Abstract
The high-lying vibrational states of the magnesium dimer (Mg2), which has been recognized as an important system in studies of ultracold and collisional phenomena, have eluded experimental characterization for half a century. Until now, only the first 14 vibrational states of Mg2 have been experimentally resolved, although it has been suggested that the ground-state potential may support five additional levels. Here, we present highly accurate ab initio potential energy curves based on state-of-the-art coupled-cluster and full configuration interaction computations for the ground and excited electronic states involved in the experimental investigations of Mg2. Our ground-state potential unambiguously confirms the existence of 19 vibrational levels, with ~1 cm−1 root mean square deviation between the calculated rovibrational term values and the available experimental and experimentally derived data. Our computations reproduce the latest laser-induced fluorescence spectrum and provide guidance for the experimental detection of the previously unresolved vibrational levels.
INTRODUCTION
The weakly bound alkaline-earth dimers (AE2) have emerged as probes of fundamental physics relevant to ultracold collisions (1), doped helium nanodroplets (2), coherent control of binary reactions (3), and even fields rarely associated with molecular science, such as optical lattice clocks (4) and quantum gravity (5). The magnesium dimer is especially important, since it has several desirable characteristics that can be useful in the above applications, such as the absence of hyperfine structure in the most abundant 24Mg isotope that facilitates the analysis of binary collisions involving laser-cooled and trapped atoms, it helps us understand heavier AE2 diatomics, and, unlike its lighter Be2 analog, it is nontoxic (6). Unfortunately, the status of Mg2 as a prototype heavier AE2 species is complicated by the fact that its high-lying vibrational levels and, consequently, the long-range part of its ground-state potential energy curve (PEC) have eluded experimental characterization for half a century. In this regard, the magnesium dimer is even more challenging than its celebrated beryllium counterpart, whose elusive 12th vibrational level near the dissociation threshold (7, 8), which we also found in (9), was confirmed in 2014 (10) after reanalyzing the spectra obtained in stimulated emission pumping experiments (11).
Experimentally, probing vibrational manifold of the magnesium dimer in its ground,
Typically, high-lying vibrational states near dissociation constitute a small fraction of the entire vibrational manifold, but this is not the case for the weakly bound magnesium dimer, which has a shallow minimum on the ground-state PEC at re = 3.89039 Å (20) and a tiny dissociation energy De of 430.472(500) cm−1 (20, 21). If the five extra levels, which have been speculated about, truly existed, they would represent more than a quarter of the entire vibrational manifold in the ground electronic state. Furthermore, without precise knowledge of the ground-state PEC of Mg2, especially its long-range part that determines the positions of the high-lying vibrational states near the dissociation threshold, one cannot accurately interpret the aforementioned ultracold and collisional phenomena involving interacting magnesium atoms. It is intriguing why a seemingly docile main group diatomic continues to challenge state-of-the-art spectroscopic techniques. The experimental difficulties in detecting the elusive v″ > 13 states of the magnesium dimer originate from several factors, including small energy gaps between high-lying vibrations that are comparable to rotational spacings (12, 25), resulting in overlapping spectral lines, and unfavorable signal-to-noise ratio in the existing LIF spectra (20). Rotational effects complicate the situation even more, since, in addition to affecting line intensities (20, 22, 25), they may render the high-lying vibrational states of Mg2 unbound. All of these and similar difficulties prompted Knöckel et al. (20, 21) to conclude that experimental work alone is insufficient and that accurate theoretical calculations are needed to guide further analysis of the ground-state PEC and rovibrational states of Mg2, especially the elusive v″ > 13 levels near the dissociation threshold.
Unfortunately, there have only been a handful of theoretical investigations attempting to determine the entire vibrational manifold of the magnesium dimer. This is, at least in part, related to the intrinsic complexity of the underlying electronic structure and difficulties with obtaining an accurate representation of the ground-state PEC using purely ab initio quantum-chemical means. At the Hartree-Fock theory level, which neglects electron correlation and dispersion interactions, Mg2 remains unbound. As demonstrated in this work, one needs to go to much higher theory levels, incorporate high-order many-electron correlation effects, including valence as well as inner-shell electrons, and use large, carefully calibrated, one-electron basis sets to accurately capture the relevant physics and obtain a reliable description of the
The initial theoretical estimates of the number of vibrational states supported by the
The call for a reliable ab initio computation of the ground-state PEC and rovibrational states of Mg2, including the v″ > 13 levels that have eluded experimentalists for decades, expressed by Knöckel et al. (20, 21), is answered in the present work. We report the highly accurate PECs for the ground,
To make our comparisons with the experiment more complete, for each of the two electronic potentials considered in this study, we examine both the most abundant 24Mg2 species and the 24Mg25Mg, 24Mg26Mg, 25Mg2, 25Mg26Mg, and 26Mg2 isotopologs (to our knowledge, rovibrational levels of the Mg2 species other than 24Mg2 have not been calculated using ab initio potentials before). We combine the above information with the
RESULTS
The most essential numerical information, generated in the present study using the computational protocol described in Materials and Methods, is summarized in Figs. 1 to 3 and Tables 1 to 3. All of the numerical data supporting the content and conclusions of this work are included in the main text and compiled in the Supplementary Materials document and data files S1 and S2 attached to it. In describing and discussing our results, we begin with the PECs and rovibrational term values characterizing the
The last experimentally observed v″ = 13 level is marked in blue, the predicted v″ = 14 to 18 levels are marked in green, and the ab initio
The
(A) Comparison of the experimental
The G(v″, J″) energies calculated using the ab initio
The G(v″, J″) energies calculated using the ab initio
All line positions are in reciprocal centimeter. The available experimental values are the actual line positions, whereas our calculated results are errors relative to experiment. If the experimentally determined line positions are not available, we provide their calculated values in square brackets. Horizontal bars indicate term values not supported by the
PECs and rovibrational states
As shown in Table 1, our ab initio
Further insights into the quality of our ab initio calculations for the ground-state PEC can be obtained by comparing the resulting rovibrational term values with their counterparts determined using the most accurate, experimentally derived, analytical forms of the
The high quality of our calculated G(v″, J″) values and spacings between them, which can also be seen in Tables 1 and 2 and Fig. 1, allows us to comment on the existence of the v″ > 13 levels that have escaped experimental detection for decades. As already alluded to above and as shown in Table 2 and Fig. 1, our ab initio
As shown in Fig. 1, where we plot the wave functions of the high-lying, purely vibrational, states of 24Mg2, starting with the last experimentally observed v″ = 13 level, along with the
As shown in Table 1 and Fig. 1, the G(v″ + 1) − G(v″) vibrational spacings rapidly decrease with increasing v″, from 47.7 cm−1 or 68.6 K for v″ = 0 to 11.7 cm−1 or 16.8 K for v″ = 12, and to 0.8 cm−1 or 1.2 K for v″ = 17, when 24Mg2 is considered. This means that at regular temperatures all vibrational levels of the magnesium dimer, which is a very weakly bound system, are substantially populated, making selective probing of the closely spaced higher-energy states, including those with v″ > 13, virtually impossible, since practically every molecular collision (e.g., with another dimer) may result in a superposition of many rovibrational states, with some breaking the dimer apart. At room temperature, for example, the cumulative population of the v″ > 13 states of 24Mg2, determined using the normalized Boltzmann distribution involving all rotationless levels bound by the
The accuracy of our ab initio description of the more strongly bound
LIF: Ab initio theory versus experiment
The most compelling evidence for the predictive power of our ab initio electronic structure and rovibrational calculations is the nearly perfect reproduction of the experimental
The notable agreement between the theoretical and experimental LIF spectra shown in Fig. 3A and Table 3, with differences in line positions not exceeding 1 to 1.5 cm−1 and with virtually identical intensity patterns, suggests that our predicted transition frequencies involving the elusive v″ > 13 states are very accurate, allowing us to provide guidance for their potential experimental detection in the future. Before discussing our suggestions in this regard, we note that owing to our ab initio calculations, we can now locate the previously unidentified P12/R10 doublets involving the v″ > 13 states within the experimental LIF spectrum reported in figure 3 of (20). Indeed, as shown in Fig. 3 and Table 3, the LIF spectrum corresponding to the
As one can see by inspecting data file S2 and Fig. 3, and consistent with the remarks made by Knöckel et al. in (20), the experimental detection of the P12/R10 doublets involving v″ > 13, when transitioning from the
Knöckel et al. (20) also suggested that the difficulties with detecting the P12/R10 doublets involving the v″ = 14 and 15 states, which have higher Franck-Condon factors than those characterizing the experimentally observed
Theory-inspired avenues for detection of elusive states
In general, our ab initio calculations carried out in this work indicate that under the constraints of the LIF experiments reported in (20, 21), where the authors populated the
In an effort to assist the experimental community in detecting the elusive v″ = 14 to 18 vibrational levels, we searched for the
DISCUSSION
We used state-of-the-art ab initio quantum-mechanical methodologies to address a half-century-old enigma regarding the v″ = 14 to 18 vibrational states of the magnesium dimer. We provided the highly accurate ground-state PEC and rovibrational term values of 24Mg2 and its less abundant 24Mg25Mg, 24Mg26Mg, 25Mg2, 25Mg26Mg, and 26Mg2 isotopologs. We demonstrated that the
We hope that this study will fuel new spectroscopic investigations of the challenging Mg2 species and its heavier group 2 analogs, which are important in a variety of phenomena at the intersection of chemistry and atomic, molecular, and optical physics. A few years ago, ab initio calculations (8) combined with spectroscopic analyses (7, 10) led to the discovery of the elusive 12th vibrational level of the beryllium dimer. By dealing with five similarly challenging states in a system three times larger than Be2, we demonstrated that the predictive power of modern ab initio quantum chemistry is no longer limited to small few-electron species.
MATERIALS AND METHODS
Ab initio electronic structure calculations
The goal of the ab initio electronic structure calculations performed in this study was to obtain highly accurate
The first term on the right-hand side of Eq. 1 denotes the total electronic energy obtained in the full CCSDT calculations correlating all electrons other than the 1s shells of the Mg monomers and using the aug-cc-pwCVQZ basis set developed in (40), abbreviated in this section and in the Supplementary Materials as AwCQZ. The second and third terms on the right-hand side of Eq. 1, which represent the difference between the frozen-core full CI and CCSDT energies obtained using the aug-cc-pV(Q + d)Z basis of (40), abbreviated in this section and in the Supplementary Materials as A(Q + d)Z, correct the nearly all-electron CCSDT/AwCQZ energy for the valence correlation effects beyond CCSDT. The A(Q + d)Z and AwCQZ basis sets were taken from the Peterson group’s website (41). We used these bases rather than their standard aug-cc-pVnZ and aug-cc-pCVnZ counterparts, since it has been demonstrated that the aug-cc-pV(n + d)Z and aug-cc-pwCVnZ basis set families, including A(Q + d)Z and AwCQZ, accelerate the convergence of bond lengths, dissociation energies, and spectroscopic properties of magnesium compounds (26, 40). The aug-cc-pV(T + d)Z, aug-cc-pwCVTZ, and aug-cc-pwCV5Z bases (40), abbreviated in this section and in the Supplementary Materials as A(T + d)Z, AwCTZ, and AwC5Z, respectively, and used in the auxiliary calculations discussed in section S1 to demonstrate the convergence of our computational protocol with respect to the basis set size (see tables S1 and S2), were taken from the Peterson group’s website (41) as well.
As shown in section S1, the AwCQZ and A(Q + d)Z bases are large and rich enough to provide spectroscopic properties of the magnesium dimer that can be regarded as reasonably well converged with respect to the basis set size, to within ~0.1 to 2 cm−1 for the experimentally observed v″ ≤ 13 levels and ~3 to 5 cm−1 for the remaining high-lying vibrational states and De (see, e.g., table S2). Ideally, one would like to improve these results further by extrapolating, for example, the nearly all-electron CCSDT energetics in Eq. 1, which are responsible for the bulk of the many-electron correlation effects in Mg2, to the complete basis set (CBS) limit. Unfortunately, a widely used two-point CBS extrapolation (42) based on the subvalence CCSDT/AwCTZ and CCSDT/AwCQZ data, which are the only CCSDT data of this type available to us, to determine the CBS counterpart of the first term on the right-hand side of Eq. 1 would not be reliable enough. As demonstrated in (26) and as elaborated on in section S1 (see table S2), a CBS extrapolation using the AwCTZ and AwCQZ basis sets worsens, instead of improving, the De, re, and vibrational term values of the magnesium dimer compared to the unextrapolated results using the AwCQZ basis. As shown in table S2, the CBS extrapolation using the AwCQZ and AwC5Z basis sets would be accurate enough, but the CCSDT/AwC5Z calculations for the magnesium dimer correlating all electrons but the 1s shells of Mg atoms are prohibitively expensive. One could try to address this concern by replacing CCSDT in Eq. 1 by the more affordable CCSD(T) approach (32), resulting in
In principle, one could extend the above composite scheme, given by Eq. 1, to the electronically excited
While the
All electronic structure calculations for Mg2 performed in this study, summarized in tables S3 to S5, were based on the tightly converged restricted Hartree-Fock (RHF) reference functions (the convergence criterion for the RHF density matrix was set up at 10−9). The valence full CI calculations for the
The grid of Mg-Mg separations r, at which the electronic energies of the
It is worth pointing out that our ab initio data points representing the
Calculations of rovibrational term values and rovibronic transitions
The rovibrational term values, including bound and quasi-bound states supported by our ab initio
The quality of the potential fits generated by LEVEL16 is very high. We illustrate it here by summarizing the results of two of the several numerical tests that we carried out for the ground-state PEC. In one of the tests, we computed the electronic energy of the
We also used LEVEL16 to determine the rovibrational term values characterizing the experimentally derived analytical X-representation potential developed in (20), which we used to assess the accuracy of our ab initio–determined
Last, but not least, we used LEVEL16 to compute the line positions of all allowed
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/14/eaay4058/DC1
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
REFERENCES AND NOTES
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