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Outer sphere electron transfer reactions: A generalized Landau-Zener model
Chemical Physics 36 (1979) 161-169 0 North-Holland Publishinp Company
OUTER SPHERE ELECTRON TRANSFER REACTIONS: A GENERALIZED LANDAU-ZENER
MODEL
Shlomo EFRIMA * and Mordechai BlXON Department of Chertzistr_v,Tel-Aril Chirersity, Ranrat Aviv. Israel Received 2 1 June
I978
A schematic model for the description of reactant pair dynamics is introduced and solved within a generalized LandauZencr proccdnrc. The reactant dynamics is characterized by the finite lifetime of a pair, and thddynsmics of the solvent tluctuations is taken to be linear in time. Evaluation of the average transition probability sho\vs that in usual realistic conditions the reactant dynamics is not an important factor for the computation of rate constants.
1. Introduction The theory of outer sphere electron transfer reactions in polar solvents has been studied extensively. Several formulations were developed, notably the classical model of Marcus [I-S], the quantum mechanical nonadiabatic models which are based on the quantized polaron picture of the solvent [4-91, and the semiclassical models [4-6, 1O-131. There is no reason to doubt the classical character of the polar solvent dynamics at room temperature. Therefore, for such systems, the semiclassical approach may be used safely. The only problem is how much of the details of the solvent dynamics can be incorporated into the theory, and what features of it are essential in order to get a full description. Different approaches, regarding pictures and details of the solvent and its coupling with the electron transfer system, have been used. Levi&, Dogonadze and co-workers 14-6, lo] used a model of harmonica1 vibrational polarization modes for the solvent. From this model they evaluated the rate of change of the interaction with the solvent at the intersection point of the electronic terms. The use of this constant rate of change enables them to apply the asymptotic Landau-Zener [ 14,151 procedure to * Present address: Department of Chemistry, University of California, Santa Barbara, California 93106, USA.
get the transition probability. By this method it is possible to get the whole spectrum of possibilities, from the nonadiabatic behavior up to the adiabatic one. Christov [l l] and Schmidt [12] use essentially the same harmonic model for the polar solvent, within somewhat different calculationa procedures. A different way to describe the interaction with the solvent is to look at the equilibrium distribution of polarization fluctuations, as proposed by Marcus [I 31. Such a model was used, within a framework of semiclassical theory, by Efrima and Bixon [ 131 to analyze the nonadiabatic rate constant. Evidently, in all the models the treatment of the solution dynamics is far from being complete. Especially the dynamics of the two molecules between which the electron is transferred is completely ignored. The aim of the present article is to incorporate the dynamics of the pair of reacting molecules into the theory. We are doing it by using a highly simplified model that schematizes the dynamics by a finite lifetirnc of the pair. Because the interaction can take place only within a finite duration, it is impossible to use the asymptotic Landau-Zener procedure and we have to generalize it. In the next section we discuss the model. In section 3 we evaluate the transition probability and in section 4 we average it and discuss the consequences.
162
S. Efrittra, hf. Bisor&.3rrer
sphere ekctrott
2. The mode1 As in our previous paper [ ~31 we are treating the system as composed of an electron (treated as a quantum mechanical system) which is parametrically coupled to a classical solvent. Its hamiltonian is written 3s H = T, + Vi(r) + V-&r) + Usi
+ 4r/,(r, K!(f)),
(1)
where T, is the electron kinetic energy operator, Vi(r) and Vf(r) are the potential energies of interaction with the donor and acceptor. respectively. Ilsi(r) is the potential of interaction between the electron and the solvent in its initial equilibrium configuration. M&r, w(r)) is the time dependent interaction of the electron with the solvent fluctuations which are characterized by w(r). An cstra ingredient that we incorporate in the model. in a highly schematic and simplified way, is the dynamics of the ions (molecules) between which the electron is transferred. As usual we assume that the electron transfer takes place only when the two ions are approximately in contact. All the terms in the hamiltonian (eq. (1)) are evaluated for contact configuration. The formation of the pair is visualized as resulting from a fast fluctuation in which one ion jumps over some distance to get into contact with a second ion. Assuming average thermal velocity, and a jump distance of the order of I-2 8, one gets values in the range 1O- 13- 1O- I ’ s for the time needed for a pair formation. During this time the interaction between the electron on one ion and the second ion grows from zero to the final value that is used in eq. (L). It is very difficult to estimate the average lifetime of a pair (T~“ir)_One way to get it iS to aSSUnKthdt the lifetime is determined by a free diffusion process. Taking a diffusion cdefficient of the order of 0.5 X 10m5 2 -1 s. cm 5 , one obtains 7*pairin the range 10-*l-lO-lo Evidently this is a very rough estimate, even the use of the macroscopic description of the diffusion process over molecular distances is questionable. In any case it is quite difficult to assume free diffusion because the reacting complexes are usually charged and there exists a strong electric repulsion-between them. As a result the lifetime of a pair should be smaller than the above estimate. In any case it seems Safe to assume that Tpair is long compared to the fast formation and dissociation
trattsfer reactions
processes. Our picture is finally schematized into a highIy simplified model. ft describes the dynamics of the ion pair a5 consisting of an instantaneous formation and a finite lifetime ‘rpair which terminates abruptly. According to this modei the interaction terms in the hamiltonian (eq. (1)) exist only during the lifetime of the pair and vanish at any other time. Let us proceed with the definition of two partial hamiltonians HOi and Hof Ifop = T, + V&)
Q = i, f.
+ c/Jr).
(3
These hamiltonians have sets of eigenfunctions {$,tj) and {$J,,J} with energies E,,i and EtJf, respectively. One should note that also in the final state: as defined here, the solvent is in its initial equilibrium configuration. The wavefunction of our system is approximated as a linear combination of the ground state functions of the partial hamiltonians G(t) = Ci(t) exp(Vl-’ Eoi r) $oi + C,.(t) esp(-ifi-’
Eo/)
$Jof
(3)
In order to get C,(t) and Cf(‘) we have to solve the time dependent Schriidinger equation with the ham& tonian specified by eq. (1). We are interested in the case where the interaction terms in Hare different from zero only for a short period. Therefore the equation has to be solved with initial conditions at a specified time which is taken conveniently to be f = 0: CJO) = 1,
C&0)=0.
(4)
These conditions are different than those employed by Landau and Zener who were interested in the long time asymptotic behavior and therefore used the asymptotic conditions Ci(-m) = 1 and Cf(-m) = 0. Schrijdinger equation in the present case is equivalent to a set of two coupled first order differential equation5
acf/at = if+{Ci
exp[-%-l(~Oi
- Eof)f] Vsj
f crv& and a similar equation for Ci. We define
(5)
S. Efrima, IV. Bison fOuter sphere electron
Q = (1 -
+-’ Wff -
vfi=(‘-sy
Hi&,. -
EofCl- @ , (6)
[f+HiiSif],
(7)
where Sir is the overlap integral
and Hpe.(Q,Q'= i,f) is the QQ’matrix element of the hamiltonian. Vii and v,are obtained from eqs. (6) and (7) by interchanging the indices i and f. The system of first order equations is transformed into the following second order differential equation
0
V (t’) dt’ ff
(TO)
and the initial conditions are
Vfi. Q&o)=& tif(o)=-fi-'
(11)
Thequantity AWfi is defined as AWfi = W/ - Wj .
(12)
W,(Q= i,f3 is the energy in the Q state where shifts due to the interaction between the reactants and to the interaction with the solvent fluctuations are included WQ= Eoo + Vop_
(13)
The treatment up to here is general and involves no approximations (except the omission of excited states). In order to simplify the analysis two assumptions are introduced. The first one is f$=o.
163
tion has a better basis than in the original LandauZener case. Vir includes the matrix element of the interreactant interaction in addition to the matrix element of the time dependent interaction with the solvent. In the time range [0, ~~~~~ the former is time independent, and it is large compared to the second interaction. The second assumption, which is similar to that of Landau and Zener, is that the interaction with the solvent fluctuations may be approximated by a linear time dependence of AWfi AW, = AW; t c&t,
(15)
where art$ is the energy difference between the initial and final states in the time r = 0. It is determined by the distribution of fluctuations at that time (AU&, w(r = 0)). The parameter (Yis the Landau-Zener factor which is defined as
where a/ is defined as a,.(t) = C+(f) exp $j (
transfer reactions
(14)
This is also one of the basic assumptions of Landau and Zener, and has been the main cause for criticizing their results [16,17]. In the present case this assurnp-
ru==AW. fi at
fi A Wri= A b$ ’
(16)
it measures the average rate of change of fluctuations in the solvent (c~= 0 means a static solvent). A discussion of this factor is given in appendix A, where the value cx< 10z5 s-* _ ISestimated for water at room temperature. In principle Q depends on AWTi. But, for small enough ranges of A$ the dependence is neglected. One should also note that for each AW> (Ymay be positive or negative with the same probability. lorl- IL2 is the characteristic time for the changes in the interaction between the electron and solvent. In the following discussion all times are expressed in this unit. For example, the duration of contact between the reactants is given according to the above estimates as Ial%
. = 10-100, p;ur
(171
[LIP% = 10cd4 is the unit of energy corresponding to the characteristic time for solvent motions. It must be stressed that the values given to 101l”~fiand lct11’2~pa~r are only very crude estimates.
S. Efiiina, 41. Bi.~otlfOiiter sphere electrorl tramfer reacriom
164
3. Derivation of the transition probability
where /35 is defined as
In order to get the transition probabi!ity one has to solve eq. (9) for the amplitude al(r). Doing so, the two possible directions of solvent motion (CY = 21~1)should be taken into account. They are distinguished in the subsequence by the extra index (It)_ Let us define the following quantities, first
~~=~,(r=O)=e’in~4Alt~j,llaj1~‘~.
1: = AlV*./la11’21i
which is an expression for the energy gap between the initial and final states at f = 0. Using the wronskian relation for the parabolic cylinder functions [IS]one cm simplify the expression in eq. (73) to obtain.
-t
+ jr~ll’~t 7
fi '
OS)
which is the expression for the energy gap between the initial and thl
s&s
at the
t and is given in units of
Jal*“fi. We shall find use also for the same quantity with a dirferent plnsc ~ =,+i7;/4
f
We also define PO as the same quantity without the phase factor fro = ilWJ~/lLYl%
l”&f,l~
(25)
= 57-’ lyl ll’(37)l’
(19)
?‘A-
x P_Zir(Zk)
Finally we write down the square of the transition rnent in terms of the characteristic units of so!vent tion 7 = / Vjf12/20&
Instead of treating directly the amplitude ujk we
(?I)
and by using eq. (9) and the above definitions one gets the following differential equation [IS] d2,vk/dzf f (+ - 7_i7-i
1’;) W, = 0.
(22)
This equation is known as the Weber equation and its solutions are expressed in terms of the parabolic cylindcr functions [la]. Using the Whittaker functions Q1(2) one gets afterportie manipulations the following solution for eq. (22) with the initial conditions in eq-(I I)
This cspression is the complete solution to our problem. It gives the transition probability as function of time, the initial energy gap ((3o)and the magnitude of the perturbation irjf. The Inndau-Zener result is obtained from it in the limits 00 = -I*+ -b 00 or PO = -y+ +_m I$
= 1 -c-
-D_2ir(--i~)D_?ir(Pt)l’~
(23)
4rlrl _
(27)
The expression (26) represents the &era1 solution, but it is not a very practical one for further use. The reason is that it is quite difficult to handle Whittaker functions of complex order and argument. In order to perform the averaging over the solvent states, and to get reasonable analytic expressions, one must resort to some approximations. This can be readily done in the limits of the nonadiabatic and the adiabatic cases. The nonadiabatic limit is characterized by a small perturbation relative to the characteristic energy of the solvent motion, IrId I.
’ ID__7iy(:_) -+ D_2iy(-P*)
- D_ziu(-‘i)D_,u(P,)I~. (26)
mo-
PO)
lc~l”~f),
o_?i-rGP=)
no-
change variables once again 3s follows w, = a f5 exp(-+iv,
(24)
(I vi+ < ICYI “‘h).
(28)
This condition was discussed by us previously [ 131 and we expect for it to hold in most outer-sphere reactions. In order to get the nonadiabatic limit of eq. (26) we need some mathematical transformations which
S. Efrirrla.M. Bi.wn/Umr sphere elecrron rramfer reacriotls are outlined in appendix B. The final result of the manipulations is the following simple expression lrz~(~)l~= 2al7l ICS(7+‘2 po, - cS(n- “‘&)I’,
(39)
where CS(x> = C’(Y)+ S(x), and C(x), S(x) are the Fresnel integrals. This solution, of course, yields in the firnit flu =y5 -+ +m the classical Landau-Zener result @I- = 47rlyl. In tfle case of tfle adiabatic limit (7 > 1) it is possible to derive from eq. (26) the following expression for the transition probability (the mathematical outline of the derivation is given in appendis B)
f (21-y\-4i)
-$P,lrl-*
arcsinh[~y,f~f-f
(0; + 8frf)t”
elf ‘8lrl)“‘]ll”.
Once again the Landau-Zener in the limit flu.= --y* -+ fM
(~0) expression is obtained
2 =2fyl fQfL X lim f3i2(exp{-arcsinh &I--
(31) &,/@/yf)‘/‘]
]I’= 1.
Eqs. (29) and (30) are reasonably simple and easy to evaluate. They give us, in the nonadiabatic (I$ < I) and in the adiabatic (fyi > 1) cases, the transition probability as a function of the initial energy gap (flu) and the time (lc~!f”~t). Its general behavior, as revealed in numerical evaluations can be summarized as follows. (1) The transition probability oscillates in time at a frequency proportional to the energy gap at the corresponding time. (2) The transition probability is small if tfte system does not cross the intersection point during the time period [0, r]. It becomes larger (especially in the adiabatic case) as the system comes closer to the intcrsection point within the above time period. (3) The transition probability becomes large if the value of&-, is such that during the period [0, r] the system passes through the intersection. The long time probability in this case grows with f&f and approaches the Landau-Zener values (477171in the nonadiabatic
163
case and 1 in the adiabatic case)_ (4) In the nonadiabatic case tfrere are some “resonances” in the tong time transition probability as functions of PO_At such values of & (approximately IpI =(t~n)~l~ the transition probability is larger than the Landau-Zener value. In order to get the rate constant one has to average the transition probability at the time rP-pair over the distribution of on. As a first stage let us took at the dependence of tfrc transition probability on PO at fixed times. Fig. 1 shows fnf(f)f$ in the nonadiabatic case for several values of ]o~f~/~t.At very short times (fuf”‘t < 1) the shape of the curves reflects the uncertainty principle and its width is proportional to f-l _ For larger times (fof ‘12t > 1) the width is proportional to time, which is a reflection of the fact that all systems with initial energy gap in the range --&II
4. The rate constant The rate constant of the electron transfer reaction is proportional to the average transition probability at the time 7pair*The averaging is over tfre solvent states and in the present model it amounts to the averaging over the distribution of CI,, the initial energy gap. We adopt Marcus model for tfle solvent and get the folfowing expression for ttle average iransition probability: Ir/ = (45iknTh)-1”
r d(e4sjf) -01
exp
(32) where X is the solvent reorganization energy. A$, is the difference between the diagonal matrix elements of the local electric potential in the initial and final states, e is the electron charge, kB is the Boltzmann constant and T is the absolute temperature. More details on the averaging procedure may be found in ref. [13].
S. Efiima,
166
Al. BixoidOtiter
sphere electron transfer reactiorts
0.20
l3
<2 0.16
0.12 3 t 0.08
004
-000
l$z. I. Thedrpendmx of tlw transition probability Won the initial rap fl,-,. at various t xc: (1) 0.1. (2) 0.5. (3) 5, (4) 10,(5) 100, (6) 200. valllrsoila1"2
_
times (ICYIInf)
in the nonadiabatic
case
(1 = 10-‘LTlle
According piwit
CA $,_ = aZ,.
the iutersection
point
,
is
(33)
is the energy gap
whcrc Gfillill stateYin ctw
lo tlic model
by the relation
the initial
between
equilibrium
the initial
configuration
and
of
s1h,,t.
Sinlply
by looking
at figs.
1 and 2 one sets that the
region, and thcrcfore it is possible to write it quite accurately as
integral
is cmfined
to the reaction
~Ei/+laIrrTpair
Evidently. if the range of integration is smal! compared IO IIIC width of the gaussian,
it is permissible to approximate the integral by the value of the integrand at the intersection point (eAGif = A+), times tfle integration range. This approximation should be quite good even when the integration range is close to the gaussian width. The integral, as written in eq. (34), is proportional to the gaussian evaluated at some intermediate energy E* which is in the range ail - I~l~rpair < E* < AEii. But at this point it is worthwhile to reconsider one simplifying assumption in our model. According to eq. (16) any initial energy gap dWTi grows linearly in time without any bound, the averaging is only over the initial energy gap. Of course all fluctuations grow only up to some finite value and then decay. Therefore only a fraction of the stateswith initial gap in the range ail - ~7p3ir B Arr~i f ~~~ can reach the point AWfi = AEjF Taking this consideration into account one has to correct the value of E* and take it closer to AE$. The conclusion of the above discussion is that if the
S. Efrima. hf. BixonlOuter
sphere
electron
167
transfer reactions
,’
0.8
/’
3
, 0.6
0.6
t
TO.4
,
0.2
I -32
I I -16 0 -pxlo
I
16
--/3x10
1.0
0.8 0.6
3 t
0.4
--pXlO
1.0
E
1.0 l-
0.8 0.6
0.6 3 to.4
0.2 L
-P
,P
0.8
z= 1 0.4
0.2 0.0
I 32
0.2 Ii
0.0 :
0.0
:
Fig. 2. The dcpcndencc of the transition probability IVon the initial gap PO, at various times (lai1’2t) in the adiabatic limit. The values
of
ld1’2t
integration
are:
(1)
0.5,
(2) I, (3) 5, (4) 10, (5) 100,
(6) 200.
range is smaller, or close to the gaussian
width one may safely approximate the average transition probability by
X
45rlrl l,l”’ ‘Tpair,
Irl g 1;
ja(“*r pair,
Irl>
1.
most coincide for the physical range of To.+ that is 10 < loll 7pair < 100. The conclusion is, of course, that the Landau-Zener theory, or the “static” approximation [ 131 describes well the electron transfer reaction in the nonadiabatic conditions_ -I I
(36)
Eq. (36) is exactly the result of Lcvich et al. [4,5] in their solution of the semiclassical Landau-Zener model. Thus, although the microscopic behavior of the system is not exactly the same as the time asymptotic behavior, the averaging over solvent configurations reduces the differences. This conclusion is confirmed by numerical evaluation of the exact solution. Fig. 3 shows the results of a claculation of the average transition probability in the nonadiabatic case (using eqs. (29) and (32)). The results of the asymptotic Landau-Zener model are given for comparison: As is evident, the two curves al-
Fig. 3. The avcrag transition probability, Av(W) as function of time, lal “*t, (I), compared to the result of Levich and Dogonadze [S], (2). Nonadiabatic limit y = lo-*, Gfi = h =IcV,T=300K.
168
S. Lyritna. df_ Bixott/Ou tcr sphere electron transfer reacrions
The fluctuation 4Fp originates in the polarization fluctuations of the solvent. The time dependence of such fluctuations is related to the dielectric relaxation properties of the system. Assuming linear behavior, the rate constant for the decay of a polarization fluctuation is given by 7-l , where r is the dielectric relaxation time. Because of the proportionality between 4sfi and a polarization fluctuation one may write (A3) and from eq. (A?-) we get OL= (ICI/fir) 45. I-i% :‘,_The :wx:y~ tramiCon probability as timction of time, to the rrsuJt of Lcvich and Dogonadze l@l ‘-1. (I ). cornp:tr~~I 151. (2). Adiabatic limit y = 1. AEfi = A= 1 cV. T= 300 R.
In fig. 3 one can see the results of a calculation for the adiabatic case (based on eq. (30)). Once again the conclusion is that for the physical range of rpair the asymptotic describes the reaction correctly. There arc considerable differences between the detailed computation and the asymptotic theory at short and long times. but they have no physical significance. In any case one should rcmentber that the present model is inadequ;lte for very short times because of the assumption of instantaneous formation of the ion pair. At long times there appears another difficulty, it is unreasonable to assume a linear time dependence such as in eq. (15) for long periods.
Xppendis h. Evaluation of the Landau-Zener
factor
In tllis appcndis we sketch a highly simplified and crude way d using the solvent model of Marcus to estimate :hc Lmdau-Zener Pdctor Q. According to this model OIW has
(Al) ‘u:‘,? and 4$,i- are defined in section 4. The time dependence enters only through the fluctuating potential difference 4qfj and therefore we may write
where
(A4)
The average square 2 is easily evaluated by using the known equiIibrium distribution of 4$n [13] to give
where X is the solvent reorganizatiounergy. To get a numerical estimate for (Y’ in the case of water at room temperature we use for h the representative value 1 eV, and for the dielectric relaxation time the value 1 X 10-l’ s [19]_ From the above values one gets 2 = 1.5 X IO” s-~, and taking the inverse fourth root we get the characteristic time ICYI-“’ = 1.6 x lo-l3 s. The above estimate for (Yis surely too high. This is because we have based our evaluation of the averaged (IIon the assumption of linear regression of fluctuations. But, as we are interested in the fluctuations far from equilibrium, we have a situation where most of the fluctuations are near their maximum values and therefore their rate of change is stower. As a result our estimate is only an upper bound, cr < 1O-‘5 se2_
Appendix B The aim of this appendix is to outline the mathematical procedures that enable us to get thenonadiabatic and adiabatic limits of eq. (27). Aft the mathematical relations we use are taken from ref. 1181. In order to get the nonadiabatic limit (Irl < 1) we first transform the Whittaker’s parabolic cylinder functions D,,(z) into Kummer functions [IX, ch. 131. Using recurrence relations between Kummer functions
S. Eftima. AI. Bi_mrl/Outer sphere electron tramfer reactions
and an integral representation obtains
[18, sec. 132.1] one
D2i,(?e-i’4iry)
L-
169
exp{+slyl
- 3 log Y(*Y, Irl) + iO(Q,
-5 i$2 Irl)) ,
(B4)
where we use the following definitions of the various functions Y(Q, 171)= cv” t 8171- -7i)“’ , N’zy, I70 = +-YY(V,
171)+ (2171- $i)
X arcsinh [ky/2(2]7l - &i)‘/2] x [I _ c)“iiyl _,,= /
d*,*tiW!jr
(B5)
WI
**ilrl +2 in eq. (B4) is related
0
Ie-io2
I = 7171 . I
?o 1~1 by
(67)
Further expansion in inverse powers of 171gives finally (I71 % 1) eq. (30). This espression is still exact. Expanding it in powers of 171and keeping only the lowest order, one gets
References [ 11 R.A. Marcus, J. Chcm. Phys. 24 (1956) 966. i2] R.A. Marcus, J. Chem. Phys. 24 (1956) 979. [3] R.A. Marcus, Ann. Rev. Phys. Chem. 15 (1964) 155. [4] V.G. Levich, in: Advances in electrochemistry and electrochemical engineering, Vol. 4 ed. P. Delahay, (1966). [S] V.G. Levich, in: Physicai chemistry - an advanced treatise. Vol. 9B eds. H. Eyring, D. Henderson and W. Jost (Academic Press, London, 1970). [6] R.R. Dogonadze, in: Reactions of moIecuIes at electrades, ed. N. Hush. [7] P.P. Schmidt, J. Chem. Phys. 58 (1973) 4384. [8) W. Schmicklcr and W. Vielstich, Electrochim. Acta 18 (1973) 883. [9] N.R. Kestner, J. Logan and J. Jortner, J. Phys. Chem. 78 (1974) 2148. [IO] R.R. Dogonadze and Z.D. Urushade, J. Electroanal.
The integrals appearing in eq. (B2) can be expressed
in terms of Fresnel integrals C(.V)and S(A-).Using the combination CS(x) = c(x) f i S(.u) we may write the final expression for ILI~(~)]~in the nonadiabatic limit as laf(r)l: = Ziilyl ICS(ir- *‘2fio) - CS(n-1’2Q2_
(B3)
To derive an expression for the adiabatic limit we need the asymptotic form Of Dzi,(Z) as 1719 1. Such asymptotic expressions can be found for other paraboIic cvlinder functions 118. ch. 191_using them we get
Chem. 32 (1971) 23.5. [ 111 S.G. Christov, Ber. Bunsenges Physik. Chem. 79 (1975)
357. [I21 P.P. Schmidt, J. Phys. Chem. 77 (1973) 488. [ 131 S. Efrima and M. Bison, J. Chcm. Phys. 64 (1976) 3639. [14] L.D. Landau, Phys. Z. Sow. 2 (1932) 46. [15] C. Zener, Proc. Roy. Sot. (London) Al37 (1932) 696. 1161 D.R. Bates, Proc. Roy. Sot. (London) A257 (1960) 22. 117 ] CA. Coulson and K. Zalewski, Proc. Roy. Sot. (London) A268 (1962) 437. [18 #I M. Abramowitz and LA. Stegun eds., Handbook ol mathematical functions (Dover Publications, New York, 1965). [I9 1 J.B. Hasted:Aqueous dielectrics (Chspman and Hall, London, 1973).
OUTER SPHERE ELECTRON TRANSFER REACTIONS: A GENERALIZED LANDAU-ZENER
MODEL
Shlomo EFRIMA * and Mordechai BlXON Department of Chertzistr_v,Tel-Aril Chirersity, Ranrat Aviv. Israel Received 2 1 June
I978
A schematic model for the description of reactant pair dynamics is introduced and solved within a generalized LandauZencr proccdnrc. The reactant dynamics is characterized by the finite lifetime of a pair, and thddynsmics of the solvent tluctuations is taken to be linear in time. Evaluation of the average transition probability sho\vs that in usual realistic conditions the reactant dynamics is not an important factor for the computation of rate constants.
1. Introduction The theory of outer sphere electron transfer reactions in polar solvents has been studied extensively. Several formulations were developed, notably the classical model of Marcus [I-S], the quantum mechanical nonadiabatic models which are based on the quantized polaron picture of the solvent [4-91, and the semiclassical models [4-6, 1O-131. There is no reason to doubt the classical character of the polar solvent dynamics at room temperature. Therefore, for such systems, the semiclassical approach may be used safely. The only problem is how much of the details of the solvent dynamics can be incorporated into the theory, and what features of it are essential in order to get a full description. Different approaches, regarding pictures and details of the solvent and its coupling with the electron transfer system, have been used. Levi&, Dogonadze and co-workers 14-6, lo] used a model of harmonica1 vibrational polarization modes for the solvent. From this model they evaluated the rate of change of the interaction with the solvent at the intersection point of the electronic terms. The use of this constant rate of change enables them to apply the asymptotic Landau-Zener [ 14,151 procedure to * Present address: Department of Chemistry, University of California, Santa Barbara, California 93106, USA.
get the transition probability. By this method it is possible to get the whole spectrum of possibilities, from the nonadiabatic behavior up to the adiabatic one. Christov [l l] and Schmidt [12] use essentially the same harmonic model for the polar solvent, within somewhat different calculationa procedures. A different way to describe the interaction with the solvent is to look at the equilibrium distribution of polarization fluctuations, as proposed by Marcus [I 31. Such a model was used, within a framework of semiclassical theory, by Efrima and Bixon [ 131 to analyze the nonadiabatic rate constant. Evidently, in all the models the treatment of the solution dynamics is far from being complete. Especially the dynamics of the two molecules between which the electron is transferred is completely ignored. The aim of the present article is to incorporate the dynamics of the pair of reacting molecules into the theory. We are doing it by using a highly simplified model that schematizes the dynamics by a finite lifetirnc of the pair. Because the interaction can take place only within a finite duration, it is impossible to use the asymptotic Landau-Zener procedure and we have to generalize it. In the next section we discuss the model. In section 3 we evaluate the transition probability and in section 4 we average it and discuss the consequences.
162
S. Efrittra, hf. Bisor&.3rrer
sphere ekctrott
2. The mode1 As in our previous paper [ ~31 we are treating the system as composed of an electron (treated as a quantum mechanical system) which is parametrically coupled to a classical solvent. Its hamiltonian is written 3s H = T, + Vi(r) + V-&r) + Usi
+ 4r/,(r, K!(f)),
(1)
where T, is the electron kinetic energy operator, Vi(r) and Vf(r) are the potential energies of interaction with the donor and acceptor. respectively. Ilsi(r) is the potential of interaction between the electron and the solvent in its initial equilibrium configuration. M&r, w(r)) is the time dependent interaction of the electron with the solvent fluctuations which are characterized by w(r). An cstra ingredient that we incorporate in the model. in a highly schematic and simplified way, is the dynamics of the ions (molecules) between which the electron is transferred. As usual we assume that the electron transfer takes place only when the two ions are approximately in contact. All the terms in the hamiltonian (eq. (1)) are evaluated for contact configuration. The formation of the pair is visualized as resulting from a fast fluctuation in which one ion jumps over some distance to get into contact with a second ion. Assuming average thermal velocity, and a jump distance of the order of I-2 8, one gets values in the range 1O- 13- 1O- I ’ s for the time needed for a pair formation. During this time the interaction between the electron on one ion and the second ion grows from zero to the final value that is used in eq. (L). It is very difficult to estimate the average lifetime of a pair (T~“ir)_One way to get it iS to aSSUnKthdt the lifetime is determined by a free diffusion process. Taking a diffusion cdefficient of the order of 0.5 X 10m5 2 -1 s. cm 5 , one obtains 7*pairin the range 10-*l-lO-lo Evidently this is a very rough estimate, even the use of the macroscopic description of the diffusion process over molecular distances is questionable. In any case it is quite difficult to assume free diffusion because the reacting complexes are usually charged and there exists a strong electric repulsion-between them. As a result the lifetime of a pair should be smaller than the above estimate. In any case it seems Safe to assume that Tpair is long compared to the fast formation and dissociation
trattsfer reactions
processes. Our picture is finally schematized into a highIy simplified model. ft describes the dynamics of the ion pair a5 consisting of an instantaneous formation and a finite lifetime ‘rpair which terminates abruptly. According to this modei the interaction terms in the hamiltonian (eq. (1)) exist only during the lifetime of the pair and vanish at any other time. Let us proceed with the definition of two partial hamiltonians HOi and Hof Ifop = T, + V&)
Q = i, f.
+ c/Jr).
(3
These hamiltonians have sets of eigenfunctions {$,tj) and {$J,,J} with energies E,,i and EtJf, respectively. One should note that also in the final state: as defined here, the solvent is in its initial equilibrium configuration. The wavefunction of our system is approximated as a linear combination of the ground state functions of the partial hamiltonians G(t) = Ci(t) exp(Vl-’ Eoi r) $oi + C,.(t) esp(-ifi-’
Eo/)
$Jof
(3)
In order to get C,(t) and Cf(‘) we have to solve the time dependent Schriidinger equation with the ham& tonian specified by eq. (1). We are interested in the case where the interaction terms in Hare different from zero only for a short period. Therefore the equation has to be solved with initial conditions at a specified time which is taken conveniently to be f = 0: CJO) = 1,
C&0)=0.
(4)
These conditions are different than those employed by Landau and Zener who were interested in the long time asymptotic behavior and therefore used the asymptotic conditions Ci(-m) = 1 and Cf(-m) = 0. Schrijdinger equation in the present case is equivalent to a set of two coupled first order differential equation5
acf/at = if+{Ci
exp[-%-l(~Oi
- Eof)f] Vsj
f crv& and a similar equation for Ci. We define
(5)
S. Efrima, IV. Bison fOuter sphere electron
Q = (1 -
+-’ Wff -
vfi=(‘-sy
Hi&,. -
EofCl- @ , (6)
[f+HiiSif],
(7)
where Sir is the overlap integral
and Hpe.(Q,Q'= i,f) is the QQ’matrix element of the hamiltonian. Vii and v,are obtained from eqs. (6) and (7) by interchanging the indices i and f. The system of first order equations is transformed into the following second order differential equation
0
V (t’) dt’ ff
(TO)
and the initial conditions are
Vfi. Q&o)=& tif(o)=-fi-'
(11)
Thequantity AWfi is defined as AWfi = W/ - Wj .
(12)
W,(Q= i,f3 is the energy in the Q state where shifts due to the interaction between the reactants and to the interaction with the solvent fluctuations are included WQ= Eoo + Vop_
(13)
The treatment up to here is general and involves no approximations (except the omission of excited states). In order to simplify the analysis two assumptions are introduced. The first one is f$=o.
163
tion has a better basis than in the original LandauZener case. Vir includes the matrix element of the interreactant interaction in addition to the matrix element of the time dependent interaction with the solvent. In the time range [0, ~~~~~ the former is time independent, and it is large compared to the second interaction. The second assumption, which is similar to that of Landau and Zener, is that the interaction with the solvent fluctuations may be approximated by a linear time dependence of AWfi AW, = AW; t c&t,
(15)
where art$ is the energy difference between the initial and final states in the time r = 0. It is determined by the distribution of fluctuations at that time (AU&, w(r = 0)). The parameter (Yis the Landau-Zener factor which is defined as
where a/ is defined as a,.(t) = C+(f) exp $j (
transfer reactions
(14)
This is also one of the basic assumptions of Landau and Zener, and has been the main cause for criticizing their results [16,17]. In the present case this assurnp-
ru==AW. fi at
fi A Wri= A b$ ’
(16)
it measures the average rate of change of fluctuations in the solvent (c~= 0 means a static solvent). A discussion of this factor is given in appendix A, where the value cx< 10z5 s-* _ ISestimated for water at room temperature. In principle Q depends on AWTi. But, for small enough ranges of A$ the dependence is neglected. One should also note that for each AW> (Ymay be positive or negative with the same probability. lorl- IL2 is the characteristic time for the changes in the interaction between the electron and solvent. In the following discussion all times are expressed in this unit. For example, the duration of contact between the reactants is given according to the above estimates as Ial%
. = 10-100, p;ur
(171
[LIP% = 10cd4 is the unit of energy corresponding to the characteristic time for solvent motions. It must be stressed that the values given to 101l”~fiand lct11’2~pa~r are only very crude estimates.
S. Efiiina, 41. Bi.~otlfOiiter sphere electrorl tramfer reacriom
164
3. Derivation of the transition probability
where /35 is defined as
In order to get the transition probabi!ity one has to solve eq. (9) for the amplitude al(r). Doing so, the two possible directions of solvent motion (CY = 21~1)should be taken into account. They are distinguished in the subsequence by the extra index (It)_ Let us define the following quantities, first
~~=~,(r=O)=e’in~4Alt~j,llaj1~‘~.
1: = AlV*./la11’21i
which is an expression for the energy gap between the initial and final states at f = 0. Using the wronskian relation for the parabolic cylinder functions [IS]one cm simplify the expression in eq. (73) to obtain.
-t
+ jr~ll’~t 7
fi '
OS)
which is the expression for the energy gap between the initial and thl
s&s
at the
t and is given in units of
Jal*“fi. We shall find use also for the same quantity with a dirferent plnsc ~ =,+i7;/4
f
We also define PO as the same quantity without the phase factor fro = ilWJ~/lLYl%
l”&f,l~
(25)
= 57-’ lyl ll’(37)l’
(19)
?‘A-
x P_Zir(Zk)
Finally we write down the square of the transition rnent in terms of the characteristic units of so!vent tion 7 = / Vjf12/20&
Instead of treating directly the amplitude ujk we
(?I)
and by using eq. (9) and the above definitions one gets the following differential equation [IS] d2,vk/dzf f (+ - 7_i7-i
1’;) W, = 0.
(22)
This equation is known as the Weber equation and its solutions are expressed in terms of the parabolic cylindcr functions [la]. Using the Whittaker functions Q1(2) one gets afterportie manipulations the following solution for eq. (22) with the initial conditions in eq-(I I)
This cspression is the complete solution to our problem. It gives the transition probability as function of time, the initial energy gap ((3o)and the magnitude of the perturbation irjf. The Inndau-Zener result is obtained from it in the limits 00 = -I*+ -b 00 or PO = -y+ +_m I$
= 1 -c-
-D_2ir(--i~)D_?ir(Pt)l’~
(23)
4rlrl _
(27)
The expression (26) represents the &era1 solution, but it is not a very practical one for further use. The reason is that it is quite difficult to handle Whittaker functions of complex order and argument. In order to perform the averaging over the solvent states, and to get reasonable analytic expressions, one must resort to some approximations. This can be readily done in the limits of the nonadiabatic and the adiabatic cases. The nonadiabatic limit is characterized by a small perturbation relative to the characteristic energy of the solvent motion, IrId I.
’ ID__7iy(:_) -+ D_2iy(-P*)
- D_ziu(-‘i)D_,u(P,)I~. (26)
mo-
PO)
lc~l”~f),
o_?i-rGP=)
no-
change variables once again 3s follows w, = a f5 exp(-+iv,
(24)
(I vi+ < ICYI “‘h).
(28)
This condition was discussed by us previously [ 131 and we expect for it to hold in most outer-sphere reactions. In order to get the nonadiabatic limit of eq. (26) we need some mathematical transformations which
S. Efrirrla.M. Bi.wn/Umr sphere elecrron rramfer reacriotls are outlined in appendix B. The final result of the manipulations is the following simple expression lrz~(~)l~= 2al7l ICS(7+‘2 po, - cS(n- “‘&)I’,
(39)
where CS(x> = C’(Y)+ S(x), and C(x), S(x) are the Fresnel integrals. This solution, of course, yields in the firnit flu =y5 -+ +m the classical Landau-Zener result @I- = 47rlyl. In tfle case of tfle adiabatic limit (7 > 1) it is possible to derive from eq. (26) the following expression for the transition probability (the mathematical outline of the derivation is given in appendis B)
f (21-y\-4i)
-$P,lrl-*
arcsinh[~y,f~f-f
(0; + 8frf)t”
elf ‘8lrl)“‘]ll”.
Once again the Landau-Zener in the limit flu.= --y* -+ fM
(~0) expression is obtained
2 =2fyl fQfL X lim f3i2(exp{-arcsinh &I--
(31) &,/@/yf)‘/‘]
]I’= 1.
Eqs. (29) and (30) are reasonably simple and easy to evaluate. They give us, in the nonadiabatic (I$ < I) and in the adiabatic (fyi > 1) cases, the transition probability as a function of the initial energy gap (flu) and the time (lc~!f”~t). Its general behavior, as revealed in numerical evaluations can be summarized as follows. (1) The transition probability oscillates in time at a frequency proportional to the energy gap at the corresponding time. (2) The transition probability is small if tfte system does not cross the intersection point during the time period [0, r]. It becomes larger (especially in the adiabatic case) as the system comes closer to the intcrsection point within the above time period. (3) The transition probability becomes large if the value of&-, is such that during the period [0, r] the system passes through the intersection. The long time probability in this case grows with f&f and approaches the Landau-Zener values (477171in the nonadiabatic
163
case and 1 in the adiabatic case)_ (4) In the nonadiabatic case tfrere are some “resonances” in the tong time transition probability as functions of PO_At such values of & (approximately IpI =(t~n)~l~ the transition probability is larger than the Landau-Zener value. In order to get the rate constant one has to average the transition probability at the time rP-pair over the distribution of on. As a first stage let us took at the dependence of tfrc transition probability on PO at fixed times. Fig. 1 shows fnf(f)f$ in the nonadiabatic case for several values of ]o~f~/~t.At very short times (fuf”‘t < 1) the shape of the curves reflects the uncertainty principle and its width is proportional to f-l _ For larger times (fof ‘12t > 1) the width is proportional to time, which is a reflection of the fact that all systems with initial energy gap in the range --&II
4. The rate constant The rate constant of the electron transfer reaction is proportional to the average transition probability at the time 7pair*The averaging is over tfre solvent states and in the present model it amounts to the averaging over the distribution of CI,, the initial energy gap. We adopt Marcus model for tfle solvent and get the folfowing expression for ttle average iransition probability: Ir/ = (45iknTh)-1”
r d(e4sjf) -01
exp
(32) where X is the solvent reorganization energy. A$, is the difference between the diagonal matrix elements of the local electric potential in the initial and final states, e is the electron charge, kB is the Boltzmann constant and T is the absolute temperature. More details on the averaging procedure may be found in ref. [13].
S. Efiima,
166
Al. BixoidOtiter
sphere electron transfer reactiorts
0.20
l3
<2 0.16
0.12 3 t 0.08
004
-000
l$z. I. Thedrpendmx of tlw transition probability Won the initial rap fl,-,. at various t xc: (1) 0.1. (2) 0.5. (3) 5, (4) 10,(5) 100, (6) 200. valllrsoila1"2
_
times (ICYIInf)
in the nonadiabatic
case
(1 = 10-‘LTlle
According piwit
CA $,_ = aZ,.
the iutersection
point
,
is
(33)
is the energy gap
whcrc Gfillill stateYin ctw
lo tlic model
by the relation
the initial
between
equilibrium
the initial
configuration
and
of
s1h,,t.
Sinlply
by looking
at figs.
1 and 2 one sets that the
region, and thcrcfore it is possible to write it quite accurately as
integral
is cmfined
to the reaction
~Ei/+laIrrTpair
Evidently. if the range of integration is smal! compared IO IIIC width of the gaussian,
it is permissible to approximate the integral by the value of the integrand at the intersection point (eAGif = A+), times tfle integration range. This approximation should be quite good even when the integration range is close to the gaussian width. The integral, as written in eq. (34), is proportional to the gaussian evaluated at some intermediate energy E* which is in the range ail - I~l~rpair < E* < AEii. But at this point it is worthwhile to reconsider one simplifying assumption in our model. According to eq. (16) any initial energy gap dWTi grows linearly in time without any bound, the averaging is only over the initial energy gap. Of course all fluctuations grow only up to some finite value and then decay. Therefore only a fraction of the stateswith initial gap in the range ail - ~7p3ir B Arr~i f ~~~ can reach the point AWfi = AEjF Taking this consideration into account one has to correct the value of E* and take it closer to AE$. The conclusion of the above discussion is that if the
S. Efrima. hf. BixonlOuter
sphere
electron
167
transfer reactions
,’
0.8
/’
3
, 0.6
0.6
t
TO.4
,
0.2
I -32
I I -16 0 -pxlo
I
16
--/3x10
1.0
0.8 0.6
3 t
0.4
--pXlO
1.0
E
1.0 l-
0.8 0.6
0.6 3 to.4
0.2 L
-P
,P
0.8
z= 1 0.4
0.2 0.0
I 32
0.2 Ii
0.0 :
0.0
:
Fig. 2. The dcpcndencc of the transition probability IVon the initial gap PO, at various times (lai1’2t) in the adiabatic limit. The values
of
ld1’2t
integration
are:
(1)
0.5,
(2) I, (3) 5, (4) 10, (5) 100,
(6) 200.
range is smaller, or close to the gaussian
width one may safely approximate the average transition probability by
X
45rlrl l,l”’ ‘Tpair,
Irl g 1;
ja(“*r pair,
Irl>
1.
most coincide for the physical range of To.+ that is 10 < loll 7pair < 100. The conclusion is, of course, that the Landau-Zener theory, or the “static” approximation [ 131 describes well the electron transfer reaction in the nonadiabatic conditions_ -I I
(36)
Eq. (36) is exactly the result of Lcvich et al. [4,5] in their solution of the semiclassical Landau-Zener model. Thus, although the microscopic behavior of the system is not exactly the same as the time asymptotic behavior, the averaging over solvent configurations reduces the differences. This conclusion is confirmed by numerical evaluation of the exact solution. Fig. 3 shows the results of a claculation of the average transition probability in the nonadiabatic case (using eqs. (29) and (32)). The results of the asymptotic Landau-Zener model are given for comparison: As is evident, the two curves al-
Fig. 3. The avcrag transition probability, Av(W) as function of time, lal “*t, (I), compared to the result of Levich and Dogonadze [S], (2). Nonadiabatic limit y = lo-*, Gfi = h =IcV,T=300K.
168
S. Lyritna. df_ Bixott/Ou tcr sphere electron transfer reacrions
The fluctuation 4Fp originates in the polarization fluctuations of the solvent. The time dependence of such fluctuations is related to the dielectric relaxation properties of the system. Assuming linear behavior, the rate constant for the decay of a polarization fluctuation is given by 7-l , where r is the dielectric relaxation time. Because of the proportionality between 4sfi and a polarization fluctuation one may write (A3) and from eq. (A?-) we get OL= (ICI/fir) 45. I-i% :‘,_The :wx:y~ tramiCon probability as timction of time, to the rrsuJt of Lcvich and Dogonadze l@l ‘-1. (I ). cornp:tr~~I 151. (2). Adiabatic limit y = 1. AEfi = A= 1 cV. T= 300 R.
In fig. 3 one can see the results of a calculation for the adiabatic case (based on eq. (30)). Once again the conclusion is that for the physical range of rpair the asymptotic describes the reaction correctly. There arc considerable differences between the detailed computation and the asymptotic theory at short and long times. but they have no physical significance. In any case one should rcmentber that the present model is inadequ;lte for very short times because of the assumption of instantaneous formation of the ion pair. At long times there appears another difficulty, it is unreasonable to assume a linear time dependence such as in eq. (15) for long periods.
Xppendis h. Evaluation of the Landau-Zener
factor
In tllis appcndis we sketch a highly simplified and crude way d using the solvent model of Marcus to estimate :hc Lmdau-Zener Pdctor Q. According to this model OIW has
(Al) ‘u:‘,? and 4$,i- are defined in section 4. The time dependence enters only through the fluctuating potential difference 4qfj and therefore we may write
where
(A4)
The average square 2 is easily evaluated by using the known equiIibrium distribution of 4$n [13] to give
where X is the solvent reorganizatiounergy. To get a numerical estimate for (Y’ in the case of water at room temperature we use for h the representative value 1 eV, and for the dielectric relaxation time the value 1 X 10-l’ s [19]_ From the above values one gets 2 = 1.5 X IO” s-~, and taking the inverse fourth root we get the characteristic time ICYI-“’ = 1.6 x lo-l3 s. The above estimate for (Yis surely too high. This is because we have based our evaluation of the averaged (IIon the assumption of linear regression of fluctuations. But, as we are interested in the fluctuations far from equilibrium, we have a situation where most of the fluctuations are near their maximum values and therefore their rate of change is stower. As a result our estimate is only an upper bound, cr < 1O-‘5 se2_
Appendix B The aim of this appendix is to outline the mathematical procedures that enable us to get thenonadiabatic and adiabatic limits of eq. (27). Aft the mathematical relations we use are taken from ref. 1181. In order to get the nonadiabatic limit (Irl < 1) we first transform the Whittaker’s parabolic cylinder functions D,,(z) into Kummer functions [IX, ch. 131. Using recurrence relations between Kummer functions
S. Eftima. AI. Bi_mrl/Outer sphere electron tramfer reactions
and an integral representation obtains
[18, sec. 132.1] one
D2i,(?e-i’4iry)
L-
169
exp{+slyl
- 3 log Y(*Y, Irl) + iO(Q,
-5 i$2 Irl)) ,
(B4)
where we use the following definitions of the various functions Y(Q, 171)= cv” t 8171- -7i)“’ , N’zy, I70 = +-YY(V,
171)+ (2171- $i)
X arcsinh [ky/2(2]7l - &i)‘/2] x [I _ c)“iiyl _,,= /
d*,*tiW!jr
(B5)
WI
**ilrl +2 in eq. (B4) is related
0
Ie-io2
I = 7171 . I
?o 1~1 by
(67)
Further expansion in inverse powers of 171gives finally (I71 % 1) eq. (30). This espression is still exact. Expanding it in powers of 171and keeping only the lowest order, one gets
References [ 11 R.A. Marcus, J. Chcm. Phys. 24 (1956) 966. i2] R.A. Marcus, J. Chem. Phys. 24 (1956) 979. [3] R.A. Marcus, Ann. Rev. Phys. Chem. 15 (1964) 155. [4] V.G. Levich, in: Advances in electrochemistry and electrochemical engineering, Vol. 4 ed. P. Delahay, (1966). [S] V.G. Levich, in: Physicai chemistry - an advanced treatise. Vol. 9B eds. H. Eyring, D. Henderson and W. Jost (Academic Press, London, 1970). [6] R.R. Dogonadze, in: Reactions of moIecuIes at electrades, ed. N. Hush. [7] P.P. Schmidt, J. Chem. Phys. 58 (1973) 4384. [8) W. Schmicklcr and W. Vielstich, Electrochim. Acta 18 (1973) 883. [9] N.R. Kestner, J. Logan and J. Jortner, J. Phys. Chem. 78 (1974) 2148. [IO] R.R. Dogonadze and Z.D. Urushade, J. Electroanal.
The integrals appearing in eq. (B2) can be expressed
in terms of Fresnel integrals C(.V)and S(A-).Using the combination CS(x) = c(x) f i S(.u) we may write the final expression for ILI~(~)]~in the nonadiabatic limit as laf(r)l: = Ziilyl ICS(ir- *‘2fio) - CS(n-1’2Q2_
(B3)
To derive an expression for the adiabatic limit we need the asymptotic form Of Dzi,(Z) as 1719 1. Such asymptotic expressions can be found for other paraboIic cvlinder functions 118. ch. 191_using them we get
Chem. 32 (1971) 23.5. [ 111 S.G. Christov, Ber. Bunsenges Physik. Chem. 79 (1975)
357. [I21 P.P. Schmidt, J. Phys. Chem. 77 (1973) 488. [ 131 S. Efrima and M. Bison, J. Chcm. Phys. 64 (1976) 3639. [14] L.D. Landau, Phys. Z. Sow. 2 (1932) 46. [15] C. Zener, Proc. Roy. Sot. (London) Al37 (1932) 696. 1161 D.R. Bates, Proc. Roy. Sot. (London) A257 (1960) 22. 117 ] CA. Coulson and K. Zalewski, Proc. Roy. Sot. (London) A268 (1962) 437. [18 #I M. Abramowitz and LA. Stegun eds., Handbook ol mathematical functions (Dover Publications, New York, 1965). [I9 1 J.B. Hasted:Aqueous dielectrics (Chspman and Hall, London, 1973).