Xenon hexafluoride (XeF6)
Gillespie and Hedberg predicted that the geometry of ground state XeF6 is a distorted octahedron with C3v point symmetry. According to prior studies, two characteristic bond lengths of Xe-F are experimentally determined to be 1.85 and 1.94 ± 0.036 Å, and the∠F-Xe-F of 114.9° and 81.0° were obtained at the self-consistent field (SCF) level. To calculate the geometries of XeF6, initial symmetries were set to C2v, C3v, and octahedral (Oh); subsequently, the bond lengths and angles were fully relaxed. These three symmetries of XeF6 were calculated using the DFT, Møller–Plesset second-order perturbation theory (MP2), and coupled cluster singles and doubles (CCSD) methods. The calculated structures by DFT and MP2 are described in Supplementary Tables S1 and S2 of the Supplementary Information (SI).
Certain calculated structures converged equivalently to the Oh molecular geometry, even though they started at different initial point groups, such as C2v or C3v, according to the DFT and MP2 results. Other calculated structures stay at their initial symmetry and tend to slightly overestimate the bond length compared with the experimentally obtained values by ~ 0.06 Å. Note that bond lengths determined using the MP2 method were closer to the experimental values than those provided by the DFT method. The angles obtained from these two methods tend to be underestimated compared with other calculated values. The C3v structure was stable at the lowest energies for all basis sets except for the Oh-converged structure. The energy differences for other symmetries compared with C3v are denoted as ΔE. The bond angles calculated using the DFT and MP2 methods deviated noticeably from other calculated values up to 9.55%.
In case DFT and MP2 gave the same answer, we additionally applied CC method as a cross-check to validate our calculation results. The XeF6 structure was determined at the CCSD level, as presented in Supplementary Table S3. At the CCSD level, the relativistic effect of Xe was not considered, and three initial symmetries were the same as those in the DFT and MP2 results. The initial symmetries were maintained during the relaxation, and C3v symmetry is predicted to be the ground state for all basis sets. For the LANL2DZ and CEP-31G basis sets, the calculated lengths agree with experimental data, with a small difference in the order of two decimal places in angstrom. The angles for the C3v symmetry differed from other calculated values only by approximately 1° for these basis sets.
In summary, the bond lengths and angles determined using the DFT and MP2 methods were overestimated and underestimated, respectively, implying stabilizing the metastable stereo-isomers was challenging. Regarding XeF6, CCSD level calculations with non-relativistic effect gave quantitative agreement with experimental data, consistent with several previous studies.
In the case of many-electron atoms, the relativistic contraction of inner-shell orbitals by screening affects the outer-shell orbitals. This may significantly impact the chemical and physical properties of heavy inert gases in the lower half of the periodic table. Therefore, the relativistic effect of Xe was further considered at the CCSD level using the DKH Hamiltonian. Supplementary Table S4 summarizes the bond length and total energies of XeF6 for the three geometries with DK3 basis sets. The relativistic effect did not significantly affect geometric parameters. The C3v structure possesses lower total energy than the Oh structure and by C2v structure by 18.88 kcal/mol and 8.28 kcal/mol, respectively using the cc-pVTZ-DK3 basis set. For aug-cc-VTZ-DK3, the overall energy differences, \(\Delta \mathrm{E}\) were lower than those of cc-VTZ-DK3, although there was a tendency to overestimate the bond length by approximately 2.06%. In the DKH calculation, the angle was reduced by approximately 5º compared with the case employing the LANL2DZ basis set in the NR calculation. Although the NR calculation produced values that were in better agreement with the experimental results for XeF6, the DKH calculation also exhibited small differences within 0.01 Å. Even when relativistic effects were not considered, the CCSD level was found to be consistent with the experiment for XeF6 (C3v). Our calculation of XeF6 parameters can be used for the following computational study to determine the effective computational level depending on initial symmetry in Rn-F chemistry.
Radon difluoride (RnF2) and radon tetrafluoride (RnF4)
The atomization energies of argon, krypton, and xenon difluoride indicate that these molecules tend to become stable with increasing atomic number. Following this trend, radon difluoride is expected to be more stable than the corresponding xenon molecules. Therefore, the simpler structures of radon difluoride (RnF2) and radon tetrafluoride (RnF4) were first determined. However, the only reported experimental values for radon molecules are those for RnF2, which Fields proposed. Therefore, first, the stability was investigated by calculating the formation energies of radon fluorides. The convex hull diagram in Supplementary Fig. S1 indicated that the lowest energy could occur between the Rn and binary F2 phases. The energy convex hull graph helps to understand that RnF6 is the most stable molecule along the thermodynamic route, while RnF2 and RnF4 may exist as metastable forms. Speculatively, as there is an experimental observation of RnF2, we can easily speculate the possible formation of RnF4 and RnF6. For structural optimization, the geometry of RnF4 and RnF6 were determined by CCSD, with relativistic effects (DKH). As presented in Table 1, the bond lengths of RnF2 and RnF4 were estimated to be 2.04–2.05 Å and 2.00–2.01 Å, depending on the basis sets employed. The optimized structures of RnF2 and RnF4 were obtained as linear and square planar, as in the case of xenon fluoride, respectively. The bond lengths of radon fluorides were longer than those of xenon fluorides, which is a predictable result considering the larger atomic size of radon. In addition, radon di- and tetrafluoride retained a similar structure to xenon fluorides.
Radon hexafluoride (RnF6)
The symmetrically optimized structures and energies for C2v, C3v, and Oh obtained using DFT are summarized in Supplementary Table S5. Previous studies predicted the bond lengths of RnF6, but they exhibited large differences in bond length depending on the calculation employed. For the DFT simulations, all basis sets converged equivalently to Oh, and the bond lengths of RnF6 differed by up to 0.09 Å compared with the experimental XeF6 structures. The stronger the attraction between bonding atoms, the shorter the bond, and the greater the atom size, the longer the bond. As expected, the bond lengths of RnF6 were greater than those of xenon, although the difference between these was only 0.09 Å. As Rn has an atomic radius of approximately 0.12 Å larger than that of Xe, the calculated bond lengths for radon fluoride were predicted to be shorter, as expected. Using the enthalpy of RnF2 formation, Rn was theoretically demonstrated to form relatively strong covalent bonds compared with other noble gases. In that study, Rn was expected to form shorter and more stable bonds with fluorine than XeF6, and these results are consistent with our calculations.
The calculated structures and relative energies for RnF6 for the MP2 method are described in Supplementary Table S6. For all converged Oh structures, the RnF6 bond length varied between 1.98 and 2.03 Å, and these values tended to be slightly underestimated compared to the DFT results.
Unlike XeF6, all converged geometries exhibited Oh symmetry in the case of RnF6, irrespective of initial symmetry. Although stabilization of the metastable C2v and C3v state was attempted, this was only possible using a fixed bond angle, as shown in Fig. 1. These results reveal the energy barrier between the Oh ground state and the other metastable states (C3v and C2v) was particularly small. Therefore, the initial states easily relaxed into the Oh symmetry state. The bond lengths and total energies that were fully relaxed at the CCSD level are presented in Table 2. The radon fluoride bond lengths obtained using CCSD were slightly shorter for all basis sets than the other two methods (DFT and MP2). Unlike the DFT and MP2 results, the CEP-31G basis set was used to predict that C3v symmetry was the most stable structure at the CCSD level, which differed from the second most stable structure of C2v by approximately 5 kcal/mol.
Comparison of the total energy (keV) for RnF6 calculated by the CCSD method. aanglefix is the fixed given angles and f symmfollow is the freely relaxed bond length.
Additionally, DKH calculations were conducted considering the relativistic effect on the structural parameters of Rn. Table 3 summarizes the bond length and total energies of RnF6 for the three initial geometries (C2v, C3v, and Oh). All initial geometries with different basis sets were converged to Oh, and the bond lengths were estimated to be approximately 1.97 Å. Note that the relativistic effect on heavy atoms destabilizes the d and f orbitals because the atom's inner (core) orbitals are strongly attracted to the nucleus. However, the outer orbitals are subject to minimal relativistic effects because the effective nuclear charge felt by the electrons decreases due to the screening effect. This usually causes the s and p orbitals to contract further and the d and f orbitals to expand, stabilizing the 6p orbitals slightly but the 6s orbital strongly for Rn. Therefore, this effect reduces the covalent radius of Rn and consequently induces a shorter radon fluoride bond length, following the typical trend of bond shrinkage in molecules. The relativistic effect leads to shorter bond lengths because of the stabilization of Rn atom inner orbitals.
Also, it is worth noting that the ground-state geometry of XeF6 is rather different from that of RnF6 when computed using the same methods and basis sets. To compare relativistic effects between Xe and Rn molecules, the molecular electrostatic potential (MEP) of RnF6 and XeF6 was mapped, as shown in Fig. 2. The molecular electrostatic potential surfaces describe the charge distribution of molecules in three dimensions. Figure 2 shows a sigma hole (σ-hole), a region with a positive surface electrostatic potential in the MEP of XeF6(C3v). The lone pair of Xe experiences an electron shielding effect that shields a partial positive charge in the Xe nucleus, inducing a σ-hole due to the repulsive force between the Xe lone pair and the three F atoms. In contrast, RnF6 has evenly distributed σ-holes in the middle of each face of the octahedral, as shown in Fig. 2b. The bond lengths of RnF6 calculated using NR and DKH calculations and the CCSD method are shown in Fig. 3. Evidently, RnF6 converged equivalently to Oh symmetry for all basis sets, except for CEP-31G, even in the case of the DKH calculations.
Molecular electrostatic potential (MEP) surface mapped (a) XeF6 (C3v) and (b) RnF6 (Oh).
The bond lengths of RnF6 including all calculations by the CCSD method. *The calculated values with relativistic effect (Rel) and non-relativistic effect (NR).
Vibrational spectra
Infrared (IR) vibrational spectra were obtained at the CCSD level, considering the relativistic effect, for the optimized structures of radon fluorides with different stoichiometries. The calculated vibrational spectra obtained with the aug-cc-pVTZ-DK3 basis set are tabulated in Table 4, wherein the calculated IR spectra are plotted against the wavenumbers. As no studies conducted thus far have reported the IR spectra of Rn molecules, these could not be compared with experimentally obtained frequencies. For RnF6, all the given initial structures converged to Oh symmetry; therefore, a frequency comparison for each structure was meaningless. In the case of XeF6, the C2v structure had one imaginary frequency, the Oh structure had a three-fold degenerate imaginary frequency, whereas Rn had no imaginary frequency. The presence of an imaginary frequency implies that the optimized structure under investigation is not stable. This suggests that the optimized structures of radon fluoride were harmonically stable on the potential energy surface of the molecules. Note that the calculated frequencies at 648.28 cm−1, consistent with the EU (double degeneracy) mode, accounted for most of the IR active frequency of RnF4. The calculated vibrational spectra for RnF6 indicated that the t1u bending mode (at 663.93 cm−1) contributed to the greatest extent to the intensity. As RnF4 and RnF6 have not been characterized experimentally, our predicted vibrational spectra can play a critical reference for future studies.
Dissociation energy
The dissociation energy of radon fluorides was predicted at CCSD levels with DKH calculations. Previous studies described the dissociation energies of xenon fluoride, which were obtained using classical thermodynamic equilibrium measurements with predicted entropies. Likewise, this study described the thermodynamic reaction and dissociation energy of MF6 (M = Xe or Rn) using the following expressions.
$${\text{MF}}_{{6}} \, \to \,{\text{MF}}_{{4}} \, + \,{\text{F}}_{{2}} ,$$
(1)
$${\text{MF}}_{{6}} \, \to \,{\text{MF}}_{{2}} \, + \,{\text{2F}}_{{2}} ,$$
(2)
$${\text{MF}}_{{6}} \, \to \,{\text{M}}\, + \,{\text{3F}}_{{2}} .$$
(3)
Previous studies have described the heat of formation calculations for XeF6 and obtained – 62.1 ± 1.4 kcal/mol at 0 K and − 64.0 ± 1.4 kcal/mol at 298 K. In this study, the dissociation energy of XeF6 (C3v) → Xe + 3F2 for the most stable structure, which was obtained consistently with the measured values, was 5.04 kcal/mol smaller than the experimentally obtained equilibrium value, and 26.94 kcal/mol lower than that from the photoionization experiment, as shown in Supplementary Fig. S2. Several previous studies reported errors in the experimental equilibrium values of 1–2 kcal/mol, which depended on structural differences. However, the difference in the total dissociation energy in this study was relatively large, depending on symmetry. For example, the total dissociation energy of the CF4+ ion exhibited a variation of approximately 50 kcal/mol according to structural differences at the same level owing to the Jahn–teller effect. Therefore, providing experimental evidence that addresses the question of energy differences according to XeF6 structures is challenging. The calculation results for XeF6 (C3v) using aug-cc-pVTZ-DK3A may be reasonable as these are similar to the equilibrium experimental values.
Dissociation energy was calculated according to various reactions of radon fluoride, and the results are presented in Supplementary Table S7; additionally, the dissociation energy of RnF6 → Rn + 3F2 is shown in Fig. 4. In general, the loss of F2 from RnF2 is slightly harder than the F2 loss from RnF4. The loss of F2 in RnF4 is much harder than that in RnF6. This is consistent with a growing steric community as more fluoride is added to the central Rn. Because convergence to Oh was achieved for all basis sets, no difference in the dissociation energy between Rn molecule structures was observed. There are no significant differences between previous studies with the MP2 method and our calculated values (aug-cc-pVTZ-DK3A: 139.62 kcal/mol and cc-pVTZ-DK3A: 122.85 kcal/mol, see Fig. 4).
The dissociation energy of RnF6 → Rn + 3F2 with relativistic calculation by CCSD method. *The calculated values16 with relativistic effect (Rel) and non-relativistic effect (NR).
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