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Starting from a microscopic tight-binding model and using second order perturbation theory, we derive explicit expressions for the intrinsic and Rashba spin-orbit interaction induced gaps in the Dirac-like low-energy band structure of an isolated graphene sheet. The Rashba interaction parameter is first order in the atomic carbon spin-orbit coupling strength ξ and first order in the external electric field E perpendicular to the graphene plane, whereas the intrinsic spin-orbit interaction which survives at E = 0 is second order in ξ. The spin-orbit terms in the low-energy effective Hamiltonian have the form proposed recently by Kane and Mele. Ab initio electronic structure calculations were performed as a partial check on the validity of the tight-binding model.

We perform a first principles calculation of the anomalous Hall effect in ferromagnetic bcc Fe. Our theory identifies an intrinsic contribution to the anomalous Hall conductivity and relates it to the k-space Berry phase of occupied Bloch states. This dc conductivity has the same origin as the well-known magneto-optical effect, and our result accounts for experimental measurement on Fe crystals with no adjustable parameters.

For metals and semiconductors the calculation of crystal wave functions is simplest in a plane wave representation. However, in order to obtain rapid convergence it is necessary that the valence electron wave functions be made orthogonal to the core wave functions. Herring satis6ed this requirement by choosing as basis functions "orthogonalized plane waves. " It is here shown that advantage can be taken of crystal symmetry to construct wave functions p which are best described as the smooth part of symmetrized Sloch functions. The wave equation satisfied by p contains an additional term of simple character which corresponds to the usual complicated orthogonalization terms and has a simple physical interpretation as an e6'ective repulsive potential, Qualitative estimates of this potential in analytic form are presented. Several examples are worked out which display the cancellation between attractive and repulsive potentials in the core region which is responsible for rapid convergence of orthogonalized plane wave calculations for s states; the slower convergence of p states is also explained. The formalism developed here can also be regarded as a rigorous formulation of the "empirical potential" approach within the one-electron framework; the present results are compared with previous approaches. The method can be applied equally well to the calculation of wave functions in molecules.

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Kleinman, Bylander, and Morrison Reply: We [1] have recently argued that because of the nonlinear nature of the local spin-density approximation (LSDA) for exchange and correlation (XC) potentials, an unphysical diminution of the XC interaction between valence electrons is present in the core region. We then performed a pseudopotential [2] calculation for a nine layer Rh(OOl) film in which all the electrons see a Hartree-Fock potential from the core electrons and an LSDA potential from the valence electrons. We found [1] that the first two layers of the film are ferromagnetic with \.Sju B per surface unit cell. To mimic our XC potential Weinert, Bliigel, and Johnson [3] constructed their all-electron VLSDA potential and essentially reproduced our results. Using the LSDA they found that the Rh(OOl) surface is paramagnetic. They found that the VLSDA resulted in an unrealistically large moment for Fe whereas the LSDA moment is in close agreement with experiment. They also invoked the Stoner model to determine that the VLSDA results in a susceptibility for Rh which is too large by a factor between 4 and 6. From these two facts they conclude that although the LSDA is not defect free, in this case its results are to be believed and the Rh(OOl) surface is paramagnetic. Although we think their model may have overestimated the VLSDA susceptibility error and underestimated the LSDA susceptibility error, we are in general agreement with their results.We do not believe that there can be any dispute over the fact that the core charge density reduces the LSDA exchange interaction between valence electrons in the core region by a factor of j (pvai/pcore) 2/3 and that this is unphysical. Correlation should result in a large reduction of the exchange splitting which both the LSDA and VLSDA correlation terms fail to do. It is the cancellation of these errors which allows the LSDA to work as well as it does, although we should not forget that it fails to predict the correct ground state of iron. Furthermore, Moruzzi and Marcus [4] have just shown that the LDA yields accurate bulk moduli for the nonmagnetic 3d and 4d transition metals. The error for Ni is actually increased while for Co and Fe it is decreased when the LSDA is used, but in all three cases it remains large. Thus it is clear that although the LSDA yields accurate spin polarizations for these metals, it fails to predict their magnetic energies. It was, and is, our feeling that because the Ad electrons of Rh are more free-electron-like than the 3d's of Fe, the error in their LSDA correlation should be smaller. Whether the LSDA or VLSDA better predict the surface properties of Rh will be decided by experiment. We have been informed [5] that Jona et ai, using spin polarized photoemission, have found a weak spin polarization (0.1 to 0.3/us) of the Rh(00l) surface. These data were taken at room temperature. If the usual linear temperature dependence of two dimensional ferromagnetism holds, a much larger polarization may be found at He temperatures.

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We show that a new pseudopotential recently introduced by Hamann, Schliiter, and Chiang can easily be extended to include all relativistic corrections to order a (not Z a').