Pseudo-classification of natural elements and atomic total energy prediction

doi:10.21203/rs.3.rs-1419605/v2

Technological development requires the search for new materials with specific properties. Moreover, the forecasting of physical and chemical compound formation and proprieties can be based on Datamining methods. In this paper, one of the unsupervised methods of Datamining is used, like Principal Component Analysis (PCA). The first goal is to make pseudo-classification of natural and synthetic elements of the periodic table based on all physicochemical and energy available properties. Likewise, the second goal is to test the effectiveness of this statistical method to estimate correlations and appreciate the relationships between the variables and properties. A pseudo-classification of periodic elements is a new way of seeing Mendeleev's table. At the end of this work, a predictive polynomial has been presented that can allow the scientific community to make atomic total energy predictions for each natural periodic element, from Z=2 to Z=103.

Physical Chemistry

Materials Chemistry

Applied Statistics

Artificial Intelligence and Machine Learning

PCA

Periodic table

physicochemical

Electronic properties

Atomic total energy

LDA

The chemical elements discovery and their classification philosophy started decades or more. Thirty-three of elements and substances were reported by Lavoisier (1743-1794) where he described elements treatment in chemical1. In 1807-1808, Davy isolated sodium, potassium, barium, strontium, and calcium through electrolysis2. Like Davy's research, Döbereiner defined the traid law to notice the correlation between bromine, iodine, chlorine, sulfur, tellurium, and selenium3.

It was just waiting for John Dalton, to know that the chemical elements are described by their atomic weights4. However, Gay-Lussac5 in 1809 and Avogadro6 in 1911 introduced notions on physicochemical properties and the differentiation of molecules or atoms. The symbolic representation of the elements was proposed by Berzélius7 in 1814

The notion of an atomic mass system and the molecule were in 1860 in the first international chemistry congress held in Karlsruhe8. In 1867, the Russian chemist Dmitry Mendeleev was selected as a professor of mineral chemistry at the University of St. Petersburg after achieving his work on the density of gases and the spectroscope of Gustav Kirchhoff 9-12. He vertically classified the natural elements already determined in his time according to atomic number and atomic mass. Thus, he predicted the properties of some missing natural elements from their adjacents using traid law. A review of forms illustrations of the periodic elements during 100 years, is reported by Mazurs13 1974. The Graphic illustration as Spiral and Blocks of the chemical elements is described by Imyanitov14 2016. The philosophy and possibility of improvement of the periodic table are reported by Scerri15 2020.

The periodicity of natural elements and the introduction of quantum mechanics and fundamental concepts of chemical elements reported by authors (Scerri15 2020; Pyykkö17 2019; Imyanitov14 2016; Mazurs13 1974; van Spronsen16 1969).

The identification of the charges of the atomic nucleus (positron) and the display of some missing chemical elements were carried out by Henry Moseley that used frequencies of the X-rays18. The periodic table of chemical elements, which corresponds to the arrangement of chemical elements, is defined to respect a specific periodicity. Currently, the periodic table has 118 chemical elements, which hasn’t a fixed limit and is updated for each new discovery or innovation. The classical arrangement of natural elements defines that the chemical elements have a same number of valence band electrons assemble in the same columns. The horizontal classification is established based on increasing numbers of electrons from top to bottom, then left to right. Mendeleev's table is probably the earliest example of the used data analysis technique in materials science.

In this work, I use the unsupervised classification, such as the principal component analysis (PCA) method for studying periodic elements of Mendeleev's table. Our goal is to analyze correlations between the physical properties of elements. In the same way, I try to see the role of unsupervised methods in searching for a new way of seeing Mendeleev's table. At the end of this paper, I extract the correlations between the atomic energies to estimate the Total Energy, only in function of atomic number that can enhance electronic structure calculation methods of elements, especially in condensed-matter physics discipline.

Arrangement is a crucial step in data analysis. It consists of an objects group of a data set into homogeneous classes19-20. To estimate the predictive potential of array natural elements classification, one of the methods of Datamining is used, such as Principal Component Analysis “PCA”.

Datamining appeared in the 90s to extract new knowledge from a database. It extracts from the database new conclusions about the data entered21-23.

The basic idea of PCA is to reduce the dimension of the data matrix, retaining as much as possible the variations present in the starting data set. This reduction will only be possible if the initial variables are not independent and have non-zero correlation coefficients. These initial variables are transformed into new variables, called principal components.

They are obtained by linear combinations of the previous ones and are well arranged with each other. The principal component analysis determines orthonormal eigenvectors and their corresponding eigenvalues of the scatter matrix of the original variables. The orthonormal eigenvectors are used to construct the principal components. The eigenvalues are the variances of the corresponding principal components24.

In this work, one of the methods of Datamining Principal Component Analysis PCA25-29 is encoded and used for extracting knowledge from natural and synthetic elements and their properties. Polynomial Regression used in the present research predicts the relationship between dependent and independent variables. The application of the polynomial Regression technique for modelling is reported30. It is possible to predict the relationship between variables in the nonlinear state by increasing the order of polynomial terms31. Extensive polynomial and trigonometric regression are applied to characterize curvilinear relationships32. Technique prediction can be enhanced following model complexity degree by using the advanced method as partial least squares, artificial intelligence and machine learning. Hence, nonlinear regression is used in the present research to predict the relationship between electronic properties, total energy and atomic number.

Figure1 presents the diagram of properties (Scores plot) and Figures (2a,2b) show a diagram of elements (loading plot), respectively that are based on experimental data33. Its allow us to detect the correlations between the 21 physical properties of the elements, which are noted Var1 to Var21 for better visibility of the diagram.

It can be noticed some correlations and anti-correlations. Some of these correlations are simple to explain. Others reflect intrinsic physical principles and relationships. In next, it can be find the relationships between the properties as follows:

-Var1 correlated to Var2: The atomic number and the atomic mass are correlated. It's completely normal, as long as the atomic mass includes the sum of the neutrons and protons masses. The number of these latter is precisely the atomic number Z.

-Var 1 and Var2 correlated to Var 16:

The heat capacity, atomic number, and atomic mass are correlated, which can explain as follows: The heat capacity expresses the ability of the material to assimilate heat. The principal mode of heat energy assimilation is the phonons or networks (lattices) vibrations.

-Var21 and Var 19 are correlated:

The melting temperature and the boiling temperature have a physical relationship. It is axiomatically, as long as the boiling occurs after fusion.

-Var19 and Var 21 have a relationship with Var20:

The melting and boiling temperatures correlate to the change in enthalpy at melting.

This last quantity expresses the energy supplied so that the material passes from the solid to the liquid state (phases behaviours).

Var13 and Var14 correlated to Var 9:

The elastic constants C11 and C12 and Young's modulus have relationships.

The two constants define the material response to deformation on the atomic scale. Young's modulus is the macroscopic response of the material to tensile deformation. Unquestionably, the macroscopic mechanical behaviour properties are expressions of the displacement of atoms on a smaller scale when applying an external constraint.

- Var4 correlated to Var5: electronegativity and the energy of the first ionization are correlated. An electronegative material tends to attract electrons from other atoms in a bond. For stronger atomic binding, the electrons of the last orbit are strongly linked and are very difficult to exit which are expressed by high ionization energy.

-Var6 correlated to Var8: The lattice parameter and the molar volume are correlated, which is logical since the interatomic spacing is function molar volume.

-Var17, Var6, and Var8 are correlated:

The entropy has a relationship with the lattice parameter. Therefore, with the molar volume. It's' implies that the increase in molar volume growths the disorder in the system.

-Var10 correlated to Var11 and Var12: the elasticity constants S11, S12, and S44 are all correlated. This fact is consistent as long as all these quantities are directly related to how the structure varies when the external stress is applied. In other words, they all arise from the interatomic bond.

On the other hand, the anti-correlations between some variables are the following:

-Var14 has an inverse correlation to Var6 and Var8:

Young's module has an inverse correlation to the lattice parameter and the molar volume. It's known from experience that Young's module is proportional to the compression module that measures the lattice compressibility. So as important as it is, it is difficult to compress the volume when the lattice parameter and the molar volume decrease, the volume per atom also decreases. As a consequence, the atom's possibility of bringing closer together decreases. It explains why the network parameter decreases the compression module as much. Hence Young's module increases.

Consequently, the decreases of electrons implied less energy required to tear them away (first ionization energy) decreases. Also, when the number of electrons increases, the electronegativity decreases, and its' nucleus doesn't attract electrons from adjacent atoms more strongly.

Figure 2a reveals the spatial distribution of elements in the cartesian planes PC1PC2.

So it can be seen that helium, hydrogen, oxygen and some rare gases (Xe, Kr) are visible on the side of the diagram.

Then, tungsten occupies a particular position that can be distinguished by some metallic elements (Ru, Rh, Ta…).

The remainders of elements have a strong correlation, with the significant part that concentrate in the centre of the diagram, in the form of a cloud of points with poor readability.

As seen in this form, the chemical elements condensate in the centre of gravity of the diagram. Indeed, as much as the diagram of properties reflects a large number of physical correlations, the diagram of variables seems condensed and difficultly to read visually. To see more clearly, I have drawn this diagram in a cleverly different way.

Figure 2b is the three-dimensional representation of the variable diagram on which each element is represented by a cube more or less distant in the perpendicular direction to allow all the elements to be visualized. It remains to be noted that some distribution of elements in the graph shows a grouping according to the closest and similar properties. Therefore, it is very likely that the large number of properties used in this analysis is responsible for the variance in the results and the distribution of items.

Figure 2b is the three-dimensional representation of the variable diagram. Each element is represented by a cube more or less distant in the perpendicular direction to allow all the chemical elements to be visualized. Although figure 2b is sufficiently clear. It remains to note that the distribution of the elements in the graph does not follow a particular logic. Hence, it is very likely, that the large number of properties used, is responsible for the variance of the results and the distribution of the elements.

The elements and their properties are collected in a database based on LDA calculations34: the total energy, coulombian, kinetic, exchange, and correlation energy for 92 elements ranging from Hydrogen to Uranium. Figure 3 represents the properties plot (score plot). It can be noticed that the atomics energies are distributed practically only along the axis PC1.

It can also remark that the energies are distributed practically only along the axis PC1. The electron nucleus electrostatic interaction energy and the total energy are on the positive side of PC1, while the kinetic energy and the electron electrostatic interaction energy are on the negative side of PC1.

The energy of exchange and correlation is also on the right side, but with a few components according to PC2. The matrix shown in Figure 4 illustrates correlation types between energies. Figure 5 represents the diagram of variables (loadings plot). A few areas are zoomed in to visualize the points cloud of elements arrangement. As it's shown in figure 5, the elements of the periodic table are aligned according to their atomic number, starting from hydrogen and up to uranium in the opposite direction of the PC1 axis. Also, according to PC2, they form a kind of bell towards the positive direction of PC2. It can explain this as follows: The negative direction of the PC1 axis is correlated with the electron-electron interaction energy and kinetic energy of the electrons, so the elements with more electrons move in this direction.

The distribution curvature of the elements shows too the heaviest elements.

The energy of Coulomb interaction counts more than the energy of exchange and correlation because there are too many electrons. It is also the case for the lightest elements, where there aren't enough electrons for energy exchange and correlation. For the middle of elements, the exchange and correlation energy matter more than the coulomb energy, so the positions of elements are shifted upwards.

E Total-Predict= (p1*z¹⁰ +p2*z⁹ +p3*z⁸ + p4*z⁷ +p5*z⁶ + p6*z⁵ +p7*z⁴ + p8*z³ + p9*z² +p10*z + p11) *10⁵ (1)

P1=3.67E-20; P2=-2.00E-17; P3= 4.51E-15; P4=-5.51E-13; P5=3.99E-11; P6=-1.76E-09; P7= 4.91E-08; P8=-1.07E-06; P9=-6.11E-06; P10=5.46E-06; P11=-6.44E-06

Table 1 shows Atomic total energy predicts using polynomial equation (1) shown in figure 6. Following the error arrange, the results match NIST Standard Reference Database33-34 from Z=2 to Z=103.

Table 1.

z	Etot predicts “ Hartree”	Etot Hartree "LDA" [13-24 ].	SD
93	-26374.2130385254	-26325.4386967855	0.18%
94	-27054.7117761243	-27002.6683849033	0.19%
95	-27745.9078872578	-27689.6834855652	0.20%
96	-28448.1151881729	-28386.4092359141	0.22%
97	-29161.6974796868	-29093.9325964725	0.23%
98	-29887.0721625673	-29811.2722013778	0.25%
99	-30624.7135474639	-30538.8250815148	0.28%
100	-31375.1557599997	-31276.6433895199	0.31%
101	-32138.9951310089	-32024.7791035710	0.36%
102	-32916.8919507090	-32783.2840569287	0.41%
103	-33709.5714538646	-33551.5868267897	0.47%

In this study, the database conception is based on the physical properties of the periodic table of chemical elements using the PCA method to analyze the database and visualize results on scores and loading plots for examining relationships between elements properties. A new way of seeing Mendeleev's table or pseudo-classification of natural and synthetic elements has been suggested in this study. In processing, principal component Analysis (PCA) is applied for analyzing a database of elements many properties, as well as physicochemical, mechanical, thermal… and electronic properties, such as kinetic, electrostatic, total, exchange and correlation energy until atomic number 92. Then, it is revealed that the kinetic energy and electron-electron interaction energy characterize the heavy elements, so the electron nucleus interaction energy and the total energy characterize the light elements. It is also shown that the energy of exchange and correlation has a particular impact on the distribution of the elements on the loading plot, which needs a more physical explanation. The polynomial predictive allows forecasting the total energy for each element of the periodic table, only in function of atomic number from Z=2 to Z=103.

Var1: Atomic number Z

Var2: Atomic mass

Var3: The states of oxidation

Var4: Electronegativity

Var5: The 1st ionization energy

Var6: The lattice parameter

Var7: The number of atoms per cell

Var8: The molar volume

Var9: Young's module

Var10: The observation of elasticity S11

Var11: The finding of elasticity S44

Var12: The observation of elasticity S12

Var13: The elasticity module C11

Var14: The elasticity module C44

Var15: Thermal conductivity

Var16: Molar capacity Cp

Var17: Standard entropy S0

Var18: The Enthalpy difference ∆H

Var19: The melting temperature

Var20: The change in enthalpy at fusion

Var21: The boiling temperature

PC1 First principal axis

PC2 Second principal axis

PC1PC2 Cartesian plane

Acknowledgements

This research was realized at Abou Bekr Belkaid University of Tlemcen, Algeria.

I express my sincere thanks to the anonymous reviewers for reviewing this research paper. I would like to thank also my thesis supervisor in engineering physics35,36 and jury members for their constructive comments.

Competing interests

The author declares no competing interests.

Lavoisier, Antoine-Laurent , Traité élémentaire de chimie. Tome 1, (1743-1794).
Davy, H., XXIII. Electro-chemical researches, on the decomposition of the earth; with observations on the metals obtained from the alkaline earth, and on the amalgam procured from ammonia, Philosophical Transactions of the Royal Society of London, 1808, Vol. 98 (1808), pp. 333-370, https://www.jstor.org/stable/107302
J. W. Dobereiner (1780–1849). Nature 163, 434–435 (1949). https://doi.org/10.1038/163434d0
Harold Hartley,” John Dalton, F.R.S. (1766-1844) and the Atomic Theory-A Lecture to Commemorate his Bicentenar”, Proceedings of the Royal Society of London. Series B, Biological Sciences Vol. 168, No. 1013 (Nov. 14, 1967), pp. 335-359 , https://www.jstor.org/stable/75700
Gay-Lussac, Thenard, recherches physico-chimiques, TOME I.
Avogadro, Amedeo; Gay-Lussac, Joseph Louis; Dalton, Foundations of the molecular theory: comprising papers and extracts,1808-1811.
Berzelius, Jöns Jacob, Théorie des proportions chimiques, et table synoptique des poids atomiques des corps simples et de leurs combinaisons les plus importantes, (1779-1848; baron), Paris, 1835. https://gallica.bnf.fr/ark:/12148/bpt6k90589s
• Osnovy himii , « Mendeleev, Dmitrij Ivanovi (1834-1907) »,Leningrad , Tome I, 1934.
Osnovy himii , « Mendeleev, Dmitrij Ivanovi (1834-1907) »,Leningrad , Tome I, 1934.
Mendeleev, D.: On the periodic regularity of the chemical elements. Ann. der Chem. Pharmacie 8(1), 133–229 (1871) Google Scholar
Mendeleev, D.: The periodic law of the chemical elements. Moniteur Scientifique 21, 691–693 (1879)
Mendeleev, D. I. "On the relation of the properties to the atomic weights of the elements." Mendeleev on the periodic law: Selected writings 1905 (1869): 16-17. Google Scholar
Mazurs: Graphic Representations of the Periodic System During One Hundred Years. 1st and 2nd eds. University of Alabama Press, Tuscaloosa (1957/1974)
Imyanitov, N.S.: Spiral as the fundamental graphic representation of the periodic law. Blocks of elements as the autonomic parts of the periodic system. Found. Chem. 16, 1530173 (2016)
Scerri, E.: How can the periodic table be improved? Philos. Trans. R. Soc. A. Accepted (2020)
Van Spronsen, J.W.: The Periodic System of Chemical Elements. Elsevier, Amsterdam (1969)
Pyykkö, P.: An essay on periodic tables. Pure Appl. Chem. 91, 1959–1967 (2019)
Egdell RG, Bruton E. Henry Moseley, X-ray spectroscopy and the periodic table. Philos Trans A Math Phys Eng Sci. 2020 Sep 18;378(2180):20190302. doi: 10.1098/rsta.2019.0302. Epub 2020 Aug 17. PMID: 32811359.
H. Hotelling,"Analysis of a complex of statistical variables into principal components",Journal of Educational Psychology, vol.24 , pp.417- 441 , 1933.
L. Ferre, "Selection of components in principal component analysis: A comparaison of methods", Computational Statistics and Data Analysis , pp.669-682, 1995 .
Pieter Adriaans and Dolf Zantinge_ Data Mining_ Addison_Wesley,1996.
Jean-Michel Franco and EDS-Institut Prometheus. Le Data Warehouse. Le Data Mining.yrolles,1997.
J- Han and M-Kamber. Data Mining: Concepts and Techniques. Morgan Kaufmann, 2000.
Jolliffe I.T., "Principal component analysis", Springer-Verlag, New York, 1986
Jolliffe I.T. Principal Components in Regression Analysis. In: Principal Component Analysis. Springer Series in Statistics. Springer, New York, NY, 1986.
BESSE, P. et FERRÉ, L. Sur l'usage de la validation croisée en analyse en composantes principales. Revue de statistique appliquée, 1993, vol. 41, no 1, p. 71-76.
ADRIAANS, P. et ZANTINGE, D. Data Mining, Addison Wesley LongmanLimited. Edinbourgh Gate, Harlow, CM20 2JE, England, 1996
FRANCO, Jean-Michel. Le data warehouse: le data mining. Eyrolles: Informatiques magazine, 1997.
HAN, Jiawei et KAMBER, Micheline. Data Mining: Concepts and Techniques, The Morgan Kaufinann Series in Data Management Systems, Jim Gray, Series Editor Morgan Kaufmann Publishers. 2000.
Dean, A., Voss, D., Draguljić, D. Polynomial Regression. In: Design and Analysis of Experiments. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-52250-0_8. (2017).
Heiberger, R.M., Neuwirth, E. Polynomial Regression. In: R Through Excel. Use R. Springer, New York, NY.https://doi.org/10.1007/978-1-4419-0052-4_11. (2009).
J.O., Pantula, S.G., Dickey, D.A. Polynomial Regression. In: Rawlings, (eds) Applied Regression Analysis. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/0-387-22753-9_8 . (1998).
MARTIENSSEN, W. et WARLIMONT, H. Springer Handbook of Condensed Matter and Materials Data. V. 1. Berlin-Heidelberg. 2005.
KOTOCHIGOVA, Svetlana, LEVINE, Zachary H., SHIRLEY, Eric L., et al. Atomic reference data for electronic structure calculations. National Institute of Standards and Technology, 2005.
Brahim Belahcene, Datamining des propriétés physiques des éléments du tableau périodique, Universite Abou Bekr Belkaid Tlemcen UABT, 2009. Google Scholar
Brahim Belahcene, Datamining de la classification périodique des éléments, Colloque National sur les Techniques de Modélisation et de Simulation en Science des Matériaux, 2009. Google Scholar

Pseudo-classification of natural elements and atomic total energy prediction

Status:

Journal Publication

Version 2

Abstract

Figures

Background

Classification method and regression technical description

Results and interpretation

Dimensional representation of the variable diagram

Correlation of the atomic energy and electronic system

Conclusion

Nomenclature

Declarations

References

Status:

Journal Publication

Version 2