Effect of an external field on the reversible reaction of a neutral particle and a charged particle in three dimensions. II. Excited-state reaction

Effect of an external field on the reversible reaction of a neutral particle and a charged particle in three dimensions. II. Excited-state reaction

Shang Yik Reigh, Kook Joe Shin, and Hyojoon Kim

Citation: The Journal of Chemical Physics 132, 164112 (2010); doi: 10.1063/1.3394894 View online: http://dx.doi.org/10.1063/1.3394894

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/132/16?ver=pdfcov Published by the AIP Publishing

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Effect of an external field on the reversible reaction of a neutral particle and a charged particle in three dimensions. II. Excited-state reaction

Shang Yik Reigh(이상익兲,¹Kook Joe Shin(신국조兲,^1,a兲 and Hyojoon Kim(김효준兲^2,b兲

1Department of Chemistry, Seoul National University, Seoul 151-747, South Korea

2Department of Chemistry, Dong-A University, Hadan-2-dong, Busan 604-714, South Korea 共Received 20 January 2010; accepted 24 March 2010; published online 27 April 2010兲

The excited-state reversible reaction of a neutral particle and a charged particle in an external electric field is studied in three dimensions. This work extends the previous investigation for the ground-state reaction关S. Y. Reighet al., J. Chem. Phys. 129, 234501共2008兲兴to the excited-state reaction with two different lifetimes and quenching. The analytic series solutions for all the fundamental probability density functions are obtained with the help of the diagonal approximation.

They are found to be in excellent agreement with the exact numerical solutions of anisotropic diffusion-reaction equations. The analytical solutions for reaction rates and survival probabilities are also obtained. We find that the long-time kinetic transition from a power-law decrease to an exponential increase can be controlled by the external field strength or excited-state decay rates or both. ©2010 American Institute of Physics.关doi:10.1063/1.3394894兴

I. INTRODUCTION

One of long standing issues in diffusion-influenced reac- tions is the external field effect^1–20 since it is common in a broad range of experiments.¹ For molecules with electric charges or dipole moments, the external electric field can be a useful controllable factor.^6–14 Similarly, the external mag- netic field has been a valuable tool for various molecular studies. Even for a nonpolar particle, the ubiquitous gravita- tional force can affect its Brownian motion and change its ultimate fate in diffusion-reaction systems.⁵From a theoret- ical point of view, however, the rigorous inclusion of the external field effect significantly increases the mathematical difficulty, especially in three dimensions共3D兲, since it breaks down the spatial symmetry and we have to solve anisotropic diffusion-reaction equations. The previous theoretical inves- tigations in 3D focused on the irreversible reactions includ- ing the ion pair recombination problem^6–8 and the reaction between a neutral and an ion particle^10–13 in the presence of an external electric field.

The recent studies for reversible effects on diffusion- influenced reactions have attracted growing interest.^15–30Ex- act analytical solutions for a variety of reversible diffusion- influenced reactions have been discovered. The ground-state solutions for the A+B↔C type reaction^21,22 were extended to the excited-state reactions with two different lifetimes and quenching.^23,24TheA+B↔C+Dtype reactions were solved for the ground-state and excited-state reactions.^25,26The trap- ping problem was successfully extended to the reversible case.²⁷ The effects of an external field on reversible diffusion-influenced reactions were exactly solved for vari- ous cases but only in one dimension 共1D兲 using the math- ematical isomorphism.^5–19The 3D analytical solutions in the

presence of a constant external field were obtained only very recently for the ground-state reversible reaction for a neutral and an ion molecule共Paper I兲.²⁰

An interesting kinetic transition at long times was pre- dicted for the excited-state reactions with two different lifetimes^28,29 and this kinetic transition behavior was con- firmed experimentally for the proton transfer reactions.³⁰On the other hand, the external field effect was found to cause a similar kinetic transition behavior.^16–19 Since the field strength is much easier to control experimentally than the excited-state lifetimes, the transition behavior can be ob- served without efforts to find reactants with appropriate lifetimes.³¹ The intriguing interplay effect between the field strength and lifetimes on the long-time kinetic transition be- haviors was reported in 1D.^16,19Therefore, the field effect on excited-state 3D reactions is of interest. In this article, we investigate the external field effect on the diffusion- influenced geminate recombination of a neutral particle and a charged particle with two different excited-state lifetimes and quenching in 3D.

This paper is organized as follows. In Sec. II, the kinetic equations of probability density functions are analytically solved and then simplified using the diagonal approximation.

The analytical expressions of reaction rates and survival probabilities are obtained. In Sec. III, the long-time kinetic transition behaviors for probability density functions and sur- vival probabilities are revealed. The irreversible association and dissociation reactions are analyzed in Sec. IV. The nu- merical results are discussed in Sec. V followed by conclud- ing remarks in Sec. VI.

II. THEORY AND SOLUTION

We consider two Brownian particles under the influence of a constant external field in 3D: One is a neutral particleA and the other is an excited particle 共B^ⴱ兲^q with a charge q.

When two particles collide, they can associate with the in-

a兲Electronic mail: shinkj@snu.ac.kr.

b兲Electronic mail: hkim@donga.ac.kr.

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trinsic rate constant ka into the excited-state bound pair 共AB^ⴱ兲^q, which can dissociate reversibly into the unbound pair of Aand共B^ⴱ兲^qwith the rate constantkd. The excited bound and unbound particles, 共AB^ⴱ兲^q and共B^ⴱ兲^q can decay to their ground-states with the unimolecular rate constants,k₀andk₀⬘^, respectively. In addition, a bimolecular quenching process with the quenching rate constantk_qcompetes with the asso- ciation reaction. Thus, the reaction mechanism can be sche- matically written as

A+共B^ⴱ兲^q

k_d k_a

共AB^ⴱ兲^q, 共2.1a兲

A+共B^ⴱ兲^q→

k_q

A+B^q, 共2.1b兲

共AB^ⴱ兲^q→

k₀

共AB兲^q, 共2.1c兲

共B^ⴱ兲^q→

k₀⬘

B^q. 共2.1d兲

The frame of reference is chosen such that the particleAis at the origin without loss of generality. We take thezaxis along the same direction of the electric fieldEifqis positive, and along the reverse direction ifq is negative, making the sys- tem symmetric about the azimuthal angle in the spherical polar coordinate.

Let␳共r,␮^,t兩ⴱ,␮0兲be the probability density function to observe the particle共B^ⴱ兲^q at a distance rand an angle ␮^at timetfor an initially bound particle共AB^ⴱ兲^qwith an angle␮0. Here,ⴱdenotes the bound state and␮^{= cos}␪ with the polar angle␪. Using similar methods to those used for an initially separated pair,^20,32 we can derive solutions for an initially bound particle. The density function satisfies the following Debye–Smoluchowski equation:¹

⳵␳共r,␮^,␶兩ⴱ,␮0兲

⳵␶ ^=ⵜ·关ⵜ␳共r,␮,␶兩ⴱ,␮0兲 +␳共r,␮^,␶兩ⴱ,␮0兲ⵜW兴

−k₀⬘␳共r,␮,t兩ⴱ,␮0兲, 共2.2兲 where␶^=Dt,D is the relative diffusion constant of the un- bound pair, and W is the potential energy multiplied by ␤

= 1/kBTwith the Boltzmann constantkBand the temperature T. It should be noted thatk₀⬘/Dis redefined ask₀⬘for simplic- ity and all the other rate constants 共ka, kd, kq, and k₀兲 are redefined similarly henceforth. We have W共r,␮兲= −2kFr, where kF=F␮^{, F=}␤兩q兩EDB/共2D兲, and DB is the diffusion constant of共B^ⴱ兲^q. The kinetic equation for the binding prob- ability function␳共ⴱ,␮^,␶兩ⴱ,␮0兲, which is the probability to observe共AB^ⴱ兲^qat an angle␮ ^{and time}␶, is given by

⳵␳共ⴱ,␮^,␶兩ⴱ,␮0兲

⳵␶ ^=ka␳共a,␮,␶兩ⴱ,␮0兲

−共kd+k₀兲␳共ⴱ,␮^,␶兩ⴱ,␮0兲. 共2.3兲 The generalized anisotropic boundary condition at the reac- tion distance a and the normalized initial condition can be written as

冏

^{⳵␳共r,␮}⳵^{,␶r兩ⴱ,␮0兲}

冏

r=a

=共ka+kq+ 2kF兲␳共a,␮,␶兩ⴱ,␮0兲

−kd␳共ⴱ,␮^,␶兩ⴱ,␮0兲, 共2.4兲

␳共ⴱ,␮,0兩ⴱ,␮0兲= 1

2␲^a2␦共␮⁻␮0兲. 共2.5兲

Note that␳共r,␮^{, 0}兩ⴱ,␮0兲= 0 by definition.

The following two transformations:

␳共r,␮^,␶兩ⴱ,␮0兲= 1

2␲^{e1/2关W共a,␮0兲−W共r,␮兲兴h共r,}␮^,␶兩ⴱ,␮0兲, 共2.6兲

␳共ⴱ,␮,␶兩ⴱ,␮0兲= 1 2␲^e1/2关

W共a,␮0兲−W共a,␮兲兴h共ⴱ,␮,␶兩ⴱ,␮0兲 共2.7兲 are useful to make Eqs.共2.2兲–共2.5兲more tractable equations as

⳵h共r,␮^,␶兩ⴱ,␮0兲

⳵␶

=

冋

⳵^⳵r22+ 2 r

⳵

⳵^r−共F2+k0⬘兲

册

^{h共r,␮,␶兩ⴱ,␮0兲}

+ 1 r²

⳵

⳵␮

冋

^{共1 −␮2兲⳵h共r,␮⳵␮,␶兩ⴱ,␮0兲}

册

^{, 共2.8兲}

⳵^h共ⴱ,␮^,␶兩ⴱ,␮0兲

⳵␶ ^=kah共a,␮,␶兩ⴱ,␮0兲

−共kd+k₀兲h共ⴱ,␮^,␶兩ⴱ,␮0兲, 共2.9兲

冏

^⳵h共r,␮⳵^{,␶r兩ⴱ,␮0兲}

冏

r=a

=共F␮+ka+kq兲h共a,␮,␶兩ⴱ,␮0兲

−kdh共ⴱ,␮^,␶兩ⴱ,␮0兲, 共2.10兲 h共ⴱ,␮,0兩ⴱ,␮0兲= 1

a²␦共␮⁻␮0兲. 共2.11兲

Then, we can separate angular and radial variables of h共r,␮^,␶兩ⴱ,␮0兲 using

h共r,␮^,␶兩ⴱ,␮0兲=

兺

l=0

⬁

A_l共␮0兲P_l共␮兲R_l共r,␶兩ⴱ兲, 共2.12兲 where the angular Legendre function Pl共␮兲 and the radial functionRl共r,␶兩ⴱ兲satisfy the following respective equations:

⳵

⳵␮

冋

^{共1 −␮2兲⳵P⳵␮l共␮兲}

册

^{+l共l+ 1兲Pl共␮兲= 0, 共2.13兲}

⳵^Rl共r,␶兩ⴱ兲

⳵␶ ⁼

冋

^{⳵⳵r22+2r⳵⳵r−}

^冉

^{F2+k0⬘+l共lr+ 12 兲}

^冊 册

⫻Rl共r,␶兩ⴱ兲. 共2.14兲

The constant Al共␮0兲 is determined by the initial condition.

164112-2 Reigh, Shin, and Kim J. Chem. Phys.132, 164112共2010兲

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Since the delta function can be expanded in terms of the Legendre polynomials as ␦共␮⁻␮0兲=兺_l=0^⬁ Al共␮0兲Pl共␮兲, we haveAl共␮0兲=共2l+ 1兲Pl共␮0兲/2.

In the Laplace-transformed 关f˜共s兲=兰₀^⬁f共␶兲e^−s␶d␶兴 space, Eq. 共2.14兲with the initial conditionRl共ⴱ, 0兩ⴱ兲= 1/a²can be written as

⳵^2R˜l

⳵^{r2 +} 2 r

⳵^R˜l

⳵^{r −}

冉

^{s+F2+k0⬘+l共l}r^{+ 1兲2}

冊

^{R˜l= 0. 共2.15兲}

Two linearly independent solutions of this homogenous dif- ferential equation can be expressed by the modified spherical Bessel functions of fractional order: y_1l共r,s兲

=兵␲/共2␴^r兲其^1/2I_l+1/2共␴^r兲 andy_2l共r,s兲=兵␲/共2␴^r兲其^1/2K_l+1/2共␴^r兲, where ␴⁼共s+F²+k₀⬘兲^1/2.³³ Considering the outer boundary condition, we have

R˜

l共r,s兩ⴱ兲=␣l共s兲y2l共r,s兲, 共2.16兲

where␣l共s兲is to be determined by the inner boundary con- dition.

By eliminating ˜h共ⴱ,␮,s兩ⴱ,␮0兲 from Eqs. 共2.9兲 and 共2.10兲, we obtain

冏

^{⳵˜h共r,␮}⳵^{,s兩rⴱ,␮0兲}

冏

r=a

=兵B共s兲+F␮其˜h共a,␮^,s兩ⴱ,␮0兲

−H共s兲␦共␮⁻␮0兲, 共2.17兲 where B共s兲=k_q+k_a−k_ak_d/共s+k_d+k₀兲 and H共s兲=k_d/兵a²共s +kd+k₀兲其. By re-expanding the right hand side of Eq.共2.17兲 in terms of Legendre polynomials such as Eq.共2.12兲with the recurrence relation共2l+ 1兲P1Pl=共l+ 1兲Pl+1+lP_l−1, we obtain the following infinite matrix formula to find␣l共s兲:

冢

^{DE110 EDE20121 ED212 EE221l Dl E2l}

冣冢

^{␣␣␣]␣]012l}

冣

⁼

冢

^HHH]H]

冣

_共2.18兲^,

where the elements are defined as Dl共s兲=B共s兲y2l共a,s兲

−y_2l⬘共a,s兲,

E_1l共␮0,s兲=F l 2l+ 1

P_l−1共␮0兲

Pl共␮0兲 y_2l−1共a,s兲,

E_2l共␮0,s兲=Fl+ 1 2l+ 1

P_l+1共␮0兲

Pl共␮0兲 y_2l+1共a,s兲,

andy_2l⬘ is the first derivative with respect tor. If we assume that E_il/D_l 共i= 1 , 2兲is small enough, we obtain␣l共s兲 as

␣l共s兲 ⬵ H共s兲

B共s兲y_2l共a,s兲−y_2l⬘共a,s兲+O

冉

^ED^ill

冊

^{. 共2.19兲}

In the small F limit, we get Eil/Dl⬃constant⫻F+O共F³兲 and H/Dl⬃constant+O共F²兲. Therefore, we can reduce the tridiagonal matrix for the initially bound state in Eq.共2.18兲

to the diagonal one in the smallFlimit. Under this diagonal approximation, we can obtain the probability density func- tions in the Laplace domain as

␳

˜共r,␮,s兩ⴱ,␮0兲= 1 4␲^a2e

F共r␮−a␮0兲

兺

l=0

⬁

共2l+ 1兲Pl共␮0兲Pl共␮兲

⫻ kd

共s+kd+k₀兲

y_2l共r,s兲 B共s兲y_2l共a,s兲−y_2l⬘共a,s兲,

共2.20兲

␳

˜共ⴱ,␮,s兩ⴱ,␮0兲= 1 4␲^a2e

Fa共␮−␮0兲

兺

l=0

⬁

共2l+ 1兲Pl共␮0兲Pl共␮兲

⫻ 1 共s+kd+k₀兲

共k_q+k_a兲y_2l共a,s兲−y_2l⬘共a,s兲 B共s兲y_2l共a,s兲−y_2l⬘共a,s兲 .

共2.21兲 The probability density functions for the initially unbound state separated byr₀can be obtained similarly. Following the procedure of our previous work,²⁰ we can obtain

␳

˜共r,␮,s兩r₀,␮0兲= 1 4␲^e

F共r␮−r0␮0兲

兺

l=0

⬁

共2l+ 1兲Pl共␮0兲Pl共␮兲

⫻¯y_1l共min共r,r0兲,s兲y2l共max共r,r0兲,s兲

␲/共2␴兲 , 共2.22兲

␳

˜共ⴱ,␮,s兩r0,␮0兲= 1 4␲^a2e

F共a␮−r0␮0兲

兺

l=0

⬁

共2l+ 1兲Pl共␮0兲Pl共␮兲

⫻ ka

s+kd+k₀

y_2l共r0,s兲 B共s兲y_2l共a,s兲−y_2l⬘共a,s兲,

共2.23兲 where ¯y_1l共r,s兲=y_1l共r,s兲−␣l共s兲y_2l共r,s兲. From the diagonal approximation, ␣l is given by ␣l⬵兵B共s兲y_1l共a,s兲

−y_1l⬘共a,s兲其/兵B共s兲y2l共a,s兲−y_2l⬘共a,s兲其. Comparison of Eq.

共2.20兲with Eq.共2.23兲leads to the detailed balance condition ka␳共r,␮^,␶兩ⴱ,␮0兲e^2Fa␮0=kd␳共ⴱ,␮0,␶兩r,␮兲e^2Fr␮. 共2.24兲 By integrating the kinetic equations and using the Gauss’s theorem, we obtain the following equation irrespec- tive of the initial condition:

−dS共␶兲

d␶ ^{= −}

冕

_⌫=⌺关ⵜ␳⁺␳ⵜW兴d⌫+k₀⬘S共␶兲, 共2.25兲 where⌺is the reaction surface andS共␶兲is the total survival probability of共B^ⴱ兲^q. Since the reaction rate is defined as the surface integral of the flux through the reaction boundary, we obtain the useful relation between the survival probability and the rate of reaction as

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−dS共␶兲

d␶ ^=K共␶兲+k₀⬘S共␶兲. 共2.26兲

This equation means that the survival probability changes due to two contributions of the association-dissociation reac- tions betweenAand共B^ⴱ兲^qand the unimolecular decay rate.

Using the above diagonal approximation, the reaction rate and the survival probability for the initially bound pair are easily derived as

K˜共s兩ⴱ,␮0兲=e^−Fa␮0

兺

l=0

⬁

共2l+ 1兲P_l共␮0兲 k_d s+k_d+k₀

⫻y_1l共a,−k₀⬘兲y_2l⬘共a,s兲−y_2l共a,s兲y_1l⬘共a,−k₀⬘兲 B共s兲y2l共a,s兲−y_2l⬘共a,s兲 ,

共2.27兲

˜S共s兩ⴱ,␮0兲= −K˜共s兩ⴱ,␮0兲

s+k₀⬘ ^{. 共2.28兲}

For the initially unbound case, we have K˜共s兩r0,␮0兲=e^−Fr0␮0

兺

l=0

⬁

共2l+ 1兲Pl共␮0兲y2l共r0,s兲

⫻B共s兲y1l共a,−k₀⬘兲−y_1l⬘共a,−k₀⬘兲

B共s兲y2l共a,s兲−y_2l⬘共a,s兲 , 共2.29兲

˜S共s兩r0,␮0兲=1 −K˜共s兩r₀,␮0兲

s+k₀⬘ ^{. 共2.30兲}

Thus, one can easily check that the following normalization condition holds:²⁴

共s+k₀⬘兲S˜共s兲+共s+k₀兲p˜共ⴱ,s兲+kq˜p共a,s兲= 1, 共2.31兲 where˜p共s兲=兰_⌺^˜␳共s兲d⌫. It should be noted that the survival probabilities can be alternatively derived from the kinetic equations without the detail calculations of the probability density functions.^20,34,35

III. ASYMPTOTIC KINETIC TRANSITION BEHAVIORS When excited molecules have different lifetimes or the system is affected by the external field, the kinetic transitions at long times have been predicted for various reversible diffusion-influenced reactions.^23–30In the present system, we have to consider both different-lifetime and external field effects that affect the pattern of the long-time kinetic transi- tion behaviors.

The diagonal approximation, which is valid in the weak field limit, allows us to obtain analytical solutions in the form of series expansions in the Laplace domain as shown above. We find that the vast majority of the contribution to the long-time dynamics comes from the lowest-order 共l= 0兲 term in the series expansions, which can be analytically in- verted to give time-domain results. For the initially unbound pair, we have

␳l=0

e 共r,␮^,␶兩r0,␮0兲

= 1

4␲^rr0

e^{F共r␮−r0␮0兲}

再 ^冑

^{41␲␶关e−共r−r0兲2/4␶+e−共r+r0− 2a兲2/4␶兴}

+

兺

i=1

3 ␴i共␴j+␴i兲共␴k+␴i兲

共␴j−␴i兲共␴k−␴i兲 ⌽i共r+r₀− 2a,␶兲

冎

^{, 共3.1兲}

where three roots␴i,␴j, and␴k共i⫽j⫽k= 1, 2, or 3兲satisfy the following relations:

␬1=␴1+␴2+␴3= −共ka+kq+ 1/a兲, 共3.2兲

␬2=␴1␴2+␴2␴3+␴3␴1=kd−F²+k₀−k₀⬘^, 共3.3兲

␬3=␴1␴2␴3=␬1␬2+kakd, 共3.4兲 and

⌽i共r,␶兲= exp共␴i2␶⁻␴ir兲erfc

冉 ^冑

^r4␶⁻␴i

冑

_␶

冊

^{, 共3.5兲}

with the complementary error function erfc共x兲. Here, we have defined the effective Green function as ␳^e=␳^{e共F2+k0⬘兲␶.} Then, the other probability density functions for the l= 0 terms can be obtained as follows:

␳l=0

e 共ⴱ,␮^,␶兩r0,␮0兲= ka

4␲^ar0

e^{F共a␮−r0␮0兲}

⫻

兺

i=1

3 ␴i

共␴j−␴i兲共␴k−␴i兲⌽i共r₀−a,␶兲, 共3.6兲

␳l=0

e 共ⴱ,␮^,␶兩ⴱ,␮0兲= − 1 4␲^a2e

Fa共␮−␮0兲

⫻

兺

i=1

3 ␴i共␴j+␴k兲

共␴j−␴i兲共␴k−␴i兲⌽i共0,␶兲, 共3.7兲 and␳_l=0^e 共r,␮^,␶兩ⴱ,␮0兲 can be readily obtained from the de- tailed balanced condition Eq. 共2.24兲.

The long-time behaviors of effective probability density functions depend on ⌽i共r,␶兲 as previously analyzed in the time domain.^15,23We can predict the long-time behaviors of probability functions straightforwardly in the Laplace do- main without detailed analysis of three roots ␴i.^36,37 In this way, we find that the long-time behaviors depend on the sign of ␬3.

When␬3⬍0, the effective probability density functions decay as ␶^−3/2 at long times as

␳l=0

e 共r,␮^,␶兩r₀,␮0兲 ⬃ e^{F共r␮−r0␮0兲}

rr₀共4␲␶兲^3/2

冋

^rr0+a2+

冉

^␬␬²3

冊

²

−共r+r₀− 2a兲

冉

^a+␬^␬23

冊 册

^{, 共3.8兲}

164112-4 Reigh, Shin, and Kim J. Chem. Phys.132, 164112共2010兲

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␳l=0

e 共ⴱ,␮^,␶兩r0,␮0兲 ⬃k_ae^{F共a␮−r0␮0兲} ar₀共4␲␶兲^3/2

1

␬3

冋

^␬␬²3

−共r0−a兲

册

^,

共3.9兲

␳l=0

e 共ⴱ,␮^,␶兩ⴱ,␮0兲 ⬃ e^{Fa共␮−␮0兲} a²共4␲␶兲^3/2

kakd

␬3

2 . 共3.10兲

When␬3= 0 or

F²+k₀⬘^=k0+kd共kq+ 1/a兲/共ka+kq+ 1/a兲, 共3.11兲 one root vanishes and the transition behavior of ␶^{−1/2 de-} crease is observed as

␳l=0

e 共r,␮^,␶兩r₀,␮0兲 ⬃ 1

4␲^rr0

冑

_␲␶^{eF共r␮−r0␮0兲, 共3.12兲}

␳l=0

e 共ⴱ,␮,␶兩r0,␮0兲 ⬃ ka

4␲^ar0␬2

冑

␲␶^e

F共a␮−r0␮0兲, 共3.13兲

␳l=0

e 共ⴱ,␮^,␶兩ⴱ,␮0兲 ⬃− 1 4␲^a2

冑

␲␶

␬1

␬2

e^{Fa共␮−␮0兲}. 共3.14兲

When ␬3⬎0, the long-time effective density functions in- crease exponentially as

␳l=0

e 共r,␮^,␶兩r0,␮0兲 ⬃ 1 2␲^rr0

␴i共␴j+␴i兲共␴k+␴i兲 共␴j−␴i兲共␴k−␴i兲

⫻e^{␴i2␶−␴i共r+r0−2a兲+F共r␮−r0␮0兲}, 共3.15兲

␳l=0

e 共ⴱ,␮^,␶兩r0,␮0兲 ⬃ ka

2␲^ar0

␴i

共␴j−␴i兲共␴k−␴i兲

⫻e^{␴i2␶−␴i共r0−a兲+F共a␮−r0␮0兲}, 共3.16兲

␳l=0

e 共ⴱ,␮^,␶兩ⴱ,␮0兲 ⬃− 1 2␲^a2

␴i共␴j+␴k兲 共␴j−␴i兲共␴k−␴i兲

⫻e^{␴i2␶+Fa共␮−␮0兲}, 共3.17兲 where␴iis the positive root.

We can change the fate of effective probability density functions by controlling the field strength or the rate con- stants. In a usual experiment, we can control the field strength more easily than the rate constants. The critical field Fcto determine kinetic transition behaviors is given by Fc

=

冑

^k0−k₀⬘^+kd共kq+ 1/a兲/共ka+kq+ 1/a兲 from Eq. 共3.11兲. The long-time effective density functions decay as ␶^{−3/2 when F}

⬍Fcand increase exponentially whenF⬎Fc. WhenF=Fc, they decay as␶^−1/2.

On the other hand, the effective survival probabilities always increase exponentially. The lowest-order terms of the effective survival probabilities in the time domain are given by

S_l=0^e 共␶兩r₀,␮0兲=e^F2␶−sinh共Fa兲e^−Fr0␮0 Fr₀

⫻

兺

i=1

5 ␴i共C0␴i2+C₁兲

共␴j−␴i兲共␴k−␴i兲共␴l−␴i兲共␴m−␴i兲

⫻⌽i共r0−a,␶兲, 共3.18兲

S_l=0^e 共␶兩ⴱ,␮0兲=kdsinh共Fa兲e^−Fa␮0 Fa

⫻

兺

i=1

5 ␴i共␴i+F␥兲

共␴j−␴i兲共␴k−␴i兲共␴l−␴i兲共␴m−␴i兲

⫻⌽i共0,␶兲, 共3.19兲

where C₀= −共␬1+F␥兲, C₁=C₀␬2−k_ak_d, and ␥^{= coth}共Fa兲. Here, we have five roots; the same three roots as in Eqs.

共3.2兲–共3.4兲 and additional two roots of ␴4= −F and ␴5=F 共i⫽j⫽k⫽l⫽m= 1, 2, 3, 4, or 5兲. At long times, the survival probabilities are given by

S_l=0^e 共␶兩r0,␮0兲

⬃

冋

^{1 +共1 −2Fre2Fa0 兲共FC−0␬共F1兲共F2+2␬+2兲␬−2兲k−akkdakde−Fr0共␮0+1兲}

册

⫻e^F2␶, 共3.20兲

S_l=0^e 共␶兩ⴱ,␮0兲 ⬃1 a

kd

共F−␬1兲共F²+␬2兲−k_ak_de^{F2␶−Fa共␮0−1兲}. 共3.21兲 It should be noted that, when k₀⬘^{= 0, S}l=0共⬁兲 converges to certain values.

IV. IRREVERSIBLE CASES

An interesting special limit of the above general results is the irreversible case since we can observe different behav- iors from those in the reversible case. In this section, we discuss the irreversible association 共kd= 0兲 and dissociation 共ka= 0兲 cases in detail. For both irreversible cases, we can simplify probability functions and survival probabilities us- ing the fact that ␬3=␬1␬2 and three roots are given by ␴1

=␬1,␴2=

冑

−␬2, and␴3= −

冑

−␬2.

Since␴2+␴3= 0, Eq.共3.1兲can be simplified to

␳l=0

e 共r,␮^,␶兩r₀,␮0兲= 1 4␲^rr0

e^{F共r␮−r0␮0兲}

再 ^冑

^{41␲␶关e−共r−r0兲2/4␶}

+e^{−共r+r0− 2a兲2/4␶}兴

+␬1⌽1共r+r₀− 2a,␶兲

冎

^{, 共4.1兲}

and this function shows a␶^−3/2 power-law decay behavior at long times as

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␳l=0

e 共r,␮^,␶兩r₀,␮0兲 ⬃共r−a+ 1/␬1兲共r0−a+ 1/␬1兲

rr₀共4␲␶兲^3/2 . 共4.2兲 On the other hand, the long-time behavior of the effective binding probability function ␳l=0

e 共ⴱ,␮^,␶兩r₀,␮0兲, which is given by Eq. 共3.6兲 but with three roots in the irreversible case, depends on the sign of␬2considering that␬3=␬1␬2and

␬1⬍0. When␬2⬎0, it shows a power-law decay behavior of

␶^{−3/2 and when} ␬2⬍0, it increases exponentially. When ␬2

= 0, two roots vanish unlike the reversible case and it simpli- fies to

␳l=0

e 共ⴱ,␮^,␶兩r0,␮0兲= −kae^{F共a␮−r0␮0兲} 4␲^ar0␬1

⫻

冋

^erfc

^冉

^r0

^冑

^4−␶a

^冊

^{−⌽1共r0−a,␶兲}

册

^,

共4.3兲 which converges to a constant of kae^{F共a␮−r0␮0兲}/共−4␲^ar0␬1兲.

Note that this long-time behavior is contrasted with a ␶^−1/2 power-law behavior in the reversible case. One can confirm that similar results can be obtained for the function

␳l=0

e 共r,␮^,␶兩ⴱ,␮0兲 using the detailed balance condition Eq.

共2.24兲.

The effective binding probability ␳^e共ⴱ,␮^,␶兩ⴱ,␮0兲 can be easily calculated from Eq.共2.21兲as

␳^e共ⴱ,␮^,␶兩ⴱ,␮0兲= 1

2␲^a2␦共␮⁻␮0兲e^−␬2␶. 共4.4兲 The global binding probability function is given by averag- ing over the initial angle ␮0 and taking the surface integral over the reaction surface⌺ asp^e共ⴱ,␶兩ⴱ兲=e^−␬2␶.

The effective survival probability Eq.共3.18兲in the irre- versible limit can be simplified to

S_l=0^e 共␶兩r0,␮0兲=e^F2␶

−C₀e^−Fr0␮0sinh共Fa兲

Fr₀

兺

i=1

3 ␴i⌽i共r₀−a,␶兲 共␴j−␴i兲共␴k−␴i兲,

共4.5兲 where␴1=␬1,␴2= −F, and␴3=F. The asymptotic behavior shows an exponential increase

S_l=0^e 共␶兩r0,␮0兲 ⬃e^F2␶

冋

^{1 −C0共e2Fr2Fa− 1兲e}0共F−^−Fr␬1⁰兲^共␮0+1兲

册

^{. 共4.6兲}

One can obtain the expressions of S_l=0^e 共␶兩ⴱ,␮0兲 in the irre- versible limit from Eqs. 共3.19兲 and 共3.21兲. Therefore, the effective irreversible survival probabilities always increase exponentially such as the reversible ones.

From Eq.共4.6兲, one can obtain a generalized field-free version of the irreversible escape probability, S_l=0共␶兩r₀兲

⬃关1 +共a/r₀兲兵共ka+kq兲/␬1其兴e^−k0⬘␶, which reduces to the well- known result of S_l=0共⬁兩r₀兲= 1 −共a/r₀兲兵ka/共ka+ 1/a兲其 in the ground-state limit. This corroborates the fact that the l= 0 term is dominant at long times. Note that, in the absorbing boundary condition limit 共ka→⬁ andkd= 0兲, we can derive

lim_␶→⬁关S共␶兩r₀,␮0兲e^k0⬘␶兴=S_g共⬁兩r₀,␮0兲from Eq.共2.30兲, where the ground-state escape probability S_g共⬁兲 was obtained in Paper I关Eq. 共5.6兲兴.

V. NUMERICAL RESULTS

Numerically exact results can be obtained by solving the infinite matrix equation in Eq.共2.18兲. After the matrix size is chosen as small as possible by checking the convergence, we take the inversion of the matrix and perform the inverse Laplace transformation.³⁸The numerical results using the di- agonal approximation are obtained by numerically inverting analytic equations in Sec. II. We verify numerically that the l= 0 term is the dominant contribution to the long-time be- havior and the higher-order terms usually affect the short- time region.²⁰Long-time results from thel= 0 term contribu- tion are compared to full numerical solutions.

The time dependence of the effective probability density function for the initially bound state, ␳^e共ⴱ,␮^,␶兩ⴱ,␮0兲 with respect to the field strengthFis shown in Fig.1. The dimen- sionless parameter values are k_a= 9.0, k_d= 1.0, a= 1.0, ␮0

= −0.5, ␮^{= −0.5, k}q= 0.5,k₀= 0.1, and k₀⬘= 0.1. For these pa- rameter values, the matrix of Eq.共2.18兲converges so rapidly that it is enough to use 10⫻10 matrix and the first ten terms in the series expansions in Eq. 共2.12兲. The solid lines show the exact numerical results with the off-diagonal terms in the matrix, the dashed lines show the diagonal approximation results. The diagonal approximation is found to produce the underestimated but qualitatively correct behaviors. Indeed, the discrepancy resulting from the diagonal approximation increases as the field strength increases. The dotted lines show the corresponding asymptotic behaviors for the l= 0 terms which are shown to be the dominant contribution in the long time limit. One can see clearly that the kinetic tran-

1 10 100

1E-5 1E-4 1E-3 0.01 0.1 1 10

F=Fc+0.2

F=F_c





e 













F=Fc-0.2

FIG. 1. The field effect: the time dependence of the effective probability density function,␳共ⴱ,␮,␶兩ⴱ,␮0兲e^共F2+k0⬘兲␶for several values ofF. It shows the usual␶^−3/2→␶^−1/2→exponential transition behavior asFis increased.

The solid lines correspond to the numerical calculations with off-diagonal terms up to ten terms in the series sum. The dashed lines correspond to the numerical inversion of the diagonal approximation关Eq.共2.21兲兴and the dot- ted lines to the asymptotic limits for thel= 0 term关Eqs.共3.10兲,共3.14兲, and 共3.17兲兴. The parameter values are ka= 9.0, kd= 1.0, a= 1.0,

␮0= −0.5,␮^{= −0.5,k}q= 0.5,k₀= 0.1, andk₀⬘= 0.1. The critical field is given byF_c=

冑

^k0−k₀⬘^+kd共k_q+ 1/a兲/共k_a+k_q+ 1/a兲= 0.3780.

164112-6 Reigh, Shin, and Kim J. Chem. Phys.132, 164112共2010兲

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