Barrierless reaction kinetics : Inertial effect on different distribution functions of relevant Brownian functionals

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by  Ashutosh Dubey

Barrierless reaction kinetics :Inertial effect on different distribution functions of
relevant Brownian functionals
Ashutosh Dubey1, Malay Bandyopadhyay1,and A. M. Jayannavar2,3
1. School of Basic Sciences, Indian Institue of Technology Bhubaneswar, Bhubaneswar, India 751007
2. Institute of Physics, Sachivalaya Marg, Sainik School PO, Bhubaneswar, India, 751005.
3.Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai-400085, India
7 We investigate effect of inertia on barrierless electronic reactions in solution by suggesting and
1 examiningdifferentprobabilitydistributionfunctions(PDF)ofrelevantBrownian functionalsasso-
0 ciated with the lifetime and reactivity of the process. Activationless electronic reaction in solution
2 can be modelled as a free Brownian motion with inertial term in the underdamped regime. In
this context we suggest several important distribution functions that can characterize the reaction
n
a kinetics. MostofthestudiesonBrownianfunctionalwhichhasvastpotentialapplicationindiverse
J fields, are confined in the overdamped regime. To the best of our knowledge, we are attempting
first time to incorporate themuch important inertial effects on thestudyof different PDFs related
2
withBrownianfunctionalsofanunderdampedBrownianmotionwithtimedependentdriftanddif-
1
fusion coefficients using celebrated backward Fokker-Planck and path decomposition methods. We
] have explored nontrivial scaling behaviour of different PDFs and calculated explicitly the critical
h exponentsrelated with theasymptotic limits in time.
c
e
PACSnumbers: 05.40-a,05.20.-y,75.10.Hk
m
-
t The picosecond and subpicosecond laser spectroscopy (typically less than the friction coefficient ), the inertial
a
t experiments open the doorway to investigate several motion influence on reaction rates is significant. It is
s
. chemically and biologically important reactions in solu- shown that the transient behaviour of the integrated ex-
t
a tion in which the reactants do not face any barrier to citedstatepopulationisverymuchsensitivetoinitialve-
m the motion along the reaction coordinate [1–3]. The dy- locityprofileofthereactioncoordinateandmaycauseto
namics of these reactions demonstrate various interest- a nonlinear dependence of viscosity on the reaction rate
-
d ing behavior such as the dependence of relaxation rate [8, 9]. In this model the motion of a reactant molecule
n on the solvent viscosity, the solvent polarity, the tem- on the reaction coordinate is modelled by a dynamical
o
perature, and the wavelength of the exciting light [1–3]. Langevin equation [11, 12]
c
[ Thedynamicsofthesebarrierlessreactionsobviouslydif-
fersignificantlyfromtraditionalchemicalreactionswhere x¨= γx˙ +η(t), (1)
−
1
reactants face activation barrier [1–3]. In general barri-
v where, x is the reaction coordinate of the molecule, m
erless reactions are very fast and recent advance spec-
5 is the mass of the molecule, γ is the friction coefficient
troscopic techniques enable us to observe the complex
3
of solventand the associatedwhite Gaussiannoise is de-
3 motion of the reacting molecules from the reactants to
noted by η(t). The gaussian noise η(t) and the asso-
3 transitionstatetoproductsovertimescalesrangingfrom
ciated friction coefficient γ will maintain the following
0 10 fs to 10 ps [1–3]. An important example of such bar-
1. rierlessreactionisthevisiontransductionprocess,which fluctuation-dissipation relation
0 involves a barrierless cis to trans transition of rhodopsin <η(t)> = 0
7 [4]. Other important examples of barrierless reactions
2K Tγ
1 are isomerization of stillbene and diphenyl butadiene in <η(t)η(t′ > = B δ(t t′) (2)
v: solution [3] and nonradiative decay of triphenyl rings [5] m −
i and many more [1–3]. The fundamental time scales of where, < > denotes the ensemble average over all
X such reactions are dictated by the nuclear or electronic thepossibler−ealizationsoftherandomforceη(t),T isthe
ar rearrangementsin the transition state. These barrierless temperature of the heat bath and kB is the Boltzmann
reactions are simplest to model theoretically and open constant.
the doorwayto observe directly the motion of a reacting ConsideraBrownianmotionwhichcanberepresented
molecule along the reaction coordinate. In such barrier- by a typical path x(τ). Thus, a Brownian functional is
less reactions, the solvent friction is the only restriction t
defined in the interval [0,t] as T = U(x(τ)). Here,
to the reaction. The first theoreticaltreatment on barri- 0
U(x(τ)) is some prescribed arbitrary function. For each
erless reactions was proposedby Oster and Nishijima [5] R
realizationof Brownian path, the quantity T will be dif-
. Later on a systematic studies on such reactions were
ferent and it is of great interest to study the pdf of T.In
made by Bagchi et. al. [6, 7]. But, all these studies
thisrespectseveralinterestingquestionsofwideinterest
are restricted in the overdamped regime. One of us [8–
can be raised such as, (i)the probability of finding the
10] extended this model to the low viscosity or inertial
system in a certain domainat a certain instant (survival
regime. In fact, for smallmolecules and overshorttimes
probability), (ii)the pdf of time P(tf x0) at which the
|
2
system exit a certain domain first time (known as first The fundamental solution of the conditional probability
passage time tf ) starting from initial point x0, (iii)the P(x,v,tx0,v0,t = 0) of the Kramers equation is well
|
pdf P(M) of the maximum value of a BM process be- knownandits expressionis too long to discuss here [11].
fore of its first passage time, and (iv)the joint probabil- Now, considering this solution one can easily obtain an
ity distribution P(M;tm) of the maximum value M and expression for the number density n(x,t) by integrating
its occurrence time t before the first passage time of out the velocity component [9]
m
the BM process. Brownian functionals are extensively
usedinthe study ofdiversefields rangingfromprobabil- n(x,t)= 1 exp [x−x0− vβ0(1−e−βt)]2 ,
ity theory [13,14] and finance [13] to disorderedsystems
2πF(t) − 2F(t)
and mesoscopic physics [13]. (cid:16) (cid:17)
(4)
All the above mentionedPDFs are calculatedand dis- where F(t)p= qβ−3(2βt 3 + 4e−βt e−2βt, β = γ,
cussed for simple Wiener and Ornstein-Uhlenbeck pro- − − m
q = βkBT. The associated Fokker-Planck equation for
cesses[13–15]aswellasinthecontextofDNAbreathing m
n(x; t) can be obtained as follows [9]
dynamics [16]. But, all these discussions are based on
constant drift and diffusion terms. More importantly,
∂n(x,t) ∂2n(x,t) ∂n(x,t)
all these discussions are restricted in the overdamped or =D(t) µ(t) , (5)
∂t ∂x2 − ∂x
high friction limit. However, the extension to a time de-
pendent drift and time dependent diffusion terms for an
The projected evolution of n(x,t) is non-Markovian in
underdamped Brownian motion is hard to solve. This is
nature,since the evolutionof n(x,t) is dependent on the
mainly because of the fact that the system has broken
initial velocity v0 which arises from the non-Markovian
both the space and time homogeneity.
process. It can be easily observedthat Eq. (5) easily re-
The main objective of the present work is to demon-
ducesto Smoluchowskiequationforafree particleinthe
strate inertial effect in different PDFs which can charac-
long time limit βt >>1. In this context, the four PDFs
terizesuchbarrierlessreactionsusingwellstudied[13–15]
P(tf x0), P(Ax0), P(M) and P(M,tm) are of great in-
Brownianfunctionals method. To the best of ourknowl- | |
terest for determining different properties of the excited
edge, it is the first attempt to incorporate inertial effect
state. ThefirstpassagetimepdfP(tf x0)willprovideus
in first passage study which is one of the important un- |
aboutthelifetimeoftheexcitedstate. Arelatedquantity
solved problem. The other objective of this work is the t
extension of the use of the recently studied backward is the survival probability C(x0,t) = 1− 0 P(tf|x0)dtf
whichcanbe inferredfromthe experimentsby usingdif-
Fokker-Planck (BFP) method [13] and the path decom- R
ferentspectroscopictechnique. Forthepathasdescribed
position (PD) method [13] in the underdamped regime
by Eq. (1), one can introduce the area under the path
or inertial regime of a Brownian motion with time de-
before first passage time as A = tf x(t′)dt′ and calcu-
pendent drift and diffusion coefficients. Both the BFP 0
andPDmethods arebasedonthe Feynman-Kacformal- late its pdf P(A|x0). This quantRity provide us the in-
formationaboutthe effective reactivityofthe barrierless
ism [17] and both of them are first time used for explor-
reaction process. Another proposed measureable quan-
ing underdamped BM process with purely time depen-
tityforquantifyingthereactivityofbarrierlessreactionis
dent drift and diffusion terms for barrierless intramolec-
the distribution of the maximum number of reactants in
ular electron-transfer reactions. Both the techniques are
theexcitedstatebeforeitsfirstpassagetime,i.e.,P(M).
extensively used in studying many aspects of classical
Since, the timescale of this reaction is very short a rel-
Brownianmotion, aswellasfor exploringdifferentprob-
evant measure for testing the reactivity is its maximum
lemsincomputerscienceandastronomy[13–15]. Forthe
number of reactants in the excited state before its de-
first time, we consider these elegant methods to study
cay. Finally, the joint probability distribution function
the Brownian functionals for a BM with purely time de-
P(M,t ) will provide the information about both the
pendent drift and diffusion in the context of barrierless m
maximum reactivity and its occurrence time.
reaction kinetics.
Transforming from (x,t) to (z,τ) by using the follow-
Model,Methods and Measures: One of the most
ing transformation equations
directwaytodemonstratetheeffectofthesolventfriction
on a reaction in solution is to study a reaction without
anactivationbarrier. Severalstudies havebeenmadeon τ = D(t)dt+A (6)
barrierless intramolecular electron transfer reaction us- Z
ing various spectroscopic techniques [2]. Equation (1)
and
can be cast into the Kramers equation for the full phase
space probability distribution function P(x,v,t), where
z =x+ µ(t)dt+B (7)
v is the velocity of the particle. The full phase space
Z
Kramers equation is given by
one can convert Eq. () in (z,τ) space as follows :
∂P γ ∂ k T ∂ ∂P ∂n(z,τ) ∂2n(z,τ)
B
= v+ P v , (3) = (8)
∂t m∂v m ∂v − ∂x ∂τ ∂z2
(cid:16) (cid:17)
3
0.01 0.03 a 1×10-3 2×10-3 b
0.008 0.02 mm==12 0.8 01 mm==12
n(t|x)f000..000046 0.0010 100 200 mmm===345 n(A|x)000..46 0 100 200 mmm===345
0.002 0.2
00 50 100 150 200 00 50 100 150 200 250
tf A
104 c 10-1 d
m=1 m=1
102 m=2 m=2
m=3 m=3
n(t)m100 mm==45 n(t)m10-2 mm==45
10-2
10-1400 101 102 10-130-3 10-2 10-1 100
tm/x02 tm/x02
FIG. 1: (color online) (a) Plot of first passage time distribution of density profile of excited state n(tf|x0) for different mass
values which are indicated in the legends and (b) the distribution of area under the curve, n(A|x0) which give us information
about reactivityof processisplotted fordifferentmass values.(c) Plot ofpdfofn(tm)before firstpassage timetf for thelarge
tm cases in the log scale. The same thingis plotted for small tm case in (d)
PDF of Brownian functional over a fixed time interval with z0 = x0+ vβ0(1−e−βt) and τ = 2βq3(2βt+4e−βt−
[0,τf] is defined as e−2βt 3). This quantity canbe taggedwith the experi-
−
τf ments. The distribution of A can be computed from Eq.
T = U(z(τ))dτ, (9) () by substituting U(z0) = z0. Using proper boundary
Z0 conditions as mentioned above one can show:
wanadrdouFrokakimer-iPslatoncckalmcueltahtoedPDasFdPis(cTus|zse0d). inUsdinegtabilasckin- Q(z0)=32/3Γ(2/3)Ai(21/3p1/3z0) (14)
Refs. [13, 14] one can obtain : Taking the inverse Laplace transform
dd2zQ2 −pU(z0)Q(z0)=0 (10) P(A(τf)|z0)= 32/32Γ1/(13/3)[A(τz0)]4/3exp − 9A2(zτ03 )
0 f (cid:20) f (cid:21)
(15)
Substituting U(z0)=1 and using proper boundary con-
Now, going back to originalvariables one can obtain the
ditions Q(z0 = 0) = 1 and Q(z0 ) = 0 we obtain
→ ∞ distribution function of A :
:
n(τf|z0)= √z40πe−τz302//24τf (11) n(A|x0,v0)= 32/32Γ1/(31/3)qβ−3([eA−]β4t/f3−1)2
Then going back to the original vafriables (x,t) we can ×[βx0+v0(1−e−βtf)]exp − 2[βx0+v90β(13A−e−βtf)]3
(cid:20) (cid:21)
obtain the first passage time distribution of the number
(16)
density in the excited state :
Using well known path decomposition method [13–15],
2β(e−βtf 1)2[βx0+v0(1 e−βtf)] one can find the joint probability density of density in
n(tf|x0,v0)=βsqπ (2βtf +−4e−βtf e−2βtf− 3)3/2 the excited state,n(M,tm) which can provide both the
− −
information: (i)the maximum number of particles avail-
exp(cid:18)− 2q(2ββ[tβfx+0+4ev−0β(t1f−−ee−−β2tβft)f]2−3)(cid:19) (12) athbele(iiin)tthheeteimxceitaetdwsthaitcehbitefworileliattstadiencaitys(mtfa)xaimsuwmellbaes-
We can also extract much more information. Another fore first passage time or decay :
importantquantityisthesurvivalprobabilityofthepop- qπ(e−βtm 1)2 ∞
ulation remaining at the excited state surface at time t n(M,tm)= β2M3− (−1)p+1p
after excitation : p=1
X
t pπ[βx0+v0(1 e−βtm)]
C(x0,v0,t) = 1− dtfn(tf|x0,v0) ×sin βM−
Z0 (cid:20) (cid:21)
= 1 1 Γ(3/2, z02, (13) exp p2π2q(2βtm+4e−βtm −e−2βtm −3) (17)
− Γ(1/2) 4τ × 4M2β3
(cid:20) (cid:21)
4
The closed form expressions of marginal distribution out M in the following two limits (i)t /x2 >> 1 and
m 0
n(t ) can also be obtained from Eq. (16) by integrating (ii)t /x2 <<1. In the large t limit one can find :
m m 0 m
β (e−βtm 1)2 [βx0+v0(1 e−βtm)]
n(t ) log2 β − − (18)
m ≈ sqπ [(2βtm+4e−βtm e−2βtm 3)]3/2
− −
and the small t limits give us
m
qβ (e−βtm 1)2
n(t )= − . (19)
m r π [βx0+v0(1−e−βtm)][(2βtm+4e−βtm −e−2βtm −3)]1/2
All the four PDFs are plotted in figure 1. The basic Langevin model provides a very good description of the
message as obtained form the figures is that the mass barrierless reaction kinetics [5–7]. We establish and in-
effect is distinctly observed in the transient regimes. As vestigateananomalousdiffusion processwhichgoverned
we go to the asymptotic limit in time we can observe by an underdamped Brownian motion with an explicit
universalscalingbehaviourofallfourPDFs. Weobserve time dependence of the diffusion and drift coefficients
nonmonotonicbehaviourofthepdfs offirstpassagetime to capture correct characteristics of barrierless reaction
P(tf x0) and area P(Ax0). In the low-friction regime, kinetics. The four PDFs distinctly show the persistent
| |
both P(tf x0) and P(Ax0) increase with time to reach inertial effects and it plays a non-negligible role in the
| |
a maximum and then crosses over to universal scaling transientregimeaswellasatreasonablylongtime. This
behaviour regime as observed by an overdamped Brow- multifacetedproblemhelpustoinvestigateinertialeffect
nian particle. In this time asymptotic limit, we observe to characterize barrierless reactions in the low friction
Pth(aAt x(i0))PDFAo−f4fi/3r,st(ipiia)sPsa(Mge)timeMP−(2tfa|xn0d)(∼iv)tfP−3(/t2m,)(ii) rmeagkimeea.dvOannctehmeeonttheinr hthaendB,rtohwenpiarnesefunntcsttioundaylhmeleptshotdo
t−3/2|. Bu∼t, the transient regim∼e shows nonuniversal be∼- to a problem of Brownian motion with an explicit time
m dependence of the diffusion and drift coefficients in th-
haviour (depends on mass) which is mainly due to the
elowfrictionorinertialregime. Ourinvestigationwillbe
inertial term of Eq. (1). P(t ) has power law behaviour
m helpfulinunderstandingfundamentaltimescalesrelated
at small tails also but the behaviour is nonuniversal and
with barrierless reactions which are dictated by the nu-
it depends explicitly on mass. We also show that in the
clearorelectronicrearrangementsinthetransitionstate.
low friction regime all four PDFs depends crucially on
the initial conditions,i.e., on the velocity (v0) and posi-
tion (x0). In the present context, we consider special
Acknowledgments
conditionofp(v0)=δ(v0)andp(x0)=δ(x 1/2),which
−
corresponds to a particle placed at the mid point with
zero velocity initially. One may consider different initial MB acknowledge the financial support of IIT
conditionsanddifferentexcitedstatepotentialforfuther BhubaneswarthroughseedmoneyprojectSP0045. AMJ
research. thanksDST,IndiaforawardofJCBosenationalfellow-
It is quite generally assumed that the overdamped ship.
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