Guanshen Fang*1 Massoud Amini*2 Hao Chen*3 福田信幸*4 細矢治夫*5 河合雅弘*6 Joseph E. LeBlanc*7 Paul G. Mezey*8 成木勇夫*9 岡田 正*10 Eric Rambo*11 Mark Spivakovsky*12 竹内 茂*13 Keith F. Taylor*14 Hongyi Wong*15
山中 聡*16 横谷正明*17 Peter Zizler *18 有本 茂*19
Mathematics and chemistry
interdisciplinary joint research and the Fukui Project XXV
Guanshen FANG, Massoud AMINI, Hao CHEN, Nobuyuki FUKUDA, Haruo HOSOYA Masahiro KAWAI, Joseph E. LEBLANC, Paul G. MEZEY, Isao NARUKI, Tadashi OKADA Eric RAMBO, Mark SPIVAKOVSKY, Shigeru TAKEUCHI, Keith F. TAYLOR, Hongyi WONG
Satoshi YAMANAKA, Masaaki YOKOTANI, Peter ZIZLER and Shigeru ARIMOTO†
This is the 25th part of the series of articles that records and further develops essentials of the Mathematics and Chemistry Interdisciplinary Symposium 2013 Tsuyama, whose main themes were symmetry, periodicity, and repetition. The symposium was held on April 5th and 6th in Tsuyama city, Okayama, Japan, in conjunction with the Fukui Project and was devoted to the memory of the late Professor Kenichi Fukui (1981 Nobel Prize) who initiated the project. The present series also provides challenging cross-disciplinary problems which are directly related to the Fukui conjecture, the Global Pattern Identification (GPI) in the Repeat Space Theory (RST), and Artificial Intelligence (AI). Some of these problems are formulated using mathematical language not well known among chemists despite the importance of these notions in elucidating additivity and high-speed asymptotic phenomena in molecules having many repeating identical moieties. The cross-disciplinary interaction between the Repeat Space Theory and the Spatial Anthropology has been discussed in connection with the Science-Art Multi- angle Network (SAM Network) Project, which seeks to bridge Science and Art (visual, audial, and conceptual) for a creative collaboration, and is an important part of the Fukui Project.
Key Words: the Fukui conjecture, Memoir of Prof. K. Fukui, Unique factorization domain (UFD), Global Pattern Identification in the Repeat Space Theory (RST), Spatial Anthropology, Artificial Intelligence (AI)
I Introduction
Okayama prefecture, Japan. The main themes of the 原稿受付 平成30年9月20日
＊1, ＊6, ＊10 ＊11, ＊16 ＊17総合理工学科
＊4 総合理工学科非常勤講師
＊2 Dept. of Math.Tarbiat Modares University, Iran
＊3 Dept. of Fund.Ed., Dalian Neusoft University of Information, China
＊5 お茶の水女子大学 理学部・元教授
＊7 School of Sciences, Humanities, and Visual Communications, Pennsylvania College of Technology, USA
＊8 Institute of Chemistry, Eotvos University of Budapest, Hungary
＊9 立命館大学 理工学部・数学物理学系・数理科学科・元教授
＊12 CNRS and Institute de Mathématiques de Toulouse, France
＊13 岐阜大学 教育学部・数学科・元教授
＊14 Dept. of Math. and Stat., Dalhousie University, Canada
＊15 School of Communication, Arts and Social Sciences, Singapore Polytechnic, Singapore
＊18 Dept. of Math., Phys., and Eng., Mount Royal University, Canada
＊19 Former Professor of NIT, Tsuyama College, Japan †Director of the Fukui Project (New Frontier Project) For correspondence, visit:
https://www.researchgate.net/profile/Shigeru_Arimoto (Links to other co-authors also available at the above website.)
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3. The Intrinsic Reaction Coordinate and a Path of Ascent to Mount Hiei:
In Memory of Professor Kenichi Fukui, a Nobel Prize Winning Chemist, and a Pioneer of
Many New Paths in Mathematical Chemistry Paul G. Mezey
Along the path of my adventures in Mathematical Chemistry, I can thank some exceptional fortune, many accidental, lucky events, and, in part, perhaps, also some rational, logical aspects of how science progresses, leading to new associations, connections among fields of science and even between scientists themselves. However, learning about an important scientific discovery, just when one needs it, has turned out to be decisive in my case: this has been the introduction of the Intrinsic Reaction Coordinate, IRC, by one of the earliest Mathematical Chemists: Professor Kenichi Fukui. My reading of his IRC paper has had a major influence on the scientific path I have followed in Mathematical Chemistry, only enhanced after I had the chance to meet him, and further strengthened by the privilege to discuss many related scientific problems and become one of his co-authors.
How did I get involved in mathematical chemistry?
As a student back in Budapest, Hungary, I have become interested in the puzzles of how those tiny things, molecules behave. My imagination has failed me completely, and this has disturbed me a great deal. After all, molecules are made up by atomic nuclei and electrons, which, to my classically motivated mind, behave utterly differently. I have been aware that for nuclei, the classical concepts of relative positions, nuclear conformations, nuclear rearrangements, as used by experimental chemists, have been still viable on the conceptual level, they have not been rendered completely useless by quantum mechanics, yet, for electrons, I could find practically nothing classical within a molecule; for imagining electrons within a molecule, my classical imagination has been useless. How is it possible, that such a rich, versatile, and incredibly diverse part of nature, the world of molecules, is based on two so very differently behaving families of objects: nuclei and electrons? Surely, in a strict sense, quantum mechanics applies to both families, but what a different way: nuclei seemingly give only grudging recognition to quantum mechanics, for example, in tunneling, or H- bonding, yet electrons within a molecule are apparently fully quantum-mechanical, and ignore any conceptual
simplifications available in classical mechanics.
What could I do? I assumed that geometry, with precise locations, with models of classical motions in geometric rearrangements, with not any sign of any relevance of the Heisenberg Uncertainty Relation, is far too classical for quantum mechanics. Yet, a lesser-used branch of mathematics, topology, often nicknamed
“rubber geometry”, with tools of homeomorphisms, continuous deformations, and algebraic simplifications, does allow for inherent, but well-defined uncertainties, or rather, undefined details when describing topological objects, and if instead of geometry, topology is chosen in a way compatible with quantum mechanics, perhaps a better understanding of the nucleus – electron duality of molecules can be achieved.
So, I have decided to study mathematics after my MSc in chemistry. A short time after getting my PhD in chemistry, I defended my mathematics MSc, all at the University of Budapest (Later, at the University of Saskatchewan, Canada, I received a DSc in Mathematics, essentially, in mathematical chemistry).
But, during my early studies, until my move to Canada, I was not aware of the ideas of an early pioneer of Mathematical Chemistry: Professor Kenichi Fukui. I understood only much later, that he could see very early, what was coming: Mathematical Chemistry
I had no idea yet that half a world away, already much earlier, a very young Japanese scientist, Kenichi Fukui has been wondering, if there is already mathematical physics, why isn’t there yet mathematical chemistry? After all, any science, if it is precise, must involve mathematics, and chemistry has become rich and very complex as a mostly experimental science, and mathematics in chemistry could possibly be used in far many more ways. Concerning the reference to Professor Fukui’s early comment about mathematical chemistry, Prof. Buckingham and Prof.
Nakatsuji have written in ref. [1]:
“Kenichi’s method of study was to read deeply a small number of selected papers, rather than a lot of literature in a wide range of fields. The field of mathematical physics was already established.
Courant & Hilbert’s The methods of Mathematical physics was one of his favourite books. He
wondered then why ‘mathematical chemistry’ did not exist and thought that the empirical nature of
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chemistry should decrease through cultivating mathematical chemistry. ‘Decreasing the empirical nature of chemistry’ was a phrase that Professor Fukui was often to use in his lectures.”
After my arrival in Canada in 1973, I started to implement some of the short notes I had made for myself during the mathematics courses I have taken. As a math student at the University of Budapest, I had the advantage of taking those classes as a chemist, and I have been amazed, during nearly every mathematics lecture, that here, again, there has been a witty mathematical result, that would fit so well to this or that chemistry problem …. Yet, to my knowledge, no such advance has been made ….. I still have some of those notes, but, I think that probably, most of those ideas have become obsolete by now ….
The first such mathematically motivated problem I looked at was the stability of the outcomes of chemical reactions, leading to the early recognition that among the very numerous choices for the description of more-or-less approximate reaction paths, available at that time, there was only one that had the mathematical rigor and usefulness at the same time: the IRC, the Intrinsic Reaction Coordinate of Professor Fukui [2]. In fact, I was surprised why anyone was still using any other definition.
Further mathematical chemistry motivation has come from the very powerful paper by Professor Tachibana and Professor Fukui, on the differential geometry of chemically reacting systems [3].
Of course, the choice was clear, the IRC was the definition I have used in the reaction path stability analysis, on “Reactive Domains of Potential Energy Hypersurfaces”, one of the first mathematical chemistry papers I have ever written [4]. It has been a strong additional motivation and a special privilege, that this study, together with a related one [5] has been quoted by Professor Fukui [6] already in the summer of 1981, actually, before he has received the news on his Nobel Prize for Chemistry, on October 19, 1981. In his classical 1981 Accounts of Chemical Research paper on IRC [6], Professor Fukui has mentioned two of these early papers on reaction topology using the topology- motivated path terminology [4,5]:
“Why the term “intrinsic” was used here was to introduce the concept of “intrinsic motion” – a quasistatic nuclear displacement. The term “intrinsic path” was suggested by Mezey instead of “intrinsic coordinate”. This may be reasonable since an IRC is a mathematical curve,
but this curve has been obtained through a number of too simplified assumptions to be named “path” of a chemical reaction.”
Professor Fukui has explicitly provided motivation for the topological approach to quantum chemistry, as distinct from the graph-theoretical approaches, although graph theory is often confused with topology by some experimental chemists. Professor Fukui has given strong support in his opening chapter, ref [7], in the book devoted to "Applied Quantum Chemistry", Proceedings of the Hawaii 1985 Nobel Laureate Symposium on Applied Quantum Chemistry, in Honor of G. Herzberg, R. S.
Mulliken, K. Fukui, W. Lipscomb, and R. Hoffman, Honolulu, HL 16-21 December 1984, Eds.: V.H. Smith, Jr., H.F. Schaefer III, and K. Morokuma, Reidel Publ. Co., 1986. In fact, referring to a topological approach, formulated in terms of potential energy surface “cells”, also expressed as “catchment regions”’ following an analogy with the very early works of Cayley [8] and Maxwell [9], Professor Fukui has stated that
“This model is helpful in obtaining a most general theoretical picture of a chemical reaction involving the tunneling process and multistep processes. In this connection, it is noteworthy that Paul Mezey [Ref13] is vigorously developing a theory of global behavior of the reaction space --- “reaction topology”, and an interesting directed graphs·theory is being developed by Oktay Sinanoglu [Ref14]. Ref13. P. Mezey, Proceedings of the Nobel Laureate Symp. on Applied Quant. Cham., Dec.
1984, Honolulu, Hawaii, and references cited therein.
Ref14. A. Fernandez and O. Sinanoglu, Theoret. Chim.
Acta (Berl.) 65, 179 (1984).”
In fact, the reference [Ref13] in the above quote by Professor Fukui is just another chapter [10] by P. G.
Mezey in the same book "Applied Quantum Chemistry", Proceedings of the Hawaii 1985 Nobel Laureate Symposium on Applied Quantum Chemistry, in Honor of G. Herzberg, R. S. Mulliken, K. Fukui, W. Lipscomb, and R. Hoffman, Honolulu, HL 16-21 December 1984, Eds.:
V.H. Smith, Jr., H.F. Schaefer III, and K. Morokuma, Reidel Publ. Co., 1986.
Some further details on the topological quantum chemistry model of reacting systems, reaction topology, have also been described, all based on the fundamental concept of IRC. These developments can be found in references [10-13].
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With major input from Professor Shigeru Arimoto, an earlier student and co-worker of Professor Fukui, alternative mathematical chemistry approaches, involving Chemical Network Systems and a family of linear operators have been successfully developed, further advancing the original motivating ideas of Professor Fukui [14-20].
On a more personal level, I am very happy to recall that I had the privilege to visit Professor Fukui at the Institute for Fundamental Chemistry in Kyoto and at the Kyoto University on many occasions. These collaborations were very fruitful, in total we have published seven papers, devoted to various aspects of mathematical chemistry. But, Professor Fukui was motivating not only by the strength of his scientific mind, but also on a more spiritual level.
I personally find that spiritual components can well serve science not only as a motivating force towards trying to understand the world with the aim of providing new, useful knowledge, but also by serving interesting and often unexpected analogies. The mental processes involved in reaching a positive thought and in thinking of a problem from a very new, interesting perspective, or thinking of a new type of motivation are, in my opinion, very similar. I have always felt a strong, stable balance of the spiritual and the scientific aspects of life from his perspective, whenever I had the chance to discuss almost any question with Professor Fukui.
I can never forget the impression I received when in 1994, on an excursion to Mount Hiei, next to Kyoto, taken with his group, I have had the chance to walk a few steps with a very special walking stick, one that Professor Fukui had received earlier as a gift from a Buddhist Monk, who had made himself the thousand-night pilgrimage down and up again to Mount Hiei. I could feel the strength of devotion, the cleansing of mind resulting from the physical effort, and I tried to imagine the new associations, and the thoughts of self-understanding this devoted Monk could have felt. During this excursion with professor Fukui and his research group, it was a path of ascent to Mount Hiei, in the physical sense, and also in a spiritual sense.
It was serious, and also cheerful, as life itself should be.
In the picture enclosed, this path of Mount Hiei is shown, with one experienced, and one less experienced pilgrim.
I also remember another hilltop excursion in 1987, on the occasion of the 1987 Budapest WATOC World Congress. Professor Fukui was the main speaker at the conference, accompanied by his wonderful wife, Tomoe.
With my wife, Justine, we had the privilege to invite them to the Budapest Castle Hilltop restaurant, near the King’s Castle, without a king, but with a beautiful hilltop view of Budapest, showing all the bridges and riverside buildings.
Of course, we did not walk there but took a taxi, however, there were other kinds of spiritual components: Mrs.
Fukui, Tomoe had always been an expert in many components of international folklore music, and she was fascinated by a Hungarian folklore orchestra playing at that restaurant. She recognized some tunes, typically pentatonic, which she has known to be components of the early childhood music education program introduced by the late Hungarian composer, Zoltan Kodaly. I had known his wife, Sarolta Kodaly, who has made major efforts to make this method known internationally, apparently successfully in Japan, and we had discussed many interesting connections between sciences and music. The very place also had some emotional connotations for me, since the restaurant, located in the Hilltop Hilton Hotel of Budapest, was a modern building, however, with some remnants of medieval walls of a Jesuit seminary. The original building at some later time had become a high school for girls. That old building had been the very high school attended by my mother for 8 years. However, during the brutal bombings of the second world war, all except one of the several hundred buildings of the entire Castle District were utterly destroyed, and the school, with arches and walls witnessing lots of medieval history, was no more. After some decades, the Hilton hotel was built there, and, although history has not, but life has changed,
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and all four of us in that great location, we could enjoy soul-pleasing folklore music, with a panoramic view of the city of Budapest, and discussions on the spirituality of music and the mathematics of chemistry, with many-many paths between them.
One may say, some paths may lead to some more, unexpected paths. And, the paths of science, history, and spirit are all intertwined.
Paul G. Mezey
References
[1] A.D. Buckingham and H. Nakatsuji: Kenichi Fukui. 4 October 1918 -- 9 January 1998: Elected F.R.S. 1989
Biogr. Mems Fell. R. Soc. 2001 47, 223-237, published 1 November 2001 [2] K. Fukui A formulation of the reaction coordinate. J. Phys. Chem.
74, 4161–4163 (1970).
[3] A. Tachibana and K. Fukui, Differential geometry of chemically reacting systems, Theor. Chim. Acta 49, 321-347 (1978).
[4] P.G. Mezey, Reactive Domains of Energy Hypersurfaces and the Stability of Minimum Energy Reaction Paths, Theor. Chim. Acta, 54, 95-111 (1980).
[5] P.G. Mezey, Catchment Region Partitioning of Energy Hypersurfaces, I, Theor. Chim. Acta, 58, 309-330 (1981).
[6] K. Fukui, The Path of Chemical Reactions – The IRC Approach, Accounts of Chem. Res. 14, 363-368 (1981).
[7] K. Fukui, Reminiscences – My Forty Years Study of Chemical Reactions, in "Applied Quantum Chemistry", Proceedings of the Hawaii 1985 Nobel Laureate Symposium on Applied Quantum Chemistry, in Honor of G. Herzberg, R. S. Mulliken, K. Fukui, W. Lipscomb, and R.
Hoffman, Honolulu, HL 16-21 December 1984, Eds.: V.H. Smith, Jr., H.F.
Schaefer III, and K. Morokuma, Reidel Publ. Co., 1986, pp. 1-25.
[8] A. Cayley, On contour and slope lines, London Edinburgh Dublin Philos.
Mag. J. Sci., Ser. 4, 18, 264–268 (1859).
[9] J. C. Maxwell, On hills and dales, London Edinburgh Dublin Philos. Mag.
J. Sci., Ser. 4, 40, 421–427 (1870).
[10] P.G. Mezey, Reaction Topology, in "Applied Quantum Chemistry", Proceedings of the Hawaii 1985 Nobel Laureate Symposium on Applied Quantum Chemistry, in Honor of G. Herzberg, R. S. Mulliken, K. Fukui, W.
Lipscomb, and R. Hoffman, Honolulu, HL 16-21 December 1984, Eds.: V.H.
Smith, Jr., H.F. Schaefer III, and K. Morokuma, Reidel Publ. Co., 1986, pp. 53- 74.
[11] P.G. Mezey, Topology of Energy Hypersurfaces, I-V, Theor. Chim. Acta, 62, 133-161 (1982), 63, 9-33 (1983), 67, 43-61 (1985), 67, 91-113 (1985), 67, 115-136 (1985).
[12] P.G. Mezey, Molecular Structure and Reaction Mechanism: A Topological Approach to Quantum Chemistry, J. Mol. Struct., Theochem, 103, 81-99 (1983). (volume dedicated to Nobel Laureate Prof. K. Fukui).
[13] P.G. Mezey, Classification Schemes of Nuclear Geometries and The Concept of Chemical Structure. Metric Spaces of Chemical Structure Sets Over Potential Energy Hypersurfaces, J. Chem. Phys., 78, 6182-6186 (1983).
[14] S. Arimoto, K. Fukui, K.F. Taylor, and P.G. Mezey, Structural Analysis of Certain Linear Operators Representing Chemical Network Systems via the Existence and Uniqueness Theorems of Spectral Resolution I, Int. J. Quantum Chem., 53, 375-386 (1995).
[15] S. Arimoto, K. Fukui, K.F. Taylor, and P.G. Mezey, Structural Analysis of Certain Linear Operators Representing Chemical Network Systems via the Existence and Uniqueness Theorems of Spectral Resolution II, Int. J. Quantum Chem., 53, 387-406 (1995).
[16] S. Arimoto, K. Fukui, H. Ohno, K.F. Taylor, and P.G. Mezey, Structural Analysis of Certain Linear Operators Representing Chemical Network Systems via the Existence and Uniquenesss Theorems of Spectral Resolution III, Int. J.
Quantum Chem., 63, 149-163 (1997).
[17] P.G. Mezey, K. Fukui, S. Arimoto, and K.Taylor, Polyhedral Shapes of Functional Group Distributions in Biomolecules and Related Similarity Measures, Int. J. Quantum Chem., 66, 99-105 (1998).
[18] S. Arimoto, K. Fukui, K.F. Taylor, and P.G. Mezey, Structural Analysis of Certain Linear Operators Representing Chemical Network Systems via the Existence and Uniquenesss Theorems of Spectral Resolution IV, Int. J. Quantum Chem., 67, 57-69 (1998).
[19] S. Arimoto, K. Fukui, P. Zizler, K.F. Taylor, and P.G. Mezey, Structural Analysis of Certain Linear Operators Representing Chemical Network Systems via the Existence and Uniqueness Theorems of Spectral Resolution V, Int. J.
Quant. Chem, 74, 633-644 (1999).
[20] P.G. Mezey, K. Fukui, and S. Arimoto, A Treatment of Small Deformations of Polyhedral Shapes of Functional Group Distributions in Biomolecules, Int. J.
Quant. Chem., 76, 756-761 (2000).
4. Challenging Problem 1*
Shigeru Arimoto
In Section 2 of the previous Part XXIV of this series, we formulated the following
Challenging Problem 1: Prove or disprove the following:
Proposition 1. Let {LN} be a fixed element of the repeat space with block-size q. Let I be a fixed closed interval on the real line such that I contains all the eigenvalues of L2N for all positive integers N. Let 1/2: I → denote the function defined by 1/2(t) = |t|1/2. Then, there exist real numbers and such that
Tr1/2(L2N) = N + + o(1) (1) as N → .
Note. For the notion of the repeat space Xr(q) with block-size q, see Eq. (C.3) of the Appendix to the previous
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Part XXIV. We retain the notation of [1]for the present Parts XXIV-XXVIII. The reader is referred to [1] for detailed definitions and explanations of symbols in Parts XXIV-XXVIII. We recommend the reader who is not familiar with the notion of the ‘function’ (M) of an n
n normal matrix to refer to Appendix to the previous Part XXIV.
In this section, we pose the following Challenging Problem 1*:
Challenging Problem 1*: Prove or disprove the following:
Proposition 1*. Let {LN} be a fixed element of the repeat space with block-size q. Let I be a fixed closed interval on the real line such that I contains all the eigenvalues of L4N for all positive integers N. Let 1/4: I → denote the function defined by 1/4(t) = |t|1/4. Then, there exist real numbers and such that
Tr1/4(L4N) = N + + o(1) (1)*
as N → .
We remark that there are several distinct approaches to the solutions of the above problem. The reader is invited to formulate similar challenging problems and to give solutions.
References
[1] S. Arimoto, New proof of the Fukui conjecture by the Functional Asymptotic Linearity Theorem, J. Math. Chem. 34 (2003) 259, and references therein.
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