Building Classes of Similar Chemical Elements from Binary

Chapter 5

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Building Classes of Similar Chemical Elements from Binary Compounds and Their Stoichiometries Guillermo Restrepo* Bioinformatics Group, Department of Computer Science, Leipzig University, Härtelstrasse 16-18, D-04107 Leipzig, Germany Laboratorio de Química Teórica, Facultad de Ciencias Básicas, Universidad de Pamplona, km 1 vía Bucaramanga, 543050 Pamplona, Colombia *E-mail: [email protected]; [email protected].

Similarity is one of the key concepts of the periodic table, which was historically addressed by assessing the resemblance of chemical elements through that of their compounds. A contemporary approach to the similarity among elements is through quantum chemistry, based on the resemblance of the electronic properties of the atoms involved. In spite of having two approaches, the historical one has been almost abandoned and the quantum chemical oversimplified to free atoms, which are of little interest for chemistry. Here we show that a mathematical and computational historical approach yields well-known chemical similarities of chemical elements when studied through binary compounds and their stoichiometries; these similarities are also in agreement with quantum chemistry results for bound atoms. The results come from the analysis of 4,700 binary compounds of 94 chemical elements through the definition of neighbourhoods for every element that were contrasted producing similarity classes. The method detected classes of elements with different patterns on the periodic table, e.g. vertical similarities as in the alkali metals, horizontal ones as in the 4th-row platinum metals and mixed similarities as in the actinoids with some transition metals. We anticipate the methodology here presented to be a starting point for more temporal and even more detailed studies of the periodic table.

© 2017 American Chemical Society Benvenuto and Williamson; Elements Old and New: Discoveries, Developments, Challenges, and Environmental Implications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


Introduction Chemistry is about substances and especially about compounds. Historically, chemical analysis has led to chemical elements, which are characterised by different properties, being of especial historical importance the atomic weight (1, 2). Through the study of how substances react with each other, it has been possible to find similarities among substances and to relate those similarities classes, which constitute the core of chemistry (3). A particular result of studying compound similarities is that it leads to classifying chemical elements by resemblance (4). The periodic table (PT) by Mendeleev resulted from combining atomic weights and similarity of chemical elements such that atomic weights were taken as the ordering principle and similarities as the source of classes (groups of the PT). Hence, ordering and similarity of an element define its position in the PT. Later on, the atomic number was the accepted ordering criterion, which brought an atomistic ontology for the concept of element: a collection of atoms with the same atomic number (number of protons) (5–7). The exponential growth of chemical substances made difficult assessing similarities through the historical approach (8). This, in combination with the emerging atomistic ontology and the advent of quantum mechanics, particularly its application to chemical elements to understand the structure of the PT, led to model similarities among chemical elements through similarities on the energetic distributions of valence shell electrons (9, 10). This is the cause of the typical contemporary overemphasised textbook introduction to the periodic table through electronic configurations, which normally correspond to those of free atoms (11). However, these configurations are rather dissimilar to those of the bound atoms present in substances, all in all the relevant species for chemistry (11–14). Jørgensen put it plainly as “There is not the slightest doubt, however, that no simple relation exists between the electron configuration of the ground state of the neutral atom and the chemistry of the element under consideration (11).” Besides the textbook oversimplification of the PT to the electronic structure of free atoms, for example in the form of the Madelung rule, authors of these books also overlook a fundamental piece of information of the historical approach, namely that Mendeleev’s studies were mainly based on compounds, particularly on oxides, hydroxides, hydrides and halides and that by studying their similarities he came up with resemblances for the chemical elements (15, 16). Mendeleev highlighted the need to rely on compounds and their proportions of combination rather than on properties of chemical elements; this is evident in his statement that “if CO2 and SO2 are two gases which closely resemble each other both in their physical and chemical properties, the reason of this must be looked for not in an analogy of sulphur and carbon, but in that identity of the type of combination, RX4, which both oxides assume (17, 18).” He adds, “the elements, which are most chemically analogous, are characterized by the fact of their giving compounds of similar form RXn.” Some few contemporary authors have stressed the importance of compounds and have elaborated on the related problem of quantifying qualitative parts of chemistry in similarity studies (19). Following Mendeleev’s ideas, a key concept 96 Benvenuto and Williamson; Elements Old and New: Discoveries, Developments, Challenges, and Environmental Implications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.

to understand similarities of chemical elements is that of valency, which is obtained by stoichiometric decomposition of compounds in chemical analysis (20). Hence, relying on chemical compounds and their stoichiometries, we discuss in the current chapter results of a mathematical and computational study of similarity of chemical elements (4).


Materials and Methods As a first approach to chemical classification of elements through compounds, binary substances were analysed (4). As usual, the binary compound was defined as any substance containing two different elements, independently of their proportions. Hence, H2O, KCl and CH4 entered in the study, but H2SO4 or C60 did not. In total, 4,700 binary compounds were analysed, which accounted for 94 chemical elements with at least one binary compound reported in the literature. As the binary compound exists, its decomposition reaction to chemical elements also exists (Figure 1A). The set of decomposition reactions gives place to a network, a hypergraph, as exemplified in Figure 1B, which, for our purposes, can be reduced to a subnetwork (Figure 1C) whose vertices are the elements produced by decomposition (21). The edges of this subnetwork, or lines between two vertices, are pairs of elements produced by a decomposition reaction. We call this network the product-network.

Figure 1. A) Two decomposition reactions ρ and ρ′. B) Modelling of ρ and ρ′ as a hypergraph. C) Product-network of B.

As the interest is analysing the similarity among chemical elements, the structural similarity of the chemical elements in the product-network is to be explored in such a way that elements linked to common elements are similar. Thus, for example, Na and K are similar for they form fluorides, chlorides and bromides, to name but a few of their common binary compounds. This is shown in Figure 2A and 2B, where Figure 2B shows that Na and K are linked to F, Cl and Br, therefore Na and K are similar; F, Cl and Br are also similar, for they are linked to Na and K. To quantify such a similarity, the neighbourhood of each element was determined, such that it contains the element in question and the elements that are connected to it in the product-network. For the three elements of Figure 1, the neighbourhoods are Nx= {x,y}, Ny= {x,y,z} and Nz= {y,z}. The corresponding neighbourhoods for the elements of Figure 2 are shown in Figure 2C. 97 Benvenuto and Williamson; Elements Old and New: Discoveries, Developments, Challenges, and Environmental Implications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


The more similar the neighbourhoods, the more similar the elements are; or the more different the neighbourhoods, the more different the elements. These differences can be calculated by counting the number of elements in the symmetric difference between pairs of neighbourhoods. The symmetric difference of two sets results from their union and the removal of their intersection. The number of elements of the corresponding symmetric differences of Figure 2 are shown in Figure 2D, where, for example, the two elements of the symmetric difference between NK and NNa are Na and K. Figure 2 shows that Na and K are similar and F, Cl and Br also form another similarity class.

Figure 2. A) Hypergraph of reactions ρ1 to ρ6, its product-network (B) and the neighbourhoods of each element (C). D) Number of elements of the symmetric difference between pairs of neighbourhoods.

Now, let us suppose that we have the hypergraph shown in Figure 3A, whose product-network is shown in Figure 3B along with the respective neighbourhoods of the elements (Figure 3C). This network indicates that H is evenly similar to F and B. But F only forms a binary compound with H while B forms by far more with H. A method to take into account this diversity of combinations, following Mendeleev’s ideas, is to consider the stoichiometry of the combinations as follows: for a compound xayb, the neighbourhood of x is given by {xa/b,yb/a}, in this way the stoichiometry of the binary combination is attached to the respective element. 98 Benvenuto and Williamson; Elements Old and New: Discoveries, Developments, Challenges, and Environmental Implications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


Hence, the neighbourhoods for the elements in Figure 3 are extended to NF= {F1/1,H1/1}, NH= {F1/1,H1/1,H6/2,H9/5,H14/10,…,B2/6,B5/9,B10/14,…}, NB= { H6/2,H9/5,H14/10,…,B2/6,B5/9,B10/14,…}, from which it can be concluded that there are more resemblance between H and B than between F and H, for there are by far more commonalities between NH and NB than the only two (F1/1,H1/1) between NF and NH.

Figure 3. A) Hypergraph of some decomposition reactions of binary compounds with H and either F or B. B) Product-network of A. C) Neighbourhoods of each element in B.

These stoichiometric neighbourhoods were determined and their differences quantified. The elements were clustered according to their neighbourhood resemblance using hierarchical cluster analysis (HCA) with the average union as grouping methodology. An illustrative example of the HCA algorithm is shown in Figure 4, which starts with the differences for the neighbourhoods shown in Figure 4A. It is found that A and B are the most similar elements; then, after grouping A with B new differences are recalculated (Figure 4B). As the grouping methodology is the average union, the difference between {A,B} and D, for example, is given by the average of the differences between A and D and that of B and D, i.e. 4 (22). With this new table of differences (difference matrix), the algorithm keeps running (Figures 4C, 4D) until all elements are grouped. A depiction of these groupings is a dendrogram (Figure 4E), which shows that A and B are the most similar elements and that they, in turn, are similar to element C. This group of three elements is similar to D and, finally, the most dissimilar element to the group of four elements is E. Another depiction of the similarities takes into account the different levels of similarity of the merging process, which are also seen in the hierarchical structure of a dendrogram. Figure 4F shows such a similarity landscape. 99

Benvenuto and Williamson; Elements Old and New: Discoveries, Developments, Challenges, and Environmental Implications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


Figure 4. A) Number of different elements between Ni and Nj. B) New difference matrix after merging A and B, C) {A,B} with C and D) {A,B,C} with D. E) Dendrogram and F) its similarity landscape.

Results and Discussion The 94 elements explored through the 4,700 binary compounds are shown in Figure 5 making use of the conventional medium-long form of the PT (9).

Figure 5. 94 elements explored through binary compounds. The similarity landscape for the chemical elements is shown in Figure 6 and the dendrogram is found in reference (4). 100 Benvenuto and Williamson; Elements Old and New: Discoveries, Developments, Challenges, and Environmental Implications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


Figure 6. Similarity landscape of 94 chemical elements.

Figure 6 shows that the most different element is H, i.e. there is no other element whose presence and proportion in binary compounds is similar to that of H. Other similar cases are found for B, C, N and O, which are evidences of the singularity principle, i.e. the chemistry of the second period elements is often different to the latter members of their respective groups (23, 24). In fact, elements on the red regions of Figure 6 are not only different from the elements of their groups but entirely different from all the other elements. This indicates that no other element forms binary compounds with the elements they combine and with the stoichiometries they have. Still on the red regions, halogens show up, right between the singularities of H, C and O and those of S, B, P and N. This indicates that halogens, as a group of the PT, have no other element behaving as they do with the elements they form binary compounds. However, halogens are very similar among themselves, as they belong to a blue region. The strongest similarity within halogens occurs between Cl and Br; a bit different is I and, as an evidence of the singularity principle, F is the least similar halogen. By increasing the similarity, Se and Te form a very similar couple and not far from them As and Sb result also similar. In fact, of the chalcogens, only Se and Te are similar, for all other members of this group show up as single classes. Likewise pnictogens behave, where only As and Sb are similar. Right in the middle of {Se,Te} and {As,Sb} Si is found; which shows that some elements which it forms binary compounds with are elements which Se and Te, and As and Sb also form with and with similar stoichiometries. 101



At a higher level of similarity, alkali metals show up constituting a similarity “bridge” between the already discussed elements, especially {As,Sb}, Si, {Se,Te}, the single pnictogen Bi and the transition metals; especially Cr and Mn. Within the very strong similarities of alkali metals, the couples {Li,Na} and {Rb,Cs} are the most similar classes. K is most similar to Rb and Cs than to Li and Na. Bi, Cr and Mn can also be regarded as elements connecting the red-orange regions of markedly differences with the green-blue regions, where most of the similar elements are gathered. At the periphery of this regions belong the remaining transition metals, forming several clusters of very strong similarities. It is found that the V-group is one of these clusters, as well as the Zn-group, this latter interestingly sharing similarities with alkaline earth metals and with {Ge,Sn,Pb}. These resemblances of the Zn-group elements indicate that they form binary compounds with several of the elements which the alkaline earth metals do with and also that the Zn-group elements combine in a similar fashion as Ge, Sn and Pb do. Hence, the Zn-group can be regarded as an intermediate group between alkaline earth metals and Ge, Sn and Pb. Another interesting transition metal cluster is {Fe,Co,Ni,Pd}, whose elements are part of the so called “platinum metals,” group VIII in the old IUPAC group numbering or VIIIB in the CAS numbering. This shows that they indeed have commonalities regarding the elements which they combine with. This reminds us that the similarity of these “platinum metals” was what led Mendeleev to group them together as noted in his claim that “Only among these metals are compounds of the type RO4 or R2O8 formed (which is why they are designated as the eighth group) (18, 25).” Although these similarities, Ru, Os, Rh, Ir and Pt are not so similar to Fe, Co, Ni and Pd. Rh and Ir, in turn, are more similar to lanthanoids and actinioids. All elements of Ti-group are similar, especially Zr and Hf, but they are also similar to two actinoids: Th and U. The similarity of Th and {Zr,Hf} was already in 1945 highlighted by Seaborg (26). Schwarz has also recently discussed the resemblance of early actinoids with some 6th-row transition metals (27). In particular, the resemblance of Zr and Hf was recognised by Goldschmidt as the effect of the lanthanoid contraction, which in modern terms is understood as a spatial shrinking of lanthanoids species as the result of the filling of 4f shells that brings a contraction of the 5p and 6s shells of the species. Such a contraction makes that, e.g. Zr(IV) and Hf(IV) have almost equal ionic radii when six-coordinated (28–30). Although the Zr-Hf similarity is well documented in the literature, it is striking to find that it is considered an exception, “due to an anomalous cancellation of relativistic effects,” of the differences between 5th- and 6th-row elements of the same group (30, 31). In the current study we found that out of the 17 possible pairs of 5th- and 6th-row elements belonging to a group, there are other five pairs sharing similarities: {Nb,Ta}, {Mo,W}, {Tc,Re}, {Ru,Os} and {Rh,Ir}. The first two were also discussed by Huheey and Huheey, who found as a cause of it the almost equal radii for 5th- and 6th-row species (29). Fricke et al. also discussed the generality of this resemblance and the similar oxidation states for 5th- and 6th-row elements (32). 102



Another cluster, which involves main group elements, is {Al,Ga,In,Tl,Cu,Ag,Au}, where the Al-group is merged with the coinage metals. By increasing the similarity level is found that the Al-group is a subcluster and {Cu,Ag} another one. It is observed that the similarity of the Al-group with the Cu-group is given by the resemblance between Tl and Au, which may be caused by relativistic effects. The largest cluster gathering similar elements is the one of lanthanoids and actinoids. Lanthanoids are more similar among them than actinoids among them. Jørgensen states that such a strong resemblance among lanthanoids, characterised by a constant dominant oxidation state III, was initially noticed by Rydberg in 1914 when mentioning the “existence of fifteen consecutive elements having almost the same chemical properties (11, 27).” Fricke et al. gave an explanation by relating ionisation energy with valency: “The ionization energy […] is a valuable quantity for theoretical predictions of the valency of the elements. […] the ionization energy becomes larger if the wave function of a new additional electron is not screened by the other electrons with the same wave function. This altogether means that the ionization energy curve for the lanthanides is very flat and, therefore, the valency is always nearly the same (32).” And they explained the differences among actinoids “because the 5f electrons are not so buried in the atom as the 4f electrons in the lanthanides and thus the valency at the beginning is larger whereas it becomes smaller at the end.” Within lanthanoids, Ce is the most different element and there is a strong resemblance of lanthanoids with Sc and Y, a similarity already found by Goldschmidt, especially for Y and Dy and Ho (28). In fact, Sc, Y and the lanthanoids constitute the so-called rare earths, which are grouped together given their chemical similarities that have found support on quantum chemical grounds (33). Scerri has discussed about the element at the beginning of the third row of the transition elements, which in some tables is La and in others Lu (9). Schwarz and Rich have stated that Lu cannot be considered a lanthanoid, for it does not fill f orbitals as they are already filled; and have suggested that Lu should be regarded as a transition metal (33). According to our results, La appears in between two clusters, one of 11 lanthanoids and another of transition metals, namely {Y,Sc}. Lu is part of the clusters of 11 lanthanoids and the smallest cluster containing it is {Ho,Er,Lu}, which shows that Lu is more similar to lanthanoids than to transition metals, while La share similarities with lanthanoids and with transition metals. Therefore La must be the element located at the beginning of the third row of transition metals if chemical resemblances is what it is to be emphasized. Besides the similarity of lanthanoids with Sc and Y, lanthanoids are also similar to some actinoids, as found in the presence of Tb and Pr in a large cluster of actinoids and noble gases. A resemblance that has been discussed by Schwarz on quantum chemical grounds (27). However, the similarities are more notorious among members of the same row for lanthanoids and actinoids, as also reported by Diwu et al. and Cary et al. (34, 35) The relaxed similarities among actinoids can be seen for example in their dominant oxidation states, which vary from II in No to VI in U; variations related to 103



the actinoid contraction, which is more irregular and not as large as the lanthanoid one (27, 32, 36). If lanthanoids have similarities with Sc and Y, actinoids have them with more transition metals, e.g. with Zr, Hf, Tc and Re. It is found, for example, that U is similar to the Ti-group of transition metals and to Th. Similarities of actinoids with transition metals have been reported by Rayner-Canham and studied by Schwarz and Rich (23, 33). The largest set of similar actinoids is Am, Cm, Bk, Cf, Es. An interesting place for Pu is found in the similarity landscape, which results alike to some actinoids {Cm,Bk,Es,Am,Cf,Ac} and to the lanthanoids Tb and Pr. Chemical singularities of Pu have been recently studied by Schwarz in a systematic study of tricarbonato-actinyl anions (27). Cary et al. argue that the special behaviour of Pu is given by its unique electronic properties, which come from the changing roles of the 5f orbitals, which among other features, allow it to equilibrate four oxidation states in solution, something not reported for any other chemical element (35). Scerri has also made the point about the element at the beginning of the fourth row of the transition elements, being Ac in some tables and Lr in others (9). Unfortunately no relevant data was available for Lr at the time of the study, therefore it is not possible to discuss its resemblance to other elements (4, 37). Ac is found in a single class, which is also similar to a class of elements with very few number of binary compounds, i.e. Ra, Kr, Xe and At (38). In general, Figure 6 shows that the most different elements are: most of the second row elements, H, halogens and alkali metals; whose trends of combination are unique and which are very different from those of transition metals, lanthanoids and actinoids. These three later sets are, in general, alike regarding the elements which they form binary compounds with and also alike in the proportion they do it.

Conclusions and Outlook Although there have been several studies on the similarity of chemical elements through different chemical, physical and physicochemical properties of the elements; the study reported in Leal et al., here further explained, is the only one where the recovery of trends in the PT has been attained by uniquely using chemical information and where more elements have been regarded (4, 39–42). This study follows Mendeleev’s ideas of devising similarities for the elements based on their compounds through a mathematical and computational approach. It is found that the method is able to detect several well-known classes of similar elements with different patterns on the PT, e.g. vertical similarities as in alkali metals, halogens, Al-group and Cu-group; horizontal ones as in 4th-row platinum metals, lanthanoids, actinoids; and some other mixed patterns as in lanthanoids and Sc and Y (rare earths); and actinoids with some transition metals. It is found that for most of these similarities, there is a quantum chemical possible explanation based on non-simplistic rules of electronic configuration fillings but on studies that take into account, e.g. relativistic effects, for heavy atoms and also the bound character of atoms in compounds. 104



The approach followed in the current chapter constitutes also an alternative way to introducing the PT to students, with a more chemical “flavour” than the traditional oversimplification relying on electronic configurations of free atoms. The ingredients of the approach here presented are fundamental concepts of chemistry such as compound, reaction and stoichiometry, which make part of the bulk level by Nelson for describing chemistry (43). However, in teaching, the concepts must be presented as simple as possible and the current approach is not that simple. A work to do to reduce the complexity of this presentation is to look for those particular regions of the neighbourhoods defined for the elements, which make that the results do not vary too much. This would also shed some light on the most relevant neighbours for keeping the similarity structure of the PT. The similarity classes found match to a big extent the chemical way of presenting the elements in some few specialised chemical books like Chemistry of the Elements, which is to be expected from books rooted in chemical reactions (44). The study here discussed was based upon binary compounds, but the current amount of chemical information, stored in electronic libraries, and the current computing capacity, allow thinking in running the study with ternary, quaternary and even with the whole set of chemical substances. An interesting example of exploration of large networks of chemicals are the studies of Grzybowski and his team (45–47). The study can also be run at different time periods to explore how the patterns on the periodic table have changed in time. At first glance, it looks like the approach to chemical similarity here discussed cannot stand the test of time, for it relies on compounds, which are especially scarce for the heavy elements. Moreover, for these elements the few compounds that are obtained are synthesised in a one-atom-at-a-time fashion, which is rather different to the bulk one, of the traditional chemistry (48, 49). This brings not only a clash of chemical traditions, but also the mixture of two different ontological levels for compounds, i.e. the bulk and the one of atomic aggregates. But the method overcomes these problems, for it is actually based, more than upon compounds, on their mathematically generality, i.e. their composition and stoichiometry. Both can be extracted from either bulk or atomic aggregate compounds; wet-lab synthesized or in a one-atom-at-a-time way or even estimated through quantum chemical approaches (50). From a methodological viewpoint there were two important steps for finding similarity classes, namely the construction of neighbourhoods and their clustering. In Bernal et al. it has been shown that those neighbourhoods can be further explored through an interesting connection with Formal Concept Analysis, a mathematical technique based on the finding of closed sets of objects (51). There, objects are characterised by their attributes and, for the particular case of binary compounds, objects are chemical elements and their attributes are those chemical elements belonging to their neighbourhoods. The use of neighbourhoods for chemical elements brings also possibilities to explore in more detail the landscape of similarities through the inclusion of basic elements of topology (41, 42, 52). Works on this direction have started with Restrepo et al. and have also found a connection with the results obtained by Formal Concept Analysis (41, 51). The richness of this subject is still to be 105



explored and constitutes an interesting field of study for mathematical chemistry (53). Regarding the clustering of elements, the algorithm used was the hierarchical cluster analysis, one of the most used classificatory techniques, particularly in chemistry (54). However, it has limitations, e.g. the stability of clusters is reduced given equidistances in the difference matrices during the merging processes. This is a problem known as ties in proximity, which was brought to the attention of the chemical community by MacCuish and which has been further explored by Leal et al. (54, 55) Leal and his team developed measures of cluster’s frequency taking into account ties; therefore it is worth exploring how frequent and stable are clusters of chemical elements given ties using these techniques. A related issue of hierarchical cluster analysis is that clusters may vary if the grouping methodology changes. The work to do is to explore if clusters obtained in this work remain stable after using several other grouping methodologies. However, previous results on chemical elements, although not using relational properties as in the current work, have shown that clusters are in general stable, especially for those around noble gases, e.g. alkali metals and halogens (41, 42). Finally, from a philosophical perspective, Schädel has stated the current, in our opinion counterhistorical, understanding of the PT, whose premise is that the atomic number and the electronic configuration of an element determine its position in the PT (49). This brings the consequence that from the electronic configuration of the position, the chemical properties arise, which leads to claim that chemical properties can be linked to trends of the electronic configurations along groups or periods. In our opinion, the premise is that similarity in chemical properties of compounds, along with atomic number lead to the position of the elements in the PT (3). This brings the consequence that from similarities in chemical properties, electronic configurations are obtained. Fricke et al. clearly stated it for the case of superheavy elements: “Theoretical predictions of the chemistry of superheavy elements can be done in two ways (32). First the behaviour of the well-known elements as a function of their chemical group and period can be extrapolated into the unknown regions. Secondly eigenvalues, wavefunctions, most stable configurations etc. can be calculated theoretically so that on this basis the predictions of the chemistry can be done in a more accurate way.” Given the stability of our approach, Schädel’s question whether the PT is still valid regarding chemical properties of the superheavy elements, from Rf onwards, has a positive answer (49).

Acknowledgments G. R. thanks the Universidad de Pamplona and the Alexander von Humboldt Foundation/Stiftung for funding this research. Eugen Schwarz is specially thanked for his valuable comments on early versions of this document and for pointing out important aspects of the quantum chemistry of heavy elements.

106 Benvenuto and Williamson; Elements Old and New: Discoveries, Developments, Challenges, and Environmental Implications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.

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13. In reference 11 Jørgensen discusses how “the very high ionization energies of atomic spectroscopy are to a large extent compensated by the electrostatic attraction of the surrounding anions” for crystals. 14. Wang, S-G.; Schwarz, W. H. E. Icon of chemistry: The periodic system of chemical elements in the new century. Angew. Chem., Int. Ed. 2009, 48, 3404–3415. 15. Schwarz, W. H. E.Which brings chemically relevant results for about 20% of all elements. Personal Communication. See reference 14. 16. Schwarz, W. H. E.; Wang, S-G. Some solved problems of the periodic system of chemical elements. Int. J. Quantum Chem. 2010, 110, 1455–1465. 17. Mendeleev, D. Principles of Chemistry; Longmans, Green & Co: London, 1905; Chapter 15. 18. Jensen, W. B. Mendeleev on the Periodic Law, Selected Writings, 1869-1905; Dover: Mineola, NY, 2005. 19. Schwarz, W. H. E. Recommended questions on the road towards a scientific explanation of the periodic system of chemical elements with the help of the concepts of quantum physics. Found. Chem. 2007, 9, 139–188. 20. Schwarz, W. H. E. Towards a physical explanation of the periodic table (PT) of chemical elements, achievements of the previous generations. In Fundamental world of quantum chemistry; Brändas, E. J., Kryachko, E. S., Eds.; Springer: Dordrecht, The Netherlands, 2004; Vol. III, pp 645−669. 21. Klamt, S.; Haus, U-U.; Theis, F. Hypergraphs and cellular networks. PLoS Comput. Biol. 2009, 5, e1000385. 22. MacCuish, J.; MacCuish, N. E. Clustering in Bioinformatics and Drug Discovery; Mathematical and Computational Biology Series; CRC Press: Boca Ratón, FL, 2011; pp 1−244. 23. Rayner–Canham, G. Periodic patterns. J. Chem. Educ. 2000, 77, 1053–1056. 24. An atomic interpretation of this effect comes from the small atomic radius of 2nd-row species regarding the bigger size of atoms in higher rows. 25. Mendeleev, D. The periodic law of the chemical elements. J. Chem. Soc. 1889, 55, 634–656. 26. Seaborg, G. T. The chemical and radioactive properties of the heavy elements. Chem. Eng. News. 1945, 23, 2190–2193. 27. Liu, J-B.; Chen, G. P.; Huang, W.; Clark, D. L.; Schwarz, W. H. E.; Li, J. Bonding trends across the series of tricarbonato-actinyl anions [(AnO2)(CO3)3]4- (An = U−Cm): The plutonium turn. Dalton Trans. 2017, 46, 2542. 28. Wedepohl, K. H. The importance of the pioneering work by V. M. Goldschmidt for modern geochemistry. Naturwissenschaften 1996, 83, 165–171. 29. Huheey, J. E.; Huheey, C. L. Anomalous properties of elements that follow “long periods” of elements. J. Chem. Educ. 1972, 49, 227–230. 30. Pyykkö, P.; Desclaux, J-P. Relativity and the periodic system of elements. Acc. Chem. Res. 1979, 12, 276–281. 31. Pyykkö, P. The physics behind chemistry and the periodic table. Chem. Rev. 2012, 112, 371–384. 108



32. Fricke, B.; Greiner, W.; Waber, J. T. The continuation of the periodic table up to Z = 172. The chemistry of superheavy elements. Theoret. Chim. Acta. 1971, 21, 235–260. 33. Schwarz, W. H. E.; Rich, R. L. Theoretical basis and correct explanation of the periodic system: Review and update. J. Chem. Educ. 2010, 87, 435–443. 34. Diwu, J.; Grant, D. J.; Wang, S.; Gagliardi, L.; Albrecht-Schmitt, T. E. Periodic trends in lanthanide and actinide phosphonates: Discontinuity between plutonium and americium. Inorg. Chem. 2012, 51, 6906–6915. 35. Cary, S. K.; Vasiliu, M.; Baumbach, R. E.; Stritzinger, J. T.; Green, T. D.; Diefenbach, K.; Cross, J. N.; Knappenberger, K. L.; Liu, G.; Silver, M. A.; DePrince, A. E.; Polinski, M. J.; Van Cleve, S. M.; House, J. H.; Kikugawa, N.; Gallagher, A.; Arico, A. A.; Dixon, D. A.; Albrecht-Schmitt, T. E. Emergence of californium as the second transitional element in the actinide series. Nat. Commun. 2015, 6, 6827. 36. Given these variations in actinoids’s similarity, Jørgensen11 suggested not referring to them as “actinides”, or actinoids, in more contemporary terms, but to 5f group elements; for, as pointed out by Schwarz and Rich,33 actinide/ actinoid means “like actinium.” 37. A recent revision of the literature shows that there are at least three binary compounds for including it in a new study. 38. For all these elements the current literature has more binary compounds, allowing running a finer study for them. 39. Zhou, X.-Z.; Wei, K.-H.; Chen, G.-Q.; Fan, Z.-X.; Zhan, J.-J. Fuzzy cluster analysis of chemical elements. Jisuanji Yu Yingyong Huaxue. 2000, 17, 167–168. 40. Sneath, P. H. A. Numerical classification of the chemical elements and its relation to the periodic system. Found. Chem. 2000, 2, 237–263. 41. Restrepo, G.; Mesa, H.; Llanos, E. J.; Villaveces, J. L. Topological study of the periodic system. J. Chem. Inf. Comput. Sci. 2004, 44, 68–75. 42. Restrepo, G.; Llanos, E. J.; Mesa, H. Topological space of the chemical elements and its properties. J. Math. Chem. 2006, 39, 401–416. 43. Nelson, P. G. Teaching chemistry progressively: From substances, to atoms and molecules, to electrons and nuclei. Chem. Educ. Res. Pract. 2002, 3, 215–228. 44. Greenwood, N. N.; Earnshaw, A. Chemistry of the Elements, 2nd ed.; Elsevier: Oxford, UK, 2005; pp v−xviii. 45. Fialkowski, M.; Bishop, K. J. M.; Chubukov, V. A.; Campbell, C. J.; Grzybowski, B. A. Architecture and evolution of organic chemistry. Angew. Chem., Int. Ed. 2005, 44, 7263–7269. 46. Bishop, K. J. M.; Klajn, R.; Grzybowski, B. A. The core and most useful molecules in organic chemistry. Angew. Chem., Int. Ed. 2006, 45, 5348–5354. 47. Grzybowski, B. A.; Bishop, K. J. M.; Kowalczyk, B.; Wilmer, C. E. The ‘wired’ universe of organic chemistry. Nat. Chem. 2009, 1, 31–36. 48. This difference is not only a matter of scale and of amount of substance, but also of the synthetic machinery and of its attached theory. According to Schädel:49 “As a single atom cannot exist in different chemical forms 109



49. 50.

51.

52. 53.

54.

55.

taking part in the chemical equilibrium at the same time, the classical law of mass action—well established for macroscopic quantities and characterizing a dynamic, reversible process in which reactants and products are continuously transformed into each other—is no longer valid. For single atoms, the concept of chemical equilibrium needs to be substituted by an equivalent expression in which concentrations, activities, or partial pressures are replaced by probabilities of finding the atom in one state or the other”. Besides formation of compounds, other properties studied at this level are volatilities, formation of complexes in aqueous solutions and their interaction with other phases.49 Interestingly, properties such as ionic radius and the stability of oxidation states are indirectly obtained by using the PT, for they result from the comparison with the known properties of lighter members of the group the element belongs to.49 Schädel, M. Chemistry of superheavy elements. Angew. Chem., Int. Ed. 2006, 45, 368–401. Pyykkö, P. A suggested periodic table up to Z ≤ 172, based on Dirac-Fock calculations on atoms and ions. Phys. Chem. Chem. Phys. 2011, 13, 161–168. Bernal, A.; Llanos, E. J.; Leal, W.; Restrepo, G. Similarity in chemical reaction networks: Categories, concepts and closures. In Advances in mathematical chemistry and applications; Basak, S. C., Restrepo, G., Villaveces, J. L., Eds.; Bentham: Sharjah, United Arab Emirates, 2015; Chapter 2, pp 24−54. Restrepo, G.; Mesa, H. Chemotopology: beyond neighbourhoods. Curr. Comput-Aided Drug Des. 2011, 7, 90–97. Restrepo, G. Mathematical chemistry, a new discipline. In Essays in the Philosophy of Chemistry; Scerri, E., Fisher, G., Eds.; Oxford University Press: New York, 2016; Chapter 15, pp 332−351. Leal, W.; Llanos, E. J.; Restrepo, G.; Suárez, C. F.; Patarroyo, M. E. How frequently do clusters occur in hierarchical cluster analysis?: A graph theoretical approach to studying ties in proximity. J. Cheminf. 2016, 8, 4. MacCuish, J.; Nicolaou, C.; MacCuish, N. E. Ties in proximity and clustering compounds. J. Chem. Inf. Comput. Sci. 2001, 41, 134–146.

110 Benvenuto and Williamson; Elements Old and New: Discoveries, Developments, Challenges, and Environmental Implications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.

Building Classes of Similar Chemical Elements from Binary

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