Sensitivity to a possible variation of the ProtontoElectron Mass Ratio of TorsionWaggingRotation Transitions in Methylamine (\ceCh_3nh_2)
Abstract
We determine the sensitivity to a possible variation of the protontoelectron mass ratio for torsionwaggingrotation transitions in the ground state of methylamine (\ceCH_3NH_2). Our calculation uses an effective Hamiltonian based on a highbarrier tunneling formalism combined with extendedgroup ideas. The dependence of the molecular parameters that are used in this model are derived and the most important ones of these are validated using the spectroscopic data of different isotopologues of methylamine. We find a significant enhancement of the sensitivity coefficients due to energy cancellations between internal rotational, overall rotational and inversion energy splittings. The sensitivity coefficients of the different transitions range from to . The sensitivity coefficients of the 78.135, 79.008, and 89.956 GHz transitions that were recently observed in the disk of a spiral galaxy located in front of the quasar PKS 1830211 [S. Muller et al. Astron. Astrophys. 535, A103 (2011)] were calculated to be for the first two and for the third transition, respectively. From these transitions a preliminary upper limit for a variation of the proton to electron mass ratio of is deduced.
pacs:
06.20.Jr, 33.15.e, 98.80.kI Introduction
Recently, it was shown that transitions between accidently degenerate levels that correspond to different motional states in polyatomic molecules are very sensitive to a possible variation of the protontoelectron mass ratio, . Kozlov et al. Kozlov et al. (2011) showed that transitions that convert rotational motion into inversion motion, and vice versa, in the different isotopologues of hydronium (\ceH_3O+) have coefficients ranging from to ^{1}^{1}1In Kozlov et al. Kozlov et al. (2011), is defined as the electrontoproton mass ratio, consequently, the sensitivity coefficient used in that work is minus times the sensitivity coefficient used here, =.. Similarly, Jansen et al. Jansen et al. (2011a, b) and Levshakov et al. Levshakov et al. (2011) showed that transitions that convert internal rotation into overall rotation in the different isotopologues of methanol have coefficients ranging from to . Here, the sensitivy coefficient, , is defined by
(1) 
For comparison, pure rotational transitions have , while pure vibrational transitions have and pure electronic transitions have .
Accidental degeneracies between different motional states in polyatomic molecules are likely to occur if the energies associated with the different types of motions are similar. In this paper, we present a calculation of the sensitivity coefficients for microwave transitions in methylamine (\ceCH_3NH_2). Methylamine is an interesting molecule for several reasons: (i) it displays two large amplitude motions; hindered internal rotation of the methyl (\ceCH3) group with respect to the amino group (\ceNH2), and tunneling associated with wagging of the amino group. The coupling between the internal rotation and overall rotation in methylamine is rather strong resulting in a strong dependence of the torsional energies on the quantum number, which is favorable for obtaining large enhancements of the coefficients Jansen et al. (2011b). (ii) Methylamine is a relatively small and stable molecule that is abundantly present in our galaxy and easy to work with in the laboratory. Recently it was also detected in the disk of a high redshift () spiral galaxy located in front of the quasar PKS 1830211Muller et al. (2011).
This paper is organized as follows. In Section II, we introduce the effective Hamiltonian used for calculating the level energies in the vibrational ground state of methylamine. In Section III we derive how the constants that appear in this Hamiltonian scale with . Finally, in Section IV we use the Hamiltonian and the scaling relations to determine the sensitivity coefficients of selected transitions.
Ii Hamiltonian and energy level structure
Methylamine, schematically depicted in Fig. 1, is a representative of molecules exhibiting two coupled largeamplitude motions, the torsional motion of a methyl group and the wagging (or inversion) motion of an amine group. A combination of intermediate heights of the potential barriers with a leading role of the light hydrogen atoms in the largeamplitude motions results in relatively large tunneling splittings even in the ground vibrational state. On the righthand side of Fig. 1, a contour plot of the potential energy is shown with the relative angle between the methyl and the amino group, , on the horizontal axis and the angle between the \ceNH_2plane and the \ceCNbond, , on the vertical axis. The methyl torsion motion is indicated with the arrow labeled by whereas the amino wagging motion is indicated with the arrow labeled by . From the contour plot, it is seen that amino wagging motion of the \ceNH_2group is accompanied by a rotation of the \ceCH_3group about the \ceCN bond with respect to the \ceNH_2 group. Consequently, the amino wagging motion is strongly coupled to the hindered methyl top internal rotation resulting in a rather complicated computational problem.
In Fig. 2 the lowest rotational levels of the ground vibrational state of \ceCH3NH2 are shown. The level ordering resembles that of a nearprolate asymmetric top molecule. In addition to the usual asymmetric splitting, every , level is split due to the different tunneling motions. The internal rotation tunneling splits each rotational level into one doubly degenerate and one nondegenerate sublevel. Each of these sublevels are further split into two due to the inversion motion. Together, this results in eight levels with overall symmetry , , , , , , and for and four levels for . The and levels in the and symmetry species, correspond to and respectively. Because of nuclearspin statistics, in the ground vibrational state the nondegenerate levels of even, are only allowed to possess the overall symmetry , , whereas levels with odd, are only allowed to possess the overall symmetry , . The doubly degenerate levels of and symmetry are denoted by levels, i.e. by , levels. The exact ordering of the different symmetry levels within a certain , level is determined by the relative contributions of the and parameters (see for example Fig. 3 of Ref. Ilyushin et al. (2010)). The internal motions are strongly coupled to the overall rotation resulting in a strong dependence of the torsionalwagging energies on the quantum number. Thus the level ordering may differ from one ladder to another. This turns out to be important for obtaining large enhancement factors, as it may result in closely spaced energy levels with a different functional dependence on which are connected by a symmetry allowed transition.
The panel on the right hand side of Fig. 2 shows an enlarged view of the , and , levels, with all symmetry allowed transitions assigned with roman numerals. Note that transitions with in the manifold are not allowed. The transitions labeled by iii,iv,vi,vii,viii and x are of particular interest as these connect the closely spaced levels of different manifolds and have an enhanced sensitivity to a variation of . A similar enhancement occurs for transitions between the , , and , levels as well as for transitions between the , , and , levels. In what follows, we will outline the procedure to calculate the sensitivities of these transitions. The resulting sensitivity coefficients are presented in Table 2 and Table 3 and discussed in Sec. IV.
In the present work, we use the grouptheoretical highbarrier tunneling formalism developed for methylamine by Ohashi and Hougen Ohashi and Hougen (1987), which is capable of reproducing observations of the rotational spectrum of the ground vibrational state of \ceCH_3NH_2 to within a few tens of kilohertz Ilyushin et al. (2005); Ilyushin and Lovas (2007). The highbarrier formalism assumes that the molecule is confined to one of equivalent equilibrium potential minima for many vibrations, but that it occasionally tunnels from one of these minima to another. The formalism fits in between the infinitebarrier approximation, where no tunneling splittings are observed, and the lowbarrier approximation, where the present formalism breaks down. A backward rotation of the whole molecule is introduced to cancel the angular momentum generated by one of the large amplitude motions – the socalled internal axis method – requiring the usage of extended group ideas. The reader is referred to Refs. Ohashi and Hougen (1987); Ilyushin et al. (2005); Ilyushin and Lovas (2007); Hougen and DeKoven (1983); Ilyushin et al. (2010) for a detailed description of the highbarrier tunneling formalism and the used Hamiltonian.
Table 1 lists the molecular constants used in our calculations. It includes three types of parameters: ‘nontunneling’ or pure rotational parameters; parameters associated with pure methyl torsion motion (odd numerical subscripts ); and parameters associated with the \ceNH2 wagging motion (even numerical subscripts ). The obtained scaling relations for the different parameters of the highbarrier tunneling formalism of methylamine are listed in the rightmost column of Table 1. In the next sections, we will discuss the scaling relations for the lowest order parameters, the scaling relations for the higher order parameters, and the problems encountered in determining these, are discussed in the supplementary online material onl .
Rotation^{1}^{1}1These parameters do not involve tunneling motions.  Inversion^{2}^{2}2These parameters arise from the \ceNH2 inversion tunneling motion.  Torsion^{3}^{3}3These parameters arise from the \ceCH3 torsional tunneling motions.  

Iii Scaling relations of the molecular parameters
We will use two different approaches for determining the dependence of the molecular constants that appear in the Hamiltonian:
(i) The first approach is based on the fact that the tunneling model essentially assumes that for each largeamplitude tunneling motion the systempoint travels along some path in coordinate space. In zeroth approximation, we may represent each large amplitude motion as a onedimensional mathematical problem after parameterizing the potential along the path and the effective mass that moves along it. Thus, for each large amplitude motion, we will set up a Hamiltonian that contains one position coordinate and its momentum conjugate. The parameters of this one dimensional Hamiltonian may be connected with the observed splittings which are fitting parameters of the highbarrier tunneling formalism. The parameters of the onedimensional Hamiltonians are functions of the moments of inertia and the potential barrier only, and their dependence can be found in a similar fashion as was done for methanol and other internal rotors Jansen et al. (2011a, b). Application of this approach is straightforward in the case of the leading tunneling parameters of methylamine but some ambiguities appear for the and dependences of the main terms, because there are several ways of representing these dependences in a one dimensional model.
(ii) In the second approach, we use the spectroscopic data of different isotopologues of methylamine to estimate the dependence of the tunneling constants. In analogy with methanol, we expect the tunneling splittings to follow the formula Jansen et al. (2011a):
(2) 
This formula originates from the semiclassical (WentzelKramersBrillouin (WKB)) approximation that assumes that the effective tunneling mass, represented by , changes with isotopic substitution, but that the barrier between different wells remains unchanged. This expression was successfully applied to the , splittings and the , splittings in methanol Jansen et al. (2011a). In methylamine, the parameters correspond to the splittings in the , due to tunneling between framework and framework (the set of frameworks represent the equivalent potential wells between which the system can tunnel), and application of the WKB approach to these parameters is straightforward. Moreover, since in fact all tunneling parameters in methylamine may be related to the same type of overlap integral as the parameters, we may expect that the isotopologue dependence of all tunneling terms can be described by Eq. (2). Unfortunately, ambiguities appear again when we apply this approach to higher order terms in the methylamine Hamiltonian. These ambiguities are connected to the fact that vibrational basis set functions localized near various minima are not orthogonal, but in fact have nonzero overlap integrals with each other. The correlation problems that arise in the highbarrier tunneling formalism due to nonorthogonality of the basis functions are discussed in some detail in Ref. Ohashi and Hougen (1985). The main consequence which affects the isotopologue approach is that there may be ‘leakage’ from one parameter to another; each fitted parameter appears as a sum of the ‘true’ parameter value plus a small linear combination of all other parameters with a coefficient that goes to zero when the overlap integral goes to zero. While this effect should be insignificant for the main tunneling parameters of methylamine, it may be important for higher order terms because even a small ‘leakage’ of the low order parameters may be comparable in magnitude with the ‘true’ values of the higher order parameter.
In order to verify the mass dependence coefficients for the parameters of the methylamine Hamiltonian, we have refitted available data on the \ceCH3ND2 Takagi and Kojima (1971), \ceCD3NH2 Kréglewski et al. (1990a) and \ceCD3ND2Kréglewski et al. (1990b) isotopologues of methylamine using the highbarrier tunneling formalism. Unfortunately, the amount of data available in the literature was rather limited; 66 transitions for \ceCH3ND2 Takagi and Kojima (1971), 41 transition for \ceCD3NH2 Kréglewski et al. (1990a) and 49 transitions for \ceCD3ND2Kréglewski et al. (1990b). Therefore, many of the higher order terms were not determined in the fits, while some low order parameters were determined with a few significant digits only. As a result, it was possible to obtain the dependence of the main tunneling parameters and only. In order to obtain information on higher order terms, we have undertaken a new investigation of the \ceCH3ND2 spectrum with the Kharkov millimeter wave spectrometer. The newly obtained dataset for \ceCH3ND2 contains 614 transitions, comparable to the number of microwave transitions available for \ceCH3NH2 (696 transitions). The \ceCH3NH2 and \ceCH3ND2 fits have an almost equal number of varied parameters and obtained similar weighted rootmeansquare deviations. The results of the \ceCH3ND2 investigation will be published elsewhere Ilyushin et al. , here we will use only those results necessary for obtaining the scaling relations.
iii.1 Pure Rotational Constants
The pure rotational or ‘nontunneling’ parameters in the model are connected to the usual moments of inertia of the molecule and to the centrifugal distortion parameters. Therefore, we will assume the same dependence for these parameters as used for methanol Jansen et al. (2011b).
iii.2 \ceCH3 torsion and the parameter
The parameter in the high barriertunneling formalism corresponds to a pure torsion motion. The quantity may be related to the usual  internal rotation splitting in a molecule that contains a group of symmetry. Assuming that the potential barrier is described by a cosine function and taking the moment of inertia of the methyl top to represent the mass that tunnels, we may set up a onedimension internal rotation Hamiltonian
(3) 
with for a threefold barrier, is the angular momentum operator associated to the internal rotation coordinate, is the internal rotation parameter and the barrier height. Using a value for derived from the molecular constants, we may fit the barrier height to the observed value for and estimate the dependence of .
In the used axis system, the offdiagonal contribution to the inertia tensor is represented by the parameter. For methylamine, this parameter is set to zero as it is not required by the fit. Thus, we may assume that the methyl top axis coincides with the principal axis , and , and , with being a conversion factor (). Using values for and (recalculated from rotational parameters) from Table 1, we obtain and (ab initio value Smeyers et al. (1996)). The value for is close to the expected one which supports the validity of the present analysis. Now, using this value for and the value for from Table 1, a fit to Eq. (3) yields the effective barrier height cm (ab initio value 708.64 cm Smeyers et al. (1996)). The onedimensional model with this value for predicts values for the first torsional band and the splitting in the first excited torsional state that are in a good agreement with the observed values (269 cm versus 264 cm Kréglewski and Wlodarczak (1992) for the band origin and 186 GHz versus 180 GHz Kréglewski and Wlodarczak (1992) for the splitting in the =1). All this indicates that the one dimensional model is physically sound and sufficiently accurate for our purposes.
Finally, we obtain the dependence of via
(4) 
where we have used the fact that scales as , i.e., we assume that the neutron mass has a similar variation as the proton mass. The numerical evaluation using Eq. (4) yields .
In the upper panel of Fig. 3, the value of the parameter is plotted as a function of the reduced moment of inertia, for 4 different isotopologues of methylamine. As mentioned, the quantity corresponds to the usual internal rotation splitting in a methyl top molecule, hence, we expect the tunneling splitting to follow Eq. (2). The solid line in the upper panel of Fig. 3, corresponds to and , obtained using the \ceCH3NH2 and \ceCH3ND2 data. The reduced moment of inertia is directly proportional to . Thus, the sensitivity coefficient is given by:
(5) 
From the above expression, we find for the parameter of \ceCH3NH2 a sensitivity coefficient of , in excellent agreement with the value found from the onedimensional Hamiltonian model.
iii.3 Inversion and the parameter
The interpretation of the parameter in terms of an effective mass moving in a onedimensional effective potential is not straightforward. For instance, ab initio calculations of the kinetic parameter for the inversion motion in the equilibrium geometry range from 9.6017 cm Smeyers et al. (1996) to 26.7291 cm Smeyers et al. (1998), while the barrier height in different studies varies from 1686 cmTsuboi et al. (1966) to 2081 cmSztraka (1987). Since the system needs to tunnel six times in order to return to its initial configuration, we will treat this large amplitude motion as a sixfold periodic well problem, following Ohashi et al. Ohashi et al. (1988). Furthermore, we assume that the potential along the path can be represented by a rapidly converging Fourier series. Thus, we use Eq. (3) with replaced by and as a zeroth order model. The effective inversiontorsion constant and barrier height can be determined from the splittings in the ground state and \ceNH2 wagging band origin (780 cm Tsuboi et al. (1964)). From this, we obtain =9.19 cm and =2322 cm, close to the values obtained by Ohashi et al. Ohashi et al. (1988). Following the same procedure as for , we obtain the dependence of , .
In the lower panel of Fig. 3, the value of the parameter is plotted as a function of the reduced moment of inertia, for 4 different isotopologues of methylamine. The solid line in Fig. 3 corresponds to and obtained using the \ceCH3NH2 and \ceCH3ND2 data. From this fit, we find for the parameter of \ceCH_3NH_2 a sensitivity coefficient equal to , again in excellent agreement with the onedimensional Hamiltonian model.
iii.4 and parameters
The linear terms and correspond to the interaction of components of the total angular momentum with the angular momentum generated in the moleculefixed axis system by the two large amplitude motions. In methylamine, and represent the interaction of the angular momentum generated by the \ceNH2 inversion and the ‘corrective’ /3 rotation of the \ceCH3 group with the and components of the total angular momentum, respectively. It can be shown in different ways that has the same dependence on as . For instance, it follows from a study of the correlations between the , and parameters carried out by Ohashi and Hougen Ohashi and Hougen (1987). In methylamine, two possible choices exist for . can be chosen such that Coriolis coupling due to the inversion plus corrective rotation is eliminated ( fixed to zero), or such that Coriolis coupling due to the internal rotation of the \ceCH3 group is eliminated ( fixed to zero). These two choices result in a difference Ohashi and Hougen (1987). Since is in both cases a (dimensionless) ratio between different moments of inertia and independent of , the above equation implies that and should have the same dependence.
From the \ceCH3ND2 isotopologue data, a sensitivity coefficient was found, in good agreement with the obtained from the onedimensional model and close to the value for . The term is expected to have the same dependence as . We were not able to check the isotopologue dependence for this term, since it was not required by the \ceCH3ND2 fit.
iii.5 Higher order terms
The dependence of the higher order terms, including the and dependences of the and parameters, was determined in a similar fashion (see the online material to this paper onl ). Unfortunately, some ambiguities and discrepancies between the different approaches appeared in the determination of the scaling relations for some higher order terms, which is reflected by the rather large error for these parameters (see Sec. IV). This is not a serious concern as the higher order tunneling parameters only marginally affect the coefficients of the considered transitions.



