J. Phys. Chem. 1995, 99, 2753-2763
2753
Rate Theory and Quantum Energy Flow in Molecules: Modeling the Effects of Anisotropic Diffusion and of Dephasing Sarah A. Schofield and Peter G. Wolynes* Department of Chemistry, University of Illinois at Urbana- Champaign, Urbana, Illinois 61801 Received: July 1, 1994; In Final Form: October 27, 1994@
We examine the effect on energy flow and unimolecular dissociation of anisotropic diffusion in quantum number space and dephasing, using a scaling approach to describe the energy flow and localization. This approach can be applied to systems with local couplings in the quantum number space of approximate normal modes. We find that anisotropy in the coefficient of diffusion on the constant energy surface can combine with anistropy in the finite extent of state space to produce results which mimic those of a lower dimensional isotropic system, due to the early saturation of rapidly relaxing modes. We discuss the diffusion which is caused by dephasing, for both delocalized and localized systems. The results for the energy flow dynamics are used to find corrections to the RRKM reaction rate caused by returns to the region of reactive states. We find that returns are enhanced near the transition to localized eigenstates, causing a greatly diminished average reaction rate. However, we also find that dephasing can cause the average reaction rate to increase over the value for an isolated system.
1. Introduction The complete mechanistic understanding of chemical reaction dynamics and spectroscopy requires a consideration of how energy flows in polyatomic molecules. Thus, it is no surprise that the study of energy flow has been a central question for chemical physicists since the founding of the subject.’ However, it is amazing how little has been definitively resolved in this quest for understanding. This is testimony both to the experimental difficulties inherent in probing a quantum mechanical system which is physically small but nevertheless complex and to the theoretical depth of the questions which are involved. The study of energy flow in molecules causes us to confront some of the deepest issues in theoretical science: the origin of irreversibility, the emergence of chaos and unpredictability from deterministic equations, and the connections between classical mechanics and quantum mechanics. The energy flow problem can therefore function as a laboratory for these fundamental problems of physics. Ultimately, the understanding of energy flow in molecules could also make possible the fulfillment of the dream of directly influencing the course of chemical reactions by controlling the motions of molecules through the use of tailored external fields.2 Two different philosophies motivate the theoreticians approaching the problem of energy flow in molecules. The first philosophy sees molecules as “individuals”. Our understanding of molecular force fields is sufficient for us to make detailed models of moderate-sized molecules, thus providing a specific Hamiltonian for computational study. At the same time, techniques for numerically solving the Schrodinger equation for the moderate dimensional systems, such as 21 mode b e n ~ e n e , ~ have advanced and continue to advance at a rapid pace. Thus, a direct computational assault on the energy flow in specific molecules has been feasible for some time. The other philosophy insists that many features of energy flow in molecules are generic and can be understood through the consideration of less detailed models which try to capture elements of the fundamental physics. Even with the successes of the direct computational route, this second philosophy is @
Abstract published in Advance ACS Abstracts, February 1, 1995.
0022-365419512099-2753$09.00/0
valuable because it helps us to globally understand energy flow in classes of molecules and allows us to develop pictures for the interpretation of such detailed computations. The latter task is often difficult because of the high dimensionality of the phase space of even a moderate-sized molecule. It also allows us to isolate and appreciate intellectually the fundamental physical issues which motivate the theoretical study of the problem. Many simple models addressing different foundational issues have been developed. The issue of irreversibility was first addressed in the context of radiationless p r o c e s ~ e s . ~Here, .~ a single quantum level was coupled to a continuum or a broadened set of discrete levels leading to transport from the initial state. This picture, while initiated for electronic transitions, has been used a great deal in the vibrational energy flow situation as well. The study of this simple model showed the importance of the magnitude of the coupling relative to the density of states in determining the possibility of irreversible energy flow in a quantum mechanical system. The origin of chaos and unpredictability has mostly been addressed in the context of classical mechanical models. In low dimensional systems, Chirikov highlighted the significance of overlapping Fermi resonances in giving rise to chaos.6 Oxtoby and Rice in pioneering work showed how the overlap of Fermi resonances was crucial in the transition to chaos in classical models of intramolecular energy flow.’ Finally, the drive to understand the quantum mechanical consequences of classical chaos led to calculations for quantized systems which are chaotic classically* and to the introduction of global random matrix models9 of chaotic quantum systems of low dimensionality.lOJ1 The study of driven systems, for example, the quantum kicked rotor, has shown the importance of the presence of more than one dimension in enabling such quantum systems with suitably chosen parameters to follow classical irregular motion, without dynamic localization. l2 Many of the intellectual themes from these earlier simple models have been brought together in the study of local random matrix models of energy flow within moderate-sized mole c u l e ~ . ~The ~ . classical ~~ chaos of moderate-sized molecules suggests their description by random matrix models. Random matrix models can be used to describe quantum dynamics in
0 1995 American Chemical Society
2754 J. Phys. Chem., Vol. 99, No. 9, 1995
the absence of external driving. Global random matrix models have previously found success in molecular rate theory4 and spectro~copy.'~ On the other hand, the detailed mechanism of energy flow through Fermi resonances suggests that certain pathways and connections in the state space are considerably more important than others.14 This leads to the locality constraint. Finally, the analysis of such local random matrix models resembles in many ways the concatenation of several stages of the simple transport into a structured continuum studied in the radiationless process t h e ~ r y . ~ The study of local random matrix models gives insight into several qualitative questions about quantum energy flow. In their initial work, Logan and Wolynes15 showed that there was a strong similarity to the problem of quantum transport in a disordered medium: the so-called Anderson localization problem.17-19 That there is a transition from localized states to extended states for which energy can flow easily throughout the molecule is an important observation for reconciling the two qualitative pictures of energy flow which dominate the thinking of experimentalists. While workers in kinetics often imagine extraordinarily rapid energy flow which allows the utilization of statistical models for rate constants (the RiceRamsperger-Kassel-Marcus (RRKM) theory*O), spectroscopists, on the other hand, imagine assigning specific quantum numbers to spectral lines, thus assuming very long-lived modes of motion. The LoganNolynes analysis suggests that a transition between these two pictures occurs at a critical value of the coupling that depends on the local density of states. This criterion echoes many qualitative criteria contained in earlier treatments of the irreversibility p r ~ b l e m .Being ~ a function of the parameters of the system, this localization differs from the dynamical localization of one-dimensional driven systems.' Later, Schofield and Wolynes16showed how one can use scaling ideas from the theory of Anderson localization18 to address questions arising from the finite but large size of the state space of a moderate-sized molecule. The scaling approach presupposes a renormalization transformation in state space with an unstable fixed point at the transition. This has the consequence that as the wave function evolves and has greater extent, thus exploring the transformation, the nature of the dynamics can evolve in time from more constrained motion to more classical motion. In application to unimolecular dissociation, the inclusion of the finite size of the state space enabled the local random matrix theory to be used to address important issues of the validity of the RRKM approximation for molecules described by local random matrix models.21 In this paper, we continue our study of local random matrix models with the intent of introducing into the discussion of local random matrix models some of the crucial realistic features that enter into the study of specific molecules. The local random matrix models which have been discussed so far have only three relevant energy scales, a coupling and the inverse of a local density of states as well as the inverse of the global density of states when we consider the finite size effects. Real molecules obviously have a variety of different energy scales. Typically, there are some high-frequency motions associated with the vibrations of the light atoms and some lower frequency motions having to do with the motions of the heavy atoms and global skeletal motions of the whole molecule. The existence of differences in energy scale means that the energy flow between these modes may be quite slow and that the transport among the high-frequency modes can be quite different from the transport between the low-frequency modes. In addition, the extent of the state space for the high-frequency modes is per force somewhat smaller than for the low-frequency modes due
Schofield and Wolynes to energy conservation, Thus, a realistic local random matrix model should take into account the anisotropy of motion that would be expected for these different modes. In this paper, we address one of the simpler models of such anisotropic motion. Here we assume, in the classical limit, the motion resembles a diffusion in the phase space with an anisotropic diffusion constant. Other very low energy scales also enter the dynamics of realistic molecules. The overall rotation of the molecules has a weak coupling to the intemal dynamics through the Coriolis effect, but at the same time, the rotations entail an associated very large density of states. Similarly, a molecule in a gas or in a condensed phase is coupled either transiently through a collision or weakly through phonon vibron coupling to the entire density of states of a macroscopic system. These lowfrequency-mode effects can be modeled as providing an external dephasing of the intramolecular vibrational dynamics. The role of dephasing was addressed earlier by Logan and Wolynes in their microscopic treatment of the many dimensional Fermi resonance m0de1.l~ Here we show how such external dephasing can be taken into account in a scaling picture. We also discuss how this external dephasing can enter into the problem of corrections to the RRKM theory of chemical reaction rates. This analysis is relevant then to the problem of intermolecular vs intramolecular energy flow in the Kramers turnover regime.** The organization of this paper is as follows: In section 2, we reprise the essential elements of the scaling perspective on energy flow and localization for systems described with isotropic local random matrices. We then use a generalization of this scaling theory to include anisotropy. This generalization is modeled upon the approaches that have been carried out in the context of the electronic conduction problem. An important result of this analysis is that high-frequency modes can effectively be frozen out of the energy diffusion process. Following this, we discuss the role of external dephasing in intramolecular energy flow. In section 3, we use these results to discuss non-RRKM corrections to the dissociation rate of isolated molecules and specifically focus on the role of external dephasing on the corrections to transition-state theory. The results for a single representative case in which the freezing out of high-frequency modes occurs are used to illustrate the conclusions of both section 2 and section 3. Finally, in section 4, we make some concluding remarks about the extension of the theory and its comparison to more realstic models.
2. Energy Flow and Localization in a Scaling Perspective In following the flow of energy from an initial state, a scaling approach may be used when the initial state is statistically equivalent to the other states to which it is coupled. These states are specified by the degree of excitation in the set of approximate normal modes of nuclear motion. The true eigenstates result from couplings between the modes, leading to coupling between the initial state and other basis states. The overlaps between the eigenstates and the basis states are complicated functions, apparently random. Treating them as random variables allows the determination of average properties such as the average of the survival probability, the probability that at a later time the wave function again overlaps the initial state. The scaling perspective on energy flow applies to systems with local couplings and of moderate size, that is, with at least four modes and a three-dimensional constant energy surface. The initial state is coupled to a few other states, each of which are in turn coupled to a few other states (which may include the initial state), and so on until the bounds set by the finite size of the molecule are reached. This results in a local random matrix
Rate Theory and Flow in Molecules
J. Phys. Chem., Vol. 99, No. 9, 1995 2755
-
argument may be used to determine the general characteristics of the variation of the flux common to a variety of systems described by local random matrix models. If single-parameter scaling is assumed,ls that is, g is the only scaling function, and if the flux at a larger length depends only on the flux at a smaller length, then the scaling behavior is embodied in the function p(g) = Figure 1. Schematic diagram of local coupling. States are indicated with thick horizontal lines and the presence of coupling by thinner connecting lines.
P
Figure 2. Scaling curve for localization.
model of the Hamiltonian. In a local random matrix model, the eigenstates can localize. The coupling between states is schematically illustrated in Figure 1. Note the absence of bottlenecks or special states, which can lead to exponential decay of the survival pr~bability,~
The transition between localized and delocalized eigenstates occurs when p = 0. This would be an unstable fixed point in a renormalization calculation, as illustrated in Figure 2. For p > 0, the flow is toward the limit of delocalized eigenstates as L and g increase. For p < 0, the flow is in the opposite direction toward the localized limit. The p curve in the two limits is determined from combining the definition of the flux with the scaling of the density of states with L,
arising through the dependence of volume V, with the known dependences of the transport time on length in the two limits. po(E) is the density of states per unit volume. By definition, df is the fractal dimension. For a molecule, its upper bound is the dimensionality of the constant energy hypersurface in quantum number space, s - 1, with s being the number of modes of nuclear motion. The “volume” V refers, in fact, to the area on this hypersurface. When the eigenstates are localized, the exponential dependence of the transport time on length leads to a linear dependence of p on In g. In contrast, when the eigenstates are delocalized, the time scales with length
t
P ( t ) = (W>
(2.2)
with the average being over different initial conditions of similar energy. This averaging smooths out oscillations about the overall decay which may be present in individual survival probabilities. As will be seen in section 3, it is the average survival probability that enters in the theory of non-RRKM corrections to average rates of unimolecular dissociation. For completeness, a brief summary of the scaling perspective on energy flow and localization in an isolated isotropic system follows. Details are contained in ref 16. To describe energy flow and localization, for a given length L in quantum number space, there is associated the function
g(L) = k(&MU
(2.3)
known as the Thouless number, which is simply the ratio of the uncertainty in energy (h/t(L))caused by energy flow over a scale L to the average spacing between energy levels (l/p(E,L)), with @ ( E )being the density of states at energy E. If g(L) > g, with g, of order unity, then the eigenstates are delocalized over a range L, whereas if g(L) < g, then the eigenstates are l o ~ a l i z e d . ~ ~In- ’principle, ~ knowing the couplings between the states enables a renormalization calculation to determine the function g(L) for a particular system. Altematively, a scaling
-
La
(2.6)
leading to a constant value of
W)= l ~ W ~ ~ l ~ ~ (2.1) ~ ~ ~ I Z The locality of the coupling and the statistical homogeneity of state space in the local random matrix model lead instead to power law decays of the average survival probability,
(2.4)
d In L
p,,=d-a
(2.7)
in the delocalized limit. The exponent a gives the degree of correlation of the random walk. When a < 2, the random walk is correlated. When a > 2 , it is anticorrelated. When a = 2, it is uncorrelated, as in Brownian motion. Reversing the procedure, the dynamics near the transition can be obtained from the scaling function p(g). Because p 0 and g gc near the transition, eq 2.3 and eq 2.4 give
-
I
t--
-
d+
L’ 0
with (2.9) a microscopic frequency. Therefore, near the localization transition, the geometrical scaling (volume with length) and the dynamical scaling (time with length) become one and the same. To determine the survival probability, we have used the ansatz16 (2.10) with V explored by the random walk as a function of time. This, in combination with eq 2.3 and eq 2.5, gives the relationship between the survival probability and the Thouless number
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2756 J. Phys. Chem., Vol. 99, No. 9, 1995
(2.11) The scaling of the survival with time is then related to the scaling of the Thouless number with length. In particular, define an exponent for the survival by d In P(t)
(2.12)
y=dlnt
Then the time scaling of the survival and the length scaling of the Thouless number are related by
Y=-
4
(2.13)
P -4
for a given instant in space and time. Thus, with the dimensionality known, the time variation of the survival can be determined from the scaling function /?,shown in Figure 2. In the delocalized limit, eq 2.7 and eq 2.13 combine to give
(2.14)
-
-
This will be referred to as diffusional scaling. Near the localization transition, when 3!, 0 and g gc, the survival varies with time according to
P(t)
-
1 ot
-
scaling description to an anisotropic system. We show that in an anisotropic system the critical scaling of the survival probability in the delocalized limit is modified due to a varying number of effective degrees of freedom with time, caused by the combined effects of anisotropy in the diffusion coefficient and anisotropy in the finite extent of state space. Extemal dephasing is another effect present in realistic situations. Because localization involves quantum phase coherence, dephasing destroys localization for times larger than the dephasing time. Thus, it causes diffusion at longer times, as described in section 2B below. A. Anisotropy. In this section, we discuss the inclusion of anistropy in the scaling description of energy flow and localization. In an anisotropic system, a random walk spreads out in different directions at different rates due to the anisotropy in the diffusion coefficient. This means that there is a family of lengths (L1, b....,LN}describing the extent of the random walk in different directions. One might think at first that this means there is a family of scaling functions pi, i = 1 - N , giving the variation of flux with length. However, previous workers in the field of electron localization in metals found that predictions based on retaining the assumption of single-parameter scaling in an anisotropic system were validated by experimental r e s ~ l t s .The ~ ~essential ~ ~ ~ point is that as a consequence of the single-parameter scaling assumption there must exist a transformed space in which the system appears isotropic. In the delocalized limit, the natural transformation,
(2.15)
(2.18)
This will be referred to as critical scaling. For large df, this is a much weaker variation with time than in the case when the eigenstates are delocalized. This was the main result of our previous scaling treatment of energy flow and localization in an isotropic system.16 On the delocalized side of the transition, because the point p = 0 in Figure 2 is an unstable fixed point, the average survival probability follows eq 2.15 for short times and eq 2.14 for longer times. In this case, the dynamics may be approximated by simply connecting these two results at a crossover time, given by (2.16) That is, on the delocalized side of the transition, when tc > 1/D, or equivalently w > D, the average survival probability is approximately
Note that it is the exponent of the power law scaling of the average survival probability with time that changes from a smaller value to a larger value. This is distinct from the variation of the rate coefficient. Since w > D, the larger rate applies at shorter times. The scaling results for the average survival probability described here apply only for intermediate times, longer than the time for the first few hops and shorter than the time to reach the finite extent of state space due to the finite molecular size. Both results, eq 2.14 and eq 2.15, apply only for t < @(E),with e(@being the global density of states. &(E) is the Poincar6 recurrence time, also known as the break time. This is the time scale for recurrences to individual quantum states, due to the discreteness of the energy. Real molecules are more properly described by anisotropic diffusion in state space. Below, in section 2A, we extend the
used in the previous work on electron localization, tracks the scaling of length with time,
L,(t)
"u
(Dit)l/a; L(t)
- (h)l/a
(2.19)
In addition, through use of the geometric mean diffusion coefficient
(2.20) the transformation ensures volume preservation:
P ( t ) = V(t)=
n
L,(t)
(2.21)
i
Single-parameter scaling then dictates that d In g/d In t be the same universal scaling function p, shown in Figure 2, as is obtained for an isotropic system. This means that near the transition
L = (Wt)'/df
(2.22)
the same variation as for an isotropic system with w given by eq 2.9. Transforming back to the original coordinates gives
Li(t) = (Wit)lIdf
(2.23)
where
wi = W ( D ~ / D ) ~ ~ I ~
(2.24)
Thus, through the dependence of wi on Di, the anisotropy of the random walk near the localization transition is governed by the anisotropy of the random walk in the limit of delocalized eigenstates.
Rate Theory and Flow in Molecules
J. Phys. Chem., Vol. 99,No. 9, 1995 2751
Applying the spatial transformation on quantum number space for an anisotropic molecule, using eq 2.19, eq 2.21, and eq 2.22, it is immediately obvious that because the average survival probability depends only on the volume (eq 2.10), the average survival probability is identical to that for an isotropic system, save for the replacement of the isotropic diffusion coefficient with the mean D. This applies only for times less than the time to reach the finite extent of state space. In molecular applications, one must incorporate also the effects caused by anisotropy in the finite extent of state space. When there is anisotropy in the state space, the most convenient expression for the survival is
(2.25) An interpretation of this equation is that the survival is the product of individual one-dimensional survivals. This interpretation is useful in determining the effect of the finite size of the state space which arises from the finite size of the molecule. This is an effect not considered in the previous work on electron localization in anisotropic systems. In the limit of delocalized eigenstates, the finite extent of state space may introduce a number of time cut-offs for the range of scaling behavior. In the original spatial frame of reference, let
log(ijt)
Figure 3. Survival probability in the delocalized limit. T h: parameters are N1 = 2, N2 = 6 , d l = d2 = 5 , a = 2 , and D I= (i) 3 0 and (ii) D. 0
-1
N
=
~
N
~
(2.26)
i
-2
be the total volume of state space. N is the total number of quantum states on the surface of constant energy. There will be a series of cut-off times, given by
-3 WP)
Ti = N:/Di
(2.27) -4
after which time the “length” for a given direction is fixed at
~ i =m( D ~~ T ~=)N~~ / ~
(2.28)
In addition, eq 2.19 for Li(t) applies only for times larger than l/D. To illustrate these time bounds and their implications for the behavior of the survival probability, consider the case when there are dl directions which share diffusion coefficient D1 and extent N1 and d2 directions which share diffusion coefficient 0 2 and extent N2. Then there are three times giving the bounds of scaling, UD, T I , and T2. The net behavior depends on the relative magnitudes of T1 and T2. In either case, let Ts be the shorter and the the longer of the two times. We take the random walk to be uncorrelated, so a = 2, and let df = d , the dimensionality of the constant energy surface in quantum number space. Then applying the time bounds, according to eq 2.28, gives i f t I1/D
i f f 2 T, This is illustrated in Figure 3, for a system with moderate anisotropy, in comparison with an isotropic system of the same diffusion coefficient as the mean. It can be seen that the main effect of anisotropy in the finite size of state space is the presence of two different exponents giving the power law dependence of the survival on time, with the smaller exponent
-5
-0.5
U
u.5
1
1.5
log(wt)
Figure 4. Survival prgbability. The parameters are N1 = 2, NZ = 4, dl = 2, d2 = 8, a = 2 D / o = (i) 1.25, (ii) 0.2, (iii) 0.002, and (iv) 6 = 1.6.
arising at later times. The exponents are obtained from the slopes of the curves in the figure. Generalizing to any total number of different diffusion coefficients, for each mode that saturates, the effective dimensionality appearing in the scaling of the survival with time is lowered by unity. As for an isotropic system, the effective dimensionality has an upper bound of s - 1, the number of modes of nuclear motion. Figure 4 illustrates the results for a system of stronger anisotropy, in which df < s - 1 through the whole time course of the scaling. The solid line shows the diffusional behavior predicted by eq 2.29 in the limit of strongly delocalized eigenstates. The other curves show behavior affected by the localization transition and are discussed in detail below. We note that the parameters chosen to generate Figures 3 and 4, although certainly reasonable for molecules, were chosen primarily to illustrate the behavior of interest. For both Figure 3 and Figure 4,a fundamental rate or frequency is not
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2158 J. Phys. Chem., Vol. 99, No. 9, 1995
chosen, because D is used to weight time (and thus need not be determined) in Figure 3 and likewise w is used to weight time in Figure 4. Near the transition to localized eigenstates, the finite extent of state space introduces one single time cut-off in the dynamics of the survival probability. This contrasts with the delocalized limit in which there is a cut-off time for each unique combination of direction-dependent diffusion coefficient and extent in state space. This is because the fundamental relation near the transition is the dependence of the volume with time, and volume elements are unchanged by the rescaling of the coordinates according to eq 2.18. The survival is, therefore, given by
I1
(2.35)
i f t Iw-l
near the transition to localized eigenstates. However, the implied time variation of the direction-dependent lengths of the random walk does change as a result of different sized bounds for different directions. Using the same simplified system for illustration as above, with dl directions with diffusion coefficient D1 and extent N1 and d2 directions with diffusion coefficient D2 and extent N2, there are only three determining times for the critical dynamics, 110, Nlw, and the earlier of the two times TI and T2 involving reaching the bounds in state space, satisfying Ti = @/mi
(2.31)
With T,, the shorter of the two times, and TI,the longer of the two times, applying the time bounds gives
L,(t) =
i f t 1 T,
(2.32)
for the random walk in the more confined directions and
[(w,t)l/d if t IT, if T, It INlw
(2.33)
if t 1 Nlw for the less confined directions. Generalizing from this example, as in the case of delocalized eigenstates, for each direction that saturates due to the finite extent of the state space, the effective dimensionality governing motion in the remaining directions is reduced by unity. Again, this applies for shorter times. At longer times, there can be a crossover to the limiting behavior either of localized or of delocalized eigenstates. On the delocalized side of the transition, for df > a,the survival probability and the random walk scalings vary from those characteristic of being near the localization transition to those of the delocalized limit. The simplest case occurs when the time bounds set by the finite size of the state space are all equal. In this case, connecting the limiting dynamics at an intermediate time to illustrate the variation, the survival probability is given by
if t < l/w if 110 < t < t,
p1 (ut)-'
if t, It IT, P(t) = { (dt)-d'2 xdS(Dlt)-d1/2 if T, It ITl 1IN if t ITl
-- P2
P(t) L(t)
L(t)
t-'
tlidf
I
if t < t,
if t > t,
(2.37)
with df being the number of active dimensions of the energy surface and having the same meaning as in an isotropic system. However, in contrast to an isotropic system, dfvaries with time, as different directions saturate due to the finite extent of the constant energy surface, which is due to the finite size of the molecule. On the localized side of the transition, the dynamics are similar to those in eq 2.30 and eq 2.22 except that the time bounds are set by the localization lengths. Define the mean localization length, (2.38) with &, the direction-dependent localization lengths. Then on the localized side of the transition, in the simplified model of two distinct classes of directions used above, the random walk lengths are given by
for the more localized directions and
g/w
I
if llw It It, (ijt,t)-dda
if t, It Ip/ij
1lN
if t L P I D
with the crossover time given by
(2.34)
(2.36)
(2.40)
if t L g / w for the less localized directions, with Ti = sp'/wi
(2.41)
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The survival probability is given by
on qtc. When qt, = 1, the postdephasing diffusion coefficient is GJ = D. When 17tc < 1, crossover to diffusion govemed by delocalized dynamics of the isolated system occurs before the dephasing time. Then, since dephasing does not change the diffusion coefficient for delocalized dynamics, when qtc < 1, @ = d. Only if qt, > 1 will the crossover be caused by the dephasing, occuring at time l/q, with eq 2.47 applying. When the eigenstates are localized in the isolated system and when the localization length is reached before the dephasing time, the postdephasing diffusion coefficient depends on the localization length E and varies lineraly with dephasing rate. That is.
This is illustrated in Figure 4,curve iv. Note the larger value of the survival probability at long time than in the delocalized cases (i, ii, and iii in Figure 4), due to the localization of the eigenstates. This result applies for b < o and 6 < " I d . B. Dephasing. In this section, we discuss the inclusion of extemal dephasing, in addition to anistropy, in the scaling description of energy flow and localization. A simple picture of dephasing is the following. Let 17 be the rate of dephasing. Up until a time 1/17, the same dynamics occur as without dephasing, whether it is diffusion or slow dynamics influenced by the localization transition. However, after the dephasing time, the dynamics are diffusive with a diffusion coefficient that depends on the length of the random walk up to the dephasing time. The diffusion coefficient is given by
a = qLa(t = 1/17)
(2.48)
Thus, when the eigenstates are localized, diffusion at long times occurs only in the presence of dephasing. Treatment of finite size effects carries over from the previous section. For example, when there is a crossover, using the same model of two classes of directions, labeled by shorter finite size time (s) and longer finite size time (0,with
(2.43)
with L(t) being the length of the random walk for an isolated system. In an anisotropic system, this procedure is applied in the rescaled quantum number space introduced in section 2.A. The equivalent procedure in the actual quantum number space is to determine the direction-dependent diffusion coefficient from
ai= qLP(t = 1/17>
a=qta
(2.44)
Then the mean, as expected in analogy with the approach in section 2.A, is
Ti = N f / a i
(ut)-'
if l/o < t < t,
f(-dya)tl-d2'l 1IN
if T, It ITl if t 1 T,
(2.49)
(2.45) In this section, we examine in detail how the postdephasing diffusion coefficient CZdepends on the dynamics of the isolated system in the absence of dephasing. In all cases, it will be assumed that the dephasing time is shorter than tmax,the time to reach the bounds set by the finite size of the system. Otherwise, the dephasing has no influence. When the isolated dynamics are entirely diffusive, the diffusion coefficient does not change after the dephasing time. That is, combining eq 2.43 with eq 2.19 gives
a=b
(2.46)
This is the simplest case. When the slow critical scaling applies over the entire scaling time range, dephasing results in a crossover to diffusion with diffusion coefficient that depends on the microscopic frequency and on the dephasing rate. That is, combining eq 2.43 with 2.19 and eq 2.22 gives (2.47) Comparing this with eq 2.35 for the crossover time to diffusion in an isolated system, it can be seen that it is the same equation, with the crossover time tc replaced by l/v and the mean diffusion coefficient D replaced by @. When there is a crossover to diffusion in the isolated system, there is also a crossover to diffusion in the presence of dephasing, with a diffusion coefficient that varies depending
3. Non-RRKM Corrections to the Average Rate In this section, the results of the preceding section are applied to the determination of dynamical corrections to the microcanonical transition state, RRKM, prediction for the average rate of unimolecular dissociation. Dynamical corrections cause the transmission coefficient, K
= k/kmKM
(3.1)
to be less than the RRKM limit of unity. As has been appreciated for some time, the assumption of a quasiequilibrium between reactant and a transition-state configuration that leads to the RRKM result is equivalent to an assumption of no dynamical retums to the set of reactive states comprising the transition state. The transition state may be defined either in configuration space or in quantum number state space. In the configuration space picture, the transition states comprised all those states on the dividing surface perpendicular to the direction in which the potential has a saddle point. This is the usual picture but can be difficult to quantize. In the traditional Kassel
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2760 J. Phys. Chem., Vol. 99, No. 9, 1995
coupling between reactant states and product states gives the res~lt*~~~~ K=-
1
(3.5)
where
(3.6)
+-’
Figure 5. Non-RRKM dynamics. A path in quantum number space which returns to the transition region, indicated by cross-hatching, is shown.
picture, which is more easily extended to the quantum domain, the transition state comprises those states which are directly coupled to product states, that is, those states which are by definition reactive. The definition can, but need not, make recourse to a dividing surface in configuration space. Instead, these states can be specified, in a quantum picture, but their coordinates in quantum number state space. To distinguish this picture, the term transition region will be used to denote the (connected) set of states in quantum number space which are reactive. The term RRKM rate will be used to denote the microcanonical average rate which results in either picture when there is a quasiequilibrium between all states of the molecule and the subset of reactive states. The non-RRKM dynamics under consideration are illustrated in Figure 5. A three-mode system is shown only for ease of illustration. The treatment of dynamical corrections presented here is restricted to the “energy diffusion” regime. That is, if z is the time scale of these energy flow dynamics, the treatment is confined to the situation when zwo > 1 , where wois the attempt frequency,
and Pr(t)is the probability to return to the transition region of reactive states due to non-RRKM dynamics. Equation 3.5 may be interpreted as the addition in the series of the RRKM resistance to reaction and the dynamical resistance to reaction, 9. where the resistance is the inverse of the transmission. It is essentially the same as the expression derived by Northrup and Hynes in the stable states theory of reactions.25 The approach taken here differs from previous approaches to dynamical corrections to the RRKM in its incorporation of manydimensional quantum energy flow via the return probability. In conjunction with the scaling results for the average survival probability of an individual state P(t) in an isotropic system, P(t), a natural approximation to the probability of returning to the transition region is
(3.7) with being the number of states in, and volume of, the transition region and being the mean time to escape from the transition region. This approximation has been used in obtaining corrections to the RRKM prediction for the average rate due to the slower dynamics embodied in P(r) as the transition to localized eigenstates is approached in an isotropic system.21 Summarizing the results for an isotropic system, combining eq 3.5 and eq 3.7 together with the scaling predictions for the survival probability gives
4=
i
dpo($)“/df[l - Fll(df - a ) D ln((GewlD)”‘d-2’/~)lw w,# ln(eN1g)lw
08
if D l o 2 llA* if 1/A* 2 DlwGaldf2 eddf/A if DlwGaldf5 11A
(3.8)
with the constant
F = -A*
in the RRKM prediction for the average reaction rate,
A
(3.9)
and the function is the equilibrium probability of occupying the transition region
P:q
(3.4)
G = exp(wF/DA*)
(3.10)
arising due to the finite extent of the state space. They depend on the constants
p(E)is the number of reactive states comprising the transition region. In addition, the treatment is confined to situations when zlhe(E) 1 so that only dynamics on time scales short compared to the PoincarC recurrence time are considered. For completeness, a brief summary of the results of previous treatment of the non-RRKM dynamics resulting from quantum energy flow and the approach to localization in an isolated isotropic system follows. A resummed expansion for the microcanonical average rate of dissociation in orders of the
and
which we introduce for convenience of notation. It is instructive
Rate Theory and Flow in Molecules
J. Phys. Chem., Vol. 99, No. 9, 1995 2761
to rewrite eq 3.8 as follows:
4 = 0 ts--[l- d
ODd-2
4=
F]
(3.19)
with
if D/w 2 1/A*
with
tlf = (Nf)ddf/D
(3.14)
the escape time in the limit of delocalized eigenstates and t i = $/w
(3.15)
the escape time near the transition to localized eigenstates. Thus, one can see that the non-RRKM correction 4 is govemed primarily by the length of time to escape from the region of reactive states, as measured against the RRKM attempt frequency wo. Note that eq 2.9 and eq 3.2 combine to give the relationship between the two microscopic frequencies,
2 = 2zg,
incorporating the effects of the finite extent of state space. Comparing with an isotropic system, the first line of eq 3.13, the form of 4 is the same but the escape time depends on the mean D in place of the isotropic D and the detailed finite size correction F changes. With the dynamics being those of a system near the localization transition, 4 for an anisotropic system does not differ from that for an isotropic system. In particular, near the transition, using eq 2.15 with eq 3.7, the probability to retum to the transition region is given by i f t < $/w if $/w It IN / o if t 2 Nlw
(3.16)
0 0
When D Av) > AM,) I 11N 1/N w,tL ln((Ge)d"d-2'wtJN$N) if l/N' 2 Allt,) > Av) I (3.29)
mati ln((Gc)d(d-2'w/~N') .2
-2.5
Figure 6. Transmission coefficient. The relevant parameters are the N = and gc = 1. same as in Figure 4. In addition @ =
account for the anisotropy in the diffusion. In addition, the finite size correction F , given in eq 3.20, is modified to account for the ansiotropy in the finite extent of state space. The variation of the transmission coefficient that results from inserting eq 3.25 in eq 3.1 is illustrated in Figure 6, for the same system whose dynamics are illustrated in Figure 4. This anisotropic system resembles an isotropic system with diffusion coefficient D1 = Dz,due to the saturation of the more rapidly relaxing degrees of freedom. The same system was chosen as in Figure 3 in order to illustrate the connection between the average survival probability and the transmission coefficient. The labels i, ii, and iii in Figure 6 each correspond to the same parameter values as the same labels in Figure 4. There is no corresponding point in Figure 6 for the system labeled by iv in Figure 4 because that system has localized eigenstates, rendering an average reaction rate an insufficient description. The results given in eq 3.25 with eq 3.20 are easily generalized beyond the simple anisotropic model used here for illustration. Only the finite size correction, incorporated in F and G, changes as a function of the interrelationship between the anistropy in the diffusion coefficient and the anisotropy of the extent of the state space. B. Dephasing. The results of including non-RRKM dynamics in determining the reaction rate, discussed above, may be extended easily to include dephasing. To start building up the result, consider the case when the dephasing time 1/11 is smaller than the crossover time between critical scaling and diffusion in an isolated system, given by eq 2.35. Consider tc to be fixed as a system parameter and vary 11, subject to vtc > 1. Then using eq 3.25, with D replaced by G, as given by eq 2.47 yields
In this case, due to a larger diffusion coefficient, t, is smaller than when eq 3.26 applies. Therefore, 4 levels off for small y~ at a larger value than in eq 3.26. Continuing in the direction of less localization, if t,w 5 @, then
When the localization length is smaller, there is a time, te = (d/w, associated with reaching the localization length. As long as fmax > tg > t* and tph < tma, then it is sensible to determine corrections to the average rate. The result is
9=
Note that
k-11,
v-+O
(3.32)
provided that the system size is at the same time increased such that v/w > 11N. Figure 7 illustrates the results given eq 3.26 (i), eq 3.29 (ii), eq 3.30 (iii), and eq 3.31 (iv) for the same anisotropic system used to generate Figures 4 and 6. Again this is to show the connection between the quantum energy flow and rate of reaction. The labels i, ii, iii, and iv of Figure 7 each correspond to the same parameter values as the same labels in Figure 4. The analysis used in this section applies for 11 < w only. For larger values of q, the reaction rate is expected to decrease with increasing dephasing. This regime requires an analysis which incorporates the dependence of the local density of states on the dephasing rate, as discussed by Logan and W01ynes.'~ 4. Concluding Remarks
ifAq) 5 1/N The function&), which appears in the conditions on applicability of the different functional forms for 9, is given by (3.27)
In this paper, we have presented the results which represent general trends concerning the effects of anisotropy and dephasing on quantum energy flow and rates of reaction of molecules whose eigenstates may localize. The scaling approach which is used models systems in which all initial conditions used in determining the average energy flow are statistically equivalent and which have local couplings between states. The results are
Rate Theory and Flow in Molecules
J. Phys. Chem., Vol. 99, No. 9, 1995 2163 return to the region of reactive states.31 This indicates that the whole distribution of reaction rates must be determined rather than solely the average reaction rate, representing a promising avenue of future research.
Acknowledgment. It is a great pleasure to dedicate this paper to a real “mensch”, Stuart Rice. His leadership in the scientific community as a whole is much appreciated. At the same time, his creative contributions to chemical physics are a beacon, showing how much intellectual depth can be found in the important problems of our field. We also acknowledge the financial support of the National Science Foundation through Grant CHE 92-23224. References and Notes
-2.5
.2
I
1
-1.5
-1
, -0.5
0
0.5
1
1.5
Wrl/od Figure 7. Transmission coefficient as a function of dephasing rate. The parameters are the same as in Figure 6.
semiquantitative. They reveal that an anisotropic system can appear to be an isotropic system of lower dimensionality. We have described the influence of dephasing in causing diffusion in systems which do not fully relax in isolation due to localization of the eigenstates. This diffusion occurs on the surface on constant energy in the quantum number space of approximate normal modes. Diffusion across this surface causing variations in total energy, which could also occur in the presence of dephasing, is not considered here. Dephasing of vibrational motion may arise through collision or due to the influence of rotations with a high density of states. Incorporating the dependence of the average rate of dissociation on the dynamics of energy flow allowed the determination of the transmission coefficient, the deviation of the average rate from the microcanonical transition-state prediction, as a function of the dephasing rate. Thus, the trends shown in the results of this paper, illustrated in Figure 7, can in principle be determined experimentally. Calculations of quantum energy flow represent another avenue of p r ~ g r e s s . ~Given a Hamiltonian, the average survival probability, the average probability to return to the region of reactive states, and the average rate of reaction can all be calculated. Calculations of the survival probability and its average for moderate, nonreactive levels of excitation of in model systems of locally coupled nonlinear vibrations have shown both a linear variation of the relaxation rate with the strength of ~ o u p l i n g , *as ~ ,predicted ~~ in ref 15, and a power law variation of the survival probability with time,28consistent with the predictions of the scaling approach. A natural goal of future calculations would be to obtain dynamical information for higher, more reactive levels of initial excitation. Due to the focus on the average rate of reaction, the approach used in this paper fails to fully describe reactions of strongly localized system. Fluctuations in state-specific reaction rates are predicted to increase dramatically near the transition to localized e i g e n ~ t a t e ssince , ~ ~ they depend on the probability to
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