Normal Unenhanced Raman Spectra of CO and CH4 Adsorbed on Cobalt(poly)

Abstract

Normal unenhanced Raman spectra (NURS) of low-polarizability CO molecules were observed for the first time on cobalt at R. T. and residual gas pressure. We assign five bands observed between 2030–2130 cm^?1 to linear chemisorbed CO species, while those observed between 1840–2010 cm^?1 have been ascribed to the 2-fold chemisorbed species. The three bands observed between 1740–1830 cm^?1 we believe are due to the 3-fold species. The corresponding fourteen Co-C stretches were observed and assigned. A model based upon electron backdonation is proposed for each of the three structures. NURS were also observed at R. T. for physisorbed CH₄ and assignments are made to the four frequencies of CH_4.     

Computing f-divergences and distances of high-dimensional probability density functions

Very often, in the course of uncertainty quantification tasks or data analysis, one has to deal with high-dimensional random variables. Here the interest is mainly to compute characterizations like the entropy, the Kullback–Leibler divergence, more general $f$-divergences, or other such characteristics based on the probability density. The density is often not available directly, and it is a computational challenge to just represent it in a numerically feasible fashion in case the dimension is even moderately large. It is an even stronger numerical challenge to then actually compute said characteristics in the high-dimensional case. In this regard it is proposed to approximate the discretized density in a compressed form, in particular by a low-rank tensor. This can alternatively be obtained from the corresponding probability characteristic function, or more general representations of the underlying random variable. The mentioned characterizations need point-wise functions like the logarithm. This normally rather trivial task becomes computationally difficult when the density is approximated in a compressed resp. low-rank tensor format, as the point values are not directly accessible. The computations become possible by considering the compressed data as an element of an associative, commutative algebra with an inner product, and using matrix algorithms to accomplish the mentioned tasks. The representation as a low-rank element of a high order tensor space allows to reduce the computational complexity and storage cost from exponential in the dimension to almost linear.