Tunneling of two interacting particles: Transition between separate and cooperativetunnelingYuri I. Dakhnovskii and Mikhail B. Semenov Citation: The Journal of Chemical Physics 91, 7606 (1989); doi: 10.1063/1.457282 View online: http://dx.doi.org/10.1063/1.457282 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/91/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Threshold separation distance for attractive interaction between dust particles AIP Conf. Proc. 1041, 289 (2008); 10.1063/1.2997134 Hydrodynamic interactions of spherical particles in Poiseuille flow between two parallel walls Phys. Fluids 18, 053301 (2006); 10.1063/1.2195992 Two neutron separation energies and phase transitions in the Interacting Boson Model AIP Conf. Proc. 638, 175 (2002); 10.1063/1.1517958 Kinetic theory of the hydrodynamic interaction between two particles J. Chem. Phys. 74, 2494 (1981); 10.1063/1.441318 Separable Schrödinger Equations for Two Interacting Particles in External Fields J. Math. Phys. 11, 2208 (1970); 10.1063/1.1665381

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Tunneling of two interacting particles: Transition between separate and cooperative tunneling

Yuri I. Dakhnovskii8 )

Institute of Chemical Physics, Academy of Sciences of the USSR, Kosygin Str. 4, 117977, Moscow GSP-J, USSR

Mikhail B. Semenov Department of Theoretical Physics, Moscow State University, Moscow, USSR

(Received 9 August 1988; accepted 22 May 1989)

Tunneling of two particles interacting with each other is considered. The semiclassical (instant on) action is calculated taking into account the interaction with an oscillator bath. It is shown that for repulsive interaction between particles, there are two regions of tunneling depending on the value of the interaction constant a. For values larger than a critical value a c ,

simultaneous transition occurs. For a fixed interaction constant, there is a critical temperature at which a change of tunneling regime takes place (from single particle to cooperative tunneling). The same analysis has also been applied to a model including the interaction with phonons. The transition to the cooperative tunneling regime of two particles moving in opposite direction should be observable in optical absorption spectra.

I. INTRODUCTION

Particle transfer by tunneling is very important in var­ious chemical and physical processes. I In chemistry, particle transfer takes place most frequently from one localized state (initial state) to another (final state) by overcoming a bar­rier either classically or by quantum tunneling. It has been observed previously that particle-medium interaction is im­portant because it affects the temperature dependence of the transfer probability.2 However, in the case of some types of phonon spectra (the Ohmic one) and for symmetric poten­tials, it has been shown that a localization of the particle in the initial state occurs, i.e., particle motion through the bar­rier becomes impossible due to a very large viscosity3-6 in­duced by the medium.

In many chemical and physical processes, the transfer of two (or more) particles is possible.7 Two-photon transfer is characterized by simultaneous (cooperative) tunneling of two particles and is accompanied by electron rearrangement (forming and breaking ofbonds).8 For example, let us con­sider a transfer of two protons in the dimer of7-azoindole.8

At first the protons are situated at the nitrogen atoms of the five-membered rings. When two molecules of 7-azoindole approach one another, forming a dimer, then cooperative two-proton transfer can occur. Two protons are placed at the nitrogen atoms of the six-membered rings. The proton transfer is accompanied by rearrangement of the electron orbitals. The simultaneous transfer of two protons is thus favored, whereas independent particle tunneling becomes energetically forbidden because of the orbital rearrange­ments.

In this paper we shall examine a simultaneous transfer of two particles that weakly interact with each other. If the

aJ Present address: Faculte des Sciences, Departement de Chimie, Universi­te de Sherbrooke, Sherbrooke (Quebec), Canada, 11K 2RI.

interaction is absent, either particle moves independently in its own double-well potential (see Fig. I). We shall study the effect of particle interaction on single and cooperative parti­cle tunneling as a function of coupling to the medium.

II. TRANSITION PROBABILITY

Let us choose the potential energies of each particle U(R 1) and U(R2) in the following form3.4 :

U(R 1) = (J/(R 1 + a)2 O( - R 1) 2

+ [AI+ w2(R~_b)2] O(R 1),

U(R2) = w2(R2 - a)2 O( - R 2)

2

+ [AI+ W2(R~+b)2] O( -R2), (1)

where 0 is a step function; the potential parameters a, b, and AI are depicted in Fig. 1. R I and R2 are the coordinates of the particles. AI is an asymmetry parameter in the double-well potential.

The particle masses are assumed to be equal as appropri­ate for proton and have the value unity. w is the harmonic "frequency" of the potential wells. Further, let us choose the interaction Vnt to have the form of an "attractive" harmonic potential

a(RI - R2)2 2

(2)

Such a potential energy can describe the following repUlsive physical situation: Two interacting charged particles are placed at a large distance Ro from each other along the x axis and Ro~a, where a is a transfer distance (intersite separa­tion) along the y axis (Fig. 2). In this case, the interaction potential energy may be expanded in a power series of the parameter (R 1y -R2y)2/R~, where R 1y and R 2y are per-

7606 J. Chem. Phys. 91 (12). 15 December 1989 0021-9606/89/247606-06$02.10 @ 1989 American Institute of PhYSics

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Y. I. Dakhnovskii and M. B. Semenov: Tunneling of two interacting particles 7607

(<<)

R,

R.

FIG. 1. Double-well potential energy. (a) Transfer of the first particle oc­curs from left to right. (b) Transfer of the second particle occurs from right to left, (the opposite direction to the first particle).

Q 14. 0/2

.- - - - - - - - - R.,

, ~0/2

6 R,y - - - - -- - - -e

FIG. 2. Ro (along the Rx axis) is the distance between the tunneling parti­cles. R I, and R 2, are tunneling coordinates.

pendicular tunneling coordinates (see Fig. 2). For the Cou­lomb repulsion between two particles in the medium with dielectric constant Eo,

e2 e2 Vrep

= EolRI Eo [R ~ + (R ly _ R2y)2] 1/2

EoRo 2 EoRo R~ (3)

Therefore

a = e2/EoR 6. (4)

The negative harmonic potential energy (second term) ap­pears therefore as an effective attractive potential even though the potential is repUlsive at all times. This negative term seems to decrease the repulsive potential from its maxi­mal value at Ro. The constant energy U(Ro) = e2 I EoRo may be included into the definition of the potential energies U(R I ) and U(R2).

The medium dynamics are described by the oscillator Hamiltonian

(5)

Each of the particles is assumed to interact linearly with the oscillator bath

V~~h (R1,Q;) = RI L C;Q;, (6a) ;

V:':~h (R 2,Q;) = R2 L C;Q;. (6b) ;

We shall be interested in the transition probability per unit time, or, strictly speaking, only its exponential which can be obtained in the Langer form9 as

r = 2TImZ/ReZ,

where for metastable levels

ir r= -2ImE E=Eo--, 2

(7a)

(7b)

Equation (7a) is obtained by generalizing expression (5b) to nonzero temperatures

r = 2.I; exp( - Eo/T)lm E;

.I; exp( - Eo;ln

2TIm.I; exp( - E;/n 2TIm Z

Re .I; exp( - E;ln Re Z (8)

Here i is the number of energy levels in the metastable state, Z is the partition function of the system, and T is the tem­perature. To calculate r, it is convenient to present Z in the form of the path integral I

The appearance of the imaginary part comes from the decay of energy levels in the initial state well. S is the action of the total system. After exact integration over phonon coordi­nates, I we obtain

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7608 Y. I. Dakhnovskii and M. 8. Semenov: Tunneling of two interacting particles

where

(II)

/3= l/kTis the inverse temperature, and Vn = 21'rn//3 is a Matsubara frequency, and finally D( T) is the phonon Green function which is equal to

D(vn ) = - I C;/(cu; + ~). (12) I

The two-dimensional semiclassical trajectory (instanton) minimizing the action functional S is determined from the equations of motion

R, + n~R, + a,Rz + Jfi

l2

dT,K(T- T,) -/312

x[R,(T,) +Rz(T,)} +cu2aB( -R,)

- cu2bB(R,) 0, (13a)

- Rz + n~Rz + aR, + J/3/2 dT,K(T- T I ) -fi12

X[RI(T,) +R2 (T j )] +cu2aB(Rz)

+ cu2bB( - Rz) = O. (l3b)

In Eq. (13), the kernel K is defined by

K(t) = ~ f 5n/v"" /3 n= 00

(14a)

where 5 n is defined by rewriting Eqs. (12):

D( vn ) = I C 7/cu7 + 5n' (14b) i

i.e., the zero-frequency term is now separated. We shall seek the solution of Eq. (13) as a Fourier sum over the frequen­cies Vn:

R -~ ~ R(') ;V,,' ,- £.. n e , /3n=-oo

R - 1 ~ R (2) Iv,,' 2-- £.. n e .

/3n=-oo

We have introduced the renormalized frequencies

n~ = cu2 - IC;/cu; - a,

i

and the renormalized interaction constant

(15)

(16)

Substituting Eq. (15) into Eq. (13), we obtain for n = 0,

R (I) + R (2) = 2ll/(a + b)E o 0 £\2 '

Uo +a j

R~I) _R~2) 2cu2a/3 + 4cu2(a + b)To

n~ -at n~ -a. ' (17)

where

E=IT,-Tzl,

To= (T, + T2)/2.

(18)

(19)

The times Tj and T z taken by the particles to go from their initial state to the top of the barrier are determined from the equations

RI(T,) =0, R2 (T2 ) =0. (20)

Equations (20) permit us to change the arguments of the B functions. Thus instead of the R I' R z dependence, we can obtain the time-dependent B function, which reduces Eqs. ( 13) to linear form. The times ± T I and ± T z correspond to instants when the particles pass semiclassically "over" the tops of the barrier.3

,4 Substituting the trajectory determined from Eqs. (15), (17), and (18) into Eq. (10), we obtain the semiclassical (instant on) action

S = 4cu4a(a + b)To _ cu4 (a + b)2t? _ 4cu4(a + b)2~

n~ - a l /3(n~ + a l ) /3(n~ - a l )

8cu4 (a+b)2 f [Sin2VnTocos2Vn£/2

/3 n=1 (~+n~-al)~

sinzvn £/2 COS2VnTo ] + (21)

(~+n~ +a1 +25n)~ , which is the exponent of the two-particle transition probabil­ity per unit time. To is the average ofthe times of passage across the barriers by the particles as given by Eqs. (19) and (20).

In organic materials, the interaction with a vibrational subsystem is often due to the interaction with one vibrational mode (a promoting mode). In this case, the sums in Eq. (21) can be easily calculated, given that the phonon Green function becomes

(22)

where cu L is a frequency of a localized vibration. Putting Eq. ( 14b) in expression (21 ) for the action S and performing the sums gives the analytic results for S. The expression for S is relegated to the Appendix.

A. Effect of particle Interaction without the medium

Firstly we consider the case of weak interaction between the particles, i.e., a/ cu2 < 1, in the absence of interaction with the medium. Then the action acquires the limiting form

S= (23)

where

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Y. I. Dakhnovskii and M. B. Semenov: Tunneling of two interacting particles 7609

, € - - - + - + ---""-=----=-:::----"------=-~ F( € a) = ( 1 1) sinh d10 sinh €W cosh d10 [cosh flo (f3 /2 - 2 To) - cosh flJ3 /2 ] w2 fl~ fl~ w3 fl~ sinh flJ3 /2 cosh €W [ cosh W (.8 /2 - 2 To) + cosh w.8 /2 ]

+ w3sinh w.8 /2

When a/ w2 < 1 and.8w -+ 00 (zero temperature), we obtain a new dimensionless function F(x) defined from

a 1-F= - -'-F(x),

w2 w3

where

(25)

F(x) = 3e- X + xe- X - 2A cosh x + 2x, (26)

and

w2

x=€w>O; A =_e- 2WTo. a

x now becomes a dimensionless parameter wiT 1 - T 21 re­flecting the difference in time each particle takes to get to the top of the barrier from its initial state. The function F(x) is shown in Fig. 3 for two values of A. A<1I2 corresponds to strong interparticle coupling a, for which one can expect cooperative effects. Thus simultaneous (cooperative trans­fer) should occur at x = 0 or TI = T 2, since both particles cross the barrier simultaneously. The general form of F for that case is illuminated in Fig. 3 from which we see that

(27)

The case x = 2wT 0 corresponds to separate or single particle transfer, since then only one particle time T 1 or T 2 is of impor­tance. Thus from Eq. (19), we have € = T 1 or T 2 and hence x = 2WTo.

The form ofF for this last case is that ofF2 (x) for which now F2(0»F2(2wTo)' Since from the definition of the ac­tion S [Eq. (23)], a minimum in F(x) is a minimum in S, then this results in a maximum transfer probability. We see from condition (27) which holds for large a and hence

2 Fix)

"

FIG. 3. The dependence F(x) on x: (l)F(O) <F(2wTo); (2) F(O) >F(2wTo). Both minima of F(x) occur at the boundaries of the interval x = (O,2WTo). x = o corresponds to simultaneous tunneling; x = 2WTo cor­responds to single particle tunneling.

(24)

~trong coupling between the particles that simultaneous par­ticle transfer is favored over single particle transfer. The weak coupling case always favors single particle transfer as expected.

Condition (27), which implies simultaneous transfer imposes, the following condition on the parameters To and a:

1 a --<-<1. 4wTo w2

(28)

Furthermore, from our previous work on particle tunnel­ing/.4 we can obtain the value of To as a function of the potential parameters b and a as

wTo=-ln -- ~1. 1 (b + a) 2 b-a

(29)

This implies (b - a)/(b + a) < 1, i.e., slight asymmetry. For an exactly symmetric potential energy, condition (28) is not valid since, in this case, To =.8/4.3.4 Then Eq. (28) should be replaced by

2 a -<-2 <1. (30) .8w w

At arbitrary temperatures, the condition for a change of tun­neling regime, i.e., of simultaneous transfer or single particle transfer is determined from the condition F( 0) = F( 2WTo) . From Eq. (24) this gives

.8w 2w2 cosh(.8w/2 - 2wTo) 3 coth - + --( cosh 2WTo - 1) --:::-----"-''--

2 a sinh .8w/2

_ 4 + 3 cosh(.8w/2 - 2WTo) - 1TTo

sinh .8w/2

+ 2wTo sinh (.8w/2 - 2wTo) .

sinh .8w/2 (31)

If the interaction constant a is assumed to be fixed and the inverse temperature.8 is assumed to be variable, then simul­taneous transfer takes place whenever

a-a .8w>4wTo -In ___ c, a c = w/4To (32)

ac

anda>ac ' This results in F(O) <F(2wTo)' Thusac plays the role of a critical constant which depends on the ratio of the frequency w of the wells and the average passage time over the barrier To given by Eq. (32).

As a -+ ac , it follows that .8w -+ 00. The inequality (32) follows from the assumption that .ow ~ 1 and 4wTo ~ 1. Thus, the critical temperature Tc dividing the tunneling regimes is determined by the magnitude of the interaction between the two particles according to the equation

KT = W (33) c 4wTo -In(a - ac)/ac

To the average passage time is obtained from Eqs. (19) and (20). Furthermore, given the value of F(O) or F(2wTo), upon substitution into Eq' (25) one can also obtain the value

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7610 Y. I. Dakhnovskii and M. B. Semenov: Tunneling of two interacting particles

of the action and thus the transition probability for the two regimes. Clearly the function Fgiven by Eq. (24), which is the E-dependent part of the semiclassical action S [Eq. (23) ] is analogous to the free energy. 10 The difference between the "times" in passing through each of the one-dimensional bar­riers (E = 11" I - 1"21 cu) is the order parameter ofthe system. The minimum of the "free energy" takes place either at the origin of the coordinates (simultaneous tunneling) x = 0, or at the boundary value of x = 2cu1"0' i.e., independent tunnel­ing.

Let us estimate the value of the critical temperature Tc accordingtoEq. (31). For this purpose, we choose a = 2ac '

therefore

KT =~= fIcu (34) c 4cu1"0 21n[ (1 + b /a)/(1 - b /a)]

For b /a= 1.01 (a small assymetry), KTc =fIcu/l1. For the case where fIcu = 3000 cm -I, we obtain Tc = 375 K. In or­ganic glass systems, this temperature should be much smaller (fIcu = 100-;-.200 cm- I

, KTc = 10-;-.20 K). The val­ue of Tc will change as a function of a. The value of a de­pends on the distance between the two particles according to Eq. (4). If (a - a c )/ac -( 1, the temperature decreases; in the opposite case it will increase.

B. Effect of the medium

The same analysis can now be performed when we con­sider the interaction with a local phonon mode. For the sake of simplicity, this interaction is also assumed to be small, i.e., C /cu2

-( 1 and C /cui -( 1. Then

(35)

where

F(E,a,C 2) = 3e- x + xe- x - 2A cosh x + 2x + D

[ e-XY 2e- x

X - +--y(1_y2) l-r

+ 3e- x + xe- X _ 2x(1y: r)], (36)

(37)

with the condition that Y# 1. Comparing Eqs. (26) and (36), we observe the modification of the action by terms proportional to D, i.e., the ratio of the phonon coupling (C 2/ cu2

) to the particle interaction (a). The case ofy = 1 will be considered separately. For simultaneous transition, we have as before 1"1 = 1"2' i.e., x = 0:

F(x,A,D,y) Ix = 0

=3-2A+D[- Y 1 r +~+3]. (38) (1- ) l-y

F now becomes independent of the tunneling times 1"1 and 1"2 since these are correlated.

For independent tunneling, only the time of one particle ( 1"1 or 1"2) is of importance. We obtain

F(x,A,D,y) Ix = 2WT"

=4cu1"0-Ae2wT,,_ D4cu1"o(1-r) y2

F is now a sensitive function of the tunneling times.

(39)

The cooperative particle transfer takes place if condition (26) applies leading to the inequality

(40)

Therefore, the criterion for medium-free cooperative transi­tion [Eq. (28)] becomes more stringent if we account for the interaction with phonons. Equation (40) was obtained under the condition of weak coupling, i.e.,

C 2/CU4 -(y2; C 2/CU4 -«(1_y2)2. (41)

In the special case of resonance, or y = 1, i.e., cu = CUV

F, Eq. (36) now takes the form

F = 3e - x + xe - x - 2A ch x + 2x

- D[eX(x2+7x+ 15) +8x]. 4

(42)

The simultaneous tunneling action now depends on the sim­ple expression

F(x)lx=0=3-2A-.!.(D, (43a)

whereas the single particle tunneling action depends on the more complex function

F(x) I _ = 3e - 2WT" + 2cu'T e - 2WT" + 4cu'T x - 2WTo 0 0

+ 14cu1"0 + 15) + 16cu1"0]' (43b)

If CU1"o> 1 then the simultaneous particle transition in the resonance case is now subject to the condition

I C 2 a --+-<--(1. (44) 4cu1" 0 cu2 cu2

In this case, Eq. (43b) reduces also to

F(2cu1"0) =4cu1"0(1-4D) _Ae2WT", (45)

which is now that part of the action for independent tunnel­ing in the symmetric well case and is to be compared to the asymmetric case (39).

III. SUMMARY AND DISCUSSION

We have shown that in the case of two particles interact­ing by an attractive potential, a qualitative change between two types of tunneling transfer takes place at a particular value of the interaction constant a c • This change is very sharp as the interaction constant is varied. The analysis of the tansition is therefore similar to that used in the Landau theory of phase transitions. 10 Thus function F, which is an E­

dependent part of the semiclassical action S [Eq. (23)] is analogous to the free energy. 10 The difference between the "times" in passing through each of the one-dimensional bar­riers (E = 11"1 - 1"2Icu) is the order parameter ofthe system. The minimum of the "free energy" takes place either at the origin of coordinates (simultaneous tunneling) or at the boundary value of x, x = 2cu1"0, i.e., independent tunneling. For weak interaction, i.e., a/ cu2

-( 1, the criterion for cooper-

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Y. I. Dakhnovskii and M. B. Semenov: Tunneling of two interacting particles 7611

ative particle tunneling is obtained at low temperatures. For a given interaction constant a, the inverse temperature Tc for the change of the regimes is predicted by Eq. (33) and the expression for the critical constant a c [Eq. (32)]. For dif­ferent cases the value of the critical temperature was estimat­ed to be equal to Tc = 10 - 400 K in accordance with Eqs. (33) and (34). In glasses, Tc may be small; in chemical reaction it should be large. Its value can be changed with the variation ofthe distance between the particles and, therefore with the change of their concentration. As a rule the transi­tion takes place when quantum tunneling is important (KTJwc,l). Therefore this effect is a quantum one in spite of the high temperature (400 K).

For weak interaction withasingiephonon, C 10/< I and T = 0, one can obtain the transfer condition exactly. For simultaneous tunneling, the attraction between particles needs to be larger, because the interaction with the medium vibrations induces an effective repulsion of the particles [see Eq. (40)].

The presence of the transition between the two different tunneling regimes may be observed from a study of the opti­cal absorption connected with two particles transfer. In the case of simultaneous transfer, the dipole moment which is,

APPENDIX

responsible for the optical transition, is equal to 0 because of inversion symmetry of the transfer. In the case of indepen­dent transfer, the net dipole moment may be nonvanishing, thus resulting in an absorption spectrum. This spectrum will disappear if the inverse temperature reaches the critical val­ue, defined by Eq. (32). The absorption will depend on the concentration of the transfer centers. Since the average dis­tance between particles is proportional to n- I

/3

, where n is the concentration of transfer centers, increasing n results in an increase of the attractive force between particles. In con­clusion it should be noted that the semiclassical action of two attractive tunneling particles has also been calculated exact­ly [Eq. (18)] for a multimode phonon medium. This leads to a more general expression for the transition discussed in this paper.

ACKNOWLEDGMENTS

The authors would like to thank A. I. Larkin (Landau Institute for Theoretical Physics, Chemogolovka, USSR) for his stimulating interest in this work as well as A. D. Bandrauk and J. McCann, Universite de Sherbrooke, for illuminating discussions and help in preparing this paper.

We present here the general expression for the instanton action [Eq. (21)]:

S = 2£v4(a2 - b 2)1"0 _ 2£v4(a + b)2 { _ psinh no(Pl2 - 1"o)sinh no1"o + p(n~ -a{ )cosh n2(P12 - 1"o)cosh n 21"0

n~ P 2n~sinh(ncP/2) 2n~ (n~ - n~ )sinh n:p/2

P( n~ - m~ )cosh n l (P 12 - 1"o)cosh n l 1"o PE PE P sinh Eno - + ---+~--~ 2n~ (n~ - n~ ) sinh nIP 12 4(m2

- 2C 2/mD 4n~ 4n~

p(n~ -mDsinhEn2 _ p(n~ -mDsinhEn l + pcoshEno [coshn (PI2-2r) -coshncP/2] + 4n~ (n~ - ni) 4n~ (n~ _ nD 4n~sinhpnol2 0 0

pent -m~)coshEn2 [coshn2(p12-21"2) +coshn:p/2] + ~(n! -m~)COShEnl

4n~ (n~ - n~ )sinhpn2/2 4n l (n( - n 2 )sinhpn l /2

X [COSh n{~ - 21"0) + cosh nt]},

where we define new frequencies by the expressions

n~ = m2 - a,

n~ = m2 + m~ + ~ (m2

- m~)2 + 8C2

2 (A2)

I A. O. Caldeira and A. J. Leggett, Ann. Phys. N. Y. 149, 374 (1983). 2V. A. Benderskii, V. I. Goldanskii, and A. A. Ovchinnikov, Chern. Phys. Lett. 73, 492 (1980).

(Al)

I 3yU. I. Oakhnovskii, A. A. Ovchinnikov, and M.·B. Semenov, Zh. Eksp. Teor. Fiz. 92, 955 (1987) [SOY. Phys. JETP 65,541 (1987) J.

'Y. I. Oakhnovskii, A. A. Ovchinnikov, and M. B. Semenov, Mol. Phys. 63,497 (1988).

50. Weiss, M. Grabert, P. Hanggi, and P. Riseborough, Phys. Rev. B 35, 9535 (1987).

"S. Chakravarty and A. J. Leggett, and Phys. Rev. Lett. 52, 5 (1984). 7Z. Smedarchina, W. Siebrand, and T. A. Wildman, Chern. Phys. Lett. 143, 395 (1988).

"K. Tokumura, Y. Watanabe, and M. Itoh, J. Phys. Chern. 90, 2362 (1986).

9J. S. Langer, Ann. Phys. N.Y. 141, 108 (1967). IOL. O. Landau and E. M. Lifshitz, Statistical Physics (Pergamon, London,

1958), p. 430.

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