General theory of multistage geminate reactions of isolated pairs of reactants. I. KineticequationsAlexander B. Doktorov and Alexey A. Kipriyanov Citation: The Journal of Chemical Physics 140, 184104 (2014); doi: 10.1063/1.4874001 View online: http://dx.doi.org/10.1063/1.4874001 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Reversible diffusion-influenced reactions of an isolated pair on some two dimensional surfaces J. Chem. Phys. 139, 194103 (2013); 10.1063/1.4830218 Theory of reversible associative-dissociative diffusion-influenced chemical reaction. II. Bulk reaction J. Chem. Phys. 138, 044114 (2013); 10.1063/1.4779476 Theory of reversible associative-dissociative diffusion-influenced chemical reaction. I. Geminate reaction J. Chem. Phys. 135, 094507 (2011); 10.1063/1.3631562 Multisite reversible geminate reaction J. Chem. Phys. 130, 074507 (2009); 10.1063/1.3074305 The integral encounter theory of multistage reactions containing association-dissociation reaction stages. III.Taking account of quantum states of reactants J. Chem. Phys. 121, 5115 (2004); 10.1063/1.1783273

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THE JOURNAL OF CHEMICAL PHYSICS 140, 184104 (2014)

General theory of multistage geminate reactions of isolated pairsof reactants. I. Kinetic equations

Alexander B. Doktorov and Alexey A. KipriyanovVoevodsky Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academyof Sciences, Novosibirsk State University, Novosibirsk 630090, Russia

(Received 11 March 2014; accepted 17 April 2014; published online 9 May 2014)

General matrix approach to the consideration of multistage geminate reactions of isolated pairs ofreactants depending on reactant mobility is formulated on the basis of the concept of “effective” par-ticles. Various elementary reactions (stages of multistage reaction including physicochemical pro-cesses of internal quantum state changes) proceeding with the participation of isolated pairs of reac-tants (or isolated reactants) are taken into account. Investigation has been made in terms of kineticapproach implying the derivation of general (matrix) kinetic equations for local and mean probabili-ties of finding any of the reaction species in the sample under study (or for local and mean concentra-tions). The recipes for the calculation of kinetic coefficients of the equations for mean quantities interms of relative coordinates of reactants have been formulated in the general case of inhomogeneousreacting systems. Important specific case of homogeneous reacting systems is considered. © 2014AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4874001]

I. INTRODUCTION

Reactions depending on spatial migration of reactants(for example, diffusion-influenced reactions) in condensedmedia play an important role in different fields of science andtechnology. Among these are, for instance, electron excitationenergy transfer,1 electron (or proton) migrations in photosyn-thetic systems,2–4 various chemical reactions occurring in col-loid or polymer solutions,5–7 in nano- and bio- systems,8–10

or trapping and detrapping problems in semiconductors11 andmany others. From this standpoint, development of the the-ory of reactions depending on mobility of reactants is crucialto understand the systems and to apply to complex problems.Many fundamental issues on theories had been addressed overmany years.3, 5, 6, 12–25 Most of the theories referred to elemen-tary bulk reactions; however, multistage reactions26–29 signif-icant in studies of biochemical reactions have also been in-vestigated, as well as transfer reactions in luminescence andchemiluminescence and photo- and electrochemistry.30 An in-creasing interest in the consideration of different multistagereactions has stimulated formulation and development ofgeneral (matrix) theories of bulk multistage reactions includ-ing all possible elementary stages (bimolecular chemical ex-change and addition reactions, dissociation and monomolec-ular transformation reactions).26–29 This makes it possible toestablish general detailed balancing principles (total balanceof particles and kinetic and thermodynamic balancing). Inthe framework of the developed approach, “internal” degreesof freedom (for instance, spin states) have been included inthe consideration, thus the reactions with the participation ofparamagnetic particles (free radicals and radical ions), in ex-ternal magnetic field also,28 have been studied.

Geminate reactions of isolated pairs of reactants aris-ing from external radiation31, 32 are another important classof reactions in solutions. Many analytical fundamental is-sues on theories of diffusion-influenced geminate reactions

had been also addressed over many years.33–43 However, theycommonly referred either to treating of elementary geminatereactions (including reversible ones), or the simplest two-stage reactions (reversible bimolecular reactions involvingmonomolecular deactivation stage of excited molecules) forhomogeneous reaction system. Spatially inhomogeneous sys-tems have been considered only by the authors of this work42

for geminate elementary reaction. Besides, in the literaturethere are no papers dealing with geminate reactions betweenidentical reactants. The presence of internal quantum degreesof freedom (for example, spin ones) has only been taken intoaccount in the examination of elementary irreversible reac-tions between radicals. At the same time, consistent consid-eration of such degrees of freedom is necessary, for instance,in studies of biological applications of multistage reactionswith the participation of radicals or other paramagnetic parti-cles (including the case of external magnetic field). Over thelast years there has been an increasing interest in the theoret-ical investigation of multistage geminate reactions of isolatedpairs.44 So it seems reasonable, similarly to the theory of bulkreactions, to develop general matrix kinetic theory of multi-stage geminate reactions of isolated pairs that allows one toconsider all the above aspects and can be a reliable basis forthe description of a wide class of specific chemically reactingsystems. This is the goal of the present contribution.

Section II deals with the description of multistage gemi-nate reactions of isolated pairs of reactants. Kinetic schemesof elementary stages of the multistage reaction under studyare given including all types of bimolecular and monomolec-ular reactions in isolated pairs. The concept of “effective”particles is introduced which serves to determine mean andlocal probabilities of the species residence in the specimen atany moment of time. Besides, integral-matrix Liouvillians de-scribing the course of reactions of different types and molec-ular motion of reactants are formed. In Sec. III, kinetic equa-tions for local and mean probabilities (or local and mean

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184104-2 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

concentrations of species) have been derived. Kinetic coef-ficients (integral kernels and inhomogeneous sources) in ki-netic equations for mean kinetic characteristics are expressedin terms of the values in relative coordinates of geminatepairs. An important and widely encountered case of homoge-neous reaction systems is considered. Section IV is devoted tothe method of generalizing of the employed approach in tak-ing into account “internal” degrees of freedom including spinstates of reactants. The main results are given in Sec. V.

II. DESCRIPTION OF MULTISTAGE GEMINATEREACTIONS OF ISOLATED PAIRS

A. Reaction schemes and “effective particles”

1. Reactions in isolated pairs

In the most general case, a multistage reaction involvesdifferent elementary (in the general case, reversible) reactions(multistage reaction stages). These are bimolecular stages ofelementary exchange reactions

Ai + Ak → Al + Am(i, k, l,m = 1, 2, . . . NA), (2.1)

where Ai—reactants (species), NA—the number of speciesparticipating in bimolecular exchange reactions. Note that re-action scheme (2.1), despite the arrow in one direction, obvi-ously takes into consideration that any elementary bimolec-ular stage in this scheme can be reversible. Elementary irre-versible recombination or addition reactions are bimolecularreactions of another type

Ai + Ak → Cζ (i, k = 1, 2, . . . NA; ζ = 1, 2, . . . NC),(2.2)

where Cζ —reactants (species) in the products of the reactionunder study, NC—the number of species formed due to suchbimolecular reactions. Dissociation reactions are the reactionsreverse to reactions (2.2)

Cζ → Ai + Ak (i, k = 1, 2, . . . NA; ζ = 1, 2, . . . NC).(2.3)

Note that the initial reactants and products of bimolecular ex-change reactions are denoted as Ai, while in recombination oraddition reactions these species are just initial reactants, andproducts cannot be initial for other bimolecular reaction, thusthey as denoted by Cζ . This is because geminate reactions ofisolated pairs of reactants are examined. In this case, the initialisolated geminate pair gives rise (as a result of recombinationor addition (2.2)) to isolated reactants. They cannot enter intobimolecular reaction, but can just dissociate to produce somepair of reactants of species Ai. At the same time, if we con-sider isolated pairs of reactants of species Cζ , dissociation ofany reactant of such a pair results in the system of three orfour particles, not an isolated pair. Exactly for this reason bi-molecular reactions between reactants Cζ are not investigated.Apart from the above reactions, reactions of monomoleculartransformation can be included in the consideration

Ai → Ak,Cζ → Cμ. (2.4)

Just as scheme (2.1), this reaction scheme, despite the arrowin one direction, takes into account that any elementary stagein this scheme can be reversible. It may be both processes

of reversible chemical transformation (for example, reversiblecis-trans isomerization), and physical excitation processes ofmolecules under the action of permanent radiation and decayof excited molecules into initial unexcited state. Processes ofmonomolecular transformation Ai ↔ Cζ are not examined,since in the framework of the consideration of reactions ofisolated pairs geminate pairs Ak + Cζ are neglected. Just withsuch restrictions it is possible to develop general (matrix) the-ory of multistage geminate reactions.

2. Vector distribution functions

Consider a many-particle reacting system (a sample) ofthe volume �sam where reactions (2.1)–(2.4) take place. Sincethese reactions have a geminate nature, the system can be de-scribed using the Gibbs ensemble of isolated pairs. Let us ex-amine an initial many-particle system (a sample) at a certaintime t. At this time, let it contain NAiAk

(t) pairs of reactantsfrom species Ai + Ak, and NCζ

(t) reactants of species Cζ . Ifthe total number of all pairs at time t is NAA, and the totalnumber of reactants from all species C is NC(t), then the num-ber

N = NAA(t) + NC(t) (2.5)

remains unchanged during the reaction. Equation (2.5) canbe given a probabilistic meaning by defining the probabilityPAiAk

(t) that a pair of reactants, from reactants of species Ai

and Ak, exists in the form of a free pair and the probabilityPCζ

(t) that they exist in a bound state,

PAiAk(t) = NAiAk

(t)

N, PCζ

(t) = NCζ(t)

N,

(2.6)NA∑

i≤k=1

PAiAk(t) +

NC∑ζ=1

PCζ(t) = 1.

The last equality expresses the property of the completenessof the introduced probabilities and is equivalent to the condi-tion of conservation of the number of particles. The state ofsome free pair corresponds to the Gibbs subensemble of suchpairs with a distribution function Fik(r1, r2, t)42 which is theprobability density of detecting a pair with coordinates r1 andr2 at time t. The bound state corresponds to a subensembleof reactants with a distribution function Fζ (r, t) which is theprobability density of finding a reactant with coordinate r attime t. Further, it is convenient to pass to vector distributionfunctions on the Gibbs ensemble. The basis for the construc-tion of such distribution functions is the concept of “effective”particles.26–29 In the framework of such concept, the sets ofreacting species are introduced

A = {Ai} = {A1, A2, . . . , ANA};

(2.7)C = {Cζ } = {C1, C2, . . . , CNC

},i.e., the species are treated as “internal” states of “effective”particles A or C. In this case, column-vectors of distributionfunctions in the pair AA and distribution functions of reac-tants C are used. Distribution vector of the pair should be spec-ified in formal collective basis |AiAk〉, that is a direct product

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184104-3 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

of single-particle basis

|AiAk〉 = |Ai〉 |Ak〉.Distribution vector �A1A2 of “effective” particles pair A1A2

should be constructed on this basis as follows. It is a col-umn of the probability densities containing N2

A elements ofthe form �ik(q1, q2, t). Here q1 and q2 are the coordinates ofthe configuration space of the corresponding effective pairs.In the general case, configuration space involves both spacecoordinates and orientation angles of reactants, as well as in-ternal degrees of freedom. However, for simplicity, first werestrict ourselves to the examination of space coordinatesr, since consideration of more complete coordinate systempresents no particular problems (see Sec. IV). Distributioncolumn-vector �C of “effective” particle C is specified in ba-sis Cζ and contains NC elements �ζ (q, t). Note that the num-ber of components of the vector �A1A2 exceeds that of pairsof reactants. This is related to the necessity of uniform de-scription of pairs of identical and non-identical reactants (seethe Appendix).

Also introduce operations T rA1A2 = T rA1T rA2 and T rCof column-vectors by which, in the case of the absence of “in-ternal” quantum degrees of freedom (for instance, spin ones),we mean summation over components of these vectors, i.e.,over the states of the corresponding “effective” particles (i.e.,over species). So multistage reaction of isolated geminatepairs (containing elementary stages (2.1)–(2.4)) can be rep-resented as reactions A + A → A + A, A + A ↔ C, A →A, and C → C of “effective” particles A and C. In the con-text of many-particle theory, the evolution of such a reactionsystem may be described by the Gibbs ensemble just by con-sidering two Fock boxes45 containing either one “effective”particle C, or two effective particles A. As mentioned above,the state of two “effective” particles A1 and A2 at moment t isdefined by the column-vector of probability densities �ik(r1,r2, t) of finding the reactant of species Ai at the point r1 ofthree-dimensional space, and the reactant of species Ak at thepoint r2. By virtue of the identity of “effective” particles A1

and A2,

�ik(r1, r2, t) = �ki(r2, r1, t). (2.8)

The state of “effective” particle C is specified by the column-vector of probability densities �ζ (r, t) of finding the reactantof species ζ at the point r.

The norm of these functions defines the probability offinding geminate reaction system at the moment of time t ei-ther in the state of pairs consisting of reactants of two species,or in the state of isolated particles of C,

PA1A2 (t) = T rA1A2

∫∫dr1dr2

2�A1A2 (r1, r2, t),

(2.9)

PC(t) = T rC

∫dr�C(r, t).

Since for the system at hand there are no other variants, there-fore we have the relation completely equivalent to the lastequality in Eq. (2.6)

PA1A2 (t) + PC(t) = 1. (2.10)

Further statistical description of the many-particle react-ing system can be conveniently performed in the so-calledthermodynamic limit46 on the basis of generalization to thesystems of “effective” particles of the approach used inRef. 42 in studies of elementary reversible geminate reaction.Now we examine bulk properties of the sample considering itin the thermodynamic limit. For this purpose, we assume thatN → ∞, �sam → ∞, however, their ratio is a finite value

n = T − lim N/�sam. (2.11)

Then boundaries of the Fock box expand to infinity, and thestate of species in these boxes should be described by non-normable functions

�Tik(r1, r2, t) = T − lim �sam�ik(r1, r2, t),

(2.12)�T

ζ (r, t) = T − lim �sam�ζ (r, t).

These functions do not go to zero, if the coordinate of speciestends to infinity. Thus, Eq. (2.9) should be replaced by

P TA1A2

(t) = limυ→∞ T rA1A2

∫∫υ

dr1dr2

2υ�T

A1A2(r1, r2, t),

(2.13)

P TC (t) = lim

υ→∞ T rC

∫υ

drυ

�TC (r, t).

Symbol υ under the integral means that integration is madeover the volume υ. Condition of completeness takes the form

P TA1A2

(t) + P TC (t) = 1. (2.14)

3. Mean and local characteristics of effective particlesand species in the sample

It follows from the above reasoning that in the thermody-namic limit mean concentrations of effective particles A andC in the sample can be represented as

[A]t = 2P TA1A2

(t)n; [C]t = P TC (t)n. (2.15)

Factor 2 in the first equality takes into account that the pairA1A2 contains two “effective” particles. As mean concentra-tions may be treated as physical observables, the equality fol-lowing from Eqs. (2.14) and (2.15),

n = [A]t /2 + [C]t (2.16)

defines theoretical parameter n (2.11) significant for fur-ther discussion and directly related to reactant density in thesample.

Mean concentrations [Ai]t and [Cζ ]t of species in physi-cal sample are also among the observables in experiment, andthey are the components of column-vectors of concentrations[A]t and [C]t of “effective” particles A and C dimensionality.Expressions of these concentrations in terms of distributionfunctions are of the form

[Ai]t = n limυ→∞

∑i1i2

∫∫υ

dr1dr2

2υ(δii1 + δii2 )�T

i1i2(r1, r2, t)

= n limυ→∞

∑k

∫∫υ

dr1dr2

υ�T

ik(r1, r2, t), (2.17)

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184104-4 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

[Cζ ]t = n limυ→∞

∑ζ ′

∫υ

drυ

δζζ ′�Tζ ′(r, t)

= n limυ→∞

∫υ

drυ

�Tζ (r, t). (2.18)

Evidently, mean concentrations of effective particles and theirspecies are related as

[A]t = T rA[A]t ; [C]t = T rC [C]t . (2.19)

In kinetic theory, local concentrations are the main ob-servable. In the system in question, these are, first of all, localconcentrations of “effective” particles nA(r, t) and nC(r, t).They are related to mean concentrations by spatial averagingprocedure

[A]t = limυ→∞

∫υ

drυ

nA(r, t); [C]t = limυ→∞

∫υ

drυ

nC(r, t).

(2.20)Local concentrations of species of “effective” particles ni(r, t)and nζ (r, t) are introduced in a similar way

[Ai]t = limυ→∞

∫υ

drυ

ni(r, t); [Cζ ]t = limυ→∞

∫υ

drυ

nζ (r, t)

(2.21)and are the components of column-vectors nA(r, t) andnC(r, t). The introduced local concentrations are related as

nA(r, t) = T rA nA(r, t); nC(r, t) = T rC nC(r, t). (2.22)

So in physical sample under study the quantities [A]t ,[C]t ; nA(r, t), nC(r, t); [Ai]t, [Cζ ]t; ni(r, t), nζ (r, t) are observ-ables. Most informative are local concentrations of species.If these quantities are known, they can be used to calculatethe rest of observables by Eqs. (2.21), (2.22), and (2.19). Thegoal of the kinetic theory is to derive kinetic equations to findquantities ni(r, t) and nζ (r, t).

Local and mean concentrations characterize the state ofthe sample in the thermodynamic limit. Further it will be con-venient to introduce local probabilities pi(r, t) and pζ (r, t)defining the state of the Gibbs ensemble of “effective” par-ticles in the thermodynamic limit that are related to local con-centrations of species as follows:

ni(r, t) = npi(r, t); nζ (r, t) = npζ (r, t). (2.23)

Comparison of Eqs. (2.17) and (2.18) with Eq. (2.21) in viewof Eq. (2.23) shows that local probabilities are related to dis-tribution functions in the thermodynamic limit as

pi(r, t) =∑

k

∫dr2�

Tik (r, r2, t); pζ (r, t) = �T

ζ (r, t).

(2.24)Obviously, spatial averaging of these probabilities gives theprobabilities of finding the species in the sample

pi(t) = limυ→∞

∫υ

drυ

pi(r, t); pζ (t) = limυ→∞

∫υ

drυ

pζ (r, t).

(2.25)

They are related to the mean concentrations of species of “ef-fective” particles in physical sample in the simplest way,

[Ai]t = npi(t); [Cζ ]t = npζ (t). (2.26)

So the next problem is to derive kinetic equations for lo-cal probabilities defined on the Gibbs ensemble of “effective”particles for multistage geminate reaction under considera-tion. This is rather easy to do even for spatially inhomoge-neous systems.

B. Reactivity and free motion operators

1. Integral operators

Most general description of reactivity and motion of reac-tants in solution is made by introducing integral-matrix oper-ators (integral-matrix Liouvillians) acting on column-vectorof distribution functions �T

A1A2(r1, r2, t) and �T

C (r, t). Con-struction of operator kernel matrices is based on elementaryrates of processes (sink terms) calculated by elementary eventtheory.12 In the described approach, they are a priori given.

In some our previous works (see Refs. 13 and 14) inconstructing any integral operators, we considered uniformlyspace and time variables on extended time axis (−∞ < t< ∞), and this made it necessary to use generalized func-tions formalism.47 In the present contribution (as in Ref. 42)and in constructing integral T-operators (Ref. 29) time func-tions will be replaced by their Laplace transforms. This willmake operations with generalized functions unnecessary, andmathematical part of derivation—more clear. Laplace trans-forms are defined in a standard way and are marked by theupper index L. For example,

�T LA1A2

(r1, r2) = �T LA1A2

(r1, r2; s)

=∞∫

0

dt �TA1A2

(r1, r2, t) exp (−st). (2.27)

As for space variables, the functions under discussionhave symmetry (2.8) in them. This means that, as in quantummechanics,48 the desired kernels are to be symmetric aboutpermutation of “effective” particles coordinates. This require-ment essentially restricts the range of acceptable forms of thedesired kernels. However, as it will be seen below, severalmathematical representations for reactivity Liouvillians de-scribing one and the same reaction can exist. Now introducethe reactivity operators for elementary stages of multistagegeminate reaction of isolated pairs of reactants.

2. Monomolecular processes

Let us consider the rates of monomolecular processes in-cluding those determined by spontaneous decay of excitedstates or by external radiation excitation to be constant quan-tities. Denote the first order rate constants corresponding tothe transition of reactants between species Ai ← Ai′ in the“effective” particle A as Kii′ , and those corresponding to thetransition of reactants between species Cζ ← Cζ ′ in C as Kζ ζ ′ .Note that “diagonal” values of these constants are equal to

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184104-5 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

zero, i.e., Kii = Kζ ζ ≡ 0. Then the matrix of integral ker-nels of Liouvillians QA and QC of monomolecular processesof “effective” particles A and C are defined by the followingmatrix elements (kernels):

(QA)i|i ′(r|r′) = δ(r − r′)

⎡⎣−δii ′

∑k =i ′

Kki ′ + Kii ′

⎤⎦ , (2.28)

(QC)ζ |ζ ′(r|r′) = δ(r − r′)

⎡⎣−δζζ ′

∑μ =ζ ′

Kμζ ′ + Kζζ ′

⎤⎦ .

(2.29)Here δ(r − r′)—Dirac δ-function, δii′—Kronecker delta. Thefirst term between square brackets is diagonal and describesthe escape of reactant from the i′th species to all other speciesduring monomolecular transformations. The second nondiag-onal term describes the transformation of reactant of the i′thspecies to reactant of the ith species. It is easily verified thatEqs. (2.28) and (2.29) give the property equivalent to the lawof “effective” particles preservation in monomolecular trans-formation processes

T rAQA = T rC QC = 0. (2.30)

Note that hereinafter by the operation Tr of Liouvillianswe mean Tr of column-vectors resulting from the action ofthese operators (Liouvillians) on any initial column-vectors(see Sec. IV). The operator QA1A2 = QA1 ⊕ QA2 acting onthe column-vector �T

A1A2(r1, r2, t) of two “effective” parti-

cles is the direct sum of operators of isolated “effective”particles.27–29

3. Exchange bimolecular processes

Bimolecular process is defined by the 4-center elemen-tary rate Rik|i′k′ (r1, r2|r′

1, r′2) of the transition (per unit time) of

a pair of reactants from species Ai′ and Ak′ residing at points r′1

and r′2 to a pair of reactants from species Ai and Ak (which are

at points r1 and r2) due to exchange bimolecular reaction.17, 29

The form of this function is a priori known and it satisfies theconditions of translation and permutation symmetries

Rik|i ′k′(r1, r2|r′1, r′

2) = Rik|i ′k′(r1+r, r2+r|r1 + r, r′2+r);

(2.31)Rik|i ′k′(r1, r2|r′

1, r′2) = Rki|k′i ′(r2, r1|r′

2, r′1),

where r—arbitrary displacement vector. The presence oftranslation symmetry is determined by the assumed spatialhomogeneity of the solvent. The existence of additional sym-metries is also possible; however, this is inessential for thederivation of equations. Note that this quantity is identicallyzero for “diagonal” values i = i′ k = k′. Also note that the useof elementary rate Rik|i′k′(r1, r2|r′

1, r′2) in “effective” particles

concept requires that it be zero at permutation just in one pairof indices (see the Appendix).

Along with elementary rates, introduce 2-centerquantities

wik|i ′k′(r′1 − r′

2) =∫∫

dr1dr2Rik|i ′k′(r1, r2|r′1, r′

2). (2.32)

For the case of different reactants from different species(i = k = i′ = k′), they are complete rates of transition of apair of reactants from the species (Ai′ , Ak′) (at points r′

1 andr′

2) to a pair of reactants from the species (Ai, Ak) at any spa-tial position of these reactants. However, if there are identi-cal reactants, the above physical meaning of the introducedquantities turns out to be inexact (due to this identity). Never-theless, it will be convenient to call these quantities completerates in this case as well.

In the Appendix, we give integral-matrix LiouvillianVb

A1A2of exchange bimolecular processes. Matrix elements

of its integral kernels are

(VbA1A2

)ik|i ′k′(r1, r2|r′1, r′

2)

= −δii ′δkk′δ(r1 − r′1)δ(r2 − r′

2)∑

lmwlm |l′k′(r′

1 − r′2)

× θlm|i ′k′ + θik|i ′k′Rik|i ′k′(r1, r2|r′1, r′

2), (2.33)

where the coefficient θ ik|i′k′ allows for the identity of effectiveparticles A1 and A2 and is defined as follows:

θik|i ′k′ = 1

1 + δi ′k′ + δik − δikδi ′k′. (2.34)

The first, diagonal term, in the definition of the matrixelement of the operator Vb

A1A2corresponds to the escape of

reactants (residing at points r′1 and r′

2) from species Ai′ andAk′ to other pairs with any of spatial position of reactants, Thesecond, nondiagonal term, describes the appearance of a pairof reactants (at points r1 and r2) of species Ai and Ak from apair of reactants (residing at points r′

1 and r′2) from species

Ai′ and Ak′ .It is easy to see that the Liouvillian Vb

A1A2satisfies the

condition equivalent to preservation of “effective” particles inthe process of exchange reactions

T rA1A2

∫∫dr1dr2

2Vb

A1A2

≡ T rA1A2

∫∫dr1dr2

2

(Vb

A1A2

)(r1, r2|r′

1, r′2) = 0,

(2.35)

where (VbA1A2

)(r1, r2|r′1, r′

2) denotes the kernel matrix of thecorresponding integral operators, and the operation T rA1A2

should be understood in the sense stated above (afterEq. (2.30)) (see also Sec. IV). Further by integration of opera-tors we shall always mean integration of their integral kernels.

4. Association processes

Processes of bimolecular association (irreversible bi-molecular reaction of recombination or addition) are de-scribed by 3-center elementary rate Rζ |lm(r|r1, r2)17, 29 of thetransition of a pair of reactants (residing at points r1 and r2)of species Al and Am of “effective” particle A to reactant (re-siding at point r) of the species Cζ of “effective” particle C.The form of this elementary rate satisfies the conditions oftranslation and permutation symmetries

Rζ |lm (r|r1, r2) = Rζ |lm(r + r′|r1 + r′, r2 + r′),(2.36)

Rζ |lm (r|r1, r2) = Rζ |ml(r|r2, r1),

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184104-6 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

where r′—arbitrary displacement vector. Here it is convenientto introduce 2-center complete rate of association of a pair ofreactants (residing at points r1 and r2) from species Al and Am

into reactant of the species Cζ at any reactant position,

wζ |lm(r1 − r2) =∫

dr Rζ |lm (r|r1, r2). (2.37)

Using the approach given in the Appendix, one can showthat association processes (as in Ref. 29) are described by twoLiouvillians: (Va

A1A2) and VCP . The first Liouvillian acts in

the space of the pair (A1A2) on the vector �TA1A2

(r1, r2) andcorresponds to the decay of reactants of species Ai and Ak

due to association. The second Liouvillian also acts in the co-ordinate space of the pair (A1A2), but it corresponds to theappearance of reactant (at point r) of the species Cζ of “ef-fective” particle C. These two Liouvillians are defined by thematrix elements

(VaA1A2

)ik|i ′k′(r1, r2|r′1, r′

2)

= −δii ′δkk′δ(r1 − r′1)δ(r2 − r′

2)∑

ζ

wζ |i ′k′(r′1 − r′

2),

(2.38)

(VCP )ζ |lm (r|r1, r2) = 1

2Rζ |lm (r|r1, r2) . (2.39)

It follows from the definitions that the Liouvillian VaA1A2

is diagonal and describes the escape from the pair (A1A2)caused by association. Due to the presence of Dirac δ-functions, it actually is 2-center one. The Liouvillian VCPdescribes coming into “effective” particle C from the pair(A1A2). It differs from similar Liouvillian in Ref. 29 by thefactor 1

2 . This is because in the present work, we use symmet-ric basis for the pair (A1A2).

As is readily seen, the condition equivalent to the lawof “effective” particles preservation in association processestakes place

T rA1A2

∫dr1dr2

2Va

A1A2+ T rC

∫dr VCP = 0. (2.40)

Here, as before, by operations T rA1A2 and T rC , we mean thesame as in the foregoing (after Eq. (2.30)) (see also Sec. IV).

5. Dissociation processes

Dissociation processes are described by 3-center elemen-tary rate Rik|ζ (r1, r2|r) of the transition of reactant (residing atpoint r) of the species Cζ of “effective” particle C to a pair ofreactants (residing at points r1 and r2) of species Ai and Ak of“effective” particles of the pair (A1A2).17, 29 These rates sat-isfy the conditions of translation and permutation symmetries

Rik|ζ (r1, r2|r) = Rik|ζ (r1 + r′, r2 + r′|r + r′),(2.41)

Rik|ζ (r1, r2|r) = Rki|ζ (r2, r1|r),

where r′—arbitrary displacement vector. Two-center com-plete rate of dissociation of reactants of the species Cζ of the“effective” particle C into a pair of reactants of species Ai and

Ak at any spatial position of the species is

wik|ζ =∫

dr1dr2Rik|ζ (r1, r2|r). (2.42)

By virtue of translational symmetry (2.41), this rate does notdepend on space coordinate of reactants of the species Cζ .

With the approach given in the Appendix, one can showthat dissociation processes (as in Ref. 29) are described bytwo Liouvillians: VPC and VC . The first Liouvillian acts onthe coordinates of “effective” particle C and transforms thefunctions φT L

ζ (r) to the functions in the coordinate space ofthe pair (A1A2). The second Liouvillian transforms the func-tions φT L

ζ (r) to the functions of the same type, i.e., acts inthe coordinate space of the particle C. These Liouvillians aredefined by the matrix elements

(VPC)ik|ζ (r1, r2|r) = Rik|ζ (r1, r2|r),(2.43)

(VC)ζ |ζ ′(r|r′) = −1

2δζζ ′δ(r − r′)

∑ik

wik|ζ ′ .

It follows from this definition that the Liouvillian VPC isnondiagonal and describes coming of reactants into the pair(A1A2) determined by dissociation. The Liouvillian VC is di-agonal and describes the escape of reactant from species Cζ ′

as a result of dissociation. This Liouvillian differs from simi-lar Liouvillian from Ref. 29 by the factor 1

2 . This difference isdue to the use of symmetric basis for the pair (A1A2) in thiswork.

It is easily verified that the Liouvillians introduced in Eq.(2.43) satisfy the property equivalent to the law of “effective”pairs preservation in dissociation processes

T rA1A2

∫dr1dr2

2VPC + T rC

∫drVC = 0. (2.44)

Here, as before, by operations T rA1A2 and T rC , we imply thesame as in the above discussion (after Eq. (2.30)) (see alsoSec. IV).

6. Liouvillian of free spatial motion

The Liouvillian of free motion is constructed on the basisof operators (in the general case, integral ones) of free mo-tion in a pair of reactants of species Ai and Ak of a pair of“effective” particles (A1A2) or reactant of the species Cζ of“effective” particle C,29, 49

Lik = Li + Lk + L′ik; Lζ , (2.45)

where Li , Lk , Lζ —individual operators of free motion ofreactants of the corresponding species (in the specific case,these can be diffusion motion operator), and L′

ik—the opera-tor describing the force interaction of these reactants in thepair A1A2. All operators of spatial migration preserve thenumber of reactants, thus they satisfy the relations∫

dr Li =∫

dr Lk =∫

dr Lζ =∫

dr1 dr2 Lik = 0.

(2.46)Matrices of integral kernels of Liouvillians of spatial mi-gration LA1A2 in the pair (A1A2) and LC of the “effective”

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184104-7 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

particle C are defined by the matrix elements

(LA1A2 )ik|i ′k′(r1, r2|r′1, r′

2)

= δii ′δkk′Lik(r1, r2|r′1, r′

2)

= δii ′ δkk′(Li(r1|r′1)δ(r2 − r′

2)

+Lk(r2|r′2)δ(r1 − r′

1) + L′ik(r1, r2|r′

1, r′2)),

(LC)ζ |ζ ′(r|r′) = δζζ ′Lζ (r|r′), (2.47)

i.e., are diagonal. Equation (2.46) yields the property analo-gous to Eqs. (2.30) and (2.35),∫

dr1 dr2

2LA1A2 = 0, T rA1A2

∫dr1 dr2

2LA1A2 = 0.

(2.48)Here, as before, by the operation T rA1A2 we mean the sameas in the foregoing (after Eq. (2.30)) (see also Sec. IV).

III. KINETIC EQUATIONS OF MULTISTAGE GEMINATEREACTIONS

A. General discussion

1. Liouville equations

The starting point for the derivation of kinetic equationsis the Liouville equations for distribution functions formu-lated on the Gibbs ensemble of “effective” particles.29 Theseequations are balance equations in the course of chemical re-actions and free motion of reactants. Obviously, the Laplacetransforms of the equations are of the form

(s − LA1A2 )�T LA1A2

= (QA1A2 + VbA1A2

+ VaA1A2

)�T LA1A2

+ VPC �T LC + �T 0

A1A2, (3.1)

(s − LC)�T LC = (QC + VC)�T L

C + VCP �T LA1A2

+ �T 0C .

(3.2)They are matrix generalization of equations used in the con-sideration of elementary associative-dissociative reaction.42

The left-hand sides of these equations correspond to free mo-tion of “effective” particles. The first terms in their right-handsides describe chemical reactions in the pair (A1A2) and re-actant from C, respectively. The second terms correspond tocoming of reactants either into the pair (A1A2), or into the“effective” particle C caused by associative-dissociative pro-cesses. The last terms are the initial conditions for Liouvilleequations.

2. Kinetic equations for local probabilities

Equations for Laplace transforms of local probabilitiesare obtained from Eqs. (3.1) and (3.2) with allowance for therelation between these transforms and Laplace transforms ofdistribution functions following from Eq. (2.24),

pLi (r) =

∑k

∫dr2�

T Lik (r, r2); pL

ζ (r) = �T Lζ (r). (3.3)

These quantities are the components of column-vectors pLA(r)

and pLC (r) related to vectors �T L

AA2(r, r2) and �T L

C (r) in a way

similar to Eq. (3.3),

pLA(r) = T rA2

∫dr2�

T LAA2

(r, r2); pLC (r) = �T L

C (r).

Using these relations, in view of Eqs. (2.45)–(2.47), we caneasily obtain the equations for the vectors of Laplace trans-forms of local probabilities from Eqs. (3.1) and (3.2),

(s − LA)pLA(r)

= QApLA(r) + T rA2

∫dr2(L′

AA2+ Vb

AA2+ Va

AA2)�T L

AA2

+ T rA2

∫dr2VPCpL

C + p0A(r), (3.4)

(s − LC)pLC (r) = (QC + VC)pL

C + VCP�T LAA2

+ p0C(r). (3.5)

Here p0A(r) and p0

C(r)—initial values of local probabilities.However, these equations cannot be treated as kinetic, sincethey involve functions �T L

AA2that are not observables. In order

to eliminate them, we turn to Eq. (3.1). Its formal solution isof the form

�T LAA2

= GLA1A2

VPCpLC + GL

A1A2�T 0

A1A2. (3.6)

Here we introduce the Laplace transform of integral-matrixpropagator (resolvent or the Green function49) of the pair(A1A2) of “effective” particles. It is defined by matrix inte-gral kernel obeying the matrix equation

(s − LA1A2 − QA1A2 − VbA1A2

− VaA1A2

)GLA1A2

(r1, r2|r′1, r′

2)

= δ(r1 − r′1)δ(r2 − r′

2)EA1A2 , (3.7)

where EA1A2 is a unit matrix (with elements(EA1A2

)ik|i ′k′

= δii ′ δkk′) of the pair (A1A2). Substitution of Eq. (3.6) inEqs. (3.4) and (3.5) gives the desired equations for Laplacetransforms of local probabilities

(s − LA1 )pLA1

(r)

= QA1 pLA1

(r) + T rA2

∫dr2

[(L′A1A2

+ VbA1A2

+ VaA1A2

)× GL

A1A2VPC + VPC

]pLC + T rA2

∫dr2

(L′A1A2

+ VbA1A2

+ VaA1A2

)GL

A1A2�T 0

A1A2+ p0

A1(r), (3.8)

(s − LC)pLC (r) = QCpL

C (r) + (VCPGLA1A2

VPC + VC)pLC

+ VCPGLA1A2

�T 0A1A2

+ p0C(r). (3.9)

These equations are matrix generalization of kinetic equationsof elementary geminate reaction42 and are linear inhomoge-neous integral equations the kernels of which are expressedin terms of the solution of Eq. (3.7) for the Green functionthat describes the evolution (chemical and dynamic) of two“effective” particles. The left-hand side of the obtained ki-netic equations describes the free evolution of local probabil-ities due to spatial motion of reactants. The first terms in theright-hand sides of Eqs. (3.8) and (3.9) describe monomolec-ular transformations. The second term in Eq. (3.8) is a “col-lision integral” which defines bimolecular and dissociation

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184104-8 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

channels of geminate reaction. The second term in the right-hand sides of Eq. (3.9) is a “collision integral” which definesdissociation channels of geminate reaction. The third terms inEqs. (3.8) and (3.9) are inhomogeneous terms of kineticequations, and, according to general principles of the kinetictheory,46 they describe the effect of initial correlations onthe evolution of local probabilities (concentrations). The forthterms define the contribution from the initial conditions.

3. Kinetic equations for mean probabilities

Unlike bulk reactions, kinetic equations (3.8)–(3.9) validin the general case of inhomogeneous systems can give, asin Ref. 42 in the consideration of elementary geminate reac-tion, a set of closed equations for mean probabilities (2.25) (ormean concentrations of species (2.26)). With this aim, we ap-ply, in view of preservation laws, spatial averaging procedureto kinetic equations (3.8)–(3.9). So for Laplace transforms ofcolumn-vectors of mean probabilities, we have

spLA1

= QA1 pLA1

+ 2T rA2�Ld pL

C − 2T rA2 ILA1A2

+ p0A1

,

(3.10)

spLC = QC pL

C − �LC pL

C + ILC + p0

C . (3.11)

Here, Laplace transforms matrices of memory functions areintroduced,

�Ld =

∫dr1dr2

2

((Vb

A1A2+ Va

A1A2

)GL

A1A2VPC

+ VPC)(r1, r2|r), (3.12)

�LC = −

∫dr

(VCPGL

A1A2VPC + VC

)(r|r0). (3.13)

Note that indication of coordinate dependence of operators(as in Eq. (2.35)) denotes their integral kernels. Besides, evenif the coordinate dependence is not indicated explicitly, spa-tial integration of operators always implies integration of theirkernels. Due to translational symmetry, Laplace transforms ofmemory functions (3.12) and (3.13) are independent of spacecoordinates.

Laplace transforms of inhomogeneous sources appearingin Eqs. (3.10) and (3.11) are defined by the equalities

ILA1A2

= − limυ→∞

∫υ

dr1dr2

2υ

[(Vb

A1A2

+ VaA1A2

)GL

A1A2�T 0

A1A2

](r1, r2), (3.14)

ILC = lim

υ→∞

∫υ

drυ

[VCPGL

A1A2�T 0

A1A2

](r). (3.15)

Often, time originals of memory functions13, 14 are general-ized time functions.47

B. Relative coordinate space

1. Operators in relative coordinates

Laplace transforms of memory functions (3.12)–(3.13)and inhomogeneous sources (3.14)–(3.15) may be expressedin terms of relative coordinates of a pair of “effective” parti-cles. For this purpose, it is convenient to introduce the projec-tion operator that transfers arbitrary finite coordinate func-tion of two reactants (r1, r2) to the function ψ(r) dependingon their relative coordinates by the rule

ψ(r) = (r1, r2) =∫

dr1dr2 δ (r − (r1 − r2)) (r1, r2).

(3.16)Obviously, the identity is valid∫

drψ(r) =∫

dr1dr2(r1, r2). (3.17)

For example, for elementary rates introduced above whichserve as a basis for the construction of reaction Liouvillians,we have

rik|i ′k′(r|r′) = Rik|i ′k′(r1, r2|r′1, r′

2)

=∫

dxRik|i ′k′(r, 0|r′ + x, x), (3.18)

rζ |lm(r′) = Rζ |lm (r|r1, r2) =∫

dxRζ |lm(0|r′ + x, x),

(3.19)

rik|ζ (r′) = Rik|ζ (r1, r2|r) =∫

dxRik|ζ(r′ + x, x|0)

.

(3.20)In relative coordinates elementary rates (3.19)–(3.20) dependsolely on relative vectors r′ in the pair (A1A2) and do not de-pend on the coordinate of the “effective” particle C by virtueof translation symmetry.

Matrix equation (3.7) for Laplace transforms of integral-matrix propagator kernels of the pair (A1A2) in relative coor-dinates takes the form(

s − LA1A2 − qA1A2 − υbA1A2

− υaA1A2

)gLA1A2

(r|r′)

= δ(r − r′)EA1A2 . (3.21)

Here quantities in relative coordinates are introduced

(LA1A2 )ik|i ′k′(r|r′) = (LA1A2 )ik|i ′k′(r1, r2|r′1, r′

2),(qA1A2

)ik|i ′k′ (r|r′) = (QA1A2 )ik|i ′k′(r1, r2|r′

1, r′2), (3.22)

(gLA1A2

)ik|i ′k′(r|r′) = (GLA1A2

)ik|i ′k′(r1, r2|r′1, r′

2).

Kernels of bimolecular reaction Liouvillians (2.33) and (2.38)in relative coordinates are(

υbA1A2

)ik|i ′k′ (r|r′) = −δii ′δkk′δ(r − r′)

∑lm

wlm|i ′k′(r′)θlm|i ′k′

+ θik|i ′k′rik i ′k′(r|r′), (3.23)

(υaA1A2

)ik|i ′k′ (r|r′) = −δii ′δkk′δ(r − r′)

∑ζ

wζ |i ′k′(r′).

(3.24)

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2. Kinetic coefficients

In relative coordinate space the expression for source(3.14) is most easily obtained. With identity (3.17) in (3.14),we have

ILA1A2

= −∫

dr2

[(υbA1A2

+ υaA1A2

)gLA1A2

�A1A2 (r′)].

(3.25)Here we introduce a column-vector of initial distributionfunctions in the pair (A1A2) in relative coordinates

φA1A2(r) = lim

υ→∞

∫υ

dr1dr2

υδ (r − (r1 − r2)) �T 0

A1A2(r1, r2)

≡ limυ→∞

1

υυ�T 0

A1A2(r1, r2). (3.26)

More complicated law of the function �T 0A1A2

(r1, r2) pro-jection on relative coordinate space is related to the fact thatinitial distribution does not fall into a class of finite func-tions. The norm of the component φik(r) of the column-vectorφA1A2 (r) is the probability P T 0

ik of finding the pair (A1A2) inthe state |AiAk〉 at the initial moment of time (compare withEq. (2.13))

P T 0ik =

∫dr2

φik(r). (3.27)

The memory function �Ld is considered by analogy. As a re-

sult, we arrive at the representation

�Ld = −

∫dr dr0

2

[((υbA1A2

+ υaA1A2

)gLA1A2

υd + υd

)](r|r0).

(3.28)Here dissociation operator υd (analog of VPC) is introducedwhich is expressed is terms of its matrix elements as follows:

(υd )ik|ζ (r|r0) = −δ (r − r0) rik|ζ (r). (3.29)

Equation (3.20) shows that it is directly related to elementarydissociation rate. Note that, as in Ref. 42, in defining this op-erator in terms of elementary rate, negative sign is chosen.

Representation of the memory function �LC in terms of

relative coordinate space quantities is

�LC = −

∫drdr0

[(υa gL

A1A2υd + υ0

)](r|r0). (3.30)

By analogy with dissociation operator, here we introduce as-sociation operator υa (analog of VCP ) defined in terms of itsmatrix elements

(υa)ζ |lm (r|r0) = −δ (r − r0) rζ |lm(r)/2. (3.31)

It is directly related to elementary association rate. The oper-ator υ0 (analog of VC) has only diagonal matrix elements thevalue of which is specified by complete rate of dissociationinto pairs with fixed r

(υ0)ζ |ζ ′ (r|r0) = −1

2δζζ ′δ (r − r0)

∑ik

rik|ζ (r). (3.32)

Now it is easy to represent the Laplace transform ILC in terms

of relative coordinate space quantities

ILC = −

∫dr

[υa gL

A1A2φA1A2

(r)]. (3.33)

3. Preservation laws

Earlier the properties of Liouvillians in the laboratorycoordinate system were formulated (see Eqs. (2.30), (2.35),(2.40), and (2.44)). Based on these properties, kinetic equa-tions (3.10) and (3.11), in view of definitions (3.12)–(3.15) ofkinetic coefficients, yield the condition of total balance:

s

(1

2T rApL

A + T rCpLC

)

= 1

2T rAp0

A + T rCp0C,

1

2T rApA(t) + T rC pC(t) = const.

(3.34)

Of interest is the formulation of Liouvillian properties in rel-ative coordinate space which is easy to do using definitionsof operators in relative coordinates. For monomolecular pro-cesses, we have

T rA1A2 qA1A2 = T rC qC = 0. (3.35)

For bimolecular processes

T rA1A2

∫dr υb

A1A2= 0. (3.36)

For association processes

1

2T rA1A2

∫dr υa

A1A2− T rC

∫dr υa = 0. (3.37)

For dissociation processes

1

2T rA1A2

∫dr υd − T rC

∫dr υ0 = 0. (3.38)

As expected, Eqs. (3.10) and (3.11), and definitions (3.25),(3.28), (3.30), and (3.33), in view of properties (3.35)–(3.38),give Eq. (3.34).

C. Spatially homogeneous systems

Spatially homogeneous reacting systems are a widely en-countered important specific case. Along with homogeneityof a solvent leading to the existence of translation symme-try for elementary event rates (2.31), (2.36), and (2.41), spa-tially homogeneous reacting systems are assumed to have ad-ditional homogeneity, i.e., independence of coordinates of alllocal initial distributions (local concentrations of all species).In this case, at any instant of time all measurable local physi-cal quantities coincide with spatially averaged quantities

[Ai]t = ni(r, t); [Cζ ]t = nζ (r, t); [A]t = nA(r, t);

[C]t = nC(r, t). (3.39)

On the Gibbs ensemble of “effective” pairs this means

pi(t) = pi(r, t); pζ (t) = pζ (r, t);

�Tik(r1, r2, t) = �T

ik(r1 − r2, 0, t); pζ (t) = �Tζ (t).

(3.40)

In particular, initial distribution vector component in relativecoordinates of the pair (A1A2) (see Eq. (3.26)) is

φik(r) = �T 0ik (r1 − r2, 0, 0); (r = r1 − r2) . (3.41)

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184104-10 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

So the kinetic equations for mean column-vector of probabil-ities coincide with the equations for local probabilities

spLA1

= QA1 pLA1

+ 2T rA2�Ld pL

C − 2T rA2 ILA1A2

+ p0A1

,

s pLC = QC pL

C − �LC pL

C + ILC + p0

C . (3.42)

With the formalism of generalized time functions47 (i.e., func-tions on extended time interval (−∞ < t < ∞)), one can re-cover the originals in Eq. (3.42). As a result, the kinetic equa-tions of multistage geminate reaction on normal (not extendedtime interval t ≥ 0) take the form

d

dtpA1 (t) = QA1 pA1 (t) + 2T rA2

t∫−0

�d (t − t0) pC (t0) dt0

− 2T rA2 IA1A2 (t), (3.43)

d

dtpC(t) = QCpC −

t∫−0

�C (t − t0) pC (t0) dt0 + IC(t)

(3.44)with the initial conditions pA1 (0) = p0

A1and pC(t) = p0

C . Notethat in the integrals over time in Eqs. (3.43) and (3.44) thelower integration limit is taken equal to −0, i.e., to the valuetending to zero from the side of negative values. The reasonis that kinetic equation kernels have a singularity at zero val-ues of the argument which must enter the integration domain.This is commonly not a problem in practical calculations ofintegral part of equations even by numerical methods. Indeed,according to Ref. 15, the kinetic equation kernels may be rep-resented as a sum of kinetic and relaxing parts. It is the firstpart that contains δ-shaped time singularity that is eliminatedby time integration assumed in the equations. Relaxing partinvolves no time singularities, and the lower integration limitis put zero. However, in analytical investigation of generalasymptotic properties, mathematical representation of the ker-nel on extended time axis seems preferable.

IV. CONSIDERATION OF “INTERNAL” DEGREESOF FREEDOM

A. Classical degrees of freedom

When deriving kinetic equations, for simplicity, we re-stricted ourselves to the examination of the dependence ofphysical quantities solely on space coordinates r. However,most of real reacting systems commonly contain additionaldegrees of freedom which we call “internal.” For example, instudies of reactions between reactants with anisotropic reac-tivity, one should take into account the dependence on Eu-ler angles � of orientation of molecular axis of reactants thatchange due to reactant rotation. Apart from such continuousvariables, discrete (classical or quantum) degrees of freedomcan also exist. As we use integral-matrix formalism, continu-ous variables will be treated as arguments of integral kernelsand physical characteristics, while discrete variables—as vari-ables defining matrix elements of the corresponding integralkernels and components of appropriate column-vectors.

So, for example, in order to allow for orientation anglesof the chosen molecular axis of reactants, it is necessary to

replace the space coordinate r of reactant by the coordinateof its configuration space q = {�, r}. Of course, all reactionLiouvillians and Liouvillians of free molecular motion musttake account of such a dependence. In particular, the Liouvil-lian of molecular motion must involve additional operators (inthe general case, integral) that describe reactant rotation.

Consideration of discrete classical “internal” variables inthe accepted concept of “effective” particles just means thatby the “state” of the “effective” particle we now imply not theentire species but only part of it corresponding to the givenvalue of discrete classical variable in reactants of this species.Total number (concentration or probability of finding) of thegiven species in solution is defined as a sum over these clas-sical variables.

B. Quantum degrees of freedom

As in Ref. 28, we shall consider quantum “internal” states(for instance, spin states of electrons and nuclei of reactants)as discrete. All these states of reactants of the given species(unlike states of “effective” particles defining species) mustbe quasi-resonance, i.e., they must cover narrow energy spec-trum of the width much less than kT (k Boltzmann constant,T absolute temperature). In this case, internal quantum transi-tions do not affect the character of molecular motion of reac-tants of the given species.28, 50 Further all quantum numbersof states of reactants of any, for example, ith species of the“effective” particle A or ζ th species of the “effective” parti-cle C will be denoted by Greek letters γ i, εi, μi, ν i or γ ζ , εζ ,μζ , νζ . Any quantum number of the set γ , ε, μ, ν (for reac-tant of any given species defined by the lower index for thesenumbers) includes a set of all quantum numbers and coversthe entire, one and the same (permissible for reactants of thegiven species) series of values.

The use of four designations of one and the same setof quantum numbers is determined by the fact that, in con-trast to discrete classical degrees of freedom, quantum “in-ternal” states cannot be simply included in the definition of“effective” particle state. The reason is that quantum evolu-tion of states of any reactant, for example, the reactant ofthe ith species is defined by the density matrix ργiεi

. In thedescription of the system in the Liouville space accepted bythe authors, its components are the components of column-vector that contain both diagonal (populations), and nondiag-onal (phase) elements. Thus the states, for example, of “ef-fective” particle A with allowance for quantum states areAiγiεi

. The corresponding distribution functions (componentsof column-vector of states �i γiεi , kμk

,νkof “effective” parti-

cle A), in principle, depend on spin states. Matrix elementsof reaction Liouvillians also depend on spin states, and thisensures, if necessary, spin selection rules (i.e., the course ofreaction only from a definite collective spin state of partners).

Now Liouvillians QA and QC of individual “effective”particles must be represented as

QA = RA + i�A + �A, QC = RC + i�C + �C, (4.1)

where Liouvillians RA and RC take into account monomolec-ular transformation but with allowance for spin states. Its ma-trix elements are defined by analogy with Eqs. (2.28) and

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184104-11 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

(2.29) with configuration space coordinates instead of spacecoordinates, and any state i of “effective” particles is re-placed by the corresponding states iγ iεi with allowance forspin variables. Liouvillians �A and �C describe dynamic evo-lution of internal quantum states of reactants of the givenspecies and corresponds to internal quantum Hamiltonian HAi

of this species reactant. Such a Hamiltonian can include,for instance, the interaction with external magnetic field (oranisotropic splitting in zero field). In this case, it can dependon orientation angles of molecular axis with respect to ex-ternal field direction (or spin state quantization axis). Matrixelements of this Hamiltonian are written in the basis of inter-nal quantum states of reactant. Liouvillians �A and �C , obvi-ously, have only diagonal matrix elements in species indices

(�A)iμiνi , iγiεi= 1

�

(HAi

εi , νiδγi ,μi

− HAiμi ,γi

δνi , εi

),

(�C)ζμiνi , ζγiεi= 1

�

(HCζ

εζ νζδγζ μζ

− HCζ

μζ γζδνζ εζ

).

(4.2)

Liouvillians �A and �C describe internal quantum state relax-ation. Just as Liouvillians �A and �C , they have only diago-nal matrix elements in species indices and satisfy the requiredselection rules for internal relaxation operators.51

In view of internal quantum states, bimolecular Liouvil-lian Vb

A1A2should be represented as

VbA1A2

= RbA1A2

+ iVinA1A2

, (4.3)

where the Liouvillian RbA1A2

takes into consideration ex-change bimolecular reaction but with allowance for spinstates. Its matrix elements are defined by analogy withEq. (2.33) with configuration space coordinates instead ofspace coordinates, and any state i of “effective” particles isreplaced by the corresponding states iγ iεi with allowancefor spin states. The Liouvillian Vin

A1A2describes dynamic

evolution of internal quasi-resonance quantum states of twoquantum-interacting reactants of the given species (for exam-ple, Ai and Ak) and corresponds to internal quantum interac-tion Hamiltonian V AiAk of these species reactants. Matrix el-ements of this Hamiltonian are written in the basis of collec-tive internal quantum states of two reactants. Evidently, theLiouvillian Vin

A1A2has only diagonal matrix elements in a pair

indices ik of the corresponding species, and is as follows:(Vin

A1A2

)iμivi , kαkβk ; iγi εi ,kλkξk

= 1

�

(V

AiAk

εiξk,viβkδγiλk,μiαk ; − V

AiAk

μiαk,γiλkδviβk,εi ξk

). (4.4)

As for other reaction Liouvillians, their matrix elements aredefined by analogy by previous formulae replacing space co-ordinates by configuration space ones, and any state i or ζ of“effective” particles A or C by the corresponding states i γ iεi

or ζ γ ζ εζ with allowance for spin variables. It is stated abovethat Liouvillians of molecular motion must involve operatorsof additional motions (for example, rotation) in the config-uration space. As to taking account of spin states, they areintroduced by replacing states i or ζ of “effective” particlesA or C by the corresponding states i γ iεi or ζ γ ζ εζ with al-lowance for spin variables. By virtue of quasi-resonance con-

dition, motion operators in this Liouvillian depend solely onthe species indices, not on internal quantum states.

Inclusion of internal quantum degrees of freedom intoconsideration affects the operation Tr. As before, it assumessummation over all components of column-vector referringto different species. However, on summation over quantumstates of any of species, it applies only to the components cor-responding to diagonal elements of density matrix, i.e., forinstance (see Eq. (2.22)),

T rAnA (q, t) =∑iγi

niγiγi(q, t),

(4.5)T rCnC (q, t) =

∑ζγζ

nζγζ γζ(q, t).

As earlier, by the operation Tr of Liouvillians we mean Trof column-vector obtained as a result of the action of theseLiouvillians on any initial column-vector (not at all the sum-mation of diagonal elements of Liouvillian matrices). Thus,for example,

T rAQA ≡ (T rAQA)kμkνk=

∑iγi

(QA)iγiγi |kμkνk(4.6)

is actually the component k μkνk of some row-vector. That iswhy, for instance, going to zero of quantities in Eqs. (2.30),(2.35), (2.40), (2.44), and (2.48) means that all components ofsuch a vector are equal to zero.

Validity of Eqs. (2.30), (2.35), (2.40), (2.44), and (2.48)necessary for the fulfillment of total balance condition, in thecase of the presence of internal quantum states, requires that,first, Tr of additional dynamic Liouvillians go to zero

T rA�A = 0, T rC�C = 0, T rA�A = 0, T rC�C = 0,

T rA1A2 VinA1A2

= 0. (4.7)

As is known (and is easily seen from Eqs. (4.2) and (4.4)), thisactually takes place. Second, it is required that relations of thetype of Eqs. (2.30), (2.35), (2.40), and (2.44) hold for reactionoperators with allowance for spin states. This imposes certainrestrictions on the properties of Liouvillian matrix elements(selection rules)28 that can be established only in consideringparticular reaction systems.

V. SUMMARY

Based on the concept of “effective” particles developedby the authors earlier, matrix kinetic equations of multistagegeminate reactions of isolated pairs of reactants have beenderived. Elementary stages of multistage reactions consid-ered involve all types of bimolecular and monomolecularphysicochemical processes possible in isolated pairs includ-ing changes in internal classical and quasi-resonance quantumdegrees of freedom. For example, taking into account internalspin states allows one to examine reactions with the partic-ipation of paramagnetic particles (radicals and radical ions),and in external magnetic fields as well. To consistently de-velop the theory of multistage geminate reactions, the prob-lem of many-particle consideration of a reacting system hasbeen reduced to the equivalent investigation of the Gibbs

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184104-12 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

ensemble of “effective” particles. Unlike most of approachesused in the literature in studies of elementary geminate re-actions of isolated pairs, the problem is formulated in termsof kinetic theory. This means that instead of survival prob-abilities of reacting pairs, we introduce definitions of meanand local probabilities of finding species in a sample (ormean and local concentrations) at any moment of time. Forgeneral matrix formulation of the results, column-vectors ofthe observed macroscopic kinetic quantities are used. Onthe basis of elementary event rates describing the course ofdifferent types of physicochemical processes and molecularmotion of reactions, integral-matrix Liouvillians have beenconstructed. Kinetic coefficients (integral kernels and inho-mogeneous sources) of the obtained kinetic equations formean kinetic characteristics are expressed in terms of quan-tities in relative coordinates of geminate pairs. An importantwidely encountered case of homogeneous reaction systems isconsidered.

ACKNOWLEDGMENTS

The authors are grateful to the Russian Foundation of Ba-sic Research for financial support (Project No. 12-03-00058).

APPENDIX: PROPERTIES OF LIOUVILLIANS

To study general properties of Liouvillians in the conceptof “effective” particles, first consider irreversible bimoleculargeminate reaction A + B → C + D defined by 4-center el-ementary rate RCD|AB(r1, r2|r′

1, r′2) of the transition (per unit

time) of a pair of reactants (residing at points r′1 and r′

1, re-spectively) of species A and B to a pair of reactants (residingat points r1 and r2, respectively) of species C and D.

The formalism of the Fock boxes was used by the authorsearlier both to describe different (bulk and geminate) bimolec-ular reactions (exchange and association), and dissociationreactions.23, 25, 42, 52 For the description of geminate reactionA + B → C + D, two Fock boxes are necessary which cor-respond to possible states of the reaction system (elementaryoutcomes): the first box contains a pair of reactants (AB), thesecond one—a pair of reactants (CD).

Statistical description of the first box (in the thermody-namic limit) is made, for example, by the probability densityFT

AB(r1, r2, t) of finding the reactant of the species A at pointr1 and reactant of the species B at point r2 at the instant oftime t. The probability P T

AB(t) of finding the reaction systemat the moment of time t in the box (AB) is

P TAB(t) = lim

υ→∞

∫υ

dr1dr2

υFT

AB(r1, r2, t). (A1)

Apparently, the function FTAB(r1, r2, t) involves full statistical

information concerning the reaction pair kinetics in the firstbox.

However, there is another description of the state of reac-tants in the first box by the probability density FT

BA(r1, r2, t)to find the reactant of the species A at point r2, and thereactant of the species B at point r1. This function is ex-pressed in terms of the previously introduced function in an

obvious way

FTBA(r1, r2, t) = FT

AB(r2, r1, t). (A2)

That is why it also contains full statistical information con-cerning the pair kinetics in the first box. Though formally bothdescriptions are different, in fact, they are equivalent and anyof them can be chosen.

Everything written above about statistical description ofthe first box also refers to the second one. So geminate re-action A + B → C + D may be described by four formallydifferent sets of densities:

(1) FTAB iFT

CD; (2) FTBA iFT

CD; (3) FTAB iFT

DC ;

(4) FTBA iFT

DC. (A3)

All these description methods are equivalent in essence; how-ever, they can lead to mathematically different forms of chem-ical interaction Liouvillians in “effective” particle formalism.

Description of reaction. Let us describe the state of thereaction system by the first density set from Eq. (A3). Chemi-cal reaction induces their time variation due to transitions be-tween the Fock boxes. Explicit form of such transitions is de-fined by the equations representing balance conditions:

d

dtF T

AB(r1, r2, t)

= −∫

dr′1dr′

2 RCD|AB(r′1, r′

2|r1, r2)FTAB(r1, r2, t),

(A4)d

dtF T

CD(r1, r2, t)

=∫

RCD|AB(r1, r2|r′1, r′

2)dr′1dr′

2 FTAB(r′

1, r′2, t).

(A5)

In concise form these equations may be written in terms ofintegral operators WCD|AB and RCD|AB acting on functionsof space coordinates. The kernels of these operators take theform

(WCD|AB)(r1, r2|r′1, r′

2)

= δ(r1 − r′1)δ(r2 − r′

2)wCD|AB(r1 − r2), (A6)

where

wCD|AB(r1 − r2) =∫

dr′1dr′

2 RCD|AB(r′1, r′

2|r1, r2),

(RCD|AB)(r1, r2|r′1, r′

2) = RCD|AB(r1, r2|r′1, r′

2). (A7)

As a result, Eqs. (A4) and (A5) are as follows:

d

dtF T

AB(r1, r2, t) = −WCD|ABFTAB, (A8)

d

dtF T

CD(r1, r2, t) = RCD ABFTAB. (A9)

Equations (A4) and (A5) (or Eqs. (A8) and (A9)) will serve asa basis for the construction of Liouvillian in effective particleformalism.

“Effective” particle formalism for the reaction A + B →C + D begins with statistical description of possible states, forexample, initial reactants. These states are the components of

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184104-13 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

column-vectors corresponding to all possible combinations ofpairs of reactants of two species determined by two identicalparticles A1 and A2 that are at spatial points r1 and r2, re-spectively. For the reaction at hand, 16 different elementaryoutcomes are obtained the description of which calls for 16�T

ik(r1, r2, t). However, chemical reaction induces time vari-ation of four functions only

�TAB(r1, r2, t); �T

CD(r1, r2, t); �TBA(r1, r2, t)

�TDC(r1, r2, t). (A10)

Thus these functions are sufficient to describe the reaction A+ B → C + D.

Relation to Fock boxes. Comparison of statistical descrip-tions in two formalisms shows that the description of the re-action in “effective” particle formalism requires that twice asmany elementary outcomes be taken into account. This is be-cause, for instance, species A in Fock formalism is markedonly by space coordinate value. In “effective” particle formal-ism, indication of belonging to one of two “effective” particlesis added to this species mark. That is why elementary out-come space of “effective” particle formalism without restric-tive properties of matrix elements of matrix-integral reactionLiouvillian is too detailed.

Nevertheless, any observables can be described in theframework of both formalisms. For example, the probabilitydensity to find at points r1 and r2 a pair of initial reactants(AB) in the Fock formalism is

dPTAB (r1, r2, t) = [

FTAB(r1, r2, t) + FT

AB(r2, r1, t)]dr1dr2

(A11)and in “effective” particle formalism

dPTAB(r1, r2, t) = [

�TAB(r1, r2, t) + �T

BA(r1, r2, t)]dr1dr2.

(A12)So the probability P T

AB(t) to find the reaction system in thestate of the pair (AB) is

P TAB(t) = lim

υ→∞

∫υ

dPTAB(r1, r2, t)

2υ. (A13)

Here integration is performed over all different pairs of pointsr1 and r2 in the volume υ. Substituting Eq. (A11) in Eq. (A13)gives Eq. (A1). Comparison between Eqs. (A11) and (A12)shows that probability densities of the two formalisms maybe identified

�TAB(r1, r2, t) = FT

AB(r1, r2, t);(A14)

�TBA(r1, r2, t) = FT

AB(r2, r1, t).

Similarly, for the pair CD

�TCD(r1, r2, t) = FT

CD(r1, r2, t);(A15)

�TDC(r1, r2, t) = FT

CD(r2, r1, t).

Description of reaction. Equations (A14) and (A15) eas-ily give the binary interaction Liouvillian in “effective” par-ticle formalism with the aid of Eqs. (A8) and (A9). Matrix

elements describing the reaction A + B → C + D are

d

dt

⎛⎜⎜⎜⎜⎝

�TAB

�TCD

�TBA

φTDC

⎞⎟⎟⎟⎟⎠ (r1, r2, t) = Vb

A1A2

⎛⎜⎜⎜⎜⎝

�TAB

�TCD

�TBA

�TDC

⎞⎟⎟⎟⎟⎠ (r1, r2, t).

(A16)Here integral-matrix operator Vb

A1A2takes the form

VbA1A2

=

⎛⎜⎜⎝

−WCD|AB 0RCD|AB 0

0

0−WDC|BA 0RDC|BA 0

⎞⎟⎟⎠ . (A17)

It is the desired interaction Liouvillian of a pair of reactantsfrom “effective” particle species.

Identity of “effective” particles. Liouvillian (A17) mustbe tested for invariance under permutation of “effective”particles. For this purpose, first, construct the operator P12

of “effective” particle permutation. New probability densi-ties �T

ik(r1, r2, t) with permuted “effective” particles are ex-pressed in terms of previous (before permutation) probabilitydensities as follows:

�TAB(r1, r2, t) = �T

BA(r2, r1, t);

�TCD(r1, r2, t) = �T

DC(r2, r1, t)(A18)

�TBA(r1, r2, t) = �T

AB(r2, r1, t);

�TDC(r1, r2, t) = �T

CD(r2, r1, t).

These equalities may be represented in a matrix form⎛⎜⎜⎜⎜⎝

�TAB

�TCD

�TBA

�TDC

⎞⎟⎟⎟⎟⎠ (r1, r2, t) = P12

⎛⎜⎜⎜⎜⎝

�TAB

�TCD

�TBA

�TDC

⎞⎟⎟⎟⎟⎠ (r1, r2, t). (A19)

Here the permutation operator is defined by the expression

P12 =(

0 E2

E2 0

)℘, (A20)

where E2—unit matrix of dimensionality 2 × 2, ℘—permutation operator of points r1 and r2. The Liouvillian˜Vb

A1A2acting on new probability densities is expressed in

terms of the previous Liouvillian (A18) in a standard way

˜Vb

A1A2= P12Vb

A1A2P12. (A21)

In explicit calculations, symmetry conditions for elementaryrate should be taken into consideration,

RCD|AB(r1, r2|r′1, r′

2) = RDC|BA(r2, r1|r′2, r′

1), (A22)

which is physically evident. As a result, we have

˜Vb

A1A2= Vb

A1A2. (A23)

Thus the constructed Liouvillian (A17) is invariant under ef-fective particle permutation.

Unambiguity of Liouvillian construction. Liouvillian(A17) has been constructed for the Fock fist formalism. The

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184104-14 A. B. Doktorov and A. A. Kipriyanov J. Chem. Phys. 140, 184104 (2014)

choice of the fourth set is easily seen to lead to the same Liou-villian. However, the choice of the second or the third set fromcomplete set (A3) gives the binary interaction Liouvillian ofthe form

VbA1A2

=

⎛⎜⎜⎝−WDC|AB 0 0 0

0 0 RCD|BA 00 0 −WCD|BA 0

RDC|AB 0 0 0

⎞⎟⎟⎠ . (A24)

Since for elementary rate the following condition is valid:

RDC|AB(r1, r2|r′1, r′

2) = RCD|BA(r2, r1|r′2, r′

1), (A25)

therefore, Liouvillian (A24) is also invariant under “effective”particle permutation. Consideration of Liouvillians as linearcombination of Liouvillians (A17) and (A24) is physicallymeaningless, because it does not correspond to the notionsof Fock boxes on the Gibbs ensemble.

So one can use either Liouvillian (A17), or Liouvillian(A24). Formally the choice is made as follows. If for differ-ent states (i = k = i′ = k′) matrix elements of bimolecularLiouvillian are specified as(

VbA1A2

)ik|i ′k′(r1, r2|r′

1, r′2) = 0, (A26)

then by definition(Vb

A1A2

)ik|k′i ′(r1, r2|r′

1, r′2) = 0. (A27)

Geminate reactions of other types (associative-dissociative and monomolecular) are considered by analogy.Unlike the case of exchange bimolecular reactions describedabove, the constructed Liouvillians have a unique form. Thusno additional conditions of the choice of (A27) type arise.

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