Dipole (hyper)polarizabilities of neutral silver clusters

At the Douglas-Kroll-Hess （DKI4） level, the B3PW91 functional along with the all-eleclron relativistic basis sets of valence triple and quadruple zeta qualities are used to determine the structure, stability, and electronic properties of the small silver clusters （Agn, n ≤ 7）. The results presented in this study are in good agreement with the experimental data and theoretical values obtained at a higher level of theory from the literature. Static polarizability and hyperpolarizability are also reported. It is verified that the mean dipole polarizability per atom exhibits an odd-even oscillation and that the polarizability anisotropy is directly related to the cluster shape. In this article, the first study of hyperpolarizabilities of small silver clusters is presented. Except for the monomer, the second hyperpolarizabilities of the silver clusters are significantly larger than those of the copper clusters.

On the Existence of Non-Intersecting Straight Lines on the Plane

In this brief note, we adduce the logical rationale that if at least one infinite straight line non-intersecting with the given straight line passes through a given point not lying on a given straight line, then it must be unique.

Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions

It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equivalent, because they have quite different observational consequences due to the different space-time structure involved.

On the Singular Effects in the Relativistic Landau Levels in Graphene with a Disclination

The effect of a pseudo Aharonov-Bohm(AB) magnetic field generated by a disclination on a two-dimensional electron gas in graphene is addressed in the continuum limit within the geometric approach. The influence of the coupling between the spinor fields and the singular conical curvature is investigated, which shows that singularities have pronounced impact in the Hall conductivity. Moreover, the degeneracy related to the Dirac valleys is broken for negative values of the angular momentum quantum numbers, ?, including ? ≡ 0. In this case, a Hall plateau develops at the null filling factor. Obtaining the Hall conductivity by summing over the positive and the negative ?′s, the null Landau level is recovered and the plateau at the null filling factor disappears. In any case, the standard plateaus, which are seen in a flat graphene are not obtained with these curvature and singular effects.