J . Phys. Chem. 1986, 90, 3466
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Fe(H20)62+-Fe(H20)63+). The thermodynamic factor is a function of the overall free-energy change AGO (or KI2) for the reaction. He used the relationship between AG*, AGO, and X to derive a particularly useful equation, now generally referred to as the "Marcus cross-relation". This equation has the form k L 2 = (kllk22KIz)1/2 when the equilibrium constant for the electrontransfer reactions (KI2)is not too large (k12is the rate constant for the cross reaction, and k l l and kz2are the self-exchange rate The advantage of the cross constants of the couples).30~41~48~53*7~~~10 relation is that, in contrast to the ab initio calculation of rates, which requires data on equilibrium bond lengths and force constants (as well as the work required to form the precursor and successor complexes), the only information required to calculate k12is the self-exchange rates of the two couples and the (workcorrected) equilibrium constant for the reaction. This has greatly facilitated the testing of various aspects of the model. The cross relation is widely used for interpreting electron-transfer rates in both homogeneous and heterogeneous (electrochemical) systems and is finding application in a variety of other reactions, including excited-state processes, atom and proton transfers, group transfers, hydride transfers, and gas-phase ion-molecule reactions. An important prediction of the Marcus theory is the existence of the inverted free-energy region (1AC"l > A), in which electron-transfer rates are predicted to decrease with increasing driving force.16,48,53,57,164,174 This prediction was originally greeted with considerable skepticism. Fortunately experimentalists were not
deterred and striking confirmation of this prediction has recently been reported. Another manifestation of the inverted region is the energy gap law of radiationless transition theory. In recent extensions163-'64,173,177,194 Marcus has considered the effect of nuclear tunneling corrections and has shown these to be small for many cases of interest. Other extensions include the effect of separation distance and the mutual orientation of the redox sites on electron-transfer rates, and applications to biological syst e m ~ . ~ The ~ redox ~ , centers ~ ~ ~in biological , ~ ~ ~systems ~ ~ are ~ ~ frequently far apart. As a consequence the electronic transmission factors are generally less than unity, Le., the reactions are electronically nonadiabatic, and the magnitudes of the transmission factors are sensitive to the distance and orientation of the two redox sites. A major strength of Marcus' contributions is that the theoretical expressions he derived are free of adjustable parameters and, because of their classical basis, they can be interpreted (and tested) in terms of physically measurable quantities. As the papers in this special issue show, Marcus' contributions to the electrontransfer area have not only stood the test of time but continue to be widely used today. They provide the framework for interpreting much ongoing research on electron-transfer reactions. Indeed, the Marcus theory of electron-transfer reactions is now so firmly established that significant departures from its predictions (at least in the normal free-energy region!) are taken as evidence of a more complicated mechanism.
Marcus' Contributions to the Semiclassical Theory of Collisions and Bound States J. N. L. Connor Department of Chemistry, University of Manchester, Manchester MI 3 9PL, England (Received: May 8, 1986) classical mechanics and, on the other hand, semiclassical quanThe research of R. A. Marcus combines a deep knowledge of tization by trajectories" and showed how to calculate vibrational quantum mechanics, classical mechanics, and statistical mechanics spectra semiclassically from classical t r a j e c t o r i e ~ . ~This ~ ~ , work '~~ with a powerful understanding of experimentally important has stimulated a large body of subsequent research by various problems. The result is that Marcus' theories have influenced workers on other and complementary methods for semiclassical almost every field of chemical kinetics and he has developed quantization, e.g., reviewed in ref 168. This bridge between theories in widespread use by other scientists. classical and quantum mechanics, and the now recognized quaMarcus first showed how action-angle variables, commonly used si-periodic to chaotic transition in classical mechanics, has spurred in various areas of physics, could be useful for collisional problems extensive current studies in quantum mechanics to find an involving rotational and vibrational energy transfer and for some analogous transition and for analyzing its effect on spectral and These variables processes involving chemical reactions.65~76~79~82~s3 chemical reaction rate properties related to intramolecular energy form a cornerstone for the semiclassical theory of inelastic and randomization. reactive collisions developed in 1970.89 For treatments of some chemical reactions, Marcus introduced In his development of semiclassical theory to bridge the gap into the literature coordinates which pass smoothly from those between classical and quantum calculations of collision dynamics, suited for reactants to those suited for p r o d u ~ t s ~ ~ ~ ~ ~ - n a t u r a l Marcus calculated the semiclassical wave function and from it, collision coordinates. These coordinates have been used by other S-matrices and collision cross sections.s9-91~93~9s-104 The method investigators to calculate accurate numerical solutions of the uses exact classical trajectory data to calculate the phase and Schriidinger equation for chemical reactions. These coordinates, amplitude of the postcollision wave function. Tunneling is allowed and the resulting Hamiltonian, have been extended by others to for by including complex valued trajectories, either analyticalpolyatomic systems ("reaction path Hamiltonian") and found or by direct c o m p ~ t a t i o n . ~ H ~ .e' introduced ~~ new ly89.90,93J04 additional use for treating the approximate dynamics of a class uniform approximations to treat the various types of wave functions of chemical reactions. that arise.91+98.'04This work has provided considerable insights Marcus' many other contributions to chemical kinetics and into the connection between quantum and classical mechanics and collisional processes include a curvilinear extension of transition into the legitimacy and limitations of using classical theories to state t h e ~ r y , " the ~ , ~theory ~ of relaxation times including those interpret modern molecular-beam and other experiments in in microwave transient experiments,'01,11"1'6a new tunneling path chemical dynamics. H2 H2 + H exchange reaction,'2s which has for the H In his semiclassical treatment of problems involving bound subsequently been used to obtain three-dimensional rate constants states, Marcus and his students developed methods for calculating for this reaction, a statistical-dynamical theory of reaction cross the phase integrals j p dq for systems with smoothly varying ~ e c t i o n s ,microcanonical ~~.~~ transition-state the placing nonseparable potentials, and they were able to implement, for the on a firmer basis of the theory of vibrationally adiabatic reacfirst time, Einstein's 1917 idea of finding eigenvalues for such t i o n ~(this ~ ~term, . ~ ~now ~ widely used, was originated by Marcuss6), systems by quantizing these phase integrals.107~113~124~144~'50~151~'68 the theory of symmetric'62 and asymmetric light-atom-transfer Marcus pointed out a connection, on the one hand, between the reactions,180~186~19~'g3 and the theory of heavy-atom b l o ~ k i n g . ~ ~ ~ , ' ~ ~ concepts of quasi-periodic behavior and the KAM theory in
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0022-3654/86/2090-3446$01.50/0
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0 1986 American Chemical Society
