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The microwave spectroscopy of trans-ethyl methyl ether in the ν29=1 excited torsional state
Journal of Molecular Spectroscopy 255 (2009) 164–171
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The microwave spectroscopy of trans-ethyl methyl ether in the torsional state
m29 ¼ 1 excited
Kaori Kobayashi a,*, Takanori Matsui a, Shozo Tsunekawa a, Nobukimi Ohashi b a b
Department of Physics, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan Kanazawa University, Kakuma, Kanazawa 920-1192, Japan
a r t i c l e
i n f o
Article history: Received 26 December 2008 In revised form 19 March 2009 Available online 5 April 2009 Keywords: trans-Ethyl methyl ether Methyl internal rotation Microwave spectroscopy Barrier height to internal rotation
a b s t r a c t The trans-ethyl methyl ether molecule has two inequivalent methyl group internal rotors. One methyl rotor is bonded to the oxygen atom and the other to the carbon atom. The internal rotations of these methyl rotors correspond to the vibrational modes, m29 and m28 , respectively. In this study, the microwave absorption spectrum in the m29 ¼ 1 excited torsional state was analyzed for the first time. Initial assignment has been carried out by making use of measurements at dry ice temperature. Over 1500 lines up to J ¼ 76 and K ¼ 5 were assigned. The spectrum involving internal rotation splittings was analyzed by using the tunneling matrix formulation developed by Hougen. Crown Copyright Ó 2009 Published by Elsevier Inc. All rights reserved.
1. Introduction The trans-ethyl methyl ether molecule is an interstellar molecule identified in W51e2 [1]. It is becoming common to detect torsionally excited saturated organic molecules like CH3 OH [2], C2 H5 CN [3], and HCOOCH3 [4,5] in the star-forming regions. These molecules are recently called ‘‘weed” because huge numbers of lines are observed everywhere. Although the intensity of transethyl methyl ether was not very strong, future progress of the highly sensitive new radio telescopes could make it possible to detect this molecule in the torsionally excited state. Precise transition frequencies are indispensable for detection. The structure of trans-ethyl methyl ether (CH3 CH2 OCH3 ) is shown in Fig. 1. One methyl group is bonded to the oxygen atom (abbreviated as O—CH3 ) and the other is bonded to the carbon atom (abbreviated as C—CH3 ). The vibrational frequencies have been studied by mid-resolution infrared spectroscopy [6,7]. The frequencies of internal rotation caused by these methyl rotors are 248 cm1 for O—CH3 torsion (m29 ) and 278 cm1 for C—CH3 torsion (m28 ), respectively, and these two inequivalent methyl rotors give rise to a complicated splitting. In addition there are three other modes below 500 cm1 : skeletal torsion (m30 115 cm1 Þ, CCO bend (m18 288 cm1 ), and COC bend (m17 475 cm1 ). These five low-lying states are well populated at room temperature and make the spectra congested.
There are many laboratory microwave studies of trans-ethyl methyl ether. Hayashi and his co-workers studied normal species and many isotopomers, and provided molecular constants, the dipole moment, and the molecular structure [8,9]. The barrier heights to the O—CH3 and C—CH3 internal rotation were determined to be 893 and 1154 cm1 , respectively, based on the analysis of the internal rotation. Assignment of torsionally excited states was also mentioned, but unfortunately neither the transition frequencies nor molecular constants were reported. Fuchs et al. extended the microwave study of the ground state up to 350 GHz and provided an analysis treating explicitly splittings due to the two internal rotors [10]. Independently, Tsunekawa et al. reported transitions in the ground state in the 24–110 GHz frequency range, which exhibit only splittings due to the O—CH3 torsion [11]. A spectral atlas of about 21 000 lines up to 200 GHz was prepared at the same time but the assigned lines were limited. To clarify the unassigned lines, which are also useful for astronomical identification, we started a series of the studies on the torsionally excited states. The first skeletal torsionally excited state was reported recently [12]. The present paper is the second in our series of studies for the low-lying torsionally excited states of transethyl methyl ether, and an analysis, including splittings due to the two inequivalent internal rotors, of the m29 ¼ 1 excited torsional state will be given. 2. Experiment
* Corresponding author. Fax: +81 76 445 6549. E-mail address: [email protected] (K. Kobayashi).
The microwave absorption spectrum of trans-ethyl methyl ether was measured in the frequency range of 24–110 GHz using
0022-2852/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2009.03.012
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j1 >;
j2 >¼ ð123Þj1 >;
j5 >¼ ð465Þj1 >;
j8 >¼ ð132Þð456Þj1 >;
Fig. 1. Molecular structure of trans-ethyl methyl ether. The arrow shows the O—CH3 torsion.
conventional Stark- and source-modulation microwave spectrometers, whose details are given in Ref. [13]. Part of the spectrum was recorded both at room temperature and at dry ice temperature in order to make easier line assignments based on intensities relative to those of the corresponding ground state spectral lines. The spectral lines measured at dry ice temperature were stronger since the sample pressure was kept the same as that at room temperature and lower energy states were more populated. The rest of the data were taken from the spectral atlas.
The present analysis for ethyl methyl ether with two inequivalent methyl internal rotors was made by means of the tunneling matrix formulation (TMF), which is valid when the barrier to internal rotation is high. Although this formalism, developed first by Hougen [14] for the analysis of the high resolution spectrum of the hydrazine molecule, has been used for the analyses of various molecules with large amplitude vibrations, for example, in the study on N,N-dimethyl acetamide [15] where three inequivalent methyl internal rotors were treated, we will describe below the outline of the formalism applied to the inequivalent-two-top internal rotation problem of ethyl methyl ether.
j4 >¼ ð456Þj1 >; j7 >¼ ð123Þð465Þ;
j9 >¼ ð132Þð465Þj1 >;
ð1Þ
where (123) and (456) are feasible operations corresponding to 120° internal rotation of the O—CH3 top and the C—CH3 top, respectively. 3.3. Symmetrized vibrational wavefunctions It is convenient to use symmetrized vibrational wavefunctions as basis functions when setting up the Hamiltonian matrix. Assuming ð23Þð56Þð78Þ j1 >¼ j1 > for the (mO—CH3 , mC—CH3 Þ ¼ ð1; 0Þ state (the operation (23)(56)(78)* corresponds to the reflection in the plane of symmetry, and can be seen to be feasible by subsequently performing a 180° over all rotation of the molecule around the axis perpendicular to the plane of symmetry.), the nine framework functions defined in Eqs. (1) produce the following symmetrized vibrational wavefunctions:
jA2 >¼ 3. Theoretical formulation
j3 >¼ ð132Þj1 >;
j6 >¼ ð123Þð456Þj1 >;
pffiffiffiffiffiffiffiffiffiffiffiffiffi NðA2 Þ½1; 1; 1; 1; 1; 1; 1; 1; 1;
ð2Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðEAÞ½1; x ; x; 1; 1; x ; x ; x; x; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jEAb >¼ NðEAÞ½1; x; x ; 1; 1; x; x; x ; x ;
jEAa >¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðAEÞ½1; 1; 1; x ; x; x ; x; x ; x; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jAEb >¼ NðAEÞ½1; 1; 1; x; x ; x; x ; x; x ;
jAEa >¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðEE1Þ½1; x ; x; x ; x; x; 1; 1; x ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jEE1b >¼ NðEE1Þ½1; x; x ; x; x ; x ; 1; 1; x; jEE1a >¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðEE2Þ½1; x ; x; x; x ; 1; x; x ; 1; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jEE2b >¼ NðEE2Þ½1; x; x ; x ; x; 1; x ; x; 1; jEE2a >¼
3.1. Character table For the purpose of specifying the symmetry species labeling used in the present paper, we show in Table 1 the character table of the permutation-inversion group G18 for the ethyl methyl ether, which has Cs symmetry in its equilibrium configuration and possesses two inequivalent methyl groups exhibiting internal rotation. The character table is equivalent to that shown in Ref. [16] for the N-methylacetamide molecule.
where x ¼ expði2p=3Þ,
½a1 ; a2 ; a3 ; . . . . . . . . . ; a9 ¼
9 X
ai ji >;
ð3Þ
i¼1
and normalization factors NðCÞðC ¼ A2 ; EA; AE; EE1; EE2Þ are given by
NðA2 Þ ¼ ½9ð1 þ 2 < 1j2 > þ2 < 1j4 > þ2 < 1j6 > þ2 < 1j7 >Þ1 ;
3.2. Vibrational framework functions Vibrational wavefunctions called framework functions, which are localized at potential minima, are used in the tunneling matrix formalism. In the present case they are written as
NðEAÞ ¼ ½9ð1 < 1j2 > þ2 < 1j4 > < 1j6 > < 1j7 >Þ1 ; NðAEÞ ¼ ½9ð1 þ 2 < 1j2 > < 1j4 > < 1j6 > < 1j7 >Þ1 ; NðEE1Þ ¼ ½9ð1 < 1j2 > < 1j4 > < 1j6 > þ2 < 1j7 >Þ1 ; NðEE2Þ ¼ ½9ð1 < 1j2 > < 1j4 > þ2 < 1j6 > < 1j7 >Þ1 :
Table 1 Character table of the permutation-inversion group G18 for the ethyl methyl ether molecule.
A1 A2 EA AE EE1 EE2
E
2(123)
2(456)
2(123)(456)
2(123)(465)
9(23)(56)(78)*
1 1 2 2 2 2
1 1 1 2 1 1
1 1 2 1 1 1
1 1 1 1 1 2
1 1 1 1 2 1
1 1 0 0 0 0
1, 2, 3 denote the protons in the O-methyl group, and 4, 5, 6 are the protons in the C-methyl group. 7 and 8 denote two hydrogen atoms in the CH2 group.
ð4Þ (We put < 1 j 1 >¼ 1.) The overlap integrals < 1j2 >, < 1j4 > etc. are included explicitly in the normalization factors given in Eqs. (4) in the present formulation since the vibrational framework functions jn > ðn ¼ 1; 2; 3; . . . ; 9Þ are not orthogonal to each other. The problem associated with the nonzero overlap integrals was not treated in the original study of Hougen [14], but it was described rather generally in Ref. [17] of Ohashi and Hougen. Suffixes a and b in Eqs. (2) denote two components of the doubly degenerate symmetry species.
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3.4. Phenomenological Hamiltonian operator
3.5.3. AE species
In the present analysis we used the following phenomenological Hamiltonian operator in Watson’s S-reduced form:
< AEa j < J;KjHjJ; K 0 > jAEa >¼< AEb j < J; KjHjJ; K 0 > jAEb > ( X ½< 1jvk j1 > þ2 < 1jvk j2 > < 1jvk j4 > ¼ 9NðAEÞ
H ¼ AJ2z þ BJ2x þ CJ2y DJ J4 DJK J2 J 2z DK J 4z þ d1 J2 ðJ 2þ þ J 2 Þ
k pffiffiffi < 1jvk j6 > < 1jvk j7 > < J;KjRk jJ; K 0 > þ 3½< 1jqj4 >
þ d2 ðJ 4þ þ J 4 Þ þ HJK J4 J 2z þ HKJ J2 J 4z þ HK J 6z þ h1 J4 ðJ 2þ þ J2 Þ þ h2 J2 ðJ 4þ þ J 4 Þ þ h3 ðJ 6þ þ J 6 Þ þ l4 ðJ 8þ þ J 8 Þ þ i q þ qJ J2 þ qK J 2z þ qJK J2 J2z J z :
ð5Þ
In Eq. (5) coefficients A; B; C; DJ ; . . . ; q; qJ ; qK ; qJK are all operators depending on the variables describing internal rotations. Only terms which were necessary for obtaining a good fit in the least squares analysis are retained in Eq. (5).
þ < 1jqJ j4 > JðJ þ 1Þþ < 1jqK j4 > K 2 þ < 1jqJK j4 > JðJ þ 1ÞK 2 KdK;K 0 pffiffiffi þ 3½< 1jqj6 > þ < 1jqJ j6 > JðJ þ 1Þþ < 1jqK j6 > K 2 pffiffiffi þ < 1jqJK j6 > JðJ þ 1ÞK 2 KdK;K 0 3½< 1jqj7 > þ < 1jqJ j7 > JðJ þ 1Þ ) þ < 1jqK j7 > K 2 þ < 1jqJK j7 > JðJ þ 1ÞK 2 KdK;K 0 :
ð9Þ
3.5.4. EE1 species 3.5. Parameterized expressions for the Hamiltonian matrix elements We set up the Hamiltonian matrix with the Hamiltonian operator given in Eq. (5) and the basis functions jC >j J; K >, where jC > are the symmetrized vibrational wavefunctions given in Eqs. (2) and jJ; K > is a symmetric-top rotational eigenfunction. The expressions for the Hamiltonian matrix elements for each symmetry species are shown below, and for convenience, the Hamiltonian operator given in Eq. (5) was rewritten as follows
X
vk Rk ¼ AJ 2z þ BJ2x þ CJ 2y DJ J4 DJK J2 J 2z DK J 4z þ d1 J2 J 2þ þ J 2
k
þ d2 J 4þ þ J 4 þ HJK J4 J 2z þ HKJ J2 J 4z þ HK J 6z þ h1 J4 J 2þ þ J 2 þ h2 J2 J 4þ þ J 4 þ h3 J 6þ þ J 6 þ l4 J 8þ þ J 8 :
H¼
X
vk Rk þ i q þ qJ J2 þ qK J 2z þ qJK J2 J 2z J z ;
ð6Þ
< EE1a j < J;KjHjJ; K 0 > jEE1a >¼< EE1b j < J;KjHjJ;K 0 > jEE1b > ( X ½< 1jvk j1 > < 1jvk j2 > < 1jvk j4 > < 1jvk j6 > ¼ 9NðEE1Þ k pffiffiffi þ2 < 1jvk j7 > < J;KjRk jJ; K 0 > þ 3½< 1jqj2 > þ < 1jqJ j2 > JðJ þ 1Þ pffiffiffi þ < 1jqK j2 > K 2 þ < 1jqJK j2 > JðJ þ 1ÞK 2 KdK;K 0 þ 3½< 1jqj4 >
þ < 1jqJ j4 > JðJ þ 1Þþ < 1jqK j4 > K 2 þ < 1jqJK j4 > JðJ þ 1ÞK 2 KdK;K 0 pffiffiffi 3½< 1jqj6 > þ < 1jqJ j6 > JðJ þ 1Þþ < 1jqK j6 > K 2 ) þ < 1jqJK j6 > JðJ þ 1ÞK 2 KdK;K 0 :
ð10Þ
3.5.5. EE2 species
< EE2a j < J; KjHjJ; K 0 > jEE2a >¼< EE2b j < J; KjHjJ; K 0 > jEE2b > ( X ½< 1jvk j1 > < 1jvk j2 > < 1jvk j4 > ¼ 9NðEE2Þ k pffiffiffi þ 2 < 1jvk j6 > < 1jvk j7 > < J; KjRk jJ; K 0 > þ 3½< 1jqj2 >
k
where vk and Rk are used as general representatives of rotational constants and the operators of the pure rotation part. 3.5.1. A1 , A2 species Since the Wang-type rotational functions are not used in the present paper, A1 and A2 symmetry species are treated together to obtain
X < A2 j < J; KjHjJ; K > jA2 >¼ 9NðA1 Þ ½< 1jvk j1 > 0
k
þ 2 < 1jvk j2 > þ2 < 1jvk j4 > þ2 < 1jvk j6 > þ2 < 1jvk j7 > < J; KjRk jJ; K 0 > :
ð7Þ
3.5.2. EA species
< EAa j < J; KjHjJ;K 0 > jEAa >¼< EAb j < J; KjHjJ; K 0 > jEAb > ( X ½< 1jvk j1 > < 1jvk j2 > þ2 < 1jvk j4 > < 1jvk j6 > ¼ 9NðEAÞ
þ < 1jqJ j2 > JðJ þ 1Þþ < 1jqK j2 > K 2 þ < 1jqJK j2 > JðJ þ 1ÞK 2 KdK;K 0 pffiffiffi 3½< 1jqj4 > þ < 1jqJ j4 > JðJ þ 1Þþ < 1jqK j4 > K 2 pffiffiffi þ < 1jqJK j4 > JðJ þ 1ÞK 2 KdK;K 0 3½< 1jqj7 > þ < 1jqJ j7 > JðJ þ 1Þ ) þ < 1jqK j7 > K 2 þ < 1jqJK j7 > JðJ þ 1ÞK 2 KdK;K 0 :
ð11Þ
The quantities < 1jvk jn > ðn ¼ 1; 2; 4; 6; 7Þ and < 1jqjm > ðm ¼ 2; 4; 6; 7Þ appearing in Eqs. (7)–(11) are called tunneling matrix elements (n ¼ 1 corresponds to the non-tunneling process). Tunneling matrix elements < 1jvk jn > ðn ¼ 3; 5; 8; 9Þ and < 1jqjm > ðm ¼ 1; 3; 5; 8; 9Þ do not appear in Eqs. (7)–(11) because of the following relations among tunneling matrix elements:
< 1jvk j2 > ¼< 1jvk j3 >; < 1jvk j4 >¼< 1jvk j5 >; < 1jvk j6 >¼< 1jvk j9 >; < 1jvk j7 >¼< 1jvk j8 >; < 1jqj1 >¼ 0; < 1jqj2 >¼ < 1jqj3 >; < 1jqj4 >¼ < 1jqj5 >< 1jqj6 >¼ < 1jqj9 >; < 1jqj7 >¼ < 1jqj8 > :
ð12Þ
k
pffiffiffi < 1jvk j7 > < J; KjRk jJ; K 0 > þ 3½< 1jqj2 > þ < 1jqJ j2 > JðJ þ 1Þ pffiffiffi þ < 1jqK j2 > K 2 þ < 1jqJK j2 > JðJ þ 1ÞK 2 KdK;K 0 þ 3½< 1jqj6 > þ < 1jqJ j6 > JðJ þ 1Þþ < 1jqK j6 > K 2 þ < 1jqJK j6 > JðJ þ 1ÞK 2 KdK;K 0 pffiffiffi þ 3½< 1jqj7 > þ < 1jqJ j7 > JðJ þ 1Þþ < 1jqK j7 > K 2 ) þ < 1jqJK j7 > JðJ þ 1ÞK 2 KdK;K 0 :
ð8Þ
3.6. Note on the least squares analysis with Eqs. (7)–(11) Since the overlap integrals < 1j2 >, < 1j4 > etc. between pairs of framework wavefunctions, which are undetermined quantities, are included in the expressions (7)–(11), tunneling matrix elements < 1jvk j1 >, < 1jvk j2 >, < 1jqj2 > etc. cannot be determined in any least squares analysis. Instead, vk11 ; vk12 ; vk14 ; vk16 ; vk17 ; q12 ; q14 ; q16 ; q17 etc. are used as adjustable parameters related to the tunneling matrix elements by the following relations:
K. Kobayashi et al. / Journal of Molecular Spectroscopy 255 (2009) 164–171
vk11 þ 2vk12 þ 2vk14 þ 2vk16 þ 2vk17 ¼ 9NðA2 Þ½< 1jvk j1 > þ2 < 1jvk j2 > þ2 < 1jvk j4 > þ 2 < 1jvk j6 > þ2 < 1jvk j7 >; vk11 vk12 þ 2vk14 vk16 vk17 ¼ 9NðEAÞ½< 1jvk j1 > < 1jvk j2 > þ2 < 1jvk j4 > < 1jvk j6 > < 1jvk j7 >;
vk11 þ 2vk12 vk14 vk16 vk17 ¼ 9NðAEÞ½< 1jvk j1 > þ2 < 1jvk j2 > < 1jvk j4 > < 1jvk j6 > < 1jvk j7 >; vk11 vk12 vk14 vk16 þ 2vk17 ¼ 9NðEE1Þ½< 1jvk j1 > < 1jvk j2 > < 1jvk j4 > < 1jvk j6 > þ2 < 1jvk j7 >;
vk11 vk12 vk14 þ 2vk16 vk17 ¼ 9NðEE2Þ½< 1jvk j1 > < 1jvk j2 >
and
q12 þ q16 þ q17 ¼ 9NðEAÞ½< 1jqj2 > þ < 1jqj6 > þ < 1jqj7 >; q14 þ q16 q17 ¼ 9NðAEÞ½< 1jqj4 > þ < 1jqj6 > < 1jqj7 >; q12 þ q14 q16 ¼ 9NðEE1Þ½< 1jqj2 > þ < 1jqj4 > < 1jqj6 >; q12 q14 q17 ¼ 9NðEE2Þ½< 1jqj2 > < 1jqj4 > < 1jqj7 >: ð14Þ (We have similar relations for qJ ; qK and qJK .) Since the overlap integrals < 1j2 >, < 1j4 > etc. are considered to be small, the parameters vk11 ; vk12 ; vk14 ; q12 ; q14 etc. are approximately equal to the tunneling matrix elements < 1jvk j1 >, < 1jvk j2 >, < 1jvk j4 >, < 1jqj2 >, < 1jqj4 > etc., respectively. Eqs. (7)–(11) for the Hamiltonian matrix elements are changed to obvious expressions involving the parameters vk11 ; vk12 ; vk14 ; vk16 ; vk17 ; q12 ; q14 ; q16 ; q17 etc. For example, Eq. (7) is changed to
X ½vk11 þ 2vk12 þ 2vk14 þ 2vk16 k
þ 2vk17 < J; KjRk jJ; K 0 > :
for C ¼ A2 ; EA; AE; EE1 and EE2. In relations (17), j ee >; j eo >; j oe > and j oo > represent rotational wavefunctions with K a ; K c ¼ even, even; even, odd; odd, even and odd, odd, respectively. In some rotational transitions obeying selection rules
EA $ EA; AE $ AE; EE1 $ EE1 and EE2 $ EE2 on the overall symmetry species, we observe ‘‘forbidden transitions”, which do not follow the selection rules shown in (17) above. Such forbidden transitions, as observed in the present investigation, were found to occur when the term iqJ z mixes two K-type doubling levels significantly in the upper and/or the lower state involved in the transition. 4. Analysis
< 1jvk j4 > þ2 < 1jvk j6 > < 1jvk j7 >; ð13Þ
< A2 j < J; KjHjJ; K 0 > jA2 >¼
167
ð15Þ
In the present analysis for the trans-ethyl methyl ether in the
m29 ¼ 1 excited torsional state, the 1 ! 6 and 1 ! 7 tunneling processes were neglected, and therefore we set < 1 j vk j 6 >¼ < 1 j vk j 7 >¼ 0, < 1 j q j 6 >¼< 1 j q j 7 >¼ 0, < 1 j qJ j 6 >¼< 1 j qJ j 7 >¼ 0, < 1 j qK j 6 >¼< 1 j qK j 7 >¼ 0, < 1 j qJK j 6 >¼< 1 j qJK j 7 >¼ 0, and accordingly, vk16 ¼ vk17 ¼ 0, q16 ¼ q17 ¼ 0, qJ16 ¼ qJ17 ¼ 0, qK16 ¼ qK17 ¼ 0, qJK16 ¼ qJK17 ¼ 0.
In the ground state, the splitting due to the O—CH3 internal rotor is larger than that of the C—CH3 internal rotor and only splitting due to the O—CH3 rotor was observed for low J, K transitions. In the m29 ¼ 1 state, larger splitting due to the O—CH3 methyl-internalrotation motion was expected. We found series of lines with reasonable intensity and splitting patterns. The spectrum observed at dry ice temperature, exhibiting the typical quartet pattern, is shown in Fig. 2. The splittings due to the two methyl-rotors were observed even at low J and K. The extension of line assignments up to high rotational quantum numbers J and K was made by using least-squares-fittings and frequency predictions based on the tunneling matrix formulation (TMF), whose outline was described in the preceding section. In total, 1507 lines up to J ¼ 76 and K ¼ 5 were assigned and part of the lines are listed in Table 2. In many cases the EE1 and EE2 species components of the internal rotation splitting pattern were overlapping as seen in Table 2, where the species labeling EE is used for the unresolved EE1 and EE2 species. In transitions with higher K quantum number, the EE1 and EE2 species were resolved. For the cases where the EE1 and EE2 transitions were unresolved, the overlapped transition frequencies were fit to the average values of the EE1 and EE2 transition frequencies. In the least squares analysis, all the lines were given weights of 1.0. Molecular parameters thus obtained are shown in Table 3. The root mean square deviation in the fit was 50 kHz. 5. Discussion Since the m29 ¼ 1 excited torsional state treated in the present study is of vðO—CH3 torsionÞ ¼ 1, vðC—CH3 torsionÞ ¼ 0, values of
3.7. Selection rules for electric dipole transitions Selection rules on tunneling-rotational symmetry species for electric dipole transitions are
A1 $ A2 ; EA $ EA; AE $ AE; EE1 $ EE1; EE2 $ EE2:
ð16Þ
From the overall selection rules shown in (16) above, and
< Cjla jC > –0;
< Cjlb jC > –0; C ¼ A2 ; EA; AE; EE1 and EE2
< Cjlc jC >¼ 0 for
(note that ð23Þð56Þð78Þ ðj1 >; j2 >; j4 >; j6 >; j7 >Þ ¼ ðj1 >; j3 >; j5 >; j9 >; j8 >Þ and ð23Þð56Þð78Þ ½la ; lb ; lc ¼ ½la ; lb ; lc ), we see that the allowed transitions are
jC > jee >$ jC > jeo >;
jC > joe >$ jC > joo >;
jC > jee >$ jC > joo >;
jC > jeo >$ jC > joe >;
ð17Þ
Fig. 2. The 1019 10010 transition of trans-ethyl methyl ether in the first O—CH3 torsionally excited state. The EE1 and EE2 transitions were overlapped in this rotational transition.
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K. Kobayashi et al. / Journal of Molecular Spectroscopy 255 (2009) 164–171
Table 2 Assigned transitions, observed transition frequencies (in MHz), and observed minus calculated residuals (in MHz) of trans-ethyl methyl ether in the first state. Upper state J
0
Ka
31 32 33 24 25 26 27 28 12 13 14 15 16 6 7 8 9 10 11 19 20 21 22 23 24 25 26 27 28 32 33 34 35 13 14 15 16 17 18 23 24 25 26 27 28 33 34 35 36 37 38 6 7 9 10 24 25 26 27 28 32 33 34 35 36 4 4 47 48 49 50 51 a b
2 2 2 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5
0
Lower state Kc
0
29 30 31 23 24 25 26 27 12 13 14 15 16 5 6 7 8 9 10 19 20 21 22 23 22 23 24 25 26 31 32 33 34 12 13 14 15 16 17 20 21 22 23 24 25 31 32 33 34 35 36 4 5 7 8 20 21 22 23 24 29 30 31 32 33 1 0 42 43 44 45 46
J
00
30 31 32 23 24 25 26 27 11 12 13 14 15 6 7 8 9 10 11 18 19 20 21 22 24 25 26 27 28 32 33 34 35 12 13 14 15 16 17 23 24 25 26 27 28 33 34 35 36 37 38 5 6 8 9 24 25 26 27 28 32 33 34 35 36 3 3 47 48 49 50 51
Ka
3 3 3 2 2 2 2 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4
Overlapped lines. Weight was set to 0.
00
AA Kc
00
28 29 30 22 23 24 25 26 11 12 13 14 15 6 7 8 9 10 11 18 19 20 21 22 23 24 25 26 27 32 33 34 35 11 12 13 14 15 16 21 22 23 24 25 26 32 33 34 35 36 37 3 4 6 7 21 22 23 24 25 30 31 32 33 34 0 0 43 44 45 46 47
AE
Observed frequency
Obs.-calc.
168 966.353 180 303.783 191 703.278 155 731.227 166 024.283 176 290.505 186 514.575 196 681.568 79 578.219 88 583.116 97 579.853 106 553.474 115 490.887 267 50.342 27 770.896 28 968.814 30 356.224 31 945.650 33 750.210 158 416.097 165 316.205 172 275.371 179 296.845 186 381.949 62 831.063 64 214.630 65 890.419 67 868.316 70 156.627 143 054.611 147 301.259 151 627.594 156 028.196 165 850.207 172 223.261 178 483.773 184 636.785 190 687.902 196 644.112 105 667.978 104 019.953 102 354.959 100 700.086 99 083.870 97 535.730 143 210.880 145 671.724 148 272.991 151 014.035 153 893.050 156 908.311 167 449.512 175 441.272 191 319.505 199 183.180a 163 442.271 162 705.905 161 855.522 160 882.110 159 777.398 167 943.187 168 385.571 168 908.334 169 518.209 170 221.872 199 119.123a
0.075 0.032 0.064 0.045 0.080 0.066 0.005 0.036 0.103 0.043 0.102 0.022 0.063 0.017 0.068 0.021 0.000 0.010 0.017 0.022 0.008 0.008 0.007 0.005 0.011 0.037 0.031 0.005 0.005 0.018 0.021 0.066 0.042 0.001 0.007 0.015 0.075 0.025 0.013 0.01 0.004 0.002 0.004 0.005 0.006 0.037 0.039 0.039 0.054 0.030 0.015 0.052 0.025 0.065 0.002 0.037 0.014 0.027 0.060 0.024 0.094 0.022 0.002 0.033 0.029 0.011
191 833.033 189 409.323 186 853.782 184 181.432 181 411.596
0.050 0.021 0.094 0.002 0.005
Observed frequency
Obs.-calc.
168 962.102 180 299.660 191 699.341 155 728.921 166 022.187a 176 288.468 186 512.688 196 679.819 79 577.182 88 582.146 97 578.924 106 552.673 115 490.149 267 51.859 27 772.400 28 970.362 30 357.813 31 947.270 33 751.905 158 416.885 165 316.923 172 276.024 179 297.493 186 382.550 62 833.880 64 217.461 65 893.205 67 871.039 70 159.339 143 060.516 147 307.325 151 633.785 156 034.487 165 853.891 172 226.888 178 487.290 184 640.232 190 691.327 196 647.599a 105 673.486 104 025.339 102 360.247 100 705.221 99 088.868 97 540.591 143 216.428 145 677.280 148 278.472a 151 019.561 153 898.658 156 914.020 167 442.855b 175 435.458 191 317.238 199 183.180a 163 450.544 162 713.826 161 863.158 160 889.439 159 784.606 167 949.621 168 391.884 168 914.561 169 524.347 170 227.952
0.028 0.020 0.037 0.024 0.139 0.053 0.008 0.042 0.055 0.003 0.043 0.028 0.070 0.049 0.059 0.018 0.007 0.015 0.006 0.014 0.015 0.005 0.055 0.070 0.018 0.006 0.005 0.009 0.014 0.034 0.001 0.050 0.059 0.001 0.009 0.028 0.032 0.013 0.143 0.004 0.002 0.024 0.006 0.009 0.006 0.004 0.014 0.134 0.004 0.002 0.105 0.278 0.035 0.063 0.038 0.052 0.009 0.005 0.015 0.003 0.021 0.001 0.008 0.031 0.001
199 129.076 191 838.531 189 414.617 186 858.881 184 186.344 181 416.306
0.007 0.084 0.020 0.076 0.014 0.025
EA
EE
Observed frequency Obs.-calc.
EE1 Observed frequency
168 893.102 180 231.543
0.000 0.021
EE2 Obs.-calc.
180 227.515 0.100 191 628.216a,b 0.284 155 689.151 165 983.697 176 251.447 186 477.342 196 646.037 79 561.659 88 567.240 97 564.624 106 539.114 115 477.381 26 771.905 27 792.532 28 990.662 30 378.120 31 967.742 33 772.481 158 429.089 165 328.335 172 286.715 179 307.353 186 391.564 62 871.446 64 253.499 65 927.776 67 904.173 70 191.023 143 127.944 147 375.758 151 703.469 156 105.264
155 691.462 165 985.862 176 253.515 186 479.235 196 647.599a,b 79 562.656 88 568.189 97 565.511 106 539.913 115 478.140 26 770.499 27 791.068 28 989.052 30 376.538 31 966.097 33 770.796 158 428.331 165 327.623 172 286.026 179 306.696 186 391.045 62 868.601 64 250.740 65 925.015 67 901.400 70 188.305 143 122.012 147 369.725 151 697.265 156 099.060 165 897.939 172 271.632 178 532.472 184 685.387 190 736.439 196 692.318 105 756.316 104 105.554 102 438.094 100 780.942 99 162.478 97 612.092 143 282.384 145 742.811 148 343.739 151 084.208 153 962.94 156 977.797 167 294.306
0.03 0.003 0.029 0.007 0.237 0.043 0.047 0.047 0.049 0.054 0.022 0.106 0.034 0.040 0.042 0.036 0.054 0.075 0.047 0.063 0.033 0.004 0.012 0.016 0.035 0.013 0.016 0.022 0.014 0.050 0.095 0.100 0.043 0.091 0.065 0.031 0.019 0.015 0.007 0.011 0.005 0.002 0.011 0.000 0.015 0.040 0.006 0.008 0.089
199 103.476 163 643.470 162 892.640 162 027.215 161 039.682 159 922.675a,b 168 018.015 168 463.111 168 987.519 169 598.476 170 302.642
0.089 0.008 0.051 0.025 0.118 0.165 0.031 0.073 0.018 0.012 0.003
199 094.215 163 661.791 162 910.270 162 043.749
0.048 0.068 0.026 0.034
159 936.159 168 021.906 168 467.486 168 992.295 169 603.463 170 307.706
0.016 0.009 0.088 0.044 0.011 0.071
199 242.647 191 937.304 189 511.754 186 954.459 184 280.472 181 508.819
0.027 0.130 0.041 0.051 0.016 0.040
199 252.593 191 943.655b 189 517.678 186 960.095 184 285.851 181 513.948
178 535.560
m29 torsionally excited
0.107
190 739.641 0.026 196 695.527 0.047 105 763.097 0.024 104 111.931 0.029 102 444.100 0.018 100 786.638 0.012 99 167.916 0.012 97 617.299 0.004 143 288.020 145 748.419 148 349.313 151 089.787 153 968.486 156 983.340
0.008 0.286 0.103 0.131 0.125 0.095
Observed frequency
Obs.-calc.
191 628.216a 0.056 0.006 0.073 0.004 0.043 0.051 0.065 0.064 0.040 0.040 0.057 0.058 0.098 0.039 0.063 0.049 0.048 0.058 0.002 0.008 0.050 0.002 0.030 0.016 0.011 0.028 0.005 0.035 0.042 0.054 165 902.324
0.102
178 536.395 184 689.202
0.088 0.083
105 760.572 104 109.924 102 442.657 100 785.487 99 167.008 97 616.605 0.036 0.027 0.012 0.037 0.028 0.037
0.027 0.037 0.041 0.009 0.001 0.031
199 124.929a 163 639.527 162 889.276 162 025.068 161 038.789 159 922.675a,b 168 026.901 168 471.280 168 995.232 169 605.838 170 309.577
0.092 0.082 0.018 0.046 0.135 0.276 0.010 0.010 0.027 0.041 0.036
199 248.443 191 941.936 189 516.426 186 959.085 184 284.959 181 513.261
0.029 0.030 0.015 0.000 0.085 0.063
0.090
169
K. Kobayashi et al. / Journal of Molecular Spectroscopy 255 (2009) 164–171 Table 3 Molecular parametersa,b (in MHz) of trans-ethyl methyl ether in the Non-tunneling parameters A11 B11 C 11 DJ11 DJK11 DK11 d1;11 d2;11
27 882.4722(17) 4153.24850(11) 3888.28791(11) 0.000972717(39) 0.0015575(20) 0.03535(16) 0.0000875604(77) 0.0000036881(14)
HKJ11 HK11 h1;11
0.000002136(87) 0.0001240(63) 0.0000000000379(15)
h3;11 l4;11
0.00000000000827(38) 0.000000000000001352(86)
a b
m29 ¼ 1 excited torsional state.
O—CH3 internal rotation tunneling parameters
C—CH3 internal rotation tunneling parameters
A12 B12
6.40153(98) 0.019241(14)
A14 B14
0.47064(85) 0.002954(15)
DJK12 DK12 d1;12 d2;12 HJK12 HKJ12 HK12
0.0004907(10) 0.065209(94) 0.0000001318(17) 0.00000041033(74) 0.00000001295(21) 0.000003902(50) 0.0002526(35)
DJK14 DK14 d1;14 d2;14 HJK14 HKJ14
0.00007361(78) 0.003750(38) 0.0000000447(20) 0.0000000703(15) 0.00000000083(20) 0.000000567(30)
h2;14
0.00000000000586(57)
h3;12
0.00000000001002(21)
q12 qJ12 qK12 qJK12
35.9747(34) 0.003277(12) 0.64265(17) 0.00006767(57)
q14 qJ14 qK14
2.8510(29) 0.0003187(34) 0.04413(16)
Numbers in parentheses are one standard uncertainty (1r) in units of the least significant digits. Note that A1n < 1jAjn >; B1n < 1jBjn > etc. (for n ¼ 1; 2; 4) as mentioned in the text.
A14 , B14 , q14; which are the parameters related to the C—CH3 internal rotation tunneling, are expected to be as small as those for the ground state. However, values of the parameters A14 , B14 and q14 determined are rather large compared to those for the ground state. (According to the new analysis made by using previously observed transition frequencies [11] for the ground state, A14 ¼ 0:01871ð52Þ MHz, B14 ¼ 0:000106ð19Þ MHz, and q14 is undetermined since it is too small.) Furthermore, contrary to the prediction that the parameters A14 and B14 take positive values as in the ground state, they were found to have negative values, which produce the inverted A/E sequence pattern regarding the C—CH3 internal rotation tunneling splitting. This inversion was further confirmed by the splitting of the EE sublevel. To understand the fact mentioned above quantitatively, we tried to carry out two evaluations described below, assuming that it is produced mainly by a kinetic O—CH3 —C—CH3 coupling as represented by the term 2F 12 pO—CH3 pC—CH3 in the internal rotation Hamiltonian. The torsional barrier heights and the molecular parameters related to the internal rotation splittings were estimated in Evaluations 1 and 2, respectively. In the present discussion, we neglect the interaction scheme involving the skeletal torsionally excited state because the energy difference between O—CH3 and C—CH3 torsionally excited states is much smaller, even though it was suggested in the previous study [12] on the skeletal torsionally excited state that the inverted A/E sequence seen in the first skeletal torsionally excited state would be produced by the O—CH3 torsion-skeletal torsion and C—CH3 torsion-skeletal torsion interactions. Evaluation 1. We estimated values of barrier heights V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ to the internal rotations by combining the Coriolislike parameters q12 and q14 obtained from the spectral analysis with the relations obtained from Principle Axis Method (PAM) first order perturbation theory [18]
pffiffiffi 3q12 ¼ j < mO—CH3 ¼ 1; mC—CH3 ¼ 0; EAjP a jmO—CH3 ¼ 1;
mC—CH3 ¼ 0; EA > j; pffiffiffi 3q14 ¼ j < mO—CH3 ¼ 1; mC—CH3 ¼ 0; AEjP a jmO—CH3 ¼ 1;
mC—CH3 ¼ 0; AE > j;
where
Pa ¼ 2F O—CH3 qa ðO—CH3 ÞpO—CH3 2F C—CH3 qa ðC—CH3 ÞpC—CH3 2F 12 qa ðC—CH3 ÞpO—CH3 2F 12 qa ðO—CH3 ÞpC—CH3 ;
ð18Þ
and the internal rotation Hamiltonian operator including the kinetic O—CH3 —C—CH3 coupling term
H ¼ F O—CH3 p2O—CH3 þ F O—CH3 p2C—CH3 þ 2F 12 pO—CH3 pC—CH3 1 1 þ V 3 ðO—CH3 Þ½1 cosð3sO—CH3 Þ þ V 3 ðC—CH3 Þ 2 2 ½1 cosð3sC—CH3 Þ:
ð19Þ
In Eqs. (18),
ka ðO—CH3 ÞIa ðO—CH3 Þ ; Ia ðmolÞ k ðC—CH3 ÞIa ðC—CH3 Þ qa ðC—CH3 Þ ¼ a Ia ðmolÞ
qa ðO—CH3 Þ ¼
where ka ðO—CH3 Þ: the a-axis component of direction cosine of the O—CH3 internal rotation axis, ka ðO—CH3 Þ: the a-axis component of direction cosine of the C—CH3 internal rotation axis. In Table 4, we give values of quantities required for the determination of the barrier heights using Eqs. (18) and (19). Those values were obtained from geometrical parameters given in the ab initio study made by Durig et al. [7] The barrier heights, V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ thus obtained for the internal rotations are given in Table 5, where the values of q12 and q14 reproduced with Eqs. (18) are shown together with those obtained in the least squares analysis of the spectrum. The barrier heights including those obtained in the previous studies [6,8,12] are compiled in Table 6. Our barrier heights agree with the IR results [6]. A relatively large discrepancy of V 3 ðC—CH3 Þ to the one obtained in the MW study [8] was noted. It could be because the barrier height in the previous study was determined for an isotopomer. Evaluation 2. We evaluated parameters A12 , A14 , B12 and B14 , which are related with contributions from internal rotation tunnel-
170
K. Kobayashi et al. / Journal of Molecular Spectroscopy 255 (2009) 164–171
Table 4 Quantities required for determination of the barrier heights V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ with the use of Eqs. (18) and (19). F O—CH3 ¼ 6:4041 cm1 F 12 ¼ 1:1545 cm1
F C—CH3 ¼ 6:3618 cm1
qa ðO—CH3Þ ¼ 0:1678 qb ðO—CH3Þ ¼ 0:00942
qa ðC—CH3 Þ ¼ 0:1584 qb ðC—CH3 Þ ¼ 0:01198
Table 5 Barrier heights V 3 ðO—CH3 Þ, V 3 ðC—CH3 Þ, and Coriolis-like parameters q12 , q14 used for the determination of the barrier heights. V 3 ðO—CH3 Þ ¼ 881:5 cm1 V 3 ðC—CH3 Þ ¼ 1070:1 cm1
q12 (MHz) q14 (MHz)
Calc.
OC
35.9747 2.8510
35.9632 2.8673
0.0115 0.0163
ings to the rotational constants A and B, by using the barrier heights V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ obtained in Evaluation 1 and the following relations obtained from PAM second order perturbation theory [18].
3A14 ¼
2
2
X j < v 1 ¼ 1; v 2 ¼ 0; AEjPa jv 0 ; v 0 ; AE > j2 0 1 2 ; 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; AEÞ Eðv 1 ; v 2 ; AEÞ 1
3B12 ¼
2
X j < v 1 ¼ 1; v 2 ¼ 0; A2 jPb jv 0 ; v 0 ; A1 > j2 1 2 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; A2 Þ Eðv 1 ; v 2 ; A1 Þ 1
2
X j < v 1 ¼ 1; v 2 ¼ 0; EAjPb jv 0 ; v 0 ; EA > j2 0 1 2 ; 0 Eð v ¼ 1; v ¼ 0; EAÞ Eð v ; v 02 ; EAÞ 1 2 0 0 1 v ;v 1
3B14 ¼
2
X j < v 1 ¼ 1; v 2 ¼ 0; A2 jPb jv 0 ; v 0 ; A1 > j2 1 2 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; A2 Þ Eðv 1 ; v 2 ; A1 Þ 1
2
X j < v 1 ¼ 1; v 2 ¼ 0; AEjPb jv 0 ; v 0 ; AE > j2 0 1 2 ; 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; AEÞ Eðv 1 ; v 2 ; AEÞ 1
ð20Þ
2
where v 1 , v 1 ’ and v 2 , v 2 ’ denote the O—CH3 torsional and the C—CH3 torsional quantum numbers, respectively, and P b is defined as Table 6 Comparison of the barrier heights.
V 3 ðO—CH3 Þ (cm1 Þ V 3 ðC—CH3 Þ (cm1 Þ
6.32290 0.47563 0.019918 0.002722
10.4 0.0396 0.0327 0.000227
6.40153 0.47064 0.019241 0.002954
v 29 ¼ 1 excited torsional state
The ground state
The
EðEAÞ EðA1 Þ ¼ 7:8 EðAEÞ EðA1 Þ ¼ 1:7 EðEE1Þ EðA1 Þ ¼ EðEE2Þ EðA1 Þ ¼ 9:5
EðEAÞ EðA2 Þ ¼ 307:0 EðAEÞ EðA2 Þ ¼ 25:9 EðEE1Þ EðA2 Þ ¼ EðEE2Þ EðA2 Þ ¼ 332:9
Pb ¼ 2F O—CH3 qb ðO—CH3 ÞpO—CH3 2F C—CH3 qb ðC—CH3 ÞpC—CH3 2F 12 qb ðC—CH3 ÞpO—CH3 2F 12 qb ðO—CH3 ÞpC—CH3
ð21Þ
kb ðO—CH3 ÞIb ðO—CH3 Þ ; Ib ðmolÞ k ðC—CH3 ÞIb ðC—CH3 Þ qb ðC—CH3 Þ ¼ b : Ib ðmolÞ
X j < v 1 ¼ 1; v 2 ¼ 0; A2 jP a jv 0 ; v 0 ; A1 > j2 1 2 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; A2 Þ Eðv 1 ; v 2 ; A1 Þ 1
Values determined in the least squares spectral analysis
qb ðO—CH3 Þ ¼
2
X j < v 1 ¼ 1; v 2 ¼ 0; EAjPa jv 0 ; v 0 ; EA > j2 0 1 2 ; 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; EAÞ Eðv 1 ; v 2 ; EAÞ 1
(MHz) (MHz) (MHz) (MHz)
Values calculated from Eqs. (20) with F 12 ¼ 0
In Eq. (21),
X j < v 1 ¼ 1; v 2 ¼ 0; A2 jP a jv 0 ; v 0 ; A1 > j2 1 2 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; A2 Þ Eðv 1 ; v 2 ; A1 Þ 1
A12 A14 B12 B14
Values calculated from Eqs. (20)
Table 8 A/E energy differences (in MHz) in the J ¼ K ¼ 0 level.
Obs.
Obs. and Calc. denote values obtained in the least squares analysis of the spectrum and those calculated from Eqs. (18), respectively.
3A12 ¼
Table 7 Comparison between A12 , A14 , B12 and B14 calculated from Eqs. (20) and those determined in the least squares analysis of the spectrum.
MW (m29 ¼ 1) This study
MW (m30 ¼ 1) Ref. [12]
MW (ground state)a Ref. [8]
IR
881.5 1070.1
858.02 —b
893 1154
913.9 1076.4
From the evaluation by using Eqs. (20) with the barrier heights V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ determined above, we obtained values for the tunneling parameters A12 , A14 , B12 and B14 as shown in Table 7, where it is seen that they agree in sign and closely in magnitude with those for values obtained from the least squares analysis of the spectrum. In Table 7, values for the tunneling parameters obtained without the kinetic coupling of the two torsional motions ( F 12 ¼ 0) are also shown for comparison, which suggests that if F 12 ¼ 0, then A14 and B14 have positive values, and the inverted A/E sequence pattern regarding the C—CH3 internal rotation does not occur. The coupling of the torsions is also mentioned in the previous studies [6,8]. The evaluations made above supports the assumption that the inverted A/E sequence pattern regarding the C—CH3 internal rotation is produced mainly by the kinetic coupling between the O—CH3 and the C—CH3 internal rotations. In Table 8, we show A/E energy differences in the J ¼ K ¼ 0 level of the ground and the v 29 ¼ 1 excited torsional states, which are evaluated with the internal rotation Hamiltonian given in Eq. (19) and F O—CH3 , F C—CH3 , F 12 , V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ values in Tables 4 and 5. It is seen that the inverted A/E order regarding the C—CH3 internal rotation occurs also in the energy levels of the v 29 ¼ 1 excited torsional state. (Note that EðAEÞ EðA2 Þ < 0 and EðEE1orEE2Þ EðEAÞ < 0 in the v 29 ¼ 1 excited torsional state.) Acknowledgments
Ref. [6]
a The V 3 ðO—CH3 Þ was obtained for CH3 CH2 OCH3 while the V 3 ðC—CH3 Þ was obtained for CH3 CH2 OCD3 . b The V 3 ðC—CH3 Þ was not obtained since the torsional splitting was not observed for the m30 ¼ 1 torsional level.
K.K. would like to thank N. Mori and Y. Hori for helping the microwave measurement. This study was partially supported by Grant-in-Aid for Young Scientists (B) by the Ministry of Education, Culture, Sports, Science and Technology of Japan (Grant No. 20740103) and National Astronomical Observatory of Japan.
K. Kobayashi et al. / Journal of Molecular Spectroscopy 255 (2009) 164–171
Appendix A. Supplementary data Supplementary data for this article are available on Science Direct (www.sciencedirect.com) and as part of the Ohio State University Molecular Spectroscopy Archives (http://library.osu. edu/sites/msa/jmsa_hp.htm). Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.jms.2009.03.012. References [1] G.W. Fuchs, U. Fuchs, T.F. Giesen, F. Wyrowski, Astronom. Astrophys. 444 (2005) 521–530. [2] F.J. Lovas, R.D. Suenram, L.E. Snyder, J.M. Hollis, R.M. Lees, Astrophys. J. 253 (1982) 149–153. [3] D.M. Mehringer, J.C. Pearson, J. Keene, T.G. Phillips, Astrophys. J. 608 (2004) 306–313. [4] K. Kobayashi, K. Ogata, S. Tsunekawa, S. Takano, Astrophys. J. 657 (2007) L17– L19.
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[5] K. Demyk, G. Wlodarczak, M. Carvajal, Astron. Astrophys. 489 (2008) 589– 600. [6] J.R. Durig, D.A.C. Compton, J. Chem. Phys. 69 (1978) 4713–4719. [7] J.R. Durig, Y. Jin, H.V. Phan, J. Liu, D.T. Durig, Struct. Chem. 13 (2002) 1–26. [8] M. Hayashi, K. Kuwada, J. Mol. Struct. 28 (1975) 147–161. [9] M. Hayashi, M. Adachi, J. Mol. Struct. 78 (1982) 53–62. [10] U. Fuchs, G. Winnewisser, P. Groner, F.C. De Lucia, E. Herbst, Astrophys. J. Suppl. 144 (2003) 277–286. [11] S. Tsunekawa, Y. Kinai, Y. Kondo, H. Odashima, K. Takagi, Molecules 8 (2003) 103–119. [12] K. Kobayashi, T. Matsui, N. Mori, S. Tsunekawa, N. Ohashi, J. Mol. Spectrosc. 251 (2008) 301–304. [13] Y. Fukuyama, H. Odashima, K. Takagi, S. Tsunekawa, Astrophys. J. Suppl. 104 (1996) 329–346. [14] J.T. Hougen, J. Mol. Spectrosc. 89 (1981) 296–327. [15] M. Fujitake, Y. Kubo, N. Ohashi, J. Mol. Spectrosc. 236 (2006) 97–109. [16] N. Ohashi, J.T. Hougen, R.D. Suenram, F.J. Lovas, Y. Kawashima, M. Fujitake, J. Pyka, J. Mol. Spectrosc. 227 (2004) 28–42. [17] N. Ohashi, J.T. Hougen, J. Mol. Spectrosc. 112 (1985) 384–400. [18] The PAM perturbation theory for the one-top problem is given in: W. Gordy, R.L. Cook, Microwave Molecular Spectra, John Wiley & Sons, 1984, pp. 587, for example. The PAM perturbation theory for the two-top problem is obtained in the same way as for the one-top problem.
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The microwave spectroscopy of trans-ethyl methyl ether in the torsional state
m29 ¼ 1 excited
Kaori Kobayashi a,*, Takanori Matsui a, Shozo Tsunekawa a, Nobukimi Ohashi b a b
Department of Physics, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan Kanazawa University, Kakuma, Kanazawa 920-1192, Japan
a r t i c l e
i n f o
Article history: Received 26 December 2008 In revised form 19 March 2009 Available online 5 April 2009 Keywords: trans-Ethyl methyl ether Methyl internal rotation Microwave spectroscopy Barrier height to internal rotation
a b s t r a c t The trans-ethyl methyl ether molecule has two inequivalent methyl group internal rotors. One methyl rotor is bonded to the oxygen atom and the other to the carbon atom. The internal rotations of these methyl rotors correspond to the vibrational modes, m29 and m28 , respectively. In this study, the microwave absorption spectrum in the m29 ¼ 1 excited torsional state was analyzed for the first time. Initial assignment has been carried out by making use of measurements at dry ice temperature. Over 1500 lines up to J ¼ 76 and K ¼ 5 were assigned. The spectrum involving internal rotation splittings was analyzed by using the tunneling matrix formulation developed by Hougen. Crown Copyright Ó 2009 Published by Elsevier Inc. All rights reserved.
1. Introduction The trans-ethyl methyl ether molecule is an interstellar molecule identified in W51e2 [1]. It is becoming common to detect torsionally excited saturated organic molecules like CH3 OH [2], C2 H5 CN [3], and HCOOCH3 [4,5] in the star-forming regions. These molecules are recently called ‘‘weed” because huge numbers of lines are observed everywhere. Although the intensity of transethyl methyl ether was not very strong, future progress of the highly sensitive new radio telescopes could make it possible to detect this molecule in the torsionally excited state. Precise transition frequencies are indispensable for detection. The structure of trans-ethyl methyl ether (CH3 CH2 OCH3 ) is shown in Fig. 1. One methyl group is bonded to the oxygen atom (abbreviated as O—CH3 ) and the other is bonded to the carbon atom (abbreviated as C—CH3 ). The vibrational frequencies have been studied by mid-resolution infrared spectroscopy [6,7]. The frequencies of internal rotation caused by these methyl rotors are 248 cm1 for O—CH3 torsion (m29 ) and 278 cm1 for C—CH3 torsion (m28 ), respectively, and these two inequivalent methyl rotors give rise to a complicated splitting. In addition there are three other modes below 500 cm1 : skeletal torsion (m30 115 cm1 Þ, CCO bend (m18 288 cm1 ), and COC bend (m17 475 cm1 ). These five low-lying states are well populated at room temperature and make the spectra congested.
There are many laboratory microwave studies of trans-ethyl methyl ether. Hayashi and his co-workers studied normal species and many isotopomers, and provided molecular constants, the dipole moment, and the molecular structure [8,9]. The barrier heights to the O—CH3 and C—CH3 internal rotation were determined to be 893 and 1154 cm1 , respectively, based on the analysis of the internal rotation. Assignment of torsionally excited states was also mentioned, but unfortunately neither the transition frequencies nor molecular constants were reported. Fuchs et al. extended the microwave study of the ground state up to 350 GHz and provided an analysis treating explicitly splittings due to the two internal rotors [10]. Independently, Tsunekawa et al. reported transitions in the ground state in the 24–110 GHz frequency range, which exhibit only splittings due to the O—CH3 torsion [11]. A spectral atlas of about 21 000 lines up to 200 GHz was prepared at the same time but the assigned lines were limited. To clarify the unassigned lines, which are also useful for astronomical identification, we started a series of the studies on the torsionally excited states. The first skeletal torsionally excited state was reported recently [12]. The present paper is the second in our series of studies for the low-lying torsionally excited states of transethyl methyl ether, and an analysis, including splittings due to the two inequivalent internal rotors, of the m29 ¼ 1 excited torsional state will be given. 2. Experiment
* Corresponding author. Fax: +81 76 445 6549. E-mail address: [email protected] (K. Kobayashi).
The microwave absorption spectrum of trans-ethyl methyl ether was measured in the frequency range of 24–110 GHz using
0022-2852/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2009.03.012
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j1 >;
j2 >¼ ð123Þj1 >;
j5 >¼ ð465Þj1 >;
j8 >¼ ð132Þð456Þj1 >;
Fig. 1. Molecular structure of trans-ethyl methyl ether. The arrow shows the O—CH3 torsion.
conventional Stark- and source-modulation microwave spectrometers, whose details are given in Ref. [13]. Part of the spectrum was recorded both at room temperature and at dry ice temperature in order to make easier line assignments based on intensities relative to those of the corresponding ground state spectral lines. The spectral lines measured at dry ice temperature were stronger since the sample pressure was kept the same as that at room temperature and lower energy states were more populated. The rest of the data were taken from the spectral atlas.
The present analysis for ethyl methyl ether with two inequivalent methyl internal rotors was made by means of the tunneling matrix formulation (TMF), which is valid when the barrier to internal rotation is high. Although this formalism, developed first by Hougen [14] for the analysis of the high resolution spectrum of the hydrazine molecule, has been used for the analyses of various molecules with large amplitude vibrations, for example, in the study on N,N-dimethyl acetamide [15] where three inequivalent methyl internal rotors were treated, we will describe below the outline of the formalism applied to the inequivalent-two-top internal rotation problem of ethyl methyl ether.
j4 >¼ ð456Þj1 >; j7 >¼ ð123Þð465Þ;
j9 >¼ ð132Þð465Þj1 >;
ð1Þ
where (123) and (456) are feasible operations corresponding to 120° internal rotation of the O—CH3 top and the C—CH3 top, respectively. 3.3. Symmetrized vibrational wavefunctions It is convenient to use symmetrized vibrational wavefunctions as basis functions when setting up the Hamiltonian matrix. Assuming ð23Þð56Þð78Þ j1 >¼ j1 > for the (mO—CH3 , mC—CH3 Þ ¼ ð1; 0Þ state (the operation (23)(56)(78)* corresponds to the reflection in the plane of symmetry, and can be seen to be feasible by subsequently performing a 180° over all rotation of the molecule around the axis perpendicular to the plane of symmetry.), the nine framework functions defined in Eqs. (1) produce the following symmetrized vibrational wavefunctions:
jA2 >¼ 3. Theoretical formulation
j3 >¼ ð132Þj1 >;
j6 >¼ ð123Þð456Þj1 >;
pffiffiffiffiffiffiffiffiffiffiffiffiffi NðA2 Þ½1; 1; 1; 1; 1; 1; 1; 1; 1;
ð2Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðEAÞ½1; x ; x; 1; 1; x ; x ; x; x; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jEAb >¼ NðEAÞ½1; x; x ; 1; 1; x; x; x ; x ;
jEAa >¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðAEÞ½1; 1; 1; x ; x; x ; x; x ; x; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jAEb >¼ NðAEÞ½1; 1; 1; x; x ; x; x ; x; x ;
jAEa >¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðEE1Þ½1; x ; x; x ; x; x; 1; 1; x ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jEE1b >¼ NðEE1Þ½1; x; x ; x; x ; x ; 1; 1; x; jEE1a >¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðEE2Þ½1; x ; x; x; x ; 1; x; x ; 1; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jEE2b >¼ NðEE2Þ½1; x; x ; x ; x; 1; x ; x; 1; jEE2a >¼
3.1. Character table For the purpose of specifying the symmetry species labeling used in the present paper, we show in Table 1 the character table of the permutation-inversion group G18 for the ethyl methyl ether, which has Cs symmetry in its equilibrium configuration and possesses two inequivalent methyl groups exhibiting internal rotation. The character table is equivalent to that shown in Ref. [16] for the N-methylacetamide molecule.
where x ¼ expði2p=3Þ,
½a1 ; a2 ; a3 ; . . . . . . . . . ; a9 ¼
9 X
ai ji >;
ð3Þ
i¼1
and normalization factors NðCÞðC ¼ A2 ; EA; AE; EE1; EE2Þ are given by
NðA2 Þ ¼ ½9ð1 þ 2 < 1j2 > þ2 < 1j4 > þ2 < 1j6 > þ2 < 1j7 >Þ1 ;
3.2. Vibrational framework functions Vibrational wavefunctions called framework functions, which are localized at potential minima, are used in the tunneling matrix formalism. In the present case they are written as
NðEAÞ ¼ ½9ð1 < 1j2 > þ2 < 1j4 > < 1j6 > < 1j7 >Þ1 ; NðAEÞ ¼ ½9ð1 þ 2 < 1j2 > < 1j4 > < 1j6 > < 1j7 >Þ1 ; NðEE1Þ ¼ ½9ð1 < 1j2 > < 1j4 > < 1j6 > þ2 < 1j7 >Þ1 ; NðEE2Þ ¼ ½9ð1 < 1j2 > < 1j4 > þ2 < 1j6 > < 1j7 >Þ1 :
Table 1 Character table of the permutation-inversion group G18 for the ethyl methyl ether molecule.
A1 A2 EA AE EE1 EE2
E
2(123)
2(456)
2(123)(456)
2(123)(465)
9(23)(56)(78)*
1 1 2 2 2 2
1 1 1 2 1 1
1 1 2 1 1 1
1 1 1 1 1 2
1 1 1 1 2 1
1 1 0 0 0 0
1, 2, 3 denote the protons in the O-methyl group, and 4, 5, 6 are the protons in the C-methyl group. 7 and 8 denote two hydrogen atoms in the CH2 group.
ð4Þ (We put < 1 j 1 >¼ 1.) The overlap integrals < 1j2 >, < 1j4 > etc. are included explicitly in the normalization factors given in Eqs. (4) in the present formulation since the vibrational framework functions jn > ðn ¼ 1; 2; 3; . . . ; 9Þ are not orthogonal to each other. The problem associated with the nonzero overlap integrals was not treated in the original study of Hougen [14], but it was described rather generally in Ref. [17] of Ohashi and Hougen. Suffixes a and b in Eqs. (2) denote two components of the doubly degenerate symmetry species.
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3.4. Phenomenological Hamiltonian operator
3.5.3. AE species
In the present analysis we used the following phenomenological Hamiltonian operator in Watson’s S-reduced form:
< AEa j < J;KjHjJ; K 0 > jAEa >¼< AEb j < J; KjHjJ; K 0 > jAEb > ( X ½< 1jvk j1 > þ2 < 1jvk j2 > < 1jvk j4 > ¼ 9NðAEÞ
H ¼ AJ2z þ BJ2x þ CJ2y DJ J4 DJK J2 J 2z DK J 4z þ d1 J2 ðJ 2þ þ J 2 Þ
k pffiffiffi < 1jvk j6 > < 1jvk j7 > < J;KjRk jJ; K 0 > þ 3½< 1jqj4 >
þ d2 ðJ 4þ þ J 4 Þ þ HJK J4 J 2z þ HKJ J2 J 4z þ HK J 6z þ h1 J4 ðJ 2þ þ J2 Þ þ h2 J2 ðJ 4þ þ J 4 Þ þ h3 ðJ 6þ þ J 6 Þ þ l4 ðJ 8þ þ J 8 Þ þ i q þ qJ J2 þ qK J 2z þ qJK J2 J2z J z :
ð5Þ
In Eq. (5) coefficients A; B; C; DJ ; . . . ; q; qJ ; qK ; qJK are all operators depending on the variables describing internal rotations. Only terms which were necessary for obtaining a good fit in the least squares analysis are retained in Eq. (5).
þ < 1jqJ j4 > JðJ þ 1Þþ < 1jqK j4 > K 2 þ < 1jqJK j4 > JðJ þ 1ÞK 2 KdK;K 0 pffiffiffi þ 3½< 1jqj6 > þ < 1jqJ j6 > JðJ þ 1Þþ < 1jqK j6 > K 2 pffiffiffi þ < 1jqJK j6 > JðJ þ 1ÞK 2 KdK;K 0 3½< 1jqj7 > þ < 1jqJ j7 > JðJ þ 1Þ ) þ < 1jqK j7 > K 2 þ < 1jqJK j7 > JðJ þ 1ÞK 2 KdK;K 0 :
ð9Þ
3.5.4. EE1 species 3.5. Parameterized expressions for the Hamiltonian matrix elements We set up the Hamiltonian matrix with the Hamiltonian operator given in Eq. (5) and the basis functions jC >j J; K >, where jC > are the symmetrized vibrational wavefunctions given in Eqs. (2) and jJ; K > is a symmetric-top rotational eigenfunction. The expressions for the Hamiltonian matrix elements for each symmetry species are shown below, and for convenience, the Hamiltonian operator given in Eq. (5) was rewritten as follows
X
vk Rk ¼ AJ 2z þ BJ2x þ CJ 2y DJ J4 DJK J2 J 2z DK J 4z þ d1 J2 J 2þ þ J 2
k
þ d2 J 4þ þ J 4 þ HJK J4 J 2z þ HKJ J2 J 4z þ HK J 6z þ h1 J4 J 2þ þ J 2 þ h2 J2 J 4þ þ J 4 þ h3 J 6þ þ J 6 þ l4 J 8þ þ J 8 :
H¼
X
vk Rk þ i q þ qJ J2 þ qK J 2z þ qJK J2 J 2z J z ;
ð6Þ
< EE1a j < J;KjHjJ; K 0 > jEE1a >¼< EE1b j < J;KjHjJ;K 0 > jEE1b > ( X ½< 1jvk j1 > < 1jvk j2 > < 1jvk j4 > < 1jvk j6 > ¼ 9NðEE1Þ k pffiffiffi þ2 < 1jvk j7 > < J;KjRk jJ; K 0 > þ 3½< 1jqj2 > þ < 1jqJ j2 > JðJ þ 1Þ pffiffiffi þ < 1jqK j2 > K 2 þ < 1jqJK j2 > JðJ þ 1ÞK 2 KdK;K 0 þ 3½< 1jqj4 >
þ < 1jqJ j4 > JðJ þ 1Þþ < 1jqK j4 > K 2 þ < 1jqJK j4 > JðJ þ 1ÞK 2 KdK;K 0 pffiffiffi 3½< 1jqj6 > þ < 1jqJ j6 > JðJ þ 1Þþ < 1jqK j6 > K 2 ) þ < 1jqJK j6 > JðJ þ 1ÞK 2 KdK;K 0 :
ð10Þ
3.5.5. EE2 species
< EE2a j < J; KjHjJ; K 0 > jEE2a >¼< EE2b j < J; KjHjJ; K 0 > jEE2b > ( X ½< 1jvk j1 > < 1jvk j2 > < 1jvk j4 > ¼ 9NðEE2Þ k pffiffiffi þ 2 < 1jvk j6 > < 1jvk j7 > < J; KjRk jJ; K 0 > þ 3½< 1jqj2 >
k
where vk and Rk are used as general representatives of rotational constants and the operators of the pure rotation part. 3.5.1. A1 , A2 species Since the Wang-type rotational functions are not used in the present paper, A1 and A2 symmetry species are treated together to obtain
X < A2 j < J; KjHjJ; K > jA2 >¼ 9NðA1 Þ ½< 1jvk j1 > 0
k
þ 2 < 1jvk j2 > þ2 < 1jvk j4 > þ2 < 1jvk j6 > þ2 < 1jvk j7 > < J; KjRk jJ; K 0 > :
ð7Þ
3.5.2. EA species
< EAa j < J; KjHjJ;K 0 > jEAa >¼< EAb j < J; KjHjJ; K 0 > jEAb > ( X ½< 1jvk j1 > < 1jvk j2 > þ2 < 1jvk j4 > < 1jvk j6 > ¼ 9NðEAÞ
þ < 1jqJ j2 > JðJ þ 1Þþ < 1jqK j2 > K 2 þ < 1jqJK j2 > JðJ þ 1ÞK 2 KdK;K 0 pffiffiffi 3½< 1jqj4 > þ < 1jqJ j4 > JðJ þ 1Þþ < 1jqK j4 > K 2 pffiffiffi þ < 1jqJK j4 > JðJ þ 1ÞK 2 KdK;K 0 3½< 1jqj7 > þ < 1jqJ j7 > JðJ þ 1Þ ) þ < 1jqK j7 > K 2 þ < 1jqJK j7 > JðJ þ 1ÞK 2 KdK;K 0 :
ð11Þ
The quantities < 1jvk jn > ðn ¼ 1; 2; 4; 6; 7Þ and < 1jqjm > ðm ¼ 2; 4; 6; 7Þ appearing in Eqs. (7)–(11) are called tunneling matrix elements (n ¼ 1 corresponds to the non-tunneling process). Tunneling matrix elements < 1jvk jn > ðn ¼ 3; 5; 8; 9Þ and < 1jqjm > ðm ¼ 1; 3; 5; 8; 9Þ do not appear in Eqs. (7)–(11) because of the following relations among tunneling matrix elements:
< 1jvk j2 > ¼< 1jvk j3 >; < 1jvk j4 >¼< 1jvk j5 >; < 1jvk j6 >¼< 1jvk j9 >; < 1jvk j7 >¼< 1jvk j8 >; < 1jqj1 >¼ 0; < 1jqj2 >¼ < 1jqj3 >; < 1jqj4 >¼ < 1jqj5 >< 1jqj6 >¼ < 1jqj9 >; < 1jqj7 >¼ < 1jqj8 > :
ð12Þ
k
pffiffiffi < 1jvk j7 > < J; KjRk jJ; K 0 > þ 3½< 1jqj2 > þ < 1jqJ j2 > JðJ þ 1Þ pffiffiffi þ < 1jqK j2 > K 2 þ < 1jqJK j2 > JðJ þ 1ÞK 2 KdK;K 0 þ 3½< 1jqj6 > þ < 1jqJ j6 > JðJ þ 1Þþ < 1jqK j6 > K 2 þ < 1jqJK j6 > JðJ þ 1ÞK 2 KdK;K 0 pffiffiffi þ 3½< 1jqj7 > þ < 1jqJ j7 > JðJ þ 1Þþ < 1jqK j7 > K 2 ) þ < 1jqJK j7 > JðJ þ 1ÞK 2 KdK;K 0 :
ð8Þ
3.6. Note on the least squares analysis with Eqs. (7)–(11) Since the overlap integrals < 1j2 >, < 1j4 > etc. between pairs of framework wavefunctions, which are undetermined quantities, are included in the expressions (7)–(11), tunneling matrix elements < 1jvk j1 >, < 1jvk j2 >, < 1jqj2 > etc. cannot be determined in any least squares analysis. Instead, vk11 ; vk12 ; vk14 ; vk16 ; vk17 ; q12 ; q14 ; q16 ; q17 etc. are used as adjustable parameters related to the tunneling matrix elements by the following relations:
K. Kobayashi et al. / Journal of Molecular Spectroscopy 255 (2009) 164–171
vk11 þ 2vk12 þ 2vk14 þ 2vk16 þ 2vk17 ¼ 9NðA2 Þ½< 1jvk j1 > þ2 < 1jvk j2 > þ2 < 1jvk j4 > þ 2 < 1jvk j6 > þ2 < 1jvk j7 >; vk11 vk12 þ 2vk14 vk16 vk17 ¼ 9NðEAÞ½< 1jvk j1 > < 1jvk j2 > þ2 < 1jvk j4 > < 1jvk j6 > < 1jvk j7 >;
vk11 þ 2vk12 vk14 vk16 vk17 ¼ 9NðAEÞ½< 1jvk j1 > þ2 < 1jvk j2 > < 1jvk j4 > < 1jvk j6 > < 1jvk j7 >; vk11 vk12 vk14 vk16 þ 2vk17 ¼ 9NðEE1Þ½< 1jvk j1 > < 1jvk j2 > < 1jvk j4 > < 1jvk j6 > þ2 < 1jvk j7 >;
vk11 vk12 vk14 þ 2vk16 vk17 ¼ 9NðEE2Þ½< 1jvk j1 > < 1jvk j2 >
and
q12 þ q16 þ q17 ¼ 9NðEAÞ½< 1jqj2 > þ < 1jqj6 > þ < 1jqj7 >; q14 þ q16 q17 ¼ 9NðAEÞ½< 1jqj4 > þ < 1jqj6 > < 1jqj7 >; q12 þ q14 q16 ¼ 9NðEE1Þ½< 1jqj2 > þ < 1jqj4 > < 1jqj6 >; q12 q14 q17 ¼ 9NðEE2Þ½< 1jqj2 > < 1jqj4 > < 1jqj7 >: ð14Þ (We have similar relations for qJ ; qK and qJK .) Since the overlap integrals < 1j2 >, < 1j4 > etc. are considered to be small, the parameters vk11 ; vk12 ; vk14 ; q12 ; q14 etc. are approximately equal to the tunneling matrix elements < 1jvk j1 >, < 1jvk j2 >, < 1jvk j4 >, < 1jqj2 >, < 1jqj4 > etc., respectively. Eqs. (7)–(11) for the Hamiltonian matrix elements are changed to obvious expressions involving the parameters vk11 ; vk12 ; vk14 ; vk16 ; vk17 ; q12 ; q14 ; q16 ; q17 etc. For example, Eq. (7) is changed to
X ½vk11 þ 2vk12 þ 2vk14 þ 2vk16 k
þ 2vk17 < J; KjRk jJ; K 0 > :
for C ¼ A2 ; EA; AE; EE1 and EE2. In relations (17), j ee >; j eo >; j oe > and j oo > represent rotational wavefunctions with K a ; K c ¼ even, even; even, odd; odd, even and odd, odd, respectively. In some rotational transitions obeying selection rules
EA $ EA; AE $ AE; EE1 $ EE1 and EE2 $ EE2 on the overall symmetry species, we observe ‘‘forbidden transitions”, which do not follow the selection rules shown in (17) above. Such forbidden transitions, as observed in the present investigation, were found to occur when the term iqJ z mixes two K-type doubling levels significantly in the upper and/or the lower state involved in the transition. 4. Analysis
< 1jvk j4 > þ2 < 1jvk j6 > < 1jvk j7 >; ð13Þ
< A2 j < J; KjHjJ; K 0 > jA2 >¼
167
ð15Þ
In the present analysis for the trans-ethyl methyl ether in the
m29 ¼ 1 excited torsional state, the 1 ! 6 and 1 ! 7 tunneling processes were neglected, and therefore we set < 1 j vk j 6 >¼ < 1 j vk j 7 >¼ 0, < 1 j q j 6 >¼< 1 j q j 7 >¼ 0, < 1 j qJ j 6 >¼< 1 j qJ j 7 >¼ 0, < 1 j qK j 6 >¼< 1 j qK j 7 >¼ 0, < 1 j qJK j 6 >¼< 1 j qJK j 7 >¼ 0, and accordingly, vk16 ¼ vk17 ¼ 0, q16 ¼ q17 ¼ 0, qJ16 ¼ qJ17 ¼ 0, qK16 ¼ qK17 ¼ 0, qJK16 ¼ qJK17 ¼ 0.
In the ground state, the splitting due to the O—CH3 internal rotor is larger than that of the C—CH3 internal rotor and only splitting due to the O—CH3 rotor was observed for low J, K transitions. In the m29 ¼ 1 state, larger splitting due to the O—CH3 methyl-internalrotation motion was expected. We found series of lines with reasonable intensity and splitting patterns. The spectrum observed at dry ice temperature, exhibiting the typical quartet pattern, is shown in Fig. 2. The splittings due to the two methyl-rotors were observed even at low J and K. The extension of line assignments up to high rotational quantum numbers J and K was made by using least-squares-fittings and frequency predictions based on the tunneling matrix formulation (TMF), whose outline was described in the preceding section. In total, 1507 lines up to J ¼ 76 and K ¼ 5 were assigned and part of the lines are listed in Table 2. In many cases the EE1 and EE2 species components of the internal rotation splitting pattern were overlapping as seen in Table 2, where the species labeling EE is used for the unresolved EE1 and EE2 species. In transitions with higher K quantum number, the EE1 and EE2 species were resolved. For the cases where the EE1 and EE2 transitions were unresolved, the overlapped transition frequencies were fit to the average values of the EE1 and EE2 transition frequencies. In the least squares analysis, all the lines were given weights of 1.0. Molecular parameters thus obtained are shown in Table 3. The root mean square deviation in the fit was 50 kHz. 5. Discussion Since the m29 ¼ 1 excited torsional state treated in the present study is of vðO—CH3 torsionÞ ¼ 1, vðC—CH3 torsionÞ ¼ 0, values of
3.7. Selection rules for electric dipole transitions Selection rules on tunneling-rotational symmetry species for electric dipole transitions are
A1 $ A2 ; EA $ EA; AE $ AE; EE1 $ EE1; EE2 $ EE2:
ð16Þ
From the overall selection rules shown in (16) above, and
< Cjla jC > –0;
< Cjlb jC > –0; C ¼ A2 ; EA; AE; EE1 and EE2
< Cjlc jC >¼ 0 for
(note that ð23Þð56Þð78Þ ðj1 >; j2 >; j4 >; j6 >; j7 >Þ ¼ ðj1 >; j3 >; j5 >; j9 >; j8 >Þ and ð23Þð56Þð78Þ ½la ; lb ; lc ¼ ½la ; lb ; lc ), we see that the allowed transitions are
jC > jee >$ jC > jeo >;
jC > joe >$ jC > joo >;
jC > jee >$ jC > joo >;
jC > jeo >$ jC > joe >;
ð17Þ
Fig. 2. The 1019 10010 transition of trans-ethyl methyl ether in the first O—CH3 torsionally excited state. The EE1 and EE2 transitions were overlapped in this rotational transition.
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K. Kobayashi et al. / Journal of Molecular Spectroscopy 255 (2009) 164–171
Table 2 Assigned transitions, observed transition frequencies (in MHz), and observed minus calculated residuals (in MHz) of trans-ethyl methyl ether in the first state. Upper state J
0
Ka
31 32 33 24 25 26 27 28 12 13 14 15 16 6 7 8 9 10 11 19 20 21 22 23 24 25 26 27 28 32 33 34 35 13 14 15 16 17 18 23 24 25 26 27 28 33 34 35 36 37 38 6 7 9 10 24 25 26 27 28 32 33 34 35 36 4 4 47 48 49 50 51 a b
2 2 2 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5
0
Lower state Kc
0
29 30 31 23 24 25 26 27 12 13 14 15 16 5 6 7 8 9 10 19 20 21 22 23 22 23 24 25 26 31 32 33 34 12 13 14 15 16 17 20 21 22 23 24 25 31 32 33 34 35 36 4 5 7 8 20 21 22 23 24 29 30 31 32 33 1 0 42 43 44 45 46
J
00
30 31 32 23 24 25 26 27 11 12 13 14 15 6 7 8 9 10 11 18 19 20 21 22 24 25 26 27 28 32 33 34 35 12 13 14 15 16 17 23 24 25 26 27 28 33 34 35 36 37 38 5 6 8 9 24 25 26 27 28 32 33 34 35 36 3 3 47 48 49 50 51
Ka
3 3 3 2 2 2 2 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4
Overlapped lines. Weight was set to 0.
00
AA Kc
00
28 29 30 22 23 24 25 26 11 12 13 14 15 6 7 8 9 10 11 18 19 20 21 22 23 24 25 26 27 32 33 34 35 11 12 13 14 15 16 21 22 23 24 25 26 32 33 34 35 36 37 3 4 6 7 21 22 23 24 25 30 31 32 33 34 0 0 43 44 45 46 47
AE
Observed frequency
Obs.-calc.
168 966.353 180 303.783 191 703.278 155 731.227 166 024.283 176 290.505 186 514.575 196 681.568 79 578.219 88 583.116 97 579.853 106 553.474 115 490.887 267 50.342 27 770.896 28 968.814 30 356.224 31 945.650 33 750.210 158 416.097 165 316.205 172 275.371 179 296.845 186 381.949 62 831.063 64 214.630 65 890.419 67 868.316 70 156.627 143 054.611 147 301.259 151 627.594 156 028.196 165 850.207 172 223.261 178 483.773 184 636.785 190 687.902 196 644.112 105 667.978 104 019.953 102 354.959 100 700.086 99 083.870 97 535.730 143 210.880 145 671.724 148 272.991 151 014.035 153 893.050 156 908.311 167 449.512 175 441.272 191 319.505 199 183.180a 163 442.271 162 705.905 161 855.522 160 882.110 159 777.398 167 943.187 168 385.571 168 908.334 169 518.209 170 221.872 199 119.123a
0.075 0.032 0.064 0.045 0.080 0.066 0.005 0.036 0.103 0.043 0.102 0.022 0.063 0.017 0.068 0.021 0.000 0.010 0.017 0.022 0.008 0.008 0.007 0.005 0.011 0.037 0.031 0.005 0.005 0.018 0.021 0.066 0.042 0.001 0.007 0.015 0.075 0.025 0.013 0.01 0.004 0.002 0.004 0.005 0.006 0.037 0.039 0.039 0.054 0.030 0.015 0.052 0.025 0.065 0.002 0.037 0.014 0.027 0.060 0.024 0.094 0.022 0.002 0.033 0.029 0.011
191 833.033 189 409.323 186 853.782 184 181.432 181 411.596
0.050 0.021 0.094 0.002 0.005
Observed frequency
Obs.-calc.
168 962.102 180 299.660 191 699.341 155 728.921 166 022.187a 176 288.468 186 512.688 196 679.819 79 577.182 88 582.146 97 578.924 106 552.673 115 490.149 267 51.859 27 772.400 28 970.362 30 357.813 31 947.270 33 751.905 158 416.885 165 316.923 172 276.024 179 297.493 186 382.550 62 833.880 64 217.461 65 893.205 67 871.039 70 159.339 143 060.516 147 307.325 151 633.785 156 034.487 165 853.891 172 226.888 178 487.290 184 640.232 190 691.327 196 647.599a 105 673.486 104 025.339 102 360.247 100 705.221 99 088.868 97 540.591 143 216.428 145 677.280 148 278.472a 151 019.561 153 898.658 156 914.020 167 442.855b 175 435.458 191 317.238 199 183.180a 163 450.544 162 713.826 161 863.158 160 889.439 159 784.606 167 949.621 168 391.884 168 914.561 169 524.347 170 227.952
0.028 0.020 0.037 0.024 0.139 0.053 0.008 0.042 0.055 0.003 0.043 0.028 0.070 0.049 0.059 0.018 0.007 0.015 0.006 0.014 0.015 0.005 0.055 0.070 0.018 0.006 0.005 0.009 0.014 0.034 0.001 0.050 0.059 0.001 0.009 0.028 0.032 0.013 0.143 0.004 0.002 0.024 0.006 0.009 0.006 0.004 0.014 0.134 0.004 0.002 0.105 0.278 0.035 0.063 0.038 0.052 0.009 0.005 0.015 0.003 0.021 0.001 0.008 0.031 0.001
199 129.076 191 838.531 189 414.617 186 858.881 184 186.344 181 416.306
0.007 0.084 0.020 0.076 0.014 0.025
EA
EE
Observed frequency Obs.-calc.
EE1 Observed frequency
168 893.102 180 231.543
0.000 0.021
EE2 Obs.-calc.
180 227.515 0.100 191 628.216a,b 0.284 155 689.151 165 983.697 176 251.447 186 477.342 196 646.037 79 561.659 88 567.240 97 564.624 106 539.114 115 477.381 26 771.905 27 792.532 28 990.662 30 378.120 31 967.742 33 772.481 158 429.089 165 328.335 172 286.715 179 307.353 186 391.564 62 871.446 64 253.499 65 927.776 67 904.173 70 191.023 143 127.944 147 375.758 151 703.469 156 105.264
155 691.462 165 985.862 176 253.515 186 479.235 196 647.599a,b 79 562.656 88 568.189 97 565.511 106 539.913 115 478.140 26 770.499 27 791.068 28 989.052 30 376.538 31 966.097 33 770.796 158 428.331 165 327.623 172 286.026 179 306.696 186 391.045 62 868.601 64 250.740 65 925.015 67 901.400 70 188.305 143 122.012 147 369.725 151 697.265 156 099.060 165 897.939 172 271.632 178 532.472 184 685.387 190 736.439 196 692.318 105 756.316 104 105.554 102 438.094 100 780.942 99 162.478 97 612.092 143 282.384 145 742.811 148 343.739 151 084.208 153 962.94 156 977.797 167 294.306
0.03 0.003 0.029 0.007 0.237 0.043 0.047 0.047 0.049 0.054 0.022 0.106 0.034 0.040 0.042 0.036 0.054 0.075 0.047 0.063 0.033 0.004 0.012 0.016 0.035 0.013 0.016 0.022 0.014 0.050 0.095 0.100 0.043 0.091 0.065 0.031 0.019 0.015 0.007 0.011 0.005 0.002 0.011 0.000 0.015 0.040 0.006 0.008 0.089
199 103.476 163 643.470 162 892.640 162 027.215 161 039.682 159 922.675a,b 168 018.015 168 463.111 168 987.519 169 598.476 170 302.642
0.089 0.008 0.051 0.025 0.118 0.165 0.031 0.073 0.018 0.012 0.003
199 094.215 163 661.791 162 910.270 162 043.749
0.048 0.068 0.026 0.034
159 936.159 168 021.906 168 467.486 168 992.295 169 603.463 170 307.706
0.016 0.009 0.088 0.044 0.011 0.071
199 242.647 191 937.304 189 511.754 186 954.459 184 280.472 181 508.819
0.027 0.130 0.041 0.051 0.016 0.040
199 252.593 191 943.655b 189 517.678 186 960.095 184 285.851 181 513.948
178 535.560
m29 torsionally excited
0.107
190 739.641 0.026 196 695.527 0.047 105 763.097 0.024 104 111.931 0.029 102 444.100 0.018 100 786.638 0.012 99 167.916 0.012 97 617.299 0.004 143 288.020 145 748.419 148 349.313 151 089.787 153 968.486 156 983.340
0.008 0.286 0.103 0.131 0.125 0.095
Observed frequency
Obs.-calc.
191 628.216a 0.056 0.006 0.073 0.004 0.043 0.051 0.065 0.064 0.040 0.040 0.057 0.058 0.098 0.039 0.063 0.049 0.048 0.058 0.002 0.008 0.050 0.002 0.030 0.016 0.011 0.028 0.005 0.035 0.042 0.054 165 902.324
0.102
178 536.395 184 689.202
0.088 0.083
105 760.572 104 109.924 102 442.657 100 785.487 99 167.008 97 616.605 0.036 0.027 0.012 0.037 0.028 0.037
0.027 0.037 0.041 0.009 0.001 0.031
199 124.929a 163 639.527 162 889.276 162 025.068 161 038.789 159 922.675a,b 168 026.901 168 471.280 168 995.232 169 605.838 170 309.577
0.092 0.082 0.018 0.046 0.135 0.276 0.010 0.010 0.027 0.041 0.036
199 248.443 191 941.936 189 516.426 186 959.085 184 284.959 181 513.261
0.029 0.030 0.015 0.000 0.085 0.063
0.090
169
K. Kobayashi et al. / Journal of Molecular Spectroscopy 255 (2009) 164–171 Table 3 Molecular parametersa,b (in MHz) of trans-ethyl methyl ether in the Non-tunneling parameters A11 B11 C 11 DJ11 DJK11 DK11 d1;11 d2;11
27 882.4722(17) 4153.24850(11) 3888.28791(11) 0.000972717(39) 0.0015575(20) 0.03535(16) 0.0000875604(77) 0.0000036881(14)
HKJ11 HK11 h1;11
0.000002136(87) 0.0001240(63) 0.0000000000379(15)
h3;11 l4;11
0.00000000000827(38) 0.000000000000001352(86)
a b
m29 ¼ 1 excited torsional state.
O—CH3 internal rotation tunneling parameters
C—CH3 internal rotation tunneling parameters
A12 B12
6.40153(98) 0.019241(14)
A14 B14
0.47064(85) 0.002954(15)
DJK12 DK12 d1;12 d2;12 HJK12 HKJ12 HK12
0.0004907(10) 0.065209(94) 0.0000001318(17) 0.00000041033(74) 0.00000001295(21) 0.000003902(50) 0.0002526(35)
DJK14 DK14 d1;14 d2;14 HJK14 HKJ14
0.00007361(78) 0.003750(38) 0.0000000447(20) 0.0000000703(15) 0.00000000083(20) 0.000000567(30)
h2;14
0.00000000000586(57)
h3;12
0.00000000001002(21)
q12 qJ12 qK12 qJK12
35.9747(34) 0.003277(12) 0.64265(17) 0.00006767(57)
q14 qJ14 qK14
2.8510(29) 0.0003187(34) 0.04413(16)
Numbers in parentheses are one standard uncertainty (1r) in units of the least significant digits. Note that A1n < 1jAjn >; B1n < 1jBjn > etc. (for n ¼ 1; 2; 4) as mentioned in the text.
A14 , B14 , q14; which are the parameters related to the C—CH3 internal rotation tunneling, are expected to be as small as those for the ground state. However, values of the parameters A14 , B14 and q14 determined are rather large compared to those for the ground state. (According to the new analysis made by using previously observed transition frequencies [11] for the ground state, A14 ¼ 0:01871ð52Þ MHz, B14 ¼ 0:000106ð19Þ MHz, and q14 is undetermined since it is too small.) Furthermore, contrary to the prediction that the parameters A14 and B14 take positive values as in the ground state, they were found to have negative values, which produce the inverted A/E sequence pattern regarding the C—CH3 internal rotation tunneling splitting. This inversion was further confirmed by the splitting of the EE sublevel. To understand the fact mentioned above quantitatively, we tried to carry out two evaluations described below, assuming that it is produced mainly by a kinetic O—CH3 —C—CH3 coupling as represented by the term 2F 12 pO—CH3 pC—CH3 in the internal rotation Hamiltonian. The torsional barrier heights and the molecular parameters related to the internal rotation splittings were estimated in Evaluations 1 and 2, respectively. In the present discussion, we neglect the interaction scheme involving the skeletal torsionally excited state because the energy difference between O—CH3 and C—CH3 torsionally excited states is much smaller, even though it was suggested in the previous study [12] on the skeletal torsionally excited state that the inverted A/E sequence seen in the first skeletal torsionally excited state would be produced by the O—CH3 torsion-skeletal torsion and C—CH3 torsion-skeletal torsion interactions. Evaluation 1. We estimated values of barrier heights V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ to the internal rotations by combining the Coriolislike parameters q12 and q14 obtained from the spectral analysis with the relations obtained from Principle Axis Method (PAM) first order perturbation theory [18]
pffiffiffi 3q12 ¼ j < mO—CH3 ¼ 1; mC—CH3 ¼ 0; EAjP a jmO—CH3 ¼ 1;
mC—CH3 ¼ 0; EA > j; pffiffiffi 3q14 ¼ j < mO—CH3 ¼ 1; mC—CH3 ¼ 0; AEjP a jmO—CH3 ¼ 1;
mC—CH3 ¼ 0; AE > j;
where
Pa ¼ 2F O—CH3 qa ðO—CH3 ÞpO—CH3 2F C—CH3 qa ðC—CH3 ÞpC—CH3 2F 12 qa ðC—CH3 ÞpO—CH3 2F 12 qa ðO—CH3 ÞpC—CH3 ;
ð18Þ
and the internal rotation Hamiltonian operator including the kinetic O—CH3 —C—CH3 coupling term
H ¼ F O—CH3 p2O—CH3 þ F O—CH3 p2C—CH3 þ 2F 12 pO—CH3 pC—CH3 1 1 þ V 3 ðO—CH3 Þ½1 cosð3sO—CH3 Þ þ V 3 ðC—CH3 Þ 2 2 ½1 cosð3sC—CH3 Þ:
ð19Þ
In Eqs. (18),
ka ðO—CH3 ÞIa ðO—CH3 Þ ; Ia ðmolÞ k ðC—CH3 ÞIa ðC—CH3 Þ qa ðC—CH3 Þ ¼ a Ia ðmolÞ
qa ðO—CH3 Þ ¼
where ka ðO—CH3 Þ: the a-axis component of direction cosine of the O—CH3 internal rotation axis, ka ðO—CH3 Þ: the a-axis component of direction cosine of the C—CH3 internal rotation axis. In Table 4, we give values of quantities required for the determination of the barrier heights using Eqs. (18) and (19). Those values were obtained from geometrical parameters given in the ab initio study made by Durig et al. [7] The barrier heights, V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ thus obtained for the internal rotations are given in Table 5, where the values of q12 and q14 reproduced with Eqs. (18) are shown together with those obtained in the least squares analysis of the spectrum. The barrier heights including those obtained in the previous studies [6,8,12] are compiled in Table 6. Our barrier heights agree with the IR results [6]. A relatively large discrepancy of V 3 ðC—CH3 Þ to the one obtained in the MW study [8] was noted. It could be because the barrier height in the previous study was determined for an isotopomer. Evaluation 2. We evaluated parameters A12 , A14 , B12 and B14 , which are related with contributions from internal rotation tunnel-
170
K. Kobayashi et al. / Journal of Molecular Spectroscopy 255 (2009) 164–171
Table 4 Quantities required for determination of the barrier heights V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ with the use of Eqs. (18) and (19). F O—CH3 ¼ 6:4041 cm1 F 12 ¼ 1:1545 cm1
F C—CH3 ¼ 6:3618 cm1
qa ðO—CH3Þ ¼ 0:1678 qb ðO—CH3Þ ¼ 0:00942
qa ðC—CH3 Þ ¼ 0:1584 qb ðC—CH3 Þ ¼ 0:01198
Table 5 Barrier heights V 3 ðO—CH3 Þ, V 3 ðC—CH3 Þ, and Coriolis-like parameters q12 , q14 used for the determination of the barrier heights. V 3 ðO—CH3 Þ ¼ 881:5 cm1 V 3 ðC—CH3 Þ ¼ 1070:1 cm1
q12 (MHz) q14 (MHz)
Calc.
OC
35.9747 2.8510
35.9632 2.8673
0.0115 0.0163
ings to the rotational constants A and B, by using the barrier heights V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ obtained in Evaluation 1 and the following relations obtained from PAM second order perturbation theory [18].
3A14 ¼
2
2
X j < v 1 ¼ 1; v 2 ¼ 0; AEjPa jv 0 ; v 0 ; AE > j2 0 1 2 ; 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; AEÞ Eðv 1 ; v 2 ; AEÞ 1
3B12 ¼
2
X j < v 1 ¼ 1; v 2 ¼ 0; A2 jPb jv 0 ; v 0 ; A1 > j2 1 2 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; A2 Þ Eðv 1 ; v 2 ; A1 Þ 1
2
X j < v 1 ¼ 1; v 2 ¼ 0; EAjPb jv 0 ; v 0 ; EA > j2 0 1 2 ; 0 Eð v ¼ 1; v ¼ 0; EAÞ Eð v ; v 02 ; EAÞ 1 2 0 0 1 v ;v 1
3B14 ¼
2
X j < v 1 ¼ 1; v 2 ¼ 0; A2 jPb jv 0 ; v 0 ; A1 > j2 1 2 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; A2 Þ Eðv 1 ; v 2 ; A1 Þ 1
2
X j < v 1 ¼ 1; v 2 ¼ 0; AEjPb jv 0 ; v 0 ; AE > j2 0 1 2 ; 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; AEÞ Eðv 1 ; v 2 ; AEÞ 1
ð20Þ
2
where v 1 , v 1 ’ and v 2 , v 2 ’ denote the O—CH3 torsional and the C—CH3 torsional quantum numbers, respectively, and P b is defined as Table 6 Comparison of the barrier heights.
V 3 ðO—CH3 Þ (cm1 Þ V 3 ðC—CH3 Þ (cm1 Þ
6.32290 0.47563 0.019918 0.002722
10.4 0.0396 0.0327 0.000227
6.40153 0.47064 0.019241 0.002954
v 29 ¼ 1 excited torsional state
The ground state
The
EðEAÞ EðA1 Þ ¼ 7:8 EðAEÞ EðA1 Þ ¼ 1:7 EðEE1Þ EðA1 Þ ¼ EðEE2Þ EðA1 Þ ¼ 9:5
EðEAÞ EðA2 Þ ¼ 307:0 EðAEÞ EðA2 Þ ¼ 25:9 EðEE1Þ EðA2 Þ ¼ EðEE2Þ EðA2 Þ ¼ 332:9
Pb ¼ 2F O—CH3 qb ðO—CH3 ÞpO—CH3 2F C—CH3 qb ðC—CH3 ÞpC—CH3 2F 12 qb ðC—CH3 ÞpO—CH3 2F 12 qb ðO—CH3 ÞpC—CH3
ð21Þ
kb ðO—CH3 ÞIb ðO—CH3 Þ ; Ib ðmolÞ k ðC—CH3 ÞIb ðC—CH3 Þ qb ðC—CH3 Þ ¼ b : Ib ðmolÞ
X j < v 1 ¼ 1; v 2 ¼ 0; A2 jP a jv 0 ; v 0 ; A1 > j2 1 2 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; A2 Þ Eðv 1 ; v 2 ; A1 Þ 1
Values determined in the least squares spectral analysis
qb ðO—CH3 Þ ¼
2
X j < v 1 ¼ 1; v 2 ¼ 0; EAjPa jv 0 ; v 0 ; EA > j2 0 1 2 ; 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; EAÞ Eðv 1 ; v 2 ; EAÞ 1
(MHz) (MHz) (MHz) (MHz)
Values calculated from Eqs. (20) with F 12 ¼ 0
In Eq. (21),
X j < v 1 ¼ 1; v 2 ¼ 0; A2 jP a jv 0 ; v 0 ; A1 > j2 1 2 0 0 v 0 ;v 0 Eðv 1 ¼ 1; v 2 ¼ 0; A2 Þ Eðv 1 ; v 2 ; A1 Þ 1
A12 A14 B12 B14
Values calculated from Eqs. (20)
Table 8 A/E energy differences (in MHz) in the J ¼ K ¼ 0 level.
Obs.
Obs. and Calc. denote values obtained in the least squares analysis of the spectrum and those calculated from Eqs. (18), respectively.
3A12 ¼
Table 7 Comparison between A12 , A14 , B12 and B14 calculated from Eqs. (20) and those determined in the least squares analysis of the spectrum.
MW (m29 ¼ 1) This study
MW (m30 ¼ 1) Ref. [12]
MW (ground state)a Ref. [8]
IR
881.5 1070.1
858.02 —b
893 1154
913.9 1076.4
From the evaluation by using Eqs. (20) with the barrier heights V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ determined above, we obtained values for the tunneling parameters A12 , A14 , B12 and B14 as shown in Table 7, where it is seen that they agree in sign and closely in magnitude with those for values obtained from the least squares analysis of the spectrum. In Table 7, values for the tunneling parameters obtained without the kinetic coupling of the two torsional motions ( F 12 ¼ 0) are also shown for comparison, which suggests that if F 12 ¼ 0, then A14 and B14 have positive values, and the inverted A/E sequence pattern regarding the C—CH3 internal rotation does not occur. The coupling of the torsions is also mentioned in the previous studies [6,8]. The evaluations made above supports the assumption that the inverted A/E sequence pattern regarding the C—CH3 internal rotation is produced mainly by the kinetic coupling between the O—CH3 and the C—CH3 internal rotations. In Table 8, we show A/E energy differences in the J ¼ K ¼ 0 level of the ground and the v 29 ¼ 1 excited torsional states, which are evaluated with the internal rotation Hamiltonian given in Eq. (19) and F O—CH3 , F C—CH3 , F 12 , V 3 ðO—CH3 Þ and V 3 ðC—CH3 Þ values in Tables 4 and 5. It is seen that the inverted A/E order regarding the C—CH3 internal rotation occurs also in the energy levels of the v 29 ¼ 1 excited torsional state. (Note that EðAEÞ EðA2 Þ < 0 and EðEE1orEE2Þ EðEAÞ < 0 in the v 29 ¼ 1 excited torsional state.) Acknowledgments
Ref. [6]
a The V 3 ðO—CH3 Þ was obtained for CH3 CH2 OCH3 while the V 3 ðC—CH3 Þ was obtained for CH3 CH2 OCD3 . b The V 3 ðC—CH3 Þ was not obtained since the torsional splitting was not observed for the m30 ¼ 1 torsional level.
K.K. would like to thank N. Mori and Y. Hori for helping the microwave measurement. This study was partially supported by Grant-in-Aid for Young Scientists (B) by the Ministry of Education, Culture, Sports, Science and Technology of Japan (Grant No. 20740103) and National Astronomical Observatory of Japan.
K. Kobayashi et al. / Journal of Molecular Spectroscopy 255 (2009) 164–171
Appendix A. Supplementary data Supplementary data for this article are available on Science Direct (www.sciencedirect.com) and as part of the Ohio State University Molecular Spectroscopy Archives (http://library.osu. edu/sites/msa/jmsa_hp.htm). Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.jms.2009.03.012. References [1] G.W. Fuchs, U. Fuchs, T.F. Giesen, F. Wyrowski, Astronom. Astrophys. 444 (2005) 521–530. [2] F.J. Lovas, R.D. Suenram, L.E. Snyder, J.M. Hollis, R.M. Lees, Astrophys. J. 253 (1982) 149–153. [3] D.M. Mehringer, J.C. Pearson, J. Keene, T.G. Phillips, Astrophys. J. 608 (2004) 306–313. [4] K. Kobayashi, K. Ogata, S. Tsunekawa, S. Takano, Astrophys. J. 657 (2007) L17– L19.
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