Fast configuration-interaction calculations for nobelium and ytterbium
aa r X i v : . [ phy s i c s . a t o m - ph ] D ec Fast configuration-interaction calculations for nobelium and ytterbium
V. A. Dzuba and V. V. Flambaum
School of Physics, University of New South Wales, Sydney 2052, Australia
M. G. Kozlov
Petersburg Nuclear Physics Institute of NRC “Kurchatov Institute”, Gatchina 188300, Russia andSt. Petersburg Electrotechnical University “LETI”, Prof. Popov Str. 5, 197376 St. Petersburg
We calculate excitation energies for low states of nobelium, including states with open 5 f subshell.An efficient version of the many-electron configuration-interaction method for treating the atom as asixteen external electrons system has been developed and used. The method is tested on calculationsfor ytterbium which has external electron structure similar to nobelium. The results for nobelium areimportant for prediction of its spectrum and for interpretation of recent measurements. Ytterbiumis mostly used to study the features of the method. PACS numbers: 31.15.A-,11.30.Er
I. INTRODUCTION
Configuration interaction (CI) method [1, 2] is one offew tools used to calculate electron structure of open-shell many-electron atoms. However, due to huge in-crease of the computational cost with the number of ex-ternal electrons, practical application is usually limitedto systems with only few (no more than four) externalelectrons above closed shells. There are no other ab ini-tio methods to deal with more complicated polyvalentsystems. On the other hand, the use of different semi-empirical approaches is questionable when experimentaldata is poor or absent. Superheavy elements (
Z > f subshell) [6, 12], W [6], Ta, Db [13], Og [14], etc.In this work we further develop the method to makeit substantially more efficient. We demonstrate that ne-glecting the difference between energies of the states ofthe same excited configuration allows one to separatesummation over projections of the total angular momen-tum of single-electron states from summation over otherquantum numbers. Since summation over projections isthe same for all similar configurations it can be performed only once and then reused for other similar configura-tions. This reduces computational time for Yb morethan twenty times while the effect on the accuracy ofthe calculations is negligible. We use Yb atom as anexample and then apply the method to nobelium. Thisallows us to predict No spectrum including states withexcitations from the 5 f subshell. It is also important,that we provide the proof of validity of previous calcula-tions used for interpretation of the experimental measure-ments. The energy, hyperfine structure and isotope shiftfor the P o1 state of several No isotopes were measured[15] and used together with atomic calculations to ex-tract nuclear parameters of these isotopes [16]. Nobeliumatom was treated in the calculations as a system withtwo valence electrons above closed shells. It is knownthat similar treatment of the P o1 state of Yb gives verypoor results due to the mixing with a close state contain-ing excitation from the 4 f subshell. This mixing cannotbe properly accounted for in the two-valence-electrons-above-closed-shells calculations. We demonstrate thatthe situation in No is different and corresponding mixingis small. Therefore, interpretation of the measurementsbased on the two-valence-electron calculations is correct.New energy levels for low states of No including thosewith open 5 f subshell have been calculated. II. FAST CONFIGURATION INTERACTIONMETHOD
Fast configuration interaction method (FCI) is a mod-ification of the CIPT (configuration interaction with per-turbation theory) method introduced in Ref. [6]. Westart from its brief description using ytterbium atom asan example. We consider Yb as a system with sixteenelectrons above closed shells. The CI Hamiltonian hasthe form H CI = X i h i + X i Nobelium is the heaviest element ( Z =102) for whichexperimental spectroscopic data are available. The fre-quency of the strong electric dipole transition from theground state to a state of opposite parity and the firstionization potential have been recently measured [15, 21].The measurements [15] include hyperfine structure andisotope shifts for three nobelium isotopes No, No,and No. The data were used to extract nuclear pa-rameters, such as nuclear radii, magnetic dipole andelectric quadrupole moments. This procedure relies onthe atomic calculations. In particular, an advancedcombination of the CI with coupled-cluster method wasused [15, 16]. Nobelium atom has the electron struc-ture similar to that of ytterbium. Its ground-state is[Ra]5 f s S . The state for which the frequenciesof the transitions were measured was [Ra]5 f s p P .The calculations treated nobelium as a system with twovalence electrons above closed shells. However, it isnot known in advance whether such calculations pro-duce good results for No. Similar calculations for the4 f s p P state of Yb give very poor results for hy-perfine structure [12] and electric dipole transition ampli-tude from the ground state [22] due to the strong mixingwith the close state of the same parity and J but withan excitation from the 4 f subshell, 4 f d s (7/2,5/2) o1 (last line of Table I). This mixing cannot be included inthe two-valence-electron calculations. However, treatingYb as a sixteen electron system with the CIPT methodleads to dramatic improvement of the results [12]. Theresults of the FCI calculations presented in Table II showthe potentially trouble-making state of the 5 f s d configuration in No (last line of Table II) is significantlyhigher on the energy scale of No than a similar state inYb. Energy interval in No is five times larger and mixingis small. The mixing in the P state of interest is 98%to 2% in No (2% admixture of the state with excitationfrom 5 f subshell) and 75% to 25% in Yb. This meansthat the mixing in No can be neglected and the 7 s p P state can be treated as a two-valence-electron state.Note the good agreement of the FCI energies withthe only known experimental value and with sophisti-cated calculations by the CI+all-order method for thetwo-valence-electron states above closed-shell core of No.There are two major sources of uncertainty in the FCIcalculations. One is neglecting core-valence correlationswith core states below 5 f . Another is the perturbativetreatment of the excited configurations. 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