Fine and hyperfine interactions in ¹⁷¹YbOH and ¹⁷³YbOH

Polar molecules have emerged as ideal systems to measure symmetry-violating physics in low energy, table top experiments.^1–4 The large internal fields and closely spaced opposite parity states present in molecules provide extreme sensitivity to both time reversal (T) violating and parity (P) violating physics.^5,6 With the enhanced coherence times resulting from laser cooling, precision measurements with heavy polar molecules have the potential to probe beyond standard model (BSM) physics at PeV scales.^3,7,8 In addition, P-violating effects are amplified in molecules when opposite parity states, such as neighboring rotational states, are tuned to near degeneracy via an external field.^9–11 Searches for P-violation in molecules can provide both precision tests of the standard model (SM) via measurement of nuclear spin-dependent parity violation (NSD-PV),^12,13 and sensitive probes of P-violating BSM physics.¹⁴ The odd isotopes of the linear triatomic molecule ytterbium monohydroxide, ¹⁷¹YbOH and ¹⁷³YbOH, have been identified as promising candidates for next generation precision measurements of P- and T-violating physics. The ¹⁷¹YbOH isotopologue provides sensitivity to NSD-PV,¹⁵ while the ¹⁷³YbOH isotopologue is sensitive to new T,P-violating hadronic BSM physics via a measurement of the ¹⁷³Yb nuclear magnetic quadrupole moment (NMQM).^7,16,17 These molecules combine advantageous parity doublets, which are absent in their diatomic analogs, with the ability to be laser cooled.^7,18

NSD-PV arises from three major sources: vector electron-axial nucleon electroweak coupling (V_eA_n), the nuclear anapole moment, and the combination of nuclear electroweak charge and normal hyperfine structure. The P-odd, effective NSD-PV Hamiltonian for a single molecular electronic state is^10,15

where κ is the measurable NSD-PV parameter encapsulating the effects of all NSD-PV sources, W_P is an effective parameter quantifying the overlap of the valence electron’s wavefunction with the nucleus, S is the valence electron spin, $Î=II$⁠, where I is the nuclear spin of the ¹⁷¹Yb nucleus, and $n̂$ is a unit vector along the molecular symmetry axis. Currently, the only non-zero measurement of NSD-PV is in atomic Cs.¹⁹ 

The coupling of new T, P-violating BSM physics to standard model particles will result in T, P-violating energy shifts in atoms and molecules.^1,3,4 The T, P-odd effective molecular Hamiltonian describing these energy shifts is²⁰ 

where k_S is the T, P-odd scalar-pseudoscalar nucleon-electron current interaction constant,^21,d_e is the electron electric dipole moment (eEDM), Q is the nuclear Schiff moment (NSM), M is the NMQM, W_S, W_d, W_Q, and W_M are coupling constants parameterizing the sensitivity of the molecule to the different T,P-violating sources, and $T̂$ is a rank 2 tensor that relates the NMQM shift to the nuclear spin orientation. Note the NMQM term is only nonzero for nuclei with $I>12$⁠. A measurement of a non-zero eEDM, NSM, or NMQM within the projected sensitivity of contemporary experiments would be a confirmation of new T,P-violating BSM physics, while a null measurement places bounds that constrain potential BSM theories. The current limit on the eEDM⁵ is $de<1.1×10−29ecm$⁠, and several upcoming molecular experiments aim to improve this limit by an order of magnitude or more.^8,18,22–24 The most stringent limit on an NSM results from measurements involving ¹⁹⁹Hg.²⁵ Several other NSM experiments in other atomic^26–29 and molecular^30,31 systems are also under way. The best bounds on an NMQM are obtained from measurements of atomic Cs.³² NMQMs are enhanced by collective effects in deformed nuclei,^20,33 and the large quadrupole deformation of the ¹⁷³Yb nucleus indicates that it should have an enhanced NMQM. A measurement of an NMQM in ¹⁷³YbOH would complement eEDM searches as the former probes BSM physics in the hadronic sector, as opposed to the leptonic sector.^20,34,35 Furthermore, an NMQM measurement would complement NSM searches since a measurement of both an NMQM and an NSM in several different systems will allow the exact source of the hadronic BSM physics to be pinpointed.

Polyatomic molecules offer several distinct advantages over diatomic molecules for precision measurements.^4,7 First, linear triatomic molecules generically have parity doublets, known as l-doublets, in their excited bending modes. In YbOH and other similar molecules, the l-doublet splitting is ∼10 MHz.⁷ In the case of T,P-violation measurements, l-doublets enable full polarization in modest electric fields, <1 kV/cm⁷, and act as internal comagnetometer states, allowing for robust control over systematic errors via reversal of the T,P-violating energy shift without switching external laboratory fields. In the case of an NSD-PV measurement, the opposite parity l-doublet states can be tuned to near degeneracy with modest magnetic fields, ∼1 to 10 mT,¹⁵ both increasing sensitivity and reducing the systematics associated with large B fields and field reversals. The electronic structure of the molecule can then be leveraged for laser cooling (if the molecule has an electronic structure amendable to laser cooling, such as YbOH) without conflicting with the internal structure needed for parity doublets. Precision measurements utilizing laser cooled and trapped molecules could increase measurement sensitivity by orders of magnitude compared to beam experiments.⁷ Laser cooling of the ¹⁷⁴YbOH isotopologue has already been demonstrated¹⁸ and could be extended to the odd isotopologues by taking hyperfine structure into consideration.

Here, we report on the characterization of the $Ã2Π1/2(0,0,0)−X̃2Σ1/2(0,0,0)$ transition of ¹⁷¹YbOH and ¹⁷³YbOH, which for convenience will be designated as $000Ã2Π1/2$-$X̃2Σ+$⁠. Our study includes both excitation spectroscopy using laser induced fluorescence (LIF) detection of a cold supersonic molecular beam (MB) sample and laser absorption measurements of a cryogenic buffer-gas cooled (CBGC) sample. The derived molecular parameters are necessary for the implementation of both NSD-PV and NMQM measurements as well as for laser cooling. In order to extract the NSD-PV and NMQM parameters, κ and M, from P- or T-violating molecular energy shifts, the values of the coupling constants W_P or W_M must be known. These coupling constants cannot be measured experimentally and instead must be calculated via ab initio or semi-empirical methods. Comparison of measured hyperfine parameters with calculated ones provides a rigorous test of the computational methods used to calculate the T,P-violating coupling constants as both sets of parameters probe the nature of the valence electron in the vicinity of the nucleus. Finally, we demonstrate a novel spectroscopic technique utilizing laser-induced chemical reactions³⁶ to both amplify the signal from the desired isotopologue and disentangle the respective isotopologue’s spectrum from that of other, more abundant, overlapping isotopologues.

A review of experimental measurements and theoretical predictions for YbOH can be found in the article describing high-resolution optical analysis performed on the even isotopologues.³⁷ More recent experimental work includes the aforementioned demonstration of Sisyphus and Doppler laser cooling of ¹⁷⁴YbOH¹⁸ as well as the determination of fluorescence branching ratios, radiative lifetimes, and transition moments.³⁸ Recently reported theoretical studies include calculations of the molecular NMQM sensitivity coefficient, W_M, of Eq. (2).^16,17 As part of these studies, the ground state magnetic hyperfine parameters, A_|| for ¹⁷¹YbOH and ¹⁷³YbOH, and the nuclear electric quadrupole coupling parameter, e²q₀Q for ¹⁷³YbOH, were predicted.

Initial experiments were performed at ASU using a molecular beam LIF spectrometer similar to that used in previous high-resolution optical studies of ^172,174YbOH.^37,39 In brief, YbOH is produced by laser ablating (532 nm, ∼10 mJ/pulse, 20 Hz) a Yb rod in the presence of a methanol/argon supersonic expansion. The resulting beam is skimmed to produce a well collimated beam with a temporal pulse width of ∼40 µs in the detection region. The molecular beam is probed by an unfocused (∼5 mm), low power (∼5 mW), single frequency cw-dye laser ∼0.5 m downstream. It is estimated that the laser probes approximately 1 × 10⁹ YbOH molecules in each molecular beam pulse. The resulting on-resonance LIF signal was viewed through a 580 ± 10 nm bandpass filter, detected by a photomultiplier tube (PMT), and processed using gated photon counting. Typically, the photon counts from 35 ablation pulses at each excitation laser frequency are summed. The absolute excitation wavelength is determined by co-recording a sub-Doppler I₂ spectrum,⁴⁰ and the relative wavelength is measured by co-recording the transmission of an actively stabilized etalon (free spectral range of 751.393 MHz).

Subsequent high-resolution spectra were recorded at Caltech by absorption spectroscopy. The apparatus was nearly identical to that used in the initial demonstration of laser-induced chemical enhancement of YbOH production.³⁶ In summary, cold YbOH molecules are produced via cryogenic buffer-gas cooling, which is described in detail elsewhere.^41–45 The molecules are produced inside a copper, cryogenic buffer-gas cell cooled to ∼4 K, with an internal cylindrical bore of 12.7 mm and a length of ∼83 mm. Helium buffer-gas is introduced into the cell by a 3.2 mm diameter copper tube. The helium then passes through a diffuser 3.2 mm from the gas inlet and exits the cell through a 5 mm aperture on the other end of the cell. All measurements presented here were performed inside the buffer-gas cell, as opposed to in the extracted beam. YbOH molecules are created by ablating (532 nm, ∼30 mJ/pulse, ∼5 Hz) solid pressed targets of Yb powder mixed with either Yb(OH)₃ or Te(OH)₆ powders. To make the Yb + Yb(OH)₃ targets, the powders were mixed to obtain a 1:1 stoichiometric ratio of Yb to OH, then ground in a mortar and pestle, passed through a 230 mesh sieve, mixed with a 4% polyethylene glycol binder (PEG8000) by weight, and pressed in an 8 mm die at ∼10 MPa for ∼15 min. A similar procedure was used to produce the Yb + Te(OH)₆ targets.

To measure the absorption spectra, three cw-laser beams are passed through the spectroscopy window: the primary tunable absorption spectroscopy beam (1 mm diameter, ∼30 μW), the normalization laser (1 mm diameter, ∼40 μW) used to monitor the shot-to-shot fluctuations in YbOH production, and the chemical enhancement laser (3 mm diameter, ∼300 mW), which increases the molecular yield by exciting atomic Yb to the metastable ³P₁ state, thereby boosting chemical reactions rates between Yb atoms and OH-containing ablation products.³⁶ 

The normalization laser is fixed to either the ^OP₁₂(2) or the ^RR₁₁(2) line of the $000Ã2Π1/2$-$X̃2Σ+$ transition of ¹⁷²YbOH at 17 322.1732 cm⁻¹ and 17 327.0747 cm⁻¹, respectively.³⁷ The frequency of the enhancement laser is fixed to the ³P₁–¹S₀ transition of the desired Yb isotope. Specifically, the ¹⁷⁴Yb(17 992.0003 cm⁻¹), ¹⁷⁶Yb(17 991.9685 cm⁻¹), F″ = 1/2 → F′ = 1/2 ¹⁷¹Yb (17 991.9292 cm⁻¹), and the F″ = 5/2 → F′ = 7/2 ¹⁷³Yb (17 991.9207 cm⁻¹) transitions⁴⁶ were used. The fixed lasers are locked to a stabilized HeNe laser via a scanning transfer cavity and active feedback (∼5 MHz resolution). The primary absorption laser is continuously scanned in frequency, and the resulting absorption is detected with a photodiode. The absolute frequency of the primary absorption laser is monitored using a digital wavemeter, and the relative frequency is tracked in a separate transfer cavity with respect to a stabilized HeNe laser (∼7 MHz resolution). The absolute transition frequencies are calibrated using known ¹⁷²YbOH and ¹⁷⁴YbOH spectral lines.³⁷ The light of the enhancement laser is switched on and off using a mechanical shutter so that both enhanced and unenhanced spectra of the desired isotopologue can be measured in successive shots.

The chemical enhancement is utilized as a spectroscopic tool to disentangle the complicated isotopologue structure. This technique is illustrated for ¹⁷¹YbOH in Fig. 1 and described below. The measured optical depth (OD) of the primary probe beam is integrated over the duration of the molecular pulse, typically ∼3 ms. This integrated OD is then normalized by the integrated OD of the normalization probe. The normalized signal at a single laser frequency without enhancement light is S_UE = S₁₇₁ + S_B, where S₁₇₁ is the measured integrated OD for a specific transition of ¹⁷¹YbOH and S_B is the background integrated OD from all other overlapping YbOH isotopologues or other molecules. The measured normalized and integrated OD with the enhancement light on and tuned to the ¹⁷¹Yb atomic resonance is S_E = ES₁₇₁ + S_B. Here, E is the enhancement factor, which is defined as N_E/N₀, the ratio of the number of molecules produced with the enhancement laser on, N_E, to the number produced with the enhancement laser off, N₀. Taking the difference between the enhanced and unenhanced signals $SE−SUE=E−1S171$ results in the spectrum from only the ¹⁷¹YbOH isotopologue. It is observed that E in the cell is typically ∼4 to 8. This technique generalizes to the other isotopologues of YbOH and other molecules as well.

A general description of the $000Ã2Π1/2$-$X̃2Σ+$ band can be found in the previous report on the analysis of the field-free, Stark and Zeeman spectroscopy of ^172,174YbOH.³⁵ Here, we focus on the odd isotopologues. There are seven naturally occurring isotopes of Yb in moderate abundance: ¹⁶⁸Yb (0.1%), ¹⁷⁰Yb (3.0%), ¹⁷¹Yb (14.3%), ¹⁷²Yb (21.8%), ¹⁷³Yb (16.1%), ¹⁷⁴Yb (31.8%), and ¹⁷⁶Yb (12.8%). The ¹⁷¹YbOH and ¹⁷³YbOH spectra are more complex than those for ^172,174YbOH, primarily due to the large ¹⁷¹Yb(I = 1/2, μ = +0.493 67 μ_N) and ¹⁷³Yb(I = 5/2, μ = −0.679 89 μ_N) magnetic hyperfine interaction and, in the case of ¹⁷³YbOH, the large nuclear electric quadrupole hyperfine interaction (Q = 280.0 ± 4.0 fm²).⁴⁷ The previously described branch designation used for the even and odd isotopologues of the (0,0)A²Π_1/2-X²Σ⁺ band of YbF⁴⁸ will also be used for the $000Ã2Π1/2$-$X̃2Σ+$ band YbOH. The even isotopologues exhibit six branches, labeled ^PP₁₁, ^QQ₁₁, ^RR₁₁, ^PQ₁₂, ^OP₁₂, and ^QR₁₂, following the ^ΔNΔJ_Fi'Fi″(N″) designation expected for a ²Π_1/2(Hund’s case (a))- ²Σ(Hund’s case (b)) band. The low rotational levels of the $X̃2Σ+$ state for the odd isotopologues exhibit a Hund’s case (b_βS) energy level pattern due to the large ¹⁷¹Yb(I = 1/2) and ¹⁷³Yb(I = 5/2) magnetic hyperfine interaction, where the electron spin angular momentum, S, is coupled to the ¹⁷¹Yb(I = 1/2) or ¹⁷³Yb(I = 5/2) nuclear spin angular momentum, I, to give an approximately good intermediate quantum number G. The angular momentum G is coupled to the rotational angular momentum, N, to give the intermediate angular momentum F_1, which is then coupled to the proton nuclear spin angular momentum I₂(=1/2) to give the total angular momentum F. The corresponding coupling limit wavefunction, $SI(Yb)G(GN)F1F1I(H)F$⁠, is useful for describing the low-rotational energy levels of the $X̃2Σ+(0,0,0)$ state. The Yb and H hyperfine interactions in the $Ã2Π1/2$ state are very small compared to the rotational and Λ-doubling spacing, and the energy level pattern is that of a molecule near a sequentially coupled Hund’s case (a_βJ) limit. The corresponding coupling limit wavefunction, $ηΛSΣJΩJI(Yb)F1F1I(H)F$⁠, is useful in describing the low-rotational energy levels of the $Ã2Π1/2(0,0,0)$ state. With the exception of broadening of the lowest rotational branch features of the molecular beam spectra, there was no evidence of proton hyperfine splitting. The six ²Π_1/2(Hund’s case (a)) - ²Σ(Hund’s case (b)) branches of the even isotopologues split and regroup into eight branches labeled ^OP_1G, ^PP_1G + ^PQ_1G, ^QQ_1G + ^QR_1G, and ^RR_1G, appropriate for a²Π_1/2(case (a_βJ))-²Σ(case (b_βS)) band with G = 0 and 1 for ¹⁷¹YbOH and G = 2 and 3 for ¹⁷³YbOH. The lines of the ^OP_1G and ^RR_1G branches (odd isotopologues) and the ^OP₁₂ and ^RR₁₁ branches (even isotopologues) form progressions in N″, with adjacent members separated by ∼4B″ extending to red and blue, respectively, and are relatively unblended. The $000Ã2Π1/2$-$X̃2Σ+$ band exhibits a blue degraded head formed by low-N″ features of the ^PQ₁₂ and ^PP₁₁ (even isotopologues) and ^PP_1G + ^PQ_1G (odd isotopologues) branches. Isotopic spectral shifts are very small because the potential energy surfaces for the $X̃2Σ+(0,0,0)$ and $Ã2Π1/2(0,0,0)$ states are very similar, causing the ^PQ₁₂, ^PP₁₁, ^PP₁₁, ^QQ₁₁, and ^QR₁₂ branches of the even isotopologues and ^PP_1G + ^PQ_1G and ^QQ_1G + ^QR_1G branches of the odd isotopes to be severely overlapped.

The observed and calculated LIF molecular beam spectra in the region of the ^OP₁₂ (3) even isotopologues branch features and the ^OP_1G(3) odd isotopologues branch features are presented in Fig. 2. The observed spectrum in Fig. 2 is similar to that of the (0,0)A²Π_1/2-X²Σ⁺ band of YbF (Ref. 48) with the exception that a small splitting (∼50 MHz) due to ¹⁹F(I = 1/2) was observed in YbF whereas the smaller H(I = 1/2) doubling in YbOH is not fully resolved. The predicted spectrum was obtained, assuming a 15 K rotational temperature, 30 MHz full width at half maximum (FWHM) Lorentzian lineshape, and optimized parameters (see below). The two components of the ^OP_1G(3) transition are widely spaced and of the opposite order because the spacing between the X²Σ⁺(0,0,0) G = 3 and G = 2 groups of levels of ¹⁷³YbOH is ∼−5660 MHz (∼3b″_F) and that between the G = 1 and G = 0 levels of ¹⁷¹YbOH is ∼+6750 MHz (∼b″_F). The spectral patterns of the G = 3 and G = 2 groups of ¹⁷³YbOH are irregular because of the large nuclear electric quadrupole interaction (e²q₀Q) in both the $X̃2Σ+(0,0,0)$ and $Ã2Π1/2(0,0,0)$ states.

The laser-induced fluorescence spectrum of a molecular beam sample in the highly congested bandhead region (i.e., between 17323.50 and 17323.85 cm⁻¹) is presented in Fig. 1 of Ref. 37. The bandhead region is dominated by low-rotational ^PP₁₁, ^QQ₁₁, ^PQ₁₂, and ^QR₁₂ branch features of the more abundant even isotopologues, which makes assignment of the ^PP_1G + ^PQ_1G and ^QQ_1G + ^QR_1G branch features of the less abundant odd isotopologues of the molecular beam sample extremely difficult. Recording the chemically enhanced absorption spectrum of a CBGC sample in the bandhead region was critical to the assignment and subsequent analysis. The observed and predicted high-resolution absorption spectra in the bandhead region of a CBGC sample are presented in Fig. 3, both with [panel (b)] and without [panel (a)] ¹⁷³YbOH chemical enhancement induced by the ¹⁷³Yb atomic excitation. There is approximately a factor of four increase in the signal upon atomic excitation. The difference with and without chemical enhancement [panels (a) and (b)] is presented in panel (c) of Fig. 3. The predicted spectrum was obtained assuming a 5 K rotational temperature, 90 MHz FWHM Lorentzian lineshape, and optimized parameters (see below). There is no evidence of proton hyperfine splitting at this spectral resolution. The CBGC spectra are greatly simplified relative to the supersonic molecular beam spectra due to both the isotopic selectivity, which is evident from Fig. 3 [panel (a) vs (c)], and the lower rotational temperature (∼5 vs ∼20 K). Even so, the spectral features shown in Fig. 3 [panel (c)] are a blend of many transitions (see below). For example, the feature marked “A” in Fig. 3 is predominantly a blend of the $F1″=3→F1′=3$ component of the ^PP₁₂ + ^PQ₁₂(1) branches and the $F1″=4→F1′=4$ component of the ^PP₁₂ + ^PQ₁₂(2) branch features of ¹⁷³YbOH. The feature marked “B” is predominantly a blend of the ^PP₁₂ + ^PQ₁₂(3) (⁠$F1″=5→F1′=5$⁠), ^PP₁₂ + ^PQ₁₂(1) (⁠$F1″=2→F1′=3$⁠), and ^PP₁₂ + ^PQ₁₂(3) (⁠$F1″=4→F1′=4$⁠) transitions.

The observed and predicted high-resolution absorption spectra in the bandhead region of the CBGC sample are presented in Fig. 4, both with [panel (b)] and without [panel (a)] ¹⁷¹YbOH chemical enhancement induced by the ¹⁷¹Yb atomic excitation. The spectrum obtained by subtracting the signals with and without chemical enhancement induced by the ¹⁷¹Yb atomic excitation is presented in panel (c). The predicted spectrum was obtained assuming a 5 K rotational temperature, 90 MHz FWHM Lorentzian lineshape, and optimized parameters (see below). The head at 17 323.55 cm⁻¹ is an unresolved blend of the ^PP₁₂ + ^PQ₁₂(1), ^PP₁₂ + ^PQ₁₂(2), and ^PP₁₂ + ^PQ₁₂(3) branch features (see below). Unlike the ¹⁷³YbOH spectrum shown in Fig. 3, there are numerous ¹⁷¹YbOH spectral features that are unblended. For example, the feature marked “A” in Fig. 4 is the $F1″=1→F1′=1$ component of ^QQ₁₁ + ^QR₁₁(1), “B” is the $F1″=5→F1′=5$ component of ^PP₁₁ + ^PQ₁₁(5), and “C” is the $F1″=2→F1′=2$ component of ^QQ₁₁ + ^QR₁₁(2) (see below).

Although the ^RR_1G branch features of the odd isotopologues are less blended than the ^PP_1G + ^PQ_1G and ^QQ_1G + ^QR_1G branch features, recording the absorption spectrum of the CBGC sample was still critical for spectral disentanglement and assignment of the molecular beam LIF spectrum. This is illustrated in Fig. 5 where the observed and predicted molecular beam LIF (left side) and CBGC absorption (right side) spectra in the region of the ^RR₁₁(2) (even isotopologues) and ^RR_1G(2) (odd isotopologues) lines are presented. Predicted molecular beam spectra assumed a rotational temperature of 20 K and an FWHM linewidth of 30 MHz, while the CBGC spectra assumed those to be 5 K and 90 MHz, respectively, and both used optimized parameters (see below). The CBGC absorption spectrum recorded without the atomic excitation is presented in panel (a). The CBGC absorption spectra recorded with the atomic excitation laser tuned to the ³P₁–¹S₀ transitions of ¹⁷⁶Yb, ¹⁷⁴Yb, and ¹⁷¹Yb are presented panels (b)–(d), respectively. The ¹⁷¹YbOH absorption spectrum in panel (e) was obtained by taking the difference of “a” and “d,” revealing a spectral feature that is obscured in the higher resolution molecular beam LIF spectrum by the much more intense ^RR₁₁(2) line of ¹⁷⁴YbOH.

The precisely measured transition wavenumber for 94 spectral features of the $000Ã2Π1/2$-$X̃2Σ+$ band of ¹⁷³YbOH, which are assigned to 124 transitions, is given in Table S1 of the supplementary material, along with the difference between the observed and calculated wavenumbers. The precisely measured transition wavenumber for 63 spectral features of the $000Ã2Π1/2$-$X̃2Σ+$ band of ¹⁷¹YbOH, which are assigned to 68 transitions, is given in Table S2 of the supplementary material, along with the difference between the observed and calculated wavenumbers. Also presented are the quantum number assignments.

The analysis of the ¹⁷¹YbOH and ¹⁷³YbOH $000Ã2Π1/2$-$X̃2Σ+$ spectra was nearly identical to that of the (0,0)A²Π_1/2-X²Σ⁺ band of ¹⁷¹YbF (Ref. 49) and ¹⁷³YbF (Ref. 48) and will not be described in detail here. In brief, the observed transition wavenumbers listed in Tables S1 and S2 of the supplementary material were used as inputs into a weighted least-squares optimization routine. The molecular beam data were given twice the weight of the more extensive CBGC data due to the high spectral resolution (30 MHz FWHM vs 90 MHz) and the fact that it was co-recorded with the I₂ wavelength calibration spectrum. The $X̃2Σ+(0,0,0)$ and $Ã2Π1/2(0,0,0)$ energies were obtained by diagonalizing 24 × 24 (=(2S + 1) (2I₁ + 1)(2I₂ + 1)) and 48 × 48 (=2(2S + 1)(2I₁ + 1) (2I₂ + 1)) matrices for ¹⁷³YbOH and 8 × 8 and 16 × 16 matrices for ¹⁷¹YbOH constructed in a sequentially coupled Hund’s case (a_βJ) basis set, $ηΛSΣJΩJI(Yb)F1F1I(H)F$⁠. Note that a Hund’s case (a_βJ) basis is used for the $X̃2Σ+(0,0,0)$ state even though the energy level pattern is that of a molecule close to a Hund’s case (b_βS). The $X̃2Σ+(0,0,0)$ states were modeled using an effective Hamiltonian^48,49 that included rotation (B and D), spin-rotation (γ), Yb and H magnetic hyperfine (Fermi contact, b_F, and dipolar, c), and ¹⁷³Yb axial nuclear electric quadrupole (e²q₀Q) parameters. The energy levels for the $Ã2Π1/2(0,0,0)$ states were modeled using an effective Hamiltonian that included the spin–orbit (A), rotation (B and D), Λ-doubling (p + 2q), Yb magnetic hyperfine (Λ-doubling type, d, and electronic orbital hyperfine, a) and ¹⁷³Yb axial nuclear electric quadrupole (e²q₀Q).⁵⁰ In the Hund’s case (a_βJ) limit, the determinable combination of magnetic hyperfine parameters is h_1/2 ≡ $a−bF2−c3$⁠, and the levels can be accurately modeled using $aÎzL̂z$ and by fixing the Fermi contact, b_F, and dipolar, c, terms to zero. The dataset, which is restricted to the $Ω=1/2$ levels of the $Ã2Π(0,0,0)$ state, is insensitive to the perpendicular component of the nuclear electric quadrupole interaction (e²q₂Q). The effective Hamiltonians are described in more detail in the supplementary material.

Analyses that floated various combinations of parameters were attempted. In all cases, the small proton magnetic hyperfine parameters for the $X̃2Σ+(0,0,0)$ state were held fixed to those determined from the analysis of the ¹⁷⁴YbOH microwave spectrum (b_F = 4.80 MHz and c = 2.46 MHz).³⁹ The spin–orbit parameter, A, of the $Ã2Π(0,0,0)$ state was constrained to the previously determined⁵² value of 1350 cm⁻¹. The analysis is relatively insensitive to the value of A, other than the obvious linear displacement, because of the large separation between the $Ã2Π1/2(0,0,0)$ and $Ã2Π3/2(0,0,0)$ states and the relatively small rotational parameter, B (i.e., the rotationally induced spin-uncoupling effect is negligible because B/A ≅ 2 × 10⁻⁴). The centrifugal correction parameters, D, for the $X̃2Σ+(0,0,0)$ and $Ã2Π(0,0,0)$ states and the small spin-rotation parameter, γ, for the $X̃2Σ+(0,0,0)$ state were constrained to the values predicted from extrapolation of the values for ¹⁷⁴YbOH using the expected isotopic mass dependence. In the end, the ¹⁷³YbOH dataset was satisfactorily fit by optimizing 10 parameters, and the ¹⁷¹YbOH dataset, by optimizing 8 parameters. The standard deviation of the ¹⁷¹YbOH and ¹⁷³YbOH fits was 27 and 25 MHz, respectively, which is commensurate with estimated weighted measurement uncertainty. The optimized parameters and associated errors are presented in Table I, along with those for the A²Π_1/2(v = 0) and X²Σ⁺(v = 0) states of ¹⁷¹YbF (Ref. 49) and ¹⁷³YbF (Ref. 48). The spectroscopic parameters are independently well determined with the largest correlation coefficient being between B″ and B′, which for the ¹⁷¹YbOH and ¹⁷³YbOH fits are 0.854 and 0.960, respectively.

Spectral predictions, such as those presented in Figs. 2–5, were essential for the assignment and analysis. The approach was identical to that used in modeling the (0,0)A²Π_1/2-X²Σ⁺ band of ¹⁷¹YbF (Ref. 49) and ¹⁷³YbF (Ref. 48). The electric dipole transition moment matrix was constructed in a sequentially coupled Hund’s case (a_βJ) basis set $ηΛSΣJΩJI(Yb)F1F1I(H)F$ and cross multiplied by the eigenvectors to produce the transition moments. The relative intensities were taken as the product of the square of the transition moment, a Boltzmann factor, and the relative isotope abundance. The relative intensities and a Lorentzian lineshape were then used to predict the spectra. The complete set of spectroscopic parameters for the six most abundant isotopologues used in these predictions is presented in Table S3 of the supplementary material. The parameter values for ¹⁷¹YbOH and ¹⁷³YbOH were taken from the present study, while those for ¹⁷⁴YbOH [$X̃2Σ+(0,0,0)$] were taken from the analysis of the microwave spectrum.³⁹ The parameters for ¹⁷⁴YbOH [$Ã2Π1/2(0,0,0)$], ¹⁷²YbOH [$X̃2Σ+(0,0,0)$], and ¹⁷²YbOH [$Ã2Π1/2(0,0,0)$] were taken from the analysis of the optical spectra.³⁷ The parameters for the ¹⁷⁰YbOH and ¹⁷⁶YbOH isotopologues were obtained by extrapolation of those for ¹⁷¹YbOH, ¹⁷²YbOH, ¹⁷³YbOH, and ¹⁷⁴YbOH, using the expected mass dependence. In all cases, the $X̃2Σ+(0,0,0)$ proton magnetic hyperfine parameters were constrained to the values of ¹⁷⁴YbOH, and those for the $Ã2Π1/2(0,0,0)$ were constrained to zero.

The major objectives of this study are to analyze the $000Ã2Π1/2$-$X̃2Σ+$ rotationally resolved spectra of the odd isotopologues, precisely determine the $Ã2Π1/2(0,0,0)$ and $X̃2Σ+(0,0,0)$ energy levels, and demonstrate the utility of chemical enhancement as a spectroscopic tool. The obtained spectral information is needed for the design and implementation of experiments under way to characterize NSD-PV and search for new T-violating BSM physics. The calculated energies for the N = 0–4 levels of the $X̃2Σ+(0,0,0)$ state and J = 0.5–5.5 levels of the $Ã2Π1/2(0,0,0)$ state for ¹⁷¹YbOH are presented in Tables S4 and S5, respectively, of the supplementary material. The calculated energies for the N = 0–4 levels of the $X̃2Σ+(0,0,0)$ state and J = 0.5–5.5 levels of the $Ã2Π1/2(0,0,0)$ state for ¹⁷³YbOH are presented in Tables S6 and S7, respectively, of the supplementary material. Spectroscopic parameters given in Table S3 can be used to accurately predict the energies of higher rotational levels of the even and odd isotopologues. A perusal of the energy level patterns for the odd isotopologues $Ã2Π1/2(0,0,0)$ states reveal that the large Λ-doubling, relative to the rotational spacing, makes the patterns more similar to that of a²Σ⁻ state than the a²Π_1/2 state. Specifically, the pattern is that of rotationally spaced J = N ± 1/2 pairs of levels (i.e., ρ-doublets), with the exception of the N = 0, J = 1/2 lowest energy level, which has negative parity. Although the energy levels could be modeled as the a²Σ⁻ state, a²Σ⁺ − ²Σ⁻ transition is electric dipole forbidden and not consistent with the observed intensities.

The predicted stick spectra for ¹⁷¹YbOH and ¹⁷³YbOH generated using the optimized spectroscopic parameters and a rotational temperature of 15 K are presented in Figs. 6 and 7, respectively. Of particular interest are the transitions that terminate at the J = 0.5, +parity level of the $Ã2Π1/2(0,0,0)$ state, to be used for photon cycling and laser cooling.¹⁸ For the even isotopologues, these are the ^PQ₁₂ (1) and ^PP₁₂ (1) transitions, and for the odd isotopologues, they are the ^PQ_1G + ^PP_1G (1) transitions. These transitions are in a highly congested region of the spectrum (see Figs. 6 and 7). It is noteworthy that the ^PQ₁₂ (1) and ^PP₁₂ (1) lines of the even isotopologues are “rotationally closed” (i.e., the excitation and fluorescence spectra only involve the N″ = 1 levels), whereas in the odd isotopologues, the branching ratio for the (J = 0.5, +) $Ã2Π1/2(0,0,0)$ → (N = 3, −) $X̃2Σ+(0,0,0)$ transition is ∼1% relative to the (J = 0.5, +) $Ã2Π1/2(0,0,0)$ → (N = 1, −) $X̃2Σ+(0,0,0)$ transitions. This is primarily caused by the mixing of the J = 0.5, + parity and the close lying (ΔE ≅ 3 GHz) J = 1.5, + parity levels of the $Ã2Π1/2(0,0,0)$ state by hyperfine terms in the effective Hamiltonian. There is a similar, yet much smaller in magnitude, hyperfine induced mixing of the N = 1 and N = 3 levels of the $X̃2Σ+(0,0,0)$ state.

The hyperfine parameters are of particular interest because they are sensitive probes of the electronic wavefunction in the region of the nuclei. The ability to accurately predict hyperfine parameters is the most direct gauge of the computational methodology used to predict W_s, W_d, W_Q, and W_M coupling constants of Eq. (2). As part of the recent calculation,¹⁶ the nuclear magnetic quadrupole interaction constants W_M for YbOH, the ground state A_|| (= b_F +$23$c) magnetic hyperfine constant for ¹⁷³YbOH and ¹⁷¹YbOH, and the axial electric quadrupole coupling constant e²Qq₀ for ¹⁷³YbOH were predicted. These predictions were performed using numerical gradients of a four-component Dirac Coulomb Hamiltonian, with electronic correlation treated using the multireference Fock-space coupled cluster method (FSCC). Similarly, a recent calculation⁵³ of W_d and W_s includes predictions for both the ground state A_|| and A_⊥(= b_F −$23$c) magnetic hyperfine constants for ¹⁷³YbOH. In this case, quasi-relativistic two-component calculations with many-body interactions treated at the level of complex generalized Hartree–Fock (cGHF) and complex generalized Kohn–Sham (cGKS) density functional theory methods were implemented. Most recently, the axial electric quadrupole coupling constant e²Qq₀ for ¹⁷³YbOH was predicted using analytic gradients formulated from spin–orbit CCSD(T) theory, computed against an atomic mean-field scalar relativistic (SFX2C-AMF) Hamiltonian.⁵⁴ The predicted hyperfine parameters for the $X̃2Σ+(0,0,0)$ state are compared with the measured values in Table II. There are no theoretical predictions for the hyperfine interactions in the $Ã2Π1/2(0,0,0)$ state. The calculated hyperfine parameters obtained using relativistic coupled cluster approaches^16,54 are in excellent agreement with the observed values, with the calculated magnetic hyperfine parameters being 2.5% (Ref. 16) larger in magnitude and the e²Qq₀ parameter being 5.5% (Ref. 16) and 5.2% (Ref. 54) larger in magnitude. Evidently, calculating the core polarization is more difficult than calculating the valence electron properties. Nonetheless, the excellent agreement suggests that the calculated W_M coupling constants for ground state ¹⁷³YbOH of Ref. 16 and the effective electric field, E_eff, of Ref. 55, which was obtained using the same computational methodology, are quantitatively accurate. The predicted E_eff is relevant to the electron electric dipole moment (eEDM) measurements for all the isotopologues of YbOH. The quasirelativistic two component calculations in Ref. 53 give magnetic hyperfine constants that are smaller in magnitude by ∼15% and 31% for the cGHF and cGKS methods, respectively.

Although the high-level relativistic electronic structure calculation¹⁶ accurately predicts the hyperfine interactions for the $X̃2Σ+(0,0,0)$ state, a less quantitative, molecular-orbital based, model using atomic information is also useful. Such a model provides chemical insight, can be used to rationalize trends among similar molecules (e.g., YbF and YbOH), and is able to predict hyperfine interactions for excited electronic states that are not readily addressed by relativistic electronic structure calculations. The $X̃2Σ+(0,0,0)$ hyperfine fitting parameters are related to various averages over the spatial coordinates of the electrons by^51,56

and

where $ŝzi$ is the spin angular momentum operators for the ith electron, δ_i(r) is a Dirac delta function, and r and θ are polar coordinates. In the case of b_F and c, the sum runs only over unpaired electrons, whereas for e²Qq₀, the sum is over all electrons. The observed ratio b_F(¹⁷¹YbOH)/b_F(¹⁷³YbOH) for the $X̃2Σ+(0,0,0)$ state (= −3.62 ± 0.04) is in excellent agreement with that of the nuclear g-factors, g_N (¹⁷¹Yb)/g_N (¹⁷³Yb) (= −3.630). Meanwhile, the ratio for the less well determined dipolar ratio c(¹⁷¹YbOH)/c(¹⁷³YbOH) (=−2.86 ± 0.57) is within two standard deviations. To a first approximation, the dominant electronic configuration for the $X̃2Σ+(0,0,0)$ state has one unpaired electron in a hybridized 6s/6p/5d σ-type, Yb⁺-centered orbital, which is polarized away from the Yb–OH bond. The hybridization is driven by the stabilization achieved from shifting the center of the charge for the unpaired electron away from the electrophilic end of the Yb⁺OH⁻ molecule. Based upon a comparison with b_F (¹⁷¹Yb⁺) (= 0.4217 cm⁻¹)⁵⁷ for Yb⁺(f¹⁴6s²S_1/2), the σ-type orbital is ∼54% 6s character. The b_F(¹⁷¹YbOH) and b_F(¹⁷³YbOH) values are ∼7% smaller in magnitude than the corresponding values for YbF. Evidently, the unpaired 6s electron is more effectively polarized by OH⁻ than F⁻ (i.e., larger 6s/6p/5d hybridization). This is consistent with the point charge electrostatic model prediction that successfully models the permanent electric dipole moments for ground states of YbOH and YbF.³⁷ The Yb⁺-centered, unpaired electron was predicted to be shifted away from the Yb–F and Yb–O bond by 0.859 and 0.969 Å, respectively, reflecting the increased 6s/6p/5d hybridization in YbOH relative to YbF.

Unlike the noted change in b_F, the ¹⁷³Yb axial nuclear electric quadrupole parameter, e²q₀Q, for the ground states of ¹⁷³YbOH and ¹⁷³YbF is very nearly identical, indicating that core polarization makes a substantial contribution relative to that of the unpaired valence electron. As evident from a comparison of Eqs. (4) and (5), the contribution from the unpaired electron to e²q₀Q is

Substitution of Q = 280 fm² and g_N of −0.271 95 gives e²Qq₀(unpaired) = −0.0456 cm^-1, which is ∼40% (−0.0456/−0.1107) of the determined e²Qq₀ value for the $X̃2Σ+(0,0,0)$ state. The remaining −0.0651 cm⁻¹ portion is due to core polarization.

An interpretation of the excited state hyperfine parameters is more difficult because b_F and c were constrained to zero. The Λ-doubling type magnetic hyperfine parameter, d, on the other hand, is well determined and is given by^51,56

where $ŝi+$ is the one electron spin angular momentum raising operator and ϕ_i is the azimuthal angle of the electron. The ratio of d for ¹⁷¹YbOH and ¹⁷³YbOH (=0.031 99/−0.008 73 ≅ 3.66) is in excellent agreement with the expected ratio of g_N values. The same rationale presented for the observed d values of YbF⁵⁸ is applicable to YbOH, and an estimate gives values for ¹⁷¹YbOH and ¹⁷³YbOH of 0.0657 and −0.0182 cm⁻¹, respectively, which are the correct signs but approximately a factor of two too large. These values were obtained by assuming that the orbital of the sole unpaired electron for the $Ã2Π1/2(0,0,0)$ state is pure Yb⁺ 6p. The more realistic assumption that the orbital of the sole unpaired electron is an admixture of Yb⁺ 6p, Yb 6p, and Yb 5d and would give values in closer agreement with the experimental values.

The magnitude of the e²q₀Q parameter decreases from 0.1107(16) to 0.0642(17) cm⁻¹ upon $X̃2Σ+(0,0,0)$ to $Ã2Π1/2(0,0,0)$ excitation. The angular expectation value for the 6p orbital contribution to the open shell σ-orbital of the $X̃2Σ+(0,0,0)$ state is $p03⁡cos2θi−1p0=4/5$ while that for the open shell π-orbital of the $Ã2Π1/2(0,0,0)$ state is $p±13⁡cos2θi−1p±1=−2/5$⁠. Accordingly, if the σ- and π-orbitals of the two states had identical 6p contribution, then e²Qq₀(unpaired) for $Ã2Π1/2(0,0,0)$ would be ∼0.0228 cm⁻¹ [i.e., half the magnitude and opposite sign of the $X̃2Σ+(0,0,0)$ state value]. Assuming that the core polarization contributions in the $Ã2Π1/2(0,0,0)$ state are the same as the $X̃2Σ+(0,0,0)$ state, then e²Qq₀ for the $Ã2Π1/2(0,0,0)$ state is predicted to be −0.0423 cm⁻¹, in qualitative agreement with observed values given the assumptions being made.

A novel spectroscopic technique utilizing laser-induced chemical reactions to distinguish features in congested and overlapped spectra was demonstrated. This technique was utilized to determine the fine and hyperfine structure of ^171,173YbOH for the first time. The derived molecular parameters are consistent with previous measurements of the even isotopologues ^172,174YbOH as well as with isoelectronic ^171,173YbF. The determined hyperfine parameters also provide an experimental comparison to check the quality of the calculated P- and T, P-violating coupling constants W_s, W_P, W_d, and W_M. Finally, the characterization of the $X̃2Σ+0,0,0−Ã2Π1/2(0,0,0)$ transition of ^171,173YbOH performed in this work will aid in the implementation of NSD-PV and NMQM measurements, laser cooling and photon cycling, and future spectroscopic investigations of these promising molecular isotopologues.

See the supplementary material for the description of the effective Hamiltonian and for the transition wavenumber for the A²Π_1/2−X²Σ⁺ (0,0,0) electronic transition of ¹⁷¹YbOH (Table S1), the transition wavenumber for the A²Π_1/2−X²Σ⁺ (0,0,0) electronic transition of ¹⁷³YbOH (Table S2), spectroscopic parameters in wavenumbers (cm⁻¹) used to model the $X̃2Σ+(0,0,0)$ and $Ã2Π1/2(0,0,0)$ states of YbOH (Table S3), the calculated low-rotational (N = 0–4) $X̃2Σ+(0,0,0)$ state energies of ¹⁷¹YbOH (Table S4), the calculated low-rotational (J = 0.5–5.5) $Ã2Π1/2(0,0,0)$ state energies of ¹⁷¹YbOH (Table S5), the calculated low-rotational (N = 0–4) $X̃2Σ+(0,0,0)$ state energies of ¹⁷³YbOH (Table S6), and the calculated low-rotational (J = 0.5–5.5) $Ã2Π1/2(0,0,0)$ state energies of ¹⁷³YbOH (Table S7).

The research at Arizona State University and Caltech was supported by grants from the Heising–Simons Foundation (ASU: Grant No. 2018-0681; Caltech: Grant No. 2019-1193). N.R.H. acknowledges support from the NIST Precision Measurement (Grant No. 60NANB18D253) and the NSF CAREER Award (Grant No. PHY-1847550). We thank Graceson Aufderheide and Richard Mawhorter (Physics and Astronomy Department, Pomona College, Pomona, CA 91711, USA) for assistance with Yb + Te(OH)₆ target fabrication, Phelan Yu for help fabricating the Yb + Yb(OH)₃ targets, and Dr. Anh Le (Chemistry Dept., Georgia Institute of Technology, Atlanta, GA 30318, USA) for assistance in recording the molecular beam spectra. We thank Ben Augenbraun and Phelan Yu for feedback on the article.

The data that support the findings of this study are available within the article and its supplementary material.