XXX Introduction To begin with It is suggested that the reader start with the following. • A study of the TREE diagram. • Read the table of contents and compare it with the TREE. • Address the question of what is your goal, i.e. why you would like to read such a book? • Choose a personal path on the TREE, the suggested itineraries may be of some help. 23 • Become acquainted with the organization of any chapter. CHAPTER ORGANIZATION Once an itinerary is chosen the student will cover different chapters. All the chap- ters have the same structure, and are divided into sections. • Where are we In this section readers are made aware of their current position on the TREE diagram. In this way, they know the relationship of the current chapter to other chapters, what chapters they are expected to have covered already, and the re- maining chapters for which the current chapter provides a preparation. The cur- rent position shows whether they should invest time and effort in studying the cur- rent chapter. Without the TREE diagram it may appear, after tedious study of the current chapter, that this chapter was of little value and benefit to the reader. • An example Here the reader is confronted with a practical problem that the current chap- ter addresses. Only after posing a clear-cut problem without an evident solution, will the student see the purpose of the chapter and how the material presented sheds light on the stated problem. • What is it all about In this section the essence of the chapter is presented and a detailed exposi- tion follows. This may be an occasion for the students to review the relationship of the current chapter to their chosen path. In surveying the subject matter of a given chapter, it is also appropriate to review student expectations. Those who have chosen a special path will find only some of the material pertinent to their needs. Such recommended paths are also provided within each chapter. 23 This choice may still be tentative and may become clear in the course of reading this book. The subject index may serve as a significant help. For example a reader interested in drug design, that is based in particular on enzymatic receptors, should cover the chapters with (those considered most important) and then those with (at the very least, intermolecular interactions). They will gain the requisite familiarity with the energy which is minimized in computer programs. The reader should then proceed to those branches of the TREE diagram labelled with . Initially they may be interested in force fields (where the above mentioned energy is approximated), and then in molecular mechanics and molecular dynamics. Our students may begin this course with only the labels. However, such a course would leave them without any link to quantum mechanics. Introduction XXXI • Why is this important There is simply not enough time for a student to cover and become familiar with all extent textbooks on quantum mechanics. Therefore, one has to choose a set of important topics, those that represent a key to an understanding of the broad domains of knowledge. To this end, it often pays to master a complex mathematical apparatus. Such mastery often leads to a generalization or sim- plification of the internal structure of a theory. Not all chapters are of equal importance. At this point, the reader has the opportunity to judge whether the author’s arguments about the importance of a current chapter are convincing. • What is needed It is extremely disappointing if, after investing time and effort, the reader is stuck in the middle of a chapter, simply for lack of a particular theoretical tool. This section covers the prerequisites necessary for the successful completion of the current chapter. Material required for understanding the text is provided in the Appendices at the end of this book. The reader is asked not to take this section too literally, since a tool may be needed only for a minor part of the material covered, and is of secondary importance. This author, however, does presuppose that the student begins this course with a basic knowledge of math- ematics, physics and chemistry. • Classical works Every field of science has a founding parent, who identified the seminal prob- lems, introduced basic ideas and selected the necessary tools. Wherever appro- priate, we mention these classical investigators, their motivation and their most important contributions. In many cases a short biographical note is also given. • The Chapter’s Body The main body of each chapter is presented in this section. An attempt is made to divide the contents logically into sub-sections, and to have these sections as small as possible in order to make them easy to swallow. A small section makes for easier understanding. • Summary The main body of a chapter is still a big thing to digest and a student may be lost in seeing the logical structure of each chapter. 24 A short summary gives the motivation for presenting the material at hand, and why one should expend the effort, what the main benefits are and why the author has attached importance to this subject. This is a useful point for reflection and consideration. What have we learnt, where are we heading, and where can this knowledge be used and applied? • Main concepts, new terms New terms, definitions, concepts, relationships are introduced. In the current chapter they become familiar tools. The reader will appreciate this section (as well as sections Whyisthisimportantand Summaries) just before an examination. 24 This is most dangerous. A student at any stage of study has to be able to answer easily what the purpose of each stage is. XXXII Introduction • From the research front It is often ill advised to present state of the art results to students. For exam- ple, what value is it to present the best wave function consisting of thousands of terms for the helium atom? The logistics of such a presentation are difficult to contemplate. There is significant didactic value in presenting a wavefunction with one or only a few terms where significant concepts are communicated. On the other hand the student should be made aware of recent progress in generat- ing new results and how well they agree with experimental observations. • Adfuturum The reader deserves to have a learned assessment of the subject matter cov- ered in a given chapter. For example, is this field stale or new? What is the prog- nosis for future developments in this area? These are often perplexing questions and the reader deserves an honest answer. • Additional literature The present text offers only a general panorama of quantum chemistry. In most cases there exists an extensive literature, where the reader will find more detailed information. The role of review articles, monographs and textbooks is to provide an up-to-date description of a particular field. References to such works are provided in this section, often combined with the author’s comments on their appropriateness for students. • Questions In this section the reader will find ten questions related to the current chap- ter. Each question is supplied with four possible answers. The student is asked to choose the correct answer. Sometimes the answer will come easily. In other cases, the student will have to decide between two or more similar possibilities that may differ only in some subtle way. In other cases the choice will come down to the truth or an absurdity (I beg your pardon for this). Life is filled with situations where such choices have to be made. • Answers Here answers to the above questions are provided. WEB ANNEX http://www.chem.uw.edu.pl/ideas The role of the Annex is to expand the readers’ knowledge after they read a given chapter. At the heart of the web Annex are useful links to other people’s websites. The Annex will be updated every several months. The Annex adds at least four new dimensions to my book: colour, motion, an interactive mode of learning and connection to the web (with a plethora of possibilities to go further and further). The living erratum in the Annex (with the names of those readers who found the errors) will help to keep improving the book after it was printed. Introduction XXXIII ACKNOWLEDGEMENTS The list of people given below is ample evidence that the present book is not just the effort of a single individual, although I alone am responsible for any remain- ing errors or problems. Special thanks are reserved for Professor Andrzej Sadlej (University of Toru ´ n, Poland). I appreciate very, very much his extraordinary work. I would like to acknowledge the special effort provided by Miss Edyta Małolepsza, who devoted all her strength and talents (always smiling) to keep the whole long- time endeavour running. I acknowledge also the friendly help of Professor Andrzej Holas from the Polish Academy of Sciences, Professors Bogumił Jeziorski and Woj- ciech Grochala from the University of Warsaw and Professor Stanisław Kucharski from the Silesian University, who commented on Chapters 1, 8, 10 and 11, as well as of Eva Jaroszkiewicz and my other British friends for their linguistic expertise. My warmest thoughts are always associated with my friends, with whom discus- sions were unbounded, and contained what we all appreciated most, fantasy and surrealism. I think here of Professor Jean-Marie André (Facultés Universitaires de Namur, Belgium) and of Professor Andrzej J. Sadlej, Professor Leszek Stolarczyk and Professor Wojciech Grochala (from the University of Warsaw). Thank you all for the intellectual glimmers in our discussions. Without my dearest wife Basia this book would not be possible. I thank her for her love and patience. Izabelin, in Kampinos Forest (central Poland), hot August 2006 SOURCES OF PHOTOGRAPHS AND FIGURES Courtesy of The Nobel Foundation, © The Nobel Foundation (pp. 5, 8, 9, 11, 12, 26, 29, 67, 70, 94, 97, 111, 113, 132, 140, 220, 252, 268, 285, 294, 395, 425, 428, 434, 511, 523, 565, 579, 617, 619, 655, 663, 663, 683, 753, 765, 765, 765, 806, 830, 832, 850). Wikipedia public domain (1, 35, 57, 97, 229, 279, 310, 380, 525, 745, 880, 898) p. 391 – courtesy of Nicolaus Copernicus University, Toru ´ n, Poland Fig. 1.14 – courtesy of Professor Akira Tonomura, Japan p. 112 – with permission from National Physical Laboratory, courtesy of Dr. R.C. McGuiness, UK p. 220 – courtesy of Professor J.D. Roberts, California Institute of Tech- nology, USA p. 287 – reused from International Journal of Quantum Chemistry (1989), Symp. 23, © 1989 John Wiley&Sons, Inc., reprinted with permission of John Wiley&Sons, Inc. p. 392 – courtesy of Mr Gerhard Hund, Germany p. 575 – courtesy of Professor Richard Bader, Canada p. 618 – reproduced from a painting by Ms Tatiana Livschitz, courtesy of Professor W.H. Eugen Schwartz, Germany p. 863 – courtesy of Bradley Uni- versity, USA p. 131 – courtesy of Alburtus Yale News Bureau (through Wikipedia) p. 219 – courtesy Livermore National Laboratory photographs by the author (377, 456, 509, 509, 311, 805). XXXIV Introduction Sources of Figures: besides those acknowledged in situ: courtesy of Wydawnictwo Naukowe PWN, Poland from “Idee chemii kwantowej”, © 1993 PWN (Figs. 1.1–1.7, 1.9, 1.10, 1.15, 2.1–2.4, 2.6, 3.1, 3.2, 3.3, 3.4, 4.1–4.6, 4.8–4.16, 4.23, 5.1–5.3, 6.1, 6.3, 6.6–6.10, 6.13, 6.14, 6.16, 7.1–7.11, 7.13, 7.14, 8.3, 8.4, 8.6–8.12, 8.18–8.27, 9.1–9.6, 9.21, 9.22, 10.2– 10.12, 12.1–12.9, 12.13, 12.14, 13.1–13.6, 13.8, 13.10–13.13, 13.18, 14.1–14.8, 14.14–14.22, 14.26, 14.27b, 15.1–15.3, 15.5, 15.7–15.10, B.1, B.2, C.1–C.7, G.1, J.1, J.2, O.1, R.1, R.2, X.1–X.4, Y.1, Y.2) Figs. 14.14, 14.15, 14.17 and Tables 14.1–14.6 reused and adapted with permission from the American Chemical Society from S. Shaik, Journal of the Amer- ican Chemical Society, 103 (1981) 3692. © 1981 American Chemical Society. Courtesy of Professor Sason Shaik. Figures 4.3, 4.4, 4.6–4.13, 4.15, 4.17–4.21, 6.12, 7.8, 8.5, 8.8, 8.9, 8.26, 10.1–10.4, 11.2, 12.5, 12.6, 14.1, 14.2, 14.22, 14.25b, O.1, R.2, T.1, Y.1 (also a part of the cover figure) have been made using the software of Mathematica by Stephen Wolfram. Quotations: from John C. Slater’s book “Solid State and Molecular Theory”, London, Wiley, 1975 (pp. 180, 329, 343, 986); from Werner Heisenberg’s “Der Teil und das Ganze” (p. 11); from Richard Feynman’s Nobel lecture, © The Nobel Foundation (p. 105); from Si- mon Schnoll’s “Gieroi i zladiei rossiyskoi nauki” (p. 524); from Richard Zare in “Chemical and Engineering News” 4 (1990) 32 (p. 766); Erwin Schrödinger’s curriculum vitae (un- published) – courtesy of the University of Wrocław, The Rector Leszek Pacholski, Poland (p. 70). Despite reasonable attempts made, we were unable to contact the owners of the copy- right of the following images: p. 850 – from S. Schnoll’s book “Geroi i zladiei rossiyskoi nauki”, Kronpress, Moscow, 1997 p. 744 – reproduced from B.W. Nekrasov, “Kurs ob- shchei khimii”, Goskhimizdat, Moscow, 1960 Figs. 13.21 and 13.22 reproduced from “Biology Today”, CRM Books, Del Mar, USA, ©1972 Communications Research Ma- chines and the copyright owners of the items found in the websites http://www-gap. dcs.st-and.ac.uk/~history (pp. 4, 8, 9, 26, 42, 43, 72, 94, 105, 110, 219, 278, 318, 329, 446, 482, 512, 575, 580, 683, 683, 849, 851, 858, 866, 866, 876, 878, 879, 903, 997) http://nuclphys.sinp.msu.ru (p. 252) http://www.phy.duke.edu, photo Lotte Meitner-Graf (p. 521) http://www.volny.cz (p. 7) http://metwww.epfl.ch (p. 438) http://osulibrary. orst.edu (p. 331) http://www.mathsoc.spb.ru (p. 329) http://www.stetson.edu (p. 311) http://www.aeiou.at (p. 93) http://www.quantum-chemistry-history.com (pp. 361, 364). If you are the copyright owner to any of the images we have used without your explicit permission (because we were unable to identify you during our search), please contact Prof. Lucjan Piela, Chemistry Department, Warsaw University, 02093 Warsaw, Poland, e-mail: piela@chem.uw.edu.pl, phone (48)-22-8220211 ext. 525. We will place the appropriate in- formation on our website http://www.chem.uw.edu.pl/ideas which represents an appendix to the present book, as well as correct the next editions of the book. Chapter 1 THE MAGIC OF QUANTUM MECHANICS Where are we? We are at the beginning of all the paths, at the base of the TREE. An example Since 1911 we have known that atoms and molecules are built of two kinds of parti- cles: electrons and nuclei. Experiments show the particles may be treated as point-like objects of certain mass and electric charge. The electronic charge is equal to −e, while the nuclear charge amounts to Ze,where e = 16 · 10 −19 CandZ is a natural num- ber. Electrons and nuclei interact according to the Coulomb law, and classical mechan- ics and electrodynamics predict that any atom or molecule is bound to collapse in a matter of a femtosecond emitting an infinite amount of energy. Hence, according to the classical laws, the complex matter we see around us (also our bodies) should simply not exist at all. However, atoms and molecules do exist, and their existence may be described in detail Charles Augustin de Coulomb (1736–1806), French military engineer, one of the founders of quantitative physics. In 1777 he constructed a torsion balance for measuring very weak forces, with which he was able to demonstrate the inverse square law for electric and magnetic forces. He also studied charge distribution on the surfaces of dielectrics. by quantum mechanics using what is known as the wave function. The axioms of quantum mechanics provide the rules for the derivation of this function and for the calculation of all the observable properties of atoms and molecules. What is it all about History of a revolution ()p.4 Postulates ()p.15 The Heisenberg uncertainty principle ()p.34 The Copenhagen interpretation ()p.37 How to disprove the Heisenberg principle? The Einstein–Podolsky–Rosen recipe ()p.38 Is the world real? ()p.40 • Bilocation 1 2 1. The Magic of Quantum Mechanics The Bell inequality will decide ()p.43 Intriguing results of experiments with photons ()p.46 Teleportation ()p.47 Quantum computing ()p.49 Any branch of science has a list of axioms, on which the entire construction is built. 1 For quantum mechanics, six such axioms (postulates) have been established. The postulates have evolved in the process of reconciling theory and experiment, and may sometimes be viewed as non-intuitive. These axioms stand behind any tool of quantum mechanics used in practical applications. They also lead to some striking conclusions concerning the reality of our world, for example, the possibilities of bilocation, teleportation, and so on. These unexpected conclusions have recently been experimentally confirmed. Why is this important? The axioms given in this chapter represent the foundation of quantum mechanics, and justify all that follows in this book. In addition, our ideas of what the world is really like acquire a new and unexpected dimension. What is needed? • Complex numbers (necessary). • Operator algebra and vector spaces, Appendix B, p. 895 (necessary). • Angular momentum, Appendix F, p. 955 (recommended). • Some background in experimental physics: black body radiation, photoelectric effect (rec- ommended). Classical works The beginning of quantum theory was the discovery, by Max Planck, of the electromag- netic energy quanta emitted by a black body. The work was published under the title: “Über das Gesetz der Energieverteilung im Normalspektrum” 2 in Annalen der Physik, 4 (1901) 553. Four years later Albert Einstein published a paper “Über die Erzeugung und Verwand- lung des Lichtes betreffenden heuristischen Gesichtspunkt”inAnnalen der Physik, 17 (1905) 132, in which he explained the photoelectric effect by assuming that the energy is absorbed by a metal as quanta of energy. In 1911 Ernest Rutherford discovered that atoms are composed of a massive nucleus and electrons: “The Scattering of the α and β Rays and the Structure of the Atom”, in Proceedings of the Manchester Literary and Philosophical Society,IV, 1 And which are not expected to be proved. 2 Or “On the Energy Distribution Law in the Normal Spectrum” with a note saying that the material had already been presented (in another form) at the meetings of the German Physical Society on Oct. 19 and Dec. 14, 1900. On p. 556 one can find the following historical sentence on the total energy denoted as U N :“Hierzu ist es notwendig, U N nicht als eine stetige, unbeschränkt teilbare, sondern als eine diskrete, aus einer ganzen Zahl von endlichen gleichen Teilen zusammengesetzte Grösse aufzufassen”, which translates as: “Therefore, it is necessary to assume that U N does not represent any continuous quantity that can be divided without any restriction. Instead, one has to understand that it as a discrete quantity composed of a finite number of equal parts. Classical works 3 55 (1911) 18. Two years later Niels Bohr introduced a planetary model of the hydrogen atom in “On the Constitution of Atoms and Molecules”inPhilosophical Magazine,Series6, vol. 26 (1913). Louis de Broglie generalized the corpuscular and wave character of any particle in his PhD thesis “Recherches sur la théorie des quanta”, Sorbonne, 1924. The first mathematical formulation of quantum mechanics was developed by Werner Heisenberg in “Über quantentheoretischen Umdeutung kinematischer und mechanischer Beziehungen”, Zeitschrift für Physik, 33 (1925) 879. Max Born and Pascual Jordan recognized matrix algebra in the formulation [in “Zur Quantenmechanik”, Zeitschrift für Physik, 34 (1925) 858] and then all three [the famous “Drei-Männer Arbeit” entitled “Zur Quantenmechanik. II.” and published in Zeitschrift für Physik, 35 (1925) 557] expounded a coherent mathematical basis for quantum mechanics. Wolfgang Pauli introduced his “two-valuedness” for the non-classical electron coordinate in “Über den Einfluss der Geschwindigkeitsabhängigkeit der Elektronenmasse auf den Zeemaneffekt”, published in Zeitschrift für Physik, 31 (1925) 373, the next year George Uhlenbeck and Samuel Goudsmit described their concept of particle spin in “Spinning Electrons and the Structure of Spectra”, Nature, 117 (1926) 264. Wolfgang Pauli published his famous exclusion principle in “Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren” which appeared in Zeitschrift für Physik B, 31 (1925) 765. The series of papers by Erwin Schrödinger “Quan- tisierung als Eigenwertproblem”inAnnalen der Physik, 79 (1926) 361 (other references in Chapter 2) was a major advance. He proposed a different mathematical formulation (from Heisenberg’s) and introduced the notion of the wave function. In the same year Max Born, in “Quantenmechanik der Stossvorgänge” which appeared in Zeitschrift für Physik,37 (1926) 863 gave an interpretation of the wave function. The uncertainty principle was dis- covered by Werner Heisenberg and described in “Über den anschaulichen Inhalt der quanten- theoretischen Kinematik und Mechanik”, Zeitschrift für Physik, 43 (1927) 172. Paul Adrien Maurice Dirac reported an attempt to reconcile quantum and relativity theories in a series of papers published from 1926–1928 (references in Chapter 3). Albert Einstein, Boris Podolsky and Natan Rosen proposed a test (then a Gedanken or thinking-experiment, now a real one) of quantum mechanics “Can quantum-mechanical description of physical reality be considered complete?” published in Physical Review, 47 (1935) 777. Richard Feynman, Ju- lian Schwinger and Shinichiro Tomonaga independently developed quantum electrodynam- ics in 1948. John Bell, in “On the Einstein–Podolsky–Rosen Paradox ”, Physics, 1 (1964) 195 reported inequalities which were able to verify the very foundations of quantum me- chanics. Alain Aspect, Jean Dalibard and Géard Roger in “Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers”, Physical Review Letters, 49 (1982) 1804 reported measurements which violated the Bell inequality and proved the non-locality or/and (in a sense) non-reality of our world. Akira Tonomura, Junji Endo, Tsuyoshi Matsuda and Takeshi Kawasaki in “Demonstration of Single-Electron Buildup of an Interference Pattern”, American Journal of Physics, 57 (1989) 117 reported direct electron interference in a two-slit experiment. Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres and William K. Wootters, in “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels”inPhysical Review Letters, 70 (1993) 1895 designed a teleportation experiment, which has subsequently been successfully accomplished by Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter and Anton Zeilinger, “Experimental Quantum Teleportation”inNature, 390 (1997) 575. 4 1. The Magic of Quantum Mechanics 1.1 HISTORY OF A REVOLUTION The end of the nineteenth century saw itself as a proud period for physics, which seemed to finally achieve a state of coherence and clarity. Physics at that time believed the world consisted of two kingdoms: a kingdom of parti- James Clerk Maxwell (1831– 1879), British physicist, pro- fessor at the University of Ab- erdeen, Kings College, Lon- don, and Cavendish Profes- sor in Cambridge. His main contributions are his famous equations for electromagnet- ism (1864), and the earlier discovery of velocity distribu- tion in gases (1860). cles and a kingdom of electromagnetic waves. Motion of particles had been de- scribed by Isaac Newton’s equation, with its striking simplicity, universality and beauty. Similarly, electromagnetic waves had been accurately described by James Clerk Maxwell’s simple and beautiful equations. Young Planck was advised to aban- don the idea of studying physics, because everything had already been discovered. This beautiful idyll was only slightly in- complete, because of a few annoying details: the strange black body radiation, the photoelectric effect and the mysterious atomic spectra.Justsomeratherexoticprob- lems to be fixed in the near future. . . As it turned out, they opened a New World. The history of quantum theory, one of most revolutionary and successful theories ever designed by man, will briefly be given below. Many of these facts have their continuation in the present textbook. Black body radiation 1900 – Max Planck Max Planck wanted to understand black body radiation. The black body may be modelled by a box, with a small hole, Fig. 1.1. We heat the box up, wait for the system to reach a stationary state (at a fixed temperature) and see what kind of electromagnetic radiation (intensity as a function of frequency) comes out of the hole. In 1900 Rayleigh and Jeans 3 tried to apply classical mechanics to this prob- lem, and calculated correctly that the black body would emit electromagnetic radi- ation having a distribution of frequencies. However, the larger the frequency the larger its intensity, leading to what is known as ultraviolet catastrophe, an absurd UV catastrophe conclusion. Experiment contradicted theory (Fig. 1.1). At a given temperature T the intensity distribution (at a given frequency ν, Fig. 1.1.b) has a single maximum. As the temperature increases, the maximum should shift towards higher frequencies (a piece of iron appears red at 500 ◦ C, but bluish at 1000 ◦ C). Just like Rayleigh and Jeans, Max Planck was unable to derive 3 James Hopwood Jeans (1877–1946), British physicist, professor at the University of Cambridge and at the Institute for Advanced Study in Princeton. Jeans also made important discoveries in astrophysics (e.g., the theory of double stars). 1.1 History of a revolution 5 Max Karl Ernst Ludwig Planck (1858–1947), German physicist, professor at the universi- ties in Munich, Kiel and Berlin, first director of the Institute of Theoretical Physics in Berlin. Planck was born in Kiel, where his father was a university professor of law. Max Planck was a universally talented school pupil, then an out- standing physics student at the University of Berlin, where he was supervised by Gustaw Kirchhoff and Hermann Helmholtz. Music was his passion throughout his life, and he used to play piano duets with Einstein (who played the violin). This hard-working, middle-aged, old- fashioned, professor of thermodynamics made a major breakthrough as if in an act of scien- tific desperation. In 1918 Planck received the Nobel Prize “for services rendered to the ad- vancement of Physics by his discovery of en- ergy quanta”. Einstein recalls jokingly Planck’s reported lack of full confidence in general rela- tivity theory: “Planck was one of the most out- standing people I have ever known, ( )Inre- ality, however, he did not understand physics. During the solar eclipse in 1919 he stayed awake all night, to see whether light bending in the gravitational field will be confirmed. If he understood the very essence of the general rel- ativity theory, he would quietly go to bed, as I did”. (Cited by Ernst Straus in “Einstein: A Cen- tenary Volume”, p. 31). John William Strutt, Lord Rayleigh (1842– 1919), British physicist, Cavendish Professor at Cambridge, contributed greatly to physics (wave propagation, light scattering theory – Rayleigh scattering). In 1904 Rayleigh re- ceived the Nobel Prize “for his investigations of the densities of the most important gases and for his discovery of argon in connection with these studies”. black body classical theory (ultraviolet catastrophe) experiment Fig. 1.1. Black body radiation. (a) As one heats a box to temperature T , the hole emits electromagnetic radiation with a wide range of frequencies. The distribution of intensity I(ν) as a function of frequency ν is given in Fig. (b). There is a serious discrepancy between the results of classical theory and the experiment, especially for large frequencies. Only after assuming the existence of energy quanta can theory and experiment be reconciled. . also the friendly help of Professor Andrzej Holas from the Polish Academy of Sciences, Professors Bogumił Jeziorski and Woj- ciech Grochala from the University of Warsaw and Professor Stanisław Kucharski from. and surrealism. I think here of Professor Jean-Marie André (Facultés Universitaires de Namur, Belgium) and of Professor Andrzej J. Sadlej, Professor Leszek Stolarczyk and Professor Wojciech Grochala. R.C. McGuiness, UK p. 220 – courtesy of Professor J.D. Roberts, California Institute of Tech- nology, USA p. 287 – reused from International Journal of Quantum Chemistry (1989), Symp. 23, © 1989