Electronic spectroscopy of the $A1̃2A′′/A2̃2A′−X̃2A′$ transitions of jet-cooled calcium ethoxide radicals: Vibronic structure of alkaline earth monoalkoxide radicals of C_s symmetry

Alkaline earth monoalkyl (MR) and monoalkoxide (MOR) free radicals, e.g., CaCH₃, CaOCH₃, SrCH₃, and SrOCH₃, are promising candidates for laser-cooling of polyatomic molecules.^1,2 In 2020, CaOCH₃, a symmetric top, became the first nonlinear molecule to be laser-cooled.³ Most recently, the ²²⁵Ra$OCH3+$ cation has been proposed as a candidate molecule for testing fundamental symmetry due to its deformed ²²⁵Ra nucleus.^4,5 Moreover, the practicality of laser-cooling asymmetric-top molecules has been investigated using ab initio calculated and experimentally determined molecular properties.⁶ Compared to diatomics and linear molecules, asymmetric-top molecules have certain advantages in optical cycling and laser cooling because they can be readily polarized in the laboratory frame due to the small asymmetry splitting in the K ≠ 0 levels. The fast advance of laser-cooling of molecules and its applications in the search for physics beyond the Standard Model^7,8 requires more detailed investigation and quantitative understanding of spin–rovibronic (spin–rotational–vibrational–electronic) energy level structure and transition intensities of metal-containing molecules, both by laser-spectroscopy measurements and by first-principles calculations.

In 2019, we reported the first quantitative spin–vibronic analysis of the lowest electronic states of an alkaline earth monoalkoxide free radical, namely, calcium methoxide (CaOCH₃).⁹ Laser-induced fluorescence (LIF) and dispersed fluorescence (DF) spectra of the $Ã2E−X̃2A1$ electronic transition of CaOCH₃ have been recorded under jet-cooled conditions. Vibronic transitions observed in the LIF/DF spectra were assigned unambiguously based on complete active space self-consistent field (CASSCF) and coupled-cluster singles and doubles (CCSD) calculations. Vibrational branching ratios (VBRs) of the $Ã2E→X̃2A1$ transition were determined using the LIF/DF spectra, combined with a more precise intensity measurement in a jet-cooled cavity ring-down (CRD) measurement.

The calculated Franck–Condon (FC) matrix for the $Ã2$E–$X̃2A1$ electronic transition of CaOCH₃ is highly diagonal. However, a number of weaker lines in the experimental spectra are attributable to vibronic transitions corresponding to off-diagonal FC matrix elements, and their intensities are reproduced by detailed calculations going beyond the Born–Oppenheimer (BO) approximation. The off-diagonal transitions include those from the vibrational ground level of the initial electronic state to the vibrational levels of the asymmetric CaOC stretch mode [ν₃(a₁)] and, most interestingly, the asymmetric CaOC bending mode [ν₈(e)] of the final electronic state. (The numbers in parentheses correspond to the Herzberg convention for labeling the modes. See Table II of Ref. 9.) Transitions from the v = 0 level of an electronic state to the v = 1 level of an asymmetric mode of another electronic state is forbidden under the diabatic BO approximation and the FC approximation but induced by the Jahn–Teller (JT) effect¹⁰ of the $Ã2$E state in this case. A spin–vibronic spectroscopy model reveals that, in addition to the JT effect, the pseudo-Jahn–Teller (pJT) interactions¹⁰ between the $Ã2$E and $B̃2A1$ states, as well as the spin–orbit (SO) interaction of the $Ã2$E state,¹¹ enhance the transition to the ν₈ = 1 level. Additionally, the Duschinsky interaction¹² induces transitions between the CaO-stretch (ν₄) mode and non-CaO-stretch modes. Finally, anharmonic vibrational terms in the ground state induce transitions that are forbidden in the harmonic-oscillator approximation. The spectroscopic model we developed accounts for all these complex interactions, which enabled accurate prediction of transition frequencies and intensities.

The JT effect in molecules such as CaOCH₃ that belong to the C_3v point group is reduced to the pJT effect in molecules of the C_2v, C_s, or C₁ point groups. The intra-molecular interactions in pJT-active molecules are even more complex than JT-active ones due to the lowered symmetry. Spin–vibronic analysis of C_s or C₁ molecules, similar to that done previously for C_3v molecules, is necessary for future laser-cooling of asymmetric-top molecules and their applications in fundamental sciences.

A common approach to the spectroscopic investigation of low-symmetry molecules is the asymmetric alkyl substitution of a symmetric-top molecule such as CaOCH₃. Using this approach, we have obtained LIF, DF, and CRD spectra of jet-cooled calcium ethoxide (CaOC₂H₅) radicals and analyzed their spin–vibronic structure. The spectroscopic analysis requires an in-depth understanding and modeling of couplings between the close-lying electronic states originating from the breaking of the degeneracy of the $Ã2$E state of CaOCH₃ into nearly degenerate $A1̃2A″$ and $A2̃2A′$ states in CaOC₂H₅. Based on the previous work on CaOCH₃, one expects that the spin–vibronic interactions between the $A1̃2A″$/$A2̃2A′$ and the $B̃2A′$ states, including both the SO and the pJT effects, also affect the vibronic structure of CaOC₂H₅.

We have developed the methodology for treating pJT and SO interactions for C_s molecules and incorporated it in the SOCJT (Spin–Orbit Coupling Jahn–Teller) program,¹¹ which was used to analyze the CaOCH₃ spectra. The upgraded SOCJT program has been employed to analyze and simulate transition frequencies and intensities for the excitation and emission spectra of CaOC₂H₅ in the present work.

Previously, Bernath and co-workers reported the LIF/DF spectra of the $Ã−X̃$ and $B̃−X̃$ electronic transitions of alkaline earth monoalkoxide radicals (MOR; M = Ca, Sr, and Ba with R being an alkoxide group).^13,14 These radicals were produced in a Broida-type oven by the reaction of a metal vapor with an oxidant (alcohol, acetone, or acetaldehyde). Paul et al. reported the DF spectra of CaOC₂H₅ by exciting its $Ã−X̃$ origin band.¹⁵ The results reported in the present paper represent the first comprehensive spectroscopic investigation of the $Ã−X̃$ electronic transition of CaOC₂H₅.

The rest of this paper is organized as follows: First, the experimental apparatuses for the jet-cooled LIF/DF and CRD measurements are described (Sec. II). Quantum chemistry calculations of related electronic states of the CaOC₂H₅ radical will then be presented (Sec. III A), followed by the ab initio vibronic calculations of the energy level structure and transition intensities that take into account both the pJT and SO interactions (Sec. III B). The spectral simulation using the calculated results will be compared with the experimentally obtained spectra for vibrational assignment in Sec. IV and discussed in Sec. V. Also to be discussed are the implications of the present work to laser-cooling of asymmetric-top molecules and detecting symmetry-violating interactions.

The LIF/DF and CRD apparatuses were described in our previous publication on the CaOCH₃ radical⁹ and are only briefly described here. In the present work, CaOC₂H₅ radicals were produced by 1064 nm laser ablation of a calcium metal rod with a Nd:YAG laser (Continuum, Powerlite Precision II 8000) in the throat of a supersonic jet of helium seeded with ethanol. Ethanol was contained in a stainless steel reservoir at room temperature and entrained in the flow by passing high-pressure helium (backing pressure = 180 psi above 1 atm) through the reservoir. The supersonic jet molecular beam was produced by expanding the seeded flow through a pinhole valve (0.3 mm diameter) into the vacuum chamber (stagnation pressure = 20 mTorr). A 12 mm thick Teflon extension with a 1.5 mm diameter orifice at its center was attached to the supersonic nozzle for additional vibrational cooling.

The excitation laser used in the LIF/DF experiment is a pulsed dye laser (Spectra-Physics, Cobra Stretch) pumped by the second harmonic of an Nd:YAG laser (Spectra-Physics, GCR-4). The laser-induced fluorescence was collected by a lens system perpendicular to both the excitation laser beam and the jet expansion. A photomultiplier tube (PMT, Hamamatsu, H10721-01) was used to detect the focused fluorescence in the LIF measurement. The spectral linewidth of the LIF spectrum is ≈0.1 cm⁻¹, limited mainly by the linewidth of the pulsed dye laser (≈0.06 cm⁻¹) and the residual Doppler width. The frequency of the pulsed dye laser was calibrated by a wavemeter (HighFinesse, W7). The accuracy of the vibronic transition frequencies in the LIF spectrum is limited to ≈1 cm⁻¹, mainly by the width of the rotational contour. For the DF experiment, the fluorescence is dispersed by a monochromator (Acton Research, SpectraPro 300i) equipped with an intensified CCD camera (Princeton Instruments, PI-MAX 512). The width of the entrance slit of the monochromator was adjusted to balance the spectral linewidth and the signal-to-noise ratio (S/N). The typical spectral resolution of DF spectra recorded in the present work is ≈20 cm⁻¹. The wavelength of the DF spectra was calibrated using a mercury arc lamp, and the frequency uncertainty is ≈5 cm⁻¹, limited mainly by the linewidth.

Intensities of the recorded LIF transitions were calibrated by the wavelength-dependent sensitivity of the PMT. Intensities of the DF spectra were calibrated by the grating efficiency of the spectrograph and the quantum efficiency of the iCCD camera, both of which are wavelength-dependent. Further details of intensity calibration can be found in our previous work on CaOCH₃.⁹ The origin-band transitions in the LIF experiment are significantly saturated so that an accurate measurement of their transition intensities is impractical. Therefore, pulsed-laser CRD spectra were also recorded to obtain accurate transition intensities. The CRD mirrors (Los Gatos Research, R ≥ 99.995%, center wavelength = 620 nm) were mounted on the two arms of the vacuum chamber to form a ring-down cavity with a length of L = 76 cm. The ring-down mirrors were purged by a nitrogen flow continuously to prevent contamination (mainly by the metal vapor). Transmission of the ring-down beam from a pulse dye laser (Sirah, PrecisionScan) through the cavity was focused onto a PMT (Hamamatsu, H10721-01). The empty-cavity ring-down time (τ₀) was about 250 µs. Ring-down signals were recorded at each wavelength with the ablation laser on and off. The “ablation laser off” CRD spectrum is subtracted from the “ablation laser on” spectrum to obtain the absorption spectrum of radicals.

In this section, we describe the computational and theoretical analyses used to assign and understand the observed spectra. There are essentially two levels to these calculations. At the first level (Sec. III A), quantum chemistry calculations for the relevant diabatic electronic states are performed, including the ground electronic $X̃2A′$ state, the first (⁠$A1̃2A″$⁠) and the second (⁠$A2̃2A′$⁠) excited electronic state originating from the $Ã2$E state of CaOCH₃, and the third electronic state (⁠$B̃2A′$⁠). These calculations at the first level conform to the BO approximation, as does the subsequent analyses of the vibrational structure within the electronic states. The second-level spin–vibronic calculations (Sec. III B) include interactions neglected in the BO approximation, which improves the quality of spectral simulations and provides insight into the pJT and SO coupling between the lowest electronic states of CaOC₂H₅.

In a previous work of ours,¹⁵ the $X̃2A′$ and $A1̃2A″$/$A2̃2A′$ states of CaOC₂H₅ were calculated using the complete active space self-consistent field (CASSCF) method with the cc-PVTZ basis set, employing the Gaussian 09 program package.¹⁶ The active space used for the calculations and the electron promotions involved in the excited-state calculations can be found therein. The $Ã1$ and $Ã2$ states of CaOC₂H₅ are nearly degenerate and strongly coupled. Therefore, a state-averaged (SA) calculation with equal weights for the two states was carried out for the geometric optimization, which converges to the minimum of the conical intersection (CI) seam. Notably, the CASSCF calculations failed to predict the symmetry (A′ or A″) of these two nearly degenerate electronic states correctly,¹⁵ probably due to the limited active space and the inaccurate weights used in the calculation.

In the present work, the coupled cluster singles and doubles (CCSD) method was implemented with the CFOUR program suite¹⁷ to calculate the lowest four electronic states mentioned above. A cc-pVTZ basis set was employed for the CCSD calculation of the $X̃2A′$ state. Properties of the $A1̃2A″$ and $A2̃2A′$ states were calculated using the equation-of-motion electron excitation coupled-cluster theory (EOMEE-CCSD/cc-pVTZ). Both the CASSCF and CCSD methods conform to the BO approximation and calculate the adiabatic states. The rationale for the new CCSD calculations is first to allow cross-checking of the previous results. More importantly, the CCSD results are used directly in the second level of calculation that involved a vibronic treatment (see Sec. III B).

Geometry optimization of all the states provides equilibrium bond lengths and bond angles, as well as rotational constants, which are summarized in Table I, along with values calculated on the CASSCF(3,6)/cc-PVTZ level of theory as reported in Ref. 15. The largest change in geometry upon the $Ã1$/$Ã2$ $←X̃$ excitation is the reduction of the CaO bond length (≈28 mÅ). Therefore, qualitatively one expects vibronic bands to the excited CaO stretch levels in the LIF/DF spectra to be the strongest non-diagonal transitions.

Quantitatively, the vibrational assignment of LIF and DF spectra in the present work is guided by the electronic structure calculations. CaOC₂H₅ has 21 vibrational modes, including 13 a′ modes and 8 a″ modes. Harmonic frequencies of the $X̃$ and $Ã$ states calculated using the CASSCF and CCSD methods are summarized in Table II. In most cases, the harmonic frequencies calculated using either the CASSCF or CCSD methods well reproduce the experimentally determined frequencies. In general, the CCSD calculations are more accurate than the CASSCF ones. Notably, they better reproduce the experimental frequencies of the in-plane and out-of-plane CaOC bending modes. The discrepancies between the CCSD-calculated harmonic frequencies are within 20 cm⁻¹ of the experimental values with the exception of the CO stretch mode (ν₈), the highest-frequency vibrational mode for which transitions have been observed in the LIF/DF spectra. For the $X̃2A′$ and $A1̃2A″$/$A2̃2A′$ states, the calculated values are 74 and 40 cm⁻¹ higher than the experimentally determined values, respectively. These discrepancies may partly be attributed to the anharmonicity of the CO stretch mode, which was also observed in the LIF and DF spectra of alkoxy radicals. However, the anharmonicity (2ω_eχ_e) of the CO stretch mode of alkoxy radicals is usually smaller than 10 cm⁻¹.^18–21 A more plausible explanation of the discrepancy is, therefore, the vibronic interaction of the high-lying $Ã$-state ν₈ level with the $B̃2A′$ state, which lowers the $Ã$-state ν₈ level (see Sec. IV A).

FC factors for both absorption and emission transitions were computed with ground- and excited-state vibrational wavefunctions from either the CASSCF or CCSD calculations. The FC calculations were performed using the ezSpectrum software.²² Both the LIF and DF spectra of the $A1̃2A′′/A2̃2A′−X̃2A′$ transition were simulated under the harmonic oscillator approximation using the calculated vibrational frequencies and FC factors [see Figs. 1(b) and 1(c) and Sec. IV for details].

In addition to the quantum chemistry calculations described in Sec. III A, a complete simulation of CaOC₂H₅ spectra was done from first principles with the values of all spectral parameters derived from the electronic structure calculations, including vibronic and SO interactions. The approach is similar to our previous work⁹ on CaOCH₃ and the prior work on CH₃O by Weichman et al.²³ However, in both these cases, the molecules are symmetric tops with nominal C_3v symmetry and a degenerate electronic state. On the other hand, CaOC₂H₅ is an asymmetric top with only C_s symmetry and non-degenerate electronic states, and so the previous treatments must be generalized.

The complete spin–vibronic Hamiltonian matrix for the CaOC₂H₅ radical in a quasi-diabatic basis includes pJT and SO couplings between the $Ã1$⁠, $Ã2$⁠, and $B̃$ electronic states. These terms were neglected in our treatment in Subsection III A because, consistent with the BO approximation, those treatments ignored spin–vibronic terms coupling different eigenstates of the electronic Hamiltonian.

The spin–vibronic Hamiltonian is written in the quasi-diabatic basis containing $Ã1$⁠, $Ã2$⁠, and $B̃$ electronic states,

where $T̂N$ is the nuclear kinetic energy and $V̂$ is the potential energy. The potential energy matrix is expanded in the normal coordinates (q_j) of the $X̃$ state (from here on referred to as R₀) and is a complex 6 × 6 matrix in the spin-electronic basis, |Λ⟩ ⊗ |Σ⟩, where Λ and Σ are the projections of the orbital angular momentum (L) and the spin (S) of the electron, respectively,

In the matrix, from left to right, the columns are labeled by the $Ã1$⁠, $Ã2$⁠, and $B̃$ vibronic basis sets with Σ = +1/2 for the first three and correspondingly Σ = −1/2 for the last three. ΔE_ii^′ (i = $Ã1$⁠, $Ã2$⁠, or $B̃$⁠) is the electronic energy difference between the ith and i′th electronic states at R₀. V_i is the diagonal term corresponding to each electronic state and contains the harmonic oscillator terms and the derivative coupling,

where $ωji$ is the harmonic frequency of the jth mode. The derivative coupling parameter, $dji$⁠, is non-zero along totally symmetric normal coordinates only and is given by

where |Λ_i⟩ is the electronic wave function of the ith state and V_n is the non-relativistic potential energy of the molecule and consists of nuclear–electron, nuclear–nuclear, and electron–electron interactions.

The pJT coupling between the $Ã1$⁠, $Ã2$⁠, and $B̃$ states is given by $VpJTii′$⁠,

The SO coupling is given as

where a_α is the SO coupling constant and α (= x, y, z) denotes the principal axes. Following our previous work on alkoxy radicals, the principal axis perpendicular to the C_s plane, the c-axis in the case of CaOC₂H₅, is defined as the y axis.^24–26 The x and z axes lie in the C_s plane. The type-I right-hand (I^r) convention is followed here, i.e., the traditional a, b, and c axes correspond to z, x, and y, respectively. For symmetry reasons, $VyÃ1Ã2$⁠, $VyÃ1B̃$⁠, $VxÃ2B̃$⁠, and $VzÃ2B̃$ vanish.

The CFOUR quantum chemistry package was used to calculate the values for $ωji$⁠, $dji$⁠, $VpJTii′$⁠, and $Vαii′$⁠. The $ωji$ and $dji$ values were calculated using finite differences of adiabatic potential energy surfaces. $Vαii′$ was calculated using the mean-field approach for EOM-CCSD wavefunctions. Table III contains the calculated derivative coupling constants and pJT coupling constants, while the SO constants between the $Ã1$⁠, $Ã2$⁠, and $B̃$ states are summarized in Table IV.

To the best of our knowledge, none of the present quantum chemistry codes are capable of including both quadratic and SO terms in the potential of a C_s molecule and, hence, the former were excluded from Eq. (2) since they are likely to be small compared to the latter. To make the calculations feasible, we have ignored any coupling between the a′ and a″ modes, thereby making the Hamiltonian block diagonal. Using the Lanczos algorithm, blocks of the Hamiltonian matrix were diagonalized separately for the a′ and a″ modes to obtain eigenvalues. The a′ calculation includes ν₈ − ν₁₃ modes with v_max{ν₈, ν₉, ν₁₀, ν₁₁, ν₁₂, ν₁₃} = {10, 7, 9, 12, 12, 16}, and the a″ calculation includes ν₁₆−ν₂₁ modes with v_max{ν₁₆, ν₁₇, ν₁₈, ν₁₉, ν₂₀, ν₂₁} = {8, 8, 10, 10, 12, 14}. The basis set for the a′ calculation contains ≈15 × 10⁶ basis functions, and the a″ calculation contains ≈11 × 10⁶ basis functions. Calculation of eigenfunctions up to 1000 cm⁻¹ above the mean of the $A1̃2A″$ and $A2̃2A′$ states was converged. Therefore, in the present work, the simulated LIF spectrum based on the spin–vibronic calculations is truncated at 1000 cm⁻¹ above the origin [dashed vertical line in Fig. 1(d)].

The calculated eigenvalues and eigenfunctions were then used to predict the transition frequencies and intensities of the LIF and DF spectra. The line strengths, S_α, are given by

where μ_α is the electric dipole moment operator and Ψ_i and Ψ_f are the spin–vibronic wavefunctions of the initial and final energy levels, respectively, obtained by solving the Hamiltonian given above [Eq. (2)]. In order to obtain S’s without storing the eigenvectors, calculations are done using Ψ_i as the seed vector in the Lanczos algorithm.²³ The calculated electronic transition dipole moments between the ground state and the lowest excited states are listed in Table V.

Comparisons between the experimental spectra and simulations based on the methods described in Sec. III allow the assignment of most of the observed vibronic transitions. This interplay of spectroscopic and computational results helps scrutinize the polyatomic asymmetric-top molecule CaOC₂H₅ as a potential candidate for laser-cooling.

Panels (a) and (a′) of Fig. 1 show the experimentally obtained LIF spectrum of the $Ã←X̃$ transition of CaOC₂H₅. As indicated in the figure, the experimental LIF spectrum features six doublets, all with a frequency interval of ∼67 cm⁻¹. The experimentally measured SO-free frequencies of the transitions, relative to the origin transition frequencies, are given in Table II. Using the harmonic frequencies given in Table II, we can assign a vibrational mode to each of the doublets [see Fig. 1(a)]. In addition, we label the lower- and higher-frequency peaks as representing the $A1̃2A′′←X̃2A′$ and $A2̃2A′←X̃2A′$ transitions, respectively. The doublets are present in the EOM-CCSD electronic structure calculations, although, as we will see later, the $A2̃2A′−A1̃2A″$ separation is more determined by the SO interaction, which is included in the complete spin–vibronic simulation.

The strongest doublet centered at 15 882 cm⁻¹ is assigned to the $A1̃2A′′←X̃2A′$ and $A2̃2A′←X̃2A′$ origin transitions. The CASSCF and EOMEE-CCSD calculations predict the origin transition frequency to be 15 390 and 15 869 cm⁻¹, respectively. In our previous work on CaOCH₃,⁹ the corresponding CCSD calculation also predicted the origin transition frequency with excellent accuracy, a predicted value of 15 918 cm⁻¹ compared to the experimental value of 15 925 cm⁻¹.

Three doublets centered at 389, 532, and 1170 cm⁻¹ to the blue of the center of the origin band were observed in the LIF spectrum. Compared to the calculated $Ã$-state vibrational frequencies (see Table II), these bands can be assigned as transitions to the v′ = 1 levels of the CaO stretch mode (⁠$1201$⁠), the OCC bending mode (⁠$1101$⁠), and the CO stretch mode (⁠$801$⁠), respectively. Two additional doublets are centered nearer to the origin at 93 and 130 cm⁻¹ to the blue. The most reasonable assignments of these doublets are transitions to the v′ = 1 levels of the CaOC bending mode, in-plane (⁠$1301$⁠) and out-of-plane (⁠$2101$⁠), respectively.

It is instructive to compare these observations to the three spectral simulations displayed in panels (b)–(d) of Fig. 1. Panel (b) contains the simulation of the LIF spectrum using the SA-CASSCF method. Although this method does predict the $Ã$-state vibrational frequencies fairly well, it is insufficient to accurately reproduce the experimental spectrum. First, the doublet splitting cannot be predicted using the SA-CASSCF method because the $Ã1−Ã2$ splitting is zero at the CI, and the calculation does not include the SO interaction. Furthermore, intensities of the vibronic bands are not accurately reproduced. The reasons for the intensity inaccuracy will be elucidated below.

Panel (c) of Fig. 1 contains the simulation resulting from the EOM-CCSD electronic structure calculations. This method clearly does better than SA-CASSCF as we now see the doublet structure of the vibrational transitions. However, the splitting of the doublets is only 17 cm⁻¹ compared to 67 cm⁻¹ observed experimentally. This is because, again, the SO coupling within the non-degenerate state vanishes, which affects the magnitude of the predicted doublet splitting.

Panel (d) of Fig. 1 is the complete spin–vibronic simulation of the LIF spectrum from first principles as described in Sec. III B. This calculation includes SO and first-order pJT couplings between the $Ã1$⁠, $Ã2$⁠, and $B̃$ states. This calculation does a quite good job of reproducing the experimentally observed spectrum. The calculated splitting of the origin doublet from first principles is ≈60 cm⁻¹, in rather good agreement with the experimental splitting of 67 cm⁻¹. In addition, the intensities of the various vibronic transitions are well predicted.

Figure 2 shows a zoomed-in plot of the 50–600 cm⁻¹ region of the LIF spectrum. First, we see the $1301$ doublet transition predicted at 90 cm⁻¹, compared to the experimental SO-free transition frequency of 93 cm⁻¹. There are also other lines calculated in the 50–300 cm⁻¹ region, the strongest of which is shown in Fig. 2 at 129 cm⁻¹. These are essentially transitions to dark states that gain intensity due to the pJT couplings. Limited by the S/N ratio, these transitions were not observed in the experiment. In the 300–600 cm⁻¹ region, the $1201$ and $1101$ bands are observed. A complete list of calculated $A1̃2A′′/A2̃2A′←X̃2A′$ transitions and their intensities is given in Table S.1 of the supplementary material.

In the 300–600 cm⁻¹ region, there are also low-intensity transitions observed to the blue of $1101$ and the red of $1201$ in the experimental spectrum [see Fig. 3(a)]. These transitions may be attributed to hot bands. Figure 3(b) contains a simulation of the LIF spectrum at a vibrational temperature of 75 K. The simulation predicts the $11011311$ hot band to the red of the $1101$ band, which matches where it is observed in the experimental spectrum. As expected, the $12011311$ hot band is also predicted to be lower in frequency than the $1201$ transition, but experimentally the observed weak transitions have higher frequencies than the $1201$ band. It is possible that higher-order couplings between the ν₁₂ (CaO stretch) and ν₁₃ (in-plane CaOC bending) modes, which are neglected in the present work (see above), cause the 12¹13¹ combination levels of the $Ã$ states to occur at higher frequencies than would otherwise be expected. Unfortunately, there are no methods presently available to calculate such higher-order couplings for quasi-degenerate electronic states and, hence, it is not possible to test our hypothesis.

Alternatively, the weak peaks at 517 cm⁻¹, i.e., 15 cm⁻¹ to the red of the $1101$ band, can be assigned to the combination band of ν₁₂ (389 cm⁻¹) and ν₂₁ (130 cm⁻¹), as indicated in Fig. 2(a). Such an assignment can be used to explain the DF spectra obtained by pumping the $1201$ transition in the LIF spectrum (see Sec. IV D).

The vibrational frequency of the $Ã$-state CO-stretch (ν₈) mode is not predicted in the present ab initio spin–vibronic calculations because the Hamiltonian diagonalization converges up to 1000 cm⁻¹ and the simulation of the $A1̃2A′′/A2̃2A′←X̃2A′$ transition is truncated as discussed in Sec. III B. The harmonic frequency of the ν₈ mode is predicted to be 74 and 40 cm⁻¹ higher than the experimental value by the CASSCF and CCSD calculations, respectively. It is expected that the pJT interaction with the $B̃2A′$ state lowers the ν₈ levels as it does to other $Ã$-state vibrational levels observed in the LIF spectrum (e.g., ν₁₁, ν₁₂, ν₁₃) (see Fig. 1). Therefore, inclusion of the spin–vibronic interactions would move the predicted ν₈ vibrational frequency closer to the experimentally observed value.

The origin transitions are power-saturated in the LIF measurements with a pulsed laser. To determine the VBRs of the $A1̃/A2̃−X̃$ electronic transitions, we need to know the degree of this saturation, and we turn to jet-cooled pulsed-laser CRD spectroscopy measurements. The LIF and CRD spectra of the origin, $1201$⁠, and $1101$ bands are compared in Fig. 4. The determined saturation factor for the origin transitions in the LIF experiment is 2.75. Therefore, FC factors for non-origin transitions determined from the LIF spectrum are reduced by a factor of 2.75 relative to the origin transitions, followed by re-normalization. The details for this process are given in Ref. 9.

The relative intensities, i.e., FC factors, of the $Ã←X̃$ transitions so determined are listed in Table VI. The relative intensities of the $A1̃2A′′←X̃2A′$ and $A2̃2A′←X̃2A′$ transitions are combined for the total FC factors. The experimental intensities of the origin transitions are estimated to have a relative error of 10% determined from the error of three repeated experimental traces. The main error source is the fluctuation of the free radical concentration in the jet expansion and the necessity to use the CRD calibration to eliminate the effects of saturation. The relative error of non-origin transitions is estimated to be 5%. The uncertainties of the relative intensities, i.e., the FC factors, have been determined using the error propagation method outlined in Sec. S.3 of the supplementary material of Ref. 9.