Download The crucial role of triplets in photoinduced charge transfer and

The crucial role of triplets in photoinduced charge transfer
and separationy
A. I. Burshtein*a and K. L. Ivanovb
a
b
Department of Chemical Physics, Weizmann Institute of Science, 76100, Rehovot, Israel
International Tomography Center, Novosibirsk State University, Novosibirsk 630090, Russia
Received 18th February 2002, Accepted 21st June 2002
First published as an Advance Article on the web 31st July 2002
We show that irreversibility of quasi-resonant bimolecular ionization in solution is due to geminate and/or
bimolecular recombination of radical ions into the triplet state of neutral products. Spin conversion in the
radical ion pair, which can be a limiting stage of their geminate recombination, is included in the integral
encounter theory of the phenomenon. Both the triplet and charge separation quantum yields are calculated in
the contact approximation and their free energy dependence is specified.
I.
Introduction
The impurity quenching of the luminescence in liquid solutions
is often carried out by electron transfer from excited active
molecules to the quenchers or vice versa. The concentration
dependence of the relative quenching quantum yield is given
by a conventional Stern–Volmer law with some quenching
constant k. The latter is often identified with the rate of electron transfer which has to obey the Free Energy Gap (FEG)
law of Marcus.1 This is the quasi-parabolic bell shaped curve
in the log k vs. DGi plot, where DGi is the contact free energy
of ionization (forward electron transfer). The ascending
branch of this curve (in the direction of more negative DG) is
located in ‘‘ normal region ’’ where |DGi| < lc , where lc is the
contact value of electron transfer reorganization energy. The
opposite descending branch of the same FEG curve covers
the highly exergonic ‘‘ inverted region ’’, where |DGi| > lc . This
is the very general conclusion of the Marcus theory based on
first principles. It was well confirmed by experimental studies
of intra-molecular electron transfer or inter-molecular transfer
in solid state, but almost never in liquid solutions.2 Since in
liquids the particles are mobile the fastest reactions between
them are limited by encounter diffusion of reactants, which
does not depend on the free energy. As a result the top of
the bell shaped FEG curve can be cut by a plateau representing
diffusion controlled electron transfer. However, from the very
first experimental study of the phenomenon carried out by
Rehm and Weller3 it becomes clear that the situation is much
more complex and paradoxical than one can expect.
First of all, the diffusional plateau appeared to be too wide,
actually unrestricted at the high exergonicity side. There was
no sign of descending branch in the whole free energy range
available for experimental study. This is the most famous
Rehm–Weller paradox: the lack of an inverted region.3 A number of attempts were made to understand and explain this
peculiarity of electron transfer in liquid state.2 Most of the
researchers attribute this phenomenon to electron transfer to
excited electronic states of the products when the transfer to
their ground state is too exergonic.4,5 The free energy of
y Electronic Supplementary Data (ESI) available: Reversible triplet
transfer (Appendix A). See http://www.rsc.org/suppdata/cp/b2/
b201784a/
DOI: 10.1039/b201784a
electron transfer accompanied by product excitation is much
smaller and the corresponding rate much higher, so that diffusion remains the limiting stage of reaction even deeply in
inverted region.
However, there is also another paradox, noticed by Rehm
and Weller, but almost forgotten by later authors. The same
set of experimental data shows that the ascending (normal)
branch of FEG curve is located at too low energies, DGi 0.
This would be a right place if the ionization were an irreversible reaction. However, at DGi & 0 the reverse electron transfer
(back to excited state) has comparable or even larger rate than
the forward one. As a result the excited state is restored either
in the primary created ion pair or in subsequent encounters of
the separated ions in the bulk. In any case the excitation is not
quenched near the resonance and k should turn to zero long
before the free energy approaches zero and becomes positive.
As this did not happen Rehm and Weller resolved the paradox
assuming that ion recombination is much faster than the
reverse electron transfer. This can make the forward transfer
effectively irreversible provided the recombination is actually
fast enough near DGi ¼ 0. To meet this condition Rehm and
Weller assumed that the recombination rate is not only high,
but equally high at any free energies.
This assumption which is in a rough contradiction with the
general FEG law was rejected in our previous work.5 In the
quasi-resonance region DGi 0 the recombination of the Radical Ion Pair (RIP) to the ground state is deeply in inverted
region and too slow to compete with the reverse electron transfer. Instead, we supposed that the fast recombination of RIPs
in this region can proceed through the triplet channel: from triplet RIP to excited triplet product. However, this channel of
geminate recombination opens only after spin conversion in
RIP (created in the singlet state) that should proceed even faster. Assuming it to be infinitely fast, we demonstrated that the
ionization is actually irreversible. However, this assumption is
neither reliable nor obligatory. Here we are going to demonstrate that even at zero spin conversion rate the triplet channel
is still working, though only in a bulk. There the free ions
which escaped geminate recombination meet with uncorrelated
spins and 3/4 of them form the triplet RIPs that are allowed to
recombine in the triplet products.
To prove this statement, the geminate and bimolecular
recombination of RIPs through a triplet channel should be
Phys. Chem. Chem. Phys., 2002, 4, 4115–4125
This journal is # The Owner Societies 2002
4115
both included in the reaction scheme of photoinduced electron
transfer:
1
WI
D þ A )
WB
j
WB
2_ þ
_ þ A_ !
* ½D
D þ 2A_ ! 1 D þ A
I0 "# tS
D
WS .& WT
½D A
3
WT .& WS
D þA
DþA
. tS
ð1:1Þ
The energy scheme of this reaction is represented in Fig. 1,
where the energies of singlet and triplet excitations are E and
ET , respectively. The Marcus rates1 of forward and backward
electron transfer, WI(r) and WB(r), as well as of singlet and triplet RIPs recombination, WS(r) and WT(r), are used in the
Integral Encounter Theory (IET) of the phenomenon as input
data.5 All of them have the same general form which obeys the
Marcus FEG law:
2ðrsÞ=L
W ðrÞ ¼ we
sffiffiffiffiffiffiffiffi
!
lc
½DG þ lðrÞ2
exp ;
lðrÞ
4lðrÞkB T
ð1:2Þ
where only DG is different for all transfer rates. We addressed
here only highly polar solvents where the free energies are rindependent, unlike the ‘‘ outer-sphere ’’ reorganization energy
s
lðrÞ ¼ lc 2 ;
r
e2 1 1
where lc ¼
:
s n2 e
ð1:3Þ
Here n is the refractive index, e is the dielectric constant and s
is the contact distance between reactants, which is assumed for
simplicity to be the same for all of them as well as lc . The preexponential factor
pffiffiffi 2
pV0
w ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi
lc kB T
ð1:4Þ
is dominated by V0 which in turn is the pre-exponent of the
tunneling matrix element which decreases with distance, exponentially with decrement L. If, in addition, the electron transfer is assisted essentially by a quantum mode the necessary
generalization of eqn. (1.2) can be easily done.2
As was shown in ref. 5, the triplet channel affords fast
enough geminate recombination, provided the singlet–triplet
conversion equilibrates the population of spin levels in the
RIPs before their separation. The inter-particle distance in
RIPs which are nominated as 1,3[D_+...A_] is modulated by
encounter diffusion of mobile ion-radicals, so that their collective (singlet and triplet) states become degenerate at relatively large separation. This allows the hyperfine interaction
or spin-orbit coupling to initiate the spin transition between
these states with a spin-conversion rate ks . As far as spin
conversion does not control the geminate recombination of
RIPs, the latter proceeds with the rate (3/4)WT during
encounter time td ¼ s2/D~ (here D~ is the diffusion coefficient
for ions).
In fact, the spin conversion is not the fastest process in the
geminate reaction: at least once it was recognized as a limiting
stage of geminate recombination.6 Immediate equilibration of
spin states is too rigid limitation that can be removed without
sacrificing the general conclusions of the theory. The triplet
channel remains the main one even when geminate recombination is controlled by the spin conversion. We will prove this
statement assuming that spin conversion is incoherent and proceeds with the rate ks from triplet to singlet and with the rate
3ks in the opposite direction.2 If ks td 1, the geminate recombination to triplet is negligible because equilibration of the spin
system in RIP is not completed. However, the ions participating in the bimolecular recombination in the bulk meet each
other in a random spin state and compose singlet and triplet
pairs in a one-to-three ratio. Therefore the triplet channel of
bimolecular recombination is not switched off even at zero spin
conversion. This makes the forward electron transfer irreversible even in the quasi resonant region and eliminates all problems with the free energy dependence of the Stern–Volmer
constant at DGI 0.
A different situation arises if one is interested in the charge
separation quantum yield or geminate production of triplet
excitations, both of them are affected by the spin conversion
in the primary RIP. Using the contact approximation in
IET, we will estimate them as well as the Stern–Volmer constant of fluorescence quenching and specify the free energy
and ks dependencies of all measurable quantities.
The outline of the paper is as follows. In the next section the
simplest (incoherent) model of spin conversion is incorporated
in IET equations of photochemical charge accumulation and
ion recombination into singlet or triplet states of neutral products. Two alternative cases of fast and slow spin conversion
are considered and compared with the results of the spinless
theory of ref. 5. In section III the system response to either stationary or instantaneous excitation is studied. The stationary
ion concentration (i.e. photocurrent), as well as the quantum
yields of the fluorescence and phosphorescence were found.
The recombination efficiency of singlet born RIPs and the
quantum yield of triplets are also calculated as functions of
spin conversion rate, sharply contrasting with their primitive
estimates within the ‘‘ exponential model ’’. In section IV the
theory is applied to key photochemical experiments. These
are the Rehm Weller study of the free energy dependence of
the Stern–Volmer constant and similar dependence of the efficiency of charge recombination going to either ground or triplet state of the neutral product. In the conclusions we
discuss two main limitations of the present theory: contact
approximation for transfer rates and rate description of spin
conversion.
II. Stochastic spin conversion in contact IET
Fig. 1 Scheme of the energy levels and electronic transitions in the
reactant pair. Dashed lines are triplet states of ion pair and neutral
reactants.
4116
Phys. Chem. Chem. Phys., 2002, 4, 4115–4125
The general integro-differential equations of IET for the density vectors rD(t) and rA(t) of the reactants D and A are of
the following form:7–9
@rD ðtÞ ^
¼ QD rD ðtÞ TrA
@t
Zt
^ ðt tÞrA ðtÞ rD ðtÞ dt; ð2:1aÞ
R
gap law as well as the corresponding rates (1.2):
"
#
Z
ðDG þ lc Þ2
3
0
k ¼ W ðrÞd r ¼ k exp :
4lc kB T
ð2:9Þ
0
@rA ðtÞ ^
¼ QA rA ðtÞ TrD
@t
Zt
^ ðt tÞrD ðtÞ rA ðtÞ dt: ð2:1bÞ
R
0
Here Q^D/A are the operators of the intra-molecular relaxation
in reactants D and A, while R^(t) is the IET kernel (memory
matrix). In the contact approximation the latter can be
expressed through the matrix of reaction rate constants, K^,
and the contact value of the free Green function of the RIP,
Ĝ0(s|s,t):8
^~ ðs j s;sÞK
^~ ðsÞ ¼ K
^ ½E
^ þG
^ 1 :
R
0
ð2:2Þ
^
~ 0 ðs j s; sÞ is the Laplace transformation of Ĝ0(s | s,t)
Here G
and Ê is an identity matrix.
The general Green function Ĝ0(r,r0 ,t) obeys the following
auxiliary equation:
^ AD ÞG
^ 0 ðr j r0 ;tÞ ¼ dðr r0 Þ dðtÞE
^:
ð@t DDr Q
4prr0
ð2:3Þ
Its contact value is
ð2:4Þ
where kD ¼ 4psD is the rate constant of the bimolecular reaction controlled by encounter diffusion with coefficient D. The
Laplace transformation of G is equal to
~ ðsÞ ¼
G
1
pffiffiffiffiffiffiffi ;
1 þ std
ð2:5Þ
where td ¼ s2/D (hereafter it is assumed that D ¼ D~).
It is convenient to calculate the IET kernel of reversible
photoionization in the following collective basis:
0
1
1 D A
B ðDþ A Þ C
B
C
ð2:6Þ
rD rA ¼B þ S C:
@ ðD A ÞT A
3 D A
On this basis the operators K^ and Q^AD take the form:
1
0
kf
kb
0
0
C
B
kb kc
0
0 C
B kf
C;
B
^
K ¼B
C
0
kt kt A
@ 0
0
B
B
^ AD ¼ B
Q
B
@
where gs ¼ G(s + 1/tS)/kD , g0 ¼ G(s)/kD , g1 ¼ G(s + 4ks)/
kD .
The general set of kinetic equations for populations of neutral and charged states can be easily gained from the matrix
equations (2.1). This set obtained in Appendix A is rather complex, but can be essentially simplified if the triplet transfer is
irreversible. This is usually the case because the energy of triplet excitation is the lowest and |DGt| kBT, that is kt kt .
Setting kt ¼ 0 it is possible to reduce the general set of equations to a much simpler form:
Z t
Z t
R ðt tÞNS ðtÞdt þ
R# ðt tÞP2 ðtÞdt
N_ S ¼ c
0
^ 0 ðs j s;tÞ ¼ 1 GðtÞ expðQ
^ AD tÞ;
G
kD
0
The Laplace transformation of the contact Green function is
0
1
gs
0
0
0
B
C
B 0 3 g0 þ 1 g1 3 ðg0 g1 Þ 0 C
B
C
^
4
4
4
~ 0 ðs j s; sÞ ¼ B
C; ð2:10Þ
G
B
C
B 0 1 ðg0 - g1 Þ 1 g0 þ 3 g1 0 C
@
A
4
4
4
0
0
0
g0
0
kt
1=tS
0
0
0
3ks
ks
0
3ks
ks
0
0
0
0
kt
1
C
0C
C:
0C
A
ð2:7Þ
0
Here the rate constants of the forward and backward electron
transfer between singlet states, kf and kb , as well as those
between triplet states, kt and kt , relate to each other according to a detailed balance principle:
kf
DGi
kt
DGt
;
;
ð2:8Þ
¼ exp ¼ exp kb
kB T
kt
kB T
while kc and kt are the rate constants of the competing recombination channels, to the ground and excited triplet state. It is
assumed that the solvent polarity is so high that there is no
electrostatic interaction between any reactants and all k are
real kinetic rate constants which obey the Marcus free energy
0
NS
þ IðtÞNG ;
tS
P_ ¼ c
Zt
Ry ðt tÞNS ðtÞdt 0
N_ T ¼ c
ð2:11aÞ
Zt
Rz ðt tÞP2 ðtÞdt;
ð2:11bÞ
0
Z
t
RQ ðt tÞNS ðtÞdt þ
0
Z
t
NT
;
tT
ð2:11cÞ
R€ ðt tÞP2 ðtÞdt 0
where NS ¼ [1D*], NT ¼ [3D*], NG ¼ [D] and P ¼ [D+] ¼
[A] and I(t) is the rate of light excitation of donors in the
ground state. Unlike even the more simple set used earlier5,9,10
this one is supplemented by a third equation for triplet excitations. The first term in it is responsible for the accumulation of
triplets due to geminate recombination of ions, while the second one is responsible for the same electron transfer, but during bimolecular ion encounters in a bulk. It is assumed that in
comparison with any excitation, the non-excited donors are
present in great excess, so that their concentration NG const
const under illumination with light of low and moderate intensity. Otherwise, the variation of NG(t) can be taken into
account as well.11,12
In the contact approximation the kernels are:
~ ðsÞ ¼ kf 4 þ g1 ð3kc þ kt Þ þ g0 ðkc þ 3kt þ 4g1 kc kt Þ ;
R
X
1 þ g1 kt
#
~
R ðsÞ ¼ kb
;
ð2:12aÞ
X
~ y ðsÞ ¼ 4kf 1 þ g1 kt ;
R
X
k
ð1
þ
4g1 kt Þ þ ð1 þ gs kf Þðkc þ 3kt þ 4g1 kc kt Þ
b
z
~ ðsÞ ¼
R
;
X
ð2:12bÞ
~ Q ðsÞ ¼ 3kt ðg0 g1 Þkf ;
R
X
ð1
þ
gs kf Þð1 þ g1 kc Þ þ g1 kb
€
~ ðsÞ ¼ 3kt
R
;
X
ð2:12cÞ
where
X ¼ ½1 þ gs kf ½4 þ g0 ðkc þ 3kt Þ þ g1 ð3kc þ kt Þ
þ 4g0 g1 kc kt þ kb ½g0 þ 3g1 þ 4g0 g1 kt :
ð2:13Þ
Phys. Chem. Chem. Phys., 2002, 4, 4115–4125
4117
More general contact expressions for these and other kernels,
which account for the reversibility of triplet transfer, are exhibited in Appendix A.
A. Fast ST-conversion
To recognize the role of spin conversion in charge recombination, one should concentrate on its unique characteristics
~ ðs þ 4ks Þ
G
1=kD
ffi;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2:14Þ
¼
kD
1 þ ðs þ 4ks Þtd
pffiffiffiffiffiffiffiffiffiffiffiffi
The very fast spin conversion, 4k3 td ! 1, turns g1 to 0 and
reduces the equality (2.13) to a much simpler form:
g1 ðsÞ ¼
ðeff Þ
Þ
Þ þ g0 kb
X ! 4½ð1 þ gs kf Þð1 þ g0 kðeff
r
¼ 4X f ;
where
1
3
krðeff Þ ¼ kc þ kt ;
4
4
1
ðeff Þ
kb ¼ kb :
4
ð2:15aÞ
ð2:15bÞ
As a result four kernels take exactly the same form as in our
previous article:5
ðeff Þ
ðeff Þ
~ ðsÞ ¼ kf 1 þ g0 kr
R
Xf
~ y ðsÞ ¼ kf ;
R
Xf
~ # ðsÞ ¼ kb ;
R
Xf
;
ðeff Þ
~ z ðsÞ ¼ kb
R
ðeff Þ
þ ð1 þ kf gs Þkr
Xf
and the rest are
~ Q ðsÞ ¼ 3 kt g0 kf ;
R
4
Xf
III. Quantum yields of fluorescence, ions and
triplets
Experimentally, the electron transfer in solutions is studied in
two different ways. One involves detection of fluorescence and
phosphorescence, as well as photocurrent or probe light
absorption of ions at stationary illumination of the sample:
0 at t < 0
IðtÞ ¼ I0 BðtÞ ¼
:
ð3:1Þ
I0 at t 0
The other one uses the modern picosecond technique to register fast evolution of the system shortly after almost instantaneous light excitation:
~ € ðsÞ ¼ 3 kt 1 þ gs kf :
R
4
Xf
The fast ST-conversion equipopulates all 4 states of the RIP.
Therefore one has to replace kb and kr in the spinless integral
theory5,9 by their effective values (2.15) which are composed of
the recombination rates, multiplied by the equilibrium weights
of the corresponding states. All the rest is the same except that
between the products of recombination there are not only
ground state particles but triplet excitations as well.
B. Slow ST-conversion
The results are quite different for slow spin conversion, which
controls the geminate electron transfer through
pffiffiffiffiffiffiffiffiffiffiffiffithe triplet
channel and can completely switch it off. If 4k3 td ! 0 then
g1 g0 and X reduces to the following:
X ! 4½1 þ g0 kt ½ð1 þ gs kf Þð1 þ g0 kc Þ þ g0 kb ¼ 4X t X 0 :
The kernels can be rewritten as follows:
~ y ðsÞ ¼ kf ;
~ ðsÞ ¼ kf 1 þ g0 kc ;
R
R
X0
X0
k
þ
k
þ
k
k
g
1
3
k
b
c
f
c
s
t
z
~ ðsÞ ¼
R
þ
:
4
4 Xt
X0
~ # ðsÞ ¼ 1 kb ;
R
4 X0
The shape of these kernels is rather obvious from a physical
point of view. The kernels R* and Ry are exactly the same as
in the spinless theory,5 where kc should be substituted for kr .
Backward electron transfer to the ground state remains the single channel of geminate ion recombination, when spin conversion during encounter is negligible. The kernel R# is 4 times
smaller than the similar kernel in the spinless case, because
only a quarter of the RIPs, originating from random encounters in the bulk, are formed in the singlet state. The last kernel,
Rz, is presented as a sum of two contributions taken with the
equilibrium weights of the singlet and triplet states. The first
of them (kb + kc + kf kcgs)/X0 is exactly the same as in the
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spinless case, but weighted with the coefficient 1/4 arising from
spin statistics. Another one, kt/Xt is responsible for the irreversible recombination of triplet RIPs into the excited triplet state
of neutral product, 3D*...A. Again, the weight 3/4 accounts for
a probability of forming triplet RIPs during random encounters of counterions in solution. As long as triplets are not
~Q ¼ 0
formed in the course of geminate ion recombination R
all of them owe their origin entirely to bimolecular ion recom~ € / 3 kt . This
bination in the bulk, represented by the kernel R
4
5
opportunity was missed in our previous work where we confined ourselves to fast spin conversion only. Due to bimolecular ion recombination to the triplet state, the fluorescence
quenching by electron transfer remains irreversible and its
quantum yield does not experience any dramatic changes when
spin conversion slows down, or even turns to zero. On the contrary, all geminate reactions are affected significantly, as will be
shown below.
Phys. Chem. Chem. Phys., 2002, 4, 4115–4125
N0
dðtÞ:
NG
IðtÞ ¼
ð3:2Þ
These two ways of getting information should be considered
separately, adjusting the general kinetic equations to their specificity.
A.
Stationary phenomena
If the permanent light is suddenly switched on as in eqn. (3.1),
the system will soon adjust to it and after a while will approach
the stationary regime. In this regime all concentrations take
their stationary values NsS , NsT , Ps which obey the equations
following from the set (2.11) and (3.1):
Z 1
Z 1
Ns
0 ¼ c
R ðtÞdt NSs þ
R# ðtÞdt P2s S þ I0 NG ;
tS
0
0
ð3:3aÞ
Z 1
Z 1
Ry ðtÞdt NSs Rz ðtÞdt P2s
ð3:3bÞ
0 ¼ c
0¼c
Z
0
1
0
RQ ðtÞdt NSs þ
Z
0
0
1
R€ ðtÞdt P2s NTs
:
tT
ð3:3cÞ
The relative fluorescence quantum yield is defined as the ratio
of stationary singlet excitation concentration in the presence of
quenchers to the same concentration in their absence:
Z¼
NSs
NSs
¼
:
s
NS jc¼0 I0 NG tS
ð3:4Þ
The solution of eqn. (3.3) provides us with the stationary
values of all concentrations:
NSs ¼
I0 NG tS
~ ð0Þ R
~ # ð0ÞR
~ y ð0Þ=R
~ z ð0Þ
1 þ ctS R
P2s ¼ c
~ y ð0Þ s
R
NS
~
Rz ð0Þ
ð3:5aÞ
ð3:5bÞ
~y ~ Q ð0Þ þ R
~ € ð0Þ R ð0Þ N s :
NTs ¼ ctT R
~ z ð0Þ S
R
ð3:5cÞ
Using NsS in definition (3.4) we confirm that the fluorescence
quantum yield obeys the Stern–Volmer law
Z¼
1
;
1 þ cktS
ð3:6Þ
with constant
k ¼ k0 ½1 wjc ;
expressed through IET kernels:
~ ð0Þ; w ¼
k0 ¼ R
ð3.7Þ
9
~ # ð0Þ
R
;
~ z ð0Þ
R
jc ¼
~ y ð0Þ
R
:
~
RH ð0Þ
As will be shown below, k0 is the rate constant of the singlet
quenching during the primary geminate process, but such a
quenching is not completely irreversible. The ions which avoid
geminate recombination and separate with a quantum yield jc
can encounter later in the bulk and restore the singlet excitation with a probability w. The product wjc is a fraction of singlets restored in subsequent encounters. Only a limited fraction
of singlets, 1 wjc , is actually quenched irreversibly, the rest
contribute to the delayed fluorescence.13
The stationary photocurrent i is determined by a sample
conductivity euPsS which is proportional to the free ion concentration Ps and their mobility u. With the notations made above
we can represent the ion concentration in the following form:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jc ck0 s
f
ð3:8Þ
Ps ¼
NS ¼
N G I0 ;
kr
kr
B. Geminate production of free ions and triplets
If the concentration of quenchers is rather low and the kinetics
of luminescence or transient absorption after instantaneous
pulse excitation (3.2) is traced in a limited time interval, then
the bimolecular recombination has no time to develop and is
not seen within this interval.6,15 Only geminate recombination
is accomplished during this short time interval leaving free ions
and triplets in the amount given by their quantum yields. To
describe the kinetics of geminate recombination and product
accumulation we need only the reduced set of eqns. (2.11)
where all terms quadratic in P are omitted as well as the slow
triplet decay:
Z t
NS
N_ S ¼ c
R ðt tÞNS ðtÞdt ;
ð3:13aÞ
tS
o
P_ ¼ c
Zt
Ry ðt tÞNS ðtÞdt;
ð3:13bÞ
0
Z
N_ T ¼ c
t
RQ ðt tÞNS ðtÞdt:
ð3:13cÞ
o
Also, there is no excitation term because the instantaneous
light pumping is better accounted for by the initial conditions:
N S ð0Þ ¼ N 0 ;
Pð0Þ ¼ N T ð0Þ ¼ 0:
ð3.14Þ
The fluorescence quantum yield is very often defined
through the relaxation kinetics of instantaneously generated
excitations:16,17
Z 1
Z¼
N S ðtÞdt=N 0 tS :
ð3.15Þ
0
where kr ¼ R~z(0) is the recombination constant of free ions
and
f ¼ cjc ;
ð3.9Þ
is their quantum yield. The latter is a product of the total ionization quantum yield
ck0 tS
ð3:10Þ
c¼
1 þ cktS
and jc , which is a fraction of the ions getting free. In the case
of
irreversible
photoionization
considered
earlier
w / R~# / kb ¼ 0, k ¼ k0 and c ¼ 1 Z, so that the general
eqn. (3.8) reduces to eqn. (5.6) obtained recently in ref. 14.
The quantum yield of phosphorescence can be defined similarly to that of fluorescence:
Y¼
NTs
:
I0 NG tS
ð3:11Þ
Using here the triplet and singlet populations (3.5a) and (3.5c)
we obtain:
ðg0 g1 Þkf
Y ¼ 3kt tT cZ
X
k0 ð1 þ gs kf Þð1 þ g1 kc Þ þ g1 kb
: ð3:12Þ
þ jc
kr
X
s¼0
The first term in the square brackets represents the geminate
contribution to the triplet concentration and phosphorescence while the second one arises from bimolecular ion
encounters and recombination in the bulk. If the spin conversion is slow the first term turns to zero and the whole triplet
production owes its origin to the formation of triplet RIPs in
the bulk. However, with time resolved techniques there is an
opportunity to study only the geminate yield of triplet products at any conversion rate. In the next subsection it will be
analyzed in line with the efficiency of geminate ion recombination.
If NS(t) is found from the general eqns. (2.11) and (3.2) and
substituted into eqn. (3.15) then the result will exactly coincide
with eqn. (3.6) which follows from the alternative definition,
eqn. (3.4). However, if the delayed fluorescence is cut off by
measuring NS(t) in a limited time range, then the solution of
the reduced eqn. (3.13a) when used in eqn. (3.15) leads to a different result:18
Z0 ¼
1
¼ 1 c0 ;
1 þ ck0 tS
ð3:16Þ
where c0 is the quantum yield of ions produced by the forward
charge transfer in primary encounters of neutral reactants. It
differs from the stationary one, (3.10), but can be obtained
from it by a simple substitution of k0 for k. As was noted,
k0 is actually a quenching constant relating to singlet excitations that will never be restored.
The number of free ions ejected into the volume after the
geminate reaction is accomplished as:
Pð1Þ ¼ N 0 c0 jc :
ð3.17Þ
The fraction of initially produced ions which avoid geminate
recombination through any of the channels, is usually represented as follows:
jc ¼
~ y ð0Þ
R
1
¼
:
~ ð0Þ 1 þ Z=D
R
ð3:18Þ
The recombination efficiency Z calculated in the contact
approximation is
kD 3g1 ðkc kt Þ þ g0 ðkc þ 3kt Þ þ 4g0 g1 kc kt
:
Z¼
16ps
1 þ g1 kt
s¼0
ð3:19Þ
An essentially new feature here is the inclusion of recombination to the triplet state which is more often ignored or
accounted for within the naı̈ve and partially wrong exponential
Phys. Chem. Chem. Phys., 2002, 4, 4115–4125
4119
model (EM).19–21 The triplet channel is represented in EM by
the permanent recombination rate kisc which combines the spin
conversion rate with that of electron transfer between triplet
states. The competition between triplet and singlet recombination (represented in EM by the rate constant kb) leads to the
following EM expression for recombination efficiency in highly
polar media:19–21
Z ¼ ðkb þ kisc Þs2 =3:
ð3.20Þ
If there is no recombination through the triplet channel
(ks ¼ kisc ¼ 0) and the ions are generated at contact distance,
then the correspondence between EM and the contact approximation established earlier2 is established by identification of
the recombination parameters:
kc ¼
4ps3
kb :
3
ð3:21Þ
Here it will suffice to analyze an alternative case of recombination through the triplet channel only as it is represented in EM
and contact approximation. Setting kc ¼ kb ¼ 0 we obtain
from (3.19) and (3.20):
8
kisc s2
>
>
exponential model;
<
3 ffiffiffiffiffiffiffiffiffiffiffi
p
Z¼
4k t
3kt
>
>
pffiffiffiffiffiffiffiffiffiffiffis d
contact approximation:
:
16ps 1 þ 4ks td þ kt =kD
ð3:22Þ
From Fig. 2, illustrating this comparison, one can easily see
that the true Z approaches the exponential model expectation
only in the limit ks ! 1 provided
3
4ps3
kisc :
ð3:23Þ
kt ¼
3
4
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
At slow diffusion, Z tends to 34 4ks s2 D. This is represented by
the ascending branches of the curves where recombination is
under mixed conversion–diffusion control. On the contrary,
at D ! 1 there is a mixed kinetic–conversion control:
3kt pffiffiffiffiffiffiffiffiffiffiffiffiffi
Z¼
4ks =D. The latter is represented by the descending
16p
branches of the same curves which are the lower the smaller
the encounter time, td , restricting the spin conversion.
The free energy dependence of EM recombination efficiency
is the Marcus-like bell-shaped curve peculiar to the kinetic
Fig. 3 Free energy dependence of the recombination efficiency in the
contact approximation at different diffusion and fixed spin conversion
rate ks ¼ 1 ns1. Here E ¼ 3.5lc , ET ¼ 2.3lc , lc ¼ 35T.
controlled backward electron transfer.2 In contact approximation the picture is much more complex (Fig. 3). At the slowest
diffusion the top of Marcus curve is cut by a plateau in which
the reaction is under mixed diffusion–conversion control (solid
line). With increasing diffusion the plateau shifts up and narrows until it disappears completely, restoring the EM dependence (dashed line). At farther increase of diffusion the
whole curve is pushed down by the mixed kinetic–conversion
control of recombination (dotted line).
The quantum yield of triplets produced by geminate recombination is evidently
~ Q ð0ÞN
~ S ð0Þ ¼ N0 c0 jT ;
NT ð1Þ ¼ cR
ð3:24Þ
where
jT ¼
~ Q ð0Þ
R
;
~ ð0Þ
R
ð3:25Þ
is an efficiency of triplet production from the geminate RIPs.
In the free energy region where the competing channel of singlet recombination is ineffective and only triplets and ions are
produced, the sum of their quantum yields should be equal
to 1. This statement is easy to prove in the contact approximation, using the definitions of these yields and eqns. (2.12) and
(2.13):
lim jc ¼ 1 jT :
kc !0
ð3:26Þ
Usually triplet production is studied at a great excess of
acceptors when c0 ! 1, so that jT is actually a ratio of the produced triplets to the initially excited donors. In the long term
the triplets decay due to phosphorescence and/or the radiationless transition with time tT td . However, if their absolute concentration at the beginning is sufficiently high they
start to annihilate in biexcitonic encounters,22–24 that were
not taken into account here.
IV. The role of triplet production in key
experiments
Fig. 2 Diffusional dependence of the recombination efficiency in the
contact approximation at different spin conversion rates ks ¼ 0.01, 0.1,
1, 10, 100, 105 ns1 (from bottom to top). The horizontal line is an
expectation of the exponential model.19–21 Here kt ¼ 103 Å3 ns1,
s ¼ 5 Å.
4120
Phys. Chem. Chem. Phys., 2002, 4, 4115–4125
The state of the art in photochemistry of electron transfer in
liquid solutions is not perfect. During the last decade the theory overcame the limitations of primitive models, but available
experimental data are scanty. The energies of triplet states
are not always known and triplets are very rarely detected
experimentally in line with fluorescence or charge separation.
The last two phenomena are often studied as functions of corresponding free energies, but to fit the theory to these dependencies one needs the reorganization energy which is not
known well enough. In similar solvents the contact reorganization energy is sometimes25 taken as 0.2 eV, sometimes20,26 as
1.6 eV and what is more it is often chosen different for forward
and backward electron transfer (ionization and recombination). At such arbitrariness in choosing fitting parameters it
is rather difficult to discriminate between different interpretations of key experiments made by Rehm and Weller3 and their
successors. Therefore we shall restrict our consideration only
to the main features of the phenomena that have to be taken
into account anyway.
A.
Irreversibility of quasi-resonant forward transfer
In our previous work we stated that fast ion recombination is
necessary to make quasi-resonant and even endothermic forward electron transfer irreversible.5 The recombination of
singlet RIPs into the ground state is not as fast at DGi 0
and in the absence of triplet recombination (kt ¼ 0), cannot
prevent an electron to return back to the excited state in this
free energy region. The latter is seen as a gap between two
curves of k(DGi) depicted in Fig. 4—one for the irreversible
and another for the reversible forward transfer. Although triplet recombination is switched off the spin conversion still
affects the recombination through the singlet channel by changing the population of the reactive state from 1 to 1/4. Speeding up the spin conversion increases a little bit the charge
Fig. 4 The Stern–Volmer constant for irreversible (thin line) and
reversible (thick line) energy quenching in the absence of triplet recombination (above) and the charge separation quantum yield under the
same conditions (below) for different spin conversion rates kstd ¼ 0,
1, 1 from bottom to top. Here E ¼ 3.5lc , ET ¼ 2.3lc , lc ¼ 35T,
k0f ¼ k0c ¼ 20kD .
separation quantum yield (see Fig. 4) when triplet RIPs do
not recombine.
The abrupt interruption of the forward transfer at the resonant gap is accompanied by an equally sharp but opposite variation of the excitation probability w. This sharp turn occurs in
the narrow free energy region where the rates of transfer to the
ground and excited singlet states are equalized. To the right of
it, the backward transfer to the excited state dominates and
ionization is essentially reversible, while to the left the overwhelming majority of RIPs recombine to the ground state
making the forward transfer irreversible. If the maxima of
both rates are the same, the border point is exactly in the middle between them.
However, the situation changes qualitatively when the
recombination through the triplet channel appears (see Fig.
5). The border point is shifted to a middle between kt and
kb , and becomes very sensitive to the ratio kt/kD . If the spin
conversion
during encounter time is practically zero
pffiffiffiffiffiffiffiffiffiffiffi
( 4ks td ¼ 0) then it can be easily derived from w ¼ R~#(0)/
R~z(0) that
w¼ "
kb ð1 þ kt =kD Þ
#
:
kf kc
kt
pffiffiffiffiffiffiffiffiffi
1þ
k b þ kc þ
kD
kD ð1 þ td =tÞ
"
#
!
kf
kc
kb
pffiffiffiffiffiffiffiffiffi
þ
1þ
þ 3kt 1 þ
kD
kD
kD ð1 þ td =tÞ
ð4:1Þ
Fig. 5 The probability of reverse excitation as a function of ionization free energy DGi at E ¼ 3.5lc , ET ¼ 2.3lc , lc ¼ 35kBT. (a)
k0t ¼ k0c ¼ k0b ¼ 2000kD (thick line); k0t ¼ k0c ¼ k0b ¼ 20kD (dashed
line). (b) The family of the curves with different k0t /kD ¼ 20; 0.1;
105; 0 (from right to left), at k0c ¼ k0b ¼ 20kD .
Phys. Chem. Chem. Phys., 2002, 4, 4115–4125
4121
In the alternative limits of fast and slow electron transfer to triplet products we obtain from here much simpler results:
8
1
>
>
"kb
#
>
>
4
>
k
>
f
>
pffiffiffiffiffiffiffiffiffi
> 4kb þ ð3kD þ 4kc Þ 1 þ
>
>
k
ð1
þ
t
=t
Þ
>
D
d
>
<
at kt =kD ! 1;
w¼
>
>
>
kb
>
>
>
"
# 1 at kt =kD ! 0:
>
>
>
kf
>
>
pffiffiffiffiffiffiffiffiffi
>
: kb þ k c 1 þ
kD ð1 þ td =tÞ
ð4:2Þ
When DGi changes from negative values to positive the rate of
reverse transfer to excited state, kb , increases as well as w.
However, within the region where kt is large its upper limit is
1/4 (the rest ions recombine to triplet products). Only at
higher free energies when triplet channel is completely switched
off all ions recombine backward to initial excited state and w
approaches 1 as shown in Fig. 5(a). However, at faster diffusion the intermediate asymptotics (1/4) is missed and w
smoothly approaches 1. In Fig. 5(b) only the latter situation
is considered but at different rates of triplet recombination,
kt . An increase of this rate significantly expands the region
of irreversible forward transfer toward and even beyond zero
free energy. This expansion results in the corresponding extension of the free energy dependence of the Stern–Volmer constant shown in Fig. 6a.
Remarkably this effect does not depend too much on the
rate of spin conversion. Even if it is zero only geminate recombination to the triplet state is switched off, while bimolecular
triplet recombination in the bulk still works well keeping forward electron transfer irreversible. Dramatic changes are seen
only in the charge separation quantum yield (Fig. 6b) which is
Fig. 6 The same as in Fig. 4, but in the presence of irreversible triplet
RIPs recombination (k0t ¼ 20kD) to excited triplet states of neutral
products. All the parameters are the same as in Fig. 4.
4122
Phys. Chem. Chem. Phys., 2002, 4, 4115–4125
essentially a geminate property. In the absence of spin conversion it is 1 in the resonant region because neither of the recombination channels work there. With increasing spin conversion,
triplet recombination reduces the fraction of survived ions.
This effect is worthy of discussion in more detail in the next
subsection in line with the triplet quantum yield.
B.
Triplet channel of backward electron transfer
It is usual practice not to present a directly measured charge
separation quantum yield jc , but the recombination efficiency
Z extracted from it by relationship (3.18). Z is the real measure
of the backward electron transfer rate, an analog of kb in the
exponential model or kc in the contact approximation. This
parameter differs from zero only at those free energies whose
absolute values are close to the contact reorganization energy:2
DG lc . If there are two competing channels of recombination, through singlet and triplet RIPs, the free energy dependence of Z consists of two bell-shaped components. The
maximum of the singlet one is placed higher than the ground
state by lc , while the maximum of another is risen up lc from
the triplet state. Therefore, two peaks in the Z(DG) dependence
are spaced a definite energy apart, as seen in Fig. 7a. This splitting is exactly the energy of the triplet excitation ET .
At zero spin conversion there is only the peak corresponding
to the singlet channel. When the conversion rate increases up
to infinity it reduces and finally becomes 4 times lower when
all spin states are equally populated. The triplet peak behavior
is the opposite: it increases with ks and finally becomes 3 times
stronger than the singlet one.
The location and shape of the triplet quantum yield jT(DG)
are similar to the triplet peak in the recombination efficiency
(Fig. 7). Such a similarity can be used for the identification
of triplet recombination channel. This is possible if the singlet
Fig. 7 Recombination efficiency (top) and triplet quantum yield (bottom) as functions of the ionization free energy at different conversion
rates kstd ¼ 0, 1, 10, 1 (from solid to dotted lines). Here E ¼ 2.8lc ,
ET ¼ 2.3lc , lc ¼ 38T, k0f ¼ k0c ¼ 20kD .
recombination is forbidden or two peaks are well separated in
free energy, as in Fig. 7. As follows from eqn. (3.26), in this
case the triplet quantum yield is uniquely related to recombination efficiency:
jT ¼ 1 jc ¼
Z
:
ZþD
ð4:3Þ
It follows from this relationship that the top of the triplet
quantum yield is more flat due to its saturation with diffusion,
but the location and the width are the same. Hence, it is very
useful if both quantities are measured simultaneously, as in ref.
34. It was found that jc < 2% while the triplet quantum yield is
much higher but only in presence of halogen substituents
which facilitate the spin conversion through the spin–orbit
coupling. The very interesting results of this work are worthy
of special consideration accounting for the heavy atom effects
and triplet exciplex formation preceding the ion dissociation.
There is also another, more recent work (ref. 20) where both
sets of data are partially available. The similarity of these sets
compared in Table 1, indicates that the quenching mechanism
is mainly and perhaps exclusively the triplet one. Although the
total quantum yield is less than 1, one should take into account
that the registration of triplets was made too late, in the microsecond time region. If there were a large number of initially
generated triplets a significant part of them can annihilate earlier in the biexcitonic encounters.22–24 Besides, the triplets are
studied in too short a free energy interval and there is a huge
arbitrariness in choosing lc and other parameters. The Marcus-like curve fitted to the observed recombination efficiency
in Fig. 4 of ref. 27 looks good, but we failed to reproduce this
result with fitting parameters reported by the authors.
The results of our own attempts shown in Fig. 8 demonstrate that the maximum of recombination to the ground state
calculated in the contact approximation is too far from that
obtained experimentally. Only taking into account the space
dependence of the tunnelling and Arrhenius factors, this curve
can be broadened and shifted in the right direction, but not
enough. Making these calculations, we incorporated in the solvent and inner modes reorganization energies the values
lc ¼ 1.1 eV and li ¼ 0.5 eV, reported in ref. 27 as giving the
best fit to the recombination efficiency. As we saw, the fit is
not good at all, but what is worse, these energies differ drastically from those found in the same work for the forward electron transfer, that is lc ¼ 0.4 eV. There is no physical reason
to take for the forward and backward transfer in the same system threefold different solvent reorganization energies, which
in fact should be the same. Choosing between 1.1 eV and
0.4 eV we prefer the latter one because with this very value
for the contact reorganization energy we got a reasonable fit
to the Rehm–Weller data in our last work.5 With this choice
the accordance between the theoretical and experimental data
becomes the worst for the singlet channel, but rather good for
the triplet one (Fig. 8). The latter is offset by the energy of triplet excitation ET from the singlet one. This energy was
reported to be 2.30 eV according to ref. 28. For better fitting
we permitted ourselves to vary it, but only a little bit and
Table 1
Quencher
DGr
jc
jT
jc + jT
Toluene
Iodobenzene
Bromobenzene
Chlorobenzene
Benzene
Cyclohexanone
Fluorobenzene
Cyclopentanone
2.33
2.33
2.49
2.63
2.65
2.65
2.71
2.77
0.06
–
0.02
0.07
0.08
0.09
0.08
0.09
0.152
0.45
0.364
0.185
0.132
0.200
0.144
0.200
0.21
–
0.38
0.26
0.21
0.29
0.22
0.29
Fig. 8 The free energy dependence of the recombination efficiency
(top) reported in ref. 27 (open circles) and theoretical curves Z(DGr),
constructed with their fitting parameters (lc ¼ 1.1 eV, lq ¼ 0.5 eV,
ho ¼ 0.136 eV) in the contact approximation (dashed line) and for
remote electron transfer (solid line). Fitting of the contact theory to
the same data (bottom), but taking much smaller solvent reorganization energy lc ¼ 0.4 eV and making account for the triplet recombination channel (right peak). The energy of triplet excitation is assumed to
be ET ¼ 1.8eV, the spin conversion is assumed to be infinitely fast
(ks ¼ 1), like in the exponential model.
finally chose ET ¼ 1.8 eV. If this is a reasonable choice, then
the electron transfer between triplet states is perhaps the main,
if not the only channel of RIPs recombination in this system.
However, this is not ever the case. If the triplet energy is not
as large, both channels can contribute comparably to the RIP
recombination in the middle of the free energy interval separating the two peaks. This is what was actually obtained in ref. 19
and 29 and shown at the bottom of Fig. 9 where only one side of
the higher peak is clearly seen. This peak was attributed in the
original works to the formation of the exciplex. This is unlikely
the case, although the existence of exciplexes was well argued.
The exciplexes are usually formed reversibly30–33 and their
decay is not essentially faster than separated excitation unless
it is facilitated by heavy atom substitution.34 Hence, normally
the exciplexes can hardly accelerate too much the RIPs recombination35 or avoid dissociation to free ions. This follows from
the general results accounting for photochemical exciplex formation and dissociation by means of the Unified Theory.2,36
Even leaving open the question of the origin of the right
peak we see a significant parallel in the data presented in
two parts of Fig. 9. The similar two hump structure is hidden
in the free energy dependence of Z observed in ref. 15. In fact,
there is the same peculiarity at the high exergonicity of recombination that can be interpreted as an intersection of the two
close peaks, one of which belongs to the triplet channel
(Fig. 9, top). This suggestion conforms with the main conclusion made in ref. 6 that the spin conversion is a limiting stage
Phys. Chem. Chem. Phys., 2002, 4, 4115–4125
4123
Fig. 9 The two hump free energy dependence of the recombination
efficiency obtained in ref. 19 (stars, bottom) and in ref. 15 (open circles,
top), approximated by the singlet and triplet recombination peaks
(lc ¼ 0.99 eV, lq ¼ 0.35 eV, hoq ¼ 0.05 eV, ks ¼ 9 ns1, ET ¼ 1.65
eV).
Fig. 10 Rehm–Weller plot for a few systems which differ by the rate
of triplet recombination. (a) The theoretical curves for k0t /kD ¼ 20;
0.1; 105; 108; 0 (from right to left). The rest of parameters are the
same as in Fig. 5(b). (b) Interpolation through experimental points
from Fig. 2 of ref. 39.
of recombination in one of the systems studied in ref. 15
(Per + N,N-dimethyl-o-toluidine). This system is represented
by a triangle seen in Fig. 9. The straight experimental evidences of triplet formation were also obtained37 for another
system (Per + N,N-dimethylaniline), marked by a diamond in
Fig. 9. Although in this case the singlet channel is more efficient than the triplet one, nothing prevented the detection of
the pronounced magnetic field effect in this very system.38
We were not going to fit perfectly our simple theory to data
under discussion. We only want to demonstrate that they
can not be understood properly when the triplet channel is
completely excluded from consideration, as in ref. 15.
When triplet RIPs recombination is dominant the variation
of its rate can give rise to the multiple Rehm–Weller plots in
the quasi-resonant region that was discovered by Jacques and
Allonas39 but was given another explanation. As was shown
above, the triplet recombination is effective even in the absence
of spin conversion and expands the region of irreversible ionization the more, the higher kt is (Fig. 5(b)). If in RIPs produced
from different reactants kt is also different, then the corresponding ascending branches of Stern–Volmer constants calculated
from eqns. (3.7) and (4.1) should be positioned at different
places, as in Fig. 10(a). For a comparison in Fig. 10(b) we represented three different Rehm–Weller plots in acetonitrile
redrawn from Fig. 2 of the original experimental work.39 The
striking similarity of the plots is in favor of our explanation.
Since the same picture was obtained later in another system40
the multiple Rehm–Weller plot seems to be a general feature
of the phenomenon in a quasi-resonant region.
V. Conclusions
4124
Phys. Chem. Chem. Phys., 2002, 4, 4115–4125
The spin conversion in RIPs and their subsequent recombination to the triplet excited state of neutral product were shown
to be necessary for making the quasi-resonant forward electron transfer irreversible. The same process is responsible for
the highly exergonic recombination and the production of triplet excitations by backward electron transfer. Their experimental study, supplementary to geminate charge
recombination and separation, is strongly recommended for
identification of the recombination mechanism and its limiting
stages.
The integral theory employed here for a quantitative
description of the phenomenon was simplified in two aspects.
First of all, the recombination, as well as creation of RIPs
was assumed to be contact. This simplification allowed us to
study analytically all the effects, but it is not actually necessary
if one uses the numerical solution of the IET equations. The
newly developed programs are for our disposal and have
already been tested in a few cases of much current interest.37,41
Another limitation is more serious: the incoherent spin conversion does not often happen and only in relatively weak magnetic fields (see Appendix in ref. 42). To overcome this
restriction one has to use not only the spin state populations
but also their coherencies and dynamic Hamiltonian, including
hyperfine interaction in RIPs.43 This is not a principal complication, but it increases the rank of operators involved in the
numerical calculations and such a generalization of IET needs
some more work to be completed.
Acknowledgements
The authors are grateful to Prof. R. Marcus for the useful reference to the article by Kikuchi.19 All numerical calculations were
performed with the program written by E. B. Krissinel.43
This work was supported by the Israel Science Foundation
(grant N6863), the INTAS (project YSF2001/2-103), as well
as by grant N151 of The 6th Competition of RAS Young
Scientists’ Projects and RFBR (grant N01-03-6300).
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