Following the observation of molecular absorption spectra emitted by stars in the 1930s, the idea that the universe is primarily atomic with only a small amount of diatomic character has shifted to the realization that the universe is significantly molecular [1]. While hydrogen and helium comprise roughly 99% of the universe, the other notable atomic citizens of the universe include O, C, Ne, N [2], Mg [3], S [4], Si [5], and Fe [6], listed roughly in decreasing order of abundance [7]. Most notably, organic molecules such as benzonitrile [8] and C 2 H 2 [9] have been detected with a few hundred others also now known [10]. Fewer inorganic and organometallic species have been observed thus far, though, but this has been shifting as additional reference data have been produced for such species [11] leading to new observations [12].
One such atom in the aforementioned list is magnesium. The twelfth element on the periodic table has been hypothesized to be a necessary building block for planet formation, evidenced by higher concentration values in planet-hosting stars and a deficit in those not occurring in at least this region of the ISM. Hence, magnesium replacement in the form of c-MgC 2 H 2 as an initial exploration is possible opening new motifs for organometallic species.
Early theoretical work has explored the lowest singlet and triplet PES of [Mg, C 2 , H 2 ] and its isomers using HF/6-31G* and MP2/3-21G* levels of theory. Therein, the linear, singlet state is determined to be the ground state [21]. Later density functional theory (DFT) calculations support the experimental observations in which laser-ablation techniques employing Mg atoms and acetylene molecules imply that following relaxation or decomposition, metastable excited Mg atoms can be inserted into the C-H bond of acetylene creating magnesaprop-2-yne (HCCMgH) [22]. Beyond this work, a binary complex consisting of Mg bound to acetylene has been theoretically studied at the BCCD(T)/6-311++G(3df,3pd) and MP2/6-311++G(3df,3pd) levels of theory and found to be in good agreement with experimental observation of a T-shaped or cyclic geometry for c-MgC 2 H 2 [23] paving the way for potential observation of c-MgC 2 H 2 if the proper spectral data can be measured.
Three isomers of the [Mg, C 2 , H 2 ] complex including these above are reported using DFT at the B3LYP/6-311++G(2d,2p) and B3LYP-D3BJ/6-311++G(2d,2p) levels of theory along with CCSD(T)/aug-cc-pCVTZ [24]. Of these three low-lying isomers, magnesaprop-2-yne (HCCMgH), magnesapropadiene (MgCCH 2 ), and magnesacyclopropene (c-MgC 2 H 2 ), the first and third have been previously reported; the second had not been as fully examined. Of these the most thermodynamically stable is the magnesaprop-2-yne isomer (HCCMgH) confirming earlier work. The ground state of the other two isomers, magnesapropadiene and magnesacyclopropene, is shown to be 1 3 𝐵 2 as they both have C 2𝑣 symmetry. The triplet state of magnesapropadiene is more polar (1.290 D) than that of magnesacyclopropene (0.099 D). The higherenergy singlet electronic states of these C 2𝑣 molecules are more polar, 5.89 D and 5.25 D, respectively. Therefore, this theoretical previous study [24] asserts that the singlet HCCMgH isomer, while the lowest in energy, would be difficult to observe using microwave spectroscopy, but both triplet C 2𝑣 structures are highly probable foci for laboratory studies or astronomical observation [24].
The observation of c-MgC 2 H 2 will require highly-accurate rotational constants and/or fundamental anharmonic vibrational frequencies depending upon the wavelength of light most fitting for the astronomical source observed. The use of fourth-order Taylor series approximations to the internuclear molecular Hamiltonian, or quartic force fields (QFFs), with high levels of quantum chemical electronic structure computations has been benchmarked to provide rotational constants typically within 0.12% of experiment for molecules of this size and fundamental anharmonic vibrational frequencies within 0.7% (roughly less than 6.0 cm -1 ) [25][26][27][28][29][30][31][32][33][34][35][36][37]. Consequently, this study employs such quantum chemical approaches to produce the necessary rovibrational spectral data in order to characterize c-MgC 2 H 2 in both its ground, triplet and excited, singlet state in order to assist potentially with its detection in the ISM. Finally, the possible formation mechanism for this molecule from acetylene and Mg will also be explored.
Unless otherwise noted, all computations make use of coupled cluster theory at the singles, doubles, and perturbative triples level [CCSD(T)] [38] within the explicitly correlated F12b formalism [39,40] and the cc-pVTZ-F12 basis set [41], henceforth abbreviated as F12-TZ, as available in the MOLPRO 2019.1 quantum chemistry program [42,43]. All computations for 1 3 𝐵 2 c-MgC 2 H 2 utilize restricted open-shell reference wave functions [44,45], while closed-shell molecules employ standard restricted Refs. [46].
All fundamental vibrational frequencies are generated through the use of QFFs in the fashion standard for these types of applications [25,26,47,48]. F12-TZ QFFs, specifically, are known to produce highly accurate vibrational frequencies at a relatively low computational cost within 5 cm -1 or so of higher-order methods and even experimental results [49][50][51]. Regardless of electronic state, the geometries of both 1 1 𝐴 1 and 1 3 𝐵 2 c-MgC 2 H 2 are first optimized with exceptionally tight convergence criteria. The reference geometry is displaced by 0.005 Å or 0.005 radians, respective of bond lengths and bond angles/dihedrals, using symmetry-internal coordinates via the INTDER program [52]. This molecule is of the same connectivity allowing for use of coordinates employed previously for similar cyclic species: cyclopropenylidene and its silanated analogue [36,53,54]. The coordinates are defined from Fig. 1 where the left-side of each structure should be understood to contain atoms C 1 and H 1 and vice versa:
Once the 1585 points for either electronic state are computed, their relative energies are fit via a least squares method to better than 10 -17 a.u. 2 The generated force constants are then transformed from symmetry-internal coordinates into Cartesian coordinates using INTDER [52] as this is more computationally expedient for use in subsequent steps. The SPECTRO [55] program then computes the spectroscopic constants and vibrational frequencies, which are produced by rotational and vibrational perturbation theory at second order (VPT2) [56][57][58]. The dipole moments for both 1 1 𝐴 1 and 1 3 𝐵 2 c-MgC 2 H 2 are computed via F12-TZ using version 2020.1 of the MOLPRO quantum chemistry program [59] using finite differences of energies in response to an external electric field. Harmonic intensities for both electronic excited states are computed using Gaussian16 via the MP2/6-31+G(d) level of theory [60][61][62] which has been shown to produce semi-quantitative agreement with higher levels of theory for far less computational cost [63,64].
The rovibrational spectra of both 1 1 𝐴 1 and 1 3 𝐵 2 c-MgC 2 H 2 contain Fermi resonances and polyads of these resonances that the SPECTRO program is capable of treating for more accurate predictions [65]. The In order to examine how this molecule may form in the ISM such that it can be observed in the first place, relaxed scans of both the lowest singlet and triplet electronic states of Mg + C 2 H 2 are The association of Mg atoms to acetylene modeled by varying the distance between the Mg atom to the midpoint of the C-C bond in acetylene denoted as point X. Mg-X distances are varied from 1.65 Bohr to 5.55 Bohr with step sizes of 0.1 Bohr, and the ∠Mg-X-C angles are fixed at 90 • . The remaining geometrical parameters are allowed to optimize at each Mg-X bond length allowing for a relaxed scan.
The 1 3 𝐵 2 state of magnesacyclopropene lies 10.28 kcal/mol (0.446 eV or 3596 cm -1 ) below the 1 1 𝐴 1 state including the anharmonic zeropoint vibrational energy correction in line with previous prediction of 10.41 kcal/mol [24]. The orbital occupation for the ground
The singlet doubly occupies the 4𝑏 2 orbital in its wavefunction. The two highest occupied molecular orbitals (MOs) in the singlet state represent the two 𝜋 orbitals of acetylene. The additional, highest singly-occupied MO of the 1 3 𝐵 2 state is a majority non-bonding/lone-pair-type orbital on the Mg atom opposite the acetylene.
The optimized F12-TZ equilibrium singlet and triplet state Mg-C bond lengths are 1.98 Å and 2.20 Å, respectively, along with H-C bond lengths that are nearly equivalent at 1.09 Å and 1.08 Å, respectively. The C-Mg-C angle is shown to be 41.3 and 34.0 degrees, respectively. Interestingly, the largest structural difference between these two electronic states lies with the ∠H-C-Mg angle. The triplet state has a bond angle of 145.3 • , roughly 25 • less than that of the singlet state given in Table 1. Consequently, when the electron is excited, this angle begins to relax displaying a more acetylenic structure.
Additionally, the structural changes, especially the C=C bond length as shown in Fig. 1 and listed in Table 1, are attributable to different occupations of the electrons, as well. From Fig. 2, the highest occupied molecular orbital (4𝑏 2 ) of the 1 1 𝐴 1 state has repulsion between the two carbon atoms leading to a longer C = C bond at 1.40 Å than the 1.29 Å C = C bond in the 1 3 𝐵 2 . Once this orbital is only singlyoccupied in the triplet state with one electron moving into the 8𝑎 1 orbital (also in Fig. 2), the repulsion decreases allowing this bond to shorten. Furthermore, the C-Mg-C angle relaxes from 41.30 • to 34.00 • , the H-C-Mg angle decreases from 169.29 • to 145.31 • , and the Mg-C bond lengthens from 1.98 Å to 2.20 Å. The electron repulsion is also, again, likely the probable cause for the H-C-Mg angle being greater in the 1 1 𝐴 1 state.
Geometries and spectroscopic constants for 1 A 1 and 3 B 2 electronic states of c-MgC 2 H 2 compared to previous theory a . These structural parameters produce rotational constants that display some prolate character with the B and C constants of the singlet state being larger than their triplet counterparts. The currently computed equilibrium geometrical values and rotational constants are also in good agreement with previous work [24]. However, Table 1 also provides the sextic distortion constants, the vibrationally-averaged (𝑅 𝛼 ) rotational constants (including A 0 , B 0 , and C 0 constants), and the vibrationally-excited rotational constants for the 24 Mg isotope that are numbered in the same order as the fundamental vibrational frequencies given in Table 2. The comparison of the dipole moments for the two electronic states examined presently, found in Table 1, corroborate the previous disparity in the dipole moments with the 1 3 𝐵 2 state's mere 0.11 D falling shockingly short of the 1 1 𝐴 1 state's much larger 5.47 D. This disproportion can be attributed to the partial charge of Mg and the electron density in each state. In the 1 1 𝐴 1 state, the partial charge of Mg is 0.64, while 0.38 in the 1 3 𝐵 2 state. Finally, the 25 Mg, 26 Mg, and D isotopologue data are giving in the supplemental information (SI).
The 1 3 𝐵 2 electronic state is shown to be the ground electronic state in Fig. 3, and contains a high intensity vibrational mode of 118 km/mol (as shown in Table 2) for the 𝜔 8 mode with a corresponding 𝜈 8 frequency of 513.5 cm -1 . While this mode may be highly observable given the large intensity value, the 1 1 𝐴 1 electronic state contains an even higher intensity mode, the 𝜔 6 mode, at 276 km/mol with a corresponding 𝜈 6 frequency at 827.8 cm -1 . Additionally, the 1 1 𝐴 1 state has three fundamentals with intensities above 100 km/mol in line with similar Mg-bearing molecules computed recently [66] while the 1 3 𝐵 2 state only contains one mode with an intensity above 100 km/mol. The 1 1 𝐴 1 state is more tightly bound to its magnesium atom, as discussed in the next section and as evidenced by the nearly doubled 𝐹 22 force constant in the singlet state as given in Tables S1 andS2 in the SI. such, the intensity of the 1 1 𝐴 1 frequencies will be influenced by the bonding while the Mg atom of the 1 3 𝐵 2 state will have significantly less influence on the induced dipole moments of this state. For both electronic states, a larger number of fundamental frequencies fall into the sub-1000 cm -1 regime as compared to most 𝑝-block molecules due to the heavier mass of and weaker bonding to the magnesium atom.
Overall, the F12-TZ harmonic frequencies for the 1 1 𝐴 1 and 1 3 𝐵 2 state of c-MgC 2 H 2 compare well with previous theory with the best agreement being less than 0.5 cm -1 for the 1 3 𝐵 2 𝜔 7 mode [24] as shown in Table 2. The worst agreement between previous and present theory is approximately 31 cm -1 for the 1 1 𝐴 1 𝜔 6 mode. This, however, is less than 5% and is within the accepted margin of error. This analysis implies that the present study's F12-TZ QFF vibrational frequency data are accurate and reliable. However, the present study goes beyond merely the harmonic frequencies to, again, produce the anharmonic fundamental frequencies as given in Table 2.
The present work proposes two possible mechanisms for the formation of 1 3 𝐵 2 c-MgC 2 H 2 that conserves spin:
In Eq. ( 10), C 2 H 2 remains in its ground electronic state. The computed singlet-to-triplet excitation energy for Mg at the F12-TZ level is 59.84 kcal/mol (2.59 eV). Conversely, in Eq. ( 11), Mg would remain in the ground electronic state with the calculated excitation energy for C 2 H 2 lying 115.84 kcal/mol (5.02 eV), almost double that of Mg. This is not surprising given that the bond order in acetylene must be reduced during such an excitation. Hence, the excitation of the Mg atom is more likely to contribute to the formation of 1 3 𝐵 2 c-MgC 2 H 2 as is borne out in the triplet dissociation in Fig. 3. However, an intersystem crossing from the 1 3 𝐵 2 state could still form the 1 1 𝐴 1 state.
Both the singlet and triplet PES for the Mg-X relaxed scan of the C 2𝑣 isomer of c-MgC 2 H 2 are depicted in Fig. 3. Both clearly exhibit Morse potentials with the triplet state being flatter than the singlet. Previous work has been conducted for protonated acetylene, which shares comparable connectivity and ligand-acetylene core interactions, exhibiting a flat PES. The results of the QFF analysis for this molecule shows agreement between the computed and experimental values to better than 0.1% implying that the current methodology produces accurate descriptions of such flat surfaces [31]. The triplet dissociation energy is 41.03 kcal/mol while the singlet is higher at 94.89 kcal/mol (relative to its own minimum) in line with roughly half the Mg-C bond strength in the comparable HMgCH 3 molecule [67]. The optimal Mg-X bond length for the 1 3 𝐵 2 is determined to be 2.10 Å and 1.90 Å in the singlet state for this scan.
The flat PES for the triplet surface brings the vibrational levels closer together for what is effectively the 𝑆 2 coordinate and 𝜈 4 fundamental in the Mg-X stretch. As such the descent through the vibrational levels could be enhanced in environments where the collision probability is relatively high. Additionally, the overlap between the two electronic states allows for a low energy excited state. The vertical 1 1 𝐴 1 ← 1 3 𝐵 2 transition is computed on this scan to be 19.82 kcal/mol or 6932 cm -1 . Adjusting for the estimated ZPVE for 𝜈 4 of roughly 224 cm -1 reduces this vertical excitation down to 6708 cm -1 or 1491 nm. Conversely, formation of the 1 1 𝐴 1 state could readily relax to the triplet surface due to the high degree of overlap as shown in Fig. 3. In either case, these two electronic states would likely coevolve to some degree implying that even if the triplet surface is preferred in Mg association with acetylene, formation of the singlet would have a relatively low energy cost if somewhat less likely pathway to access.
Vibrational frequencies (cm -1 ), and IR intensities (km/mol), given in parenthesis, for 1 A 1 and 3 B 2 electronic states of c-MgC 2 H 2 compared to previous theory and experiment. a Contributes more than 90% to the dominant components of a coordinate unless specified otherwise. b Previous theory for 1 1 𝐴 1 utilizes ae-CCSD(T)/aug-cc-pCVTZ, and 1 3 𝐵 2 utilizes ae-ROCCSD(T)/aug-cc-pCVTZ from Ref. [24].
Radiative association (RA) appears to be the lowest energy mechanism for any gas phase formation of c-MgC 2 H 2 . Several attempts to compute reaction schema for known Mg-containing interstellar molecules with acetylene returned high barriers as well as thermodynamically and kinetically disfavorable results. Mostly, the Mg-C bond energy of c-MgC 2 H 2 is weaker than the Mg-C and Mg-N bonds of known interstellar molecules. While RA is notoriously slow in the gas phase, c-MgC 2 H 2 has underlying properties that may increase the likelihood of observation for the more intense rotational and vibrational transitions of the 1 1 𝐴 1 state.
The more intense vibrational transitions and the larger dipole moment imply that the 1 1 𝐴 1 state of c-MgC 2 H 2 is the more likely candidate than the 1 3 𝐵 2 state for potential observation by the James Webb Space Telescope in the infrared and also for radioastronomical spectroscopic observation from the ground. The 5.47 D dipole moment for the 1 1 𝐴 1 state of c-MgC 2 H 2 will be more observable than its lowerlying 1 3 𝐵 2 counterpart (0.11 D) in the submillimeter regime. While the two states are more similar in their infrared intensities, the singlet state has three fundamentals with intensities above 100 km/mol while the triplet has only one. Additionally, the formation of this molecule appears to require radiative association, but the relatively flat triplet PES likely facilitates faster association of the triplet state than most cases invoking RA and both states will likely coevolve due to the relative low energy separating them along the Mg-X coordinate. In any case, the spectral data provided in this work will no doubt assist with any potential laboratory observations of this molecule paving the way for possible later astronomical observation.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.