The formula for tunaelling between tm objects is fundamental to undw standing the scanning tunnelling microscope. Although an exact expression for the one-electron turmelling m n t is known, its derivation hss previously been complex.Here we present a new derivation which is relatively simple and based on physical arguments. We go on to use OUT methods to show for the first time the connee tion between the e x a d formula and the widely used transfer Hamiltonian method of Bardeen. Finally we emphasize the importance of choosing a form& that minimizes the computational &ort requimd to rastm the STM tip across a d a c e .

A fully relativistic first-principles electronic structure calculation method is presented for magnetic materials. The method is based on the local spin moment density concept for relativistic Hamiltonians. In order to obtain manageable Kohn-Sham-Dirac equations including magnetic fields, the orbital contribution to the four-current density is omitted. The starting point is Takeda's relativistic generalisation of the augmented spherical wave method for non-magnetic crystals (RASW). In its basic form, the proposed method for magnetic crystals is only slightly more involved as RASW, and still takes all relativistic and spin polarisation effects into account, from first principles (including the Delta l=2 coupling). The treatment of relativistic and spin polarisation effects can be called 'on equal footing'. In both relevant limits the method is exact (within the mentioned framework). Furthermore, a more elaborate scheme is suggested, which is a systematic improvement of the basic scheme. A comparison is made with other recently published methods. Finally, results of self-consistent calculations for ferromagnetic Ni and Gd, performed with the basic scheme, are compared with previous calculations and experimental data from the literature. For Ni, the results are in good agreement both with previous calculations and with experiment. For Gd interesting new results have been obtained concerning the spectroscopic splitting factor g. The influence of the choice for an explicit exchange and correlation functional is studied as well as the influence of the coupling between l and l+2 levels.

Fully relativistic first-principles electronic structure calculations of both the average structure and a supercell approximation of silver-free incommensurately modulated calaverite (AuTe2) are presented. The differences between the results of both calculations are relatively small for the occupation numbers and the density of states, but quite dramatic for the shape of the Fermi surface. From the occupation numbers it is concluded that a previously proposed idea for explaining the modulation, based on mixed valencies for the gold atoms, is probably not applicable. The calculated Fermi surface of the average structure shows that the modulation cannot be understood in terms of Fermi-surface nesting either. The density of states in the supercell approximation compares very favourably with recently obtained X-ray photoelectron spectroscopy data. A rigid-potential calculation shows that the integral of the one-electron valence energies for the supercell is substantially more negative than the corresponding energy for the average structure, while the electrostatic energy difference has the opposite sign but is much smaller. This provides a qualitative indication of the electronic instability of the average structure with respect to the modulation of the supercell. Finally the authors conclude that Te s-like states a (Te p-Au d)-like complex dominate the energetics of the modulation.

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