There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

The electronic ground state of a periodic system is usually described in terms of extended Bloch orbitals, but an alternative representation in terms of localized "Wannier functions" was introduced by Gregory Wannier in 1937. The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness. Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wannier functions, accomplishing the solid-state equivalent of constructing localized molecular orbitals, or "Boys orbitals" as previously known from the chemistry literature. These developments are reviewed here, and a survey of the applications of these methods is presented. This latter includes a description of their use in analyzing the nature of chemical bonding, or as a local probe of phenomena related to electric polarization and orbital magnetization. Wannier interpolation schemes are also reviewed, by which quantities computed on a coarse reciprocal-space mesh can be used to interpolate onto much finer meshes at low cost, and applications in which Wannier functions are used as efficient basis functions are discussed. Finally the construction and use of Wannier functions outside the context of electronic-structure theory is presented, for cases that include phonon excitations, photonic crystals, and cold-atom optical lattices.

Maximally-localized Wannier Functions: Theory and Applications

Nano-optics of surface plasmon polaritons

A grid-based real-space implementation of the projector augmented wave (PAW) method of Bl\"ochl [Phys. Rev. B 50, 17953 (1994)] for density functional theory (DFT) calculations is presented. The use of uniform three-dimensional (3D) real-space grids for representing wave functions, densities, and potentials allows for flexible boundary conditions, efficient multigrid algorithms for solving Poisson and Kohn-Sham equations, and efficient parallelization using simple real-space domain-decomposition. We use the PAW method to perform all-electron calculations in the frozen core approximation, with smooth valence wave functions that can be represented on relatively coarse grids. We demonstrate the accuracy of the method by calculating the atomization energies of 20 small molecules, and the bulk modulus and lattice constants of bulk aluminum. We show that the approach in terms of computational efficiency is comparable to standard plane-wave methods, but the memory requirements are higher.

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Real-space grid implementation of the projector augmented wave method

The diffraction of electromagnetic waves from the rough surface of a material of finite permittivity is examined for the case where the wavelength of the incident radiation is comparable to the dimensions of the surface roughness. Two methods of calculation studied are the Rayleigh method and the extinction-theorem integral-equation method. The latter is shown to have a unique exact solution. This property is, in turn, used to show how the Rayleigh method can be modified to give convergent results. The extinction theorem is also used to reduce the Rayleigh equations to a simpler form. These reduced equations, which are extremely convenient to use in the case of small roughness, are applied in this case to find perturbative expressions for the reflected field and for the surface-plasmon dispersion relation.

Optical properties of rough surfaces: General theory and the small roughness limit

A new procedure for calculating the frequency- and wave-vector-dependent dielectric response function is described. It is based on decoupling and solving the equations of motion for the Green's functions of the charge-density operators by a moment-conserving method which is discussed. By use of this method an expression for the dielectric function in the static limit ($\ensuremath{\omega}\ensuremath{\rightarrow}0$) is obtained; it depends on a function $G(k)$, for which numerical values are calculated and tabulated. Evidence that the procedure described here leads to reliable values of $G(k)$ for small, intermediate, and large values of $k$ is presented.

Calculation of the Dielectric Function for a Degenerate Electron Gas with Interactions. I. Static Limit

This article reviews the role of reparametrization invariance (the invariance of the properties of a system with respect to the choice of the co-ordinate system used to describe it) in deriving stochastic equations that describe the growth of surfaces. By imposing reparametrization invariance on a system, the authors identify the physical origin of many of the terms in its growth equations. Both continuum-growth equations for interfaces and equations for the coarse-grained evolution of discrete-lattice models are derived with this method. A detailed analysis of the discrete-lattice case and its small-gradient expansion provides a physical basis for terms found in commonly studied growth equations. The reparametrization-invariant formulation of growth processes also has the advantage of allowing one to model shadowing effects that are lost in the no-overhang approximation and to conserve underlying symmetries of the system that are lost in a small-gradient expansion. Finally, a knowledge of the full equation of motion, beyond the lowest-order gradient expansion, may be relevant in problems where the usual perturbative renormalization methods fail. [S0034-6861(96)00104-3]

Stochastic growth equations and reparametrization invariance

The structural, electronic, and magnetic properties of small Rh-N clusters (N = 2-8,10,12,13, and 19) are studied using the discrete-variational local-spin-density-functional method. The groundstate structures of Rh-2, Rh-3, and Rh-4 are obtained and found to exhibit a tendency toward higherdimensional geometries with longer bond lengths and more nearest-neighbor bonds. The equilibrium bond lengths in the chosen geometry for the Rh-5-Rh-8, Rh-10, Rh-12, and Rh-13 clusters are determined and show 4-6% bond length contractions compared with the bulk interatomic spacing. A. complex size dependence of the magnetic properties of the Rhry clusters is found to be consistent with a recent experiment. The average magnetic moment per atom of the Rh-N clusters varies from 0 mu(B) to 2 mu(B). The clusters with magnetic ground states have ferromagnetic interactions, while-the local moments of the 5s and 5p often align antiferromagnetically with that of the 4d. The calculated magnetic moments of the Rh-10, Rh-12, Rh-13, and Rh-19 clusters are compared and discussed with the experimental ones. Icosahedral growth is suggested for Rh clusters. The reactivity of the Rh-N clusters toward H-2, N-2, and CO molecules is predicted. The densities of states, exchange splittings, valence band widths, and ground-state electronic configurations are presented for all the clusters. Finally an energy difference is identified which may be used as the criterion for the existence of multiple magnetic solutions in local-spin-density-functional calculations.

Structural, electronic, and magnetic properties of small rhodium clusters.

The physical effects associated with the implementation of two different types of thermal ~or stochastic! walls that could be used in computer simulations are considered. The effects of these boundary conditions are first demonstrated using the ideal-gas model and then their influence on interacting particles ~with an example of hard spheres! is discussed. @S1063-651X~98!50501-6#

Flavio Toigo

Papers

Optical properties of rough surfaces: General theory and the small roughness limit

Calculation of the Dielectric Function for a Degenerate Electron Gas with Interactions. I. Static Limit

Stochastic growth equations and reparametrization invariance

Structural, electronic, and magnetic properties of small rhodium clusters.

Thermal walls in computer simulations