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Chemical Physics Letters 447 (2007) 162–167

Activationless dynamic solvent effect

Serguei Feskov a, Vladislav Gladkikh b, Anatoly I. Burshtein b,*

a Department of Physics, Volgograd State University, University Avenue, 100, Volgograd 400062, Russiab Weizmann Institute of Science, Rehovot 76100, Israel

Received 15 June 2007; in final form 30 August 2007Available online 6 September 2007

Abstract

The dynamic solvent effect (DSE) is studied numerically at any free energy of electron transfer. In the activationless case the exactresult known for an irreversible transfer is confirmed but the commonly accepted approximate results for reversible transfer werecorrected. The general rate dependence on electron coupling is shown to fit the Zusman interpolation between the perturbation theoryand DSE, not only in the normal and inverted region but in the activationless case as well. DSE is responsible for the saturation of theelectron transfer at short distances where the electron coupling becomes strong.� 2007 Published by Elsevier B.V.

1. Introduction

The electron transfer between the molecules separatedby distance r is mainly determined by the electron couplingbetween the donor and acceptor parabolic terms, V. Sincethe tunnelling length L is short compared to the contactdistance r, the coupling

V ¼ V 0e�2ðr�rÞ=L; ð1:1Þsharply decreases with inter-particle separation. At rela-tively long r it is rather small and the perturbation theoryquadratic in coupling leads to the famous Marcus rate ofelectron transfer [1]:

W PTðrÞ ¼V 2

�h

ffiffiffippffiffiffiffiffiffikTp exp �ðDGþ kÞ2

4kT

!; ð1:2Þ

where DG is the free energy and k is the reorganizationenergy of electron transfer (kB = 1).

At shorter distances the coupling becomes too large andthe weak non-adiabatic transfer gives way to a strong one[2]. The latter known as the dynamical solvent effect (DSE)can be also considered as adiabatic passage over a cuspedbarrier separating the wells. The transition from the weak

0009-2614/$ - see front matter � 2007 Published by Elsevier B.V.

doi:10.1016/j.cplett.2007.08.090

* Corresponding author. Fax: +972 89344123.E-mail address: cfbursh@wisemail.weizmann.ac.il (A.I. Burshtein).

to strong non-adiabatic transfer is given by the generalZusman interpolation derived for highly activated transfer[3,4]:

W Z ¼W PT

1þ W PT=W DSE

¼W PT at V ! 0;

W DSE at V !1:

�ð1:3Þ

The DSE rate

W DSE ¼1

sexp �ðDGþ kÞ2

4kT

!ð1:4Þ

is inverse in s which is the time of reaching the crossingpoint by diffusional motion along the reaction coordinate.The latter is proportional to sL which is the longitudinalrelaxation time of polar solvent assisting the transfer. Forhighly activated transfer in either the normal (�DG < k)or inverted Marcus region (�DG > k) there is the followingrelationship between these times established by Zusman [3]:

1

s¼ 1

sL

ffiffiffiffiffiffiffiffiffiffiffi4pkTp jDGþ kjjDG� kj

jDGþ kj þ jDG� kj : ð1:5Þ

The resonant transfer (DG = 0) is really highly activatedwhen its activation energy k/4� T. According to theZusman formula (1.5) its rate is

1

s0

¼ 1

4sL

ffiffiffiffiffiffik

pT

r: ð1:6Þ

S. Feskov et al. / Chemical Physics Letters 447 (2007) 162–167 163

A more general and accurate result (valid also for the lowerfriction and parabolic barrier) was derived later by Calefand Wolynes [5]. Applied to the highest friction and cuspbarrier it takes the simplest form [6]:

1

s¼ 1

s0J 0ðkÞat DG ¼ 0; ð1:7Þ

which is identical to the Zusman Eq. (1.6) providedJ0(k) = 1. What actually happens is that [6]

J 0ðkÞ ¼ �i

ffiffiffiffiffiffipk4T

re�k=4T erfði

ffiffiffiffiffiffiffiffiffiffiffik=4T

pÞ

� 1þ 2Tkþ 3

2Tk

� �2

þ � � � ð1:8Þ

turns to 1 only when k/T!1.Even at high k the Zusman expression (1.5) is a reason-

able approximation only in the normal and inverted Mar-cus regions where the transfer is highly activated, butcompletely incorrect in between, at �DG = k, where thetransfer is activationless (Fig. 1). This particular case wasinvestigated by Burshtein and Kofman [7] earlier than allothers assuming that the transfer products are very unsta-ble, so unstable that the reverse electron transfer from themto reactants is negligible. When diffusion to the crossingpoint is a limiting stage, the dissipation of initial state pop-ulation was shown to be [7]

N ¼ 2p�1 arcsin e�t=sL ! 2p�1e�t=sL at t� sL: ð1:9ÞHence the rate of the long time asymptotic decay of suchan irreversible activationless transfer was found to be:

1

s¼ 1

sL

at � DG ¼ k: ð1:10Þ

The non-exponential decay law (1.9) was then confirmed anumber of times and even used to introduce the effectiverate of transfer kd ¼

R10 NðtÞdt ¼ 1

sL ln 2[8] which differs

from the asymptotic value of the transfer rate (1.10). None-theless we restrict our discussion to only this parameter,1/s(DG).

Fig. 1. The thermally activated electron transfer from the donor well(thick line) in either the normal region (to the well shown by the dashedline) or inverted region (to the well shown by the dash-dotted line) and theactivationless transfer (to the well shown by the solid thin line A).

The limitations of the electron transfer activation andirreversibility were overcome by Rips and Jortner thatmade an attempt to find the true s(DG) dependence validat any DG and k [9]. As follows from their generalEq. (3.4) for the rate for the reversible activationlesstransfer,

1

s¼ 1

sL

1ffiffiffiffiffiffiffiffiffiffiffipT=k

pþ ln 2

! 1

sL ln 2ð1:11Þ

as k!1, making the reverse transfer negligible. Sincethere is an odd factor ln2 compared to exact solution(1.10) for irreversible transfer, the rest of the resultsbecome also questionable and have to be inspected.

With this object in view, we will solve the problemnumerically in the next section calculating the reversibleelectron transfer rate at any DG. In Section 3 we will spec-ify the free energy dependence of 1/s and use it for gettingWDSE as well as WZ. Obtained with the Zusman interpola-tion, the latter will be confirmed with the straightforwardnumerical calculation of the transfer rate at any DG anddifferent couplings. In Section 4 the rate dependence onelectron coupling will be visualized for the normal, activa-tionless and inverted regions. We will also indicate thelimitations of the theory due to the adiabatic cut off of elec-tron transfer at the largest coupling.

2. Dynamic solvent effect

The general set of equations for the densities of reac-tants and products, n1(q, t) and n2(q, t), takes the followingform [3,9]:

_n1 ¼ �adðq� qyÞ n1 � n2½ � þ 1

sL

1þ qo

oqþ D2 o2

oq2

� �n1

ð2:1aÞ_n2 ¼ adðq� qyÞ n1 � n2½ �

þ 1

sL

1þ ðq� 2kÞ o

oqþ D2 o

2

oq2

� �n2; ð2:1bÞ

where

a ¼ 2pV 2=�h; D ¼ffiffiffiffiffiffiffiffi2kTp

and the transfer is assumed to be located at the crossingpoint q = q�. This is the commonly accepted rate descrip-tion of the population transfer between two energy levelsstarted from 1980 [4,10]. To the activationless transferthe rate Eq. (2.1a) were applied even earlier being derivedfrom the general density matrix equations (see Appendixto Ref. [7]).

In solving Eq. (2.1a) an assumption of initial thermalequilibrium in donor state is usually invoked while theproduct state is empty:

n1ð0Þ ¼expð�q2=2D2Þffiffiffiffiffiffiffiffiffiffiffi

2pD2p ; n2ð0Þ ¼ 0: ð2:2Þ

As a matter of fact, the initial distribution is not ever ther-mal and the start is taken not necessarily from the bottom

164 S. Feskov et al. / Chemical Physics Letters 447 (2007) 162–167

of the well but this is a separate problem that will be con-sidered elsewhere.

The general Eqs. (2.1) were solved numerically using thefollowing technique. Let G1(q, tjq0) and G2(q, tjq0) be theGreen functions for Eqs. (2.1) in the absence of electronictransitions (V = a = 0) when these equations take the form:

o

ot� bLk

� �Gkðq; tjq0Þ ¼ 0 ðk ¼ 1; 2Þ; ð2:3Þ

where bLk are the operators of the free diffusion in thecorresponding well. They should be solved with the initialconditions Gk(q, t = 0jq0) = d(q � q0). The explicit form ofthese functions for the case of the parabolic wells is knownto be

Gkðq; tjq0Þ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2pD2ðtÞq� exp � ½q� qmin

k � ðq0 � qmink Þe�t=sL �2

2D2ðtÞ

!:

ð2:4Þ

Here D2ðtÞ ¼ D2ð1� e�2t=sLÞ, and qmin1 ¼ 0; qmin

2 ¼ 2k are thepotential minima positions in the reactant and productwells, correspondingly.

When V 6¼ 0, the set of the coupled Eqs. (2.1) has to besolved assuming that initially the system is localized atpoint q0 on one or another term (k0 = 1,2). Such a solutioncan be presented as a vector, composed from the corre-sponding Green functions, �gk0

ðq; tjq0Þ:

�gk0ðq; tjq0Þ ¼

g1;k0ðq; tjq0Þ

g2;k0ðq; tjq0Þ

!;

�gk0ðq; t ¼ 0jq0Þ ¼

dðq� q0Þd1;k0

dðq� q0Þd2;k0

� �; ð2:5Þ

where di,j is the Kronecker symbol. The components of thisvector obey the integral equations

g1;k0ðq; tjq0Þ¼ d1;k0

G1ðq; tjq0Þ�Z

dt0Z

dq0Kk0ðq0; t0Þ

dðq0 �qyÞG1ðq; t� t0jq0Þ;

g2;k0ðq; tjq0Þ¼ d2;k0

G2ðq; tjq0ÞþZ

dt0Z

dq0Kk0ðq0; t0Þ

dðq0 �qyÞG2ðq; t� t0jq0Þ;

ð2:6Þ

whose kernel is Kk0ðq; tÞ ¼ aðg1;k0

� g2;k0Þ. The integration

of Eq. (2.6) over q 0 yields

g1;k0ðq; tjq0Þ ¼ d1;k0

G1ðq; tjq0Þ

�Z

dt0Kk0ðqy; t0ÞG1ðq; t � t0jqyÞ;

g2;k0ðq; tjq0Þ ¼ d2;k0

G2ðq; tjq0Þ

þZ

dt0Kk0ðqy; t0ÞG2ðq; t � t0jqyÞ:

ð2:7Þ

Taking the Laplace transformation of these equations

~g1;k0ðq; sjq0Þ ¼ d1;k0

eG1ðq; sjq0Þ � eK k0ðqy; sÞeG1ðq; sjqyÞ;

~g2;k0ðq; sjq0Þ ¼ d2;k0

eG2ðq; sjq0Þ þ eK k0ðqy; sÞeG2ðq; sjqyÞ;

ð2:8Þ

and substituting them into the definition of eK k0ðq; sÞ, we

can resolve the equation obtained regarding eK k0ðqy; sÞ:

eK k0ðqy; sÞ ¼ d1;k0

aeG1ðqy; sjq0Þ � d2;k0aeG2ðqy; sjq0Þ

1þ a½eG1ðqy; sjqyÞ þ eG2ðqy; sjqyÞ�: ð2:9Þ

Using the last result in Eq. (2.8), we can now proceedfurther making inverse Laplace transformation of themand getting the vector �gk0

ðq; tjq0Þ from Eq. (2.5). This canbe done numerically using several different computationaltechniques (see, e.g. [11, 12]). We however found that theEuler summation scheme [13] of Abate and Whitt givesprobably the best compromise between the precision andefficiency of the numerical calculations.

In principle, having known the Green function (2.5), onecan directly calculate the densities of particles in each wellat any time t as an integral over their initial distributions

nkðq; tÞ ¼Z

dq0½n1ðq0; 0Þgk;1ðq; tjq0Þ þ n2ðq0; 0Þgk;2ðq; tjq0Þ�:

ð2:10Þ

The difficulty here is the numerical evaluation of theLaplace transformation eGðq; sjq0Þ, which presents a com-putational problem. This difficulty can be avoided withthe following minor modification of the algorithm. Let usintroduce a grid of N points qi on the reaction coordinateaxis and denote Xlk(qjjqi) = gl,k(qj,Dtjqi). Then, the proba-bility to fall inside the [qj,qj+1] interval on the lth electronicterm, for the particle being initially at qi on the kth term, bythe time Dt is approximated by Xlk(qjjqi)Dqj. HereDqj = qj+1 � qj is the mesh width of the grid. If the timestep Dt is small (Dt� sL), the parabolic potentials nearthe qi point can be replaced by straight lines with the slopesA1 = qi/2k and A2 = qi/2k � 1. After such a linearizationthe Laplace transformation of the Green function,eG lin

k ðq; sjq0Þ, can be obtained in the close analytic form

eG link ðq; sjq0Þ ¼

sLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

k þ 4bsp� exp � ak

2bq� q0ð Þ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

k þ 4bsp

2bjq� q0j

!;

ð2:11Þ

ak ¼2ksL

Ak; b ¼ D2

sL

:

Using eG link ðq; sjq0Þ instead of eGkðq; sjq0Þ in Eqs. (2.8) and

(2.9), and the Abate–Whitt method for the numericalLaplace inversion, we calculated the quantities Xlk(qjjqi) forthe fixed time interval Dt. Then replacing the integration

Fig. 2. The free energy dependence of the inverse s calculated numericallyat k = 1 eV (solid line) and approximated by the Zusman analyticalexpression (1.5) (dashed line) and by the Calef–Wolynes estimate of theresonance transfer pre-exponent (1.7), shown by the triangle on theordinate axis. The Burshtein–Kofman result for the irreversible activa-tionless transfer (at �DG = k) is shown by the square point above themaximum.

S. Feskov et al. / Chemical Physics Letters 447 (2007) 162–167 165

in Eq. (2.10) by summation, we obtained the following sim-ple computational scheme:

n1ðqi; t þ DtÞ ¼X

j

Dqj½X11ðqijqjÞn1ðqj; tÞ þ X12ðqijqjÞn2ðqj; tÞ�;

n2ðqi; t þ DtÞ ¼X

j

Dqj½X21ðqijqjÞn1ðqj; tÞ þ X22ðqijqjÞn2ðqj; tÞ�:

ð2:12ÞWe tested this algorithm in some limiting cases. Goodagreement between the numerical and analytic results wasfound. The QM2L simulation software is available on theInternet (http://physics.volsu.ru/feskov/software.htm).

Let’s follow the time evolution of the total populationsin two wells,

NðtÞ ¼Z

n1ðq; tÞdq; MðtÞ ¼Z

n2ðq; tÞdq; ð2:13Þ

approaching over time the equilibrium values Ns = N(1)and Ms = M(1). We are mainly concerned with highlyexothermic transfer, when

Ns � Ms � 1 since � DG� T : ð2:14ÞSetting Ns = 0 one can expect that in such a situation equil-ibration proceeds exponentially at long times:

NðtÞ ¼ 1�MðtÞ ’ e�Wt at t!1: ð2:15ÞObtained as an approximate solution of Eq. (2.1), N(t) pro-vides us with information about the magnitude and the freeenergy dependence of the transfer rate W(DG).

3. Zusman rate

As follows from the Zusman interpolation (1.3), WZ

approaches its maximum, when the coupling increases:

W Z ! W DSE ¼1

sexp �ðDGþ kÞ2

4kT

!at V !1: ð3:1Þ

Taking rather large V we calculated numerically the dy-namic solvent effect rate as well as s which is different atdifferent DG. The free energy dependence 1/s(DG) is shownin Fig. 2. It looks similar to that presented in Fig. 1 of Ripsand Jortner’s work [9] where the ratio s(0)/s(DG) is plottedversus DG. However, there is a significant differencebetween these two presentations. As seen from ourFig. 2, the absolute value 1/s obtained numerically differsfrom that given by the analytic Zusman approximation(1.5) even at (DG + k)2� 4kT where the latter is valid. Thisdifference is masked when the ratio s(0)/s(DG) is consideredinstead of 1/s(DG). For instance at resonant transfer(DG = 0) the difference in absolute values of 1/s is about5%, while the relative values are indistinguishable beingboth equal to 1.

This difference is worthy of special attention. The reso-nant transfer is out of the region (2.14) where the reactionis highly exergonic and therefore irreversible. In case of res-onance the potential wells after equilibration areequipopulated,

Ns ¼ Ms ¼ 1=2 at DG ¼ 0 ð3:2Þand the system approaches this distribution non-exponen-tially. However, within certain time limits at the very begin-ning, the equilibration kinetics develops linearly in timeallowing to specify W:

N ¼ 1

2½1þ e�2Wt� � 1� Wt at Wt� 1: ð3:3Þ

A completely different type of situation occurs in thecase of activationless transfer (DG + k = 0) to which theZusman approximation (1.5) is inapplicable. In thisapproximation W = 1/s = 0 contrary to the Burshtein–Kofman prognosis (1.10) given for an activationless irre-versible transfer: s = sL. For the reversible transfer of thissort the result is different but also finite as follows fromthe Rips–Jortner estimate (1.11). However, this expressionis questionable because it does not reduce to its irreversibleanalog (1.9). As seen from Eq. (2.1b) the flux from thecrossing point to the bottom of the product well is propor-tional to 2k/sL. In the vicinity of the crossing point this wellshould be completely exhausted when k!1ð limk!1

n2ð0Þ ¼ 0Þ so that the reverse transfer is excluded.

The rate of irreversible transfer is the greatest possible.At our k = 1 eV it exceeds the maximal 1/s for reversibletransfer (calculated numerically) by 17%.

Although expression (1.11) overestimates the upper limitfor the transfer rate its functional form is reasonable. Beingrepresented as a linear relationship,

ssL

¼ affiffiffiffiffiffiffiffiffiffiffipT =k

pþ b; ð3:4Þ

it was inspected numerically. The linearity was confirmed(Fig. 3) but the tangent a is 0.7 instead of 1 expected in

Fig. 3. Approaching the irreversible transfer limit, limk!1

=sL ¼ 1. The

points are the numerical results interpolated by a straight line. The dottedline indicates s/sL = ln2.

166 S. Feskov et al. / Chemical Physics Letters 447 (2007) 162–167

(1.11) and b = 1 instead of ln2. The numerical results con-firm the identity (1.10) which is an upper limit for the max-imal s reached at the largest k (square point in Fig. 2).

For the simplest approximate estimation of 1/s it wasrecommended in Ref. [9] to set it equal to the Zusman valuefor resonant transfer, 1/s0. This average value shown by thedotted line in Fig. 2, differs from the real ones by not morethan 40%. Being used in Eq. (1.4) for approximate estima-tion of WDSE, it leads to the following universal formulafor the Zusman rate:

W Z �W PT

1þ V 2

�h

ffiffippffiffiffiffikTp s0

¼V 2

�h

ffiffippffiffiffiffikTp

1þ 4pV 2

�hk sL

exp �ðDGþ kÞ2

4kT

!: ð3:5Þ

Fig. 4. The electron transfer rates obtained with the Zusman interpolation(1.3) using the numerically calculated DSE rate (upper thick line) andperturbation theory for different couplings: V0 = 2.5 · 10�4 eV (a);V0 = 7.5 · 10�3 eV (b); V0 = 2.5 · 10�3 eV (c). The points are the samerates but numerically calculated for some free energies.

This approximation was used in Ref. [14] to specify therates of two parallel electron transfer channels: quasi-reso-nant and highly exothermic.

Having numerically obtained solid data for 1/s we werenow ready to make the calculations of the rate at anyelectron coupling. Using the exact estimation of WDSE

from Eq. (1.4) along with WPT from Eq. (1.2), we obtainedthe Zusman transfer rates (1.3) for a few different electroncouplings (Fig. 4). The results were confirmed by thestraightforward numerical calculations of a few points inthe normal, activationless and inverted regions. This is aconvincing demonstration that the Zusman interpolation(1.3) is valid everywhere, even in the activationless caseto which his estimation of s, Eq. (1.4), is inapplicable.

4. The rate dependence on electron coupling

In any vertical cross-section of Fig. 4 at a given DG, weobtain the rate dependence on coupling V, which is qua-dratic in V until the perturbation theory holds but finallysaturates approaching the DSE limit. In Fig. 5 we demon-strate the transfer rate saturation with V in three differentcross-sections: normal, activationless and inverted. Every-where the linearity of lnW in lnV (with a slope 2) indicatesthe region where the perturbation theory is valid, while thehorizontal plateau represents the dynamic solvent effect.

As a matter of fact, this plateau gives way to theKramers adiabatic cut off at even larger V. This effectwas studied separately for the resonant transfer [6] andhighly exergonic transfer in the inverted region [15]. Inboth cases there are the limitations of the Zusman interpo-lation separating it from the highest coupling (Kramers)limit. In the case of resonant transfer it follows from Eq.(2.7) of Ref. [6]:

V � 2pTJ 20ðkÞ < 2pT : ð4:1Þ

Fig. 5. The electron coupling dependence of the Zusman transfer rate inthe activationless case A (jDGj = k = 1 eV), normal region, N (jDGj/k = 0.85), inverted region, I (jDGj/k = 1.3). All the points were numeri-cally calculated.

S. Feskov et al. / Chemical Physics Letters 447 (2007) 162–167 167

Such an upper boundary for V is shown by the verticaldashed line in Fig. 5. In fact, it confines the present theoryto the situation where the splitting in the crossing point isless than the thermal energy. At very small k 0.2 �0.5 eV there is also one more limitation: the splitting shouldbe smaller than the barrier which is k/4 at resonancetransfer. At larger coupling the barrier disappears andthe exciplexes are formed [16–18].

5. Conclusions

As seen from Eq. (1.1) the coupling changes exponen-tially with inter-particle separation, reaching the maximalvalue at the closest approach distance. If this value, V0, islow enough the DSE plateau is not reachable and the per-turbation theory result, Eq. (1.2), holds at any r. This is thecommonly used approximation for weak transfer [1,19]. Ifthe coupling turns out to be strong at short distances, theZusman interpolation (1.3) should be used instead as inRef. [14] (see Figs. 1 and 2). Here we proved that this inter-polation accounts well for the DSE at any DG, provided sis correctly estimated, either analytically or numerically.

Acknowledgements

FSV gratefully acknowledges the Weizmann Institute ofScience for the hospitality during his stay in Israel, and the

Russian Foundation for Basic Research for support(Grants Nos. 04-03-95502 and 05-03-32680).

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