence of a Hamiltonian tunneling matrix element. This contribution is contained in a Wigner
DðJÞ ðvpn ; hpn ; /pn ÞK 0 ;K function where the three angles vpn ; hpn , and
/pn depend on the path being considered, the letter p identifying
the path connecting configurations m and n.
In the case of mono deuterated methyl formate, there are three
non-superimposable configurations. These three configurations are
illustrated in Fig. 7 where they are identified by their configuration
ðnÞ
number n, with n ¼ 1, 2, and 3, and by aeq , the value of the torsional angle around which the corresponding reference function
is centered. This value is the value of a for one of the minima of
the potential energy function. The reference basis functions
[14,15] for configuration n is defined as:
WJK cn ¼ wn ðaÞ jJK ci;
ð5Þ
L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
59
In agreement with the IAM formalism [14,15], we seek to evaluate Hamiltonian matrix elements between any two reference
functions of Eq. (5):
HJK cm;
JK 0 c0 n
¼ hWJK cm jHjWJK 0 c0 n i;
ð9Þ
where m; n ¼ 1, 2, or 3 and H is the effective rotation–torsion Hamiltonian. The three non-tunneling matrix elements with m ¼ n in
this equation correspond to the pure rovibrational energy. If
0
cc0 ð1ÞKþK ¼ þ1, both rotational wavefunctions in Eq. (9) belong
to the same C S symmetry and, using symmetry considerations, the
non-tunneling matrix elements can be cast in the following way:
HJK c1;JK 0 c0 1 ¼ HJK c2;JK 0 c0 2 ¼ hJK cjout Hr jJK 0 c0 i þ Ed dK;K 0 ;
HJK c3;JK 0 c0 3 ¼ hJK cjin Hr jJK 0 c0 i;
ð10Þ
where out Hr and in Hr are the rotational Hamiltonian of the D-out of
plane and D-in plane configurations, respectively, and Ed is the en0
ergy difference defined in Eq. (8). If cc0 ð1ÞKþK ¼ 1, both rotational wavefunctions in Eq. (9) belong to different C S symmetry,
the matrix elements with m ¼ n are then zero because both rotational Hamiltonians in Eqs. (10) belong to the A0 symmetry species.
Due to symmetry relations, the six tunneling matrix elements with
m–n in Eq. (9) fulfill the following relations:
HJK c1;JK 0 c0 2 ¼ HJK c2;JK 0 c0 1 ;
HJK c1;JK 0 c0 3 ¼ HJK c2;JK 0 c0 3 ;
Fig. 7. The three non-superimposable configurations of mono deuterated methyl
formate. Circled numbers 2 and 3 indicate the position of the two hydrogen atoms
of the methyl group. The circled D letter indicates the position of the deuterium
atom of the methyl group. Each configuration is identified by its number n equal to
1, 2, or 3. Configurations 1 and 2 are the two isoenergetic D-out of plane
ðnÞ
configurations. Configuration 3 is the C S -symmetry D-in plane configuration. aeq
written below each configuration is the value of a, in degrees, around which the
corresponding reference function is centered.
where n ¼ 1, 2, and 3; wn ðaÞ is a vibrational function; and jJK ci, with
K P 0 and c ¼ 1, is the rotational wavefunction defined in Eq. (12)
of Ref. [21]. Using Table VIII of Ref. [18] and Eq. (13) of Ref. [21]
shows that these rotational wavefunctions belong to the symmetry
species A0 or A00 of C S when cð1ÞJþK is equal to +1 or 1, respectively. The exact expression of the vibrational functions in Eq. (5)
is not required when using the IAM approach [14,15]. However,
they must be centered around the appropriate value of a. For
instance, we can take:
h
i
w1 ðaÞ / exp k12 1 cos a að1Þ
;
eq
h
i
;
w2 ðaÞ / exp k12 1 cos a að2Þ
eq
h
i
w3 ðaÞ / exp k3 1 cos a að3Þ
;
eq
ð6Þ
where k12 and k3 are two positive constants. With this choice, we
have the following transformation properties for the reference functions of Eq. (5) under the ð23Þ permutation-inversion operation:
ð23Þ WJK c1 ¼ cð1ÞJþK WJK c2 and ð23Þ WJK c3 ¼ cð1ÞJþK WJK c3 ;
ð7Þ
obtained using Table VIII of Ref. [18] and Eq. (13) of Ref. [21]. If only
the vibrational functions of Eqs. (6) are considered, Eqs. (7) leads to:
hw1 jHt jw1 i ¼ hw2 jHt jw2 i ¼ hw3 jHt jw3 i þ Ed ;
ð8Þ
where Ht is the pure torsional part of the Hamiltonian and Ed is a
constant representing the vibrational energy difference between
reference functions 1 and 3. This energy difference was computed
making use of the theoretical results of Quade and Lin [22] concerning internal rotation of an asymmetrical CH2 D methyl group. Using
the slightly altered structure of Ref. [20] described in Section 3.1
and the potential energy function of Eq. (4), a value of 9.3 cm1
was obtained.
ð11Þ
HJK c3;JK 0 c0 1 ¼ HJK c3;JK 0 c0 2 ;
where the upper (lower) sign is to be used if both rotational wavefunctions belong to the same (different) C S symmetry. These equations show that only three different tunneling matrix elements
arise. They will be evaluated using Eq. (5) of Ref. [15].
The tunneling matrix element in the first of Eqs. (11) corresponds to the 1 ! 2 tunneling path allowing us to go from
configuration 1 to configuration 2 through a 120.4° rotation of
the methyl group. According to the IAM approach [14,15], the
rotational dependence of this tunneling matrix element is
parameterized by three Eulerian-type angles, denoted here
v2 ; h2 , and /2 , that are to be computed solving Eqs. (49) of Ref.
[14]. For the 1 ! 2 tunneling path, there arises a relation
between atom positions along the tunneling path because the
xz plane is a plane of symmetry. Using the results in Section
4.A of Ref. [15] shows that the relation v2 ¼ /2 þ p is then fulfilled. This relation and other results in this reference allow us
to write the tunneling matrix elements as:
h
ðJÞ
HJK c1;JK 0 c0 2 ¼h2 ð1ÞK d ðh2 ÞK;K 0 cosðK þ K 0 Þ/2 þ
i
c0 dðJÞ ðh2 ÞK;K 0 cosðK K 0 Þ/2 ;
ð12Þ
when both rotational wavefunctions belong to the same C S symmetry and:
h
ðJÞ
K
HJK c1;JK 0 c0 2 ¼ih2 ð1Þ d ðh2 ÞK;K 0 sinðK þK 0 Þ/2
i
ðJÞ
ð13Þ
þc0 d ðh2 ÞK;K 0 sinðK K 0 Þ/2 ;
when both rotational wavefunctions belong to different C S symmetry. In these two equations, h2 is a constant representing the vibrational integral in Eq. (5) of Ref. [15]. Using the slightly altered
structure of Ref. [20] described in Section 3.1 and Eqs. (49) of Ref.
[14], numerical values for h2 and /2 were found to be 4.647° and
85.144°, respectively.
The tunneling matrix element in the second of Eqs. (11) corresponds to the 1 ! 3 tunneling path allowing us to go from configuration 1 to configuration 3 through a 119.8° rotation of the
methyl group. As in the case of the 1 ! 2 tunneling path, the
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L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
Table 1
Matrix elementsa arising for the 1 ! 2 and 1 ! 3 tunneling paths.
Wþ
JK 0 c0
Wþ
JK c
W
JKc
out
hJK cj
W
JK 0 c0
0 0
HJK c1;JK c0 2
Hr jJK c i þ Ed dK;K 0 þ HJK c1;JK 0 c0 2
hJK cjout Hr jJK 0 c0 i þ Ed dK;K 0 HJKc1;JK 0 c0 2
pffiffiffi
2HJK c1;JK 0 c3
HJKc1;JK 0 c0 2
pffiffiffi
2HJK c3;JK 0 c1
WJK c3
WJK 0 c0 3
pffiffiffi
2HJK c1;JK 0 c0 3
pffiffiffi
2HJKc3;JK 0 c0 1
0
hJK cjin Hr jJK 0 c0 i
0
a
Submatrix arising between basis set functions of Eq. (20) such that cc0 ð1ÞKþK ¼ 1. In the body of the table, matrix elements are to be obtained using Eqs. (10) and (12)–
(18).
rotational dependence of this tunneling matrix element is parameterized by three Eulerian-type angles, denoted v3 ; h3 , and /3 , that
are also to be computed solving Eqs. (49) of Ref. [14]. Using the
results of Ref. [15] allows us to write the tunneling matrix
elements in the following way:
3.3. Tunneling sublevels
h
0
ðJÞ
HJK c1;JK 0 c0 3 ¼h3 ð1ÞKþK d ðh3 ÞK;K 0 cosðK/3
WJK c ¼ ðWJK c1 WJK c2 Þ= 2;
pffiffiffi
i
ðJÞ
þK 0 v3 Þ þ c0 d ðh3 ÞK;K 0 cosðK/3 K 0 v3 Þ ;
ðJÞ
i
þK 0 v3 Þ þ c0 d ðh3 ÞK;K 0 sinðK/3 K 0 v3 Þ ;
ð14Þ
ð15Þ
ð16Þ
ð17Þ
when both rotational wavefunctions belong to the same C S symmetry and:
HJK c3;JK 0 c0 1 ¼ HJK 0 c0 1;JK c3 ;
WJKc ;
WJK c3 ;
ð20Þ
where cð1ÞJþK ¼ þ1ð1Þ for the submatrix corresponding to the
A0 ðA00 Þ symmetry species. Table 1 shows the matrix elements arising
for either submatrix between basis set functions of Eq. (20).
Preliminary analyses of the microwave data showed that it was
not possible to evidence effects dues to the 1 ! 3 and 3 ! 1 large
amplitude motions. As a result the corresponding tunneling matrix
elements were set to zero. Because of the form of Table 1, rotational levels arising from the D-in plane configuration become then
those of an isolated vibrational state. Rotational-tunneling levels
arising from the the D-out of plane configurations are to be
obtained using a smaller basis set than the one given in Eq. (20),
which does not contain the WJK c3 function. The matrix elements
of the smaller submatrices arising in this case are given in Table 2.
The form of Table 2 suggests that rotational-tunneling sublevels
arising from the D-out of plane configurations should be labeled
using the usual rotational quantum numbers JK a K c and the signs
+ and . For small J-values, this labeling scheme can be easily
understood as the rotational-tunneling energy can be written as:
E ðJK a K c Þ ¼ Ed þ Er ðJK a K c Þ h2 ;
ð21Þ
r
Inserting these relations in Eqs. (14) and (15) leads to:
HJK c3;JK 0 c0 1 ¼ HJK 0 c0 1;JK c3 ;
00
which belong to the symmetry species A ðA Þ when
cð1ÞJþK ¼ þ1ð1Þ. Such linear combinations allow us to block
diagonalize the tunneling-rotational Hamiltonian into two submatrices corresponding to the A0 and A00 symmetry species. The basis
set functions to be used are then:
WþJK c ;
when both rotational wavefunctions belong to different C S symmetry. In these two equations, h3 is a constant having the same meaning as h2 in Eqs. (12) and (13). Using the same procedure as in the
previous paragraph, numerical values for v3 ; h3 and /3 were found
to be 111.050°, 6.093°, and 259.203°, respectively.
The tunneling matrix element in the last of Eqs. (11) corresponds to the 3 ! 1 tunneling path allowing us to go from configuration 3 to configuration 1 through a 119:8 rotation of the
methyl group. As in the case of the two previous tunneling paths,
the rotational dependence of this tunneling matrix element is
parameterized by three Eulerian-type angles, denoted v03 ; h03 , and
/03 , that are also to be computed solving Eqs. (49) of Ref. [14].
Because the 3 ! 1 tunneling motion is just the 1 ! 3 tunneling
motion carried out backward, the vibrational integral for these
0
two paths, h3 and h3 , are equal. There also arises a relation between
atom positions along these two paths. Using the same ideas as in
Section 4 of Ref. [15], it can be shown that the following relations
arise between both sets of Eulerian-type angles:
v03 ¼ p /3 ; h03 ¼ h3 ; and /03 ¼ p v3 :
ð19Þ
0
when both rotational wavefunctions belong to the same C S symmetry and:
h
0
ðJÞ
HJK c1;JK 0 c0 3 ¼ ih3 ð1ÞKþK d ðh3 ÞK;K 0 sinðK/3
In order to block diagonalize the tunneling-rotational Hamiltonian matrix, we are led to consider the following symmetry
adapted combinations of reference functions:
ð18Þ
when both rotational wavefunctions belong to different C S
symmetry.
where E ðJK a K c Þ is the pure rotational energy. This result was obtained approximating /2 and h2 by, respectively, p=2 and 0 in Eqs.
(12) and (13). Since h2 is negative, Eq. (21) means that the + sublevels is below the sublevel. For high J-values, the rotation-tunneling levels obtained after diagonalization of the two submatrices
were assigned rotation-tunneling quantum numbers making sure
that the variation of the tunneling splitting with the rotational
quantum numbers is a smooth one.
Anticipating the analysis results in Section 4.1 and taking for the
spectroscopic parameters needed in Section 3.2 the values obtained
in the analysis of the microwave data, rotational-tunneling energy
Table 2
Matrix elementsa arising for the 1 ! 2 tunneling path only.
Wþ
JK c
W
JKc
Wþ
JK 0 c0
W
JK 0 c0
hJK cjout Hr jJK 0 c0 i þ Ed dK;K 0 þ HJK c1;JK 0 c0 2
HJKc1;JK 0 c0 2
HJK c1;JK 0 c0 2
hJK cjout Hr jJK 0 c0 i þ Ed dK;K 0 HJKc1;JK 0 c0 2
0
KþK
a
0
Submatrix arising between the Wþ
¼ 1. In the body of the table, matrix elements are to be obtained using
JK c and WJKc basis set functions of Eq. (20) such that cc ð1Þ
Eqs. (10), (12), and (13).
61
L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
configurations, distortion terms were added to the tunneling splitting using as in Section 4.C of Ref. [24] a rotational operator D with
matrix elements given by Eqs. (12) and (13), where h2 is set to 1.
With the operator D, tunneling matrix element for the 1 ! 2 tunneling motion are then given by:
1
HJK c1;JK 0 c0 2 ¼ hJK cjfh2 þ h2k J 2z þ h2j J2 þ f2 ðJ 2þ þ J 2 Þ þ s2xz fJ x ; J z g
2
þh2kk J 4z þ h2kj J 2z J2 þ h2jj J4 þ f2k fJ 2þ þ J 2 ; J 2z g=2
þf2j ðJ 2þ þ J 2 ÞJ2 ; DgjJK 0 c0 i;
Fig. 8. Plots of the tunneling splitting, E ðJK a K c Þ Eþ ðJK a K c Þ, in MHz, as a function
of J for the three following pairs of rotational levels: J 2;J2 and J3;J2 , solid lines, J6;J2
and J 7;J2 , dashed lines, and J10;J2 and J11;J2 , dotted lines. Each pair of rotational
levels is characterized by the same K c -value. For high enough J-values, the
tunneling splitting is the same for both component of a given pair.
levels were computed using Eqs. (10), (12), and (13) and diagonalizing the Hamiltonian matrix with the help of Table 2. Fig. 8 shows
plots of the tunneling splittings for several pairs of rotational levels.
In agreement with Eq. (21), the tunneling splitting is close to
2h2 ¼ 82:9 MHz for small J-values. For higher J and K a -values, the
tunneling splitting displays a complicated behavior because of its
rotational dependence, described by Eqs. (12) and (13), and because
of the interaction of this rotational dependence with the K-type doubling. For large enough J-values, the tunneling splitting only depends
on K c as both components of all pairs of curves in Fig. 8 merge
together. This results can be confirmed using the same idea as in Section 3.D of Ref. [23].
4. Analyses
The section reports the results of the analyses of the frequencies
of the rotation-tunneling transitions and of the hyperfine structure
of 25 such transitions.
4.1. Rotation-tunneling transitions analysis
The rotational Hamiltonian of the D-in plane and D-out of plane
configurations in Eqs. (10) were written using the Ir reduction and
Watson’s A-set of distortion parameters. For the two D-out of plane
ð22Þ
where h2k ; h2j ; f2 ; s2xz ; h2kk ; h2kj ; h2jj ; f2k , and f2j are distortion parameters; J; Jx ; J y , and J z are the total angular momentum and its components; J ¼ Jx iJ y ; and {,} is the anticommutator. Distortion
parameters in JðJ þ 1Þ were also added to h2 and /2 so that in Eqs.
(12) and (13) these two angles should be replaced by
h2 þ h2j JðJ þ 1Þ and /2 þ /2j JðJ þ 1Þ, respectively.
The microwave data recorded in the present investigation were
analyzed calculating the tunneling-rotational energy with the
theoretical approach described in the previous section. Experimental frequencies were introduced in a least-squares fit procedure
where they were given a weight equal to the inverse of the square
of their experimental uncertainty. Unresolved doublets were treated as follows: assuming that such a doublet corresponds to the
J 00 K 00a1 K 00c1 ; and J 0 K 0a2 K 0c2 ; two transitions J 0 K 0a1 K 0c1 ; 00 00
00
J K a2 K c2 ; with calculated frequencies F 1 and F 2 , respectively.
The doublet was treated in the analysis as a single transition with
a calculated frequency equal to ðF 1 þ F 2 Þ=2. In the present data set,
usual unresolved K-type doublets arise for all configurations. For
the D-out of plane configurations, additional doublets arise
because of unresolved tunneling components.
The data set contains 335 a-type and 153 b-type transitions for
the D-in plane configuration, and 642 a-type and 258 b-type transitions for the D-out of plane configurations. Table 3 for the center frequency of the 25 hyperfine patterns recorded from 4 to 17 GHz, Table
4 for the 332 transitions measured in the 7–80 GHz region, list
assignments, observed frequencies, and observed minus calculated
differences. For the 1031 transitions measured in the 140–660 GHz
region, Table 5 lists the same quantities but only for transitions with
a frequency between 528 and 582 GHz. The whole list of transitions
is available in ASCII format from the authors or from the editorial
office. For the 1388 fitted transitions, the root-mean-square (RMS)
value of the observed minus calculated residuals is 0.13 MHz and
the unitless standard deviation of the analysis is 1.5. Table 6 gives
the number of fitted lines and RMS values for each data subset and
Table 3
Assignments,a experimental frequencies,b and experimental minus calculated differencesc for the center frequencies of the hyperfine patterns.
J0
K 0a
K 0c
2
2
2
3
3
3
4
1
2
2
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
1
2
2
2
3
1
2
2
1
1
0
+
+
+
+
J 00
K 00a
K 00c
2
2
2
3
3
3
3
0
1
1
0
0
1
1
1
1
1
1
1
2
0
1
1
0
0
0
2
2
2
3
3
3
2
0
1
1
0
0
1
+
+
+
+
Expb
Diffc
J0
K 0a
K 0c
J 00
K 00a
K 00c
Expb
Diffc
4232.845
4695.127
4696.041
8460.586
9381.415
9383.233
9803.205
11419.513
11857.711
11858.087
11894.975
11895.265
13351.110
1
0
1
3
0
1
2
2
2
3
0
1
2
1
4
2
2
4
4
2
4
4
3
3
3
1
1
1
1
1
1
1
1
1
1
1
1
0
3
1
1
3
3
1
3
3
2
2
2
+
1
4
2
2
4
4
2
3
3
3
3
3
0
1
0
0
1
1
0
2
2
0
0
0
1
4
2
2
4
4
2
2
2
3
3
3
+
13352.325
14074.551
15061.910
15063.463
15589.884
15592.883
16433.145
17283.270
17283.925
17888.376
17890.515
18897.054
2
2
1
2
1
2
3
5
4
2
4
1
+
+
+
+
+
+
+
+
a
Transitions are assigned with the usual rotational quantum numbers of the upper and lower levels. For transitions within the D-out of plane configurations, the + or signs identify tunneling components.
b
Exp is the center frequency m0 in MHz obtained in Section 4.2 and reported in Tables 9 and 11. For this data set, an uncertainty of 3 kHz was taken.
c
Diff is the experimental minus calculated frequency in kHz corresponding to the constants in Tables 7 and 8.
62
L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
Table 4
Assignments,a observed frequencies,b and observed minus calculated differencesc for transitions recorded in the 7–80 GHz region.
J0
K 0a
K 0c
10
13
13
17
17
25
25
21
21
19
28
15
38
38
7
10
10
34
24
30
30
14
14
38
26
26
18
18
22
22
20
29
7
7
16
39
39
1
2
12
12
11
40
5
5
5
32
32
16
16
28
28
22
31
20
20
24
24
9
9
5
5
7
6
6
3
4
4
16
5
5
18
8
7
21
3
4
4
5
5
7
7
6
6
5
7
4
10
10
2
3
3
9
6
8
8
4
4
9
7
7
5
5
6
6
5
7
2
2
4
10
10
1
1
3
3
3
9
1
1
1
8
8
4
4
7
7
5
7
5
5
6
6
2
2
2
2
2
2
2
1
2
2
5
1
1
4
2
2
5
7
9
9
12
12
18
18
15
15
14
21
11
28
28
5
7
7
25
18
22
22
10
10
29
19
19
13
13
16
16
15
22
5
5
12
29
29
1
1
9
9
9
31
4
4
4
24
24
12
12
21
21
17
24
15
15
18
18
7
7
3
3
5
4
4
3
2
2
12
4
4
14
6
5
16
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
J 00
K 00a
K 00c
10
13
13
17
17
25
25
21
21
19
28
15
38
38
7
10
10
34
24
30
30
14
14
38
26
26
18
18
22
22
20
29
7
7
16
39
39
0
1
12
12
10
40
5
5
5
32
32
16
16
28
28
22
31
20
20
24
24
9
9
5
5
6
6
6
2
4
4
15
4
4
18
7
7
21
3
4
4
5
5
7
7
6
6
5
7
4
10
10
2
3
3
9
6
8
8
4
4
9
7
7
5
5
6
6
5
7
2
2
4
10
10
0
1
3
3
4
9
0
0
0
8
8
4
4
7
7
5
7
5
5
6
6
2
2
1
1
3
1
1
1
1
1
6
2
2
4
3
1
5
8
10
10
13
13
19
19
16
16
15
22
12
29
29
6
8
8
26
19
23
23
11
11
30
20
20
14
14
17
17
16
23
6
6
13
30
30
0
0
10
10
6
32
5
5
5
25
25
13
13
22
22
18
25
16
16
19
19
8
8
4
4
4
5
5
2
3
3
9
3
3
15
5
6
17
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Obsb
Diffc
J0
K 0a
K 0c
7138.919(150)
7830.510(200)
7833.049(200)
8435.890(200)
8439.009(200)
8496.609(150)
8500.829(150)
8620.979(300)
8624.189(300)
9357.729(150)
9660.898(150)
10771.490(150)
10982.280(150)
10985.670(200)
11205.720(150)
11338.510(200)
11342.410(200)
11940.550(150)
11965.740(200)
12762.160(150)
12766.810(150)
12823.550(150)
12828.100(150)
13204.070(150)
13368.880(150)
13374.320(150)
13534.130(300)
13539.110(150)
13665.750(150)
13671.290(150)
14324.580(150)
14489.530(150)
14546.010(150)
14549.460(150)
16398.570(150)
16608.310(500)
16612.800(500)
24925.710(200)
25355.100(500)
26034.500(150)
26041.900(150)
26282.280(500)
26998.320(150)
27465.980(150)
27864.730(150)
27869.190(150)
28044.730(500)
28053.740(500)
28505.820(150)
28514.620(150)
29273.240(150)
29282.960(150)
29555.400(150)
29583.660(150)
29702.330(150)
29712.300(150)
29887.020(150)
29897.010(150)
30841.930(150)
30849.020(150)
31377.850(150)
31379.900(150)
31400.300(150)
31540.950(150)
31543.530(150)
32078.740(150)
32288.850(200)
32290.910(200)
32317.240(150)
32472.040(200)
32472.040(200)
32858.870(150)
33095.290(150)
33133.620(150)
40893.370(150)
139
318
202
223
213
233
337
389
146
91
327
122
130
183
221
338
118
175
86
209
59
111
62
158
78
97
344
49
71
14
160
55
150
99
72
298
123
96
233
181
144
255
115
155
226
22
103
23
124
302
132
178
260
218
216
243
93
262
221
209
135
122
91
61
146
135
218
12
156
50
215
275
23
102
19
34
8
12
25
11
11
35
35
31
31
15
15
27
27
19
19
23
23
21
5
2
3
8
8
4
4
2
2
4
9
2
2
2
1
2
3
3
7
3
3
36
3
3
6
2
2
4
2
27
3
6
6
6
3
13
7
5
14
3
8
8
37
4
4
4
8
12
12
3
17
17
6
33
32
15
8
2
3
6
3
3
9
9
8
8
4
4
7
7
5
5
6
6
5
1
1
0
2
2
1
1
1
1
1
2
0
0
0
1
1
0
0
2
1
1
8
2
0
1
1
1
0
1
6
0
1
1
2
2
3
2
2
4
1
2
2
9
2
0
0
2
4
4
2
4
4
1
8
7
4
26
6
9
19
8
8
26
26
23
23
11
11
20
20
14
14
17
17
16
4
2
3
6
6
3
3
2
2
3
8
2
2
2
1
1
3
3
5
3
3
28
1
3
5
2
2
4
2
21
3
5
5
4
2
10
5
3
11
2
6
6
28
2
4
4
6
8
8
1
13
13
5
25
25
12
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
J 00
K 00a
K 00c
34
8
12
25
11
11
35
35
31
31
15
15
27
27
19
19
23
23
21
5
1
2
8
8
4
4
1
1
4
8
1
1
1
0
1
2
2
7
2
2
36
3
2
6
1
1
3
1
27
2
6
6
6
2
13
7
5
13
2
8
8
37
4
3
3
8
11
11
3
17
17
5
33
32
14
8
2
3
6
3
3
9
9
8
8
4
4
7
7
5
5
6
6
5
1
1
1
2
2
0
0
1
1
0
3
0
0
0
0
1
1
1
1
1
1
8
1
0
0
0
0
1
0
6
0
0
0
1
2
3
1
1
5
1
1
1
9
1
1
1
1
5
5
1
4
4
2
8
7
5
27
7
10
20
9
9
27
27
24
24
12
12
21
21
15
15
18
18
17
5
1
2
7
7
4
4
1
1
4
5
1
1
1
0
0
2
2
6
2
2
29
2
2
6
1
1
3
1
22
2
6
6
5
1
11
6
4
8
1
7
7
29
3
3
3
7
7
7
2
14
14
4
26
26
9
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Obsb
Diffc
16890.470(150)
17269.830(150)
17646.030(150)
17761.090(150)
17789.040(150)
17794.440(150)
18043.190(300)
18048.670(300)
19274.210(150)
19281.260(150)
19687.370(150)
19694.100(150)
20195.670(150)
20203.150(150)
20572.180(150)
20579.590(150)
20679.310(150)
20687.020(150)
21006.630(150)
21026.330(150)
21428.350(150)
21642.060(200)
21930.870(150)
21936.270(100)
22086.790(150)
22089.360(150)
22225.250(200)
22225.250(200)
22508.850(150)
22572.330(150)
22734.090(150)
23644.300(150)
23644.300(150)
23681.130(150)
24249.750(150)
24742.860(150)
24742.860(150)
33136.940(150)
33250.210(150)
33250.210(300)
33483.630(150)
33829.280(200)
33842.730(150)
33883.930(150)
34011.260(150)
34011.260(150)
34225.060(150)
34934.520(150)
35063.550(150)
35110.290(150)
35264.480(150)
35270.270(150)
35457.170(150)
35687.350(200)
35962.260(150)
35971.110(150)
36021.340(150)
36087.870(150)
36306.410(150)
36423.780(150)
36428.170(150)
37194.060(150)
37328.290(150)
37684.680(200)
37684.680(200)
37828.710(150)
38124.460(150)
38129.210(150)
38991.000(200)
39193.130(150)
39204.280(150)
39220.810(150)
39287.410(500)
40248.320(150)
47474.040(200)
118
298
7
341
84
76
251
140
95
58
167
179
4
123
74
90
23
166
33
207
299
80
189
89
210
250
117
380
248
194
45
299
234
178
174
58
7
169
12
384
259
39
292
197
174d
174d
32
177
215
49
264
236
127
124
161
33
34
174
61
48
66
121
32
373
47
208
294
330
108
187
308
46
154
284
26
63
L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
Table 4 (continued)
J0
K 0a
K 0c
J 00
K 00a
K 00c
Obsb
Diffc
J0
K 0a
K 0c
J00
K 00a
K 00c
Obsb
21
10
10
9
9
9
3
3
4
21
21
14
10
10
3
11
4
4
3
16
4
17
37
4
4
4
4
10
4
5
28
3
14
14
4
4
12
11
7
7
13
13
7
5
5
6
6
12
5
5
5
13
10
5
5
18
13
6
9
16
14
5
6
9
5
7
7
14
14
8
5
12
5
8
6
6
5
2
2
2
2
2
2
2
1
11
11
4
3
3
1
2
1
1
1
9
0
5
8
2
2
0
0
2
2
0
6
2
3
3
2
2
3
3
3
3
3
3
1
2
2
2
2
2
1
1
0
3
3
2
0
5
3
2
3
4
3
2
0
1
1
2
2
3
3
3
1
2
2
3
1
1
16
8
8
7
7
7
2
2
4
10
11
10
7
7
3
9
4
4
3
8
4
13
29
3
3
4
4
8
2
5
22
2
11
11
3
3
9
8
4
4
10
10
6
4
4
5
5
10
5
5
5
10
7
4
5
14
11
5
6
13
11
3
6
8
4
6
6
11
11
5
5
10
3
6
6
6
+
+
21
10
10
9
9
9
3
3
3
22
22
13
9
9
2
11
3
3
2
17
3
16
37
3
4
3
3
10
3
4
28
3
14
14
3
3
12
11
7
7
13
13
6
5
5
6
6
12
4
4
4
13
10
4
4
17
12
6
9
15
14
4
5
9
4
7
7
14
14
8
4
12
4
8
5
5
5
2
2
1
1
1
1
1
1
10
10
5
4
4
0
2
1
1
0
8
0
6
8
2
1
0
0
1
2
1
6
1
3
3
2
2
2
2
2
2
2
2
2
0
0
1
1
2
1
1
0
2
2
2
0
6
4
1
2
5
2
2
1
1
1
1
1
2
2
2
0
1
2
2
0
0
17
9
9
8
8
8
3
3
3
13
12
9
6
6
2
10
3
3
2
9
3
10
30
2
4
3
3
9
1
4
23
3
12
12
2
2
10
9
5
5
11
11
5
5
5
6
6
11
4
4
4
11
8
3
4
11
8
6
7
10
12
2
5
9
3
7
7
12
12
6
4
11
2
7
5
5
+
+
40905.480(150)
41082.890(150)
41091.820(150)
41235.950(150)
41589.410(150)
41595.370(150)
42487.280(200)
42491.260(200)
42659.640(150)
42969.320(150)
42969.320(150)
43064.050(150)
43072.780(200)
43072.780(200)
43617.830(150)
43839.610(150)
44181.480(150)
44181.480(150)
44279.090(200)
44435.360(150)
44661.640(150)
44685.190(150)
45072.040(150)
45595.680(150)
45771.450(150)
46191.890(150)
46191.890(150)
46332.280(150)
46610.900(150)
46718.550(150)
46725.420(150)
46930.750(150)
47353.340(150)
47364.680(150)
47466.080(150)
47466.080(150)
53952.790(150)
54154.020(150)
54285.440(150)
54288.090(150)
54330.840(150)
54337.410(150)
54358.420(150)
54542.032(150)
54547.420(150)
54907.130(150)
54913.620(150)
54973.770(150)
55006.090(150)
55006.090(150)
55153.540(150)
55386.850(150)
55685.300(150)
56863.090(150)
56875.020(150)
56946.960(150)
57680.820(150)
58046.420(150)
58150.630(150)
58277.870(150)
58669.540(150)
58803.360(150)
58908.700(150)
59400.050(150)
60110.650(150)
60712.320(150)
60719.670(150)
60817.450(300)
60826.160(300)
61091.210(150)
61592.700(150)
61605.550(150)
61742.570(150)
69740.650(150)
70348.840(150)
70348.840(150)
309
279
288
209
159
200
5
130
2
267
229
58
203
67
211
268
77d
77d
187
73
84
31
255
159
53
73d
73d
182
68
246
241
234
327
476
29d
29d
76
156
114
65
210
233
154
166
31
126
38
8
23d
23d
198
174
131
50
48
43
56
19
40
114
280
136
133
91
271
200
182
417
383
13
241
23
42
47
180d
180d
10
10
11
11
6
6
8
8
4
4
9
9
9
4
5
5
12
12
5
4
8
8
18
18
24
11
20
14
4
4
34
4
5
25
25
22
5
6
5
14
15
15
18
7
7
7
6
16
16
14
7
5
16
6
6
9
7
6
17
17
13
7
7
6
6
8
8
6
6
8
6
8
8
6
6
11
3
3
3
3
1
1
2
2
1
2
3
3
2
2
2
2
3
3
0
1
3
3
4
4
5
2
6
3
1
1
8
1
1
13
13
5
2
0
1
3
4
4
5
2
1
1
1
4
4
4
3
3
3
3
3
2
2
3
4
4
4
3
3
0
0
2
2
2
3
2
3
1
3
1
1
4
7
7
8
8
5
5
6
6
3
2
6
6
7
3
4
4
9
9
5
3
5
5
14
14
19
9
15
12
4
4
26
4
5
12
13
17
3
6
4
12
11
11
13
6
6
6
6
12
12
10
4
3
14
4
4
7
6
3
13
13
9
5
5
6
6
7
7
5
4
7
3
7
6
5
5
7
+
+
+
+
10
10
11
11
5
5
7
7
3
3
9
9
8
4
5
5
12
12
4
3
8
8
18
18
24
11
19
13
3
3
34
3
4
26
26
22
4
5
4
13
15
15
17
7
6
6
5
16
16
14
7
5
15
6
6
8
7
6
17
17
13
7
7
5
5
8
8
5
5
8
5
7
8
5
5
11
2
2
2
2
2
2
3
3
1
2
2
2
3
1
1
1
2
2
1
1
2
2
4
4
5
2
7
4
0
0
8
0
1
12
12
5
2
1
1
4
3
3
6
1
2
2
1
3
3
3
2
2
4
2
2
3
0
2
3
3
3
2
2
0
0
1
1
2
3
1
3
2
2
1
1
3
8
8
9
9
4
4
5
5
2
1
7
7
6
4
5
5
10
10
4
2
6
6
15
15
20
10
12
9
3
3
27
3
4
15
14
18
2
5
3
9
12
12
12
7
5
5
5
13
13
11
5
4
11
5
5
6
7
4
14
14
10
6
6
5
5
8
8
4
3
8
2
6
7
4
4
8
+
+
+
+
47628.420(150)
277
47630.960(300)
385
47879.780(150)
206
47883.470(150)
14
47973.570(150)
95d
47973.570(150)
95d
48203.010(500)
735
48203.010(500)
328
48273.510(150)
92
48851.870(150)
260
48942.760(150)
34
48945.000(150)
80
49491.250(150)
40
49867.230(150)
45
49914.730(150)
25
49920.090(150)
81
50041.500(150)
276
50046.390(150)
215
50378.930(150)
109
50391.480(150)
48
51349.860(150)
174
51352.170(300)
242
51529.560(150)
273
51542.860(150)
435
52211.000(150)
217
52389.260(150)
389
52518.470(150)
253
52549.370(150)
217
52688.840(150)
164d
52688.840(150)
164d
53017.500(1500)
318
53096.290(150)
187
53158.420(150)
26
53442.570(150)
71d
53442.570(150)
71d
53948.070(150)
14
61744.220(150)
274
62623.020(150)
97
62653.670(150)
225
62698.270(150)
128
62751.940(150)
195
62754.480(150)
41
63060.270(150)
241
63275.170(150)
15
63553.400(150)
51
63555.340(150)
1
63569.690(150)
150
63578.000(150)
314
63581.950(150)
181
63957.700(150)
27
64046.650(150)
210
64054.740(150)
88
64555.620(150)
202
65286.590(150)
373
65292.520(150)
80
65644.560(300)
198
66495.630(150)
218
66630.910(150)
152
66727.950(300)
397
66733.670(300)
501
66747.400(150)
149
67146.140(150)
32
67152.090(150)
10
67250.550(150)
125d
67250.550(150)
125d
67263.960(300)
395
67272.420(300)
341
68044.020(150)
29
68964.420(150)
80
69222.700(300)
390
69230.800(150)
200
69530.980(150)
6
69734.250(150)
58
74650.460(300)
306
74652.660(300)
382
74687.250(300)
360
(continued on next page)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Diffc
64
L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
Table 4 (continued)
J0
K 0a
K 0c
7
6
3
3
6
6
6
18
18
18
6
13
24
9
9
16
7
7
15
9
9
0
2
2
2
2
3
3
4
4
4
3
3
7
3
3
3
1
3
4
2
2
7
5
2
2
4
4
4
14
14
14
3
10
18
7
7
13
7
5
11
8
8
+
+
+
+
+
J 00
K 00a
K 00c
6
5
2
2
5
5
5
18
18
18
5
12
23
9
9
16
6
7
15
9
9
1
2
1
1
2
3
3
3
3
3
3
4
8
2
2
3
1
2
3
1
1
6
4
1
1
3
3
3
15
15
15
2
9
15
8
8
14
6
6
12
9
9
+
+
+
+
+
Obsb
Diffc
J0
K 0a
K 0c
J00
K 00a
K 00c
Obsb
Diffc
70674.670(150)
70714.060(150)
71042.060(150)
71045.410(150)
71201.920(150)
71952.770(150)
71952.770(150)
72171.590(300)
72330.050(300)
72338.160(300)
72407.920(150)
72576.770(150)
72872.350(300)
73129.030(150)
73136.350(300)
73250.520(300)
73894.810(150)
74264.410(150)
74296.560(150)
74471.160(150)
74480.810(300)
67
167
76
138
160
19d
19d
316
387
458
155
74
229
12
250
339
184
170
54
211
128
11
6
6
7
30
16
8
7
3
3
10
10
7
14
10
10
9
8
8
7
4
2
2
0
6
4
3
1
2
2
3
3
0
4
4
4
3
1
1
2
7
4
4
7
24
12
6
7
1
1
8
8
7
10
6
6
7
7
7
6
+
+
11
5
5
6
30
15
8
6
2
2
10
10
6
14
10
10
9
7
7
6
3
2
2
0
6
5
2
1
1
1
2
2
0
3
3
3
2
2
2
2
8
3
3
6
25
11
7
6
2
2
9
9
6
11
7
7
8
6
6
5
+
+
74689.120(150)
74813.620(150)
74815.870(150)
75335.970(150)
75462.810(150)
75759.570(150)
76246.550(150)
76336.240(150)
76457.880(150)
76462.620(150)
77373.050(150)
77381.150(300)
77461.980(150)
77768.890(150)
78601.050(150)
78604.400(150)
78891.270(150)
78924.840(300)
78927.450(150)
79123.590(150)
81
170
62
260
335
203
106
208
181
129
32
225
203
160
121
27
7
282
140
120
+
+
+
+
+
+
+
+
a
Transitions are assigned with the usual rotational quantum numbers of the upper and lower levels. For transitions within the D-out of plane configurations, the + or signs identify tunneling components.
b
Obs is the observed frequency in MHz. The uncertainty is given in parentheses in kHz.
c
Diff is the observed minus calculated frequency in kHz corresponding to the constants in Tables 7 and 8.
d
Line was treated as an unresolved doublet.
for the D-in plane and D-out of plane configurations. Table 7 for the
D-in plane configuration and Table 8 for the D-out of plane configurations give the value of the parameters determined in the analysis
as well as their uncertainty.
4.2. Hyperfine structure analysis
The hyperfine splittings observed for several transitions of the
D-in plane and D-out of plane configurations are due to quadrupole
Table 5
Assignments,a observed frequencies,b and observed minus calculated differencesc for transitions recorded in the 140–660 GHz region.
J0
K 0a
K 0c
J 00
K 00a
K 00c
Obsb
Diffc
J0
K 0a
K 0c
J 00
K 00a
K 00c
Obsb
Diffc
50
50
50
50
51
51
53
53
47
47
47
47
47
47
47
47
48
48
48
48
49
49
49
49
50
50
50
50
51
51
51
1
2
1
2
0
0
0
1
5
5
6
6
5
5
6
6
4
5
4
5
3
4
3
4
2
3
2
3
1
2
1
49
49
49
49
51
51
53
53
42
42
42
42
42
42
42
42
44
44
44
44
46
46
46
46
48
48
48
48
50
50
50
+
+
+
49
49
49
49
50
50
52
52
46
46
46
46
46
46
46
46
47
47
47
47
48
48
48
48
49
49
49
49
50
50
50
1
2
1
2
0
0
0
1
6
6
6
6
5
5
5
5
4
5
4
5
3
4
3
4
2
3
2
3
1
2
1
48
48
48
48
50
50
52
52
41
41
41
41
41
41
41
41
43
43
43
43
45
45
45
45
47
47
47
47
49
49
49
+
+
+
528864.299(200)
528864.299(200)
528865.653(200)
528865.653(200)
530117.637(29)
530117.637(29)
533817.639(200)
533817.639(200)
534673.110(29)
534673.110(29)
534676.568(29)
534676.568(29)
534679.017(29)
534679.017(29)
534682.500(29)
534682.500(29)
535611.372(29)
535611.372(29)
535612.270(29)
535612.270(29)
536704.125(29)
536704.125(29)
536705.566(29)
536705.566(29)
537885.021(29)
537885.021(29)
537886.682(29)
537886.682(29)
539115.163(29)
539115.163(29)
539116.506(29)
56
56
63
63
3d
3d
2
2
1d
1d
21d
21d
11d
11d
8d
8d
8d
8d
11d
11d
17
17
5
3
29
29
22
22
60
60
25
51
52
52
52
52
48
48
48
48
48
48
48
48
45
49
49
49
49
55
55
55
55
54
47
47
28
49
49
48
48
2
0
0
1
1
5
5
6
6
5
5
6
6
9
5
4
5
4
2
1
1
2
4
12
12
7
8
8
15
15
50
52
52
52
52
43
43
43
43
43
43
43
43
36
45
45
45
45
54
54
54
54
50
35
35
21
41
41
34
33
+
+
+
+
+
+
+
50
51
51
51
51
47
47
47
47
47
47
47
47
44
48
48
48
48
54
54
54
54
53
46
46
27
48
48
47
47
2
1
0
0
1
6
6
6
6
5
5
5
5
9
5
4
5
4
2
1
1
2
4
12
12
5
9
9
15
15
49
51
51
51
51
42
42
42
42
42
42
42
42
35
44
44
44
44
53
53
53
53
49
34
34
22
40
40
33
32
+
+
+
+
+
+
+
539116.506(29)
540372.070(29)
540372.070(29)
540372.070(29)
540372.070(29)
544878.972(29)
544878.972(29)
544881.000(29)
544881.000(29)
544882.480(29)
544882.480(29)
544884.522(29)
544884.522(29)
544947.672(100)
545837.421(29)
545837.421(29)
545838.236(29)
545838.236(29)
580088.250(200)
580088.250(200)
580089.658(200)
580089.658(200)
580229.404(29)
580604.361(29)
580604.361(29)
581617.872(29)
581919.677(29)
581920.650(29)
581925.931(29)
581937.822(200)
25
5d
5d
5d
5d
21d
21d
30d
30d
8d
8d
14d
14d
21
20d
20d
21d
21d
38
38
20
20
22
9d
9d
34
8
5
149
180
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
a
Transitions are assigned with the usual rotational quantum numbers of the upper and lower levels. For transitions within the D-out of plane configurations, the + or signs identify tunneling components. Only transition with a frequency between 528 and 582 GHz appear in this table. The whole list of transitions is available in ASCII format
from the authors or from the editorial office.
b
Obs is the observed frequency in MHz. The uncertainty is given in parentheses in kHz.
c
Diff is the observed minus calculated frequency in kHz corresponding to the constants in Tables 7 and 8.
d
Line was treated as an unresolved doublet.
65
L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
Table 6
Results summarya of the analysis of the microwave data.
Data
D-in plane
Molecular beamb
Stark modulationc
Submillimeter waved
All data
a
b
c
d
D-out of plane
All configurations
N
RMS
N
RMS
N
RMS
7
121
360
488
0.002
0.175
0.063
0.103
18
211
671
900
0.002
0.210
0.124
0.147
25
332
1031
1388
0.002
0.198
0.106
0.133
For each data subset and for the D-in plane and D-out of plane configurations, the number of fitted lines N and the RMS value, in MHz, are given.
This data subset consists of the center frequencies m0 obtained in Section 4.2 and reported in Tables 9 and 11.
This data subset consists of transitions measured in the 7–80 GHz region using Stark modulation techniques.
This data subset consists of transitions measured in the 140–660 GHz region using BWO’s.
Table 8
Parametersa for the D-out of plane configurations.
Table 7
Parametersa for the D-in plane configuration.
Parameter
Value
A
B
C
Parameter
0.617 649 745(39)
0.224 495 075(10)
0.172 284 266(10)
DKK 106
DKJ 106
DJJ 106
dK 106
dJ 106
1.931
0.493
0.206
0.055
0.065
861(250)
271(53)
993(16)
155(110)
653(4)
12
HKKK 10
HKKJ 1012
HKJJ 1012
HJJJ 1012
hKK 1012
hKJ 1012
hJJ 1012
LKKJJ 1015
LKJJJ 1015
LJJJJ 1015
lJJJ 1015
Value
29.043
3.028
1.836
0.121
13.320
6.095
0.057
999(590000)
171(190000)
152(56000)
672(7300)
006(360000)
695(28000)
441(2300)
0.310
0.189
0.006
0.002
813(26000)
672(8800)
612(1100)
403(420)
coupling arising from the deuterium atom. This hyperfine structure
was analyzed using a least-squares fit procedure in which the following components of the effective quadrupole coupling tensors
were determined:
eQq1yy ;
eQq3xx ;
Value
Parameter
4.827 711(5300)
85.184 427(8500)
A
B
C
h2j 105
/2j 105
0.530 857(280000)
3.014 810(740000)
h2 103
1.413 652(3000)
6
a
Parameters are in cm1. Numbers in parentheses are one standard deviation in
the same units as the last digit. These parameters are involved in the pure rotational
Hamiltonian of the lower lying D-in plane configuration in the second of Eqs. (10).
eQq1xx ;
Parameter
h2
/2
and eQq3yy ;
ð23Þ
h2k 10
h2j 106
f2 106
s2xz 106
0.281
0.099
0.017
0.137
430(51000)
152(5500)
663(2900)
075(9100)
h2kk 109
h2kj 109
h2jj 109
f2k 109
f2j 109
0.443
0.414
0.022
0.096
0.023
103(200000)
390(63000)
692(2900)
254(38000)
064(3300)
h2kkk 1012
h2kkj 1012
h2kjj 1012
f2kk 1012
f2kj 1012
f2jj 1012
0.521
0.678
0.195
0.459
0.148
0.003
257(140000)
336(69000)
591(19000)
405(55000)
727(17000)
035(830)
Value
0.664 512 489(45)
0.213 990 231(8)
0.166 924 411(8)
DKK 106
DKJ 106
DJJ 106
dK 106
dJ 106
2.595
0.647
0.164
0.118
0.049
568(770)
750(44)
309(7)
333(110)
400(3)
HKKK 1012
HKKJ 1012
HKJJ 1012
HJJJ 1012
hKK 1012
hKJ 1012
hJJ 1012
40.885
22.621
0.443
0.065
6.973
2.756
0.041
185(4600000)
072(200000)
105(59000)
646(1300)
409(730000)
970(21000)
471(590)
LKKKK 1015
LKKKJ 1015
LKKJJ 1015
LKJJJ 1015
27.622
1.604
0.624
0.035
955(8700000)
851(140000)
670(56000)
507(4100)
a
Parameters are in cm1 except h2 ; /2 ; h2j , and /2j which are in degrees. Numbers
in parentheses are one standard deviation in the same units as the last digit.
Parameters are defined in Eq. (22) or are involved in the pure rotational Hamiltonian of the upper lying D-out of plane configurations in the first of Eqs. (10).
where superscripts 1 and 3 indicate components for the D-out of
plane and D-in plane configurations, respectively. Due to the large
amplitude motion, non-diagonal components of the effective quadrupole coupling tensor eQ q1 can in principle be also determined, as
in the case of the hydrazine molecule [25]. In the present analysis,
such an attempt was unsuccessfull as the values obtained were illdefined.
Hyperfine patterns were recorded for seven rotational transitions
of the D-in plane configuration and for 18 rotation-tunneling transitions of the D-out of plane configurations. In Table 9, assignments,
Table 9
Assignments,a observed frequencies and observed minus calculated differences in the quadrupole hyperfine spectrum of the D-in plane configuration.
F0
101
0
2
1
000
211
1
3
2
3
1
202
a
b
c
F 00
Obsb
Diffc
1
1
1
11419.4345
11419.5064
11419.5519
1.2
1.1
0.1
2
2
2
3
1
16433.0899
16433.1053
16433.1214
16433.1493
16433.1698
1.2
3.3
0.2
2.8
0.8
F0
F 00
211
2
3
212
312
3
4
2
303
312
3
4
2
313
Obsb
Diffc
F0
413
3
5
4
322
413
4
5
3
414
2
3
4232.8368
4232.8470
0.2
0.2
3
4
2
18897.0398
18897.0570
18897.0674
0.0
1.9
1.8
3
4
2
8460.5800
8460.5861
8460.5942
1.8
3.0
1.2
Hyperfine components are assigned with the F quantum number.
Obs is the observed frequency of the hyperfine components in MHz.
Diff is the observed minus calculated frequency in kHz corresponding to the constants in Tables 10 and 13.
F 00
Obsb
Diffc
2
4
3
9803.1867
9803.1913
9803.2341
2.6
2.9
0.3
4
5
3
14074.5427
14074.5521
14074.5596
0.4
2.1
2.5
66
L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
Table 10
Center frequencies
m0 from the analysis of the hyperfine structure of the D-in plane configuration.
J0
K 0a
K 0c
J 00
K 00a
K 00c
m0a
RMSb
Nc
1
2
2
3
3
4
4
0
1
1
1
1
1
1
1
1
1
2
2
3
3
0
2
2
3
3
3
4
0
0
1
0
1
2
1
0
2
2
3
3
2
4
11419.5131(11)
16433.1453(9)
4232.8448(13)
18897.0541(11)
8460.5864(11)
9803.2048(11)
14074.5513(11)
0.9
2.0
0.2
1.5
2.2
2.3
1.9
3
5
2
3
3
3
3
a
b
c
The center frequency is given in MHz. Its uncertainty is given in parentheses in the same units as the last digit.
For each rovibrational transition, the RMS deviation is given in kHz.
N is the number of hyperfine components.
Table 11
Assignments,a observed frequencies and observed minus calculated differences in the quadrupole hyperfine spectrum of the D-out of plane configurations.
F0
F 00
101 ; þ
1
2
0
000 ; þ
1
1
1
101 ; 1
2
0
000 ; 1
1
1
110 ; þ
1
0
2
1
2
101 ; þ
0
1
2
2
1
110 ; 1
0
2
1
2
101 ; 0
1
2
2
1
211 ; þ
1
3
2
202 ; þ
1
3
2
a
b
c
Diffc
F0
11895.2451
11895.2691
11895.3033
0.6
0.6
1.2
211 ; 1
3
2
202 ; 1
3
2
11894.9557
11894.9796
11895.0149
0.3
0.2
0.5
211 ; þ
3
2
Obsb
13352.3048
13352.3048
13352.3191
13352.3408
13352.3408
13351.0892
13351.0892
13351.1038
13351.1255
13351.1255
15063.4402
15063.4591
15063.4851
0.4
2.0
1.7
0.4
0.6
0.2
2.1
1.9
0.5
0.4
2.9
1.5
1.5
F 00
Diffc
F0
Obsb
Diffc
15061.8877
15061.9056
15061.9311
2.1
1.3
0.8
312 ; 2
4
3
303 ; 2
4
3
17888.3620
17888.3714
17888.3926
0.9
0.8
0.1
212 ; þ
3
2
4696.0365
4696.0586
0.8
0.8
312 ; þ
2
4
3
313 ; þ
2
4
3
9383.2168
9383.2287
9383.2497
1.0
2.3
1.3
211 ; 3
2
212 ; 3
2
4695.1230
4695.1450
0.9
0.9
202 ; þ
2
3
111 ; þ
1
2
312 ; 2
4
3
313 ; 2
4
3
9381.3991
9381.4106
9381.4319
0.9
2.0
1.2
11857.6893
11857.7178
0.3
0.3
202 ; 2
3
1
111 ; 1
2
0
413 ; þ
3
5
4
414 ; þ
3
5
4
15592.8701
15592.8791
15592.8997
0.3
2.3
2.0
312 ; þ
2
4
3
303 ; þ
2
4
3
413 ; 3
5
4
414 ; 3
5
4
15589.8703
15589.8787
15589.9020
0.8
1.2
0.4
413 ; þ
4
3
413 ; 4
3
322 ; þ
3
2
322 ; 3
2
17283.2595
17283.2773
1.5
1.5
17283.9146
17283.9326
1.6
1.6
Obsb
11858.0653
11858.0951
11858.1087
17890.5012
17890.5111
17890.5317
0.3
1.6
1.3
1.1
1.2
0.1
F 00
Hyperfine components are assigned with the F quantum number.
Obs is the observed frequency of the hyperfine components in MHz.
Diff is the observed minus calculated frequency in kHz corresponding to the constants in Tables 12 and 13.
observed frequencies, and observed minus calculated differences
are listed for hyperfine components. For each hyperfine pattern,
Table 10 gives values for the center frequencies, the RMS deviations
and the number of hyperfine components. For the D-out of plane
configurations, Tables 11 and 12 display the same results. Values
obtained for the components of the effective quadrupole coupling
tensors are listed in Table 13 where they are compared to values calculated from the structure using the eQq value reported by Wofsy
et al. [26] for CH3 D and to values obtained through ab initio calculations. In these calculations 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 employed to calculate the effective quadrupole
coupling tensor. Gaussian 03 [30] with the split-valence basis set
6-311+G(3df,2pd) as implemented in Gaussian 03 was also
employed in these calculations.
5. Discussion
Internal rotation in the mono deuterated species of methyl formate HCOOCH2 D is investigated in this paper both experimentally
and theoretically. The torsion displayed by this molecule is theoretically challenging since it is due to an asymmetrical CH2 D
methyl group.
The rotation–torsion energy levels of HCOOCH2 D are accounted
for making use of the IAM-like approach developed in Refs. [14,15].
Applying this approach requires first studying the equilibrium configurations and the tunneling paths of the molecule. Three equilibrium configurations arise in the case of HCOOCH2 D and they are
illustrated in Fig. 7. One of them, the one with C S symmetry,
referred to as the D-in plane configuration, is below the two other
energetically equivalent ones, referred to as the D-out of plane configurations. The tunneling motions correspond to rotations through
67
L. Margulès et al. / Journal of Molecular Spectroscopy 254 (2009) 55–68
Table 12
Center frequencies
m0 from the analysis of the hyperfine structure of the D-out of plane configurations.
J0
K 0a
K 0c
J00
K 00a
K 0c
m0a
RMSb
Nc
1
1
1
1
2
2
2
2
2
2
3
3
3
3
4
4
4
4
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
2
2
2
2
2
2
3
3
3
3
+
+
+
+
+
+
+
+
+
0
0
1
1
2
2
2
2
1
1
3
3
3
3
4
4
3
3
0
0
0
0
0
0
1
1
1
1
0
0
1
1
1
1
2
2
0
0
1
1
2
2
2
2
1
1
3
3
3
3
4
4
2
2
+
+
+
+
+
+
+
+
+
11895.2645(11)
11894.9754(11)
13352.3252(8)
13351.1098(8)
15063.4634(11)
15061.9101(11)
4696.0410(13)
4695.1274(13)
11857.7114(13)
11858.0874(11)
17890.5154(11)
17888.3761(11)
9383.2325(11)
9381.4147(11)
15592.8834(11)
15589.8841(11)
17283.2696(13)
17283.9248(13)
0.9
0.4
1.2
1.3
2.1
1.5
0.8
0.9
0.3
1.2
0.9
0.7
1.6
1.5
1.8
0.9
1.5
1.6
3
3
5
5
3
3
2
2
2
3
3
3
3
3
3
3
2
2
a
b
c
The center frequency is given in MHz. Its uncertainty is given in parentheses in the same units as the last digit.
For each rovibrational transition, the RMS deviation is given in kHz.
N is the number of hyperfine components.
Table 13
Effective quadrupole coupling tensors components.a
Component
Valueb
Calculatedc
Ab initiod
0.0028
eQq1xx
0.0031(11)
0.0014
eQq1yy
0.0769(13)
0.0839
0.0848
eQq3xx
0.0610(19)
0.0822
0.0671
eQq3yy
0.0938(20)
0.0955
0.1061
a
Values are in MHz. Numbers in parentheses are one standard deviation in the
same units as the last digit. Superscripts 1 and 3 indicate the D-out of plane and Din plane configurations, respectively.
b
Fitted values obtained in Section 4.2 are reported.
c
Calculated values are based on the geometry of the D-in plane and D-out of
plane configurations and on the eQq value reported in Ref. [26] for CH3 D.
d
Calculated with the 6-311+G(3df,2pd) basis set as implemented in Gaussian 03
[30].
roughly 120° of the methyl group. However, unlike in the case of a
symmetrical CH3 group, two types of tunneling paths should be
considered depending on whether they connect the two energetically equivalent D-out of plane configurations or one of these configurations and the C S symmetry D-in plane configuration. Using
the IAM-like approach of Refs. [14,15], the rotational dependence
of the Hamiltonian tunneling matrix elements were computed
and are given in Eqs. (12)–(15), (17), (18). For J ¼ 0, three tunneling
sublevels arise. The lowest one has A0 symmetry and is about
10 cm1 below the two other close lying ones having A0 and A00
symmetry.
These theoretical results were used to analyze the microwave
data measured in this work. Although the effects of the tunneling
path connecting the two D-out of plane configurations could be
seen, as they lead to a splitting of the microwave lines, the effects
of the tunneling path connecting one of the D-out of plane configurations to the D-in plane configuration could not be evidenced. In
the analysis, the rotational-tunneling levels arising from the two
D-out of plane configurations were, therefore, treated independently from those arising from the D-in plane configuration. Rotational levels arising from the latter configuration were calculated
using a standard Watson-type Hamiltonian. Rotational-tunneling
levels arising from the former configurations were treated using
the IAM-type approach. For this subset of data, the IAM approach
accounts for the complicated rotational dependence of the tunneling splitting, illustrated in Fig. 8, and made it possible to reproduce
the microwave data with an unitless standard deviation of 1.5.
Table 6 emphasizes that for the submillimeter wave data subset
and for the D-out of plane configurations an RMS value of
0.124 MHz was achieved, which does not compare quite favorably
with an experimental uncertainty of 30 kHz. It is believed that this
unsatisfactory result is due to neglection of the tunneling matrix
elements arising from the tunneling path connecting one of the
D-out of plane configurations to the D-in plane configuration.
The angles h2 and /2 involved in the rotational dependence of
the tunneling splitting and defined in Eqs. (12) and (13) were also
determined in the analysis and their values are listed in Table 8.
These values are within 0.2° and 1°, respectively, from the ones
computed from the structure in Section 3.2. This satisfactory agreement suggests that the nature of the large amplitude tunneling
motion taking place in mono deuterated methyl formate is well
understood. The results of the hyperfine structure analysis are also
consistent with the geometry of the three configurations and with
ab initio calculations, as emphasized by Table 13.
Acknowledgments
Financial support from the French program for astrophysical
chemistry (PCMI) is gratefully acknowledged. Part of this work is
also supported by the French ANR-08-BLAN TOPMODEL grant.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at doi:10.1016/j.jms.2008.12.007.
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