Laser-induced fluorescence/dispersed fluorescence (LIF/DF) and cavity ring-down spectra of the $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2212X\u03032A\u2032$ electronic transition of the calcium ethoxide (CaOC_{2}H_{5}) radical have been obtained under jet-cooled conditions. An essentially constant $A\u03032\u2212A\u03031$ energy separation for different vibronic levels is observed in the LIF spectrum, which is attributed to both the spin–orbit (SO) interaction and non-relativistic effects. Electronic transition energies, vibrational frequencies, and spin–vibrational eigenfunctions calculated using the coupled-cluster method, along with results from previous complete active space self-consistent field calculations, have been used to predict the vibronic energy level structure and simulate the recorded LIF/DF spectra. Although the vibrational frequencies and Franck–Condon (FC) factors calculated under the Born–Oppenheimer approximation and the harmonic oscillator approximation reproduce the dominant spectral features well, the inclusion of the pseudo-Jahn–Teller (pJT) and SO interactions, especially those between the $A1\u03032A\u2033$/$A2\u03032A\u2032$ and the $B\u03032A\u2032$ states, induces additional vibronic transitions and significantly improves the accuracy of the spectral simulations. Notably, the spin–vibronic interactions couple vibronic levels and alter transition intensities. The calculated FC matrix for the $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2212X\u03032A\u2032$ transition contains a number of off-diagonal matrix elements that connect the vibrational ground levels to the levels of the *ν*_{8} (CO stretch), *ν*_{11} (OCC bending), *ν*_{12} (CaO stretch), *ν*_{13} (in-plane CaOC bending), and *ν*_{21} (out-of-plane CaOC bending) modes, which are used for vibrational assignments. Transitions to the *ν*_{21}(*a*″) levels are allowed due to the pJT effect. Furthermore, when LIF transitions to the $A\u0303$-state levels of the CaOC-bending modes, *ν*_{13} and *ν*_{21}, are pumped, $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2192X\u03032A\u2032$ transitions to the combination levels of these two modes with the *ν*_{8}, *ν*_{11}, and *ν*_{12} modes are also observed in the DF spectra due to the Duschinsky mixing. Implications of the present spectroscopic investigation to laser cooling of asymmetric-top molecules are discussed.

## I. INTRODUCTION

Alkaline earth monoalkyl (MR) and monoalkoxide (MOR) free radicals, e.g., CaCH_{3}, CaOCH_{3}, SrCH_{3}, and SrOCH_{3}, are promising candidates for laser-cooling of polyatomic molecules.^{1,2} In 2020, CaOCH_{3}, a symmetric top, became the first nonlinear molecule to be laser-cooled.^{3} Most recently, the ^{225}Ra$OCH3+$ cation has been proposed as a candidate molecule for testing fundamental symmetry due to its deformed ^{225}Ra 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 (CaOCH_{3}).^{9} Laser-induced fluorescence (LIF) and dispersed fluorescence (DF) spectra of the $A\u03032E\u2212X\u03032A1$ electronic transition of CaOCH_{3} 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 $A\u03032E\u2192X\u03032A1$ 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 $A\u03032$E–$X\u03032A1$ electronic transition of CaOCH_{3} 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 [*ν*_{3}(*a*_{1})] and, most interestingly, the asymmetric CaOC bending mode [*ν*_{8}(*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) effect^{10} of the $A\u03032$E state in this case. A spin–vibronic spectroscopy model reveals that, in addition to the JT effect, the pseudo-Jahn–Teller (pJT) interactions^{10} between the $A\u03032$E and $B\u03032A1$ states, as well as the spin–orbit (SO) interaction of the $A\u03032$E state,^{11} enhance the transition to the *ν*_{8} = 1 level. Additionally, the Duschinsky interaction^{12} induces transitions between the CaO-stretch (*ν*_{4}) 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 CaOCH_{3} that belong to the *C*_{3v} point group is reduced to the pJT effect in molecules of the *C*_{2v}, *C*_{s}, or *C*_{1} 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 *C*_{1} 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 CaOCH_{3}. Using this approach, we have obtained LIF, DF, and CRD spectra of jet-cooled calcium ethoxide (CaOC_{2}H_{5}) 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 $A\u03032$E state of CaOCH_{3} into nearly degenerate $A1\u03032A\u2033$ and $A2\u03032A\u2032$ states in CaOC_{2}H_{5}. Based on the previous work on CaOCH_{3}, one expects that the spin–vibronic interactions between the $A1\u03032A\u2033$/$A2\u03032A\u2032$ and the $B\u03032A\u2032$ states, including both the SO and the pJT effects, also affect the vibronic structure of CaOC_{2}H_{5}.

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 CaOCH_{3} spectra. The upgraded SOCJT program has been employed to analyze and simulate transition frequencies and intensities for the excitation and emission spectra of CaOC_{2}H_{5} in the present work.

Previously, Bernath and co-workers reported the LIF/DF spectra of the $A\u0303\u2212X\u0303$ and $B\u0303\u2212X\u0303$ 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 CaOC_{2}H_{5} by exciting its $A\u0303\u2212X\u0303$ origin band.^{15} The results reported in the present paper represent the first comprehensive spectroscopic investigation of the $A\u0303\u2212X\u0303$ electronic transition of CaOC_{2}H_{5}.

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 CaOC_{2}H_{5} 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.

## II. EXPERIMENTAL

The LIF/DF and CRD apparatuses were described in our previous publication on the CaOCH_{3} radical^{9} and are only briefly described here. In the present work, CaOC_{2}H_{5} 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 CaOCH_{3}.^{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 (*τ*_{0}) was about 250 *µ*s. 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.

## III. CALCULATIONS

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 $X\u03032A\u2032$ state, the first ($A1\u03032A\u2033$) and the second ($A2\u03032A\u2032$) excited electronic state originating from the $A\u03032$E state of CaOCH_{3}, and the third electronic state ($B\u03032A\u2032$). 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 CaOC_{2}H_{5}.

### A. Quantum chemistry calculations

In a previous work of ours,^{15} the $X\u03032A\u2032$ and $A1\u03032A\u2033$/$A2\u03032A\u2032$ states of CaOC_{2}H_{5} 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 $A\u03031$ and $A\u03032$ states of CaOC_{2}H_{5} 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 suite^{17} to calculate the lowest four electronic states mentioned above. A cc-pVTZ basis set was employed for the CCSD calculation of the $X\u03032A\u2032$ state. Properties of the $A1\u03032A\u2033$ and $A2\u03032A\u2032$ 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 $A\u03031$/$A\u03032$ $\u2190X\u0303$ 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.

. | CASSCF^{a}
. | CCSD^{b}
. | EOM-CCSD^{c}
. | |
---|---|---|---|---|

$X\u0303$ . | $A\u03031/A\u03032$ . | $X\u0303$ . | $A\u03031/A\u03032$ . | |

r_{CaO} (Å) | 2.014 | 1.984 | 2.115 | 2.087 |

r_{OC} (Å) | 1.372 | 1.375 | 1.417 | 1.424 |

r_{CC} (Å) | 1.523 | 1.521 | 1.532 | 1.530 |

∠CaOC (deg) | 179 | 179 | 179 | 179 |

∠OCC (deg) | 112 | 112 | 112 | 112 |

A (cm^{−1}) | 0.816 | 0.821 | 0.778 | 0.773 |

B (cm^{−1}) | 0.066 | 0.067 | 0.062 | 0.063 |

C (cm^{−1}) | 0.063 | 0.064 | 0.059 | 0.059 |

$\Delta EA\u0303\u2212X\u0303$ (cm^{−1})^{d} | 15 390 | 15 869 |

. | CASSCF^{a}
. | CCSD^{b}
. | EOM-CCSD^{c}
. | |
---|---|---|---|---|

$X\u0303$ . | $A\u03031/A\u03032$ . | $X\u0303$ . | $A\u03031/A\u03032$ . | |

r_{CaO} (Å) | 2.014 | 1.984 | 2.115 | 2.087 |

r_{OC} (Å) | 1.372 | 1.375 | 1.417 | 1.424 |

r_{CC} (Å) | 1.523 | 1.521 | 1.532 | 1.530 |

∠CaOC (deg) | 179 | 179 | 179 | 179 |

∠OCC (deg) | 112 | 112 | 112 | 112 |

A (cm^{−1}) | 0.816 | 0.821 | 0.778 | 0.773 |

B (cm^{−1}) | 0.066 | 0.067 | 0.062 | 0.063 |

C (cm^{−1}) | 0.063 | 0.064 | 0.059 | 0.059 |

$\Delta EA\u0303\u2212X\u0303$ (cm^{−1})^{d} | 15 390 | 15 869 |

^{a}

At the CASSCF(3,6)/cc-PVTZ level of theory. Calculated values from Ref. 15.

^{b}

At the CCSD/cc-PVTZ level of theory.

^{c}

At the EOM-CCSD/cc-PVTZ level of theory.

^{d}

Defined as the energy separation between the $X\u0303$ state and the center of the $A1\u03032A\u2033$ and $A2\u03032A\u2032$ states. Compared to the experimental value of 15 882 cm^{−1}.

Quantitatively, the vibrational assignment of LIF and DF spectra in the present work is guided by the electronic structure calculations. CaOC_{2}H_{5} has 21 vibrational modes, including 13 *a*′ modes and 8 *a*″ modes. Harmonic frequencies of the $X\u0303$ and $A\u0303$ 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 (*ν*_{8}), the highest-frequency vibrational mode for which transitions have been observed in the LIF/DF spectra. For the $X\u03032A\u2032$ and $A1\u03032A\u2033$/$A2\u03032A\u2032$ 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 (2*ω*_{e}*χ*_{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 $A\u0303$-state *ν*_{8} level with the $B\u03032A\u2032$ state, which lowers the $A\u0303$-state *ν*_{8} level (see Sec. IV A).

. | . | $X\u0303$ . | $A\u03031/A\u03032$ . | . | ||||
---|---|---|---|---|---|---|---|---|

Mode . | Symmetry . | CASSCF . | CCSD . | Expt. . | CASSCF . | EOM-CCSD . | Expt. . | Description . |

ν_{1} | a′ | 3198 | 3118 | 3202 | 3136 | |||

ν_{2} | a′ | 3141 | 3064 | 3145 | 3052 | |||

ν_{3} | a′ | 3080 | 3001 | 3096 | 3010 | |||

ν_{4} | a′ | 1658 | 1556 | 1658 | 1520 | |||

ν_{5} | a′ | 1610 | 1526 | 1609 | 1495 | |||

ν_{6} | a′ | 1557 | 1444 | 1558 | 1426 | |||

ν_{7} | a′ | 1514 | 1415 | 1517 | 1393 | |||

ν_{8} | a′ | 1294 | 1244 | 1170 | 1299 | 1210 | 1170 | CO stretch |

ν_{9} | a′ | 1174 | 1123 | 1178 | 1109 | |||

ν_{10} | a′ | 974 | 942 | 916 | 983 | 930 | CC stretch | |

ν_{11} | a′ | 514 | 522 | 514 | 558 | 528 | 532 | OCC bending |

ν_{12} | a′ | 367 | 393 | 386 | 411 | 391 | 389 | CaO stretch |

ν_{13} | a′ | 96 | 97 | 93 | 114 | 90 | 93 | CaOC bending (in-plane) |

ν_{14} | a″ | 3201 | 3124 | 3206 | 3141 | |||

ν_{15} | a″ | 3084 | 3002 | 3104 | 3045 | |||

ν_{16} | a″ | 1597 | 1516 | 1598 | 1480 | |||

ν_{17} | a″ | 1431 | 1342 | 1437 | 1304 | |||

ν_{18} | a″ | 1289 | 1207 | 1291 | 1177 | |||

ν_{19} | a″ | 861 | 817 | 863 | 806 | |||

ν_{20} | a″ | 302 | 308 | 303 | 288 | Methyl torsion | ||

ν_{21} | a″ | 140 | 136 | 129 | 161 | 118 | 130 | CaOC bending (out-of-plane) |

. | . | $X\u0303$ . | $A\u03031/A\u03032$ . | . | ||||
---|---|---|---|---|---|---|---|---|

Mode . | Symmetry . | CASSCF . | CCSD . | Expt. . | CASSCF . | EOM-CCSD . | Expt. . | Description . |

ν_{1} | a′ | 3198 | 3118 | 3202 | 3136 | |||

ν_{2} | a′ | 3141 | 3064 | 3145 | 3052 | |||

ν_{3} | a′ | 3080 | 3001 | 3096 | 3010 | |||

ν_{4} | a′ | 1658 | 1556 | 1658 | 1520 | |||

ν_{5} | a′ | 1610 | 1526 | 1609 | 1495 | |||

ν_{6} | a′ | 1557 | 1444 | 1558 | 1426 | |||

ν_{7} | a′ | 1514 | 1415 | 1517 | 1393 | |||

ν_{8} | a′ | 1294 | 1244 | 1170 | 1299 | 1210 | 1170 | CO stretch |

ν_{9} | a′ | 1174 | 1123 | 1178 | 1109 | |||

ν_{10} | a′ | 974 | 942 | 916 | 983 | 930 | CC stretch | |

ν_{11} | a′ | 514 | 522 | 514 | 558 | 528 | 532 | OCC bending |

ν_{12} | a′ | 367 | 393 | 386 | 411 | 391 | 389 | CaO stretch |

ν_{13} | a′ | 96 | 97 | 93 | 114 | 90 | 93 | CaOC bending (in-plane) |

ν_{14} | a″ | 3201 | 3124 | 3206 | 3141 | |||

ν_{15} | a″ | 3084 | 3002 | 3104 | 3045 | |||

ν_{16} | a″ | 1597 | 1516 | 1598 | 1480 | |||

ν_{17} | a″ | 1431 | 1342 | 1437 | 1304 | |||

ν_{18} | a″ | 1289 | 1207 | 1291 | 1177 | |||

ν_{19} | a″ | 861 | 817 | 863 | 806 | |||

ν_{20} | a″ | 302 | 308 | 303 | 288 | Methyl torsion | ||

ν_{21} | a″ | 140 | 136 | 129 | 161 | 118 | 130 | CaOC bending (out-of-plane) |

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 $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2212X\u03032A\u2032$ 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].

### B. *Ab initio* spin–vibronic calculations

In addition to the quantum chemistry calculations described in Sec. III A, a complete simulation of CaOC_{2}H_{5} 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 work^{9} on CaOCH_{3} and the prior work on CH_{3}O 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, CaOC_{2}H_{5} 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 CaOC_{2}H_{5} radical in a quasi-diabatic basis includes pJT and SO couplings between the $A\u03031$, $A\u03032$, and $B\u0303$ 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 $A\u03031$, $A\u03032$, and $B\u0303$ electronic states,

where $T\u0302N$ is the nuclear kinetic energy and $V\u0302$ is the potential energy. The potential energy matrix is expanded in the normal coordinates (*q*_{j}) of the $X\u0303$ state (from here on referred to as *R*_{0}) 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 (

**) of the electron, respectively,**

*S*In the matrix, from left to right, the columns are labeled by the $A\u03031$, $A\u03032$, and $B\u0303$ vibronic basis sets with Σ = +1/2 for the first three and correspondingly Σ = −1/2 for the last three. Δ*E*_{ii′} (*i* = $A\u03031$, $A\u03032$, or $B\u0303$) is the electronic energy difference between the *i*th and *i*′th electronic states at *R*_{0}. *V*_{i} is the diagonal term corresponding to each electronic state and contains the harmonic oscillator terms and the derivative coupling,

where $\omega ji$ is the harmonic frequency of the *j*th 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 *i*th 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 $A\u03031$, $A\u03032$, and $B\u0303$ states is given by $VpJTii\u2032$,

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 CaOC_{2}H_{5}, 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, $VyA\u03031A\u03032$, $VyA\u03031B\u0303$, $VxA\u03032B\u0303$, and $VzA\u03032B\u0303$ vanish.

The CFOUR quantum chemistry package was used to calculate the values for $\omega ji$, $dji$, $VpJTii\u2032$, and $V\alpha ii\u2032$. The $\omega ji$ and $dji$ values were calculated using finite differences of adiabatic potential energy surfaces. $V\alpha ii\u2032$ 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 $A\u03031$, $A\u03032$, and $B\u0303$ states are summarized in Table IV.

. | . | . | Derivative coupling $dji$ . | pJT coupling $VpJTii\u2032$ . | ||||
---|---|---|---|---|---|---|---|---|

Mode number . | Mode symmetry . | Harmonic frequency . | $A\u03031$ . | $A\u03032$ . | $B\u0303$ . | $A\u03031\u2212A\u03032$ . | $A\u03031\u2212B\u0303$ . | $A\u03032\u2212B\u0303$ . |

ν_{1} | a′ | 3118.3 | −13.22 | −4.19 | 6.35 | 0 | 0 | 15.2 |

ν_{2} | a′ | 3064.3 | 5.79 | 9.56 | −1.97 | 0 | 0 | 6.1 |

ν_{3} | a′ | 3001.2 | −28.59 | −28.78 | −6.69 | 0 | 0 | 12 |

ν_{4} | a′ | 1555.8 | −37.96 | −29.42 | −26.06 | 0 | 0 | −3 |

ν_{5} | a′ | 1526 | 15.91 | −20.85 | −11.69 | 0 | 0 | −22.6 |

ν_{6} | a′ | 1443.7 | −49.68 | −59.14 | −24.66 | 0 | 0 | 7.3 |

ν_{7} | a′ | 1415 | −32.73 | −32.27 | −25.84 | 0 | 0 | −10.7 |

ν_{8} | a′ | 1243.6 | −162.86 | −163.02 | −125.73 | 0 | 0 | −7.2 |

ν_{9} | a′ | 1123.4 | −32.07 | −29.73 | −34.29 | 0 | 0 | −20.2 |

ν_{10} | a′ | 942.6 | 69.16 | 73.15 | 73.13 | 0 | 0 | −3.6 |

ν_{11} | a′ | 521.9 | 166.04 | 166.4 | 184.19 | 0 | 0 | 34.8 |

ν_{12} | a′ | 393.2 | 140.16 | 137.87 | 136.8 | 0 | 0 | −71.7 |

ν_{13} | a′ | 97.4 | −12.87 | −8.15 | −3.25 | 0 | 0 | −228.6 |

ν_{14} | a″ | 3123.9 | 0 | 0 | 0 | 4.2 | −16 | 0 |

ν_{15} | a″ | 3001.5 | 0 | 0 | 0 | −1 | −18.6 | 0 |

ν_{16} | a″ | 1515.6 | 0 | 0 | 0 | 9.4 | −16 | 0 |

ν_{17} | a″ | 1342 | 0 | 0 | 0 | −8.4 | −11.4 | 0 |

ν_{18} | a″ | 1207.8 | 0 | 0 | 0 | −5.9 | −5.4 | 0 |

ν_{19} | a″ | 817.2 | 0 | 0 | 0 | −3.8 | 0.2 | 0 |

ν_{20} | a″ | 307.5 | 0 | 0 | 0 | −6.5 | −53.4 | 0 |

ν_{21} | a″ | 136.4 | 0 | 0 | 0 | 2.3 | −229.6 | 0 |

. | . | . | Derivative coupling $dji$ . | pJT coupling $VpJTii\u2032$ . | ||||
---|---|---|---|---|---|---|---|---|

Mode number . | Mode symmetry . | Harmonic frequency . | $A\u03031$ . | $A\u03032$ . | $B\u0303$ . | $A\u03031\u2212A\u03032$ . | $A\u03031\u2212B\u0303$ . | $A\u03032\u2212B\u0303$ . |

ν_{1} | a′ | 3118.3 | −13.22 | −4.19 | 6.35 | 0 | 0 | 15.2 |

ν_{2} | a′ | 3064.3 | 5.79 | 9.56 | −1.97 | 0 | 0 | 6.1 |

ν_{3} | a′ | 3001.2 | −28.59 | −28.78 | −6.69 | 0 | 0 | 12 |

ν_{4} | a′ | 1555.8 | −37.96 | −29.42 | −26.06 | 0 | 0 | −3 |

ν_{5} | a′ | 1526 | 15.91 | −20.85 | −11.69 | 0 | 0 | −22.6 |

ν_{6} | a′ | 1443.7 | −49.68 | −59.14 | −24.66 | 0 | 0 | 7.3 |

ν_{7} | a′ | 1415 | −32.73 | −32.27 | −25.84 | 0 | 0 | −10.7 |

ν_{8} | a′ | 1243.6 | −162.86 | −163.02 | −125.73 | 0 | 0 | −7.2 |

ν_{9} | a′ | 1123.4 | −32.07 | −29.73 | −34.29 | 0 | 0 | −20.2 |

ν_{10} | a′ | 942.6 | 69.16 | 73.15 | 73.13 | 0 | 0 | −3.6 |

ν_{11} | a′ | 521.9 | 166.04 | 166.4 | 184.19 | 0 | 0 | 34.8 |

ν_{12} | a′ | 393.2 | 140.16 | 137.87 | 136.8 | 0 | 0 | −71.7 |

ν_{13} | a′ | 97.4 | −12.87 | −8.15 | −3.25 | 0 | 0 | −228.6 |

ν_{14} | a″ | 3123.9 | 0 | 0 | 0 | 4.2 | −16 | 0 |

ν_{15} | a″ | 3001.5 | 0 | 0 | 0 | −1 | −18.6 | 0 |

ν_{16} | a″ | 1515.6 | 0 | 0 | 0 | 9.4 | −16 | 0 |

ν_{17} | a″ | 1342 | 0 | 0 | 0 | −8.4 | −11.4 | 0 |

ν_{18} | a″ | 1207.8 | 0 | 0 | 0 | −5.9 | −5.4 | 0 |

ν_{19} | a″ | 817.2 | 0 | 0 | 0 | −3.8 | 0.2 | 0 |

ν_{20} | a″ | 307.5 | 0 | 0 | 0 | −6.5 | −53.4 | 0 |

ν_{21} | a″ | 136.4 | 0 | 0 | 0 | 2.3 | −229.6 | 0 |

i − i′
. | $Vxii\u2032$ . | $Vyii\u2032$ . | $Vzii\u2032$ . | V^{ii′}
. |
---|---|---|---|---|

$A1\u03032A\u2033\u2212A2\u03032A\u2032$ | 13.68 | 0.0 | 58.32 | 59.90 |

$A1\u03032A\u2033\u2212B\u03032A\u2032$ | 43.08 | 0.0 | 10.20 | 44.27 |

$A2\u03032A\u2032\u2212B\u03032A\u2032$ | 0.0 | 43.74 | 0.0 | 43.74 |

i − i′
. | $Vxii\u2032$ . | $Vyii\u2032$ . | $Vzii\u2032$ . | V^{ii′}
. |
---|---|---|---|---|

$A1\u03032A\u2033\u2212A2\u03032A\u2032$ | 13.68 | 0.0 | 58.32 | 59.90 |

$A1\u03032A\u2033\u2212B\u03032A\u2032$ | 43.08 | 0.0 | 10.20 | 44.27 |

$A2\u03032A\u2032\u2212B\u03032A\u2032$ | 0.0 | 43.74 | 0.0 | 43.74 |

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 *ν*_{8} − *ν*_{13} modes with *v*_{max}{*ν*_{8}, *ν*_{9}, *ν*_{10}, *ν*_{11}, *ν*_{12}, *ν*_{13}} = {10, 7, 9, 12, 12, 16}, and the *a*″ calculation includes *ν*_{16}−*ν*_{21} modes with *v*_{max}{*ν*_{16}, *ν*_{17}, *ν*_{18}, *ν*_{19}, *ν*_{20}, *ν*_{21}} = {8, 8, 10, 10, 12, 14}. The basis set for the *a*′ calculation contains ≈15 × 10^{6} basis functions, and the *a*″ calculation contains ≈11 × 10^{6} basis functions. Calculation of eigenfunctions up to 1000 cm^{−1} above the mean of the $A1\u03032A\u2033$ and $A2\u03032A\u2032$ 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.

Transition . | μ_{x}
. | μ_{y}
. | μ_{z}
. | μ
. |
---|---|---|---|---|

$X\u03032A\u2032\u2212A1\u03032A\u2033$ | 0.0 | 2.28375 | 0.0 | 2.28375 |

$X\u03032A\u2032\u2212A2\u03032A\u2032$ | 0.51931 | 0.0 | 2.21532 | 2.27537 |

$X\u03032A\u2032\u2212B\u03032A\u2032$ | 1.85519 | 0.0 | 0.41683 | 1.90144 |

Transition . | μ_{x}
. | μ_{y}
. | μ_{z}
. | μ
. |
---|---|---|---|---|

$X\u03032A\u2032\u2212A1\u03032A\u2033$ | 0.0 | 2.28375 | 0.0 | 2.28375 |

$X\u03032A\u2032\u2212A2\u03032A\u2032$ | 0.51931 | 0.0 | 2.21532 | 2.27537 |

$X\u03032A\u2032\u2212B\u03032A\u2032$ | 1.85519 | 0.0 | 0.41683 | 1.90144 |

## IV. SPECTRAL SIMULATIONS

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 CaOC_{2}H_{5} as a potential candidate for laser-cooling.

### A. LIF spectrum of the $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2190X\u03032A\u2032$ transition

Panels (a) and (a′) of Fig. 1 show the experimentally obtained LIF spectrum of the $A\u0303\u2190X\u0303$ transition of CaOC_{2}H_{5}. 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 $A1\u03032A\u2032\u2032\u2190X\u03032A\u2032$ and $A2\u03032A\u2032\u2190X\u03032A\u2032$ transitions, respectively. The doublets are present in the EOM-CCSD electronic structure calculations, although, as we will see later, the $A2\u03032A\u2032\u2212A1\u03032A\u2033$ 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 $A1\u03032A\u2032\u2032\u2190X\u03032A\u2032$ and $A2\u03032A\u2032\u2190X\u03032A\u2032$ 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 CaOCH_{3},^{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 $A\u0303$-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 $A\u0303$-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 $A\u03031\u2212A\u03032$ 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 $A\u03031$, $A\u03032$, and $B\u0303$ 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 $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2190X\u03032A\u2032$ 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 *ν*_{12} (CaO stretch) and *ν*_{13} (in-plane CaOC bending) modes, which are neglected in the present work (see above), cause the 12^{1}13^{1} combination levels of the $A\u0303$ 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 *ν*_{12} (389 cm^{−1}) and *ν*_{21} (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 $A\u0303$-state CO-stretch (*ν*_{8}) 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 $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2190X\u03032A\u2032$ transition is truncated as discussed in Sec. III B. The harmonic frequency of the *ν*_{8} 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 $B\u03032A\u2032$ state lowers the *ν*_{8} levels as it does to other $A\u0303$-state vibrational levels observed in the LIF spectrum (e.g., *ν*_{11}, *ν*_{12}, *ν*_{13}) (see Fig. 1). Therefore, inclusion of the spin–vibronic interactions would move the predicted *ν*_{8} vibrational frequency closer to the experimentally observed value.

### B. CRD spectrum of the $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2190X\u03032A\u2032$ transition and experimental determination of the FC factors

The origin transitions are power-saturated in the LIF measurements with a pulsed laser. To determine the VBRs of the $A1\u0303/A2\u0303\u2212X\u0303$ 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 $A\u0303\u2190X\u0303$ transitions so determined are listed in Table VI. The relative intensities of the $A1\u03032A\u2032\u2032\u2190X\u03032A\u2032$ and $A2\u03032A\u2032\u2190X\u03032A\u2032$ 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.

. | . | Expt. . | Calc. . | ||||
---|---|---|---|---|---|---|---|

$A\u03031/A\u03032$-state vibrational level . | Relative energy (cm^{−1})
. | $A\u03031\u2190X\u0303$ . | $A\u03032\u2190X\u0303$ . | Total . | Calc. 1^{a}
. | Calc. 2^{b}
. | Calc. 3^{c}
. |

0^{0} | 0 | 0.460 ± 0.030 | 0.382 ± 0.029 | 0.841 ± 0.042 | 0.906 | 0.89 | 0.884 |

13^{1} | 93 | 0.018 ± 0.001 | 0.013 ± 0.001 | 0.031 ± 0.002 | 0.003 | 0.011 | |

21^{1} | 130 | 0.025 ± 0.002 | 0.019 ± 0.001 | 0.045 ± 0.002 | |||

12^{1} | 389 | 0.018 ± 0.001 | 0.025 ± 0.002 | 0.044 ± 0.002 | 0.034 | 0.043 | 0.055 |

11^{1} | 532 | 0.018 ± 0.001 | 0.014 ± 0.001 | 0.032 ± 0.002 | 0.038 | 0.036 | 0.045 |

8^{1} | 1170 | 0.003 ± <0.001 | 0.005 ± <0.001 | 0.008 ± <0.001 | 0.011 | 0.006 | 0.006^{d} |

. | . | Expt. . | Calc. . | ||||
---|---|---|---|---|---|---|---|

$A\u03031/A\u03032$-state vibrational level . | Relative energy (cm^{−1})
. | $A\u03031\u2190X\u0303$ . | $A\u03032\u2190X\u0303$ . | Total . | Calc. 1^{a}
. | Calc. 2^{b}
. | Calc. 3^{c}
. |

0^{0} | 0 | 0.460 ± 0.030 | 0.382 ± 0.029 | 0.841 ± 0.042 | 0.906 | 0.89 | 0.884 |

13^{1} | 93 | 0.018 ± 0.001 | 0.013 ± 0.001 | 0.031 ± 0.002 | 0.003 | 0.011 | |

21^{1} | 130 | 0.025 ± 0.002 | 0.019 ± 0.001 | 0.045 ± 0.002 | |||

12^{1} | 389 | 0.018 ± 0.001 | 0.025 ± 0.002 | 0.044 ± 0.002 | 0.034 | 0.043 | 0.055 |

11^{1} | 532 | 0.018 ± 0.001 | 0.014 ± 0.001 | 0.032 ± 0.002 | 0.038 | 0.036 | 0.045 |

8^{1} | 1170 | 0.003 ± <0.001 | 0.005 ± <0.001 | 0.008 ± <0.001 | 0.011 | 0.006 | 0.006^{d} |

^{a}

At the CASSCF(3,6)/cc-PVTZ level of theory.

^{b}

At the (EOM)-CCSD/cc-PVTZ level of theory.

^{c}

Complete spin–vibronic calculations. The transition intensity for $A\u03031$ and $A\u03032$ components of each vibrational transition is given in Table S.1 of the supplementary material.

^{d}

Fixed to the value in the EOM-CCSD calculations.

### C. $A\u0303\u2192X\u0303$ DF spectra obtained by pumping the origin transitions and VBRs

Figures 5(a) and 5(c) illustrate the DF spectra obtained by pumping the $A1\u03032A\u2032\u2032\u2190X\u03032A\u2032$ and $A2\u03032A\u2032\u2190X\u03032A\u2032$ origin transitions, respectively. DF spectra obtained under slightly different experimental conditions were reported in a previous publication of ours.^{15} In that paper, vibrational assignments were made by comparing the experimental transition frequencies and intensities to those calculated at the CASSCF(3,6)/cc-PVTZ level of theory [see Fig. 5(e)]. The vibrational assignment has been confirmed by the CCSD calculations in the present work [see Figs. 5(b) and 5(d)]. Frequencies and intensities of $A\u0303(v\u2032=0)\u2192X\u0303$ vibronic transitions calculated on the EOM-CCSD/cc-PVTZ level of theory are listed in Table S.2.

In the present work, we mainly report the FC factors and VBRs of the DF spectra. It is worth noting that VBRs for the emission spectroscopy are determined by not only FC factors but also the transition frequencies (*ν*) because the Einstein *A* coefficient is proportional to *ν*^{3}. The experimental determination of the FC factors or VBRs for the *origin* transitions from the DF spectra is difficult due to the contamination of the scattering of the excitation laser. The origin-transition FC factors are, therefore, fixed to the values determined in the LIF and CRD experiments (see Table VI). FC factors of the other transitions are determined by the maintenance of their experimentally determined intensity ratios scaled by *ν*^{3} while keeping the total FC factor normalized. VBRs are calculated by scaling the FC factors of all observed DF peaks by *ν*^{3} followed by re-normalization. The VBRs for the emission so determined are listed in Table VII.

$X\u03032A\u2032$ . | . | Expt. . | Calc. 1^{a}
. | Calc. 2^{b}
. | Calc. 3^{c}
. | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

-state . | Relative . | $A\u030312A\u2032\u2032\u2192X\u03032A\u2032$ . | $A\u030322A\u2032\u2192X\u03032A\u2032$ . | Averaged . | . | . | . | . | $A\u030312A\u2032\u2032\u2192X\u03032A\u2032$ . | $A\u030322A\u2032\u2192X\u03032A\u2032$ . | Averaged . | |||

vibrational . | energy . | . | . | . | FC . | . | FC . | . | FC . | . | FC . | . | FC . | . |

level . | (cm^{−1})
. | VBR . | VBR . | VBR . | Factor . | VBR . | Factor . | VBR . | Factor . | VBR . | Factor . | VBR . | Factor . | VBR . |

0_{0} | 0 | 0.858 ± 0.013 | 0.845 ± 0.014 | 0.855 ± 0.009 | 0.918 | 0.928 | 0.891 | 0.901 | 0.877 | 0.888 | 0.882 | 0.894 | 0.879 | 0.890 |

13_{1} | 93 | 0.007 ± 0.001 | 0.007 ± 0.001 | 0.003 ± <0.001 | 0.003 | 0.003 | 0.014 | 0.014 | 0.007 | 0.007 | 0.011 | 0.011 | ||

12_{1} | 386 | 0.059 ± 0.006 | 0.062 ± 0.006 | 0.061 ± 0.001 | 0.034 | 0.032 | 0.081 | 0.076 | 0.055 | 0.052 | 0.055 | 0.052 | 0.055 | 0.052 |

11_{1} | 514 | 0.058 ± 0.006 | 0.067 ± 0.006 | 0.062 ± 0.001 | 0.038 | 0.035 | 0.019 | 0.017 | 0.044 | 0.040 | 0.045 | 0.041 | 0.044 | 0.040 |

10_{1} | 916 | 0.007 ± 0.001 | 0.007 ± 0.001 | 0.007 ± <0.001 | 0.002 | 0.002 | 0.001 | 0.001 | ||||||

8_{1} | 1170 | 0.012 ± 0.001 | 0.013 ± 0.001 | 0.013 ± 0.001 | 0.001 | 0.001 | 0.006 | 0.005 | 0.008 | 0.006 | 0.008 | 0.006 | 0.008 | 0.006 |

$X\u03032A\u2032$ . | . | Expt. . | Calc. 1^{a}
. | Calc. 2^{b}
. | Calc. 3^{c}
. | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

-state . | Relative . | $A\u030312A\u2032\u2032\u2192X\u03032A\u2032$ . | $A\u030322A\u2032\u2192X\u03032A\u2032$ . | Averaged . | . | . | . | . | $A\u030312A\u2032\u2032\u2192X\u03032A\u2032$ . | $A\u030322A\u2032\u2192X\u03032A\u2032$ . | Averaged . | |||

vibrational . | energy . | . | . | . | FC . | . | FC . | . | FC . | . | FC . | . | FC . | . |

level . | (cm^{−1})
. | VBR . | VBR . | VBR . | Factor . | VBR . | Factor . | VBR . | Factor . | VBR . | Factor . | VBR . | Factor . | VBR . |

0_{0} | 0 | 0.858 ± 0.013 | 0.845 ± 0.014 | 0.855 ± 0.009 | 0.918 | 0.928 | 0.891 | 0.901 | 0.877 | 0.888 | 0.882 | 0.894 | 0.879 | 0.890 |

13_{1} | 93 | 0.007 ± 0.001 | 0.007 ± 0.001 | 0.003 ± <0.001 | 0.003 | 0.003 | 0.014 | 0.014 | 0.007 | 0.007 | 0.011 | 0.011 | ||

12_{1} | 386 | 0.059 ± 0.006 | 0.062 ± 0.006 | 0.061 ± 0.001 | 0.034 | 0.032 | 0.081 | 0.076 | 0.055 | 0.052 | 0.055 | 0.052 | 0.055 | 0.052 |

11_{1} | 514 | 0.058 ± 0.006 | 0.067 ± 0.006 | 0.062 ± 0.001 | 0.038 | 0.035 | 0.019 | 0.017 | 0.044 | 0.040 | 0.045 | 0.041 | 0.044 | 0.040 |

10_{1} | 916 | 0.007 ± 0.001 | 0.007 ± 0.001 | 0.007 ± <0.001 | 0.002 | 0.002 | 0.001 | 0.001 | ||||||

8_{1} | 1170 | 0.012 ± 0.001 | 0.013 ± 0.001 | 0.013 ± 0.001 | 0.001 | 0.001 | 0.006 | 0.005 | 0.008 | 0.006 | 0.008 | 0.006 | 0.008 | 0.006 |

^{a}

At the CASSCF(3,6)/cc-PVTZ level of theory.

^{b}

At the (EOM)-CCSD/cc-PVTZ level of theory.

^{c}

Complete spin–vibronic calculations.

### D. $A\u0303\u2192X\u0303$ DF spectra obtained by pumping other vibronic bands

DF spectra obtained by pumping strong non-origin vibronic bands in the LIF spectrum, namely, $1301$, $2101$, $1201$, $1101$, and $801$, are represented in Figs. 6(b)–6(f) and compared to those obtained by pumping the origin band [Fig. 6(a)]. As a general rule (but with exceptions), DF transitions from a vibronic level of the $A\u0303$ state to those $X\u0303$-state vibrational levels of the pumped mode gain intensity because of favorable FC factors. Additionally, because of the Duschinsky mixing,^{12} transitions to combination levels of the pumped mode and other modes may also gain intensities. Following these principles and guided by the *ab initio* calculated vibrational frequencies and transition intensities, the assignment of most peaks observed in the DF spectra is straightforward (see Fig. 6). Individual DF spectra are discussed below.

When the $1301$ transition in the LIF spectrum is pumped, the DF spectrum [Fig. 6(b)] is dominated by transitions to the *ν*_{13} fundamental level of the $X\u03032A\u2032$ state and combination levels of the *ν*_{13} mode with *ν*_{12}, *ν*_{11}, and *ν*_{8}. These $A\u0303\u2192X\u0303$ transitions gain intensities through the Duschinsky mixing between *ν*_{13} and the other three FC-favored modes.

Similarly, when the $2101$ transition in the LIF spectrum is pumped, transitions to the fundamental level of *ν*_{21} and combination levels of 12_{1}21_{1}, 11_{1}21_{1}, and 8_{1}21_{1} of the $X\u03032A\u2032$ state are observed in the DF spectrum, although the transitions to the combination levels are quite weak [see Fig. 6(c)]. Surprisingly, a strong transition to the 11_{1}13_{1} level is observed in the DF spectrum obtained by pumping the $2101$ transition. The *ν*_{13} and *ν*_{21} modes are the in-plane and out-of-plane CaOC bending modes, respectively. Both modes are pJT-active. The 21^{1} and 11_{1}13_{1} levels have the *a*″ and *a*′ vibrational symmetries, respectively. Therefore, the transition between these two levels ($11101310$ and $2101$) is FC-forbidden under the BO approximation but allowed with the pJT interaction that mixes the 21^{1} level of the $A\u0303$ state with its (totally symmetric) vibrational ground level.

When the $1201$ transitions in the LIF spectrum are pumped, the only strong transitions in the DF spectra are those to the 12_{1} levels of the $X\u03032A\u2032$ state [see Fig. 6(d)].

The DF spectra obtained by pumping the $1101$ (OCC bending) transitions [Fig. 6(e)] deserve special attention. One would expect strong transitions to the 11_{1} level of the $X\u03032A\u2032$ state in the DF spectra. However, the strong peaks labeled “X_{1}” and “X_{2}” in Fig. 6(e) do not match the energies of the 11_{1} level of the $X\u03032A\u2032$ state but are redshifted from the 11_{1} level by ∼12 cm^{−1}. We recall that the 11^{1} levels of the $A\u0303$ state are only ∼15 cm^{−1} above its 12^{1}21^{1} combination levels as determined in the LIF spectrum (see Sec. IV A). Therefore, we attribute the aforementioned redshift of peaks “X_{1}” and “X_{2}” from the 11_{1} level to collision-induced relaxation from the $A\u0303$-state 11^{1} level to its 12^{1}21^{1} level following the LIF excitation. After this non-radiative decay, the excited-state molecules decay to the nearly degenerate 11_{1} (514 cm^{−1}) and 12_{1}21_{1} (386 + 129 cm^{−1}) levels of the $X\u03032A\u2032$ state. The overall redshift for the DF transition compared to the pump frequency is, therefore, the energy difference between the 11^{1} and 12^{1}21^{1} levels of the $A\u0303$ state (15 cm^{−1}) combined with the vibrational energy of the 11_{1} and 12_{1}21_{1} level of the $X\u03032A\u2032$ state, hence the extra redshift with respect to the 11_{1} level.

Two other strong bands, labeled “Y_{1}” and “Y_{2}” in Fig. 6(e), are observed in the DF spectra obtained by pumping the $1101$ band in the LIF spectrum. The “Y_{1}” band appears at 1160 cm^{−1} redshift when the transition to the upper SO component (*ν*_{00} + 568 cm^{−1}) is pumped, while “Y_{2}” appears at 1130 cm^{−1} redshift when the transition to the lower SO component (*ν*_{00} + 502 cm^{−1}) is pumped. These two transitions lie significantly higher than the expected transitions to the 11_{2} level (fundamental frequency = 514 cm^{−1}) and are not lined up with strong transition around 1170 cm^{−1}, assigned to the 8_{1} transitions (see Fig. 6). Due to the near degeneracy between the 11_{2} and 12_{2}21_{2} levels, one expects strong Fermi-resonance between these vibrational levels. However, the vibrational assignment of these two peaks is not possible without high-resolution spectra or further spin–vibronic calculations.

Finally, when the $801$ transitions in the LIF spectrum are pumped, the only transitions observed in the DF spectra are those to the 8_{1} levels of the $X\u03032A\u2032$ state [see Fig. 6(f)].

The vibrational structure of the recorded DF spectra can be better illustrated by blueshifting them by the ground-state frequencies of the pumped modes. The resulting spectra are displayed in Fig. 7. The shifted spectra show that the CaO stretch mode (*ν*_{12}), the OCC bending mode (*ν*_{11}), and the CaO stretch mode (*ν*_{8}) are mixed with both pJT-active modes, the in-plane CaOC bending (*ν*_{13}) and the out-of-plane CaOC bending (*ν*_{21}), although mixing with *ν*_{21}(*a*″) is significantly weaker.

## V. DISCUSSION

### A. The $A2\u03032A\u2032$–$A1\u03032A\u2033$ separation

The present work on the calcium ethoxide radical provides the first comprehensive experimental and computational study of the spin–vibronic structure of an alkaline earth monoalkoxide radical of *C*_{s} symmetry. It is a natural extension of our previous spectroscopic investigation of the calcium methoxide radical.^{9} The most significant difference between these two radicals is that, in CaOCH_{3}, the $A\u0303$ state is split by only the SO interaction and retains its ^{2}E symmetry, while in CaOC_{2}H_{5}, it splits into two nearly degenerate states with different symmetries, $A1\u03032A\u2033$ and $A2\u03032A\u2032$, and the separation between these two states ($\Delta EA\u03032\u2212A\u03031$) is due to both the SO interaction and the non-relativistic effects (Δ*E*_{0}). The non-relativistic effects separating these two electronic states include the “difference potential”^{27} (*ε*_{2}) and the difference between their zero-point energies (ΔZPE).^{28–30} The overall separation of the $A1\u03032A\u2033$ and $A2\u03032A\u2032$ states can be calculated as^{26}

The spin–vibronic calculations of the $A\u03031$ and $A\u03032$ states based on the CCSD calculations predict the splitting between the vibrationless levels of these two states to be 62 cm^{−1}, which is rather close to the observed value. The calculated value, 62 cm^{−1}, is the root-sum-square of the non-relativistic separation between the $A\u03031$ and $A\u03032$ states (17 cm^{−1}) and the SO interaction (60 cm^{−1}) [see Eq. (8)]. The present work, therefore, exemplifies the importance of the SO interaction in the study of low-symmetry alkaline earth monoalkoxides and similar free radicals subject to both the SO and pJT interactions.

The SO splitting and the non-relativistic splitting cannot be determined independently in the current spin–vibronic analysis of the experimental LIF spectrum because both effects separate the $A1\u03032A\u2033$ and $A2\u03032A\u2032$ states. However, simulation and fitting of future rotationally resolved $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2190X\u03032A\u2032$ LIF spectrum using continuous-wave (cw) lasers or laser amplifiers can be used to differentiate and quantize these two mechanisms because they affect the rotational and fine structure differently.^{26} This has been demonstrated in our previous work on alkoxy radicals with *C*_{s} symmetry,^{24,25,31,32} which have nearly degenerate $X\u0303$ and $A\u0303$ states with *A*′ and *A*″ symmetries that are coupled by both SO and pJT effects. Analysis of the rotational and fine structure of experimentally observed $B\u0303\u2190A\u0303/X\u0303$ vibronic transitions unraveled the interplay between the pJT interaction and the SO interaction. Furthermore, the magnitude of the mixing between the *A*′ and *A*″ states was determined in fitting the $B\u0303\u2190A\u0303$ and $B\u0303\u2190X\u0303$ spectra simultaneously. Recently, the spectroscopic model for analyzing nearly degenerate electronic states of open-shell molecules has been extended to the case of *C*_{1} molecules^{26} and implemented in the analysis of large straight-chain alkoxy radicals.^{33–35}

### B. Comparison between experiment and calculations

In the present work, vibrational assignments were made by comparing experimental spectra to the SA-CASSCF and (EOM-)CCSD calculated harmonic vibrational frequencies and FC factors. In general, the EOM-CCSD calculations can target each of the two excited states ($A1\u03032A\u2033$ and $A2\u03032A\u2032$) and provide significantly more accurate frequency and intensity values compared to the experiment than the SA-CASSCF calculations (see Table II). However, there is still a considerable discrepancy between the calculated $A\u0303$-state vibrational frequencies and experimentally determined ones [see Figs. 1(a) and 1(c)].

Introducing the pJT couplings between the $A1\u03032A\u2033$and $A2\u03032A\u2032$ states improved the calculated vibronic structure significantly so that the predicted $A\u0303\u2190X\u0303$ transition frequencies match the experimental ones very well. Furthermore, as described in Sec. V A, the inclusion of the $A\u03032$-$A\u03031$ SO interaction reproduces the observed splitting of the two states quite well (62 cm^{−1} vs 67 cm^{−1} in experiment). Including the pJT and SO couplings with the $B\u03032A\u2032$ state slightly improves the quality of the vibronic simulation in terms of SO-free transition frequencies and intensities. In addition, the inclusion of coupling with the $B\u0303$ state in the calculation quenches the SO splitting of the zero-point levels by ≈5%, to 60 cm^{−1}. The simulated LIF spectrum is compared to the experimental spectrum in Fig. 1(d) vs Fig. 1(a)′. Tables S.1 and S.2 list the frequencies and intensities of the $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2190X\u03032A\u2032$ vibronic transitions calculated excluding and including the pJT and SO couplings with the $B\u0303$ state, respectively. There are quite a few lines in the 50–250 cm^{−1} region that, while remaining weak, do gain some intensity due to the coupling with the $B\u0303$ state as can be seen by comparing Tables S.1 and S.2. The enhancement of transition intensities of “dark states” in this low-frequency region is mainly attributed to strong $B\u03032A\u2032$-$A2\u03032A\u2032$ pJT interaction via the *ν*_{13} mode, which has a calculated $A\u0303$-state harmonic frequency of 90 cm^{−1} (see Table III).

Transitions to levels of the *ν*_{21} (out-of-plan CaOC bending) mode of CaOC_{2}H_{5} is FC-forbidden under the BO approximation because of its *a*″ symmetry. Such transitions gain intensities from the pJT coupling as well as the SO interaction. Previously, transitions to the levels of the out-of-plane mode (*ν*_{8}) of CaOCH_{3} were also observed in its LIF/DF spectra. Similarly, FC-forbidden vibronic transitions have also been observed in the $B\u0303\u2190A\u0303/X\u0303$ transitions of alkoxy radicals with *C*_{s} symmetry, including isopropoxy^{32} and cyclohexoxy.^{25,31} As demonstrated therein, analysis and simulation of the rotational and fine structure of experimentally obtained spectra enable the determination of the pJT and SO coupling strengths.

### C. Implications to direct laser cooling of CaOC_{2}H_{5}

The excitation scheme of laser-cooling molecules is mainly determined by FC factors and VBRs. A highly diagonal FC matrix for vibronic transitions between the involved electronic states is preferred to avoid population leakage and achieve the number of photon scatterings necessary for laser cooling.^{36} Previously, FC factors for the $A\u0303$-$X\u0303$ vibronic transitions of CaOC_{2}H_{5} were calculated using the density functional theory method on the B3LYP/def2-TZVPP level of theory.^{6} FC factors of the origin transition and the transition from the vibrational ground level of the $A\u0303$ state to the *v*″ = 1 level of the CaO stretch mode ($1201$) were found to be 0.892 and 0.102, respectively. These values are consistent with the calculated results without vibronic interactions reported in the present work (See Table VII). As demonstrated in the present calculations, spin–vibronic interactions induce off-diagonal FC matrix elements with considerable magnitudes, which have to be taken into account in future direct laser cooling experiments.

In the future laser-cooling experiment, pre-cooled CaOC_{2}H_{5} molecules are excited from the $X\u0303$ state to the $A\u03031$ state. Following the excitation of the origin transition, spontaneous emission to excited vibrational levels of the ground electronic state occurs due to off-diagonal transitions, causing loss of population from the cooling cycle. Therefore, a reasonable number of repump lasers are used to send the population on $X\u0303$-state vibrational levels back to the cooling cycle and limit the loss of population for laser cooling. Based on the spectroscopic investigation of the present work, in addition to the spontaneous emission to the vibrational ground level of the $X\u0303$ state, VBRs for transitions to both the 11_{1} (OCC bending) and 12_{1} (CaO stretch) levels are also significant. Two repumping lasers are, therefore, needed to return the population from these two ground-state vibrational levels to the cooling cycle. The combined VBR of the spontaneous decay to the 0_{0}, 11_{1}, and 12_{1} levels of the $X\u03032A\u2032$ state is 0.975. As a result, a molecule will experience on average ≈40 scattering events before it decays to vibrational “dark” states that are not addressed by the repumping lasers. If a third repumping laser is added to return the population on the 8_{1} level of the $X\u03032A\u2032$ state to the cooling cycle, the combined VBR will be increased to 0.987 so that a molecule will experience on average 77 scattering events before decaying to vibrational dark states.

## VI. CONCLUSIONS

We report vibrationally resolved LIF, DF, and CRD spectra of the $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2212X\u03032A\u2032$ transition of the calcium ethoxide radical. The vibrational assignment has been made based on vibrational frequencies and FC factors calculated using the CASSCF and CC methods. The calculations predict the vibrational frequencies to a significant degree of accuracy and the intensities of the allowed transitions quasi-quantitatively. However, the *ab initio* calculations do not reproduce weak transitions in the LIF spectrum induced by the pJT effect, including transitions to the *ν*_{21} (out-of-plane CaOC bending with the *a*″ symmetry) level and combination levels of this pJT-active mode. In the DF spectra obtained by pumping the origin band, strong transitions to the *ν*_{12} (CaO stretch) and *ν*_{11} (OCC bending) levels of the $X\u03032A\u2032$ state were observed, with weaker transitions to a few other vibrational levels. When vibronic bands other than the origin are pumped, the obtained DF spectra contain off-diagonal transitions attributed to the Duschinsky rotation. In the case of the DF spectra obtained by the $1101$ transitions, a non-radiative decay to the nearby 12_{1}21_{1} combination level precedes the $A\u0303\u2192X\u0303$ radiative decay.

The current spin–vibronic calculation does not include quadratic coupling terms and is truncated at 1000 cm^{−1} due to convergence issues. Development of quantum chemistry codes that include both the quadratic coupling terms and SO interaction is expected to help further improve the accuracy of spin–vibronic calculations on CaOC_{2}H_{5} and other open-shell molecules with *C*_{s} symmetry. High-resolution spectra with resolved rotational and fine structure are desired for experimental determination of the SO and non-relativistic interactions that separate the $A1\u03032A\u2033$ and $A2\u03032A\u2032$ states. They will also help confirm certain tentative vibrational assignments in the present work.

Relative intensities of $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2190X\u03032A\u2032$ and $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2192X\u03032A\u2032$ vibronic transitions were determined in LIF and DF spectra, respectively. However, the excitation transitions of the origin band are power saturated in the current LIF measurement using a pulsed dye laser, while the fluorescence signal in the DF measurement of the origin transitions is contaminated by the laser scattering. Therefore, CRD measurements were performed to determine the saturation factor. Future LIF/DF and CRD spectroscopy measurements with cw lasers can determine the FC factors and VBRs with better accuracy. The S/N ratio of the current experiment is mainly limited by the shot-to-shot fluctuation and long-term variation of the concentration of free radicals produced by laser ablation. In future experiments, the S/N ratio can be improved by monitoring the signals of the origin transition and the vibronic transitions simultaneously, followed by intensity normalization.

## SUPPLEMENTARY MATERIAL

See the supplementary material for frequencies and intensities of the $A1\u03032A\u2032\u2032/A2\u03032A\u2032\u2190X\u03032A\u2032$ vibronic transitions calculated excluding and including the pJT and SO couplings with the $B\u0303$ state.

## ACKNOWLEDGMENTS

This work was supported by the National Science Foundation under Grant Nos. CHE-1454825 and CHE-1955310. K.S. gratefully acknowledges a Terry A. Miller Postdoctoral Fellowship from the Ohio State University. T.A.M. acknowledges support from the Ohio Supercomputer via Project No. PAS0540.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.