Critical evaluation of classical trajectory surface ... - ACS Publications

J . Phys. Chem. 1987,91, 459-466

459

Critical Evaluation of Classical Trajectory Surface-Hopping Methods As Applied to the H, System H,+

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J. R. Stine M-1 MS C920, Los Alamos National Laboratory, Los Alamos. New Mexico 87545

and J. T. Muckeman* Chemistry Department, Brookhaven National Laboratorty, Upton, New York 1 1 973 (Received: August 18, 1986)

We analyze the classical trajectory surface-hopping methods of Stine and Muckerman, Tully and Preston, and Blais and Truhlar to determine the appropriateness of each for treating nonadiabatic behavior in Hz++ H2 collisions. The emphasis is on testing the assumptions and evaluating the detailed prescriptions associated with each surface-hopping decision in the three methods.

I. Introduction The classical trajectory method has been successfully applied to numerous systems in which the motion of the nuclei is governed by a single adiabatic potential energy surface. The treatment of dynamical systems with close-lying electronic states is more difficult because the traditional concept of a single adiabatic potential energy surface breaks down. Tully and Preston’ were the first to propose a classical trajectory surface-hopping (CTSH) model, as a “phenomenological extension of the classical trajectory approach”, in which the motion of the system is governed by the forces derived from a single adiabatic surface until a region of strong nonadiabatic coupling is encountered. At some well-defined time during the transversal of this region, the system is allowed to “hop“ to another adiabatic surface, which alone governs the motion of the system until the next hop. The probability of hopping to another surface is determined by static and dynamic properties of the system. The appropriateness of using any CTSH method to model nonadiabatic processes is dependent upon the assumption that Occurrences of strong nonadiabatic coupling are localized and well separated in time along a trajectory.ls2 If this assumption is not valid for a given physical system, then no such method should be used to describe its dynamical behavior. Systems satisfying the localization assumption may be treated by any CTSH method that provides three essential features in a physically reasonable manner, Le., in general accord with those of an “exact” semiclassical solution.z These features are (1) a prescription for specifying the points in time at which transitions may occur, i.e., for determining the localized nonadiabatic regions; (2) a prescription for determining the probability of a transition at such a point; and (3) a prescription for specifying the “unique direction” along which the momentum of the dynamical system is adjusted to conserve energy after a transition. Clearly, CTSH methods may differ according to the various “prescriptions” adopted. More than one method can be valid so long as the differing prescriptions lead to essentially the same (correct) choices of where a transition can occur, its probability, and the unique direction. On the other hand, if two methods differ significantly in any of these choices, then at least one of the methods is inappropriate. Ultimately, the selection of a CTSH method must be guided by the extent to which the model conforms to the properties of the system being studied. Recently, Mead and Truhlar3 raised questions regarding the validity and generality of a CTSH method (the so-called SM method) developed by Stine and M ~ c k e r m a nand ~ , ~applied to ~

~

~

~

~

~~

~

(1) Tully, J. C.; Preston, R. K. J . Chem. Phys. 1971, 55, 562. (2) Miller, W. H.; George, T. F. J. Chem. Phys. 1972, 56, 5637. (3) Mead, C. A.; Truhlar, D. G. J . Chem. Phys. 1986,84, 1055.

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electronically nonadiabatic collision processes in the H2+ H2 system. We elaborate here on our response6to their criticism with a detailed examination of the validity of the prescriptions employed in the SM method as applied to the H4+system. In addition, we consider the methods of Tully and Preston’ (TP) and of Blais and Truhlar’ (BT). The H4+system provides a particularly good test case for the comparison and evaluation of CTSH methods because (1) it is typical of a large class of systems that exhibit nonadiabatic behavior associated with an asymptotic degeneracy (vide infra); (2) at least the lower electronic energy surfaces are described reasonably well by a diatomics-in-molecules (DIM) treatment, which allows the analytic evaluation of the gradient and nonadiabatic coupling parameters; (3) it is a system for which considerable experimental data e x i ~ t for * ~ comparison; ~ and (4) at large Hz+to Hz separations, it can be analyzed intuitively and analytically. In section 11, we describe the features of the three CTSH methods-particularly regarding the requisite three prescriptions mentioned above. In section I11 we apply these methods to isolated encounters by a trajectory of regions of strong nonadiabatic coupling in Hz+ Hz collisions and compare the results. We draw a number of conclusions from our study in section IV.

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11. Theory

We consider here the general features for the application of several CTSH methods to a system containing N nuclei and, therefore, m = 3N - 6 internal degrees of freedom. The original CTSH method of Tully and Preston,’ like the S M m e t h ~ d , was ~-~ described in the context of its application to systems that exhibit a particular type of asymptotic behavior. This behavior might be termed an “asymptotic interfragment ( m - 1)-dimensional excitation-exchange degeneracy” involving the electronic states of two fragments of the total system.’O In such a system the vertical deexcitation energy of an excited electronic state of one fragment becomes degenerate with the excitation energy of the (4) Stine, J. R.; Muckerman, J. T. J . Chem. Phys. 1976, 65, 3975. (5) Stine, J. R.; Muckerman, J. T. J . Chem. Phys. 1978, 68, 185. (6) Stine, J. R.; Muckerman, J. T. J . Chem. Phys. 1986, 84, 1056. (7) Blais, N. C.; Truhlar, D. G. J . Chem. Phys. 1983, 79, 1334. (8) See, e.g.: (a) Krenos, J. R.; Lehman, K. K.; Tully, J. C.; Hierl, P. M.; Smith, G. P. Chem. Phys. 1976,16, 109. (b) Koyano, I.; Tanaka, K. J. Chem .Phys. 1980, 72, 4858. (c) Anderson, S. L.; Houle, F. A.; Gerlich, D.; Lee, Y. T. J. Chem. Phys. 1981, 75, 2153. (9) Shao, J. D.; Ng, C . Y . Chem. Phys. Lett. 1985, 118, 481.

(10) Some explanation of the term “(m- 1)-dimensional” in this context is perhaps necessary. Here we consider systems where for sufficiently large interfragment separation there exists an infinite ( m- 1)-dimensionalsubspace of the m-dimensional coordinate space of the total system in which the two electronic potential energy surfaces are degenerate.

0022-3654/87/2091-0459$01.50/0 0 1987 American Chemical Society

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The Journal of Physical Chemistry, Vol. 91, No. 2, 1987

other on an (m - 1)-dimensional surface in an asymptotic (Le., “infinitely” separated fragments) region. Consider, e.g., the three-atom system A* BC. If at infinite A to BC separation BC* vertical excitation there is some RBcat which the BC energy exactly corresponds to the A A* excitation energy, then the fragment states A* + BC and A BC*, corresponding to interfragment excitation exchange, are degenerate on the twodimensional surface defined by RBc = constant. The degeneracy becomes an avoided intersection as the isolated fragments are brought together. This type of behavior is quite common in ion-molecule reactions (e.g., H+ + H2 and H2+ + H2) where electronically adiabatic motion in the near-asymptotic region would result in (unphysical) instantaneous charge transfer between widely separated species. Such systems therefore exhibit most of their nonadiabatic behavior in the near-asymptotic region, where the potential energy depends very weakly on the interfragment, but strongly on the intrafragment, degrees of freedom. The T P method requires that a “seam” in m-dimensional internal-coordinate space be defined prior to the dynamical calculation. TP accomplished this by demonstrating for their prototypical system that the locus of the “ridge” of large magnitude of the nonadiabatic coupling element in coordinate space was well represented by an analytic expression derived from the asymptotic (m - 1)-dimensional excitation-exchange degeneracy of the system. In the calculation of surface-hopping trajectories in the TP method, the hops are restricted to those times at which a trajectory passes through the (m - 1)-dimensional seam. Usually the LandauZener (LZ) approximation for the probability of a hop is employed;” the only component of the velocity upon which this depends is that perpendicular to the seam a t such a point. It is also along the direction perpendicular to the seam that the momentum is adjusted to conserve energy after a hop; i.e., the unique direction is normal to the seam. An otherwise allowed hop is rejected if there is insufficient momentum along the prescribed direction to accommodate energy conservation in an upward transition. Because the seam (and the direction normal to it) is defined solely in terms of the m internal degrees of freedom, the total angular momentum, the components of which are conjugate to the Euler angles specifying the orientation of the system relative to some inertial frame, is automatically conserved. The S M method is closely related to the TP method but extends it somewhat by removing both the requirement that the seam be represented by an analytic expression and that it be defined in advance of the dynamical calculation. This is accomplished by using the basic concept of Tully and Preston of an (m - 1)-dimensional seam to derive alternative criteria for when hops may occur and the unique direction associated with a transition. A feature of an (m- 1)-dimensional avoided intersection involving only two electronic states is that the difference (W) between the adiabatic potential energy surfaces ( W, and W,) behaves like a “trough” as a function of the internuclear distances; there is a unique direction normal to the surface of avoided intersection along which W exhibits a minimum, but Wis constant along the m 1 directions lying on that surface. The essential feature of the S M method is to locate this (m- 1)-dimensional surface of avoided intersection by examining W ( t )along a trajectory. Whenever a trajectory passes through such an avoided intersection, W(t) along that trajectory must exhibit a local minimum. Because of the possibility of a classical turning point near an (m- 1)-dimensional seam, not all local minima of W(t) must correspond to passes through the seam, but the cases of interest can be unambiguously identified by a vanishing gradient in coincidence with a zero time derivative of W. A question arises in the application of both the T P and S M methods? To what extent is the idealized (m - 1)-dimensional avoided intersection model [based on asymptotic behavior) valid as the fragments begin to interact? Clearly, since it is usually the case that the adiabatic splitting a t the seam increases as the fragments are brought together, Wcannot depend solely on in-

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(11) See, e.g.: (a) Chapman, S . ; Preston, R.K.J. Chem. Phys. 1974, 60, 650. (b) Chapman, S. J . Chem. Phys. 1985, 82, 4033.

Stine and Muckerman trafragment coordinates. On the other hand, if the true surface of avoided or actual intersection were of dimension m - 2 or lower, a trajectory would rarely (if ever) encounter it and a surfacehopping approach may be inappropriate. The S M method, by design, attempts to map the physical system onto the idealized model only when it is reasonable to do so, Le., when the qualitative features of the physical system are locally well represented by the “trough” model. This “trough” model is a multidimensional analogue of the Landau-Zener one-dimensional two-state model Le., in a region of strong nonadiabatic coupling, a suitable transformation exists such that Wcan be expressed locally as a function of only one coordinate.6 We repeat here only those elements of the S M method required for clarity in the present discussion. Let to be a time at which W(t)along a trajectory goes through some local minimum value, and let ro = r(to). Transform the coordinates r to a set of coordinates x that diagonalize the Hessian and j3 matrix at ro, Le., x = cT (r-ro) with j3c = ch, A,, = is the Hessian matrix with elements (8W/&, The expansion of W ( x ) through second order is then

A similar expansion for the idealized model would have all (dW/dx,), = 0 and only one nonzero A,, say A,, that is positive. For the SM method to be applicable, the real system must have properties similar to this model; i.e., we require that A, >> lAil for i I2 and that IVWo( be small compared to (A, Wo)’/2. The latter condition is the so-called “gradient test” associated with the SM method. The three parameters of the model (To, A,, and C1) are evaluated by calculating the corresponding local properties of the actual potential energy surfaces. When these conditions are satisfied, W(x)depends very weakly on all coordinates other than x,, and we consider the avoided crossing to occur on a surface of “pseudodimension” m - 1. We note that we use x1 as an approximation to the direction of the coupling vector, (+21V+l), which is important in the theory of nonadiabatic transitions. Here +1 and +2 are the wave functions corresponding to the lower and upper adiabatic potential energy surfaces, respectively. We prefer to use x1 instead of the coupling vector because it relates to the local topology of the interacting surfaces, it is directly related to the coupling vector in a region of strong nonadiabatic c o ~ p l i n gand , ~ it can be readily evaluated analytically for our systems. As in the TP method, the probability of a transition at to is determined by the LZ approximation along the direction normal to the surface of avoided intersection (Le., along nl), and the component of momentum along 2 , is adjusted to conserve energy after a transition. As the surface of avoided intersection and the unique direction are defined solely by the m internal degrees of freedom, total angular momentum is automatically conserved. An explicit condition for the applicability of the SM (and an implicit one for the TP method) is that there will never be more than one large eigenvalue of j3 in a region of strong nonadiabatic coupling; Le., the avoided intersection will remain of pseudodimension m - 1 from the near-asymptotic region into the interaction region far enough to describe the important nonadiabatic behavior of the system. Put another way, it is necessary that the strength of the nonadiabatic coupling decrease more rapidly with decreasing interfragment separation than the breakdown of the (m - 1)dimensional model for the avoided intersection. To monitor the strength of the nonadiabatic coupling, S M calculated the Massey parameter

along each surface-hopping trajectory. When this parameter has a (large) value indicating negligible probability for a transition at some point where dW/dt was zero, the surface-hopping decision (and all of the associated testing) can be bypassed and the trajectory continued on the “current” surface.

The Journal of Physical Chemistry, Vol. 91, No. 2, 1987 461

Classical Trajectory Surface-Hopping Methods It is of interest to compare certain features of the S M method with formal results derived by Miller and George (MG) in their “exact” semiclassical treatment of electronically nonadiabatic transitions2 The S M criterion of localization at a minimum of W ( t ) is precisely the criterion arising from a stationary-phase approximation to the “exact” solution. While no significant difference between the S M and TP criteria for localization would occur so long as the idealized (m - 1)-dimensional avoided intersection model is applicable, the S M criterion is appropriate even when that model no longer adequately describes the system because the MG result does not depend on such a model. Also, as MG point out, the dimensionality of an actual intersection of potential surfaces in complex coordinate space is m - 1. Stine and MuckermanI2 have previously demonstrated that the TP method and the “exact” semiclassical method of MG yield the same probability associated with a given hopping decision in the high-energy limit when pairs of surfaces interact locally and the splitting can be expressed as a function of only xl. The gradient and &eigenvalue spectrum tests in the S M method ensure that W(x) is primarily a function of only x1 so that the complex (m - 1)-dimensional surface of intersection is essentially normal to Im (xI). This being the case, the L Z expression for the transition probability is expected to be reliableI2except near threshold, where the high-energy limit assumption obviously breaks down. We have shown elsewhere that the M G approach can be used to derive an expression for the transition probability for the L Z model (i.e., hyperbolic form for W ( x l ) )without invoking the high-energy limit assumption.I3 This expression, in the notation of the present paper, is

P = exp[-2 Im (4)/h]

(3)

where Im (6) .. , = 1

3 105 + -7-2 + -7-4 +y“ + ... 32 3072 196608

with

= 2E/Wo and so is the diabatic velocity along R1. Two important features of this expression are that it is always smaller than the L Z probability and that it rapidly approaches the L Z result as E increases above threshold. Blais and Truhlar’ have presented a CTSH method that is quite different from those of TP and S M in several respects. The BT method is designed to treat a more general class of nonadiabatic systems-not only those well represented by the “troughlike” model discussed above. It also does not require that only two electronic states be strongly interacting at a given nuclear geometry. Like the S M method, BT do not require the nonadiabatic regions to be prescribed before the dynamical calculation is carried out. In the BT method, a semiclassical probability for being on each surface is calculated along with the classical trajectory on the “current” surface. Any time at which the probability for being on the “current” surface drops below (or to) 0.5, a new surface is randomly selected according to the instantaneous set of probabilities. The additional equations governing the probabilities are those derived by Nikitin,14 with the choice of “effective potential” being the “current” surface. For the twestate case (generalization to more states is straightforward) these are irl = azQ exp(-iS/h)

(4)

and ir2 = a l a exp(+iS/h)

where ~~

(12) Stine, J. R.; Muckerman, J. T. Chem. Phys. Lett. 1976, 44, 46.

(13) Stine, J. R.; Muckerman, J. T., to be submitted for publication. 0 4 ) Nikitin, E. E. In Chemische Elementarprozesse; Hartmann, H., Ed.; Springer-Verlag: West Berlin, 1968.

and

The quantity P,(t) = la,(t)I2is taken to indicate the probability for the trajectory being on potential surface i at time t . After an attempted hop, the probabilities are reset-to unity for the “new” surface, which can be the previous “current” surface, and to zero for all other surfaces. The unique direction in the BT method is taken to be along VW(to), where to is the time at which the hopping decision is made. While the localization of a transition is defined in a time-dependent manner, the choice of OW,as the unique direction allows a (discontinuous) effective potential for a transition to be defined solely in terms of the m internal degrees of freedom, so that total angular momentum is conserved.2 There are a few aspects of the BT method that warrant some discussion. The first is that BT arbitrarily chose the unique direction in their method to be along V W,. Whereas in the TP and S M methods this direction is defined naturally by the seam, its choice in the BT method, which contains no concept of a seam, appears unrelated to its prescriptions for the localization of and probability for a transition. In the case of pseudo-(m - 1)-dimensional avoided intersections, this vector changes both magnitude and direction rapidly as a trajectory passes through a region of strong coupling. A seemingly better choice would be the vector ($21V$l), which appears in eq 5 in exactly the appropriate context. The second aspect is that the choice of the “current” surface as the effective potential is arbitrary. Indeed, for the two-state case, the effective potential used in the estimation of the transition probability might better be taken as some average of the two adiabatic potential surfaces. This would, of course, considerably complicate the trajectory calculations because this trajectory would have to be calculated separately from that appropriate for the “current” surface. Also, prescribing the time at which a transition can occur as the time when the probability for the “current” surface falls to some specified value leaves the choice of that value arbitrary. This prescription differs from TP’s use of eq 4 and 5 to test the L Z expression for the probability of a transition upon the complete traversal by the trajectory through a nonadiabatic region.’ Both the quantum and semiclassical transition probabilities are defined only for such a complete traversal., Also, resetting the integrated probabilities in a region of strong coupling is both arbitrary and quite possibly erroneous. Finally, and also associated with resetting the probabilities after a transition, there is a possibility of multiple hops during a single pass through a region of strong nonadiabatic coupling. Such behavior, should it occur, would violate the “localized and well separated in time” criterion for the applicability of any CTSH method.

111. Results and Discussion In this section we apply the CTSH methods discussed above to an isolated encounter of a region of nonadiabatic coupling by each of four representative H2++ H, trajectories. We choose the initial state of the H2 molecule to be (n=O,j=O) and that for H2+ to be (n=l,j=O). The initial separation of the centers of mass of the two reactant molecules, R,, is taken to be 11.0, 8.0, 7.5,or 7.0 ao. The orientation of the molecular axes, the vibrational phases, and the impact parameter are randomly selected. The initial relative translational energy in each of the four trajectories is 0.1 eV. We distinguish the four identical nuclei by the labels A, B, C, and D so that the initial configuration of the system is AB+ + CD. By use of our previous convention for identifying the six atomic distances, the AB separation is given by rl and the C D separation by r,. At infinite separation of the reactant fragments, an interfragment excitation-exchange degeneracy occurs whenever rl = r,: for rl > r, the lowest adiabatic potential energy surface corresponds to AB+ CD, and for r, < r, the lowest surface corresponds to AB + CD’. In this asymptotic region, it is intuitively clear that the trajectory must “hop” from one adiabatic surface to the other (i.e., remain on the same

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462

The Journal of Physical Chemistry, Vol. 91, No. 2, 1987

Stine and Muckerman

03-< 1

b

a

25 -

uLr -6

7-

-7.0

-

20 -I

01

.v

I

>

3 ”

_ _

t

00

I

I

1

1

s a

15 -

10-

Lu C

-0 1

-.p

c

5-

c

a

-02

0 0

-0 3

6

7

8

9

10

11

12

13

-7.5 -

0-

I

R (a,)

Figure 1. Contour plot of the adiabatic splitting, W, as a function of the

intrafragment coordinate u and the interfragment separation, R. The contours are in increments of 0.1 eV. The bold line is a segment of the representative trajectory with R, = 11.0 ao, and the dot indicates the location of the first minimum of W ( t )along that trajectory. The dashed line is the rl = r6 ( u = 0) specification of the seam.

diabatic surface) as it passes through the five-dimensional hyperplane defined by r l - r6 = 0 to avoid transferring the charge to the other fragment. If either fragment is vibrationally excited, the system repeatedly passes through this hyperplane in the asymptotic region. When the two fragments are relatively far apart, a contour plot of W(rl,r6)for fixed (large and unequal) values of r2 through rs shows a “troughlike” behavior (see, e.g., Figure 4 in ref 3). Of course, only the dependence of W o n the two intrafragment degrees of freedom is shown in such a plot; a more revealing picture is obtained by including an interfragment coordinate. We define two new coordinates, u and u, by u = (rl

+ r6)/2

I

-101 -3

,

,

-2

-I rj

,

-

,

,

‘

0

I

2

3

rb

(0.1

Figure 2. Typical cross section of the eight adiabatic potential energy surfaces arising from a DIM treatment of the H,’ system in the region of strong nonadiabatic coupling (a). The lowest two surfaces are shown on an expanded scale in (b). -2:

rTq II . , . _

-50

:25 -50

I1

1

-75

-75

u = (rl - r6)/2

such that the u axis lies in the r I = r6 hyperplane and the u axis is perpendicular to it. Figure 1 shows a contour plot of W(u,R) where R is the distance between the centers of mass of the two diatomic fragments. It is apparent that this W surface is approximately a trough, with the deviation becoming more pronounced with decreasing R. In the T P method the definition of the seam would be prescribed from the asymptotic region, and in the present case would be given by u = 0. This is the simplest of the three CTSH methods to implement because one need only test for when the two diatomic distances become equal along the trajectory. Even though the T P definition is based on asymptotic conditions, Figure 1 clearly shows the “floor” of the perturbed trough is very well described by u = 0. Using this specification for the seam, we now examine a cut through the adiabatic potential energy surfaces perpendicular to it. Figure 2 shows such a cut for R = 8 a. from which it may be concluded that the condition of only two electronic states avoiding an intersection is met for this system provided the total energy is below that required to dissociate the system into its constituent atoms. We now focus on the properties relevant to the application of the TP, S M , and BT methods in the vicinity of the first pass of each of the four representative trajectories through the TP seam. These occur at Ro values of 10.47, 7.47, 6.97, and 6.46 ao. The first such occurrence is shown in Figure 1, where the bold line is a segment of the trajectory and the dot is the corresponding point along the trajectory where W ( t )exhibits a local minimum. Before proceeding with this analysis, it is instructive to examine the strength and localization of the nonadiabatic coupling along each trajectory in the vicinity of its first pass through the reference seam. Denoting to as the time at which the trajectory intersects the seam, we plot in Figure 3a,b the nonadiabatic coupling term, Q, as a function of t - to for all four cases. It is seen that at large intermolecular separation this coupling is extremely localized, Le., that the duration of the nonadiabatic coupling is small compared to the time between successive encounters of a nonadiabatic region.

I

1

0.0

--._

0.0

Figure 3. Selected scalar properties along each of the four representative trajectories in the vicinity of their first encounter of the TP seam as a function of time: (A) the nonadiabatic coupling term, Q ;(C) the adiabatic splitting, W; (E) the Massey parameter, w; (G) the probability of remaining on surface 1, P I . Panels B, D, F, and H are respectively the same as Panels A, C, E, and G except for an expanded time axis. The solid, dashed, dot-dash, and dot-dot-dash lines indicate the trajectories with R,values of 11.0,8.0,7.5, and 7.0 ao, respectively. The zero of time in each case is taken to be the time of intersection with the seam.

Typically, the duration of the nonadiabatic region is about 0.05 time units (see footnote to Table I for the precise definition of units) and the incidence of the next avoided intersection is about one vibrational period, or about one time unit. As the intermolecular distance decreases, the nonadiabatic region becomes less localized, but the splitting between the surfaces also increases and the width of the avoided intersection broadens (see Figure 3c,d). When the coupling is strong, the splittin& is small and the CTSH approach is justified. As the coupling becomes less localized, the splitting gets larger and the system tends to behave more and more adiabatically. These qualitative arguments are quantified by the Massey parameter, w , which has a value much less than unity only along those portions of a trajectory in which the tendency for nonadiabatic behavior is strong. Figure 3e,f shows the Massey parameter as a function of time in the region of the first intersection of each of the four trajectories with the u = 0 seam. The “decision points”

Classical Trajectory Surface-Hopping Methods

The Journal of Physical Chemistry, Vol. 91, No. 2, 1987 463

TABLE I: Scalar Properties of the H2++ H2System at the First Encounter of Four Representative Trajectories with the TP Seama$

Ro

w

IVul

10.47 7.47 6.97 6.46

0.0123 0.1681 0.2555 0.3857

5.97 (-3) 7.67 (-2) 1.14 (-1) 1.69 (-1)

dW/dt 6.08 (-3) 7.97 (-2) 1.21 (-1) 1.85 (-1)

la1I2

XI

PLZ

A2

0.507 852.6 4.7 (-4) 0.583 94.8 4.6 (-2) 0.624 62.7 6.7 (-2) 0.682 41.8 9.6 (-2)

0.9995 0.919 0.822 0.634

"The units employed in all calculations are based on the units of mass, length, and energy being defined as 1 amu, 1 ao, and 1 eV, respectively. Using this convention, the unit of time is 5.3873 fs and h is 0.122 179. bTrajectories were started with R, values of 11.0, 8.0, 7.5, and 7.0 a, and relative translational energy of 0.1 eV (see text).

TABLE 11: Scalar Properties of the H2++ H2System at the First Minimum of W ( t )along Each of the Four Representative Trajectories from Table I R,

w

11.0 8.0 7.5 7.0

0.0123 0.1681 0.2555 0.3855

low 7.81 9.73 1.44 2.10

(-3) (-2) (-1) (-1)

AI 852.62 94.84 62.73 41.86

XZ

4.3 4.6 6.7 9.6

(-3) (-2) (-2) (-2)

la1I2

PLZ

0.507 0.591 0.635 0.696

0.9995 0.919 0.822 0.635

vd

-3.0 -4.8 -1.1 -2.3

(-6) (-4) (-3) (-3)

in all three surface-hopping methods being discussed always occur within the time interval in which w C 1. In the S M method, nonadiabatic transitions can occur only at times where W(t)goes through a local minimum. Figure 3c,d shows that in each case the minimum of W(t)corresponds very closely to the TP-type prescription for the localization of transitions and occurs slightly before the trajectory intersects the TP seam. The BT method allows transitions to occur when the probability for being on the "current" surface drops below (or to) 0.5. Figure 3g,h shows the behavior of Pl(t-to), where the "current" surface is always taken to be W,, near the four encounters of a trajectory with the u = 0 hypersurface. It is readily apparent that as the intermolecular separation decreases, the BT prescription localizes transitions further and further from both the TP and S M decision points, and always after the trajectory passes through the u = 0 seam. The important features in the Massey parameter plot are (1) that the duration of Occurrences of strong nonadiabatic behavior (-0.1 time unit) is short compared to the time between such occurrences (- 1.O time unit), (2) that the duration of strong coupling gets longer as the interfragment distance becomes smaller, and (3) that

the minimum value of the Massey parameter becomes larger (corresponding to weaker nonadiabatic behavior) as the interfragment distance becomes smaller. Nonadiabatic behavior would not be important at interfragment separations much smaller than 6.46a. because the value of the Massey parameter at its minimum is increasing rapidly there (e.g., 0.06at 6.97 a. vs. 0.13 at 6.46 ao). The transition becomes energetically forbidden for R C 6.1 a0* Table I presents a number of scalar properties of the trajectories and potential energy surfaces at the first intersection of each trajectory and the u = 0 hyperplane (TP prescription). Because in applying the TP method one analyzes the characteristics of the system and verifies that the assumptions are reasonable before trajectories are actually calculated, the properties listed in Table I serve for comparison with those from other methods that do not require such a preanalysis. Tables I1 and I11 present similar information at the time when dW/dt = 0 and P1= 0.5 (SM and BT prescriptions), respectively. Table IV lists the components of the unique direction in r space associated with the S M and BT methods at each intersection of a trajectory with the TP seam, and Tables V and VI give the unique direction in the SM and BT methods at the S M and BT decision points, respectively. The last column in Tables IV-VI specifies the angle between the unique direction in one of the other two methods and that in the TP method. We will make frequent reference to these tables in the ensuing discussion. The basic requirements for the applicability of the SM method are that whenever W(t) exhibits a minimum (other than one caused by a turning point) in a region of strong nonadiabatic coupling, the expansion of W(r) must closely resemble a similar expansion for the idealized model which has all (8W/t3xi),= 0 and only one nonzero Xi, say XI, that is positive. As stated above, this requires that lVWol be small compared to (XI Wo)1/2and that XI be large compared to lXil for i L 2. The three parameters of the model (Wo, X1, and 2 , ) are evaluated by calculating the corresponding local properties of the actual potential energy surfaces. It is possible in the application of the S M method to compute all the elements of the Hessian matrix at ro by numerical differentiation of W(r) and to calculate its eigenvalue spectrum and eigenvectors. In our previous application of the method this was not done; rather, we assumed (by noting the shape of the contours in Figure 1) the Hessian

TABLE III: Scalar Properties of the H2++ H2System at the First Occurrence of P, = 0.5 along Each of the Four Representative Trajectories from Table I R, 11.0 8.0 7.5 7.0

W 0.0123 0.1711 0.2661 0.4260

IV ul 4.93 (-2) 6.81 (-1) 1.04 1.60

d W/dt 5.97 (-2) 9.07 (-1) 1.45 2.36

A1

A2

PLZ

vd

852.56 90.44 56.01 31.75

1.7 (-2) 4.6 (-2) 6.5 (-2) 9.1 (-2)

0.9995 0.915 0.799 0.528

2.6 (-5) 5.1 (-3) 1.2 (-2) 3.0 (-2)

TABLE I V Unique Directions in SM and BT Methods at the First Crossing of the TP Seam, i.e., Trajectories from Table I

Y

= 0, along the Four Representative angle from ?, deg

unique direction in r space

method

R, = 11.0 a. SM SM(approx) BT

0.707 1 0.7071 0.053

-6 (-7) -1 (-6) -0.506

7 (-8) -5 (-7) -0.748

SM SM(approx)

BT

0.707 1 0.7071 0.043

-1 (-4) -1 (-4) -0.496

-5 (-5) -5 (-5) -0.741

SM SM(approx) BT

0.7071 0.7071 0.042

-3 (-4) -3 (-4) -0.495

-1 (-4) -1 (-4) -0.733

SM SM(approx) BT

0.7071 0.7071 0.047

-5 (-4) -5 (-4) -0.494

-2 (-4) -2 (-4) -0.733

0.0 0.0

1 (-6) 9 (-7) -0.321

-0.7071 -0.707 1 -0.053

9 (-5) 9 (-5) -0.334

-0.7071 -0.7071 -0.043

0.0 86.5

8 (-5) 8 (-5)

2 (-4) 2 (-4) -0.337

-0.7071 -0.707 1 -0.041

0.0 0.0 86.6

2 (-4)

4 (-4) 4 (-4) -0.340

-0.707 1 -0.707 1 -0.053

0.0

5 (-7) 3 (-7) -0.275

85.7

R, = 8.0 a. 4 (-5) 3 (-5) -0.299

0.0

R, = 7.5 a. -0.305

R, = 7.0 a,, 2 (-4) -0.3 12

0.0 85.9

464

Stine and Muckerman

The Journal of Physical Chemistry, Vol. 91, No. 2, 1987

TABLE V Unique Directions in S M and BT Methods at the First Minimum of W ( t )along Each of the Four Representative Trajectories for Table I angle from unique direction in r space i., deg method

R. = 11.0 a,, SM BT

0.7071 -0.4573

-2 (-6) -0.3870

-1 (-6) -0.5723

SM BT

0.7076 -0.4368

-3 (-4) -0.3912

-3 (-4) -0.5835

SM

0.7083 -0.4295

-6 (-4) -0.3941

-6 (-4) -0.58 7 3

0.7096 -0.4214

-1 (-3) -0.3968

-1 (-3) -0.591 3

-8 1-8) -0.2 103

3 (-7) -0.2453

-0.7071 0.4574

0.0 49.7

-1 (-5) -0.2633

-0.7066 -0.4368

0.0 51.8

-2 (-5) -0.2681

-0.7059 0.4294

0.1 52.6

-4 (-5) -0.2741

-0.7046 0.4208

0.2 53.5

R, = 8.0 a. -6 (-5) -0.2358

R, = 7.5 a,, BT

-1 (-4) -0.243 1

R, = 7.0 a.

SM BT

-3 (-4) -0.25 13

TABLE VI: Unique Directions in the SM and BT Methods at the First Occurrence of P I = 0.5 along Each of the Four Representative Trajectories for Table I angle from unique direction in r space i , deg model

R. = 11.0 a. SM BT

0.7071 0.7019

8 (-6) -0.0613

1 (-5) -0.0906

0.7014 0.7000

2 (-3) -0.0554

2 (-3) -0.0824

0.6930 0.6965

0.0035 -0.0532

0.0056 -0.0790

0.6695 0.6871

0.0086 -0.0495

0.0133 -0.0730

5 (l6) -0.03 33

7 (-6) -0.0388

-0.7071 -0.7019

0.0 7.0

1 (-3) -0.0370

-0.7 128 -0.705 5

0.5 6.4

0.0028 -0.0356

-0.7209 -0.7096

1.2 6.1

0.0066 -0.0328

-0.7426 -0.7198

3.1 5.8

R, = 8.0 a. SM BT

1 (-3) -0.03 32

R, = 7.5 a. SM BT

0.0024 -0.0324

R, = 7.0 .. SM BT

matrix to have the properties it would have in the idealized ( m

- 1)-dimensional model for the avoided intersection. This allowed the use of a factorization formula to evaluate the approximate expressions for AI and x1

(7) where the label j denotes any intrafragment coordinate involved in the asymptotic degeneracy relation. Table I1 shows that the value of lVWl at the S M decision points is generally small, indicating that the gradient test in the S M method is easily satisfied. In all cases (X,Wo)1/2 4. The S M assumption of only one large eigenvalue of is also seen to be generally valid: at Ro = 10.47 a. the largest eigenvalue is over 6 orders of magnitude greater than the next largest, and a t Ro = 6.46 a. this ratio is still over 400. The distance, ud, from the TP seam is quite small (negative values indicate the SM decision point occurs before the trajectory reaches the seam) although it does increase with decreasing interfragment separation. The SM Landau-Zener transition probabilities are in excellent agreement with those at the TP seam. This is because the unique direction in the SM method is associated with the eigenvector corresponding to the largest eigenvalue of the Hessian matrix, and that vector changes very slowly in the vicinity of the seam, as is apparent from Tables IV-VI. Two entries appear in Table IV for the unique direction in the SM method. The first is the eigenvector corresponding to the largest eigenvalue of the exact Hessian matrix; the second, labeled “SM(approx)”, is the approximation used in our previous work and given by eq 7. There is no significant difference between the two entries at any of the intermolecular distances listed in Table IV, and in all cases both correspond quite closely to the TP direction. The S M unique direction at the SM decision points is also in excellent agreement with the TP prescription; the largest deviation is 0.2 deg at the smallest interfragment separation. Even when evaluated at the BT decision points, which occur after the trajectories encounter the seam, the largest deviation of the SM unique direction from that of TP is

a,

OZ059 -0.0 304

only 3.1 deg (see Table VI) and the S M Landau-Zener transition probabilities are in reasonable agreement with those at the appropriate decision points (see Table 111). The momentum correction is virtually identical in the T P and SM methods; Le., the intrafragment coordinate v drives the transition. In applying the BT method along the four trajectories, we encounter a problem with the implementation of the BT criterion for identifying a possible transition point. The numerical integration routine we use has the option of stopping a trajectory when any given function of the integration variables has a zero value, so a trajectory can be stopped precisely when lull2- 0.5 = 0. Doing so, however, would produce the unphysical result of electron transfer occurring at arbitrarily large intermolecular separation with a probability of 0.5. On the other hand, as is evident from Figure 3g,h, defining the possible transition point to be wherever the last (fixed or variable) time step in the integration that leaves Iu,I2 < 0.5 happens to fall would appear to be dependent on the step size, the integration method being used, and even the values of the tolerance parameters in the integrator. The step size employed by BT in their CTSH study of Na* + H, was 0.02 in the time units employed here. This is approximately the same time interval over which PIchanges from 0.5 to almost 0 at the largest of the four intermolecular separations under consideration. At this largest distance, the point at which PI = 0.5 is very close to the T P seam (see Table 111), and integrating up to one step past the seam would still produce a substantial probability for remaining on the same adiabatic surfabe. In this example, the vector V W changes rapidly as the trajectory passes through the T P seam. At the seam it is essentially orthogonal to the unique directions in the TP and S M methods (see Table IV). Recalling that lVWl is very small in the vicinity of the seam at large separations, the unique direction in the BT method using P I = 0.5 as the criterion for a decision point is in part determined by those components of the gradient that are considered negligible in the SM method. This would lead to some adjustment of the momentum vector in the direction of relative translational, rather

The Journal of Physical Chemistry, Vol. 91, No. 2, 1987 465

Classical Trajectory Surface-Hopping Methods

coupling, but this would undoubtedly occur at shorter distances than those considered here. Our results, therefore, do not address the possible ramifications of that aspect of the BT method.

TABLE VII: Comparison of Transition Probabilities along Each of the Four Representative Trajectories from Table I‘ 11.0 8.0 7.5 7.0

0.9995 0.919 0.822

0.9995 0.919 0.819

0.9996 0.937

0.635

0.620

0.727

IV. Conclusions

0.864

‘The LZ and eq 3 probabilities are evaluated at the SM decision points; the semiclassical probabilities correspond to integrating eq 4 and 5 completely through the region of strong coupling with V,, = W , (see text). than purely fragment vibrational, motion. If the trajectory is integrated past the time at which P , = 0.5, the direction of the gradient becomes closer to the unique directions of the TP and SM methods. Once again, however, the precise direction would be step size dependent. Of course, the further from the TP seam the BT decision point is, the greater the splitting between the two adiabatic surfaces and the more difficult it becomes to satisfy the energy conservation condition for an upward transition. At the smaller values of the intermolecular distance, the duration of the nonadiabatic region becomes longer than any reasonable value of the integration step size. In these cases, the nominal P , = 0.5 decision point occurs further from the TP seam so that V W lies somewhat closer to the TP unique direction. We note in passing that if ( Ic/21VIc/I)were used as the unique direction in the BT method, more consistent results would be obtained. In their method this vector must be calculated anyway and, in contrast to V W, is intimately involved in the equations involving nonadiabatic transitions (Le., eq 4 and 5). For example, along the trajectory beginning at R, = 7.0 ao, (Ji21VIc/1)deviates from the TP unique direction by 0.2, 0.2, and 1.7 deg at the SM, TP, and BT decision points, respectively. The estimation of transition probabilities warrants some discussion. All LZ probabilities quoted in Tables 1-111 are microscopically reversible because they were computed by using the velocity corresponding to the “symmetrized” potential [ Wl W 2 ] / 2at the decision point.4 The L Z values are expected to be reliable unless the adiabatic splitting is sufficiently large to cause the upward transition to be near threshold for a given amount of kinetic energy along the unique direction. In all cases studied, the BT transition probability is at least 0.5, but may be greater depending upon how far beyond the nominal decision point a step in the integration happens to take the trajectory. If each trajectory is integrated completely through the region of strong nonadiabatic coupling (as indicated by the la,l values becoming constant), the properly defined semiclassical transition probabilities for a given choice of effective potential are obtained. These are compared in Table VI1 with the L Z probabilities and those from eq 3 for the trajectories started at R, = 11.0, 8.0, 7.5, and 7.0 ao, respectively. Here W , is used for the effective potential. For the trajectory initiated at R, = 11.O ao, the semiclassical probability is essentially unity. Presumably, because of the small splitting, this would be the case for any choice of the effective potential. This is in excellent agreement with the L Z and eq 3 probabilities at the S M decision point, but shows that the BT method’s use of instantaneous probabilities can lead to unphysical results. In the present example, the unphysical result is a step size dependent, overly large charge-transfer cross section (which would actually be divergent if the P I = 0.5 criterion were employed precisely). For transitions at smaller interfragment distances, the semiclassical probabilities are somewhat larger than both the corresponding L Z and eq 3 values presented in Table VII. This is most likely a consequence of the choice of effective potential; as the splitting gets larger and less localized, different effective potentials will yield differing probabilities. The choice of Wl is likely to result in too large a probability because the corresponding kinetic energy is quite large. That the eq 3 probabilities are only slightly smaller than the LZ values shows how rapidly that more general expression approaches the high-energy limit. In no case in the present study did the “current” instantaneous probability in the BT method fail to fall below 0.5 as a trajectory passed through a region of strong

+

+

We have demonstrated through application to the H2+ H, system that the phenomenological Tully-Preston model for CTSH collisions provides physically reasonable prescriptions for the localization of transitions, the probability of transitions, and the unique direction associated with the momentum change after a transition under certain conditions. These conditions are that it is possible to define an ( m - 1)-dimensional surface of intersection from asymptotic degeneracy considerations and that such a surface becomes a surface of avoided intersection of pseudodimension m - 1 throughout the region of strong nonadiabatic coupling. We have further shown that the Stine-Muckerman method is equivalent to the TP method for the idealized model of an ( m 1)-dimensional surface of avoided intersection, but is more closely related to the “exact” semiclassical description2 of electronically nonadiabatic collisions in real systems. The TP and S M methods were shown to yield almost identical results for H,+ + H2collisions, and the S M cross section^,'^ as well as those obtained by Eaker and Schatz using a simplified variant of the S M method,16 are in reasonable agreement with e ~ p e r i m e n tas ~ ~a ?function ~ of both collision energy and initial H2+vibrational state. Greater insight into the applicability of both the T P and S M models has been provided by an examination of V W, the eigenvalue spectrum of ,8, and 2, (the eigenvector corresponding to the largest eigenvalue of 8) in the vicinity of each of several intersections of a trajectory with the TP seam. These results demonstrated that the S M criterion of a pseudo-(m - 1)-dimensional surface of avoided intersection is satisfied throughout the domain of significant nonadiabatic behavior. We have also examined the applicability of the Blais-Truhlar method for CTSH collisions. Although this method was not applied by its originators to systems of the type considered here, it was presented as a more general method than those of TP and SM. The results presented in the previous section indicate, however, that the BT method has several deficiencies when applied to the H4+system. Neither the transition probability nor the unique direction (along which the system momenta are adjusted after a transition) is precisely defined for a given pass of a trajectory through a region of strong coupling; both are dependent on the integration step size employed. Furthermore, there is very poor agreement between the BT transition probability and those computed with the TP and S M methods, and the BT unique direction is only in rough agreement with those from the other two methods. This situation is not improved by specifying the decision point in the BT method more precisely. If the decision point is defined to be the point along the trajectory at which P 1 , the probability of being on the current surface, is exactly 0.5, the probability of a transition is independent of the interfragment separation (among other problems, this leads to a divergent charge-transfer cross section if either H2+or H2 is vibrationally excited), and the unique direction has its maximum deviation from those of the other methods. It is not clear from this work whether or not the BT method is more appropriate for treating the nonadiabatic behavior associated with conical intersections (as it was originally applied) than for the type of system considered here. We note, however, that a DIM potential energy surface for H3 has only one large eigenvalue of /3 in the vicinity of the D3* electronic degeneracy but unlike the present system the gradient test is not satisfied ((XlW)’/2 N lVWl for H3). We see no reason to expect the BT method to be more appropriate for such a system than for the present. For a system with a conical avoided intersection, the g matrix would have two nonzero eigenvalues and the contours of W would be elliptical. In the limit of one of these eigenvalues going to zero these contours become increasingly eccentric ellipses and approach the troughlike contours of the ( m (15) Muckerman, J. T. Theor. Chem. (N.Y.)1981, 6, 1. (16) Eaker, C . W.; Schatz, G . C.J . Phys. Chem. 1985, 89, 2612.

J . Phys. Chem. 1987, 91, 466-412

466

- 1)-dimensional avoided intersection model. Indeed, our system can be considered as a conical avoided intersection with extremely eccentric ellipses for the contours of W. Although the TP and SM methods, as originally proposed, cannot be applied to conical avoided intersections and hence cannot serve for comparison with methods that have been applied to such systems, it would be of interest to examine these methods as the conical avoided intersection is deformed into the troughlike one. Finally, we point out that there may be systems which do not exhibit an asymptotic ( m - 1)-dimensional interfragment excitation-exchange degeneracy for which the SM and TP models are nevertheless appropriate. Any system that exhibits a pseudo-(m - 1)-dimensional avoided intersection throughout the entire domain of strong nonadiabatic coupling-no matter where that domain is located-may be treated by the SM and TP methods. In such

an application, the S M method has obvious advantages because the analytic specification of the ( m - 1)-dimensional seam cannot be deduced from asymptotic considerations. In the SM method, the gradient test, eigenvalue spectrum of @, and the Massey parameter can be used as diagnostics for its applicability. The “cost” of this increased flexibility, of course, is additional testing along each trajectory which is not required in the TP method.

Acknowledgment. This research was carried out at Brookhaven National Laboratory under Contract De-AC02-76H00016 with the US.Department of Energy and supported by its Division of Chemical Sciences, Office of Basic Energy Sciences, and at Los Alamos National Laboratory under Contract W-7405-eng-36 with the University of California. Registry No. H,, 1333-74-0; H2+, 12184-90-6.

Homogeneous and Heterogeneous Electron Transfer to Benzyl Phenyl Sulfide Maria Gabriella Severin: Maria Carmen ArCvalo,f Giuseppe Farnia,t and Eli0 Vianello*t Dipartimento di Chimica Fisica, Universitd di Padova, 351 31 Padova, Italy, and Departamento de Quimica Fisica, Universidad de La Laguna, Tenerife. Spain (Received: February 26, 1986)

The thermodynamic and kinetic parameters of the electron transfer (ET) to benzyl phenyl sulfide in DMF have been determined by cyclic voltammetry. The heterogeneous ET has been studied at three different electrodes: Hg,Pt, and glassy carbon. The activation parameters, namely the standard rate constant and the transfer coefficient, corrected for the double-layer contribution, show no significant difference for the three materials. The rate constants for the homogeneous ET to benzyl phenyl sulfide from eight electrogeneratedanion radicals have been determined and compared with the corresponding reaction free energies, according to the current theories of the ET process. The results indicate that the ET is an endergonic process driven by a fast bond breaking of the ET product. Both the homogeneous and the heterogeneous reactions are consistent with an outer-sphere adiabatic electron-exchangeprocess, characterized by a relatively high activation free energy. To account for the latter, inner reorganization energies must be considered, together with solvent reorganization. This is in agreement with the easy breaking of the C-S bond in the primary ET product.

Increasing interest is being paid to the study of electron-transfer (ET) reactions. As a matter of fact, E T is one of the basic processes in many chemical fields, including organic and inorganic chemistry, photochemistry, electrochemistry, biochemistry, etc. Furthermore, ET is probably the only reaction for which consistent, even if simplified, theoretical models have been proposed, allowing predictions to be made of the kinetic parameters. As to the theoretical model, the fundamental works by Marcus,’ Hush,Z L e ~ i c hand , ~ Dogonadze4 have continued to receive improvements and reexamination5 on the basis of both new experimental results and the most recent theoretical acquisitions. The comparison of theories and experiments has been limited to the inspection of several possible correlations, for example, between kinetic and thermodynamic aspects, thermic and photochemical, and homogeneous and heterogeneous processes. However, increasing attention is being given to the ability of most recent theories to describe absolute rate constants in terms of molecular structural Actually, the development of this latter aspect has been almost exclusively devoted to reactions involving transition-metal complexes,6 for which abundance of both kinetic data and structural studies on ET reagents and products is available. As to the organic compounds, a lot of kinetic studies are reported mainly concerning particular classes such as quinones, aromatic hydrocarbons, nitroaromatic, etc., but only some of the correlations suggested by the theories have been attempted, and these tests have often led to disagreeing results. In particular, simple classical models such as Marcus’ were often claimed not ‘Universitl di Padova. Universidad de La Laguna. f

0022-3654/87/2091-0466$01.50/0

to apply well to nonspherical organic molecules. Only recently has the whole potentiality of the current theories in predicting kinetic aspects been properly However, this approach is strongly limited by the present paucity of structural data for the products of organic ET reactions. For other compounds, such as halides, sulfides, etc., the instability of the ET products, owing to subsequent decay reactions, has deterred authors from attempting extensive studies and correlations, and often the attention has been mainly centered on the decay rather than on the ET reactions. ET data have been reported for several organic halides and compared with theoretical p r e d i c t i o n ~ . ~ JAlso ~ in this case, (1) Marcus, R. A. Ann. Rev. Phys. Chem. 1964,15, 155; J . Chem. Phys. 1965, 43, 679.

(2) Hush, N. S. Trans. Faraday SOC.1966, 57, 557. (3) Levich, V. G. Adv. Electrochem. Electrochem. Eng. 1966, 4 , 249. (4) Dogonadze, R. R. In Reactions of Molecules at Electrodes; Hush, N. S., Ed.; Wiley Interscience: New York, 1971; p 135. (5) For recent reviews, see: (a) Schmidt, P. P. Spec. Period. Rep., Electrochem. 1975,5, 1. (b) Ulstrup, J. Charge Transfer Processes in Condensed Media; Springer-Verlag: Berlin, 1979. (c) Newton, M. D.; Sutin, N. Annu. Rev. Phys. Chem. 1984, 35, 437. (6) For example, see: (a) Brunschwig, B. S,; Creutz, C.; McCartney, D. Sutin, N. Discuss. Faraday SOC.1982, 74, 113. (b) Hupp, H.; Sham, T.-K.; J. T.; Lin, H.Y.; Farmer, J. K.; Gennett, T.; Weaver, M. J. J . Electroanal. Chem. 1984, 168, 313. (c) Conner, K. A.; Gennett, T.; Weaver, M. J.; Walton, R. A. J . Electroanal. Chem. 1985, 196, 69. (7) Grampp, G.; Jaenicke, W. J. Chem. SOC.,Faraday Trans. 2 1985,81, 1035. (8) Harrer, W.; Grampp, G.; Jaenicke, W. Chem. Phys. Lett. 1984, 112. 263. Fischer, S. F.; Van Duyne, R. P. Chem. Phys. 1977, 26, 9. (9) Andrieux, C. P.; Blocman, C.; Dumas-Bouchiat, J. M.; Saveant, J. M. J. Am. Chem. SOC.1979, 101, 3431. Andrieux, C. P.; Blocman, C.; Dumas-Bouchiat, J. M.; M’Halla, F.; SavEant, J. M. J. Am. Chem. SOC.1980, 102, 3806. Rusling, J. F.; Arena, J. V. J. EIectroanaL Chem. 1985, 186, 2 2 5 .

0 1987 American Chemical Society