Ab initio electronic structure calculations

The goal of the ab initio electronic structure calculations performed in this study was to obtain highly accurate X 1 Σ g + and A 1 Σ u + PECs of the magnesium dimer and the corresponding X 1 Σ g + − A 1 Σ u + transition dipole moment function μ z ( X − A ) ( r ) involved in the photoabsorption and LIF experiments reported in (12, 20–22). In the case of the ground-state PEC, we combined the numerically exact description of the valence electron correlation effects provided by full CI with the high-level description of subvalence correlations involving all electrons but the 1s shells of Mg atoms obtained using CCSDT (34, 35). Thus, the X 1 Σ g + PEC of Mg 2 reported in this work was obtained by adopting the composite scheme E X 1 Σ g + = E X 1 Σ g + ( CCSDT / AwCQZ ) + ( E X 1 Σ g + ( Full CI / A ( Q + d ) Z ) − E X 1 Σ g + ( CCSDT / A ( Q + d ) Z ) ) (1)

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 D e (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 Mg 2 , 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 D e , r e , 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 E ˜ X 1 Σ g + = E X 1 Σ g + ( CCSD ( T ) / AwCQZ ) + ( E X 1 Σ g + ( Full CI / A ( Q + d ) Z ) − E X 1 Σ g + ( CCSD ( T ) / A ( Q + d ) Z ) ) (2)but, as explained in section S2, the computational protocol defined by Eq. 2 is not sufficiently accurate for the spectroscopic considerations reported in this work due to the inadequate treatment of triples by the baseline CCSD(T) approximation (cf. fig. S1). For all these reasons, we have to rely on Eq. 1, in which we use CCSDT, not CCSD(T), and finite (albeit large and carefully optimized) AwCQZ and A(Q + d)Z basis sets rather than the poor-quality CBS extrapolation from the CCSDT/AwCTZ and CCSDT/AwCQZ information.

In principle, one could extend the above composite scheme, given by Eq. 1, to the electronically excited A 1 Σ u + state by replacing CCSDT in Eq. 1 with its EOMCCSDT counterpart (37, 38), but the nearly all-electron full EOMCCSDT calculations using the large AwCQZ basis set turned out to be prohibitively expensive for us. To address this problem, we resorted to one of the CR-EOMCCSD(T) approximations to EOMCCSDT, namely, CR-EOMCCSD(T),IA (39), which is capable of providing highly accurate excited-state PECs of near-EOMCCSDT quality at the small fraction of the cost. Thus, our composite scheme for the calculations of the A 1 Σ u + PEC was defined as E A 1 Σ u + = E A 1 Σ u + ( CR-EOMCCSD ( T ) , IA / AwCQZ ) + ( E A 1 Σ u + ( Full CI / A ( Q + d ) Z ) − E A 1 Σ u + ( CR-EOMCCSD ( T ) , IA / A ( Q + d ) Z ) ) (3)where the first term on the right-hand side of Eq. 3 is the total electronic energy of the A 1 Σ u + state obtained in the CR-EOMCCSD(T), IA/AwCQZ calculations correlating all electrons other than the 1s shells of the Mg monomers and the next two terms correct the nearly all-electron CR-EOMCCSD(T),IA/AwCQZ calculations for the valence correlation effects beyond the CR-EOMCCSD(T),IA level using the difference of the full CI and CR-EOMCCSD(T),IA energies obtained with the A(Q + d)Z basis. Before deciding on the use of CR-EOMCCSD(T),IA, we tested other CR-EOMCC schemes (43) by comparing the resulting A 1 Σ u + potentials obtained using Eq. 3 and the corresponding rovibrational term G(v′, J′) values with the available experimentally derived data reported in (21, 44). Although all of these schemes worked well, the computational protocol defined by Eq. 3, with the CR-EOMCCSD(T),IA approach serving as a baseline method, turned out to produce the smallest maximum unsigned errors and RMSD values relative to experiment.

While the X 1 Σ g + and A 1 Σ u + PECs obtained in this study appear to be accurate enough for reproducing and interpreting the experimental A 1 Σ u + → X 1 Σ g + LIF spectra reported in (20, 21), one might wonder whether the neglect of the post–Born-Oppenheimer and relativistic effects in our ab initio calculations could substantially affect our main conclusions. According to (20, 21), the non-adiabatic Born-Oppenheimer corrections (BOCs) for the X 1 Σ g + and A 1 Σ u + states and the mass-dependent adiabatic BOC for the X 1 Σ g + state are negligible. The adiabatic BOC for the A 1 Σ u + state, as defined in (21), may have to be accounted for, but, based on the numerical data reported in (21), its magnitude is well within the uncertainty of the ab initio calculations reported in this work. According to (30), special relativity reduces the dissociation energy D e characterizing the X 1 Σ g + PEC by 4.3 cm−1, i.e., the relativistic effects change the D e by about 1%. However, our preliminary analysis using the modified version of the ab initio protocol adopted in the present work, in which the valence full CI and CCSDT calculations using the A(Q + d)Z basis set and the nearly all-electron CCSDT/AwCQZ computations are replaced by their scalar-relativistic counterparts employing the third-order Douglas-Kroll (DK) Hamiltonian (45, 46) and the triple-ζ aug-cc-pV(T + d)Z-DK and aug-cc-pwCVTZ-DK bases (40), demonstrates that the number of bound vibrational states supported by the relativity-corrected X 1 Σ g + potential is exactly the same as in the case of the analogous nonrelativistic calculations using the A(T + d)Z and AwCTZ bases [the small negative differences between the relativity-corrected and nonrelativistic rotationless G(v″) values vary from <1 cm−1 or 0.8% for v″ = 0 to 2 to ~1% for the highest vibrational states near the corresponding dissociation thresholds]. Similar applies to the A 1 Σ u + PEC, where the effect of relativity on the D e value, estimated using the triple-ζ DK analog of the quadruple-ζ nonrelativistic computational protocol adopted in this work, is 0.2%, but the total number of bound vibrational states supported by the nonrelativistic and relativity-corrected potentials remains the same. The ab initio vibrational spectra corresponding to the X 1 Σ g + and A 1 Σ u + electronic states obtained using the triple-ζ DK modification of the nonrelativistic protocol used in the present study also show that the effects of relativity on the rotationless G(v″ + 1) − G(v″) and G(v′ + 1) − G(v′) energy spacings do not exceed 0.4 cm−1 in the former case and 0.3 cm−1 in the case of the latter energy differences. Thus, while our preliminary findings regarding the small, but nonnegligible, effects of relativity need a thorough reexamination using both the larger basis sets, such as aug-cc-pV(Q + d)Z-DK and aug-cc-pwCVQZ-DK, and the various truncations in the DK Hamiltonian expansions, which may influence the calculated spectra too (46), and we will return to these issues in the future work, the X 1 Σ g + and A 1 Σ u + PECs obtained in the present study are sufficiently accurate to interpret and analyze the LIF spectra reported in (20, 21) and to comment on the corresponding rovibrational manifolds, especially for the ground electronic state.

All electronic structure calculations for Mg 2 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 X 1 Σ g + and A 1 Σ u + states were performed using the GAMESS package (47), whereas the valence and subvalence CCSDT computations for the X 1 Σ g + state were carried out with NWChem (48). The valence and subvalence CR-EOMCCSD(T),IA calculations for the A 1 Σ u + state were executed using the RHF-based CR-EOMCCSD(T) routines developed in (39), which take advantage of the underlying ground-state CC codes described in (49) and which are part of GAMESS as well. The GAMESS RHF-based CC routines (49) were also used to perform the CCSD(T) calculations needed to explore the basis set convergence and the viability (or the lack thereof) of the alternative to the CCSDT-based composite scheme given by Eq. 1, defined by Eq. 2 (see sections S1 and S2, especially table S2 and fig. S1). The convergence thresholds used in the post-RHF steps of the CC and EOMCC computations reported in this work were set up at 10−7 for the relevant excitation amplitudes and 10−7 hartree (0.02 cm−1) for the corresponding electronic energies. The default GAMESS input options that were used to define our full CI calculations guaranteed energy convergence to 10−10 hartree.

The grid of Mg-Mg separations r, at which the electronic energies of the X 1 Σ g + and A 1 Σ u + states reported in this study (cf. tables S3 and S4) were determined, was as follows: 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, 3.2, 3.3, 3.4, 3.6, 3.7, 3.8, 3.9, 4.0, 4.1, 4.2, 4.4, 4.6, 4.8, 5.0, 5.2, 5.4, 5.6, 5.8, 6.0, 6.4, 6.8, 7.2, 7.6, 8.0, 8.4, 8.8, 9.2, 9.6, 10.0, 11.0, 12.0, 13.0, 15.0, 20.0, 25.0, 30.0, and 100.0 Å. We adopted the same set of r values to determine the electronic transition dipole moment function μ z ( X − A ) ( r ) between the X 1 Σ g + and A 1 Σ u + electronic states, needed to calculate LIF line intensities using the Einstein coefficients. The μ z ( X − A ) ( r ) calculations reported in this work were performed using the valence full CI approach, as implemented in GAMESS, adopting the A(Q + d)Z basis set of (40) (see fig. S2 and table S5).