Oganesson: A Noble Gas Element That Is Neither Noble Nor a Gas

Abstract Oganesson (Og) is the last entry into the Periodic Table completing the seventh period of elements and group 18 of the noble gases. Only five atoms of Og have been successfully produced in nuclear collision experiments, with an estimate half‐life for 294118 Og of 0.69+0.64-0.22  ms.[1] With such a short lifetime, chemical and physical properties inevitably have to come from accurate relativistic quantum theory. Here, we employ two complementary computational approaches, namely parallel tempering Monte‐Carlo (PTMC) simulations and first‐principles thermodynamic integration (TI), both calibrated against a highly accurate coupled‐cluster reference to pin‐down the melting and boiling points of this super‐heavy element. In excellent agreement, these approaches show Og to be a solid at ambient conditions with a melting point of ≈325 K. In contrast, calculations in the nonrelativistic limit reveal a melting point for Og of 220 K, suggesting a gaseous state as expected for a typical noble gas element. Accordingly, relativistic effects shift the solid‐to‐liquid phase transition by about 100 K.

the seventh period of elements and group 18 of the noble gases.O nly five atoms of Og have been successfully produced in nuclear collision experiments,w ith an estimate half-life for 294 118 Og of 0.69 þ0:64 À0:22 ms. [1] With such as hort lifetime,c hemical and physical properties inevitably have to come from accurate relativistic quantum theory.Here, we employt wo complementary computational approaches, namely parallel tempering Monte-Carlo (PTMC) simulations and first-principles thermodynamic integration (TI), both calibrated against ah ighly accurate coupled-cluster reference to pin-down the melting and boiling points of this super-heavy element. In excellent agreement, these approaches showOgto be as olid at ambient conditions with am elting point of % 325 K. In contrast, calculations in the nonrelativistic limit reveal am elting point for Og of 220 K, suggesting ag aseous state as expected for atypical noble gas element. Accordingly, relativistic effects shift the solid-to-liquid phase transition by about 100 K.
Six new elements (Nh, Fl, Mc,L v, Ts and Og) have been added into the Periodic Table of Elements over the past 20 years,c ompleting the 7p shell and the 7th period. [2] These exotic short-lived superheavy elements can only be created at aone-atom-at-a-time scale with production rates of one atom per week or even less.Experiments to explore their chemistry is thus very limited, [3][4][5][6][7] and only accurate computational approaches based on either wavefunction or density functional theory can give ad etailed glimpse into their physical and chemical properties.T hese superheavy elements show very unusual behavior compared to their lighter congeners due to strong relativistic effects. [8][9][10] Fore xample Cn and Fl are predicted to be chemically inert [7,11,12] due to the relativistic 7s shell contraction for Cn and the large spinorbit splitting of the 7p shell, resulting in ac losed 7p 1/2 shell for Fl.
In contrast to all other noble-gas solids,O gw as recently predicted to be as emiconductor. [13] Further,t he electron localization function for the Og atom shows au niform electron-gas-like behavior in the valence region, accompanied by alarge dipole polarizability. [8] These findings indicate that for the interaction between Og atoms,3 -body effects might become more important than for the lighter noble gases.I ndeed, this was recently confirmed by calculations, which also revealed as tark increase in the many-body interaction due to relativistic effects. [14] Based on such am any body expansions derived rigorously from relativistic coupled cluster theory,t he melting temperature of the noble gases from Ne to Rn were obtained through parallel tempering Monte Carlo (PTMC), resulting in deviations of not more than afew Kelvin compared to experimental results. [15][16][17] For ag eneral review on rare gas solids we refer to ref. [18].
Considering the unusually strong attractive interaction in the Og dimer, [14] one might speculate that Og is as olid at room temperature,a lthough different extrapolations lead to contradictory results. [19][20][21] In order to resolve this longstanding controversy,w ee mploy two complementary approaches to calculate the melting temperature of Og. Firstly,weuse PTMC simulations with direct sampling of the bulk using periodic boundary conditions,a sw ell as magic number icosahedral clusters where the melting temperature is obtained from extrapolation to the bulk limit. [15][16][17] These PTMC calculations employ 2-and 3-body potentials derived from relativistic coupled-cluster (CC) calculations. [14] Secondly,t ov erify these results and moreover to determine the boiling point, we use thermodynamic integration (TI) based on relativistic dispersion-corrected density-functional theory to calculate absolute Gibbs energies of solid and liquid Og, while gaseous Og is modeled as non-interacting (ideal) gas. [7,12] Subsequently,linear extrapolation to the intersections between the solid, liquid and gaseous Gibbs energies eventually provides the melting and boiling points.Adetailed description of the methods used can be found in the supplementary material.
We start with the discussion on the PTMC results for finite clusters in the canonical ensemble.M elting simulations were performed for Mackay icosahedral magic clusters of size N = 13, 55, 147, 309, 561, 923 and 1415 atoms considering 2-body interactions only,w ith additional simulations including 3body interactions up to ac luster size of N = 923. Heat capacities of the icosahedral clusters as af unction of the simulated temperature are shown in Figure 1. Here,t he highest peak for as pecific cluster size corresponds to the solid-to-liquid phase transition of the entire cluster,w hereas smaller peaks are associated with structural transitions (socalled pre-melting). [22] Theb ulk melting temperatures were determined by extrapolation of the finite cluster results to the bulk value with inverse cluster radius,e quivalent to N À1/3 .2body melting temperatures were obtained by extrapolation of the clusters N = 147-1415 and 3-body corrections were taken as the difference in melting temperature when extrapolating clusters of size N = 147-923 including 2-versus 2 + 3-body interactions,s ee Figure 1f or the extrapolation and Table 1 for as ummary of the results.
Let us now move to the results obtained for bulk cells with periodic boundary conditions,w hich are simulated in the isobaric-isothermal ensemble at 1atm pressure.S ince the solid-to-liquid phase transition temperature is known to converge to the superheated melting temperature T SH with increasing cell size,t he melting temperatures extracted from the bulk simulations are corrected using the expression T m = T SH /1.231. [17,23] Due to the high computational cost of the 3body corrections,this is accomplished in two steps:Firstly,for the largest cell (N = 864) using the 2-body potential, and secondly including the influence of 3-body effects for asmaller (N = 256) cell. Ther esults of the periodic bulk and finite cluster calculations are collected in Table 1.
Inspection of Table 1r eveals excellent agreement between the periodic and cluster simulations at all levels. Forthe relativistic MP at the 3-body level, they provide 324 K and 320 K, respectively.R elativistic effects significantly increase the cohesion between the atoms,w hich is due to strong increase of the attractive 2-body interactions overcompensating aw eaker increase of the repulsive 3-body interactions compared to the non-relativistic potential.
Accordingly,w ef ind the relativistic contributions to the interaction potential to have alarge influence on the melting transition. While calculations in the non-relativistic limit show that Og would be aliquid or agas at room temperature with am elting temperature of % 220 K, as expected for at ypical noble gas element, relativistic effects shift the solid-to-liquid phase transition by about 100 K, which is in equal parts due to scalar-relativistic effects (48 K) and spin-orbit coupling (56 K).
Concerning the many-body decomposition, we find 3body contributions to be much larger compared to all other noble gases (DT m 3Àbody for Xe % 20 K, for Rn % 50 K). [16,17] However,w ed on ot expect 4-body contributions to exert as ignificant influence since their contributions merely increase the cohesive energy by 1.4 percent. This is in contrast to the 3-body contributions,which lowers the cohesive energy by 30 percent. [14] Nonetheless,the 4-body contributions are of attractive type and therefore the calculated melting temperature should be interpreted as al ower bound to the true melting temperature.
Having discussed the results of the PTMC calculations,let us now move to the free-energy calculations employing TI. Fort his,w eu se spin-orbit relativistic DFT with projectoraugmented wave (PAW )pseudo-potentials as well as parameters for the DFT-D3 dispersion correction introduced in previous work. [12,14] To establish the relation between the different functionals and resulting potentials,F igure 2c ompares the potential energy curves of solid Og obtained with the 4-body potential derived from relativistic coupled-cluster theory (CC) to that obtained with spin-orbit relativistic DFT with the PBE-D3, PBEsol and SCANfunctionals.
Since the melting point is very sensitive to the shape and depth of the potential energy surface for the bulk material, various density functionals can give quite different results. [24] Hence,a lthough at first glance SCANp rovides the closest match to the CC cohesive energy,the functional with the best agreement concerning for the general shape of the potential curve is the dispersion-corrected PBE-D3. This becomes evident when the relative depth of the potential is corrected by linear scaling of the Hamiltonian (that is,i nteraction strength, all energies and forces) with af actor of l = 0.776. This can be seen as as howcase for the importance of using ad ispersion correction with DFT,a nd we have thus selected PBE-D3 for our study (a detailed description and discussion of this scaling is provided in ref. [25]).
After determining the equilibrium volume of the condensed phases at as imulation temperature of T = 500 K,  which corresponds to an effective temperature of lT = 388 K, we calculate their Gibbs energies via TI. [25] Fort he liquid, represented by a6 1a tom configuration, we begin from the non-interacting atoms at the liquid equilibrium volume and integrate to the fully interacting liquid at the scalar-relativistic DFT/PBE-D3 level of theory.S ubsequently,t hermodynamic perturbation theory (TPT) is employed to include explicit spin-orbit coupling and converge the numerical accuracy (details can be found in the supplementary). Forthe fcc solid represented by a3 6a tom configuration, we start from the crystal at 0K,c alculate the Gibbs energy in the harmonic approximation, and eventually integrate to the scalar-relativistic anharmonic solid. Similar to the liquid, spin-orbit coupling and numerical convergence are included using TPT.L inear extrapolation of the liquid and solid Gibbs energies to their intersection provides amelting point of T m = 425 AE 14 K, which appears far too high compared to the PTMC results at first glance.However,this is due to the aforementioned over-binding of the PBE-D3 functional, which can be corrected by means of l-scaling.T his provides av alue of lT m = 330 AE 11 K, and thus is in excellent agreement with the results of the PTMC method.
Since TI provides absolute Gibbs energies of the liquid, we can in addition to the melting temperature also determine the normal boiling point (NBP). To locate the intersection with the gas phase,g aseous Og is modeled as an ideal gas at normal pressure.T his is ag ood approximation even at low temperatures,a se vident from the negligible 2-body virial correction ranging from 0.2 to 1meV/atom from 500 to 200 K, which affects the NBP by less than 1K.T his approach predicts aN BP of Og is located at 453 K( 562 Kw ithout lscaling), meaning Og has an atypical large liquid range of 125 Kfor anoble gas.
Note that it was unfortunately not possible to obtain ad etailed breakdown of the impact of relativistic effects in the DFT calculations due to technical issues.S pecifically, while any consistent scalar-relativistic treatment would require at ime-consuming scalar-relativistic re-parametrization of the DFT-D3 correction, calculations in the non-relativistic limit were prevented by convergence issues during the generation of the respective PAW pseudo-potential.
Let us now discuss the calculated transition temperatures in the background of the previous literature.Grosse estimated the melting point by extrapolation of the critical temperature with the period number. On the basis that T m /T c = 0.55 approximately holds for the noble gases up to Xe,the melting temperature of Og was estimated as 258 K. [19,26] An extrapolation of the critical temperature with atom number Z would have perhaps been ab etter choice and leads to am elting temperature of 360 K. However,t here is no theoretical justification for such linear relations.T os ee whether the melting temperature for Og follows the noble gas melting trend or not, one must first understand which quantity correlates with the melting temperature based on theoretical foundations rather than empirical observations. Recently,w eh ave shown that for as ystem with interaction strength scaled by af actor of l,a lso the melting temperature increases by af actor l,w hich has already been applied to correct the results of the TI above.T hus,t he melting point directly correlates with the interaction energy, i.e.[Eq. (1)] Fore xample for aL ennard-Jones system g = 12.16, as obtained from our PTMC simulations of an ideal LJ system in good agreement with previous results, [27][28][29][30][31][32] and f LJ = 0.116 obtained from Lennard-Jones-Ingham coefficients for the fcc crystal. [33] Upon scaling of the 2-body potentials (or equivalently the 2-body cohesive energy curve) of the noble gases,these are all of near identical shape. [14] The2 -body melting point of Og is therefore expected to follow the linear scaling trend between melting point and cohesive energy,a ss et by the lighter congeners.I ndeed, the noble gas melting temperatures obtained by simulation with the 2-body potentials follow the predicted linear trend and, irrespective of the level of relativistic treatment, the 2-body melting temperatures of Og fall on this line,asshown in Figure 3. Forthe lighter noble gas elements,t he 3-body and higher-order contributions to the potential are small, such that the experimental melting temperatures as afunction of cohesive energy also fall on the interpolated line obtained from the 2-body scaling. However, the potential energy surface is altered significantly as 3-body contributions in the heavier noble gases are enhanced by relativistic effects in the heavier noble gases,c hanging the pre-factor g in the equation above.A saconsequence,t he 3body corrected melting temperatures (and consequently the experimental melting temperatures) deviate substantially from the interpolated line for Og at ar elativistic level of theory.
Regarding the boiling point, Nash gave an estimation from extrapolation between the NBPs of the noble gases and their atomic polarizabilities to lie between 320-380 K (according to the same principle the NBP of Rn was estimated at 178-221 K, [21] in agreement with the experimental value of 211 K [16] ). Rescaling this estimate based on the latest highlevel value for the polarizability of a = 58 AE 6a.u. [14] leads to Figure 2. Comparison of the potential energy curves obtained with the 2 + 3 + 4body CC potentiala nd various density functionals along the stretching coordinateo ffcc bulk Og. DFT curves scaled (l)tothe same potential depth as the 2 + 3 + 4-body potential at equilibrium distance are also shown (dashed lines).
aNBP between 360-420 K. This is in contrast to our TI-based result of 450 AE 2K,which moreover translates into an atypical large liquid range of almost 125 K( cf.R n9 .5 K). Although this result might appear surprising and can not be confirmed by the PTMC approach, it should be pointed out that the employed Gibbs energy based approach has recently been tested for ar epresentative set of elements of the periodic table,f or which it provided boiling points in excellent agreement with experimental references with < 2% mean absolute deviation, and < 1% deviation for Xe. [25] Based on the deviation in this comprehensive test, we provide af inal estimate of 450 AE 10 Kf or the NBP.I nf act, the large liquid range can be rationalized by the unusual large 3-body effect for Og compared to other noble gases,a nd moreover by the fact that Og was recently predicted to be asemiconductor. [13] Semiconductors are known to have al arger temperature range for the liquid phase compared to the noble gases. Nevertheless,f uture work should include more accurate abinitio potential energy surfaces for the 3-and 4-body contributions to confirm our predictions from DFT,w hich will be computationally challenging.
Finally,having discussed the phase transitions,let us move to the density of Og. From the periodic 3-body corrected PTMC simulation, we obtain before melting starts as olid density of 1 319K s = 7.2 gcm À3 ,w hich decreases to 1 327K l = 6.6 gcm À3 for the liquid phase.T he spin-orbit relativistic PBE-D3 calculations provide densities of 1 390K s = 7.38 gcm À3 and 1 390K l = 7.10 gcm À3 for solid and liquid Og,r espectively. Applying the many-body potentials including lattice vibrations to the fcc lattice,weobtain anearest neighbor distance of r nn = 4.396 ,g iving ad ensity for 294 Og of 1 0K s = 8.126 gcm À3 using the recommended isotopic mass of M a = 294.214 amu. Grosse et al. estimated the density of Og by linear extrapolation of the atomic volume of the lighter noble gases as afunction of period number using an atomic mass of 314 u. This resulted in abulk density of 1 0K s = 6.29 gcm À3 and ad ensity of the liquid at the melting point of 1 256K l = 4.92 gcm À3 . [19,26] Rescaling to an atomic mass of 294.214 ua nd extrapolating with the Zv alue instead, provides densities of 1 0K s = 5.89 gcm À3 and 1 256K l = 4.61 gcm À3 for the solid and liquid phase,respectively,and thus much lower than our predictions,asexpected for such empirical estimates.
In summary,wehave calculated the melting temperature of Og by means of parallel-tempering Monte Carlo (PTMC) simulations based on an ab-initio potential derived from highlevel relativistic coupled-cluster theory,and through thermodynamic integration (TI) based on relativistic density-functional theory.I ne xcellent agreement with each other, these complementary approaches predict melting points of 324 K (Periodic PTMC), 320 K( Cluster PTMC) and 330 K( TI), which we combine to afinal estimate of 325 AE 15 K. Accordingly,w ec onclude that Og is as olid at ambient conditions. Moreover,b ased on the absolute Gibbs energy obtained via TI, we predict aNBP of 450 AE 10 K, meaning that Og exhibits alarge liquid range of 125 K. Although the large liquid range as well as the solid aggregate state are rather unusual for an oble gas element, they fall into place beside as eries of further atypical properties.A ltogether,t his raises the question if Og should still be seen as anoble gas element, or if that title should be handed to the Group 12 element copernicium. [12] Concerning periodic trends,w eo bserve that the results obtained for the melting point of Og with the PTMC method using ah ypothetical 2-body potential limited to 2body interactions are in line with the lighter congeners.Only through inclusion of 3-body effects,w hich are mostly of relativistic nature and thus larger for Og than for the lighter noble gases,abreakdown of periodic trends and relations can be observed.