RAY HEFFERLIN, PHD

Cavanaugh, R., R. Marsa, J. Robertson, and R. Hefferlin, Adjacent-DIM-isoelectronic Molecules and Chemical Similarity among Triatomics. J. Mol. Structure, 382, 137-145 (1996).

Hefferlin, R. and Matus, M.T., Molecular Similarity for Small Species: Refining the Isoelectronic Index. J. Chem. Inf. Comp. Sci. 41, 484-494 (2001).

Hefferlin, R., Systematics of Diatomic Molecular Transition Moments: Encouraging Progress -- II. J. Quant. Spectrosc. Radiat. Transfer 16, 1101-1108, 1976.

Hefferlin, R. et al, The Periodic Table of Diatomic Molecules. -- I. An Algorithm for Retrieval and Predication of Spectrophysical Properties. J. Quant. Spectrosc. Radiat. Transfer 21, 315-336, 1979.

Hefferlin, R. et al, A Periodic Table Of Free Diatomic Molecules. II. Predicted Internuclear Separations from Curve-Fitted Data. J. Quant. Spectrosc. Radiat. Transfer 21, 337-354, 1979.

Hefferlin, R. and H. Kuhlman, The Periodic System for Free Diatomic Molecules -- III. Theoretical Articulation. J. Quant. Spectrosc. Radiat. Transfer 24, 379-383, 1980.

Hefferlin, R. and M. Kutzner, Systematics of Ground-State Potential Minima between two Main-Group Atoms of Ions. J. Chem. Phys. 75, 1035-1036, 1981.

Hefferlin, R. et al, Periodic Systems of N-atom Molecules. J. Quant. Spectrosc. Radiat. Transfer 32, 257-268, 1984.

Hefferlin, R., Periodic Systems, in B.H. Banathy, Proceedings of the Society for General Systems Research, Vol. II, Applications. International Conference, May 27-31, 1985, Los Angeles, Intersystems Publications, Seaside, CA, 1985, pp. 573-582.

Hefferlin, R., Periodic Systems of Molecules and their Relation to the Systematic Analysis of Molecular Data, Edwin Mellin Press, Lewiston, New York, 1989.

Hefferlin, R., Molecule, in McGraw-Hill Yearbook of Science and Technology, McGraw-Hill, 1989, pp. 224-228 (no abstract).

Hefferlin, R., Matrix-product Periodic Systems of Molecules. J. Chem. Inf. Comp. Sci. 34, 314-317, 1994.

Hefferlin, R., Comment on "Periodicity and Peculiarity in 120 First- and Second-row Diatomic Molecules." J. Phys. Chem. 99, 5222, 1995.

Latysheva, V. A.and Hefferlin, R. Periodic Systems of Molecules as Elements of Shchukarev’s “Supermatrix”, i.e. Chemical Element Periodic System. J. Chem. Inf. Comput. Sci. 44, 1202, 2004.

Hefferlin, R. Mapping out PQ=QP Redundancies in N X N Data Arrays, Int. J. Chem. Model. 1, 415-420 (2009). [A molecular periodic system mapped onto Moebius strips.]

Hefferlin, R. Periodic Systems of Small Molecules, Wikipedia. No Abstract.

Burdick, Gary W. and Hefferlin, Ray, Data Location in a Four-Dimensional Periodic system of Diatomic Molecules, in Putz, Mihai V, Editor, Chemical Information and Computational Challenges, Nova Science Publishers, 2011.

Hefferlin, R. Barrow, J. Do Triatomic Molecules Echo Atomic Periodicity? viXra Chemistry 1403.0946 second revision June 26, 2014.

Hefferlin, R. Molecular Taxonomy: string, quark, hadron, nuclear, atomic and chemical-molecular or invariant systems.  Lambert Academic Publishers, Saarbrueken, Germany, 2011.

Ding-Yu Chung and Ray Hefferlin,The Higgs Boson in the Periodic System of Elementary Particles, J. Mod. Phys. 4, 21-26, 2013.

Hefferlin, R.  Molecular Taxonomy, in Basak, S. C., Restrepo, G., Villaveces, J. L., Editors, Advances  in Mathematical Chemistry and Applications, Vol. 1, Bentham, 2014, pp. 308-321.  In press as of July 10, 2014

Zhuvikin, G. V. and Hefferlin, R. Periodicheskaya Sistema Dvykhatomnikh Molekul: Teoretiko-Gruppovoj Podkhod, Vestnik Leningradsckovo Universiteta, No. 16, 1983, pp. 10-16.  No abstract. 

Hefferlin, R., Classification of molecules and group theory, in Y. Saint-Aubin and L. Vinet, XVIIth International Colloquium on Group Theoretical Methods in Physics, June 27 to July 2, 1988, Sainte-Adele, Canada, World Scientific, Singapore, 1989, pp. 652-655.

Zhuvikin, G.V. and R. Hefferlin, Bosonic Symmetry and Periodic Systems of Molecules, in MA del Olmo, M. Santander, and J.M. Guilarte, Group Theoretical Methods in Physics. Proceedings of the XIX International Colloquium, Salamanca, Spain, 1992, vol. II. Anales de Fisica, Monographias 1, Real Sociedad Espanola de Fisica, 1992, pp. 358-361.

Zhuvikin, G.V. and R. Hefferlin, R., Symmetry Principles for Periodic Systems of Molecules, Joint Report #1 of the Physics Departments of Southern College, Collegedale, TN, USA and St. Petersburg University, St. Petersburg, Russia, Sept. 11, 1994.

Carlson, C.M., R. Hefferlin, and G.V. Zhuvikin, Analysis of Group Theoretical Periodic Systems of Molecules using Tabulated Data, Joint Report #2 of the Physics Departments of Southern College, Collegedale, TN,USA and St. Petersburg University, St. Petersburg, Russia, March, 1995.

Carlson, C.M., R.J. Cavanaugh, R.A. Hefferlin, R.A., and G.V. Zhuvikin, Complete, Verified, Group-dynamic Molecular Periodic Systems, in A. Arima, T. Eguchi, and N. Nakanishi, Yamada conference XL: XX International colloquium on group theoretical methods in physics, World Scientific, Singapore, 1995, pp. 211-214.

Carlson, C.M., R.J. Cavanaugh, R.A. Hefferlin, and G.V. Zhuvikin Periodic Systems of Molecular States from the Boson Group Dynamics of SO(3)xSU(2)s. J. Chem. Inf. Comp. Sci. 36, 396-398 (1996).

Hefferlin, R.A., A.V. Sepman, and G.V. Zhuvikin, Molecular Multiplets of Alkaline Atoms. J. Chem. Inf. Comp. Sci. 36, 399-401 (1996).

Hefferlin, R., Field Theory for Chemical Spaces, in V. Alexiades and G. Siopsis, Trends in Mathematical Physics. AMS/IP Studies in Advanced Mathematics vol. 13, American Mathematical Society, Providence, RI, and International Press, Cambridge, MA, 1999, pp. 279-284, 1999.

Hefferlin, R., Numerical Solutions of the Laplace Equation in Chemical Space, in Hansen, P., Fowler, P. and Zheng, M.L., Discrete Mathematical Chemistry (DIMACS Workshop, Proceedings of the Workshop on Discreet Mathematical Chemistry, Dimacs Center, Piscataway, NJ, March 23-24, 1998), Series in Discrete Mathematics and Computer Science, vol. 51; the American Mathematical Society, 2000, pp. 199-201.

Hefferlin, R., The Textures of Chemical Spaces, J. Mol. Struct. THEOCHEM 506, 71-86, 2000. 

Hefferlin, R. and Knoll, J., On the Use of Laplace's Equation for Global Predictions of Internuclear Separation and Dissociation Energy, J. Math. Chem. 28, 169-192, 2000.

Luk, K., R. Hefferlin and W.B. Davis How Deep in Molecular Space can Periodicity be Found?, WSEAS 9th CSCC Multiconference, Vouliagmeni, Athens, July 11-16, 2005 [ISSN 1790-0832; CD ISBN 960-8457-29-7, paper 497-338].

Hefferlin, R. and Ken Luk, A Periodicity-Sensitive Vector Index for Small Molecules, Croat. Chem. Acta 79, 559-571, 2006.

Hefferlin, R. Hasse Diagrams and their Relation to Molecular Periodicity, in R. Brueggemann and L. Carlsen, Partial Order in Environmental Sciences and Chemistry, Springer, Berlin, 2006, pp. 27-33.

Hefferlin, R., Two Descriptors for Series of Congeneric Molecules, Croat. Chem. Acta 82, 595-598, 2009. [This article was accepted in 2007; it is a follow-on to "A Periodicity-Sensitive Vector Index for Small Molecules," Croat. Chem. Acta, 79, 559-571, 2006]

Hefferlin, R., Discrete Vector Calculus on Periodic Systems of Atoms and Molecules, J. Math. Chem., 43, 386-394, 2007.

Gary W. Burdick and Ray Hefferlin, "Data Location in a Four-Dimensional Periodic System of Diatomic Molecules," Chapter 7, Pages 197-212 of Mahai Putz, Editor, Chemical Information and Computational Challenges in the 21rst Century, a Celebration of the 2011 International Year of Chemistry, Nova Publishers, January 11, 2013 [ISBN 978-1-61209-712,1].

Ray Hefferlin, Jonathan Sackett, Jeremy Tatum, Why do Molecules echo Atomic Periodicity?Int. J. Quant. Chem., DOI 10.1002/qua.24469, 2013

Ray Hefferlin and Joshua Barrow, Do Triatomic Molecules echo Atomic Periodicity? viXra 1403.0946, 2014.

James, B., K. Caviness, J. Geach, C. Walters, and R. Hefferlin, Global Molecular Identification from Graphs. Neutral and Ionized Main-Group Diatomic Molecules, J. Chem. Inf. Comp. Sci. 42, 1-7, 2002.

Geach, J., C. Walters, B. James, K.E. Caviness, and R.A. Hefferlin, Global Molecular Identification from Graphs. Main-Group Triatomic Molecules, Croatica Chemica Acta 75, 383-400, 2002.

Walters, C.J., K.E. Caviness, and R. Hefferlin,Global Molecular Identification from Graphs. IV. Molecules with Four Closed p-Shell Atoms and beyond, Croat. Chem. Acta, 77, 65-71, 2004.

Hefferlin, R. Algebraic Characterization of Simple Transition-Metal Molecules, Croat. Chem. Acta, 80, 227-232, 2007.

Vukicevic, D., Hefferlin, R. A Graph-Theory Approach to Global Determination of Octet Molecules, MATCH Comm. Math. Comput. Chem., 57, 557-572, 2007

Hefferlin, R. and W. Innis, The Differential Coefficient (ðP/ðn)ne for Properties of Diatomic Molecules and Atoms. J. Quant. Spectrosc. Radiat. Transfer 24, 97-112, 1983.

Hefferlin, R., B. Davis, and B. Laing, The Learning and Prediction of Triatomic Molecular Data with Neural Networks, in Demidov, V., IAS'97. Murmansk. Proceedings of the Second International Arctic Seminar, Physics and Mathematics, Murmansk State Pedagogical Institute, Murmansk, Russia, 1997, pp. 31-36.

Carlson, C., J. Gilkeson, K. Linderman, S. LeBlanc, and R. Hefferlin,   Estimation of Properties of Triatomic Molecules from Tabulated Data Using Least-squares Fitting. Croatica Chemica Acta 70, 479-508 (1997).

Wohlers, J., W.B. Laing, R. Hefferlin, and W.B. Davis, Least-squares and Neural-network Forecasting from Critical Data: Diatomic Molecular re and Triatomic DeltaHa and IP, in Carbo-Dorca, R. and P. G. Mezey, Advances in Molecular Similarity, vol. 2, 1998, JAI Press, Stamford, CT, 1998, pp. 265-287.

Hefferlin, R. and Kuznetsova, L.A., Systematics of Diatomic Molecular Transition Moments, J. Quant. Spectrosc. Radiat. Transfer, 62, 765-774 (1999).

Hefferlin, R., Davis W.B., J. Ileto, An Atlas of Forecasted Molecular Data. 1. Internuclear Seperations of Main-Group and Transition-Metal Neutral Gas-Phase Diatomic Molecules in the Ground State, J. Chem. Inf. Comp. Sci. 43, 622-628, 2003.

Davis, W. B. and R. Hefferlin, Neural Networks in Mining Molecular Properties from Tabulated Data, WSEAS Transactions on Information Science and Applications, vol. 2, 951-953, 2005 [ISBN 1790-0832].

W.B. Davis and R. Hefferlin, An Atlas of Forecasted Molecular Data. 2. Vibration Frequencies of Main-Group and Transition-Metal Neutral Gas-Phase Diatomic Molecules in the Ground State, J. Chem. Inf. Comp. Sci. 46, 820-825, 2006.

R. Hefferlin, Vibration Frequencies using Least squares and Neural Networks for 50 new s and p Electron Diatomics, J. Quant. Spectr. Radiat. Transf. 111, 71-77, 2010; [Advanced Web publication, DOI information 10.1016/j.jqsrt.2009.08.004]

R. Hefferlin,Internuclear Separations using Least squares and Neural Networks for 46 new s and p Electron Diatomics, Int. J. Mol. Model, 4 (#1), 13-20, 2012.  Draft available at http://viXra.org/abs/1208.0026..

Ray Hefferlin, Jonathan Sackett and Jeremy Tatum Systematics and Prediction in Franck-Condon Factors, Chapter 8, Pages 179-191, in Kiyoshi Nishikawa, Jean Maruani, Erkki J. Braendas, Gerardo Delgado-Barrio and Piotr Piecuch, Editors, “Quantum Systems in Chemistry and Physics,” Nova Publishers, January 11, 2013 [ISBN 978-94-007-5296-2]. 

Hefferlin, R., The 50th Anniversary of work on Molecular Periodic Systems: Broad Scientific and Human-interest Aspects, Southern Scholars Journal, College Press, Collegedale, TN, 1987, pp. 38-44.

Hefferlin, R., E. V. Babaev, and G.W. Burdick, Periodic Systems of Molecules: Physical and Chemical, Report #1 of the Physics Department, Southern College, Collegedale, TN, USA, Nov. 17, 1994.

Hefferlin, R and G.U.[should be W.] Burdick, Fizicheskie i khimicheskie periodicheskie sistemy molekul, Zhurnal Obshchei Xhimii 64, 1870-1885, 1994 (English translation: Periodic Systems of Molecules: Physical and Chemical, Russ. J. Gen. Chem. 64, 1659-1674,1994)

Babaev, E. V. and R. Hefferlin, The Concepts of Periodicity and Hyper-periodicity: from Atoms to Molecules, in Rouvray, D.H. and E.C. Kirby, Concepts in Chemistry, Research Studies Press Limited, Taunton, Somerset, England, 1996.

Hefferlin, R., The Periodic Systems of Molecules: Presuppositions, Problems, and Prospects, in Baird, D., Scerri, E, and McIntyre, L., Philosophy of Chemistry, Boston Studies in the Philosophy of Science, Vol. 242, Kluwer Academic Publishers, Springer, Dodrecht, the Netherlands, pp. 221-243, 2006. Fig. 7 on pg. 237 is incorrect (though the caption is not); please contact the author for a correct figure that you can paste in.

Hefferlin, R., Kronecker-Product Periodic Systems of Small Gas-Phase Molecules and the Search for Order in Atomic Ensembles of Any Phase, Comb. Chem. High Through. Screen. 11, 690-706, 2008.

 Systematics of Diatomic Molecular Transition Moments: Encouraging Progress -- II

Literature absolute transition moments [R_e] for A-X(0-0) bands have been compiled and used to obtain "best" values of R_e for some 90 transitions. Correlations were sought between R_e and various molecular parameters such as multiplicity, average internuclear separation, etc. None was found. The A-X (and related) R_e have been plotted for isoelectronic molecules against the nulclear charge difference and fitted with simple curves which seem to approach the R_e axis with zero slope. Both of these results are in accord with theory. The concept of isovalence has been clarified to allow for the orbital structure of molecules. It appears that the heavier of two isovalent hydrides has the greater R_e. Graphs of R_e for groups of isovalent molecules can be fitted with simple curves.

Back

To Top

Related periodicities in properties of diatomic molecules are well known, and periodic tables have already been constructed for some classes of molecules. The major difficulty is that two orthogonal periodicities are superposed, at 45^° on the Z₁,Z₂ plane, on two others. Our proposed complete table is formed by cutting the plane along the Z₁ and Z₂ directions and stacking areas for similar molecules into 15 blocks. Evidence is presented that 15 is the correct number and that the stacking orders are optimal. It is shown that generalization to ionized molecules requires a fouth dimension, but that this 4-d architecture can be mapped into three dimensions. Computer curve-fitting of data has yielded initial results for r_e in three blocks. 116 data fitted to r_e = A + ln √χ (where χ is area number, and A depends on the compartment, in the block) differ from tabulated data by  <σ>=3%. 254 predicted r_e are given, some of laser interest, some of superheavy (quasi) molecules, and some for molecules with superheavy atoms.

Back

To Top

Recent theory substantiates the stacking order of the areas in the recently proposed periodic table of the free diatomic molecules. The periodic table suggests curve-fitting algorithms for r_e which, after semi-empirical perturbations, yield ground-state values for r_e that agree with literature values and with those from an alternate algorithm from 5 to 8%. 800 predicted r_e extend to superheavy qausimolecules and to molecules with superheavy atoms, and many pertain to molecules of laser interest.

Back

To Top

A periodic system for free neutral diatomic molecules, previously proposed on the basis of tabulated data, is shown related topologically to the chart of the atoms. The relation is generalized to ionized atoms or ionized molecules. Support is drawn from tabulated data.

Back

To Top

The featured figure in this paper was prepared for three purposes: (1) to demonstrate the vivid periodicities among tabulated internuclear separations r_e of diatomic molecules and quasimolecules formed from main-group atoms and ions, (2) graphically to compare the known r_e for van der Waals molecules to see what might be expected in future experiments on other van der Waals molecules, and (3) to ascertain the extent to which isoelectronic diatomic molecules along the homonuclear direction of the Z₁, Z₂ plane (e.g., LiCl^- and BeAr⁺) have constant r_e.

Back

To Top

The coefficient (ðP/ðn)_ne, where n is the stage of ionization and n_e is the total number of electrons, has been investigated for atoms and for diatomic molecules by the use of tabulated data. The P are (a) orbital configurations, (b) state symbols, (c) ionization potentials, (d) internuclear separations of ground state molecules, (e) crystal radii of ionized atoms, (f) relativistic HF radii of neutral atoms, (g) dissociation potentials of molecules, and (h) the logarithms of the transition moments of electron transitions from the first excited to the ground states of atoms and molecules. The differentials for atoms in cases (a), (c), and (h) add to form those for the diatomic molecules, agreeing with a prediction based on the matrix formulation of the periodic system of diatomic molecules.

Back

To Top

The atoms have long been classified into a periodic system, which is now based on quantum mechanics and group theory. A classification of molecules containing any number (N) of atoms is proposed. It is an extension of the periodic system of the atoms. The approach in this paper is that of group theory, although the proposed system has been subjected to exhaustive comparison with experimental and ab initio computational results for diatomic molecules, and conforms to the commonly known behaviors of molecules with larger N. Orthonormal transformations are performed so that the molecules can be arranged according to their numbers of electrons and to the differences of atomic numbers of the constituent atoms. These arrangements parallel the physical reality of atomic bonding and permit partial three-dimensional models of the systems to be constructed for molecules with as many as four atoms.

Back

To Top

We show how vast amounts of information have been, or can be, formulated into arrays with periodic and monotonic axes. For example, we propose a Periodic System of nuclei for the first time. We consider two cases where Periodic Systems in the natural world can be bootstrapped from objects of one size to objects of the next larger size. The mathematical methods in the two cases are essentially equivalent. We apply the methods to linguistics in an exploratory way. Then we construct a Periodic System of all the Periodic Systems previously discussed.

Back

To Top

As the result of rapidly accelerating work begun 50 years ago, there now exists an understanding of the Periodic Law for Molecules, and there now exist numerous forms of Periodic Systems of Molecules. This work is descried in broad strokes, with attention to its human interest aspects. (We have since found that Grimm in 1935 was not the first to construct some sort of molecular classifications scheme)

Back

To Top

 [a list of errata is available from the author]

Preface

Captions for Photographs

Information on Microfiche Inserts

Part I: Systematics, Periodicity, and Special Groups of Molecules

Chapter 1: Fundamentals
Chapter 2: The System of Elements from the Group-Theoretic Viewpoint, by A. I. Fet
Chapter 3: The Periodic Law for Molecules
Chapter 4: Isoelectronic Molecules

Part II. Graphical Methods

Chapter 5: Graphs on portions of the Z₁, Z₂, … space
Chapter 6: Graphs and Statistical Analyses for Isovalent Molecules; a Generalization of the Periodic Law of Molecules

Part III: Methods for Self-Consistent Smoothing and Predicting of Molecular Data

Chapter 7: Least-Squares Fitting and Prediction for Neutral Molecules
Chapter 8: Ionized Molecules
Chapter 9: Quarks in MoleculesChapter
Chapter 10: The General Periodic System of Molecules
Chapter 11: An Alternative Periodic Table for Triatomic Molecules, by F.-A. Kong
Chapter 12: A History of Periodic Systems

Part IV: Multiplicity and Additivity

Chapter 13: Can Atomic Data be Combined in Mathematically Simple Ways to Predict Molecular Data?

Part VI: Applications to Other Areas

Chapter 14: Taxonomy and Evolutionary Study of Technology, by A. I. Steinhaus
Chapter 15: Periodicity as a Paradign

Notes Added in Proof

List of Scientists Known to be Involved in Related Work

Index

Back 

To Top

We review the group-theoretical investigation into the periodicity of the elements and then the researches into the major and the minor periodicities of molecules. The results agree with the periodic law and allow building of periodic systems of molecules.

Back 

To Top

The phenomenological group-theoretical interpretation of the periodic system of chemical elements, given by Rumer and Fet (1971) and by Barut (1971), prompted us to work on development of the periodic systems of molecules based on the same interpretation [Zhuvikin, Hefferlin (1982), Hefferlin et al (1984), Hefferlin (1989)]. Our new concept, presented in this paper, is to join to this group-theoretical approach for molecular systems the formalism suitable for bosons. Thus, the periodic chart of the elements, instead of being multiplied by itself into higher-dimensional spaces, serves as a kind of template for multiplets of molecular state vectors created by Lie algebras.

Back

To Top

Direct products of the periodic chart of the elements, considered as a matrix, with itself produce 2N-dimensional periodic systems of N-atomic molecules. It is shown that these periodic systems subsume a large number of published molecular periodic systems.

Back

To Top

Extensive research, most of it inspired by fundamental-particle physics, has been done on the group-dynamic foundations of the classification of atoms known as the periodic system. Also, a very great deal of work has been carried out for a century on varieties of ways to classify molecules and to predict the properties of new ones in a systematic way. Much of this work has been motivated by the need for small-molecule astrophysical data and for pharmaceutical data. It is the purpose of this paper to employ the methods of group dynamics, as they have been applied to atoms, for the molecular classification problem. We begin with known bosonic creation and annihilation operators, since any number of identical atoms may be present in a molecule. Then we define state vectors in the space H(N) of N-atomic molecules. We require that this space be decomposed into irreducible representations (IR's) of a group with demonstrated applicability to the classification of atoms; the basis vectors of these IR's form multiplets of molecules which bear a great deal of similarity to multiplets of atoms. Finally, we show how to formulate observable operators of various orders and in several approximations; these operators provide expectation values expressed as functions of state quantum numbers. (In a later paper we show how tabulated experimental data for various properties, substituted into these expectation values and plotted one multiplet at a time, produce surfaces which are periodic, are often well-behaved, and allow estimation of molecular data.) In the paper we list the symmetry groups to be treated and describe the role which each of them plays in classifications of atoms. We take up compact multiplets in conformal symmetry: SO(3), SO(3)xSU(2)s , SO(3)xSU(2)d , SO(3)xSU(2)s xSU(2)d , SO(4)xSU(2)s , and SU(2)j ; for each are given the group chains, their quantum numbers, the operators and their relationships in Lie algebras, and examples of molecular multiplets. Then we consider noncompact multiplets in the conformal symmetry SO(2,1), and multiplets in the unitary symmetries SU(n) and SU(2), following a similar outline. The molecular state vectors are often composed of linear superpositions of molecular symbols; this situation is analogous to the linear superpositions of quark combinations which make up hadron states in the standard model.

Back

To Top

This paper is a review of research on molecular periodic systems, a developed field of research. The distinguishing characteristics of Physical Periodic Systems are that they include only molecules with a given number of atoms, e.g., diatomic molecules (N=2), and that all relevant molecules with that number of atoms are or can be included. Physical Periodic systems are defined as follows: a Physical Periodic System for N-atom molecules is the outer N-product of an originating atomic periodic table, or can be generated from such a product system by use of slicing or projection techniques to include only molecules with a given number of atoms, e.g., diatomic molecules (N=2); they include recent classifications based on group dynamics. The distinguishing characteristic of Chemical Periodic Systems is that they contain molecules with differing numbers of atoms. It is also a characteristic that they include only molecules stable under a well-known set of conditions (such as atmospheric-pressure wet chemistry). Work related to each kind of system is described chronologically, whether or not the work was identified as a periodic system by its author. All publications claiming to present molecular periodic systems (and of which the authors are aware) are listed, in an effort to achieve a complete review of the rapidly growing field. Finally, a recommendation is made that future classification systems be identified as Physical or Chemical periodic systems, in accordance with the distinguishing characteristics. Care should be taken to ensure that a system truly displays periodicity, and is not just an interesting lattice.

Back

To Top

This paper is a review of research on molecular periodic systems, a developed field of research. The distinguishing characteristics of physical periodic systems are that they include only molecules with a given number of atoms, e.g., diatomic molecules (N=2), and that all relevant molecules with that number of atoms are or can be included. Physical periodic systems are defined as follows: a phyical periodic system for N-atom molecules is the outer N-product of an originating atomic periodic table, or it can be generated from such a product system by the use of slicing or projection techniques to include only molecules with a given number of atoms, e.g., diatomic molecules (N=2); they include recent classifications based on group dynamics. The distinguishing characteristic of chemical periodic systems is that they contain molecules with differing numbers of atoms. It is also a characteristic that they include only molecules stable under a well-known set of conditions (such as atmospheric-pressure wet chemistry). Work related to each kind of system is described chronologically, whether or not the work was identified as a periodic system by its author. All publications claiming to present molecular periodic systems (and of which the authors are aware) are listed, in an effort to achieve a complete review of the raplidly growing field. Finally, a recommendation is made that future classification systems be identified as physical or chemical periodic systems in accordance with the distinguishing characteristics. Care should be taken to ensure that a system truly displays periodicity and is not just an interesting lattice.

Back

To Top

The second of a series, this paper substantiates the theory given previously by substituting tabulated data for observable operator expectation values of molecular-state-vector multiplets corresponding to the groups SO(3)xSU(2)s , SO(4)xSU(2)s , and SO(2,1). We plot the data on the principle axes of the multiplets using various orders of observable operators and expectation value identities whose particular merits are discussed. The generally well-behaved plots for the first and third symmetries demonstrate a similarity between various N-atomic species, one example of invariance for different properties, periodicity with respect to changes in the row numbers of the constituent atoms, and inclusive similarity in many cases where the independent variable ranges of one multiplet lie within those of another multiplet. We examine advantages and disadvantages of several expectation value identities and find that the group SO(4)xSU(2)s produces poorly behaved plots.

Back

To Top

(Part of first paragraph; no abstract)
The purpose of this comment is to put the article in context so that there may be more awareness of the large and growing, world-wide field of periodicity and periodic systems.

Back

To Top

We present periodic systems of molecules which (1) agree with the periodicities observed in molecular data, (2) classify molecules so those with similar data are located "closer" together than those with dissimilar data, which (3) are analogous to the periodic chart of the elements, (4) are very much similar to it in form, (5) allow the estimation of data for some neutral, ground-state, gas-phase molecules, and (6) are rigorously based on group dynamics with boson operators. A complete wall-chart periodic system for all diatomic molecules in symmetry the SO(3)xSU(2)s is available on request from the third or fourth authors. The wall chart shows a schematic of the periodic system for triatomic molecules (also under the same symmetry). This presentation explains why the periodic systems are needed, refers to the earlier use of group dynamics as an alternate theoretical basis of the periodic chart of the elements using the symmetry SO(4,2)xSU(2)s, outlines the general approach used in the construction of the periodic systems of molecules, refers to instructions on reading the wall chart, presents some of the graphical evidence that the systems agree with tabulated molecular data, and alludes to the strategy we are employing to approach the construction of molecular periodic systems with the general symmetry, SO(4,2)xSU(2)s, of the chart of the elements.

Back

To Top

An overview of the principles of group-dynamic periodic systems is given. The process by which molecular multiplets are obtained is then described. Plots of observable operator expectation values for SO(3)xSU(2)s and SO(2,1) multiplets clearly demonstrate periodicity. To construct molecular periodic systems, these multiplets are situated in coordinates which are their chemical quantum numbers along with the principal quantum number coming from the SO(4,2)xSU(2)s group chain. Equivalence classes of substitutable (isomorphic) molecular multiplets are then defined for the two symmetry groups. The structures of the resulting molecular periodic systems are shown using plots of the number of multiplets located at given chemical-quantum-number coordinates.

Back

To Top

Molecules formed from alkali-metal atoms are classified with the apparatus of group dynamics. The groups SO(1,2), SO(3), and SU(?) are used in turn. Ionization potentials (IP) of diatomic combinations of Li-Cs display symmetry-breaking consistent with the group chain SU(?)?SU(1). Known IP of triatomic combinations of Li-K are used to predict values of three unknown species: LiLiK, LiNaK, and LiKK have IP of 3.9, 3.8, and 3.6 eV, respectively.

Back

To Top

(Table of contents) 

I. ORIGIN OF PERIODICITY

II. THE PERIODIC TABLE

III. PERIODIC SYSTEMS IN OTHER SCIENCES

IV. MOLECULAR PERIODICITY

V. THE ART AND THE LOGIC OF EQUALIZATION: Classification of isosteric ensembles

VI. THE HYPER-PERIODICITY PATTERN: Classification of isovalent ensembles

VII. SPECIAL TYPES OF CHARTS: Diatomic molecules

VIII. CONCLUSION

Acknowledgments

Back

To Top

Generations of Soviet scientists contributed invaluable insights into molecular classification. Unfortunately, this research is little appreciated in much of the world. Among these workers, S. A. Shchukarev was of great importance. His and his followers’ legacy includes a host of graphical displays showing enthalpies of formation of gaseous molecules from free atoms ΔH_a, and standard enthalpies of formation of substances ΔH^of plotted on the atomic number of the central elements, on their oxidation states, their internuclear separations, and other variables for a wide range of molecules. These graphs serve as databases, from which data can be extracted, to moderate precision, visually. We discuss graphs for one very limited set, or “pleiade” (gas-phase oxides of nitrogen), and for three much broader sets, or subsystems, (gas-phase fluorides of all main subgroup atoms, and oxides of transition-metal atoms in gas-phase and in STP conditions). When dissolved in water, molecules lose their identities but periodicity is echoed in the acids and aquocations that are formed. We show, as an example in tabular form, that redox potentials of high-oxygen acids containing S, S_e, and T_e change concomitantly with ΔH_a and ΔH_f of their hexafluorides. We present graphical evidence that three properties for cations of groups 1 to 3 (in the short version of the periodic chart) behave similarly and share the periodicity of the elements. One of the properties is related to the ionization potential, which is shown in a tabular example to vary concomitantly with energy of hydration. It was the ultimate goal of S. A. Shchukarev that the transformation of any one graphical database into any other, having different molecules under different conditions, would be made mathematically.

Back

To Top

Isoelectronic molecules and isovalent molecules commonly have chemical similarity. This paper adds a third distribution of molecules and demonstrates that it exhibits chemical similarity in the case of linear/bent triatomics: the similarities of same-period triatomics which have equal C₁ + 2C₂ + C₃ ≡ g(C) where C_i is the number of valence electrons of atom i and where i = 2 is the middle atom. It is shown, by graphical and statistical analyses of critically analyzed data for several observables of main-group molecules, that molecules with constant g(C) are at least as similar as isoelectronic molecules and are vastly more similar than are molecules chosen at random. It is suggested that since g(C) can be written additively as (C₁ + C₂) + (C₂ + C₃) a theoretical basis for the new tool might be constructed using a "diatomics-in-molecules" (DIM or similar theory in which the 1-3 bond is ignored. The generalization to molecules with more than three atoms is presented.

Back

To Top

(First section; no abstract)
Systematizations of tabulated data for small molecules hold out hope for a better understanding molecular periodicity and molecular periodic system, and hence for more enhanced comprehension of chemistry by students and more efficient preparation of computer data-bases, and also for relating that understanding to the observed periodicities (and corresponding periodic tables) of nuclei and nucleons and of hadrons and quarks. Importantly, they make it possible to forecast approximate data for large numbers of molecules. Neural networks (NN) can "learn" data and predict many new data without a smoothing equation. This report describes their use on gas-phase, acyclic, neutral, ground-state triatomic molecular data, and comparison of the results with least-squares (LS) predictions based on largely similar selection of tabulated data.

Back

To Top

This paper shows that it is feasible to make rapid forecasts of data for large numbers of molecules by using least-squares smoothing of tabulated data, though the forecasts are not asa precise as those from quantum-chemical computation packages which deal with one molecule at a time. The molecules' properties were chosen to be of value in plasma and astronomical physics. The work begins with the graphical analysis of critically-analyzed data for ground states of neutral, acyclic, main-group, row 2 to row 6, triatomic molecules to infer a least-squares smoothing equation. The equation is quadratic in a function (R₁R₂ + R₂R₃) of the atomic period numbers, quadratic in the group number of the central atom, and cubic in the total number of valence electrons. The coefficients of the equation (some of them zero for some properties) were obtained from high-quality tabulated data for heat of atomization, ionization potential, log of the partition function at 1000K, and log of the partial-pressure equilibrium constant for the constituent atoms over the diatomic molecules at 1000K. The equation with its coefficients were tested by comparison with data, from the same tabulations, for a few molecules not in the original set. Finally, values were forecasted for 164, 145, 107, and 164 additional molecules, for four the properties listed above and in the same order.

Back

To Top

Multiple regression was used to predict 299 diatomic internuclear separations using atomic period and group numbers as a basis. Van der Waals molecules were excluded. The standard deviation of the differences of predictions from 150 tabulated data is 4.128%. Neural networks, one with van der Waals molecules in the learning set and one without, each predicted the property for 2,145 real and non-redundant molecules; of the differences of the predictions from 342 and 316 tabulated data are 25.00% and 8.63%. For comparable cases, the least-squares technique was more accurate. Multiple regression has been used to predict 205 triatomic ionization potentials using a 3-d basis consisting of combinations of atomic period and group numbers. The of differences from 80 tabulated data is 14.65%. Neural networks using the same 3-d basis and using a 6-d period and group number basis have predicted the IP for 2,596 and for 5,148 molecules; the of the differences from 69 and 92 tabulated data are 12.35% and 10.97%. The neural network method was more accurate than the least-squares technique. Neural networks using the same two bases have predicted the bonding energies for 16,324 and for 5,418 molecules; the of the differences from 79 and 117 tabulated data are 22.67% and 15.13%.

Back

To Top

The periodic chart of the elements, and periodic systems of molecule, are constructed in what are known as chemical spaces. Properties of target atoms or molecules can be interpolated one by one (the triad principle) or they can be iteratively interpolated all at once for points inside well-selected boundaries (the numeric solution to Laplace's equation). Examples will be shown for both atomic and molecular data. The second derivatives in the Laplace equation describe the curvatures of the data along the appropriate axes, and so if the curvature is known in along one or more axes, then the curvature along the remaining axis may be predicted. Examples of this phenomenon will be presented. This connection of the triad principle with Laplacian field theory is not only interesting in that it connects a simple concept in two disparate disciplines, but also it is useful in the important tasks of molecular data analysis and molecular design.

Back

To Top

Back

To Top

The triad principle states that numerical values of properties of atoms and molecules can be obtained by interpolation in their chemical spaces. This principle is associated with the numerical solution to the Laplace equation.

Back

To Top

The classification of small molecules and the forecasting of small-molecular data has advanced far in recent years, stimulated by needs in astrophysical and laboratory plasmas. The systematization of larger species has also progressed, prompted by the demands of environmental, toxicological, and pharmaceutical chemistry. Here is presented a first known thorough analysis of the chemical spaces in which molecules are placed in the course of these systematizations. The approach is to ask "What can we learn about the chemical spaces on the basis of the gas-phase spectroscopic and thermodynamic data displayed by the molecules placed in them?" In other words, this paper is a first attempt to look heuristically at data as if derived from the properties of chemical spaces rather than as descended from quantum mechanics or group dynamics. It is found that there are in some spaces salient lines, planes, and hyperplanes on which molecules become homonuclear and homovalent; on which species become isoelectronic or at least have the same numbers of valence electrons; and on which molecules take on extreme and least-descent intermediate data values. The spaces have certain symmetries or broken symmetries with respect to the first of these loci. An algebraic formulation of Lewis-bond structures leads to the discovery of salient lines, planes, and hyperplanes on which molecules assume closed-shell covalent or ionic structures. A field theory of chemical spaces is reviewed, it being pointed out the extent to which the discretized Laplace equation applies. New data have been forecasted by this equation. Molecular ions, and alloys and Bertheloids, lie at non-integer positions in chemical spaces.

Back

To Top

This paper tests a chemical-similarity model against properties of gas-phase neutral triatomic and four-atom molecules. The model is a variant of the Diatomics-in-Molecules (DIM) picture, which considers a molecule to be the superposition of all diatomic molecules that could be formed from adjacent atoms in the molecule. The variant is that adjacent atoms are counted as a diatomic molecule only if they are bonded. The tests consist of investigating whether molecules with the same number of electrons, computed by the adjacent-DIM model, have data more similar than do molecules selected at random. The tests vindicate the model for the heat of atomization and for the equilibrium constant for formation, they agree with the model with lesser confidence for the entropy and the partition function, and they show that the model fails for the ionization potential. The model applies with most confidence to molecules with the more electronegative atoms from rows 2 or 3 (those of greatest interest in organic chemistry), and with lesser confidence otherwise. For these properties and these molecules, the model passes graphical and statistical tests at least as well as does the traditional isoelectronic model. Thus, this work refines what may be called the isoelectronic index.

Back

To Top

In this book, the term “molecule” is generalized: it refers to normal molecules, to nuclei, and to hadrons and leptons, all of which we take to be composed of smaller objects. The topics begin in this descending order: stand-alone molecular and atomic classifications; the distribution of atomic nuclei on the Segré chart; the beautiful symmetries of baryon and mesons; the elegant quark diagrams; and manifestations of strings. Then in ascending order: string representations emerging as gauge bosons, quarks, leptons and hadrons; quark diagrams recreating the meson and baryon charts; the Segré chart being sculpted into a periodic system of nuclei; the periodic chart of the elements blossoming to create solidly supported periodic systems of molecules. Similarities in the algebraic and graphical constructions of the periodic systems are highlighted.

Back

To Top

Properties of target species can be estimated by various means including interpolations in periodic charts. Interpolation is equivalent to numerical solution of the Laplace equation. A test of this equivalence, within some confidence level, for any N-atomic molecule surrounded by 4N nearest neighbors: the sum of the second differences of the data in all directions must be zero. Since very few molecules have 4N neighbors with known data, the test becomes: the sum of the averages of the second differences must be zero. The validity of these tests is explored. For radii of main-group atoms, and for internuclear separations of their diatomic combinations, the averages are different from zero and the sums of the averages are zero to within one σ if second-nearest neighbors are used. Dissociation potentials pass the tests but with larger scatter. Predictions for dissociation potentials, using iterative interpolation within boundaries on which there are known data, are reviewed.

Back

To Top

A taste of the philosophical and historical aspects of the scientific enterprise for professional chemists, and a glimpse of scientists’ ontology and epistemology for philosophers and historians. The narrative is based on a limited endeavor with most of whose living participants the author has a personal acquaintance. The endeavor consists of creating, testing, and utilizing organizations of molecular symbols in architectures of two or more dimensions. These periodic systems play the same role for molecules as the chart of the elements does for individual atoms. The presuppositions, on several levels, that contributed to the creation of molecular periodic systems are explored. Problems faced by some systems are exposed, including the difficulty of visualizing them in multiple dimensions and the challenge of finding not molecular symbols, but linear combinations of symbols, in the compartments. Prospects for further maturation and usefulness of the periodic systems are presented.

Back

To Top

Diophantine equations and inequalities are presented for main-group closed-shell diatomic molecules. Specifying various bond types (covalent, dative, ionic, van der Waals) and multiplicities, it becomes possible to identify all possible molecules. While many of the identified species are probably unstable under normal conditions, they are interesting and present a challenge for computational or experimental analysis. Ionized molecules with net charges of -1, 1, and 2 are also identified. The analysis applies to molecules with atoms from periods 2 and 3 but can generalized by substituting isovalent atoms. When closed-shell neutral diatomics are positioned in the chemical space (with axes enumerating the numbers of valence electrons of the free atoms), it is seen that they lie on a few parallel isoelectronic series.

Back

To Top

It is required that molecules with a given graph and with covalent, coordinate-covalent, and ionic bonding contain closed-shell atoms. This requirement results in an equation for each atom, which states that the number of valence electrons pertaining to it before bonding, plus those made available to it in the bonding processes, close its valence shell. Solving the equations results in identifying the atoms and the bond orders of the Lewis diagrams. The algebraic procedure can identify new species. Some of them may be considered impossible (for instance, with high steric strain), or may be transitory, or may be found only under the most unusual conditions. Lists of triatomic molecules, clusters, and resonances found by solving the equations is presented. The code for the computer program that identifies the species is listed. Closed-shell molecules lie on parallel planes in their chemical spaces, namely those on which isoelectronic molecules are located.

Back

To Top

The identification of main-group molecules having atoms with closed valence p shells, i.e., having atoms with eight valence electrons, is continued into the realm of four-atom molecules. All possible covalently-bonded species, obtained from two independent computer programs, are shown. The method for generalizing to molecules in which some or all atoms have closed valence s shells, i.e., having atoms with two valence electrons, is recalled. A list of all prototype linear/bent four-atom molecules, with dative bonds in addition to covalent and/or van der Waals bonds, is presented (permutations having been culled out). A program code, lists of molecules based on other graphs, and lists of five- and covalently-bonded six- atom species, are available on the Web. For molecules derived from the two other four-vertex graphs, their vast numbers require extensive indexing schemes for the results to be useful. The paper concludes with some preliminary observations concerning the stabilities of four-atom molecules having atoms with closed shells.

Back

To Top

Needed spectroscopic data on diatomic molecules can often be found in the superb critical tables of Huber and Herzberg¹ or in the literature published since 1979. Unfortunately, these sources apply to only a fraction of the diatomic species that can exist and so investigators have had to rely on interpolation, additivity, or ad hoc rules to estimate needed values, all of which require other information that is often lacking. This Atlas presents 1001 additional internuclear separations for use until critical tables are available to fill the needs more precisely. The Atlas was produced by mining the data from Huber and Herzberg for trends with least-squares analysis and with neural network software. There are 162 molecules about whose data Huber and Herzberg had no qualifications and whose data were employed for this work; 248 copies of data with low and high magnitudes were added so that the neural network software might train well throughout the range of magnitudes. Internuclear separations for 1001 species not found in Huber & Herzberg are presented, and least-squares predictions supplement some of them. The results, i.e., the Atlas, are presented as an appendix in Table A. The average error, based on the average of the absolute differences between the predicted values and tabulated values for the molecules having Huber and Herzberg data, is 0.074 Å; if each error is expressed as a percent of the forecast to which it pertains, the average of these errors is 2.94%. There are 21 "questionable" data from the Huber and Herzberg, not used in the preparation of the Atlas, for which predictions are included in the Atlas. Of these, 14 agree with the predicted internuclear separations to within the stated errors and five more agree to within twice the stated errors. Additional atlases for other properties of diatomic molecules are in preparation.

Back

To Top

This atlas of diatomic-molecular vibration frequencies parallels the previously offered Atlas of Internuclear Separations. The Atlas was produced by mining the data from Huber and Herzberg and training neural network software to forecast new data. New protocols were employed with the powerful software, which was originally designed for forecasting the financial markets. The Atlas presents 1920 additional vibration frequencies for use until critical tables are available to fill the needs more precisely. The precision of the predictions is characterized by the average fractional 1% confidence limit, that is, 10.66%. The accuracies of the predictions are determined in two ways. First, 221 of the 224 Huber and Herzberg data values used for training and validation fall within the prediction confidence limits or fall outside by less than 10% of the Huber and Herzberg values, and 181 values agree (within the limits). Second, 87 of 101 comparison data values, consisting of literature data and some additional Huber and Herzberg values, fall within the prediction confidence limits or fall outside by less than half the prediction values, and 44 of the 101 values agree (within the limits).

Back

To Top

The use of neural networks for the study of molecular data is not new, however the protocols are still imperfect. We discuss frequency compensation, choice of error measure, and use of a genetic algorithm for variable selection. We show how the determination of an optimal number of models was made both by culling from a greater set those with large errors and by starting from models with low error and few nodes and adding more models until the average, over all molecules, of the error standard deviation reached a minimum.

Back

To Top

We find occasional echoes of periodicity, i.e. the trends found in the chart of the elements, in several-atom (up to 32) molecules and use it to make forecasts for molecular data, some of which have been confirmed.

Back

To Top

In the vastness of molecular space there are many series X, XY, …, XY_n…XY_N, where N is lies between 3 and less than say 50, whose data for a given property and phase are approximately linear with respect to n. We develop a vectorial representation of the tabulated data in a series, and a vector index to describe the series. We start with X as a metallic atom and with the property as heat of atomization, and show that the vector index manifests periodicity. We then move to cases where X is itself a molecule and where the properties are enthalpies of formation, entropy, retention index, hydrophobility, and boiling point. The vector index is a two-dimensional vector whose upper element describes the value of the property for the atom or molecule X and whose lower element describes the abscissa difference of any two members of the series after the data have been fitted, in least-squares fashion, to a standard, linear with n, series A, AL, … AL_n, … AL_N. Matrices can transform the data vectors of any series of species whatsoever to any other series of the same imensionality. Matrices can also transform the vector index for property data of any approximately linear series, in any phase, to the vector index for any other approximately linear series.

Back

To Top

Hasse diagrams are applied to molecules and radiation phenomena. Then the relation of these diagrams to periodicity in atoms is noted. The possibility is raised that Hasse diagrams can also be related to the growing body of evidence that periodicity exists in molecules with two, three, and four atoms; in binary inorganic molecules; and in some organic molecules.

Back

To Top

Data in a square nxn (n even) array, symmetric about the major diagonal, may be mapped into a new array of dimensionality (n+1)x(n+1) which has minimal redundant data and is uniformly populated. The few remaining redundant data vanish when the array is rolled into a Möbius strip. If the data are periodic modulo n, then several Möbius strips can form a periodic system of diatomic molecules. A heuristic mapping procedure is described in detail.

Back

To Top

This work extends previous studies of molecules with main-group atoms to those also containing transition-metal atoms. The maximum oxidation states of group 3 to 7 atoms are succinctly described by the same algebraic equation that applies to ionically-bonded main-group atoms in molecules. Covalently bonded transition-metal molecules are characterized by the same equations that apply to main-group species, except that the octet rule is supplemented by the rule of 18, and a proposed rule of 12. Examples are given of organo-metallics with simple ligands and of transition-metal atoms in p-ring sandwiches. The solutions for a special case of the same equations apply to the gas-phase diatomic species which are of such importance in stellar and planetary atmospheres and in combustion devices.

Back

To Top

Set theory is used to forecast the existence of all possible octet-rule cyclic and acyclic molecules formed from main-group atoms and having ionic and/or covalent bonding with orders up to three

Back

To Top

Two quantitative characterizations are compared for series, each of which consists of a central atom or molecule and successively attached functional groups. The vector index component V₂ brings out the periodicity related to the central entity and to the ligands (if they are atoms) or brings out the periodicity related to the substituents (if they are atoms) put into an originating molecule. The Zenkevich parameters also show the periodicity seen in V₂ but are valuable for treating data farther from linearity with respect to the number of attached groups provided that there is not an inflection point. An algebraic comparison shows that it is not possible to derive the vector index components from the Zenkevich parameters or visa versa except when the data are exactly linear.

Back

To Top

In this paper we combine atomic and molecular data, which are displayed in their periodic systems, in such a way as to take the discrete gradient. Then we act on the resulting vector field with the discrete vector divergence. The curl is obviously zero; the scalar field is conservative. We act on the original data with the discrete Laplacian operator (the iterated average utilizing only data on a border which contains the extreme values). The properties considered are atomic electronegativity, ionization potential and radius; and diatomic-molecular dissociation potential and internuclear separation. The
calculus should work well to highlight the local energy minima of the nuclear valley of stability.

Back

To Top

The periodic law, manifested in the chart of the elements, is so fundamental in chemistry and related areas of physics that the question arises “Might periodicity among molecules also be embodied in a periodic system?” This review paper details how a particular periodic system of gas-phase diatomic molecules, allowing for the forecasting of thousands of new data, was developed. It can include ionized and even quarked-nuclei molecules and it coincides with locality (averaging) and the additivity found in some data; it has interesting vector properties, and it may be related in challenging ways to partial order. The review then explains how periodic systems for triatomic and four-atomic species are evolving along a similar path. The systems rest largely upon exhaustive comparisons of tabulated data, relate to some extent to the octet rule, and include reducible representations of the dynamic group SO(4) in higher spaces. Finally, the paper shows how periodicity can be quantified in data for larger molecules. Data for properties of homologous or substituted molecules, in any phase, are quantified with a vector index, and the index for one set can be transformed into that for another set.

Back

To Top

Least-squares forecasts for vibration frequencies of diatomic molecules, most with 10–12 valence electrons, are combined with those obtained from neural networks, both trained on critical data. It is required that the standard deviation bounds of the one prediction lie within the bounds of the other; this requirement results in 69 molecules,  50 of which may not have been studied before. The composite standard deviations of the composite predictions average 5.9%, so there is a 68% chance that each of these 50 predictions will prove to be within 5.9% of its ultimately correct value. As a test, 28 literature values, for 12 of the molecules, were found; of these 28 values, 78.6% fall between the lower and upper composite standard deviation limits.

Back

To Top

Combined least-squares and neural-network forecasts for internuclear separations of diatomic molecules, most with from 9 to 12 atomic valence electrons, are presented. We start with 108 forecasts from least-squares based on Handbook of Physics and Chemistry 2009 data, and 1001 forecasts from neural networks based on Huber and Herzberg 1979 data, and then require that the standard-deviation bounds of the forecasts overlap each other; this requirement leaves 67 molecules, of which 46 seem not to have been studied previously. The composite errors average 0.1036Å % on either side of the composite predictions. There is agreement with 33 of 41 independent test data (80.5%), and those not in agreement fall outside the composite error limits by an average of 1.83%.

Back

To Top

This work was not done to rival the highest precisions that can be reached by computations or experiment today, but rather to attain the best precision attainable by methods that employ independent variables based on molecular periodicicty. The independent variables used for both methods are the period numbers and group numbers of the two atoms; the training data were taken from Huber and Herzberg [1] and from the Handbook of Physics and Chemistry [2]. The neural networks were built using PREDICT; the least squares forecasts in limited regions of the transition- metal/main-group space were made with a quadratic polynomial of the independent variables. Forecasts for 65 molecules were made using both methods. Thecomposite predictions and error measures were found; the average of the error measures is 37.7+/-39.3 cm^-1 (11.0+/-8.5% of the prediction [magnitudes--added]). The accuracies of these predictions were validated by data in the two sources used for training: 8 out of 13 (61.5%) agreed within the error bars and the remainder were outside by an average of 2.5 cm^-1. An extensive serach for independent literature values is in prgress. There are 46 molecules for which the error measures of the two methods are nested or at least overlap. Those molecules whose data are not in either training set are listed here; the asterisks indicate that independent literature data have been found. Two of three fall within the error bars of the predictions.

Back

To Top

It is the hypothesis of this chapter that diatomic molecular Franck-Condon factor tables echo the periodicities of constituent atoms. This means that in isoelectronic series, entire Deslandres tables for molecules that are one proton-shift away from rare-gas molecules have distinctive behavior relative to other Deslandres tables in the series. An example is in the 21-electron sequence where BeCl, whose chlorine atom is next to the closed-shell magic-number atom argon. The periodicity is established quantitatively and indeed allows for prediction of the vibration frequency for a hypothetical 2Π upper state for Cl.

Back

To Top

The chart of the elements provides a striking representation of the periodic law. The construction of an analogous periodic system of diatomic molecules has tantalized several investigators over the span of a century. There have been different approaches to the construction. In this chapter, while not denigrating other approaches, we consider the one which is solidly based on the chart of the elements. In its most general form it is a direct product of the element chart (considered as a matrix) with itself; it can be represented therefore as a four-dimensional architecture. Finding data imprinted in the element chart is trivial; finding digital data in a complex four-dimensional geometrical structure is not difficult with indexing, but in this chapter we present a semi-visual method to supplement the indexing process. We also submit a semi-visual representation of a search for new data based on surface fitting of fixed-group diatomics. The presentation of the semi-visual method begins with a very simple case in which the element chart is imagined to have just four elements and then extends to its development in the hyperstructure with main-group and transition-metal molecules. The addition of lanthanoid and actinoid molecules would present no serious difficulty.

Back

To Top

It is proposed that the observed Higgs Boson at the LHC is the Standard Model Higgs boson that adopts the existence of the hidden lepton condensate. The hidden lepton is in the forbidden lepton family, outside of the three lepton families of the Standard Model. Being forbidden, a single hidden lepton cannot exist alone; so it must exist in the lepton condensate as a composite of μ' and μ'^± hidden leptons and their corresponding antileptons. The calculated average mass of the hidden lepton condensate is 128.8 GeV in good agreements with the observed 125 or 126 GeV. The masses of the hidden lepton condensate and all elementary particles including leptons, quarks, and gauge bosons are derived from the periodic system of elementary particles. The calculated constituent masses are in good agreement with the observed values by using only four known constants: the number of the extra spatial dimensions in the eleven-dimensional membrane, the mass of electron, the mass of Z boson, and the fine structure constant.

Back

To Top

Franck-Condon factors are investigated for sequences of free main-group diatomic molecules.  Theory-based Condon loci (parabolas) and Morse-potential loci are plotted on Deslandres tables to verify if they indeed follow the largest Franck-Condon factors.  Then the inclination angles of the Condon loci are determined. Thus, entire band systems are quantified by one variable, the angle.  For all available isoelectronic sequences this angle increases from a central minimum toward magic-number molecular boundaries.  The theory for the Condon locus gives the angle in terms of the ratio of the upper-state to the lower-state force constants.  It is concluded that the periodicity is caused due to the fact that this ratio becomes larger as rare-gas molecules are approached, a trend that probably points to the extreme cases of the rare-gas molecules themselves. Thus, molecular periodicity echoes atomic periodicity in that data plots have extrema at molecules with magic-number atoms, yet it does not echo the details of atomic periodicity in series between those molecules.

Back

To Top

The objective of this work is to test whether vibration frequencies ν₁ of free, ground-state, main-group triatomic molecules manifest a periodicity similar to those of atomic spectroscopic constants. This test and an earlier test on energies of atomization underscore the role of the periodic law as a foundation of chemistry.  Using data from four data bases and from computation, we have collected and have mapped ν₁ data in mathematical spaces of fixed-period molecules.  These spaces are 8×8×8 atom cubes with rare-gas molecules on each face.  The ν₁ collected from various sources might be of use in optical or near-optical searches for cold triatomics in interstellar space.

Back

To Top

This chapter is for those in the field of mathematical chemistry who would like to practice their skills in other than normal molecules; for colleagues in the physics community with a curiosity about periodicities of particles from molecules to strings; and for specialists in informatics. Similarities are shown to exist in the constructions of periodic systems for five orders of particles.

Back

To Top