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The rare-earth metals have high magnetic moments and a diverse range of magnetic structures 1 . Their magnetic properties are determined by the occupancy of the strongly localized 4f electronic shells, while the outer s±d electrons determine the bonding and other electronic properties 2 . Most of the rare-earth atoms are divalent, but generally become trivalent in the metallic state. In some materials, the energy difference between these valence states is small and, by changing some external parameter (such as pressure), a transition from one to the other occurs. But the mechanism underlying this transition and the reason for the differing valence states are not well understood. Here we report ®rst-principles electronic-structure calculations that enable us to determine both the valency and the lattice size as a function of atomic number, and hence understand the valence transitions. We ®nd that there are two types of f electrons: localized core-like f electrons that determine the valency, and delocalized band-like f electrons that are formed through hybridization with the s±d bands and which participate in bonding. The latter are found only in the trivalent systems; if their number exceeds a certain threshold, it becomes energetically favourable for these electrons to localize, causing a transition to a divalent ground state.Here we report a systematic theoretical investigation of the rareearth elements and their sulphides using ab initio electronicstructure methods. These go well beyond standard calculations in which the 4f shell is described by an atomic model, while an itinerant picture for the s±d electrons is implemented 3,4 . We include a self-interaction correction (SIC) which removes the spurious interaction of each electron with itself that occurs in conventional band-structure theory 5 . Our approach has the advantage of describing both the bonding s±d electrons and the f electrons on an equal footing. The SIC has a negligible effect on the bonding s±d electrons, but is substantial for the f electrons 6 . Application of the SIC to the f electrons provides a de®nition of valency of the metallic rare-earth materials. Here we associate the valency with the number of states available for the electron to propagate through the solid, namely N valency Z 2 N core 2 N SIC where Z is the atomic number of the rare earth and N core is the number of atomic core electrons. The quantity N SIC is the number of letters to nature 756 NATURE | VOL 399 | 24 JUNE 1999 | www.nature.com Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb -2 0 2 4 E II -E III (eV) Figure 1 The energy difference (in eV) between the divalent and trivalent state of rare-earth materials. The dashed line shows the`experimental' values for the rare-earth metals 3 . The open circles and the crosses show the calculated values for the rare-earth metals and the rare-earth sulphides, respectively. We see that the divalent and trivalent energy difference is large and positive at the beginning of the series, indicating that the trivalent state is well favoured. The en...

We apply the self-interaction corrected local spin density approximation to study the electronic structure and magnetic properties of the spinel ferrites MnFe2O4, Fe3O4, CoFe2O4, and NiFe2O4. We concentrate on establishing the nominal valence of the transition metal elements and the ground state structure, based on the study of various valence scenarios for both the inverse and normal spinel structures for all the systems. For both structures we find all the studied compounds to be insulating, but with smaller gaps in the normal spinel scenario. On the contrary, the calculated spin magnetic moments and the exchange splitting of the conduction bands are seen to increase dramatically when moving from the inverse spinel structure to the normal spinel kind. We find substantial orbital moments for NiFe2O4 and CoFe2O4.

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Using a new bunched-beam technique in the GSI heavy-ion experimental storage ring (ESR), we performed precision laser spectroscopy on relativistic heavy ions in the hitherto inaccessible infrared optical region. We determined the wavelength of the M1 transition between the F 1 ͑t ഠ 50 ms͒ and F 0 hyperfine states of the 1s ground state of hydrogenlike 207 Pb 811 . Comparing the result of 1019.7(2) nm with very recent theoretical predictions concerning QED and nuclear size contributions, a disagreement of 4.5 nm is found. Since the nucleus of 207 Pb 811 is well described by the single-particle shell model, uncertainties in nuclear corrections are expected to be small. [S0031-9007(98)07624-8] PACS numbers: 32.30.Jc, 12.20.Fv, 21.10.Ky The hyperfine splitting (HFS) of the 1s ground state of one-electron, two-body (hydrogenlike) system is the simplest and most basic magnetic interaction in atomic physics. In hydrogen the splitting is measured to thirteen significant figures, considerably more precise than the six-digit precision of the theoretical calculations of this quantity [1]. These calculations solve the Dirac equation and then add corrections for the effects of the finite size of the nuclear charge and magnetization as well as for the QED effects of self-energy and vacuum polarization. While the QED contributions are of the order of 10 26 to 10 25 for a single proton, these corrections are several percent in hydrogenlike ions of large Z in which the electron experiences exceptionally intense electric and magnetic fields. Thus measurements of the spectra of these systems can stringently test theoretical calculations of QED and nuclear effects.Recently the 1s ground state transitions in high-Z, hydrogenlike ions have become accessible to optical spectroscopy at the experimental storage ring (ESR) at GSI-Darmstadt and at the electron beam ion trap Super-EBIT at Lawrence Livermore National Laboratory. Measurements of the ground state hyperfine splittings of 209 Bi 821 at GSI [2] and 165 Ho 661 at LLNL [3] have stimulated a large number of theoretical calculations of the wavelengths of these transitions [4][5][6][7][8][9][10][11][12][13][14][15]. Discrepancies are fond between theory and experiment for both 209 Bi 821 and 165 Ho 661 .The calculations for bismuth yield a value 1 nm ͑5 3 10 23 ͒ larger than the measured value. On the basis of the precisions assigned to the corrections this discrepancy is significant, but corrections for the nuclear effects vary considerably depending upon how much the nuclear core is assumed to be polarized. For holmium, a smaller discrepancy between the calculated and measured values is reported [3], but the theoretical analysis did not take into account nuclear polarization [15] which is expected to contribute significantly.In view of this unsatisfactory situation we measured the 1s ground state hyperfine transition of 207 Pb 811 . We chose this nucleus because it is well described by the single-particle model. The magnetic moment has been measured with high precision in the ato...

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The electronic and magnetic structure of bulk NiO and the NiO͑100͒ surface is calculated using densityfunctional theory ͑DFT͒ in the local-spin-density ͑LSD͒ approximation including self-interaction corrections. We calculate the exchange coupling constants in bulk NiO and at the NiO͑100͒ surface and show that in the case of bulk they agree better with experiment than the standard DFT calculations in the LSD approximation. We develop a model for the exchange interactions at the NiO͑100͒ surface and discuss how they change from the surface to bulk.