3 Topics in Current Chemistry Fortschritte der Chemischen Forschung Van der Waals Systems Springer-Verlag Berlin Heidelberg New York 1980 This series presents critical reviews of the present position and future trends in modern chemical research. It is addressed to all research and industrial chemists who wish to keep abreast of ad- vances in their subject. As a rule, contributions are specially commissioned. The editors and publishers will, however, always be pleased to receive suggestions and supplementary information. Papers are accepted for "Topics in Current Chemistry" in English. ISBN 3-540-10058-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10058-X Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloging in Publication Data. Main entry under title: Van der Waals systems. (Topics in current chemistry; 93) Bibliography: p. Includes index. .1 Van der Waals forces -- Addresses, essays, lectures. 2. Molecular orbitals -- Addresses, essays, lectures. 3. Infra-red spectrometry -- Addresses, essays, lectures. I. Series. QD1. F58 vol. 93 [QD461] 540s [541.2'26] 80-19620 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting~ reproduction by photocopying machine or similar means, and storage in data banks. Under 54 § of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in GDR The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific state~e.rl~, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 2152/3020-543210 Contents Ab Initio Studies of the Interactions in Van der Waals Molecules Ad van der Avoird, Paul E. .S Wormer, Fred Mulder and R. M. Berns Van der Waals Systems: Molecular Orbitals, Physical Properties, Thermodynamics of Formation and Reactivity Pavel Hobza and Rudolf Zahradnik 53 Intermolecular Interactions and Anesthesia: Infrared Spectroscopic Studies Ginette Trudeau, Paul Dupuis, Camille Sandorfy, Jean-Max Dumas and Maurice Gu6rin 19 Author Index Volumes 50-93 721 Editorial Board: Prof. Dr. Michael J. S. Dewar Department of Chemistry, The University of Texas. Austin, TX 78712, USA Prof. Dr. Klaus Hafner Institut fiir Organische Chemie der TH PetersenstraBe 15, D-6100 Darmstadt Prof. Dr. Edgar Heilbronner Physikalisch-Chemisches Institut der Universit/it KlingelbergstraBe 80, CH-4000 Basel Prof. Dr. Sh6 It6 Department of Chemistry, Tohoku University, Sendal, Japan 980 Prof. Dr. ~ Jean-Marie Lehn Institut de Chimie, Universit6 de Strasbourg, ,1 rue Blaise Pascal, B. P. 296/R8, F-67008 Strasbourg-Cedex Prof. Dr. Kurt Niedenzu University of Kentucky, College of Arts and Sciences Department of Chemistry, Lexington, KY 40506, USA Prof. Dr. Charles W. Rees Hofmann Professor of Organic Chemistry, Department of Chemistry, Imperial College of Science and Techno- logy, South Kensington, London SW7 2AY, England Prof. Dr. Klaus Schiifer Institut ffir Physikalische Chemic der Universit/it Im Neuenheimer Feld 253, D-6900 Heidelberg 1 Prof. Dr. Georg Wittig Institut f(ir Organische Chemie der Universit~it Im Neuenheimer Feld 270, D-6900 Heidelberg 1 Managing Editor: Dr. Friedrich L. Boschke Springer-Verlag, Postfach 501 280, D-6900 Heidelberg 1 Springer-Verlag Postfach 501 280 - D-6900 Heidelberg 1 Telephone (06221) 487-1 • Telex 04-61723 Heidelberger Platz 3 - D-1000 Berlin 33 Telephone (030) 822001 • Telex 01-83319 Springer-Verlag 175, Fifth Avenue • New York, NY 10010 New York Inc. Telephone 4 77-82 00 Ab Initio Studies of the Interactions in Van der Waals Molecules Ad van der Avoird, Paul E. S. Wormer, Fred Mulder, and Rut M. Berns University of Nijmegen, Institute of Theoretical Chemistry, Toernooiveld, Nijmegen, Netherlands Table of Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Mechanisms of Van der Waals Interactions; Distance and Orientational Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Distance and orientational dependence ............... 4 2.2 Model potentials ........................ 6 2.3 Contributions to the interaction energy ............... 8 2.3.1 Electrostatic; long range multipole interactions, penetration effects 8 2.3.2 Induction, dispersion; muitipole interactions, penetration effects 10 2.3.3 Exchange . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Interactions from supermolecule calculations . . . . . . . . . . . . 13 2.5 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Quantitative Ab Initio Calculations . . . . . . . . . . . . . . . . . . . 16 3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.1 Molecular wave functions and propexties . . . . . . . . . . . 16 3.1.2 Isotropic long range interactions (second order) . . . . . . . . 17 3.1.3 Anisotropic long range interactions . . . . . . . . . . . . . . 20 3.1.4 Intermediate range interactions . . . . . . . . . . . . . . . . 20 3.2 Illustrative results . . . . . . . . . . . . . . . . . . . . . . . . 22 4 Structure of Van der Waais Molecules . . . . . . . . . . . . . . . . . 29 4.1 Analytical representation of the intermolecular potential; fitting of the ab initio results; atom-atom potentials . . . . . . . . . . . . . . . 30 4.1.1 (C2 H4) 2 . . . . . . . . . . . . . . . . . . " . . . . . . . . 30 4.1.2 (N2) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Potential surfaces of Van der Waals molecules; (N2) 2 and (C2H4) 2 . 37 5 Some Properties of Van der Waals Molecules . . . . . . . . . . . . . . 40 5.10rientational dependence . . . . . . . . . . . . . . . . . . . . . 40 5.2 Interaction dipole moments . . . . . . . . . . . . . . . . . . . . 41 5.3 Pair polarizabilities . . . . . . . . . . . . . . . . . . . . . . . 43 Ad van der Avoird et al. 6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Ab Initio Studies of the Interactions ni naV der Waals seluceloM 1 Introduction Van der Waals molecules are complexes of molecules (or atoms) which are not held together by chemical bonding, as "normal" molecules, but by weaker Van der Waals forces. One of the main reasons to study experimentally the structure and spectra of Van der Waals molecules ~1 is to extract (rather detailed) information about the Van der Waals interactions between the constituent molecules, information which can be used for a better understanding and description of the properties of molecular gases, liquids and solids. For small molecules, up to about ten light atoms, even more detailed information about these Van der Waals interactions 1 can presently be obtain- ed from ab initio calculations, i.e. directly from the approximate solution of Schr6dinger's equation by variational or perturbational methods. The accuracy of the results is still a matter of concern, since the interactions are very small relative to the molecular total or (internal) binding energies and calculational errors which are larger than the Van der Waals binding energies are easily introduced. The collaboration between theory and experiment is here very useful: the experiment can serve as a check on the accuracy of the calculations. On the other hand, theoretical results can help in the interpretation of the spectra. Thus, experiment and theory can both be improved and the combined experience for small molecules can lead to physically justified, empirically parametrized model potentials for molecules larger than those for which the ab initio calculations are feasible. Several books and review articles 2-1a~ are concerned with Van der Waals inter- actions. In the present survey, we shall first describe which are the interaction mechanisms that hold Van der Waals molecules together and we shall concentrate on the dependence of these interactions on the orientations of the constituent molecules (at longer and shorter distances, sect. 2). Then, we outline some ab initio methods enabling reasonably accurate quantitative calculations of these interactions and we discuss possible sources of errors (sect. 3), some of which can be serious. Next, in section 4, we look at some results of ab initio calculations and their bearing on the structure and dynamics of Van der Waals molecules, after first dealing with some problems occurring in the representation of the ab initio results by analytical model potentials. The final section 5 describes the effect of intermolecular interactions on some properties of Van der Waals molecules other than the energy, and the quantitative calculation of these properties. Specifically, we discuss the interaction dipole moment and the interaction induced change in the polarizability, which are of importance for the intensities in infrared absorption and (inelastic) light scattering (Raman spectra); these are connected with the inter- and intramolecular vibrations in Van der Waals molecules. We shall compare our (ab initio) results with experimental data, but we shall not deal in this paper with empirical or semiempirical determinations of Van der Waals interactions, since these are extensively described in the other surveys 1-13). In name the texts, many naV interactions der Waals ni reserved for forces range tong attractive the between (neutral) and, molecules often, one only includes the -6 leading R term ni the inter- action energy; ew esu the name ni a broader sense, meaning all the attractive and repulsive snoitcaretni non-bonding chemically between selucelom sect. (cf. .)2 Ad van der Avoird et al. 2 Mechanisms of Van der Waals Interactions; Distance and Orientationai Dependence The forces between the dosed shell molecules in their electronic ground states (which are the constituents of most Van der Waals molecules studied at present) are of Coulombic origin. By this we mean that they originate from the Coulomb operator, describing the interaction between the electrons and nuclei in the complex. Even for (light) open-shell molecules, such as NO or 02 , the interactions between the magnetic spin and orbital momenta are expected to be smaller by several orders of magnitude than the electrostatic forces .)41 Relativistic (retardation) effects can be neglected for the distances of interest in Van der Waals molecules .MT Therefore, the system of interacting molecules can be described by the time-independent non-relativistic Schr6dinger equation. Practically always, when no electronice xcitations or chemical reactions are considered, one can solve this Schr6dinger equation in the Born-Oppen- heimer approximation, i.e. one first obtains an effective potential for the nuclei by solving the equation for theel ectronic motion in the clamped nuclei approximation and then one calculates the nuclear (vibrational and rotational) states in this potential. Finally, in most work on intermolecular forces the rigid molecule approximation is made, i.e. it is assumed that the forces holding the nuclei together within one molecule are so much stronger than the intermolecular forces that the intramolecular and the intermolecular nuclear motions can be decoupled. Looking for example at ethylene (C2I-L) molecules (sect. 4), the frequencies of the internal vibrations range from 3100 cm -1 (C--H stretch) to 810 cm -~ (out of plane bending), while we expect the C2H4--C2H 4 vibrations in a Van der Waals molecule to lie below 051 cm -1. 2.1 Distance orientationai and dependence Consider two rigid molecules A and B, both of arbitrary shape. Let R = (R, 5) = (R, ®, ~) be the vector pointing from the center of mass of A to the center of mass of B. The coordinates of i~ are measured with respect to a space-fixed frame. Let the orientation of molecule A bdee scribed by the Euler angles A~-- = (~A' ')AY'A3[ which are the angles associated with an (active) rotation of the molecule from an initial position in which a reference frame fixed on A is parallel to the space-fixed frame, to its present position. Similarly, the orientation of B is determined by the Euler angles B~ = ,Bt¢( ,B~ 7B). The interaction energy between A and B is most generally described by the following expansion .)61,51 AEAB (~A, ~B, R) = ,,~ AEA(R) A^(~A , ~a, f~) (I a) A where A---(LA, KA, ,BL KB, L) is the combination of "quantum numbers" applicable to the system A--B of interest, se table .1 The angular functions are defined as: __)O(AA ,A ~)O ,B ~) ---~ ALA, ,BL ,AK ,BK ,A)O(L ,B)O-- )~___ (LA LB L) ,AALD )A)O(AK , ,BMD BL)B~(BK * C~(~). (lb) ,n..,,AM'--~" M~_ M A MB -- Ab Initio Studies of the Interactions in Van der Waals Molecules Table 1. Angles and quantum numbers specifying the orientational dependence of the interaction energy AE m, formula (1) A B angular coordinates" volume of quantum numbers b ,Am-( ~B, ~) angular A coordinate space a V general general aA, 'A/~ ,A~ %, ,S~I ,B~ O, O 256n 5 ,AL ,AK ,~L K ,S L molecule molecule linea: general %t, ,A~I %, ,~31 ,Bt2 O, ~ 128r: L ,A I m, K s, L(K = A 0) molecule molecule atom general %, ,n~I ,e/~ ®, ~ 3t123 Ls, s K (L = A A = K 0, molecule L = L )e linea# linea: ,^xc ,A31 %, ,B~ O, • 64~ 3 L ,A L ,B L (K A = S = K 0) molecule molecule atom linear" %, ,B~I O, * 16n 2 B L (L = A = A K s K = 0; molecule L = )BL atom atom O, • 4n - (LA = I%, = L~ = sK =L=0) One can choose a special coordinate system such that, for instance, ® .= * = % = 0. So one needs a maximum of 5 (internal) angles in order to fix the (relative) orientations of the molecules in a dimer AB. This reduces the volume V by a factor of 8a 2 (or 4n in the atom--atom case) b From the behaviour of the angular functions (1 b) under inversion of the total system, it follows that the summation (la) over the quantum numbers A can be restricted to even values of (L^ + B + L L). If the molecules A or B have finite symmetry groups, A can be further restricted. For instance, if they have a center of inversion only terms with even A L or e L contribute. IfA and B are identical molecules one can derive that: BL+AL .BLEA ,AL ,BK ,AK L = (--1) AELA, .AK ,BL ,BK • L Relevant information can be found also in refs) )5 and )73 For linear molecules the remaining Euler angles can be chosen such that they coincide with the polar angles: t0 A = ~A' A31 =- % 0A; ----- ~B' ~31 = 03 AL The functions {DMA, ,)A)O_(AK M A = -L A ..... L ,A K = A -L A ..... LA} constitute a (2L s + 1)-dimensional matrix )A__)o(ALD which represents the rotation 3o_ A of molecule A. The set of these matrices forms a (2L A + 1)-dimensional irreducible representation of the rotation group SO(3) 17). In the active rotation convention, which we are using, the rotation matrices are given by 17' ts): LA e-i~tAMA LA ' DMA ' = KA(f.OA) dMA, ) e -I~'AKA KA([~A (2) A L )A~(AK is where dMA" a Wigner d-function .~71 The rotation matrices of molecule B are defined analogously. The symbol MAMa stands for a 3-j coefficient and Ad van der et Avoird .la cL(~_) is a Racah spherical harmonic, in the phase of Condon and Shortley, which can also be written as a special (K = 0) rotation matrix~7): )~f_(EC = ( 4n ~1/2 YE(O, )b¢ = D ,E o(~, O, 0). )3( \2L + lJ This property (3) and the presence of the 3-j symbol in 1( b) makes the angular functions scalar, i.e. invariant under rotations of the total system (see Appendix). They span the complete space of scalar functions depending on ~A, bor- and ~_ due to the completeness of the rotation matrices in the Hilbert space L2[SO(3)] (Peter-Weyl theorem19)). The expansion, (1), is most convenient for molecular scattering calcula.- tions as well as for the solution of the nuclear motion problem in Van der Waals molecules (in terms of coupled translational and rotational vibrations of the rigid molecules), since it leads to a maximum separation of variables in the differential equations to be solved and it allows the power of angular momentum techniques to be employed. For linear molecules A and B, where the interaction energy does not depend on the Euler angles A7 and 7s, only terms with A = K B = K 0 contribute and one can use (3) to obtain a simplified expression for (1) )°2 (see also table 1): BAEA ,A0( ,A~ ,n0 ,aC )R -- ~ ,ALEA ,nL )R(L ,ALA ,31-1 ,A0(L 'AC 'B0 'B)~ O, )O (4a) L L ,, A B L with angular functions: ALA = 'BL 'A0(L ' ~A' 'B0 'B~t O' ~) "~'),AM ~ ) M M ( B AM AL BL, Mn BL × ,A0(AMC LA )A~ ,B0(BMC )B~ CE(O, O). (4b) If one of the molecules, say A, is an atom in an S-state only the terms with A = L 0 (L = )BL remain in (1) or (4). The same holds if we want to average over all orientations of one molecule, or, equivalently, put one molecule in the "unperturbed" rotational J = 0 state. When averaging over the orientations of both molecules, of course, only the isotropic contribution AE BA - -- AEo, ,o ,0 ,0 0(R) remains. isot ropic In practical calculations of the intermolecular interaction potential one often chooses a special coordinate system with the z-axis parallel to ,P and such that B~ = 0, which simplifies the angular functions 1( b) and (4b), while still retaining all the dynamical coefficients AEA(R ). This simplification is easily introduced remembering thatlT): CE(0, 0) = ,M6 o (Kronecker delta), independently of L. )5( 2.2 Model potentials ,)R(AEA The dynamical coefficients which are functions of the intermolecular distance only, fully determine the orientational dependence of the interaction potential. If one wishes to derive these functions from experimental data one has to replace them by