Element 118: The First Rare Gas with an Electron Affinity

VOLUME 77, NUMBER 27 PHYSICAL REVIEW LETTERS 30 DECEMBER 1996 Element 118: The First Rare Gas with an Electron Affinity Ephraim Eliav and Uzi Kaldor School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel Yasuyuki Ishikawa Department of Chemistry, University of Puerto Rico, P.O. Box 23346, San Juan, Puerto Rico 00931–3346 Pekka Pyykkö Department of Chemistry, P.O. Box 55 (A.I. Virtasen aukio 1), FIN-00014 University of Helsinki, Finland (Received 27 August 1996) The electron affinity of the rare gas element 118 is calculated by the relativistic coupled cluster method based on the Dirac-Coulomb-Breit Hamiltonian. A large basis set (34s26p20d14f9g6h4i) of Gaussian-type orbitals is used. The external 40 electrons are correlated. Inclusion of both relativity and correlation yields an electron affinity of 0.056 eV, with an estimated error of 0.01 eV. Nonrelativistic or uncorrelated calculations give no electron affinity for the atom. [S0031-9007(96)02040-6] PACS numbers: 31.30.Jv, 27.90.+b, 31.25.Eb Studies of superheavy elements have been partially motivated by the hope of finding exotic, unexpected electronic properties, due to the relativistic effects, such as the stabilization of s shells or the destabilization of high-l (d and f) shells. We have earlier shown that the groundstate d 10 s1 electron configuration of the lighter coinage metals (Cu, Ag, Au) is replaced by d 9 s2 for ekagold, element 111 [1], and that analogous changes occur for ekamercury, element 112 [2]. Even the next element, the main-group ekathallium E113, has a chance of behaving as a transition metal [3]. Not only the occupied but also the initially empty s levels are stabilized by relativity. As an example, the yellow color of PbCl22 6 was attributed to the relativistic stabilization of the low-lying, empty a1 molecular orbital [4]. We now consider the possibility that the stabilization of the 8s shell would be large enough to give an electron affinity to ekaradon, element 118. The first known case of a closed-shell atom with an electron affinity was that of Ca [5–7]. The added electron there had the same principal quantum number as the valence electrons, with the configuration ns2 np. Here we consider adding an electron with a higher principal quantum number, yielding a ns2 np 6 sn 1 1ds configuration. Long-lived Xe2 has been reported [8], but no state assignments were made, and it was not determined whether the observation corresponded to a bound state or a resonance. Calculations by Nicolaides and Aspromallis [9] found no bound state of Xe2 . The method we employ is relativistic coupled cluster (RCC) with single and double excitations. This method has been applied to a series of heavy elements, including Au [10], Pr31 and U41 [11], Yb and Lu [12], Hg [2], Tl [3], and Ra [13], as well as the superheavy elements Lr [12], 104 [14], 111 [1], 112 [2], and 113 [3]. The properties calculated are primarily transition energies, 5350 0031-9007y96y77(27)y5350(3)$10.00 including ionization potentials, excitation energies, and electron affinities. Good agreement with experimental values is obtained when the latter are known; in other cases predictions may be made regarding order and separation of electronic states. The RCC method with single and double excitations includes relativistic and correlation effects simultaneously to high order. A detailed description of the method may be found in earlier papers [10,11], and only a brief account is given here. We start from the projected DiracCoulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian [15–18] ∑X 1 H1 ­ L fcai ? pi 1 c2 sbi 2 1d i ∏ X 1 Vnuc sidg 1 V si, jd L1 . (1) i,j The nuclear potential Vnuc includes the effect of finite nuclear size. L1 is a product of projection operators onto the positive energy states of the Dirac Hamiltonian. The Hamiltonian H1 has normalizable, bound-state solutions. Equation (1) is the no-virtual-pair approximation, with virtual electron-positron pairs not allowed in intermediate states. The two-electron potential in Coulomb gauge, correct to second order in the fine-structure constant a, is the Coulomb-Breit potential [16,17,19] V ­ 1 1 2 2 fa1 ? a2 1 sa1 ? r12 d sa2 ? r12 dyr12 g, r12 2r12 (2) where the second term is the frequency-independent Breit interaction. Using the second quantization formalism, the DCB Hamiltonian H1 is rewritten in terms of normal-ordered products of the spinor operators, hr 1 sj and hr 1 s1 utj © 1996 The American Physical Society VOLUME 77, NUMBER 27 PHYSICAL REVIEW LETTERS [17,20] H ­ H1 2 k0jH1 j0l ­ X frs hr 1 sj 1 rs 1 X krsjjtul hr 1 s1 utj , 4 rstu (3) where frs and krsjjtul are, respectively, elements of oneelectron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac fourcomponent spinors. The effect of the projection operator L1 is now taken over by normal ordering, denoted by the curly braces in the equation above, which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive energy state, and the negative energy states are ignored. The no-pair approximation leads to a natural and straightforward extension of the nonrelativistic open-shell CC theory. The multireference valence-universal Fock space coupled-cluster approach is employed here, which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. According to Lindgren’s formulation of the open-shell CC method [21], the effective Hamiltonian has the form Heff ­ PHVP, V ­ hexpsSdj , (4) where V is the normal-ordered wave operator, and the excitation operator S is defined with respect to a closedshell reference determinant. In addition to the traditional decomposition into terms with different total (l) number of excited electrons, S is partitioned according to the number of valence holes (m) and valence particles (n) to be excited with respect to the reference determinant, ! √ X X X sm,nd Sl . (5) S­ m$0 n$0 l$m1n In the present application we use the sm, nd ­ s0, 0d and (0,1) sectors. The lower index l is truncated at l ­ 2. The resulting coupled-clusters-singles-anddoubles (CCSD) scheme involves the fully self-consistent, iterative calculation of all one- and two-body virtual excitation amplitudes, and sums all diagrams with these excitations to infinite order. Here we start by solving the all-electron Dirac-Fock-Coulomb (DFC) or DiracFock-Breit (DFB) equations for the closed-shell neutral E118 atom, which defines the (0,0) sector. This state is correlated by CCSD; an electron is then added in the 8s orbital, recorrelating the whole system. The correlated orbitals include the 5f, 6spd, and 7sp shells, 40 electrons in all; the 78 electrons of fXeg4f 14 5d 10 are treated as the core. To avoid “variational collapse” [22,23], the Gaussian spinors in the basis are made to satisfy kinetic 30 DECEMBER 1996 balance [24]. They also satisfy relativistic boundary conditions associated with a finite nucleus, described here as a sphere of uniform proton charge [20]. We use an atomic mass of 302, and the speed of light c is set at 137.035 99 atomic units. The universal basis set of Malli et al. [25] is employed. It consists of Gaussian-type orbitals, with exponents given by the geometric series zn ­ a 3 b sn21d , a ­ 106 111 395.371 615 , b ­ 0.486 752 256 286 . (6) The largest basis included 34 s functions (n ­ 1 34), 26 p (n ­ 9 34), 20 d (n ­ 13 32), 14 f (n ­ 17 30), 9 g (n ­ 21 29), 6 h (n ­ 24 29), and 4 i orbitals (n ­ 25 28). The orbitals were left uncontracted. Virtual orbitals with energies higher than 80 hartree were omitted. The RCC calculation gave an electron affinity (EA) of 512 cm21 , or 0.063 eV. The Breit interaction has a negligible effect, changing the EA by 3 cm21 . Nonrelativistic CC yields no electron affinity. The orbital energy of the 8s Dirac-Fock orbital is positive, so that the Koopmans’ EA is also negative. This causes the (unbound) orbital to “escape” to the most diffuse functions available in the basis and raises the question of its suitability as a starting point for the RCC EA calculation. To study this question a series of tests were carried out, where the unoccupied orbitals were computed in different electronic fields, obtained by assigning partial charges to some of the external shells, thus leading to a bound 8s orbital. These artificial fields were compensated by appropriate correction of the perturbation term. Assigning a charge of 0.8e to the 7p3y2 electrons gave an EA of 454 cm21 ; a charge of 0.75e on the 7s electrons yielded 449 cm21 ; and putting 0.9e on all 7s and 7p electrons yielded an electron affinity of 437 cm21 . These results are quite close to each other and not too far from the 512 cm21 quoted above; we regard them as more reliable than the latter. Several other tests were performed, all with a charge of 0.8e on the 7p3y2 electrons, to estimate the stability and reliability of the calculated EA. To check the dependence on the nuclear mass A, it was changed from 302 to 283; the effect on the EA was only 3 cm21 . A larger difference, with an EA of 618 cm21 , was obtained with a point nucleus; this is, however, not a very realistic model for such a heavy atom. Finally, to assess the basis set convergence, the f, g, and h limits of the EA were calculated; they came out as 427, 447, and 452 cm21 , respectively. The convergence is satisfactory. Our value for the electron affinity of element 118 is 0.056 eV. The estimated error bounds are about 0.01 eV. Similar calculations gave no 2S bound state for Rn2 . The computations reported above were carried out at TAU. Research at TAU was supported by the U.S.-Israel Binational Science Foundation and by the Israeli Ministry of Science. Y. I. was supported by the National Science 5351 VOLUME 77, NUMBER 27 PHYSICAL REVIEW LETTERS Foundation through Grant No. PHY–9008627. P. P. was supported by The Academy of Finland. [1] E. Eliav, U. Kaldor, P. Schwerdtfeger, B. A. Hess, and Y. Ishikawa, Phys. Rev. Lett. 73, 3203 (1994). [2] E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. A 52, 2765 (1995). [3] B. D. El-Issa, P. Pyykkö, and H. M. Zanati, Inorg. Chem. 30, 2781 (1991). [4] E. Eliav, U. Kaldor, Y. Ishikawa, M. Seth, and P. Pyykkö, Phys. Rev. A 53, 3926 (1996). [5] C. Froese Fischer, J. B. Lagowski, and S. H. Vosko, Phys. Rev. Lett. 59, 2263 (1987). [6] D. J. Pegg, J. S. Thompson, R. N. Compton, and G. D. Alton, Phys. Rev. Lett. 59, 2267 (1987). [7] S. Salomonson, H. Warston, and I. Lindgren, Phys. Rev. Lett. 76, 3092 (1996). [8] H. Haberland, T. Kolar, and T. Reiners, Phys. Rev. Lett. 63, 1219 (1989). [9] C. A. Nicolaides and G. Aspromallis, Phys. Rev. A 44, 2217 (1991). [10] E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. A 49, 1724 (1994). [11] E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. A 51, 225 (1995). [12] E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. A 52, 291 (1995). 5352 30 DECEMBER 1996 [13] E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. A 53, 3050 (1996). [14] E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. Lett. 74, 1079 (1995). [15] G. E. Brown and D. G. Ravenhall, Proc. R. Soc. London Sect. A 208, 552 (1951). [16] M. H. Mittleman, Phys. Rev. A 4, 893 (1971); 5, 2395 (1972); 24, 1167 (1981). [17] J. Sucher, Phys. Rev. A 22, 348 (1980); Phys. Scr. 36, 271 (1987). [18] W. Buchmüller and K. Dietz, Z. Phys. C 5, 45 (1980). [19] I. Lindgren, in Many-Body Methods in Quantum Chemistry, edited by U. Kaldor, Lecture Notes in Chemistry Vol. 52 (Springer-Verlag, Heidelberg, 1989), p. 293; Nucl. Instrum. Methods Phys. Res., Sect. B 31, 102 (1988). [20] Y. Ishikawa and H. M. Quiney, Phys. Rev. A 47, 1732 (1993); Y. Ishikawa, ibid. 42, 1142 (1990); Y. Ishikawa, R. Barrety, and R. C. Binning, Chem. Phys. Lett. 121, 130 (1985). [21] I. Lindgren and J. Morrison, Atomic Many-Body Theory (Springer-Verlag, Berlin, 1986), 2nd ed. [22] F. Mark and W. H. E. Schwarz, Phys. Rev. Lett. 48, 673 (1982). [23] W. Kutzelnigg, Int. J. Quantum Chem. 25, 107 (1984). [24] R. E. Stanton and S. Havriliak, J. Chem. Phys. 81, 1910 (1984). [25] G. L. Malli, A. B. F. Da Silva, and Y. Ishikawa, Phys. Rev. A 47, 143 (1993).