Introduction to Molecular Energy Transfer James T. Yardley Corporate Research Center Allied Chemical Corporation M orris town, New Jersey 1980 ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers New York London Toronto Sydney San Francisco COPYRIGHT © 1980, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 7DX Library of Congress Cataloging in Publication Data Yardley, James Τ Introduction to molecular energy transfer. Includes bibliographical references and index. 1. Molecular dynamics. 2. Chemical reaction, Conditions and laws of. 3. Energy transfer. I. Title. QD461.Y36 541.2'2 80-10898 ISBN 0-12-768550-2 PRINTED IN THE UNITED STATES OF AMERICA 80 81 82 83 9 8 7 6 5 4 3 2 1 Preface The past decade has seen great advances in our understanding of energy transfer phenomena. These advances are to a considerable extent the result of the development of spectroscopic "state-to-state" methods for examining energy flow, particularly those which give explicitly the time behavior of specified quantum states. The advances also result from a growing need for energy transfer information in many diverse fields such as biochemistry, applied physics, electrical engineering, laser physics, astronomy, and aero­ nautical engineering. This book is intended to provide an elementary introduction to the phenomena of molecular energy transfer and relaxation involving vibrational, electronic, and rotational degrees of freedom. It is unique in that it attempts to utilize a simplified unified theoretical approach for exploration of a wide variety of processes. A number of modern spectroscopic experimental methods have been described in detail. This work should be useful not only to graduate students of chemistry and physics who are concerned with energy transfer phenomena, but also to research workers in related fields who need an introduction to this exciting area of research. Included are many equations that should be of use for practical estimates or for the development of simple models. I have tried to select modern examples in an effort to provide some feeling for the current state-of-the-art. I would hope that many of the general principles enumerated will remain valid for some time, although our experience in this field is still rapidly expanding. Chapter 1 simply provides some background information in time- dependent quantum mechanics, simple collision theory, and interaction of IX χ PREFACE radiation with matter. I have used cgs units consistently in the book because most workers generally find them more convenient. Chapters 2-6 are concerned with vibrational energy transfer. Chapter 2 reviews traditional descriptions of vibrational motion in molecules, using carbon dioxide and propynal for explicit examples. Chapter 3 describes several important experimental methods which have been used extensively to explore vibra­ tional energy transfer. Many of the methods discussed may be also applied in studies of other forms of energy transfer. For example, Section 3.4 discusses various forms of Stern-Volmer behavior, extensively applied to electronic energy transfer as well as vibrational transfer. In Chapter 4 vibration-to- translation (V-T) energy transfer is discussed. First some experimental data for relaxation of hydrogen and carbon monoxide are presented. Then several simple approaches to understanding V-T processes are explored, including the theory of Schwartz, Slawsky, and Herzfeld (SSH theory). These theories are discussed with respect to the data on hydrogen and carbon monoxide as well as other selected experimental data. In Chapter 5 vibration-to-vibration (V-V) energy transfer is examined with respect to simple (SSH) theories. Energy transfer resulting from long-range dipole-dipole interactions is examined theoretically and experimentally. Highly excited systems are discussed, including a derivation of the Treanor-Teare distribution for the steady state. Finally, some principles governing V-V relaxation in poly­ atomic molecules are presented, with specific application to CH3F and to CH . Chapter 6 covers vibrational relaxation phenomena in condensed 4 media, including liquids and cryogenic matrices. Chapters 7-9 deal with electronic energy transfer. Chapter 7 reviews the chemist's simple pictures for electronic excitation. In Chapter 8 electronic transfer resulting from intermolecular interaction is examined, including curve crossing and resonant phenomena. Electronic-to-vibrational (E-V) energy transfer is discussed. Electronic transfer is also examined, including a detailed discussion of Forster transfer in liquids. Chapter 9 gives experi­ mental evidence for intramolecular electronic energy transfer. It provides some simple physical models and then attempts an exposition of general principles governing radiationless electronic transitions in polyatomic mole­ cules. Finally, Chapter 10 discusses rotational energy transfer, including brief reviews of rotational energy levels, experiments, and simple theories. Acknowledgments I am deeply indebted to all of my professional colleagues for providing insight and intuition. To Professor C. Bradley Moore I express particular thanks for his personal and professional encouragement, for his extensive physical insight, and for many well-executed and clearly written accounts of research in molecular energy transfer. I have benefitted greatly from interactions with Dr. Don Heller, Professor M. J. Berry, Professor W. H. Flygare, and Professor R. A. Marcus. I also acknowledge the important contributions of my graduate students at the University of Illinois as well as those students who participated in my special topics course on molecular energy transfer, and the many other students with whom I have had interaction. I thank the University of Illinois for providing a stimulating setting in which the initial phases of this work were carried out. I would particularly thank Professor W. M. Gelbart, Professor C. Bradley Moore, Professor George Flynn, Dr. Louis Brus, and Professor Herschel Rabitz for reading and criticizing certain parts of this manuscript. I would not have attempted this work without the skill and cooperation of Dale Myers who often skillfully typed course notes on very short notice, that of Gayle Wise who cheerfully took on this enormous typing task in addition to her other extensive duties, and that of Dot Putkowski who pitched in enthusiastically when additional help was needed. I thank Allied Chemical Corporation for providing time and facilities for the completion of this work. Finally, I express my gratitude to Serena Yardley who sacrificed much during the developmental years for the work presented here and to Anne Yardley for her patience, encouragement, and understanding during the completion of this work. xi CHAPTER ι Foundations for Molecular Energy Transfer 1.1 POSTULATES OF QUANTUM MECHANICS The quantum nature of the internal states of molecules makes an under­ standing of quantum mechanics essential for most discussions of molecular energy transfer. This is particularly true for experiments in which molecular energy transfer is examined spectroscopically. It will therefore be beneficial to review briefly a set of postulates from which most quantum-mechanical observations may be rationalized. POSTULATE I. Dynamical variables from classical mechanics may be replaced with linear operators. An operator transforms one function into another and will be indicated by writing / = Hg where H is the operator, g is the function being operated upon, and / is the resulting function. A linear operator obeys the laws (1) H(/ + g) = Hf + Hg and (2) (H + J)f = Hf + Jf. Note that HJ Φ JH. If the dynamical variable describing a single particle is expressed in cartesian coordinates and is of the form \_f(x, y,z) + g(Px>P,Pz)]> where x, y, and ζ are the coordinates and p, p, and p are the y x y z corresponding momenta, then Table 1-1 gives the rules for replacement. Note that if the dynamical variable has the form xp, application of the x above rules would be ambiguous since classically xp = px, but the quan­ x x tum operator deduced from Table 1-1 for xp differs from the operator px. x x ι 2 1 FOUNDATIONS FOR MOLECULAR ENERGY TRANSFER TABLE 1-1 Correspondence Rules for Classical Mechanical Dynamical Variables with Quantum-Mechanical Operators Quantity Dynamical variable Linear operator Cartesian coordinates χ, y, ^ x, y, ζ Time t t Linear momentum -ihd/dx, —ihd/dy, - ihd/dz Px, Py, Pz Energy Ε ihd/dt In this case the classical variable must be symmetrized to the form\{xp + px)1 x x before the application of the above rules. Rules for other situations exist but will not be of use here. Note that the linear operators are complex and may operate on complex functions. POSTULATE II. Measurable physical quantities are represented by her- mitian operators and are called observables. A single measurement of the observable represented by operator R may yield only an eigenvalue of the operator R. Hermitian operators are those for which fafiRcfrj) dz = \(Κφυ*φ]٢ζ where </> and φ are continuous square-integrable orthonormal f ] functions over the space upon which R operates, and dz is an appropriate volume element. A square integrable function φ is one for which the integral §ψ*ψ٢ζ exists. Orthonormal functions are those for which §φfφdz = τ j ij7 where o = 0, i φ j, and = 1, / = j. {j The eigenvalues and eigenfunctions of an operator R are a full set of solutions to the eigenvalue problem R<l>j= Rrfj, (1.1-1) where Rj is some constant and where φ are continuous single valued, square ] integrable, and orthonormal. An important theorem states that the eigen­ functions of a hermitian operator form a complete set; i.e., any continuous square-integrable function / may be expressed as / = ^· αφ where / and } φ are related to the space in which R operates and satisfy the same boundary ] conditions. POSTULATE III. Any quantum-mechanical system may be described by a (complex) state function Φ, which is single valued, continuous, and square integrable with norm unity over the appropriate coordinates; i.e., |φ*Φί/τ=1. (1.1-2) 1 A. Messiah, ''Quantum Mechanics," Vol. I p. 70. Wiley, New York, 1964. 1.2 QUANTUM MECHANICS OF CONSERVATIVE SYSTEMS 3 The state function is normally a function of time and of the coordinates necessary to describe the system. Note that the state function may be an eigenfunction of some particular operator, but it does not have to be. POSTULATE IV. If a quantum-mechanical system is described by Φ, then the average value of a sequence of measurements on an observable S is given by <S> = jV(SO)dT. (1.1-3) As a consequence of this last postulate it m2ay be concluded that for a system described by state function Φ, Φ*Φ = |Φ| represents the probability per unit "volume" (in the appropriate space) of finding the system at some "point" in the space. Spectroscopic experiments usually examine energies of systems, thus the hamiltonian operator Η is of particular importance. The postulates state that any (discrete) measurement of the energy of a system has to yield one of the eigenvalues E of the eigenvalue problem HI/ZJ = Ejtyj. The state function j which describes a system, however, may or may not be an eigenfunction of H. If it is not, it still may be written as a linear combination of hamiltonian eigenfunctions Φ = Y^jCijXJjj. This is a consequence of the theorem that the eigenfunctions of a hermitian operator form a complete set of functions in the appropriate space. In this case, the average value for many measurements of the energy would be 1 Μ <£> = f Φ*ΗΦΛ = Σ «7«* UjH^dx = Σ \4%· ( ) k 2 jk j The quantity |oy| then gives the weighting coefficient for eigenvalue Ej and may thus be interpreted as the probability for obtaining eigenvalue Ej or the probability amplitude for state j. It may also be noted that if Φ is given, the coefficient a may be determined from cij = §ψ*Φ٢τ. } 1.2 QUANTUM MECHANICS OF CONSERVATIVE SYSTEMS For isolated systems, energy must be conserved. In this case the classical hamiltonian can have no explicit time dependence and can be written strictly in terms of coordinates and momenta. We may thus write Η = H(p, q), where ρ and q refer to momenta and coordinates, respectively. Although energy transfer in some sense cannot occur in an isolated system it still is instructive to examine the quantum mechanics for such a situation. If we measure the 4 1 FOUNDATIONS FOR MOLECULAR ENERGY TRANSFER energy of such a system, we must obtain one of the eigenvalues Η ψ ί ( ^ . « ) / 9 · ) = (1-2-1) In the above we consider the possibility that the eigenfunction may be time dependent even if the energy is not. Noting from the postulates that Η = ihd/dt, it also must be true that Η ψ9 0 h= i f) 2) ( L 2 ( έ' ) ^ ' Jt - " Equations (1.2-1) and (1.2-2) are coupled partial differential equations for Ψj(q,t), which we may try to solve by separation of variables by setting Ψ/<7,ί) = i/jj(q)Uj(t). Substituting into Eqs. (1.2-1) and (1.2-2), we find ift-w,(t) = EjUj(t), (1.2-3) ΗφΜ) = Εβ/,Μ). (1.2-4) Equation (1.2-4) presumably may be solved directly to give the usual time- independent eigenvalues and eigenfunctions. The solution for Eq. (1.2-3) gives Uj(t) = Wj(io)exp[ — iEj(t — t)/h~]. It is seen that the time dependence of 0 Ψ, is Ψ,-Οζ,ί) = *A/(<z)exp[ — iEj(t — t)/h~\. The time dependence of Ψ · exists 0 7 only in a phase factor which oscillates at angular frequency Ej/h. We shall define coy = Ej/h. This is the angular frequency associated with state j. Then Ψ; = φεχρ[-ίω(ί - ί )]· ] ] 0 We might consider the situation in which the hypothetical system is described by state function Φ = Ψ/^, t). The expectation value of the energy is of course time independent as it should be: <£> = |Φ*ΗΦί/τ = Ej. Furthermore the expectation value of any other time-independent quantity is time independent, i.e., <K> = $Φ*(ΚΦ)٢τ = |Ψ?(ΚΨ ) ٢τ which is indepen­ 7 dent of time. Also the probability per unit volume of finding the system at any point in the appropriate "space" is time independent since Ψ*Ψ; = Φ*Φ y Thus, when the state function for a system is an eigenfunction of the molec­ ular hamiltonian H, the system is said to be in a stationary α state. On the other hand, the state function might be Φ = Σ] /^ j- The expecta­ tion value of the energy is still time independent, as it should be a2 J J <£> = Γφ*ΗΦ</τ - X afa f Ψ*ΗΨ*٣/τ = Σ \k\E. k k jk k However, other properties are not necessarily independent of time. As a trivial example, 2suppose Φ = αΨ + αΨ where α,α,φ, and φ are real. ι 1 2 2 ι 2 1 2 In this case, |Φ| = α\φ\ + α\φ\ + 2α α *ΑιΦ cos[co i(i ~ ίο)] where ω = 12 2 2 2ί 1.3 QUANTUM MECHANICS OF NONCONSERVATIVE SYSTEMS 5 ω2 — ω and is the transition frequency between states 2 and 1. It is seen 1 that the probability per unit volume of finding the system at any point oscillates at the transition frequency ω . To be more specific, consider a 1χ particle of mass m confined to free motion between χ = 0 and χ =12 /L. This is just the particle-in-a-box problem. The eigen2fu2n2ction2s are (2/L) sin(mrx/L) for η = 1, 2, 3,. .. and the eigenvalues areι /2n nh/2mL. Con1siβder, for exam­ ple, a state function Φ defined by Φ = 2~ φ exp^'c^i) Η- 2~φ exp(ia)i), ί 2 2 then Fig. 1-1 shows the probability per centimeter of finding the particle at different parts of the box at different times. It is seen indeed that at different times the particle may be found in a particular part of the box with varying probability. N2o2te th2at in a measurement of energy, the probability of getting eigenvalue nh/2mL i2s 2inde2pendent of time and equal to 02.52 whi2le the probability of getting 4n h/2mL is 0.5. This gives <£> = 2.5n h/2mL. In cases where the state function Φ for a conservative system is not an eigen­ function of the time-independent hamiltonian, the system is in a nonstationary state. 1.3 QUANTUM MECHANICS OF NONCONSERVATIVE SYSTEMS For studies in molecular energy transfer we shall often encounter situa­ tions in which the system under consideration is perturbed by some outside