It is the great honor and the privilege for us to pay the tribute to the remarkable person, Professor Ilya G. Kaplan. One of us, E.S.K., had the privilege to be partly among Kaplan's many former graduate students and three, O.N., J.S., and J.L., have fruitfully collaborated with him. We all have been good friends of Ilya and the admirers of his work. We decided to present our tribute in a quite unusual format, the format of dialogue where Ilya Kaplan shares his memoirs of his path to and in science.

**International Journal of Quantum Chemistry**

# Introduction

## Abstract

It is a great honor and privilege for the guest editors to pay tribute to the remarkable Professor Ilya G. Kaplan. One of us, E.S.K., had the privilege to be partly among Kaplan's many former graduate students and three, O.N., J.S., and J.L., have fruitfully collaborated with him. We all have been good friends of Ilya and the admirers of his work. Here, we present our tribute in a quite unusual format: a dialogue where Ilya Kaplan shares his memoirs of his path to and in science.

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## Dialogue

**Ilya G. Kaplan:** The path to science—Dotted Biography

In 1955, I graduated from Department of theoretical physics at the University of Saratov, in the former Soviet Union, but was assigned to work as an engineer in electrical installations in an aircraft factory in Saratov, which produces the famous fighters MiG (MIG). For the reason of secrecy, the plant was officially named as that of the harvesters. But, it was a disclosed secret. Among people, it was called as the plant of flying harvesters. Having worked at this plant for two years, I decided to try to attend the PhD graduate school of the Academy of Sciences (AS) in Moscow.

Using few months in the evenings for preparation, I took a vacation from work and came to the Moscow Institute of Chemical Physics (ICP), which was led by then Nobel laureate Academician N. N. Semenov. I called the theoretical Department and was immediately connected to its Head, A. S. Kompaneets.

Kompaneets was one of the first students of Lev D. Landau (or shortly, Dau) (Alexander Solomonovich Kompaneets (1914-1974) was the “first pupil of L. D. Landau” (see B. Gorobets, The Landau's Circle, Moscow, Summer Garden, 2006. Chapter 3, p. 323; E. S. Kryachko. This Issue; A. S. Kompaneets, Personalia, Uspekhi Fiz. Nauk 114, 687-688 (1974).) because he was the first who completely passed the famous Landau's theorerical minimum). Kompaneets came out to meet me (at that time anyone cannot enter any Institute of AS without permission) and I told him that I am working at the plant and I want to be his PhD student. He immediately examined me, asking questions in mathematics and physics, mainly to evaluate my intelligence, and concluded that my knowledge was not enough, but the way of thinking is definitely good and so he agreed to allow me to pass the PhD entrance exams, and I urgently need to be prepared for the mathematics, classical mechanics, and field theory. I replied that I had studied mechanics using the book “Analytical Mechanics” by Frenkel.

Immediately, I first encountered the reaction of perfect opposition to the works by Frenkel from (students) pupils of Dau that originated from the Teacher, who, incidentally, began his scientific career at the Leningrad Physico-Technical Institute in the Theoretical Department headed by Frenkel. Later on, I many times came across with this hardly explained negative attitude of the Landau school to the outstanding physicist Frenkel, always full of profound physical ideas in physics. Kompaneets then said me sharply: “Forget about the book by Frenkel, the mechanics should be taught only with the book by Landau and Piatigorskii” (in the later editions of E. M. Lifshitz replaced Piatigorskii as a co-author).

Few months later I again arrived in Moscow, successfully passed the exams and was enrolled in the post graduate school of the AS of USSR at the ICP. Although, during the exams, I was almost failed in the compulsory examination on the history of the Communist Party of the Soviet Union when was given a score of 3 (out of 5 possible), because I have inaccurately mentioned the date of one of the numerous Congresses of the Communist Party, and unsatisfactorily, as the examiners said, replied to the question why the Lenin's work “One Step Forward, Two Steps Back” is named like that. Apparently, I was lucky enough since a score of 3 had no impact on my admission.

At our first meeting after my PhD enrolment, Kompaneets told me that to have knowledge necessary for theoretical physicist, I must pass the so called theoretical minimum by Landau* in the Institute for Physical Problems (IPP), where Dau headed the Theoretical Department. He gave me a year for taking this theoretical minimum and warned that within a year I should not come to him and ask him any questions.

Landau's theoretical minimum was quite famous in the Soviet Union and in Moscow in particular: anyone could take it regardless the level of education and PhD status. To begin with suffices to call Dau and tell him about your desire to take these exams. Usually, Dau appointed the next Thursday, 15 min before the start of his famous seminar, that is, precisely at 10:45 am. The first exam, on math, was given directly by Dau himself. Dau offered the person taking this exam to sit in the room next to the seminar hall and offered only the problems of handling some complicated integral, of lowering the order of differential equation and so on. To pass this exam successfully, one had have to solve preliminarily all the problems from the known at that time book of problems in mathematical analysis by Gyunter i Kuzmin for Physics Faculties of the USSR Universities. There were only two exam's scores: “passed” or “failed.” The first problem that one cannot afford to solve became the last one. However, one was allowed to take this exam several times.

I solved all the problems in my first coming, and Dau took me to his apartment to make a record in the list for the date of the next exam. I recall as I was struck by his two-store apartment with a fireplace, which was situated in a two-store annex building, adjacent to the main building of the Institute, similar to the apartments of the other leading members of the IPP. The matter is that the institute was designed by its director, Peter Kapitza who worked for many years with Rutherford in Cambridge, when he arrived in the Soviet Union in 1934 and was not allowed to return abroad.

At that time, Dau's theoretical minimum consisted of nine exams: Math I, Math II, Mechanics, Field theory including the Gravitation Field, Non-relativistic Quantum Mechanics, Statistical Physics, Quantum Electrodynamics of Continuous Media and Physical Kinetics. It was recommended to use the appropriate book of the Landau—Lifshitz course in “Theoretical Physics,” with some sections excluded. During the examination, it was necessary to respond to any question and to solve several problems from Exercises at the end of Sections of the appropriate book.

During the year, I passed six exams. In addition to the exams on Math I, taken directly from Dau, the other ones were given by his collaborators (his early students), working in the Theoretical Department. Mathematics II and Mechanics were given by Abrikosov, the future Nobel laureate, Field Theory and Statistical Physics by Khalatnikov, and Quantum Mechanics by E. M. Lifshits, the Landau co-author of the course in theoretical physics. One year of my PhD study passed and I had have to work on my Thesis. Then, I got an unpleasant surprise. My supervisor Kompaneets told me that he now works exclusively on the secret topics which are related to the nuclear weapons and he would not want to draw me into this topic (afterwards I was very grateful him for that). However, he does not have any “open” project for me.

The following year I began to work in a very good library of the Physics Department of Moscow State University, to deal with different issues that seemed to me interesting. In particular, I was interested in the works by Racah, after reviewing all his papers on the addition of angular momenta and the transition matrices between the different schemes of addition (Klebsch-Gordan and Racah coefficients, etc.), which subsequently appeared to be very useful to me. I also studied the theory of permutation groups. But, the project for my PhD still had not been established. This was rather natural for the young man who had never been engaged in scientific work. I recall, once I asked my University friend Lev Pitaevskii, who was then the PhD graduate student of Dau, whether he could look the PhD theme for me in Theoretical Department. Pitaevski told me that they have a special book of problems led by Dau, but there are those problems which (he) Dau cannot solve himself.

All this time I regularly attended the Moscow Dau Seminar, visited by the majority of theoretical physicists of Moscow. Usually a large Seminar Hall of the Institute of Physical Problems was crowded. In the front row sat Dau, the next to him was E. M. Lifshitz, further sat the prominent physicists from school of Dau—Pomeranchuk, Ginzburg, Migdal, and often threefold Hero of Socialist Labor Yakov Zel'dovich, awarded for his work on the atomic and hydrogen bombs, when he arrived in Moscow from a secret facility Arzamas, sometimes purposely to take part in this seminar. Questions to the speaker were usually asked by Dau and only few participants, among which were Pomeranchuk, Migdal, and Zel'dovich. E. M. Lifshitz asked his questions not to the speaker but rather to next-sitting Dau. Frequently Dau asked the speaker at the beginning of his talk to present the summarized result and sometimes, he immediately declared that the result is incorrect for this and that reasons, that affect us, the young scientists by his instant insight into the problem, and it happened that after some answers of the speaker to Dau, the latter stopped the Seminar, saying that the speaker was not prepared. This definitely considerably hurt the speaker. This style was further accepted by many Dau students. For instance, my supervisor Kompaneets, being dissatisfied with my answers to his question interrupted my talk at the seminar in his theoretical department several times. He gave 10 min to think and said that if I can not find the correct answer during this time, he would stop the seminar. So I passed a pretty hard school of scientific seminars, which steeled me for the future.

A deep knowledge of Dau in different areas of physics are often baffled by reputable scientists. I recall Pekar, one of the great physicist from Kiev who developed the theory of polarons in solids, was mixed with Dau's question of whether there exists a connection between some effect in a solid with a certain physical phenomena observed in nuclei.

I attended the memorable seminar, when in 1958, N. N. Bogolyubov was invited by Dau to give a talk on his original theory of superconductivity. That was the year after publication of the BCS theory of superconductivity. Dau began immediately interrupt him, saying that his statements are false and ill-defined. This lasted until Bogolyubov abruptly replied on one of the critics by Dau that a similar statement Dau can read in his own book on a certain page. Dau did not respond and then Bogolyubov quietly finished his talk.

I must say that Dau has been completely and unselfishly devoted to science, he was extremely honest and always said what he thought. But, he did not spare anyone, if he thought that this work is wrong. For the majority of scientists, I am not talking about young people, the immediate impression was that Dau is a genius. Once in winter evening I met Dau in the Moscow downtown. He was in an unbuttoned fur coat with head held high. I thought that his eyes emanate some glow. I then could not long escape from the feeling that his eyes shone.

In general, Landau school has taught me to a strict foundation of any statement, and allowed me to work quite efficiently in various areas of physics and physical chemistry. What else is important, it is never to enter as a co-author of the article if you do not think you made a substantial contribution.

The third year of my PhD has embarked on. The dissertation project has been still absent. I performed two theoretical works, unrelated to each other, and even published them in the leading physics journal in the USSR, the “Journal of Experimental and Theoretical Physics,” or shortly “ZhETP,” and then, suddenly arrived from Vilnius I. B. Levinson, co-author, together with Vanagas and Yucis, of a very good book on the theory of angular momenta, to give a talk at the Kompaneets seminar. As I mentioned, in the past I studied the Racah's theory of addition of angular momenta and was also quite aware of the theory of permutation groups. Levinson, who was aware that I still do not have a project for my PhD Thesis, offered me the problem of how to construct wave functions with arbitrary permutational symmetry in the central field. He emphasized that knowledge of the permutation group among physicists was quite rare at that time.

I took this problem and quickly developed a method of the construction of the coordinate wave functions in a multi-shell model of the nucleus, which, in the absence of isotopic spin, describe the atoms as well. To do this, I had to introduce the transformation matrix of the permutation group, which, as it turned out, were analogous to the transformation matrix of the rotation group, studied in the works by Wigner and Racah. This became the foundation of my PhD Thesis.

After finishing the post-graduate study, *aspirantura* of the ICP of AS in 1961, I began to work at the Institute of Radiation Chemistry, which was a branch of the Moscow Karpov Physical-Chemical Institute, in the suburb town Obninsk. At that time I was sure that I was unable to pose new tasks, new ideas, though I was able to solve quite complicated problems, formulated by others. Years later, though very soon, I figured out that I was wrong in this self-assessment. Having engaged in radiation chemistry, I have critically revised some accepted concepts and developed the important role of the triplet states in radiation chemistry, as well as several other original approaches, novel at that time, including the theory of photoionization of molecules in the X-ray and gamma regions. Together with my co-worker Markin, we have calculated relativistic photoeffect on the hydrogen molecule for the first time and predicted the interference phenomenon in the angular distribution of electrons from oriented molecules. This effect has been experimentally confirmed only after 38 years.

Simultaneously, I began to study the application of the permutation group to quantum chemistry and developed a new approach called quantum chemistry without spin. The works on the application of group theory to various problems of quantum mechanics and quantum chemistry, I summarized in a monograph on the symmetry of many-electron systems, published in Moscow, in Russian, in 1969. The book was accepted by the Moscow Publishing House on Physics and Mathematics literature thanks to the recommendation by Kompaneets. In the same year, I defended the second degree—Doctor of Physical-Mathematical Sciences in my Alma Mater—the ICP. My entrance to science could be considered to be finally completed.

The extended version of my first book was translated into English by Joe Gerratt who actively worked in the same area, and was published by Academic Press in 1975. Pretty soon the book became broadly known. It was quoted many times, even currently, although more than 30 years after the English edition has been already passed. I recall that 10 years ago during one of the March Meetings of the American Physical Society, after seeing a sign with my name, I was approached by the unknown person who asked whether I am the author of the book on the symmetry of many-electron systems? After my affirmative reply, he told me that he worked in Los Alamos Lab in the Group of Hans Bethe, and Bethe highly recommended my book on the symmetry of many-electron systems to all members. I was very delighted to hear the appreciation of my book by Bethe, one of the founders of quantum mechanics.

I have to say that I have always been interested in the issues of symmetry in quantum mechanics. So, for many years I was occupied with the question of why, according to the Pauli principle, in Nature only the antisymmetric or symmetric states with respect to permutations of identical particles are realized, while the Schrödinger equation is satisfied by any permutation symmetry. This and related issues I discuss in Section 4 of the article, published in this issue, and now I want to recall, as I reported my study of the principle of Pauli in Zurich at the Institute for Theoretical Physics, where Pauli worked for many years. I went to the Institute along Wolfgang Pauli Strasse and lectured in the audience with a large portrait of Pauli on the wall. At first, I felt some inconvenient tightness from what I am saying about Pauli to the members of the Institute, where he worked. But, then it became clear that my talk is very far from their interests, and quite unusual for them. Pursuing the practical tasks in solid-state physics, they did not think about the issues of the foundations of quantum mechanics, which agitated at the time their famous countryman. Well, everything in the world undergoes change, and the priorities are also changed. Ancient Greek philosophers were right saying that you can not twice enter in the same river.

In 1959 during his post-graduate study, Ilya Kaplan married Larisa Popova. Recently Ilya and Larisa celebrated their “golden” wedding. Photo 5 shows this couple at their “silver” wedding.

Prof. Ilya Kaplan has been fruitfully working in such different fields of physics and physical chemistry as application of group-theoretical methods to quantum mechanics and primary processes in radiation chemistry; the neutrino rest mass problem and the electronic structure of superconductors, and so forth. Below we shortly describe his main achievements in the six scientific fields† and present his CV.

## Ilya G. Kaplan: Scientific Carrier—Milestones

### Application of group theoretical methods to quantum mechanics of many-particle systems and to quantum chemistry

In 1961, for the first time Ilya Kaplan introduced the transformation matrices of the permutation group.[48] As was shown later, these matrices are the analogue of the 3*n–j* symbols of the three-dimensional (3D) rotation group studied by Wigner and Racah. The transformation matrices allowed to develop the general method of calculation of the matrix elements of Hamiltonian for nuclear and atomic multishell configurations in states with a definite total spin **S** (see Ref. [49] in Ilya G. Kaplan: List of Publications). The connection between the permutation groups and the space symmetry groups was used to elaborate the general method of classifying quantum states allowed by the Pauli principle for arbitrary systems of identical fermions or bosons.[46]

During many years Prof. Kaplan studied the foundations of the Pauli principle.[17, 21, 151, 182] He demonstrated that, on the one hand, the Pauli exclusion principle cannot be derived from other fundamental principles of quantum mechanics and on the other hand, it cannot be treated as a postulate, since all other symmetry options for the total wave function of identical particles, except the 1D representations, are prohibited due to contradictions with the nonindistinguishability of identical particles in quantum mechanics and their independency from each other (see Kaplan's paper in this Issue).

The decomposition of the total wave function on coordinate and spin wave functions allowed Kaplan, independently on Matsen, to create the original approach to quantum chemistry, which was named “Quantum chemistry without spin” (1963–1973). Application of this approach to quantum chemistry allows to obtain a deeper understanding of the nature of the chemical bond and to develop the efficient methods of classification and calculation of molecular electronic states. These works were summarized in the monograph (see text footnote).

In the scientific literature, such terms as “The Kaplan transformation matrices,” “Kaplan's formula,” “Kaplan's algorithm” have been generally accepted and used (see, e.g., V. G. Neudatchin et al., Adv. Nucl. Phys., Vol. 11, p. 1 (1979); R. Pauncz “Spin Eigenfunctions”, Plenum, N.Y. (1979), and so on).

### Neutrino rest mass and quantum chemistry

Ilya Kaplan for the first time applied quantum chemical approach to the fundamental problem of contemporary physics—the problem of the neutrino rest mass. The method of taking into account the influence of the molecular structure of a β-source on the β-electron energy was elaborated for this problem (see e.g. [120, 132, 135] and review [34]). This method had been widely used in Laboratories measured the neutrino rest mass on β-spectrometers.

### Penetration of charge particles through molecular medium and primary processes of radiation chemistry

In series of studies, the theory of degradation of ionising radiation in matter was developed and the important role of triplet states was established.[52] The analysis of delocalization processes of energy absorption in molecular medium was performed. The limitations of such conception as “track” and its absence in polymers have been formulated.[131]

The method of the computer experiment giving possibility to simulate the primary stage of radiolysis from physical stage and track formation to the chemical kinetic stage was developed.[133, 142] As a result the peculiarities of radiation chemical reaction in tracks of charge particles of different nature were revealed.[30, 33]

In 1988–1994, Prof. Kaplan was a member (from Russia) of the collaborative research program “Atomic and Molecular Data for Radiotherapy and Radiation Research” sponsored by IAEA (Vienna, Austria), see report.[10]

### Theory of photoelectron spectroscopy and compton scattering

In 1968, in collaborations with Markin the pioneer calculations of photoionization cross sections of many-atomic molecules were performed.[69] These works stimulated a great number of investigations in many laboratories. Then, the theory of molecular photoionization in X- and γ-region was developed and firstly in the world the relativistic photoeffect on the hydrogen molecule was calculated[83] In transition matrix element the exponential operator, taking into account all multipoles, was used. It has been shown that chemical bonding must be taken into account even for the photon energy exceeding the energy of chemical bond by factor of thousand or more. This theoretical prediction was confirmed experimentally in University of Oregon by Bernd Crasemann and co-workers and was used in astrophysics for determination of the molecular hydrogen concentration in galaxy space.

In 1969, the oscillation behavior of photoelectron angular distribution from an oriented molecule in X-ray region was predicted.[71] This effect was confirmed in experiment only after 38 years [Science **949**, 318 (2007)]. For development of the theory of photoelectron spectroscopy Prof. Kaplan was awarded by the USSR State Prize in 1985.

In 1975, the theory of the Compton effect on a bound electron was elaborated.[95] Based on these studies, the experimental scheme in which the full 3D electron momentum density can be detected was suggested,[102] in contrary with the Compton profile experiments related to the 2D electron momentum density. Recently Kaplan returned to the problem of Compton scattering on molecules obtaining the triple differential Compton cross section for electron correlated systems in terms of the Dyson orbitals.[185]

In 2002–2005, in collaboration with J. V. Ortiz's group, the Kaplan's approach to the photoionization in X-ray region[71] was reformulated in the electron propagator formalism using the Dyson orbitals.[186, 191] This allowed taking into account the final-state orbital relaxation as well as the electron correlation effects.

### Theory of intermolecular forces and nonadditivity

The application of the symmetry-adapted perturbation theory and quantum-chemical methods for calculations of intermolecular forces in the wide range of distances were analysed so far as the influence of nonadditive interactions on the effective atom-atomic potentials. These results were generalized by Kaplan in 1982, in published in Russian monograph,[7] the extended version of which was translated in English by Eugene Kryachko, (see this issue).

After Ilya Kaplan started to work in Mexican National University (UNAM), in collaboration with Octavio Novaro and his group the general approach for calculating nonadditive forces in many-particle systems was elaborated and closed formula for the energy of *n*-body interactions was obtained.[156] The study of small silver clusters revealed the pronounced “size effect” and the importance of not only three-, but also four- and five-body forces in the many-body expansion.[155] The method of construction of many-body model potentials with parameters fitting with *ab initio* potential surfaces and an application of these model potentials in molecular dynamics simulation codes were developed.[166, 169, 181] Important studies of the nature of the bonding in the alkaline-earth clusters, built with atoms without valence electrons, were performed during his staying in Jerzy Leszczynski's laboratory in Jackson University[43, 179] and revisited in the last publication.[204]

The contemporary state of the theory of intermolecular interactions was analyzed and systematized by Ilya Kaplan in his recent monograph,[4] published by Wiley in 2006. The modernized and corrected English edition of this book has been translated into Russian and published by BINOM, Moscow, 2012. At present, the book is translating into Chinese in Peking University and will be published in China by Chemical Industry Press.

### Electronic structure of superconductors and statistics of charge carriers

For study of the charge and spin distributions in superconductors (SC) in collaboration with Jacques Soullard and Hernandez-Cobos, the embedded cluster method, taking into account the electron correlation and reproducing the Madelung potential on each cluster sites, was elaborated.[42] The elaborated method allows studying the electronic structure and the impurity effects in SC. It was applied to study the pure and Zn- and Ni-doped Y123 ceramics,[171, 183] and also the pure and Ti-doped unconventional SC Sr_{2}RuO_{4}.[196, 198] The study of doped Y123 ceramics revealed the considerable breaking of the holes pairs in the vicinity of impurity. It was shown that in the case of nonmagnetic Zn impurity all nearest CuO_{2} units around Zn lose their holes.[183] The results of study[183] obtained a large citation factor; they were used by the Nobel Laureate Abrikosov for one of his theories.

The comparative study of statistics of Cooper's electron pairs (charge carriers in low-temperature SC) and coupled hole pairs (charge carriers in high-temperature SC) was performed, see review.[18] The analysis of trilinear commutation relations for the Cooper pair operators reveals that they correspond to the modified parafermi statistics of rank *p* = 1.[187] The exact commutation relations of the hole-pair operators in quasi-impulse representation correspond to a modified parafermi statistics of rank *M* (*M* is the number of sites in the lattice formed by the centers of mass of each hole pair).[173, 177] This statistics is the same, as was revealed in 1976 by Kaplan[98] for molecular excitons and magnons. It was also shown that in spite of the non-bosonic behavior of coupled hole pairs, there is no statistical prohibition on the Bose–Einsten condensation.

In the last years, Ilya Kaplan has been involved in calculations of the electron structure of transition-metal dimers.[198, 199] Although this field is new for him, his studies are carried out at the state-of-the-art level.[19, 202, 203]

^{*}Theoretical minimum of Landau comprises of 10 exams on the basic fields of mathematics and theoretical physics. It includes the following exams: Mathematics-I, Mechanics, Field Theory, Mathematics II, Quantum Electrodynamics, Statistical Physics I, Mechanics of Continuous Media, Electrodynamics of Continuous Media, Statistical Physics II, and Physical Kinetics. The updated list of those who passed Landau theoretical minimum is stored elsewhere: http://theorminimum.itp.ac.ru/ and http://physics.stackexchange.com/questions/13861/lev-landaus-theoretical-minimum, http://banan000.xanga.com/695334609/landaus-theoretical-minimum/. See also: B. Ioffe, ArXiv:hep-ph/0204295, 2002.

^{†}References are taken from the list of Kaplan's publications.