Research paperCoupled cluster study of spectroscopic constants of ground states of heavy rare gas dimers with spin–orbit interaction
Graphical abstract
The CCSD(T) approach with spin–orbit interaction just included in the coupled cluster iteration based on two-component relativistic effective core potential provides the accurate spectroscopic constants of ground states of Kr2, Xe2 and Rn2 dimers compared to the available experimental values and benchmark results.
Introduction
The electronic configuration of heavy rare gas (Rg) atom is (n−1)s2(n−1)p6(n−1)d10ns2np6. The dimers Rg2 are weakly bonded diatomic molecules with electronic configuration , using non-relativistic notation. The spectroscopy of Rg2 (and Rg3) have been reported in theoretical [1], [2] and experimental [3] researches. Theoretically, the highly precise theoretical approaches are necessary to obtain accurate spectroscopic constants of Rg2.
It is well known that density functional theory (DFT) is a popular and low-cost computational protocol [4]. Unfortunately, the advantage of DFT is describing the molecules containing covalent bond and light element [5] although the van der Waals interaction based on Kohn Sham theory has been considered in some previous papers to improve the performance of DFT [5], [6], [7], [8]. Thus DFT cannot provide the benchmark results.
Another very interesting theoretical method, namely range separated DFT, pioneered by Savin (see Ref. [9] and references therein) can be adopted to study the spectroscopy of Rg2. In this approach, wave function theory (WFT) is employed to treat the long range region (lrWFT) and DFT is used to treat the short range region (srDFT) [10], [11], [12], [13], [14], [15]. In one recent paper, [16] based on the eXact 2-Component (X2C) molecular-mean field Hamiltonian, the spectroscopic constants of Xe2, Rn2 (and (E118)2) obtained from lrCCSD(T)-srLDA approach by Saue et al. are close to their own benchmark results obtained from CCSD(T) approach. Although the lrWFT-srDFT approach can in some extent remedy the deficiency of DFT in the long range region, it is still inappropriate to be used to provide the benchmark results.
The coupled cluster theory at the CCSD(T) level is the ‘gold standard’ of quantum chemistry for the electronic ground state with a dominant single reference character. Hobza et al. has indicated that CCSD(T) approach combined with extrapolation to the complete basis set (CBS) limit can provide chemical accuracy of 1 kcal/mol for such systems [17] in spite of its high-cost. Furthermore, both scalar relativistic effect (SRE) and spin–orbit interaction (SOI) [18] are important to obtain accurate spectroscopic constants of heavier Rg2. An efficient method to treat SRE and SOI is adopting the two-component relativistic effective core potential (2c-RECP) [19]. In this approach, the SOI operator is taken from the 2c-RECP and simplified to a one-electron operator. Besides, it has been indicated that considering SOI only in the post Hartree Fock (HF) iteration is an efficient and highly precise treatment for SOI based on 2c-RECP [19]. Firstly, the HF section and the integral transformation is the same as in the nonrelativistic or scalar relativistic calculations. Secondly, both the molecular orbitals (MOs) and the two-electron integrals in the MO representation are real and can be classified according to the irreducible representation of the molecular single point group. The above two issues have been explored and the coupled cluster theory with SOI just included in the coupled cluster iteration (SOI-CC) based on 2c-RECP has been achieved [19]. Moreover, Lee et al. confirmed that such a two-step procedure in which SOI is included only in the post HF procedure works better with coupled cluster than configuration interaction since the former is better at describing the orbital relaxation [20]. The authors think that the SOI-CC approach based on 2c-RECP should be widely utilized since its high efficiency and precision. Thus the SOI-CC approach at the CCSD(T) level (SOI-CCSD(T)) based on 2c-RECP is adopted to study the spectroscopic constants of ground states of Kr2, Xe2 and Rn2 in present work.
To authors’ best knowledge, the spectroscopic constants of ground state of Rn2 have no experimental reports yet and only a limited number of theoretical studies have been reported [16], [21], [22], [23], [24]. Runeberg and Pyykkö have employed CCSD(T) method based on relativistic large core pseudopotentials to study the spectroscopic constants of ground states of Xe2 and Rn2 [21]. A similar computational protocol was employed by Nash to obtain the corresponding values for Rn2 (and (E118)2) [22]. The spectroscopic constants of ground states of He2-(E118)2 are reported by Kullie and Saue using lrMP2-srDFT approach based on the four-component relativistic Dirac-Coulomb (DC) Hamiltonian [24]. Most recently, as mentioned above, Saue et al. performed the CCSD(T) calculations based on the X2C molecular-mean field Hamiltonian to provide the benchmark results of spectroscopic constants of ground states of Xe2, Rn2 (and (E118)2) [16].
It is worthwhile to mention that, for dissociation energy of Rn2, Runeberg and Pyykkö’s value is 222.6 cm−1 [21], Nash’s value is 129.1 cm−1 [22], and Kullie and Saue’s value is 323.9 cm−1 [24]. Nevertheless, the recent benchmark results by Saue et al. is 282.80/281.41 cm−1 [16]. Thus the motivation of present work is to confirm that the SOI-CCSD(T) approach based on 2c-RECP developed by Wang et al. [19] can provide reliable dissociation energy (and equilibrium bond length, harmonic frequency) of ground state of Rn2 (and Kr2, Xe2) compared to the benchmark results [16] (and the available experimental values).
This paper is organized as follows: essential computational details such as the treatment of SOI and extrapolation to the CBS limit of electronic correlation energy are given in Section 2. In Section 3, we present and discuss our calculated values of spectroscopic constants including equilibrium bond length, harmonic frequency and dissociation energy of ground states of Kr2, Xe2 and Rn2 obtained from both SOI-CCSD(T) and CCSD(T) approaches based on 2c-RECP, compared to the available experimental values and other group’s theoretical values. The conclusion of present work is given in Section 4.
Section snippets
Computational details
The theory of SOI-CCSD(T) approach based on 2c-RECP employed in present work is given in Ref. [19] and references therein, thus it will not be repeated here. However, it is worthwhile to stress the treatment of SOI. In the pseudopotential potential (PP) approximation, the SOI part which has the form of one-electron operator is written as [25], [26]
In Eq. (1), and are, respectively, spin and
Results and discussions
The calculated spectroscopic constants of ground states of Kr2, Xe2 and Rn2 obtained from both CCSD(T) and SOI-CCSD(T) approaches based on 2c-RECP in the finite basis sets, namely cc-pwCVXZ-PP (X = Q, 5) and the CBS limit, namely cc-pwCV∞Z-PP, are list in Table 1. The outer-core and valence atomic orbitals (n−1)s(n−1)p(n−1)dnsnp which are not contained in the RECPs are correlated. The available experimental values and other group’s calculated results are also given in Table 1 for comparison.
For Kr
Conclusion
The spectroscopic constants of ground states of Kr2, Xe2 and Rn2 are obtained from both CCSD(T) and SOI-CCSD(T) approaches based on 2c-RECP. The SOI is simplified to a one-electron operator and just included in the post HF iteration, i.e. in the coupled cluster iteration. The atomic orbitals (n−1)s(n−1)p(n−1)dnsnp which are not contained in RECPs are correlated. Thus the present coupled cluster calculations are low-cost compared to the all-electron relativistic calculations. The finite basis
Acknowledgements
Zhe-Yan Tu thanks Prof. Fan Wang of Sichuan University for helpful discussions about the SOI-CCSD(T) calculations. We thank Dr. Yao-Heng Su of Xi’an Polytechnic University and Dr. Dong-Dong Yang of Ludong University for useful comments and suggestions on our manuscript. This work is supported financially by the National Natural Science Foundation of China (Grant Nos. 21503153, 21301134 and 51402230), the Natural Science Foundation of Shaanxi Province (Grant Nos. 2014JM1025 and 2015JM6282), the
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