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

General theory of multistage geminate reactions of isolated pairs of reactants. I. Kinetic equations Alexander 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 pairs of reactants. I. Kinetic equations Alexander B. Doktorov and Alexey A. Kipriyanov Voevodsky Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of 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 of reactants depending on reactant mobility is formulated on the basis of the concept of “effective” particles. Various elementary reactions (stages of multistage reaction including physicochemical processes of internal quantum state changes) proceeding with the participation of isolated pairs of reactants (or isolated reactants) are taken into account. Investigation has been made in terms of kinetic approach implying the derivation of general (matrix) kinetic equations for local and mean probabilities of finding any of the reaction species in the sample under study (or for local and mean concentrations). The recipes for the calculation of kinetic coefficients of the equations for mean quantities in terms of relative coordinates of reactants have been formulated in the general case of inhomogeneous reacting systems. Important specific case of homogeneous reacting systems is considered. © 2014 AIP 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 condensed media play an important role in different fields of science and technology. Among these are, for instance, electron excitation energy transfer,1 electron (or proton) migrations in photosynthetic systems,2–4 various chemical reactions occurring in colloid or polymer solutions,5–7 in nano- and bio- systems,8–10 or trapping and detrapping problems in semiconductors11 and many others. From this standpoint, development of the theory of reactions depending on mobility of reactants is crucial to understand the systems and to apply to complex problems. Many fundamental issues on theories had been addressed over many years.3, 5, 6, 12–25 Most of the theories referred to elementary bulk reactions; however, multistage reactions26–29 significant in studies of biochemical reactions have also been investigated, as well as transfer reactions in luminescence and chemiluminescence and photo- and electrochemistry.30 An increasing interest in the consideration of different multistage reactions has stimulated formulation and development of general (matrix) theories of bulk multistage reactions including all possible elementary stages (bimolecular chemical exchange and addition reactions, dissociation and monomolecular transformation reactions).26–29 This makes it possible to establish general detailed balancing principles (total balance of particles and kinetic and thermodynamic balancing). In the framework of the developed approach, “internal” degrees of freedom (for instance, spin states) have been included in the consideration, thus the reactions with the participation of paramagnetic particles (free radicals and radical ions), in external magnetic field also,28 have been studied. Geminate reactions of isolated pairs of reactants arising from external radiation31, 32 are another important class of reactions in solutions. Many analytical fundamental issues on theories of diffusion-influenced geminate reactions 0021-9606/2014/140(18)/184104/14/$30.00

had been also addressed over many years.33–43 However, they commonly referred either to treating of elementary geminate reactions (including reversible ones), or the simplest twostage reactions (reversible bimolecular reactions involving monomolecular deactivation stage of excited molecules) for homogeneous reaction system. Spatially inhomogeneous systems have been considered only by the authors of this work42 for geminate elementary reaction. Besides, in the literature there are no papers dealing with geminate reactions between identical reactants. The presence of internal quantum degrees of freedom (for example, spin ones) has only been taken into account in the examination of elementary irreversible reactions between radicals. At the same time, consistent consideration of such degrees of freedom is necessary, for instance, in studies of biological applications of multistage reactions with the participation of radicals or other paramagnetic particles (including the case of external magnetic field). Over the last years there has been an increasing interest in the theoretical investigation of multistage geminate reactions of isolated pairs.44 So it seems reasonable, similarly to the theory of bulk reactions, to develop general matrix kinetic theory of multistage geminate reactions of isolated pairs that allows one to consider all the above aspects and can be a reliable basis for the description of a wide class of specific chemically reacting systems. This is the goal of the present contribution. Section II deals with the description of multistage geminate reactions of isolated pairs of reactants. Kinetic schemes of elementary stages of the multistage reaction under study are given including all types of bimolecular and monomolecular reactions in isolated pairs. The concept of “effective” particles is introduced which serves to determine mean and local probabilities of the species residence in the specimen at any moment of time. Besides, integral-matrix Liouvillians describing the course of reactions of different types and molecular motion of reactants are formed. In Sec. III, kinetic equations for local and mean probabilities (or local and mean

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concentrations of species) have been derived. Kinetic coefficients (integral kernels and inhomogeneous sources) in kinetic equations for mean kinetic characteristics are expressed in terms of the values in relative coordinates of geminate pairs. An important and widely encountered case of homogeneous reaction systems is considered. Section IV is devoted to the method of generalizing of the employed approach in taking into account “internal” degrees of freedom including spin states of reactants. The main results are given in Sec. V.

of reversible chemical transformation (for example, reversible cis-trans isomerization), and physical excitation processes of molecules under the action of permanent radiation and decay of excited molecules into initial unexcited state. Processes of monomolecular transformation Ai ↔ Cζ are not examined, since in the framework of the consideration of reactions of isolated pairs geminate pairs Ak + Cζ are neglected. Just with such restrictions it is possible to develop general (matrix) theory of multistage geminate reactions.

II. DESCRIPTION OF MULTISTAGE GEMINATE REACTIONS OF ISOLATED PAIRS

2. Vector distribution functions

A. Reaction schemes and “effective particles”

1. Reactions in isolated pairs

In the most general case, a multistage reaction involves different elementary (in the general case, reversible) reactions (multistage reaction stages). These are bimolecular stages of elementary exchange reactions Ai + Ak → Al + Am (i, k, l, m = 1, 2, . . . NA ),

(2.1)

where Ai —reactants (species), NA —the number of species participating in bimolecular exchange reactions. Note that reaction scheme (2.1), despite the arrow in one direction, obviously takes into consideration that any elementary bimolecular stage in this scheme can be reversible. Elementary irreversible recombination or addition reactions are bimolecular reactions 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 reaction under study, NC —the number of species formed due to such bimolecular reactions. Dissociation reactions are the reactions reverse to reactions (2.2)

Consider a many-particle reacting system (a sample) of the volume sam where reactions (2.1)–(2.4) take place. Since these reactions have a geminate nature, the system can be described using the Gibbs ensemble of isolated pairs. Let us examine an initial many-particle system (a sample) at a certain time t. At this time, let it contain NAi Ak (t) pairs of reactants from species Ai + Ak , and NCζ (t) reactants of species Cζ . If the total number of all pairs at time t is NAA , and the total number of reactants from all species C is NC (t), then the number N = NAA (t) + NC (t)

(2.5)

remains unchanged during the reaction. Equation (2.5) can be given a probabilistic meaning by defining the probability PAi Ak (t) that a pair of reactants, from reactants of species Ai and Ak , exists in the form of a free pair and the probability PCζ (t) that they exist in a bound state, PAi Ak (t) = NA

NAi Ak (t) , N

PAi Ak (t) +

i≤k=1

NC

PCζ (t) =

NCζ (t) , N

(2.6)

PCζ (t) = 1.

ζ =1

Cζ → Ai + Ak (i, k = 1, 2, . . . NA ; ζ = 1, 2, . . . NC ). (2.3) Note that the initial reactants and products of bimolecular exchange reactions are denoted as Ai , while in recombination or addition reactions these species are just initial reactants, and products cannot be initial for other bimolecular reaction, thus they as denoted by Cζ . This is because geminate reactions of isolated pairs of reactants are examined. In this case, the initial isolated geminate pair gives rise (as a result of recombination or addition (2.2)) to isolated reactants. They cannot enter into bimolecular reaction, but can just dissociate to produce some pair of reactants of species Ai . At the same time, if we consider isolated pairs of reactants of species Cζ , dissociation of any reactant of such a pair results in the system of three or four particles, not an isolated pair. Exactly for this reason bimolecular reactions between reactants Cζ are not investigated. Apart from the above reactions, reactions of monomolecular transformation can be included in the consideration

The last equality expresses the property of the completeness of the introduced probabilities and is equivalent to the condition of conservation of the number of particles. The state of some free pair corresponds to the Gibbs subensemble of such pairs with a distribution function Fik (r1 , r2 , t)42 which is the probability density of detecting a pair with coordinates r1 and r2 at time t. The bound state corresponds to a subensemble of reactants with a distribution function Fζ (r, t) which is the probability density of finding a reactant with coordinate r at time t. Further, it is convenient to pass to vector distribution functions on the Gibbs ensemble. The basis for the construction of such distribution functions is the concept of “effective” particles.26–29 In the framework of such concept, the sets of reacting species are introduced

Ai → Ak , Cζ → Cμ .

i.e., the species are treated as “internal” states of “effective” particles A or C. In this case, column-vectors of distribution functions in the pair AA and distribution functions of reactants C are used. Distribution vector of the pair should be specified in formal collective basis |Ai Ak , that is a direct product

(2.4)

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

A = {Ai } = {A1 , A2 , . . . , ANA }; (2.7) C = {Cζ } = {C1 , C2 , . . . , CNC },


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of single-particle basis |Ai Ak = |Ai |Ak . Distribution vector A1 A2 of “effective” particles pair A1 A2 should be constructed on this basis as follows. It is a column of the probability densities containing NA2 elements of the form ik (q1 , q2 , t). Here q1 and q2 are the coordinates of the configuration space of the corresponding effective pairs. In the general case, configuration space involves both space coordinates and orientation angles of reactants, as well as internal degrees of freedom. However, for simplicity, first we restrict ourselves to the examination of space coordinates r, since consideration of more complete coordinate system presents no particular problems (see Sec. IV). Distribution column-vector C of “effective” particle C is specified in basis Cζ and contains NC elements ζ (q, t). Note that the number of components of the vector A1 A2 exceeds that of pairs of reactants. This is related to the necessity of uniform description of pairs of identical and non-identical reactants (see the Appendix). Also introduce operations T rA1 A2 = T rA1 T rA2 and T rC of column-vectors by which, in the case of the absence of “internal” 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 geminate pairs (containing elementary stages (2.1)–(2.4)) can be represented as reactions A + A → A + A, A + A ↔ C, A → A, and C → C of “effective” particles A and C. In the context of many-particle theory, the evolution of such a reaction system may be described by the Gibbs ensemble just by considering 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 is defined by the column-vector of probability densities ik (r1 , r2 , t) of finding the reactant of species Ai at the point r1 of three-dimensional space, and the reactant of species Ak at the point 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 columnvector of probability densities ζ (r, t) of finding the reactant of species ζ at the point r. The norm of these functions defines the probability of finding geminate reaction system at the moment of time t either in the state of pairs consisting of reactants of two species, or in the state of isolated particles of C, dr1 dr2 PA1 A2 (t) = T rA1 A2 A1 A2 (r1 , r2 , t), 2 (2.9) PC (t) = T rC drC (r, t). Since for the system at hand there are no other variants, therefore we have the relation completely equivalent to the last equality in Eq. (2.6) PA1 A2 (t) + PC (t) = 1.

(2.10)

Further statistical description of the many-particle reacting system can be conveniently performed in the so-called thermodynamic limit46 on the basis of generalization to the systems of “effective” particles of the approach used in Ref. 42 in studies of elementary reversible geminate reaction. Now we examine bulk properties of the sample considering it in the thermodynamic limit. For this purpose, we assume that N → ∞, 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 the state of species in these boxes should be described by nonnormable functions Tik (r1 , r2 , t) = T − lim sam ik (r1 , r2 , t), (2.12) Tζ (r, t) = T − lim sam ζ (r, t). These functions do not go to zero, if the coordinate of species tends to infinity. Thus, Eq. (2.9) should be replaced by dr1 dr2 T T A1 A2 (r1 , r2 , t), PA1 A2 (t) = lim T rA1 A2 υ→∞ 2υ υ

PCT (t)

= lim T rC υ→∞

(2.13) dr T (r, t). υ C

υ

Symbol υ under the integral means that integration is made over the volume υ. Condition of completeness takes the form PAT 1 A2 (t) + PCT (t) = 1.

(2.14)

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

It follows from the above reasoning that in the thermodynamic limit mean concentrations of effective particles A and C in the sample can be represented as [A]t = 2PAT 1 A2 (t)n; [C]t = PCT (t)n.

(2.15)

Factor 2 in the first equality takes into account that the pair A1 A2 contains two “effective” particles. As mean concentrations may be treated as physical observables, the equality following from Eqs. (2.14) and (2.15), n = [A]t /2 + [C]t

(2.16)

defines theoretical parameter n (2.11) significant for further discussion and directly related to reactant density in the sample. Mean concentrations [Ai ]t and [Cζ ]t of species in physical sample are also among the observables in experiment, and they 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 distribution functions are of the form dr1 dr2 (δii1 + δii2 )Ti1 i2 (r1 , r2 , t) [Ai ]t = n lim υ→∞ 2υ i i 1 2

υ

dr1 dr2 = n lim Tik (r1 , r2 , t), υ→∞ υ k

(2.17)

υ


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dr δζ ζ Tζ (r, t) [Cζ ]t = n lim υ→∞ υ ζ

= n lim

υ→∞

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

υ

[Ai ]t = np¯ i (t);

dr T (r, t). υ ζ

(2.18)

υ

Evidently, mean concentrations of effective particles and their species are related as [A]t = T rA [A]t ;

[C]t = T rC [C]t .

υ

(2.20) Local concentrations of species of “effective” particles ni (r, t) and nζ (r, t) are introduced in a similar way dr dr [Ai ]t = lim ni (r, t); [Cζ ]t = lim nζ (r, t) υ→∞ υ→∞ υ υ υ

υ

(2.21) and are the components of column-vectors nA (r, t) and nC (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 observables. Most informative are local concentrations of species. If these quantities are known, they can be used to calculate the rest of observables by Eqs. (2.21), (2.22), and (2.19). The goal of the kinetic theory is to derive kinetic equations to find quantities ni (r, t) and nζ (r, t). Local and mean concentrations characterize the state of the sample in the thermodynamic limit. Further it will be convenient to introduce local probabilities pi (r, t) and pζ (r, t) defining the state of the Gibbs ensemble of “effective” particles in the thermodynamic limit that are related to local concentrations 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 view of Eq. (2.23) shows that local probabilities are related to distribution functions in the thermodynamic limit as dr2 Tik (r, r2 , t); pζ (r, t) = Tζ (r, t). pi (r, t) = k

(2.24) Obviously, spatial averaging of these probabilities gives the probabilities of finding the species in the sample dr dr pi (r, t); p¯ ζ (t) = lim pζ (r, t). p¯ i (t) = lim υ→∞ υ→∞ υ υ υ

(2.26)

So the next problem is to derive kinetic equations for local probabilities defined on the Gibbs ensemble of “effective” particles for multistage geminate reaction under consideration. This is rather easy to do even for spatially inhomogeneous systems.

(2.19)

In kinetic theory, local concentrations are the main observable. In the system in question, these are, first of all, local concentrations of “effective” particles nA (r, t) and nC (r, t). They are related to mean concentrations by spatial averaging procedure dr dr [A]t = lim nA (r, t); [C]t = lim nC (r, t). υ→∞ υ→∞ υ υ υ

[Cζ ]t = np¯ ζ (t).

υ

(2.25)

B. Reactivity and free motion operators

1. Integral operators

Most general description of reactivity and motion of reactants in solution is made by introducing integral-matrix operators (integral-matrix Liouvillians) acting on column-vector of distribution functions TA1 A2 (r1 , r2 , t) and TC (r, t). Construction of operator kernel matrices is based on elementary rates of processes (sink terms) calculated by elementary event theory.12 In the described approach, they are a priori given. In some our previous works (see Refs. 13 and 14) in constructing any integral operators, we considered uniformly space and time variables on extended time axis (−∞ < t < ∞), and this made it necessary to use generalized functions formalism.47 In the present contribution (as in Ref. 42) and in constructing integral T-operators (Ref. 29) time functions will be replaced by their Laplace transforms. This will make operations with generalized functions unnecessary, and mathematical part of derivation—more clear. Laplace transforms are defined in a standard way and are marked by the upper index L. For example, TAL1 A2 (r1 , r2 ) = TAL1 A2 (r1 , r2 ; s) ∞ =

dt TA1 A2 (r1 , r2 , t) exp (−st). (2.27) 0

As for space variables, the functions under discussion have symmetry (2.8) in them. This means that, as in quantum mechanics,48 the desired kernels are to be symmetric about permutation of “effective” particles coordinates. This requirement essentially restricts the range of acceptable forms of the desired kernels. However, as it will be seen below, several mathematical representations for reactivity Liouvillians describing one and the same reaction can exist. Now introduce the reactivity operators for elementary stages of multistage geminate reaction of isolated pairs of reactants. 2. Monomolecular processes

Let us consider the rates of monomolecular processes including those determined by spontaneous decay of excited states or by external radiation excitation to be constant quantities. Denote the first order rate constants corresponding to the transition of reactants between species Ai ← Ai in the “effective” particle A as Kii , and those corresponding to the transition 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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zero, i.e., Kii = Kζ ζ ≡ 0. Then the matrix of integral kerˆ C of monomolecular processes ˆ A and Q nels of Liouvillians Q of “effective” particles A and C are defined by the following matrix elements (kernels): ⎤ ⎡ ˆ A )i|i (r|r ) = δ(r − r ) ⎣−δii Kki + Kii ⎦ , (2.28) (Q k =i

⎡ ˆ C )ζ |ζ (r|r ) = δ(r − r ) ⎣−δζ ζ (Q

⎤ Kμζ + Kζ ζ ⎦ .

μ =ζ

(2.29) Here δ(r − r )—Dirac δ-function, δ ii —Kronecker delta. The first term between square brackets is diagonal and describes the escape of reactant from the i th species to all other species during monomolecular transformations. The second nondiagonal term describes the transformation of reactant of the i th species to reactant of the ith species. It is easily verified that Eqs. (2.28) and (2.29) give the property equivalent to the law of “effective” particles preservation in monomolecular transformation processes ˆ A = T rC Q ˆ C = 0. T rA Q

(2.30)

Note that hereinafter by the operation Tr of Liouvillians we mean Tr of column-vectors resulting from the action of these operators (Liouvillians) on any initial column-vectors ˆ A1 ⊕ Q ˆ A2 acting on A1 A2 = Q (see Sec. IV). The operator Q T the column-vector A1 A2 (r1 , r2 , t) of two “effective” particles is the direct sum of operators of isolated “effective” particles.27–29 3. Exchange bimolecular processes

Bimolecular process is defined by the 4-center elementary rate Rik|i k (r1 , r2 |r1 , r2 ) of the transition (per unit time) of a pair of reactants from species Ai and Ak residing at points r1 and r2 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 the conditions of translation and permutation symmetries Rik|i k (r1 , r2 |r1 , r2 ) = Rik|i k (r1 +r, r2 +r|r1 + r, r2 +r); (2.31) Rik|i k (r1 , r2 |r1 , r2 ) = Rki|k i (r2 , r1 |r2 , r1 ), where r—arbitrary displacement vector. The presence of translation symmetry is determined by the assumed spatial homogeneity of the solvent. The existence of additional symmetries is also possible; however, this is inessential for the derivation of equations. Note that this quantity is identically zero for “diagonal” values i = i k = k . Also note that the use of elementary rate Rik|i k (r1 , r2 |r1 , r2 ) in “effective” particles concept requires that it be zero at permutation just in one pair of indices (see the Appendix). Along with elementary rates, introduce 2-center quantities dr1 dr2 Rik|i k (r1 , r2 |r1 , r2 ). (2.32) wik|i k (r1 − r2 ) =

For the case of different reactants from different species (i = k = i = k ), they are complete rates of transition of a pair of reactants from the species (Ai , Ak ) (at points r1 and r2 ) to a pair of reactants from the species (Ai , Ak ) at any spatial position of these reactants. However, if there are identical reactants, the above physical meaning of the introduced quantities turns out to be inexact (due to this identity). Nevertheless, it will be convenient to call these quantities complete rates in this case as well. In the Appendix, we give integral-matrix Liouvillian ˆb V A1 A2 of exchange bimolecular processes. Matrix elements of its integral kernels are ( VbA1 A2 )ik|i k (r1 , r2 |r1 , r2 ) = −δii δkk δ(r1 − r1 )δ(r2 − r2 )

lm

wlm |l k (r1 − r2 )

× θlm|i k + θik|i k Rik|i k (r1 , r2 |r1 , r2 ),

(2.33)

where the coefficient θ ik|i k allows for the identity of effective particles A1 and A2 and is defined as follows: θik|i k =

1+δ

i k

1 . + δik − δik δi k

(2.34)

The first, diagonal term, in the definition of the matrix ˆb element of the operator V A1 A2 corresponds to the escape of reactants (residing at points r1 and r2 ) from species Ai and Ak to other pairs with any of spatial position of reactants, The second, nondiagonal term, describes the appearance of a pair of reactants (at points r1 and r2 ) of species Ai and Ak from a pair of reactants (residing at points r1 and r2 ) from species Ai and Ak . ˆb It is easy to see that the Liouvillian V A1 A2 satisfies the condition equivalent to preservation of “effective” particles in the process of exchange reactions dr1 dr2 ˆ b VA1 A2 T rA 1 A 2 2 dr1 dr2 ˆ b

≡ T rA 1 A 2 VA1 A2 (r1 , r2 |r1 , r2 ) = 0, 2 (2.35) ˆ b )(r1 , r2 |r , r ) (V A1 A2 1 2

denotes the kernel matrix of the where corresponding integral operators, and the operation T rA1 A2 should be understood in the sense stated above (after Eq. (2.30)) (see also Sec. IV). Further by integration of operators we shall always mean integration of their integral kernels. 4. Association processes

Processes of bimolecular association (irreversible bimolecular reaction of recombination or addition) are described by 3-center elementary rate Rζ |lm (r|r1 , r2 )17, 29 of the transition of a pair of reactants (residing at points r1 and r2 ) of species Al and Am of “effective” particle A to reactant (residing at point r) of the species Cζ of “effective” particle C. The form of this elementary rate satisfies the conditions of translation 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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where r —arbitrary displacement vector. Here it is convenient to introduce 2-center complete rate of association of a pair of reactants (residing at points r1 and r2 ) from species Al and Am into reactant of the species Cζ at any reactant position, (2.37) wζ |lm (r1 − r2 ) = dr Rζ |lm (r|r1 , r2 ). Using the approach given in the Appendix, one can show that association processes (as in Ref. 29) are described by two ˆ CP . The first Liouvillian acts in ˆ a ) and V Liouvillians: (V A1 A2 the space of the pair (A1 A2 ) on the vector TA1 A2 (r1 , r2 ) and corresponds to the decay of reactants of species Ai and Ak due to association. The second Liouvillian also acts in the coordinate space of the pair (A1 A2 ), but it corresponds to the appearance of reactant (at point r) of the species Cζ of “effective” particle C. These two Liouvillians are defined by the matrix elements ˆ aA A )ik|i k (r1 , r2 |r1 , r2 ) (V 1 2 = −δii δkk δ(r1 − r1 )δ(r2 − r2 )

Ak at any spatial position of the species is wik|ζ = dr1 dr2 Rik|ζ (r1 , r2 |r).

(2.42)

By virtue of translational symmetry (2.41), this rate does not depend on space coordinate of reactants of the species Cζ . With the approach given in the Appendix, one can show that dissociation processes (as in Ref. 29) are described by ˆ C . The first Liouvillian acts on ˆ PC and V two Liouvillians: V the coordinates of “effective” particle C and transforms the functions φζT L (r) to the functions in the coordinate space of the pair (A1 A2 ). The second Liouvillian transforms the functions φζT L (r) to the functions of the same type, i.e., acts in the coordinate space of the particle C. These Liouvillians are defined by the matrix elements ˆ PC )ik|ζ (r1 , r2 |r) = Rik|ζ (r1 , r2 |r), (V ˆ C )ζ |ζ (r|r ) = − 1 δζ ζ δ(r − r ) wik|ζ . (V 2 ik

(2.43)

wζ |i k (r1 − r2 ),

ζ

(2.38) ˆ CP )ζ |lm (r|r1 , r2 ) = 1 Rζ |lm (r|r1 , r2 ) . (V 2

(2.39)

â It follows from the definitions that the Liouvillian V A1 A2 is diagonal and describes the escape from the pair (A1 A2 ) caused by association. Due to the presence of Dirac δˆ CP functions, it actually is 2-center one. The Liouvillian V describes coming into “effective” particle C from the pair (A1 A2 ). It differs from similar Liouvillian in Ref. 29 by the factor 12 . This is because in the present work, we use symmetric basis for the pair (A1 A2 ). As is readily seen, the condition equivalent to the law of “effective” particles preservation in association processes takes place dr1 dr2 ˆ a ˆ CP = 0. (2.40) VA1 A2 + T rC dr V T rA 1 A 2 2 Here, as before, by operations T rA1 A2 and T rC , we mean the same as in the foregoing (after Eq. (2.30)) (see also Sec. IV). 5. Dissociation processes

Dissociation processes are described by 3-center elementary rate Rik|ζ (r1 , r2 |r) of the transition of reactant (residing at point r) of the species Cζ of “effective” particle C to a pair of reactants (residing at points r1 and r2 ) of species Ai and Ak of “effective” particles of the pair (A1 A2 ).17, 29 These rates satisfy 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 complete rate of dissociation of reactants of the species Cζ of the “effective” particle C into a pair of reactants of species Ai and

ˆ PC is It follows from this definition that the Liouvillian V nondiagonal and describes coming of reactants into the pair ˆ C is di(A1 A2 ) determined by dissociation. The Liouvillian V agonal and describes the escape of reactant from species Cζ as a result of dissociation. This Liouvillian differs from similar Liouvillian from Ref. 29 by the factor 12 . This difference is due to the use of symmetric basis for the pair (A1 A2 ) in this work. 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 dr1 dr2 ˆ ˆ C = 0. (2.44) T rA 1 A 2 VPC + T rC drV 2 Here, as before, by operations T rA1 A2 and T rC , we imply the same as in the above discussion (after Eq. (2.30)) (see also Sec. IV). 6. Liouvillian of free spatial motion

The Liouvillian of free motion is constructed on the basis of operators (in the general case, integral ones) of free motion in a pair of reactants of species Ai and Ak of a pair of “effective” particles (A1 A2 ) or reactant of the species Cζ of “effective” particle C,29, 49 Lˆ ik = Lˆ i + Lˆ k + Lˆ ik ; Lˆ ζ ,

(2.45)

k , L ζ —individual operators of free motion of where Lˆ i , L reactants of the corresponding species (in the specific case, these can be diffusion motion operator), and Lˆ ik —the operator describing the force interaction of these reactants in the pair A1 A2 . All operators of spatial migration preserve the number of reactants, thus they satisfy the relations dr Lˆ i = dr Lˆ k = dr Lˆ ζ = dr1 dr2 Lˆ ik = 0. (2.46) Matrices of integral kernels of Liouvillians of spatial migration Lˆ A1 A2 in the pair (A1 A2 ) and Lˆ C of the “effective”


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particle C are defined by the matrix elements

similar to Eq. (3.3), L L (r, r2 ); pA (r) = T rA2 dr2 TAA 2

(Lˆ A1 A2 )ik|i k (r1 , r2 |r1 , r2 ) =

δii δkk Lik (r1 , r2 |r1 , r2 )

Using these relations, in view of Eqs. (2.45)–(2.47), we can easily obtain the equations for the vectors of Laplace transforms of local probabilities from Eqs. (3.1) and (3.2),

= δii δkk (Li (r1 |r1 )δ(r2 − r2 ) +Lk (r2 |r2 )δ(r1 − r1 ) + Lik (r1 , r2 |r1 , r2 )), (Lˆ C )ζ |ζ (r|r ) = δζ ζ Lζ (r|r ),

pLC (r) = TC L (r).

(2.47)

i.e., are diagonal. Equation (2.46) yields the property analogous to Eqs. (2.30) and (2.35), dr1 dr2 ˆ dr1 dr2 ˆ LA1 A2 = 0, T rA1 A2 LA1 A2 = 0. 2 2 (2.48) Here, as before, by the operation T rA1 A2 we mean the same as in the foregoing (after Eq. (2.30)) (see also Sec. IV).

(s − Lˆ A )pLA (r)

ˆ A pLA (r) + T rA2 =Q

L ˆ AA + V ˆ bAA + V ˆ aAA )TAA dr2 (L 2 2 2 2

+ T rA 2

ˆ PC pLC + p0A (r), dr2 V

(3.4)

L ˆC +V ˆ C )pLC + V ˆ CP TAA (s − Lˆ C )pLC (r) = (Q + p0C (r). (3.5) 2

A. General discussion

Here p0A (r) and p0C (r)—initial values of local probabilities. However, these equations cannot be treated as kinetic, since L that are not observables. In order they involve functions TAA 2 to eliminate them, we turn to Eq. (3.1). Its formal solution is of the form

1. Liouville equations

L ˆ LA A V ˆ LA A TA0A . ˆ PC pLC + G TAA =G 2 1 2 1 2 1 2

III. KINETIC EQUATIONS OF MULTISTAGE GEMINATE REACTIONS

(3.6)

The starting point for the derivation of kinetic equations is the Liouville equations for distribution functions formulated on the Gibbs ensemble of “effective” particles.29 These equations are balance equations in the course of chemical reactions and free motion of reactants. Obviously, the Laplace transforms of the equations are of the form

Here we introduce the Laplace transform of integral-matrix propagator (resolvent or the Green function49 ) of the pair (A1 A2 ) of “effective” particles. It is defined by matrix integral kernel obeying the matrix equation

ˆ A1 A2 + V ˆ bA A + V ˆ aA A )TALA (s − Lˆ A1 A2 )TAL1 A2 = (Q 1 2 1 2 1 2

(3.7)

where EA1 A2 is a unit matrix (with elements EA1 A2 ik|i k = δii δkk ) of the pair (A1 A2 ). Substitution of Eq. (3.6) in Eqs. (3.4) and (3.5) gives the desired equations for Laplace transforms of local probabilities

ˆ PC TC L + TA0A , +V 1 2

(3.1)

ˆC +V ˆ C )TC L + V ˆ CP TALA + TC 0 . (s − Lˆ C )TC L = (Q 1 2 (3.2) They are matrix generalization of equations used in the consideration of elementary associative-dissociative reaction.42 The left-hand sides of these equations correspond to free motion of “effective” particles. The first terms in their right-hand sides describe chemical reactions in the pair (A1 A2 ) and reactant from C, respectively. The second terms correspond to coming of reactants either into the pair (A1 A2 ), or into the “effective” particle C caused by associative-dissociative processes. The last terms are the initial conditions for Liouville equations. 2. Kinetic equations for local probabilities

Equations for Laplace transforms of local probabilities are obtained from Eqs. (3.1) and (3.2) with allowance for the relation between these transforms and Laplace transforms of distribution functions following from Eq. (2.24), piL (r) = dr2 TikL (r, r2 ); pζL (r) = Tζ L (r). (3.3) k

These quantities are the components of column-vectors pLA (r) L (r, r2 ) and TC L (r) in a way and pLC (r) related to vectors TAA 2

ˆ A1 A2 − Q ˆ A1 A2 − V ˆ bA A − V ˆ aA A )GLA A (r1 , r2 |r1 , r2 ) (s − L 1 2 1 2 1 2 = δ(r1 − r1 )δ(r2 − r2 )EA1 A2 ,

(s − Lˆ A1 )pLA1 (r)

ˆ A1 pLA (r) + T rA2 =Q 1

ˆ bA A + V ˆ aA A dr2 Lˆ A1 A2 + V 1 2 1 2

ˆ LA A V ˆ PC pLC + T rA2 ˆ PC + V ×G 1 2

ˆ bA A dr2 Lˆ A1 A2 + V 1 2

L ˆ A A TA0A + p0A (r), ˆ aA A G +V 1 2 1 2 1 2 1

(3.8)

ˆ C )pLC (r) = Q ˆ C pLC (r) + (V ˆ CP G ˆ C )pLC ˆ LA A V ˆ PC + V (s − L 1 2 ˆ CP G ˆ LA A TA0A + p0C (r). +V 1 2 1 2

(3.9)

These equations are matrix generalization of kinetic equations of elementary geminate reaction42 and are linear inhomogeneous integral equations the kernels of which are expressed in terms of the solution of Eq. (3.7) for the Green function that describes the evolution (chemical and dynamic) of two “effective” particles. The left-hand side of the obtained kinetic equations describes the free evolution of local probabilities due to spatial motion of reactants. The first terms in the right-hand sides of Eqs. (3.8) and (3.9) describe monomolecular transformations. The second term in Eq. (3.8) is a “collision integral” which defines bimolecular and dissociation


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channels of geminate reaction. The second term in the righthand sides of Eq. (3.9) is a “collision integral” which defines dissociation channels of geminate reaction. The third terms in Eqs. (3.8) and (3.9) are inhomogeneous terms of kinetic equations, and, according to general principles of the kinetic theory,46 they describe the effect of initial correlations on the evolution of local probabilities (concentrations). The forth terms define the contribution from the initial conditions. 3. Kinetic equations for mean probabilities

Unlike bulk reactions, kinetic equations (3.8)–(3.9) valid in the general case of inhomogeneous systems can give, as in Ref. 42 in the consideration of elementary geminate reaction, a set of closed equations for mean probabilities (2.25) (or mean concentrations of species (2.26)). With this aim, we apply, in view of preservation laws, spatial averaging procedure to kinetic equations (3.8)–(3.9). So for Laplace transforms of column-vectors of mean probabilities, we have ˆ A1 p¯ LA + 2T rA2 Ld p¯ LC − 2T rA2 ILA A + p¯ 0A , s p¯ LA1 = Q 1 1 2 1 (3.10)

ˆ C p¯ LC − LC p¯ LC + ILC + p¯ 0C . s p¯ LC = Q

(3.11)

Here, Laplace transforms matrices of memory functions are introduced,

L dr1 dr2 ˆ b ˆ PC Â A V ˆ aA A G VA1 A2 + V Ld = 1 2 1 2 2

ˆ PC (r1 , r2 |r), +V (3.12) LC = −

ˆ CP G ˆ C (r|r0 ). ˆ LA A V ˆ PC + V dr V 1 2

(3.13)

Note that indication of coordinate dependence of operators (as in Eq. (2.35)) denotes their integral kernels. Besides, even if the coordinate dependence is not indicated explicitly, spatial integration of operators always implies integration of their kernels. Due to translational symmetry, Laplace transforms of memory functions (3.12) and (3.13) are independent of space coordinates. Laplace transforms of inhomogeneous sources appearing in Eqs. (3.10) and (3.11) are defined by the equalities dr1 dr2 ˆ b VA1 A2 ILA1 A2 = − lim υ→∞ 2υ υ

L ˆ A A TA0A (r1 , r2 ), ˆ aA A G +V 1 2 1 2 1 2

(3.14)

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 expressed in terms of relative coordinates of a pair of “effective” particles. For this purpose, it is convenient to introduce the projecˆ that transfers arbitrary finite coordinate function operator tion of two reactants (r1 , r2 ) to the function ψ(r) depending on their relative coordinates by the rule ˆ ψ(r) = (r1 , r2 ) = dr1 dr2 δ (r − (r1 − r2 )) (r1 , r2 ). (3.16) Obviously, the identity is valid drψ(r) = dr1 dr2 (r1 , r2 ).

(3.17)

For example, for elementary rates introduced above which serve as a basis for the construction of reaction Liouvillians, we have ˆ ik|i k (r1 , r2 |r1 , r2 ) rik|i k (r|r ) = R = dxRik|i k (r, 0|r + x, x), ˆ ζ |lm (r|r1 , r2 ) = rζ |lm (r ) = R ˆ ik|ζ (r1 , r2 |r) = rik|ζ (r ) = R

(3.18)

dxRζ |lm (0|r + x, x),

dxRik|ζ

(3.19)

r + x, x|0 .

(3.20) In relative coordinates elementary rates (3.19)–(3.20) depend solely on relative vectors r in the pair (A1 A2 ) and do not depend on the coordinate of the “effective” particle C by virtue of translation symmetry. Matrix equation (3.7) for Laplace transforms of integralmatrix propagator kernels of the pair (A1 A2 ) in relative coordinates takes the form

s − Lˆ A1 A2 − qˆ A1 A2 − υˆ bA1 A2 − υˆ aA1 A2 gLA1 A2 (r|r ) = δ(r − r )EA1 A2 .

(3.21)

Here quantities in relative coordinates are introduced ˆ A1 A2 )ik|i k (r1 , r2 |r1 , r2 ), ˆ L (Lˆ A1 A2 )ik|i k (r|r ) = (

ˆ A1 A2 )ik|i k (r1 , r2 |r1 , r2 ), (3.22) ˆ Q qˆ A1 A2 ik|i k (r|r ) = ( ˆ LA A )ik|i k (r1 , r2 |r1 , r2 ). ˆ G (ˆgLA1 A2 )ik|i k (r|r ) = ( 1 2 Kernels of bimolecular reaction Liouvillians (2.33) and (2.38) in relative coordinates are b

υˆ A1 A2 ik|i k (r|r ) = −δii δkk δ(r − r ) wlm|i k (r )θlm|i k lm

ILC

= lim

υ→∞

dr ˆ ˆ L VCP GA1 A2 TA01 A2 (r). υ

+ θik|i k rik i k (r|r ),

(3.23)

(3.15)

υ

Often, time originals of memory functions13, 14 are generalized time functions.47

a

υˆ A1 A2 ik|i k (r|r ) = −δii δkk δ(r − r ) wζ |i k (r ). ζ

(3.24)


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2. Kinetic coefficients

3. Preservation laws

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

dr b L υˆ A1 A2 + υˆ aA1 A2 gˆ LA1 A2 A1 A2 (r ) . IA 1 A 2 = − 2 (3.25) Here we introduce a column-vector of initial distribution functions in the pair (A1 A2 ) in relative coordinates dr1 dr2 φ A1 A2 (r) = lim δ (r − (r1 − r2 )) TA01 A2 (r1 , r2 ) υ→∞ υ

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

1 T rA p¯ LA + T rC pLC s 2

υ

≡ lim

υ→∞

1 ˆ υ TA01 A2 (r1 , r2 ). υ

(3.26)

More complicated law of the function TA01 A2 (r1 , r2 ) projection on relative coordinate space is related to the fact that initial distribution does not fall into a class of finite functions. The norm of the component φ ik (r) of the column-vector φA1 A2 (r) is the probability PikT 0 of finding the pair (A1 A2 ) in the state |Ai Ak at the initial moment of time (compare with Eq. (2.13)) dr T0 Pik = φik (r). (3.27) 2 The memory function Ld is considered by analogy. As a result, we arrive at the representation

dr dr0 b Ld = − υˆ A1 A2 + υˆ aA1 A2 gˆ LA1 A2 υˆ d + υˆ d (r|r0 ). 2 (3.28) ˆ PC ) is introduced Here dissociation operator υˆ d (analog of V which is expressed is terms of its matrix elements as follows: (υˆ d )ik|ζ (r|r0 ) = −δ (r − r0 ) rik|ζ (r).

(3.29)

Equation (3.20) shows that it is directly related to elementary dissociation rate. Note that, as in Ref. 42, in defining this operator 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 υˆ a gˆ LA1 A2 υˆ d + υˆ 0 (r|r0 ). (3.30) By analogy with dissociation operator, here we introduce asˆ CP ) defined in terms of its sociation operator υˆ a (analog of V matrix elements (υˆ a )ζ |lm (r|r0 ) = −δ (r − r0 ) rζ |lm (r)/2.

(3.31)

It is directly related to elementary association rate. The operˆ C ) has only diagonal matrix elements the ator υˆ 0 (analog of V value of which is specified by complete rate of dissociation into pairs with fixed r 1 (υˆ 0 )ζ |ζ (r|r0 ) = − δζ ζ δ (r − r0 ) rik|ζ (r). (3.32) 2 ik Now it is easy to represent the Laplace transform ILC in terms of relative coordinate space quantities ILC = − dr υˆ a gˆ LA1 A2 φ A1 A2 (r) . (3.33)

=

1 1 T rA p¯ 0A + T rC p0C , T rA p¯ A (t) + T rC p¯ C (t) = const. 2 2 (3.34)

Of interest is the formulation of Liouvillian properties in relative coordinate space which is easy to do using definitions of operators in relative coordinates. For monomolecular processes, we have T rA1 A2 qˆ A1 A2 = T rC qC = 0. For bimolecular processes T rA1 A2 dr υˆ bA1 A2 = 0. For association processes 1 a T rA1 A2 dr υˆ A1 A2 − T rC dr υˆ a = 0. 2 For dissociation processes 1 T rA1 A2 dr υˆ d − T rC dr υˆ 0 = 0. 2

(3.35)

(3.36)

(3.37)

(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 encountered important specific case. Along with homogeneity of a solvent leading to the existence of translation symmetry for elementary event rates (2.31), (2.36), and (2.41), spatially homogeneous reacting systems are assumed to have additional homogeneity, i.e., independence of coordinates of all local initial distributions (local concentrations of all species). In this case, at any instant of time all measurable local physical 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 p¯ i (t) = pi (r, t);

p¯ ζ (t) = pζ (r, t);

Tik (r1 , r2 , t) = Tik (r1 − r2 , 0, t);

p¯ ζ (t) = Tζ (t). (3.40)

In particular, initial distribution vector component in relative coordinates of the pair (A1 A2 ) (see Eq. (3.26)) is φik (r) = Tik0 (r1 − r2 , 0, 0);

(r = r1 − r2 ) .

(3.41)


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So the kinetic equations for mean column-vector of probabilities coincide with the equations for local probabilities ˆ A1 pLA + 2T rA2 Ld pLC − 2T rA2 ILA A + p0A , spLA1 = Q 1 1 2 1 ˆ C pLC − LC pLC + ILC + p0C . s pLC = Q

(3.42)

With the formalism of generalized time functions47 (i.e., functions on extended time interval (−∞ < t < ∞)), one can recover the originals in Eq. (3.42). As a result, the kinetic equations of multistage geminate reaction on normal (not extended time interval t ≥ 0) take the form d ˆ A1 pA1 (t) + 2T rA2 pA (t) = Q dt 1

t d (t − t0 ) pC (t0 ) dt0 −0

− 2T rA2 IA1 A2 (t), d ˆ C pC − pC (t) = Q dt

replace the space coordinate r of reactant by the coordinate of its configuration space q = {, r}. Of course, all reaction Liouvillians and Liouvillians of free molecular motion must take account of such a dependence. In particular, the Liouvillian of molecular motion must involve additional operators (in the general case, integral) that describe reactant rotation. Consideration of discrete classical “internal” variables in the accepted concept of “effective” particles just means that by the “state” of the “effective” particle we now imply not the entire species but only part of it corresponding to the given value of discrete classical variable in reactants of this species. Total number (concentration or probability of finding) of the given species in solution is defined as a sum over these classical variables.

(3.43)

t C (t − t0 ) pC (t0 ) dt0 + IC (t) −0

(3.44) with the initial conditions pA1 (0) = p0A1 and pC (t) = p0C . Note that in the integrals over time in Eqs. (3.43) and (3.44) the lower integration limit is taken equal to −0, i.e., to the value tending to zero from the side of negative values. The reason is that kinetic equation kernels have a singularity at zero values of the argument which must enter the integration domain. This is commonly not a problem in practical calculations of integral part of equations even by numerical methods. Indeed, according to Ref. 15, the kinetic equation kernels may be represented as a sum of kinetic and relaxing parts. It is the first part that contains δ-shaped time singularity that is eliminated by time integration assumed in the equations. Relaxing part involves no time singularities, and the lower integration limit is put zero. However, in analytical investigation of general asymptotic properties, mathematical representation of the kernel on extended time axis seems preferable. IV. CONSIDERATION OF “INTERNAL” DEGREES OF FREEDOM A. Classical degrees of freedom

When deriving kinetic equations, for simplicity, we restricted ourselves to the examination of the dependence of physical quantities solely on space coordinates r. However, most of real reacting systems commonly contain additional degrees of freedom which we call “internal.” For example, in studies of reactions between reactants with anisotropic reactivity, one should take into account the dependence on Euler angles of orientation of molecular axis of reactants that change due to reactant rotation. Apart from such continuous variables, discrete (classical or quantum) degrees of freedom can also exist. As we use integral-matrix formalism, continuous variables will be treated as arguments of integral kernels and physical characteristics, while discrete variables—as variables defining matrix elements of the corresponding integral kernels and components of appropriate column-vectors. So, for example, in order to allow for orientation angles of the chosen molecular axis of reactants, it is necessary to

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) must be quasi-resonance, i.e., they must cover narrow energy spectrum of the width much less than kT (k Boltzmann constant, T absolute temperature). In this case, internal quantum transitions do not affect the character of molecular motion of reactants of the given species.28, 50 Further all quantum numbers of states of reactants of any, for example, ith species of the “effective” particle A or ζ th species of the “effective” particle C will be denoted by Greek letters γ i , εi , μi , ν i or γ ζ , εζ , μζ , ν ζ . Any quantum number of the set γ , ε, μ, ν (for reactant of any given species defined by the lower index for these numbers) includes a set of all quantum numbers and covers the entire, one and the same (permissible for reactants of the given species) series of values. The use of four designations of one and the same set of quantum numbers is determined by the fact that, in contrast to discrete classical degrees of freedom, quantum “internal” states cannot be simply included in the definition of “effective” particle state. The reason is that quantum evolution of states of any reactant, for example, the reactant of the ith species is defined by the density matrix ργi εi . In the description of the system in the Liouville space accepted by the authors, its components are the components of columnvector that contain both diagonal (populations), and nondiagonal (phase) elements. Thus the states, for example, of “effective” particle A with allowance for quantum states are Aiγi εi . The corresponding distribution functions (components of column-vector of states i γi εi , kμk ,νk of “effective” particle A), in principle, depend on spin states. Matrix elements of reaction Liouvillians also depend on spin states, and this ensures, if necessary, spin selection rules (i.e., the course of reaction only from a definite collective spin state of partners). ˆ C of individual “effective” ˆ A and Q Now Liouvillians Q particles must be represented as ˆ A + i Â =R ˆ A + ˆ A , Q

ˆC =R ˆ C + i ˆ C + ˆ C , Q

(4.1)

ˆ A and R ˆ C take into account monomolecwhere Liouvillians R ular transformation but with allowance for spin states. Its matrix elements are defined by analogy with Eqs. (2.28) and


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(2.29) with configuration space coordinates instead of space coordinates, and any state i of “effective” particles is replaced by the corresponding states iγ i εi with allowance for ˆ C describe dynamic evoˆ A and spin variables. Liouvillians lution of internal quantum states of reactants of the given species and corresponds to internal quantum Hamiltonian Hˆ Ai of this species reactant. Such a Hamiltonian can include, for instance, the interaction with external magnetic field (or anisotropic splitting in zero field). In this case, it can depend on orientation angles of molecular axis with respect to external field direction (or spin state quantization axis). Matrix elements of this Hamiltonian are written in the basis of interˆ C , obviˆ A and nal quantum states of reactant. Liouvillians ously, have only diagonal matrix elements in species indices

dition, motion operators in this Liouvillian depend solely on the species indices, not on internal quantum states. Inclusion of internal quantum degrees of freedom into consideration affects the operation Tr. As before, it assumes summation over all components of column-vector referring to different species. However, on summation over quantum states of any of species, it applies only to the components corresponding to diagonal elements of density matrix, i.e., for instance (see Eq. (2.22)), niγi γi (q, t), T rA nA (q, t) = iγi

T rC nC (q, t) =

(4.5) nζ γζ γζ (q, t).

ζ γζ

ˆ A )iμi νi , (

iγi εi

ˆ C ) ζ μ i νi , (

ζ γi εi

A

Hεi i, νi δγi ,μi − HμAii ,γi δνi , εi , C

C = 1 Hεζζνζ δγζ μζ − Hμζζ γζ δνζ εζ .

=

1

(4.2)

Liouvillians ˆ A and ˆ C describe internal quantum state relaxˆ C , they have only diagoˆ A and ation. Just as Liouvillians nal matrix elements in species indices and satisfy the required selection rules for internal relaxation operators.51 In view of internal quantum states, bimolecular Liouvilˆb lian V A1 A2 should be represented as ˆ bA A = R ˆ bA A + i V ˆ in V A1 A2 , 1 2 1 2

(4.3)

ˆb where the Liouvillian R A1 A2 takes into consideration exchange bimolecular reaction but with allowance for spin states. Its matrix elements are defined by analogy with Eq. (2.33) with configuration space coordinates instead of space coordinates, and any state i of “effective” particles is replaced by the corresponding states iγ i εi with allowance ˆ in for spin states. The Liouvillian V A1 A2 describes dynamic evolution of internal quasi-resonance quantum states of two quantum-interacting reactants of the given species (for example, Ai and Ak ) and corresponds to internal quantum interaction Hamiltonian Vˆ Ai Ak of these species reactants. Matrix elements of this Hamiltonian are written in the basis of collective internal quantum states of two reactants. Evidently, the ˆ in has only diagonal matrix elements in a pair Liouvillian V A1 A2 indices ik of the corresponding species, and is as follows: in

Â A V 1 2 iμi vi , kαk βk ; iγi εi ,kλk ξk =

1 Ai Ak k Vεi ξk ,vi βk δγi λk ,μi αk ; − VμAi iαAk ,γ . δ i λk vi βk ,εi ξk

(4.4)

As for other reaction Liouvillians, their matrix elements are defined by analogy by previous formulae replacing space coordinates by configuration space ones, and any state i or ζ of “effective” particles A or C by the corresponding states i γ i εi or ζ γ ζ εζ with allowance for spin variables. It is stated above that Liouvillians of molecular motion must involve operators of additional motions (for example, rotation) in the configuration space. As to taking account of spin states, they are introduced by replacing states i or ζ of “effective” particles A or C by the corresponding states i γ i εi or ζ γ ζ εζ with allowance for spin variables. By virtue of quasi-resonance con-

As earlier, by the operation Tr of Liouvillians we mean Tr of column-vector obtained as a result of the action of these Liouvillians on any initial column-vector (not at all the summation of diagonal elements of Liouvillian matrices). Thus, for example, ˆ A )iγi γi |kμk νk ˆ A ≡ (T rA Q ˆ A )kμk νk = (Q (4.6) T rA Q iγi

is actually the component k μk ν k of some row-vector. That is why, for instance, going to zero of quantities in Eqs. (2.30), (2.35), (2.40), (2.44), and (2.48) means that all components of such 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 the case of the presence of internal quantum states, requires that, first, Tr of additional dynamic Liouvillians go to zero ˆ A = 0, T rA

ˆ C = 0, T rC

ˆ in T rA 1 A 2 V A1 A2 = 0.

T rA ˆ A = 0,

T rC ˆ C = 0, (4.7)

As is known (and is easily seen from Eqs. (4.2) and (4.4)), this actually takes place. Second, it is required that relations of the type of Eqs. (2.30), (2.35), (2.40), and (2.44) hold for reaction operators with allowance for spin states. This imposes certain restrictions on the properties of Liouvillian matrix elements (selection rules)28 that can be established only in considering particular reaction systems. V. SUMMARY

Based on the concept of “effective” particles developed by the authors earlier, matrix kinetic equations of multistage geminate reactions of isolated pairs of reactants have been derived. Elementary stages of multistage reactions considered involve all types of bimolecular and monomolecular physicochemical processes possible in isolated pairs including changes in internal classical and quasi-resonance quantum degrees of freedom. For example, taking into account internal spin states allows one to examine reactions with the participation of paramagnetic particles (radicals and radical ions), and in external magnetic fields as well. To consistently develop the theory of multistage geminate reactions, the problem of many-particle consideration of a reacting system has been reduced to the equivalent investigation of the Gibbs


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ensemble of “effective” particles. Unlike most of approaches used in the literature in studies of elementary geminate reactions of isolated pairs, the problem is formulated in terms of kinetic theory. This means that instead of survival probabilities of reacting pairs, we introduce definitions of mean and local probabilities of finding species in a sample (or mean and local concentrations) at any moment of time. For general matrix formulation of the results, column-vectors of the observed macroscopic kinetic quantities are used. On the basis of elementary event rates describing the course of different types of physicochemical processes and molecular motion of reactions, integral-matrix Liouvillians have been constructed. Kinetic coefficients (integral kernels and inhomogeneous sources) of the obtained kinetic equations for mean kinetic characteristics are expressed in terms of quantities in relative coordinates of geminate pairs. An important widely encountered case of homogeneous reaction systems is considered. ACKNOWLEDGMENTS

The authors are grateful to the Russian Foundation of Basic Research for financial support (Project No. 12-03-00058). APPENDIX: PROPERTIES OF LIOUVILLIANS

To study general properties of Liouvillians in the concept of “effective” particles, first consider irreversible bimolecular geminate reaction A + B → C + D defined by 4-center elementary rate RCD|AB (r1 , r2 |r1 , r2 ) of the transition (per unit time) of a pair of reactants (residing at points r1 and r1 , respectively) of species A and B to a pair of reactants (residing at points r1 and r2 , respectively) of species C and D. The formalism of the Fock boxes was used by the authors earlier both to describe different (bulk and geminate) bimolecular reactions (exchange and association), and dissociation reactions.23, 25, 42, 52 For the description of geminate reaction A + B → C + D, two Fock boxes are necessary which correspond to possible states of the reaction system (elementary outcomes): the first box contains a pair of reactants (AB), the second one—a pair of reactants (CD). Statistical description of the first box (in the thermodynamic limit) is made, for example, by the probability density T (r1 , r2 , t) of finding the reactant of the species A at point FAB r1 and reactant of the species B at point r2 at the instant of T (t) of finding the reaction system time t. The probability PAB at the moment of time t in the box (AB) is dr1 dr2 T T PAB (t) = lim (A1) FAB (r1 , r2 , t). υ→∞ υ υ T (r1 , r2 , t) involves full statistical Apparently, the function FAB information concerning the reaction pair kinetics in the first box. However, there is another description of the state of reacT (r1 , r2 , t) tants in the first box by the probability density FBA to find the reactant of the species A at point r2 , and the reactant of the species B at point r1 . This function is expressed in terms of the previously introduced function in an

J. Chem. Phys. 140, 184104 (2014)

obvious way T T (r1 , r2 , t) = FAB (r2 , r1 , t). FBA

(A2)

That is why it also contains full statistical information concerning the pair kinetics in the first box. Though formally both descriptions are different, in fact, they are equivalent and any of them can be chosen. Everything written above about statistical description of the first box also refers to the second one. So geminate reaction A + B → C + D may be described by four formally different sets of densities: T T T T T T i FCD ; (2) FBA i FCD ; (3) FAB i FDC ; (1) FAB T T (4) FBA i FDC .

(A3)

All these description methods are equivalent in essence; however, they can lead to mathematically different forms of chemical interaction Liouvillians in “effective” particle formalism. Description of reaction. Let us describe the state of the reaction system by the first density set from Eq. (A3). Chemical reaction induces their time variation due to transitions between the Fock boxes. Explicit form of such transitions is defined by the equations representing balance conditions: d T F (r1 , r2 , t) dt AB T = − dr1 dr2 RCD|AB (r1 , r2 |r1 , r2 )FAB (r1 , r2 , t), (A4) d T F (r1 , r2 , t) dt CD T = RCD|AB (r1 , r2 |r1 , r2 )dr1 dr2 FAB (r1 , r2 , t). (A5) In concise form these equations may be written in terms of integral operators Wˆ CD|AB and Rˆ CD|AB acting on functions of space coordinates. The kernels of these operators take the form (Wˆ CD|AB )(r1 , r2 |r1 , r2 ) = δ(r1 − r1 )δ(r2 − r2 )wCD|AB (r1 − r2 ),

(A6)

where wCD|AB (r1 − r2 ) = dr1 dr2 RCD|AB (r1 , r2 |r1 , r2 ), (Rˆ CD|AB )(r1 , r2 |r1 , r2 ) = RCD|AB (r1 , r2 |r1 , r2 ).

(A7)

As a result, Eqs. (A4) and (A5) are as follows: d T T , (A8) F (r1 , r2 , t) = −Wˆ CD|AB FAB dt AB d T T F (r1 , r2 , t) = Rˆ CD AB FAB . (A9) dt CD Equations (A4) and (A5) (or Eqs. (A8) and (A9)) will serve as a basis for the construction of Liouvillian in effective particle formalism. “Effective” particle formalism for the reaction A + B → C + D begins with statistical description of possible states, for example, initial reactants. These states are the components of


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J. Chem. Phys. 140, 184104 (2014)

column-vectors corresponding to all possible combinations of pairs of reactants of two species determined by two identical particles A1 and A2 that are at spatial points r1 and r2 , respectively. For the reaction at hand, 16 different elementary outcomes are obtained the description of which calls for 16 Tik (r1 , r2 , t). However, chemical reaction induces time variation of four functions only TAB (r1 , r2 , t);

TCD (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 descriptions in two formalisms shows that the description of the reaction in “effective” particle formalism requires that twice as many elementary outcomes be taken into account. This is because, for instance, species A in Fock formalism is marked only by space coordinate value. In “effective” particle formalism, indication of belonging to one of two “effective” particles is added to this species mark. That is why elementary outcome space of “effective” particle formalism without restrictive properties of matrix elements of matrix-integral reaction Liouvillian is too detailed. Nevertheless, any observables can be described in the framework of both formalisms. For example, the probability density to find at points r1 and r2 a pair of initial reactants (AB) in the Fock formalism is T T dPTAB (r1 , r2 , t) = FAB (r1 , r2 , t) + FAB (r2 , r1 , t) dr1 dr2 (A11) and in “effective” particle formalism dPTAB (r1 , r2 , t) = TAB (r1 , r2 , t) + TBA (r1 , r2 , t) dr1 dr2 . (A12) T (t) to find the reaction system in the So the probability PAB state of the pair (AB) is dPTAB (r1 , r2 , t) T . (A13) (t) = lim PAB υ→∞ 2υ υ

Here integration is performed over all different pairs of points r1 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 may be identified TAB (r1 , r2 , t)

=

T FAB (r1 , r2 , t);

(A14) T (r2 , r1 , t). TBA (r1 , r2 , t) = FAB

elements describing the reaction A + B → C + D are ⎛ T ⎞ ⎛ T ⎞ AB AB ⎜ T ⎟ ⎜ T ⎟ ⎟ ⎜ CD ⎟ d ⎜ ⎟ ⎜ T ⎟ (r1 , r2 , t). ⎜ CD ˆb T ⎟ (r1 , r2 , t) = VA1 A2 ⎜ ⎟ ⎜ dt ⎝ BA ⎠ ⎝ BA ⎠ T φDC TDC (A16) ˆb takes the form Here integral-matrix operator V A1 A2 ⎛ ⎞ −Wˆ CD|AB 0 0 ⎜ ˆ ⎟ ⎟. ˆ bA A = ⎜ RCD|AB 0 V (A17) 1 2 ⎝ −Wˆ DC|BA 0⎠ 0 Rˆ DC|BA 0 It is the desired interaction Liouvillian of a pair of reactants from “effective” particle species. Identity of “effective” particles. Liouvillian (A17) must be tested for invariance under permutation of “effective” particles. For this purpose, first, construct the operator Pˆ 12 of “effective” particle permutation. New probability densi˜ Tik (r1 , r2 , t) with permuted “effective” particles are exties pressed in terms of previous (before permutation) probability densities as follows: ˜ TAB (r1 , r2 , t) = TBA (r2 , r1 , t); ˜ TCD (r1 , r2 , t) = TDC (r2 , r1 , t) (A18) ˜ TBA (r1 , r2 , t) = TAB (r2 , r1 , t); ˜ TDC (r1 , r2 , t) = TCD (r2 , r1 , t). These equalities may be represented in a matrix form ⎛ T ⎞ ⎛ T ⎞ ˜ AB AB ⎜ ˜T ⎟ ⎜ T ⎟ ⎜ CD ⎟ ⎜ ⎟ ⎜ T ⎟ (r1 , r2 , t) = Pˆ 12 ⎜ CD ⎟ ⎜ ⎜ T ⎟ (r1 , r2 , t). (A19) ˜ BA ⎟ ⎝ ⎠ ⎝ BA ⎠ ˜ TDC TDC Here the permutation operator is defined by the expression

0 E2 ˆ ℘ , (A20) P12 = E2 0 where E2 —unit matrix of dimensionality 2 × 2, ℘— ˆ permutation operator of points r1 and r2 . The Liouvillian b ˆ A A acting on new probability densities is expressed in V 1 2 terms of the previous Liouvillian (A18) in a standard way b Â V

1 A2

ˆ bA A Pˆ 12 . = Pˆ 12 V 1 2

In explicit calculations, symmetry conditions for elementary rate should be taken into consideration, RCD|AB (r1 , r2 |r1 , r2 ) = RDC|BA (r2 , r1 |r2 , r1 ),

Similarly, for the pair CD

(A21)

(A22)

which is physically evident. As a result, we have

T TCD (r1 , r2 , t) = FCD (r1 , r2 , t);

(A15) T (r2 , r1 , t). TDC (r1 , r2 , t) = FCD

Description of reaction. Equations (A14) and (A15) easily give the binary interaction Liouvillian in “effective” particle formalism with the aid of Eqs. (A8) and (A9). Matrix

b Â A = V ˆ bA A . V 1 2 1 2

(A23)

Thus the constructed Liouvillian (A17) is invariant under effective particle permutation. Unambiguity of Liouvillian construction. Liouvillian (A17) has been constructed for the Fock fist formalism. The


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J. Chem. Phys. 140, 184104 (2014)

choice of the fourth set is easily seen to lead to the same Liouvillian. However, the choice of the second or the third set from complete set (A3) gives the binary interaction Liouvillian of the form ⎞ ⎛ −Wˆ DC|AB 0 0 0 ⎜ 0 0 Rˆ CD|BA 0⎟ ⎟. ˆ bA A = ⎜ (A24) V 1 2 ⎝ 0 0 −Wˆ CD|BA 0⎠ Rˆ DC|AB 0 0 0 Since for elementary rate the following condition is valid: RDC|AB (r1 , r2 |r1 , r2 ) = RCD|BA (r2 , r1 |r2 , r1 ),

(A25)

therefore, Liouvillian (A24) is also invariant under “effective” particle permutation. Consideration of Liouvillians as linear combination of Liouvillians (A17) and (A24) is physically meaningless, because it does not correspond to the notions of 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 different states (i = k = i = k ) matrix elements of bimolecular Liouvillian are specified as b

Â A (r1 , r2 |r1 , r2 ) = 0, (A26) V 1 2 ik|i k then by definition b

Â A V (r1 , r2 |r1 , r2 ) = 0. 1 2 ik|k i

(A27)

Geminate reactions of other types (associativedissociative and monomolecular) are considered by analogy. Unlike the case of exchange bimolecular reactions described above, the constructed Liouvillians have a unique form. Thus no additional conditions of the choice of (A27) type arise.

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