J. Phys. Chem. 1988, 92, 3711-3725 3711 loss associated with the transition between states is offset by a For the Re( MQ+)-based complexes an additional complication gain in energy in the intramolecular vibrations (eq 3). to intramolecular electron transfer exists arising from a "flattening" On the other hand, if (AGe,,2 + A0,2) > (AG,,l + A,,,) (Figure of the relative orientations of the two rings of the MQ+ ligand 4B) and nuclear tunneling effects in the librational modes are when it accepts an However, the analysis presented unimportant, intramolecular electron transfer can not occur in here does provide a general explanation for the role of free energy a frozen environment. The situation is the same as in Figure 3A. change in environments where dipole motions are restricted. For Even direct Re(1) - MQ' excitation (process a in Figure 4B) a directed, intramolecular electron transfer such as reaction 1, must either lead to the ground state by emission or to the Re- the change in electronic distribution between states will, in general, (11)-bpy'--based MLCT state by reverse electron transfer: lead to a difference between Ao,l and A0,2. This difference creates a solvent dipole barrier to intramolecular electron transfer. It [ (bpy)Re"(CO) 3( MQ')] 2+* -+ (bpy'-)Re"( CO) 3(MQ')] 2'* follows from eq 4 that if light-induced electron transfer is to occur (5) in a frozen environment, the difference in solvent reorganizational The inversion in the sense of the electron transfer in the frozen energy, A& = bS2- A,,, must be compensated for by a favorable environment is induced energetically by the greater solvent re- free energy change, A(AG) = - AGes,l,as organizational energy for the lower excited state, A0,2 > Ao,l - Aces,~- AGes.2 > Ao-2 - &,I (AGes,2 - AGes,l)* or In a macroscopic sample, a distribution of solvent dipole en- vironments exists around the assembly of ground states. For some -A(AG) > A& of the high-energy distributions, vertical Re(1) - bpy excitation Presumably, this condition is met in the reaction center." There may lead to or beyond the intersection point of the Re(II)(bpy'-) the free energy change is favorable and the surrounding dipole and Re(II)(MQ') potential energy curves shown in Figure 4B. reorganizational energies are small. The low polarity of the For such cases, unusual excitation energy effects may exist for surrounding environment minimizes the magnitude of Ai,,. chromophores surrounded by solvent dipole distributions of suf- ficiently high energy. Such phenomena are difficult to observe Acknowledgment. Acknowledgement is made to the National by emission measurements in [(bpy)Re(CO),(MQ+)I2+. For this Science Foundation for financial support under Grant CHE- complex, the energy difference between the two excited states is 8503092. relatively small and the Re"-bpy'--based emission is far more intense than Re"-MQ'-based emission. However, excitation (15) (a) Swiatkowski, G.;Menzel, R.; Rapp, W. J. Lumin. 1987, 37, 183. energy effects are observed for [ (4,4'-(NH2)2-bpy)Re(C0)3- (b) Yamaguchi, S.; Yoshimizu, N.; Maeda, S. J. Phys. Chem. 1978,8t, 1078. (16) (a) Knyazhanski, M. I.; Feigelman, V. M.; Tymyanski, Y. R. J. (MQf)12'. For the amino derivative, a residual high-energy Lumin. 1987,37, 215. (b) Barker, D. J.; Cooney, R. P.; Summers, L. A. J. Re11-[4,4'-(NH2)2-bpy'-]-basedemission can be observed, but Raman Spectrosc. 1987,18,443. (c) Hester, R. E.; Suzuki, S. J. Phys. Chem. only at relatively high excitation energies (Figure 2C). The 1982, 86, 4626. excitation leading to Re11-[4,4'-(NH2)2-bpy'-]emission is shown (17) (a) Parson, W. W. In Photosynfhesis;Amesz, J., Ed.; Elsevier: Am- sterdam, 1987; Chapter 3, p 43. Won, y.; Friesner, R. A. Proc. Natl. Acad. as process c in Figure 4A. At lower excitation energies, the Sci. U.S.A. 1987, 84, 5511. (c) Creighton, S.; Hwang, J.-K.; Warshel, A.; high-energy emission disappears and only the broad, lower energy Parson, W. W.; Norris, J. Biochemistry 1988, 27, 774. (d) Gunner, M. R.; emission from the Re"-MQ'-based excited state is observed. Robertson, D. E.; Dutton, P. L. J. Phys. Chem. 1986, 90, 3787. FEATURE ARTICLE Classical and Modern Methods in Reaction Rate Theory Bruce J. Berne,* Michal Borkovec,t and John E. Straubt Department of Chemistry, Columbia University, New York, New York 10027 (Received: February 9, 1988) The calculation of chemical reaction rate constants is of importance to much of chemistry and biology. Here we present our current understanding of the physical principles determining reaction rate constants in gases and liquids. We outline useful theoretical methods and numerical techniques for single- and many-dimensional systems, both isolated and in solvent, for weak and strong collision models and discuss connections between different theories from a unified point of view. In addition, we try to indicate the most important areas for future work in theory and experiment. 1. Introduction influence the rate of chemical reactions is beginning to emerge. The influence of solvents on rate constants of chemical reactions This feature article presents our personal view of classical, as well has been studied for over a century and has recently received renewed attention.'-5 A clear physical picture of how solvents (1) Kramers, H. A. Physica 1940, 7, 284. (2) Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J. Phys. Chem. 1983, 87, Present address: Institut fur Physikalische Chemie, Universitat Basel, 2664. Klingelbergstr. 80, 4056 Basel, Switzerland. (3) Hynes, J. T. Annu. Rev. Phys. Chem. 1985, 36, 573. 'Present address: Department of Chemistry, Harvard University, Cam- (4) Hynes, J. T. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; bridge, MA 02138. CRC Press: Boca Raton, FL, 1985; p 171. 0022-3654/88/2092-3711$01.50/0 0 1988 American Chemical Society 3712 The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 Berne et al. 01 bl C) 4=0 PI0ePIt0 A0fC A +0C Figure 1. Schematic representation of the reaction coordinate potential for (a) isomerization, (b) recombination, and (c) atom-transfer reactions. The arrow denotes the dividing surface. as modern, theories of chemical reaction rate constants. It is useful to classify common chemical reactions into three types: isomerization reactions, dissociation-recombination re- actions, and atom-transfer reactions (see Figure 1). Isomerization reactions are unimolecular whereas atom-transfer reactions are bimolecular. In dissociation-recombination reactions, the dis- sociation step is unimolecular but the recombination step is bi- molecular. The potential energy surfaces, as a function of the reaction coordinate, have a characteristic structure for each re- action type. Isomerization reactions correspond to transitions between metastable wells in a double or multiple minima potential (Figure 1a). Dissociation-recombination reactions involve a transition between a metastable well and isolated species in a single minimum potential (Figure 1b). In atom-transfer reactions, the potential energy surface has a barrier as a function of the reaction coordinate but no metastable wells (Figure IC). We can distinguish reactants from products by asking whether they lie to the left or right of a dividing surface (arrows in Figure 1) known as the transition state.6 The precise position of this 10-2 10-1 1 10 102 surface is unimportant for the exact calculation of the rate constant Y provided the barrier separating the metastable species is high enough. Reactants of unimolecular reactions are bound, Le., Figure 2. Log-log plot of the transition-state theory normalized rate constant as a function of the static friction constant y typical of (a) an surrounded by potential walls, whereas reactants undergoing isomerization or dissociation-recombination reaction and (b) an atom- bimolecular reaction are not. The ratio of the rate constant for transfer reaction. the forward reaction to the rate constant for the backward reaction is related to the equilibrium constant by detailed balance. to products. The resulting rate constant, which is an upper bound To make this more explicit, let us mention a few prototypical on the true rate constant, depends on the properties of the reactant examples. Trans-gauche isomerization of butane represents an and on the solvent density only through the potential of mean force. isomerization Here the reaction coordinate is the Transition-state theory (TST) does not explain the behavior dihedral angle which moves on a periodic tristable potential curve of unimolecular reactions at low pressures; Lindemann proposed and the dividing surface is located at the trans-gauche barrier. a mechanism for the activation of the reactantI3 which describes In the iodine dissociation-recombination reaction the reaction coordinate is the bond lengthlo," which moves on a Morse-like A + M A* + M activation potential curve; the dividing surface is located at a distance of kb a few equilibrium bond lengths. Finally, a typical atom-transfer k reaction is a hydrogen exchange reaction.12 In the simplest case, A* - P barrier crossing the reaction coordinate is the difference in the bond lengths moving on a Porter-Karplus-like surface and the dividing surface is a unimolecular reaction proceeding in two steps. First, a reactant characterized by equal bond lengths. A is activated by a collision with a solvent molecule M; i.e., it Important classical theories for calculating rate constants can acquires enough energy to cross the barrier. Second, the activated be divided into three groups: transition-state theory: unimolecular reactant A* crosses the barrier region to become product P. rate theory in gases,13 and the theory of diffusion-controlled re- Assuming that [A*] reaches a steady state, the rate constant action~.~*'~Transition-state theory assumes that there is equi- for the formation of the product is librium between the reactant A and an activated transition state A* which decomposes with a characteristic vibrational frequency (5) Hanggi, P.