- Email: [email protected]
Friction dependence of reaction rates in simple barrierless reactions
8 April 1994
CHEMICAL PHysK=s LETTERS ELSEVIER
Chemical Physics Letters 220 (1994) 353-358
Friction dependence of reaction rates in simple barrierless reactions G.V. Raviprasad, A.M. Jayannavar Instituteof Physics,SachivalayaMarg, Bhubaneswar751005, India
Received 8 November 1993; in fina form 10 February 1994
AhStllWt
We have investigated the dynamics of simple chemical reactions which proceed without an activation barrier along the reaction coordinate. In the absence of the barrier the solvent friction is the only impediment to the reactive motion. By numerical simulation we show that in the low-friction limit the reaction rate increases as a fractional power of the friction coefftcient. The power law dependence is sensitive to the initial conditions of the reactive coordinate. After exhibiting a maximum, the behaviour crosses over to that of inverse friction dependence in the high-friction limit. We have compared our results with earlier approximate analytical treatments and differences are pointed out.
1. IntrodIlction
Recent advances in ultrafast laser spectroscopy have made it possible to probe directly the dynamical processes involved in reaction dynamics of molecular systems [ 11. Spectroscopic methods now allow us to observe the complex motion of the reacting molecules from the reactants to transition state to products over time scales ranging from 10 fs to 10 ps. These fundamental time scales are dictated by the rapidity of the nuclear or electronic rearrangements in the transition state. Most reactions of importance in chemistry and biology occur in a liquid medium. The host solvent provides energy to the reacting molecules as well as impeding the motion of the reactants along the reaction coordinate. The influence of solvent on the reaction dynamics can be incorporated via solvent friction and concomitant fluctuations arising due to the exchange of momentum and energy between the reactants and their neighbouring host molecules. The role of the solvent friction on chemical reactions in liquids has been studied extensively over several decades.
One of the most direct ways of studying the influence of solvent friction on a reaction in solution is to study a reaction that has no activation barrier. Several barrierless intramolecular electron-transfer reactions have been studied by using various spectroscopic techniques [2,3]. These reactions are the simplest to treat theoretically as compared to traditional chemical reactions which involve the motion of the reaction coordinate over the barrier [ 4-71. Moreover, they offer the possibility of observing directly the motion of a reacting molecule along the reaction coordinate. In the absence of a barrier, solvent friction is the only impediment to the reaction. These reactions are very fast as there is no time-scale separation between the reactive motion and the rate of reaction. The earliest theoretical treatment of barrierless reactions was due to Oster and Nishijima [ 8 1, motivated to explain the observed viscosity dependence in the non-radiative decay of triphenyl rings. This model assumes that the reactive motion is the rotational diffusion of the reactants on a flat excitedstate potential energy surface, i.e. it assumes the rotational diffusion of phenyl rings. The non-radiative
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDZOOO9-2614(94)00196-W
354
G, K Raviprasad,A.M. Jayannavar/Chemical PhysicsLetters220 (I 994) 353-358
decay of the excited state occurs only from the definite conformation of phenyl rings. This implies that the phenyl rings must diffuse a certain distance from their initial configuration to reach a final configuration which is strongly non-radiatively coupled to the ground state. The final configuration on a reaction coordinate acts as a photochemical sink where decay occurs from the excited-state surface with unit probability. The systematic theoretical studies of barrierless reactions were initiated by Bagchi et al. [ 41, who introduced certain solvable models with definitive predictions which can be verified experimentally. They carried out a detailed analytical and numerical study of zero-barrier reactions by solving different models of the sink. The treatment has also been extended to the case where the reaction surface is a harmonic potential and for finite decay at a reactive sink which is placed at a position depending on the details of the ground-state and excited-state potential. The predictions of these models differ from each other most dramatically in the temporal form of the excited-state population decay, rate constants and fluorescence quantum yield. So far, most theoretical treatments have been in the high-friction (y) limit. In this regime the reaction rate (k) decreases as 1/y. However, the absolute value of the reaction rate is sensitive to the initial distribution of the reaction coordinate. In the Oster-Nishijima model the dynamics are given by a one-dimensional diffusion equation for the reaction coordinate x, namely
(1) where P(x, t) is the probability distribution function and D is the diffusion constant (D= k,T/ y, y the relevant friction coefficient). Eq. ( 1) has to be solved in the presence of two absorbing boundaries. Assuming the absorbing boundaries to be located at x=0 and at x= a, one imposes the Smoluchowski boundary condition [4-71, namely P(x=O, t)=P(x=a, t) ~0. If the initial position of the reaction coordinate is x0, then the time (t) evolution of the excited state population is given by a series expansion,
(2) The integrated excited-state population as a function of time is given by sin( a(2n]: 1)x0) P(t)=
$ox(2n4+1)
X exp (
?rZ(2n+l)ZDt a2
>*
Eq. (3) essentially leads to a two-parameter theory. Initially the decay is non-exponential (or a sum of exponentials) which crosses over to the pure exponential form. There is only one time scale related to the decay rate, namely the time required to diffuse a distance of length a. The dependence of temperature and viscosity on the population decay or reaction rate is expressed in terms of a simple scaling function f(kBT/j%.x2). The long time decay rate is inversely proportional to the viscosity coefficient, i.e. the decay time varies linearly with viscosity. Ben-Amotz and Harris [ 2,3] have shown that the excited-state population decay observed in the crystal violet can be better fitted to Eq. (3). However, the decay rate law (3) is not consistent with the observed linear dependence of viscosity on decay time for low viscosities with a turnover to non-linear viscosity dependence at high viscosities. Motivated from the fact that the Oster-Nishijima model agrees with some experimental data, one of us extended this model to the low-viscosity (or inertial) regime [ 91. For small molecules, over short times typically less than the inverse of the friction coefficient, a significant amount of free or inertial motions are possible and can influence reaction rates nontrivially. In fact it has been shown (via an approximate treatment valid over the entire friction domain) that the transient behaviour of the integrated excited-state population is very sensitive to the initial velocity profile of the reaction coordinate and can lead to a non-linear dependence of viscosity on the reaction rate [ 9 1. In the above-mentioned treatment the motion of a particle is modelled by a dynamical evolution governed by the Langevin equation [ 10 1,
G. K Ravipmsad, A.M. Jayannavar I Chemical Physics Letters 220 (I 994) 353-358
m.f= -yi+f(t)
.
(4)
The friction coefficient y and the concomitant Gaussian white noisef( t) are related via the fluctuationdissipation theorem as (5a)
=O, (f(t)f(t’)>
==B7,@(f-f)
,
(5b)
where ( ) represents the ensemble average over all the realisations of a random force f(t), T the temperature, m the mass and kB the Boltzmann constant. Eq. (4) can be converted into the equation for a full phase-space probability distribution function P(x, u, t), where u is the velocity of the particle (~=a). The full phase-space Kramers equation is given by
(6) Using the well-known solution to the above equation one can readily write the evolution equation for the marginal probability distribution J”P(x, v, t) dv. By imposing an absorption boundary condition on the marginal probability distribution function at x= 0 and at x= a, the time evolution of the excited-state population has been obtained and is given by [ 91
0x9 t) =[i $, sin(y) Xsin(y)exp(-%?)I X exp 8-‘v0(l-e-@) ( F(t) x exp -/3%;(
(x--x0)
1-e-Br)2 2F(t) >’
> (7)
where F(t)=qj?-3(2gt-3+4e-8t-e-ZB’), jl=y/m and q= j?kBT/m. It should be noted that the time evolution of P(x, t) depends on the initial velocity v. of the particle. This feature stems from the non-Markovian property of the marginal distribution function. Unlike Eq. (2), Eq. (7) is valid over the entire friction domain. In the high-friction regime, i.e. /3tZ+ 1, one recovers Eq. ( 2 ) from Eq. ( 7 ) . Now one can obtain the required integrated excited-state population P(t) from Eq. (7) as
P(t)=
j,
355
4 dv,j:dxP(x,r,x,,,vo)
0
0
--oo
xP(~oMxo)
9
(8)
where p(xo) and p( vo) are the probability densities of the initial position and velocity of the reaction coordinate, respectively, at time t= 0. As there is no clear separation of time scales between the motion in the reactive region and the rest of the particle surface, Bag&i et al. [4] have introduced two rate constants. The integrated or time averaged rate constant kI is defined as 00 kc’=
I
P(t) dt,
(9)
0
where P(t) is the total population remaining at the excited-state surface at time t after excitation. The long time rate constant kL is defined through the long time limit of P( t) as (10)
The two rate constants can differ significantly from each other. In our analysis we work with a rate constant as defined in Eq. ( 9 ) . In our present work we have carried out a numerical simulation of the Langevin dynamics of a particle evolving according to Eq. ( 4) in the presence of two sinks. These absorbing sinks are placed at a distance ‘a’ apart and act as perfect absorbers, in the sense that the reaction event takes place as soon as the reactant reaches either of the sink positions for the first time (perfect absorbers). We obtain the non-monotonic relationship between the rate and the friction coefficient. We show that in the low-friction regime the reaction rate depends sensitively on the initial conditions, i.e. on the velocity and position distribution of the reactant. In this regime the reaction rate increases as rd, 6> 0 and depends on the initial conditions. After exhibiting a maximum the behaviour crosses over to 1/yin the high friction limit. We compare our results with earlier analytical calculations. In an earlier study Bagchi et al. [ 111 have carried out some quantitative numerical calculations for the case of a particle motion in a harmonic well. They have solved the FokkerPlanck equation with the sink by generating the sto-
G. K Raviprasad,A.M. Jayannavar/ ChemicalPhysicsLetters220 (1994) 353-358
356
chastic trajectories. They have also shown that at low viscosities, the rate exhibits a non-monotonic dependence on viscosity for a Gaussian sink. Our treatment differs from Bagchi et al., where we emphasize the role of the initial velocity distribution of the reaction coordinate on the rate constant and we invoke no potential at all along the reaction coordinate. In section 2 we briefly discuss our numerical procedure and in section 3 we present our results.
( = 10000) and N(t) is the number of trajectories crossing either of the boundaries (which are at a distance a apart) for the first time at time t. For simplicity in all our calculations we have set units of mass ( m ) , temperature ( T) and distance (a) between the two absorbers to be unity. We have also compared the results in the case of a potential-free region in onedimension without boundaries with the known solution of the Fokker-Planck equation [ lo] and verilied that the numerical simulation is in good agreement with the exact analytical solution.
2. Numerical procedure We have solved the Langevin equation in one-dimension without approximation using an iteration method. The Langevin equation in a potential-free region is given by Eq. (4). We have followed exactly the same procedure as that of Wada et al. [ 121. Numerical integration of the Langevin equation is performed over a very small time step T. The change in the position and momentum are given by t+? x(t+r)-x(t)=
dti,
(11)
m[i(t+.r)-i(t)1
=
I*
dt'
[
--yi(t’)+f(t’)]
,
*
& I dt’N(t’)
,
(13)
0
where No is the total
number
In Figs. l-3 we have plotted the reaction rate versus friction coefficient, both on the logarithmic scale for three different initial conditions. We have assumed for simplicity the initial distribution of position (x0) and velocity ( vo) to be decoupled and independently given by: case (i) p(xo)=6(x-fa), p( v. ) = 6( vo), which corresponds to a particle placed initially at the mid-point with zero velocity, case (ii) p(xo)=l/a and p(vo)=S(vo), corresponding to a situation where the particle is initially at rest and can be found anywhere in the region 0
(12)
where f( t’) is a computer generated uncorrelated Gaussian stochastic process with statistics as given in Eqs. (5a) and (5b). We first fix an initial position and velocity of the particle consistent with the chosen initial distribution function. We have taken three different cases as mentioned below for this distribution. We then iteratively follow the trajectory of the particle. As soon as the particle reaches either of the boundaries after a time t, we terminate the event. In a sense we have calculated the first passage time distribution after averaging over 10000 realizations for each case. The normalized survival probability P(t) as a function of time is given by P(t)=l-
3. Results and discussion
of realizations
0.01
/
0.001
““‘I’,
,“““I
0.01
“““‘I
3 ,a”“1
0.1
1
f
10
“““‘1
100
r/r* Fii 1. The dimensionless rate constant (k/k*) is plotted against the dimensionless friction parameter (y/y*) for case (i), where p(&=d(x-$a) and p(b)=d(g). Here +&%/a and k*= y*/m.
G. V. Raviprasad,A.M. Jayannavar/Chemical PhysicsLetters220 (1994) 353-358
0.01
I
1
0.001
llrrml
lllliil,
1111111,
0.01
0.1
1
1
Il1811/,
/
10
I
llillll,
100
T/Y’ Fig. 2. The dimensionless rate constant (k/k*) is plotted against the dimensionless friction parnmeter (y/y*) for case (ii), where p(~)=l/aandp(u,,)=6(u,-,).Herey*=&%/aandk*=y*/ m.
Fig. 3. The dimensionless rate constant (k/k’) is plotted against the dimensionless f&ion parameter (y/y* ) for case (iii ) , where p&)=1/a and p(u,,)=,/wexp(-mv$12kBT). Here y*=,&%/a and k*=y*/m.
probability, case (iii) corresponds to an equal probability for the particle to be found in the region with Odxo
357
mal equilibrium velocity distribution, i.e. p(va)=,/wexp( -mv$/2kJ). In the figures, the asterisks and the dashed lines represent results based on our numerical and analytical work, Eqs. ( 7)- (9 ) , respectively, and the full lines represents the earlier analytical results based on Eq. (3). Both numerical and analytical results agree onl in the highfriction limit (y larger than typically P kB Tm/a). In this regime the reaction rate decreases as k- y - ‘. In the low-friction limit k increases as y”, where 6 equals 0.3, 0.28 and 0.065 for cases (i), (ii) and (iii), respectively. This is the leading behaviour at low fiio tion apart from an additive constant. This constant takes a value zero for case (i) and (ii). In the limit y-+0, the reaction coordinate decouples from the surrounding medium and for a situation where the initial velocity of the particle is zero the coordinate remains at rest for all times and consequently the reaction rate tends to zero as y+ 0. For case (iii), there is a finite value for k as y+ 0, i.e. the particle is mobile in the absence of coupling to the surrounding medium and can decay on reaching the boundaries. After exhibiting a maximum, the rate crosses over to the inverse friction dependence in the high-friction limit. Our earlier analytical treatment valid over the entire friction domain gives somewhat higher rates as compared to our numerical results, but still shows the nonmonotonic relationship between the rate and the fiiction coefficient. However, results based on the Smoluchowski equation (3) are found to overestimate the rate significantly in the low-friction region and fail to predict the non-monotonic dependence of rate on the friction coefficient. In conclusion, we have shown that the reaction rate in a simple barrierless chemical reaction in the lowfriction limit depends sensitively on the initial position and velocity profile of the reacting particles. In the low-friction limit the reaction rate increases as y6, 6>0 and depends on the initial conditions. In our treatment we have assumed the photochemical sinks to be perfect ones, where the reaction or decay of an excited state occurs with unit probability as soon as the reaction coordinate reaches the sink position. Even in the high-friction limit earlier results have predicted a fractional power dependence of the rates on the friction in the presence of imperfect sinks [ 41. In these models every time the reaction coordinate comes into contact with the sink the particle decays
358
G. V. Raviprasad, A.M. Jayannavar /Chemical Physics Letters 220 (1994) 353-358
at a rate b( &HX corresponds to our instantaneous death model), and for intermediate values of ko fractional behaviour has been predicted. We expect the barrierless reactions to exhibit rich and diverse dynamical behaviour in the presence of imperfect absorbers in the low-friction limit. These results will be different from the high-barrier reactions.
References [ 1 ] Special issue on dynamics of molecular systems, Phys. Today 43, No. 5 (1990). [2] D. Ben-Amotz and C.B. Harris, J. Chem. Phys. 86 (1987) 4856.
[ 31 D. Ben-Amotx and C.B. Harris, J. Chem. Phys. 86 (1987) 5433. [4] B. Bag&i, G.R. Fleming and D.W. Oxtoby, J. Chem. Phys. 78 (1983) 7375. [5] B. Bag&i, Chem. Phys. Letters 135 (1987) 558. [6] B. Bagchi, J. Chem. Phys. 87 (1987) 5393. [ 71 B. Bagchi and G.R. Fleming, J. Phys. Chem. 94 (1990) 9. [8] G. Oster and Y. Nishijima, J. Am. Chem. Sot. 78 (1956) 1581. [ 91 A.M. Jayannavar, Chem. Phys. Letters 199 ( 1992) 149. [lo] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. [ 111 B. Bagchi, S. Singer and D.W. Oxtoby, Chem. Phys. Letters 99 (1983) 225. [ 121 T. Wada, N. Carjan and Y. Abe, Proceedings of the Tours Symposium on Nuclear Physics, France (World Scientific, Singapore, 1991) p. 136; T. Wada, N. Carjan and Y. Abe, Nucl. Phys. A 538 (1992) 283~.
CHEMICAL PHysK=s LETTERS ELSEVIER
Chemical Physics Letters 220 (1994) 353-358
Friction dependence of reaction rates in simple barrierless reactions G.V. Raviprasad, A.M. Jayannavar Instituteof Physics,SachivalayaMarg, Bhubaneswar751005, India
Received 8 November 1993; in fina form 10 February 1994
AhStllWt
We have investigated the dynamics of simple chemical reactions which proceed without an activation barrier along the reaction coordinate. In the absence of the barrier the solvent friction is the only impediment to the reactive motion. By numerical simulation we show that in the low-friction limit the reaction rate increases as a fractional power of the friction coefftcient. The power law dependence is sensitive to the initial conditions of the reactive coordinate. After exhibiting a maximum, the behaviour crosses over to that of inverse friction dependence in the high-friction limit. We have compared our results with earlier approximate analytical treatments and differences are pointed out.
1. IntrodIlction
Recent advances in ultrafast laser spectroscopy have made it possible to probe directly the dynamical processes involved in reaction dynamics of molecular systems [ 11. Spectroscopic methods now allow us to observe the complex motion of the reacting molecules from the reactants to transition state to products over time scales ranging from 10 fs to 10 ps. These fundamental time scales are dictated by the rapidity of the nuclear or electronic rearrangements in the transition state. Most reactions of importance in chemistry and biology occur in a liquid medium. The host solvent provides energy to the reacting molecules as well as impeding the motion of the reactants along the reaction coordinate. The influence of solvent on the reaction dynamics can be incorporated via solvent friction and concomitant fluctuations arising due to the exchange of momentum and energy between the reactants and their neighbouring host molecules. The role of the solvent friction on chemical reactions in liquids has been studied extensively over several decades.
One of the most direct ways of studying the influence of solvent friction on a reaction in solution is to study a reaction that has no activation barrier. Several barrierless intramolecular electron-transfer reactions have been studied by using various spectroscopic techniques [2,3]. These reactions are the simplest to treat theoretically as compared to traditional chemical reactions which involve the motion of the reaction coordinate over the barrier [ 4-71. Moreover, they offer the possibility of observing directly the motion of a reacting molecule along the reaction coordinate. In the absence of a barrier, solvent friction is the only impediment to the reaction. These reactions are very fast as there is no time-scale separation between the reactive motion and the rate of reaction. The earliest theoretical treatment of barrierless reactions was due to Oster and Nishijima [ 8 1, motivated to explain the observed viscosity dependence in the non-radiative decay of triphenyl rings. This model assumes that the reactive motion is the rotational diffusion of the reactants on a flat excitedstate potential energy surface, i.e. it assumes the rotational diffusion of phenyl rings. The non-radiative
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDZOOO9-2614(94)00196-W
354
G, K Raviprasad,A.M. Jayannavar/Chemical PhysicsLetters220 (I 994) 353-358
decay of the excited state occurs only from the definite conformation of phenyl rings. This implies that the phenyl rings must diffuse a certain distance from their initial configuration to reach a final configuration which is strongly non-radiatively coupled to the ground state. The final configuration on a reaction coordinate acts as a photochemical sink where decay occurs from the excited-state surface with unit probability. The systematic theoretical studies of barrierless reactions were initiated by Bagchi et al. [ 41, who introduced certain solvable models with definitive predictions which can be verified experimentally. They carried out a detailed analytical and numerical study of zero-barrier reactions by solving different models of the sink. The treatment has also been extended to the case where the reaction surface is a harmonic potential and for finite decay at a reactive sink which is placed at a position depending on the details of the ground-state and excited-state potential. The predictions of these models differ from each other most dramatically in the temporal form of the excited-state population decay, rate constants and fluorescence quantum yield. So far, most theoretical treatments have been in the high-friction (y) limit. In this regime the reaction rate (k) decreases as 1/y. However, the absolute value of the reaction rate is sensitive to the initial distribution of the reaction coordinate. In the Oster-Nishijima model the dynamics are given by a one-dimensional diffusion equation for the reaction coordinate x, namely
(1) where P(x, t) is the probability distribution function and D is the diffusion constant (D= k,T/ y, y the relevant friction coefficient). Eq. ( 1) has to be solved in the presence of two absorbing boundaries. Assuming the absorbing boundaries to be located at x=0 and at x= a, one imposes the Smoluchowski boundary condition [4-71, namely P(x=O, t)=P(x=a, t) ~0. If the initial position of the reaction coordinate is x0, then the time (t) evolution of the excited state population is given by a series expansion,
(2) The integrated excited-state population as a function of time is given by sin( a(2n]: 1)x0) P(t)=
$ox(2n4+1)
X exp (
?rZ(2n+l)ZDt a2
>*
Eq. (3) essentially leads to a two-parameter theory. Initially the decay is non-exponential (or a sum of exponentials) which crosses over to the pure exponential form. There is only one time scale related to the decay rate, namely the time required to diffuse a distance of length a. The dependence of temperature and viscosity on the population decay or reaction rate is expressed in terms of a simple scaling function f(kBT/j%.x2). The long time decay rate is inversely proportional to the viscosity coefficient, i.e. the decay time varies linearly with viscosity. Ben-Amotz and Harris [ 2,3] have shown that the excited-state population decay observed in the crystal violet can be better fitted to Eq. (3). However, the decay rate law (3) is not consistent with the observed linear dependence of viscosity on decay time for low viscosities with a turnover to non-linear viscosity dependence at high viscosities. Motivated from the fact that the Oster-Nishijima model agrees with some experimental data, one of us extended this model to the low-viscosity (or inertial) regime [ 91. For small molecules, over short times typically less than the inverse of the friction coefficient, a significant amount of free or inertial motions are possible and can influence reaction rates nontrivially. In fact it has been shown (via an approximate treatment valid over the entire friction domain) that the transient behaviour of the integrated excited-state population is very sensitive to the initial velocity profile of the reaction coordinate and can lead to a non-linear dependence of viscosity on the reaction rate [ 9 1. In the above-mentioned treatment the motion of a particle is modelled by a dynamical evolution governed by the Langevin equation [ 10 1,
G. K Ravipmsad, A.M. Jayannavar I Chemical Physics Letters 220 (I 994) 353-358
m.f= -yi+f(t)
.
(4)
The friction coefficient y and the concomitant Gaussian white noisef( t) are related via the fluctuationdissipation theorem as (5a)
==B7,@(f-f)
,
(5b)
where ( ) represents the ensemble average over all the realisations of a random force f(t), T the temperature, m the mass and kB the Boltzmann constant. Eq. (4) can be converted into the equation for a full phase-space probability distribution function P(x, u, t), where u is the velocity of the particle (~=a). The full phase-space Kramers equation is given by
(6) Using the well-known solution to the above equation one can readily write the evolution equation for the marginal probability distribution J”P(x, v, t) dv. By imposing an absorption boundary condition on the marginal probability distribution function at x= 0 and at x= a, the time evolution of the excited-state population has been obtained and is given by [ 91
0x9 t) =[i $, sin(y) Xsin(y)exp(-%?)I X exp 8-‘v0(l-e-@) ( F(t) x exp -/3%;(
(x--x0)
1-e-Br)2 2F(t) >’
> (7)
where F(t)=qj?-3(2gt-3+4e-8t-e-ZB’), jl=y/m and q= j?kBT/m. It should be noted that the time evolution of P(x, t) depends on the initial velocity v. of the particle. This feature stems from the non-Markovian property of the marginal distribution function. Unlike Eq. (2), Eq. (7) is valid over the entire friction domain. In the high-friction regime, i.e. /3tZ+ 1, one recovers Eq. ( 2 ) from Eq. ( 7 ) . Now one can obtain the required integrated excited-state population P(t) from Eq. (7) as
P(t)=
j,
355
4 dv,j:dxP(x,r,x,,,vo)
0
0
--oo
xP(~oMxo)
9
(8)
where p(xo) and p( vo) are the probability densities of the initial position and velocity of the reaction coordinate, respectively, at time t= 0. As there is no clear separation of time scales between the motion in the reactive region and the rest of the particle surface, Bag&i et al. [4] have introduced two rate constants. The integrated or time averaged rate constant kI is defined as 00 kc’=
I
P(t) dt,
(9)
0
where P(t) is the total population remaining at the excited-state surface at time t after excitation. The long time rate constant kL is defined through the long time limit of P( t) as (10)
The two rate constants can differ significantly from each other. In our analysis we work with a rate constant as defined in Eq. ( 9 ) . In our present work we have carried out a numerical simulation of the Langevin dynamics of a particle evolving according to Eq. ( 4) in the presence of two sinks. These absorbing sinks are placed at a distance ‘a’ apart and act as perfect absorbers, in the sense that the reaction event takes place as soon as the reactant reaches either of the sink positions for the first time (perfect absorbers). We obtain the non-monotonic relationship between the rate and the friction coefficient. We show that in the low-friction regime the reaction rate depends sensitively on the initial conditions, i.e. on the velocity and position distribution of the reactant. In this regime the reaction rate increases as rd, 6> 0 and depends on the initial conditions. After exhibiting a maximum the behaviour crosses over to 1/yin the high friction limit. We compare our results with earlier analytical calculations. In an earlier study Bagchi et al. [ 111 have carried out some quantitative numerical calculations for the case of a particle motion in a harmonic well. They have solved the FokkerPlanck equation with the sink by generating the sto-
G. K Raviprasad,A.M. Jayannavar/ ChemicalPhysicsLetters220 (1994) 353-358
356
chastic trajectories. They have also shown that at low viscosities, the rate exhibits a non-monotonic dependence on viscosity for a Gaussian sink. Our treatment differs from Bagchi et al., where we emphasize the role of the initial velocity distribution of the reaction coordinate on the rate constant and we invoke no potential at all along the reaction coordinate. In section 2 we briefly discuss our numerical procedure and in section 3 we present our results.
( = 10000) and N(t) is the number of trajectories crossing either of the boundaries (which are at a distance a apart) for the first time at time t. For simplicity in all our calculations we have set units of mass ( m ) , temperature ( T) and distance (a) between the two absorbers to be unity. We have also compared the results in the case of a potential-free region in onedimension without boundaries with the known solution of the Fokker-Planck equation [ lo] and verilied that the numerical simulation is in good agreement with the exact analytical solution.
2. Numerical procedure We have solved the Langevin equation in one-dimension without approximation using an iteration method. The Langevin equation in a potential-free region is given by Eq. (4). We have followed exactly the same procedure as that of Wada et al. [ 121. Numerical integration of the Langevin equation is performed over a very small time step T. The change in the position and momentum are given by t+? x(t+r)-x(t)=
dti,
(11)
m[i(t+.r)-i(t)1
=
I*
dt'
[
--yi(t’)+f(t’)]
,
*
& I dt’N(t’)
,
(13)
0
where No is the total
number
In Figs. l-3 we have plotted the reaction rate versus friction coefficient, both on the logarithmic scale for three different initial conditions. We have assumed for simplicity the initial distribution of position (x0) and velocity ( vo) to be decoupled and independently given by: case (i) p(xo)=6(x-fa), p( v. ) = 6( vo), which corresponds to a particle placed initially at the mid-point with zero velocity, case (ii) p(xo)=l/a and p(vo)=S(vo), corresponding to a situation where the particle is initially at rest and can be found anywhere in the region 0
(12)
where f( t’) is a computer generated uncorrelated Gaussian stochastic process with statistics as given in Eqs. (5a) and (5b). We first fix an initial position and velocity of the particle consistent with the chosen initial distribution function. We have taken three different cases as mentioned below for this distribution. We then iteratively follow the trajectory of the particle. As soon as the particle reaches either of the boundaries after a time t, we terminate the event. In a sense we have calculated the first passage time distribution after averaging over 10000 realizations for each case. The normalized survival probability P(t) as a function of time is given by P(t)=l-
3. Results and discussion
of realizations
0.01
/
0.001
““‘I’,
,“““I
0.01
“““‘I
3 ,a”“1
0.1
1
f
10
“““‘1
100
r/r* Fii 1. The dimensionless rate constant (k/k*) is plotted against the dimensionless friction parameter (y/y*) for case (i), where p(&=d(x-$a) and p(b)=d(g). Here +&%/a and k*= y*/m.
G. V. Raviprasad,A.M. Jayannavar/Chemical PhysicsLetters220 (1994) 353-358
0.01
I
1
0.001
llrrml
lllliil,
1111111,
0.01
0.1
1
1
Il1811/,
/
10
I
llillll,
100
T/Y’ Fig. 2. The dimensionless rate constant (k/k*) is plotted against the dimensionless friction parnmeter (y/y*) for case (ii), where p(~)=l/aandp(u,,)=6(u,-,).Herey*=&%/aandk*=y*/ m.
Fig. 3. The dimensionless rate constant (k/k’) is plotted against the dimensionless f&ion parameter (y/y* ) for case (iii ) , where p&)=1/a and p(u,,)=,/wexp(-mv$12kBT). Here y*=,&%/a and k*=y*/m.
probability, case (iii) corresponds to an equal probability for the particle to be found in the region with Odxo
357
mal equilibrium velocity distribution, i.e. p(va)=,/wexp( -mv$/2kJ). In the figures, the asterisks and the dashed lines represent results based on our numerical and analytical work, Eqs. ( 7)- (9 ) , respectively, and the full lines represents the earlier analytical results based on Eq. (3). Both numerical and analytical results agree onl in the highfriction limit (y larger than typically P kB Tm/a). In this regime the reaction rate decreases as k- y - ‘. In the low-friction limit k increases as y”, where 6 equals 0.3, 0.28 and 0.065 for cases (i), (ii) and (iii), respectively. This is the leading behaviour at low fiio tion apart from an additive constant. This constant takes a value zero for case (i) and (ii). In the limit y-+0, the reaction coordinate decouples from the surrounding medium and for a situation where the initial velocity of the particle is zero the coordinate remains at rest for all times and consequently the reaction rate tends to zero as y+ 0. For case (iii), there is a finite value for k as y+ 0, i.e. the particle is mobile in the absence of coupling to the surrounding medium and can decay on reaching the boundaries. After exhibiting a maximum, the rate crosses over to the inverse friction dependence in the high-friction limit. Our earlier analytical treatment valid over the entire friction domain gives somewhat higher rates as compared to our numerical results, but still shows the nonmonotonic relationship between the rate and the fiiction coefficient. However, results based on the Smoluchowski equation (3) are found to overestimate the rate significantly in the low-friction region and fail to predict the non-monotonic dependence of rate on the friction coefficient. In conclusion, we have shown that the reaction rate in a simple barrierless chemical reaction in the lowfriction limit depends sensitively on the initial position and velocity profile of the reacting particles. In the low-friction limit the reaction rate increases as y6, 6>0 and depends on the initial conditions. In our treatment we have assumed the photochemical sinks to be perfect ones, where the reaction or decay of an excited state occurs with unit probability as soon as the reaction coordinate reaches the sink position. Even in the high-friction limit earlier results have predicted a fractional power dependence of the rates on the friction in the presence of imperfect sinks [ 41. In these models every time the reaction coordinate comes into contact with the sink the particle decays
358
G. V. Raviprasad, A.M. Jayannavar /Chemical Physics Letters 220 (1994) 353-358
at a rate b( &HX corresponds to our instantaneous death model), and for intermediate values of ko fractional behaviour has been predicted. We expect the barrierless reactions to exhibit rich and diverse dynamical behaviour in the presence of imperfect absorbers in the low-friction limit. These results will be different from the high-barrier reactions.
References [ 1 ] Special issue on dynamics of molecular systems, Phys. Today 43, No. 5 (1990). [2] D. Ben-Amotz and C.B. Harris, J. Chem. Phys. 86 (1987) 4856.
[ 31 D. Ben-Amotx and C.B. Harris, J. Chem. Phys. 86 (1987) 5433. [4] B. Bag&i, G.R. Fleming and D.W. Oxtoby, J. Chem. Phys. 78 (1983) 7375. [5] B. Bag&i, Chem. Phys. Letters 135 (1987) 558. [6] B. Bagchi, J. Chem. Phys. 87 (1987) 5393. [ 71 B. Bagchi and G.R. Fleming, J. Phys. Chem. 94 (1990) 9. [8] G. Oster and Y. Nishijima, J. Am. Chem. Sot. 78 (1956) 1581. [ 91 A.M. Jayannavar, Chem. Phys. Letters 199 ( 1992) 149. [lo] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. [ 111 B. Bagchi, S. Singer and D.W. Oxtoby, Chem. Phys. Letters 99 (1983) 225. [ 121 T. Wada, N. Carjan and Y. Abe, Proceedings of the Tours Symposium on Nuclear Physics, France (World Scientific, Singapore, 1991) p. 136; T. Wada, N. Carjan and Y. Abe, Nucl. Phys. A 538 (1992) 283~.