J. Phys. Chem. 1993,97, 13 107-1 3 1 1 6
13107
Adiabaticity Factor for Electron Transfer in the Multimode Case: An Energy Velocity Perspective Atsuo Kuki Department of Chemistry, Baker Laboratory, Cornell University, Ithaca, New York 14850 Received: July 1, 1993; In Final Form: October 15, 1993’
The multimode description of outer-sphere electron transfer in the Landau-Zener perspective leads to a picture of the electron-transfer event as occurring at the intersection of two N-dimensional potential energy surfaces. At this crossing seam, the behavior of the reacting system is controlled by the transmission coefficient. The usual one-dimensional Landau-Zener result with thermal equilibrium velocities can be extended to the multimode case. Dogonadze in particular has considered the Landau-Zener transmission coefficient expressions in this multimode case and derived the corresponding frequency factor in the nonadiabatic limit. The transmission coefficient for electron transfer coupled to an arbitrary number of classical molecular solvent modes is explored here in the case of general adiabaticity, and the significance of the resultant dimensionless adiabaticity parameter and its effective solvent frequency is clarified. A common feature of most electron-transfer theories is the assumption of a single outer-sphere mode. The Landau-Zener analysis, however, leads to a direct prescription for a single effective outer-sphere mode, and the nature of this reduction from the N underlying mechanical modes is ascertained. By pursuing an energy velocity perspective, the relation of the multimode Landau-Zener result to the standard single-mode result is used to illustrate general properties of dimensionless adiabaticity parameters. Modifications for the inclusion of quantum inner-sphere modes and the deep inverted region are also discussed.
I. Introduction One of the most intriguing quantum mechanical effects on chemical, photochemical, photophysical, and biophysical rate processes is nonadiabaticity, or the breakdown of the BornOppenheimer perspective, in which the coupled electron-nuclear dynamics no longer proceeds only on a single Born-Oppenheimer adiabatic potential energy surface.’-5 The full Born-Oppenheimer theory with its higher order expansion is completely quantum mechanical, and hence the off-diagonal nonadiabatic coupling can in principle be computed in rigorous quantum mechanical terms.6 Nevertheless, particularly in the case of large polyatomic or biomolecular problems, certain more approximate analyses are highly desirable and successful. In particular the LandauZener (LZ) perspective on the problem highlights the branching process which occurs when two zeroth-order diabatic curves The essence of this branching viewpoint is a classical, localized approximation for the nuclear coordinates, which allows the mixing to be described in terms of trajectories which pass through the crossingregion. As a result, the branching perspective succeeds in localizing the control of the interesting phenomena to within A of the crossing point, where A is the first (or higherl0Jl) order electronic coupling between the two zero-order electronic states. This advantage relative to a fully quantum treatment is largely retained in the extensions to the semiclassical treatment of the nuclear coordinates and momenta.12 When more than one nuclear coordinate is involved, the crossing point becomes a crossing seam between two diabatic potential energy surfaces. Particularly in this general multimode case, the fully quantum mechanical surface mixing becomes more and more cumbersome as it is necessarily more nonlocal in its dependence upon the full potential energy surfaces. While some relief can be provided by certain quantum path integral approaches to the nonadiabatic rate probleml3-15 which also employ trajectories over nuclear coordinates rather than delocalized energy eigenstates, we will pursue here an elaboration of what can be done strictly within the LZ framework. *Abstract published in Advance ACS Absrracrs, November 15, 1993.
The LZ expressions9J6J7are most valuable to the practicing kineticist in giving a rather accurate description of the turnover from the very small coupling regime to larger coupling regimes and in providing the direct formula for the electronictransmission coefficient (KEL, see below). In particular the validity of the commonly used one-dimensional LZ transmission coefficient formula when applied to outer-sphere modes will be examined here. When the effective outer sphere mode is explicitly formulated from the underlying multimode potential energy surfaces, formulas with unambiguous expressionsfor the effective one-mode parameters are found. This result turns out to be rigorous and quite simple and can be called the multimode LZ adiabaticity formula. This multimode approach to the adiabaticity transition is most relevant for the understanding the relation of a single “characteristic” solvation frequency to the actual set of many solvent oscillators which constitute the molecular solvent. Recent advances’*have provided new insight on how to reduce the timedependent statistical mechanics of molecular liquids to equivalent multimode oscillator representations. Thus it may become no longer routinely necessary to simplify solvent dynamics to a single effective mode, in which case the full multimode molecular solvation information can be included within the multimode adiabaticity parameter discussed here. In earlier work, Dogonadze obtained the nonadiabatic limit of the multimode LZ transmission coefficientleZ0 and expressed his result in terms of an effective frequency derived from a transition-state theory argument. The Dogonadze effective frequency is indeed a fundamental quantity (eq 30, below) when the surfaces are purely harmonic and when the dynamical assumptions of transition state theory are valid. We will explore and extend this investigation into multimode adiabaticity by developing an alternative energy velocity perspective. The most centrally important result of any LZ analysis can be expressed in terms of a dimensionless adiabaticity parameter, which we will call y . Even non-LZ approaches to adiabaticity share certain ubiquitous features of the LZ adiabaticity parameter,
0022-3654/93/2097-13 107$04.00/0 0 1993 American Chemical Society
Kuki
13108 The Journal of Physical Chemistry, Vol. 97, No. 50, 1993
namely the essential dimensionless form
Y = C,A2/~~,,f (1) Here we see that the adiabaticity parameter y is controlled by the ratio of the electronic coupling squared to an effective energy velocity u , ~ where , an energy velocity is defined as the rate of change of the energy gap, AV, between the two diabatic surfaces, d(AV)/dt. This dimensionless adiabaticity parameter controls the dependence of the rate upon A, by determining whether an adiabatic (y >> 1) or a nonadiabatic (y 0.25, direct numerical integration is easiest. Often, one will be satisfied with an interpolation formula such as eq 12a or eq 12b, since the slope in the nonadiabatic limit and the unity asymptote are accurate. The LZ form (eq 12a) and the rational function form (eq 12b), both evaluated with the exact y (eq AI l), are all superimposable at small y , then bracket the exact curve (eq A9) at large y with LZ going above and the rational functiondropping below. The calculated nonadiabatic limit is linear in 7 or quadratic in Az, as expected, though the second derivative with respect to J at the origin is undefined. (This property is the same for either the single modeor multimodethermally averaged (K L Z ) , Equation A1 gives worse behavior, returning instead an unphysical 7 In J as a leading term.) The leading linear term in eq A10 must now be multiplied by 2 to give thedouble-passresult in the nonadiabatic limit; this provides us with the correctly scaled adiabaticity parameter y =~GJ: KEL-Y
4n3I2A2 hgM [2k,T] ' I 2
nonadiabatic limit
(A1 1)
which when checked for consistency by eq 13 (section I11 and eq 3 1) is in perfect agreement with the exact quantum mechanical nonadiabatic limit.
Acknowledgment. This work is supported by the NSF Presidential Young Investigator program (CHEM-8958514). Valuable discussions with Gautam Basu, William Newman, and Ralph Young are also gratefully acknowledged. References and Notes (1) Marcus, R. A,; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (2) Hush, N. S.Electrochim. Acta 1968, 13, 1005. Hush, N. S. In Mechanistic Aspects of Inorganic Reactions; Rorabacher, D. B., Endicott, J. F., Eds.; American Chemical Society: Washington, D.C., 1982; p 301. (3) Nikitin, E. E. In Chemische Elementarprozesse;Hartmann, H., Ed.; Springer: New York, 1968; p 43. (4) Holstein, T. Philos. Mag. B 1978, 37, 49. (5) Frauenfelder, H.; Wolynes, P. G. Science 1985, 229, 337. (6) Born, M.; Huang, K. Dynamical Theory of Crystal hrrices; Oxford University Press: Oxford, 1985; pp 166, 402. (7) Landau, L. D. Phys. 2.Sowjet 1932, 2,46.
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