# Contact and Distant Luminescence Quenching in Solutions

Authors: Anatoly I. Burshtein
Journal: Advances in Physical Chemistry
 (2008)

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## Abstract

The limitations and advantages of modern encounter theories of remote transfer are discussed, as well as their application to particular transfer reactions assisted by encounter diffusion. Comparison is made with contact multiparticle theories, Brownian dynamic simulations, and the actual experimental data requiring a distant description of energy and/or electron transfer.


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## Body

## 1. Introduction

Non-Markovian chemical kinetics started a century ago
based on the famous work of Smoluchowski [1] who studied the encounter diffusion of spherical
particles reacting at any contact. The model implied that the reaction is
accomplished as soon as the intersphere distance r reduces to the size of their diameter, σ. A typical example of such a reaction is the irreversible quenching of excited
molecule A* by the impurity B, stopping abruptly the luminescence of A*:(1)The time-dependent reaction constant established by
Smoluchowski,(2)kA(t)=4πσD[1+σ2πDt],
makes the differential kinetic equation non-Markovian:
(3)N˙A*=−kA(t)cNA*−NA*τA. It becomes the conventional
Markovian equation of chemical kinetics only over long times when kA(t) approaches the constant (stationary) value limt→∞kA(t)=4πσD≡kD. This is the diffusional rate constant linear
in encounter diffusion coefficient D.In reality, not any contact of reactants results in
complete quenching ofA*.
Being less efficient, such contacts are characterized by a phenomenological
parameter ka,
known as the kinetic rate constant. It was first introduced in the extended
contact theory of Collins and Kimball [2] wherein the Markovian (stationary) limit of kA(t) is(4)k=limt→∞kA(t)=kakDka+kD.
However, such a generalization
did not overcome the contact nature of the theory that could be scarcely
applied to the distant electron transfer reactions and even more so to the long
distant energy transfer.Only an original differential encounter theory (DET)
elaborated in the late 1960s overcame the contact limitations of the
Smoluchowski-Collins-Kimball model [3–5]. Instead of the kinetic rate constant ka,
it incorporates into the theory the noncontact, space-dependent rate WA(r).
The latter determines ka as well as the whole non-Markovian kinetics of
quenching which becomes exponential (Markovian) at later time. The stationary
quenching constant k=4πRQD and its effective quenching radius RQ are also determined by WA(r) and D.
DET had a tremendous success in the treatment of luminescence quenching by
irreversible energy or electron transfer [6]. Assuming an exponential shape of W(r) [7], the theory attempted to fit the real viscosity
dependence of the quenching radius RQ(D) [8].Although DET covers only the irreversible transfer, it
is known to be exact in the particular case of the target problem (immobileA* quenched by independently moving point B's) [9]. In the simplest case the time-dependent reaction
constant of DET substituted for its contact analog (2) has the following
definition:(5)kA(t)=∫WA(r)ν(r,t)et/τAd3r.Here τA is the excitation life time and ν(r,t) is the pair correlation function of the
reactants, obeying the equation for encounter diffusion accompanied by a
distant reaction with the rate WA(r):(6)ν˙=−(1τA+WA)ν+DΔν.At later times, the theory
becomes Markovian, that is, kA(t)→k=4πRQD which is the conventional (time independent)
rate constant.At the same time, DET failed to describe the
reversible reaction between the meta-stable reactants,A* and B*.
When their lifetimes are different (τA≠τB), one of the DET “rate constants”
diverges. To avoid such a divergency, the alternative integral encounter theory
(IET) was developed [10–12] and applied to the reaction of quasiresonance energy
transfer [13–15]. Since IET is a kind of
memory function formalism, the time-dependent rate constants of DET give way to
the memory functions (the kernels of the integral terms) of IET equations not
ever turning to the Markovian ones even at later times. On the other hand, the
stationary reaction characteristics such as the fluorescence yield or the
yields of reaction products are even easier to
calculate with IET. The Laplace transformation turns  the integral equations into algebraic ones which can be
solved analytically. However, IET was known from the very beginning [13–16] as a theory for the lowest concentration of
reactants, NA and NB. Some of the multiparticle effects accounted for by DET are lost in IET.To minimize this drawback, a modified version of the
integral theory (MET) was developed [17–23]. Creation of MET extended IET to higher
concentrations keeping its advantages in accounting for the reaction
reversibility but eliminating some of the weaknesses demerits in the asymptotic
description of the long time kinetics. The contemporary matrix versions of
IET [14, 15] and MET [24–26] allow considering reactions
of any complexity and arbitrary number of internal electronic states of the
reactants and products including their spin states.In parallel, great progress was made in extending the
contact reaction model to higher concentrations of reacting spheres. A number
of the multiparticle Kernel theories, MPK1, MPK2, MPK3,
were developed [27–29] as well as the superposition approximation [30–33] (SA) and some others.
More recently, the relaxation time approximation
(RTA) was developed which revives the archaic Smoluchowski approach by
accounting for triple encounters: the pair AB with one more particle B
(“bachelor” intervening in the family business). Although original,
the self-consistent RTA (SCRTA) [34] does not add very much to more fundamental
multiparticle theories in describing the high concentration effects. Even less
warranted are the latest attempts to find better phenomenological constants for
RTA, verifying the choice by how it agrees with Brownian dynamic (BD)
simulations of reaction kinetics [35].The principle difference between encounter theories
and the inferior Smoluchowski-like approach is in their ability to account for
the internal states of the reactants (including spin-multiplicity of the pairs)
and the distant dependence of the transfer rates or corresponding Hamiltonian.
To emphasize this point this review has the following structure. A brief
overview of the integral theory of intra- and intermolecular transfer and later
IET modifications is given in Section2. Also the long time reaction kinetics
of contact reactions studied with IET, MET and competing multiparticle theories
is compared there with the latest BD simulations. Then we consider the quantum
yields of the luminescence, reversible exciplex formation and
electroluminescence (Section 3), focusing attention on their concentration
dependence and difference between δ-pulse and stationary excitation. Finally, we
turn to the spin assisted reactions (Section 4) and the remote electron
transfer (Section 5) using the integral and original unified theory (UT) for
fitting the actual experimental data collected in these sections.


## 2. Integral Encounter Theories

Among numerous
applications of the integral theory, the reversible reactions of intermolecular
or intramolecular energy transfer are the simplest ones. To demonstrate clearly
the main features of the integral (memory function) formalism it is better to
start from them.


### 2.1. Intermolecular Energy Transfer

The transfer
first studied with IET had the following reaction scheme [13]:
(7)
To make the problem even easier
let us assume that the nonexcited partners are present in great excess ([A]=NA≫NA*=[A*] and [B]=NB≫NB*=[B*]). Then the terms in the kinetic equations
which are second order in the excited state concentrations, NA*(t) and NB*(t),
are negligible and the set of two IET equations can be linearized in the
particle densities [15]:(8)N˙A*=NA∫0t〈S(t−τ)〉NB*(τ)dτ−NB∫0t〈R(t−τ)〉NA*(τ)dτ−NA*τA,N˙B*=−NA∫0t〈S(t−τ)〉NB*(τ)dτ+NB∫0t〈R(t−τ)〉NA*(τ)dτ−NB*τB.This set should be solved with
the initial conditions(9)NA*(0)=N0,NB*(0)=0,provided that A* was created by δ-pulse excitation at t=0.In any memory function formalism the main problem is
how to define the kernels “memory functions” of integral equations.
The great advantage of IET is the absence of such a problem. The space-averaged
kernels of the integral equations (8) are defined from first principles by
their Laplace transformations:(10)〈S˜(s)〉=(s+1τB)∫[WB(r)μ˜3(r,s)−WA(r)ν˜3(r,s)]d3r,〈R˜(s)〉=(s+1τA)∫[WA(r)ν˜2(r,s)−WB(r)μ˜2(r,s)]d3r.The auxiliary equations for the
dyads of pair distributions ν3,μ3,
and ν2,μ2 are the same:(11)ν˙=−(1τA+WA)ν+WBμ+DΔν,μ˙=−(1τB+WB)μ+WAν+DΔμ.Only the initial conditions for
them are different:(12)ν2(r,0)=1,μ2(r,0)=0;ν3(r,0)=0,μ3(r,0)=1.The global structure of IET
formalism is similar to that of DET. Although the kinetic equations are
integral, their kernels are defined via the same transfer rates, WA(r) and WB(r),
and the pair distribution functions obey the auxiliary equations similar to
that of DET.It is very easy to solve (11) in the contact
approximation when(13)WA(r)=kaδ(r−σ)4πσ2,WB(r)=kbδ(r−σ)4πσ2.In highly polar solvents,(14)ka=∫WA(r)d3r,kb=∫WB(r)d3rare the kinetic rate constants
of the forward and backward transfer. They relate both the
kernels(15)〈R˜(s)〉=kaΣ^(s),〈S˜(s)〉=kbΣ^(s),to a single
factor(16)Σ˜(s)=[1+ka/kD1+(s+1/τA)τd+kb/kD1+(s+1/τB)τd]−1=1F(s),where τd=σ2/D is the encounter time. With these kernels, the
general IET equations (8) take the particular form specific only to the contact
reactions [15]:(17)N˙A*=kbNA∫0tΣ(t−τ)NB*(τ)dτ−kaNB∫0tΣ(t−τ)NA*(τ)dτ−NA*τA,N˙B*=−kbNA∫0tΣ(t−τ)NB*(τ)dτ+kaNB∫0tΣ(t−τ)NA*(τ)dτ−NB*τB.At a great excess of theB molecule, one may neglect the reencounters of B* with A in the bulk. In this particular case, which
attracted the most attention, there is actually only a single A surrounded by Bs. In such a situation, the general set (17)
reduces to even a simpler one:
(18a)N˙A*−kaNB∫0tΣ(t−τ)NA*(τ)dτ−NA*τA,(18b)N˙B*=+kaNB∫0tΣ(t−τ)NA*(τ)dτ−NB*τB.
These equations describe the
quenching of the initially excited A,
as well as accumulation of energy on B,
its reverse transfer to A,
and subsequent natural decay after the end of the encounter (Figure 1). From here
on we should discriminate between the qualitatively different situations.
(1)
Energy quenching. This is a case when
either the transfer is irreversible kb=0 or τB→0 so that nothing can be transferred backward.(2)
Equal times. In a border case, τA=τB=τ,
the decay of excited states and energy redistribution between them proceed
independently.(3)
Energy storing. At τA<τB the initial quenching of A* by the forward energy transfer results in
energy storing by B's and the backward transfer from them to the
short-lived A* during the same encounter.Figure 1
The decay of the initially excited molecule,NA*(t) and (A) accumulation/dissipation of
transferred energy, NB*(t),
(B) reflected receipt and reimbursement of energy by a partner molecule
[14].To describe the energy quenching kinetics one can
transform the integral equation (18a) to a differential one, (3), used in DET.
This procedure is very delicate, especially since there is a power decay of the
memory function. It is valid in a restricted time interval, where the main part
of the quenching proceeds quasiexponentially with the time-dependent rate
constant [14],(19)kA(t)=ka∫0tΣ(t′)et′/τAdt′,that monotonously decreases up
to the stationary (Markovian) value(20)ks=limt→∞kA=kaΣ˜(−1τA).In fact, this value is unattainable because the
reduction of IET to DET does not hold so long. The asymptotic behavior ofNA* in IET was shown to be the following [14]:(21)NA*(t)NA*(0)={exp(−c∫0tkA(t′)dt′−tτA)att→0,(τdt)3/2exp(−tτA)att→∞.However, the power time
asymptotic quenching in IET was shown to be false at least at τB≤τA and was essentially corrected by the
multiparticle theories reviewed below.Qualitatively different are the results in the
opposite, energy storing situation whenτB>τA. The rate constant (19) changes the sign with time and diverges while the
“stationary constant” (20) becomes a complex quantity. In contrast, the
effective IET analog of (19),(22)keff(t)=−dln(NA*et/τA)cdt,can be specified using the
solution to the original IET equation (18a). It does not diverge although it
becomes negative at long times (Figure 2). This is just a result of changing the
sign of the energy flux first directed from A to B but then turning back from B to A.
In other words, the initial decay of the excited state is faster
than the natural one but finally becomes slower and
obeys the asymptotic power time dependence over very long times [14]:(23)NA*N0=(τdt)3/2e−t/τBatτA<τB.It should be noted that a
similar power law time dependence is an exact result of the geminate reaction
of reversible dissociation, AB⇄A+B,
first discovered by Berg [36] and re-derived afterwards [37]. This law was confirmed
experimentally [38] and
essentially generalized later on [39–41].Figure 2
The sign
alteration of the effective reaction constant of IET (dashed line) and that of
DET (solid line), in the case of energy storing (atτB=∞, τA=2τd) and kacτd=0.1 [14].


### 2.2. Intramolecular Energy Transfer

Even a simpler
example of a reversible reaction of the same sort provides the intersystem
crossingA*⇄B* induced by an inert partner C according to the following
scheme:
(24)Initially excited A* as well as B* are two electronic states of the same particle A reacting with C given in concentration c=const.
In the contact approximation the IET equations take the form [26](25)dNA*dt=−kac∫0tΣ(t−τ)NA*(τ)dτ+kbc∫0tΣ(t−τ)NB*(τ)dτ−NA*τA,dNB*dt=kac∫0tΣ(t−τ)NA*(τ)dτ−kbc∫0tΣ(t−τ)NB*(τ)dτ−NB*τB,where the Laplace transformation
of Σ(t) is given by (16). This set of equations is
equivalent to the set (17) at NA=NB=c but the reduction to (18a), (18b) is impossible and
the backward transfer during reencounters is inevitable.However, the numerical solution of (25) was
an easy matter. This was done in [26] with the initial conditions NA*(0)=N0, NB*(0)=0 corresponding to the instantaneous (δ-pulse) excitation of state A.
The results are shown in Figure 3. In the absence of C there is only a spontaneous exponential decay
of A* (solid line in Figure 3(a)). The intramolecular
energy transfer to B* modulates this process. If the transfer is
irreversible (kb=0), it only facilitates the decay. This is the
energy quenching proceeding exponentially but faster than the natural decay
(dashed line in Figure 3(a)). At kb≠0,
the energy comes back due to reencounters with C,
supporting the delayed fluorescence from A* if τB>τA.
Under such a condition, the dashed-dotted line representing NA*(t) short after rapid quenching goes down much
slower than the spontaneous decay. The smaller kb,
the later, weaker and slower should be the delayed fluorescence. This energy
storing is the most pronounced at τB=∞.Relaxation of
state populations after instantaneous excitation ofA at τB=∞ and ckaτd=1.0.
(a) Spontaneous (exponential) decay of A* with lifetime τA=2τd (thick straight line) and its dissipation
accompanied by intersystem crossing with the backward rates kb=5.0kD (dotted line), kb=kD (dashed-dotted line), and kb=0 (dashed line). (b) Accumulation and dissipation
of the initially empty state B*,
which decays through A* faster the higher is kb,
or does not decay at all if kb=0 [26].
(a)(b)This picture is supplemented with the corresponding
kinetics of energy accumulation inB* and the subsequent decay of this state. The
latter takes place even at τB=∞ due to the collision induced reverse transfer
to A* (Figure 3(b)). Such a decay is absent only at kb=0 and is faster, the higher is kb.
At any τA<τB,
the energy is stored in a more stable excited state B* but finally comes back to the short-lived
excited state A*,
causing the delayed fluorescence.Between the opposite limits of energy quenching (τA>τB) and energy storing (τA<τB) there is a border case of equal times that
deserves special attention. This exceptional case allows the exact
multiparticle solution of the problem obtained in [42]. When τA=τB=τ,
the natural decay of both states can be excluded from consideration by a simple
substitution:(26)NA*(t)=𝒫A(t)e−t/τ,NB*(t)=𝒫B(t)e−t/τ.The relative populations of
excited states, 𝒫A(t) and 𝒫B(t), tend to their equilibrium values as t→∞:(27)𝒫A(∞)=11+K=PA,𝒫B(∞)=K1+K=PB,where K=ka/kb is the equilibrium constant in a system of two
excited states. The general solution of the problem can be represented
as(28)𝒫A(t)=11+K+K1+K𝒫eff(t)=1−𝒫B(t),where 𝒫eff(t) was the subject of calculations in [42]. After the exact averaging over the multiparticle
distribution of C it was shown to be(29)𝒫eff(t)=exp(−c∫0tk(t′)dt′)with k(t) obeying the original DET equations (5) and (6)
but with substituting WA(r)+WB(r) for WA(r).
In the contact approximation this leads to the conventional
Smoluchowski-Collins-Kimball solution but with ka+kb substituted for ka.


### 2.3. Intermolecular Electron Transfer

Let us now turn
to the reversible reaction of electron transfer,A*+D⇄A−+D+ accompanied by charge recombination. After
charge separation, the free ions recombine in the bulk to either the ground
state or backward to the excited neutral products, which contribute again to
the total fluorescence:
(30)
Such a complete scheme enables
studying the luminescence quenching proceeding from left to right, as well as
the electroluminescence (resulting from the recombination of injected ions)
going from right to left.The complex kinetics of the luminescence switched byδ-excitation of A in reaction (30) was studied in [43] using the general IET
equations:(31)N˙A*=−c∫0tR*(τ)NA*(t−τ)dτ+∫0tR♯(τ)P2(t−τ)dτ−NA*(t)τA+I0NA,P˙=c∫0tR†(τ)NA*(t−τ)dτ−∫0tR‡(τ)P2(t−τ)dτ,where P(t)=[A−]=[D+] is the free-ion concentration and c=[D]=const is the concentration of donors present in
great excess. When the geminate recombination is completed and gives
way to ion recombination, the latter restores either the ground or excited
state of the neutral reactants. The fluorescence of the excited ones is
delayed. In fact this is an electroluminescence though not from injected ions
but from those which escaped the geminate recombination after photo separation.
The density of restored excitations at a later stage of their decay is
quadratic in the free-ion concentration that should be large enough to make the
delayed fluorescence detectable. To reach this goal one has to use as strong
pumping as possible and choose the fluorophors with rather long-lived excited
states. Then their fluorescence at times t>τA will be stronger than in the absence of charge
recombination.The long time asymptote of the delayed decay can be
described by the reduced equation (31), where the light pumping is absent and
charge recombination is considered as irreversible in view of a negligible
concentration of restored A* [43]:(32)N˙A*=−NA*τA−c∫0tR*(τ)NA*(t−τ)dτ+∫0tR♯(τ)P2(t−τ)dτ,P˙=−∫0tR‡(τ)P2(t−τ)dτ. The initial condition for free ions is given by the
following equation:(33)P(0)=NA*(0)(1−η)φ¯=P0,where the yield of
photo-generated ions is 1−η and the yield of their separation φ¯ is defined in (67).Since ions are stable particles, those terms in the
integral equations (32) which describe the ion recombination may be transformed
at t≫τd into their differential and even Markovian analogs
[13]. The same can be
done with the remaining ionization term, but at a much longer time, t≫τA,
when the concentration of excitations levels 	decreases slowly following charge recombination. Hence, the delayed fluorescence
can be described by the following set of differential (Markovian) equations:(34)N˙A*=−[1τA+cκ0]NA*(t)+k*P2,P˙=−k×P2, where k*=R˜♯(0) and k×=R˜‡(0). The delayed fluorescence can be obtained from the quasistationary solution of
(34):(35a)NA*(t)=k*P2[1/τA+cκ0]→At2,(35b)P=P01+k×P0t→1k×t,where(36)A=k*[1/τA+cκ0](k×)2was calculated in the contact
approximation in [43].
In the same paper the full kinetics of luminescence was shown to have a long
tail approaching the delayed fluorescence asymptote (35a) (Figure 4). The latter
screens the intermediate asymptote of the multiparticle quenching (21) which is
known to be false in IET.Figure 4
The false asymptote of geminate excitation quenching
(long dashed line) in comparison to the second power asymptote (dashed line) of
true excitation decay (solid line) [43].


### 2.4. Modifications of Integral Theory

Unfortunately, the IET is the theory keeping only the
lowest order terms in reactant concentrations. To account for the higher ones
by MET, one only has to modify the kernels of the integral equations. In
[24, 25] this was recommended to be
done by the substitution of the generalized decay rates(37)1τA*=1τA+kaNBΣ˜(s),1τB*=1τB+kbNAΣ˜(s),for the inverse times 1/τA and 1/τB figuring in  (16). Such an obtained MET was
shown to correct IET even better than the first multiparticle kernel (MPK1)
theory [27], which at
first corrected the false IET asymptote of quenching. However, the authors of
the MPK1 intuitively neglected some three-particle correlations in comparison
with others, while MET accounts explicitly for all of them [24, 25]. An excellent analysis of the drawback in MPK1 and
its consequences was recently presented by Ivanov [44]. Fortunately this drawback
was later overcome by the same authors [29]
making their last theory, MPK3, almost equivalent to MET [25].MET essentially corrects IET over long times [44]. Figure 5 demonstrates the
difference in time behavior between IET and MET effective quenching constants
for the forward and reverse transfer,(38)kfeff=ka∫0tΣ(t−τ)NA*(τ)dτNA*(t),kreff=kb∫0tΣ(t−τ)NB*(τ)dτNB*(t).The former is the analog of (22)
which is known from IET to become negative with time (Figures 2 and 5(a)) but in
MET turns to be positive again at t≫τd) (Figure 5(b)). At these long times the transfer
proceeds not between geminate partners that have already separated but with
many other B's that, being in the ground state, act as the
fresh quenchers. As to the quenching constant for stable excitation B*,
it remains positive at all times in both theories but only in MET approaches
the truly stationary value kbΣ(0).The time effective rate “constants” for the
forward and reverse transfer,kf(t)=kfeff/ka (solid lines) and kr(t)=kreff/ka (dashed lines), calculated with IET (a) and
MET (b) at τB=∞,
equal concentrations of A and B and equal kinetic constants kfeff(0)=kreff(0)=ka=10kD.
The thin solid line shows the stationary value of kreff(∞) [44].
(a)(b)In the case of irreversible quenching (kb=0), the kernel modification can also be done as
was recommended earlier [19–23], by the straightforward substitution(39)Σ(t)→Σ(t)exp(−ckt),where k is the stationary (Markovian) constant of the
irreversible transfer (4). The difference between the two recipes of
modification originates from the way in which the “point encounter
approximation” was used to simplify the three-particle terms; only in
space or also in the time domain [24]. However, this choice rather insignificantly affects
the results. Performing point encounter approximation in the coordinate space
one implies that the reaction pair evolution proceeds at distances larger than
the reaction zone which can be ignored (see [24, Equations (6.15) and (6.16)]). This allows to derive the
binary kinetic equation by the simplification of three-particle terms in
concentration expansion keeping only their dependence on relative mobility of
the reactants.An original matrix reformulation of IET and its
modification similar to (39) were presented in [45]. Such an obtained MET is
applicable to the reversible reactions of any complexity and is in fact
identical to that used in [17–21]. However, the authors did not restrict themselves to
the integral form of the theory but transformed their matrix
integrodifferential equation into a set of two coupled differential equations
which can be solved easier numerically. They also developed a general computer code
(available on the Internet [45]) and demonstrated it application to the well-known
Lotka-Volterra reaction which is oscillatory with time [46].


### 2.5. Asymptotic Excitation Decay

The pre-exponential power time dependence ofNA*(t)/NA*(0) indicated in (21) is known to be the false
IET asymptote for the long time quenching. The latter can be easily seen in the
case of irreversible transfer, rather than in the reversible case.Assuming that the transfer fromA* to B is irreversible (kb=0), the quenching kinetics was studied in
[45] by solving the
original differential equations equivalent to either IET or MET. As can be seen
from Figure 6, the difference between the curves representing these theories is
insignificant within the validity limits for IET established in [17, 18]:
(40)t≪ln(ξ−1)/kc,where ξ=(4/3)πσ3c.
The validity region for MET is known to be much wider:
(41)t≪ξ−1kc. This was confirmed in [45] by a straightforward
comparison of the MET solution with that of DET:(42)PA*=NA*exp(tτA)=exp[−c∫0tkA(t′)dt′].The latter which is exact for
the target problem is not distinguishable from MET, represented by the solid
line in Figure 6. The difference appears to be less than the precision of the
numerical calculations, provided P*(t)>e−1/ξ,
where ξ<1.
On the contrary, the dashed curve representing IET significantly deviates from
the exact result when the time exceeds the border (40) indicated by the
vertical line in this figure. The false asymptote of IET is to the right of it.
In other words the IET is valid until P*(t)>ξ.Figure 6
The quenching kinetics obtained with IET (dashed line)
and MET (solid line). The vertical strip denotes the upper boundary of the
region of IET validity [45].As has been reported, the reversible intramolecular
transfer between the meta-stable states(43)
results in either energy quenching,
if kC>kA,
or in energy storing,
if kA>kC. However, inspecting the accuracy of different ABCD theories, one cannot use DET
as a standard because this theory is not applicable to the energy storing limit (due to the divergency of the
time-dependent forward rate constant) [13–16]. The only alternative is to compare the theoretical
results with those obtained by the BD simulation of the transfer kinetics.Since these simulations are always done only for
contact reactions between chemically isotropic hard spheres, they can be
compared only with contact IET and MET, as well as with MPK, SA, SCRTA, and
others, applied to the same Smoluchowsci-like model. Even though in such a
primitive model only the concentration and time dependence ofNA* can be inspected it is yet useful and
instructive. The numerous modifications of SCRTA were the main interest of the
authors of the last BD simulations but some other theories were also included
into comparison shown in Figure 7 [35]. The two upper panels are represented as they were
published (in black and white), while the lower ones are just the colored
originals of them placed at our disposal by one of the authors.The time dependence of the survival probability,NA*(t),
of initially excited A* for several values of the unimolecular decay
constants, kA and kC (in ns−1,
one is zero and the other is indicated in the figure). The parameters are the
same everywhere, except that the concentrations in the left panels are low (cB=cD=10−3Å−4) being large in the right ones (cB=cD=10−3Å−3). In the colored panels, the BD simulations
are presented by solid lines, while all the rest are depicted by symbols
indicated in these panels.
(a)(b)(c)(d)We can see from this figure that the two variants of
BD simulations, depicted by the open symbols: triangles (algorithm 1) and circles
(algorithm 2), are actually indistinguishable. It can also be seen from the
black and white panels that they agree perfectly with MPK3 and SCRTA but not as
well as with MPK1 and even worse with IET. The black and white
presentations of data
are suitable to give the impression that IET is the
worst among the other theories. To refute this conclusion it is enough to
glance at the colored version of the same data.IET deviates from the exact results only at a high
concentration (right panels) where it is known to be inapplicable. In this
sense, it is actually “the inferior to the MET” as stated in [35] but only in the same sense
as the theory of ideal gases is the inferior to the Van der Waals theory.
Moreover, even at large concentrations IET only fails to describe the energy quenching, kC=10(kA=0),
but not the energy-storing case, kA=10(kC=0), where the false IET asymptote is screened (as in Figure 4) by delayed
fluorescence. Of course, the coincidence of the IET results with BD simulations
is excellent in any case when the concentration is low. Moreover, the expected
imperfection of IET at high concentration is totally corrected by implication
of MET ≡ MPK3.In short, the false asymptote of IET comes to light
only at high concentrations and only in the energy quenching case. This
drawback of IET is completely removed by MET which accounts for the transfer
not only to the partner in a pair but also to other surrounding reactants. By
this way, MET corrects the high concentration behavior of all the quantum
yields considered in Section3.


## 2.1. Intermolecular Energy Transfer

The transfer
first studied with IET had the following reaction scheme [13]:
(7)
To make the problem even easier
let us assume that the nonexcited partners are present in great excess ([A]=NA≫NA*=[A*] and [B]=NB≫NB*=[B*]). Then the terms in the kinetic equations
which are second order in the excited state concentrations, NA*(t) and NB*(t),
are negligible and the set of two IET equations can be linearized in the
particle densities [15]:(8)N˙A*=NA∫0t〈S(t−τ)〉NB*(τ)dτ−NB∫0t〈R(t−τ)〉NA*(τ)dτ−NA*τA,N˙B*=−NA∫0t〈S(t−τ)〉NB*(τ)dτ+NB∫0t〈R(t−τ)〉NA*(τ)dτ−NB*τB.This set should be solved with
the initial conditions(9)NA*(0)=N0,NB*(0)=0,provided that A* was created by δ-pulse excitation at t=0.In any memory function formalism the main problem is
how to define the kernels “memory functions” of integral equations.
The great advantage of IET is the absence of such a problem. The space-averaged
kernels of the integral equations (8) are defined from first principles by
their Laplace transformations:(10)〈S˜(s)〉=(s+1τB)∫[WB(r)μ˜3(r,s)−WA(r)ν˜3(r,s)]d3r,〈R˜(s)〉=(s+1τA)∫[WA(r)ν˜2(r,s)−WB(r)μ˜2(r,s)]d3r.The auxiliary equations for the
dyads of pair distributions ν3,μ3,
and ν2,μ2 are the same:(11)ν˙=−(1τA+WA)ν+WBμ+DΔν,μ˙=−(1τB+WB)μ+WAν+DΔμ.Only the initial conditions for
them are different:(12)ν2(r,0)=1,μ2(r,0)=0;ν3(r,0)=0,μ3(r,0)=1.The global structure of IET
formalism is similar to that of DET. Although the kinetic equations are
integral, their kernels are defined via the same transfer rates, WA(r) and WB(r),
and the pair distribution functions obey the auxiliary equations similar to
that of DET.It is very easy to solve (11) in the contact
approximation when(13)WA(r)=kaδ(r−σ)4πσ2,WB(r)=kbδ(r−σ)4πσ2.In highly polar solvents,(14)ka=∫WA(r)d3r,kb=∫WB(r)d3rare the kinetic rate constants
of the forward and backward transfer. They relate both the
kernels(15)〈R˜(s)〉=kaΣ^(s),〈S˜(s)〉=kbΣ^(s),to a single
factor(16)Σ˜(s)=[1+ka/kD1+(s+1/τA)τd+kb/kD1+(s+1/τB)τd]−1=1F(s),where τd=σ2/D is the encounter time. With these kernels, the
general IET equations (8) take the particular form specific only to the contact
reactions [15]:(17)N˙A*=kbNA∫0tΣ(t−τ)NB*(τ)dτ−kaNB∫0tΣ(t−τ)NA*(τ)dτ−NA*τA,N˙B*=−kbNA∫0tΣ(t−τ)NB*(τ)dτ+kaNB∫0tΣ(t−τ)NA*(τ)dτ−NB*τB.At a great excess of theB molecule, one may neglect the reencounters of B* with A in the bulk. In this particular case, which
attracted the most attention, there is actually only a single A surrounded by Bs. In such a situation, the general set (17)
reduces to even a simpler one:
(18a)N˙A*−kaNB∫0tΣ(t−τ)NA*(τ)dτ−NA*τA,(18b)N˙B*=+kaNB∫0tΣ(t−τ)NA*(τ)dτ−NB*τB.
These equations describe the
quenching of the initially excited A,
as well as accumulation of energy on B,
its reverse transfer to A,
and subsequent natural decay after the end of the encounter (Figure 1). From here
on we should discriminate between the qualitatively different situations.
(1)
Energy quenching. This is a case when
either the transfer is irreversible kb=0 or τB→0 so that nothing can be transferred backward.(2)
Equal times. In a border case, τA=τB=τ,
the decay of excited states and energy redistribution between them proceed
independently.(3)
Energy storing. At τA<τB the initial quenching of A* by the forward energy transfer results in
energy storing by B's and the backward transfer from them to the
short-lived A* during the same encounter.Figure 1
The decay of the initially excited molecule,NA*(t) and (A) accumulation/dissipation of
transferred energy, NB*(t),
(B) reflected receipt and reimbursement of energy by a partner molecule
[14].To describe the energy quenching kinetics one can
transform the integral equation (18a) to a differential one, (3), used in DET.
This procedure is very delicate, especially since there is a power decay of the
memory function. It is valid in a restricted time interval, where the main part
of the quenching proceeds quasiexponentially with the time-dependent rate
constant [14],(19)kA(t)=ka∫0tΣ(t′)et′/τAdt′,that monotonously decreases up
to the stationary (Markovian) value(20)ks=limt→∞kA=kaΣ˜(−1τA).In fact, this value is unattainable because the
reduction of IET to DET does not hold so long. The asymptotic behavior ofNA* in IET was shown to be the following [14]:(21)NA*(t)NA*(0)={exp(−c∫0tkA(t′)dt′−tτA)att→0,(τdt)3/2exp(−tτA)att→∞.However, the power time
asymptotic quenching in IET was shown to be false at least at τB≤τA and was essentially corrected by the
multiparticle theories reviewed below.Qualitatively different are the results in the
opposite, energy storing situation whenτB>τA. The rate constant (19) changes the sign with time and diverges while the
“stationary constant” (20) becomes a complex quantity. In contrast, the
effective IET analog of (19),(22)keff(t)=−dln(NA*et/τA)cdt,can be specified using the
solution to the original IET equation (18a). It does not diverge although it
becomes negative at long times (Figure 2). This is just a result of changing the
sign of the energy flux first directed from A to B but then turning back from B to A.
In other words, the initial decay of the excited state is faster
than the natural one but finally becomes slower and
obeys the asymptotic power time dependence over very long times [14]:(23)NA*N0=(τdt)3/2e−t/τBatτA<τB.It should be noted that a
similar power law time dependence is an exact result of the geminate reaction
of reversible dissociation, AB⇄A+B,
first discovered by Berg [36] and re-derived afterwards [37]. This law was confirmed
experimentally [38] and
essentially generalized later on [39–41].Figure 2
The sign
alteration of the effective reaction constant of IET (dashed line) and that of
DET (solid line), in the case of energy storing (atτB=∞, τA=2τd) and kacτd=0.1 [14].


## 2.2. Intramolecular Energy Transfer

Even a simpler
example of a reversible reaction of the same sort provides the intersystem
crossingA*⇄B* induced by an inert partner C according to the following
scheme:
(24)Initially excited A* as well as B* are two electronic states of the same particle A reacting with C given in concentration c=const.
In the contact approximation the IET equations take the form [26](25)dNA*dt=−kac∫0tΣ(t−τ)NA*(τ)dτ+kbc∫0tΣ(t−τ)NB*(τ)dτ−NA*τA,dNB*dt=kac∫0tΣ(t−τ)NA*(τ)dτ−kbc∫0tΣ(t−τ)NB*(τ)dτ−NB*τB,where the Laplace transformation
of Σ(t) is given by (16). This set of equations is
equivalent to the set (17) at NA=NB=c but the reduction to (18a), (18b) is impossible and
the backward transfer during reencounters is inevitable.However, the numerical solution of (25) was
an easy matter. This was done in [26] with the initial conditions NA*(0)=N0, NB*(0)=0 corresponding to the instantaneous (δ-pulse) excitation of state A.
The results are shown in Figure 3. In the absence of C there is only a spontaneous exponential decay
of A* (solid line in Figure 3(a)). The intramolecular
energy transfer to B* modulates this process. If the transfer is
irreversible (kb=0), it only facilitates the decay. This is the
energy quenching proceeding exponentially but faster than the natural decay
(dashed line in Figure 3(a)). At kb≠0,
the energy comes back due to reencounters with C,
supporting the delayed fluorescence from A* if τB>τA.
Under such a condition, the dashed-dotted line representing NA*(t) short after rapid quenching goes down much
slower than the spontaneous decay. The smaller kb,
the later, weaker and slower should be the delayed fluorescence. This energy
storing is the most pronounced at τB=∞.Relaxation of
state populations after instantaneous excitation ofA at τB=∞ and ckaτd=1.0.
(a) Spontaneous (exponential) decay of A* with lifetime τA=2τd (thick straight line) and its dissipation
accompanied by intersystem crossing with the backward rates kb=5.0kD (dotted line), kb=kD (dashed-dotted line), and kb=0 (dashed line). (b) Accumulation and dissipation
of the initially empty state B*,
which decays through A* faster the higher is kb,
or does not decay at all if kb=0 [26].
(a)(b)This picture is supplemented with the corresponding
kinetics of energy accumulation inB* and the subsequent decay of this state. The
latter takes place even at τB=∞ due to the collision induced reverse transfer
to A* (Figure 3(b)). Such a decay is absent only at kb=0 and is faster, the higher is kb.
At any τA<τB,
the energy is stored in a more stable excited state B* but finally comes back to the short-lived
excited state A*,
causing the delayed fluorescence.Between the opposite limits of energy quenching (τA>τB) and energy storing (τA<τB) there is a border case of equal times that
deserves special attention. This exceptional case allows the exact
multiparticle solution of the problem obtained in [42]. When τA=τB=τ,
the natural decay of both states can be excluded from consideration by a simple
substitution:(26)NA*(t)=𝒫A(t)e−t/τ,NB*(t)=𝒫B(t)e−t/τ.The relative populations of
excited states, 𝒫A(t) and 𝒫B(t), tend to their equilibrium values as t→∞:(27)𝒫A(∞)=11+K=PA,𝒫B(∞)=K1+K=PB,where K=ka/kb is the equilibrium constant in a system of two
excited states. The general solution of the problem can be represented
as(28)𝒫A(t)=11+K+K1+K𝒫eff(t)=1−𝒫B(t),where 𝒫eff(t) was the subject of calculations in [42]. After the exact averaging over the multiparticle
distribution of C it was shown to be(29)𝒫eff(t)=exp(−c∫0tk(t′)dt′)with k(t) obeying the original DET equations (5) and (6)
but with substituting WA(r)+WB(r) for WA(r).
In the contact approximation this leads to the conventional
Smoluchowski-Collins-Kimball solution but with ka+kb substituted for ka.


## 2.3. Intermolecular Electron Transfer

Let us now turn
to the reversible reaction of electron transfer,A*+D⇄A−+D+ accompanied by charge recombination. After
charge separation, the free ions recombine in the bulk to either the ground
state or backward to the excited neutral products, which contribute again to
the total fluorescence:
(30)
Such a complete scheme enables
studying the luminescence quenching proceeding from left to right, as well as
the electroluminescence (resulting from the recombination of injected ions)
going from right to left.The complex kinetics of the luminescence switched byδ-excitation of A in reaction (30) was studied in [43] using the general IET
equations:(31)N˙A*=−c∫0tR*(τ)NA*(t−τ)dτ+∫0tR♯(τ)P2(t−τ)dτ−NA*(t)τA+I0NA,P˙=c∫0tR†(τ)NA*(t−τ)dτ−∫0tR‡(τ)P2(t−τ)dτ,where P(t)=[A−]=[D+] is the free-ion concentration and c=[D]=const is the concentration of donors present in
great excess. When the geminate recombination is completed and gives
way to ion recombination, the latter restores either the ground or excited
state of the neutral reactants. The fluorescence of the excited ones is
delayed. In fact this is an electroluminescence though not from injected ions
but from those which escaped the geminate recombination after photo separation.
The density of restored excitations at a later stage of their decay is
quadratic in the free-ion concentration that should be large enough to make the
delayed fluorescence detectable. To reach this goal one has to use as strong
pumping as possible and choose the fluorophors with rather long-lived excited
states. Then their fluorescence at times t>τA will be stronger than in the absence of charge
recombination.The long time asymptote of the delayed decay can be
described by the reduced equation (31), where the light pumping is absent and
charge recombination is considered as irreversible in view of a negligible
concentration of restored A* [43]:(32)N˙A*=−NA*τA−c∫0tR*(τ)NA*(t−τ)dτ+∫0tR♯(τ)P2(t−τ)dτ,P˙=−∫0tR‡(τ)P2(t−τ)dτ. The initial condition for free ions is given by the
following equation:(33)P(0)=NA*(0)(1−η)φ¯=P0,where the yield of
photo-generated ions is 1−η and the yield of their separation φ¯ is defined in (67).Since ions are stable particles, those terms in the
integral equations (32) which describe the ion recombination may be transformed
at t≫τd into their differential and even Markovian analogs
[13]. The same can be
done with the remaining ionization term, but at a much longer time, t≫τA,
when the concentration of excitations levels 	decreases slowly following charge recombination. Hence, the delayed fluorescence
can be described by the following set of differential (Markovian) equations:(34)N˙A*=−[1τA+cκ0]NA*(t)+k*P2,P˙=−k×P2, where k*=R˜♯(0) and k×=R˜‡(0). The delayed fluorescence can be obtained from the quasistationary solution of
(34):(35a)NA*(t)=k*P2[1/τA+cκ0]→At2,(35b)P=P01+k×P0t→1k×t,where(36)A=k*[1/τA+cκ0](k×)2was calculated in the contact
approximation in [43].
In the same paper the full kinetics of luminescence was shown to have a long
tail approaching the delayed fluorescence asymptote (35a) (Figure 4). The latter
screens the intermediate asymptote of the multiparticle quenching (21) which is
known to be false in IET.Figure 4
The false asymptote of geminate excitation quenching
(long dashed line) in comparison to the second power asymptote (dashed line) of
true excitation decay (solid line) [43].


## 2.4. Modifications of Integral Theory

Unfortunately, the IET is the theory keeping only the
lowest order terms in reactant concentrations. To account for the higher ones
by MET, one only has to modify the kernels of the integral equations. In
[24, 25] this was recommended to be
done by the substitution of the generalized decay rates(37)1τA*=1τA+kaNBΣ˜(s),1τB*=1τB+kbNAΣ˜(s),for the inverse times 1/τA and 1/τB figuring in  (16). Such an obtained MET was
shown to correct IET even better than the first multiparticle kernel (MPK1)
theory [27], which at
first corrected the false IET asymptote of quenching. However, the authors of
the MPK1 intuitively neglected some three-particle correlations in comparison
with others, while MET accounts explicitly for all of them [24, 25]. An excellent analysis of the drawback in MPK1 and
its consequences was recently presented by Ivanov [44]. Fortunately this drawback
was later overcome by the same authors [29]
making their last theory, MPK3, almost equivalent to MET [25].MET essentially corrects IET over long times [44]. Figure 5 demonstrates the
difference in time behavior between IET and MET effective quenching constants
for the forward and reverse transfer,(38)kfeff=ka∫0tΣ(t−τ)NA*(τ)dτNA*(t),kreff=kb∫0tΣ(t−τ)NB*(τ)dτNB*(t).The former is the analog of (22)
which is known from IET to become negative with time (Figures 2 and 5(a)) but in
MET turns to be positive again at t≫τd) (Figure 5(b)). At these long times the transfer
proceeds not between geminate partners that have already separated but with
many other B's that, being in the ground state, act as the
fresh quenchers. As to the quenching constant for stable excitation B*,
it remains positive at all times in both theories but only in MET approaches
the truly stationary value kbΣ(0).The time effective rate “constants” for the
forward and reverse transfer,kf(t)=kfeff/ka (solid lines) and kr(t)=kreff/ka (dashed lines), calculated with IET (a) and
MET (b) at τB=∞,
equal concentrations of A and B and equal kinetic constants kfeff(0)=kreff(0)=ka=10kD.
The thin solid line shows the stationary value of kreff(∞) [44].
(a)(b)In the case of irreversible quenching (kb=0), the kernel modification can also be done as
was recommended earlier [19–23], by the straightforward substitution(39)Σ(t)→Σ(t)exp(−ckt),where k is the stationary (Markovian) constant of the
irreversible transfer (4). The difference between the two recipes of
modification originates from the way in which the “point encounter
approximation” was used to simplify the three-particle terms; only in
space or also in the time domain [24]. However, this choice rather insignificantly affects
the results. Performing point encounter approximation in the coordinate space
one implies that the reaction pair evolution proceeds at distances larger than
the reaction zone which can be ignored (see [24, Equations (6.15) and (6.16)]). This allows to derive the
binary kinetic equation by the simplification of three-particle terms in
concentration expansion keeping only their dependence on relative mobility of
the reactants.An original matrix reformulation of IET and its
modification similar to (39) were presented in [45]. Such an obtained MET is
applicable to the reversible reactions of any complexity and is in fact
identical to that used in [17–21]. However, the authors did not restrict themselves to
the integral form of the theory but transformed their matrix
integrodifferential equation into a set of two coupled differential equations
which can be solved easier numerically. They also developed a general computer code
(available on the Internet [45]) and demonstrated it application to the well-known
Lotka-Volterra reaction which is oscillatory with time [46].


## 2.5. Asymptotic Excitation Decay

The pre-exponential power time dependence ofNA*(t)/NA*(0) indicated in (21) is known to be the false
IET asymptote for the long time quenching. The latter can be easily seen in the
case of irreversible transfer, rather than in the reversible case.Assuming that the transfer fromA* to B is irreversible (kb=0), the quenching kinetics was studied in
[45] by solving the
original differential equations equivalent to either IET or MET. As can be seen
from Figure 6, the difference between the curves representing these theories is
insignificant within the validity limits for IET established in [17, 18]:
(40)t≪ln(ξ−1)/kc,where ξ=(4/3)πσ3c.
The validity region for MET is known to be much wider:
(41)t≪ξ−1kc. This was confirmed in [45] by a straightforward
comparison of the MET solution with that of DET:(42)PA*=NA*exp(tτA)=exp[−c∫0tkA(t′)dt′].The latter which is exact for
the target problem is not distinguishable from MET, represented by the solid
line in Figure 6. The difference appears to be less than the precision of the
numerical calculations, provided P*(t)>e−1/ξ,
where ξ<1.
On the contrary, the dashed curve representing IET significantly deviates from
the exact result when the time exceeds the border (40) indicated by the
vertical line in this figure. The false asymptote of IET is to the right of it.
In other words the IET is valid until P*(t)>ξ.Figure 6
The quenching kinetics obtained with IET (dashed line)
and MET (solid line). The vertical strip denotes the upper boundary of the
region of IET validity [45].As has been reported, the reversible intramolecular
transfer between the meta-stable states(43)
results in either energy quenching,
if kC>kA,
or in energy storing,
if kA>kC. However, inspecting the accuracy of different ABCD theories, one cannot use DET
as a standard because this theory is not applicable to the energy storing limit (due to the divergency of the
time-dependent forward rate constant) [13–16]. The only alternative is to compare the theoretical
results with those obtained by the BD simulation of the transfer kinetics.Since these simulations are always done only for
contact reactions between chemically isotropic hard spheres, they can be
compared only with contact IET and MET, as well as with MPK, SA, SCRTA, and
others, applied to the same Smoluchowsci-like model. Even though in such a
primitive model only the concentration and time dependence ofNA* can be inspected it is yet useful and
instructive. The numerous modifications of SCRTA were the main interest of the
authors of the last BD simulations but some other theories were also included
into comparison shown in Figure 7 [35]. The two upper panels are represented as they were
published (in black and white), while the lower ones are just the colored
originals of them placed at our disposal by one of the authors.The time dependence of the survival probability,NA*(t),
of initially excited A* for several values of the unimolecular decay
constants, kA and kC (in ns−1,
one is zero and the other is indicated in the figure). The parameters are the
same everywhere, except that the concentrations in the left panels are low (cB=cD=10−3Å−4) being large in the right ones (cB=cD=10−3Å−3). In the colored panels, the BD simulations
are presented by solid lines, while all the rest are depicted by symbols
indicated in these panels.
(a)(b)(c)(d)We can see from this figure that the two variants of
BD simulations, depicted by the open symbols: triangles (algorithm 1) and circles
(algorithm 2), are actually indistinguishable. It can also be seen from the
black and white panels that they agree perfectly with MPK3 and SCRTA but not as
well as with MPK1 and even worse with IET. The black and white
presentations of data
are suitable to give the impression that IET is the
worst among the other theories. To refute this conclusion it is enough to
glance at the colored version of the same data.IET deviates from the exact results only at a high
concentration (right panels) where it is known to be inapplicable. In this
sense, it is actually “the inferior to the MET” as stated in [35] but only in the same sense
as the theory of ideal gases is the inferior to the Van der Waals theory.
Moreover, even at large concentrations IET only fails to describe the energy quenching, kC=10(kA=0),
but not the energy-storing case, kA=10(kC=0), where the false IET asymptote is screened (as in Figure 4) by delayed
fluorescence. Of course, the coincidence of the IET results with BD simulations
is excellent in any case when the concentration is low. Moreover, the expected
imperfection of IET at high concentration is totally corrected by implication
of MET ≡ MPK3.In short, the false asymptote of IET comes to light
only at high concentrations and only in the energy quenching case. This
drawback of IET is completely removed by MET which accounts for the transfer
not only to the partner in a pair but also to other surrounding reactants. By
this way, MET corrects the high concentration behavior of all the quantum
yields considered in Section3.


## 3. Concentration Phenomena

### 3.1. Luminescence after Pulse Excitation

The quantum
yield of the fluorescence following pulse excitation is(44)η=∫0∞NA*(t)dtNA*(0)τAprovided that the light excites
instantaneously (at t=0) only A and that the luminescence comes only
from this particle. The concentration dependence of the yield is always
represented by the Stern-Volmer law(45)η=N˜A*(0)N0τA=11+NBkqτA,where kq is the quenching constant which is to be
investigated.Let us start doing this from the reversible energy
transfer studied in Section2.1. Using the Laplace transformation of the IET
equations (17), one can easily find from (45) that(46)kq=KAB1+KBANAτB.Here KAB=kaΣ˜(0) and KBA=kbΣ˜(0),
that is,(47)KABka=11+(ka/kD)/(1+τd/τA)+(kb/kD)/(1+τd/τB)=KBAkb.This result obtained by means of
IET is concentration-independent but known to be valid at only the lowest
concentrations of A and B [13].


### 3.2. Irreversible Geminate Reactions

To demonstrate
how DET and MET, as well as other multiparticle theories, correct this result
at higher concentrations, let us focus upon the irreversible transfer setting(48)KBA=kb=0,NB=c.In such a case, the theory
becomes universal that is equally good for any irreversible quenching,
including the parallel transfer of energy and electron [47]
(49)
or the double-channel electron
transfer studied in [48, 49]:
(50)
All of them can be briefly
represented by the unified scheme of the irreversible energy quenching: A*+B→A+B.The quantum yield of the irreversible quenching obeys
the Stern-Volmer dependence on concentrationc=NB represented in (45). As in the original
Stern-Volmer law, the quenching constant obtained from (46) and (47) is concentration-independent:(51)kq=ka1+ka/kD(1+τd/τA)=κi.However, contrary to κi,
the experimentally found kq≡κ is concentration-dependent and this is a
challenge for the theory to find out how the true Stern-Volmer constant κ(c) differs from its IET analog κi=limc→0κ(c).MET solves this problem substitutingτA in (51) by τA*(0) from (37):(52)κ=ka1+ka/kD(1+(τd/τA)[1+cκiτA]).In general, this κ is concentration-dependent unless the transfer
is strongly under kinetic control when κ=ka≪kD. In the alternative limit of diffusional
transfer, MET gives(53)κ=kD(1+τdτA[1+cκiτA]).In [50] this result was compared
with that of IET and other theories: SA [30–33] and DET (Figure 8). The Stern-Volmer constant of MET
and SA differs a bit from the exact result represented by DET but all of them
except IET increase with the dimensionless concentration ξ=cv (v=(4/3)πσ3). At moderate concentrations this dependence
is always linear,(54)κkD=κκi+Aξ=1+τdτA+Aξ,but the slope of it appears to
be different. For MET obtained by modification (37), A=(3/2)τA/τd+3/2 while in the old MET originated from another
modification, (39), A=(3/2)τA/τd.
The former coincides with that of SA while the latter (shown in figure) is a
bit smaller. The true DET value is in between A=(3/2)τA/τd+3(1−2/π) [50]. Since the concentration correction in (54) is
only actual for large τA≫τd,
the constant term in A is insignificant. In the opposite case, the
whole correction is negligible compared to the preceding IET term. Hence the
difference between the concentration corrections at small ξ is not essential.Figure 8
The
Stern-Volmer constants as functions of the dimensionless concentrationξ=4πσc3/3 obtained in the contact approximation and
under diffusional control at τd/τA=0.01.
The thick line represents DET which is exact for the target problem (immobile
donors and independently moving acceptors). The rest of the curves are obtained
with SA, MET, and IET [50].However, the comparison of all the theories (of
contact multiparticle quenching by point particlesB) was done later for any ξ (Figure 9) [47]. Theoretically, κ increases with concentration from the IET
value κi up to the kinetic rate constant k0≡ka, though experimentally available is only the lower (left) part of the graph, cv<1. Contact MET, which is identical to MPK3 developed later [29], underestimates the exact κ(c) represented by DET. The latter is equivalent
to the irreversible version of the first multiparticle kernel theory, MPK1
[27]. The intermediate
version of this theory [28],
MPK2, as well as the latest model theory SCRTA [34], almost coincide with DET =
MPK1 (irreversible), unlike the linearized extended superposition
approximation [51]
(LESA) which overestimates κ(c).Figure 9
The concentration dependence of the irreversible
quenching Stern-Volmer constantκ in units of k0≡ka for a number of contact theories, provided ka=3.7⋅1010M−1s−1 is the same for all of
them [47] and [48, Figure 3.88].


### 3.3. IET of Reversible Geminate Reaction

Let us now turn
to the geminate reaction similar to that included in scheme (49) but carried
out by reversible electron transfer:
(55)
Here D is the donor of an electron, while A is its acceptor. The excitation of A by the short light pulse, I(t),
resulting in charge separation, produces the free ions, D+ and A−,
with the yield ϕ.
This is the fraction φ¯ of ions initially born in amount 1−η but separated escaping recombination to the
ground state with the rate WR and the backward electron transfer to excited
products with the rate WB.IfA's are present in very low concentration, the
density of the free ions is small as well, so that their recombination in the
bulk is negligible during a bounded time domain comparable with encounter time τd~σ2/D.
The IET equations (31) can be reduced for this case by omitting the bulk terms
quadratic in P,
as well as the pumping term:(56)N˙A*=−c∫0tR*(τ)NA*(t−τ)dτ−NA*(t)τA,P˙=c∫0tR†(τ)NA*(t−τ)dτ. Instead of the pumping term one has to add the initial
conditions for these equations which represent the instantaneous excitation of A:(57)NA*(0)=N0,P(0)=0.The luminescence of the pulse excitedA* quenched in a limited time has the yield
specified by (44) which completely neglects the subsequent delayed
fluorescence resulting from the bulk recombination. Such a yield obeys the
Stern-Volmer law (45) but with the geminate quenching constant kq≡κg.
The latter is concentration-independent unlike its analog in (46)
accounting for the backward energy transfer during bulk encounters.Hence, the luminescence quenched by reversible
ionization after instantaneous excitation has the yield calculated from
(45) and (56) [52]:(58)η=11+cκgτA,where the geminate Stern-Volmer
constant is [43, 52](59)κg=R˜*(0)=κi1+κi/K(kD+kc).Here κi is the Stern-Volmer constant for irreversible
transfer (51) while(60)K=kakb=exp(−ΔGiT)is an equilibrium constant for
ionization, with free energy ΔGi (here and further on the Boltzmann constant kB=1). The kinetic reaction constants of the
outer-sphere electron transfer(61)ka=ka′e−ua/T,kb=kb′e−ub/Tobey the Arrhenius law with
activation energies satisfying the free energy gap (FEG) law [9]:(62)ua=ub+ΔGi=(ΔGi+λc)24λc,where λc is the contact reorganization energy of the
polar media. The charge recombination constant also obeys the FEG law:(63)kc=kc′e−uc/Twithuc=(ΔGr+λc)24λcbut with different free energy −ΔGr=ℰ+ΔGi where ℰ is the excitation energy of A*.For highly exergonic charge separation (−ΔGi>T) when K→∞ the reaction (55) becomes irreversible, that
is, κg→κi which is the same for the irreversible
transfer of either energy or electron. In the opposite limit (K≲1), the transfer can also be irreversible
provided the charge separation or their geminate recombination is fast: either kD→∞ or kc→∞.
Otherwise, the distribution between neutral and charged reactants is
equilibrated and the luminescence having the Stern-Volmer constant κg≈K(kD+kc) disappears simultaneously with the ion pair
that either separates or recombines.


### 3.4. Stationary Luminescence

The pumping light intensity isI=(N0/NA)δ(t) in the case of pulse excitation but I=I0=const when the fluorescence is an induced and
studied stationary. In the former case, the quantum yield η has to be calculated from formula (45) while
in the latter case it is defined in another way [53, 54]:(64)η=NAsI0NAτA=11+cκ0τA,where NAs is the stationary density of the excited
states. If the fluorescence is quenched by reversible intramolecular transfer
according to scheme (24), then both recipes were shown to give the same
result [42]. A different
situation arises when the quenching is performed by the intermolecular transfer
whose charged products recombine in the bulk restoring the excitation according
to scheme (30). In such a case, NAs can be found from the set of corresponding IET
equations (31) setting N˙A*=P˙=0.
Substituting the stationary solution thus obtained into (64) we get the
corresponding Stern-Volmer constant(65)κ0=R˜*(0)[1−R˜♯(0)R˜‡(0)×R˜†(0)R˜*(0)]=κg[1−χ⋅φ¯].It is smaller than the geminate
one (κg=R˜*(0)) because not all the excitations are quenched
forever at first encounter. Some of them are restored with an efficiency(66)χ=R˜♯(0)R˜‡(0)in the subsequent bulk
encounters of the free ions, which are separated with the yield(67)φ¯=R˜(0)R˜*(0).All the components of (65)
are well defined via the IET kernels.Calculated in the contact approximation, they reduce
expression (65) to the following one [43, 54]:(68)κ0=κi1+κi/Kkc.The principle difference between
this result and the geminate one, (59), is the absence of kD in the denominator. Diffusional ion pair
separation cannot make the stationary energy quenching irreversible. The charge
separation does not put an end to the reversible reaction, though interrupts it
for a while. Only the irreversible recombination to the ground state of the
neutral products proceeding with rate constant kc causes this to happen. If kc→∞,
the ionization is fully irreversible, that is, κ0≡κi Figure 12.On the contrary, atkc=0 the quenching is reversible, that is, the
fluorescence is not quenched at all (κ0=0). Almost the same is true when K≫κi/kc,
where K and kc were specified in (60) and (63). The
quasireversible ionization is controlled by RIP recombination which proceeds
with the rate constant(69)κ0≈Kkc=kc′e−U/T.The total activation energy of
such a reaction,(70)U=ΔGi+uc,becomes negative at highly
exergonic transfer (when |ΔGi|>uc).


### 3.5. Association/Dissociation of the Exciplex

A different
situation arises when the luminescence is interrupted by the reversible
association ofA* with impurity (B) present in great excess, resulting in
exciplex (C*) formation [55]:
(71)
Here τC and τA are the decay times of bound and unbound
excitation that may be either equal or different. The densities of the excited
particles obey the set of IET equations:(72)dNA*dt=−cka∫0tΣ(t−τ)NA*(τ)dτ+kc∫0tΣ(t−τ)NC*(τ)dτ−NA*τA,dNC*dt=cka∫0tΣ(t−τ)NA*(τ)dτ−kc∫0tΣ(t−τ)NC*(τ)dτ−NC*τC.
The luminescence quantum yields
of A* and C* are(73)ηAA=1τA∫0∞NA*(t)dt/N0,ηCA=1−ηAA,provided that only A is subjected to instantaneous light
excitation: NA*(0)=N0, NC*(0)=0.The conventional Stern-Volmer law,(74)1ηAA=1+cκτAκ=kaΣ˜(0)1+kcτCΣ˜(0),has the Stern-Volmer constant κ,
which depends on the concentration via Σ˜(0,c).
Only its minimal value Σ˜0=Σ˜(0,0) calculated with IET from (73) and (72) is
concentration-independent:(75)κ0=limc→0κ(c)=kaΣ˜01+kcτCΣ˜0Σ˜0=[1+kakD(1+τd/τA)]−1.For the irreversible binding (kc=0) this κ0=kaΣ˜0 coincides with κi given in (51).Equations (72) as well as the general κ definition given in (74) are common for all
multiparticle theories. They differ only in Σ˜(s) that were collected in [55, Tables 1 and 2] for the target
and trap problem (only A is moving between immobile traps). All Σ˜(0) monotonously increase with c approaching 1. Simultaneously the upper limit
of the Stern-Volmer constant is achieved:(76)κm=ka1+kcτC.Since only a few theories deal
with τA≠τC,
in Figure 10, only the equal lifetimes case is examined. It is easy to see that
all MPK theories and CA give similar results, unlike SCRTA and LESA which
deviate from them into opposite sides. LESA was independently shown to give an
inappropriate description of reversible transfer at equal times [42]. SCRTA in its turn strongly
overestimates the difference between the target and trap problems as compared
to MPK3≡ MET. As to the concentration-independent IET
result, it is always reproduced but only as c→0.
At higher concentration it is better to replace IET with MET.The Stern-Volmer constant of the reversible exciplex
formation atτA=τC=τ as a function of the dimensionless
concentration of B's. In a wide range of concentration (a) all
curves increase from the minimal (IET) value κ0 up to the maximal one, κm.
At low concentrations (b) the difference between them is more pronounced [55].
(a)(b)Quite recently the kinetics of energy quenching by
exciplex formation and resulting free energy dependence of Stern-Volmer
constant were thoroughly investigated with IET and compared with available
experimental data [56].


### 3.6. Electroluminescence

The ions injected from electrodes recombine to either
the ground or excited state of the neutral products. The latter can be detected
by their luminescence and the quantum yield of such an electroluminescence
is(77)ϕecl=ϕeϕes.The quantum yield of
excitations, ϕes,
can be extracted from this relationship since the emission quantum yield from
the excited state, ϕe,
is usually known. In [57–59] the ϕes dependence on the free energy of ionization, ΔGi=−ΔGb,
was measured for a number of systems. To specify this dependence we have to
calculate(78)ϕes=∫0∞NA*(t)dtP(0)τA,borrowing NA*(t) from the solution of (31) where we
set I0=0 and use the appropriate initial conditions
created by the external injection of ions into solution:(79)NA*(0)=0,P(0)=P0.Making the Laplace transformation of  (31) we obtain from (78)(80)ϕes=N˜A*(0)P0τA=1R˜‡(0)/R˜♯(0)+cτA[R˜*(0)R˜‡(0)/R˜♯(0)−R˜†(0)].Using the expressions for all
the kernels obtained in contact approximation in [53] we get the contact analog
of this equation [43]:(81)ϕes=11+(1+cτAκi)kcK/κi.This expression reduces to a
much simpler one provided the recombination into the excited state is
irreversible as was assumed in [57–59]. If this is really the
case, then κi=ka=0 and K/κi=K/ka=1/kb so that (81) takes the form used in these
works:(82)ϕes=11+kc/kb.Using the contact estimates of
the kinetic rate constants given in (61) and (63) we obtain the following final
result:(83)ϕes=[1+kc′kb′exp(−ℰ(ΔGi−λc+ℰ/2)2λcT)]−1. This is the
stepwise function approaching unity when ΔGi increases making the recombination to the
excited state more favorable than to the ground one. Finally, the excitation
becomes the unique reaction product since the recombination to the ground state
is switched off.Being calculated with (81), which accounts for the
transfer reversibility (ionization of excited state), this function appears to
be different from the simplest ones, (82) and (83), suited for
irreversible recombination. Shown in Figure 11 these functions, although they are
different, resemble the experimental results obtained in [57–59]. The correct accounting for the reaction
reversibility is the main but not only advantage of IET, compared to DET and
Markovian chemical kinetics. Taking into account the space dependence of the
rates one should use (80) without contact simplifications and it was really
employed in [43].
Moreover, in the next work the spin states of the free ions and radical ion
pairs (RIPs) formed from them were also taken into account, as well as the spin
conversion in the RIP and recombination to triplet products [60]. Even after that, the full
correspondence with the experimental findings was not reached: the height of
the true plateau remains lower than 1 for unknown reasons.Figure 11
The quantum yield of the excited states,ϕes,
calculated in the contact approximation with (dashed line) and without
(dashed-dotted line), taking into account their ionization [43].The Rehm-Weller plot for a few systems which differ by
their triplet RIP recombination rate. (a) The theoretical curves forkt′/kD=0;10−8;10−5;0.1;20 at kc′=kb′=20kD.
(b) Interpolation through experimental points from [61, Figure 2].
(a)(b)


## 3.1. Luminescence after Pulse Excitation

The quantum
yield of the fluorescence following pulse excitation is(44)η=∫0∞NA*(t)dtNA*(0)τAprovided that the light excites
instantaneously (at t=0) only A and that the luminescence comes only
from this particle. The concentration dependence of the yield is always
represented by the Stern-Volmer law(45)η=N˜A*(0)N0τA=11+NBkqτA,where kq is the quenching constant which is to be
investigated.Let us start doing this from the reversible energy
transfer studied in Section2.1. Using the Laplace transformation of the IET
equations (17), one can easily find from (45) that(46)kq=KAB1+KBANAτB.Here KAB=kaΣ˜(0) and KBA=kbΣ˜(0),
that is,(47)KABka=11+(ka/kD)/(1+τd/τA)+(kb/kD)/(1+τd/τB)=KBAkb.This result obtained by means of
IET is concentration-independent but known to be valid at only the lowest
concentrations of A and B [13].


## 3.2. Irreversible Geminate Reactions

To demonstrate
how DET and MET, as well as other multiparticle theories, correct this result
at higher concentrations, let us focus upon the irreversible transfer setting(48)KBA=kb=0,NB=c.In such a case, the theory
becomes universal that is equally good for any irreversible quenching,
including the parallel transfer of energy and electron [47]
(49)
or the double-channel electron
transfer studied in [48, 49]:
(50)
All of them can be briefly
represented by the unified scheme of the irreversible energy quenching: A*+B→A+B.The quantum yield of the irreversible quenching obeys
the Stern-Volmer dependence on concentrationc=NB represented in (45). As in the original
Stern-Volmer law, the quenching constant obtained from (46) and (47) is concentration-independent:(51)kq=ka1+ka/kD(1+τd/τA)=κi.However, contrary to κi,
the experimentally found kq≡κ is concentration-dependent and this is a
challenge for the theory to find out how the true Stern-Volmer constant κ(c) differs from its IET analog κi=limc→0κ(c).MET solves this problem substitutingτA in (51) by τA*(0) from (37):(52)κ=ka1+ka/kD(1+(τd/τA)[1+cκiτA]).In general, this κ is concentration-dependent unless the transfer
is strongly under kinetic control when κ=ka≪kD. In the alternative limit of diffusional
transfer, MET gives(53)κ=kD(1+τdτA[1+cκiτA]).In [50] this result was compared
with that of IET and other theories: SA [30–33] and DET (Figure 8). The Stern-Volmer constant of MET
and SA differs a bit from the exact result represented by DET but all of them
except IET increase with the dimensionless concentration ξ=cv (v=(4/3)πσ3). At moderate concentrations this dependence
is always linear,(54)κkD=κκi+Aξ=1+τdτA+Aξ,but the slope of it appears to
be different. For MET obtained by modification (37), A=(3/2)τA/τd+3/2 while in the old MET originated from another
modification, (39), A=(3/2)τA/τd.
The former coincides with that of SA while the latter (shown in figure) is a
bit smaller. The true DET value is in between A=(3/2)τA/τd+3(1−2/π) [50]. Since the concentration correction in (54) is
only actual for large τA≫τd,
the constant term in A is insignificant. In the opposite case, the
whole correction is negligible compared to the preceding IET term. Hence the
difference between the concentration corrections at small ξ is not essential.Figure 8
The
Stern-Volmer constants as functions of the dimensionless concentrationξ=4πσc3/3 obtained in the contact approximation and
under diffusional control at τd/τA=0.01.
The thick line represents DET which is exact for the target problem (immobile
donors and independently moving acceptors). The rest of the curves are obtained
with SA, MET, and IET [50].However, the comparison of all the theories (of
contact multiparticle quenching by point particlesB) was done later for any ξ (Figure 9) [47]. Theoretically, κ increases with concentration from the IET
value κi up to the kinetic rate constant k0≡ka, though experimentally available is only the lower (left) part of the graph, cv<1. Contact MET, which is identical to MPK3 developed later [29], underestimates the exact κ(c) represented by DET. The latter is equivalent
to the irreversible version of the first multiparticle kernel theory, MPK1
[27]. The intermediate
version of this theory [28],
MPK2, as well as the latest model theory SCRTA [34], almost coincide with DET =
MPK1 (irreversible), unlike the linearized extended superposition
approximation [51]
(LESA) which overestimates κ(c).Figure 9
The concentration dependence of the irreversible
quenching Stern-Volmer constantκ in units of k0≡ka for a number of contact theories, provided ka=3.7⋅1010M−1s−1 is the same for all of
them [47] and [48, Figure 3.88].


## 3.3. IET of Reversible Geminate Reaction

Let us now turn
to the geminate reaction similar to that included in scheme (49) but carried
out by reversible electron transfer:
(55)
Here D is the donor of an electron, while A is its acceptor. The excitation of A by the short light pulse, I(t),
resulting in charge separation, produces the free ions, D+ and A−,
with the yield ϕ.
This is the fraction φ¯ of ions initially born in amount 1−η but separated escaping recombination to the
ground state with the rate WR and the backward electron transfer to excited
products with the rate WB.IfA's are present in very low concentration, the
density of the free ions is small as well, so that their recombination in the
bulk is negligible during a bounded time domain comparable with encounter time τd~σ2/D.
The IET equations (31) can be reduced for this case by omitting the bulk terms
quadratic in P,
as well as the pumping term:(56)N˙A*=−c∫0tR*(τ)NA*(t−τ)dτ−NA*(t)τA,P˙=c∫0tR†(τ)NA*(t−τ)dτ. Instead of the pumping term one has to add the initial
conditions for these equations which represent the instantaneous excitation of A:(57)NA*(0)=N0,P(0)=0.The luminescence of the pulse excitedA* quenched in a limited time has the yield
specified by (44) which completely neglects the subsequent delayed
fluorescence resulting from the bulk recombination. Such a yield obeys the
Stern-Volmer law (45) but with the geminate quenching constant kq≡κg.
The latter is concentration-independent unlike its analog in (46)
accounting for the backward energy transfer during bulk encounters.Hence, the luminescence quenched by reversible
ionization after instantaneous excitation has the yield calculated from
(45) and (56) [52]:(58)η=11+cκgτA,where the geminate Stern-Volmer
constant is [43, 52](59)κg=R˜*(0)=κi1+κi/K(kD+kc).Here κi is the Stern-Volmer constant for irreversible
transfer (51) while(60)K=kakb=exp(−ΔGiT)is an equilibrium constant for
ionization, with free energy ΔGi (here and further on the Boltzmann constant kB=1). The kinetic reaction constants of the
outer-sphere electron transfer(61)ka=ka′e−ua/T,kb=kb′e−ub/Tobey the Arrhenius law with
activation energies satisfying the free energy gap (FEG) law [9]:(62)ua=ub+ΔGi=(ΔGi+λc)24λc,where λc is the contact reorganization energy of the
polar media. The charge recombination constant also obeys the FEG law:(63)kc=kc′e−uc/Twithuc=(ΔGr+λc)24λcbut with different free energy −ΔGr=ℰ+ΔGi where ℰ is the excitation energy of A*.For highly exergonic charge separation (−ΔGi>T) when K→∞ the reaction (55) becomes irreversible, that
is, κg→κi which is the same for the irreversible
transfer of either energy or electron. In the opposite limit (K≲1), the transfer can also be irreversible
provided the charge separation or their geminate recombination is fast: either kD→∞ or kc→∞.
Otherwise, the distribution between neutral and charged reactants is
equilibrated and the luminescence having the Stern-Volmer constant κg≈K(kD+kc) disappears simultaneously with the ion pair
that either separates or recombines.


## 3.4. Stationary Luminescence

The pumping light intensity isI=(N0/NA)δ(t) in the case of pulse excitation but I=I0=const when the fluorescence is an induced and
studied stationary. In the former case, the quantum yield η has to be calculated from formula (45) while
in the latter case it is defined in another way [53, 54]:(64)η=NAsI0NAτA=11+cκ0τA,where NAs is the stationary density of the excited
states. If the fluorescence is quenched by reversible intramolecular transfer
according to scheme (24), then both recipes were shown to give the same
result [42]. A different
situation arises when the quenching is performed by the intermolecular transfer
whose charged products recombine in the bulk restoring the excitation according
to scheme (30). In such a case, NAs can be found from the set of corresponding IET
equations (31) setting N˙A*=P˙=0.
Substituting the stationary solution thus obtained into (64) we get the
corresponding Stern-Volmer constant(65)κ0=R˜*(0)[1−R˜♯(0)R˜‡(0)×R˜†(0)R˜*(0)]=κg[1−χ⋅φ¯].It is smaller than the geminate
one (κg=R˜*(0)) because not all the excitations are quenched
forever at first encounter. Some of them are restored with an efficiency(66)χ=R˜♯(0)R˜‡(0)in the subsequent bulk
encounters of the free ions, which are separated with the yield(67)φ¯=R˜(0)R˜*(0).All the components of (65)
are well defined via the IET kernels.Calculated in the contact approximation, they reduce
expression (65) to the following one [43, 54]:(68)κ0=κi1+κi/Kkc.The principle difference between
this result and the geminate one, (59), is the absence of kD in the denominator. Diffusional ion pair
separation cannot make the stationary energy quenching irreversible. The charge
separation does not put an end to the reversible reaction, though interrupts it
for a while. Only the irreversible recombination to the ground state of the
neutral products proceeding with rate constant kc causes this to happen. If kc→∞,
the ionization is fully irreversible, that is, κ0≡κi Figure 12.On the contrary, atkc=0 the quenching is reversible, that is, the
fluorescence is not quenched at all (κ0=0). Almost the same is true when K≫κi/kc,
where K and kc were specified in (60) and (63). The
quasireversible ionization is controlled by RIP recombination which proceeds
with the rate constant(69)κ0≈Kkc=kc′e−U/T.The total activation energy of
such a reaction,(70)U=ΔGi+uc,becomes negative at highly
exergonic transfer (when |ΔGi|>uc).


## 3.5. Association/Dissociation of the Exciplex

A different
situation arises when the luminescence is interrupted by the reversible
association ofA* with impurity (B) present in great excess, resulting in
exciplex (C*) formation [55]:
(71)
Here τC and τA are the decay times of bound and unbound
excitation that may be either equal or different. The densities of the excited
particles obey the set of IET equations:(72)dNA*dt=−cka∫0tΣ(t−τ)NA*(τ)dτ+kc∫0tΣ(t−τ)NC*(τ)dτ−NA*τA,dNC*dt=cka∫0tΣ(t−τ)NA*(τ)dτ−kc∫0tΣ(t−τ)NC*(τ)dτ−NC*τC.
The luminescence quantum yields
of A* and C* are(73)ηAA=1τA∫0∞NA*(t)dt/N0,ηCA=1−ηAA,provided that only A is subjected to instantaneous light
excitation: NA*(0)=N0, NC*(0)=0.The conventional Stern-Volmer law,(74)1ηAA=1+cκτAκ=kaΣ˜(0)1+kcτCΣ˜(0),has the Stern-Volmer constant κ,
which depends on the concentration via Σ˜(0,c).
Only its minimal value Σ˜0=Σ˜(0,0) calculated with IET from (73) and (72) is
concentration-independent:(75)κ0=limc→0κ(c)=kaΣ˜01+kcτCΣ˜0Σ˜0=[1+kakD(1+τd/τA)]−1.For the irreversible binding (kc=0) this κ0=kaΣ˜0 coincides with κi given in (51).Equations (72) as well as the general κ definition given in (74) are common for all
multiparticle theories. They differ only in Σ˜(s) that were collected in [55, Tables 1 and 2] for the target
and trap problem (only A is moving between immobile traps). All Σ˜(0) monotonously increase with c approaching 1. Simultaneously the upper limit
of the Stern-Volmer constant is achieved:(76)κm=ka1+kcτC.Since only a few theories deal
with τA≠τC,
in Figure 10, only the equal lifetimes case is examined. It is easy to see that
all MPK theories and CA give similar results, unlike SCRTA and LESA which
deviate from them into opposite sides. LESA was independently shown to give an
inappropriate description of reversible transfer at equal times [42]. SCRTA in its turn strongly
overestimates the difference between the target and trap problems as compared
to MPK3≡ MET. As to the concentration-independent IET
result, it is always reproduced but only as c→0.
At higher concentration it is better to replace IET with MET.The Stern-Volmer constant of the reversible exciplex
formation atτA=τC=τ as a function of the dimensionless
concentration of B's. In a wide range of concentration (a) all
curves increase from the minimal (IET) value κ0 up to the maximal one, κm.
At low concentrations (b) the difference between them is more pronounced [55].
(a)(b)Quite recently the kinetics of energy quenching by
exciplex formation and resulting free energy dependence of Stern-Volmer
constant were thoroughly investigated with IET and compared with available
experimental data [56].


## 3.6. Electroluminescence

The ions injected from electrodes recombine to either
the ground or excited state of the neutral products. The latter can be detected
by their luminescence and the quantum yield of such an electroluminescence
is(77)ϕecl=ϕeϕes.The quantum yield of
excitations, ϕes,
can be extracted from this relationship since the emission quantum yield from
the excited state, ϕe,
is usually known. In [57–59] the ϕes dependence on the free energy of ionization, ΔGi=−ΔGb,
was measured for a number of systems. To specify this dependence we have to
calculate(78)ϕes=∫0∞NA*(t)dtP(0)τA,borrowing NA*(t) from the solution of (31) where we
set I0=0 and use the appropriate initial conditions
created by the external injection of ions into solution:(79)NA*(0)=0,P(0)=P0.Making the Laplace transformation of  (31) we obtain from (78)(80)ϕes=N˜A*(0)P0τA=1R˜‡(0)/R˜♯(0)+cτA[R˜*(0)R˜‡(0)/R˜♯(0)−R˜†(0)].Using the expressions for all
the kernels obtained in contact approximation in [53] we get the contact analog
of this equation [43]:(81)ϕes=11+(1+cτAκi)kcK/κi.This expression reduces to a
much simpler one provided the recombination into the excited state is
irreversible as was assumed in [57–59]. If this is really the
case, then κi=ka=0 and K/κi=K/ka=1/kb so that (81) takes the form used in these
works:(82)ϕes=11+kc/kb.Using the contact estimates of
the kinetic rate constants given in (61) and (63) we obtain the following final
result:(83)ϕes=[1+kc′kb′exp(−ℰ(ΔGi−λc+ℰ/2)2λcT)]−1. This is the
stepwise function approaching unity when ΔGi increases making the recombination to the
excited state more favorable than to the ground one. Finally, the excitation
becomes the unique reaction product since the recombination to the ground state
is switched off.Being calculated with (81), which accounts for the
transfer reversibility (ionization of excited state), this function appears to
be different from the simplest ones, (82) and (83), suited for
irreversible recombination. Shown in Figure 11 these functions, although they are
different, resemble the experimental results obtained in [57–59]. The correct accounting for the reaction
reversibility is the main but not only advantage of IET, compared to DET and
Markovian chemical kinetics. Taking into account the space dependence of the
rates one should use (80) without contact simplifications and it was really
employed in [43].
Moreover, in the next work the spin states of the free ions and radical ion
pairs (RIPs) formed from them were also taken into account, as well as the spin
conversion in the RIP and recombination to triplet products [60]. Even after that, the full
correspondence with the experimental findings was not reached: the height of
the true plateau remains lower than 1 for unknown reasons.Figure 11
The quantum yield of the excited states,ϕes,
calculated in the contact approximation with (dashed line) and without
(dashed-dotted line), taking into account their ionization [43].The Rehm-Weller plot for a few systems which differ by
their triplet RIP recombination rate. (a) The theoretical curves forkt′/kD=0;10−8;10−5;0.1;20 at kc′=kb′=20kD.
(b) Interpolation through experimental points from [61, Figure 2].
(a)(b)


## 4. Spin-Assisted Complex Reactions

Until now we
considered only the simplest reactions, which are sometimes termed as ABC when
dealing with exciplex formation (71) or ABCD, when addressing either A*BAB*,
or A*BA+B− reactions (7), (49),
and (50). It is rather
easy to study the spinless reactants especially when transfer is contact and
irreversible. However, the real chemistry deals with much more complex reactions
than ABC, ABCD, and so on. They include the reactants with a few internal
states and radical ions subjected to spin conversion and reverse recombination
to their precursors and/or neutral products. Here we confine ourselves to
spin-assisted contact reactions leaving noncontact effect for the next section.Any realistic theory should discriminate between the
singlet and triplet states of the radical-ion pair (RIP) and account for
reversible transitions between them (Figure13), either coherent (Hamiltonian) or
incoherent with the model rate ks.
In general, the same is true for excited states of A:
singlet (A1*) and triplet (A3*), but the internal conversion is neglected here. If there are no
triplet quenchers and the triplets are generated in a low concentration making
their annihilation negligible, then the reaction scheme of reversible triplet
production is the following one:
(84)
where τA and τT are the singlet and triplet life times. The
forward and reverse electron transfer, to and from the excited triplet acceptor A3*, has the rate constants kt and k−t, respectively, which fit the detailed balance principle
(85)k−t=ktexp(−ΔGtT),
where ΔGt is the free energy of triplet ionization.
There are two parallel ways of spin conversion: either within the geminate RIP or
through the bulk where 1/4 of the meeting-free ions associate into the singlet
RIP and 3/4 into the triplet one.Figure 13
The energetic scheme of reversible ionization of
singlet and triplet excitations,A1* and A3*.The corresponding set of integral equations for the
singlet and triplet populations,NS=[1A*] and NT=[3A*],
and the concentration of charges, P=[A−]=[D+],
is the most complex one:
(86)N˙S=−NSτA−c∫0tR*(t−τ)NS(τ)dτ+∫0tR♯(t−τ)P2(τ)dτ+c∫0tR♣(t−τ)NT(τ)dτ,P˙=c∫0tR†(t−τ)NS(τ)dτ−∫0tR‡(t−τ)P2(τ)dτ+c∫0tR⋄(t−τ)NT(τ)dτ,N˙T=−NTτT+c∫0tR♥(t−τ)NS(τ)dτ+∫0tR♠(t−τ)P2(τ)dτ−c∫0tR⋆(t−τ)NT(τ)dτ.
There are 9 different kernels
(memory functions), having separate IET definitions via different auxiliary
functions obeying equations similar to (11). All of them were solved in the
contact approximations. The kernels obtained were published in the appendix of
[54] and later on in
[62], where the general
solution of the problem is given for both singlet and triplet luminescence.


### 4.1. Zero Spin Conversion

When the spin
conversion in geminate pairs is rather slow, it can be neglected compared to
the parallel track of triplet production; via bulk recombination of free ions
into triplet RIPs:(87)Under permanent illuminationI(t)=I0=const,
there are stationary populations of all species, NSs, Ps, NTs,
which obey the set of equations following from (86):(88)NSsτA−I0NA=−c∫0tR*(t−τ)NSs(τ)dτ+∫0tR♯(t−τ)Ps2(τ)dτ+c∫0tR♣(t−τ)NTs(τ)dτ,0=c∫0tR†(t−τ)NSs(τ)dτ−∫0tR‡(t−τ)Ps2(τ)dτ+c∫0tR⋄(t−τ)NTs(τ)dτ,NTsτT=+c∫0tR♥(t−τ)NSs(τ)dτ+∫0tR♠(t−τ)Ps2(τ)dτ−c∫0tR⋆(t−τ)NTs(τ)dτ. After Laplace transformation,
they become the algebraic equations for NSs and other populations that can be easily
found. Using the former in (64) instead of NAs,
one can reproduce not only the general Stern-Volmer law but specify its
constant (65) as well. The components of the latter are well defined through
the Laplace transformations of the memory functions (kernels of (86)) [62].At zero spin conversion (ks=0), they are(89)κg=R˜*(0),χ=R˜♯(0)R˜‡(0)⋅R˜⋆(0)R˜‡(0)R˜⋆(0)R˜‡(0)−R˜♠(0)R˜⋄(0),φ¯=R˜†(0)R˜*(0). Calculating
them in contact approximation, neglecting triplet decay, the following
Stern-Volmer constant was obtained [62]:(90)κ0=κi1+κi/Kkc={κi=ka1+ka/kD*askc→∞(ionizationcontrol),kcKaskc→0(recombinationcontrol),where kD*=kD/(1+τd/τA).
The reversible production of stable triplets does not affect this dependence
identical to the spinless theory result (68).As always one should discriminate between the
ionization and recombination-controlled quenching. Under ionization control,
the free energy dependence oflnκ0 reproduces that of lnκi,
which coincides with the bell-shaped FEG curve, lnka=−ua(ΔGi)/T,
the top of which is cut by the diffusional plateau, lnκi=lnkD*.
With increasing ΔGi,
the irreversible singlet ionization becomes quasireversible and gives a way to
recombination control, which turns down the free energy dependence of the
Stern-Volmer constant and, the earlier, the slower is singlet RIP
recombination (Figure 14).  After the permanent illumination is suddenly switched off (as
well as after δ-pulse excitation), the luminescence goes out
together with singlet excitations, NS(t).
However, their population temporarily restores by recombination of ions into
singlet: until the latter are reproduced by reverse electron transfer from the
long-lived triplets. In this time domain, the reaction (87) proceeds from the
right to left backing delayed luminescence.Figure 14
The experimental results from [63] for the Stern-Volmer
constants κ0(▲,•) and triplet quenching constants kQ(Δ,∘) fitted in [62]. The black lines are κ0 and red ones are kQ.
The triangles and upper curves, calculated for strong electron transfer (ka=kt=30kD),
are related to Lumicrome (LC) quenched by aromatic donors in methanol. The
circles and the lower curves, obtained for weak kinetic controlled transfer (ka=kt=0.5kD), belong to the LC quenched by aliphatic
amines in the same solvent. The blue line shows the Stern-Volmer constant for
irreversible ionization of singlet LC, κi,
at the very same parameters.Since the decay of singlets is the fastest process,
the recombination of triplets proceeds quasistationary, so that,NS(t)∝P2(t)∝NT(t) and(91)N˙T=−NTτT−ckQNT,wherekQ=R˜⋆(0)[1−R˜♠(0)R˜‡(0)].In the contact approximation [62],(92)kQ=k−t1+κi/kcK1+κi/kcK+(k−t+4kt)/kD+3kt/kc.The quasi-exponential quenching
of triplets was actually detected recently in line with the luminescence
quenching [63]. Some
of the data obtained are shown in Figure 14. The circles and the lower curves
obtained for quenching of Lumicrome (LC) by aliphatic amines relate to the
kinetic-controlled transfer following classical FEG law which is the same for
the Stern-Volmer and triplet quenching constants. The triangles and upper
curves, obtained for LC quenched by aromatic donors relate to stronger electron
transfer represented by diffusion by diffusional plateau which is higher for
singlet quenching subjected to transient effect. Unfortunately, both plateaus
are too long extending even into endothermic
region. This paradox was solved only recently assuming that the quenching is
due to exciplex formation [56].


### 4.2. Double-Channel Geminate Recombination/Separation

If the spin
conversion is efficient, the geminate reaction followed pulse excitation
produces a number of ions and triplets. The first experimental study of such a
double-channel reaction [64] raised a few questions about the accumulation
kinetics and quantum yields of its products in solvents of different
viscosities. The reaction scheme for the double-channel geminate reaction is
easy to get from a general one, (84), by omitting all the bulk
reactions:
(93)The yields of triplet productsφT and charge separation yield φ¯ are two measurable quantities, while the yield
of the ground state recombination products, φS,
can be easily calculated from them: φS=1−φT−φ¯ [64]. All yields are obtained from the limiting values of
excitation, ions and triplet populations, which obey the following set of
reduced equations:(94)N˙S=−NSτA−c∫0tR*(t−τ)NS(τ)dτ,NS(0)=N0,P˙=c∫0tR†(t−τ)NS(τ)dτ,P(0)=0,N˙T=−NTτT+c∫0tR♥(t−τ)NS(τ)dτ,NT(0)=0. In reality, τT is the longest time that may be set infinite,
when the geminate reaction is studied. In such a case, one can easily find from
the Laplace transformed equations (88) that(95)N˜S(0)=N0τA1+cτAR˜*(0),P(∞)=lims→0sP˜(s)=cR˜†(0)N˜S(0)=N0ψφ¯,NT(∞)=lims→0sN˜T(s)=cR˜♥N˜S(0)=N0ψφT,where the total yield of ions(96)ψ=1−η=cκgτA1+cκgτA.Here κg=R˜*(0) is the same as in (89) while [62](97)φ¯=R˜†(0)R˜*(0),φT=R˜♥(0)R˜*(0).If the spin conversion is
negligible,(98)φ¯=11+kc/kD,φT=0are exactly the same as in the
spinless theory [9].
How these quantities are affected by spin-conversion, either coherent or
incoherent, will be discussed later in the frame of the unified theory (Section
5.5).Equations (88) describe the whole geminate reaction
followed instantaneous creation of singlet excitation at t=0.
This reaction is composed from two sequential stages: accumulation and
dissipation of ions. Another situation appears when the ions are created by a
straightforward pulse excitation at the moment t=0.
If the highly positioned singlet excitation is out of game (ka=kb=0), the set of (94) is reduced to the
following one:(99)P˙=c∫0tR†(t−τ)NS(τ)dτ,P(0)=P0,N˙T=−NTτT+c∫0tR♥(t−τ)NS(τ)dτ,NT(0)=0.The natural triplet decay is
accompanied by their quenching by ionization and subsequent recombination of
ions to either ground or triplet states.


### 4.3. Magnetic Field Effect in Double-Channel RIP Recombination

If besides that the ionization of triplet is also
negligible (k−t=0), then the geminate reaction scheme (93)
reduces to the simplest one:
(100)
Since any electron transfer in
this scheme is irreversible there is an alternative to IET how to find the
solution of the problem by conventional methods of quantum chemistry [65]. In contact approximation,
the total populations of charged and triplet products may be expressed via pair
correlation functions of singlet and triplet RIPs, mS(r,t) and mT(r,t):(101)P(t)=∫[mS(r,t)+mT(r,t)]d3r,N˙T=ktmT(0,t).The rate equations for these
densities account for spin conversion in ion pair born at r=r0:(102)m˙S=−3ksmS+ksmT+L^mS,mS(r,0)=δ(r−r0)4πr2,m˙T=3ksmS−ksmT+L^mT,mT(r,0)=0,where(103)L^=D˜1r2∂∂rr2e−U(r)/kBT∂∂reU(r)/kBTis an operator of the encounter
diffusion (with coefficient (D˜)) in the interparticle potential U(r).
For an ion-radical pair this is the Coulomb interaction U(r)=−rc/r,
with Onsager radius rc=e2/ϵkBT (at temperature T and dielectric constant ϵ). This is the simplest but widely used
elementary spin model (ESM) of incoherent spin-conversion in RIP with a single
phenomenological parameter (rate) ks [9, 16].The ion recombination into ground and triplet states
can be taken into account via boundary conditions to (102):(104)4πσ2D˜erc/r∂∂re−rc/rmS|σ=kcmS,4πσ2D˜erc/r∂∂re−rc/rmT|σ=ktmT. They represent the double-channel irreversible
recombination at contact.Equations (102) may be also written in operator form
for two-component vector (mSmT) where mT=m−+m0+m+ is actually a sum of three triplet components
populations. For any particular mechanism of incoherent spin-conversion, the
true rate equations should be written for either four-component
vector(105)(mSm0m−m+)ordensitymatrixρ=(mS*****m0*****m−*****m+)
with account of real transitions
between the components peculiar to the chosen mechanism. For instance, the
stochastic Liouville equation for the density matrix of Ruthenium complex was
specified in [66, 67] for
the Δg-mechanism of spin conversion governed by the
spin Hamiltonian(106)H=12(ω+0000−ω+00000ω−00ω−0). Here(107)ω±=(g+±g−)βℏH,β is the Bohr magneton, g+ and g− are g-factors of positive and negative ions and H is the magnetic field. The relaxation
superoperator that has the rank 16×16 in the Liouville space was specified in
[66, 67] as well.For moderate magnetic field and relatively fast
transversal relaxation rate(108)ω−=ΔgβHℏ≪1T2, the quasistationary
solution for all off-diagonal elements is allowed [9, Section VIII A]. It reduces the coherent
description of density matrix evolution to the incoherent one which substitutes
the phenomenological ESM set (102) by the more appropriate one [9, 68, 69]:(109)m˙S=(k0+1T2−12T1)m0−(k0+1T2+12T1)mS+m++m−2T1+L^mS,m˙0=(k0+1T2−12T1)mS−(k0+1T2+12T1)m0+m++m−2T1+L^m0,m˙+=mS+m02T1−m+T1+L^m+,m˙−=mS+m02T1−m−T1+L^m−,where the ab-initio-derived rate
of spin-conversion is(110)k0=ω−2T2.The rest depends on the relationship between the
transversal (T2) and longitudinal (T1) relaxation times. At T1=∞ the general set (109) reduces to only two
equations for S and T0 components [68] which are converting with the rate(111)ks=k0+1T2.On the other hand, for T2=T1=T0 the ESM equations (102) are approved but with
another conversion rate [68]:(112)ks=12T0+13k0.It was proven by the exact
analytic solution of the general set (109) for a contact born pair (r0=σ). The comparison of exact double-channel
solution from [70]
with that of ESM obtained in [64] showed that they are exactly the same only at zero
magnetic field (k0=0) when conversion is carried on by the
transversal relaxation but differ a bit even in the lowest order approximation
with respect to(113)k0T0=(ω−T0)2≪1,which is quadratic in magnetic
field (ω−∝H).To remove any limitations, on magnetic field or other
strength of spin conversion, one should give up the incoherent (rate)
description of such a process and dill with the original operator equations in
Liouville space. This was first done for a very special case of
“spin-independent recombination” when RIP recombination through both
channels is the same:kc=kt [71]. The authors considered simultaneous action of Δg and HFI mechanisms of spin-conversion. The former was
considered separately later on setting kt=0 that is leaving only single (singlet)
recombination channel [69]. The double recombination via Δg-mechanism, at any kc and kt,
was considered rather recently [72, 73]. The exact analytical solution of this problem essentially
corrects the results obtained earlier with incoherent approximation even within
the limits of their validity. This indicates the main weakness of the rate
theory first reducing the coherent spin conversion to incoherent and only then
accounting for the encounter diffusion and recombination of radicals. The exact
theory does the opposite: first solves the problem by simultaneously taking
into account the relative motion, recombination and conversion and only then
turn to a limit where the incoherent approximation is assumed to be right.
Unfortunately, this is true for only zero field case while the magnetic field
effect is reproduced by ESM only qualitatively (Figure 15) and only for law fields
limited by inequality (113).Figure 15
The field dependence of the MFE at contact start in
the exact theory (solid line) and in the elementary spin model (dashed-dotted
parabolic line). The vertical line separates the low field (incoherent)Ω-dependence from the high field MFE,
originating from the coherent spin conversion. The latter is well interpolated
by the empirical formula shown as the dashed curve approaching the exact result
from above. The highest field asymptotic behavior and its limit, −Γ,
are shown by the dotted lines below. The rate of contact recombination, zs,
and other parameters are the same as in the previous figure while D=10−6cm2/s. (ω−=2Ω.)


### 4.4. Viscosity Dependence of the Double-Channel Recombination Yields

The coherent HFI-induced spin conversion was studied
even earlier for a simplified model of a single-proton spinI^ interacting with the electron spin S^1 of that ion-radical where the proton is
located [74]. The
corresponding term in spin Hamiltonian of RIP, AI^S^1,
contains the HFI constant A which is the fitting parameter of the system
instead of ks.
Accounting for this interaction and electron spin exchange at the closest
distance, this problem was also solved analytically for zero field, and
viscosity dependence of the yields (97) was specified at different starting
distance r0. The diffusional dependence of φT compared with its ESM analogs was shown to be
very similar (see [74, Figure 1]). The closest similarity was reached after special
investigation of this problem in [75] accounting for Coulomb interaction between the ions.
The triplet yield dependence on Onsager radius as well as on encounter
diffusion was shown to be practically the same provided the phenomenological
spin conversion rate ks relates to HFI constant A as follows:(114)ks=A32.The diffusional dependencies of
charge separation and triplet production yields calculated with a model
(incoherent) theory and its coherent (HFI) analog are shown also in Figure 16.
Their coincidence under condition (114) is almost perfect, but the advantage of
coherent theory is the possibility to get A independently, from the ESR spectroscopy, and
thus to verify the fitting objectively if A is the same.Figure 16
The charge separation,φ¯,
and triplet quantum yield, φT,
as functions of diffusion started from r0=7.55Å>σ (σ=7.5Å). At zero magnetic field the recombination
rates through singlet and triplet channels are kc=4800Å3/ns and kt=50000Å3/ns while the phenomenological rate of spin
conversion is ks=0.15ns−1 versus HFI constant A=4.8ns−1.
The solid lines result from the coherent theory of HFI-induced spin
conversion, while the dashed lines are obtained with the model (incoherent)
theory at ks=A/32.


## 4.1. Zero Spin Conversion

When the spin
conversion in geminate pairs is rather slow, it can be neglected compared to
the parallel track of triplet production; via bulk recombination of free ions
into triplet RIPs:(87)Under permanent illuminationI(t)=I0=const,
there are stationary populations of all species, NSs, Ps, NTs,
which obey the set of equations following from (86):(88)NSsτA−I0NA=−c∫0tR*(t−τ)NSs(τ)dτ+∫0tR♯(t−τ)Ps2(τ)dτ+c∫0tR♣(t−τ)NTs(τ)dτ,0=c∫0tR†(t−τ)NSs(τ)dτ−∫0tR‡(t−τ)Ps2(τ)dτ+c∫0tR⋄(t−τ)NTs(τ)dτ,NTsτT=+c∫0tR♥(t−τ)NSs(τ)dτ+∫0tR♠(t−τ)Ps2(τ)dτ−c∫0tR⋆(t−τ)NTs(τ)dτ. After Laplace transformation,
they become the algebraic equations for NSs and other populations that can be easily
found. Using the former in (64) instead of NAs,
one can reproduce not only the general Stern-Volmer law but specify its
constant (65) as well. The components of the latter are well defined through
the Laplace transformations of the memory functions (kernels of (86)) [62].At zero spin conversion (ks=0), they are(89)κg=R˜*(0),χ=R˜♯(0)R˜‡(0)⋅R˜⋆(0)R˜‡(0)R˜⋆(0)R˜‡(0)−R˜♠(0)R˜⋄(0),φ¯=R˜†(0)R˜*(0). Calculating
them in contact approximation, neglecting triplet decay, the following
Stern-Volmer constant was obtained [62]:(90)κ0=κi1+κi/Kkc={κi=ka1+ka/kD*askc→∞(ionizationcontrol),kcKaskc→0(recombinationcontrol),where kD*=kD/(1+τd/τA).
The reversible production of stable triplets does not affect this dependence
identical to the spinless theory result (68).As always one should discriminate between the
ionization and recombination-controlled quenching. Under ionization control,
the free energy dependence oflnκ0 reproduces that of lnκi,
which coincides with the bell-shaped FEG curve, lnka=−ua(ΔGi)/T,
the top of which is cut by the diffusional plateau, lnκi=lnkD*.
With increasing ΔGi,
the irreversible singlet ionization becomes quasireversible and gives a way to
recombination control, which turns down the free energy dependence of the
Stern-Volmer constant and, the earlier, the slower is singlet RIP
recombination (Figure 14).  After the permanent illumination is suddenly switched off (as
well as after δ-pulse excitation), the luminescence goes out
together with singlet excitations, NS(t).
However, their population temporarily restores by recombination of ions into
singlet: until the latter are reproduced by reverse electron transfer from the
long-lived triplets. In this time domain, the reaction (87) proceeds from the
right to left backing delayed luminescence.Figure 14
The experimental results from [63] for the Stern-Volmer
constants κ0(▲,•) and triplet quenching constants kQ(Δ,∘) fitted in [62]. The black lines are κ0 and red ones are kQ.
The triangles and upper curves, calculated for strong electron transfer (ka=kt=30kD),
are related to Lumicrome (LC) quenched by aromatic donors in methanol. The
circles and the lower curves, obtained for weak kinetic controlled transfer (ka=kt=0.5kD), belong to the LC quenched by aliphatic
amines in the same solvent. The blue line shows the Stern-Volmer constant for
irreversible ionization of singlet LC, κi,
at the very same parameters.Since the decay of singlets is the fastest process,
the recombination of triplets proceeds quasistationary, so that,NS(t)∝P2(t)∝NT(t) and(91)N˙T=−NTτT−ckQNT,wherekQ=R˜⋆(0)[1−R˜♠(0)R˜‡(0)].In the contact approximation [62],(92)kQ=k−t1+κi/kcK1+κi/kcK+(k−t+4kt)/kD+3kt/kc.The quasi-exponential quenching
of triplets was actually detected recently in line with the luminescence
quenching [63]. Some
of the data obtained are shown in Figure 14. The circles and the lower curves
obtained for quenching of Lumicrome (LC) by aliphatic amines relate to the
kinetic-controlled transfer following classical FEG law which is the same for
the Stern-Volmer and triplet quenching constants. The triangles and upper
curves, obtained for LC quenched by aromatic donors relate to stronger electron
transfer represented by diffusion by diffusional plateau which is higher for
singlet quenching subjected to transient effect. Unfortunately, both plateaus
are too long extending even into endothermic
region. This paradox was solved only recently assuming that the quenching is
due to exciplex formation [56].


## 4.2. Double-Channel Geminate Recombination/Separation

If the spin
conversion is efficient, the geminate reaction followed pulse excitation
produces a number of ions and triplets. The first experimental study of such a
double-channel reaction [64] raised a few questions about the accumulation
kinetics and quantum yields of its products in solvents of different
viscosities. The reaction scheme for the double-channel geminate reaction is
easy to get from a general one, (84), by omitting all the bulk
reactions:
(93)The yields of triplet productsφT and charge separation yield φ¯ are two measurable quantities, while the yield
of the ground state recombination products, φS,
can be easily calculated from them: φS=1−φT−φ¯ [64]. All yields are obtained from the limiting values of
excitation, ions and triplet populations, which obey the following set of
reduced equations:(94)N˙S=−NSτA−c∫0tR*(t−τ)NS(τ)dτ,NS(0)=N0,P˙=c∫0tR†(t−τ)NS(τ)dτ,P(0)=0,N˙T=−NTτT+c∫0tR♥(t−τ)NS(τ)dτ,NT(0)=0. In reality, τT is the longest time that may be set infinite,
when the geminate reaction is studied. In such a case, one can easily find from
the Laplace transformed equations (88) that(95)N˜S(0)=N0τA1+cτAR˜*(0),P(∞)=lims→0sP˜(s)=cR˜†(0)N˜S(0)=N0ψφ¯,NT(∞)=lims→0sN˜T(s)=cR˜♥N˜S(0)=N0ψφT,where the total yield of ions(96)ψ=1−η=cκgτA1+cκgτA.Here κg=R˜*(0) is the same as in (89) while [62](97)φ¯=R˜†(0)R˜*(0),φT=R˜♥(0)R˜*(0).If the spin conversion is
negligible,(98)φ¯=11+kc/kD,φT=0are exactly the same as in the
spinless theory [9].
How these quantities are affected by spin-conversion, either coherent or
incoherent, will be discussed later in the frame of the unified theory (Section
5.5).Equations (88) describe the whole geminate reaction
followed instantaneous creation of singlet excitation at t=0.
This reaction is composed from two sequential stages: accumulation and
dissipation of ions. Another situation appears when the ions are created by a
straightforward pulse excitation at the moment t=0.
If the highly positioned singlet excitation is out of game (ka=kb=0), the set of (94) is reduced to the
following one:(99)P˙=c∫0tR†(t−τ)NS(τ)dτ,P(0)=P0,N˙T=−NTτT+c∫0tR♥(t−τ)NS(τ)dτ,NT(0)=0.The natural triplet decay is
accompanied by their quenching by ionization and subsequent recombination of
ions to either ground or triplet states.


## 4.3. Magnetic Field Effect in Double-Channel RIP Recombination

If besides that the ionization of triplet is also
negligible (k−t=0), then the geminate reaction scheme (93)
reduces to the simplest one:
(100)
Since any electron transfer in
this scheme is irreversible there is an alternative to IET how to find the
solution of the problem by conventional methods of quantum chemistry [65]. In contact approximation,
the total populations of charged and triplet products may be expressed via pair
correlation functions of singlet and triplet RIPs, mS(r,t) and mT(r,t):(101)P(t)=∫[mS(r,t)+mT(r,t)]d3r,N˙T=ktmT(0,t).The rate equations for these
densities account for spin conversion in ion pair born at r=r0:(102)m˙S=−3ksmS+ksmT+L^mS,mS(r,0)=δ(r−r0)4πr2,m˙T=3ksmS−ksmT+L^mT,mT(r,0)=0,where(103)L^=D˜1r2∂∂rr2e−U(r)/kBT∂∂reU(r)/kBTis an operator of the encounter
diffusion (with coefficient (D˜)) in the interparticle potential U(r).
For an ion-radical pair this is the Coulomb interaction U(r)=−rc/r,
with Onsager radius rc=e2/ϵkBT (at temperature T and dielectric constant ϵ). This is the simplest but widely used
elementary spin model (ESM) of incoherent spin-conversion in RIP with a single
phenomenological parameter (rate) ks [9, 16].The ion recombination into ground and triplet states
can be taken into account via boundary conditions to (102):(104)4πσ2D˜erc/r∂∂re−rc/rmS|σ=kcmS,4πσ2D˜erc/r∂∂re−rc/rmT|σ=ktmT. They represent the double-channel irreversible
recombination at contact.Equations (102) may be also written in operator form
for two-component vector (mSmT) where mT=m−+m0+m+ is actually a sum of three triplet components
populations. For any particular mechanism of incoherent spin-conversion, the
true rate equations should be written for either four-component
vector(105)(mSm0m−m+)ordensitymatrixρ=(mS*****m0*****m−*****m+)
with account of real transitions
between the components peculiar to the chosen mechanism. For instance, the
stochastic Liouville equation for the density matrix of Ruthenium complex was
specified in [66, 67] for
the Δg-mechanism of spin conversion governed by the
spin Hamiltonian(106)H=12(ω+0000−ω+00000ω−00ω−0). Here(107)ω±=(g+±g−)βℏH,β is the Bohr magneton, g+ and g− are g-factors of positive and negative ions and H is the magnetic field. The relaxation
superoperator that has the rank 16×16 in the Liouville space was specified in
[66, 67] as well.For moderate magnetic field and relatively fast
transversal relaxation rate(108)ω−=ΔgβHℏ≪1T2, the quasistationary
solution for all off-diagonal elements is allowed [9, Section VIII A]. It reduces the coherent
description of density matrix evolution to the incoherent one which substitutes
the phenomenological ESM set (102) by the more appropriate one [9, 68, 69]:(109)m˙S=(k0+1T2−12T1)m0−(k0+1T2+12T1)mS+m++m−2T1+L^mS,m˙0=(k0+1T2−12T1)mS−(k0+1T2+12T1)m0+m++m−2T1+L^m0,m˙+=mS+m02T1−m+T1+L^m+,m˙−=mS+m02T1−m−T1+L^m−,where the ab-initio-derived rate
of spin-conversion is(110)k0=ω−2T2.The rest depends on the relationship between the
transversal (T2) and longitudinal (T1) relaxation times. At T1=∞ the general set (109) reduces to only two
equations for S and T0 components [68] which are converting with the rate(111)ks=k0+1T2.On the other hand, for T2=T1=T0 the ESM equations (102) are approved but with
another conversion rate [68]:(112)ks=12T0+13k0.It was proven by the exact
analytic solution of the general set (109) for a contact born pair (r0=σ). The comparison of exact double-channel
solution from [70]
with that of ESM obtained in [64] showed that they are exactly the same only at zero
magnetic field (k0=0) when conversion is carried on by the
transversal relaxation but differ a bit even in the lowest order approximation
with respect to(113)k0T0=(ω−T0)2≪1,which is quadratic in magnetic
field (ω−∝H).To remove any limitations, on magnetic field or other
strength of spin conversion, one should give up the incoherent (rate)
description of such a process and dill with the original operator equations in
Liouville space. This was first done for a very special case of
“spin-independent recombination” when RIP recombination through both
channels is the same:kc=kt [71]. The authors considered simultaneous action of Δg and HFI mechanisms of spin-conversion. The former was
considered separately later on setting kt=0 that is leaving only single (singlet)
recombination channel [69]. The double recombination via Δg-mechanism, at any kc and kt,
was considered rather recently [72, 73]. The exact analytical solution of this problem essentially
corrects the results obtained earlier with incoherent approximation even within
the limits of their validity. This indicates the main weakness of the rate
theory first reducing the coherent spin conversion to incoherent and only then
accounting for the encounter diffusion and recombination of radicals. The exact
theory does the opposite: first solves the problem by simultaneously taking
into account the relative motion, recombination and conversion and only then
turn to a limit where the incoherent approximation is assumed to be right.
Unfortunately, this is true for only zero field case while the magnetic field
effect is reproduced by ESM only qualitatively (Figure 15) and only for law fields
limited by inequality (113).Figure 15
The field dependence of the MFE at contact start in
the exact theory (solid line) and in the elementary spin model (dashed-dotted
parabolic line). The vertical line separates the low field (incoherent)Ω-dependence from the high field MFE,
originating from the coherent spin conversion. The latter is well interpolated
by the empirical formula shown as the dashed curve approaching the exact result
from above. The highest field asymptotic behavior and its limit, −Γ,
are shown by the dotted lines below. The rate of contact recombination, zs,
and other parameters are the same as in the previous figure while D=10−6cm2/s. (ω−=2Ω.)


## 4.4. Viscosity Dependence of the Double-Channel Recombination Yields

The coherent HFI-induced spin conversion was studied
even earlier for a simplified model of a single-proton spinI^ interacting with the electron spin S^1 of that ion-radical where the proton is
located [74]. The
corresponding term in spin Hamiltonian of RIP, AI^S^1,
contains the HFI constant A which is the fitting parameter of the system
instead of ks.
Accounting for this interaction and electron spin exchange at the closest
distance, this problem was also solved analytically for zero field, and
viscosity dependence of the yields (97) was specified at different starting
distance r0. The diffusional dependence of φT compared with its ESM analogs was shown to be
very similar (see [74, Figure 1]). The closest similarity was reached after special
investigation of this problem in [75] accounting for Coulomb interaction between the ions.
The triplet yield dependence on Onsager radius as well as on encounter
diffusion was shown to be practically the same provided the phenomenological
spin conversion rate ks relates to HFI constant A as follows:(114)ks=A32.The diffusional dependencies of
charge separation and triplet production yields calculated with a model
(incoherent) theory and its coherent (HFI) analog are shown also in Figure 16.
Their coincidence under condition (114) is almost perfect, but the advantage of
coherent theory is the possibility to get A independently, from the ESR spectroscopy, and
thus to verify the fitting objectively if A is the same.Figure 16
The charge separation,φ¯,
and triplet quantum yield, φT,
as functions of diffusion started from r0=7.55Å>σ (σ=7.5Å). At zero magnetic field the recombination
rates through singlet and triplet channels are kc=4800Å3/ns and kt=50000Å3/ns while the phenomenological rate of spin
conversion is ks=0.15ns−1 versus HFI constant A=4.8ns−1.
The solid lines result from the coherent theory of HFI-induced spin
conversion, while the dashed lines are obtained with the model (incoherent)
theory at ks=A/32.


## 5. Remote Transfer

No matter how
perfect is the fitting of the contact theories to the BD simulations, these
theories are not good enough to treat the real experimental data. The main
advantage of the encounter theories compared to their contact alternatives is
their accounting for the true space dependence of the transfer rates. In the
case of electron transfer, the most common is the usage of the so-called Marcus
transfer rate, which is actually the perturbation theory estimate of the
transition rate between the parabolic terms [76–78]:
(115)W(r)=Uexp(−(ΔG+λ)24λT),where ΔG is constant in highly polar liquids while(116)U(r)=V02ℏexp(−2(r−σ)L)πλT,λ(r)=λ0(2−σr).The contact reorganization
energy,(117)λ0=e2σ(1n2−1ϵ),depends on the static dielectric
constant ϵ and the refraction index n which are usually known but specific for any
particular solvent.The electron couplingV0 and tunneling length L are the main fitting parameters of noncontact
theory, instead of a single contact constant ∫W(r)d3r.
However, the highly exergonic transfer is accompanied by vibrational
excitations of the final state, so that such a multichannel transfer has the
rate(118)W(r)=U(r)∑0∞e−SSnn!exp[−(ΔG+λ+ℏωn)24λT],where S=λq/ℏω,
while ω is the frequency and λq is the reorganization energy of a
single-assisted quantum mode.One should discriminate between the transfer in the
normal Marcus region (−ΔG<λ) and in the inverted one (−ΔG>λ). In the former the rate decreases
monotonously and quasiexponentially [79] with distance (Figure 17), while in the latter it is
bell-shaped and remote but shifts backward to contact if the transfer becomes
multichannel (Figure 18). Two approximate models are used to simplify these
alternative dependencies:(119)W(r)≈{Wcexp(−2(r−σ)l)normalregion,W0ch2(r−RΔ)invertedregion.The exponential model contains
the effective tunneling length l whose analog is Δ in the bell-shaped W(r) proposed in [80]. Keeping it the same, the
bell-shaped curve is sometimes substituted by the rectangular one [16, 81, 82].Figure 17
The distance
dependence of the Marcus transfer rate (115) in the normal region (ΔG=−0.59 eV) with L=1.65Å and λ0=1.3 eV (thick line) in comparison to its
exponential approximations for short (dotted line) and long (dashed-dotted line)
distances. From [79].Figure 18
The rates of ionization in the inverted Marcus region
(−ΔG=2λ0) in a polar solvent (rc=7Å, λ0=1.4ev, σ=5Å, T=300K, ℏω/T=6). The single channel reaction (S=0) is exhibited by the right curve while the
multichannel reaction (S=2) is represented by the left one. The latter
is decomposed in contributions related to different channels (dashed lines)
nominated by the numbers of vibronic states n.


### 5.1. Transient Quenching Kinetics

The
nonexponential kinetics of irreversible energy quenching is represented in DET
by the solution of (3):(120)NA*=N0e−c∫0tkA(t′)dt′.The non-Markovian rate constant kA(t) given in the contact approximation by (2)
diverges at the very beginning: kA(0)=∞.
The same quantity in the encounter theory, (5), tends to kinetic rate constant
at t→0:(121)kA(0)=∫WA(r)d3r=ka,sothatlnNA*N0=−ckat.Only in the opposite limit (at
long times) does the general asymptotic expression for kA(t) have the same shape as (2), except that the
effective reaction radius RQ is substituted for σ.
Integrating this expression in (120), the following long time asymptote for
quenching kinetics is obtained:(122)lnNA*N0=−c[4πRQDt+8RQ2πDt]ast→∞.The first term in the rhs of
this equation represents the final exponential energy quenching with stationary
rate constant k=4πRQD,
while the second one describes the transient effect: the preceding
nonexponential development, though determined by the same RQ.The first attempt to extractRQ from the experimentally studied quenching of
pheophytin a by toluquinone [8] was not flawless. Since the
short times were hardly available for the techniques of those times, the
detected long time quenching was thought to be exponential. Thus obtained the
pseudostationary rate constant k was greatly overestimated, as well as RQ=k/4πD.
Its diffusional dependence was specified using 40 solvents of different
viscosities but the tunneling length extracted from fitting DET to the thus
obtained RQ(D) dependence was also significantly
overestimated.Only 10 years later, the quenching of Rhodamine 3B by
N.N-dimethyleaniline was first recognized as a nonstationary one andRQ was obtained from fitting the asymptote (122)
to better data and in a wider time interval [79]. As can be seen from Figure 19, the current kA(t) is essentially larger
in its asymptotic value ka,
even at the longest attainable time t=4 ns.
Hence, the whole quenching kinetics available in this study is nonstationary
and should be fitted with an appropriate asymptotic expression, (122). The
single quenching parameter RQ was obtained from this fit, employing 7
solvents of different viscosity and such obtained RQ(D) dependence, compared with the theoretical
predictions of the contact theory and DET.Figure 19
The fit of the non-stationary electron transfer
kinetics (thick curve) to the experimental data obtained in [83]. The thin line represents
the tangent to the kinetic curve at the largest time available (t=4 ns). Its slope is the time dependent rate of
ionization, ckA(t),
which differs essentially from that of the purely exponential decay, with the
stationary rate constant ka=4πRQD (dashed straight line). (From [79]).The results of this comparison are shown in Figure20(a).
At fast diffusion (under kinetic control), there is an agreement with all
theories but at slow diffusion the experimental points lay far above the
predictions of Collins-Kimball model following from (4):(123)RQ=σkaka+4πσD.Much better fitting is achieved
with the exponential rate model (119), using the exact analytic expression for RQ(D) available in DET [7]. At fast and slow diffusion
this expression has the following simple asymptotes:(124)RQ={σka4πσDatRQ<σ(kineticcontrol),σ+l2ln(γ2Wcl24D)atRQ>σ(diffusionalcontrol),where γ=exp(C) and C is the Euler constant. However, the real
points are in between these limits and are well fitted with only the general
solution for exponential WA(r) (Figure 20). The latter is also just a short
distance approximation for the true Marcus rate. At larger distances the latter
is also exponential but with true L (Figure 17). The substitution of L≈2l for l in (124) makes the slope of RQ(lnD) at the highest viscosities almost twice as
large but this region was not attainable with the solvents studied (Figure 20(b)).
However, even at moderate viscosities the excitations never reach contact if
ionization is under diffusional control. They are quenched farther apart, at RQ>σ,
and this is a noncontact reaction accessible for only DET or IET of the remote
transfer.The dependence
on diffusion of the effective electron transfer radius,RQ.
(a) The experimental data, indicated by circles, is approximated by a thin line
representing the contact Smoluchowski-Collins-Kimball approach and by thick
line depicting the same dependence, but for the exponential transfer rate from
(119) with l=0.85Å and Wc=180ns−1.
(b) The same but in a wider diffusion diapason. The dashed-dotted line at the
bottom represents the high viscosity asymptote of the same dependence but for
the Marcus transfer rate. (From [79]).
(a)(b)By now, the best fitting of the transient effect and
the whole ionization kinetics was made in [48] where the subpicosecond kinetics of the irreversible
double-channel electron transfer (50) was studied. This is the highly exergonic
quenching of Perylene (Per) by tetracyanoethylene (TCNE) producing the ion
pairs in their ground and excited states. The pulse-induced quenching kinetics NA*(t) was accurately studied in three different time
intervals. The shortest and the longest ones were fitted with asymptotes (121)
and (122), respectively, for specifying ka and RQ while from the middle one the relative
strength of the parallel channels was inferred. The whole kinetics NA*(t) appeared to be fitted perfectly, within the
accuracy of experimental data (Figure 21).Fitting the double channel ionization kineticsNA*=PA*exp(−t/τA) at the shortest time (above) as well as at the
longest and medium ones (below). From [48].
(a)(b)


### 5.2. Stern-Volmer Constant

The concentration dependence of the quantum yields or the Stern-Volmer constant is
actually the central problem under investigation with numerous contact
theories. Yet the noncontact DET corrects essentially even this dependence
obtained with contact DET or its Smoluchowski-like analogs. To make this point
clear a straightforward fitting of the remote transfer theory to the data
presented by Stevens and Biver III [84] was undertaken in [47].The highly exergonic fluorescence quenching of
9,10-dicyanoanthracene (DCNA) by N,N,N′,N′-tetramethyl-p-phenylenediamine (TMPD) was studied in
[84] in the broad
range of TMPD concentrations, at three different temperatures (T=0°,22°,50°C) in acetonitrile solutions. Neither a contact
nor exponential approximation of the ionization rate WI(r) is good for the real fitting of this data.
Instead, one should use the true Marcus rate of electron transfer accounting
for the energy balance and the properties of the solution. The results of the
best fit are shown in Figure 22. The theoretical constants (thick curves),
depending on the dimensionless concentration cv=ξ,
are in good agreement with the experimental points in [84]. The agreement could even be better if a number of
additional factors were taken into account: the liquid structure near the
contact, the spacial dependence of the diffusional coefficients [85], and the chemical
anisotropy which is sometimes essential [86, 87]. Neglecting all these factors, only two fitting
parameters were used, the tunneling length L and the electron coupling element V0,
which are found to be [47]: L=1.6 Å, V0=4.07⋅10−2 eV. However, even the simplest theory of the
remote transfer fits the actual experimental data much better than its contact
analog (thin curves in Figure 22).Figure 22
The experimental concentration dependence of the
Stern-Volmer constant for three different temperatures (points) fitted by DET
with the single-channel Marcus transfer rate (thick lines). The thin lines
represent the contact analogs of the above curves at the same temperatures
(decreasing from top to bottom). From [47].It is also illustrated in Figure23 borrowed from
[48]. The fluorescence
quantum yield obtained in pulse experiment was compared in this work [48] with that measured
stationary (Figure 23(a)). Generally, they are in conformity with each other as
well as with that calculated theoretically. The latter was obtained by
integrating in (44) the best fitted NA*(t) from Figure 21. This integration can be
restricted by the duration of the pulse experiment or extended to infinity
taking into account the delayed fluorescence (contributed into the stationary
luminescence). The Stern-Volmer constants subtracted from all the data are
shown in Figure 23(b). There is only a small difference between the concentration
dependence of κ obtained from the time-resolved (•) and the stationary (⋆) experiments. The coincidence between this
data is best at the highest concentration where the quenching is accomplished
within the credibility time interval. The theoretical description of the whole
concentration dependence is quite satisfactory and this is a great success of
DET that it accounts for distant transfer.The non-linear
Stern-Volmer law (a) and the Stern-Volmer constants (b) obtained from the
time-resolved (•) and stationary (⋆) data in comparison with the theoretical
predictions, following from the quenching kinetics integrated over the
credibility intervals (□) and up to infinite time (red lines). From [48].
(a)(b)Contrary to DET, the contact
Smoluchowski-Collins-Kimball approach is completely unable to fit the same data
even at zero concentration. The ideal (IET) Stern-Volmer constantκ0=κ(0) is underestimated despite the usage of the
same kinetic constant of the double channel reaction (50): ka=k0+k1.
This indicates that the reaction is under diffusional control when the contact κ0con≈kD=4πσD is smaller than its distant analog κ0≈4πRQD,
because RQ>σ.
Indeed, the accurately calculated RQ appears to be 8.6Å  while σ=4Å  [48].There is also an independent confirmation that this is
actually the case. The Per/TCNE reaction is the most exergonic among the
systems collected by Rehm and Weller when they inspected the applicability of
the FEG law [88]. As
was pointed out by Marcus and Siders [89], the top of the parabolic FEG dependence has to be
cut by the diffusional plateau: κ0≈kD.
The Per/TCNE system is located at the left edge of this plateau, confusing
everybody (Figure 24). There the descending branch representing the inverted
region was expected to be detected instead of a plateau. A number of
explanations were proposed for this paradox [37, 90–94] but nobody took the liberty of suggesting that the
most exergonic ionization is still so fast that the reaction remains under
diffusional control.Figure 24
The Rehm-Weller
plot of the Stern-Volmer constant of the double channel ionization,κ0(ΔG),
with a cross (×) marking the position of the Per/TCNE system.
The bell-shaped FEG dependencies of the partial kinetic constants of the ground
state and excited ions production, k0 and k1,
split on the value of the ion pair excitation energy E±*.
From [95].After fitting in [48] the double-channel ionization kinetics, NA*(t),
at all times, we needed no suggestions: with an obtained
set of parameters, the partial constants k0=∫W0d3r and k1=∫W1d3r were calculated without adding any new fitting
parameters [95]. Not
only the sum of these constants but even each of them appear to be larger than kD at the point where Per/TCNE is located (Figure 24). Such a relationship
proves that the ionization is really under diffusion control in this system.
This is the final resolution of the old paradox.If there is no time-resolved data aboutkA(t) in the shortest time diapason but the good
value of k=kA(∞)=4πRQD is obtained, then the quantities RQ and D can be separated, fitting the theory to the
Stern-Volmer constants available from the stationary experiments. This program
was successfully realized in a very recent paper [96]. The lack of information
was filled in by relating the Stern-Volmer constant κ(c) to the DET rate constant, kA(t):(125)11+cκτA=∫0∞e−c∫0tkA(t′)dt′−t/τAdt/τA.Unfortunately, the theoretical κ(c,D) dependence that has to be fitted to the
experimental data strongly depends on how accurately kA(t) is related to the transfer rate, WA(r),
and encounter diffusion D considered as a fitting parameter. This
relationship is generally more complex than (5):(126)kA(t)=∫WA(r)n(r,t)d3r,where n(r,t)=ν(r,t)et/τA obeys the auxiliary diffusional equation
alternative to (6):(127)n˙(r,t)=−WAn(r,t)+1r2∂∂rr2D(r)g(r)∂∂rn(r,t)g(r).The latter has to be solved with
the reflecting boundary condition and initial condition n(r,0)=g(r).
Here g(r) is the equilibrium pair distribution function
accounting for the internal structure of the liquid. Besides, the hydrodynamic
effect is taken into account, which results in the space dependence of the
diffusion coefficient at short distances between reactants. This dependence is
represented by the simple expressions taken from either the Northrup and Hynes
or Deutch and Felderhof models [96]:(128)DNH(r)=D(1−12e−(r−σ)/σ)orDDF(r)=D(1−3σ4r).In fact, the authors of [96] tested 6 different models
of kA(t) calculations (with and without accounting for g(r) and D(r)) and selected only two of them which account
simultaneously for the liquid structure and the space dependence of D(r) (one or another). 150 experimental data points
for κ(c,D) were obtained from the fluorescence quantum
yield studied in the 8 solvents of different compositions (viscosities). Not
only the rate parameters, L and V0,
were found from best fitting the theory to this database, but also the
diffusion coefficient was allowed to vary individually for each solution,
during the fitting procedure. The obtained relationships between diffusion and
viscosity, D(η),
for the models (128) were shown to be very close to the Stokes-Einstein models
for the stick and slip friction coefficient. Specifying D in this way, the authors of [96] got RQ=k/4πD from k=kA(∞) and demonstrated that such obtained RQ(D) dependence is in good agreement with its
theoretical prediction having the asymptotes (124).


### 5.3. Products Accumulation

If the transfer
occurs only at contact, then the products of the reaction (say RIPs) also
appear only there, creating the contact initial condition for the subsequent
geminate evolution. In contrast, the remote transfer proceeding from any
distance creates the broad distribution of charged products around the contact
that can be specified by means of DET. For this goal the differential theory
should be completed by adding an equation for the free ion concentration to
(3):(129a)N˙A*=−kA(t)cNA*−NA*τA,(129b)P˙=kA(t)cNA*,where kA(t) was defined in (5). This set of equations is
the irreversible analog of the IET equations (31).The total number of transfer productsP is related to their pair distribution as
usual:(130)P(t)=c∫m(r,t)d3r.It follows from (129b) and
(5) that(131)m˙−WAνet/τANA*=0,m(r,t)=WA(r)∫0tν(r,t′)et′/τANA*(t′)dt′.The final distribution of
charged products and its normalized version are(132)m(r,∞)=WA(r)∫0∞ν(r,t)et/τANA*(t)dt=m0(r),f0=m0(r)∫m0(r)d3r.This distribution is weakly
concentration dependent but changes dramatically with solvent viscosity
(diffusion).The family of such distributions obtained for
exponentialWA(r) from (119) is shown in Figure 25. At slow
diffusion the distribution looks like a quasisymmetric hill shifted out of
contact. With increasing diffusion it draws nearer to the contact, only
slightly changing shape until ionization remains diffusional. As soon as it
becomes kinetic the distribution approaches the contact, taking the shape of
the ionization rate (exponential in our example). This phenomenon peculiar for
only remote transfer was first studied theoretically [97, 98] and then
specified in a few experimental works [99, 100].Figure 25
The distribution of ion pairs resulting from the
ionization with exponential transfer rate (Wc=29.12ns−1, l=0.81Å) at different encounter diffusion of neutral
precursor: D=10−4,10−6,10−7,10−8,10−9cm2/s (from left to right).


### 5.4. Charge Separation

However, the
accumulated ions are not frozen but start to separate and recombine immediately
after their appearance at starting pointr0.
Their total number, summarizing the contributions from any r0,
decreases with time as(133)P(t)=c∫Ω(t∣r0)m(r0,t)d3r0.Here m(r0,t) is defined in (131), while Ω(t∣r0)=∫m(r,r0,t)d3r is the share of ions starting from distance r0 but having escaped recombination up to time t.
The latter obeys the conjugate diffusional equation [16, 101] for geminate recombination
with distant tunneling rate WR(r) and ion diffusion coefficient D˜:(134)∂∂tΩ(t∣r)=−WR(r)Ω(t∣r)+D˜e−rc/rr2∂∂rr2erc/r∂∂rΩ(t∣r), where rc=e2/ϵT is the Onsager radius of the Coulomb well in
the solvent with dielectric constant ϵ.
Equation (134) should be solved with the following initial and boundary
conditions:(135)Ω(0∣r)=1,∂Ω(t∣r)∂r|r=σ=0,Ω(t∣∞)=1.The quantum yield of charge separation from the
particular starting distancer0 is(136)φ(r0)=Ω(∞∣r0)=11+Z(r0)/D˜,while the total yield of the
free ions is(137)ϕ=P(∞)=c∫φ(r0)m0(r0)d3r0=ψφ¯,where the yield of ionization(138)ψ=c∫m0(r0)d3r0=1−η,and the charge separation yield(139)φ¯=∫φ(r0)f0(r0)d3r0=11+Z/D˜.The last expressions in (136)
and (139) relate the charge separation yields to the partial and total
recombination efficiencies, Z(r0) and Z.
Such a diffusional dependence of the yields is natural for the archaic
“exponential model” which implied that Z=z=const [9, 16]. Though incorporated into
the encounter theory, the recombination efficiency Z(r0) becomes the function of not only D˜ but also r0,
while Z appears to be dependent on both D˜ and D.Two factors determine the type of this dependence: the
shape of the recombination layer and the location of the starting point. The
layer may be adjacent to the contact or shifted out depending on the form of
the distant recombination rateWR(r),
while r0 can be located either inside or outside this
layer or be distributed around it in accordance with f0(r0).For the exponential and rectangular models ofWR(r),
the diffusional dependence of Z(r0) is demonstrated in Figure 26. For starts taken
from contact (or deeply inside the reaction zone), the efficiency of recombination
monotonously decreases with diffusion, separating the ions. In contrast, the
same diffusion accelerates the recombination bringing together the ions born
outside the recombination zone. If initially ions were born inside the reaction
layer but close to its outer boundary, then the initial inhibition of
recombination finally gives way to its acceleration. Only as D˜→∞ the efficiency of recombination, Z∞,
is the same wherever the ions start from.The partial efficiency of geminate recombination,Z(r0),
related to its value at infinite diffusion, Z∞,
for the exponential (a) and rectangular (b) models of the reaction rate.
Contact starts (dashed lines), non-contact starts from reaction zones (solid
lines) and start from outside of the rectangular zone (dotted line).
(a)(b)As a matter of fact, one cannot change the ion
diffusion by varying solvent viscosity but keeping fixed and unique the initial
separationr0.
The diffusion of neutral reactants changes simultaneously with that of ions, D≈D˜∝1/η,
where η is the viscosity varying with solvent
composition or pressure. As a result, the distribution over r0 gradually approaches the contact when
viscosity decreases (Figure 25) and r¯0=∫r0f0(r0)d3r0→σ.
The total efficiency Z shown in Figure 27 is averaged over the current
distributions and therefore depends not only on D˜ but also on D.
When the diffusion is slowest, the start is more remote than recombination and
efficiency increases, with diffusion controlling the reaction. At faster
diffusion the situation changes to the opposite and a further increase in
diffusion reduces the recombination efficiency [102].Figure 27
The diffusional dependence of the recombination
efficiencyZ in the contact approximation (dotted line) at
starting distance r0=1.124σ and the same for the remote recombination in
the normal (solid line) and inverted (dashed line) Marcus regions, in highly
polar solvents. The horizontal dashed-dotted line represents the exponential
model result, Z=z=const.
From [102].However, the maximum which separates these regions is
not sometimes seen. The recombination layer having the shape of the Marcus rate
(115) is broad and far from contact, if the recombination proceeds in the
inverted Marcus region (Figure 18) but attached to contact and narrow in the
normal Marcus region (Figure 17). In the former case the ascending branch of Z(D) gives way to the descending one much earlier
than in the latter and does not give at all if recombination is contact (Figure
27). Only in the latter case is the reaction always under classic diffusional
control (dotted line). Otherwise efficiency passes through the maximum before
approaching its kinetic (i.e., “exponential model”) limit, z.Unfortunately, just detecting the monotonously growingZ(D˜) does not guarantee that the recombination is
contact or at least controlled by diffusion. Simultaneously with diffusion
(viscosity), the variation of solvent composition changes also n and ϵ,
as well as the contact reorganization energy (117) determined by these
parameters. As a result, WR(r) becomes different at different compositions,
as well as the curves Z(D˜),
corresponding to the fixed λ0 particular for the given composition. These
curves are shown by dashed and dotted lines in Figure 28 for the highest and
lowest values of λ0 that were achieved in the system studied by Wolf
et al. [103]. This was
the ruthenium complex quenched by methylviologen (MV++), according to the reaction
scheme
(140)
This reaction was studied in
ionic solutions, when an electrolyte is dissolved in an inert solvent, like
acetonitrile. The viscosity changed with mixture composition.Figure 28
The experimental dependence of the recombination
efficiencyZ on the diffusion coefficient D˜ shown by triangles. The solid line was
calculated theoretically, taking into account that the reorganization energy
changes along with diffusion. The dashed line corresponds to the calculation
with fixed but maximal λ0 reached at the fastest diffusion D˜=1.072×10−5cm2/s.
The dotted line was also calculated with a fixed but slowest diffusion D˜=6.67×10−7cm2/s.
From [102].It can be seen easily from Figure28 that both curves,
calculated at fixed λ0,
pass through the maximum and their viscosity dependence at faster diffusion is
the opposite to that obtained experimentally in this system. The ascending
branches of these curves can be attributed to the diffusion delivery of remote
RIPs into the recombination zone but the experimental points are situated on
the opposite, descending branches of them. Due to relatively fast diffusion,
the forward electron transfer in this system is always under kinetic control
and the RIPs are initially produced almost completely inside the recombination
layer. Hence, the diffusional delivery of RIPs into this layer cannot be
responsible for a pronounced increase in Z with diffusion, as was thought earlier
[104, 105]. It is due to the
collateral change in λ0 and D˜ with solvent composition. All experimental points
belong to the descending branches of the curves which are intermediate between
the dotted and dashed ones. The points shift up when viscosity goes down. The
resulting increase in Z just imitates the situation of
diffusion-controlled recombination.


### 5.5. Unified Theory

More transparent results were obtained with
dimethylsulfoxide-glycerol mixtures, which allow for a wide variation of
solvent viscosity, without changing the parameters which affectλ0.
The system studied in such mixtures was the photo-excited perylene quenched by
aromatic amines:(141)Being detailed, this reaction is
represented by the scheme (93) accounting for the spin states of the reactants
and RIPs. The spinless theory of Section 5.4 is
inadequate to calculate the yields of the free ions and triplet products of the
geminate recombination. The general unified theory (UT) should be employed for
reaching this goal.The original UT links together the geminate RIP
recombination and RIP accumulation accounted for by the DET equation (131)
[97, 98]. Discriminating between singlet
and triplet RIPs, one should consider their pair densities, mS(r,t) and mT(r,t),
governed by a set of equations identical to [9, equations (9.6)]:(142)m˙S−WAνet/τANA*=−WS(r)mS+D˜1r2∂∂rr2erc/r∂∂re−rc/rmS+ksmT−3ksmS,m˙T=−WT(r)mT+D˜1r2∂∂rr2erc/r∂∂re−rc/rmT−ksmT+3ksmS. They have to be solved using the reflecting boundary
conditions and assuming that initially all RIP states were empty: mS(0)=mT(0)=0.
The right-hand side of the set (142) describes the geminate
recombination/separation of RIPs subjected to spin conversion, while the
left-hand side accounts for their generation, via the same pumping term as in
(131). This term composed from the DET quantities, ν and NA*,
should be specified by the preliminary solution of DET equations (6) and (3).
Using these solutions in the UT equations (142) and solving them, one obtains
all that is necessary to calculate the quantum yields of the singlet (ground
state) and triplet products:(143)ϕS(r′)=∫0∞∫WS(r)mS(r,t)d3rdt=ψφ¯S,ϕT(r′)=∫0∞∫WT(r)mT(r,t)d3rdt=ψφ¯T,where ψ is the total RIP yield (138).The yields of singlet and triplet products determine
the efficiencies of recombination through these channels,ZS and ZT:(144)φ¯S=ZSZ+D˜,φ¯T=ZTZ+D˜,while the charge separation
yield is defined, as in (139):(145)φ¯=1−φ¯S−φ¯T=D˜Z+D˜,where Z=ZS+ZT.
In view of the conservation law (145) it suffices to consider only two of the
yields, φ¯ and φ¯T,
which are both available for the experimental detection.In the contact approximation, the UT equations are
available for the analytic solution obtained in [107, 108]. To do the same for remote
transfer, the program Qyield developed by Dr. Krissinel (see
http://www.fh.huji.ac.il/~krissinel/software.html) can be employed. If the
exponential models for both the ionization and recombination rates are used, it
should somehow reflect the fact that the recombination rates are usually more
extended, due to the larger exergonicity of the backward electron transfer.
Accounting for this feature in our recent work we assumed [106] that lR>lI thus makes the
extension of recombination larger than ionization. Under this condition the
total efficiency of remote recombination Z is a nonmonotonous function of diffusion
contrary to what is specific to the contact approximation (Figure 29). This is due
to diffusional deceleration, following the diffusional acceleration of the
recombination. At fastest diffusion the initial ion distribution coincides in
shape with WI(r) which is narrower than WS,T(r).
Therefore, the weaker the backward transfer is, the faster the ions get rid of
the recombination layer interior. Passing the maximum, the total recombination
efficiency Z falls off with D˜,
approaching the plateau Z∞≈z=kc/4πσ.Figure 29
The total (above) and triplet (below) efficiencies of
recombination obtained with contact (dashed lines) and exponential (solid
lines) approximations of the recombination layer. From [106].The efficiency of triplet production also has a
maximum, separating the diffusional control of recombination from the
conversion control that turnsZT to zero, when too fast diffusion leaves no time
for spin conversion. However, this maximum is usually not achievable in the
experimentally available interval of diffusion variation. Fortunately, the
principle difference between remote and contact recombination is seen even in
these limits and is worthy of special attention.The diffusional dependencies of both recombination
efficiencies were subjected to experimental study in [108]. The extremum in Z(D˜) first obtained there was later termed as
Angulo effect. In Figure 30 we clearly see this feature. It gained a preliminary
interpretation within the rectangular model of the recombination rate [102, 108], substituted later by the exponential one [16, 106]. For fitting better the
preexponent, Ws=WS(σ) was not assumed to be equal to Wt=WT(σ) but the relationship Ws=1.2⋅Wt was preferred. Though at the greatest
diffusion, the experimental points deviate a bit from the theoretical curves,
all the rest are fitted quite well.Figure 30
Fitting the theoretical efficiencies (solid lines) to
the real experimental data for total (▲) and triplet (■) recombination efficiencies, using the
exponential models for both the ionization and recombination rates. Parameters
obtained from the best fit are the following: Wi=29.12ns−1, Ws=77ns−1=1.2⋅Wt, lI=0.81Å, lR=1.24Å, ks=0.75ns−1, σ=7.5Å.
Reproduced from Figure 3.75 of the recent review [16].These results were essentially improved later on by
the substitution of the real Marcus rates for their exponential approximations
as well as the true (coherent) Hamiltonian description of spin-conversion for
its rate (incoherent) model [109]. Assuming the HFI induced spin-conversion mechanism
is responsible for the singlet-triplet mixing the true HFI constant A was extracted from the best fitting.
Neglecting the Coulomb interaction between solvated ions the viscosity
dependence of the free ions and triplet quantum yields were specified as well
as their magnetic field dependence. The only demerit was too large fitting
value of A≃9ns−1=505×10−4 T compared to that
expected from ESR data 23×10−4 T. However, it has been shown recently that an
accounting for the Coulomb interaction between the counter ions allows to get
as good fitting but with much smaller A.
Even in contact approximation, A=47ns−1 was obtained at moderate Onsager radius rc=12Å while for the remote electron transfer, A=0.9ns−1 is expected [75].


### 5.6. Adding the Reaction Coordinate

Usually all the encounter theories considering any
kind of the electron transfer between the reactants ignore completely the
internal structure of the latter, either
electronic or vibrational. In fact, the intramolecular relaxation accompanied
the electron transfer is responsible for the hot (transient)
recombination and equilibration of transfer
products, forming the transfer rates for subsequent thermal
relaxation [49]. The
thermal transfer rates of the forward and backward transfer, W(r)=W(x,y,z),
are usually the input data for the encounter theories. In actual fact, one
should start earlier from the Hamiltonian and intraparticle relaxation that
shape W(r).
Nobody can ignore the fact that the reaction develops not only in space but
also along the reaction coordinate q,
towards the term crossing points where it actually proceeds. In fact, only
4-dimensional W(r,q) should be used from the very beginning.The transient processes take some time and develop
coincidentally with the encounter diffusion inr space. The conventional encounter theories
presume the time separation: the faster formation of W(r) compared to its modulation by space diffusion.
This is never the case and the deep advancement of the reaction even during
short times should not be excluded. Both processes were first considered
simultaneously in DET assuming that the ionization proceeds with the rate W(r,q)=A(r)δ(q−q0) in a single crossing point q0 [110]. Next time the UT was extended correspondingly,
taking into consideration not only double-channel ionization at a few crossing
points (q1) but also the hot recombination in even more
numerous intersections (q2) between the ground and excited states of Per/TCNE and Per+/TCNE− (Figure 31) [49]. It was shown experimentally that the hot
recombination preceding the recombination of thermalized RIPs leaves for the
latter only 10% of the initially produced ion pairs (Figure 32).
This phenomenon is well proved by the 4-dimensional UT, although it predicts a
bit slower accumulation than it is in reality. This demerit has been removed
just recently by correcting the reorganization energy of the process making ion
pair accumulation activationless and thus almost instantaneous [111]. Alternatively, it might
be a consequence of the straightforward excitation of excited RIP as
happens sometimes in stable complexes [112]. If this is an effect of
light pumping it should be taken into account as it was done a few times for
other purposes [113, 114].Figure 31
Ground and
excited electronic states of the reactants,DA and D*A,
and charged products, D+A− and D+*A−.
The dotted lines represent vibrational excited states (n=1,2,3), the crossing points for forward transfer
are marked by circles (∘) and for backward transfer by bullets (•). The arrows show the direction of the
intra-molecular vibrational relaxation. From [49].Figure 32
The best fit of
the ion pairs accumulation/recombination kinetics (dots), using the
multi-channel model of backward transfer with fixedD and ϵ (solid lines) and accounting for their r-dependencies [49].The 4-dimensional IET, accounting for the
reversibility of all the transitions included in Figure31, was developed in
[115]. Not only the
reaction coordinate was included in this consideration but also the
off-diagonal elements of transfer Hamiltonian instead of the rates of electron
transfer. Quite recently, the charge transfer in the donor-acceptor complex was
studied, accounting for even two reaction coordinates: the fast and slow modes
of the polar medium [116]. The implementation of this mechanism into the encounter
theory will make it 5-dimensional, allowing one to correctly account for the
biexponential dynamic solvent effect. The synthesis of the elementary act with
an encounter diffusion of reactants within the framework of multidimensional
encounter theory opens a new horizon for the chemical kinetics of liquid phase
reactions.


## 5.1. Transient Quenching Kinetics

The
nonexponential kinetics of irreversible energy quenching is represented in DET
by the solution of (3):(120)NA*=N0e−c∫0tkA(t′)dt′.The non-Markovian rate constant kA(t) given in the contact approximation by (2)
diverges at the very beginning: kA(0)=∞.
The same quantity in the encounter theory, (5), tends to kinetic rate constant
at t→0:(121)kA(0)=∫WA(r)d3r=ka,sothatlnNA*N0=−ckat.Only in the opposite limit (at
long times) does the general asymptotic expression for kA(t) have the same shape as (2), except that the
effective reaction radius RQ is substituted for σ.
Integrating this expression in (120), the following long time asymptote for
quenching kinetics is obtained:(122)lnNA*N0=−c[4πRQDt+8RQ2πDt]ast→∞.The first term in the rhs of
this equation represents the final exponential energy quenching with stationary
rate constant k=4πRQD,
while the second one describes the transient effect: the preceding
nonexponential development, though determined by the same RQ.The first attempt to extractRQ from the experimentally studied quenching of
pheophytin a by toluquinone [8] was not flawless. Since the
short times were hardly available for the techniques of those times, the
detected long time quenching was thought to be exponential. Thus obtained the
pseudostationary rate constant k was greatly overestimated, as well as RQ=k/4πD.
Its diffusional dependence was specified using 40 solvents of different
viscosities but the tunneling length extracted from fitting DET to the thus
obtained RQ(D) dependence was also significantly
overestimated.Only 10 years later, the quenching of Rhodamine 3B by
N.N-dimethyleaniline was first recognized as a nonstationary one andRQ was obtained from fitting the asymptote (122)
to better data and in a wider time interval [79]. As can be seen from Figure 19, the current kA(t) is essentially larger
in its asymptotic value ka,
even at the longest attainable time t=4 ns.
Hence, the whole quenching kinetics available in this study is nonstationary
and should be fitted with an appropriate asymptotic expression, (122). The
single quenching parameter RQ was obtained from this fit, employing 7
solvents of different viscosity and such obtained RQ(D) dependence, compared with the theoretical
predictions of the contact theory and DET.Figure 19
The fit of the non-stationary electron transfer
kinetics (thick curve) to the experimental data obtained in [83]. The thin line represents
the tangent to the kinetic curve at the largest time available (t=4 ns). Its slope is the time dependent rate of
ionization, ckA(t),
which differs essentially from that of the purely exponential decay, with the
stationary rate constant ka=4πRQD (dashed straight line). (From [79]).The results of this comparison are shown in Figure20(a).
At fast diffusion (under kinetic control), there is an agreement with all
theories but at slow diffusion the experimental points lay far above the
predictions of Collins-Kimball model following from (4):(123)RQ=σkaka+4πσD.Much better fitting is achieved
with the exponential rate model (119), using the exact analytic expression for RQ(D) available in DET [7]. At fast and slow diffusion
this expression has the following simple asymptotes:(124)RQ={σka4πσDatRQ<σ(kineticcontrol),σ+l2ln(γ2Wcl24D)atRQ>σ(diffusionalcontrol),where γ=exp(C) and C is the Euler constant. However, the real
points are in between these limits and are well fitted with only the general
solution for exponential WA(r) (Figure 20). The latter is also just a short
distance approximation for the true Marcus rate. At larger distances the latter
is also exponential but with true L (Figure 17). The substitution of L≈2l for l in (124) makes the slope of RQ(lnD) at the highest viscosities almost twice as
large but this region was not attainable with the solvents studied (Figure 20(b)).
However, even at moderate viscosities the excitations never reach contact if
ionization is under diffusional control. They are quenched farther apart, at RQ>σ,
and this is a noncontact reaction accessible for only DET or IET of the remote
transfer.The dependence
on diffusion of the effective electron transfer radius,RQ.
(a) The experimental data, indicated by circles, is approximated by a thin line
representing the contact Smoluchowski-Collins-Kimball approach and by thick
line depicting the same dependence, but for the exponential transfer rate from
(119) with l=0.85Å and Wc=180ns−1.
(b) The same but in a wider diffusion diapason. The dashed-dotted line at the
bottom represents the high viscosity asymptote of the same dependence but for
the Marcus transfer rate. (From [79]).
(a)(b)By now, the best fitting of the transient effect and
the whole ionization kinetics was made in [48] where the subpicosecond kinetics of the irreversible
double-channel electron transfer (50) was studied. This is the highly exergonic
quenching of Perylene (Per) by tetracyanoethylene (TCNE) producing the ion
pairs in their ground and excited states. The pulse-induced quenching kinetics NA*(t) was accurately studied in three different time
intervals. The shortest and the longest ones were fitted with asymptotes (121)
and (122), respectively, for specifying ka and RQ while from the middle one the relative
strength of the parallel channels was inferred. The whole kinetics NA*(t) appeared to be fitted perfectly, within the
accuracy of experimental data (Figure 21).Fitting the double channel ionization kineticsNA*=PA*exp(−t/τA) at the shortest time (above) as well as at the
longest and medium ones (below). From [48].
(a)(b)


## 5.2. Stern-Volmer Constant

The concentration dependence of the quantum yields or the Stern-Volmer constant is
actually the central problem under investigation with numerous contact
theories. Yet the noncontact DET corrects essentially even this dependence
obtained with contact DET or its Smoluchowski-like analogs. To make this point
clear a straightforward fitting of the remote transfer theory to the data
presented by Stevens and Biver III [84] was undertaken in [47].The highly exergonic fluorescence quenching of
9,10-dicyanoanthracene (DCNA) by N,N,N′,N′-tetramethyl-p-phenylenediamine (TMPD) was studied in
[84] in the broad
range of TMPD concentrations, at three different temperatures (T=0°,22°,50°C) in acetonitrile solutions. Neither a contact
nor exponential approximation of the ionization rate WI(r) is good for the real fitting of this data.
Instead, one should use the true Marcus rate of electron transfer accounting
for the energy balance and the properties of the solution. The results of the
best fit are shown in Figure 22. The theoretical constants (thick curves),
depending on the dimensionless concentration cv=ξ,
are in good agreement with the experimental points in [84]. The agreement could even be better if a number of
additional factors were taken into account: the liquid structure near the
contact, the spacial dependence of the diffusional coefficients [85], and the chemical
anisotropy which is sometimes essential [86, 87]. Neglecting all these factors, only two fitting
parameters were used, the tunneling length L and the electron coupling element V0,
which are found to be [47]: L=1.6 Å, V0=4.07⋅10−2 eV. However, even the simplest theory of the
remote transfer fits the actual experimental data much better than its contact
analog (thin curves in Figure 22).Figure 22
The experimental concentration dependence of the
Stern-Volmer constant for three different temperatures (points) fitted by DET
with the single-channel Marcus transfer rate (thick lines). The thin lines
represent the contact analogs of the above curves at the same temperatures
(decreasing from top to bottom). From [47].It is also illustrated in Figure23 borrowed from
[48]. The fluorescence
quantum yield obtained in pulse experiment was compared in this work [48] with that measured
stationary (Figure 23(a)). Generally, they are in conformity with each other as
well as with that calculated theoretically. The latter was obtained by
integrating in (44) the best fitted NA*(t) from Figure 21. This integration can be
restricted by the duration of the pulse experiment or extended to infinity
taking into account the delayed fluorescence (contributed into the stationary
luminescence). The Stern-Volmer constants subtracted from all the data are
shown in Figure 23(b). There is only a small difference between the concentration
dependence of κ obtained from the time-resolved (•) and the stationary (⋆) experiments. The coincidence between this
data is best at the highest concentration where the quenching is accomplished
within the credibility time interval. The theoretical description of the whole
concentration dependence is quite satisfactory and this is a great success of
DET that it accounts for distant transfer.The non-linear
Stern-Volmer law (a) and the Stern-Volmer constants (b) obtained from the
time-resolved (•) and stationary (⋆) data in comparison with the theoretical
predictions, following from the quenching kinetics integrated over the
credibility intervals (□) and up to infinite time (red lines). From [48].
(a)(b)Contrary to DET, the contact
Smoluchowski-Collins-Kimball approach is completely unable to fit the same data
even at zero concentration. The ideal (IET) Stern-Volmer constantκ0=κ(0) is underestimated despite the usage of the
same kinetic constant of the double channel reaction (50): ka=k0+k1.
This indicates that the reaction is under diffusional control when the contact κ0con≈kD=4πσD is smaller than its distant analog κ0≈4πRQD,
because RQ>σ.
Indeed, the accurately calculated RQ appears to be 8.6Å  while σ=4Å  [48].There is also an independent confirmation that this is
actually the case. The Per/TCNE reaction is the most exergonic among the
systems collected by Rehm and Weller when they inspected the applicability of
the FEG law [88]. As
was pointed out by Marcus and Siders [89], the top of the parabolic FEG dependence has to be
cut by the diffusional plateau: κ0≈kD.
The Per/TCNE system is located at the left edge of this plateau, confusing
everybody (Figure 24). There the descending branch representing the inverted
region was expected to be detected instead of a plateau. A number of
explanations were proposed for this paradox [37, 90–94] but nobody took the liberty of suggesting that the
most exergonic ionization is still so fast that the reaction remains under
diffusional control.Figure 24
The Rehm-Weller
plot of the Stern-Volmer constant of the double channel ionization,κ0(ΔG),
with a cross (×) marking the position of the Per/TCNE system.
The bell-shaped FEG dependencies of the partial kinetic constants of the ground
state and excited ions production, k0 and k1,
split on the value of the ion pair excitation energy E±*.
From [95].After fitting in [48] the double-channel ionization kinetics, NA*(t),
at all times, we needed no suggestions: with an obtained
set of parameters, the partial constants k0=∫W0d3r and k1=∫W1d3r were calculated without adding any new fitting
parameters [95]. Not
only the sum of these constants but even each of them appear to be larger than kD at the point where Per/TCNE is located (Figure 24). Such a relationship
proves that the ionization is really under diffusion control in this system.
This is the final resolution of the old paradox.If there is no time-resolved data aboutkA(t) in the shortest time diapason but the good
value of k=kA(∞)=4πRQD is obtained, then the quantities RQ and D can be separated, fitting the theory to the
Stern-Volmer constants available from the stationary experiments. This program
was successfully realized in a very recent paper [96]. The lack of information
was filled in by relating the Stern-Volmer constant κ(c) to the DET rate constant, kA(t):(125)11+cκτA=∫0∞e−c∫0tkA(t′)dt′−t/τAdt/τA.Unfortunately, the theoretical κ(c,D) dependence that has to be fitted to the
experimental data strongly depends on how accurately kA(t) is related to the transfer rate, WA(r),
and encounter diffusion D considered as a fitting parameter. This
relationship is generally more complex than (5):(126)kA(t)=∫WA(r)n(r,t)d3r,where n(r,t)=ν(r,t)et/τA obeys the auxiliary diffusional equation
alternative to (6):(127)n˙(r,t)=−WAn(r,t)+1r2∂∂rr2D(r)g(r)∂∂rn(r,t)g(r).The latter has to be solved with
the reflecting boundary condition and initial condition n(r,0)=g(r).
Here g(r) is the equilibrium pair distribution function
accounting for the internal structure of the liquid. Besides, the hydrodynamic
effect is taken into account, which results in the space dependence of the
diffusion coefficient at short distances between reactants. This dependence is
represented by the simple expressions taken from either the Northrup and Hynes
or Deutch and Felderhof models [96]:(128)DNH(r)=D(1−12e−(r−σ)/σ)orDDF(r)=D(1−3σ4r).In fact, the authors of [96] tested 6 different models
of kA(t) calculations (with and without accounting for g(r) and D(r)) and selected only two of them which account
simultaneously for the liquid structure and the space dependence of D(r) (one or another). 150 experimental data points
for κ(c,D) were obtained from the fluorescence quantum
yield studied in the 8 solvents of different compositions (viscosities). Not
only the rate parameters, L and V0,
were found from best fitting the theory to this database, but also the
diffusion coefficient was allowed to vary individually for each solution,
during the fitting procedure. The obtained relationships between diffusion and
viscosity, D(η),
for the models (128) were shown to be very close to the Stokes-Einstein models
for the stick and slip friction coefficient. Specifying D in this way, the authors of [96] got RQ=k/4πD from k=kA(∞) and demonstrated that such obtained RQ(D) dependence is in good agreement with its
theoretical prediction having the asymptotes (124).


## 5.3. Products Accumulation

If the transfer
occurs only at contact, then the products of the reaction (say RIPs) also
appear only there, creating the contact initial condition for the subsequent
geminate evolution. In contrast, the remote transfer proceeding from any
distance creates the broad distribution of charged products around the contact
that can be specified by means of DET. For this goal the differential theory
should be completed by adding an equation for the free ion concentration to
(3):(129a)N˙A*=−kA(t)cNA*−NA*τA,(129b)P˙=kA(t)cNA*,where kA(t) was defined in (5). This set of equations is
the irreversible analog of the IET equations (31).The total number of transfer productsP is related to their pair distribution as
usual:(130)P(t)=c∫m(r,t)d3r.It follows from (129b) and
(5) that(131)m˙−WAνet/τANA*=0,m(r,t)=WA(r)∫0tν(r,t′)et′/τANA*(t′)dt′.The final distribution of
charged products and its normalized version are(132)m(r,∞)=WA(r)∫0∞ν(r,t)et/τANA*(t)dt=m0(r),f0=m0(r)∫m0(r)d3r.This distribution is weakly
concentration dependent but changes dramatically with solvent viscosity
(diffusion).The family of such distributions obtained for
exponentialWA(r) from (119) is shown in Figure 25. At slow
diffusion the distribution looks like a quasisymmetric hill shifted out of
contact. With increasing diffusion it draws nearer to the contact, only
slightly changing shape until ionization remains diffusional. As soon as it
becomes kinetic the distribution approaches the contact, taking the shape of
the ionization rate (exponential in our example). This phenomenon peculiar for
only remote transfer was first studied theoretically [97, 98] and then
specified in a few experimental works [99, 100].Figure 25
The distribution of ion pairs resulting from the
ionization with exponential transfer rate (Wc=29.12ns−1, l=0.81Å) at different encounter diffusion of neutral
precursor: D=10−4,10−6,10−7,10−8,10−9cm2/s (from left to right).


## 5.4. Charge Separation

However, the
accumulated ions are not frozen but start to separate and recombine immediately
after their appearance at starting pointr0.
Their total number, summarizing the contributions from any r0,
decreases with time as(133)P(t)=c∫Ω(t∣r0)m(r0,t)d3r0.Here m(r0,t) is defined in (131), while Ω(t∣r0)=∫m(r,r0,t)d3r is the share of ions starting from distance r0 but having escaped recombination up to time t.
The latter obeys the conjugate diffusional equation [16, 101] for geminate recombination
with distant tunneling rate WR(r) and ion diffusion coefficient D˜:(134)∂∂tΩ(t∣r)=−WR(r)Ω(t∣r)+D˜e−rc/rr2∂∂rr2erc/r∂∂rΩ(t∣r), where rc=e2/ϵT is the Onsager radius of the Coulomb well in
the solvent with dielectric constant ϵ.
Equation (134) should be solved with the following initial and boundary
conditions:(135)Ω(0∣r)=1,∂Ω(t∣r)∂r|r=σ=0,Ω(t∣∞)=1.The quantum yield of charge separation from the
particular starting distancer0 is(136)φ(r0)=Ω(∞∣r0)=11+Z(r0)/D˜,while the total yield of the
free ions is(137)ϕ=P(∞)=c∫φ(r0)m0(r0)d3r0=ψφ¯,where the yield of ionization(138)ψ=c∫m0(r0)d3r0=1−η,and the charge separation yield(139)φ¯=∫φ(r0)f0(r0)d3r0=11+Z/D˜.The last expressions in (136)
and (139) relate the charge separation yields to the partial and total
recombination efficiencies, Z(r0) and Z.
Such a diffusional dependence of the yields is natural for the archaic
“exponential model” which implied that Z=z=const [9, 16]. Though incorporated into
the encounter theory, the recombination efficiency Z(r0) becomes the function of not only D˜ but also r0,
while Z appears to be dependent on both D˜ and D.Two factors determine the type of this dependence: the
shape of the recombination layer and the location of the starting point. The
layer may be adjacent to the contact or shifted out depending on the form of
the distant recombination rateWR(r),
while r0 can be located either inside or outside this
layer or be distributed around it in accordance with f0(r0).For the exponential and rectangular models ofWR(r),
the diffusional dependence of Z(r0) is demonstrated in Figure 26. For starts taken
from contact (or deeply inside the reaction zone), the efficiency of recombination
monotonously decreases with diffusion, separating the ions. In contrast, the
same diffusion accelerates the recombination bringing together the ions born
outside the recombination zone. If initially ions were born inside the reaction
layer but close to its outer boundary, then the initial inhibition of
recombination finally gives way to its acceleration. Only as D˜→∞ the efficiency of recombination, Z∞,
is the same wherever the ions start from.The partial efficiency of geminate recombination,Z(r0),
related to its value at infinite diffusion, Z∞,
for the exponential (a) and rectangular (b) models of the reaction rate.
Contact starts (dashed lines), non-contact starts from reaction zones (solid
lines) and start from outside of the rectangular zone (dotted line).
(a)(b)As a matter of fact, one cannot change the ion
diffusion by varying solvent viscosity but keeping fixed and unique the initial
separationr0.
The diffusion of neutral reactants changes simultaneously with that of ions, D≈D˜∝1/η,
where η is the viscosity varying with solvent
composition or pressure. As a result, the distribution over r0 gradually approaches the contact when
viscosity decreases (Figure 25) and r¯0=∫r0f0(r0)d3r0→σ.
The total efficiency Z shown in Figure 27 is averaged over the current
distributions and therefore depends not only on D˜ but also on D.
When the diffusion is slowest, the start is more remote than recombination and
efficiency increases, with diffusion controlling the reaction. At faster
diffusion the situation changes to the opposite and a further increase in
diffusion reduces the recombination efficiency [102].Figure 27
The diffusional dependence of the recombination
efficiencyZ in the contact approximation (dotted line) at
starting distance r0=1.124σ and the same for the remote recombination in
the normal (solid line) and inverted (dashed line) Marcus regions, in highly
polar solvents. The horizontal dashed-dotted line represents the exponential
model result, Z=z=const.
From [102].However, the maximum which separates these regions is
not sometimes seen. The recombination layer having the shape of the Marcus rate
(115) is broad and far from contact, if the recombination proceeds in the
inverted Marcus region (Figure 18) but attached to contact and narrow in the
normal Marcus region (Figure 17). In the former case the ascending branch of Z(D) gives way to the descending one much earlier
than in the latter and does not give at all if recombination is contact (Figure
27). Only in the latter case is the reaction always under classic diffusional
control (dotted line). Otherwise efficiency passes through the maximum before
approaching its kinetic (i.e., “exponential model”) limit, z.Unfortunately, just detecting the monotonously growingZ(D˜) does not guarantee that the recombination is
contact or at least controlled by diffusion. Simultaneously with diffusion
(viscosity), the variation of solvent composition changes also n and ϵ,
as well as the contact reorganization energy (117) determined by these
parameters. As a result, WR(r) becomes different at different compositions,
as well as the curves Z(D˜),
corresponding to the fixed λ0 particular for the given composition. These
curves are shown by dashed and dotted lines in Figure 28 for the highest and
lowest values of λ0 that were achieved in the system studied by Wolf
et al. [103]. This was
the ruthenium complex quenched by methylviologen (MV++), according to the reaction
scheme
(140)
This reaction was studied in
ionic solutions, when an electrolyte is dissolved in an inert solvent, like
acetonitrile. The viscosity changed with mixture composition.Figure 28
The experimental dependence of the recombination
efficiencyZ on the diffusion coefficient D˜ shown by triangles. The solid line was
calculated theoretically, taking into account that the reorganization energy
changes along with diffusion. The dashed line corresponds to the calculation
with fixed but maximal λ0 reached at the fastest diffusion D˜=1.072×10−5cm2/s.
The dotted line was also calculated with a fixed but slowest diffusion D˜=6.67×10−7cm2/s.
From [102].It can be seen easily from Figure28 that both curves,
calculated at fixed λ0,
pass through the maximum and their viscosity dependence at faster diffusion is
the opposite to that obtained experimentally in this system. The ascending
branches of these curves can be attributed to the diffusion delivery of remote
RIPs into the recombination zone but the experimental points are situated on
the opposite, descending branches of them. Due to relatively fast diffusion,
the forward electron transfer in this system is always under kinetic control
and the RIPs are initially produced almost completely inside the recombination
layer. Hence, the diffusional delivery of RIPs into this layer cannot be
responsible for a pronounced increase in Z with diffusion, as was thought earlier
[104, 105]. It is due to the
collateral change in λ0 and D˜ with solvent composition. All experimental points
belong to the descending branches of the curves which are intermediate between
the dotted and dashed ones. The points shift up when viscosity goes down. The
resulting increase in Z just imitates the situation of
diffusion-controlled recombination.


## 5.5. Unified Theory

More transparent results were obtained with
dimethylsulfoxide-glycerol mixtures, which allow for a wide variation of
solvent viscosity, without changing the parameters which affectλ0.
The system studied in such mixtures was the photo-excited perylene quenched by
aromatic amines:(141)Being detailed, this reaction is
represented by the scheme (93) accounting for the spin states of the reactants
and RIPs. The spinless theory of Section 5.4 is
inadequate to calculate the yields of the free ions and triplet products of the
geminate recombination. The general unified theory (UT) should be employed for
reaching this goal.The original UT links together the geminate RIP
recombination and RIP accumulation accounted for by the DET equation (131)
[97, 98]. Discriminating between singlet
and triplet RIPs, one should consider their pair densities, mS(r,t) and mT(r,t),
governed by a set of equations identical to [9, equations (9.6)]:(142)m˙S−WAνet/τANA*=−WS(r)mS+D˜1r2∂∂rr2erc/r∂∂re−rc/rmS+ksmT−3ksmS,m˙T=−WT(r)mT+D˜1r2∂∂rr2erc/r∂∂re−rc/rmT−ksmT+3ksmS. They have to be solved using the reflecting boundary
conditions and assuming that initially all RIP states were empty: mS(0)=mT(0)=0.
The right-hand side of the set (142) describes the geminate
recombination/separation of RIPs subjected to spin conversion, while the
left-hand side accounts for their generation, via the same pumping term as in
(131). This term composed from the DET quantities, ν and NA*,
should be specified by the preliminary solution of DET equations (6) and (3).
Using these solutions in the UT equations (142) and solving them, one obtains
all that is necessary to calculate the quantum yields of the singlet (ground
state) and triplet products:(143)ϕS(r′)=∫0∞∫WS(r)mS(r,t)d3rdt=ψφ¯S,ϕT(r′)=∫0∞∫WT(r)mT(r,t)d3rdt=ψφ¯T,where ψ is the total RIP yield (138).The yields of singlet and triplet products determine
the efficiencies of recombination through these channels,ZS and ZT:(144)φ¯S=ZSZ+D˜,φ¯T=ZTZ+D˜,while the charge separation
yield is defined, as in (139):(145)φ¯=1−φ¯S−φ¯T=D˜Z+D˜,where Z=ZS+ZT.
In view of the conservation law (145) it suffices to consider only two of the
yields, φ¯ and φ¯T,
which are both available for the experimental detection.In the contact approximation, the UT equations are
available for the analytic solution obtained in [107, 108]. To do the same for remote
transfer, the program Qyield developed by Dr. Krissinel (see
http://www.fh.huji.ac.il/~krissinel/software.html) can be employed. If the
exponential models for both the ionization and recombination rates are used, it
should somehow reflect the fact that the recombination rates are usually more
extended, due to the larger exergonicity of the backward electron transfer.
Accounting for this feature in our recent work we assumed [106] that lR>lI thus makes the
extension of recombination larger than ionization. Under this condition the
total efficiency of remote recombination Z is a nonmonotonous function of diffusion
contrary to what is specific to the contact approximation (Figure 29). This is due
to diffusional deceleration, following the diffusional acceleration of the
recombination. At fastest diffusion the initial ion distribution coincides in
shape with WI(r) which is narrower than WS,T(r).
Therefore, the weaker the backward transfer is, the faster the ions get rid of
the recombination layer interior. Passing the maximum, the total recombination
efficiency Z falls off with D˜,
approaching the plateau Z∞≈z=kc/4πσ.Figure 29
The total (above) and triplet (below) efficiencies of
recombination obtained with contact (dashed lines) and exponential (solid
lines) approximations of the recombination layer. From [106].The efficiency of triplet production also has a
maximum, separating the diffusional control of recombination from the
conversion control that turnsZT to zero, when too fast diffusion leaves no time
for spin conversion. However, this maximum is usually not achievable in the
experimentally available interval of diffusion variation. Fortunately, the
principle difference between remote and contact recombination is seen even in
these limits and is worthy of special attention.The diffusional dependencies of both recombination
efficiencies were subjected to experimental study in [108]. The extremum in Z(D˜) first obtained there was later termed as
Angulo effect. In Figure 30 we clearly see this feature. It gained a preliminary
interpretation within the rectangular model of the recombination rate [102, 108], substituted later by the exponential one [16, 106]. For fitting better the
preexponent, Ws=WS(σ) was not assumed to be equal to Wt=WT(σ) but the relationship Ws=1.2⋅Wt was preferred. Though at the greatest
diffusion, the experimental points deviate a bit from the theoretical curves,
all the rest are fitted quite well.Figure 30
Fitting the theoretical efficiencies (solid lines) to
the real experimental data for total (▲) and triplet (■) recombination efficiencies, using the
exponential models for both the ionization and recombination rates. Parameters
obtained from the best fit are the following: Wi=29.12ns−1, Ws=77ns−1=1.2⋅Wt, lI=0.81Å, lR=1.24Å, ks=0.75ns−1, σ=7.5Å.
Reproduced from Figure 3.75 of the recent review [16].These results were essentially improved later on by
the substitution of the real Marcus rates for their exponential approximations
as well as the true (coherent) Hamiltonian description of spin-conversion for
its rate (incoherent) model [109]. Assuming the HFI induced spin-conversion mechanism
is responsible for the singlet-triplet mixing the true HFI constant A was extracted from the best fitting.
Neglecting the Coulomb interaction between solvated ions the viscosity
dependence of the free ions and triplet quantum yields were specified as well
as their magnetic field dependence. The only demerit was too large fitting
value of A≃9ns−1=505×10−4 T compared to that
expected from ESR data 23×10−4 T. However, it has been shown recently that an
accounting for the Coulomb interaction between the counter ions allows to get
as good fitting but with much smaller A.
Even in contact approximation, A=47ns−1 was obtained at moderate Onsager radius rc=12Å while for the remote electron transfer, A=0.9ns−1 is expected [75].


## 5.6. Adding the Reaction Coordinate

Usually all the encounter theories considering any
kind of the electron transfer between the reactants ignore completely the
internal structure of the latter, either
electronic or vibrational. In fact, the intramolecular relaxation accompanied
the electron transfer is responsible for the hot (transient)
recombination and equilibration of transfer
products, forming the transfer rates for subsequent thermal
relaxation [49]. The
thermal transfer rates of the forward and backward transfer, W(r)=W(x,y,z),
are usually the input data for the encounter theories. In actual fact, one
should start earlier from the Hamiltonian and intraparticle relaxation that
shape W(r).
Nobody can ignore the fact that the reaction develops not only in space but
also along the reaction coordinate q,
towards the term crossing points where it actually proceeds. In fact, only
4-dimensional W(r,q) should be used from the very beginning.The transient processes take some time and develop
coincidentally with the encounter diffusion inr space. The conventional encounter theories
presume the time separation: the faster formation of W(r) compared to its modulation by space diffusion.
This is never the case and the deep advancement of the reaction even during
short times should not be excluded. Both processes were first considered
simultaneously in DET assuming that the ionization proceeds with the rate W(r,q)=A(r)δ(q−q0) in a single crossing point q0 [110]. Next time the UT was extended correspondingly,
taking into consideration not only double-channel ionization at a few crossing
points (q1) but also the hot recombination in even more
numerous intersections (q2) between the ground and excited states of Per/TCNE and Per+/TCNE− (Figure 31) [49]. It was shown experimentally that the hot
recombination preceding the recombination of thermalized RIPs leaves for the
latter only 10% of the initially produced ion pairs (Figure 32).
This phenomenon is well proved by the 4-dimensional UT, although it predicts a
bit slower accumulation than it is in reality. This demerit has been removed
just recently by correcting the reorganization energy of the process making ion
pair accumulation activationless and thus almost instantaneous [111]. Alternatively, it might
be a consequence of the straightforward excitation of excited RIP as
happens sometimes in stable complexes [112]. If this is an effect of
light pumping it should be taken into account as it was done a few times for
other purposes [113, 114].Figure 31
Ground and
excited electronic states of the reactants,DA and D*A,
and charged products, D+A− and D+*A−.
The dotted lines represent vibrational excited states (n=1,2,3), the crossing points for forward transfer
are marked by circles (∘) and for backward transfer by bullets (•). The arrows show the direction of the
intra-molecular vibrational relaxation. From [49].Figure 32
The best fit of
the ion pairs accumulation/recombination kinetics (dots), using the
multi-channel model of backward transfer with fixedD and ϵ (solid lines) and accounting for their r-dependencies [49].The 4-dimensional IET, accounting for the
reversibility of all the transitions included in Figure31, was developed in
[115]. Not only the
reaction coordinate was included in this consideration but also the
off-diagonal elements of transfer Hamiltonian instead of the rates of electron
transfer. Quite recently, the charge transfer in the donor-acceptor complex was
studied, accounting for even two reaction coordinates: the fast and slow modes
of the polar medium [116]. The implementation of this mechanism into the encounter
theory will make it 5-dimensional, allowing one to correctly account for the
biexponential dynamic solvent effect. The synthesis of the elementary act with
an encounter diffusion of reactants within the framework of multidimensional
encounter theory opens a new horizon for the chemical kinetics of liquid phase
reactions.


## 6. Conclusions

There are two
dyads of encounter theories: DET/UT and IET/MET. DET is applicable only to
irreversible reactions but at any quencher concentrations, while UT is the most
convenient approach for geminate recombination/separation of charged products.
The integral theories are suitable for reversible reactions of any complexity,
at the lowest (IET) and moderate (MET) concentrations. Neither of the encounter
theories is universal but all of them together constitute a firm basis for
studying any transfer reactions in liquids.Both IET/MET and DET/UT are capable of accounting for
the complex electronic structure of reactants, including their spin states. Any
encounter theory considers energy and/or electron transfer as a remote
reaction, resorting to the contact approximation only for simplification.



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*Source: 214219-2008-12-28.xml*