Stable copernicium hexafluoride (CnF₆) with an oxidation state of VI+

Abstract

As the heaviest group 12 element known currently, copernicium (Cn) often presents the oxidation states of I+, II+, and rarely IV+ as in its homologue mercury. In this work we systematically studied the stability of some oxides, fluorides, and oxyfluorides of Cn by two-component relativistic calculations and found that the CnF₆ molecule with an oxidation state of VI+ has an extraordinary stability. CnF₆ may decompose into CnF₄ by conquering an energy barrier of about 34 kcal mol⁻¹ without markedly releasing heat. Our results indicate that CnF₆ may exist under some special conditions.

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1 Introduction

The elements of group 12 with the electronic configuration (n − 1)d¹⁰ns² usually contribute only the two ns valence electrons for bonding, and consequently their highest oxidation state (OS) is II+. Due to the stability of fully occupied (n − 1) d orbitals, they are commonly not considered to be transition metal elements by chemists. This perspective has been partly changed since the HgF₄ molecule with an OS of IV+ was experimentally verified,¹ which is distinguished from Zn and Cd where their outer d-orbitals are too inert to form a chemical bond; very recently, the kinetically stable HgF₆ molecule was also predicted in theory, where the mercury atom presents an OS of VI+.²

Moving to the heavier element copernicium (Cn with Z = 112) in group 12, the half-life of its longest-lived known isotope ²⁸⁵Cn being as long as 34 seconds,³ it was experimentally confirmed that Cn has more metallic character than early expected noble-gas-like behavior via the interaction with a gold surface,⁴ whereas modern first-principle simulations showed that Cn is a solid semiconductor⁵ or a volatile liquid.^6,7 Because of the large relativistic 7s contraction associated with the 6d expansion, the electronic configuration of Cn at its ground state is 6d_3/2⁴7s²6d_5/2⁶ at the full relativistic level⁸ and becomes 6d_3/2⁴7s²6d_5/2⁵ for Cn⁺ and 6d_3/2⁴7s²6d_5/2⁴ for Cn²⁺.⁹ Therefore, the extended 6d_5/2 spinors may be involved in chemical bonding which makes Cn adopt some high oxidation states in suitable conditions.⁸ Previous theoretical calculations have shown that CnF⁴ (IV+) is more stable than CnF₂ (II+) in the decomposition CnF₄ → CnF₂ + F₂,^8,10,11 but this is contrary to CnCl₄.^10,11 Inspired by HgF₆ previously studied by us,² it raises an interesting question of whether Cn can form a compound with a high OS of VI+. In this study, we have examined the stability of a series of Cn oxides, fluorides, and oxyfluorides and explored the roles of Cn 6d and 7s orbitals in the stabilization.

2 Computational details

For super-heavy elements the total spin S is not a good quantum number anymore due to a very strong spin–orbit coupling, and therefore two-component relativistic methods are performed throughout this study, except some analyses of properties.

In the two-component relativistic density functional theory (DFT) calculations, the PBE0 functional^12,13 is combined with either the spin–orbit relativistic effective core potentials (SOECP) to replace 92 (for Cn) and 2 (O and F) core electrons^14,15 or the all-electron exact two-component (X2C) relativistic Hamiltonian^16,17 with the atomic mean-field (AMF) approach for two-electron SO integrals¹⁸ and the atomic block approximation for unitary transformation (AU).¹⁹ AU replaces the molecular unitary transformation in the full X2C by a series of atomic ones, which is nearly a lossless approximation to X2C for light atom–light atom and heavy atom–light atom types of bonds, and may speed up analytic derivative calculations.¹⁹ The corresponding basis sets are ECP92MDFQ-TZVP (Cn)¹⁵ and Pitzer's cc-pVTZ-PP (O and F)²⁰ for SOECP/PBE0, whereas re-contracted Dyall-VTZP²¹ (Cn; see the ESI†) and x2c-TZVPall-2c²² (F) are adopted for X2C-AU/PBE0. All the atomic and molecular systems in this study are closed-shell ones,‡ except O, F, O₂, and CnF. For these open-shell systems, their closed-shell monoanions (or dianions) are calculated instead using two-component DFT and then corrected using the first (or second) ionization energies by the two-component Fock-space CCSD (coupled cluster with single and double excitation) method.²³

In order to capture electronic correlations more accurately, the two-component CCSD(T) method (CCSD with perturbative triple excitations) combined with SOECP is also used to optimize the geometries of Cn fluorides analytically,²⁴ where all the electrons, except the core ones replaced by SOECP,^14,15 are correlated. The adopted valence basis sets are ECP92MDFQ-TZVP for Cn¹⁵ and the re-contracted cc-pVTZ-PP for F.²⁵ At the optimized geometries, a two-point extrapolation scheme²⁶ with triple- and quadruple-ζ basis sets^15,25 is carried out to estimate the complete basis set (CBS) limit (see the ESI† for details).

The DIRAC 18²⁷ and CFOUR 2.0β²⁸ program packages are used for SOECP/PBE0 and SOECP/CCSD(T) calculations, respectively, whereas the X2C-AU/PBE0 calculations are performed using the BDF program²⁹ to utilize the analytic derivative algorithm for geometry optimization and vibrational frequency calculations.¹⁹ The oxidation states are estimated using the LOBA (localized orbital bonding analysis) procedure³⁰ implemented in the Multiwfn³¹ program.

3 Results

3.1 Stability of Cn oxides, fluorides, and oxyfluorides

Table 1 collects the decomposition energies (ΔE) of oxides, fluorides, and oxyfluorides of Cn calculated using the two-component relativistic DFT with the PBE0 hybrid functional^12,13 and SOECP, which are compared with literature data.^10,11,32 The zero-point energies (ZPEs) are not included in these results. In the Cn fluorides (decomposition reactions 1–5), CnF₄ is more stable than CnF₂, being consistent with early studies.^8,11 More interestingly, the energy change in reaction 5 is quite small, indicating that CnF₆ is more likely to exist than the lighter homologue HgF6 by comparing their reaction energies.² The decomposition of CnF₆ will be comprehensively studied later by more accurate methods. In the decomposition of Cn oxides (reactions 6–10), the dissociation energy of CnO is only 5.7 kcal mol⁻¹, being close to the one of 4.6 kcal mol⁻¹ reported by Titov et al.³² and comparable to the value of 4.7 kcal mol⁻¹ for the homologue HgO.³³ CnO₂ has a considerable vertical excitation energy of 1.44 eV by SOECP/TDDFT for the first excited spinor state (I)0_u⁻, and it is 2 and 73 kcal mol⁻¹ more stable in energy than CnO and CnO₃, respectively, according to reactions 7 and 9. However, the exothermic reaction 8 indicates that CnO₂ is not thermodynamically stable even if it exists. As for the Cn oxyfluorides CnOF₂ and CnO₂F₂ (reactions 11–14), they are thermodynamically unstable and could decompose spontaneously into CnF₂ and O₂, so there is no possibility of their existence.

Table 1 Decomposition energies (without ZPE correction) of Cn oxides, fluorides, and oxyfluorides by SOECP/PBE0

No.	Reactant	Symmetry	Product	ΔE (kcal mol^−1)
				TW^a	Reference
a This work. b 4c-Dirac–Fock MP2 results from ref. 10. c 2c-ZORA/B3LYP/STO-QZ4P results from ref. 11. d Kramers-unrestricted SOECP/PBE0 results from ref. 32.
1	CnF	C ∞v	Cn + F	18.0	13.4^b
2	CnF2	D ∞h		63.3
3			Cn + F2	62.3	60.3^b, 63.7^c
4	CnF4	D 4h	CnF2 + F2	28.2	23.1,^b 27.4^c
5	CnF6	O h	CnF4 + F2	−1.4
6	CnO	C ∞v	Cn + O	5.7	4.6^d
7	CnO2	D ∞h		2.2
8			Cn + O2	−50.7
9	CnO3	D 3h		−73.0
10			CnO + O2	−70.8
11	CnOF2	C 2v	CnO + F2	35.2
12				−79.9
13	CnO2F2	C 2v	CnO2 + F2	4.6
14			CnF2 + O2	−108.3

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To further confirm the stability of CnF₆, the decomposition reaction (I) CnF₆ → CnF₄ + F₂ (i.e. reaction 5 in Table 1) is studied by X2C-AU/PBE0. Furthermore, reaction (II) CnF₆ → CnF₄ + 2F not included in Table 1 is also computed at the same level since our early study of HgF₆ showed that the homologue reaction HgF₆ → HgF₄ + 2F is the rate-determining step.² The two reactions are demonstrated in Fig. 1, whereas the barrier heights at transition states (TS) and reaction energies are collected in Table 2 and are compared with the results by SOECP/PBE0 as well as the more accurate SOECP/CCSD(T)/CBS method.

Fig. 1 Decomposition paths of CnF₆. The red and blue dotted curves represent the reaction paths of (I) CnF₆ → CnF₄ + F₂ and (II) CnF₆ → CnF₄ + 2F, respectively. Values in parenthesis are ΔE^ZPE computed by X2C-AU/PBE0 or SOECP/CCSD(T)/CBS (underlined).

Table 2 Relative energies (in kcal mol⁻¹) of the decomposition reactions (I) CnF₆ → CnF₄ + F₂ and (II) CnF₆ → CnF₄ + 2F

Reaction		Method^a	ΔE	ΔE^ZPE	ΔH^298.15	ΔG^298.15
a (A) X2C-AU/PBE0, (B) SOECP/PBE0 and (C) SOECP/CCSD(T)/CBS. b ZPE and thermochemical quantities are computed by X2C-AU/PBE0.
(I)		(A)	−0.1	−1.7	−1.1	−13.0
		(B)^b	−1.4	−3.0	−2.4	−14.3
		(C)^b	3.2	1.6	2.2	−9.7
	TS1	(A)	82.9	81.3	81.6	78.6
(II)		(A)	35.7	32.5	34.0	15.7
		(B)^b	35.7	32.5	34.0	15.7
		(C)^b	36.8	33.6	35.1	16.8
	TS2	(A)	36.4	33.6	34.7	28.3

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Reaction (I) is slightly exothermic by X2C-AU/PBE0 and SOECP/PBE0 but endothermic by SOECP/CCSD(T)/CBS in enthalpies at 298.15 K (ΔH^298.15); nevertheless, after including the entropic contributions (mainly due to the translational entropy), the negative Gibbs free energy changes at 298.15 K (ΔG^298.15) at the three levels of theory imply that this reaction is exothermic at high temperatures. Because of a very high energy barrier at TS₁ of about 81 kcal mol⁻¹ to be conquered, reaction (I) seems very difficult to take place. Reaction (II) is intensely endothermic with a lower barrier of about 34 kcal mol⁻¹ at TS₂. From the spin-free (sf) perspective, TS₂ is mainly caused by the avoided crossing between the closed-shell ground state and the repulsive first triplet state (F˙⋯CnF₄⋯F˙). It should be noted that the two isolated fluorine atoms in the products may be severely exothermic by forming a fluorine molecule, leading to the same products as in reaction (I), and moreover this process may be accelerated if there are other isolated fluorine atoms detached from adjacent CnF₆ molecules. Consequently, the final products of CnF₆ decomposition are CnF₄ and F₂, which is protected by a barrier of about 34 kcal mol⁻¹ at TS₂. Utilizing the transition state theory with ΔG^298.15 = 28.3 kcal mol⁻¹ given in Table 2, the estimated rate constant of CnF₆ decomposition is 1.1 × 10⁻⁸ s⁻¹, leading to a half-life of 2.0 years. It can be expected accordingly that CnF₆ is both thermodynamically and dynamically stable at least at low temperatures under gas phase or rare-gas matrix isolation conditions. This result corrects also Malli's early study of CnF₆ (probably the first and the only study of CnF₆ published before this work) where CnF₆, as well as HgF₄ and CnF₄, was considered “unbound” by Dirac–Hartree–Fock calculations.³⁴ This contradiction is mainly due to the lack of important electronic correlations in the latter: a comparison between the SOECP/HF and SOECP/CCSD(T) energies shows that the contributions of electronic correlations are about +25 (HgF₄), +17 (CnF₄), and +31 kcal mol⁻¹ (CnF₆) to ΔE.

3.2 Properties of Cn fluorides

Some structural and electronic properties of Cn fluorides ²⁸⁵CnF_n (n = 1, 2, 4, and 6) are summarized in Table 3, including point group symmetry, Cn–F bond length (R_Cn–F), energy gap (ε_HOMS–LUMS) between the highest occupied molecular spinor (HOMS) and the lowest unoccupied molecular spinor (LUMS), vertical excitation energy of the first excited spinor state (T¹_v), generalized Mayer's bond order (BO) of the Cn–F bond, natural electron configuration and NPA atomic charge (Q) of Cn by natural bond orbital (NBO) analysis³⁵ implemented in the NBO 6.0 program,³⁶ Cremer's adiabatic force constant^37,38 of local Cn–F bond stretching (k^a_Cn–F), and harmonic vibrational frequencies (ω) with the corresponding irreducible representation. Mulliken population analyses on spin-free and spin-dependent wavefunctions are consistent within 10%, and the deviation of spin-free NBO analysis may be expected to be within this range.

Table 3 Properties of CnF_n (n = 1, 2, 4, and 6) by X2C-AU/PBE0

	CnF	CnF2	CnF4	CnF6
a By SOECP/PBE0. b By SOECP/CCSD(T). c Spin-free calculations.
Symmetry	C ∞v	D ∞h	D 4h	O h
R Cn–F (Å)	2.069	1.920	1.927	1.933
	2.108^a	1.926^a	1.929^a	1.932^a
	—	1.958^b	1.948^b	1.951^b
OSCn	I+	II+	IV+	VI+
ε HOMS–LUMS (eV)	4.06	7.19	6.22	5.68
T ^1v (eV)		2g: 4.69	A 1g: 3.23	E g: 2.92
BOCn–F	0.51	0.81	0.82	0.81
Configuration^c	6d^9.77s^1.8	6d^9.47s^1.4	6d^8.97s^1.0	6d^8.57s^0.8
Q Cn (|e|)^c	+0.44	+1.11	+1.97	+2.57
k ^aCn–F (mDyn Å^−1)	1.92	4.77	4.63	4.50
ω (cm^−1)	σ ^+: 427	πu: 227	b 2g: 167	t 2g: 163
		σ g ^+: 673	b 2u: 212	t 2u: 230
		σ u ^+: 676	a 2u: 243	t 1u: 259
			e u: 248	e g: 621
			b 1g: 639	a 1g: 644
			a 1g: 659	t 1u: 690
			e u: 684

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The closed-shell ground states of the polyatomic CnF_n molecules may be confirmed by the large HOMS–LUMS gaps and the considerable first excitation energies, and the 3(n + 1)-6 real frequencies without an imaginary one guarantee these high-symmetric minima. The BO values of Cn–F show that it is basically a single bond in CnF₂, CnF₄, and CnF₆ but a half bond in CnF, which are in qualitative agreement with the trends of Cn–F bond lengths and force constants. The half bond in CnF may be well explained by the three-electron in two-orbital model:³⁹ the covalent interactions between the Cn 6d_σ + 7s hybrid orbital and F 2p_σ shift the former remarkably to form a singly occupied antibonding orbital, leading to a simplified electronic configuration of (σ)²(σ*)¹ as in the case of the lighter homologue HgF, and therefore the empirical bond order is (2 − 1)/2 = 0.5. This means that the Nalewajski–Mrozek bond order of 1.0 for HgF in ref. 2 might be too rough.

With the help of natural localized molecular orbital analysis, the atomic composition of each Cn–F natural bond orbital in CnF_n may be estimated, as provided in Table 4. It can be seen that the bond orbitals are mainly due to the Cn 6d–F 2p_σ covalent interactions, accompanied by some secondary Cn 7s–F 2p_σ ones, being contrary to the cases of HgF_n² due to the exchanging order of (n)s and (n − 1)d. The (n)p hybridizations also play important roles in lighter homologues HgF₂, HgF₄, and HgF₆² but are negligible in CnF₂, CnF₄, and CnF₆, indicating that Cn 6d is much easier to bond than Hg 5d without the involvement of Cn 7p.

Table 4 NBO analyses of bonding between Cn and F in CnF_n

Species	Composition of each natural bond orbital
CnF	54% Cn (40% 7s + 60% 6d), 46% F (1% 2s + 99% 2pσ)
CnF2	27% Cn (49% 7s + 50% 6d), 73% F (8% 2s + 92% 2pσ)
CnF4	33% Cn (34% 7s + 65% 6d), 67% F (5% 2s + 95% 2pσ)
CnF6	38% Cn (24% 7s + 76% 6d), 61% F (4% 2s + 96% 2pσ)

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It is just very apparent that the NPA atomic charges of Cn are almost equal to those of the lost electrons on the 6d7s valence orbitals, which significantly increase from +0.44 |e| for CnF (I+) to +1.11 |e| for CnF₂ (II+), +1.97 |e| for CnF₄ (IV+), and +2.57 |e| for CnF₆, so the oxidation state of Cn in CnF₆ should be the highest one in these species. On the other hand, six bonding orbitals for the six Cn–F single bonds of CnF₆ have been formed, as plotted in Fig. 2. These two proofs suggest that the oxidation state of Cn in CnF₆ can be as large as VI+, which is broadly in line with the ratio of NPA atomic charges of Cn in CnF₄ and CnF₆, i.e. Q^IV+_Cn : Q^VI+_Cn = 0.76 : 1 ≈ OS^IV+_Cn : OS^VI+_Cn = 0.67 : 1. As a result of validation, the formal oxidation states of Cn in the four fluorides are estimated by LOBA, being respectively I+, II+, IV+, and VI+ as we expect.

Fig. 2 Occupied bonding orbitals of Cn–F in CnF₆.

In experiments, the oxidation states of target atoms may be probed by the Mössbauer isomer shifts, which are closely related to the contact densities or effective contact densities, i.e. the total electron densities of occupied s and p_1/2 spinors around the nucleus.^40,41 We have computed the effective contact densities of Cn in the fluorides by X2C-AU/PBE0 as well as the spin-free variant (see also the sf-X2C-AU/CCSD(T) results in the ESI†). The density changes (Δρ) relative to the reference density of the isolated Cn atom are shown in Fig. 3 and are compared with the results of mercury fluorides. Contrary to the non-monotonic decrease of density in mercury fluorides,^40,41 Δρ in copernicium fluorides decreases monotonically in a greater ratio, being proportional to the oxidation state of Cn. The two different trends of Δρ are not surprising since the screening effect of Cn 6d_5/2 (or 6d in the spin-free case) is relatively weaker than the one of Hg 5d_5/2 (or 5d) considering E(6d_3/2) < E(7s) < E(6d_5/2) (or E(7s) < E(6d)).

Fig. 3 Effective contact density changes of Hg and Cn in fluorides.

4 Conclusions

In this study we reported a stable compound of copernicium, CnF₆. Comprehensive calculations show that the CnF₆ molecule is protected by a considerable energy barrier of about 34 kcal mol⁻¹ in the decomposition CnF₆ → CnF₄ + F₂, being almost as stable as CnF₄, and therefore CnF₆ may exist under cryogenic conditions in rare-gas matrices as the experimental ones of HgF₄.¹ Various analyses of properties indicate that the Cn atom in CnF₆ possesses a rarely high oxidation state of VI+, which may be attributed to the relativistic contraction of 7s and expansion of 6d. In the oxides and oxyfluorides of Cn, however, the oxidation states of both IV+ and VI+ are not stable due to the weaker electronegativity of oxygen relative to fluorine, and thus CnF₄ and CnF₆ are probably the only two species of Cn with higher oxidation states than II+. We hope that our results will deepen the understanding of group 12 chemistry.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant No. 21976014 and 22073072) and was also supported by The Double First-class University Construction Project of Northwest University.

Notes and references

1. X. Wang, L. Andrews, S. Riedel and M. Kaupp, Angew. Chem., Int. Ed., 2007, 46, 8371–8375.
2. C. Gao, S.-X. Hu, H. Han, G. Guo, B. Suo and W. Zou, Comput. Theor. Chem., 2019, 1160, 14–18.
3. Y. T. Oganessian, V. K. Utyonkov, Y. V. Lobanov, F. S. Abdullin, A. N. Polyakov, I. V. Shirokovsky, Y. S. Tsyganov, G. G. Gulbekian, S. L. Bogomolov, B. N. Gikal, A. N. Mezentsev, S. Iliev, V. G. Subbotin, A. M. Sukhov, A. A. Voinov, G. V. Buklanov, K. Subotic, V. I. Zagrebaev, M. G. Itkis, J. B. Patin, K. J. Moody, J. F. Wild, M. A. Stoyer, N. J. Stoyer, D. A. Shaughnessy, J. M. Kenneally, P. A. Wilk, R. W. Lougheed, R. I. Il'kaev and S. P. Vesnovskii, Phys. Rev. C, 2004, 70, 064609.
4. R. Eichler, N. V. Aksenov, A. V. Belozerov, G. A. Bozhikov, V. I. Chepigin, S. N. Dmitriev, R. Dressler, H. W. Gäggeler, A. V. Gorshkov, M. G. Itkis, F. Haenssler, A. Laube, V. Y. Lebedev, O. N. Malyshev, Y. T. Oganessian, O. V. Petrushkin, D. Piguet, A. G. Popeko, P. Rasmussen, S. V. Shishkin, A. A. Serov, A. V. Shutov, A. I. Svirikhin, E. E. Tereshatov, G. K. Vostokin, M. Wegrzecki and A. V. Yeremin, Angew. Chem., Int. Ed., 2008, 47, 3262–3266.
5. N. Gaston, I. Opahle, H. W. Gäggeler and P. Schwerdtfeger, Angew. Chem., Int. Ed., 2007, 46, 1663–1666.
6. J.-M. Mewes, O. R. Smits, G. Kresse and P. Schwerdtfeger, Angew. Chem., Int. Ed., 2019, 58, 17964–17968.
7. J.-M. Mewes and P. Schwerdtfeger, Angew. Chem., Int. Ed., 2021, 60, 7703–7709.
8. M. Seth, P. Schwerdtfeger and M. Dolg, J. Chem. Phys., 1997, 106, 3623.
9. D. C. Hoffman, D. M. Lee and V. Pershina, in The Chemistry of the Actinide and Transactinide Elements, 4th edn, ed. L. R. Morss, N. M. Edelstein and J. Fuger, Springer, Dordrecht, 2010, p. 1672.
10. K. G. Dyall, Theor. Chem. Acc., 2011, 129, 603–613.
11. A. Ghosh and J. Conradie, Eur. J. Inorg. Chem., 2016, 2989–2992.
12. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868.
13. C. Adamo and V. Barone, J. Chem. Phys., 1998, 110, 6158–6170.
14. L. F. Pacios and P. A. Christiansen, J. Chem. Phys., 1985, 82, 2664–2671.
15. T. Hangele and M. Dolg, J. Chem. Phys., 2013, 138, 044104.
16. W. Kutzelnigg and W. Liu, J. Chem. Phys., 2005, 123, 241102.
17. W. Liu and D. Peng, J. Chem. Phys., 2009, 131, 031104.
18. J. Liu and L. Cheng, J. Chem. Phys., 2018, 148, 144108.
19. W. Zou, G. Guo, B. Suo and W. Liu, J. Chem. Theory Comput., 2020, 16, 1541–1554.
20. J.-P. Blaudeau, S. R. Brozell, S. Matsika, Z. Zhang and R. M. Pitzer, Int. J. Quant. Chem., 2000, 77, 516–520.
21. K. G. Dyall, Theor. Chem. Acc., 2011, 129, 603–613.
22. P. Pollak and F. Weigend, J. Chem. Theory Comput., 2017, 13, 3696–3705.
23. L. Visscher, E. Eliav and U. Kaldor, J. Chem. Phys., 2001, 115, 9720–9726.
24. F. Wang and J. Gauss, J. Chem. Phys., 2008, 129, 174110.
25. D. Deng, Y. Lian and W. Zou, Chem. Phys. Lett., 2017, 685, 251–258.
26. A. Halkier, T. Helgaker, P. Jørgensen, W. Klopper, H. Koch, J. Olsen and A. K. Wilson, Chem. Phys. Lett., 1998, 286, 243–252.
27. T. Saue, R. Bast, A. S. P. Gomes, H. J. A. Jensen, L. Visscher, I. A. Aucar, R. Di Remigio, K. G. Dyall, E. Eliav, E. Fasshauer, T. Fleig, L. Halbert, E. D. Hedegård, B. Helmich-Paris, M. Iliaš, C. R. Jacob, S. Knecht, J. K. Laerdahl, M. L. Vidal, M. K. Nayak, M. Olejniczak, J. M. H. Olsen, M. Pernpointner, B. Senjean, A. Shee, A. Sunaga and J. N. P. van Stralen, J. Chem. Phys., 2020, 152, 204104.
28. D. A. Matthews, L. Cheng, M. E. Harding, F. Lipparini, S. Stopkowicz, T.-C. Jagau, P. G. Szalay, J. Gauss and J. F. Stanton, J. Chem. Phys., 2020, 152, 214108.
29. Y. Zhang, B. Suo, Z. Wang, N. Zhang, Z. Li, Y. Lei, W. Zou, J. Gao, D. Peng, Z. Pu, Y. Xiao, Q. Sun, F. Wang, Y. Ma, X. Wang, Y. Guo and W. Liu, J. Chem. Phys., 2020, 152, 064113.
30. A. J. W. Thom, E. J. Sundstrom and M. Head-Gordon, Phys. Chem. Chem. Phys., 2009, 11, 11297–11304.
31. T. Lu and F. Chen, J. Comput. Chem., 2012, 33, 580–592.
32. A. Oleynichenko, A. Zaitsevskii, S. Romanov, L. V. Skripnikov and A. V. Titov, Chem. Phys. Lett., 2018, 695, 63–68.
33. B. C. Shepler and K. A. Peterson, J. Phys. Chem. A, 2003, 107, 1783–1787.
34. G. L. Malli, in Fundamental World of Quantum Chemistry: A Tribute to the Memory of Per-Olov Löwdin, ed. E. J. Brändas and E. S. Kryachko, Springer, Dordrecht, 2004, vol. III, pp. 334–348.
35. A. E. Reed, R. B. Weinstock and F. Weinhold, J. Chem. Phys., 1985, 83, 735–746.
36. E. D. Glendening, J. K. Badenhoop, A. E. Reed, J. E. Carpenter, J. A. Bohmann, C. M. Morales, C. R. Landis and F. Weinhold, NBO version 6.0, Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, 2013.
37. Z. Konkoli and D. Cremer, Int. J. Quant. Chem., 1998, 67, 1–9.
38. E. Kraka, W. Zou and Y. Tao, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2020, 10, 1480.
39. D. Cremer, E. Kraka and M. Filatov, ChemPhysChem, 2008, 9, 2510–2521.
40. S. Knecht, S. Fux, R. Van Meer, L. Visscher, M. Reiher and T. Saue, Theor. Chem. Acc., 2011, 129, 631–650.
41. H. Zhu, C. Gao, M. Filatov and W. Zou, Phys. Chem. Chem. Phys., 2020, 22, 26776–26786.

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Footnotes

† Electronic supplementary information (ESI) available: Computational details, optimized structures by SOECP/PBE0, X2C-AU/PBE0, and SOECP/CCSD(T), and harmonic vibrational frequencies by X2C-AU/PBE0. See DOI: 10.1039/d1cp04360a

‡ Although CnO₂ is an open-shell system in the spin-free case with a configuration of π_g², it becomes a closed-shell one in the two-component case where the configuration is π_1/2g²π_3/2g⁰ with a spin–orbit splitting of about 3.0 eV.

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