Using calculations from first principles, we herein consider the bond made between thiolat e with a range of different Au clusters, with a particular focus on the spin moments inv olved in each case. For odd number of gold atoms, the clusters show a
spin moment of 1.~ $mu_B$. The variation of spin moment with particle size is particularly dramatic, with t he spin moment being zero for even numbers of gold atoms. This variation may be linked w ith changes in the odd-even oscillations that occur with the number of gold atoms, and is associated with the formation of a S-Au bond. This bond leads to the presence of an extra electron that is mainly sp in character in the gold part. Our results sugg est that any thiolate-induced magnetism that occurs in gold nanoparticles may be locali zed in a shell below the surface, and can be controlled by modifying the coverage of the thiolates.
When comparing 2D shapes, a key issue is their normalization. Translation and scale are easily taken care of by removing the mean and normalizing the energy. However, defining and computing the orientation of a 2D shape is not so simple. In fact, alt
hough for elongated shapes the principal axis can be used to define one of two possible orientations, there is no such tool for general shapes. As we show in the paper, previous approaches fail to compute the orientation of even noiseless observations of simple shapes. We address this problem. In the paper, we show how to uniquely define the orientation of an arbitrary 2D shape, in terms of what we call its Principal Moments. We show that a small subset of these moments suffice to represent the underlying 2D shape and propose a new method to efficiently compute the shape orientation: Principal Moment Analysis. Finally, we discuss how this method can further be applied to normalize grey-level images. Besides the theoretical proof of correctness, we describe experiments demonstrating robustness to noise and illustrating the method with real images.
