The expansion of chemical space in 1826 and in the 1840s prompted the convergence to the periodic system

The number and diversity of substances constituting the chemical space triggered, in two important steps, the convergence of the periodic system toward a stable backbone structure eventually unveiled in the 1860s. The first step occurred in 1826, and the second was between 1835 and 1845. Interestingly, the salient features of the periodic system of the 1860s can be detected as early as the 1840s, even when considering the effect of disagreement regarding the determination of atomic weights. The methods presented here become instrumental to study the further evolution of the periodic system and to ponder its current shape.

The periodic system, which intertwines order and similarity among chemical elements, arose from knowledge about substances constituting the chemical space. Little is known, however, about how the expansion of the space contributed to the emergence of the system—formulated in the 1860s. Here, we show by analyzing the space between 1800 and 1869 that after an unstable period culminating around 1826, chemical space led the system to converge to a backbone structure clearly recognizable in the 1840s. Hence, the system was already encoded in the space for about two and half decades before its formulation. Chemical events in 1826 and in the 1840s were driven by the discovery of new forms of combination standing the test of time. Emphasis of the space upon organic chemicals after 1830 prompted the recognition of relationships among elements participating in the organic turn and obscured some of the relationships among transition metals. To account for the role of nineteenth century atomic weights upon the system, we introduced an algorithm to adjust the space according to different sets of weights, which allowed for estimating the resulting periodic systems of chemists using one or the other weights. By analyzing these systems, from Dalton up to Mendeleev, Gmelin’s atomic weights of 1843 produce systems remarkably similar to that of 1869, a similarity that was reinforced by the atomic weights on the years to come. Although our approach is computational rather than historical, we hope it can complement other tools of the history of chemistry.

Implementation of Laplace Transformed MP2 for Periodic Systems With Numerical Atomic Orbitals

We present an implementation of the canonical and Laplace-transformed formulation of the second-order Møller–Plesset perturbation theory under periodic boundary conditions using numerical atomic orbitals. To validate our approach, we show that our results of the Laplace-transformed MP2 correlation correction for the total energy and the band gap are in excellent agreement with the results of the canonical MP2 formulation. We have calculated the binding energy curve for the stacked trans-polyacetylene at the Hartree–Fock + MP2 level as a preliminary application.

Challenges for the Periodic Systems of Elements: Chemical, Historical and Mathematical Perspectives

We celebrate 150 years of periodic systems that reached their maturity in the 1860s. They began as pedagogical efforts to project corpuses of substances on the similarity and order relationships of the chemical elements. However, these elements are not the canned substances wrongly displayed in many periodic tables, but rather the abstract preserved entities in compound transformations. We celebrate the systems, rather than their tables or ultimate table. The periodic law, we argue, is not an all-encompassing achievement, as it does not apply to every property of all elements and compounds. Periodic systems have been generalised as ordered hypergraphs, which solves the long-lasting question on the mathematical structure of the systems. In this essay, it is shown that these hypergraphs may solve current issues such as order reversals in super-heavy elements and lack of system predictive power. We discuss research in extending the limits of the systems in the super-heavy-atom region and draw attention to other limits: the antimatter region and the limit arising from compounds under extreme conditions. As systems depend on the known chemical substances (chemical space) and such a space grows exponentially, we wonder whether systems still aim at projecting knowledge of compounds on the relationships among the elements. We claim that systems are not based on compounds anymore, rather on 20th century projections of the 1860s systems of elements on systems of atoms. These projections bring about oversimplifications based on entities far from being related to compounds. A linked oversimplification is the myth of vertical group similarity, which raises questions on the approaches to locate new elements in the system. Finally, we propose bringing back chemistry to the systems by exploring similarity and order relationships of elements using the current information of the chemical space. We ponder whether 19th century periodic systems are still there or whether they have faded away, leaving us with an empty 150^th celebration.

Formal structure of periodic system of elements

For more than 150 years, the structure of the periodic system of the chemical elements has intensively motivated research in different areas of chemistry and physics. However, there is still no unified picture of what a periodic system is. Herein, based on the relations of order and similarity, we report a formal mathematical structure for the periodic system, which corresponds to an ordered hypergraph. It is shown that the current periodic system of chemical elements is an instance of the general structure. The definition is used to devise a tailored periodic system of polarizability of single covalent bonds, where order relationships are quantified within subsets of similar bonds and among these classes. The generalized periodic system allows envisioning periodic systems in other disciplines of science and humanities.

Chemistry of the 5g Elements: Relativistic Calculations on Hexafluorides

A Periodic System was proposed for the elements 1-172 by Pyykkö on the basis of atomic and ionic calculations. In it, the elements 121-138 were nominally assigned to a 5g row. We now perform molecular, relativistic four-component DFT calculations and find that the hexafluorides of the elements 125-129 indeed enjoy occupied 5g states.

100th anniversary of Bohr's model of the atom

In the fall of 1913 Niels Bohr formulated his atomic models at the age of 27. This Essay traces Bohr's fundamental reasoning regarding atomic structure and spectra, the periodic table of the elements, and chemical bonding. His enduring insights and superseded suppositions are also discussed.

SENSITIVITY ANALYSIS FOR OSCILLATING DYNAMICAL SYSTEMS

Boundary value formulations are presented for exact and efficient sensitivity analysis, with respect to model parameters and initial conditions, of different classes of oscillating systems. Methods for the computation of sensitivities of derived quantities of oscillations such as period, amplitude and different types of phases are first developed for limit-cycle oscillators. In particular, a novel decomposition of the state sensitivities into three parts is proposed to provide an intuitive classification of the influence of parameter changes on period, amplitude and relative phase. The importance of the choice of time reference, i.e., the phase locking condition, is demonstrated and discussed, and its influence on the sensitivity solution is quantified. The methods are then extended to other classes of oscillatory systems in a general formulation. Numerical techniques are presented to facilitate the solution of the boundary value problem, and the computation of different types of sensitivities. Numerical results are verified by demonstrating consistency with finite difference approximations and are superior both in computational efficiency and in numerical precision to existing partial methods.