Projection of the Dynamics of Electron Transfer Reaction in Dual

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Projection of the Dynamics of Electron Transfer Reaction in Dual Space onto the One-dimensional Slower Reaction Coordinate Axis Aniket Patra, Kanagala Ajay Acharya, and Alok Samanta J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b02415 • Publication Date (Web): 06 Jul 2015 Downloaded from http://pubs.acs.org on July 7, 2015

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The Journal of Physical Chemistry

Projection of the Dynamics of Electron Transfer Reaction in Dual Space onto the One-dimensional Slower Reaction Coordinate Axis Aniket Patra Center for Materials Therory, Rutgers University, Piscataway, NJ 08854, USA Kanagala Ajay Acharya Theoretical Chemistry Section, Bhabha Atomic Research Centre, Mumbai-400085, India Alok Samanta* Theoretical Chemistry Section, Bhabha Atomic Research Centre, Mumbai-400085, India *

Email: [email protected] Phone:91-22-25590295

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ABSTRACT: We have derived here for the first time an exact dynamical equation within the domain of classical mechanics for the time dependent density distribution function of onedimensional reaction coordinate (RC) in condensed phase for electron transfer reaction by projecting the dynamics of slower modes in multidimensional Liouville space starting with a given set of coordinates of the faster modes. After ensemble averaging the faster modes, the dynamics of the whole system solely depends on the slower RC. In order to simplify the complicated equation into a tractable form, benchmark approximations are employed to reduce the formally exact equation into an equation similar to Smoluchowski equation with a delocalised sink term. As a test case, a Hamiltonian for the solute-solvent system modelled by quadratic functions for fast-relaxing vibrational and slow-relaxing polarization modes respectively has been considered. Interestingly, our simplified

kinetic equation

corresponding to this model Hamiltonian is transformed into the well known phenomenological Sumi-Marcus equation.

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1. INTRODUCTION Investigation on electron transfer (ET) reaction in condensed phase has been of utmost importance1-15 to the experimentalists as well as theoreticians over the several decades. In the Marcus theory of ET reaction; it is assumed that the polarization fluctuation plays a key role for the electron transfer to occur. The rate of electron transfer reaction is then calculated based on transition state theory (TST). However, this theory did not give any explicit definition of the RC. Later on Tachiya and many others7-10 derived the rate expression by assuming a particular form of the RC, which is identical to the results obtained by Marcus. According to the Marcus theory of electron transfer, rate constant of ET reaction follows inverted parabolic dependence as a function of change in Gibbs free energy. This prediction of the Marcus theory has been verified experimentally

10

. It is assumed in the TST that the

states from which ET reaction occurs is in thermal equilibrium. In sufficiently viscous solvents, relaxation time for the solvent molecules can be long; the actual rate of an ET reaction becomes smaller than the one calculated using TST. So one needs a kinetic equation for the RC corresponding to the ET reactions that will retain the non-equilibrium aspect in Liouville space. Many authors proposed the phenomenological kinetic equation11-16 for the RC over the last two decades. The limitation of these kinetic equation are the following: (i) These existing kinetic equations are not derived from the first principles, i.e. none of them start with the most fundamental Liouville equation within domain of classical mechanics, (ii) The excitation wavelength that creates the initial non-equilibrium state was not properly taken into account, (iii) only the polarization mode is considered for most of the cases. Recently4-6 an exact dynamical equation for the RC by projecting the dynamics involving multi-dimensional Liouville space into a one-dimensional RC space has been derived. This approach manages to circumnavigate the limitations described before. However, this kinetic equation can be further simplified if the relaxations of all the modes are very slow. We must

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emphasize, that even a few of the modes not relaxing sufficiently slowly, presents great difficulty for the aforementioned simplification procedure. Sumi and Marcus16 proposed a phenomenological theory to understand the effect of fast relaxing vibrational and slow relaxing polarization modes. The microscopic picture for this system depicts that the faster vibrational mode is equilibrated after a very short time and remains in equilibrium during the progress of reaction. Thus the dynamics of the whole system solely depends on the slowly varying polarization mode. In terms of dynamics, the rapidly varying part of the dynamics brings the system to the lower-dimensional space

where the dynamics is slower. The

limitations of the Sumi-Marcus theory is that it is not derived based on first principle theory and restricted to harmonic potential. A theoretical description of the ET reactions for this complex system is difficult due to the complicated potential surface in multi-dimensional space. The purpose of the current work is to circumvent these difficulties by developing a conceptually simpler theory for the ET reaction in condensed phase. To the best of our knowledge, precise kinetic equation for the probability distribution of one-dimensional RC in condensed phase for ET reaction has not been derived for the complex system mentioned above, based on first principle theory. In our present theory, the multiple sequential steps involved are the following: We first assume that the whole Liouville space can be decomposed into two separate spaces; where one space, 1 , consists of all fast relaxing modes and the other one,  2 , consists of all slow relaxing modes. We also assume that the full probability density can be written as a product of two separate density functions corresponding to the slow and fast variables. The relaxation of the 1 space is so rapid that, in this subspace equilibrium is achieved for any value of  2 . Thus we start with equilibrium distribution in the 1 subspace. Due to thermal fluctuations of 1 and  2 , a critical configuration A(1 )  A(2 )  l * is reached and then the reaction proceeds with intrinsic rate

4


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k . Now, the Liouville equation in slow variable space  2 can be cast into a modified equation of the form  ( 2 , t )  [iL ( 2 )  k ( A(1 )  A( 2 )  l * )] ( 2 , t ) . t

(1)

In the above equation, L(2 ) is the Liouville operator in  2 space. The density distribution function

 (, t )

in

combined

 (, t )  eq (1 )  (2 , t ), with



space

at

time

t

can

be

written

as

L(1 ) eq (1 )  0 , for which the formal solution of

  (2 , t ) can be written as  ( 2 , t )  eiLt  ( 2 , 0) . Here L  L(2 )  ik ( A(1 )  A(2 )  l * )

and  ( 2 , 0) is the initial probability distribution of the slow variable at initial time t=0. The average value  2 (t ) , of any general phase space function G ( 2 ) is defined in terms of

 (, t ) , i.e. in terms of eq  1  and  ( 2 , t ) as  2 (t )    eq  1  d 1  d  2  (  2 , t )G (  2 ).

(2)

Therefore, given the time dependence of the distribution function

 ( 2 , t ) , in principle

 2 (t ) can be obtained. On the other hand,  2 (t ) can also be evaluated using the following equation:  2 (t )    eq  1  d 1  d  2  (  2 , 0)G (  2 , t ) ,

(3)

† where G( 2 , t )  eiL t G( 2 , 0) and L†  L(2 )  ik ( A(1 )  A(2 )  l * ) . In our system, the

experimentally measurable quantity p(t ) , which is the concentration of reactant at time t, is defined as p (t )    eq (1 ) d 1   (  2 , t )d  2 .

(4)

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However, obtaining an analytical expression for p(t ) is very difficult, because it extremely difficult to obtain an expression for  ( 2 , t ) from the Eq.(1). Therefore, we try to project the problem onto a lower dimensional space where the phase function A( 2 ) lives. The reason behind this procedure will be clear subsequently as we consider a specific model Hamiltonian as an example. We define the probability distribution g ( a , t ) of the phase function A( 2 ) corresponding to an arithmetical value a at time t as g ( a , t )   d  ( 2 , t ) ( A( 2 )  a ) .

(5)

g (a, t ) can also be defined as

The quantity †

g(a,t) 

 d  ( ,0)G(a,t),

where

2

†

G(a, t )  eiL t ( A( 2 )  a)  eiL t G(a, 0) . Now we can rewrite p(t ) in terms of g ( a , t ) as p(t) 

 d       da g(a,t). 1

eq

(6)

1

Thus in the above equation we managed to reduce the multi-dimentional integration over  2 space to a single integration over a. In the subsequent sections we first develop a theoretical formalism to calculate g(a,t) and finally we apply this to a specific model Hamiltonian. 2. PROJECTION OF DYNAMICS IN  2 ONTO ONE-DIMENSIONAL SPACE: A PROJECTION OPERATOR FORMALISM

We first define the ensemble average D , E    ( 2 ) D ( 2 ) E * ( 2 ),

(7)

where  ( 2 ) is the metric in the Hilbert space satisfying iL(2 ) (2 )  0 . Now, writing

 ( 2 , 0)

as

 ( 2 ) ( 2 , 0)

and

iL( 2 ) ( 2 )  0,

using

6


one

can

show

that

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



 ( 2 , t )   ( 2 )eiLt ( 2 , 0)   ( 2 ) ( 2 , t ) ,where, ( 2 , t )  e  iLt ( 2 , 0) .Using iL( 2 ) ( 2 )  0, L can be shown to satisfy the property

  L† D, E D, LE

(8)

We then employ the methodology developed by Zwanzig20, Garcia et al21 non reactive system and generalised further by us4 for a reactive system.

In our earlier work4, we

projected the reaction dynamics in Liouville space onto the surface A(1 )  A( 2 )  a . The projection of the reaction dynamics onto the surface A(1 )  A( 2 )  a . leads to a kinetic equation where the memory kernel can not be simplified any further because of appearance of the fast variable 1 . To tackle this problem we project the whole dynamics in the phase space onto the surface A( 2 )  a instead of the same on the surface A(1 )  A( 2 )  a . We first write the projection operator is written as

PG   db

 ,  ( A( 2 )  b)

 ( A(2 )  b)

 ( A( 2 )  b),

(9)

where  ( A( 2 )  b )    d  2 ( 2 ) ( A( 2 )  b ) . Before deriving the exact kinetic equation in one-dimensional space for ET reaction, we first write the following properties for G (b, 0) and PG , which will be used in the subsequent discussions frequently, viz

 ( A(2 )  l ),  ( A(2 )  m)  [G(l ,0)] (l  m),

(10)

PG2  PG , PGG(a,0)  G(a,0),

(11)

D, PG E  PG D, E .

(12)

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Now, before obtaining

the expression for

g (a, t ) ,

g (a, t )  G(a, t ), ( 2 , 0)

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we rewrite Eq. (5) as (13)

and break G ( a, t ) into two parts

G(a, t )  PGG(a, t )  QGG(a, t )   dbC(a, t / b)G(b,0)  QGG(a, t )

where C ( a, t / b) 

G ( a, t ), G (b, 0) [G (b, 0)]

(14)

. and [G (b, 0)]  G (b, 0)

A kinetic equation for C (a, t / b) can now be derived as follows

 dG (a, t )  , G (b, 0)   (b, 0)  G (a, t ), iLG dC (a, t / b)  dt    dt [G(b, 0)] [G (b, 0)]





(15)

 (b,0) into two parts one spanning in G (b, 0) and the other in the space Again we break iLG orthogonal to it, viz.  (b, 0)  P iLG  (b, 0)  Q iLG  (b, 0) iLG G G

(16)

*  (b, 0) ,  PG (iLG (b, 0))  kG (a1 , 0)G (b, 0)   QG iLG

where a1  A(1 )  l * . It is interesting to note that all the phase functions appears in the *

Eq.(16) are functions of the fast variable 1 . The term PG (iLG(b,0)) can be expressed as PG (iLG (b , 0))   dc i (b, c )G ( c , 0).

(17)

The frequency i ( b, c ) in the above equation is given by

i  (b , c ) 

iLG (b, 0), G (c, 0) [G (c, 0)]

.

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*

The term PG {kG (a1 , 0)G (b, 0)} in Eq. (16) can also be written as PG {kG (a1 , 0)G (b, 0)}  k (a1  b)G (b, 0) *

*

(18)

Using Eqs. (15) - (18) finally we obtain get dC (a, t / b) i{(b, c)}*[G (c, 0)]   dc C ( a, t / c ) dt [G (b, 0)]  (b, 0) G (a, t ), QG iLG

 k (a1  b)C (a, t / b)  *

[G (b, 0)]

. (19)

The last term of right hand side of Eq.(19) can be expressed as  (b, 0) G ( a , t ), QG iLG



[G (b, 0)]

2  G ( a, t ), QG iLG (b, 0)



[G (b, 0)]

 (b, 0) QG G ( a , t ), QG iLG [G (b, 0)]

,

(20)

 (b, 0) and QGG(a, t ) respectively as where we simplify QG iLG

 (b, 0)  Q iLG (b, 0) QG iLG G

(21a)

d * (QG G(a, t ))  QG iLG(a, t )  kG(a1 , 0)G(a, t )  dt

(21b)

*

The quantity PG {G (a1 , 0)G (a, t )} can be simplified as

PG {G (a1 , 0)G (a, t )}  G (c, 0)C (a, t / a1 )G (a1 , 0)  *

*

*

1 L2 (a, t ) k

(22)

Hence, we have

kQG {G (a1 , 0)G (a, t )}  kG (a1 , 0)G (a, t )  kC (a, t / a1 )G (a1 , 0)  L1 (a, t )  L2 (a, t ) *

*

*

*

(23) and finally we get

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d (QGG(a, t ))  QG iLQGG(a, t )  iLPGG(a, t )  ( L1 (a, t )  L2 (a, t )) dt

(24)

After some algebra we get t

t

0

0

QG G(a, t )   dc  dsC (a, t  s / c) F (c, s )   dsU ( s )[( L1 (a, t  s )  L2 (a, t  s ))] where

the

position

dependent

random

force

F ( c, t )

can

be

(25) written

as

F (c, t )  U (t ){QGiLG(c,0)} with U (t )  exp QGiLt  . From Eqs. (20) and (25), we get

QG G(a, t ), F (b, 0)

G(b, 0) t

  dc  dsC (a, t  s / c) K (c, b, s )  M Fb 0

  dsU (s)[(L (a, t  s)  L (a, t  s))] t

1

0

2

(26)

where the operator M Fb and the memory kernel K (c, b, s) can be written respectively as

M Fb h() 

h(), F (b, 0) F (c, s), F (b, 0) . and K (c, b, s)  G(b, 0) G(b, 0)

Using Eqs. (19) and (26), we arrive at the kinetic equation for C ( a , t / b) as t dC ( a , t / b ) *   dcC ( a, t / c )i (c, b)  k ( a1  b )C ( a, t / b )   dc  dsC ( a, t  s / c ) K (c, b, s ) 0 dt (27)

 M Fb

  dsU (s)[(L (a, t  s)  L (a, t  s))] . t

0

1

2

Using Eqs (14 ) and (25), we get t

G ( a, t )   dbC ( a, t / b)G (b, 0)   db  dsC ( a, t  s / b) F (b, s ) 0

t

  dsU ( s )[( L1 ( a, t  s )  L2 ( a, t  s ))], 0

10


(28)

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Using Eqs. (27) and (28) and after some algebra we obtain the following kinetic equation:

dG(a, t ) *  Z (a, t )G(a, t )  k (a1  a)G(a, t )  F (a, t )  E (a, t ), dt

(29)

where Z (a, t ) and E (a, t ) can respectively be written as

t

Z ( a, t ) f ( a, t )   dc (i ( a, c )) f (c, t )   dc  ds 0

   f (c, t  s )  G (c, 0) K (a, c, s )    a c  G (c, 0)  

E (a, t )    db  dsM Fb U ( s )[( L1 (a, t  s )  L2 (a, t  s))] G (b, 0) t

0

t

  db  dsF (b, t  s) M Fb 0

  drU (s  r)(L (a, r)  L (a, r)) s

1

0

(30)

2

t s   db  dsC g1 (a, t  s / b)  drU ( s  r )[( L1 (b, r )  L2 (b, r ))] 0

0

Here Cˆ 1g (c, s / b) represents Laplace transform of C g1 ( a, t / b) and is defined as follows: 





* C g1 ( a , / b )   dt exp   t C g1 ( a , t / b )  i (c, b )  k ( a1  c ) (c  b )  Kˆ (c, b,  )   (c  b ) . 0

Using Eqs. (13), (29) and (30) we write

dg (a, t )  Z (a, t ) g (a, t )  k (a1*  a) g (a, t )   E (a, t ), (, 0)  dt

(31)

Here we have used  F (a, t ), (, 0)   0 . The Eq.(31) is a newly developed exact equation for the time-dependent probability distribution function in ‘ a ’ space. It is interesting to note that the coordinate of the fast variable 1 appears in the Eq.(31) through the sink function

 (a1*  a )  a1*  A(1 )  a*  and E (a, t ) through L1 (a, t ) and L2 (a, t ) . In our earlier theory4, faster  1  and slower modes   2  are present in all equations as  A(1 )  A( 2 )  except in the sink function. The inner product involving E (a, t ) in Eq.(31) vanishes if one assumes

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Page 12 of 20

PG G ( a , t )  G ( a , t ) , which is exact at small times. Hence under this assumption, Eq. (31)

takes the following form:

dg (a, t )  Z (a, t ) g (a, t )  k (a1*  a) g (a, t ). dt

(32)

Difficulty in evaluating the memory kernel K ( a , a , s ) arises because of the appearance of the projection operator in the exponential term. It is important to note that Z (a, t ) does not contain any fast variable 1

  A( )  l  . *

1

the quantity

but it appears in the sink function  (a1*  a) through a1*

Because of the slowly varying nature of the phase space function A   2 , t  ,

A   2 , t  t

is expected to be small. Under this approximation the kinetic

equation can be simplified further as: g  a, t  t



t      g  a, t  s    v a g a , t      ds  G (a, 0)  a; s     a a  a  G (a, 0)   0

(33)

 k (a  a1 ) g  a, t  *

Here,

 t )A(0)  G (a, 0)   a; t   A(  t )  A(  t )  v(a) A(  G (a, 0) v(a )  A(0) (33a) It is worthwhile to emphasize that presence of both the modes ( 1 and  2 ) in the memory kernel K (a, c, t ) (Eq.(44) of Patra et al.4) developed by us for the reaction co-ordinate prevents us to simplify the kernel. The incompleteness of the earlier theory4 have prompted

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us to develop a new theory. In this approach, absence of fast variable 1 in the memory kernel K (a, c, t ) allow us to simplify the kernel into a tractable form (Eq.(33)). To obtain p(t ) (Eq.(6)) one needs to average the distribution function g  a, t  weighted by the equilibrium distribution function eq  1  . After doing the averaging of the Eq.(33) one obtains: g  a, t  t



t     g  a, t  s    v a g a , t      ds  G(a, 0)  a; s     a a a   G(a, 0)   0

 k  (a  a1 ) g  a, t  *

(34)

where

g  a, t  =  eq  1  d 1 g  a, t 

(35)

 (a  a1* )   d 1 eq  1   (a  l *  A(1 ))

(36)

It is interesting to note that  - sink appear in Eq.(34) broaden after averaging over the fast variable. It is interesting to note that, Now p (t ) can be obtained from g  a, t  as p (t )   dag  a, t 

3. MODEL

(37)

HAMILTONIAN: DERIVATION OF THE TIME DEPENDENT

EQUATION FOR THE DENSITY DISTRIBUTION OF THE REACTION COORDINATE 'a'

A simple theoretical description for describing non-equilibrium processes consisting of a slow solvent polarisation and fast vibrational modes in multi-dimensional space is considered here . We first focus on a solute [DA(donor-acceptor)]+solvent model system. An ion pair

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D + A - is formed due to formation of a critical state l *  A(1 ) arising from solvent fluctuation. Subsequently, solvent molecules relax through mutual interactions and move downhill in the multidimensional potential manifold of the product configuration until they reach the equilibrium configuration. The model Hamiltonians6 of the neutral solute DA and ion pair D + A - in a solvent are defined respectively as 1 2 H DA   g (r ) Por (r ).Por (r )d  Qv 2 HD



A

  g (r ) Por (r ).Por (r )d   Por (r ).E (r )d 

1 2  Qv  Q0   G 2

(38)

where E (r ) represents the electric field at the point r due to D + A - . Here G represents Gibbs free energy of the ET reaction. Finite size effect of the solvent molecules is taken care of by the function g(r). The phase function A  1  and l * for the

above mentioned

Hamiltonian can defined as

A  1   QQ0 , l *      G 

(39)

Finally we get the expression for the average sink function and  G(a, 0)  as    a  v  G   exp    (a  a )  4v 4v 

2

* 1

G(a, 0) 

  

  2  exp   a  4s 4  s  

(41)

where

s 

(40)

1 E (r ).E (r ) d  4 g (r )

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Now we do a series of simplifications. First we substitute Eq.(40) and Eq.(41) into the Eq.(34). After that we apply Markovian approximation and the identify v  a  =0 in Eq.(34). 

We further assume that memory kernel   a  (  dt  a, t ) depends weakly on a i.e. 0

 a 

2s



. After all the above approximations finally we are left with the following

simplified equation. g  a, t  t

 2   2 g  a, t  1   ag  a, t    S   2  S a    S  a    a  v  G   exp   k 4v 4v 

2

  g  a, t  

(42)

which is identical to the phenomenological equation proposed by Sumi and Marcus. Here

 S and v respectively represent polarization relaxation and vibrational reorganization energy. 4. CONCLUSION

A multidimensional Liouville space based description of ET reaction in condensed phase is very complex and evaluation of the analytical expression for (2, t) is difficult. Hence a simpler one-dimensional description as presented here is an important advancement in the theoretical understanding of the ET reaction. This is the first time an exact kinetic equation (Eq.(31)) for an arbitrary potential in one-dimensional RC space has been derived by mapping the dynamics in multidimensional space spanned by two distinct modes whose time scale of relaxation are widely different. We have also shown the necessity of projecting the reaction dynamics in the full phase space onto the surface A( 2 )  a instead of the same on the surface4 A(1 )  A( 2 )  b . In order to simplify the complicated equation, we employ

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standard approximations which reduce it to the Smolochowski-type equation (Eq.(34)). We consider a model Hamiltonian. The simplified kinetic equation corresponding to the model Hamiltonian defined in Eq.(38) takes a form identical to the well known phenomenological Sumi-Marcus equation (Eq.(42)). Thus the theory developed here provides a rigorous basis for the phenomenological model.

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REFERENCES

(1) Marcus, R. A. Electrostatic Free Energy and Other Properties of States Having Nonequilibrium Polarization. I. J. Chem. Phys. 1956, 24, 979-989. (2) Marcus, R. A. Free Energy of Non equilibrium polarizations. III. Statistical Mechanics of Homogeneous and Electrode Systems. J. Chem. Phys. 1963, 39, 1734-1740 (3) Marcus, R. A. Free energy of nonequilibrium polarization systems. II. Homogeneous and electrode systems. II. J. Chem. Phys. 1963, 38, 1858-1862. (4) Patra, A.; Samanta A.; Ghosh, S. K. Mapping the reaction dynamics in Liouville space onto a reaction coordinate space: A first-principle- based theory. Phys. Rev. E 2011, 83, 026104. (5) Dhole, K.; Modak , B.; Samanta A., . Ghosh, S.K. Theory of reversible electron transfer reactions in a condensed phase. Phys. Rev. E 2010, 82, 016110 (6) Dhole, K.; Jena, N.; Samanta, A.; Ghosh, S.K. Electron transfer reactions in condensed phase: Effect of reversibility. Phys. Rev. E . 2012, 85, 026105 (7) Tachiya, M. Generalization of the Marcus Equation for the Electron-Transfer Rate. J. Phys. Chem. 1993, 97, 5911-5916 . (8) Tachiya, M. Reaction coordinate in electron transfer: What physical quantity should we use for it? J. Chem. Phys. 2008, 129, 066102. (9) Zhu, J.; Rasaiah, J. C. Reaction coordinates for electron transfer reactions J. Chem. Phys. 2008, 129, 214503.

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(10) Miller C.R., Calcaterra L.T. & Closs G.L. J. Am. Chem. Soc. 1984, 106, 3047-3049 (11) Tachiya, M. & Murata, S. Non-Marcus energy gap dependence of back electron transfer in contact ion pairs. J. Am. Chem. Soc. 116, 2434-2436 (1994) (12) Zhu, J.; Rasaiah, J. C. Dynamics of reversible electron transfer reactions. J. Chem. Phys. 1951, 95, 3325-3340. (13) Roy, S. ; Bagchi B. Time dependent solution of generalized Zusman model of outer sphere electron transfer reactions: Applications to various experimental situations. J. Chem. Phys. 1994, 100, 8802-8816. (14) Chandra, A. A theoretical study of outer sphere electron transfer reactions in electrolyte solutions. J. Chem. Phys. 1999, 110, 1569-1580 . (15) Hynes, J.T. Outer-Sphere Electron Transfer Reactions and Frequency -Dependent Friction. J. Phys. Chem. 1986, 90, 3701-3706. (16) Sumi, H.; Marcus, R. A. Dynamical Effects in electron transfer reactions. J. Chem. Phys, 1986, 84, 4894-4914. (1 7) Calef, D.F.; Wolynes, P.G. Classical solvent dynamics and electron transfer. II. Molecular aspects. J. Chem. Phys 1983, 78, 470-482. (1 8) Zusman, L.D. Outer Sphere Electron transfer in polar solvent. Chem. Phys., 1980, 49 295-304. (19) Jortner, J.; Bixon, M. (Edited) Electron Transfer--From Isolated Molecules to Biomolecules. Part One Adv. in Chem. Phys. 1999, 106. (20) Zwanzig, R. Memory Effects in Irreversible Thermodynamics. Phys. Rev. 1961, 124, 983-992. 18


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(21) Garcia, L. S.; Garcia-Colin ; J. L. Rio.

A Unified Method for Deriving Kinetic

Equations in the Statistical Mechanics of Irreversible Processes I. Exact Results, J. Stat. Phys., 1977 , 16, 235-258.

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For Table of Content only

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Projection of the Dynamics of Electron Transfer Reaction in Dual

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