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Solvent slow-mode influence on chemical reaction dynamics: a multidimensional Kramers-theory treatment
Volume 172,number 3,4
CHEMICALPHYSICSLETTERS
7 September 1990
Solvent slow-mode influence on chemical reaction dynamics: a multidimensional Kramers-theory treatment A.M. Berezhkovskii and V.Yu. Zitserman KarpovInstituteof PhysicalChemistry, UI. Obukha 10.103064 Moscow4, USSR
Received 26 September 1989;in final form 24 April 1990
A theory treating the influence of the heat-bath slow mode on condensed-phase chemical-reactiondynamics is suggested.We have shown that this mode becomes a reaction coordinate in such,situations when friction along this mode becomes extremely highand relaxationenormouslyslow.For calculationof the rate constantin such situations, we have worked out a generalization of the multidimensional Kramers theory for the case of drasticallyanisotropic friction. Basedon this generalizationa new expression for the rate constant is derived. In this expression both the activation energy and the preexponent factor are different from the corresponding quantities in the conventional expression which is inapplicablein this case.
1. Introduction
The influence of the medium on chemical-reaction dynamics is one of the central problems of condensedphase chemical-reaction theory. The common treatment of this phenomenon is based on the assumption of the possibility of division of the solvent with a couple of reacting moleculesin it (a single molecule in the case of isomerization reactions) into a reaction subsystemand a heat bath [ l-31, The reaction subsystem includes a few degrees of freedom of the reagents responsible for chemical conversion. The reagents’other degrees of freedom and all the solvent’s degrees of freedom form the heat bath. Heat-bath influence on reaction subsystem dynamics is taken into account by means of two forces: the fiiction force which, generally speaking, is time-dependent, and a zeio-mean-value random force connected with the friction force by the fluctuation-dissipation theorem. Moreover, in solution a bare potential is substituted for the potential of mean forces. As a result, generalized Langevin equations (GLE) describing the reaction subsystem dynamics arise. In the majority of the papers devoted to the problem under discussion,the authors believe that the reaction subsystemconsists of a single degree of freedom, viz., a reaction coordinate. When heat-bath modes instantly adjust themselves to the reaction-coordinate motion and it is possible to neglect the friction-force retardation, then the rate constant is determined by the familiar Kramers equation (in this paper we shall discuss the case of intermediate and high friction only) [ 1]
Here V(X) is the double-wellpotential along the reaction coordinate x, AV is the difference between the potential energies at the barrier top (situated at the point xb) and at the reagent’s well bottom (situated at the point ;c,), F”’(xb) and V”(xr) are the second derivatives of the potential V(x), T is the temperature expressed in energy units and hKris the only positive root of the equation mxh*t?fxhtV”(X~)=O,
(2)
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where m, and Q are the mass and friction coefficients corresponding to the reaction coordinate. If heat-bath modes adjust themselveswith delay, then the rate constant is determined by the Grote and Hynes equation [ 41 which we can write in the form emphasizing its resemblance to the Framers equation ( 1)
Here h,, is the only positive root of the equation m,h2tqx(h)ht
v(xb)=o,
(4)
where rj,(h) is the Laplace transform of the friction kernel q,.(t), which takes into account the heat-bath adjustment retardation in GLE describing reaction-coordinate motion
1
rj,(h)= rldOexp(-ht) dt
(5)
0
According to the Grote and Hynes theory (GHT), the heat-bath influence on the subsystem reaction dynamics can be entirely taken into account by frequency-dependent friction. At first glance in this theory there is no restriction to the character of heat-bath relaxation. However, this is not quite the same. GHT is based on the assumption that the reaction coordinate is invariant and independent of the heat-bath relaxation processes. In this case, the potential over this coordinate and the process activation energy are also assumed invariant. Further, we shall show that this assumption can fail when there is an extremely slowmode in the heat bath. The existence of such a mode was recently discovered in several molecular-dynamicssimulationsof condensed-phasereaction dynamics [ 5-71. Although quite different subjects were studied in these papers, namely motion of a silver cation in a-AgI in ref. [ 61, the SN2reaction Cl- t CH3Cl+ClCH3t Cl in water in ref. [ 51, and condensed phase isomerization in ref. [ 71, molecular-dynamicssimulations reveal, however, a surprising common peculiarity of these processes. It turns out that in contrast to thermal solute motion when the solvent is a rapid subsystem, solvent adjustment to chemical rearrangementof the reagentproceeds extremelyslowly,so slowlythat the authors speak about frozen media. Such a slow-downof heat-bath adjustment substantiallyinfluences the reaction dynamics. It manifests itself, first of all, in a significant increase in back barrier crossingsby particles which have accomplished transition from the reagent well to the product, When there is a slow mode in a heat bath, the idea arises of considering the mode dynamics on an equal basic with the chemical-coordinate dynamics. This is the path that we shall follow in the present paper. The basic mathematicaltool, which is convenient to employ for this purpose,is the multidimensionalFokker-Planck equation (FPE). In so doing we shall assume that the slownessof the heat-bath mode introduced into central consideration is ensured by extremely great friction along it. The rate-constant calculation with such an approach to the problem is equivalent to the solution of the multidimensional Kramers problem with a highly anisotropic friction. The conventional approach to the rate-constant calculation, based on the multidimensionalFPE, is founded on the following assumptions about the qualitative picture of the process: ( 1) Particles escape from the well via the saddle region. (2) The process is limited by the passagethrough this region, i.e. the barrier dynamics. (3) As this takes place, the Maxwell-Boltzmann equilibrium is maintained in the larger part of the reagent well. As a result, one comes to the followingequation for the rate constant [ 8,9] #I: “’ Equivalenceof GHT based on the conceptof timedependent friction and of the multidimensional consideration leading to eq. (6) was demonstrated in ref. [lo].
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K=
CHEMICAL PHYSICS LE-ITERS
~(&)liZHexp( - y).
7 September1990
(6)
Here AL’is the activation energy which is equal to the difference between the potential energies at the saddle point and at the well bottom, PSpand pr are the matrices of the second derivatives of the multidimensional double-wellpotential at these points, H is the only positive root of the equation (7)
where ti and f are the tensors of the mass and friction coefficients, respectively. However, the multidimensionaltheory leading to eq. (6) for the rate constant (and, consequently,its equivalence to GHT) is not alwaysapplicable. In particular, this theory is inapplicable in the case which is the most interesting for us; when the heat-bath adjustment proceeds slowlyenough but chemical conversion is possible with an unadjusted bath. This has been demonstrated in ref. [ 111, In refs. [ 11,121 we show that the conventional-processpicture, according to which particle flow escapesthe well across the saddle region and is limited by passageof this region, takes place only if friction anisotropy is not too high. When the anisotropy is high enough, the qualitative process picture changes.The flow escapes the well before reaching the saddle and is limited by reaching the transition region, i.e. well dynamics. In this case, the rate constant proves to be much smaller than the conventional equation (6) predicts. In considering a particle-escape from the reagent well in ref. [ 111, we assume that the product well is extremely deep and ignore back crossingsof particles, which have escaped the reagent well. If the second well is not too deep a particle can travel repeatedly between the wells due to the motion along the rapid coordinates for every value of the slow coordinate. As a result, the force acting on a particle along the slow coordinate is averaged and the effective double-wellpotential along this coordinate arises. In such a situation, the multidimensional Kramen problem is reduced to an effective one-dimensionalproblem. In this case, the role of the reaction coordinate is played by the slow coordinate. The effective double-wellpotential along this coordinate has a different height than the activation energy AV of the initial multidimensional potential. It is just this limiting case, corresponding to extremely hindered motion along the slow coordinate, which will be considered in the present paper. More general theory including intermediate situations in particular, those in which the conventional-processpicture takes place, will be considered in our subsequent papers.
2. Effective equation Let us consider the relaxation of the distribution originally localized in one of the wells (we shall call it the reagent well) of the multidimensionaldouble-wellpotential K Let us assumethat on the potential surface there is the only saddle point whose energy counted from the bottom of the reagentswell,AV,is large compared with T.For discussion of the effects determined by friction anisotropy, it is convenient to choose a system of COordinates in which the tensors of the mass and friction coefficients are diagonal. Suppose one of the matrix elements of the tensor fi is larger than the others. Let us designate it by q,,.The coordinate y is slow whereas the other coordinates, whose totality shall be designated as {x}, are fast. With drastically anisotropic friction, the decay of the metastable state proceeds as follows:The fast modes become adjusted to the slow one, and the motion along the latter determines the process kinetics. This means that from the general multidimensional FPE, one can exclude the fast modes reducing it to the effective onedimensional equation. Indeed, with each magnitude of the slowcoordinate y, the distribution over {x}and {a} relaxes towards the Maxwell-Boltzmanndistribution in the potential V((x}, y=const ) t The effective equation is obtained as a result of averagingthe initial multidimensional FPE over such local-equilibriumdistributions and has the form 237
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(8) Here F(y, I, 1) is the distribution function over y and i at instant t, my is the mass corresponding to the ycoordinate, K&y) = -d V&y)/dy is the effective force equal to the force - W( {x},y)/ay averaged over {x} with the Boltzmann weight, i.e. (9) It can be shown that the potential V&(y) is a double-wellcurve and, consequently, calculation of the rate constant is reduced to solution of the one-dimensional Kramers problem in this potential. In the case under consideration, the initial multidimensional potential V({x},y) has such a form that, with the actual y-coordinate values, the surfaces V({x}, y=const) are double-wellsurfaces in the subspace of the {x}-coordinate.Further, we shall assumethat the height of the barrier separatingboth wellsis much larger than T. Let us designate the energies of the first and second wellsas V,(y) and V2(y), respectively. The potentials VI(y) and V,(y) represent one-wellcurves for which the y-coordinate of the minimum of the potential V,(y) coincideswith the y-coordinate at the bottom of the reagent welly,, the coordinate of minimum of the potential V,(y) coinciding with the y-coordinate at the bottom of the product well yP It can be shown by integration of eq. (9)) that in such a potential V({A-}, y), the effective potential has the form V,(y)=-Tln[$$exp(-y)+
&exp(-y)].
(10)
Here K,(y) and K~(y) are the determinants of matrices of second derivatives of the potential V({x},y) over the variables {x} at the bottom of the first and second wells, respectively. The quantity r?fixes the reference zero for energy in the effective potential. It should be noted that in the case under consideration, there are three different time scales which characterize the course of the process. The first scale,the smallest,describesthe fast modes’adjustment to the slow one. The whole multidimensional potential V((x}, y) works on this time scale. On the larger time scales,this potential is replaced by the effective one-dimensionalpotential V,,(y). The adjustment rapidity is guaranteed by high friction anisotropy. If the anisotropy is not high enough, then particle-escapefrom the well keeps its multidimensional character and is described by the conventional rate constant (6). The second time scale characterizes the process of adjustment of the quasistationary distribution in the reagent well of the effective potential V&y). This distribution decays according to the exponential law exp( - Kf), where K is the rate constant which will be calculated in the present paper. The characteristic time K-’ describes the third, largest, time scale. Both the slownessof decay and its single-exponentialcharacter are ensured by the high barrier which separates the reagent well from the product well in the effective potential Vdf(Y).
It will be shown below that the rate constant obtained as a result of solution of the one-dimensionalKramers problem with the potential ( 10) does not coincide either qualitatively or quantitatively with the asjmptotics of eq. (6 ). In particular, the activation energy, the preexponent factor and its relation to the friction coefficients vary. It is necessary to point out that the conventional-processpicture and eq. (6j for the rate constant are correct with any friction anisotropy for those initial multidimensional potentials V({x},y) which have singlewell surfaces V({x), y= const) in the {x}-subspaceat any y-value.
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3. The rate constant calculated on thebasis of the effective equation
The rate constant obtained as a result of solution of the one-dimensionalKramers problem with the potential V,,(Y) has the form [ 1 ]
(11) h=&
[Jtt:+4m,IKk(Yb)I -%I *
(12)
Y
We believe that with Y-valuesclose to Y, the contribution of the second well to the potential V,(Y) ( 10) is infinitesimal. The minimum of the effective potential (corresponding to the initial reagents) is found at the point Y,.The coordinate Ybcorresponds to the barrier top and is obtained from the equation
dV,W + _TdlnK,(y) 0 2 dv
(131
obtained as a result of assumingthe derivative of the potential V,(y) to be equal to zero. The activation energy in the effective potential A&(y) is expressed by the formula Am,
= k’&~b) - K&Y,) v,ca,,v,(,))
=-,ln[Jgexp(-
+ JGGcxp(_
K(h);UYr))]*
(14)
For illustration, let us examine two particular cases: We shall begin with the case of a symmetric potential corresponding to the reactions of the type ABt A-+At BA [ 131.Here the effective potential is symmetric,just as is the initial one, and the quantity Ybcoincides with the magnitude of the y-coordinate at the saddle point y,,. The effective activation energy according to (14) has the form
(15) It is smaller than the activation energy in the conventional expression for the rate constant (6) by the height of the barrier which has to be overcome by the particle in order to escape from the well due to the movement along {x) with y= ySPThis decrease in the activation energy does not mean, however, that the rate constant ( 11) exceeds the asymptotics tty--rco of the conventional expression (6). This problem will be discussed in section 4. In the second case, while considering an arbitrary asymmetric potential, we shall make a number of simplifying assumptions in the course of analysis of eq. (13). Firstly, we shall take into account the fact that the preexponent factors are appreciably smoother functions of y than the corresponding exponents and we shall determine the position of the point yb from the approximate eqUatiOII (16)
V,(Y)=V,(Y) 3
obtained as a result of the requirement that the exponents in eq. (9) be equal. Assuming,moreover, that the potentials V,(y) and V,(Y) have the form (17) i.e. by neglecting the difference between the force constants, we shall find that (fig. 1) h=Yr+f4’-Q/kAY
>
(18)
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Fig. 1.Dependences of the energies correspondingto the bottom of the firstV, (y) and the second V2( y) wells of the section V( (x}, y=const) on the slowcoordinate y.
where AY=Y,-J+,
Q= VI(Y~)-~;(Y,) .
(19)
As a result, we obtain the following equation for the activation energy:
4. Discussion Comparison of our equation for the rate constant ( 11) with the conventional one (6) showsthat both the activation energy and the preexponent factor have changed. Accordingto eq. ( 1 1), the rate constant tends to zero as 1/Q with QMXLThis displays a striking contrast between the new equation ( 11) and the conventional equation ( 6 ) taking the finite value as vY-+co. In order to understand in which casesthe conventional equation (6) and in which the new equation ( 11) for the rate constant should be used, we shall recall the qualitative picture of the process in a situation when the asymptotics tly+co of eq. (6 ) is valid [ 121. In this case, a Boltzmann distribution over y in the potential well V,(y) takes place. This means that the relaxation has enough time for restoration of the distribution over y, disturbed by the outgoing flow of particles caused by the motion along {x}. Thus, the quantity q,, should not be too large if eq. (6) is to apply, In those situations when formula ( 11) is valid, the qualitative picture of the process is quite different. The particle before it leaves the reagent well in the effective potential, repeatedly passes - thanks to the motion along {x} - from the first well into the second one with each value of the coordinate y. This is ensured by an extremely slow movement along y, and is achieved only with large enough magnitudes of q,,.Eq. ( 1 1 ), therefore, is valid only when it predicts values of the rate constant which are definitely smaller than the asymptotics qY+03of eq. (6). It is interesting, in our opinion, to compare our results with GHT. For this reason we note, first of all, that in the literature there exists proof of the equivalence between GHT and the conventional multidimensional 240
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approach which leads to eq. (6) for the rate constant [ lo]. This means that our new solution differs from the GHT results. The difference is caused by a change in the effective reaction coordinate when there is an extremely slow mode in the heat bath. As a result, the parameters of the barrier, the overcoming of which is the essence of the process, change. This fact is taken into account in our approach but is overlooked by GHT. Note that the process picture used in our study reproduces the qualitative aspects of reaction dynamics discovered in moleculardynamics simulations [ 3,571. In particular, it includes multiple transitions between wells in {x}subspaceat fixed y-values due to the motion along the fast {xj-coordinates. Note that our equation ( 11) confirms the fact that the rate constant is independent of friction along the coordinates describing, as a matter of fact, chemical conversion, which was discovered in ref. [ 6 1. In conclusion, let us compare the results of our investigation with those in ref. [ 141 in which the influence of friction anisotropy on the process kinetics is also discussed. The authors of this paper have studied the rate processes in the two-dimensionaldiffusion case. They focus their attention on analysisof the saddle-point dynamics. Since they do so, the possibility of the global process-picture reorganization under which reaction coordinate, activation energy, etc., change, does not come to the authors’attention. The rate-constant equation obtained in ref. [ 141 is a special case of the conventional equation (6) which fails in the limit ‘ty+cc as has been shown above. A critical discussion of ref. [ 141 is presented in our Comment [ 151.
5. Conclusion In this paper we attempt a theoretical study of the influence of the heat-bath slow mode discovered by moleculardynarnic simulation of the condensed-phase reaction dynamics [ 3,571. We consider it expedient to include the heat-bath slow mode in the reaction subsystem and to treat it on equal grounds with those modes which describe, as a matter of fact, the chemical conversion of the reagents. When motion along this heat-bath mode proceeds extremely slowly,it is this mode that becomes an effective one-dimensionalreaction coordinate because in this case the initial multidimensional problem is reduced to the one-dimensional problem about the motion along this coordinate. The effective double-wellpotential ( 10) resulting in this way has a smaller activation energy than the initial multidimensional potential, This fact undoubtedly suggeststhat such a phenomenon has been overlooked by the GHT which, probably, correctly describes the heat-bath influence on reaction dynamics in the case when there is no extremely slow mode in the heat bath. Solvingthe multidimensional Kramers problem, we obtain a new equation for the rate constant ( 11) which is valid for drastically anisotropic friction. It supplements the conventional equation (6) since it operates in a region of parameter values where the latter is inapplicable. According to eq. ( 11 ), the rate constant tends zero like 1iv,, whereas eq. (6) predicts a finite value of the rate constant which is not correct. Let us enumerate, once more, the assumptions which were made in the course of the derivation of a new equation for the rate constant. Firstly, there is the assumption about strong friction anisotropy which allows us to pass from the multidimensional FPE to the effective one-dimensional equation ( 8). Secondly, we have the assumption about the form of the multidimensional potential surface. Accordingto this assumption, a particle can escape from the potential well only due to the motion along the fast coordinates with a fixed value of the slowcoordinate. Finally, our third assumption is also concerned with the form of the potential. It consists of the requirement that the product well should not be too deep. Under this condition, a particle has enough time for repeated passagesbetween wells before it displaces along the slow coordinate. When the second well is deep enough, a particle has no time for return. A corresponding theory was considered in ref. [ 131.
Acknowledgment The authors thank Professor M.V. Basilevskyfor useful discussions on the subject. 241
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References [ 11H. Kramers, Physica 7 ( 1940) 284. [ 21 J.T. Hynes, in: The theory of chemical reactions, Vol. 4, ed. M. Baer (CRC Press, Boca Raton, 1985) p. 171. [3] S.A.Adelman, J. Mol. Liquids 39 ( 1988) 265. [4] R.F. Grote and J.T. Hynes, J. Chem. Phys. 73 ( 1980) 2715. [5]J.P.Bergsma,B.J.Gertner,K.R. WilsonandJ.T.Hynes,J.Chem.Phys.86 (1987) 1356. [6] M.A. Olson and S.A. Adelman, I. Chem. Phys. 83 ( 1985) 1865. [7] S.-B.Zhu and G.W. Robinson, Chem. Phys. Letters 153 (1988) 539. [8] J.S. Langer, Ann Phys. 54 (I 969) 258. [ 91 H.A. Weidenmtiherand Zhang Jing-Shang,J. Stat. Phys. 34 ( 1984) 19I. [lo] E. Guardia, F. Marchesoni and M. San Miguel,Phys. Letters A 100 ( 1984) 15. [ 111A.M. Berezhkovskii,L.M. Berezhkovskiiand V.Yu.Zitserman, Chem. Phys. 130 (1989) 55. [ 121A.M. Berezhkovskiiand V.Yu.Zitserman, Chem. Phys. Letters 158 (1989) 369. [ 131A.M. Berezhkovskiiand V.Yu.Zitserman, Dokl. Akad. Nauk USSR 308 (1989) 1163. [ 141MM. KJosek-Dygas,B.M. Hoffman, B.J. Matkowsky,A. Nitzan, M.A. Ratner andZ. Schuss,J. Chem. Phys. 90 (1989) 1141. [ 15] A.M. Berezhkovskiiand V.Yu. Zitserman, J. Chem. Phys., submitted for publication.
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Solvent slow-mode influence on chemical reaction dynamics: a multidimensional Kramers-theory treatment A.M. Berezhkovskii and V.Yu. Zitserman KarpovInstituteof PhysicalChemistry, UI. Obukha 10.103064 Moscow4, USSR
Received 26 September 1989;in final form 24 April 1990
A theory treating the influence of the heat-bath slow mode on condensed-phase chemical-reactiondynamics is suggested.We have shown that this mode becomes a reaction coordinate in such,situations when friction along this mode becomes extremely highand relaxationenormouslyslow.For calculationof the rate constantin such situations, we have worked out a generalization of the multidimensional Kramers theory for the case of drasticallyanisotropic friction. Basedon this generalizationa new expression for the rate constant is derived. In this expression both the activation energy and the preexponent factor are different from the corresponding quantities in the conventional expression which is inapplicablein this case.
1. Introduction
The influence of the medium on chemical-reaction dynamics is one of the central problems of condensedphase chemical-reaction theory. The common treatment of this phenomenon is based on the assumption of the possibility of division of the solvent with a couple of reacting moleculesin it (a single molecule in the case of isomerization reactions) into a reaction subsystemand a heat bath [ l-31, The reaction subsystem includes a few degrees of freedom of the reagents responsible for chemical conversion. The reagents’other degrees of freedom and all the solvent’s degrees of freedom form the heat bath. Heat-bath influence on reaction subsystem dynamics is taken into account by means of two forces: the fiiction force which, generally speaking, is time-dependent, and a zeio-mean-value random force connected with the friction force by the fluctuation-dissipation theorem. Moreover, in solution a bare potential is substituted for the potential of mean forces. As a result, generalized Langevin equations (GLE) describing the reaction subsystem dynamics arise. In the majority of the papers devoted to the problem under discussion,the authors believe that the reaction subsystemconsists of a single degree of freedom, viz., a reaction coordinate. When heat-bath modes instantly adjust themselves to the reaction-coordinate motion and it is possible to neglect the friction-force retardation, then the rate constant is determined by the familiar Kramers equation (in this paper we shall discuss the case of intermediate and high friction only) [ 1]
Here V(X) is the double-wellpotential along the reaction coordinate x, AV is the difference between the potential energies at the barrier top (situated at the point xb) and at the reagent’s well bottom (situated at the point ;c,), F”’(xb) and V”(xr) are the second derivatives of the potential V(x), T is the temperature expressed in energy units and hKris the only positive root of the equation mxh*t?fxhtV”(X~)=O,
(2)
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where m, and Q are the mass and friction coefficients corresponding to the reaction coordinate. If heat-bath modes adjust themselveswith delay, then the rate constant is determined by the Grote and Hynes equation [ 41 which we can write in the form emphasizing its resemblance to the Framers equation ( 1)
Here h,, is the only positive root of the equation m,h2tqx(h)ht
v(xb)=o,
(4)
where rj,(h) is the Laplace transform of the friction kernel q,.(t), which takes into account the heat-bath adjustment retardation in GLE describing reaction-coordinate motion
1
rj,(h)= rldOexp(-ht) dt
(5)
0
According to the Grote and Hynes theory (GHT), the heat-bath influence on the subsystem reaction dynamics can be entirely taken into account by frequency-dependent friction. At first glance in this theory there is no restriction to the character of heat-bath relaxation. However, this is not quite the same. GHT is based on the assumption that the reaction coordinate is invariant and independent of the heat-bath relaxation processes. In this case, the potential over this coordinate and the process activation energy are also assumed invariant. Further, we shall show that this assumption can fail when there is an extremely slowmode in the heat bath. The existence of such a mode was recently discovered in several molecular-dynamicssimulationsof condensed-phasereaction dynamics [ 5-71. Although quite different subjects were studied in these papers, namely motion of a silver cation in a-AgI in ref. [ 61, the SN2reaction Cl- t CH3Cl+ClCH3t Cl in water in ref. [ 51, and condensed phase isomerization in ref. [ 71, molecular-dynamicssimulations reveal, however, a surprising common peculiarity of these processes. It turns out that in contrast to thermal solute motion when the solvent is a rapid subsystem, solvent adjustment to chemical rearrangementof the reagentproceeds extremelyslowly,so slowlythat the authors speak about frozen media. Such a slow-downof heat-bath adjustment substantiallyinfluences the reaction dynamics. It manifests itself, first of all, in a significant increase in back barrier crossingsby particles which have accomplished transition from the reagent well to the product, When there is a slow mode in a heat bath, the idea arises of considering the mode dynamics on an equal basic with the chemical-coordinate dynamics. This is the path that we shall follow in the present paper. The basic mathematicaltool, which is convenient to employ for this purpose,is the multidimensionalFokker-Planck equation (FPE). In so doing we shall assume that the slownessof the heat-bath mode introduced into central consideration is ensured by extremely great friction along it. The rate-constant calculation with such an approach to the problem is equivalent to the solution of the multidimensional Kramers problem with a highly anisotropic friction. The conventional approach to the rate-constant calculation, based on the multidimensionalFPE, is founded on the following assumptions about the qualitative picture of the process: ( 1) Particles escape from the well via the saddle region. (2) The process is limited by the passagethrough this region, i.e. the barrier dynamics. (3) As this takes place, the Maxwell-Boltzmann equilibrium is maintained in the larger part of the reagent well. As a result, one comes to the followingequation for the rate constant [ 8,9] #I: “’ Equivalenceof GHT based on the conceptof timedependent friction and of the multidimensional consideration leading to eq. (6) was demonstrated in ref. [lo].
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K=
CHEMICAL PHYSICS LE-ITERS
~(&)liZHexp( - y).
7 September1990
(6)
Here AL’is the activation energy which is equal to the difference between the potential energies at the saddle point and at the well bottom, PSpand pr are the matrices of the second derivatives of the multidimensional double-wellpotential at these points, H is the only positive root of the equation (7)
where ti and f are the tensors of the mass and friction coefficients, respectively. However, the multidimensionaltheory leading to eq. (6) for the rate constant (and, consequently,its equivalence to GHT) is not alwaysapplicable. In particular, this theory is inapplicable in the case which is the most interesting for us; when the heat-bath adjustment proceeds slowlyenough but chemical conversion is possible with an unadjusted bath. This has been demonstrated in ref. [ 111, In refs. [ 11,121 we show that the conventional-processpicture, according to which particle flow escapesthe well across the saddle region and is limited by passageof this region, takes place only if friction anisotropy is not too high. When the anisotropy is high enough, the qualitative process picture changes.The flow escapes the well before reaching the saddle and is limited by reaching the transition region, i.e. well dynamics. In this case, the rate constant proves to be much smaller than the conventional equation (6) predicts. In considering a particle-escape from the reagent well in ref. [ 111, we assume that the product well is extremely deep and ignore back crossingsof particles, which have escaped the reagent well. If the second well is not too deep a particle can travel repeatedly between the wells due to the motion along the rapid coordinates for every value of the slow coordinate. As a result, the force acting on a particle along the slow coordinate is averaged and the effective double-wellpotential along this coordinate arises. In such a situation, the multidimensional Kramen problem is reduced to an effective one-dimensionalproblem. In this case, the role of the reaction coordinate is played by the slow coordinate. The effective double-wellpotential along this coordinate has a different height than the activation energy AV of the initial multidimensional potential. It is just this limiting case, corresponding to extremely hindered motion along the slow coordinate, which will be considered in the present paper. More general theory including intermediate situations in particular, those in which the conventional-processpicture takes place, will be considered in our subsequent papers.
2. Effective equation Let us consider the relaxation of the distribution originally localized in one of the wells (we shall call it the reagent well) of the multidimensionaldouble-wellpotential K Let us assumethat on the potential surface there is the only saddle point whose energy counted from the bottom of the reagentswell,AV,is large compared with T.For discussion of the effects determined by friction anisotropy, it is convenient to choose a system of COordinates in which the tensors of the mass and friction coefficients are diagonal. Suppose one of the matrix elements of the tensor fi is larger than the others. Let us designate it by q,,.The coordinate y is slow whereas the other coordinates, whose totality shall be designated as {x}, are fast. With drastically anisotropic friction, the decay of the metastable state proceeds as follows:The fast modes become adjusted to the slow one, and the motion along the latter determines the process kinetics. This means that from the general multidimensional FPE, one can exclude the fast modes reducing it to the effective onedimensional equation. Indeed, with each magnitude of the slowcoordinate y, the distribution over {x}and {a} relaxes towards the Maxwell-Boltzmanndistribution in the potential V((x}, y=const ) t The effective equation is obtained as a result of averagingthe initial multidimensional FPE over such local-equilibriumdistributions and has the form 237
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(8) Here F(y, I, 1) is the distribution function over y and i at instant t, my is the mass corresponding to the ycoordinate, K&y) = -d V&y)/dy is the effective force equal to the force - W( {x},y)/ay averaged over {x} with the Boltzmann weight, i.e. (9) It can be shown that the potential V&(y) is a double-wellcurve and, consequently, calculation of the rate constant is reduced to solution of the one-dimensional Kramers problem in this potential. In the case under consideration, the initial multidimensional potential V({x},y) has such a form that, with the actual y-coordinate values, the surfaces V({x}, y=const) are double-wellsurfaces in the subspace of the {x}-coordinate.Further, we shall assumethat the height of the barrier separatingboth wellsis much larger than T. Let us designate the energies of the first and second wellsas V,(y) and V2(y), respectively. The potentials VI(y) and V,(y) represent one-wellcurves for which the y-coordinate of the minimum of the potential V,(y) coincideswith the y-coordinate at the bottom of the reagent welly,, the coordinate of minimum of the potential V,(y) coinciding with the y-coordinate at the bottom of the product well yP It can be shown by integration of eq. (9)) that in such a potential V({A-}, y), the effective potential has the form V,(y)=-Tln[$$exp(-y)+
&exp(-y)].
(10)
Here K,(y) and K~(y) are the determinants of matrices of second derivatives of the potential V({x},y) over the variables {x} at the bottom of the first and second wells, respectively. The quantity r?fixes the reference zero for energy in the effective potential. It should be noted that in the case under consideration, there are three different time scales which characterize the course of the process. The first scale,the smallest,describesthe fast modes’adjustment to the slow one. The whole multidimensional potential V((x}, y) works on this time scale. On the larger time scales,this potential is replaced by the effective one-dimensionalpotential V,,(y). The adjustment rapidity is guaranteed by high friction anisotropy. If the anisotropy is not high enough, then particle-escapefrom the well keeps its multidimensional character and is described by the conventional rate constant (6). The second time scale characterizes the process of adjustment of the quasistationary distribution in the reagent well of the effective potential V&y). This distribution decays according to the exponential law exp( - Kf), where K is the rate constant which will be calculated in the present paper. The characteristic time K-’ describes the third, largest, time scale. Both the slownessof decay and its single-exponentialcharacter are ensured by the high barrier which separates the reagent well from the product well in the effective potential Vdf(Y).
It will be shown below that the rate constant obtained as a result of solution of the one-dimensionalKramers problem with the potential ( 10) does not coincide either qualitatively or quantitatively with the asjmptotics of eq. (6 ). In particular, the activation energy, the preexponent factor and its relation to the friction coefficients vary. It is necessary to point out that the conventional-processpicture and eq. (6j for the rate constant are correct with any friction anisotropy for those initial multidimensional potentials V({x},y) which have singlewell surfaces V({x), y= const) in the {x}-subspaceat any y-value.
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3. The rate constant calculated on thebasis of the effective equation
The rate constant obtained as a result of solution of the one-dimensionalKramers problem with the potential V,,(Y) has the form [ 1 ]
(11) h=&
[Jtt:+4m,IKk(Yb)I -%I *
(12)
Y
We believe that with Y-valuesclose to Y, the contribution of the second well to the potential V,(Y) ( 10) is infinitesimal. The minimum of the effective potential (corresponding to the initial reagents) is found at the point Y,.The coordinate Ybcorresponds to the barrier top and is obtained from the equation
dV,W + _TdlnK,(y) 0 2 dv
(131
obtained as a result of assumingthe derivative of the potential V,(y) to be equal to zero. The activation energy in the effective potential A&(y) is expressed by the formula Am,
= k’&~b) - K&Y,) v,ca,,v,(,))
=-,ln[Jgexp(-
+ JGGcxp(_
K(h);UYr))]*
(14)
For illustration, let us examine two particular cases: We shall begin with the case of a symmetric potential corresponding to the reactions of the type ABt A-+At BA [ 131.Here the effective potential is symmetric,just as is the initial one, and the quantity Ybcoincides with the magnitude of the y-coordinate at the saddle point y,,. The effective activation energy according to (14) has the form
(15) It is smaller than the activation energy in the conventional expression for the rate constant (6) by the height of the barrier which has to be overcome by the particle in order to escape from the well due to the movement along {x) with y= ySPThis decrease in the activation energy does not mean, however, that the rate constant ( 11) exceeds the asymptotics tty--rco of the conventional expression (6). This problem will be discussed in section 4. In the second case, while considering an arbitrary asymmetric potential, we shall make a number of simplifying assumptions in the course of analysis of eq. (13). Firstly, we shall take into account the fact that the preexponent factors are appreciably smoother functions of y than the corresponding exponents and we shall determine the position of the point yb from the approximate eqUatiOII (16)
V,(Y)=V,(Y) 3
obtained as a result of the requirement that the exponents in eq. (9) be equal. Assuming,moreover, that the potentials V,(y) and V,(Y) have the form (17) i.e. by neglecting the difference between the force constants, we shall find that (fig. 1) h=Yr+f4’-Q/kAY
>
(18)
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Fig. 1.Dependences of the energies correspondingto the bottom of the firstV, (y) and the second V2( y) wells of the section V( (x}, y=const) on the slowcoordinate y.
where AY=Y,-J+,
Q= VI(Y~)-~;(Y,) .
(19)
As a result, we obtain the following equation for the activation energy:
4. Discussion Comparison of our equation for the rate constant ( 11) with the conventional one (6) showsthat both the activation energy and the preexponent factor have changed. Accordingto eq. ( 1 1), the rate constant tends to zero as 1/Q with QMXLThis displays a striking contrast between the new equation ( 11) and the conventional equation ( 6 ) taking the finite value as vY-+co. In order to understand in which casesthe conventional equation (6) and in which the new equation ( 11) for the rate constant should be used, we shall recall the qualitative picture of the process in a situation when the asymptotics tly+co of eq. (6 ) is valid [ 121. In this case, a Boltzmann distribution over y in the potential well V,(y) takes place. This means that the relaxation has enough time for restoration of the distribution over y, disturbed by the outgoing flow of particles caused by the motion along {x}. Thus, the quantity q,, should not be too large if eq. (6) is to apply, In those situations when formula ( 11) is valid, the qualitative picture of the process is quite different. The particle before it leaves the reagent well in the effective potential, repeatedly passes - thanks to the motion along {x} - from the first well into the second one with each value of the coordinate y. This is ensured by an extremely slow movement along y, and is achieved only with large enough magnitudes of q,,.Eq. ( 1 1 ), therefore, is valid only when it predicts values of the rate constant which are definitely smaller than the asymptotics qY+03of eq. (6). It is interesting, in our opinion, to compare our results with GHT. For this reason we note, first of all, that in the literature there exists proof of the equivalence between GHT and the conventional multidimensional 240
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approach which leads to eq. (6) for the rate constant [ lo]. This means that our new solution differs from the GHT results. The difference is caused by a change in the effective reaction coordinate when there is an extremely slow mode in the heat bath. As a result, the parameters of the barrier, the overcoming of which is the essence of the process, change. This fact is taken into account in our approach but is overlooked by GHT. Note that the process picture used in our study reproduces the qualitative aspects of reaction dynamics discovered in moleculardynamics simulations [ 3,571. In particular, it includes multiple transitions between wells in {x}subspaceat fixed y-values due to the motion along the fast {xj-coordinates. Note that our equation ( 11) confirms the fact that the rate constant is independent of friction along the coordinates describing, as a matter of fact, chemical conversion, which was discovered in ref. [ 6 1. In conclusion, let us compare the results of our investigation with those in ref. [ 141 in which the influence of friction anisotropy on the process kinetics is also discussed. The authors of this paper have studied the rate processes in the two-dimensionaldiffusion case. They focus their attention on analysisof the saddle-point dynamics. Since they do so, the possibility of the global process-picture reorganization under which reaction coordinate, activation energy, etc., change, does not come to the authors’attention. The rate-constant equation obtained in ref. [ 141 is a special case of the conventional equation (6) which fails in the limit ‘ty+cc as has been shown above. A critical discussion of ref. [ 141 is presented in our Comment [ 151.
5. Conclusion In this paper we attempt a theoretical study of the influence of the heat-bath slow mode discovered by moleculardynarnic simulation of the condensed-phase reaction dynamics [ 3,571. We consider it expedient to include the heat-bath slow mode in the reaction subsystem and to treat it on equal grounds with those modes which describe, as a matter of fact, the chemical conversion of the reagents. When motion along this heat-bath mode proceeds extremely slowly,it is this mode that becomes an effective one-dimensionalreaction coordinate because in this case the initial multidimensional problem is reduced to the one-dimensional problem about the motion along this coordinate. The effective double-wellpotential ( 10) resulting in this way has a smaller activation energy than the initial multidimensional potential, This fact undoubtedly suggeststhat such a phenomenon has been overlooked by the GHT which, probably, correctly describes the heat-bath influence on reaction dynamics in the case when there is no extremely slow mode in the heat bath. Solvingthe multidimensional Kramers problem, we obtain a new equation for the rate constant ( 11) which is valid for drastically anisotropic friction. It supplements the conventional equation (6) since it operates in a region of parameter values where the latter is inapplicable. According to eq. ( 11 ), the rate constant tends zero like 1iv,, whereas eq. (6) predicts a finite value of the rate constant which is not correct. Let us enumerate, once more, the assumptions which were made in the course of the derivation of a new equation for the rate constant. Firstly, there is the assumption about strong friction anisotropy which allows us to pass from the multidimensional FPE to the effective one-dimensional equation ( 8). Secondly, we have the assumption about the form of the multidimensional potential surface. Accordingto this assumption, a particle can escape from the potential well only due to the motion along the fast coordinates with a fixed value of the slowcoordinate. Finally, our third assumption is also concerned with the form of the potential. It consists of the requirement that the product well should not be too deep. Under this condition, a particle has enough time for repeated passagesbetween wells before it displaces along the slow coordinate. When the second well is deep enough, a particle has no time for return. A corresponding theory was considered in ref. [ 131.
Acknowledgment The authors thank Professor M.V. Basilevskyfor useful discussions on the subject. 241
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References [ 11H. Kramers, Physica 7 ( 1940) 284. [ 21 J.T. Hynes, in: The theory of chemical reactions, Vol. 4, ed. M. Baer (CRC Press, Boca Raton, 1985) p. 171. [3] S.A.Adelman, J. Mol. Liquids 39 ( 1988) 265. [4] R.F. Grote and J.T. Hynes, J. Chem. Phys. 73 ( 1980) 2715. [5]J.P.Bergsma,B.J.Gertner,K.R. WilsonandJ.T.Hynes,J.Chem.Phys.86 (1987) 1356. [6] M.A. Olson and S.A. Adelman, I. Chem. Phys. 83 ( 1985) 1865. [7] S.-B.Zhu and G.W. Robinson, Chem. Phys. Letters 153 (1988) 539. [8] J.S. Langer, Ann Phys. 54 (I 969) 258. [ 91 H.A. Weidenmtiherand Zhang Jing-Shang,J. Stat. Phys. 34 ( 1984) 19I. [lo] E. Guardia, F. Marchesoni and M. San Miguel,Phys. Letters A 100 ( 1984) 15. [ 111A.M. Berezhkovskii,L.M. Berezhkovskiiand V.Yu.Zitserman, Chem. Phys. 130 (1989) 55. [ 121A.M. Berezhkovskiiand V.Yu.Zitserman, Chem. Phys. Letters 158 (1989) 369. [ 131A.M. Berezhkovskiiand V.Yu.Zitserman, Dokl. Akad. Nauk USSR 308 (1989) 1163. [ 141MM. KJosek-Dygas,B.M. Hoffman, B.J. Matkowsky,A. Nitzan, M.A. Ratner andZ. Schuss,J. Chem. Phys. 90 (1989) 1141. [ 15] A.M. Berezhkovskiiand V.Yu. Zitserman, J. Chem. Phys., submitted for publication.
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