Hyperfine Structure of the State and Predictions of Optical Cycling Behavior in the transition of TlF
The rotational and hyperfine spectrum of the transition in TlF molecules was measured using laser-induced fluorescence from both a thermal and a cryogenic molecular beam. Rotational and hyperfine constants for the state are obtained. The large magnetic hyperfine interaction of the Tl nuclear spin leads to significant mixing of the lowest state rotational levels. Updated, more precise measurements of the vibrational branching fractions are also presented. The combined rovibrational branching fractions allow for the prediction of the number of photons that can be scattered in a given TlF optical cycling scheme.
The state of thallium monofluoride (TlF) has long been a platform for precision measurements of parity- and time-reversal symmetry violation Hinds and Sandars (1980); Wilkening et al. (1984); Cho et al. (1991), with high potential for discovery of new physics Khriplovich and Lamoreaux (1997). In particular, the high mass of Tl coupled with the high polarizability of the molecule make this system ideal for measuring the Schiff moment of the Tl nucleus Schiff (1963).
Optical cycling Shuman et al. (2009) is a potentially powerful tool for enhancing the signal in a symmetry violation measurement. Unit-efficiency detection of the internal state is possible when the number of photons scattered per molecule exceeds the reciprocal of the total light collection efficiency (geometric and detector efficiencies). Optical cycling also allows for the application of large optical forces, which can be useful for precision measurements. Transverse cooling can decrease beam divergence (and increase downstream flux) Shuman et al. (2010), and longitudinal slowing and trapping can increase effective interaction times Barry et al. (2014), leading to improved energy resolution.
Optical cycling requires the ability to optically couple a subspace of ground and excited states, such that the excited states rarely decay to uncoupled ground states. Recently, we proposed the TlF transition (where () is the ground (excited) state vibrational quantum number) as a candidate for optical cycling and laser cooling Hunter et al. (2012), as this transition has a sufficiently short excited state lifetime ( ns) and favorable vibrational branching fractions to form a quasi-closed optical cycle. In TlF, the state was expected to have resolved – and potentially very large – hyperfine (HF) structure Hunter et al. (2012). The HF interaction in the excited state can lead to mixing of rotational states with different quantum numbers ; this in turn can break the usual rotational selection rules, leading to branching to additional ground rotational levels which must be coupled to the optical cycle. Hence, it is crucial to characterize the rotational and HF structure of the excited state to understand and control rotational branching.
High-resolution microwave spectroscopy Hoeft et al. (1970) has provided a detailed and precise understanding of the state HF and rotational energies. Low-resolution spectroscopy with a pulsed UV laser by Wolf and Tiemann Wolf and Tiemann (1987) allowed for determination of rovibrational energies of the state. In this paper, we present high-resolution laser spectroscopy of the transition. We clearly identify rotational lines associated with states from – 70. The state HF structure is fully resolved for and is fit to a standard Hamiltonian to determine the parameters describing the HF interaction. These data allow a full characterization of the HF structure in the state, including effects of HF-induced rotational state mixing. In addition, we present improved vibrational branching measurements from . Together, these data are used to quantitatively predict the number of photons that may be scattered on the transition of TlF for various cycling schemes.
Ii Experimental Details
To allow access to and identification of a large range of rotational states, we make observations using both a thermal beam source and a cryogenic buffer gas beam source. The thermal oven source is the same as in Ref. (Hunter et al., 2012). It is described in brief here. The measurements are made in a stainless steel molecular beam apparatus maintained at a pressure of about Torr. The beam itself is created by heating a stainless steel oven containing TlF to temperatures of 688-733 K. The beam emerges from the oven through four ceramic tubes. The tubes precollimate the beam, which is then further collimated by an aperture located about 30 cm from the oven.
The cryogenic buffer gas beam source is nearly identical to that of Barry et al. (2011). A solid target is made by melting TlF powder in a copper crucible under vacuum. The crucible is mounted inside a copper cell and held at roughly 4 K by a pulse tube refrigerator. TlF molecules are produced by laser ablation of the target with 10 ns, 25 mJ pulses of 1064 nm light, and are extracted from the cell by a flow of typically 5 sccm of cryogenic helium buffer gas. The molecular beam then propagates through a region held at roughly Torr.
The transition occurs at wavelength 271.7 nm. A CW, single-frequency, tunable 1087 nm fiber laser is frequency doubled twice to produce roughly 20 mW of CW 271.7 nm light, using commercial resonant bow-tie cavities. The fiber laser frequency is locked by monitoring its transmission through a scanning Fabry-Perot cavity; the Fabry-Perot cavity length is in turn stabilized by simultaneously monitoring the transmission of a frequency-stabilized helium-neon laser.
In both molecular beam setups, the 271.7 nm laser light is directed to intersect the molecular beam perpendicular to the direction of molecule motion. Laser-induced fluorescence is collected onto a photomultiplier tube (PMT) in photon counting mode. In the thermal source, the fluorescence is collimated, passed through an interference filter, and then spatially filtered and collected. In the cryogenic source, the fluorescence is transferred to the PMT by a light pipe, followed by an interference filter.
Iii Hyperfine and Rotation Hamiltonian
iii.1 Quantum Numbers in the Spectrum
Thallium has two common isotopes, Tl (30 % natural abundance) and Tl (70 %), both with nuclear spin Lide (1997). Fluorine’s only isotope is F, also with nuclear spin . We describe the state of TlF using the Hund’s case (c) basis and the coupling scheme:
where is the total angular momentum of the molecule less nuclear spin. Hence each rotational state with quantum number has associated HF states with and , , , and .
The Hund’s case (c) basis kets are:
Here, is the projection of on the internuclear axis, is the projection of in the lab frame, and is the state parity. Following the convention of Herzberg Herzberg (1950), we refer to states with as -parity and as -parity states. As described below, the large HF interactions in the state mix neighboring rotational levels. We use rotational quantum number to describe states in the basis of Eq. 2, and label energy eigenstates in the case of large mixing by (i.e. in the absence of HF mixing). We denote the rotational quantum number in the ground state by .
iii.2 Magnetic Hyperfine
The largest HF effect is expected to be due to the magnetic HF interaction, described by the Hamiltonian :
where or ; is the total electron orbital angular momentum; and is the total electron spin. The lower-case subscript corresponds to coordinates in the molecule fixed-frame, with along the internuclear axis. In the limit of negligible coupling to other electronic states via , we may write the effective Hamiltonian in the form
Here, , where corresponds to the expectation value of the operators in the electronic state of interest. For ,
Similarly for ,
iii.3 Nuclear Spin-Rotation
The effective nuclear spin-rotation Hamiltonian is of the form , where again or . This arises from two physical mechanisms. The first is the coupling of the rotational motion of the nuclei to the nuclear spin magnetic moments. For an electronic state which is not strongly perturbed by other nearby states, this contribution dominates, and is quite small (for example, (Tl)=126.03 kHz and (F)=17.89 kHz in the state in TlF Hinds and Sandars (1980)). The second contribution arises from second-order rotational coupling to other electronic states Okabayashi et al. (2012). This contribution likely dominates in the state, where levels with and character are predicted to lie near in energy Balasubramanian (1985). In such a situation, can be significantly larger than when the main contribution comes from the first mechanism; for example, in PtF, (Pt)=3.11 MHz Okabayashi et al. (2012).
The nuclear spin-rotation matrix elements can be written as follows. For :
Because there are discrepancies in the literature about the explicit form of these HF matrix elements, we derive them in Appendix C.
iii.4 Additional Terms
Equations 5 and 6 are only strictly valid for an isolated electronic level. Perturbations by a nearby level can lead to extra terms in the effective Hamiltonian with the -dependence of centrifugal terms, modeled by substituting Okabayashi and Tanimoto (2003). However, the diagonal matrix elements of the centrifugal magnetic HF and nuclear spin-rotation Hamiltonians are identical, and the effects of off-diagonal elements are too small to be discernable in our data. We choose to constrain , and treat the fit constants (Tl) and (F) as empirical combinations of the two effects.
Rotational energy is modeled using the standard effective Hamiltonian Herzberg (1950):
with diagonal matrix elements
The state HF/rotation structure is expected to depend on the isotopologue and the / parity. We therefore fit the parity states and each Tl isotope separately. This is equivalent to the analysis of Ref. Okabayashi et al. (2012), which used the substitution and and , with the upper (lower) sign used for the - (-) parity states.
Iv Observed Spectral Features
iv.1 Line Identification
The HF structure of the ground state is unresolved ( 100 kHz) in our optical spectra. In the excited state, we expect 8 well-split isotope/HF sublevels for each rotational level and parity , corresponding to , , and the two Tl isotopes. We label the 7 splittings between the 8 lines as as shown in Fig. 1. In most cases, we easily identify lines associated with the two isotopologues, since their intensities are proportional to the isotopic abundance: we associate splittings with Tl; with Tl; and with the gap between the isotopologues. For a given isotopologue, the largest splitting (associated with the Tl nuclear spin projection) is or . The doublets separated by () then correspond to the F nuclear spin projection for TlF (TlF). As described in Section IV.2 below, in only, the separation of the levels due to the Tl spin projection is larger than the isotope shift; we use the convention of labeling the splitting between the highest TlF HF level and the lowest TlF HF level as , and thus for transitions to .
The and branch transitions are spectroscopically isolated and hence relatively easy to identify for . For these states, 8-line multiplets similar to that displayed in Fig. 1 are found approximately centered on the locations predicted from the Dunham coefficients of Tiemann Tiemann (1988). Our measured splittings for the various values of -parity excited states are listed in Table 1.
Assignment of the branch transitions is more challenging. Due to the near equality of the rotational constants of the and states, the branch transitions are generally not clearly separated. All of the lines between and (approximately 480 individual HF transitions) are contained in a frequency range of about 21 GHz (see Appendix D). We initially identified many of the branch multiplets for – using rotational constants , , and from Tiemann Tiemann (1988), then fine-tuned these parameters to obtain good agreement with all the identifiable branch lines (–). Because of their lower abundance and correspondingly smaller signals, no similar identification was possible for the majority of the TlF branch lines. Assuming the same rotational constants (scaled for reduced mass and including an isotopic offset), we have made some tentative assignments of TlF –.
The HF splitting is larger than the separation of branch transitions for –, and clear identification of the lines again becomes problematic. Ignoring the splittings which are obscured by overlapping branch lines eliminates nearly all – data (except and ). However, superposing the data taken in this spectral region from both the thermal beam and the cryogenic beam provides additional information and leads to the tentative assignments shown in Fig. 2a. The resulting splittings for -parity excited states are listed in Table 2.
We now discuss a number of unusual features in the state HF structure. Many of these observations are attributed to an unusually large value of (Tl) (found to be 29 GHz), which we believe may be the largest observed HF interaction in any diatomic molecule.
iv.2 Large Hyperfine Splitting for
The large value of (Tl) is best illustrated by the enormous splitting in (Fig. 2b). While the HF structure of covers GHz, all higher branch lines to observable in the cryogenic source ( to roughly ) are contained in only roughly 2 GHz. The large splitting in means two of the four lines are actually lower in frequency than .
iv.3 J-mixing and “Extra” Lines
In addition to the expected , and branch lines, we observe additional quartets of lines near several low and branch lines (Fig. 3). These are assigned as nominal transitions ( and branch, respectively). The presence of these lines is an indication that the state rotational levels are strongly mixed by the magnetic HF interaction. If (Tl) (F), as expected since HF structure is enhanced in heavier species, then this -mixing is only significant between states with the same values of and . The eigenstates can be written in the form
Note that is a special case which does not experience -mixing. For all other excited states, -mixing leads to electric dipole-allowed and branch transitions. Because the rotational constants of the and states are nearly identical, the lines appear very close to , and the lines appear very close to . We label the splittings as for the -branch and for the -branch (see Fig. 3).
The and branches target the excited state -parity levels. We check our line assignments by comparing splittings in with those in , which share a common excited state and thus should have the same observed HF splittings. The , , and branches target the excited state -parity levels. The branch HF structure must be compared to both and branches, as these lines address only one value of in the excited state due to the selection rule .
As an aside, we note that the F magnetic HF interaction mixes states with the same , and thus it is in principle possible to drive electric dipole transitions with , which we call the and branches, respectively. For nearly equal ground and excited state rotational constants, we expect two lines – one for each isotope, and split by for – to appear very close to (and similarly, lines near ). Because the majority of the molecules produced in our cryogenic buffer gas beam source are in , we searched for and lines around and , respectively. However, we failed to detect any such transitions.
iv.4 Inverted Doublet for
By examining the patterns in the data we can infer the ordering of the energy levels within different quartets of lines. The splitting of the branch lines is consistent with the splitting (upper fluorine doublet splitting) of the branch, indicating that lies higher in energy than for all observed branch lines, corresponding to –5. However, the splitting for is also consistent with the splitting of . Combined with the fact that for all lines connected to , but for , we conclude that the ordering of levels is regular only in , and inverted for 2. Also of note is that the splitting is larger in than in . Naively, the -scaling of diagonal matrix elements of (Eq. 5) would have led us to expect the splitting to only be larger in than in .
|This Work||Ref. Tiemann (1988)|
These peculiar features can be explained by an exceptionally large value of (Tl). Consider a simplified system where only (Tl) (Eq. 4) and are present. In the basis of states with , the interaction is characterized by the Hamiltonian
By diagonalizing , we find the energy of eigenstate is given by
The energy difference between states vanishes for when (see Fig. 4). The energy ordering of the doublets reverses for for all , but not for . For , the splitting of is much greater than that of all other states. These features are consistent with our observations and our fit value .
The best-fit values for the rotational and HF parameters are provided in Table 3. The state rotational parameters are obtained by subtracting off the precisely-known state rotational energy from the weighted center of the HF quartet for each isotope, then fitting to a polynomial in . For the -parity, we perform a cubic fit to –. For the -parity, we fit to –, and only the linear () term is statistically significant. The lines are excluded from the fit as they are observed to deviate strongly from the rotational progression as discussed above. The -parity level constants are observed to match the expected scaling with molecular reduced mass : , while .
The HF fit parameters are obtained by diagonalizing the effective Hamiltonian for each value of . The HF parameters for the - ()-parity states were fit to the observed branch (average of the and branch) splitting, weighted by their assigned uncertainties. In all cases, we find (F) to be consistent with zero within an uncertainty of 0.1 MHz.
The uncertainty in the splittings measured in the thermal beam is 3 MHz. For the cryogenic beam data, the Fabry-Perot ramp was less linear, and we assign an uncertainty of 8 MHz to these splittings. We fit to the thermal beam data if available, and the cryogenic beam data when not. In the instances where lines were measured in both setups, the agreement is typically within 10 MHz. As a check of our branch splitting assignments where one or both lines are degenerate with other branch lines (Table 2 bold data), a fit to the -parity levels was performed excluding these data. However, this did not change the fit parameters within the assigned uncertainty. The excellent agreement between the data and the line centers predicted by the best-fit Hamiltonian (Fig. 2a) provides additional support for our tentative branch line assignments.
There is fair agreement between the TlF and TlF isotopologues on the value of the F doublet splittings. The magnetic HF parameter (Tl) should be proportional to the nuclear -factor , and we find excellent agreement for -parity: Tl)/Tl)=0.9904(2), while Lide (1997). Hyperfine anomalies are known to occur at the level in neutral Tl atoms Mårtensson-Pendrill (1995); Richardson et al. (2000), but is outside the precision of this study.
Finally, we extract the electronic isotope shift for the transition Knöckel and Tiemann (1984). By averaging the shift for each HF transition for all and branch lines and subtracting off the calculated rovibrational contributions, we find MHz. Because we only measure two Tl isotopes, we are unable to deconvolve the mass- and field-shift contributions to this quantity Knöckel and Tiemann (1984).
V Optical Cycling
v.1 Applications and Requirements
Nominal State Label
The ability to scatter many photons per molecule can be used for efficient detection and for application of optical forces. The number of optical cycles required before leaking into an uncoupled state depends on the application. For instance, simple state-selective detection of molecules in a beam often does not require a highly closed cycle. A typical laser-induced fluorescence setup can achieve overall detection efficiency for emitted photons (including detector quantum efficiency) of –10% Barry et al. (2014). Hence, to detect molecules with near-unit efficiency, we only require an optical cycling scheme closed to (very roughly) –100 cycles. Applying significant optical forces requires a more closed cycle: optical cycles are required to achieve transverse Doppler cooling of a molecular beam to milliKelvin temperatures Shuman et al. (2010), and optical cycles are needed to slow a molecular beam to a near stop Barry et al. (2012) or to magneto-optically trap molecules Barry et al. (2014).
In this section, we consider vibrational and rotational branching effects which can limit the number of photons scattered on a given spectral line of the TlF transition. We denote the vibrational branching fraction by , and rotational branching fraction (irrespective of vibrational quantum number) by . We then propose a few optical cycling schemes which can be used in TlF for specific applications.
v.2 Rotational Branching
As discussed in Section IV.3, HF interactions mix states of the same total angular momentum . In Table 4, we present the eigenstates of = in the basis of Eq. 2 calculated from the best-fit spectroscopic constants in Table 3. With the state admixture in hand, it is straightforward to calculate rotational branching fractions using the line strengths Townes and Schawlow (1955) for each excited basis state.
v.3 Vibrational Branching
Previous measurements of vibrational branching in TlF found favorable branching fractions for cycling from the state: = , , and = Hunter et al. (2012). Measurements of branching fractions to 3 were limited by experimental sensitivity and the availability of narrow bandpass interference filters at the needed wavelengths. Predicted values for these branching fractions using Morse and Rydberg-Klein-Rees (RKR) potentials, given in Table 5, were uncertain at the level required to determine which of these transitions could be neglected during longitudinal cooling. Here we present precision measurements of , , , and .
|=0 (271.7 nm)||0.9892(3)||0.989(2)||0.989(2)|
|=1 (275.3 nm)||0.0003(2)||0.0005(3)|
|=2 (278.8 nm)||0.0104(2)||0.010(2)|
|=3 (282.5 nm)||0.00000(1)||0.0003|
|=4 (286.2 nm)||0.00013(1)||0.0002|
|=5 (290.0 nm)||0.00000(1)||0.0003|
|=6 (293.8 nm)||0.0002|
Measurements of the branching fractions are done using a modified version of the thermal beam in order to take advantage of the better molecular beam intensity stability. Pulsed excitation is used, and fluorescence is only detected after the laser pulse, eliminating scattered-light backgrounds. The exciting laser light is produced by a pulsed dye laser with Coumarin 540A dye pumped by a 355 nm Nd:YAG laser with pulse duration of 10 ns and 10 Hz repetition rate. The output of the dye laser is frequency doubled to produce 271.7 nm light. Measurements are made with the broadband pulsed system tuned to the largest fluorescence signal that occurs, near a large pile-up of rotational transitions between and with a bandhead that occurs at about . Background measurements are made by tuning to the high frequency side of this bandhead where virtually no fluorescence occurs.
Bandpass interference filters centered at the wavelengths corresponding to the –6 transitions are placed before the “signal” PMT (above the vacuum chamber) to isolate fluorescence from the different vibrational bands. A second “normalization” PMT (below the chamber) contains only 271.7 nm interference filters and is used to measure fluorescence from the main transition ( ) at all times. This allows us to eliminate fluctuations in laser and molecular beam intensity from our measurements. To avoid saturation problems with the photon counting, we limit the average number of photons collected by either PMT to be per laser pulse.
To measure a particular branching ratio we alternate between the corresponding filters, and count photons after subtracting off background fluorescence, normalizing with respect to the calibration signal, and taking into account the filter transmissions. Measurement of the ratio is accomplished with a modest molecular beam flux by dividing the fluorescence signals when alternately filtering for and . However, the small rate of the decay to requires higher molecular beam fluxes to achieve adequate statistical precision. To avoid saturating the normalization PMT from these larger fluorescence signals, additional attenuating filter are added. To avoid saturating the signal detector at these higher fluxes, all branching fractions for are measured by comparing the normalized signal for to that of . Their measured ratio is then multiplied by as measured above to determine . The attenuation on the normalization transition is chosen so that the statistical significance of the measurement remains dominated by the number of photons counted in the signal detector.
We calculate the experimental branching fractions displayed in Table 5 using the assumption that the measured branching fractions account for all of the significant vibrational branches from . It is worth noting that the uncertainty associated with is larger than that quoted in our earlier publication Hunter et al. (2012). The increased uncertainty is due to the identification of a systematic error associated with a changing amount of light reflected into the normalization PMT when the interference filter in the signal PMT is switched between monitoring the and transitions. This effect was likely also present in our earlier measurement.
v.4 Optical Cycling Schemes
We now examine a few specific, useful examples of optical cycling (which we abbreviate below as ). First consider (Fig. 5a). As there are no other states, the upper state of this transition is unmixed and all quantum numbers are exact to a very high extent (Table 4). Electric dipole and parity selection rules then dictate that the excited state can only decay to .
Now consider (Fig. 5b). This transition may be of interest in applications where a high photon scattering rate is desirable, as the higher excited state degeneracy is expected to allow for roughly higher photon scattering rate than the case Norrgard et al. (2016); Norrgard (2016). However, the excited state has fractional character (Table 4). Decays from go to the desired (2/5 of the time) or to (3/5 of the time). Hence, only photons can be scattered before molecules are lost to the uncoupled state. This loss is inconsequential for molecule detection applications, but is unacceptable for laser slowing and cooling. This population could be recovered by repumping with an additional laser tuned to the transition. In this case, all electric dipole decay paths to = 0 would again be optically coupled.
Several possible optical cycling schemes exist which are closed to photon scatters, sufficient for laser slowing or trapping. We present one such scheme in Fig. 5c. The main cycling transition is to minimize rotational branching. Lasers (corresponding to the transition ) may be added to the optical cycle in order of decreasing importance (, , , ), until the system is sufficiently closed for the intended application. The resolved excited state HF structure allows the three strongest off-diagonal vibronic decays to be repumped through to take advantage of the near-unity branching fraction. We calculate that with only the laser, 90 optical cycles may be achieved before molecules decay into a higher vibrational level. This should be sufficient for high-efficiency detection. The addition of one repump laser should be sufficient to achieve transverse cooling (closed to cycles). All four lasers shown in Fig. 5c would be necessary to achieve laser slowing or trapping. In this scheme, we expect to scatter photons before decaying to unpumped levels.
In addition to spectroscopically identifying the cycling transition of TlF, we have characterized the HF structure of the state. Of particular note is the large magnetic HF interaction of the Tl nuclear spin. With the exception of the lowest rotational and HF sublevel, the HF interactions significantly mix neighboring rotational levels of the state, and lead to additional rotational branching.
TlF appears to be an excellent molecule for optical cycling. In particular, rotational and vibrational branching fractions presented here indicate optical cycles may be achieved using a single laser. Efficient detection with a single laser should be possible in a symmetry violation measurement using TlF. In addition, laser cooling and trapping should be feasible with four or fewer lasers.
While the measurements presented here show that the system of TlF should allow for highly closed optical cycling, a number of considerations should be accounted for when choosing an appropriate cycling scheme in order to achieve a high photon scattering rate. We plan to detail these considerations in a future paper.
Financial support was provided by the Army Research Office, the John Templeton Foundation, the Heising-Simons Foundation, and by NSF Grants No. PHY1205824 and PHY1519265. The authors thank M. Kozlov and T. Steimle for helpful discussions. S.S.A. and N.W. would like to thank Amherst College for Summer Research Awards.
vi.1 Vibrational Branching Further Details
In measuring the branching ratios in Section V.3, we use a number of interference filters to isolate fluorescence from individual vibronic bands. Table 6 provides further details on the filters used to monitor each transition.
|Transition Wavelength||Part Number||Transmission at||Peak Wavelength||FWHM|
|271.7 nm||R214-01||17.9%||271.8 nm||10.3 nm|
|R214-02||17.0%||271.3 nm||10.1 nm|
|275.3 nm||T144-02||14.1%||275.4 nm||1.3 nm|
|R088-01||13.8%||275.3 nm||1.3 nm|
|278.8 nm||S078-02||9.9%||279.3 nm||2 nm|
|282.5 nm||T236-04||12.9%||282.7 nm||1.4 nm|
|T236-05||12.4%||282.8 nm||1.5 nm|
|286.2 nm||T338-01||11.4%||286.5 nm||1.6 nm|
|T335-10||11%||290.9 nm||10.7 nm|
|290.0 nm||T338-02||11%||290.3 nm||1.6 nm|
|T335-10||17.8%||290.9 nm||10.7 nm|
|293.8 nm||T335-14||11.7%||297.9 nm||10.8 nm|
|T335-15||10.0%||298.7 nm||11.6 nm|
vi.2 Racah Algebra Identities
Here we provide a number of useful identities and their equation numbers in Ref. Brown and Carrington (2003). In the following, and are general angular momenta which couple to form angular momentum . Rank- tensor operators and act on and , respectively. Tensors are represented in the spherical basis, with index () for the lab frame (molecule-fixed frame) coordinate system.
Transformation from molecule-fixed frame to lab frame: