triggered a flood ofpublications.However,onlyconsiderationof the most recent theoretical chemicalstudies on the PSE offers deeper insights,whichareoutlineinthefollowingsections.The most important points are: 1) Theorigin of the elements periodicity is theexceptional position of the 1s and npshells (n=2–6). 2) The groups before andafter the noble gases form the backboneof the PSE. 3) The PSE is complete withPeriod 7. 4) The electron configurationsof the d elements of Group g with chargestate q+ are nd

/pdf/icon-of-chemistry-the-periodic-system-of-chemical-elements-2lhcnl84xi.pdf

Icon of Chemistry: The Periodic System of Chemical Elements in the New Century

In 1916, Lewis and Kossel laid the empirical ground for the electronic theory of valence, whose quantum theoretical foundation was uncovered only slowly. We can now base the classification of the various traditional chemical bond types in a threefold manner on the one- and two-electron terms of the quantum-physical Hamiltonian (kinetic, atomic core attraction, electron repulsion). Bond formation is explained by splitting up the real process into two physical steps: (i) interaction of undeformed atoms and (ii) relaxation of this nonstationary system. We aim at a flexible bond energy partitioning scheme that can avoid cancellation of large terms of opposite sign. The driving force of covalent bonding is a lowering of the quantum kinetic energy density by sharing. The driving force of heteropolar bonding is a lowering of potential energy density by charge rearrangement in the valence shell. Although both mechanisms are quantum mechanical in nature, we can easily visualize them, since they are of one-electron type. They are however tempered by two-electron correlations. The richness of chemistry, owing to the diversity of atomic cores and valence shells, becomes intuitively understandable with the help of effective core pseudopotentials for the valence shells. Common conceptual difficulties in understanding chemical bonds arise from quantum kinematic aspects as well as from paradoxical though classical relaxation phenomena. On this conceptual basis, a dozen different bond types in diatomic molecules will be analyzed in the following article. We can therefore examine common features as well as specific differences of various bonding mechanisms. © 2006 Wiley Periodicals, Inc. J Comput Chem 28: 411–422, 2007

https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.20531

Toward a physical understanding of electron‐sharing two‐center bonds. I. General aspects

Phylogenetic tree comparison metrics are an important tool in the study of evolution, and hence the definition of such metrics is an interesting problem in phylogenetics. In a paper in Taxon fifty years ago, Sokal and Rohlf proposed to measure quantitatively the difference between a pair of phylogenetic trees by first encoding them by means of their half-matrices of cophenetic values, and then comparing these matrices. This idea has been used several times since then to define dissimilarity measures between phylogenetic trees but, to our knowledge, no proper metric on weighted phylogenetic trees with nested taxa based on this idea has been formally defined and studied yet. Actually, the cophenetic values of pairs of different taxa alone are not enough to single out phylogenetic trees with weighted arcs or nested taxa. For every (rooted) phylogenetic tree T, let its cophenetic vector φ(T) consist of all pairs of cophenetic values between pairs of taxa in T and all depths of taxa in T. It turns out that these cophenetic vectors single out weighted phylogenetic trees with nested taxa. We then define a family of cophenetic metrics dφ,p by comparing these cophenetic vectors by means of L
 p
 norms, and we study, either analytically or numerically, some of their basic properties: neighbors, diameter, distribution, and their rank correlation with each other and with other metrics. The cophenetic metrics can be safely used on weighted phylogenetic trees with nested taxa and no restriction on degrees, and they can be computed in O(n2) time, where n stands for the number of taxa. The metrics dφ,1 and dφ,2 have positive skewed distributions, and they show a low rank correlation with the Robinson-Foulds metric and the nodal metrics, and a very high correlation with each other and with the splitted nodal metrics. The diameter of dφ,p, for , is in O(n(p+2)/p), and thus for low p they are more discriminative, having a wider range of values.

/pdf/cophenetic-metrics-for-phylogenetic-trees-after-sokal-and-31rwyu6qto.pdf

Cophenetic metrics for phylogenetic trees, after Sokal and Rohlf

The Robinson-Foulds RF distance is the most popular method of evaluating the dissimilarity between phylogenetic trees. In this paper, we define and explore in detail properties of the Matching Cluster MC distance, which can be regarded as a refinement of the RF metric for rooted trees. Similarly to RF, MC operates on clusters of compared trees, but the distance evaluation is more complex. Using the graph theoretic approach based on a minimum-weight perfect matching in bipartite graphs, the values of similarity between clusters are transformed to the final MC-score of the dissimilarity of trees. The analyzed properties give insight into the structure of the metric space generated by MC, its relations with the Matching Split MS distance of unrooted trees and asymptotic behavior of the expected distance between binary n-leaf trees selected uniformly in both MC and MS źn3/2.

/pdf/on-a-matching-distance-between-rooted-phylogenetic-trees-3hzq9vgqg9.pdf

On a matching distance between rooted phylogenetic trees

The results of a recent survey of 194 metalloid lists are consistent with a three-criterion description of metalloids published over 35 years ago. The classifications of selenium, polonium, and astatine, and other metalloid-like elements, are briefly reviewed in this light.

Which Elements Are Metalloids

We carried out a mathematical study of 72 chemical elements taking advantage of the chemotopological method. We selected 128 properties to define the elements (physico-chemical, geochemical and chemical properties). Then, we looked for correlated properties and we reduced the number of them to 90. In this way we defined each element as a 90-tuple. Afterwards, we applied principal component analysis and cluster analysis (4 similarity functions and 5 grouping methodologies). Then, we calculated a consensus tree for the 20 dendrograms generated by the CA. Afterwards, we extracted the similarity relationships from the consensus tree and built up a basis for a topology on the set of chemical elements. Finally, we calculated some topological properties (closures, derived sets, boundaries, interiors and exteriors) of several subsets of chemical elements. We found that alkali metals, alkaline earth metals and noble gases appear not related to the rest of the elements. Also, we found that the boundary of non-metals are the semimetals with a stair-shape on the periodic table

Topological space of the chemical elements and its properties

We discussed three dissimilarity measures between dendrograms defined over the same set, they are triples, partition, and cluster indices. All of them decompose the dendrograms into subsets. In the case of triples and partition indices, these subsets correspond to binary partitions containing some clusters, while in the cluster index, a novel dissimilarity method introduced in this paper, the subsets are exclusively clusters. In chemical applications, the dendrograms gather clusters that contain similarity information of the data set under study. Thereby, the cluster index is the most suitable dissimilarity measure between dendrograms resulting from chemical investigation. An application example of the three measures is shown to remark upon the advantages of the cluster index over the other two methods in similarity studies. Finally, the cluster index is used to measure the differences between five dendrograms obtained when applying five common hierarchical clustering algorithms on a database of 1000 molecules.

Three Dissimilarity Measures to Contrast Dendrograms

We have shown in several papers the importance of using topology, particularly set-point topology, to deal with chemical questions related to the concept of similarity. The procedure developed has been called "chemotopology" and it has been applied to different chemical sets e.g. chemical elements, benzimidazoles, sterorids, amino acids and hydrides. The idea behind chemotopology is to run a hierarchical cluster analysis study on a set of objects characterised by different attributes. From this study a dendrogram is obtained, which gathers similarity neighbourhoods for the set of objects. By using a mathematical characterisation of a dendrogram it is possible to select a collection of objects' neighbourhoods which in turn become a basis for a topology. With this basis at hand different properties of subsets of objects can be calculated, all of them related to the concept of similarity e.g. closures, derived sets, boundaries, interiors and exteriors. We have also shown the chemical meaning of each one of these properties. In this manuscript, we review the foundations of the chemotopological method as well as its different applications to chemical sets. By means of examples we illustrate how the method can be used as a versatile tool for drug discovery. We also study the relationship between the topologies generated from dendrograms of a given set of objects and the dendrograms that can be obtained for particular topologies on the set of objects.

Chemotopology: beyond neighbourhoods.

We describe a mathematical methodology to endow a set of chemical interest with a topology. The procedure starts from a hierarchical classification of the set in a dendrogram (complete binary tree). Then, we cut “branches” of the tree by means of a mathematical procedure and we build up a basis for a topology with these branches. Finally, we show the way to calculate some topological properties, such as; closure, derived set, boundary, interior and exterior of subsets of chemical interest within the particular chemical set chosen at the beginning as object of study

On the Topological Sense of Chemical Sets

Chemical Elements: A Topological Approach

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