Determination of CaOH and CaOCH₃ vibrational branching ratios for direct laser cooling and trapping

Detailed quantum mechanical understanding of dynamics, interactions, and reactions for complex molecules requires precisely targeted preparation of internal molecular states and relative kinetic motions in the gas-phase solvent-free environment [1–3]. Supersonic molecular beams have allowed groundbreaking achievements in experimental studies of inelastic and reactive molecular interactions, as well as opened the path to coherent control of internal molecular states [4–9]. With some crucial exceptions, however, large velocity in the laboratory frame usually leads to short experimental coherence times and averaging over multiple internal and motional quantum states. The emergence of cryogenic buffer-gas beams (CBGBs) led to a significant reduction in the forward velocity but the highest molecular fluxes still maintained ≳10 K kinetic energy in the laboratory frame [10]. Therefore, other experimental techniques have emerged in particular those aimed at reducing the forward velocity of molecular beams to the degree sufficient for eventual three-dimensional confinement inside conservative electromagnetic traps, enabling seconds-long coherence times [11]. Molecular trapping also opens opportunities for accurate collisional studies at very low temperatures.

The use of electric, magnetic, optical and mechanical techniques for molecular beam slowing have all been experimentally demonstrated, with published comprehensive reviews providing excellent survey of the field [11, 12]. Recent advances in direct molecular cooling as well as coherent molecule formation from unbound atoms have led to the emergence and rapid development of a new vibrant field of cold and ultracold chemistry [3, 13]. The unprecedented opportunity to completely control the internal molecular degrees of freedom, as well as external motion, of diverse molecular species has yielded exciting observations on the role of quantum statistics in molecular reactive collisions as well as confining geometry of the reactants, among other results [3]. However, with few crucial exceptions [6, 9, 14, 15], the studies have been limited to the exploration of diatomic molecules which can be prepared from associated laser-cooled atomic samples.

Extension of the available experimental techniques to production of a chemically diverse set of polyatomic molecules will aid in benchmarking state-of-the-art ab initio calculations as well as shed light on novel reaction mechanisms at work. Given the recent rapid experimental progress on laser cooling and trapping of diatomic molecules [16–20], we consider the extension of these techniques to polyatomic species. The internal quantum complexity of polyatomic molecules grows significantly upon transitioning from polar diatomic radicals with a single vibrational mode to nonlinear polyatomic molecules with multiple rotational and vibrational degrees of freedom. Nevertheless, a large number of different monovalent alkaline earth derivatives have been proposed to be amenable to laser cooling [21, 22], with pioneering experimental work on the laser-cooling of the triatomic SrOH molecule [23].

In polyatomic molecules, the absence of strict angular momentum selection rules for transitions between totally symmetric vibrational modes presents a new challenge for photon cycling [24], a necessary ingredient for laser cooling. Additionally, higher-order perturbation mechanisms present only in polyatomic molecules such as Fermi resonance as well as Jahn–Teller and Renner–Teller interactions can lead to Born–Oppenheimer breakdown and coupling between vibrational modes, which can result in loss channels otherwise forbidden by the symmetries of the original unperturbed states. Spin-rotation and hyperfine splittings are in the MHz regime and can be efficiently addressed by adding radio-frequency sidebands to the cycling laser; rotational splittings are in the GHz range and can be coupled to the 'bright' rotational state with microwaves; however, vibrational transitions are in the THz range and would require additional laser systems, significantly increasing experimental complexity. Thus, precise measurements of Franck–Condon factors (FCFs) and vibrational branching ratios (VBRs) are crucial in determining the experimental feasibility of laser cooling.

While there is plenty of theoretical work on the estimation of FCFs and VBRs for both diatomic [25–27] and polyatomic molecules [21, 22, 28], accurate experimental studies have been limited primarily to diatomic species [29–32] and, recently, SrOH [33]. Accurate theoretical predictions of FCFs for polyatomic molecules remain a challenge because their multidimensional nature not only introduces interactions between the degrees of freedom on each potential surface but also between the coordinates of the two electronic states. The pair of molecules we have chosen to study here represent the simplest (by geometric structure) members of the alkaline earth monoalkoxide radicals (MOR) family [34, 35], a large class of polyatomic molecules that has previously been proposed for laser cooling applications [21]. A simple 'triatomic' model of the MOR molecules has been used in order to estimate branching ratios and indicated that laser cooling of relatively large MOR molecules with up to 15 constituent atoms could be possible. Our precise measurements of VBRs for CaOH and CaOCH₃ described below provide the first experimental validation of the theoretical model of 'shielding' of the R ligand group (e.g. CH₃) vibrational modes by the intermediate oxygen atom [21]. Increased density of vibrational states for larger polyatomic molecules could potentially lead to internal vibrational redistribution among multiple vibrational normal modes leading to enhanced loss probabilities and, therefore, necessitating accurate experimental studies.

CaOH is one of the first polyatomic molecules to be proposed suitable for laser cooling and trapping and detailed theoretical ab initio calculations have been performed of the FCFs [22] as well as energy levels [36]. Our work provides an important comparison between the theoretical and experimental results, benchmarking theoretical calculations by Isaev and Berger [22]. Our precise measurements of the branching ratios for a more complex symmetric-top molecule (STM) CaOCH₃ provide, to the best of our knowledge, the first accurate experimental study of the VBRs for a nonlinear radical suitable for direct laser cooling. Particularly, our results indicate the necessity to consider pseudo Jahn–Teller couplings in the excited electronic levels of polyatomic molecules [37]. Perhaps surprisingly, despite the presence of Jahn–Teller interaction in the electronically degenerate E symmetry state of the C_3v symmetric top, using the lowest allowed ${\tilde{X}}^{2}{A}_{1}\leftrightarrow {\tilde{A}}^{2}{E}_{1/2}$ electronic excitation will lead to ∼16 scattered photons without any additional repumping lasers. Based on our experimental measurements and theoretical analysis, we propose a feasible route for producing large ultracold samples (∼10³–10⁶ molecules) of both radicals via laser cooling on the lower spin–orbit (SO) branch of the $\tilde{X}\mbox{--}\tilde{A}$ electronic transition. Performing a detailed study of inelastic and reactive collisions on increasingly complex members of the CaOR molecule family would provide a unique window into the scattering properties of fundamental ligand groups in organic chemistry.

Since the details of the experimental apparatus have previously been described in other publications [33], we provide only a brief account of the experimental configuration employed. As a source of cold molecules we used a pulsed supersonic molecular beam with argon as a carrier gas. Both CaOH and CaOCH₃ beams were created by flowing argon over a container filled with liquid methanol (CH₃OH) maintained at room temperature. Room temperature vapor pressure of methanol (∼10 kPa) together with the backing pressure of argon (∼4000 kPa) was enough to seed a sufficient number of gas-phase molecules into the carrier gas to observe large, stable yields of both molecular radicals of interest following laser ablation of a calcium metal target. The peak fluorescence signal per single rovibrational level was approximately of similar magnitude for CaOH and CaOCH₃, which allowed for accurate studies of both radicals using the same beam source. Molecules in a specific rotational state were excited using a single frequency cw-dye laser. Dispersed fluorescence data was collected with a spectrometer that included 0.67 m focal length, high-efficiency Czerny–Turner-type monochromator with a low-dispersion grating and a cooled, gated intensified charge-coupled device camera. The relative sensitivity as a function of wavelength for the spectrometer was precisely calibrated beforehand. The wavelength resolution of the emission spectrum was controlled by adjusting the width of the monochromator input aperture and absolute calibration was performed with the argon emission lamp. The measured emission wavelengths agreed to ≲0.1 nm with theoretical predictions which allowed for unambiguous identification of the vibrational loss channels. The narrow linewidth (≲1 MHz) of the excitation dye laser allowed accurate rotationally-resolved excitation. A Burleigh high-resolution wavemeter was used as a rough frequency reference (<1 GHz), additionally confirmed by co-recording with some Doppler I₂ spectrum [38]. For each dispersed laser induced fluorescence (DLIF) measurement, three separate datasets were taken: (i) signal dataset with ablation Nd:YAG laser on and dye laser on, (ii) scattered background dataset with no Nd:YAG laser but dye laser on, and (iii) spurious ablation glow dataset with Nd:YAG laser present but no dye laser. The resulting measurement was obtained by subtracting datasets (ii) and (iii) from (i) in order to remove the scattered light offset and reduce the background ablation glow (e.g. metastable calcium emission).

In order to eliminate rotational branching during the photon cycling process the use of $J^{\prime\prime} \to J^{\prime} -1$ type angular momentum transitions has been proposed [39, 40] and for the first time demonstrated with diatomic SrF in [41] and linear triatomic SrOH in [42]. Rotationally-resolved high-resolution spectroscopy of low-J rotational levels of CaOH and CaOCH₃ has been previously performed and assigned for both $\tilde{X}\mbox{--}\tilde{A}$ and $\tilde{X}\mbox{--}\tilde{B}$ electronic bands [43–49]. The availability of prior spectroscopic measurements, low rotational temperatures of the skimmed supersonic molecular beam as well as the narrow linewidth of the cw dye laser, allowed us to deliberately address only the rotational transitions which can be used for optical cycling and laser cooling. Additionally, the absence of state-changing collisions in the probing region eliminated any possible systematic errors.

While both CaOH and CaOCH₃ have spin-rotation and hyperfine splittings arising from the unpaired electron and hydrogen spins, correspondingly, for the purposes of the experiments performed here we label the states using rotational quantum numbers $\left|N,K\right\rangle$ for non-degenerate quantum states (i.e. Σ⁺ states for linear molecules and A₁ states for STMs) and $\left|J,K\right\rangle$ states for degenerate states (i.e. Π electronic states for linear molecules and E states for STMs) with K = 0 for linear molecules. Upon the electronic excitation to a specific rotational level, the spontaneous emission rate for a dipole-allowed transition is governed by the Einstein A coefficient [50]:

$$\begin{eqnarray}&&{A}_{J^{\prime} \to J^{\prime\prime} }=\displaystyle \frac{16{\pi }^{3}{\nu }^{3}{S}_{J^{\prime} J^{\prime\prime} }}{3{\epsilon }_{0}{{hc}}^{3}\left(2J^{\prime} +1\right)},\end{eqnarray} \tag{ 1 }$$

where the molecular line strength ${S}_{J^{\prime} J^{\prime\prime} }\equiv {\sum }_{M^{\prime} ,M^{\prime\prime} }{\left|\left\langle J^{\prime} M^{\prime} \left|{\boldsymbol{\mu }}\right|J^{\prime\prime} M^{\prime\prime} \right\rangle \right|}^{2}$ is approximately given as

$$\begin{eqnarray}&&{S}_{J^{\prime} J^{\prime\prime} }={q}_{{\bf{v}}^{\prime} -{\bf{v}}^{\prime\prime} }{\left|{{\bf{R}}}_{e}\right|}^{2}{S}_{J^{\prime\prime} }^{\bigtriangleup J}\end{eqnarray} \tag{ 2 }$$

under the Born–Oppenheimer separation ${\psi }_{\mathrm{tot}}={\psi }_{\mathrm{el}}{\psi }_{\mathrm{vib}}{\psi }_{\mathrm{rot}}$. Therefore, the intensity of different vibrational bands will be proportional to

$$\begin{eqnarray}&&{\rm{VBR}}=\displaystyle \frac{{\nu }^{3}{q}_{{\bf{v}}^{\prime} -{\bf{v}}^{\prime\prime} }}{{\displaystyle \sum }_{i}{\nu }_{i}^{3}{q}_{{\bf{v}}^{\prime} -{\bf{v}}^{\prime\prime} }}\end{eqnarray} \tag{ 3 }$$

since the same electronic transition dipole moment $\left|{{\bf{R}}}_{e}\right|$ and Hönl–London rotational factor are shared by all the vibronic emission bands. Additionally, the rotational branching is limited by the $\bigtriangleup J=0,\pm 1$ selection rule and all J branches contribute identically for all the vibrational bands. For polyatomic molecules, the FCF is multidimensional and given as

$$\begin{eqnarray}&&{q}_{{\bf{v}}^{\prime} -{\bf{v}}^{\prime\prime} }={q}_{{v}_{1}^{\prime} -{v}_{1}^{\prime\prime} }{q}_{{v}_{2}^{\prime} -{v}_{2}^{\prime\prime} }\ldots ={\left|\int {\psi }_{{v}_{1}^{\prime} }^{* }{\psi }_{{v}_{1}^{\prime\prime} }{{\rm{d}}{Q}}_{1}\right|}^{2}{\left|\int {\psi }_{{v}_{2}^{\prime} }^{* }{\psi }_{{v}_{2}^{\prime\prime} }{{\rm{d}}{Q}}_{2}\right|}^{2}\ldots \end{eqnarray} \tag{ 4 }$$

By comparing the integrated areas under the dispersed fluorescence emission peaks, we can determine the relative branching ratios for decays from a given electronic, vibrational and rotational state.

3.1. CaOH measurements

Two electronic transitions have been studied for CaOH that have potential to be used in the laser cooling and trapping applications. Previously, laser cooling of the isoelectronic CaF and SrOH molecules has been demonstrated using either the ${\tilde{X}}^{2}{{\rm{\Sigma }}}^{+}\mbox{--}{\tilde{A}}^{2}{{\rm{\Pi }}}_{1/2}$ or ${\tilde{X}}^{2}{{\rm{\Sigma }}}^{+}\mbox{--}{\tilde{B}}^{2}{{\rm{\Sigma }}}^{+}$ electronic transition and, therefore, we have performed detailed measurements on both bands for CaOH as well. Additionally, in order to increase the photon cycling rate it is favorable to avoid coupling multiple excitation lasers to the same vibronic level, thus requiring separation of the main cycling and repumping lasers to address different electronic levels (e.g. main cycling on $\tilde{X}\mbox{--}\tilde{A}$ while repumping on $\tilde{X}\mbox{--}\tilde{B}$ or vise versa). As was previously demonstrated both theoretically [51] and experimentally [52], when multiple ground state sub-levels n_vg from different vibrational levels v are coupled to the same excited state sub-levels n_0e, the effective scattering rate for molecules is reduced by a factor of $2{n}_{0e}/\left({n}_{0e}+{\sum }_{v}{n}_{{vg}}\right)$, resulting in a lower radiation pressure force (F_rad). Therefore, the scattering rate and F_rad is maximized for decoupling all repumpers from the main cycling laser, resulting in a maximal possible scattering rate of ${R}_{\mathrm{scat},\max }={{\rm{\Gamma }}}_{\mathrm{sp}}{n}_{0e}/\left({n}_{0e}+{n}_{0g}\right)$, where Γ_sp = 1/τ_sp and τ_sp is the spontaneous lifetime of the excited electronic state.

Figures 1 and 2 display the results of the DLIF for CaOH excited using the ${P}_{1}\left(N^{\prime\prime} =1\right)$ transition on the $\tilde{X}\mbox{--}{\tilde{A}}^{2}{{\rm{\Pi }}}_{1/2}$ (at 15 964.38 cm⁻¹) and $\tilde{X}\mbox{--}\tilde{B}$ (at 18 021.58 cm⁻¹) bands, correspondingly. Similarly to the previous measurements with isoelectronic diatomic CaF [53] and triatomic SrOH [33], the intensity of the off-diagonal vibrational bands decreases rapidly, indicating the suitability of using either electronic transition for optical cycling. High sensitivity of our measurement allowed us to observe very weak decays to $\left({02}^{0}0\right)$ and $\left({01}^{1}0\right)$ excited bending vibrational states as shown in figures 1(b) and 2(b). Symmetry forbidden $\bigtriangleup l\ne 0$ vibronic decays in alkaline earth monohydroxides occur in second order due to H_RT × H_SO Hamiltonian term. The dipolar term of the Renner–Teller (RT) perturbation operator connects basis functions with v₂ = ±1 and $\bigtriangleup l=-\bigtriangleup {\rm{\Lambda }}=\pm 1$ [54, 55] and together with the off-diagonal parts of the SO operator leads to coupling between $\tilde{B}\left(000\right)\sim \tilde{B}\left({01}^{1}0\right)$ states [56]. Decay to the $\left({02}^{0}0\right)$ vibrational state is symmetry allowed and observed in our measurements (figure 1(b)), but the more precise determination of this band's intensity is limited by the contamination signal arising due to the 657 nm emission from the metastable atomic calcium created during the laser ablation process. Additionally, it is important to note that for both $\tilde{A}\to \tilde{X}$ and $\tilde{B}\to \tilde{X}$ emission bands, decay to vibrational levels with v ≥ 2 is significantly suppressed to below 0.3%. This rapid decrease in the intensity of off-diagonal vibrational decays should enable experimentally accessible laser cooling and trapping.

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3.2. CaOCH₃ measurements

The presence of $K\ne 0$ states for nonlinear molecules requires additional considerations for achieving effective photon cycling beyond those examined previously for diatomic and linear polyatomic molecules. Reference [21] outlined how to use $K^{\prime\prime} =1\to K^{\prime} =0$ transition for the perpendicular $\tilde{X}\mbox{--}\tilde{A}$ transition and $K^{\prime\prime} =0\to K^{\prime} =0$ transition for the parallel $\tilde{X}\mbox{--}\tilde{B}$ to achieve rotationally closed excitations. Figures 1 and 3 provide a comparison of the DLIF for CaOH and CaOCH₃. Both of the molecules were excited on the $\tilde{X}\mbox{--}\tilde{A}$ electronic transitions around 630nm and in a rotationally resolved manner to $J^{\prime} =1/2$ state. As can be clearly seen from the data presented, both CaOH and CaOCH₃ have a very small number of vibrational decay channels despite significantly increased structural complexity in going from a triatomic to a hexatomic molecule. While CaOCH₃ has eight distinct vibrational modes (4 of a₁ symmetry and 4 degenerate pairs of e symmetry), only two are optically active at the three part per thousand level as seen in the figure. Our measurements, performed on a cold molecular beam with a narrow-band dye laser exciting only the rotational level suitable for laser cooling, confirm that even nonlinear CaOR molecules behave effectively like triatomics, showing great promise for laser cooling and trapping.

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In order to account for possible systematic effects associated with multiple-photon cycling in our measurements, we have also used excitation to the upper SO component of the $\tilde{A}$ state (${}^{2}{{\rm{\Pi }}}_{3/2}$ for CaOH and ${}^{2}{E}_{3/2}$ for CaOCH₃) which does not support rotationally-closed excitations. For all measurements performed, we did not add any additional laser sidebands, thus addressing only a single spin-rotation component. In the presence of optical saturation of the transitions where more than one photon is scattered per single molecule, we anticipate to see $\approx \tfrac{3}{5}$ less signal in the ${0}_{0}^{0}$ emission band when using a non-rotationally closed transition (e.g. R or Q branch). However, for both molecules the measured branching ratios for excitation to either the lower or the upper SO component of the $\tilde{A}$ state are essentially consistent with each other (see tables 5 and 6), providing strong evidence that the measured branch intensities are dominated by the actual FCF values and setting an upper limit on the associated systematic error of 1%. In order to confirm our measurements for the $\tilde{X}\to \tilde{B}$ transitions we used R-type branch to excite molecules in rotational levels N'' > 1, thus preventing optical cycling. Additional data for such measurements is provided in the appendix.

We have also used dispersed LIF to explore the possibility of using the $\tilde{X}\mbox{--}\tilde{B}$ electronic transition in CaOCH₃ around 566 nm for laser cooling. Figure 4 shows the DLIF results indicating two additional features not present in the $\tilde{A}\to \tilde{X}$ emission spectrum: (i) emission to 1 quanta of the Ca–O–C bending mode and (ii) a new feature around 1750 cm⁻¹ away from the excitation band. Since $\tilde{B}$ is the second excited electronic level, there is a possibility of mixing with the excited vibrational levels of the $\tilde{A}$ state. The proposed intensity borrowing mechanism is shown in figure 5, indicating the resulting decays nominally forbidden by the symmetry arguments. As clearly seen in the data presented in figure 4, a relatively strong emission to the band offset by 1751 cm⁻¹ from the diagonal vibronic transition is observed in the $\tilde{B}\to \tilde{X}$ decay of CaOCH₃, making laser cooling on this electronic transition more challenging compared to $\tilde{X}\to {\tilde{A}}^{2}{E}_{1/2}$.

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Any nonlinear molecule (e.g. CaOCH₃) in an orbitally degenerate electronic state (e.g. ${}^{2}E$ state) will always distort in a way such as to lower the symmetry and remove degeneracy. This so-called Jahn–Teller effect (JTE) nullifies the $\bigtriangleup {v}_{i}=\pm 2,\pm 4,\ldots$ selection rule for non-symmetric vibrations in electronic transitions [50]. JTE in the $\tilde{A}$ electronic state as well as near-degeneracy between the excited combination band in the $\tilde{A}$ state and the ground vibrational level of the $\tilde{B}$ electronic state lead to Born–Oppenheimer approximation breakdown. Since the eigenstates of the new Hamiltonian are a mixture of the unperturbed eigenstates of the $\tilde{B}\left(000\right)$ and $\tilde{A}\left(111\right)$ levels, forbidden decays are observed in the emission spectrum. The appearance of the pseudo-JTE in the presence of accidentally near-degenerate states (as depicted in figure 5) is carefully described in [37]. The perturbing operator of the polynomial form in the normal coordinate Q_i about the molecular configuration Q₀ [37]

$$\begin{eqnarray}&&H^{\prime} ={\left(\displaystyle \frac{\partial H}{\partial {Q}_{i}}\right)}_{{Q}_{0}}{Q}_{i}+\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{\partial }^{2}H}{\partial {Q}_{i}^{2}}\right)}_{{Q}_{0}}{Q}_{i}^{2}+\ldots \end{eqnarray} \tag{ 5 }$$

is responsible for mixing of the zeroth-order Born–Oppenheimer vibrational and electronic wavefunctions. Careful theoretical understanding of vibronic coupling in the excited $\tilde{B}$ state is currently in progress and beyond the scope of the present paper.

CaOH three-dimensional potential energy surfaces have been previously mapped out using ab initio calculations for both the ground [36] as well as excited electronic states [57]. While table 1 provides a comparison between such ab initio results from different studies, we further focus our comparison on the calculations performed by Isaev and Berger [22] since, to our knowledge, they provide the only previous study of CaOH FCFs relevant to laser cooling prospects. Even though there are no previous ab initio calculations of FCFs for CaOCH₃, the use of isotopically substituted methoxy groups has allowed accurate experimental determination of the radical's geometry in both $\tilde{X}$ and $\tilde{A}$ states [45].

Table 1.  Calculated and measured normal vibrational modes (in units of cm⁻¹) for CaOH. TI and TD stand for time-independent and time-dependent density functional theory calculations, respectively. For doubly-degenerate bending vibrational modes (Π symmetry), the two specified frequencies are for individual $A^{\prime} /A^{\prime\prime}$ states.

State	Mode (symmetry)	TI	TD	Measured [58]	[22]	Combined [References]
$\tilde{X}$	Bend (Π)	356.72	344.89/345.29	352.93	411/408	347.6 [36]
$\tilde{X}$	Ca–O stretch (Σ)	625.22	603.84	609.02	622	612.1 [36]
$\tilde{X}$	O–H stretch (Σ)	3816.28	3819.96	3778	4276	3873.9 [36]
$\tilde{A}$	Bend (Π)	368.41	370.37/371.13	361.36	402/386	349.7/389.8 [57]
$\tilde{A}$	Ca–O stretch (Σ)	625.18	603.84	630.68	646	622.2 [57]
$\tilde{A}$	O–H stretch (Σ)	3816.06	3819.96	—	4279	3989.5 [57]

Because of the highly diagonal FCF matrices for both CaOH and CaOCH₃ molecules, only a few excited vibrational modes were present in our DLIF data as seen in figures 1–4. Other than the unexpected peak seen in the 1751 nm overtone band for the $\tilde{B}\to \tilde{X}$ emission, the frequencies of other vibrational modes have been previously measured and tabulated which allowed for unambiguous vibrational character assignment. Theoretical analysis of the multidimensional FCFs for all vibrational modes—including those not observed in the experiment—depends on the frequencies of the corresponding vibrational motions. We therefore initialized our calculations with ab initio molecular geometries and vibrational frequencies using the ORCA quantum chemistry program, which is discussed in [59]. Tables 1–3 provide a summary of the calculated normal vibrational modes for CaOH and CaOCH₃. We observe excellent agreement between our ORCA calculations with previous ab initio calculations for CaOH by Isaev and Berger [22] as well as previously experimentally determined values for the vibrational normal modes. In the case of MOCH₃ molecules, there are 12 normal modes, 4 of a₁ symmetry and 4 degenerate pairs of e₁ symmetry. Thus, while there are 12 internal coordinates, we reduced the problem to 8 symmetry coordinates listed in table 4.

Table 2.  Calculated and measured normal vibrational modes (in units of cm⁻¹) for the $\tilde{X}$ electronic state of CaOCH₃.

State	Mode (symmetry)	TI	TD	Measured [60]
$\tilde{X}$	Ca–O–C bend (e)	138.59/140.51	133.11/142.09	144 ± 5
$\tilde{X}$	Ca–O stretch (a1)	482.90	472.09	488 ± 5
$\tilde{X}$	Sym. CH3 bend and C–O stretch (a1)	1132.20	945.19	1156 ± 5
$\tilde{X}$	Asymmetric combination (e)	1134.30/1134.61	1109.93/1120.49	—
$\tilde{X}$	Sym. CH3 bend and C–O stretch (a1)	1425.76	1385.36	—
$\tilde{X}$	Asymmetric combination (e)	1444.06/1444.42	1425.37/1434.73	—
$\tilde{X}$	Sym. CH3 stretch (a1)	2879.18	2886.36	—
$\tilde{X}$	Asymmetric stretch (e)	2922.11/2922.61	2933.60/2933.75	—

Table 3.  Calculated and measured normal vibrational modes (in units of cm⁻¹) for the $\tilde{A}$ electronic state of CaOCH₃.

State	Mode (symmetry)	TI	TD	Measured [60]
$\tilde{A}$	Ca–O–C bend (e)	141.43/141.54	129.76/141.42	145 ± 5
$\tilde{A}$	Ca–O stretch (a1)	482.66	472.13	501.48
$\tilde{A}$	Sym. CH3 bend and C–O stretch (a1)	1132.24	1119.84	1140 ± 5
$\tilde{A}$	Asymmetric combination (e)	1134.54/1134.62	1206.80/1269.47	—
$\tilde{A}$	Sym. CH3 bend and C–O stretch (a1)	1425.48	1 425.10	—
$\tilde{A}$	Asymmetric combination (e)	1 444.08/1444.13	1672.82/1741.32	—
$\tilde{A}$	Sym. CH3 stretch (a1)	2879.54	2886.29	—
$\tilde{A}$	Asymmetric stretch (e)	2923.32/2923.36	2931.02/2934.70	—

The details of the numerical computation of CaOH and CaOCH₃ FCFs are described in the appendix. The VBRs were computed by weighting each FCF by the cube of the transition frequency from the vibrationless excited electronic state to the vibrational state in the ground electronic state and normalizing. Results of these calculations are shown in tables 5 and 6. We obtain excellent agreement between the measured and calculated VBRs for CaOH and a reasonable agreement for a more complex CaOCH₃ molecule. Incorporating possible couplings between different vibrational modes as well as inclusion of the anharmonic terms in the vibrational potential could resolve the slight discrepancy between the measured and predicted VBRs for CaOCH₃. Notice that our theoretical predictions for FCFs are much closer to the measured values we have observed compared to the ab initio values predicted by Isaev and Berger [22]. By using available experimental vibrational frequencies to determine the force constants as well as incorporating experimentally measured bond lengths and structure into our analysis, we were able to circumvent some of the challenges associated with predictions of FCFs with sufficient accuracy for designing a molecular laser cooling scheme [61]. As seen in table 5, purely ab initio calculations for even triatomic FCFs can potentially underestimate the feasibility of direct laser cooling, while calculations relying on molecular parameters extracted from previous spectroscopic data can provide a more accurate guide to developing a successful laser cooling experiment for a new molecule with desired properties (e.g. new physics sensitivity [62]).

Table 5.  Summary of the measured and calculated branching ratios for CaOH molecule.

Band	${\lambda }_{v^{\prime} ,v^{\prime\prime} }$, nm	Calc. FCFs	[22] FCFs	Calc. VBRs	Obs. VBRs
${0}_{0}^{0}{\tilde{A}}^{2}{{\rm{\Pi }}}_{1/2}\mbox{--}{\tilde{X}}^{2}{{\rm{\Sigma }}}^{+}$	626.5	0.9521	0.9213	0.9570	0.957 ± 0.002
${1}_{1}^{0}{\tilde{A}}^{2}{{\rm{\Pi }}}_{1/2}\mbox{--}{\tilde{X}}^{2}{{\rm{\Sigma }}}^{+}$	651.0	0.0459	0.0763	0.0410	0.043 ± 0.002
${2}_{2}^{0}{\tilde{A}}^{2}{{\rm{\Pi }}}_{1/2}\mbox{--}{\tilde{X}}^{2}{{\rm{\Sigma }}}^{+}$	654.3	3 × 10−4	—	3 × 10−4	${3}_{-2}^{+1}\times {10}^{-3}$
${0}_{0}^{0}{\tilde{A}}^{2}{{\rm{\Pi }}}_{3/2}\mbox{--}{\tilde{X}}^{2}{{\rm{\Sigma }}}^{+}$	623.9	0.9521	0.9213	0.959	0.959 ± 0.005
${1}_{1}^{0}{\tilde{A}}^{2}{{\rm{\Pi }}}_{3/2}\mbox{--}{\tilde{X}}^{2}{{\rm{\Sigma }}}^{+}$	648.3	0.0459	0.0763	0.041	0.041 ± 0.005
${0}_{0}^{0}{\tilde{B}}^{2}{{\rm{\Sigma }}}^{+}\mbox{--}{\tilde{X}}^{2}{{\rm{\Sigma }}}^{+}$	555.2	0.9711	—	0.9742	0.975 ± 0.001
${1}_{1}^{0}{\tilde{B}}^{2}{{\rm{\Sigma }}}^{+}\mbox{--}{\tilde{X}}^{2}{{\rm{\Sigma }}}^{+}$	574.4	0.0270	—	0.0244	0.022 ± 0.001
${2}_{1}^{0}{\tilde{B}}^{2}{{\rm{\Sigma }}}^{+}\mbox{--}{\tilde{X}}^{2}{{\rm{\Sigma }}}^{+}$	566.2	0	—	0	0.003 ± 0.001

Table 6.  Summary of the measured and calculated branching ratios for CaOCH₃ molecule.

Band	${\lambda }_{v^{\prime} ,v^{\prime\prime} }$, nm	Calc. FCFs	Calc. VBRs	Obs. VBRs
${0}_{0}^{0}{\tilde{A}}^{2}{E}_{1/2}\mbox{--}{\tilde{X}}^{2}{A}_{1}$	629.9	0.944	0.950	0.931 ± 0.003
${1}_{1}^{0}{\tilde{A}}^{2}{E}_{1/2}\mbox{--}{\tilde{X}}^{2}{A}_{1}$	649.5	0.050	0.045	0.063 ± 0.003
${3}_{1}^{0}{\tilde{A}}^{2}{E}_{1/2}\mbox{--}{\tilde{X}}^{2}{A}_{1}$	678.8	0.003	0.002	0.006 ± 0.003
${0}_{0}^{0}{\tilde{A}}^{2}{E}_{3/2}\mbox{--}{\tilde{X}}^{2}{A}_{1}$	627.1	0.944	0.954	0.945 ± 0.004
${1}_{1}^{0}{\tilde{A}}^{2}{E}_{3/2}\mbox{--}{\tilde{X}}^{2}{A}_{1}$	646.6	0.050	0.046	0.055 ± 0.004
${0}_{0}^{0}{\tilde{B}}^{2}{A}_{1}\mbox{--}{\tilde{X}}^{2}{A}_{1}$	565.9	0.941	0.947	0.910 ± 0.002
${1}_{1}^{0}{\tilde{B}}^{2}{A}_{1}\mbox{--}{\tilde{X}}^{2}{A}_{1}$	581.6	0.049	0.045	0.057 ± 0.002
${2}_{1}^{0}{\tilde{B}}^{2}{A}_{1}\mbox{--}{\tilde{X}}^{2}{A}_{1}$	570.3	0	0	0.006 ± 0.002
${1}_{1}^{0}{2}_{1}^{0}{3}_{1}^{0}{\tilde{B}}^{2}{A}_{1}\mbox{--}{\tilde{X}}^{2}{A}_{1}$	628.2	0	0	0.026 ± 0.002

Our experimental measurements and theoretical calculations confirm the possibility of achieving direct laser cooling for CaOH and CaOCH₃ with a few repumping lasers. Furthermore, our measurements provide the quantitative guidance to choosing the most experimentally efficient route to identifying the main cooling and all the repumping transition bands. Using the measured values for the branching ratios, we have determined the optimal laser cooling schemes for CaOH and CaOCH₃ depicted in figures 6 and 7, respectively. Because of the relatively light mass, CaOH can be slowed down to a complete stop from 100 m s⁻¹ CBGB using only 9000 photons which can be obtained with three repumping lasers indicated. Thus, the technical complexity of laser slowing CaOH is comparable to experiments with diatomic molecules SrF and CaF.

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While additional structural complexity for CaOCH₃ results in less diagonal FCFs, with the repumping scheme in figure 7 and assuming ∼10⁹ molecules in a single rovibrational quantum state at 100 m s⁻¹, one can obtain between 10⁴ and 10⁷ slowed molecules in the trapping region, assuming 1 part per 1000 or 1 part per 3000 loss probability to unaddressed vibrational levels.

Because of the strongly ionic nature of the Ca–O bond in CaOR molecules, the electronic transitions involve non-bonding Ca⁺ orbitals perturbed by an RO⁻ ligand even for asymmetric rotor molecules (ARMs). Since the Ca–O–C bond angle remains 180^◦, the local symmetry near the calcium metal is linear and therefore many characteristics of electronic transitions carry over from linear CaOH to more complex CaOR symmetric top and asymmetric rotor radicals [35, 63]. For example, there is 'SO' splitting of about the same size, and the electronic orbital angular momentum remains essentially unquenched. However, the symmetry reduction in going from ${C}_{\infty v}$ (CaOH) and C_3v (CaOCH₃) to molecules with lower symmetry properties like C_S and no symmetry has potential to impact the VBRs in a significant manner. For linear (${C}_{\infty v}$) and symmetric top (C_3v) molecules, Ca–O–C bending vibrations have π or e symmetry, correspondingly, and therefore only $\bigtriangleup v=\pm 2,\pm 4,\ldots$ branches appear in the electronic decays under the Born–Oppenheimer approximation⁶ . Such a bending mode becomes non-degenerate for molecules with lower symmetries than C_3v, therefore leading to allowed decays of comparable intensity to both one quanta in the Ca–O stretching and Ca–O–C bending modes as has been previously observed for calcium monoalkoxides [35]. While this presents a technical challenge, requiring an additional laser for repumping lost molecules from such a decay channel, it does not pose a fundamental threat to achieving photon cycling in ARMs like CaOC₂H₅ or CaOCH₂CH₃.

Depending on the orientation of the transition dipole moment relative to the principal axes of the molecule, vibrational and electronic spectra of ARMs are characterized as a-type, b-type or c-type transitions with the selection rules [50]:

• a-type bands: $\bigtriangleup {K}_{a}=0$, $\bigtriangleup {K}_{c}=\pm 1$ and $\bigtriangleup J=0,\pm 1$, except for ${K}_{a}^{{\prime} }=0\leftarrow {K}_{a}^{{\prime\prime} }=0$ for which $\bigtriangleup J=\pm 1$
• b-type bands: $\bigtriangleup {K}_{a}=\pm 1$, $\bigtriangleup {K}_{c}=\pm 1$ and $\bigtriangleup J=0,\pm 1$
• c-type bands: $\bigtriangleup {K}_{c}=0$, $\bigtriangleup {K}_{a}=\pm 1$ and $\bigtriangleup J=0,\pm 1$, except for ${K}_{c}^{{\prime} }=0\leftarrow {K}_{c}^{{\prime\prime} }=0$ for which $\bigtriangleup J=\pm 1$.

Therefore, as can be seen from the selection rules above, for near-oblate or near-prolate ARMs, the overall structure of the emission bands resembles that of parallel or perpendicular transitions for STMs. Choosing either a-type or c-type transitions and driving either ${K}_{c}^{{\prime\prime} }=1\to {K}_{c}^{{\prime} }=0$ or ${K}_{a}^{{\prime\prime} }=1\to {K}_{a}^{{\prime} }=0$, correspondingly will resemble optical cycling transitions for STMs as described above. Additionally, using quantum states such as ${K}_{a}^{{\prime\prime} }=0$ or ${K}_{c}^{{\prime\prime} }=0$ for a-type and c-type transitions, correspondingly, will restrict J selection rules to $\bigtriangleup J=\pm 1$ only (similarly to $\bigtriangleup N=\pm 1$ transitions used for laser cooling diatomic and linear molecules). Thus, while reduced symmetry for ARMs leads to novel challenges in repumping vibrational levels, carefully chosen rotational transitions can significantly limit (or even eliminate) rotational branching. As a first step in achieving photon cycling with ARMs, it is necessary to perform high resolution spectroscopy of the relevant electronic transitions and identify suitable bands for achieving optical cycling. While we have already completed such work for CaOCH₃ as described above, similar spectroscopy on more complex CaOR radicals (e.g. CaOCH₂CH₃) can be performed in the future. A natural chiral candidate for laser cooling is a chiral analog of calcium monomethoxide, CaOCHDT. A recent theoretical and experimental work by Liu and coworkers [66] on calcium ethoxide (CaOC₂H₅) confirms that Ca–O stretching mode is the dominant vibrational loss channel for this molecule and indicates its potential suitability for optical cycling and laser cooling using a scheme outlined here for ARMs.

While the high density of vibronic states for complex polyatomic molecules can potentially inhibit photon cycling using the $\tilde{X}\mbox{--}\tilde{B}$ electronic transitions, our measurements provide a strong case that using the lowest excited electronic level ($\tilde{A}$) will enable photon cycling and laser cooling despite the presence of SO and Jahn–Teller interactions. In addition to precise measurements of the VBRs for both $\tilde{X}\mbox{--}\tilde{A}$ and $\tilde{X}\mbox{--}\tilde{B}$ electronic excitations, we have also performed theoretical calculations that agree quite well for the $\tilde{X}\mbox{--}\tilde{A}$ band, where harmonic oscillator approximation remains valid for calculating the multidimensional FCFs. Using the analytical integral expressions presented by Sharp and Rosenstock [67], we have calculated the relevant FCFs using previously available experimental data for geometries, vibrational frequencies, and corresponding force fields. The presence of strong optically accessible electronic transitions, as well as ability to scatter multiple photons from a single molecule, opens the possibility for internal state molecular manipulation and control of the molecular motion in the laboratory frame. While radiative force slowing and cooling requires multiple thousands of repeated absorption-emission cycles [68], Sisyphus cooling processes as well stimulated emission slowing methods can provide significant gains even with a limited number of optical cycles [69, 70]. Stimulated optical forces arising from polychromatic light beams [71, 72] might be especially effective for manipulating MOR molecules with reduced symmetries and increased complexity where scattering tens of thousands of photons would require a large number of vibrational repumping lasers. Additionally, rapid optical cycling of even tens of photons can significantly improve the rate of opto-electrical cooling methods for STMs and ARMs, which have so far relied on using vibrational transitions with long lifetimes and therefore low scattering rates [73, 74].

Laser cooled polyatomic molecules provide an ideal starting point for producing ultracold fundamental radicals of chemical interest like OH, CH₃ and OCH₃. While a dedicated theoretical exploration is necessary for identifying the exact route for efficient photodissociation of polyatomic molecules into underlying constituents with minimal energy released, this approach seems feasible with the use of the STIRAP, which is the reverse of the process employed for forming ultracold diatomic molecules from ultracold alkali atoms [75]. Release of as little as 100 nK of additional kinetic energy into the motion of dissociation products has been achieved with ultracold Sr₂ molecules [76], and an efficient pathway for SrOH zero-kinetic-energy dissociation is currently being developed using ab initio molecular potentials [77]. Moreover, modern theoretical approaches to molecular quantum scattering calculations are reaching the regime of larger polyatomic molecules [78] and, therefore, experimental results on collisions would play a crucial role in further developing the field.

Furthermore, our measurements indicate the possibility of multiple photon cycling for other polyatomic species including polyatomic molecular ions. Internal state cooling and control of polyatomic molecular ions can significantly benefit from laser-induced optical cycling. Polyatomic molecular ions ScOH⁺, YOH⁺, as well as ScOCH₃⁺ and YOCH₃⁺, are isoelectronic to the corresponding calcium and strontium compounds [79, 80] and should be highly suitable for direct optical manipulation and internal state cooling using methods previously demonstrated with diatomic ions [81]. A few particularly interesting candidates to consider are AcOH⁺ and AcOCH₃⁺, which should be similar in electronic structure to RaOH and RaOCH₃ and are sensitive to new physics beyond the standard model [22], but with a significantly longer half-life of the actinium-227 nucleus compared to radium-227 [82].

Another interesting class of molecules to explore is mixed hypermettalic neutral and ionic oxides of the MOM' type, where M and M' refer to two different metal elements. Recently, quantum state controlled synthesis of BaOCa⁺ has been demonstrated [83] and theoretical work on neutral MOM (same metal) molecules has been performed motivated by the potential to search for fine structure constant α variation [84]. Neutral polyatomic molecules functionalized with optical cycling centers like Ca or Sr could also enable interesting applications in quantum sciences [85, 86].

The work at Harvard has been supported by the grants from AFOSR and NSF while the work at ASU has been supported by the HSF grant #2018-0681. We would like to thank L Baum and B Augenbraun for insightful discussions. While preparing this manuscript we have learned of another recent work on dispersed LIF of CaOCH₃ [87]. However, the details of the measurement method, analysis and resulting conclusions are vastly distinct.

A.1. Optimized geometry

In tables A1 and A2 we provide a comparison between the measured and calculated geometries for CaOH and CaOCH₃ in the ground and excited electronic states.

Table A1.  Optimized geometry for CaOH.

State	Coordinate	TI	TD	Measured [88]	[22]
$\tilde{X}$	Ca–O	1.9698	1.9656	1.9746	2.0038
$\tilde{X}$	O–H	0.9644	0.964 60	0.9562	0.9333
$\tilde{X}$	∠Ca–O–H	180.00	180.00	180.00	179.97
$\tilde{A}$	Ca–O	1.9698	1.9652	1.9532	1.9769
$\tilde{A}$	O–H	0.9648	0.9646	0.9572	0.9332
$\tilde{A}$	∠Ca–O–H	180.00	180.00	180.00	179.97

Table A2.  Optimized geometry for CaOCH₃.

State	Coordinate	TI	TD	Measured [45]
$\tilde{X}$	Ca–O	1.9726	1.9636	1.962 ± 0.004
$\tilde{X}$	O-C	1.3989	1.4011	1.411 ± 0.007
$\tilde{X}$	C–H	1.1069	1.1062	1.0937a
$\tilde{X}$	∠O–C–H	111.545	111.3	111.3 ± 0.2
$\tilde{A}$	Ca–O	1.972 75	1.9636	1.941 93
$\tilde{A}$	O–C	1.3989	1.401	1.4106
$\tilde{A}$	C–H	1.1068	1.1061	1.0923
$\tilde{A}$	∠O–C–H	111.545	111.3	111.0004b

^aUsing methanol tabulation. ^bCalculated from the available data for the H–C–H angle.

A.2. Details of FCF calculations

To compute the FCFs for CaOH and CaOCH₃, we used Wilson's GF matrix method and the Sharp–Rosenstock expansion of the FC overlap integral using a harmonic potential. In depth discussion and derivation of these methods are extensively addressed elsewhere [67, 89, 90]. Importantly, the Sharp–Rosenstock expansion depends on the vibrational coordinates of the initial and final states, denoted ${\bf{Q}}^{\prime}$ and ${\bf{Q}}$ respectively. The relationship between these two coordinate systems is:

$$\begin{eqnarray}&&{\bf{Q}}^{\prime} ={\bf{JQ}}+{\bf{K}},\end{eqnarray} \tag{ 6 }$$

where K is the difference between the equilibrium geometry of the two states in terms of the initial vibrational coordinates, while J accounts for coordinate mixing, also known as the Duchinsky effect.

Wilson's GF matrix method [91] is the extension of eigenvalue vibration problems to molecular vibrations, and can be used to find the relationship between Q and ${\bf{Q}}^{\prime}$. For an N-atom molecule, we define a set of 3N − 6 internal coordinates $\left\{{S}_{t}\right\}$ transformed by 3N coefficients B_ti from the set of 3N Cartesian components $\left\{{\xi }_{i}\right\}$ ⁷

$$\begin{eqnarray}&&{S}_{t}=\displaystyle \sum _{i=1}^{3N}{B}_{{ti}}{\xi }_{i}\hspace{0.5cm}t=1,2,3...,3N-6.\end{eqnarray} \tag{ 7 }$$

For convenience, the internal coordinates are often rewritten as N vectors ${{\bf{s}}}_{t\alpha }=({B}_{{ti}},{B}_{{ti}+1},{B}_{{ti}+2})$ for each atom given fixed $t\in [1,3N-6]$. The elements of the G (geometry) matrix are derived from mass-weighting the summed scalar product of each of the 3N − 6 internal coordinates over all atoms, while the F (force) matrix elements are derived using a harmonic approximation—each element of the F matrix is the second derivative of the potential energy surface with respect to the internal coordinates.

$$\begin{eqnarray}&&{G}_{{tt}^{\prime} }=\displaystyle \sum _{i=1}^{3N}\displaystyle \frac{1}{{m}_{\alpha }}{B}_{{ti}}{B}_{t^{\prime} i}=\displaystyle \sum _{\alpha =1}^{N}\displaystyle \frac{1}{{m}_{\alpha }}{{\bf{s}}}_{t\alpha }\cdot {{\bf{s}}}_{t^{\prime} \alpha }\qquad {F}_{{tt}^{\prime} }=\displaystyle \frac{{\partial }^{2}V}{\partial {S}_{t}\partial {S}_{t^{\prime} }}\end{eqnarray} \tag{ 8 }$$

From the G matrix, it is possible to derive the kinetic energy, and from the F matrix it is possible to derive the potential energy.

$$\begin{eqnarray}&&2T=\displaystyle \sum _{{tt}^{\prime} }({G}_{{tt}^{\prime} }^{-1}){\dot{S}}_{t}{\dot{S}}_{t^{\prime} }\qquad 2V=\displaystyle \sum _{{tt}^{\prime} }{F}_{{tt}^{\prime} }{S}_{t}{S}_{t^{\prime} }.\end{eqnarray} \tag{ 9 }$$

Writing the Lagrangian of the system, we end up with a determinant problem:

$$\begin{eqnarray}| {\bf{F}}-{{\bf{G}}}^{-1}\lambda | =\left|\begin{array}{ccc}{F}_{11}-{\left({G}^{-1}\right)}_{11}\lambda & \cdots & {F}_{1n}-{\left({G}^{-1}\right)}_{1n}\lambda \\ {F}_{21}-{\left({G}^{-1}\right)}_{21}\lambda & \cdots & {F}_{2n}-{\left({G}^{-1}\right)}_{2n}\lambda \\ \vdots & \vdots & \vdots \\ {F}_{n1}-{\left({G}^{-1}\right)}_{n1}\lambda & \cdots & {F}_{{nn}}-{\left({G}^{-1}\right)}_{{nn}}\lambda \end{array}\right|=0,\end{eqnarray} \tag{ 10 }$$

where $\lambda =4{\pi }^{2}{\nu }^{2}$ and ν is the frequency of a molecular vibration. Multiplying by the determinant of G, we get a diagonalization problem: $| {\bf{G}}| | {\bf{F}}-{{\bf{G}}}^{-1}\lambda | =| {\bf{GF}}-\lambda {\mathbb{I}}| =0$, where the corresponding eigenvectors are the vibrations (normal modes) in terms of the internal coordinates. Thus, the linear transformation L formed by the eigenvectors maps from vibrational normal mode coordinates to internal coordinates. By repeating the GF analysis for the initial and final electronic states, we end up with L matrices for both states, and thus can define Q and ${\bf{Q}}^{\prime}$.

Specifically, J is defined as the product of the inverse of the initial state L matrix and and the final state L, while K is the transformation of the equilibrium difference in internal coordinates to normal coordinates

$$\begin{eqnarray}&&{\bf{J}}={\left({\bf{L}}^{\prime} \right)}^{-1}{\bf{L}}\qquad {\bf{K}}={\left({\bf{L}}^{\prime} \right)}^{-1}({{\bf{R}}}_{\mathrm{eq}}-{{\bf{R}}}_{\mathrm{eq}}^{\prime} ).\end{eqnarray} \tag{ 11 }$$

Due to vibrational degeneracies, it will often be preferable to reduce the dimensions of the GF problem by concatenating the internal coordinates into symmetry coordinates shown in table 4.

To summarize, these Franck–Condon calculations required knowledge of the molecule geometry, vibrational frequencies, and the potential energy surface in the form of harmonic force constants defined over an internal coordinate system. Suitable internal and symmetry coordinate systems were analytically derived using literature on methanol (HOCH₃) [92], and the force constants were fit using PGOPHER software to molecular geometry and spectroscopic data [93].

The force constant fitting was initialized by first reducing the dimensionality of the GF problem to a triatomic situation by treating the CH₃ methyl group as a single atom. Given an ordering of the eigenvalues and neglecting off-diagonal F matrix interactions, the 3 × 3 GF eigenvalue problem is completely determined without knowledge of the diagonal matrix terms. We determine the correct ordering by comparing calculated FC values with known measurements and prior theory, initializing the PGOPHER force constant fitting using force values for the M–O stretch, O–CH₃ stretch, and M–O–C bend symmetry coordinates. To prevent misidentification of normal modes with frequencies, we separately fit the force constants for vibrations of different symmetries, which should be completely decoupled in such a way that the L matrix is block-diagonal. We then computed the FC overlap integrals using the Sharp–Rosenstock expansion with the matrices J, K, and ${\boldsymbol{\Gamma }}$, a diagonal matrix consisting of i reduced vibrational frequencies $4{\pi }^{2}{\nu }_{i}/h$, for both the initial and final electronic states. The FCFs were derived by calculating and normalizing the overlap integral for each vibration up to the third quantum number.

A.3. Additional supporting measurements

In this Section we provide additional supporting data in figures 8–10.

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