Franck-Condon factor.4 We show herein that, indeed, K is only weakly dependent ..... rations used in this study, and to William H. Miller for help- ful discussions.
Path-integral calculation electron transfer
of the tunnel splitting in aqueous ferrous-ferric
Massimo Marchi and David Chandler Department of Chemistry, University of California, Berkeley, California 94720
(Received 26 November 1990;accepted27 March 1991) We examine path-integral methods for computing electronic coupling matrix elementsrelevant to long-rangedelectron transfer. Formulas are derived that generalizethose already found in the literature. These extensionsallow for efficient computation, especiallyfor complex systems where there is either no inherent symmetry, or that symmetry is difficult to ascertain a priori. The usefulnessof the computational methods basedon these formulas is demonstratedby carrying out calculations for the ferrous-ferric exchange.
I. INTRODUCTION
The aqueous ferrous-ferric system is a prototype for electron-transfer reactions, a class of reactions which is of fundamental importance in chemistry and biology. In this paper we extend earlier computer modeling of this system’-3 and consider explicitly the nature of electron paths contributing to the rate of exchange.At fixed interionic separation, the rate constant k, is well describedby the golden rule formula k = +-K’(FC),
(1)
where 2K is twice the electronic matrix element or tunnel splitting, treated as a constant, and (FC) denotes the Franck-Condon factor.4 We show herein that, indeed, K is only weakly dependentupon solvent configuration. But that is not our main purposein this work. Rather, we are mostly concerned with methodology-the demonstration that path-integral calculations can provide a quantitative estimate of K in eIectron-transfer systems. Ceperley and JacucciS(a) demonstrated the feasibility and utility of the path-integral method for the case of exchange in solid helium. Alexandrou and Negele5’b’further demonstratedits feasibility studying a model of fission. Kuki and Wolynes5(c) describedits qualitative use in a biological electron-transfer system. To make the analysis of electrontransfer quantitative, however, further development seems required, and that is the subject of this paper. We derive two different expression for K in terms of imaginary-time Feynman path integrals. One is essentially the sameas that of Alexandrou and Negele,5’b’which slightly generalizesthe results of Cepereleyand Jacucci.5(a)We also derive a secondexpression,which we believe is new. In either case, evaluation is facilitated by the discretization of the path integrals.6It is then possibleto perform the required path integrals by Monte Carlo7 (MC) or molecular dynamics (MD>.8 The application of free-energy methods’with thesesimulation techniquesis also pertinent as discussedin Sets. II and III. Within statistical errors, the methods we discuss are exact. To demonstratethe approach, we have computed K for two Fe2.’ -Fe’ + distances.Our calculation was performed on configurations of hydrated ions involving two equivalent Fe3+ ions and severalhundred water molecules.Each conJ. Chem. Phys. 95 (2), 15 July 1991
0021-9606/91/
figuration used was representativeof the reaction transition state where the electronic charge is equally distributed on both ions. Along with computing K, we have identified the most probable tunneling paths betweenthe redox centers. This article is arrangedas follows. In Sets. II and III, we derive a path-integral expressionfor the tunnel splitting in a double-well potential and outline a convenient statisticalmechanicsfree-energytechnique to carry out its calculation. Details of the simulations of the ferrous-ferric exchangereaction are given in Sec.IV. In Sec.V we presentour calculations of the tunnel splitting for two constrained interionic distances.The paper ends with a brief discussionin Sec.VI. II. TUNNELING AND PATH INTEGRALS
Let us assumean electron with position r experiencesa static and symmetric or nearly symmetric double-well potential, Y(r). In the present context, the two wells in I’(r) are located at the redox centersA and B in an electron-transfer system. If uncoupled, the stable ground states in the A and B regionsof V(r) are degenerateor nearly degenerate.A weak coupling between the wells will produce resonant behavior. The resonanceenergy is the coupling K. Since weak coupling between IA > and IB ) corresponds to a high-energybarrier in V(r), the Euclidean time spent in the barrier region during a transition from the left to the right well is very short. This rapid processor instanton corresponds to a kink in the quantum path of the quanta1particle.5*6The imaginary time associatedwith the kink is the instanton time. We usethe notation of Ref. 5 (a), and denote this time by flP+i.To say it is a “small” time is to say P,fi
