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The structural and elastic properties of orthorhombic black phosphorus have been investigated using first-principles calculations based on density functional theory. The structural parameters have been calculated using the local density approximation (LDA), the generalized gradient approximation (GGA), and with several dispersion corrections to include van der Waals interactions.It is found that the dispersion corrections improve the lattice parameters over LDA and GGA in comparison with experimental results. The calculations reproduce well the experimental trends under pressure and show that van der Waals interactions are most important for the crystallographic b-axis, in the sense that they have the largest effect on the bonding between the phosphorus layers.The elastic constants are calculated and are found to be in good agreement with experimental values. The calculated C 22 elastic constant is significantly larger than the C 11 and C 33 parameters, implying that black phosphorus is stiffer against strain along the a-axis than along the b-and

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...

The heavy rare earth elements crystallize into hexagonally close packed (h.c.p.) structures and share a common outer electronic configuration, differing only in the number of 4f electrons they have. These chemically inert 4f electrons set up localized magnetic moments, which are coupled via an indirect exchange interaction involving the conduction electrons. This leads to the formation of a wide variety of magnetic structures, the periodicities of which are often incommensurate with the underlying crystal lattice. Such incommensurate ordering is associated with a 'webbed' topology of the momentum space surface separating the occupied and unoccupied electron states (the Fermi surface). The shape of this surface-and hence the magnetic structure-for the heavy rare earth elements is known to depend on the ratio of the interplanar spacing c and the interatomic, intraplanar spacing a of the h.c.p. lattice. A theoretical understanding of this problem is, however, far from complete. Here, using gadolinium as a prototype for all the heavy rare earth elements, we generate a unified magnetic phase diagram, which unequivocally links the magnetic structures of the heavy rare earths to their lattice parameters. In addition to verifying the importance of the c/a ratio, we find that the atomic unit cell volume plays a separate, distinct role in determining the magnetic properties: we show that the trend from ferromagnetism to incommensurate ordering as atomic number increases is connected to the concomitant decrease in unit cell volume. This volume decrease occurs because of the so-called lanthanide contraction, where the addition of electrons to the poorly shielding 4f orbitals leads to an increase in effective nuclear charge and, correspondingly, a decrease in ionic radii.

The self-interaction-corrected local-spin-density approximation is used to describe the electronic structure of dioxides, REO 2 , and sesquioxides, RE 2 O 3 , for the rare earths, RE=Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy and Ho. The valencies of the rare earth ions are determined from total energy minimization. We find Ce, Pr, Tb in their dioxides to have the tetravalent configuration, while for all the sesquioxides the trivalent groundstate configuration is found to be the most favourable. The calculated lattice constants for these valency configurations are in good agreement with experiment.Total energy considerations are exploited to show the link between oxidation and f -electron delocalization, and explain why, among the dioxides, only the CeO 2 , PrO 2 , and TbO 2 exist in nature.Tetravalent NdO 2 is predicted to exist as a metastable phase -unstable towards the formation of hexagonal Nd 2 O 3 .