Journal of Molecular Spectroscopy xxx (2014) xxx–xxx

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Journal of Molecular Spectroscopy

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Sensitivity to a possible variation of the proton-to-electron mass ratio oftorsion–rotation transitions in acetone (CH3)2CO

http://dx.doi.org/10.1016/j.jms.2014.03.0040022-2852/� 2014 Elsevier Inc. All rights reserved.

⇑ Fax: +380 57 706 1415.E-mail address: ilyushin@rian.kharkov.ua

Please cite this article in press as: V.V. Ilyushin, J. Mol. Spectrosc. (2014), http://dx.doi.org/10.1016/j.jms.2014.03.004

Vadim V. Ilyushin ⇑Institute of Radio Astronomy of NASU, Chervonopraporna 4, 61002 Kharkov, Ukraine

a r t i c l e i n f o a b s t r a c t

Article history:Available online xxxx

Keywords:AcetoneLarge-amplitude torsional motionThe proton-to-electron mass ratio variation

The notion that certain transitions of molecules with intramolecular large amplitude motions exhibit anenhanced sensitivity to the proton-to-electron mass ratio variation is changing the paradigm for search-ing for drifting constants from the optical to the microwave domain. It also stimulates a search for thenew molecules that possess enhanced sensitivities to a variation of the proton-to-electron mass ratio.In this paper, we calculate the sensitivity coefficients for acetone, a molecule of astrophysical interesthaving a C2v equilibrium configuration with two equivalent methyl tops undergoing internal rotation.The calculations were done for rotational transitions belonging to the ground and the first excited(m12 � 75 cm�1) torsional states of acetone, since both these states were detected during observationsof interstellar molecular clouds. For rotational transitions with linestrengths above 1.0 and lower-levelrotation excitation energies below 10 cm�1 the sensitivity coefficients in this molecule range fromKl = �0.69 to �1.23 for the ground state and from Kl = �1.98 to +6.07 for the first excited torsional state.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

The Standard Model of particle physics that describes nature atthe fundamental level does not prohibit the spatial and temporalvariation of physical constants. Moreover, a number of moderntheories beyond the Standard Model foresee such variation (seefor example [1,2]). Discovery of variations of the Standard Modelconstants in space or in time would be an unambiguous indicationof a new physics and therefore new experimental data on the var-iation (or constraints on the variation) of fundamental constantsare of great interest for the development of new theoretical modelsbeyond the Standard Model. Recently it was shown that micro-wave spectra of molecules with large amplitude motions are highlysensitive to a proton-to-electron mass ratio variation. This findingstimulated an active search for the molecules which possess en-hanced sensitivities to a variation of the proton-to-electron massratio l. For example, the following polyatomic molecules withlarge amplitude motions were considered from this point of view:NH3 [3], H2O2 [4], H3O+ [5], CH3OH [6,7], CH3NH2 [8], CH3SH [9]and some of their isotopologues. In this paper we will consider ace-tone ((CH3)2CO), a molecule of astrophysical interest with two

equivalent methyl tops which internal rotation is hindered by arelatively low potential barrier. Such type of molecules hasnot been considered yet from the point of view of a possibleenhancement of sensitivity to a proton-to-electron mass ratiovariation.

The search for drifting constants on large spatial and temporalscales is carried out by observing spectra of the interstellar gasboth in our Galaxy and at large redshifts. To a considerable extent,the interest in this topic is stimulated by the increasing capabilitiesof the new astronomical instruments to measure spectral linesfrom very distant sources with very high precision. A variation ofl will manifest itself as a change in the observed spectrum linepositions. The response of a transition to a variation of l is charac-terized by the so-called sensitivity coefficient, Kl, which is definedas the proportionality constant between the relative frequencyshift of the transition and the relative shift in l value:

Dmm¼ Kl

Dll

ð1Þ

where m is the transition frequency, l = mp/me is the proton-to-elec-tron mass ratio, and Dm is an expected shift in the transition fre-quency that corresponds to the shift Dl in proton-to-electronmass ratio. For a sensitive test, one needs to study molecular tran-sitions that are observed with a good signal to noise ratio and that

Fig. 1. Energy level diagram of the lowest torsional states of acetone with up to twotorsional quanta excited [23]. Each level is splitted into four sublevels that arelabeled according to irreducible representations of the G36 group.

2 V.V. Ilyushin / Journal of Molecular Spectroscopy xxx (2014) xxx–xxx

exhibit different Kl values. As it was noticed by a number of authors[10–13], the sensitivity to a variation of a fundamental constant isenhanced when an accidental degeneracy occurs between levelswhich have substantially different dependences on this constant.In acetone, the enhancement of sensitivity may occur due to aninterplay between the torsional motion of the two equivalentmethyl groups and the rotational motion of the molecule as awhole.

Acetone is a relatively small and stable molecule that is abun-dantly present in our galaxy and easy to work with in the labora-tory. It was first detected in interstellar medium in 1987 towardSgr B2 [14]. In 2005 the first detection of acetone toward the highmass star forming region Orion-KL in its ground and vibrationallyexcited states was reported [15]. Recently the detection of vibra-tionally excited acetone was also reported toward Sgr B2 [16].The microwave, millimeter and sub-millimeter wave spectra ofacetone have been extensively studied for more than 50 years, asdocumented in a series of papers by Groner and collaboratorsfocusing on the two-dimensional torsional potential function[17], as well as on the rotational spectrum in the vibrational andtorsional ground state [18], in the first excited state of the lowertorsional mode m12 [19], and in the first excited state of the highertorsional mode m17 [20]. The fits of the rotational transitions inRefs. [18–20] were carried out using the ERHAM program[21,22], which fits the rotational levels of individual torsionalstates separately, without explicitly introducing a potentialfunction. Recently, a new program that makes use of an explicittwo-dimensional potential function and carries out a global fit ofrotational transitions in all three torsional states simultaneouslywas successfully applied to the analysis of the availableexperimental data [23]. Namely this model [23] will be used inthe calculations of the sensitivity coefficients to a possible variationof the proton-to-electron mass ratio presented in this paper.

The rest of the paper is organized as follows. In Section 2 thetheoretical model used in the calculations of the sensitivity coeffi-cients for the acetone spectrum is reviewed. In Section 3 the resultsof the Kl calculations are presented and discussed. In Section 4some conclusions are given.

2. Hamiltonian and energy level structure

Acetone, depicted on the right-hand side of Fig. 1, is a represen-tative of the two-equivalent-methyl-rotor molecules with C2v

point-group symmetry at equilibrium exhibiting two coupledtorsion large-amplitude motions. The intermediate potential bar-rier height combined with the leading role of the light hydrogenatoms in the large-amplitude motions result in relatively largetunneling splittings (up to �0.5 GHz in the ground state) in thespectrum of the molecule. On the left-hand side of Fig. 1, the low-est torsional states up to two torsional quanta excited in the tor-sional modes are depicted. Each torsional state is splitted intofour sublevels: one nondegenerate, two doubly degenerate andone four degenerate. The sublevels in Fig. 1 are labeled accordingto irreducible representations of the G36 permutation-inversion(PI) group [24] which is an appropriate PI group for acetone typemolecules.

To obtain reliable estimates of the sensitivity coefficients Kl ofdifferent molecular transitions we need to set up a model, i.e. aHamiltonian, which gives an adequate description of the consid-ered molecular spectrum. The Hamiltonian model used forthe calculations in the present work is based on the G36

permutation-inversion group-theoretical considerations. It makesuse of an explicit two-dimensional potential function and allowsa simultaneous fitting of rotational transitions belonging todifferent torsional states of the molecule [23]. The basic lowest or-der Hamiltonian can be written as follows:

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H¼AJ2z þBJ2

x þCJ2y þFðp2

Aþp2BÞþF 0ðpApBþpBpAÞ

þ12

V3ð1�cosð3aAÞþ1�cosð3aBÞÞ

þ12

V33ðcosð3aAÞcosð3aBÞ�1Þ

þ12

V 033 sinð3aAÞsinð3aBÞþ2FqxJxðpAþpBÞþ2FqzJzðpA�pBÞ ð2Þ

where Jx, Jy, Jz are projections on the x, y, z axes of the total angularmomentum J; pA, pB are the angular momenta conjugate to theinternal rotation angles aA and aB of the two methyl tops; A, B, Care the rotational constants; F, F0 are the kinetic parameters of thetorsional problem; V3, V33, V 033 are the parameters of the potentialfunction, and qx, qz are the coupling parameters between theangular momentum of the internal rotation and that of the globalrotation. For the details of computer realization of this model thereaders are referred to Ref. [23].

The energy levels of acetone have been calculated using thePAM_C2v_2tops code [23] and the set of 40 molecular parameterswhich was obtained from the fit of 1733 lines (1720 rotational fre-quencies from [18–20] and 13 FIR transitions taken from Table 2 of[17]). This set of parameters results from a global fit of rotationaltransitions belonging to the ground, first (m12 = 1) and second(m17 = 1) excited torsional states of acetone. For more details ofthe fit and the dataset the reader is referred to Ref. [23]. Here wewill only note that the overall weighted standard deviation of0.94 was achieved in this fit, showing an adequate description ofthe considered molecular spectrum. The parameter values are gi-ven in Table 1.

In order to calculate the Kl coefficients of transitions in acetone,we need to know the energies of all levels and their dependence onl. This translates into a problem of knowing how the molecularconstants scale with l. The sensitivities of the parameters to a l

://dx.doi.org/10.1016/j.jms.2014.03.004

Table 1Molecular parameters (in cm�1) of acetone [23] and their sensitivities KP

l to a variation of the proton-to-electron mass ratio l.

Operatora ParameterbKP

lValuec

pA2 + pB

2 F l�1 5.58995(89)pA pB + pB pA F0 l�1 �0.18326(86)(1/2)(1�c3aA + 1�c3aB) V3 l0 422.95(28)(1/2)(c3aAc3aB�1) V33 l0 180.86(32)(1/2)s3aAs3aB V033 l0 �176.406(37)(1/2)(1�c6aA + 1�c6aB) V6 l0 �7.982(14)(1/2)(pA � pB)Jz 4Fqz l�1 1.29039(27)(1/2)(pA + pB)Jx 4Fqx l�1 0.58231(13)

J2z

A�(1/2)(B + C) l�1 0.1462357(87)

J2 (1/2)(B + C) l�1 0.2282710(13)

J2x�J2

y(1/2)(B�C) l�1 0.0644708(12)

�J4 DJ l�2 0.19927(69) � 10�6

�J2 J2z

DJK l�2 �0.4492(39) � 10�6

�Jz4 DK l�2 0.10104(78) � 10�5

�2J2(J2x � J2

y ) dJ l�2 0.8654(34) � 10�7

�{J2z ,(J2

x � J2y )} dK l�2 0.969(17) � 10�7

(1/2)(pA2 + pB

2) J2 FJ l�2 �0.2977(33) � 10�5

(1/2)(pA2 + pB

2) J2z

FK l�2 �0.351(10) � 10�4

(1/2)(pA pB + pB pA) J2 F0 J l�2 0.405(15) � 10�5

(1/2)(pA pB + pB pA) J2z

F0K l�2 0.591(11) � 10�4

(1/2)(1�c3aA + 1�c3aB)J2z

V3K l�1 0.5593(10) � 10�2

(c3aAc3aB�1) J2 V33J l�1 0.114543(87) � 10�2

s3aAs3aB J2 V033J l�1 0.6979(36) � 10�3

s3aAs3aB J2z

V033K l�1 �0.2240(36) � 10�2

(1/2)(pA4 + pB

4) Fm l�2 �0.152(10) � 10�3

(1/2)(pA2 pB

2 + pB2 pA

2) F0m l�2 �0.3092(56) � 10�2

(1/4)(c3aA � c3aB)(JzJx + JxJz) V3zx l�1 0.1727(11) � 10�2

(1/4)(s3aA + s3aB)(JzJy + JyJz) V03zy l�1 �0.714(19) � 10�2

(1/4)(s3aA � s3aB)(JxJy + JyJx) V03xy l�1 �0.3456(65) � 10�2

(1/4)({pB,s3aA}�{pA,s3aB}) Jy V03my l�1 0.667(27) � 10�2

(c3aAc3aB�1)(J2x � J2

y ) V33xy l�1 0.97400(80) � 10�3

s3aAs3aB(J2x � J2

y ) V033xy l�1 0.5331(33) � 10�3

(1/2)(c3aBs3aA + c3aAs3aB)(JzJy + JyJz) VV033zy l�1 0.574(18) � 10�2

(1/2)(pA�pB)Jz3 qzK l�2 �0.1050(12) � 10�4

(1/2)(pA + pB)J3x

qxx l�2 �0.2876(52) � 10�5

(1/4)(pA + pB)(J2z Jx + JxJ

2z ) qxK l�2 0.431(28) � 10�5

(1/4)({pA2,pB}�{pB

2,pA})Jz qzm0 l�2 0.793(16) � 10�3

(1/4)(c3aA�c3aB)(JzJx + JxJz) J2 V3zxJ l�2 0.186(13) � 10�6

(1/4)(s3aA�s3aB)(JxJy + JyJx) J2 V03xyJ l�2 0.239(11) � 10�7

(1/4)({pA2,pB}�{pB

2,pA})Jz3 qzm

0K l�3 0.1002(48) � 10�7

a Operator which the parameter multiplies in the program, where {A, B} � AB + BA, cx = cosx and sx = sinx.b Parameter nomenclature follows the spirit of [17] and subscript procedures of [25].c Values are in cm�1. Statistical uncertainties are shown as one standard uncertainty in the last two digits given in parentheses.

V.V. Ilyushin / Journal of Molecular Spectroscopy xxx (2014) xxx–xxx 3

variation were determined in the same manner as it was done formethanol [6,26]. First, it is expected that, if the proton-to-electronmass ratio varies, the neutron-to-electron mass ratio will change inthe same way. Therefore, the atomic masses and hence the mo-ments of inertia are directly proportional to l. This gives us anopportunity to obtain the sensitivities of the rotational and tor-sional kinetic constants A, B, C, F, F0 and q, which are explicit func-tions of the different moments of inertia associated with themolecule (the corresponding expressions can be found in Ref.[27]). Second, within the Born–Oppenheimer approximation thetorsional potential parameters V3, V33, V033 are independent of themass of the nuclei and hence of l. The higher order constants areconsidered as products of the 8 second order constants in the Ham-iltonian. The sensitivities KP

l of the Hamiltonian parameters usedto calculate the Kl coefficients of transitions in acetone are givenin the third column of Table 1.

3. Sensitivity of selected transitions

A qualitative physical picture for the source of the enhancedtransition sensitivities may be obtained from a simple expression

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(easily derived from Eq. (1)) that connects the sensitivity coeffi-cients KEi

l , KEjl of the energy levels Ei, Ej with the sensitivity coeffi-

cient Kmijl of a transition frequency mij = Ei � Ej (here the energies

of the levels are given in frequency units):

Kmijl ¼ KEi

lEi

mij� K

Ejl

Ej

mij¼ KEi

l þ KEil � K

Ejl

� � Ej

mijð3Þ

From this expression it is obvious that if the sensitivity coeffi-cients are equal for the upper and lower levels KEi

l ¼ KEjl , then the

resulting sensitivity coefficient Kmijl for the transition will be equal

to the sensitivity coefficient of the levels. For example, this is whathappens in the case of pure rotational transitions of a rigid mole-cule. In the rigid rotor approximation the rotational energies areproportional to the sum of rotational constants (multiplied by cor-responding rotational quantum numbers) which are inversely pro-portional to the moments of inertia of the molecule. The momentsof inertia depend linearly on l, giving the result that all rotationalenergy levels have sensitivity coefficients KEi

l ¼ �1. This, accordingto (3), leads to the well known fact that the sensitivity coefficientsfor pure rotational transitions in semirigid molecules have the va-lue Kl = �1.

://dx.doi.org/10.1016/j.jms.2014.03.004

Table 2Sensitivity coefficients Kl of the rotational transitions of the ground vibrational state of acetone ((CH3)2CO) below 30 GHz with a linestrength S > 1.0 and a lower-level rotationexcitation energy Elow below 10 cm�1.

SYMa J KA KC SYMa J KA KC rArBb OBSERVED(UNC)c (MHz) Elow (cm�1) Sl2

ed (D2) Kl

E4 1 1 0 E4 1 0 1 11 5253.0543(0.0018) 0.6004 0.770E+02 �1.01G 1 1 0 G 1 0 1 01 5269.0715(0.0008) 0.5242 0.205E+03 �1.02E2 1 1 0 E1 1 0 1 12 5270.9096(0.0009) 0.6006 0.511E+02 �1.03A4 1 1 0 A2 1 0 1 00 5276.0565(0.0011) 0.4480 0.129E+03 �1.02E3 2 2 0 E3 2 1 1 11 7343.7663(0.0034) 1.7910 0.345E+02 �1.01A1 2 2 0 A3 2 1 1 00 7398.9389(0.0024) 1.6403 0.104E+03 �1.03G 2 2 0 G 2 1 1 01 7399.9695(0.0019) 1.7157 0.273E+03 �1.04E1 2 2 0 E2 2 1 1 12 7453.2120(0.0032) 1.7915 0.661E+02 �1.10E2 3 3 1 E1 3 2 1 12 9316.3024(0.0126) 3.5891 0.122E+02 �0.72G 3 3 1 G 3 2 1 01 9707.4679(0.0056) 3.5142 0.210E+02 �0.93E4 3 2 1 E4 3 1 2 11 10731.3416(0.0018) 3.2301 0.135E+03 �1.00E1 3 2 1 E2 3 1 2 12 10749.9332(0.0015) 3.2305 0.901E+02 �1.01G 3 2 1 G 3 1 2 01 10751.6656(0.0013) 3.1556 0.362E+03 �1.01A2 3 2 1 A4 3 1 2 00 10762.7138(0.0018) 3.0811 0.227E+03 �1.01E4 3 3 0 E4 3 2 1 11 11073.9541(0.0050) 3.5880 0.106E+03 �1.00A4 3 3 0 A2 3 2 1 00 11199.2838(0.0050) 3.4401 0.175E+03 �1.04E3 2 1 1 E3 2 0 2 11 11252.5016(0.0018) 1.4157 0.226E+02 �0.99G 3 3 0 G 3 2 1 01 11257.7317(0.0044) 3.5142 0.260E+03 �1.08E2 2 1 1 E1 2 0 2 12 11265.2550(0.0017) 1.4158 0.451E+02 �1.00G 2 1 1 G 2 0 2 01 11272.4615(0.0014) 1.3397 0.181E+03 �1.00A3 2 1 1 A1 2 0 2 00 11286.1522(0.0022) 1.2639 0.680E+02 �1.00E2 3 3 0 E1 3 2 1 12 11468.3125(0.0085) 3.5891 0.581E+02 �1.19E3 4 3 1 E3 4 2 2 11 11525.0613(0.0045) 5.6235 0.626E+02 �1.01G 4 3 1 G 4 2 2 01 11569.9412(0.0020) 5.5507 0.500E+03 �1.02E2 4 3 1 E1 4 2 2 12 11583.0881(0.0025) 5.6243 0.124E+03 �1.04A3 4 3 1 A1 4 2 2 00 11585.2383(0.0026) 5.4777 0.188E+03 �1.02E4 5 4 1 E4 5 3 2 11 14140.7891(0.0068) 8.6034 0.208E+03 �1.00G 5 4 1 G 5 3 2 01 14263.9831(0.0039) 8.5327 0.546E+03 �1.04A2 5 4 1 A4 5 3 2 00 14276.7876(0.0057) 8.4616 0.346E+03 �1.03E1 5 4 1 E2 5 3 2 12 14348.4618(0.0065) 8.6046 0.132E+03 �1.10E1 4 4 1 E2 4 3 1 12 14913.2749(0.0209) 6.0107 0.185E+02 �0.70E2 1 1 1 E1 0 0 0 12 15038.5126(0.0016) 0.1529 0.341E+02 �0.99E3 1 1 1 E3 0 0 0 11 15064.9286(0.0013) 0.1528 0.172E+02 �1.00G 1 1 1 G 0 0 0 01 15074.0771(0.0012) 0.0763 0.137E+03 �1.00A3 1 1 1 A1 0 0 0 00 15096.3316(0.0014) 0.0000 0.515E+02 �1.01E1 2 2 1 E2 2 1 2 12 15615.7316(0.0045) 1.4304 0.278E+02 �0.95G 4 4 1 G 4 3 1 01 15627.5448(0.0104) 5.9367 0.596E+02 �0.90E4 2 2 1 E4 2 1 2 11 15736.2406(0.0026) 1.4306 0.429E+02 �1.00G 2 2 1 G 2 1 2 01 15750.8578(0.0022) 1.3547 0.113E+03 �0.99A2 2 2 1 A4 2 1 2 00 15827.7439(0.0031) 1.2791 0.715E+02 �1.02E3 4 4 0 E3 4 3 1 11 16474.9990(0.0074) 6.0079 0.324E+02 �0.99A1 4 4 0 A3 4 3 1 00 16689.8268(0.0083) 5.8642 0.965E+02 �1.04G 4 4 0 G 4 3 1 01 16912.3785(0.0079) 5.9367 0.198E+03 �1.11E4 5 3 2 E4 5 2 3 11 16920.8754(0.0027) 8.0390 0.188E+03 �1.00G 5 3 2 G 5 2 3 01 16937.6893(0.0025) 7.9677 0.503E+03 �1.00E2 5 3 2 E1 5 2 3 12 16943.4969(0.0033) 8.0395 0.125E+03 �1.01A4 5 3 2 A2 5 2 3 00 16943.5360(0.0037) 7.8964 0.315E+03 �1.00E1 4 4 0 E2 4 3 1 12 17281.5627(0.0132) 6.0107 0.460E+02 �1.18E2 3 3 1 E1 3 2 2 12 18046.0638(0.0102) 3.2979 0.378E+02 �0.87G 3 3 1 G 3 2 2 01 18392.1897(0.0044) 3.2245 0.164E+03 �0.97E3 3 3 1 E3 3 2 2 11 18448.9866(0.0040) 3.2986 0.219E+02 �1.00A3 3 3 1 A1 3 2 2 00 18580.9892(0.0045) 3.1510 0.658E+02 �1.02E3 4 2 2 E3 4 1 3 11 18630.9748(0.0041) 5.0020 0.391E+02 �0.99E1 4 2 2 E2 4 1 3 12 18651.6549(0.0036) 5.0022 0.782E+02 �1.00G 4 2 2 G 4 1 3 01 18654.3022(0.0027) 4.9285 0.314E+03 �1.00A1 4 2 2 A3 4 1 3 00 18667.5965(0.0042) 4.8550 0.118E+03 �1.00G 3 3 0 G 3 2 2 01 19942.4535(0.0043) 3.2245 0.113E+02 �1.05E4 3 1 2 E4 3 0 3 11 20392.0975(0.0037) 2.5499 0.581E+02 �0.99E2 3 1 2 E1 3 0 3 12 20404.2345(0.0031) 2.5499 0.388E+02 �1.00G 3 1 2 G 3 0 3 01 20426.4912(0.0025) 2.4742 0.155E+03 �1.00A4 3 1 2 A2 3 0 3 00 20454.9313(0.0040) 2.3988 0.972E+02 �1.00E2 5 5 1 E1 5 4 1 12 21152.5865(0.0299) 9.0832 0.146E+02 �0.69E1 4 4 1 E2 4 3 2 12 21273.6688(0.0151) 5.7985 0.392E+02 �0.84G 4 4 1 G 4 3 2 01 21845.8807(0.0082) 5.7292 0.164E+03 �0.94E4 4 4 1 E4 4 3 2 11 22083.0075(0.0059) 5.8012 0.783E+02 �1.00G 5 5 1 G 5 4 1 01 22186.7926(0.0150) 9.0085 0.729E+02 �0.89A2 4 4 1 A4 4 3 2 00 22268.5271(0.0064) 5.6588 0.130E+03 �1.02E1 3 2 2 E2 3 1 3 12 22346.2799(0.0031) 2.5525 0.362E+02 �0.99E3 3 2 2 E3 3 1 3 11 22367.8727(0.0029) 2.5525 0.181E+02 �1.00G 3 2 2 G 3 1 3 01 22413.0522(0.0024) 2.4769 0.145E+03 �1.00A1 3 2 2 A3 3 1 3 00 22468.8896(0.0039) 2.4015 0.544E+02 �1.01A1 2 0 2 A3 1 1 1 00 22793.2800(0.0022) 0.5036 0.622E+02 �1.00E3 2 0 2 E3 1 1 1 11 22794.3192(0.0025) 0.6553 0.208E+02 �1.00

4 V.V. Ilyushin / Journal of Molecular Spectroscopy xxx (2014) xxx–xxx

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Table 2 (continued)

SYMa J KA KC SYMa J KA KC rArBb OBSERVED(UNC)c (MHz) Elow (cm�1) Sl2

ed (D2) Kl

G 2 0 2 G 1 1 1 01 22800.3771(0.0021) 0.5792 0.166E+03 �1.00E1 2 0 2 E2 1 1 1 12 22820.8319(0.0024) 0.6545 0.414E+02 �1.01E4 5 5 0 E4 5 4 1 11 22938.3641(0.0098) 9.0751 0.911E+02 �0.99G 4 4 0 G 4 3 2 01 23130.7143(0.0082) 5.7292 0.451E+02 �1.09A4 5 5 0 A2 5 4 1 00 23231.5526(0.0110) 8.9378 0.151E+03 �1.04E2 4 3 2 E1 4 2 3 12 23352.2443(0.0041) 5.0196 0.638E+02 �0.97E4 4 3 2 E4 4 2 3 11 23426.9908(0.0033) 5.0197 0.962E+02 �1.00G 4 3 2 G 4 2 3 01 23469.2363(0.0028) 4.9464 0.256E+03 �1.00A4 4 3 2 A2 4 2 3 00 23549.0307(0.0041) 4.8733 0.160E+03 �1.01G 5 5 0 G 5 4 1 01 23620.9580(0.0101) 9.0085 0.169E+03 �1.10E1 4 4 0 E2 4 3 2 12 23641.9565(0.0167) 5.7985 0.134E+02 �1.18E2 5 5 0 E1 5 4 1 12 23997.9460(0.0153) 9.0832 0.457E+02 �1.13E2 2 1 2 E1 1 0 1 12 24876.5508(0.0022) 0.6006 0.515E+02 �1.00E4 2 1 2 E4 1 0 1 11 24888.2504(0.0022) 0.6004 0.772E+02 �1.00G 2 1 2 G 1 0 1 01 24899.5091(0.0021) 0.5242 0.206E+03 �1.00A4 2 1 2 A2 1 0 1 00 24916.4902(0.0022) 0.4480 0.129E+03 �1.00E1 5 4 2 E2 5 3 3 12 25078.5162(0.0079) 8.1028 0.826E+02 �0.94E3 5 4 2 E3 5 3 3 11 25305.5334(0.0045) 8.1034 0.427E+02 �1.00G 5 4 2 G 5 3 3 01 25325.2259(0.0040) 8.0327 0.338E+03 �0.99A1 5 4 2 A3 5 3 3 00 25464.1777(0.0051) 7.9626 0.128E+03 �1.02E2 5 5 1 E1 5 4 2 12 25468.0751(0.0164) 8.9393 0.442E+02 �0.87G 5 5 1 G 5 4 2 01 26115.0027(0.0107) 8.8775 0.160E+03 �0.94E3 5 5 1 E3 5 4 2 11 26526.3858(0.0080) 8.9475 0.282E+02 �1.00A3 5 5 1 A1 5 4 2 00 26772.7468(0.0087) 8.8120 0.842E+02 �1.03G 5 5 0 G 5 4 2 01 27549.1681(0.0122) 8.8775 0.653E+02 �1.11E2 5 5 0 E1 5 4 2 12 28313.4346(0.0252) 8.9393 0.126E+02 �1.23E4 5 2 3 E4 5 1 4 11 28629.9139(0.0052) 7.0840 0.110E+03 �0.99E1 5 2 3 E2 5 1 4 12 28643.1242(0.0046) 7.0840 0.733E+02 �1.00G 5 2 3 G 5 1 4 01 28671.0639(0.0036) 7.0114 0.293E+03 �1.00A2 5 2 3 A4 5 1 4 00 28705.7396(0.0057) 6.9389 0.184E+03 �1.00E3 4 1 3 E3 4 0 4 11 29844.6883(0.0043) 4.0065 0.192E+02 �0.99E2 4 1 3 E1 4 0 4 12 29848.7297(0.0041) 4.0065 0.383E+02 �1.00G 4 1 3 G 4 0 4 01 29898.9174(0.0033) 3.9312 0.153E+03 �1.00A3 4 1 3 A1 4 0 4 00 29951.0895(0.0053) 3.8560 0.576E+02 �1.01

a Species in G36, using the species notation and characters in Table A-28 of [24].b Symmetry species in G9 (see Table 2 of [23]).c Estimated uncertainties in MHz are given in parentheses following the calculated frequencies. Asymmetric rotor quantum numbers for upper and lower states are given

on the left.d This column lists the transition strength S multiplied by the electric dipole moment le squared.

V.V. Ilyushin / Journal of Molecular Spectroscopy xxx (2014) xxx–xxx 5

On the other hand, if the upper and lower energy levels havesomewhat different sensitivity coefficients KEi

l –KEjl then we may

expect some ‘‘resonance’’ enhancement in the sensitivity coeffi-cient K

mijl of the transition frequency mij in the case of close lying en-

ergy levels (i.e. when mij is small). In acetone the difference in KEil

coefficients of the rotational energy levels is provided by the cou-pling between the internal rotation of the methyl tops and overallrotation of the molecule, which have different dependence on l.From (3) it is clear that the transitions with ‘‘resonance’’ enhance-ment in their sensitivity coefficients will become more numeroustowards lower frequencies since this minimizes the denominatorof the ‘‘resonance term’’ in (3).

Using the scaling relations for the Hamiltonian parameters dis-cussed in previous section we now able to calculate numericallythe sensitivity coefficient of any desired transition within the rangeof applicability of the Hamiltonian model. In order to do numericalcalculations, we can rewrite Eq. (1) as

Kmijl ¼

mþij � m�ij2emij

ð4Þ

with mij being the transition frequency between states i and j for thepresent value of l and m�ij being the transition frequency when l isreplaced by lð1� eÞ with e being a number much smaller than 1. mij

is calculated using values for the molecular constants as listed inTable 1, and m�ij are calculated using the molecular constants scaledaccording to the relations that were determined in the previoussection.

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It may be shown that another way to calculate the sensitivitycoefficients for the desired transitions is to use an equation that di-rectly connects the sensitivity coefficient of a transition with thesensitivity coefficients of the Hamiltonian parameters [8]:

Kmijl ¼

1mij

Xs

KPsl Ps

@Ei

@Ps� @Ej

@Ps

� �ð5Þ

where @Ei@Ps¼ ih jOs ij i is the derivative of the energy level Ei with re-

spect to the Hamiltonian parameter Ps obtained using the Hell-mann–Feynman theorem, and KPs

l is the sensitivity coefficient ofthe s-th Hamiltonian parameter. Eq. (5) is based on the assumptionthat the energy of state ij i may be represented as Ei ¼

PsPs ih jOs ij i

which is valid when the Hamiltonian depends linearly on theparameters, i.e., when the Hamiltonian may be written asH ¼

PsPsOs. This assumption holds for the current model used for

the calculations of acetone spectrum and we use this procedure tocalculate the Kl coefficients in the present work.

From Eq. (5) it is seen that contributions to Kmijl from different

terms in the Hamiltonian are proportional to the relative contribu-tions of these terms to the transition frequency. Therefore, it isobvious that the resulting sensitivity coefficients are mainly deter-mined by the largest terms in the Hamiltonian, which physicalinterpretation and dependence on l are rather clear. Eq. (5) alsoillustrates the fact that the largest enhancement may be obtainedfor transitions that connect two near degenerate levels (i.e. whenmij is small) which have substantially different dependences on l.The different dependence on l is provided when the two levels

://dx.doi.org/10.1016/j.jms.2014.03.004

Table 3Sensitivity coefficients Kl of the rotational transitions of the first excited torsional state of acetone ((CH3)2CO) below 30 GHz with a linestrength S > 1.0 and a lower-level rotationexcitation energy Elow below 10 cm�1.

SYMa J KA KC SYMa J KA KC rArBb OBSERVED(UNC)c (MHz) Elow (cm�1) Sl2

ed (D2) Kl

E1 1 1 1 E2 1 0 1 12 685.5900(0.0408) 75.0679 0.948E+01 5.99G 1 1 1 G 1 0 1 01 1371.5082(0.0065) 75.6186 0.142E+02 0.08G 3 2 2 G 3 1 2 01 1778.8580(0.0087) 78.2439 0.960E+01 �0.08E2 2 2 1 E1 2 1 1 12 1797.9723(0.1071) 76.2771 0.223E+02 6.07E1 4 3 2 E2 4 2 2 12 2283.3782(0.1229) 80.1268 0.250E+02 5.45E1 3 3 1 E2 3 2 1 12 3440.4018(0.1944) 78.1137 0.182E+02 5.82G 2 2 1 G 2 1 1 01 3678.9737(0.0154) 76.8110 0.606E+02 0.57E2 5 4 2 E1 5 3 2 12 4167.0308(0.2101) 83.1443 0.343E+02 5.18G 4 3 2 G 4 2 2 01 4346.8589(0.0179) 80.6361 0.484E+02 0.22A2 1 1 0 A4 1 0 1 00 5105.4921(0.0029) 76.1962 0.129E+03 �0.90G 1 1 0 G 1 0 1 01 5690.6586(0.0076) 75.6186 0.181E+03 �1.46E4 1 1 0 E4 1 0 1 11 5758.3626(0.0371) 75.0640 0.664E+02 �1.45E1 1 1 0 E2 1 0 1 12 6398.0199(0.0457) 75.0679 0.420E+02 �1.83G 3 3 1 G 3 2 1 01 6763.5390(0.0241) 78.6174 0.804E+02 0.81A3 2 2 0 A1 2 1 1 00 7063.7970(0.0047) 77.3775 0.105E+03 �0.86G 5 4 2 G 5 3 2 01 7783.4160(0.0259) 83.6208 0.111E+03 0.54E3 2 2 0 E3 2 1 1 11 8089.7772(0.0519) 76.2594 0.302E+02 �1.43G 2 2 0 G 2 1 1 01 8634.0992(0.0118) 76.8110 0.203E+03 �1.69E2 2 2 0 E1 2 1 1 12 10274.6172(0.0852) 76.2771 0.461E+02 �1.98A4 3 2 1 A2 3 1 2 00 10519.8341(0.0059) 78.8087 0.224E+03 �0.94A2 3 3 0 A4 3 2 1 00 10569.4863(0.0085) 79.1596 0.179E+03 �0.82G 4 4 1 G 4 3 1 01 10706.0646(0.0351) 81.0592 0.485E+02 0.88A1 2 1 1 A3 2 0 2 00 11074.3564(0.0070) 77.0081 0.668E+02 �0.95A1 4 3 1 A3 4 2 2 00 11174.5339(0.0057) 81.1848 0.189E+03 �0.89G 3 2 1 G 3 1 2 01 11197.9745(0.0089) 78.2439 0.339E+03 �1.26E3 2 1 1 E3 2 0 2 11 11299.7167(0.0075) 75.8825 0.226E+02 �1.02E4 3 2 1 E4 3 1 2 11 11338.1034(0.0453) 77.6962 0.127E+03 �1.29G 2 1 1 G 2 0 2 01 11350.9462(0.0052) 76.4324 0.177E+03 �1.07E4 3 3 0 E4 3 2 1 11 11698.2554(0.0439) 78.0744 0.994E+02 �1.17E1 2 1 1 E2 2 0 2 12 11907.8506(0.0326) 75.8799 0.428E+02 �1.30E2 3 2 1 E1 3 1 2 12 12117.4490(0.0603) 77.7095 0.804E+02 �1.58G 4 3 1 G 4 2 2 01 12683.0508(0.0155) 80.6361 0.430E+03 �1.50E3 4 3 1 E3 4 2 2 11 12692.2680(0.0831) 80.0941 0.557E+02 �1.45G 3 3 0 G 3 2 1 01 13410.2121(0.0138) 78.6174 0.192E+03 �1.58E2 2 2 1 E1 2 1 2 12 13492.5537(0.0637) 75.8870 0.273E+02 �0.43A4 5 4 1 A2 5 3 2 00 13544.6184(0.0088) 84.1426 0.352E+03 �0.84E1 1 1 1 E2 0 0 0 12 14115.6467(0.0401) 74.6200 0.280E+02 �0.66E1 4 3 1 E2 4 2 2 12 14415.8061(0.0982) 80.1268 0.979E+02 �1.82E1 3 3 1 E2 3 2 2 12 14609.1704(0.0828) 77.7412 0.459E+02 �0.30G 2 2 1 G 2 1 2 01 14632.7983(0.0115) 76.4456 0.102E+03 �0.68G 1 1 1 G 0 0 0 01 14714.5616(0.0045) 75.1736 0.126E+03 �0.87A1 1 1 1 A3 0 0 0 00 14926.8508(0.0031) 75.7499 0.515E+02 �0.97G 5 5 1 G 5 4 1 01 15023.0705(0.0470) 84.1656 0.151E+02 0.90E3 1 1 1 E3 0 0 0 11 15159.4506(0.0073) 74.6230 0.172E+02 �1.01A4 2 2 1 A2 2 1 2 00 15318.9277(0.0086) 77.0217 0.715E+02 �0.90E4 5 4 1 E4 5 3 2 11 15356.4282(0.0809) 83.0938 0.190E+03 �1.32A3 4 4 0 A1 4 3 1 00 15681.8377(0.0132) 81.5576 0.993E+02 �0.80E1 3 3 0 E2 3 2 1 12 15796.0534(0.1015) 78.1137 0.478E+02 �1.78E4 2 2 1 E4 2 1 2 11 16157.6782(0.0286) 75.9019 0.422E+02 �1.09G 3 3 1 G 3 2 2 01 16182.6554(0.0160) 78.3032 0.153E+03 �0.53G 5 4 1 G 5 3 2 01 16334.8644(0.0151) 83.6208 0.423E+03 �1.57A2 5 3 2 A4 5 2 3 00 16725.9794(0.0103) 83.5847 0.309E+03 �0.97E3 4 4 0 E3 4 3 1 11 16965.6781(0.0572) 80.5175 0.312E+02 �1.01E4 5 3 2 E4 5 2 3 11 17268.7938(0.0298) 82.5177 0.185E+03 �1.13G 5 3 2 G 5 2 3 01 17274.7014(0.0086) 83.0445 0.486E+03 �1.14E2 4 4 1 E1 4 3 2 12 17703.0920(0.0799) 80.2030 0.566E+02 �0.44A1 3 3 1 A3 3 2 2 00 17880.0459(0.0099) 78.8714 0.661E+02 �0.89E1 5 3 2 E2 5 2 3 12 18276.4160(0.0651) 82.5347 0.117E+03 �1.41A3 4 2 2 A1 4 1 3 00 18416.4866(0.0120) 80.5705 0.116E+03 �0.97E3 4 2 2 E3 4 1 3 11 18452.3308(0.0203) 79.4786 0.395E+02 �0.95G 4 2 2 G 4 1 3 01 18680.5050(0.0101) 80.0130 0.308E+03 �1.03G 1 1 0 G 0 0 0 01 19033.7120(0.0057) 75.1736 0.108E+02 �1.11E3 3 3 1 E3 3 2 2 11 19034.4498(0.0408) 77.7814 0.215E+02 �1.10G 4 4 1 G 4 3 2 01 19042.2565(0.0183) 80.7811 0.194E+03 �0.55E2 5 4 1 E1 5 3 2 12 19091.9331(0.1235) 83.1443 0.988E+02 �1.80G 4 4 0 G 4 3 1 01 19438.0084(0.0172) 81.0592 0.198E+03 �1.36E2 4 2 2 E1 4 1 3 12 19438.5429(0.0436) 79.4784 0.759E+02 �1.24G 2 2 0 G 2 1 2 01 19587.9237(0.0142) 76.4456 0.148E+02 �1.36A2 3 1 2 A4 3 0 3 00 20075.6669(0.0126) 78.1390 0.959E+02 �0.95E4 3 1 2 E4 3 0 3 11 20259.5492(0.0223) 77.0204 0.583E+02 �0.95G 3 1 2 G 3 0 3 01 20327.7572(0.0098) 77.5658 0.154E+03 �0.99E1 3 1 2 E2 3 0 3 12 20868.3642(0.0266) 77.0134 0.385E+02 �1.11A4 4 4 1 A2 4 3 2 00 21322.0574(0.0120) 81.3488 0.132E+03 �0.87

6 V.V. Ilyushin / Journal of Molecular Spectroscopy xxx (2014) xxx–xxx

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Table 3 (continued)

SYMa J KA KC SYMa J KA KC rArBb OBSERVED(UNC)c (MHz) Elow (cm�1) Sl2

ed (D2) Kl

E1 4 3 2 E2 4 2 3 12 21451.8550(0.0682) 79.4874 0.621E+02 �0.61E2 5 4 2 E1 5 3 3 12 21454.2987(0.0975) 82.5677 0.842E+02 �0.42E2 3 2 2 E1 3 1 3 12 21777.3896(0.0301) 77.0148 0.357E+02 �0.84A2 5 5 0 A4 5 4 1 00 21865.4634(0.0175) 84.5944 0.155E+03 �0.81A3 3 2 2 A1 3 1 3 00 21885.9263(0.0123) 78.1414 0.540E+02 �0.93G 3 2 2 G 3 1 3 01 22034.9432(0.0078) 77.5682 0.143E+03 �0.92E2 4 4 0 E1 4 3 1 12 22037.4587(0.0940) 80.6076 0.514E+02 �1.54G 4 3 2 G 4 2 3 01 22549.4432(0.0110) 80.0289 0.246E+03 �0.81E3 2 0 2 E3 1 1 1 11 22599.0785(0.0165) 75.1287 0.199E+02 �0.95E3 3 2 2 E3 3 1 3 11 22703.6269(0.0237) 77.0241 0.181E+02 �1.05A3 2 0 2 A1 1 1 1 00 22792.3377(0.0041) 76.2478 0.629E+02 �1.00E4 4 4 1 E4 4 3 2 11 22809.0694(0.0531) 80.3009 0.768E+02 �1.09G 3 3 0 G 3 2 2 01 22829.3286(0.0215) 78.3032 0.312E+02 �1.54E1 5 5 1 E2 5 4 2 12 22835.7066(0.0722) 83.2833 0.594E+02 �0.65A2 4 3 2 A4 4 2 3 00 22865.3924(0.0116) 80.5861 0.160E+03 �0.92G 2 0 2 G 1 1 1 01 23024.2873(0.0063) 75.6644 0.158E+03 �1.06G 5 4 2 G 5 3 3 01 23337.7677(0.0163) 83.1019 0.309E+03 �0.65E4 5 5 0 E4 5 4 1 11 23547.1938(0.0784) 83.6060 0.882E+02 �0.99G 5 5 1 G 5 4 2 01 23574.5188(0.0231) 83.8804 0.224E+03 �0.69E2 2 0 2 E1 1 1 1 12 23655.4078(0.0348) 75.0908 0.380E+02 �1.17E4 4 3 2 E4 4 2 3 11 23930.5081(0.0339) 79.5027 0.955E+02 �1.07E1 2 1 2 E2 1 0 1 12 24554.2670(0.0185) 75.0679 0.496E+02 �0.91A3 5 4 2 A1 5 3 3 00 24612.6181(0.0113) 83.6419 0.128E+03 �0.90A2 2 1 2 A4 1 0 1 00 24748.6298(0.0041) 76.1962 0.129E+03 �0.98G 2 1 2 G 1 0 1 01 24792.9172(0.0041) 75.6186 0.202E+03 �0.98E4 2 1 2 E4 1 0 1 11 25118.4600(0.0152) 75.0640 0.760E+02 �1.04A1 5 5 1 A3 5 4 2 00 25551.0076(0.0153) 84.4629 0.855E+02 �0.86G 5 5 0 G 5 4 1 01 25781.3864(0.0203) 84.1656 0.216E+03 �1.20E3 5 4 2 E3 5 3 3 11 25999.5313(0.0472) 82.6020 0.422E+02 �1.09E3 5 5 1 E3 5 4 2 11 27390.7915(0.0662) 83.4693 0.277E+02 �1.07G 4 4 0 G 4 3 2 01 27774.2002(0.0307) 80.7811 0.262E+02 �1.68A4 5 2 3 A2 5 1 4 00 28222.2490(0.0167) 82.6433 0.181E+03 �0.96E1 5 5 0 E2 5 4 1 12 28283.4185(0.0823) 83.7812 0.541E+02 �1.39E4 5 2 3 E4 5 1 4 11 28454.7138(0.0301) 81.5686 0.110E+03 �0.95G 5 2 3 G 5 1 4 01 28518.4528(0.0128) 82.0933 0.292E+03 �0.99E2 5 2 3 E1 5 1 4 12 29135.4830(0.0318) 81.5629 0.734E+02 �1.09A1 4 1 3 A3 4 0 4 00 29311.2476(0.0173) 79.5928 0.570E+02 �0.94G 4 1 3 G 4 0 4 01 29710.5662(0.0112) 79.0220 0.153E+03 �0.98A4 4 2 3 A2 4 1 4 00 29769.2411(0.0169) 79.5931 0.945E+02 �0.93A1 5 3 3 A3 5 2 4 00 29852.3877(0.0154) 82.6462 0.106E+03 �0.93E3 4 1 3 E3 4 0 4 11 29962.6418(0.0209) 78.4791 0.193E+02 �1.00

a Species in G36, using the species notation and characters in Table A-28 of [24].b Symmetry species in G9 (see Table 2 of [23]).c Estimated uncertainties in MHz are given in parentheses following the calculated frequencies. Asymmetric rotor quantum numbers for upper and lower states are given

on the left.d This column lists the transition strength S multiplied by the electric dipole moment le squared.

V.V. Ilyushin / Journal of Molecular Spectroscopy xxx (2014) xxx–xxx 7

contain nonequal contributions from different types of motions inthe molecule (i.e. nonequal contributions from Hamiltonian termsthat have different dependence on l). In that case, a transition‘‘converts’’ one superposition of rotation–torsion motion to an-other superposition of rotation–torsion motion. A significantenhancement is obtained when a ‘‘cancellation’’ takes place, i.e.,when two levels have nearly the same total energy due to quanti-tatively different contributions from various types of motion in themolecule [6,26].

The calculations of the Kmijl coefficients were done for all rota-

tional transitions in the ground and first excited (m12 = 1) torsionalstates of acetone with J 6 30, Ka 6 15 and mij below 300 GHz. Forboth vibrational states only the transitions with the linestrengthsabove 1.0 and involving levels that have rotational excitation en-ergy below 10 cm�1 were considered (i.e. for the first excited(m12 = 1) torsional state with the band origin of �75 cm�1 the rota-tional levels with energies below �85 cm�1 were considered).These limits on the linestrength and on the excitational energycome from the astrophysical observational constraints on the tran-sitions that are detectable in the interstellar medium. Although thecalculations were done for J 6 30, Ka 6 15, it appeared that onlytransitions with J < 10 satisfy the linestrength and excitation

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energy constraints mentioned above. Despite all these constraintsthe resulting list of transitions is rather lengthy and therefore onlytransitions with mij below 30 GHz are given here, since namely lowfrequency transitions are expected to exhibit the largest enhance-ment in sensitivity. In Table 2 the transitions belonging to theground torsional state of acetone are presented, whereas Table 3gives the rotational transitions belonging to the first excited(m12 = 1) torsional state of the molecule. The full lists of transitionsthat satisfy the criteria mentioned above are given as the Supple-mentary material with this article.

Inspection of Tables 2 and 3 shows that taking into account theconstraints mentioned above the obtained sensitivity coefficientsin acetone molecule range from Kl = �0.69 to �1.23 for the groundstate and from Kl = �1.98 to +6.07 for the first excited torsionalstate. The wider range of the variation of the sensitivity coefficientsin the case of torsionally excited state may be explained by thestronger coupling between torsional and rotational motions in thisstate in comparison with the ground state. As it was already notedthe enhanced sensitivity to the variation of a fundamental constantmay be obtained when the two close lying levels have a substan-tially different dependence on the fundamental constant. In ace-tone the difference in Kl coefficients of the rotational energy

://dx.doi.org/10.1016/j.jms.2014.03.004

8 V.V. Ilyushin / Journal of Molecular Spectroscopy xxx (2014) xxx–xxx

levels is provided by the coupling between the internal rotation ofthe methyl tops and overall rotation of the molecule. The couplingbetween the two motions is represented by the 2FqxJxðpA þ pBÞ and2FqzJzðpA � pBÞ terms in the Hamiltonian (2) and it is seen that theamount of mixing between internal and overall rotations dependsboth on the rotational state and on the torsional state, since matrixelements of the angular momenta conjugate to the internal rota-tion angles aA and aB of the two methyl tops will be different fordifferent torsional states. This mechanism gives rise to differentcontributions of the internal rotation energy to the rotational en-ergy levels with different K quantum numbers, and therefore levelswith different K quantum numbers have different Kl coefficients.This difference becomes larger in the excited torsional state be-cause of the larger amount of coupling between the torsional androtational motions and this in turn provides the wider variabilityin Kl coefficients in the torsional excited state.

Although molecules with large amplitude torsional motionshave attracted considerable attention as a possible source of tran-sitions with enhanced sensitivity to the proton-to-electron massratio variation, only ground vibrational states in these moleculeshave been considered up to now [6–9,26], since those states aremost relevant for astrophysical searches. Nevertheless, for a num-ber of molecules transitions belonging to excited torsional stateswere observed in the interstellar medium (e.g. methanol [28], acet-aldehyde [16], methyl formate [29]), and, as is seen from the re-sults obtained for acetone, transitions from excited torsionalstates may be even more sensitive to the proton-to-electron massratio variation than corresponding transitions in the ground vibra-tional states. It is interesting to note that sensitivity coefficients inthe second excited torsional state of acetone m17 = 1 range from�7.39 to +11.22, providing even larger variability than in the firstexcited state m12 = 1 (the m17 = 1 calculations are not consideredin detail here since no interstellar observation of the m17 = 1 statehas been reported). Therefore, even though interstellar linesbelonging to excited torsional states are less preferable from anobservational point of view, it is worth considering transitionsfrom excited torsional states as secondary test lines in the searchfor a possible proton-to-electron mass ratio variation.

4. Conclusions

In this paper, the sensitivity coefficients for transitions betweenlow-lying rotation levels of the ground and first excited torsionalstates in acetone have been calculated. The reported sensitivitiesspan a range from Kl = �0.69 to �1.23 for the ground state andfrom Kl = �1.98 to +6.07 for the first excited torsional state. Thisenhancement in sensitivity occurs due to an interplay of the energycontributions associated with the torsional and rotational degreesof freedom of the molecule. Comparison of acetone ((CH3)2CO) casewith such molecules as methanol (CH3OH), methylamine(CH3NH2), and methyl mercaptan (CH3SH), which have sensitivitycoefficients in the range from �42 to +53 [6], from �19 to +24[8], and from �14.8 to +12.2 [9] respectively, shows that acetonecan be recommended only as a supplementary probe moleculefor astrophysical tests on a possible l variation. Nevertheless, itwould be interesting to make corresponding calculations of thesensitivity coefficients for another astrophysical molecule of the

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same type as acetone, namely, dimethylether ((CH3)2O). Thismolecule is also abundantly present in the interstellar mediumbut it has a higher value of the q parameter that describes the cou-pling between the internal rotation and the global rotation in themolecule (0.217 versus 0.063 in acetone). The higher value of theq parameter in the molecules with internal rotation provides morechances to get enhancement in the sensitivity of rotational transi-tions [26] although in dimethylether this factor is hampered by arather high barrier to internal rotation. The direct calculations ofthe sensitivity coefficient will show which factor of these two isdominating in dimethylether case.

Acknowledgments

The author is deeply indebted to Isabelle Kleiner for reading andextensive advice on the first draft of this paper.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.jms.2014.03.004.

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