TY - JOUR
AU - Fortenberry, Ryan C.
AB - Abstract A more fine-tuned method for probing planet-forming regions, such as protoplanetary discs, could be rovibrational molecular spectroscopy observation of particular premineral molecules instead of more common but ultimately less related volatile organic compounds. Planets are created when grains aggregate, but how molecules form grains is an ongoing topic of discussion in astrophysics and planetary science. Using the spectroscopic data of molecules specifically involved in mineral formation could help to map regions where planet formation is believed to be occurring in order to examine the interplay between gas and dust. Four atoms are frequently associated with planetary formation: Fe, Si, Mg and O. Magnesium, in particular, has been shown to be in higher relative abundance in planet-hosting stars. Magnesium oxide crystals comprise the mineral periclase making it the chemically simplest magnesium-bearing mineral and a natural choice for analysis. The monomer, dimer and trimer forms of (MgO)n with n = 1–3 are analysed in this work using high-level quantum chemical computations known to produce accurate results. Strong vibrational transitions at 12.5, 15.0 and 16.5 μm are indicative of magnesium oxide monomer, dimer and trimer making these wavelengths of particular interest for the observation of protoplanetary discs and even potentially planet-forming regions around stars. If such transitions are observed in emission from the accretion discs or absorptions from stellar spectra, the beginning stages of mineral and, subsequently, rocky body formation could be indicated. astrochemistry, molecular data, planets and satellites: individual: protoplanetary discs, infrared: planetary systems, radio lines: planetary systems 1 INTRODUCTION The crossover in chemical classification from gaseous to solid matter is a difficult question to answer in terms of astrophysics and astrochemistry. The low pressures and various temperature extremes of diverse astrophysical environments often inhibit the creation of anything molecular beyond the gas phase. Yet, rocky bodies are composed of solid matter. How and when a cluster of monomer molecules can begin to be classified as a mineral solid, especially with regards to those that comprise bodies such as rocky planets, is still not well understood (Gail & Sedlmayr 1999; McWilliams et al. 2012). The main hang-up for this analysis is largely centred around what molecular species can be observed as indicators of changes in the region being observed and mapped. Most of the interstellar molecules detected thus far have been gaseous and have atomic compositions from the upper right-hand side of the periodic table (McCarthy & Thaddeus 2001; Fortenberry 2017) since carbon, nitrogen and oxygen are among the most abundant elements in the universe (Savage & Sembach 1996). While many of these have pertinence to planetary atmospheres, origins of life and have even been detected in protoplanetary discs (Qi et al. 2003), few have been tied directly to the formation of geologically related solids and their constituent minerals. As such, reason exists to shift towards the other side of the periodic table for searches related to rocky planets and how/where they might form. Most notably, [Al]/[Fe], [Si]/[Fe], [Sc]/[Fe], [Ti]/[Fe] and [Mg]/[Fe] ratios have been found to be statistically significantly higher in the spectra of planet-hosting star systems (Adibekyan et al. 2012). Consequently, these atoms are likely important in small-mass planet formation, and the higher the [Mg]/[Si] ratio taken in the spectrum of a star or in that of a protoplanetary disc, the more likely planets are to be found (Adibekyan et al. 2015). Since magnesium is also one of the ten-most abundant atoms in the universe (Savage & Sembach 1996) molecules containing magnesium should be fairly common. As a result, magnesium almost certainly plays a role in solid mineral and subsequent rocky body formation, and molecules containing it could be utilized as indicators of such processes. Molecules containing one atom of magnesium have been previously detected in the interstellar medium (ISM). The carbon rich star IRC +10 216 has played host to a significant portion of the known, unique interstellar molecules, and those containing Mg are no exception. Notably, these include: HMgNC detected in 2013 (Cabezas et al. 2013), MgNC in 1993 (Kawaguchi et al. 1993), MgCN in 1995 (Ziurys et al. 1995) and MgCCH in 2014 (Agúndez, Cernicharo & Guélin 2014). While these forms have been detected, there seems to be no direct tie between them and mineral formation. Alternatively, the spectrum of the as-of-yet undetected magnesium oxide (MgO) clusters would be a natural place to begin such remote sensing. In addition to the abundance of magnesium, the presence of atomic oxygen in space is quite prevalent, as it is the third most abundant element in the universe (Savage & Sembach 1996). Although the formation of an Mg−O bond is unlikely in hydrogen rich environments (Kohler, Gail & Sedlmayr 1997), the formation of an ionic bond between oxygen and magnesium is inevitable due the atoms’ abundances and large electrostatic attraction. Furthermore, on Earth, magnesium oxide is the chemically simplest mineral containing magnesium since it contains a repeating pattern of only two atoms. In its cubic form, magnesium oxide creates the mineral periclase. This mineral occurs naturally in contact metamorphic rocks and also in the form of ferropericlase (Fe,Mg)O in conjunction with iron making up about 20 per cent of the Earth's mantle (Ohta et al. 2017). Magnesium oxide has become well known as a refractory material and is remarkably resistant to changes when put under extremely high temperatures and pressures (Koker 2010). In its solid form, magnesium oxide is not magnetically conductive, but its superheated liquid phase is magnetic (Coppari et al. 2013). This liquid phase may exist as melt in the interiors of Super-Earths. Melt in terrestrial planet interiors can enable magnetic fields of the planet due to their electrically conductive characteristics (Coppari et al. 2013). In order for planetary formation to occur, all minerals must form from molecules, which must form from atoms in the gas-phase. These gas-phase molecules aggregate eventually to form planetesimals which mark the beginning of rocky planet formation (Pollack et al. 1996). To determine where these gas-phase molecules may be found and at what stage of planet formation they aggregate, highly accurate spectroscopic data are needed for increasingly larger numbers of MgO clusters. Delineations in column densities or local abundances observed in protoplanetary discs for the monomer would indicate gas phase chemistry, while increased presence of the dimer, trimer and larger clusters would give a sign of a trend towards the solid and mineral phase, especially for spectral features for each larger molecular cluster. Such could be a marker for the initiation of mineral or solid material formation, but the unique features of each molecule must be available for reference. Even though, the spectral features of magnesium oxide monomer were originally observed by Herzberg over 50 yr ago (Herzberg 1966), the dimer, trimer and higher n-mers have been little studied save for some anion photodetachment analysis (Gutowski et al. 2000). The relative energies and structural data for the gaseous clusters of (MgO)n (n = 1–40) have been previously calculated with density functional theory (Chen, Xu & Zhang 2008; Chen, Felmy & Dixon 2014; Feitoza et al. 2017), showing patterns of aggregation. Most notably, the face-centred cubic patterns of (MgO)n begin to emerge at n = 4 as a cube built from two stacked (MgO)2 structures. However, depending upon the n value, cubic and hexagonal isomers vie for the minimum energy structure (Chen et al. 2014). In any case, spectral data for the smallest of these clusters will assist in the characterization of magnesium oxide forms in the laboratory and potentially even in mapping the physical properties of protoplanetary discs. 2 COMPUTATIONAL DETAILS In the current work, the necessary vibrational frequencies and rotational constants have been computed quantum chemically via a quartic force field, a fourth-order Taylor series expansion of the internuclear Hamiltonian (Fortenberry et al. 2013) and is of the form   \begin{eqnarray} V&=&\frac{1}{2}\sum _{ij} F_{ij} \Delta _i \Delta _j + \frac{1}{6}\sum _{ijk} F_{ijk} \Delta _i \Delta _j \Delta _K \nonumber\\ &&+\, \frac{1}{24}\sum _{ijkl} F_{ijkl} \Delta _i \Delta _j \Delta _k \Delta _l, \end{eqnarray} (1)in which the Fij… represents the force constants and the Δi terms describe the displacements. In the past, these types of calculations produced vibrational fundamental frequencies to within as good as 1.0 cm−1 and rotational constants within 50 MHz of experimental data (Fortenberry et al. 2011, 2012; Huang, Fortenberry & Lee 2013; Fortenberry et al. 2014a,b; Zhao, Doney & Linnartz 2014; Fortenberry, Lee & Müller 2015; Morgan & Fortenberry 2015a,b; Fortenberry, Roueff & Lee 2016; Kitchens & Fortenberry 2016; Bizzocchi et al. 2017). With these results, accurate rovibrational molecular spectra can be produced, such that small clusters of (MgO)n can be detected. Coupled cluster theory (Crawford & Schaefer 2000; Shavitt & Bartlett 2009) at the singles, doubles and perturbative triples [CCSD(T)] levels (Raghavachari et al. 1989) as well as the explicitly correlated second-order Møller–Plesset perturbation theory MP2-F12 levels (Møller & Plesset 1934; Raghavachari et al. 1989; Werner, Adler & Manby 2007; Hill & Peterson 2010) is utilized to determine the spectroscopic properties of the magnesium oxide clusters. The CCSD(T) method used to calculate the spectroscopic data in this work is regarded as the ‘gold standard’ of quantum chemistry (Helgaker et al. 2004). The lower level MP2 approach is significantly less expensive than CCSD(T), which allows it to treat larger systems, such as the magnesium oxide trimer here. Additionally, MP2 is known to produce surprisingly accurate results for a relatively low level of theory due to the presence of a fortuitous cancellation of errors called a ‘Pauling point’ in honour of Linus Pauling (Zheng, Zhao & Truhlar 2009; Sherrill 2009; Fink 2016). The molpro2015.1 program (Werner et al. 2015, 2012) as well as the psi4 program (Turney et al. 2012; Parrish et al. 2107) is used to perform the quantum chemical computations. 2.1 CCSD(T)-level Computations for MgO and (MgO)2 In order to form a quartic force field (QFF) for the closed shell molecules, MgO and (MgO)2, a restricted Hartree–Fock CCSD(T)/aug-cc-pV5Z (Dunning 1989; Kendall, Dunning & Harrison 1992; Peterson & Dunning 1995; Prascher et al. 2011) geometry optimization is performed to compute the initial geometry. The Martin–Taylor (MT) (Martin & Taylor 1994) core correlating basis set is also used to modify the geometry to include core orbitals, 1s for oxygen and 1s2s2p for magnesium. The reference geometries for these molecules are determined by combining the CCSD(T)/aug-cc-pV5Z optimized geometry and the differences in the CCSD(T)/MT optimized geometry with and without the core orbitals. This geometry optimization is performed in order to construct a structure, which is as close as possible to the true minimum. From the reference geometries, the energy points for the anharmonic internuclear potential, which, subsequently, is used to produce the spectroscopic data, are defined. The MgO monomer QFF requires in total 9 points, and the (MgO)2 QFF requires in total 233 points. The displacements of these points, the Δi terms from equation (1), are described as 0.005 Å for bond length coordinates and 0.005 rad for all bond angles and torsions. The displacements are then used to define the Taylor series expansion and refine the true minimum energy structure. Only one symmetry-internal coordinate defines the MgO molecule:   \begin{equation} S_{1}(\sigma ) = (\mathrm{O}-\mathrm{Mg}), \end{equation} (2)where the above coordinate is simply the distance between the oxygen and magnesium atoms. To fourth order, there are four steps of positive displacements, four steps of negative displacements and the reference minimum producing in total nine points. The dimer, (MgO)2, requires in total 233 points described by the following symmetry-internal coordinates:   \begin{equation} S_{1}(a_g) = \frac{1}{\sqrt{2}}[(\mathrm{O}_1-\mathrm{O}_2)+(\mathrm{Mg}_1-\mathrm{Mg}_2)], \end{equation} (3)  \begin{equation} S_{2}(a_g) = \frac{1}{\sqrt{2}}[(\mathrm{O}_1-\mathrm{O}_2)-(\mathrm{Mg}_1-\mathrm{Mg}_2)], \end{equation} (4)  \begin{eqnarray} S_{3}(b_{1g}) &=& \frac{1}{2}[(\mathrm{O}_1-\mathrm{Mg}_1)-(\mathrm{O}_1-\mathrm{Mg}_2)-(\mathrm{O}_2-\mathrm{Mg}_1)\nonumber\\ &&+\,(\mathrm{O}_2-\mathrm{Mg}_2)], \end{eqnarray} (5)  \begin{equation} S_{4}(b_{1u}) = \tau [(\mathrm{O}_1-\mathrm{Mg}_1-\mathrm{O}_2-\mathrm{Mg}_2)], \end{equation} (6)  \begin{eqnarray} S_{5}(b_{3u}) &=& \frac{1}{2}[(\mathrm{O}_1-\mathrm{Mg}_1)+(\mathrm{O}_1-\mathrm{Mg}_2)-(\mathrm{O}_2-\mathrm{Mg}_1)\nonumber\\ &&-\,(\mathrm{O}_2-\mathrm{Mg}_2)], \end{eqnarray} (7)  \begin{eqnarray} S_{6}(b_{2u}) &=& \frac{1}{2}[(\mathrm{O}_1-\mathrm{Mg}_1)-(\mathrm{O}_1-\mathrm{Mg}_2)+(\mathrm{O}_2-\mathrm{Mg}_1)\nonumber\\ &&-\,(\mathrm{O}_2-\mathrm{Mg}_2)], \end{eqnarray} (8)where the atom numbers are given in Fig. 1. Figure 1. View largeDownload slide The CcCR optimized geometry of planar, D2h (MgO)2. Figure 1. View largeDownload slide The CcCR optimized geometry of planar, D2h (MgO)2. At each point, the energy produced is determined from CCSD(T). Using the aug-cc-pVTZ, aug-cc-pVQZ and aug-cc-pV5Z basis sets, the CCSD(T) energy can be extrapolated using a three-point complete basis set (CBS) limit (Martin & Lee 1996). The energy differences between the CCSD(T) computations retaining core orbitals and those lacking core orbitals again utilizing the MT basis set are added to the CCSD(T)/CBS energies. Additionally, the scalar relativistic (Douglas & Kroll 1974) corrections are also added from computations utilizing the cc-pVTZ-DK basis set for relativity included and excluded. Using these techniques to compute exceedingly accurate energies are defined as the CcCR QFF (Huang & Lee 2008, 2009; Fortenberry et al. 2011; Huang, Taylor & Lee 2011). The ‘C’ represents the CBS energy, the ‘cC’ represents the core correlation calculated as the difference between the MT and MT with core electrons included (MTc) energies and the ‘R’ represents scalar relativity. As mentioned above, these CcCR QFF computations are highly accurate models that can physically describe spectroscopic data. 2.2 MP2-F12 for (MgO)2 and (MgO)3 The methodology for the MP2-F12 level has congruencies with the CCSD(T) level. To begin, the initial geometry is optimized using the MP2-F12/aug-cc-pVDZ. This optimized geometry is then used to define the necessary energy points to form the anharmonic potential. Again, the (MgO)2 QFF requires 233 total energy points, but the (MgO)3 QFF requires 3789 total energy points. The displacements are still 0.005 Å for bond length coordinates and 0.005 rad for all bond angles and torsions. The coordinates for the dimer are the same as those defined above for the CcCR QFF. The trimer, (MgO)3, is of D3h symmetry, but will be run in C2v for ease in constructing the points and in the force constants analysis. The magnesium oxide trimer's QFF is defined by 12 symmetry-internal coordinates:   \begin{equation} S_{1}(a_1) = \frac{1}{\sqrt{2}}[(\mathrm{Mg}_1-\mathrm{O}_2)+(\mathrm{Mg}_1-\mathrm{O}_3)]. \end{equation} (9)  \begin{equation} S_{2}(a_1) = \frac{1}{\sqrt{2}}[(\mathrm{Mg}_1-\mathrm{Mg}_2)+(\mathrm{Mg}_1-\mathrm{Mg}_3)], \end{equation} (10)  \begin{equation} S_{3}(a_1) = (\mathrm{Mg}_1-\mathrm{O}_1), \end{equation} (11)  \begin{equation} S_{4}(a_1) = \frac{1}{\sqrt{2}}[(\mathrm{O}_2-\mathrm{Mg}_1-\mathrm{Mg}_2)+(\mathrm{O}_3-\mathrm{Mg}_1-\mathrm{Mg}_3)], \end{equation} (12)  \begin{equation} S_{5}(a_1) = \frac{1}{\sqrt{2}}[(\mathrm{O}_2-\mathrm{Mg}_1-\mathrm{O}_1)+(\mathrm{O}_3-\mathrm{Mg}_1-\mathrm{O}_1)], \end{equation} (13)  \begin{equation} S_{6}(b_2) = \frac{1}{\sqrt{2}}[(\mathrm{Mg}_1-\mathrm{O}_2)-(\mathrm{Mg}_1-\mathrm{O}_3)], \end{equation} (14)  \begin{equation} S_{7}(b_2) = \frac{1}{\sqrt{2}}[(\mathrm{Mg}_1-\mathrm{Mg}_2)-(\mathrm{Mg}_1-\mathrm{Mg}_3)], \end{equation} (15)  \begin{equation} S_{8}(b_2) {=}\frac{1}{\sqrt{2}}[\angle (\mathrm{O}_2-\mathrm{Mg}_1-\mathrm{Mg}_2){-}\angle (\mathrm{O}_3-\mathrm{Mg}_1-\mathrm{Mg}_3)], \end{equation} (16)  \begin{equation} S_{9}(b_2) =\frac{1}{\sqrt{2}}[(\mathrm{O}_2-\mathrm{Mg}_1-\mathrm{O}_1)-(\mathrm{O}_3-\mathrm{Mg}_1-\mathrm{O}_1)], \end{equation} (17)  \begin{eqnarray} S_{10/x}(b_1) &=& \frac{1}{\sqrt{2}}\tau [(\mathrm{O}_3-\mathrm{Mg}_1-\mathrm{Mg}_3-\mathrm{O}_1)\nonumber\\ &&-\,(\mathrm{O}_2-\mathrm{Mg}_1-\mathrm{Mg}_2-\mathrm{O}_1)], \end{eqnarray} (18)  \begin{equation} S_{11/y}(b_1) = \tau (\mathrm{O}_2-\mathrm{Mg}_1-\mathrm{O}_3-\mathrm{O}_1), \end{equation} (19)  \begin{eqnarray} S_{12/z}(a_2) &=& \frac{1}{\sqrt{2}}\tau [(\mathrm{O}_3-\mathrm{Mg}_1-\mathrm{Mg}_3-\mathrm{O}_1)\nonumber\\ &&+\,(\mathrm{O}_2-\mathrm{Mg}_1-\mathrm{Mg}_2-\mathrm{O}_1)]. \end{eqnarray} (20)At each point, MP2-F12/aug-cc-pVDZ energies are computed, and this QFF is only defined by these terms. The atom labels are shown in Fig. 2. Figure 2. View largeDownload slide The MP2-F12/aug-cc-pVDZ optimized geometry of planar, D3h (MgO)3. Figure 2. View largeDownload slide The MP2-F12/aug-cc-pVDZ optimized geometry of planar, D3h (MgO)3. 2.3 Forming the QFF with CCSD(T) and MP2-F12 methods After obtaining the set of energies, the QFF is fitted. A least-squares fitting of the energy points (whether CcCR or MP2-F12) produces results with minimal error as indicated by the sum of squared residuals of less than 10−16 au2 (10−18 for the monomer and dimer) for all levels of theory used and molecules examined. This fitting produces the equilibrium geometry. Refitting once more zeroes the gradients and produces the force constants needed to define the potential portion for the internuclear Hamiltonian, the QFF (equation 1). The intder (Allen et al. 2005) program transforms the computed force constants into Cartesian coordinates, which are more readily translated for subsequent generic analysis. Rotational theory and vibrational second-order perturbation theory (VPT2) (Mills 1972; Watson 1977; Papousek & Aliev 1982) are employed by combining the previously computed potential energy and the new kinetic energy via the spectro (Gaw et al. 1991) program. These analyses provide the anharmonic vibrational frequencies and spectroscopic data. (MgO)2 possesses a 2ν6 = ν1 type-1 Fermi resonance as well as ν3/ν2 and ν5/ν4 C−type Coriolis resonances. (MgO)3 exhibits dozens of type-1 and type-2 Fermi, Coriolis and Darling–Denison resonances, but none perturbs the vibrational frequencies by more than 2 cm−1, which is well within the expected accuracy of the MP2-F12/aug-cc-pVDZ approach utilized here. All vibrational intensities are computed within the double-harmonic approximation and with MP2/6-31+G* in the gaussian09 program (Møller & Plesset 1934; Frisch et al. 2009; Hehre, Ditchfeld & Pople 1972). This approximation has been shown to be in good agreement with higher level computations for the prediciton of such properties (Fortenberry et al. 2014c; Yu et al. 2015). 3 RESULTS AND DISCUSSION 3.1 MgO singlet and triplet In order to determine accurate rovibrational spectroscopic data for (MgO)n as a potential marker of mineral formation in protoplanetary discs, the analysis begins with the monomer, MgO. The CcCR symmetry-internal harmonic force constants for $$\tilde{X}\ ^1\Sigma ^+$$ MgO and A 3Π MgO are listed in Table 1, and define the QFF. The singlet is more tightly bound in the F11 value indicating a higher bond order. Magnesium is less inclined to form double bonds because this atom has a [Ne]3s2 electron configuration. Sharing or outright removal of those s orbitals are much more likely than any notable p orbital population, which is a characteristic of a double bond. In fact, the triplet has a strength of merely 63 per cent to that of the singlet. The implication is that $$\tilde{X}\ ^1\Sigma ^+$$ MgO is more ionic in character since one of the electrons from the magnesium atom now occupies a lone pair orbital on the carbon in order to create the spin-pairing of the singlet. This relationship is somewhat unexpected since it is opposite of that noted for the isoelectronic MgCH2 molecule recently analysed in our group where the triplet state is the ground state (Bare et al. 2000; Bassett & Fortenberry 2017). However, this has been known in MgO for over 40 yr (Ikeda et al. 1977) and the more contemporary 2551.9713 cm−1$$a ^3\Pi \leftarrow \tilde{X} ^1\Sigma ^+$$ transition frequency (Mürtz et al. 1994, 1995) is very close to the 2385 cm−1 CcCR adiabatic excitation computed here. Regardless, MgO monomer will almost exclusively exist in the $$\tilde{X}\ ^1\Sigma ^+$$ state in protoplanetary discs at distances where rocky planets will form since the excitation energy is equivalent to an ambient temperature of roughly 3700 K. Table 1. The MgO CcCR force constants in mdyn/Ån for n, m = 0, 1, 2, 3, 4. $$\tilde{X}\ ^1\Sigma ^+$$ MgO  A 3Π MgO  F11  3.763 052  F11  2.384 934  F111  −22.118 502  F111  −11.858 582  F1111  66.966 674  F1111  52.025 584  $$\tilde{X}\ ^1\Sigma ^+$$ MgO  A 3Π MgO  F11  3.763 052  F11  2.384 934  F111  −22.118 502  F111  −11.858 582  F1111  66.966 674  F1111  52.025 584  View Large The CcCR QFF VPT2 bond lengths and spectroscopic constants are displayed in Table 2. The O–Mg bond is longer in the triplet state than in the singlet state, as a result of the smaller F11 value in the triplet. While $$\tilde{X}\ ^1\Sigma ^+$$ MgO is more ionic than A 3Π MgO, the monomer can be more tightly bound since both electrons can be involved in the bonding. However, in the triplet, the electron is retained by the magnesium but lies on the side opposite the carbon. Therefore, it is unavailable for bonding, weakening the Mg–O interaction. The $$\tilde{X}\ ^1\Sigma ^+$$ MgO bond length is comparable to the experimental bond length of 1.7490 Å (Irikura 2007), and even the excited triplet state is within 0.02 Å of experiment at 1.869 286 Å (Mürtz et al. 1995). The variance in the bond lengths, rotational constants and sextic distortion constants between the singlet and triplet states showcases that these two forms of MgO will be easily distinguished rotationally as well as electronically or by ambient temperature. Table 2. The MgO Singlet and MgO triplet CcCR QFF VPT2 bond lengths, spectroscopic constants, and vibrational frequencies (with intensities in km mol−1 in parentheses)     $$\tilde{X}\ ^1\Sigma ^+$$ MgO  A 3Π MgO      MgO  25MgO  26MgO  Mass average  Experimenta  Mass average  Experimentb  re (O-Mg)  Å  1.738 981  –  –  –  1.7490  1.876 550  1.869 286  Be  MHz  17416.0  17136.9  16880.3  17329.2  –  14881.5  –  Est. B0  MHz        17242.5  17233.31  14807.1  15072.5  De  kHz  35.331  34.208  33.191  35.005  36.937  34.9574  35.92  He  mHz  −29.838  −28.426  −27.268  −29.440  –  −8.933  –  ω1  cm−1  814.8 (1127)  809.3  803.2  813.0  785.2  647.9 (61)  650.223      $$\tilde{X}\ ^1\Sigma ^+$$ MgO  A 3Π MgO      MgO  25MgO  26MgO  Mass average  Experimenta  Mass average  Experimentb  re (O-Mg)  Å  1.738 981  –  –  –  1.7490  1.876 550  1.869 286  Be  MHz  17416.0  17136.9  16880.3  17329.2  –  14881.5  –  Est. B0  MHz        17242.5  17233.31  14807.1  15072.5  De  kHz  35.331  34.208  33.191  35.005  36.937  34.9574  35.92  He  mHz  −29.838  −28.426  −27.268  −29.440  –  −8.933  –  ω1  cm−1  814.8 (1127)  809.3  803.2  813.0  785.2  647.9 (61)  650.223  Notes.aIrikura (2007) with D from Mürtz et al. (1995). bMürtz et al. (1995). View Large However, the computed rotational constant for $$\tilde{X}\ ^1\Sigma ^+$$ MgO does not match that from experiment 17233.31 MHz (Irikura 2007). Magnesium has three major isotopes, unlike oxygen, carbon and hydrogen where one dominates. 24Mg has a natural abundance of 79 per cent, 25Mg is 10 per cent and 26Mg is 11 per cent. By averaging the equilibrium rotational constants, Be, for each isotopologue, 17329.2 MHz is produced, in much better agreement. Full vibrational-averaging for diatomics is problematic for the available software. However, estimates in the perturbation between Be and the more experimentally meaningful B0 values from data taken below for (MgO)2 give a rotational constant within 10 MHz of experiment at 17242.5 MHz. Hence, our theoretical methods should be similarly accurate for the dimer species below. Case in point is the quartic distortion constant (D), which is computed to be 35.004 kHz and in experiment 36.937 kHz (Mürtz et al. 1995). The A 3Π bond length, rotational constant, D value and even vibrational frequency are in excellent agreement with experiment (Mürtz et al. 1995) showcasing the accuracy of this computational approach. The vibrational frequencies of MgO are also listed in Table 2. The singlet state has an incredibly bright intensity (1127 km mol−1) for the fundamental mode due to an active charge transfer, which is a hallmark of ionic species. Again, anharmonic computations are problematic for diatomics, but mass-averaging lowers the ω1 fundamental frequency somewhat to 813.0 cm−1. Considering an anharmonic effect of the order of 10–20 cm−1 puts the estimated frequency within 10 cm−1 of the experimentally observed value at 785.2 cm−1. Consequently, any computational studies of magnesium must involve all three major isotopes in order for meaningful comparison to be had for physically observable properties whether in planet-forming regions or in the laboratory. 3.2 (MgO)2 The dimer, (MgO)2, requires significantly more force constants than the monomer. These force constants are produced in Table 1 of the Supporting Information (SI). The higher D2h symmetry means that the bond strengths arise from linear combinations of the force constants. As a result, the bonding in (MgO)2 is somewhat weaker between individual Mg−O bonds, but the cross-bonding between like atoms in the ring and the cyclic stabilization create a more stable structure once two MgO pieces are brought together. In fact, the CcCR separation energy for (MgO)2 → 2 MgO is −132.0 kcal mol−1 indicating a strong stabilization for creation of the dimer from the monomer. Granted some barrier will likely be present in any such reaction, but the thermodynamics will ultimately lead to the larger clusters. Each MgO unit contains 20 electrons, meaning that each successive n-mer will be adding a factor of 20 electrons to the system, making for simple addition. The vibrational frequencies of (MgO)2 are shown in Table 3. The anharmonic corrections are well behaved within our tight fitting, giving no reason to doubt the accuracy of these values as within the 10 cm−1 or less range established for MgO above and previously for p-block molecules. The mass averaging is of supreme importance here for observations of molecules mixed through a bulk. While the (24MgO)2 isotopologue will make up 62 per cent of the observable amount of (MgO)2, 24MgO25MgO and 24MgO26MgO will comprise 7.9 per cent and 8.7 per cent of (MgO)2. Hence, these and the three other combinations of magnesium masses are included in the averaged values given in the furthest column of Table 3. Table 3. The (MgO)2 CcCR and MP2-F12/aug-cc-pVDZ QFF VPT2 frequencies (in cm−1) with MP2/6-31+G* double harmonic intensities (in km mol−1 in parentheses). Units  CcCR  MP2-F12  24Mg25Mg  25Mg 25Mg  25Mg 26Mg  26Mg 26Mg  24Mg26Mg  Mass average  ω1(Ag)  681.8  650.1  679.4  676.6  674.4  672.1  677.2  673.1  ω2(B2u)  673.6 (179)  641.1  671.0  668.2  665.7  663.2  668.7  664.5  ω3(B3u)  615.0 (201)  593.1  612.4  610.0  607.7  605.4  610.0  607.7  ω4(B1g)  580.7  558.8  577.8  575.0  572.3  569.8  574.9  572.8  ω5(Ag)  453.4  441.7  450.5  447.6  444.7  441.9  447.6  446.8  ω6(B1u)  286.9 (141)  278.6  285.7  284.5  283.5  282.4  284.6  283.7  ν1(Ag)  672.2  641.2  669.7  667.2  665.1  662.8  667.7  663.7  ν2(B2u)  661.8  630.4  659.3  656.6  654.3  651.8  657.1  653.1  ν3(B3u)  605.0  583.2  602.5  600.2  597.9  559.4  563.8  587.4  ν4(B1g)  569.4  548.1  566.5  563.9  561.3  558.8  563.8  561.7  ν5(Ag)  450.3  439.0  447.4  444.6  441.7  438.9  444.6  443.8  ν6(B1u)  284.4  276.7  283.2  282.1  281.1  280.0  282.2  281.4  2ν2  1320.4  1257.8  1315.3  1197.1  1248.7  930.3  1310.9  1225.8  2ν3  1206.7  1163.1  1201.8  1310.0  1192.6  1149.4  1197.1  1202.9  2ν6  568.2  553.1  565.9  563.6  561.5  559.4  563.8  562.2  Zero-point  1639.14  1575.90  1631.88        1625.13    Units  CcCR  MP2-F12  24Mg25Mg  25Mg 25Mg  25Mg 26Mg  26Mg 26Mg  24Mg26Mg  Mass average  ω1(Ag)  681.8  650.1  679.4  676.6  674.4  672.1  677.2  673.1  ω2(B2u)  673.6 (179)  641.1  671.0  668.2  665.7  663.2  668.7  664.5  ω3(B3u)  615.0 (201)  593.1  612.4  610.0  607.7  605.4  610.0  607.7  ω4(B1g)  580.7  558.8  577.8  575.0  572.3  569.8  574.9  572.8  ω5(Ag)  453.4  441.7  450.5  447.6  444.7  441.9  447.6  446.8  ω6(B1u)  286.9 (141)  278.6  285.7  284.5  283.5  282.4  284.6  283.7  ν1(Ag)  672.2  641.2  669.7  667.2  665.1  662.8  667.7  663.7  ν2(B2u)  661.8  630.4  659.3  656.6  654.3  651.8  657.1  653.1  ν3(B3u)  605.0  583.2  602.5  600.2  597.9  559.4  563.8  587.4  ν4(B1g)  569.4  548.1  566.5  563.9  561.3  558.8  563.8  561.7  ν5(Ag)  450.3  439.0  447.4  444.6  441.7  438.9  444.6  443.8  ν6(B1u)  284.4  276.7  283.2  282.1  281.1  280.0  282.2  281.4  2ν2  1320.4  1257.8  1315.3  1197.1  1248.7  930.3  1310.9  1225.8  2ν3  1206.7  1163.1  1201.8  1310.0  1192.6  1149.4  1197.1  1202.9  2ν6  568.2  553.1  565.9  563.6  561.5  559.4  563.8  562.2  Zero-point  1639.14  1575.90  1631.88        1625.13    View Large Only three of the six fundamental vibrational frequencies are infrared-active (IR-active) due to symmetry, ν2 (the oxygen atom translation), ν3 (the magnesium atom translation) and ν6 (the out-of-plane ‘book’ motion), as given in Table 3. All three are notably bright frequencies with intensities above 100 km mol−1. Most molecules have intensities around a few dozen km mol−1. As a result, the magnesium oxide dimer should be detectable even in reduced amounts as a consequence of Beer's law. The 15–17 μm region at the beginning of the far-IR and approaching the THz region is well within the range of the James Webb Space Telescope (JWST) making this upcoming observatory a natural choice for future observation of the ν2 fundamental vibrational frequency of (MgO)2. The 281.4 cm−1 (35.5 μm or 8.44 THz) out-of-plane motion will be below what JWST can observe, but this is the least bright fundamental of the three. The frequencies for the two-quanta overtones for each of these fundamentals are also given in Table 3. The differences between the CcCR and MP2-F12/aug-cc-pVDZ calculations differ by 3–5 per cent. Hence, MP2-F12/aug-cc-pVDZ is well behaved, but undershoots the prediction of the frequencies by as much as 30 cm−1. However, this is close enough use in conjunction with isolated laboratory experiments for larger molecules where such anharmonic computations cannot be undertaken with the CcCR QFF. Even though, (MgO)2 is non-polar, the CcCR QFF structures and spectroscopic constants are included in Table 4 for completeness. The vibrationally averaged (Rα) bond lengths for (MgO)2 differ slightly between the CcCR and MP2-F12/aug-cc-pVDZ methods, as expected for different methods and basis sets. However, the spectroscopic D and H constants are consistent between the two approaches with the primary rotational constants roughly similar. The near-oblate nature of this molecule means that the A and B constants are closer to one another in magnitude and greater than the C. Even without a permanent dipole moment, rotational transitions of vibrationally excited states of molecules have been detected in the ISM (Turner 1987; Cernicharo et al. 2008), making the present predictions for the A, B and C constants of the 2, 3 and 6 states potentially of value for observations with the Atacama Large Millimeter Array (ALMA). Table 4. The CcCR QFF Zero-point (Rα vibrationally averaged) and equilibrium structures, rotational constants and quartic and sextic distortion constants of (MgO)2 and for the 25Mg and 26Mg isotopologues.   Units  CcCR  MP2-F12  24Mg25Mg  25Mg 25Mg  25Mg 26Mg  26Mg 26Mg  24Mg 26Mg  Mass Average  r0(O–Mg)    1.859 944  1.887 08  1.859 905  1.859 893  1.859 883  1.859 846  1.859 867  1.859 890  ∠(Mg–O–Mg)    79.999  79.918  79.998  79.999  79.999  79.998  79.997  79.998  A0  MHz  7796.0  7564.7  7796.1  7796.3  7796.4  7796.5  7796.3  7796.3  B0  MHz  7384.9  7186.1  7235.9  7089.4  6952.5  6817.6  7096.8  7096.2  C0  MHz  3787.4  3680.4  3747.9  3708.2  3670.4  3632.5  3710.2  3709.4  A1  MHz  7776.9  7544.8  7777.2  7777.6  7777.9  7778.3  7777.5  7777.6  B1  MHz  7369.1  7170.9  7220.5  7074.2  6937.7  6803.1  7081.9  7081.1  C1  MHz  3779.0  3671.9  3740.0  3700.2  3662.9  3624.9  3703.4  3701.7  A2  MHz  7796.2  7563.4  7796.5  7796.9  7797.2  7797.5  7796.8  7796.9  B2  MHz  7355.9  7158.8  7207.7  7061.8  6925.6  6791.3  7069.4  7068.6  C2  MHz  3774.3  3667.3  3734.6  3695.3  3657.5  3620.0  3696.3  3696.3  A3  MHz  7761.3  7530.5  7761.7  7761.9  7762.2  7762.4  7762.1  7761.9  B3  MHz  7403.0  7203.7  7253.2  7106.0  6968.4  6833.0  7113.3  7112.8  C3  MHz  3777.5  3670.5  3738.2  3698.7  3661.1  3623.4  3700.7  3699.9  A4  MHz  7770.3  7538.9  7770.6  7770.8  7771.0  7771.1  7770.9  7770.8  B4  MHz  7376.2  7177.4  7227.4  7081.2  6944.5  6809.9  7088.5  7088.0  C4  MHz  3773.7  3666.8  3734.3  3694.7  3657.1  3619.3  3696.7  3696.0  A5  MHz  7811.9  7580.8  7811.8  7811.8  7811.7  7811.6  7811.7  7811.8  B5  MHz  7384.9  7185.0  7236.0  7089.5  6952.6  6817.8  7097.0  7096.3  C5  MHz  3784.9  3677.8  3745.3  3705.7  3667.9  3630.0  3707.7  3706.9  A6  MHz  7780.4  7548.8  7780.4  7780.3  7811.7  7811.6  7780.3  7790.8  B6  MHz  7361.2  7162.3  7213.2  7067.5  6952.6  6817.8  7074.9  7081.2  C6  MHz  3791.0  3683.7  3751.4  3711.6  3667.9  3630.0  3713.6  3710.9  ee(O–Mg)  Å  1.853 962  1.8807 92  1.853 962  1.853962  1.853 962  1.853 962  1.853 962  1.853 962  ∠(Mg–O–Mg)  °  80.026  79.951  80.026  80.026  80.026  80.026  80.026  80.026  Ae  MHz  7835.43  7605.11  7835.43  7835.43  7835.43  7835.43  7835.43  7835.43  Be  MHz  7414.35  7215.57  7264.66  7117.37  6979.82  6844.33  7124.89  7124.24  Ce  MHz  3809.54  3702.61  3769.63  3729.58  3691.46  3653.21  3731.64  3730.84  DJ  kHz  3.296  3.1755  3.1542  3.0201  2.9003  2.7862  3.028  3.0308  DJK  kHz  −3.1362  −2.6168  −2.8306  −2.559  −2.3299  2.7862  −2.5735  −1.7738  DK  kHz  6.6563  6.2184  6.4924  6.355  6.2456  6.154  6.3615  6.3775  d1  kHz  −1.6033  −1.5492  −1.5363  −1.4714  −1.4123  −1.355  −1.4753  −1.4756  d2  kHz  −0.226  −0.2331  −0.2253  −0.2229  −0.2195  −1.355  −0.2231  −0.4120  HJ  Hz  0.00664  0.0062  0.006 04  0.005 49  0.005 03  0.004 61  0.005 52  0.005 56  HJK  Hz  −0.0302  −0.027 13  −0.026 83  −0.023 86  −0.021 42  −0.019 25  −0.024 07  −0.024 27  HKJ  Hz  0.004 59  −0.000 95  0.001 22  −0.001 65  −0.003 87  −0.005 77  −0.001 41  −0.001 15  HK  Hz  0.033 31  0.0362  0.033 92  0.034 37  0.034 61  0.034 76  0.0343  0.0342  h1  Hz  0.003  0.002 84  0.002 82  0.002 66  0.002 51  0.002 36  0.002 67  0.002 67  h2  Hz  −0.000 13  −0.000 11  −0.000 01  0.000 08  0.000 15  0.000 21  0.000 08  0.000 06  h3  Hz  0.0002  0.000 16  0.000 18  0.000 17  0.000 16  0.000 15  0.000 17  0.000 17    Units  CcCR  MP2-F12  24Mg25Mg  25Mg 25Mg  25Mg 26Mg  26Mg 26Mg  24Mg 26Mg  Mass Average  r0(O–Mg)    1.859 944  1.887 08  1.859 905  1.859 893  1.859 883  1.859 846  1.859 867  1.859 890  ∠(Mg–O–Mg)    79.999  79.918  79.998  79.999  79.999  79.998  79.997  79.998  A0  MHz  7796.0  7564.7  7796.1  7796.3  7796.4  7796.5  7796.3  7796.3  B0  MHz  7384.9  7186.1  7235.9  7089.4  6952.5  6817.6  7096.8  7096.2  C0  MHz  3787.4  3680.4  3747.9  3708.2  3670.4  3632.5  3710.2  3709.4  A1  MHz  7776.9  7544.8  7777.2  7777.6  7777.9  7778.3  7777.5  7777.6  B1  MHz  7369.1  7170.9  7220.5  7074.2  6937.7  6803.1  7081.9  7081.1  C1  MHz  3779.0  3671.9  3740.0  3700.2  3662.9  3624.9  3703.4  3701.7  A2  MHz  7796.2  7563.4  7796.5  7796.9  7797.2  7797.5  7796.8  7796.9  B2  MHz  7355.9  7158.8  7207.7  7061.8  6925.6  6791.3  7069.4  7068.6  C2  MHz  3774.3  3667.3  3734.6  3695.3  3657.5  3620.0  3696.3  3696.3  A3  MHz  7761.3  7530.5  7761.7  7761.9  7762.2  7762.4  7762.1  7761.9  B3  MHz  7403.0  7203.7  7253.2  7106.0  6968.4  6833.0  7113.3  7112.8  C3  MHz  3777.5  3670.5  3738.2  3698.7  3661.1  3623.4  3700.7  3699.9  A4  MHz  7770.3  7538.9  7770.6  7770.8  7771.0  7771.1  7770.9  7770.8  B4  MHz  7376.2  7177.4  7227.4  7081.2  6944.5  6809.9  7088.5  7088.0  C4  MHz  3773.7  3666.8  3734.3  3694.7  3657.1  3619.3  3696.7  3696.0  A5  MHz  7811.9  7580.8  7811.8  7811.8  7811.7  7811.6  7811.7  7811.8  B5  MHz  7384.9  7185.0  7236.0  7089.5  6952.6  6817.8  7097.0  7096.3  C5  MHz  3784.9  3677.8  3745.3  3705.7  3667.9  3630.0  3707.7  3706.9  A6  MHz  7780.4  7548.8  7780.4  7780.3  7811.7  7811.6  7780.3  7790.8  B6  MHz  7361.2  7162.3  7213.2  7067.5  6952.6  6817.8  7074.9  7081.2  C6  MHz  3791.0  3683.7  3751.4  3711.6  3667.9  3630.0  3713.6  3710.9  ee(O–Mg)  Å  1.853 962  1.8807 92  1.853 962  1.853962  1.853 962  1.853 962  1.853 962  1.853 962  ∠(Mg–O–Mg)  °  80.026  79.951  80.026  80.026  80.026  80.026  80.026  80.026  Ae  MHz  7835.43  7605.11  7835.43  7835.43  7835.43  7835.43  7835.43  7835.43  Be  MHz  7414.35  7215.57  7264.66  7117.37  6979.82  6844.33  7124.89  7124.24  Ce  MHz  3809.54  3702.61  3769.63  3729.58  3691.46  3653.21  3731.64  3730.84  DJ  kHz  3.296  3.1755  3.1542  3.0201  2.9003  2.7862  3.028  3.0308  DJK  kHz  −3.1362  −2.6168  −2.8306  −2.559  −2.3299  2.7862  −2.5735  −1.7738  DK  kHz  6.6563  6.2184  6.4924  6.355  6.2456  6.154  6.3615  6.3775  d1  kHz  −1.6033  −1.5492  −1.5363  −1.4714  −1.4123  −1.355  −1.4753  −1.4756  d2  kHz  −0.226  −0.2331  −0.2253  −0.2229  −0.2195  −1.355  −0.2231  −0.4120  HJ  Hz  0.00664  0.0062  0.006 04  0.005 49  0.005 03  0.004 61  0.005 52  0.005 56  HJK  Hz  −0.0302  −0.027 13  −0.026 83  −0.023 86  −0.021 42  −0.019 25  −0.024 07  −0.024 27  HKJ  Hz  0.004 59  −0.000 95  0.001 22  −0.001 65  −0.003 87  −0.005 77  −0.001 41  −0.001 15  HK  Hz  0.033 31  0.0362  0.033 92  0.034 37  0.034 61  0.034 76  0.0343  0.0342  h1  Hz  0.003  0.002 84  0.002 82  0.002 66  0.002 51  0.002 36  0.002 67  0.002 67  h2  Hz  −0.000 13  −0.000 11  −0.000 01  0.000 08  0.000 15  0.000 21  0.000 08  0.000 06  h3  Hz  0.0002  0.000 16  0.000 18  0.000 17  0.000 16  0.000 15  0.000 17  0.000 17  View Large 3.3 (MgO)3 Creating the magnesium oxide trimer from three monomers will also produce 268.0 kcal mol−1 indicating that the trimer is even more stable than the dimer since half of this stabilization energy is 134.0 kcal mol−1, 2.0 kcal mol−1 more than the dimer. Even though the two are computed with different methods (MP2-F12/aug-cc-pVDZ versus CcCR), the relative energies are still directly comparable. The slightly larger harmonic, diagonal (MgO)3 force constants, displayed in Table 2 of the SI, also corroborate the growth in stabilization for larger clusters of (MgO)n. Since periclase or bulk magnesium oxide is a stable ionic lattice, such an increase in stabilization for larger clusters is fully expected. As a result, the larger clusters are behaving more like the mineral with each addition of a monomer unit (Chen et al. 2014). While there are 10 possible combinations of magnesium isotopes possible in (MgO)3, only the six that contain at least one 24Mg will have natural abundances of greater than 0.4 per cent. Hence, the isotopes included in Tables 5 and 6 will make up more than 98 per cent of any observable (MgO)3. Only these are included in the mass averaging. Table 5. The MP2-F12/aug-cc-pVDZ QFF VPT2 magnesium oxide timer frequencies (in cm−1) with MP2/6-31+G* double harmonic intensities (in km mol−1 in parentheses). Units  24Mg24Mg24Mg  24Mg24Mg25Mg  24Mg 25Mg 25Mg  24Mg 26Mg 26Mg  24Mg 24Mg 26Mg  26Mg 24Mg 25Mg  Mass Average  $$\omega _1(A_2^{\prime })$$  782.3  780.2  777.8  774.6  778.9  776.3  778.3  ω2(E΄)  757.6 (321)  757.4  754.6  751.0  757.1  753.7  755.2      753.0  751.2  745.4  748.3  747.4  749.1  ω3(E΄)  600.9 (1)  599.7  598.3  595.9  598.6  597.2  598.4      599.4  598.1  595.5  598.0  596.7  597.5  $$\omega _4(A_1^{\prime })$$  513.3  512.6  511.8  510.5  511.9  511.1  511.9  $$\omega _5(A_1^{\prime })$$  355.0  353.1  351.2  347.7  351.3  349.5  351.3  $$\omega _6(A_2^{\prime \prime })$$  269.2 (220)  268.4  267.7  266.4  267.8  267.1  267.8  ω7(E΄)  188.4 (82)  188.1  187.2  186.1  187.8  186.8  187.4      187.0  186.2  184.2  185.7  185.1  185.6  ω8(E΄)  165.9  165.9  165.3  164.8  165.9  165.2  165.5      164.8  164.3  162.9  163.8  163.5  164.2  $$\nu _1(A_2^{\prime })$$  767.8  765.0  763.3  759.5  764.6  761.8  763.7  ν2(E΄)  751.2  748.4  747.4  744.1  751.9  748.4  748.6      749.6  746.0  740.2  740.3  740.2  743.2  ν3(E΄)  597.8  594.5  594.2  591.8  596.6  594.6  594.9      598.5  596.1  593.4  593.9  593.2  595.0  $$\nu _4(A_1^{\prime })$$  514.1  513.0  512.1  510.3  512.3  511.3  512.2  $$\nu _5(A_1^{\prime })$$  374.9  372.3  370.7  366.7  371.7  369.1  370.9  $$\nu _6(A_2^{\prime \prime })$$  314.2  312.9  312.1  310.1  312.8  311.4  312.2  ν7(E΄)  216.4  204.7  209.1  208.1  221.2  214.7  212.4      226.0  219.5  216.5  207.4  211.9  216.3  ν8(E΄)  194.3  197.5  195.3  194.5  192.6  193.4  194.6      189.8  190.6  188.8  193.3  191.1  190.7  Zero-Point  2718.69  2710.1  2702.1  2686.8  2703.2  2694.7  2702.6  Units  24Mg24Mg24Mg  24Mg24Mg25Mg  24Mg 25Mg 25Mg  24Mg 26Mg 26Mg  24Mg 24Mg 26Mg  26Mg 24Mg 25Mg  Mass Average  $$\omega _1(A_2^{\prime })$$  782.3  780.2  777.8  774.6  778.9  776.3  778.3  ω2(E΄)  757.6 (321)  757.4  754.6  751.0  757.1  753.7  755.2      753.0  751.2  745.4  748.3  747.4  749.1  ω3(E΄)  600.9 (1)  599.7  598.3  595.9  598.6  597.2  598.4      599.4  598.1  595.5  598.0  596.7  597.5  $$\omega _4(A_1^{\prime })$$  513.3  512.6  511.8  510.5  511.9  511.1  511.9  $$\omega _5(A_1^{\prime })$$  355.0  353.1  351.2  347.7  351.3  349.5  351.3  $$\omega _6(A_2^{\prime \prime })$$  269.2 (220)  268.4  267.7  266.4  267.8  267.1  267.8  ω7(E΄)  188.4 (82)  188.1  187.2  186.1  187.8  186.8  187.4      187.0  186.2  184.2  185.7  185.1  185.6  ω8(E΄)  165.9  165.9  165.3  164.8  165.9  165.2  165.5      164.8  164.3  162.9  163.8  163.5  164.2  $$\nu _1(A_2^{\prime })$$  767.8  765.0  763.3  759.5  764.6  761.8  763.7  ν2(E΄)  751.2  748.4  747.4  744.1  751.9  748.4  748.6      749.6  746.0  740.2  740.3  740.2  743.2  ν3(E΄)  597.8  594.5  594.2  591.8  596.6  594.6  594.9      598.5  596.1  593.4  593.9  593.2  595.0  $$\nu _4(A_1^{\prime })$$  514.1  513.0  512.1  510.3  512.3  511.3  512.2  $$\nu _5(A_1^{\prime })$$  374.9  372.3  370.7  366.7  371.7  369.1  370.9  $$\nu _6(A_2^{\prime \prime })$$  314.2  312.9  312.1  310.1  312.8  311.4  312.2  ν7(E΄)  216.4  204.7  209.1  208.1  221.2  214.7  212.4      226.0  219.5  216.5  207.4  211.9  216.3  ν8(E΄)  194.3  197.5  195.3  194.5  192.6  193.4  194.6      189.8  190.6  188.8  193.3  191.1  190.7  Zero-Point  2718.69  2710.1  2702.1  2686.8  2703.2  2694.7  2702.6  View Large Table 6. The (MgO)3 MP2-F12/aug-cc-pVDZ QFF zero-point (Rα vibrationally averaged) and equilibrium structures, rotational constants, and quartic and sextic distortion constants, including some of the 25Mg and 26Mg isotopologues.   Units  24Mg24Mg24Mg  24Mg24Mg25Mg  24Mg 25Mg 25Mg  24Mg 26Mg 26Mg  24Mg 24Mg 26Mg  26Mg 24Mg 25Mg  Mass Average  r0(O–Mg)  Å  1.848 368  1.848 348  1.848 34  1.848 318  1.848 356  1.848 336  1.848 344  ∠(Mg–O–Mg)  °  73.643  73.643  73.642  73.643  73.644  73.644  73.643  A0  MHz  2579.2  2572.0  2559.7  2526.7  2563.0  2538.0  2556.4  B0  MHz  2561.2  2531.2  2506.5  2469.0  2505.2  2493.1  2511.0  C0  MHz  1284.0  1274.7  1265.4  1247.7  1265.8  1256.6  1265.7  re(O–Mg)  Å  1.844 533  1.844 533  1.844 533  1.844 533  1.844 533  1.844 533  1.844 533  ∠(Mg–O–Mg)  °  73.622  73.622  73.622  73.622  73.622  73.622  73.622  Ae  MHz  2586.77  2580.08  2567.29  2534.96  2571.21  2546.28  2564.43  Be  MHz  2569.74  2539.14  2514.7  2476.11  2512.79  2500.35  2518.81  Ce  MHz  1289.11  1279.72  1270.36  1252.59  1270.83  1261.55  1270.69  DJ  kHz  0.3046  0.3101  0.2947  0.2898  0.3155  0.3009  0.3026  DJK  kHz  0.9528  0.8783  0.942  0.9253  0.8091  0.885  0.899  DK  kHz  −0.2212  −0.1596  −0.2237  −0.2365  −0.1042  −0.1919  −0.1895  d1  kHz  −0.2444  −0.2424  −0.2357  −0.2314  −0.241  −0.2368  −0.2386  d2  kHz  −0.1123  −0.1073  −0.1081  −0.1059  −0.103  −0.1059  −0.1071  HJ  Hz  −0.0001  0.000 14  −0.00063  −0.000 62  0.000 53  −0.000 11  −0.000 13  HJK  Hz  0.0026  −0.001 25  0.0127  0.012 56  −0.008 51  0.003 01  0.003 52  HKJ  Hz  −0.001 42  0.008 19  −0.028 98  −0.028 66  0.0273  −0.003 13  −0.004 45  HK  Hz  0.000 24  −0.005 99  0.018 77  0.018 48  −0.018 65  0.001 48  0.002 39  h1  Hz  0.000 22  0.000 22  0.000 25  0.000 24  0.000 19  0.000 22  0.000 22  h2  Hz  0.000 37  0.000 31  0.000 47  0.000 46  0.0002  0.000 35  0.000 36  h3  Hz  0.0001  0.000 16  −0.000 09  −0.000 09  0.000 27  0.000 08  0.000 07    Units  24Mg24Mg24Mg  24Mg24Mg25Mg  24Mg 25Mg 25Mg  24Mg 26Mg 26Mg  24Mg 24Mg 26Mg  26Mg 24Mg 25Mg  Mass Average  r0(O–Mg)  Å  1.848 368  1.848 348  1.848 34  1.848 318  1.848 356  1.848 336  1.848 344  ∠(Mg–O–Mg)  °  73.643  73.643  73.642  73.643  73.644  73.644  73.643  A0  MHz  2579.2  2572.0  2559.7  2526.7  2563.0  2538.0  2556.4  B0  MHz  2561.2  2531.2  2506.5  2469.0  2505.2  2493.1  2511.0  C0  MHz  1284.0  1274.7  1265.4  1247.7  1265.8  1256.6  1265.7  re(O–Mg)  Å  1.844 533  1.844 533  1.844 533  1.844 533  1.844 533  1.844 533  1.844 533  ∠(Mg–O–Mg)  °  73.622  73.622  73.622  73.622  73.622  73.622  73.622  Ae  MHz  2586.77  2580.08  2567.29  2534.96  2571.21  2546.28  2564.43  Be  MHz  2569.74  2539.14  2514.7  2476.11  2512.79  2500.35  2518.81  Ce  MHz  1289.11  1279.72  1270.36  1252.59  1270.83  1261.55  1270.69  DJ  kHz  0.3046  0.3101  0.2947  0.2898  0.3155  0.3009  0.3026  DJK  kHz  0.9528  0.8783  0.942  0.9253  0.8091  0.885  0.899  DK  kHz  −0.2212  −0.1596  −0.2237  −0.2365  −0.1042  −0.1919  −0.1895  d1  kHz  −0.2444  −0.2424  −0.2357  −0.2314  −0.241  −0.2368  −0.2386  d2  kHz  −0.1123  −0.1073  −0.1081  −0.1059  −0.103  −0.1059  −0.1071  HJ  Hz  −0.0001  0.000 14  −0.00063  −0.000 62  0.000 53  −0.000 11  −0.000 13  HJK  Hz  0.0026  −0.001 25  0.0127  0.012 56  −0.008 51  0.003 01  0.003 52  HKJ  Hz  −0.001 42  0.008 19  −0.028 98  −0.028 66  0.0273  −0.003 13  −0.004 45  HK  Hz  0.000 24  −0.005 99  0.018 77  0.018 48  −0.018 65  0.001 48  0.002 39  h1  Hz  0.000 22  0.000 22  0.000 25  0.000 24  0.000 19  0.000 22  0.000 22  h2  Hz  0.000 37  0.000 31  0.000 47  0.000 46  0.0002  0.000 35  0.000 36  h3  Hz  0.0001  0.000 16  −0.000 09  −0.000 09  0.000 27  0.000 08  0.000 07  View Large Of the 12 total fundamental vibrational frequencies, 4 are in degenerate pairs, making a total of 8 fundamental frequencies. Even though the QFF is computed in the lower symmetry C2v point group, the vibrational frequencies are computed in full D3h. Of the eight fundamental frequencies, four are vibrationally active, but the ν3 motion is nearly unobservable. As a result and like with (MgO)2, the magnesium oxide trimer has three observable fundamental vibrational frequencies (with atom labels from Fig. 2): ν2(E΄), the Mg2+Mg3/O2+O3 stretch akin to the vibrationally active mode in H3+; $$\nu _6(A_2^{\prime \prime })$$, the Mg triangle and O triangle out-of-plane separation; and ν7(E΄), the Mg1+O1 stretch. The anharmonic shifts are small for the first four fundamentals. After that, the anharmonicities are fairly large and actually increase the fundamental frequency. Such behaviour is not typical but not uncommon. These modes are low frequency in the same range where most positive anharmonicities are reported in other highly symmetric systems. However, ν5 − ν8 should still be viewed as less trustworthy than the typically behaving fundamentals in ν1−ν4. Follow-up experiments should be able to determine the ν6 and ν7 peak positions due to the large intensities of these motions. Regardless, these frequencies are not observable with JWST anyway. The only observable fundamental frequency for (MgO)3 in JWST’s range is ν7, which VPT2 with the MP2-F12/aug-cc-pVDZ QFF reports the mass averaged fundamental to be at 748.6 cm−1. Taking the benchmark for the performance of the dimer and upscaling this value by 5 per cent to perform more like the CcCR QFF puts the fundamental for this bright transition at roughly 785 cm−1, 12.7μm or 23.5 THz. This is in the exact range as the MgO monomer. Hence, their vibrational spectra will likely rest at nearly the same frequency. According to Table 6, the mass averaged bond length in this molecule is 1.848 Å, shorter than the 1.860 Å mass averaged bond length in (MgO)2 giving further indication of stronger bonding as discussed previously. This bond length does not vary significantly as the molecule includes the 25 and 26Mg isotopes such that the standard isotopologue and the other masses have the same structure for all intents and purposes. As with the dimer, (MgO)3 has no dipole moment and will not be rotationally observable. However, the rotational constants and spectroscopic data are provided for completeness. Magnesium oxide trimer is even more nearly oblate than dimer as implied by the rotational constants and as expected by the structure. Again, the rotational spectra of some of the vibrationally excited states could be observed. The accuracies for such data with MP2-F12/aug-cc-pVDZ would not be as good as desired, but the pure ground vibrational state rotational data should be a good first-order approximation for their values. 4 CONCLUSIONS Since magnesium is known to be in higher relative abundances in stars that host planets and magnesium is believed to comprise a significant portion of terrestrial-type planets, molecules containing element-12 could serve as sentinel markers in planet-forming regions where gas phase chemistry is giving way to the creation of solids, minerals most notably. The simplest magnesium mineral, magnesium oxide or periclase, is appropriate to begin with in such analyses. The advent of JWST will bring high-resolution to the observation of protoplanetary discs and planet forming regions around stars. This work provides data for the detection of the (MgO)n molecular clusters for n = 1–3. Such molecules are the building blocks of larger magnesium oxide crystals. Hence, if the n ≥ 2 molecular clusters can be detected in the gas-phase in such planet-forming environments as protoplanetary discs, they will be clear indicators of larger mineral formation moving from the gas to solid (mineral) phases. This work has shown that magnesium oxide clusters are more strongly bound as monomer units are added. Such is expected for premineral molecules that ultimately build larger clusters and eventually ionic lattices in solid materials. Additionally, the excellent monomer CcCR QFF benchmarks show that the equivalently computed dimer spectral features should be equivalently accurate. Therefore, scaling-up the trimer MP2-F12/aug-cc-pVDZ results will provide similar results, especially for the higher frequency fundamentals. In general, transitions at 12.5 μm (monomer & trimer), 15.0 μm (dimer) and 16.5 μm (dimer) will be clear indicators of initial periclase formation from these MgO clusters. These transitions also fall within a relatively clear window of observed spectra for known protoplanetary discs (Williams & Cieza 2011). They are to the red of a dominant silicate feature at 10 μm making these peaks observable as shoulders or simply separate, lower frequency peaks especially if the resolving power of JWST can be brought to bear. Since the 785 cm−1/12.5 μm band is exhibited for both the monomer and the trimer, the tetramer will likely have a transition very close to this range. As a result, mapping the 12.5 μm band in protoplanetary discs while also mapping the growth of the dimer's 16.5 μm band and/or the reduction in the monomer's rotational signal could be indicators of the very first stages of where magnesium oxide turns from a gas into a proto-mineral. Consequently, the provided spectral features may allow us for detailed and fine-tuned analysis of planet-forming regions and specifically protoplanetary discs. 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TI - Gas-phase spectra of MgO molecules: a possible connection from gas-phase molecules to planet formation
JF - Monthly Notices of the Royal Astronomical Society
DO - 10.1093/mnras/stx2912
DA - 2017-11-13
UR - https://www.deepdyve.com/lp/oxford-university-press/gas-phase-spectra-of-mgo-molecules-a-possible-connection-from-gas-sLqaZcfVff
SP - 2055
EP - 2063
VL - 474
IS - 2
DP - DeepDyve
ER -