The van der Waals interactions in systems involving superheavy elements: the case of oganesson (Z = 118)

Abstract

This work presents a study involving dimers composed of He, Ne, Ar, Kr, Xe, Rn, and Og noble gases with oganesson, a super-heavy closed–shell element (Z = 118). He–Og, Ne–Og, Ar–Og, Kr–Og, Xe–Og, Rn–Og, and Og–Og ground state electronic potential energy curves were calculated based on the 4-component (4c) Dirac–Coulomb Hamiltonian and were counterpoise corrected. For the 4c calculations, the electron correlation was taken into account using the same methodology (MP2-srLDA) and basis set quality (Dyall's acv3z and Dunning's aug-cc-PVTZ). All calculations included quantum electrodynamics effects at the Gaunt interaction level. For all the aforementioned dimers the vibration energies, spectroscopic constants (ω_e, ω_ex_e, ω_ey_e, α_e, and γ_e), and lifetime as a function of the temperature (which ranged from 200 to 500 K) were also calculated. The obtained results suggest that the inclusion of quantum electrodynamics effects reduces the value of the dissociation energy of all hetero-nuclear molecules with a percentage contribution ranging from 0.48% (for the He–Og dimer) to 9.63% (for the Rn–Og dimer). The lifetime calculations indicate that the Og–He dimer is close to the edge of instability and that Ng–Og dimers are relatively less stable when the Gaunt correction is considered. Exploiting scaling laws that adopt the polarizability of involved partners as scaling factors, it has also been demonstrated that in such systems the interaction is of van der Waals nature (size repulsion plus dispersion attraction) and this permitted an estimation of dissociation energy and equilibrium distance in the Og–Og dimer. This further information has been exploited to evaluate the rovibrational levels in this symmetric dimer and to cast light on the macroscopic properties of condensed phases concerning the complete noble gas family, emphasizing some anomalies of Og.

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1 Introduction

Research involving relativistic quantum chemistry of super-heavy atoms is attracting a lot of attention from the scientific community with the emergence of new elements in the periodic table. For instance, Eliav et al.¹ calculated the electron affinity of oganesson (Og), the most recently discovered noble gas, to be Z = 118,² by using the relativistic coupled-cluster method based on the Dirac–Coulomb–Breit Hamiltonian. The obtained value was 0.056 eV (with an estimated error of 0.01 eV) when relativistic corrections were included. However, no Og electron affinity was found when non-relativistic or uncorrelated calculations were performed.¹ This fact shows that relativistic theoretical chemistry plays an extremely relevant role, often being the exclusive chemical information source useful in describing the properties of systems involving super-heavy elements.³

Since few isotopes of the transactinide elements can be obtained, what is known about their properties comes mostly from theory. A theoretical study becomes very important, especially when the chemical/physical properties of the heaviest elements usually cannot be predicted from extrapolations of known characteristics in the groups or periods of the Periodic Table.³ In fact, the higher the nuclear charge of an atom, the faster the speed of the innermost electrons, which reach speeds corresponding to a considerable fraction of the light speed, and substantially affect the atomic behaviour. Thus, the relativistic mass of the electrons increases and causes the contraction and stabilization of the orbitals. This fact allows other outermost orbitals to expand and destabilize. Therefore, theoretical calculations for super-heavy elements need to include the special relativity theory for a correct description of the atomic properties of the heaviest elements.⁴

In the systems involving super-heavy atoms, the effects of quantum electrodynamics (QED) must be included in the rationalization of their behavior. For instance, the ionization potential of the lawrencium atom has recently been determined and the obtained result agreed with the experimental value (4.96 eV) only when QED effects were included. On the other hand, the understanding of the nature of the main components determining non-covalent interactions is very important since it allows us to rationalize and predict a large number of chemical/physical phenomena.^5–9 In the literature, however, there are a limited number of experimental/theoretical studies available that focus on characterization of the weak interaction, vibrational energies, spectroscopy, and lifetime of the dimers involving Ng and super-heavy atoms.

Based on these facts, the present paper reports on the characterization of the interaction in Ng–Og (Ng = He, Ne, Ar, Kr, Xe, Rn, and Og) systems considering relativistic effects at the 4-component level and with the inclusion of QED effects at the Gaunt interaction level and basis set superposition error (BSSE).¹⁰ More precisely, our work aimed to obtain the Ng–Og ground state electronic potential energy curves (PECs) by applying a combination of the second-order Moller–Plesset long-range theory (MP2-srDFT) with the DFT short-range approach within the framework of the 4-component relativistic method. These PECs have then been used to calculate the vibrational energies, spectroscopic constants, and lifetime of the Ng–Og dimers.

A further effort, addressed to better define the nature, strength and range of the leading interaction components, determining binding energy and equilibrium distance in systems involving Og, has also been performed fully exploiting all the obtained theoretical results. In particular, it has been demonstrated that the obtained interaction potentials are of van der Waals type, depending exclusively on the polarizability value of the involved partners,¹¹ which is a collective property of all electrons (mostly the external ones and partially the internal ones) defining both size repulsion and dispersion attraction. For the Og atom, a polarizability value consistent with that recently proposed by Jerabek et al.¹² has been adopted. The use of correlation formulas,¹¹ providing the potential well features of van der Waals interactions in terms of polarizability, also allowed the prediction of binding features and of the strength of the long-range attraction in the Og₂ dimer. Exploiting this information, the dependence of melting (T_mp) and boiling (T_bp) temperature on the two-body interaction components has been analyzed for the complete family of noble gas atoms. With respect to the general trend of all other noble gas atoms, some deviations of Og have been appropriately emphasized. In particular, an increased role of multi-body interaction contributions extends in an anomalous way the stability of the oganesson liquid phase, probably due to the highest relativistic effects, as recently proposed by Smits et al.¹⁴

2 Methodologies

The determination for any system of energies and rovibrational spectroscopic constants, involves an accurate fitting of radial dependence of the interaction energy, indicated as the potential energy curve (PEC). This fitting is necessary to characterize the dissociation energy, D_e, the equilibrium distance, R_e, and the shape of the potential well. For these purposes, two analytical forms were used. The first option to fit the PEC of the studied systems was the adoption Improved Lennard Jones (ILJ) function. This model, defined in terms of only three parameters, has been extensively used in several molecular systems and it has proved to be very efficient to describe van der Waals-type interactions.^15–19 For neutral-neutral systems, the ILJ form assumes the following expression:^20–22where and the β parameter describes the softness/hardness of the elements involved in the molecular system. It is of relevance to stress that the shape of the potential well, obtained from the ILJ formulation, is of great significance for the present study and is in agreement with that provided by other important potential models.^23–25 A detailed probe of the ILJ formulation, applied to the noble gas interactions and performed on high-resolution scattering experiments, has been discussed by Pirani et al.²⁰ In particular, the analysis of the velocity dependence of the integral cross section represented a critical test of the radial dependence of the long-range attraction strength, arising from the critical balance of appropriately damped long-range dispersion coefficients. At the same time, the observed interference effects have been exploited to control basic features of the potential well. Moreover, for a He–He molecule a direct comparison of the ILJ potential formulation with results of ab initio calculations and of other more sophisticated potential models²⁶ has been discussed in ref. 15 and 27.

The second analytical form was the extended-Rydberg²⁸ function, which has been used successfully to describe various molecular systems^29–31 and it is described according to the equation:

where c_i are adjustable coefficients. Once the analytical forms that fit the electronic energies are known, one can solve the nuclear Schrödinger equation (within the Born–Oppenheimer approximation) and determine the vibrational energies of the molecules under study. In the present study, this equation was solved employing the Discrete Variable Representation method.³² Once the vibrational energies are known, the rovibrational spectroscopic constants, such as ω_e, ω_ex_e, ω_ey_e, α_e, and γ_e, can be calculated using the following expressions:²⁹where E_v,j represents the rovibrational energy of a level (ν, j), and ν and j represent the vibrational and rotational quantum numbers, respectively.

As far as we know, there is no experimental data for the systems studied here. Thus, it is necessary to determine the spectroscopic constants using another methodology to ensure the reliability of the current results. The Dunham method,³³ which depends on the derivatives of the potential energy curves in the equilibrium position, was chosen as a second option to calculate the rovibrational spectroscopic constants.

To determine the lifetime as a function of temperature for each Ng–Og dimer within Slater theory,^34,35 the following equation was employed:

where R_g represents the universal gas constant, T is the temperature, E_0,0 is the ground state rovibrational energy, ω_e is the vibrational harmonic spectroscopic constant, and D_e is the dissociation energy.

3 Computational details

All calculations were performed using the DIrac17³⁶ program package, were based on 4-component (4c) Dirac-Coulomb Hamiltonian^37,38 and were counterpoise corrected.¹⁰ In the 4c calculations, the (SS|SS) two electron integrals were neglected and replaced by a charge³⁹ using Dirac's default settings. Since the Dirac program can include the Gaunt interaction only at the Dirac–Hartree–Fock level, it was treated additively.^40,41

For the 4c calculations, the electron correlation was taken into account using the long-range Moller–Plesset perturbation Theory MP2-lrDFT⁴² and its short-range srLDA version^43,44 since it has a better performance than MP2-srPBE for rare gas dimers.⁴² For all systems, at least 60 different distances were calculated from the repulsion region (from around 2.5–3.4 Å) up to the dissociation limit. The electrons correlated were the valence (n − 1)dnsnp and the energy threshold for inclusion of virtual orbitals for MP2-srLDA calculations was set to +20 hartrees due to computational limitations. For He, Ne and Ar atoms we employed the aug-CC-PVTZ basis sets of Dunning and coworkers.^45–48 For heavier atoms (Kr, Xe, Rn and Og), we employed the acv3z basis set of Dyall.^49,50

Finally, the obtained electronic energies were fitted with both the ILJ model and extended-Rydberg analytical form (with ten coefficients) via Powell's method.⁵¹ In the fitting process, the equilibrium distance and the dissociation energy were kept fixed. With the potential energy curves in hand, calculations of rovibrational energies, spectroscopic constants and the lifetime were carried out for each system.

4 Results and discussion

We found it appropriate to present the obtained results with their discussion in four subsections in order to better emphasize their relevance and to exhibit their general implications.

4.1 Dissociation energy and equilibrium distance

Tables S1–S7 of the ESI† show the calculated ground state electronic energies of the He–Og, Ar–Og, Kr–Og, Xe–Og, Rn–Og, and Og–Og dimers for several different internuclear distances R, considering the inclusion of quantum electrodynamics effects at the Gaunt interaction level (QEEGIL), without inclusion of QEEGIL, and with inclusion of QEEGIL + BSSE correction. Table 1 presents the dissociation energies (D_e) and equilibrium distances (R_e) for each diatomic system. From this table, it is observed that the inclusion of quantum electrodynamics effects reduces the value of the dissociation energy of these molecules, which may be attributed to retardation since it could be significant in dispersion interactions.⁵² The percentage contribution of the quantum electrodynamics effect on the dissociation energy ranged from 0.56% (for the He–Og dimer) to 9.63% (for the Rn–Og dimer), that is, as the atomic number of the noble gases increases, the more significant is the contribution of the Gaunt correction. When the BSSE correction is considered, there is a decrease of the dissociation energy of the dimers. The greatest decrease (1.53 meV) occurred for the Og–Ne system. On the other hand, the equilibrium distance remains practically unchanged when the BSSE correction is taken into account. The D_e results found for the Og–Og system were compared with those available in the literature. The differences between the values determined by Gaunt + BSSE and predicted by correlation formulas with those obtained by Saue et al. were 0.0272 eV (0.6270 kcal mol⁻¹) and 0.0236 eV (0.5447 kcal mol⁻¹), respectively. When a comparison is made taking into account the results of Jerabek et al., these values are 0.02685 eV (0.6191 kcal mol⁻¹) and 0.02394 eV (0.5526 kcal mol⁻¹), respectively. It can also be inferred from Table 1 that D_e increases as the noble gas atomic mass also increases, but the same trend does not happen with the R_e. The He–Og equilibrium distance is comparable (or even larger) with respect to that of the largest among all the systems. This feature can be explained by exploring the fact that the R_e of a van der Waals complex is determined through the balance between the terms of repulsion (exchange) and of attraction (dominated by dispersion effects). Both these terms depend on the electronic polarizability of the interacting partners as a measure of their size and provides also the probability of induced dipole formation. Knowing that Og shows the larger size, being more polarizable than He, Ne, Ar, Kr, Xe, and Rn atoms, then it is expected that the size-repulsion (term of exchange) is practically the same for the He–Og, Ne–Og, Ar–Og, Kr–Og, Xe–Og, and Rn–Og molecule and this determines equilibrium distances of comparable value.

Table 1 He–Og, Ne–Og, Ar–Og, Kr–Og, Xe–Og, Rn–Og, and Og–Og dissociation energies (D_e) and internuclear equilibrium distances (R_e) calculated without inclusion of quantum electrodynamics effects at the Gaunt interaction level (QEEGIL), with inclusion of QEEGIL, and with inclusion of QEEGIL + BSSE correction. The D_e, R_e, and asymptotic dispersion coefficient (C₆) predicted by correlation formulas (see Cambi et al.¹¹) were determined using for Og the polarizability value of 8.0 ų, a value consistent with that reported by Jerabek et al.¹² and an effective number of electrons equal to 24 (see Cambi et al.¹¹). The estimated uncertainty in the C₆ is about 10–15% for lighter and 15–20% for heavier systems

Systems	D e (meV)	R e (Å)	C6 (eV Å^6)	Source
He–Og	1.77	4.62		(Without Gaunt)
He–Og	1.76	4.62		(With Gaunt)
He–Og	1.61	4.55		(Gaunt + BSSE)
He–Og	2.91	4.37	20.2	(Predicted by correlation formulas)
Ne–Og	6.38	4.20		(Without Gaunt)
Ne–Og	6.39	4.20		(With Gaunt)
Ne–Og	4.86	4.30		(Gaunt + BSSE)
Ne–Og	6.73	4.33	44.3	(Predicted by correlation formulas)
Ar–Og	19.40	4.33		(Without Gaunt)
Ar–Og	19.25	4.33		(With Gaunt)
Ar–Og	19.00	4.30		(Gaunt + BSSE)
Ar–Og	20.13	4.40	145.9	(Predicted by correlation formulas)
Kr–Og	26.17	4.39		(Without Gaunt)
Kr–Og	25.57	4.40		(With Gaunt)
Kr–Og	24.98	4.40		(Gaunt + BSSE)
Kr–Og	27.34	4.46	215.8	(Predicted by correlation formulas)
Xe–Og	34.94	4.48		(Without Gaunt)
Xe–Og	33.44	4.49		(With Gaunt)
Xe–Og	32.76	4.50		(Gaunt + BSSE)
Xe–Og	35.53	4.56	320.7	(Predicted by correlation formulas)
Rn–Og	55.14	4.40		(Without Gaunt)
Rn–Og	51.34	4.40		(With Gaunt)
Rn–Og	50.50	4.40		(Gaunt + BSSE)
Rn–Og	43.44	4.63	426.7	(Predicted by correlation formulas)
Og–Og	104.58	4.25		(Gaunt + BSSE)
Og–Og	53.79	4.76	626.5	(Predicted by correlation formulas)
Og–Og	77.73	4.31		Jerabek et al.^12
Og–Og	77.39	4.33		Saue et al.^13

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Indeed, all R_e values are confined within 10% of their average value, and any small reduction, such as that observed when passing from He–Og to Ne–Og, and the general small increase along the Ng–Og family is due to the fine balance of size repulsion with dispersion attraction. On the other hand, the De value critically depends on the strength of the global attraction at R_e which is related to the polarizability product of involved partners. Since the polarizability of the He atom is lower than that of the other noble gases, the associated D_e is the smallest one and its value increases regularly along the family by a factor larger than 20. Therefore, the observed behavior is typical of systems whose repulsion component, controlling R_e, is mostly dominated by the size of the most polarizable atom (Og), while D_e varies along a homologous family (Ng–Og) according to the polarizability change of the Ng partner.

Fig. S1 and S2 (ESI†) show the ab initio (with and without Gaunt interaction) and ILJ adjusted PEC of the He–Og, Ne–Og, Ar–Og, Kr–Og, Xe–Og, and Rn–Og molecules, while Fig. S3 and S4 (ESI†) presents the ab initio (with and without Gaunt interaction) and Rydberg adjusted PEC of the same systems. Through these figures one notes that the Og–He electronic energies have small oscillations. Ab initio (with Gaunt + BSSE correction) and ILJ adjusted PEC of the He–Og, Ne–Og, Ar–Og, Kr–Og, Xe–Og, and Rn–Og dimers are shown in Fig. S5 (ESI†), while the ab initio (with Gaunt + BSSE correction) and Rydberg adjusted PEC of the same complexes are represented in Fig. S6 (ESI†). From these figures it is possible to verify that, even when the BSSE correction is taken into account, the small oscillations in the electronic energies of the Og–He system continue.

Tables 2 and 3 present the fitted values of the β ILJ parameter and the extended-Rydberg analytic coefficients, respectively. The β fitted values, depending on the softness of the interaction partners, ranged between 5.80 and 9.58. For this range, the ILJ analytical form works very well for various complexes formed by neutral–neutral and ion–neutral species, i.e., they provided an adequate description of the role of the van der Waals interaction component in the formation of weak hydrogen and halogen intermolecular bonds.^53,54Tables 2 and 3 also show that the RMSD are very small, evidencing the good quality of the He–Og, Ar–Og, Kr–Og, Xe–Og, Rn–Og, and Og–Og electronic energy fits with Gaunt + BSSE correction and the Ne–Og electronic fits with only Gaunt interaction.

Table 2 Fitting values of the β ILJ parameter and root mean square deviation (RMSD)

Systems	RMSD (Hartree)	β parameter
He–Og	2.0976 × 10^−6	9.5808
Ne–Og	2.7833 × 10^−6	7.2151
Ar–Og	1.0030 × 10^−5	8.0185
Kr–Og	1.2681 × 10^−5	6.6153
Xe–Og	1.2135 × 10^−5	6.9770
Rn–Og	6.3491 × 10^−6	6.8909
Og–Og	1.0493 × 10^−4	5.8012

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Table 3 Adjustable parameters (c_n (Å⁻ⁿ)) of the extended-Rydberg analytic form of degree 10 and root mean square deviation (RMSD)

c n (Å^−n)	He–Og	Ne–Og	Ar–Og	Kr–Og	Xe–Og	Rn–Og	Og–Og
C 1	1.47710955	1.38672315	1.79983909	3.39536153	2.77583010	1.70196661	1.75418289
C 2	−1.12088589	−0.81105960	−0.45115730	3.94296832	2.15135216	−0.24350070	−0.00576175
C 3	1.18081113	0.23626192	0.47373096	2.03562284	0.75799095	0.23677149	0.21663296
C 4	0.51099513	−0.25504175	0.23110898	0.77943221	0.19755360	0.01311309	0.05863362
C 5	−1.30890474	0.50599162	−0.35933640	2.41706278	0.97262535	0.00125084	0.06468921
C 6	0.13378930	−0.30124010	0.12987214	1.02490653	0.06621025	−0.02737260	−0.03353552
C 7	0.62722811	0.03474181	0.00452929	−1.44895344	−0.68265120	0.02038624	0.00327290
C 8	−0.38639035	0.02387602	−0.01121581	0.28371066	0.41795297	−0.00607865	0.00397981
C 9	0.08922130	−0.00815757	0.00216509	0.18217579	−0.08852640	0.00084895	−0.00112308
C 10	−0.00739282	0.00074162	−0.00012930	−0.04479104	0.00542335	−0.00004265	0.00010223
RMSD	1.5224 × 10^−6	1.1589 × 10^−6	8.5612 × 10^−6	6.3886 × 10^−6	1.2681 × 10^−5	6.3491 × 10^−6	1.0493 × 10^−4

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Fig. S7 (ESI†) shows the Og–Og ab initio electronic energies (with Gaunt + BSSE correction) and those adjusted by both the ILJ and Rydberg forms. It can be seen from this figure that the ILJ analytical form does not adjust well the ab initio energies at the anharmonic region of the PEC. This fact suggests that the results of rovibrational energies, spectroscopic constants and the lifetime of the Og–Og molecule are more reliable when the Rydberg analytical function is used.

4.2 Nature of the interatomic interaction

As anticipated in Section 4.1, for all systems under investigation the potential energies are determined by a critical balance of size repulsion with dispersion attraction. This means that they exhibit the nature of typical van der Waals interactions. Further confirmation of this general aspect comes from the prediction of correlation formulas, representing both R_e and D_e for all systems in terms of polarizability (α) of the involved partners.¹¹ It is interesting to note that all correlation formulas here exploited, defined in terms of the electronic polarizability α of the interacting partners and first proposed 37 years ago,⁵⁵ have been suggested on the phenomenological ground.¹¹ Moreover, the one defining R_e, showing a dependence on α^1/7 values, has been recently confirmed by a refined quantum mechanical treatment.⁵⁶ Therefore, the two approaches provide coincident-internally consistent results for symmetric and asymmetric noble gas dimers, respectively. Predicted values of the basic D_e and R_e quantities are given in Table 1, where they are compared with the results of the present ab initio calculations. Considering the good agreement between the calculated and predicted values, the same correlation formulas have been exploited to estimate R_e and D_e also for the Og–Og dimer. For all systems, the same correlation formulas are given and also the C₆ coefficient that provides the leading term of the dispersion attraction due to induced dipole–induced dipole interaction.

Considering the high number of internal and external electrons, with associated relativistic effects, the fundamental properties of the heavy Og monomer are not easy to evaluate. Therefore, for the Og–Og dimer, the D_e value predicted by the correlation formulas and that obtained by the present calculations (see Table 1) can be reasonably taken as the lower and the upper limit of the dissociation energy. In particular, considering the mutual-combined uncertainty, they are in reasonable agreement, within ≈0.5 kcal mol⁻¹ (1 kcal mol⁻¹ = 43.32 meV) with theoretical estimates given in ref. 12 and 13 (only for D_e predicted by correlation formulas does the uncertainty amount to 20–30%). Note also that the long-range C₆ attraction coefficients, evaluated for Og–Og inserting D_e and R_e of Table 1 in the asymptotic behavior of ILJ, and defined as C₆ = D_eR_e⁶, agree within 20% with the results of correlation formulas.

4.3 Some speculations on the macroscopic properties

We also researched a possible correlation between the macroscopic properties of noble gas atoms in their condensed phases and the intermolecular forces involved. In particular, the focus has been on the temperatures T_mp and T_bp of melting and boiling points of prototype van der Waals systems provided by noble gases, whose solid phase is usually classified as typical of close (compact)-packing. It is well known that for such systems the repulsive components are mainly responsible for melting while attractive components are responsible for boiling. For a family of systems, where many-body components directly relate to two-body contributions, it can be expected that the balance of their attraction and repulsion components affects the cohesive energy. Eqn (1) and (2) suggest that at R = R_e both components scale approximately with D_e. Therefore, we have attempted to find a possible correlation between T_mp and T_bp with D_e. All involved values are given in Table 4, while Fig. 1 shows the obtained plot. For all systems from He to Rn, the reported T_mp and T_bp are experimental values while for Og they have been recently proposed by Smits et al.¹⁴ The first important feature to be emphasized is that for all systems, including the super-heavy Og, T_mb scales almost linearly with D_e. The second important feature concerns T_bp which follows an approximately linear dependence on D_e for all systems, except the super-heavy Og. In particular, for most systems, the existence zone of the liquid phase is confined in the range of a few K, while Og exhibits a range of liquid larger than 100 K. This anomalous behavior probably arises from the largest many-body contributions which are triggered by relativistic effects.

Table 4 Correlation between dissociation energy D_e, due to two bodies interaction provided by different sources, with temperatures of the melting point T_mp and the boiling point T_bp. Estimated uncertainty in D_e values starts from the fraction meV for Ne–Ne up to 1 meV for Xe–Xe (in all these systems accurate experimental information is available) and increases to 3–4 meV for Rn–Rn and to 5–6 meV for Og–Og. Note that except for Og all other noble gas T_mp and T_bp are taken from ref. 57

System	T mp (K)	T bp (K)	D e (meV)	Source
Ne–Ne	24.5	27.1	4.28	Predicted by correlation formulas
			3.66	Exp.^15
			3.75	Calculations^15
Ar–Ar	83.8	87.5	11.61	Predicted by correlation formulas
			12.37	Exp.^15
			12.36	Calculations^15
Kr–Kr	116.6	120.9	17.61	Predicted by correlation formulas
			17.30	Exp.^15
			17.05	Calculations^15
Xe–Xe	161.3	166.1	25.36	Predicted by correlation formulas
			24.20	Exp.^15
			24.31	Calculations^15
Rn–Rn	202.2	211.4	36.20	Predicted by correlation formulas
			30.32	Calculations^15
Og–Og	335 ± 30	350 ± 30	53.79	Predictions-two body extrapolation ^14
	325 ± 15	450 ± 10

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Fig. 1 Dependence, for the complete family of Ng, of T_mp and T_bp temperatures on the dissociation energy D_e generated by the two-body interaction. The horizontal error bars represent the uncertainty in the D_e value, which is larger for Rn and Og because of the absence of experimental information, which is instead available for all other noble gases.

4.4 Rovibrational levels of dimers

Vibrational energy values (not shown in this work) obtained for both Og–He ILJ and Rydberg PEC with Gaunt and without Gaunt interaction are very similar, suggesting that gaunt correction is not important for this particular system. The PEC's of both Ne–Og and Ar–Og with and without Gaunt corrections have very close vibrational energy values (a total of 6 and 16, respectively) indicating that the Gaunt correction is also not very important for these dimers. It was also found that the Kr–Og ILJ PEC with and without Gaunt correction have the same number of vibrational levels (a total of twenty-six levels). Moreover, the vibrational energies calculated with Gaunt correction are smaller than those calculated without Gaunt correction. The Xe–Og dimer presents thirty-seven and thirty-six vibrational levels with and without Gaunt corrections, respectively, while the Rn–Og system contains fifty-five and fifty-two levels. Also, for both systems, the values of Gaunt vibrational energies are smaller than without Gaunt correction. These features clearly show that the Gaunt correction becomes more important as the atomic number of atoms increases.

Tables 5 and 6 present the He–Og, Ar–Og, Kr–Og, Xe–Og, and Rn–Og pure vibrational (j = 0) and rovibrational (j = 1) energies for the ILJ PEC with Gaunt + BSSE correction. For the Og–Ne system, both the used ILJ and Rydberg PEC were obtained only with the Gaunt interaction for the reasons explained in Section 4.1. From these tables, it can be seen that the He–Og system has only a single vibrational level considering both the ILJ and Rydberg PEC with Gaunt + BSSE correction (Tables S8 and S9 of the ESI†). Vibrational energies are very sensitive to PEC adjustments. In fact, small differences in adjustments can produce changes in the amount of vibrational levels within the well of each adjusted PEC. This fact was observed for some systems involved in the present study. For example, for the Og–Ar, Og–Kr, Og–Xe, and Og–Rn systems, 16, 26, 36, and 52 vibrational levels were obtained for the PEC ILJ, respectively, while for the PEC Rydberg these values were 14, 20, 27, and 50. Og–Og vibrational energies (80 levels) obtained via ILJ PEC adjusted from Gaunt + BSSE electronic energies are shown in Table 7, while the Rydberg PEC contains 107 vibrational levels (Table S10 of the ESI†). With the values of R_e and D_e estimated (correlation formulas) for the Og–Og dimer, it was also possible to construct the ILJ PEC of this system.

Table 5 He–Og, Ne–Og, Ar–Og, Kr–Og, Xe–Og and Rn–Og pure vibrational energies calculated through ILJ potential energy curves

ν	j	He–Og	Ne–Og	Ar–Og	Kr–Og	Xe–Og	Rn–Og
0		8.087589	7.557380	11.719211	9.145629	8.856483	9.512609
1		—	20.082742	33.622236	26.770172	26.089224	28.181499
2		—	29.220193	53.505590	43.507955	42.684451	46.377093
3		—	35.116381	71.388952	59.358726	58.642633	64.099541
4		—	38.161975	87.296729	74.322389	73.964313	81.349010
5		—	39.171675	101.260169	88.399103	88.650133	98.125690
6		—	—	113.320322	101.589424	102.700861	114.429800
7		—	—	123.532024	113.894481	116.117435	130.261593
8		—	—	131.968954	125.316218	128.901003	145.621367
9		—	—	138.729297	135.857706	141.052978	160.509470
10		—	—	143.940704	145.523552	152.575108	174.926314
11		—	—	147.762342	154.320420	163.469549	188.872385
12		—	—	150.382299	162.257693	173.738967	202.348253
13		—	—	152.011351	169.348290	183.386644	215.354596
14		—	—	152.876593	175.609621	192.416619	227.892208
15		—	—	153.227384	181.064657	200.833844	239.962023
16		—	—	—	185.742987	208.644363	251.565141
17		—	—	—	189.681710	215.855529	262.702847
18		—	—	—	192.925912	222.476230	273.376646
19		—	—	—	195.528566	228.517142	283.588290
20		—	—	—	197.549798	233.990987	293.339822
21		—	—	—	199.055775	238.912788	302.633612
22	0	—	—	—	200.117520	243.300080	311.472406
23		—	—	—	200.809866	247.173075	319.859375
24		—	—	—	201.210565	250.554727	327.798170
25		—	—	—	201.411283	253.470702	335.292986
26		—	—	—	—	255.949227	342.348622
27		—	—	—	—	258.020855	348.970548
28		—	—	—	—	259.718178	355.164974
29		—	—	—	—	261.075536	360.938917
30		—	—	—	—	262.128754	366.300264
31		—	—	—	—	262.914910	371.257832
32		—	—	—	—	263.472121	375.821406
33		—	—	—	—	263.839330	380.001781
34		—	—	—	—	264.056097	383.810769
35		—	—	—	—	264.169466	387.261196
36		—	—	—	—	—	390.366872
37		—	—	—	—	—	393.142551
38		—	—	—	—	—	395.603860
39		—	—	—	—	—	397.767234
40		—	—	—	—	—	399.649835
41		—	—	—	—	—	401.269475
42		—	—	—	—	—	402.644549
43		—	—	—	—	—	403.793971
44		—	—	—	—	—	404.737111
45		—	—	—	—	—	405.493744
46		—	—	—	—	—	406.083994
47		—	—	—	—	—	406.528272
48		—	—	—	—	—	406.847220
49		—	—	—	—	—	407.061634
50		—	—	—	—	—	407.192395
51		—	—	—	—	—	407.263120

---

Table 6 He–Og, Ne–Og, Ar–Og, Kr–Og, Xe–Og and Rn–Og Rovibrational energies calculated through ILJ potential energy curves with the rotational quantum number j = 1

ν	j	He–Og	Ne–Og	Ar–Og	Kr–Og	Xe–Og	Rn–Og
0		8.405667	7.649373	11.770146	9.172055	8.874696	9.526304
1		—	20.164716	33.671288	26.796031	26.107153	28.195047
2		—	29.290414	53.552666	43.533230	42.702090	46.390492
3		—	35.172352	71.433944	59.383401	58.659977	64.112788
4		—	8.200502	87.339514	74.346444	73.981355	81.362104
5		—	39.191075	101.300607	88.422519	88.666865	98.138628
6		—	—	113.358254	101.612177	102.717277	114.442579
7		—	—	123.567271	113.916547	116.133527	130.274212
8		—	—	132.001316	125.337570	128.916762	145.633822
9		—	—	138.758560	135.878315	141.068397	160.521759
10		—	—	143.966642	145.543384	152.590176	174.938435
11		—	—	147.784726	154.339439	163.484257	188.884333
12		—	—	150.400893	162.275861	173.753304	202.360027
13		—	—	152.025912	169.365564	183.400598	215.366192
14		—	—	152.886857	175.625958	192.430179	227.903622
15		—	—	153.233754	181.080007	200.846995	239.973253
16		—	—	—	185.757302	208.657093	251.576182
17		—	—	—	189.694938	215.867824	262.713695
18		—	—	—	192.938003	222.488073	273.387297
19		—	—	—	195.539469	228.528517	283.598740
20		—	—	—	197.559463	234.001879	293.350066
21		—	—	—	199.064152	238.923178	302.643646
22	1	—	—	—	200.124558	243.309952	311.482225
23		—	—	—	200.815512	247.182410	319.868974
24		—	—	—	201.214765	250.563508	327.807544
25		—	—	—	201.414366	253.478910	335.302129
26		—	—	—	—	255.956845	342.357528
27		—	—	—	—	258.027866	348.979211
28		—	—	—	—	259.724563	355.173389
29		—	—	—	—	261.081278	360.947077
30		—	—	—	—	262.133834	366.308164
31		—	—	—	—	262.919310	371.265464
32		—	—	—	—	263.475821	375.828765
33		—	—	—	—	263.842309	380.008860
34		—	—	—	—	264.058337	383.817561
35		—	—	—	—	264.171150	387.267694
36		—	—	—	—	—	390.373070
37		—	—	—	—	—	393.148442
38		—	—	—	—	—	395.609439
39		—	—	—	—	—	397.772493
40		—	—	—	—	—	399.654768
41		—	—	—	—	—	401.274076
42		—	—	—	—	—	402.648811
43		—	—	—	—	—	403.797887
44		—	—	—	—	—	404.740675
45		—	—	—	—	—	405.496948
46		—	—	—	—	—	406.086831
47		—	—	—	—	—	406.530736
48		—	—	—	—	—	406.849302
49		—	—	—	—	—	407.063326
50		—	—	—	—	—	407.193690
51		—	—	—	—	—	407.264090

---

Table 7 Og–Og Rovibrational energies calculated through ILJ potential energy curves

ν	j	Og–Og	ν	j	Og–Og	ν	j	Og–Og	ν	j	Og–Og
0		12.501425	40		702.818996	0		12.514080	40		702.827310
1		37.221281	41		712.188881	1		37.233850	41		712.197050
2		61.563577	42		721.182957	2		61.576058	42		721.190979
3		85.527891	43		729.803055	3		85.540284	43		729.810928
4		109.113793	44		738.050734	4		109.126098	44		738.058455
5		132.320846	45		745.927666	5		132.333061	45		745.935235
6		155.148607	46		753.436238	6		155.160732	46		753.443653
7		177.596626	47		760.580076	7		177.608659	47		760.587334
8		199.664447	48		767.364268	8		199.676388	48		767.371365
9		221.351612	49		773.795154	9		221.363461	49		773.802085
10		242.657659	50		779.879770	10		242.669413	50		779.886529
11		263.582122	51		785.625209	11		263.593782	51		785.631790
12		284.124537	52		791.038169	12		284.136101	52		791.044568
13		304.284438	53		796.124843	13		304.295905	53		796.131057
14		324.061362	54		800.891129	14		324.072732	54		800.897155
15	0	343.454852	55	0	805.342998	15	1	343.466122	55	1	805.348837
16		362.464454	56		809.486880	16		362.475624	56		809.492530
17		381.089724	57		813.329935	17		381.100793	57		813.335396
18		399.330227	58		816.880194	18		399.341194	58		816.885466
19		417.185546	59		820.146578	19		417.196409	59		820.151660
20		434.655274	60		823.138828	20		434.666032	60		823.143718
21		451.739031	61		825.867378	21		451.749683	61		825.872074
22		468.436455	62		828.343216	22		468.447000	62		828.347716
23		484.747218	63		830.577753	23		484.757653	63		830.582053
24		500.671020	64		832.582698	24		500.681345	64		832.586794
25		516.207603	65		834.369970	25		516.217816	65		834.373860
26		531.356753	66		835.951643	26		531.366852	66		835.955321
27		546.118305	67		837.339899	27		546.128290	67		837.343362
28		560.492154	68		838.547020	28		560.502022	68		838.550263
29		574.478257	69		839.585378	29		574.488006	69		839.588399
30		588.076645	70		840.467439	30		588.086274	70		840.470234
31		601.287437	71		841.205765	31		601.296943	71		841.208331
32		614.110862	72		841.813012	32		614.120244	72		841.815344
33		626.547268	73		842.301919	33		626.556525	73		842.304015
34		638.597113	74		842.685305	34		638.606241	74		842.687162
35		650.260928	75		842.976041	35		650.269926	75		842.977655
36		661.539322	76		843.187028	36		661.548188	76		843.188397
37		672.433049	77		843.331171	37		672.441781	77		843.332291
38		682.943163	78		843.421348	38		682.951758	78		843.422215
39		693.071161	79		843.472547	39		693.079617	79		843.473215

---

Table 8 shows the calculated values for the rovibrational spectroscopic constants of all systems under study. An important fact that can be observed from this table is the good agreement of the results obtained with both methods, i.e., eqn (3) (named with the DVR method) and Dunham's method. With the exception of the He–Og dimer, there is a good agreement between the values of the rovibrational spectroscopic constants obtained with the ILJ and extended-Rydberg PEC. It can be seen that it was not possible to determine the spectroscopic constants for the He–Og system (for both ILJ and Rydberg PECs) viaeqn (3). The reason lies in the fact that this equation can only be used if there are at least 4 vibrational levels within the well of the potential energy curve. Table 8 shows also the Og–Og spectroscopic constant results available in the literature. When the results of ω_e obtained with Dunham-Rydberg and correlation formulas are compared with Jerabek's results, the differences are 2.05 cm⁻¹ and 5.45 cm⁻¹, respectively. When we make the same comparisons for the ω_ex_e vibrational constant, we have the following differences 0.001 cm⁻¹ and 0.21 cm⁻¹, respectively. When the ω_e harmonic spectroscopic constants calculated with Dunham-Rydberg and Correlation formulas are compared with the value obtained by Saue, we have the following differences 2.70 cm⁻¹ and 4.80 cm⁻¹, respectively. From these comparisons, it can be noted that the values found in the present work for the Og–Og spectroscopic constants (mainly those obtained with Dunham-Rydberg) agree well with the results of Jerabek and Saue.

Table 8 Rovibrational spectroscopic constants for the Ng–Og (Ng = He, Ne, Ar, Kr, Xe, Rn, and Og) systems using ILJ and Rydberg potential energy curves

System	Methods	ω e (cm^−1)	ω e x e (cm^−1)	ω e y e (cm^−1)	α e (cm^−1)	γ e (cm^−1)
He–Og	DVR-ILJ	—	—	—	—	—
	Dunham-ILJ	20.88670	—	—	—	—
	DVR-Rydberg	—	—	—	—
	Dunham-Rydberg	22.14117	—	—	—	—
	DVR-ILJ	15.96782	1.73275	0.00403	4.14 × 10^−3	−4.33 × 10^−4
Ne–Og	Dunham-ILJ	16.05381	1.80394	0.02444	4.40 × 10^−3	−2.45 × 10^−4
	DVR-Rydberg	16.46278	1.93427	0.04158	3.65 × 10^−3	−6.00 × 10^−4
	Dunham-Rydberg	15.75879	1.28925	−0.12236	3.93 × 10^−3	−3.12 × 10^−4
	DVR-ILJ	23.94156	1.02459	0.00328	8.94 × 10^−4	−2.35 × 10^−5
Ar–Og	Dunham-ILJ	23.93896	1.02207	0.00236	8.99 × 10^−4	−1.94 × 10^−5
	DVR-Rydberg	24.71511	1.29686	0.02626	9.23 × 10^−4	−2.55 × 10^−5
	Dunham-Rydberg	24.66753	3.39022	−1.61570	9.96 × 10^−4	9.27 × 10^−4
	DVR-ILJ	18.51106	0.44319	−0.00004	2.76 × 10^−4	−3.94 × 10^−6
Kr–Og	Dunham-ILJ	18.51139	0.44315	−0.00006	2.76 × 10^−4	−3.47 × 10^−6
	DVR-Rydberg	19.50191	0.41346	−0.01355	1.72 × 10^−4	−2.14 × 10^−5
	Dunham-Rydberg	19.47647	0.47659	−0.09941	1.70 × 10^−4	3.35 × 10^−5
	DVR-ILJ	17.87071	0.31911	0.00008	1.39 × 10^−4	−1.40 × 10^−6
Xe–Og	Dunham-ILJ	17.87086	0.31906	0.00006	1.39 × 10^−4	−1.29 × 10^−6
	DVR-Rydberg	18.28701	0.27636	−0.00593	1.12 × 10^−4	−4.67 × 10^−6
	Dunham-Rydberg	18.27687	0.31902	−0.11668	1.12 × 10^−4	3.31 × 10^−5
	DVR-ILJ	19.14233	0.23676	0.00002	7.23 × 10^−5	−4.97 × 10^−7
Rn–Og	Dunham-ILJ	19.14251	0.23674	0.00002	7.27 × 10^−5	−4.67 × 10^−7
	DVR-Rydberg	19.16661	0.24821	0.00018	7.37 × 10^−5	−5.68 × 10^−7
	Dunham-Rydberg	19.16455	0.28186	−0.10395	7.37 × 10^−5	2.04 × 10^−5
	DVR-ILJ	25.09701	0.18846	−0.00007	4.33 × 10^−5	−1.84 × 10^−7
Og–Og	Dunham-ILJ	25.09726	0.18846	−0.00007	4.29 × 10^−5	−1.73 × 10^−7
	DVR-Rydberg	24.44471	0.20151	−0.00024	4.54 × 10^−5	−2.68 × 10^−7
	Dunham-Rydberg	24.44251	0.21228	−0.05290	3.85 × 10^−5	6.75 × 10^−6
Og–Og (correlation formulas)	DVR-ILJ	16.9382	0.1740	0.00001	4.41 × 10^−5	2.48 × 10^−7
Og–Og (correlation formulas)	Dunham-ILJ	16.9384	0.1740	0.00001	4.44 × 10^−5	2.37 × 10^−7
Og–Og (Jerabek et al.^12)		22.39	0.2132	0.00017	1.8 × 10^−4	—
Og–Og (Saue et al.^13)		21.74	—	—	—	—

---

Fig. 2 shows the behavior of lifetime as a function of temperature (which ranged from 200 to 500 K) for each molecule studied. This figure reveals that the lifetime determined by both ILJ and Rydberg PEC essentially has the same behavior for the entire temperature range considered. The He–Og lifetime obtained via Rydberg PEC and ILJ PEC are slightly above 1.0 picosecond and they are the ones with the lowest lifetime values between 200 and 500 K. All other systems have a lifetime above 1.0 picosecond within the temperature range of 200–500 K. Thus, according to Wolfgang's study,⁵⁸ a lifetime of a complex above 1.0 picoseconds means that the well of the potential energy is deep enough to assure its stability and the interacting complex must be considered stable. Therefore, based on this condition, the Og–He system is close to the edge of instability and it is expected, as this system has only one vibrational level inside of the well of ILJ and Rydberg PEC. On the other hand, the Og–Og dimer has the longest lifetime in the entire temperature range considered, confirming its great stability.

Fig. 2 Lifetime as a function of temperature for the Ng–Og (Ng = He, Ne, Ar, Kr, Xe, Rn, and Og) systems using ILJ PEC, (a) and (c), and Rydberg PEC, (b) and (d).

5 Conclusions

Using the relativistic 4-component Dirac–Fock method and with the inclusion of quantum electrodynamics effects at the Gaunt interaction level and with BSSE correction, this work presents the ground state electronic potential energy curves, the vibrational energies, spectroscopic constants, and lifetime of the He–Og, Ne–Og, Ar–Og, Kr–Og, Xe–Og, Rn–Og, and Og–Og dimers.

The inclusion of quantum electrodynamics effects reduces the value of the dissociation energy of the studied dimers, which can be attributed to the fact that interactions between electrons are no longer treated as instantaneous. Furthermore, the percentage contribution of the quantum electrodynamics effect on the dissociation energy ranged from 0.48% (for the He–Og dimer) to 9.63% (for the Rn–Og dimer). This fact is also reflected in the number of vibrational levels, mainly in the dimers composed of the heaviest noble gases.

The current results obtained for the Og–Og spectroscopic constants (mainly those obtained with Dunham-Rydberg) agree well with the theoretical results of Jerabek and Saue. The lifetime as a function of temperature (which ranged from 200 to 500 K) indicates that the Og–He molecule is close to the edge of instability as expected, because this dimer has only one vibrational level inside both ILJ and Rydberg PEC. It is expected that the results of this study can motivate future spectroscopy experiments involving the He, Ne, Ar, Kr, Xe, and Rn noble gases with the oganesson (Z = 118) super-heavy element.

In the present work, the ILJ function has been adopted to fit the ab initio points. However, the obtained D_e and R_e are not only fitting parameters, but they exhibit an appropriate meaning since they are related to the fundamental physical properties of the interacting partners. This is confirmed (see Table 1) by the good agreement (except for Og–Og) between the fitting values and predictions of correlation formulas that relate the value of the basic potential parameters with the electronic polarizability of the involved partners. Moreover, ILJ provides an asymptotic attraction, where the leading C₆ coefficient, given by C₆ = D_eR_e⁶ scales along with the Ng–Og family as those predicted by correlation formulas, and for any system, their absolute values are within ≈20%.

An accurate analysis of basic intermolecular force components involved in all Ng–Og (Ng = He, Ne, Ar, Kr, Xe, and Rn) family of dimers confirmed the nature of van der Waals of the global interaction. This is also proved by the behavior of the equilibrium distance R_e whose values, along the Ng–Og family, remain confined in the restricted range of R_e = 4.50 ± 0.25 Å, while the dissociation energy D_e changes by a factor of 40± 20. This is typical behavior of systems whose repulsion component, controlling R_e, is effectively dominated by the size of the most polarizable atom (Og), while D_e varies according to the polarizability change of the Ng partner. The use of correlation formulas, defined in terms of polarizability, which represents the basic physical properties useful to scale both size repulsion and dispersion attraction, permitted us to evaluate not only the C₆ dispersion coefficient for Ng–Og systems but also the basic R_e, D_e, and C₆ interaction features of the Og–Og dimer. It has been also emphasized that with respect to the other Ng, Og exhibits an unexpected high T_bp value which determines a range of temperature, defining the liquid stability, significantly larger than 100 K. This anomalous behavior probably arises from non-conventional many-body attractive interaction contributions controlled by high relativistic effects.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors gratefully acknowledge the financial support from the Brazilian Research Councils: CAPES, CNPq and FAPDF. LGMM would like to acknowledge Desenvolvimento Científico e Tecnologico (CNPQ, under grant 408085/2021-5) and also Universidade Federal de Sao João del Rei (CCO-UFSJ).

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2cp04456k

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