756 13. Intermolecular Interactions Fig. 13.21. Bacteriophage T represents a supramolecular construction that terrorizes bacteria. The hexagonal “head” contains a tightly packed double helix of DNA (the virus genetic material) wrapped in a coat build of protein subunits. The head is attached to a tube-like molecular connector built of 144 contractible protein molecules. On the other side of the connector there is a plate with six spikes pro- truding from it as well as six long, kinked “legs” made of several different protein molecules. The legs represent a “landing aparatus” which, using intermolecular interactions, attaches to a particular recep- tor on the bacterium cell wall. This reaction is reversible, but what happens next is highly irreversible. First, an enzyme belonging to the monster makes a hole in the cell wall of the bacterium. Then the 144 protein molecules contract probably at the expense of energy from hydrolysis of the ATP molecule (adenosine triphosphate – a universal energy source in biology), which the monster has at its disposal. This makes the head collapse and the whole monster serves as a syringe. The bacteriophage’s genetic material enters the bacterium body almost in no time. That is the end of the bacterium. • Perturbational method has limited applicability: – at long intermolecular separations what is called the polarization approximation may be used, – at medium distances a more general formalism called the symmetry adapted perturba- tion theory may be applied, – at short distances (of the order of chemical bond lengths) perturbational approaches are inapplicable. • One of the advantages of a low-order perturbational approach is the possibility of dividing the interaction energy into well defined physically distinct energy contributions. • In a polarization approximation approach, the unperturbed wave function is assumed as a product of the exact wave functions of the individual subsystems: ψ (0) 0 =ψ A0 ψ B0 .The corresponding zero-order energy is E (0) 0 =E A0 +E B0 . • Then, the first-order correction to the energy represents what is called the electrostatic interaction energy: E (1) 0 = E elst =ψ A0 ψ B0 |Vψ A0 ψ B0 , which is the Coulombic in- teraction (at a given intermolecular distance) of the frozen charge density distributions of the individual, non-interacting molecules. After using the multipole expansion E elst can be divided into the sum of the multipole–multipole interactions plus a remainder, called the penetration energy. A multipole–multipole interaction corresponds to the permanent multipoles of the isolated molecules. An individual multipole–multipole interaction term (2 k -pole with 2 l -pole) vanishes asymptotically as R −(k+l+1) , e.g., the dipole–dipole term decreases as R −(1+1+1) =R −3 . Summary 757 • In the second order we obtain the sum of the induction and dispersion terms: E (2) = E ind +E disp . • The induction energy splits into E ind (A →B) = n B |ψ A0 ψ Bn B |Vψ A0 ψ B0 | 2 (E B0 −E Bn B ) which pertains to polarization of molecule B by the unperturbed molecule A and E ind (B →A) = n A |ψ An A ψ B0 |V |ψ a0 ψ b0 | 2 (E A0 −E An A ) with the roles of the molecules exchanged. The induction energy can be represented as the permanent multipole – induced multipole interaction, where the interaction of the 2 k -pole with the 2 l -pole vanishes as R −2(k+l+1) . • The dispersion energy is defined as E disp = n A n B |ψ An A ψ Bn B |Vψ A0 ψ B0 | 2 (E A0 −E An A ) +(E B0 −E Bn B ) and represents a result of the electronic correlation. After applying the multipole expan- sion, the effect can be described as a series of instantaneous multipole – instantaneous multipole interactions, with the individual terms decaying asymptotically as R −2(k+l+1) . The most important contribution is the dipole–dipole (k =l =1), which vanishes as R −6 . • The polarization approximation fails for medium and short distances. For medium sepa- rations we may use symmetry-adapted perturbation theory (SAPT). The unperturbed wave function is symmetry-adapted, i.e. has the same symmetry as the exact function. This is not true for the polarization approximation, where the product-like ϕ (0) does not exhibit the proper symmetry with respect to electron exchanges between the interacting molecules. The symmetry-adaptation is achieved by a projection of ϕ (0) • Symmetry-adapted perturbation theory reproduces all the energy corrections that appear in the polarization approximation (E elst E ind E disp ) plus provides some exchange-type terms (in any order of the perturbation except the zeroth). • The most important exchange term is the valence repulsion appearing in the first-order correction to the energy: E (1) exch = ψ A0 ψ B0 V ˆ P AB ψ A0 ψ B0 −ψ A0 ψ B0 |Vψ A0 ψ B0 ψ A0 ψ B0 ˆ P AB ψ A0 ψ B0 +O S 4 where O(S 4 ) represents all the terms decaying as the fourth power of the overlap inte- gral(s) or faster, ˆ P AB stands for the single exchanges’ permutation operator. • The interaction energy of N molecules is not pairwise additive, i.e. is not the sum of the interactions of all possible pairs of molecules. Among the energy corrections up to the second order, the exchange and, first of all, the induction terms contribute to the non- additivity. The electrostatic and dispersion (in the second order) contributions are pair- wise additive. • The non-additivity is highlighted in what is called the many-body expansion of the interac- tion energy, where the interaction energy is expressed as the sum of two-body, three-body, etc. energy contributions. The m-body interaction is defined as that part of the interaction energy that is non-explicable by any interactions of m <mmolecules, but explicable by the interactions among m molecules. 758 13. Intermolecular Interactions • The dispersion interaction in the third-order perturbation theory contributes to the three- body non-additivity and is called the Axilrod–Teller energy. The term represents a corre- lation effect. Note that the effect is negative for three bodies in a linear configuration. • The most important contributions: electrostatic, valence repulsion, induction and disper- sion lead to a richness of supramolecular structures. • The electrostatic interaction plays a particularly important role, because it is of a long- range character as well as very sensible to relative orientation of the subsystems. The hydrogen bond X–H Y represents an example of the domination of the electrostatic interaction. This results in its directionality, linearity and a small (as compared to typical chemical bonds) interaction energy of the order of 5 kcal/mol. • Also the valence repulsion is one of the most important energy contributions, because it controls how the interacting molecules fit together in space. • The induction and dispersion interactions for polar systems, although contributing signif- icantly to the binding energy, in most cases do not have a decisive role and only slightly modify the geometry of the resulting structures. • In aqueous solutions the solvent structure contributes very strongly to the intermolecular interaction, thus leading to what is called the hydrophobic effect. The effect expels the non-polar subsystems from the solvent, thus causing them to approach, which looks like an attraction. • A molecule may have such a shape that it fits that of another molecule (synthons, small valence repulsion and a large number of attractive atom–atom interactions). • In this way molecular recognition may be achieved by the key-lock-type fit (the molecules non-distorted), template fit (one molecule distorted) or by the hand-glove-type fit (both molecules distorted). • Molecular recognition may be planned by chemists and used to build complex molecular architectures, in a way similar to that in which living matter operates. Main concepts, new terms interaction energy (p. 684) natural division (p. 684) binding energy (p. 687) dissociation energy (p. 687) dissociation barrier (p. 687) catenans (p. 688) rotaxans (p. 688) endohedral complexes (p. 688) supermolecular method (p. 689) basis set superposition error (BSSE) (p. 690) ghosts (p. 690) polarization perturbation theory (p. 692) electrostatic energy (p. 693) induction energy (p. 694) dispersion energy (p. 694) multipole moments (p. 698) permanent multipoles (p. 701) SAPT (p. 710) function with adapted symmetry (p. 711) symmetry forcing (p. 716) polarization collapse (p. 717) symmetrized polarization approximation (p. 717) MS–MA perturbation theory (p. 717) Jeziorski–Kołos perturbation theory (p. 717) valence repulsion (p. 718) Padé approximants (p. 722) Pauli blockade (p. 722) exchange–deformation interaction (p. 722) interaction non-additivity (p. 726) many-body expansion (p. 727) SE mechanism (p. 733) TE mechanism (p. 734) polarization catastrophe (p. 738) three-body polarization amplifier (p. 738) Axilrod–Teller dispersion energy (p. 741) van der Waals radius (p. 742) van der Waals surface (p. 742) supramolecular chemistry (p. 744) hydrogen bond (p. 746) From the research front 759 hydrophobic effect (p. 748) amphiphilicity (p. 749) nanostructures (p. 749) leucine-valine zipper (p. 749) synthon (p. 750) template (p. 751) “key-lock” interaction (p. 751) “hand-glove” interaction (p. 751) From the research front Intermolecular interactions influence any liquid and solid state measurements. Physicoche- mical measurement techniques giveonly some indications of the shape of a molecule, except NMR, X-ray and neutron analyses, which provide the atomic positions in space, but are very expensive. This is why there is a need for theoretical tools which may offer such information in a less-expensive way. For very large molecules, such an analysis uses the force fields de- scribed in Chapter 7. This is currently the most powerful theoretical tool for determining the approximate shape of molecules with numbers of atoms even of the order of hundreds of thousands. To obtain more reliable information about intermolecular interactions we may perform calculations within a supermolecular approach, necessarily of an ab initio type, be- cause other methods give rather poor quality results. The DFT method popular nowadays fails at its present stage of development, because the intermolecular interactions area, espe- cially the dispersion interaction, is a particularly weak point of the method. If the particular method chosen is the Hartree–Fock approach (currently limited to about 300 atoms), we have to remember that it cannot take into account any dispersion contribution to the inter- action energy by definition. 91 Ab initio calculations of the correlation energy still represent a challenge. Good quality calculations for a molecule with a dozen atoms may be carried out using the MP2 method. Still more time consuming are the CCSD(T) or SAPT calculations, which are feasible for only a few atom systems, but offer an accuracy of 1 kcal/mol required for chemical applications. Ad futurum. . . No doubt the computational techniques will continue to push the limits mentioned above. The more coarse the method used, the more spectacular this pushing will be. The most difficult to imagine would be a great progress in methods using explicitly correlated wave functions. It seems that pushing the experimental demands and calculation time required will cause experimentalists (they will perform the calculations 92 ) to prefer a rough estima- tion using primitive methods rather than wait too long for a precise result (still not very appropriate, because the results are obtained without taking the influence of solvent, etc. into account). It seems that in the near future we may expect theoretical methods exploiting the synthon concept. It is evident that a theoretician has to treat the synthons on an equal footing with other atoms, but a practice-oriented theoretician cannot do that, otherwise he would wait virtually forever for something to happen in the computer, while in reality the reaction takes only a picosecond or so. Still further into the future we will see the planning of hierarchic multi-level supramolecular systems, taking into account the kinetics and com- petitiveness among such structures. In the still more distant future, functions performed by such supramolecular structures, as well as their sensitivity to changing external conditions, will be investigated. 91 The dispersion energy represents an electronic correlation effect, absent in the Hartree–Fock en- ergy. 92 What will theoreticians do? My answer is given in Chapter 15. 760 13. Intermolecular Interactions Additional literature J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, “Molecular Theory of Gases and Liquids”, Wiley, New York, 1964. A thick “bible” (1249 pages) of intermolecular interactions. We find there everything before the advent of computers and of the symmetry-adapted perturbation theory. H. Margenau, N.R. Kestner, “Theory of Intermolecular Forces”, Pergamon, Oxford, 1969. A lot of detailed derivations. There is also a chapter devoted to the non-additivity of the interaction energy – a rara avis in textbooks. A.J. Stone, “The Theory of Intermolecular Forces”, Oxford Univ. Press, Oxford, 1996. The book contains the basic knowledge in the field of intermolecular interactions given in the language of perturbation theory as well as the multipole expansion (a lot of use- ful formulae for the electrostatic, induction and dispersion contributions). This very well written book presents many important problems in a clear and comprehensive way. “Molecular Interactions”, ed. S. Scheiner, Wiley, Chichester, 1997. A selection of articles written by experts. P. Hobza, R. Zahradnik, “Intermolecular Complexes”, Elsevier, Amsterdam, 1988. Many useful details. I.G. Kaplan, “Intermolecular Interactions”, Wiley, Chichester, 2006. A very well written book, covering broad and fundamental aspects of the field. Questions 1. In order to avoid the basis set superposition error using the counter-poise method: a) all quantities have to be calculated within the total joint atomic basis set; b) the energy of the total system has to be calculated within the total joint atomic basis set, while the subsystem energies have to be calculated within their individual atomic basis sets; c) the total energy has to be calculated within a modest quality basis set, while for the subsystems we may allow ourselves a better quality basis set; d) all the atomic orbitals have to be centred in a single point in space. 2. In the polarization approximation the zero-order wave function is: a) the sum of the wave functions of the polarized subsystems; b) the product of the wave functions of the polarized subsystems; c) a linear combination of the wave functions for the isolated subsystems; d) the product of the wave functions of the isolated subsystems. 3. Please find the false statement: a) dissociation energy depends on the force constants of the normal modes of the mole- cule; b) interaction energy calculated for the optimal geometry is called the binding energy; c) the absolute value of the dissociation energy is larger than the absolute value of the binding energy; d) some systems with energy higher than the dissociation limit may be experimentally observed as stable. Answers 761 4. Please find the false statement. The induction energy (R stands for the intermolecular distance): a) represents an electronic correlation effect; b) for two water molecules decays asymptotically as R −6 ; c) is always negative (attraction); d) for the water molecule and the argon atom vanishes as R −6 . 5. Dispersion energy (R stands for the intermolecular distance): a) is non-zero only for the noble gas atom interaction; b) is equal to zero for the interaction of the two water molecules calculated with the minimal basis set; c) represents an electronic correlation effect; d) decays asymptotically as R −4 . 6. A particle has electric charge equal to q and Cartesian coordinates xy z.Pleasefind the false statement: a) the multipole moments of the particle depend in general on the choice of the coordi- nate system; b) the operator of the z component of the dipole moment (μ z ) of the particle is equal to qz; c) a point-like particle does not have any dipole moment, therefore μ z =0; d) the octupole moment operator of the particle is a polynomial of the third degree. 7. In symmetry adapted perturbation theory for He 2 : a) we obtain the dispersion energy in the first order of the perturbation theory; b) the electronic energy exhibits a minimum as a function the interatomic distance; c) the binding energy is non-zero and is caused by the induction interaction; d) the unperturbed wave function is asymmetric (do not mix with antisymmetry) with respect to every electron permutation. 8. Please choose the false statement. In symmetry adapted perturbation theory: a) the zeroth order wave function is a product of the wave functions of the individual molecules; b) the electrostatic energy together with the valence repulsion appears in the first order; c) the exchange corrections appear in all (non-zero) orders; d) the Pauli exclusion principle is taken into account. 9. The non-additivity of the interaction energy has the following features: a) the electrostatic energy is additive, while the induction energy is not; b) the valence repulsion is additive for single exchanges; c) the dispersion interaction being a correlation effect is non-additive; d) the Axilrod–Teller interaction is additive. 10. Typical distances X Y and H Y for a hydrogen bond X–H Y are closest to the following values (Å): a) 1.5, 1; b) 2.8, 1.8; c) 3.5, 2.5; d) 2.8, 1. Answers 1a, 2d, 3c, 4a, 5c, 6c, 7b, 8a, 9a, 10b Chapter 14 INTERMOLECULAR MOTION OF ELECTRONS AND NUCLEI: C HEMICAL REACTIONS Where are we? We are already picking fruit in the crown of the TREE. Example Why do two substances react and another two do not? Why does increasing the temperature often start a reaction? Why does a reaction mixture change colour? As we know from Chap- ter 6, this tells us about some important electronic structure changes. On the other hand the products (as opposed to the reactants) tell us about profound changes in the positions of the nuclei that take place simultaneously. Something dramatic is going on. But what? What is it all about How atom A eliminates atom C from diatomic molecule BC? How can a chemical reac- tion be described as a molecular event? Where does the reaction barrier come from? Such questions will be discussed in this chapter. The structure of the chapter is the following. Hypersurface of the potential energy for nuclear motion () p. 766 • Potential energy minima and saddle points • Distinguished reaction coordinate (DRC) • Steepest descent path (SDP) • Our goal • Chemical reaction dynamics (pioneers’ approach) AB INITIO APPROACH p. 775 Accurate solutions for the reaction hypersurface (three atoms) () p. 775 • Coordinate system and Hamiltonian • Solution to the Schrödinger equation • Berry phase APPROXIMATE METHODS p. 781 Intrinsic Reaction Coordinate (IRC) or statics () p. 781 Reaction path Hamiltonian method () p. 783 762 Why is this important? 763 • Energy close to IRC • Vibrationally adiabatic approximation • Vibrationally non-adiabatic model • Application of the reaction path Hamiltonian method to the reaction H 2 +OH →H 2 O +H Acceptor–donor (AD) theory of chemical reactions () p. 798 • Maps of the molecular electrostatic potential • Where does the barrier come from? • MO, AD and VB formalisms • Reaction stages • Contributions of the structures as the reaction proceeds • Nucleophilic attack H − +ethylene → ethylene + H − • Electrophilic attack H + +H 2 →H 2 +H + • Nucleophilic attack on the polarized chemical bond in the VB picture • What is going on in chemist’ flask? • Role of symmetry • Barrier means a cost of opening the closed-shells Barrier for the electron transfer reaction () p. 828 • Diabatic and adiabatic potential • Marcus theory We are already acquainted with the toolbox for describing the electronic structure at any position of the nuclei. It is time now to look at possible large changes of the electronic structure at large changes of nuclear positions. The two motions: of the electrons and nuclei will be coupled together (especially in a small region of the configurational space). Our plan consists of four parts: • In the first part (after using the Born–Oppenheimer approximation, fundamental to this chapter), we assume that we have calculated the ground-state electronic energy, i.e. the potential energy for the nuclear motion. It will turn out that the hypersurface has a char- acteristic “drain-pipe” shape, and the bottom in the central section, in many cases, exhibits a barrier. Taking a three-atom example, we will show how the problem could be solved, if we were capable of calculating the quantum dynamics of the system accurately. • In the second part we will concentrate on a specific representation of the system’s energy that takes explicitly into account the above mentioned reaction drain-pipe (“reaction path Hamiltonian”). Then we will focus on describing how a chemical reaction proceeds. Just to be more specific, an example will be shown in detail. • In the third part (acceptor–donor theory of chemical reactions) we will find the answer to the question, of where the reaction barrier comes from and what happens to the electronic structure when the reaction proceeds. • The fourth part will pertain to the reaction barrier height in electron transfer (a subject closely related to the second and the third parts). Why is this important? Chemical reactions are at the heart of chemistry, making possible the achievement of its ultimate goals, which include synthesizing materials with desired properties. What happens 764 14. Intermolecular Motion of Electrons and Nuclei: Chemical Reactions in the chemist’s flask is a complex phenomenon 1 which consists of an astronomical number of elementary reactions of individual molecules. In order to control the reactions in the flask, it would be good to first understand the rules which govern these elementary reaction acts. What is needed? • Hartree–Fock method (Chapter 8, necessary). • Conical intersection (Chapter 6, necessary). • Normal modes (Chapter 7, necessary). • Appendices M (recommended), E (recommended), Z (necessary), I (recommended), G (just mentioned). • Elementary statistical thermodynamics or even phenomenological thermodynamics: en- tropy, free energy (necessary). Classical works Everything in chemistry began in the twenties of the twentieth century. The first publications that considered conical intersection – a key concept for chemi- cal reactions – were two articles from the Budapest schoolmates: Janos (John) von Neu- mann and Jenó Pál (Eugene) Wigner “Über merkwürdige diskrete Eigenwerte” published in Physikalische Zeitschrift, 30 (1929) 465 and “Über das Verhalten von Eigenwerten bei adi- abatischen Prozessen” which also appeared in Physikalische Zeitschrift, 30 (1929) 467. Then a paper “The Crossing of Potential Surfaces” by their younger schoolmate Edward Teller was published in the Journal of Chemical Physics, 41 (1937) 109. A classical theory of the “reaction drain-pipe” with entrance and exit channels was first proposed by Henry Eyring, Harold Gershinowitz and Cheng E. Sun in “Potential Energy Surface for Linear H 3 ”, the Journal of Chemical Physics, 3 (1935) 786, and then by Joseph O. Hirschfelder, Henry Eyring and Bryan Topley in an article “Reactions Involving Hydrogen Molecules and Atoms” in Journal of Chemical Physics, 4 (1936) 170 and by Meredith G. Evans and Michael Polanyi in “Inertia and Driving Force of Chemical Reactions” which appeared in Transactions of the Faraday Society, 34 (1938) 11. Hugh Christopher Longuet-Higgins, U. Öpik, Maurice H.L. Pryce and Robert A. Sack in a splendid paper “Studies of the Jahn–Teller Effect”, Pro- ceedings of the Royal Society of London, A244 (1958) 1 noted for the first time, that the wave function changes its phase close to a conical intersection, which later on became known as the Berry phase. The acceptor–donor description of chemical reactions was first pro- posed by Robert S.J. Mulliken in “Molecular Compounds and their Spectra”, Journal of the American Chemical Society, 74 (1952) 811. The idea of the intrinsic reaction coordinate (IRC) was first given by Isaiah Shavitt in “The Tunnel Effect Corrections in the Rates of Reac- tions with Parabolic and Eckart Barriers”, Report WIS-AEC-23, Theoretical Chemistry Lab., University of Wisconsin (1959) as well as by Morton A. Eliason and Joseph O. Hirschfelder in the Journal of the Chemical Physics, 30 (1959) 1426 in an article “General Collision Theory Treatment for the Rate of Bimolecular, Gas Phase Reactions”. The symmetry rules allow- ing some reactions and forbidding others were first proposed by Robert B. Woodward and Roald Hoffmann in two letters to the editor: “Stereochemistry of Electrocyclic Reactions” and “Selection Rules for Sigmatropic Reactions”, Journal of American Chemical Society,87 (1965) 395, 2511 as well as by Kenichi Fukui and Hiroshi Fujimoto in an article published 1 See Chapter 15. Classical works 765 in the Bulletin of the Chemical Society of Japan, 41 (1968) 1989. The concept of the steepest descent method was formulated by Kenichi Fukui in “A Formulation of the Reac- tion Coordinate”, which appeared in the Jour- nal of Physical Chemistry, 74 (1970) 4161, al- though the idea seems to have a longer his- tory. Other classical papers include a sem- inal article by Sason S. Shaik “What Happens to Molecules as They React? Valence Bond Ap- proach to Reactivity”inJournal of the Ameri- can Chemical Society, 103 (1981) 3692. The Hamiltonian path method was formulated by William H. Miller, Nicolas C. Handy and John E. Adams, in an article “Reaction Path John Charles Polanyi (born 1929), Canadian chemist of Hungarian origin, son of Mic- hael Polanyi (one of the pio- neers in the field of chemical reaction dynamics), professor at the University of Toronto. John was attracted to chem- istry by Meredith G. Evans, who was a student of his fa- ther. Three scholars: John Polanyi, Yuan Lee and Dudley Herschbach shared the 1986 Nobel prize “ for their contribu- tions concerning the dynam- ics of chemical elementary processes ”. Hamiltonian for Polyatomic Molecules”inthe Journal of the Chemical Physics, 72 (1980) 99. The first quantum dynamics simula- tion was performed by a PhD student George C. Schatz (under the supervision of Aron Kupperman) for the reaction H 2 + H → H + H 2 , reported in “Role of Direct and Res- onant Processes and of their Interferences in the Quantum Dynamics of the Collinear H + H 2 Exchange Reaction”, in Journal of Chem- ical Physics, 59 (1973) 964. John Polanyi, Dudley Herschbach and Yuan Lee proved that the lion’s share of the reaction energy is Yuan T. Lee is a native of Tai- wan, called by his colleagues “a Mozart of physical chem- istry”. He wrote that he was deeply impressed by a biog- raphy of Mme Curie and that her idealism decided his own path. delivered through the rotational degrees of freedom of the products, e.g., J.D. Barn- well, J.G. Loeser, D.R. Herschbach, “Angu- lar Correlations in Chemical Reactions. Statis- tical Theory for Four-Vector Correlations” pub- lished in the Journal of Physical Chemistry,87 (1983) 2781. Ahmed Zewail (Egypt/USA) developed an amazing experimental tech- nique known as femtosecond spectroscopy, which for the first time allowed the study of the reacting molecules at different stages of an ongoing reaction (“Femtochemistry – Ultrafast Dynamics of The Chemical Bond”, vol. I and II, A.H. Zewail, World Scientific, New Jersey, Singapore (1994)). Among others, Josef Michl, Lionel Salem, Donald G. Truhlar, Robert E. Wyatt, and W. Ronald Gentry contributed to the theory of chemical reactions. Dudley Herschbach writes in his CV, that he spent his child- hood in a village close to San Jose, picking fruit, milk- ing cows, etc. Thanks to his wonderful teacher he became interested in chemistry. He graduated from Harvard Uni- versity (physical chemistry), where as he says, he has found “an exhilarating acad- emic environment”. In 1959 he became professor at Uni- versity of California at Berke- ley. In 1967 the group was joined by Yuan Lee and con- structed a “supermachine” for studying crossing molecular beams and the reactions in them. One of the topics was the alkali metal atom – io- dine collisions. These inves- tigations were supported by John Polanyi, who studied the chemiluminescence in IR, i.e. the heat radiation of chemical reactions. . (born 1929), Canadian chemist of Hungarian origin, son of Mic- hael Polanyi (one of the pio- neers in the field of chemical reaction dynamics), professor at the University of Toronto. John was attracted. powerful theoretical tool for determining the approximate shape of molecules with numbers of atoms even of the order of hundreds of thousands. To obtain more reliable information about intermolecular. wave function is: a) the sum of the wave functions of the polarized subsystems; b) the product of the wave functions of the polarized subsystems; c) a linear combination of the wave functions for