Microwave and submillimeter molecular transitions and their dependence on fundamental constants

# Microwave and submillimeter molecular transitions and their dependence on fundamental constants

M. G. Kozlov    S. A. Levshakov Petersburg Nuclear Physics Institute, 188300 Gatchina St. Petersburg Electrotechnical University “LETI”, Prof. Popov Str. 5, 197376 St. Petersburg Ioffe Physical-Technical Institute, Polytekhnicheskaya Str. 26, 194021 St. Petersburg
###### Abstract

Microwave and submillimeter molecular transition frequencies between nearly degenerated rotational levels, tunneling transitions, and mixed tunneling-rotational transitions show an extremely high sensitivity to the values of the fine-structure constant, , and the electron-to-proton mass ratio, . This review summarizes the theoretical background on quantum-mechanical calculations of the sensitivity coefficients of such transitions to tiny changes in and for a number of molecules which are usually observed in Galactic and extragalactic sources, and discusses the possibility of testing the space- and time-invariance of fundamental constants through comparison between precise laboratory measurements of the molecular rest frequencies and their astronomical counterparts. In particular, diatomic radicals CH, OH, NH, and a linear polyatomic radical CH in electronic ground state, polyatomic molecules NH, ND, NHD, NHD, HO, HO, CHOH, and CHNH in their tunneling and tunneling-rotational modes are considered. It is shown that sensitivity coefficients strongly depend on the quantum numbers of the corresponding transitions. This can be used for astrophysical tests of Einstein’s Equivalence Principle all over the Universe at an unprecedented level of sensitivity of , which is a limit three to two orders of magnitude lower as compared to the current constraints on cosmological variations of and :  ,  .

## I Introduction

The fundamental laws of particle physics, in our current understanding, depend on 28 constants including the gravitational constant, , the mass, , and charge, , of the electron, the masses of six quarks, , , , , , and , the Planck constant, , the Sommerfeld constant , the coupling constants of the weak, , and strong, , interactions, etc. The numerical values of these constants are not calculated within the Standard Model and remain, as Feynman wrote about the fine structure constant in 1985, “one of the greatest mysteries of physics” 29 (). However, it is natural to ask whether these constants are really constants, or whether they vary with the age of the universe, or over astronomical distances.

The idea that the fundamental constants may vary on the cosmological time scale has been discussing in different forms since 1937, when Milne and Dirac argued about possible variations of the Newton constant during the lifetime of the universe 72 (); 23 (). Over the past few decades, there have been extensive searches for persuasive evidences of the variation of physical constants. So far, there was found no one of them. The current limits for dimensionless constants such as the fine structure constant, , and the electron to proton mass ratio, , obtained in laboratory experiments and from the Oklo natural reactor are on the order of one part in  17 (); 83 (); 90 () and one part in  93 (); 10 (); 28 () per year, respectively. The detailed discussion of ideas behind laboratory experiments can be found in a review 30 ().

Assuming that the constants are linearly dependent on the cosmic time, the same order of magnitude constraints on the fractional changes in and in are stemming from astronomical observations of extragalactic objects at redshifts  73 (); 2 (); 46 (); 59 (); 58 (). Less stringent constraints at a percent level have been obtained from the cosmic microwave background (CMB) at  57 (); 69 (); 78 () and big bang nucleosynthesis (BBN) at  31 (); 18 (). We note that space and/or time dependence of based on optical spectra of quasars and discussed in the literature (99, , and references therein) is still controversial and probably caused by systematic effects since independent radio-astronomical observations, which are more sensitive, show only null results for both and  87 (); 4 ().

Surprisingly, it looks as if the Einstein heuristic principle of local position invariance (LPI) — the outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed — is valid all over the universe, i.e., at the level of neither no deviate from their terrestrial values for the passed yr. In the Milky Way, it was also found no statistically significant deviations of  from zero at even more deeper level of  61 (); 60 (); 27 ().

However, the violation of the LPI was predicted in some theoretical models such as, for example, the theory of superstrings which considers time variations of , , and the QCD scale (i.e., since ) and thereby opening a new window on physics beyond the Standard Model (19, , and references therein). If the fundamental constants are found to be changing in space and time, then they are not absolute but dynamical quantities which follow some deeper physical laws that have to be understood. Already present upper limits on the variation of the fundamental constants put very strong constraints on the theories beyond the Standard Model (97, , and references therein). This motivates the need for more precise laboratory and astronomical tests of the LPI. Of course, there are also other attempts to look for the new physics. For example the electric dipole moments (EDMs) of the elementary particles are very sensitive to the different extensions of the Standard Model. Present limit on the EDM of the electron significantly constrains supersymmetrical models and other theories 88 (); 34 ().

In this review we will consider tests of LPI which are based on the analysis of microwave and submillimeter111The frequency range 1 GHz  GHz is usually referred to as a microwave range. Molecular transitions below 1 GHz (wavelength cm) are from a low-frequency range which is restricted by the ionospheric cut-off at 10 MHz ( m). astronomical spectra and which are essentially more sensitive to small variations in and than the test based on optical spectral observations of quasars.

## Ii Differential measurements of Δα/α and Δμ/μ from atomic and molecular spectra of cosmic objects

Speaking about stable matter, as, for example, atoms and molecules, we have only seven physical constants that describe their spectra 33 ():

 G,ΛQCD,α,me,mu,md,ms.

The QCD scale parameter and the masses of the light quarks u, d, and s contribute to the nucleon mass (with ) and, thus, the electron-to-proton mass ratio is a physical constant characterizing the strength of electroweak interaction in terms of the strong interaction.

In the nonrelativistic limit and for an infinitely heavy pointlike nucleus all atomic transition frequencies are proportional to the Rydberg constant, , and the ratios of atomic frequencies do not depend on any fundamental constants. Relativistic effects cause corrections to atomic energy, which can be expanded in powers of and , the leading term being , where is atomic number. Corrections accounting for the finite nuclear mass are proportional to , but for atoms they are much smaller than relativistic corrections.

Astronomical differential measurements of the dimensionless constants and are based on the comparison of the line centers in the absorption/emission spectra of cosmic objects and the corresponding laboratory values. It follows that the uncertainties of the laboratory rest frequencies and the line centers in astronomical spectra are the prime concern of such measurements. It is easy to estimate the natural bounds set by these uncertainties on the values of  and .

Consider the dependence of an atomic frequency on in the comoving reference frame of a distant object located at redshift  24 (); 25 ():

 ωz=ω+qx+O(x2),x≡(αz/α)2−1. (1)

Here and are the frequencies corresponding to the present-day value of and to a change at a redshift . In this relation, the so-called factor is an individual parameter for each atomic transition.

If , the quantity in (1) differs from zero and the corresponding frequency shift is given by

 Δωω=QΔαα, (2)

where is the dimensionless sensitivity coefficient and is the fractional change in . Here we assume that . The condition leads to a change in the apparent redshift of the distant object :

 Δωω=−Δz1+z≡Δvc , (3)

where is the Doppler radial velocity shift.

If is the observed frequency from the distant object, then the true redshift is given by

 1+z=ωzω′, (4)

whereas the shifted (apparent) value is

 1+~z=ωω′. (5)

Now, if we have two lines of the same element with the apparent redshifts and and the corresponding sensitivity coefficients and , then

 ΔQΔαα=~z1−~z21+z=Δvc . (6)

Here is the difference of the measured radial velocities of these lines, and is the corresponding difference between their sensitivity coefficients. By comparing the apparent redshifts of two lines with different sensitivity coefficients we can study variation of on a cosmological timescale.

Unfortunately, optical and UV transitions of atoms and molecules are not very sensitive to changes in and . The sensitivity coefficients of atomic resonance transitions of usually observed in quasar spectra chemical elements (C, N, O, Na, Mg, Al, Si, S, Ca, Ti, Cr, Mn, Fe, Co, Ni, Zn) are very small,  6 (). The same order of magnitude sensitivity coefficients to variations have been calculated for the UV transitions in the Lyman and Werner bands of molecular hydrogen H 98 (); 71 (); 96 (), and for the UV transitions in the 4 positive band system of carbon monoxide CO 91 ().

Small values of and put tough constraints on optical methods to probe  and . Let us consider an example of Fe ii lines arising from the ground state . In quasar spectra we observe 7 resonance transitions ranging from 1608 Å to 2600 Å with both signs sensitivity coefficients: and for transitions with Å (85, , note a factor of two difference in the definition of the coefficients with the present work). This gives us the maximum value of which is known with an error of %. From (6) it follows that a variance of would induce a velocity offset km s between the 1608 Å line and any of the line with Å. We may neglect uncertainties of the rest frame wavelengths since they are km s 79 (). If both iron line centers are measured in quasar spectra with the same error , then the error of the offset is . The error is a statistical estimate of the uncertainty of , and, hence, it should be less than the absolute value of . This gives us the following inequality to adjust parameters of spectral observations required to probe  at a given level:

 σv<ΔQ√2Δααc . (7)

At , the required position accuracy should be km s. A typical error of the line center of an unsaturated absorption line in quasar spectra is about 1/10 of the pixel size (the wavelength interval between pixels) 63 (). Current observations with the UV-Visual Echelle Spectrograph (UVES) at the ESO Very Large Telescope (VLT) provide a pixel size Å, i.e., at Å the expected error should be km s, which is comparable to the velocity offset due to a fractional change in at the level of . Such a critical relationship between the ‘signal’ (expected velocity offset ) and the error hampers measuring at the level of from any absorption system taking into account all imperfections of the spectrograph and the data reduction procedure. Systematic errors exceeding 0.5 km s are known to be typical for the wavelength calibration in both the VLT/UVES and Keck/HIRES spectrographs 2 (); 35 (); 101 (); 1 (). At this level of the systematic errors an estimate of  from any individual absorption-line system must be considered as an upper limit but not a ‘signal’. Otherwise, a formal statistical analysis of such values may lead to unphysical results (examples can be found in the literature).

The UV molecular spectra of H and CO observed at high redshifts in the optical wavelength band encounter with similar difficulties and restrictions. The maximum difference between the sensitivity coefficients in case of H is , the rest frame wavelength uncertainties are negligible,  92 (), and with the current spectral facilities at giant telescopes it is hard to get estimates of  at a level deeper than . For carbon monoxide such measurements have not been done so far but the expected limit on  should be since CO lines are much weaker than H 80 () and therefore their line centers are less certain. The analogue of Eq.(6) for the -estimation from a pair of molecular lines is 64 ():

 Δμμ=ΔvcΔQ=v1−v2c(Q2−Q1) , (8)

and for a given level of , molecular line centers should be measured with an error

 σv<ΔQ√2Δμμc . (9)

This means that at , the required position accuracy should be km s, or the pixel size Å at 4000 Å. This requirement was realized in the VLT/UVES observations of the quasar Q0347–383 100 () where a limit on  of was set.

At present the only way to probe variation of the fundamental constants on the cosmological timescale at a level deeper than is to switch from optical to far infrared and microwave bands. In the microwave, or submillimeter range there are a good deal of molecular transitions arising in Galactic and extragalactic sources. Electronic, vibrational, and rotational energies in molecular spectra are scaled as . In other words, the sensitivity coefficients for pure vibrational and rotational transitions are equal to and , respectively. Besides, molecules have fine and hyperfine structures, -doubling, hindered rotation, accidental degeneracy between narrow close-lying levels of different types, which have a specific dependence on the physical constants. The advantage of radio observations is that some of these molecular transitions are approximately 100-1000 times more sensitive to variations of and/or than optical and UV transitions.

In the far infrared waveband also lie atomic fine-structure transitions, which have sensitivity to -variation  55 (). We can combine observations of these lines and rotational molecular transitions to probe a combination  62 (). Besides, radio-astronomical observations allow us to measure emission lines from molecular clouds in the Milky Way with an extremely high spectral resolution (channel width km s) leading to stringent constraints at the level of  61 (). The level is a natural limit for radio-astronomical observations since it requires the rest frequencies of molecular transitions to be known with an accuracy better than 100 Hz. At the moment only ammonia inversion transitions and 18 cm OH -doublet transitions have been measured in the laboratory with such a high accuracy 56 (); 41 ().

In the next sections we consider in more detail the sensitivities of different types of molecular transitions to changes in and . We are mainly dealing with molecular lines observed in microwave and submillimeter ranges in the interstellar medium, but a few low-frequency transitions with high sensitivities are also included in our analysis just to extend the list of possible targets for future studies at the next generation of large telescopes for low-frequency radio astronomy.

## Iii Diatomic radicals in the Π ground state: CH, OH, and NH+

We start our analysis of the microwave spectra of molecules from the simplest systems — diatomic molecules with nonzero projection of the electronic angular momentum on the molecular axis. Several such molecules are observed in the interstellar medium. Here we will mostly focus on the two most abundant species — CH and OH. Recently it was realized that -doublet transitions in these molecules have high sensitivity to the variation of both and  15 (); 20 (); 51 (). There are also several relatively low frequency transitions between rotational levels of the ground state doublet and with sensitivities, which are significantly different from the typical rotational ones 22 (). Then we will briefly discuss the NH radical222NH has not yet been detected in space, its fractional abundance in star-forming regions is estimated (NH)/(H)  82 ()., which is interesting because it has very low lying excited electronic state . This leads to an additional enhancement of the dimensionless sensitivity coefficients  5 (). The latter are defined as follows:

 Δωω=QαΔαα+QμΔμμ. (10)

### iii.1 Λ-doubling and Ω-doubling

Consider electronic state with nonzero projection of the orbital angular momentum on the molecular axis. The spin-orbit interaction couples electron spin to the molecular axis, its projection being . To a first approximation the spin-orbit interaction is reduced to the form . Total electronic angular momentum has projection on the axis, . For a particular case of and we have two states and and the energy difference between them is: .

Rotational energy of the molecule is described by the Hamiltonian:

 Hrot =B(J−Je)2 (11a) =BJ2−2B(JJe)+BJ2e, (11b)

where is the rotational constant and is the total angular momentum of the molecule. The first term in expression (11b) describes conventional rotational spectrum. The last term is constant for a given electronic state and can be added to the electronic energy.333Note that this term contributes to the separation between the states and . This becomes particularly important for light molecules, where the constant is small. The second term describes -doubling and is known as the Coriolis interaction .

If we neglect the Coriolis interaction, the eigenvectors of Hamiltonian (11) have definite projections and of the molecular angular momentum on the laboratory axis and on the molecular axis respectively. In this approximation the states and are degenerate, . The Coriolis interaction couples these states and removes degeneracy. New eigenstates are the states of definite parity  11 ():

 |J,M,Ω,p⟩ =(|J,M,Ω⟩+p(−1)J−S|J,M,−Ω⟩)/√2. (12)

The operator can only change quantum number by one, so the coupling of states and takes place in the order of the perturbation theory in .

The -doubling for the state happens already in the first order in the Coriolis interaction, but has additional smallness from the spin-orbit mixing. The operator can not directly mix degenerate states and because it requires changing by two. Therefore, we need to consider spin-orbit mixing of the and states:

 |Ω=12⟩ =|1,−12,12⟩+ζ|0,12,12⟩, (13)

where

 ζ∼A/(EΠ−EΣ), (14)

and then

 ⟨Ω=12|HCor|Ω=−12⟩=2ζB(J+12)⟨Λ=1|Lx|Λ=0⟩. (15)

Note that depends on the non-diagonal matrix element (ME) of the spin-orbit interaction and Eq. (14) is only an order of magnitude estimate. It is important, though, that non-diagonal and diagonal MEs have similar dependence on fundamental constants. We conclude that -splitting for the level must scale as . The -doubling for state takes place in the third order in the Coriolis interaction. Here has to mix first states with and with before ME (15) can be used. Therefore, the splitting scales as .

The above consideration corresponds to the coupling case , when . In the opposite limit the states and are strongly mixed by the Coriolis interaction and spin decouples from the molecular axis (coupling case ). As a result, the quantum numbers and are not defined and we only have one quantum number . The -splitting takes place now in the second order in the Coriolis interaction via intermediate states. The scaling here is obviously of the form . Note that in contrast to the previous case , the splitting here is independent on .

We can now use found scalings of the - and -doublings to determine sensitivity coefficients (10). We only need to recall that in atomic units and . We conclude that for the case the -doubling spectrum has following sensitivity coefficients:

 State2Π1/2:Qα=−2,Qμ=1, (16a) State2Π3/2:Qα=−2,Qμ=3. (16b) For the case b, when S is completely decoupled from the axis, the Λ-doubling spectrum has following sensitivity coefficients: StateΠ:Qα=0,Qμ=2. (16c)

When constant is slightly larger than , the spin is coupled to the axis only for lower rotational levels. As rotational energy grows with and becomes larger than the splitting between states and , the spin decouples from the axis. Consequently, the -doubling is transformed into -doubling. Equations (16) show that this can cause significant changes in sensitivity coefficients. The spin-orbit constant can be either positive (CH molecule), or negative (OH). The sign of the -doubling depends on the sign of , while -doubling does not depend on at all. Therefore, decoupling of the spin can change the sign of the splitting. In Sec. III.2 we will see that this can lead to a dramatic enhancement of the sensitivity to the variation of fundamental constants.

### iii.2 Intermediate coupling

The -doubling for the intermediate coupling was studied in detail in many papers, including 70 (); 13 (); 12 () (see also the book 11 ()). Here we use the effective Hamiltonian from 70 () in the subspace of the levels and , where upper sign corresponds to the parity in Eq. (12). The operator includes spin-rotational and hyperfine parts

 Heff =Hsr+Hhf. (17)

Neglecting third order terms in the Coriolis and spin-orbit interactions, we get the following simplified form of the spin-rotational part:

 ⟨Π1/2,J,p|Hsr|Π1/2,J,p⟩ =−12A+B(J+12)2+p(S1+S2)(2J+1), (18a) ⟨Π3/2,J,p|Hsr|Π3/2,J,p⟩ =+12A+B(J+12)2−2B, (18b) ⟨Π3/2,J,p|Hsr|Π1/2,J,p⟩ =[B+pS2(J+12)]√(J−12)(J+32). (18c)

Here in addition to the parameters and we have two parameters which appear in the second order of perturbation theory via intermediate state(s) . The parameter corresponds to the cross term of the perturbation theory in the spin-orbit and Coriolis interactions, while the parameter is quadratic in the Coriolis interaction. Because of this scales as and scales as . It is easy to see that the Hamiltonian describes limiting cases and considered in Sec. III.1.

The hyperfine part of the effective Hamiltonian is defined in the lowest order of perturbation theory and has the form:

 ⟨Π1/2,J,p|Hhf|Π1/2,J,p⟩ =CF[2a−b−c+p(2J+1)d], (19a) ⟨Π3/2,J,p|Hhf|Π3/2,J,p⟩ =3CF[2a+b+c], (19b) ⟨Π3/2,J,p|Hhf|Π1/2,J,p⟩ =−CF√(2J−1)(2J+3)b, (19c) CF ≡[F(F+1)−J(J+1)−I(I+1)][8J(J+1)]−1.

Here we assume that only one nucleus has spin and include only magnetic dipole hyperfine interaction.

The effective Hamiltonian described by Eqs. (18,19) has 8 parameters. We use NIST values 66 () for the fine structure splitting , rotational constant , and magnetic hyperfine constants , , , . Remaining two parameters and are found by minimizing the rms deviation between theoretical and experimental -doubling spectra.

In order to find sensitivity coefficients we calculate transition frequencies for two values of near its physical value . The similar procedure is applied to at the physical value of the electron-to-proton mass ratio, . We use scaling rules discussed above to recalculate parameters of the effective Hamiltonian for different values of fundamental constants. Then we use numerical differentiation to find respective sensitivity coefficient.

### iii.3 Sensitivity coefficients for Λ-doublet transitions in CH and OH

In Ref. 51 (), the method described in the previous section was applied to OH, CH, LiO, NO, and NO. The molecules CH and NO have ground state (), while OH and LiO have ground state (). The ratio changes from 2 for CH molecule 21 (), to 7 for OH 68 (), and to almost a hundred for LiO and NO. Therefore, LiO and NO definitely belong to the coupling case . For OH molecule we can expect transition from case for lower rotational states to case for higher ones. Finally, for CH we expect intermediate coupling for lower rotational states and coupling case for higher states.

Let us see how this scheme works in practice for the effective Hamiltonian (18,19). Fig. 1 demonstrates -dependence of the sensitivity coefficients for CH and OH molecules. Both of them have only one nuclear spin . For a given quantum number , each -doublet transition has four hyperfine components: two strong transitions with and (for there is only one transition with ) and two weaker transitions with . The hyperfine structure for OH and CH molecules is rather small and sensitivity coefficients for all hyperfine components are very close. Because of that Fig. 1 presents only averaged values for strong transitions with .

We see that for large values of the sensitivity coefficients for both molecules approach limit (16c) of the coupling case . The opposite limits (16a,16b) are not reached for either molecule even for smallest values of . So, we conclude that the coupling case is not realized. It is interesting that in Fig. 1 the curves for the lower states are smooth, while for upper states there are singularities. For CH molecule this singularity takes place for the state near the lowest possible value . A singularity for OH molecule takes place for the state near .

These singularities appear because -splitting turns to zero. As we saw above, the sign of the splitting for the coupling case depends on the sign of the constant . The same sign determines which state , or lies higher. As a result, for the lower state the sign of the splitting is the same for both limiting cases, but decoupling of the electron spin for the upper state leads to the change of sign of the splitting. Of course, these singularities are most interesting for our purposes, as they lead to large sensitivity coefficients which strongly depend on the quantum numbers. Note, that when the frequency of the transition is small, it becomes sensitive to the hyperfine part of the Hamiltonian and the sensitivity coefficients for hyperfine components may differ significantly. The sensitivity coefficients of all hyperfine components of such -lines are given in Table 1. We can see that near the singularities all sensitivity coefficients are enhanced.

In addition to -doublet transitions and purely rotational transitions there are also mixed transitions between rotational states of and states. The transition energy here includes the rotational and the fine structure parts. Because of that, such transitions may have different sensitivities to the variation of fundamental constants 22 (). As an example, Fig. 2 shows mixed transitions in CH molecule. The sensitivity coefficients are given in Table 2. The isotopologue CD has mixed transitions of lower frequencies and higher sensitivities 22 (). Similar picture takes place for OH molecule.

The molecule NH is isoelectronic to CH and also has ground state . However, there is an important difference: for NH the first excited state lies only 340 cm above the ground state 47 (); 40 (). The spin-orbit interaction between these states leads to strong perturbations of the rotational structure and of the -doublet splittings and to an additional enhancement of the sensitivity coefficients 5 (). The spectrum of NH is shown in Fig. 3. The effective Hamiltonian is similar to the one considered above with two additional terms describing interaction between the and states 47 ():

 ⟨2Π3/2,J,p|Hso|4Σ−3/2,J,p⟩ =−12ζ3/2, (20a) ⟨2Π1/2,J,p|Hso|4Σ−1/2,J,p⟩ =−12√3ζ1/2. (20b)

Obviously, the parameters and scale as . As mentioned above, for the NH molecule the splitting between and states is only about 340 cm. This splitting includes three contributions: the non-relativistic electronic energy difference, the relativistic corrections () and the difference in the zero point vibrational energies for the two states (). Note that the accidental degeneracy of these levels for NH means that the first contribution is anomalously small. Because of that, the other two contributions can not be neglected and modify the scaling of with fundamental constants. This effect has to be taken into account in the calculations of the sensitivity coefficients 5 ().

## Iv Linear polyatomic radicals in the Π ground state: C3H

The linear form of the molecule CH (-CH) is similar to the molecule NH: it also has the ground state and two closely lying states and . Here the quasi degeneracy of the and states is not accidental, but is caused by the Renner-Teller interaction. In the following section we briefly recall the theory of the Renner-Teller effect in polyatomic linear molecules 89 (); 39 ().

### iv.1 Renner-Teller effect

The total molecular angular momentum of the polyatomic molecule includes the vibrational angular momentum associated with the twofold degenerate bending vibration mode(s): , where describes rotation of the molecule as a whole and is perpendicular to the molecular axis . Other momenta have nonzero -projections: , , , and .

Suppose we have electronic state and vibrational state of a bending mode . All together there are 4 states . We can rewrite them as one doublet state and states and . In the adiabatic approximation all four states are degenerate. Renner 89 () showed that the states with the same quantum number strongly interact, so the and states repel each other, while the doublet in the first approximation remains unperturbed. We are particularly interested in the case when one of the levels is pushed close to the ground state . This is what takes place in the -CH molecule 104 (); 45 (); 14 ().

Consider a linear polyatomic molecule with the unpaired electron in the state in the molecular frame . Obviously, the bending energy is different for bendings in and in planes: (here is the supplement to the bond angle). That means that the electronic energy depends on the angle between the electron and nuclear planes:

 H′=V′cos2ϕ, (21)

where . There is no reason for to be small, so a.u. and to a first approximation does not depend on and .

As long as interaction (21) depends on the relative angle between the electron and the vibrational planes, it changes the angular quantum numbers as follows: and . This is exactly what is required to produce splitting between the and states with as discussed above.

Interaction (21) also mixes different vibrational levels with . Thus, we have, for example, the nonzero ME between states . Such mixings reduce effective value of the quantum number and, therefore, reduce the spin-orbital splitting between the and states 81 (),

 Hso ≡AeffΛΣ,Aeff=AΛeff/Λ. (22)

Let us define the model more accurately. Following 81 () we write the Hamiltonian as:

 H =He+Tv+ALζSζ. (23)

Here the “electronic” part includes all degrees of freedom except for the bending vibrational mode and spin. For -CH there are two bending modes, but for simplicity we include the second bending mode in too. Electronic MEs in the basis have the form:

 ⟨±1|He|±1⟩ =V++V−2=k2χ2, (24a) ⟨±1|He|∓1⟩ =k′2χ2exp(∓2iϕ). (24b)

Here and are the vibrational coordinates for the bending mode. Kinetic energy in these coordinates has the form:

 Tv =−12MR2(∂2∂χ2+1χ∂∂χ+1χ2∂2∂ϕ2). (25)

We can use the basis set of 2D harmonic functions in polar coordinates and for the mass and the force constant :

 ψv,l(ρ,ϕ) =Rv,l(ρ)1√2πexp(ilϕ). (26)

It is important that the radial functions are orthogonal only for the same :

 ⟨Rv′,l|Rv,l⟩=δv′,v. (27)

This allows for the nonzero MEs between states with different quantum number . By averaging operator (23) over vibrational functions we get:

 ⟨v′,l′|He+Tv|v,l⟩=[ωv(v+1)+AΛSζ]δv′,vδl′,l+12⟨Rv′l′|k′χ2|Rvl⟩exp(∓2iϕ)δl′,l±2. (28)

The exponent here ensures the selection rule for the quantum number when we calculate MEs for the rotating molecule.

### iv.2 Molecule l-C3H

We solve the eigenvalue problem for Hamiltonian (23) using the basis set of the 2D-harmonic oscillator. Our model Hamiltonian has only 3 parameters, namely , , and the dimensionless Renner-Teller parameter : . The values for and for -CH are given in 81 (). We varied the Renner-Teller parameter to fit five lowest levels for the given bending mode: , , , , and . The optimal value appeared to be . The results are presented in Table 3. The first two columns give nominal vibrational quantum number and its actual average value. We see that the Renner-Teller term in (28) strongly mixes vibrational states. This mixing also affects and decreases spin-orbital splittings as explained by Eq. (22).