PHYSICAL REVIEW A VOLUME 52, NUMBER 4 OCTOBER 1995 Transition energies of mercury and ekamercury (element 112) by the relativistic coupled-cluster method Ephraim Eliav and Uzi Kaldor School of Chemistry, Tel Aviv University, 69978 Tel AvivIs, rael Yasuyuki Ishikawa Department of Chemistry, University of Puerto Rico, P. O. Box 23346, San Juan, Puerto Rico 00931-3346 (Received 24 May 1995) The relativistic coupled-cluster method is used to calculate ionization potentials and excitation energies of Hg and element 112, as well as their mono- and dications. Large basis sets are used, with l up to 5, the Dirac-Fock or Dirac-Fock-Breit orbitals found, and the external 34 electrons of each atom are correlated by the coupled-cluster method with single and double excitations. Very good agreement with experiment is obtained for the Hg transition energies, with the exception of the high () 12 eV) excitation energies of the dication. As in the case of element 1 1 1 [Eliav et al. , Phys. Rev. Lett. 73, 3203 (1994)], relativistic stabilization of the 7s orbital leads to the ground state of 112+ being 6d 7s, rather than the d' s ground states of the lighter group 12 elements. The 112 + ion shows very strong mixing of the d s, d s, and d' configurations. The lowest state of the dication is 6d 7s J=4, with a very close (0. 05 eV) J=2 state with strong d s and d s mixing. No bound states were found for the anions of the two atoms. PACS number(s): 31. 30. Jv, 31. 50. +w r. n TRODVCTION II. METHOD Excitation energies and ionization potentials of the mer- cury atom have been the subject of many calculations in the past, using a variety of methods. Hafner and Schwartz [1] used a relativistic model potential fitted to the low excitation energies of Hg+ and determined a large number of excitation energies of the neutral atom. Another model potential was used by Mohan and Hibbert [2] in the framework of the configuration-interaction (CI) method to calculate the 6s 'Sp — + 6s6p ' P ] transitions. The latter have also been stud- ied by Migdalek and co-workers [3 — 5] using multiconfigu- ration Hartree-Fock and CI methods and by Chou and Huang [6] with the relativistic random-phase approximation. More recently, Haussermann et al. [7] treated several states of Hg and its cation by the multireference CI method. All these calculations involved some form of model or pseudopoten- tial. Haussermann et al. [7] also carried out all-electron Dirac-Fock calculations for comparison purposes, but such calculations do not include correlations and cannot be ex- pected to yield results comparable to experiment. We are not aware of any all-electron calculations of the ionization po- tential or excitation energies of mercury, which includes both relativistic and correlation effects. An accurate theoretical prediction of transition energies in heavy atoms requires high-order inclusion of both relativistic and correlation effects. An ab initio relativistic coupled- cluster (RCC) method incorporating both effects has been applied recently to a series of heavy atoms, including gold [8], several lanthanides and actinides [9,10], and elements 104 [11]and 111 [12]. Calculated transition energies were in very good agreement with known experimental values, usu- ally within a few hundred wave numbers. Even higher accu- racy was obtained for fine-structure splittings. The method is applied here to atomic mercury and ekamercury (element 112). H+ =Hp+ U, where (in a.u. ) Ho= + A, hD(i)A, ", (2) hD(i) = c n;. p;+ c (P; — 1) + V„„, (i) + U(i), (3) v=g A, A, (v„, )„A, A, gA;U(i)A, . — Here hD is the one-electron Dirac Hamiltonian. An arbitrary potential U is included in the unperturbed Hamiltonian Hp and subtracted from the perturbation V. This potential is cho- sen to approximate the effect of the electron-electron inter- action; in particular, it may be the Dirac-Fock self- consistent-field potential. The nuclear potential V„„, includes the effect of finite nuclear size. A, are projection operators onto the positive energy states of the Dirac Hamiltonian hD. Because of their presence, the Hamiltonian H+ has nor- malizable, bound-state solutions. This approximation is known as the no-(virtual)-pair approximation, since virtual electron-positron pairs are not allowed in intermediate states. The form of the effective potential V, & depends on the gauge used. In Coulomb gauge it becomes (in a.u. , correct to sec- ond order in the fine-structure constant tx) [15] The relativistic coupled-cluster method has been de- scribed in our previous publications [8, 13] and only a brief review is given here. We start from the projected Dirac- Coulomb (or Dirac-Coulomb-Breit) Hamiltonian advocated by Sucher [14], 1050-2947/95/52(4)/2765(5)/$06. 00 52 2765 1995 The American Physical Society