Cage effect dynamics

Cage effect dynamics Igor Khudyakov, Anatoly A. Zharikov, and Anatoly I. Burshtein Citation: J. Chem. Phys. 132, 014104 (2010); doi: 10.1063/1.3280027 View online: http://dx.doi.org/10.1063/1.3280027 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v132/i1 Published by the American Institute of Physics.

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THE JOURNAL OF CHEMICAL PHYSICS 132, 014104 共2010兲

Cage effect dynamics Igor Khudyakov,1 Anatoly A. Zharikov,2 and Anatoly I. Burshtein3,a兲 1

Bomar Specialties, Torrington, Connecticut 06790, USA Physik Department T-38, Technische Universität München, D-85748 Garching, Germany 3 Weizmann Institute of Science, Rehovot 76100, Israel 2

共Received 18 October 2009; accepted 3 December 2009; published online 5 January 2010兲 The kinetics of recombination/dissociation of photogenerated radical pairs 共RPs兲 is described with a generalized model 共GM兲, which combines exponential models 共EMs兲 and contact models 共CMs兲 of cage effect dynamics. The main assumption of EM is the irreversible dissociation of RP as a first-order reaction. CM takes into account repetitive contacts of radicals in the pair or reversible diffusional separation of radicals. The present GM accounts for both the recombination of RP within the cage of outcome and their diffusional separation in a free space. GM predicts that a short initial time RP decay is described by EM, and RP kinetics at longer times of observation is described by CM. This theory successfully describes available experimental data. Solvent viscosity effect on the cage effect values is also analyzed. © 2010 American Institute of Physics. 关doi:10.1063/1.3280027兴 I. INTRODUCTION

The cage effect is an important phenomenon accompanying free radical reactions in the liquid phase. Cage effect values under thermolysis and photolysis of organic molecules into free radicals have been studied since the 1930s 共see, for review, Ref. 1兲. The development of time-resolved techniques with high resolution allowed studying not only cage effect values but also the dynamics of cage effect or kinetics of geminate recombination of photoinduced radicals, cf., e.g., recent publications.2–4 The analysis of geminate recombination provides better understanding of the dynamics of the liquid phase fast reactions. The excited molecule dissociation into radicals and their subsequent diffusional separation is usually described with either exponential or contact models of geminate reaction. The former implies that the hopping escape from the cage 共with the rate U兲 is irreversible, while the latter assumes that association of radicals 共into parent molecules兲, proceeding at contact with the rate constant ka, is irreversible as well. In actual fact this is one and the same reversible reaction, only proceeding in opposite directions. When it starts from the cage escape it is practically irreversible at the very beginning. The exponential model 共EM兲 describes well only the initial 共exponential兲 development of the process, while the contact one only its long time 共nonexponential兲 asymptotes. To get them together we develop here a new approach similar to one proposed earlier.5 They both account for the hopping particle exchange between reactive and adjacent layers, followed by diffusional separation of radicals. Basically the survival probability of radicals n共t兲 being 1 at t = 0 tends to the quantum yield of the free radicals,

␸ = lim n共t兲, t→⬁

so that the kinetics of the radical recombination is given by the function a兲

Electronic mail: [email protected].

0021-9606/2010/132共1兲/014104/6/$30.00

R=

n共t兲 − ␸ . 1−␸

共1.1兲

In the EM, nc is the survival probability of radicals located in the cage, where they either recombine or jump out with the rates W and U, respectively. Consequently, the kinetics of cage devastation is a simple exponent 关see Eq. 共3.7兲 in the review兴6 共1.2兲

R共t兲 = e−共W+U兲t .

In the alternative model of out of cage but contact recombination, the radical pairs are distributed over their separation r with a density ␮共r兲 and their total amount is n0共t兲 =

冕

␮共r,t兲d3r.

共1.3兲

If the radicals instantaneously escaped the cage, appearing outside its spherical border of radius ␴, then their initial distribution would be the following:

␮共r,0兲 =

␦共r − ␴兲 . 4␲r2

共1.4兲

In this particular case of contact start, the kinetics of contact radical recombination is given by n0共t兲 ⬅ ⍀共t兲, but R共t兲 introduced in Eq. 共1.1兲 and specified by Eq. 共5.25兲 in the same review6 is different R共t兲 = et/t0 erfc 冑t/t0 .

共1.5兲

Here the recombination time t0 = ␶d␸20

共1.6兲

is proportional to the separation 共encounter兲 time of the neutral particles 共radicals兲,

␶d = ␴2/D,

共1.7兲

where D is the encounter diffusion coefficient and 132, 014104-1

© 2010 American Institute of Physics


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Khudyakov, Zharikov, and Burshtein

␮˙ = D⌬␮ .

0

ln(C/Co), ln(R(t))

-1 -2

共2.2兲

Specific to the present problem is only the boundary condition

b

冏

-3

j␮ = 4␲␴2D

-4

⳵␮ ⳵r

冏

-5

a

-6 -7 0

1

2

t, µs

3

FIG. 1. Simulations of experimentally observed pair recombination kinetics R共t兲, with 共a兲 EM and 共b兲 CM 共cf. the text兲. The best fit was obtained with t0 = 0.029 ␮s in Eq. 共1.5兲. Experimental kinetics 共䊊兲 was obtained under laser flash photolysis of benzophenone in the presence of 4-methylphenol in glycerol at 253 K 共cf. Ref. 7 for details兲.

t→⬁

1 , 1 + ka/kD

共1.8兲

is the free radical yield in the present model. The latter implies the diffusional radical separation with the rate constant kD = 4␲␴D, which competes with a contact absorption 共radicals association兲 with the rate constant ka.6,7 The experimental R共t兲共points in Fig. 1兲 was calculated from n共t兲 = C共t兲 / C0, obtained in Ref. 8, where the total radical population C共t兲 was studied and normalized with its initial value, C共0兲 = C0. Unfortunately neither of the theoretical models, represented by Eqs. 共1.2兲 or 共1.5兲, describe the whole kinetics. As was expected, the EM represented by a straight line is good only at the beginning, while the diffusional one describes only the very end of the process, while at t = 0 its rate diverges R˙共t → 0兲 = −

␮共r,0兲 = 0.

and

共2.3兲

共2.4兲

To get the closed kinetic equation for ␮共t兲, we have to employ the well known Green function for irreversible contact quenching obeying the diffusional equation p˙ = D⌬p

␸0 = lim ⍀共t兲 =

= ka␮共␴ . t兲 − Unc ,

which accounts for both radical pair association and dissociation. The initial conditions for such a reaction, starting from the cage excitation, are evidently nc共0兲 = 1

4

r=␴

p共r,r⬘,0兲 =

at

␦共r − r⬘兲 4␲r2

共2.5兲

but with homogeneous boundary condition

冏

jp = 4␲␴2D

⳵p ⳵r

冏

r=␴

= ka p共␴ . t兲,

共2.6兲

which accounts for only the absorbing term ka p共␴ . t兲, analogous to ka␮共␴ . t兲 in Eq. 共2.3兲. As to the pumping term in the latter equation Unc, it has to be used for the definition of the source term for ␮ acting at contact g共r,t兲 = Unc共t兲

␦共r − ␴兲 . 4␲r2

共2.7兲

Adding it to Eq. 共2.2兲 we obtain after integration,

冕 ⬘冕冕 t

␮共r,t兲 =

dt

p共r,r⬘,t − t⬘兲g共r⬘,t⬘兲d3r⬘

0

t

p共r, ␴,t − t⬘兲nc共t⬘兲dt⬘ .

=U

共2.8兲

0

1

冑␲tt0 → ⬁. III. GENERAL SOLUTION OF THE PROBLEM

Applying the Laplace transformation to Eqs. 共2.1兲 and 共2.8兲 with initial conditions 共2.4兲, we get

II. BACKGROUND: CAGE RECOMBINATION AND CAGE ESCAPE

Let us consider the radical pair in a cage 关R1 ¯ R2兴 created by the photodissociation and subjected to recombination 共with the rate W兲 into the ground state of the precursor molecule. This pair is also subjected to a jumplike dissociation and backward association competing with diffusional separation of free radicals W

˜ c共s兲 − 1 = − 共W + U兲n ˜ c共s兲 + ka␮ ˜ 共␴,s兲, sn

共3.1兲

˜ 共r,s兲 = Up ˜ 共r, ␴,s兲n ˜ c共s兲. ␮

共3.2兲

By using Eq. 共3.2兲 at r = ␴ in Eq. 共3.1兲 one easily finds ñc共s兲 =

U

关R1R2兴⇐ 关R1 ¯ R2兴 R1 + R2 .

s + W + U关1 − ka˜p共␴, ␴,s兲兴

.

共3.3兲

The Laplace transformation of the total free radical population is

ka

The cage population nc共t兲 obeys the kinetic equation n˙c = ka␮共␴,t兲 − 共W + U兲nc ,

1

共2.1兲

where U is the rate of radicals escape 共out of cage transitions兲 in 1/s, ka is their backward association 共cage penetration兲 rate constant 共in cm3 / s兲, while the pair radical density in a bulk obeys the free diffusion equation

˜ 共s兲 = ⍀

冕

˜ 共r,s兲d3r = Un ˜ c共s兲 ␮

冕

˜p共r, ␴,s兲d3r.

共3.4兲

˜ 共r , ␴ , s兲d3r = ñ0共s兲 is the Laplace It should be noted that 兰p transformation of the contact recombination kinetics n0共t兲 defined in Eq. 共1.3兲, which obeys relation 共1.1兲 with R共t兲 specified in Eqs. 共1.5兲 and 共1.6兲.


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It is shown in Appendix A 共Ref. 9兲 that 1 − ka˜p共␴, ␴,s兲 = s

冕

˜p共r, ␴,s兲d3r =

␸0共1 + 冑s␶d兲

共1 + ␸0冑s␶d兲

.

共3.5兲

Using this result in Eqs. 共3.3兲 and 共3.4兲 we obtain after some rearrangements the final results ñc共s兲 =

˜ 共s兲 = ⍀

1 + ␸0冑s␶d

s + W + U␸0 + 共s + W + U兲␸0冑s␶d

,

U␸0共1 + 冑s␶d兲

. s关s + W + U␸0 + 共s + W + U兲␸0冑s␶d兴

共3.6兲

共3.7兲

Now we can see that at the end the cage is completely exhausted ˜ c共s兲 = 0. nc共⬁兲 = lim sn s→0

共3.8兲

In the EM, the radicals which escaped from the cage have never come back, i.e., ka = 0 共␸0 = 1兲 and nc共t兲 = e−共W+U兲t is identical to R共t兲 defined in Eq. 共1.2兲. In reality, the backward transfer 共association兲 is not zero 共ka ⫽ 0兲. As a result there is a retardation of cage devastation predicted by the general expression 共3.6兲 that can be specified by its Laplace inversion. The inverse Laplace transformation of the expression 共3.7兲 represents the kinetics of out-cage radical pairs accumulation and dissipation/separation, ⍀共t兲. Here it should be noted that all the above results can be approved or even deduced from the more general but rather formal approach, dividing the reactants on bounded 关nc = p共ⴱ , t 兩 ⴱ兲兴 and unbounded 关⍀ = S共t 兩 ⴱ兲兴 pairs, assuming that the latter are excited 共decay with the rate k0⬘ and are quenched with the rate kq兲.10 Setting k0⬘ = kq = 0 one can reproduce the above results but we preferred to use an original approach, which seems to be easier and more transparent in this particular case. The more essential difference with this and other investigations of reversible and contact geminate recombination11 is the formulation of the problem: We are looking not for the bound and unbound fractions of reactants but for their sum, n共t兲 = nc共t兲 + ⍀共t兲,

共3.9兲

which represents the actually measured quantity in the experiments under discussion. IV. RADICAL SEPARATION QUANTUM YIELD

s→0

共4.1兲

Substituting into this definition the results obtained in Eqs. 共3.7兲, we obtain 1 ␸= . 1 + W/U␸0

ka␮eq = Uneq c . This is true if Uv/ka = e−V/kBT .

共4.2兲

Our present situation is different: the exchange of radicals between the in-cage and out-cage regions is reversible.

共4.3兲

Following the conventional EM, we will assume V = 0 making no difference between the in-cage and out-cage space. Then it follows from Eqs. 共4.3兲 and 共1.8兲 that

␸0 =

1 , 1 + Uv/kD

共4.4兲

and from Eq. 共4.2兲 that

␸=

1 , 1 + W共1/U + 1/ksep兲

共4.5兲

where ksep = kD/v =

3D ␴2

is the usual parameter of EM. If the separation is slower than the rate of cage escape, then neglecting 1 / U, we reduce Eq. 共4.5兲 to the conventional results of either exponential or contact models

␸=

1 1 ⬅ , 1 + W/ksep 1 + kr/kD

共4.6兲

where kr = Wv = 4␲␴z is the contact reaction constant and z is the recombination efficiency.6 Though the yields in this particular limit are the same in both models the kinetics of radical separations is expected to be qualitatively different: the exponent in EM versus asymptote 共1.5兲 in the contact recombination model 共Fig. 1兲. In the opposite limit 共U Ⰶ ksep兲

␸=

After instantaneous photodissociation, the total survival probability of radicals in the system reduces from 1 to the free radical quantum yield12 ˜ 共s兲. ␸ = n共⬁兲 = ⍀共⬁兲 = lim s⍀

In the absence of cage recombination 共W = 0兲 there can be an equilibrium density of radicals in the whole space, which is ␮eq out of cage and neq c / v inside the spherical cage of the volume v = 4␲␴3 / 3. If the cage is actually a rectangular potential well, these densities relate to each other as neq c /v = eV/kBT␮eq, where V ⱖ 0 is a cage depth. On the other hand, Eq. 共2.1兲 turns at equilibrium 共n˙c = W = 0兲 into following relationship:

1 , 1 + W/U

共4.7兲

that is, the exponential 共hopping兲 cage escape controls the radical separation. Such a control is most probably realized in the case of a very deep well 共V Ⰷ kBT兲, which is not considered here.

V. KINETICS OF RADICAL PAIR RECOMBINATION

The Laplace transformation of the recombination kinetics 共1.1兲 expressed via the total radical population 共3.9兲 can now be performed using the results obtained in Eqs. 共3.6兲 and 共3.7兲


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J. Chem. Phys. 132, 014104 共2010兲


冑冑 ˜R共s兲 = U␸0共1 − ␸0兲 s␶d/s + 1 + ␸0 s␶d . s + W + U␸0 + 共s + W + U兲␸0冑s␶d

共5.1兲

Using asymptotical behavior ˜R共s兲 at large s, ˜R共s兲 ⬇ 1 − 共W + U␸0兲 s s2

共s → ⬁兲,

共5.2兲

共s → 0兲,

共5.3兲

and at small s ˜R共s兲 ⬇ ␸共1 − ␸ 兲冑␶ /s d 0

we get the following asymptotic behavior at short and large times: R共t兲 ⬇ 1 − 共W + U␸0兲t

共t ⬇ 0兲, 共5.4兲

R共t兲 ⬇ ␸共1 − ␸0兲

1

冑␲t/␶d

共t → ⬁兲.

The short time behavior is similar to that of an EM. On the contrary, the long time behavior is typical for diffusional contact recombination. The full analytical time form of the kinetics is as follows: 3

R共t兲 = 兺 ␤ie␥i 共W+U␸0兲t erfc共␥i冑共W + U␸0兲t兲, 2

共5.5兲

i=1

where the parameters ␥i are the roots of the cubic equation

␥3 − a␥2 + b␥ − a = 0

共5.6兲

with the coefficients a=

1

,

W+U ␸ = 1 + 共1 − ␸0兲 . b= W + U␸0 ␸0

共5.7兲

共5.8兲

The amplitude ␤i is given as 1 − ␥i/a 3 − 2b␥i/a + ␥2i

1 ␤2,3 = . 2

.

␥2,3 = ⫾ i,

共5.11兲

Since ␤1 = 0 the pure imaginary roots only contribute to the kinetics, which reduces to the exponential one given by Eq. 共1.2兲. A similar situation takes place in the limiting case 共U␶d Ⰷ 1兲 ␸0 → kD / Uv Ⰶ 1. The amplitude ␤1 tends to zero, so the kinetics can be described by the following set of parameters:9

␤1 = 0,

␤2,3 =

冉

冊

共5.12兲

1 ␰ = ␸冑共W + U␸0兲␶d . 2

共5.13兲

1 ␰ 1⫾i 冑1 − ␰2 , 2

Such a situation is realized in the case of U␶d Ⰷ 1, where U␸0 ⬇ 3/␶d,

␸⬇

1 1 = . 1 + W␶d/3 1 + kr/kD

共5.14兲

VI. COMPARISON WITH EXPERIMENT

共5.9兲

Equation 共5.6兲 is a particular case 共kq = k0⬘ = 0兲 of the cubic Eq. 共3.6兲 in Ref. 10, exhaustively studied there. The main modifications are a dimensionless form of the cubic equation and the use of the opposite signs for the coefficients at even degrees of ␥. In the Appendix B 共Ref. 9兲 presenting roots of the cubic equation we give analytical expressions for them as well as a general qualitative analysis. Here we discuss two particular situations. In the special case ␸0 = 1 共b = 1兲, the roots and amplitudes are the following:

␥1 = a,

␤1 = 0,

␥2,3 = ␰ ⫾ i冑1 − ␰2,

␸0冑共W + U␸0兲␶d

␤i =

FIG. 2. Simulations of the experimental data 共䊊兲 on geminate recombination at 253 K 共Ref. 8兲. Solid lines correspond to the extreme cases of ␸0 = 0. The parameters used are W + 共3 / ␶d兲 = 2.2 ␮s, ␶d = 100, 30, and 10 ␮s 共for the curves a, b, d, respectively兲. The dashed line c corresponds to the best fit with ␶d = 30 ␮s and ␸0 = 0.05. The pair consists of benzophenone free radical and 4-methylphenoxyl radical 共Ref. 8兲.

共5.10兲

The proposed analytical solution allows us to compare the theory and experiment shown in Fig. 1. The kinetics R共t兲 has a pronounced exponential behavior with the rate constant 2.2 ␮s−1 at initial times. From this behavior represented in Eq. 共5.4兲, we estimated the parameter W + U␸0 ⬇ 2.2 ␮s−1. The quantum yield ␸ ⬇ 0.1 could be obtained from n共t兲, assuming the latter obeys the EM. However, the real yield should be smaller due to a slower long time 共diffusional兲 approach to ␸ 关see Eq. 共5.4兲兴. Therefore, we fit the whole kinetics n共t兲 rather than R共t兲. In Fig. 2 the theoretical kinetics at different ␶d is compared to the experimental at T = 253 K. To keep a oneparametric picture, we considered the limiting situation given by Eqs. 共5.5兲 and 共5.12兲–共5.14兲. From this comparison, we can extract the following parameters of the system:


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Cage effect dynamics

The cage effect not only increases with viscosity, but the cage 共geminate兲 recombination becomes longer, which is easier to be detected in a moderate time interval. VIII. CONCLUSIONS

Above slow bulk recombination of the radicals, their geminate prehistory is represented by a short and sharp quasiexponential peculiarity at the very beginning of the process, not ever seen in the available time region. However, with increasing viscosity it becomes deeper and lasts longer. Hence, the cage effect becomes more pronounced and detectable in more viscous solutions.

FIG. 3. Simulations of the experimental data 共䊊兲 on geminate recombination 共Ref. 8兲. Solid lines are the simulations of the experimental data with Eq. 共3.9兲共Ref. 8兲 at 293 K, assuming ␶d = 1.1 ␮s 共the top curve兲 and ␶d = 1.2 ␮s 共the bottom one兲.

APPENDIX A: GREEN FUNCTIONS FOR THE CONTACT RECOMBINATION

There is the well known relationship between the Green functions of encounter diffusion with, ˜p共r , ␴ , s兲, and without ˜ 共r , ␴ , s兲 reaction, G 0

␶d ⬇ 30 ␮s,

␸ ⬇ 0.05,

W ⬇ 2.1 ␮s−1 .

共6.1兲

Something very interesting and specific to the system under study is the strong dependence of the viscosity with the temperature. The temperature change from T = 253 K to T = 293 K, causes about a 60 times decrease in the viscosity.8 This means that the parameter ␶d should reveal a strong temperature dependence. In Fig. 3 we compare the experimental recombination kinetics n共t兲 at T = 293 K and the theoretical with the assumption that the kinetic parameters W , U remain the same. About a 30 times decrease in ␶d correlates with the change in viscosity. In addition, we can conclude that the parameter U is sufficiently larger than W. VII. CAGE EFFECT VALUES

As a matter of fact, the radical recombination is often studied at times when the geminate recombination has already been accomplished. The free radicals recombination in the bulk proceeds with the rate constant kr N共t兲 =

N 0␸ . 1 + N 0␸ k rt

共7.1兲

Here N0 is the volume density of initially 共photochemically兲 created radicals, while N0␸ is the part of them survived in geminate recombination and started to recombine in a bulk while another part N0共1 − ␸兲 = N0␾ has been recombined during the precursor geminate stage. The “cage effect” is measured by 1 1 = , ␾= 1 + U␸0/W 1 + U/W关1 + U␴2/3D兴

共7.2兲

where ␸0 given in Eq. 共4.4兲 varies with diffusion in the interval 0 ⬍ ␸0 ⬍ 1, while 1 ⬎ ␾ ⬎ 1/共1 + U/W兲. Hence the cage effect decreases but saturates with D → ⬁ approaching its minimum value that can be hardly seen experimentally 共Fig. 3兲. the fact of the pronounced increase in the cage effect with rising viscosity was disclosed long ago.1

˜p共r, ␴,s兲 =

˜ 共r, ␴,s兲 G 0 . ˜ 共␴, ␴,s兲 1+k G a

共A1兲

0

Taking into account that 兰G0共r , ␴ , s兲d3r = 1 / s we obtain, after space integration, s

冕


1 ˜ 共␴, ␴,s兲 1 + k aG 0

.

共A2兲

The Green function of free diffusion 共in absence of any reaction兲 is ˜ 共␴, ␴,s兲 = G 0

1

kD关1 + 冑s␶d兴

共A3兲

,

˜ 共r , r⬘ , s兲, given in as follows from the general definition of G 0 Eq. 共4.11兲 of Ref. 13 Substituting Eq. 共A3兲 into Eq. 共A2兲, we obtain the relationship identical to the rhs of Eq. 共3.5兲 s

冕


␸0共1 + 冑s␶d兲 1 + ␸ 0 冑s ␶ d

,

共A4兲

which is expressed via the encounter time 共1.7兲 and the quantum yield, ␸0, defined in Eq. 共1.8兲.14 On the other hand setting r = ␴ in Eq. 共A1兲 we obtain from there ˜ 共␴, ␴,s兲 = G 0

˜p共␴, ␴,s兲 . 1 − ka˜p共␴, ␴,s兲

共A5兲

Substituting this result into Eq. 共A2兲 we obtain the equality identical to the lhs of Eq. 共3.5兲 s

冕

˜p共r, ␴,s兲d3r = 1 − ka˜p共␴, ␴,s兲.

共A6兲

APPENDIX B: THE ROOTS OF THE CUBIC EQUATION

In the general case 共ka ⫽ 0 , ␸0 ⬍ 1兲, the roots ␥i of Eq. 共5.6兲 can be analytically calculated by use of Cardano’s solution:15


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J. Chem. Phys. 132, 014104 共2010兲


a + u+ + u− , 3

␥1 =

␥2,3 =

共B1兲

冑3 a 1 − 共u+ + u−兲 ⫾ i共u+ − u−兲 , 2 3 2

共B2兲

So, one real and two complex conjugate roots is the typical situation for the problem under study. In the limit a , b → ⬁共b / a = const兲 the roots can be found as follows:

where u⫾ = 共q ⫾ 冑⌬兲1/3 and q=a

⌬=

冉

冊

+

冉冉

␥2,3 ⬇

b ⫾i 2a

冑

1−

b2 . 4a2

共B4兲

1 a2 + 1 − 共b − 1兲共b + 19兲 8

冊

冊

2

1 共b − 1兲共9 − b兲3 . 64

共B5兲

The quantity ⌬ is the discriminant of the cubic equation. If ⌬ ⬎ 0, then the equation has one real root and a pair of complex conjugate roots. If ⌬ ⬍ 0, then all roots are real. From Eq. 共B5兲 we can conclude, that the discriminant is always positive at 1 ⱕ b ⬍ 9. In the case of b ⱖ 9 the discriminant can be negative if the magnitude of the parameter a satisfies the conditions 1 1 共b − 1兲共b + 19兲 − 1 − 冑共b − 1兲共b − 9兲3 8 8 1 1 ⱕ a2 ⱕ 共b − 1兲共b + 19兲 − 1 + 冑共b − 1兲共b − 9兲3 . 8 8

共B7兲

T. Koenig and H. Fischer, in Free Radicals, edited by J. K. Kochi 共Wiley, New York, 1973兲, Vol. 1, p. 157. 2 A. B. Oelkers and D. R. Tyler, Photochem. Photobiol. Sci. 7, 1386 共2008兲. 3 I. V. Khudyakov, P. Levin, and V. Kuzmin, Photochem. Photobiol. Sci. 7, 1540 共2008兲. 4 A. B. Oelkers, L. F. Scatene, and D. R. Tyler, J. Phys. Chem. 111, 5353 共2001兲. 5 A. I. Burshtein and B. I. Yakobson, Int. J. Chem. Kinet. 12, 261 共1980兲. 6 A. I. Burshtein, Adv. Chem. Phys. 114, 419 共2000兲. 7 A. I. Burshtein, Adv. Chem. Phys. 129, 105 共2004兲. 8 P. P. Levin, I. V. Khudyakov, and V. A. Kuzmin, J. Phys. Chem. 93, 208 共1989兲. 9 See supplementary material at http://dx.doi.org/10.1063/1.3280027 for Green functions for the contact recombination and the roots of the cubic equation. 10 I. V. Gopich and N. Agmon, J. Chem. Phys. 110, 10433 共1999兲. 11 H. Kim and K. J. Shin, Phys. Rev. Lett. 82, 1578 共1999兲. 12 A. I. Burshtein and E. Krissinel, J. Phys. Chem. A 102, 7541 共1998兲. 13 A. A. Zharikov and A. I. Burshtein, J. Chem. Phys. 93, 5573 共1990兲. 14 A. I. Burshtein, A. A. Zharikov, N. V. Shokhirev, O. B. Spirina, and E. B. Krissinel, J. Chem. Phys. 95, 8013 共1991兲. 15 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1972. 1

a2 b 1 − + , 27 6 2

1 27

␥1 ⬇ a,

共B3兲

共B6兲
