In this study, we report the lowest energy structure of bare Cu13 nanoclusters as a pair of enantiomers at room temperature. Moreover, we compute the enantiomerization energy for the interconversion from minus to plus structures in the chiral putative global minimum for temperatures ranging from 20 to 1300 K. Additionally, employing nanothermodynamics, we compute the probabilities of occurrence for each particular isomer as a function of temperature. To achieve that, we explore the free energy surface of the Cu13 cluster, employing a genetic algorithm coupled with density functional theory. Moreover, we discuss the energetic ordering of isomers computed with various density functionals. Based on the computed thermal population, our results show that the chiral putative global minimum strongly dominates at room temperature.

Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants.In the scalar case, this leads to restrictions for the central limit theorems, as observed by F. Oravecz.We characterize the property of independent constants from a combinatorial point of view, based on cumulants and set partitions. This characterization also holds for the operator-valued extension.Our considerations lead rather directly to very mild variations of boolean and monotone cumulants, where constants are now independent. These alternative probability theories are closely related to the usual notions. Hence, an important part of the boolean/monotone probability theories can be imported directly.We describe some standard combinatorial aspects of these variations (and their cyclic versions), such as their Möbius functions, which feature well-known combinatorial integer sequences.The new notions with independent constants seem also more strongly related to the operator-valued extension of c-free probability.

The boolean and monotone notions of independence lack the property of independent constants. We address this problem from a combinatorial point of view (based on cumulants defined from weights on set-partitions, in the general framework of operator-valued probability spaces). We show that if the weights are singleton inductive (SI), then all higher-order cumulants involving constants vanish, just as in the free and classical case. Our combinatorial considerations lead rather directly to mild variations of boolean and monotone probability theories which are closely related to the usual notions. The SI-boolean case is related to c-free and Fermi convolutions. We also describe some standard combinatorial aspects of the SI-boolean and cyclic-boolean lattices, such as their Möbius functions, featuring well-known combinatorial integer sequences.

performed calculations and analysis, drafted the manuscript, and performed the data analyses, AdNDP; S.P.: performed calculations and analysis, drafted the manuscript, performed the data analyses, investigation, and revised and wrote the manuscript; J.L.C.: conception of the study, software development, design and validation, performed calculations and analysis, drafted the manuscript, performed the data analyses, investigation, resources, and revised and wrote the manuscript. All authors have read and agreed to the published version of the manuscript.

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We calculate the distribution with respect to the vacuum state of the distance-k graph of a d-regular tree. From this result we show that the distance-k graph of a d-regular graphs converges to the distribution of the distance-k graph of a regular tree. Finally, we prove that, properly normalized, the asymptotic distributions of distance-k graphs of the d-fold free product graph, as d tends to infinity, is given by the distribution of P k (s), where s is a semicircle random variable and P k is the k-th Chebychev polynomial.

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Let G be a finite connected graph and let G [ N,k] be the distance k-graph of the N -fold star power of G. For a fixed k ≥ 1, we show that the large N limit of the spectral distribution of G [ N,k] converges to a centered Bernoulli distribution, 1/2δ −1 + 1/2δ 1 . The proof is based in a fourth moment lemma for convergence to a centered Bernoulli distribution. arXiv:1408.5682v1 [math.PR]