Intermolecular tunnelling between diffusing spherical potential wells : a first approach to direct proton transfer in solutions

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Intermolecular tunnelling between diffusing spherical potential wells : a first approach to direct proton

transfer in solutions

E. Belorizky, P.H. Fries

To cite this version:

E. Belorizky, P.H. Fries. Intermolecular tunnelling between diffusing spherical potential wells : a first approach to direct proton transfer in solutions. Journal de Physique, 1988, 49 (5), pp.727-738.

�10.1051/jphys:01988004905072700�. �jpa-00210749�

Intermolecular tunnelling ^between diffusing spherical potential ^{wells :}

a first approach ^{to direct} proton ^{transfer in solutions}

E. Belorizky (1) and P. H. Fries (2)

(1) Laboratoire de Spectrométrie Physique, ^associé

^C.N.R.S., Université Scientifique, Technologique ^et

Médicale de Grenoble, ^B.P. 87, 38402 Saint-Martin d’Hères Cedex, ^France

(2) Centre d’Etudes Nucléaires de Grenoble, D.R.F., Laboratoires de Chimie, ⁸⁵ X (*), 38041 Grenoble Cedex, ^France

(Requ le 2 novembre 1987, accept6 ^{le 28} janvier 1988)

Résumé.

On propose

modèle théorique pour décrire le transfert intermoléculaire d’un proton ^{dans la} réaction AH+ + B ~ A + HB+ entre des molécules identiques ^{A et B}

solution. On suppose que le proton

se

trouve initialement dans l’état fondamental lié d’un potentiel sphérique ^à trois dimensions (3D) ^localisé

la molécule A. Lorsque les molécules A et B, qui ^ont

^{mouvement de} diffusion, ^entrent

collision, ^le proton peut franchir la barrière de potentiel 3D ^{et être} piégé ^{dans le} potentiel sphérique equivalent relatif à la molécule B. Ce mécanisme correspond ^à

processus ^résonnant dans

double puits ^de potentiel ^{3D. Une} expression analytique ^{du taux} de transfert peut être obtenue

évaluant la fréquence de résonance du double

puits dans le cadre de l’approximation ^{LCAO, et}

adoptant

description classique de la diffusion relative des molécules. On donne des valeurs typiques ^{du taux} de transfert par effet tunnel ^et l’on montre que

mécanisme peut dominer le processus normal d’activation à basse température.

Abstract.

A theoretical model for the intermolecular proton ^{transfer AH+ + B ~ A +} HB+ between identical molecules A and B in solution is proposed. ^The proton ^is initially ^{assumed to be in the} ground ^bound

state of

three dimensional (3D) spherical potential ^well belonging ^to molecule A. When the diffusing

molecules A and B collide, the proton

^tunnel through

^3D potential barrier and be trapped ⁱⁿ

equivalent ^3D spherical ^well belonging ^to molecule B. This corresponds ^to

^resonant process in

3D double

potential ^{well. An} analytic expression of the transfer rate

be obtained by evaluating ^the

frequency of the double well within the LCAO approximation, ^and by using

^classical description ^{of the}

relative diffusion of the molecules. Typical values of the tunnelling ^{transfer rate}

provided and it is shown that this mechanism may dominate the normal activated process

the temperature ^{is lowered.}

Classification

Physics ^Abstracts

05.40

66.10

82.20

1. Introduction.

Several theoretical studies have been devoted to the treatment of intermolecular proton ^{transfer in} liquid

solutions [1-4]. ^This topic is of much interest because it is involved in many chemical and biological

processes [2].

The kinetics of proton transfer is usually ^described by assuming three successive steps : ^the approach ^of

the reactants up ^{to a} suitable distance, the reactive

phase during ^{which the} proton ^is transferred, ^and

the separation ^{of the} products. ^These steps ^are treated independently and the slowest one deter- mines the reaction rate. This rate is either diffusion

controlled [5, 6] ^{when the} proton is transferred

instantaneously during each molecular collision, ^or independent of diffusion [2] when this transfer becomes very slow. However, it is clear that the actual proton transfer is generally ^{a more} complex

process where the diffusion of the ^{reactants as} well

as the proton and electron dynamics ^are intimately coupled. Besides the well known activation _process, where the proton transfer is diffusion controlled with

a rate proportional ^to the diffusion coefficient of the relative translational motion of the molecules, ^there

is a quantum mechanical mechanism of direct tunnel-

ling through ^the potential barrier between the ac-

ceptor and donor sites. This barrier is modulated in

height ^{and width} by the relative random motion of the diffusive reactants. Generally, ^this quantum

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004905072700

effect is discussed as a tunnel correction to the classical activated process and is limited to transfers with linear geometry [1, ^3, 7].

Here, ^{we want to} present ^a quantitative ^{model for}

a three dimensional diffusion induced tunnelling.

The mechanism of direct tunnelling proton ^transfer has already ^{been treated} by Kloffler and Brickmann

[1] ^{under the} following hypotheses :

(i) ^{the two} interacting ^molecules perform ^{a classi-}

cal random Brownian translational motion amidst all the other particles ^{which are} approximated ^{as an}

inert continuum and this motion is not affected by

the proton tunnelling ;

(ii) ^the proton transfer is independent ^{of the}

molecular rotations ;

(iii) ^the proton ^{transfer is treated as a} quantum tunnelling ^{between two identical one} dimensional

(1D) potential wells which simulate the attractive

bonding ^{forces of the} proton with each molecule.

More precisely, ^{in their treatment the} potential

barrier between the two wells was modeled either by

the intersection of two 1D harmonic oscillator poten- tials or by ^a rectangular ^{wall. For a} system ^formed by

a proton moving ^{in a double} potential well, it is well known [1, 8] ^{that the} proton transition probability

per unit time between the two wells is directly

correlated to the resonant frequency AE/h, ^where

AE is the energy splitting ^{of the two} lowest levels of this system. ^This splitting vanishes when the two wells are very distant and, obviously, ^the correspond- ing ground ^state doublet energy is that of the proton in each individual potential ^well. Here, ^{AE is a}

random function of the relative position ^{of the} interacting ^molecules.

In this paper, ^we present ^two improvements ^{to the} original model of Kloffler and Brickmann. First, ^we

consider three-dimensional (3D) wells in order to treat the spatial extension of the proton ^wave functions in a more realistic way. Second, ^{the wells}

are assumed to be spherical ^{with a} finite and constant depth. Qualitatively ^{our first} assumption

will considerably ^{reduce the} overlap ^{between the}

wave functions of the two individual wells, ^thus decreasing the energy splitting ^AE previously ^defined

in comparison ^{with t’he 1D} model. This results in a

noticeable reduction of the proton ^{transfer rate. Our} second hypothesis avoids the unrealistic presence of

a sharp peak ^at the intersection of two harmonic oscillator potentials, and allows a much more accu- rate calculation of the energy splitting ^{AE as a}

function of the distance R of the centers of the wells.

The corresponding ^two dimensional model can be

easily pictured by proton jumps ^{between two diffus-} ing spherical holes of finite depth.

The above theory may be applied, ^for example, ^to

the proton transfer between amines and the corre-

sponding ammonium ions generated by ^{a small}

amount of a strong ^acid (HCI04) ⁱⁿ aprotic ^solvents.

After a presentation ^{of our} model, the energy

splitting ^{AE of the two} lowest levels of the double

sphere potential well is calculated in section 2 within the LCAO approximation. ^The proton ^{rate transfer} will then be evaluated in section 3, using ^{two differ-}

ent descriptions of the relative diffusion of the molecules. The magnitude of the effect will be estimated and discussed in section 4.

2. The double spherical potential ^well.

The attractive potential ^{of the} proton ^{with the} molecule A is assumed to be an isotropic ^{3D well of}

finite constant depth

Notice that it is quite reasonable to have V (r )

⁰

for large ^{distances because there is no Coulomb}

interaction between the neutral molecule A and the proton ^H+. The short ranged ^{form of} V (r) ^is obviously ^{a first} approximation ^and usually Vo ^will

be of the order of the proton bonding energy in the AH+ complex. In the presence of ^a diffusing

molecule B identical to A, ^a tunnelling process may

occur for the proton between the two identical wells.

The two spherical potential wells located on the

interacting ^{molecules are shown in} figure ^{1 where R}

is the relative position ^{of the centers} of the wells.

This relative position depends ^on the translational and rotational motions of the two diffusing

molecules. For sake of simplicity it will be however assumed that the random motion of the two wells

can be described by the usual translational diffusion

equation ^{with an} effective relative diffusion constant D.

Fig. ^1.

^Proton tunnelling between the two spherical potential ^wells VA, VB ^{of constant} depth - Vo ^{and of}

radius a, which

located

equivalent positions ^{of the}

two identical molecules A and B pictured by ^{the dashed}

lines. Despite of the fact that the two molecules cannot

interpenetrate because of the electron cloud repulsion,

take V

0 outside the two potential ^wells

^first approximation. ^This neglects ^not only the Coulomb inter- action of the proton ^{with the} electronic clouds between the two spheres, but also the impossibility ^{for the} proton ^to penetrate well inside the molecules. Thus V (r) appears like

average effective potential ^{for the} proton.

2.1 ENERGY LEVELS OF THE SPHERICAL WELL.

The quantization of the three dimensional spherical

square potential ^{well is a standard} problem ^of quantum ^mechanics [9].

Setting

where M is the mass of the proton, the energy levels

corresponding ^{to s states} (f

0 ) ^are given by ^the

solutions of equation (3)

with tg ^{a a 0. It follows that there is a bound state}

2 MV o

only ^{if ,} j ² MV 1’2 ^{a 7T}

We define

where _al, vo, ^u, y are dimensionless reduced quan- tities and ao, Ry, ^{me are the Bohr radius, the}

Rydberg ^constant, and the electron mass respect- ively. ^{The s} ground ^state corresponds ^{to the} highest

solution yo of the equation

with

Defining a o and {3o the values of

and {3 correspond- ing to y

yo, ^we have

The associated s wave function is

with

A being ^a normalization constant given by

The research of the excited states can be treated in a

similar way [9].

In this paper, it will be assumed that there is only

one bound state given by equations (5). As shown in

section 3, for reasonable values of the radius a and of the depth Vo ^{of the} potential well, ^{C defined} by equation (5c) ranges between 1.8 and 2.3 for the proton, ^so that there is no excited state. Indeed excited s levels are only present ^{for C}

3 zr /2 ^while

an excited p level appears for C >

However, ^for

sake of completeness, the relative influence of an

excited state (for ^C

7T) ^{on the resonant} proton transfer will be estimated in section 4 and shown to be negligible.

2.2 ENERGY SPLITTING OF THE DOUBLE SPHERICAL WELL. - Now we consider the double spherical potential ^well represented ⁱⁿ figure ^{1 and we denote} by ^R the distance of the centres of the two spheres.

We have R > b

2 _a, where b is a minimal distance of approach which is characteristic of the shape ^and

size of the two identical molecules.

Let rA and rB be the distances of the proton ^with the centres of the spheres belonging ^to molecules A and B, ^{as shown in} figure 2, ^{and let} 4’A(rA), 4’B(rB) ^{be the wave} functions of the proton ^{in the} ground ^{s state of each} potential well taken separ-

ately. 4/A ^and 4/B ^are given by equations (7) by replacing r by rA and rB respectively.

Fig. ^2.

^Various distances involved in the calculations of the matrix elements. uA and bB

the volume of the two

spheres ^{where the} proton potential ^{value is}

Vo, ^and

^is

the volume outside these spheres where this potential ^{is 0.}

For the double potential well, ^the Hamiltonian of the system ^becomes

where VA ^and VB ^{are still} given by equation (l)-by replacing r by rA and rB.

For solving ^the Schrodinger equation

R gi (r)

E 0 (r) describing the motion of the proton in the double potential well, ^{we use} the variation method which is commonly used in the treatment of the Hydrogen molecule ion [10a], ^{i.e. a} trial function

t/I = C 1 t/I A + C 2 t/lB. ^It is well known that two

energy levels Es ^and EA ^are obtained, ^{with corre-} sponding ^{states which are} symmetrical ^and antisym-

metrical combinations of 4fA ^and 4’B-

The energy splitting ^{between these two levels is} given [10a, 11] by

where S is the overlap integral

Expression (10) gives the energy splitting ^with good accuracy only ^{if S 1. Now we have to}

evaluate the different terms involved in

equation (10a). All these terms can be calculated

analytically. Details of the calculations are given ⁱⁿ appendix ^A.

We obtain :

where A is given by equation (8) and « o, 130 by equations (6).

We see that this matrix element, which will be shown later to be the main term, ^{has a R behaviour} which is in e - 13oR.

Similarly ^{we obtain}

where f/I A I VA I ql B) ^{is given by} equation (11) ^and

Finally,

where E2 ^and E3 ^are exponential integrals [12a].

The behaviour of the proton energy splitting

AE = EA - Es given by equation (10a) ^{has been} computed ^{as a} function of R for typical ^{values of}

C = 2 corresponding ^to reasonable values of the

potential parameters (a

0.2 ao, 0.6 eV , Vo --

1 eV) and for values of the minimal distance of

approach b varying ^between 6 a and 12 a associated to barrier widths [2, 13], b - ^{2 a,} between 0.4 and 1 A. The splitting ^{t1E has been} compared ^{with its} principal contribution - 2 t/I A I V A I t/I B) and it is found that their relative difference is always ^lower

than 3 %. According ^to the crudeness of our model for the proton potential, from now AE will simply ^be approximated by - 2 t/I A V A I t/lB) ^{which follows}

from equation (11). ^{We shall write}

with

3. Resonant proton ^{transfer rate.}

The energy splitting AE(R) ^{is modulated} by ^the

random relative motion of the molecules A and B on

which are located the potential wells. In order to determine the resonant proton ^{transfer rate we shall} follow a procedure very similar ^to that used by

K16ffler and Brickmann [1].

The tunnelling through ^the potential ^{barrier will}

be considered phenomenologically ^{as a random} perturbation H(t) ^{on a} degenerate ^{two level} system

I A) ^and I B) corresponding ^{to the} ground ^{states of}

the two spherical potential wells when they ^are

remote, ^{so that} IA) ^and I B) ^are orthogonal.

Each actual state of the system ^{can be} formally

described by ^a normalized state

Taking ^the degenerate ^{zero order} energies ^of I A) and I B) ^{as the} origin of the energy, the Hamiltonian matrix of the system ^is

where the random variable Aye(7?) ^is given by ^the approximate ^{value of} expression (l0a) :

It must be noted that, in reference [1], ^{the factor}

1/2 is missing. ^The approximation ^{made in} taking

this simple ^two-level system ^is essentially ^to neglect

terms of the order of S, ^{but this is} precisely ^the approximation which has been made by taking AE(R) ^as expressed by equation (16) instead of

equation (10).

The time evolution operator ^{for our model with} the Hamiltonian of equation (15) ^becomes [14].

where i stands for the normalized eigenstates ^of H(t) ^and AEi/2 ^{are the} corresponding eigenvalues

-t AE/2.

Defining

we obtain

If at t

0 the proton is in the sphere A, ^the probability ^to find it in the sphere ^{B at time t is} simply

and the probability ^{of a} proton transition _{per unit} time is

The average value WAB of the transition prob- ability ^{taken over a} statistical ensemble of pairs ^of interacting molecules will be

The splitting AE (R (t)) given by equation (14) exponentially decreases with the interpotential ^dis-

tance R with a characteristic length l/j3o? ^which, according ^to equation (6b), ^is

An upper limit of the phase angle ^cp (t ) given by equation (18) ^is

where AE (b ) is the maximal energy splitting ^{for the}

minimal distance of approach ^{b of the two wells, and}

where t, is the lifetime of the tunnelling process. An estimate of tc ^is given by the time of the diffusion process during which the relative distance of the two wells increases from b to b + 1/ f3o, ^where 1/ f30 ^is given by equation (23).

Using the Einstein relation, ^{we have}

In table I we give ^{numerical results} concerning ^the

various quantities appearing ^{in the} theory. ^Three

different positions ^{of the} ground ^state energy level

Eo ^{of the} individual potential well, corresponding ^to typical heights ^{of the} potential barrier, ^{are con-}

sidered :

(i) Eo I

^0.02 eV : low barrier height ;

(ii) Eo I ~ 0.08 eV : intermediate barrier height ; (iii) [ Eo I = 0.2 eV : significant ^barrier height.

These values were obtained by taking ^{a fixed} spherical potential ^well radius for each proton

a

0.2 ao

0.1 A, ^which corresponds ^{to an} average value [15] ^{of the classical vibration} amplitude ^{of the} proton, a - (hi MúJO)1/2, ^with h (a 0

^{3 000} cm - 1,

and by taking reasonable values of Vo ranging ^from

0.6 eV to 1 eV.

We have chosen D = 10-5 cm2 S-1 for medium sized molecules and b was fixed [2-13] ^{to an inter-}

mediate value b = 8 a = 0.85 A, ^which corresponds

to a barrier width of 6 a

0.6 A.

First, table I shows that the argument 2 cp (t) ⁱⁿ

the sine function in equation (22) is very small, except ^{in case} (i) for the minimal distance of

approach where it remains 1. Thus, ^{the sine}

Table I.

Proton transfer rates, k, for typical ^values of ^the spherical potential depth vo ^{with a} potential

well radius a

0.2 ao, ^a relative diffusion ^constant

D = 10- 5 cm2 sl, a barrier width b - 2 a = 6 a =

1.2 ao and a number density of proton acceptors

N B 1021cm-3 ^V

function can be linearized, ^{so that} according ^to equations (18) ^and (22)

As signaled above, the energy splitting ^{AE in this}

3D model is reduced with respect ^{to the} correspond- ing linear model. With a potential barrier of the

same height ^{and width as} in the intermediate

case (ii) (Vo

^{0.75 eV, a}

^0.2 ao), ^{a 1D double} potential well defined by

leads to identical ground ^state energies ^{for the}

individual wells as for each spherical ^{well and to a}

value Aye(6

8 a )

^{6.8 x} 10- 3 eV, ^{which is an}

order of magnitude larger than for the spheres.

Thus, the linearization from equation (22) ^to equation (26) ^{is much more} justified in this model than in the 1D proton ^transfer. Taking ^{into account}

the stationary character of the correlation function in equation (26) ^we may write

where P 2 (Ro, R, t) ^{is the} joint probability density

for the relative motion of the two spherical ^wells.

Notice that we have replaced the upper time inte-

gration ^limit by ^{oo since we calculate} WAB ^{for times}

which are much greater ^than tc.

In order to evaluate WAB given by equation (27)

we can use two different methods. In the first _one, followed by Abragam [16] and Kloffler and Brickmann [1], P2 (Ro, R, t ) is the usual Gaussian solution of the diffusion equation

where D is the relative diffusion constant of the

potential wells and where V is the sample ^{volume. In} equation (27), replacing P 2 (Ro, R, t) by ^{the inte-} gral of its Fourier transformation with respect ^to

R - Ro, ^{we obtain} [16]

where b is the minimal distance of approach ^{of the}

two spheres. Notice that equation (29) is similar to

equation (34) of reference [1]. However, ^{in this}

reference ^a factor 4

is missing, D is the absolute

diffusing ^constant instead of the relative one here and AE is twice the energy splitting.

From equation (14a) ^{we can write}

where AE (b) ^{is the} maximal energy splitting ^corre- sponding ^{to R}

^b. Using expression (30) ⁱⁿ equation (29), ^{we obtain}

and after elementary integration,

At first sight ^{the main} approximation used in this method is to assume that despite of the hard sphere

condition at R

b, the Gaussian expression (28) ^of

the joint probability density P2 remains valid for

R

b. As generally ^shown by Ayant et ^al. [17] ^this

leads to rather good results for large ^times (t ^>

tc, ^{which is our case} here), ^{but to} very inaccurate

predictions ^{for short times} (t tc).

The second method [17], in which the correct

Laplace ^transform P2 (Ro, R, u) ^{of the} conditional

probability ^is used, ^is presented ⁱⁿ Appendix ^{B and} surprisingly ^{leads to} exactly ^{the same} expression (29)

of WAB. ^This result is in fact related to the spherical symmetry of the function AE(R) and would not occur for an anisotropic coupling ^like dipolar ^interac-

tions in liquids.

As a conclusion of this discussion, WAB ^is given by equation (32) ^{for an energy splitting} AE(R) given by equation (30), when the Brownian motion of the

potential ^{wells is described} by ^{the usual diffusion} equation. ^As WAB represents the average transition

probability per unit time for the transfer of a proton

within a single pair of molecules A and B, the total transfer rate is immediately given by

l

NA, NB ^are the numbers of molecules A (donnor)

and B (acceptors) respectively. ^The rate constant of proton tunnelling ^is then, ^from equations (32) ^and (33)

This result will be discussed in the following

section.

4. Discussion and conclusion.

It can be seen from equation (34) ^{that when} 130 b > 1 (short range coupling) ^{we have}

This behaviour is identical to that of the 1D model

[1], apart from the fact that the energy splitting Aye(6) is much lower in the 3D case.

For the long range coupling 8 0 b ,- 1, ^{the be-}

haviour of k is given by

and is significantly different from the 1D case [1],

where b 2/ f3 õ ^is replaced by f3 05. However, ^{it must}

be remembered that in the long range coupling ^case, tc given by equation (25) increases, 0. ^{is no more a}

small angle ^{and the} linearization process from

equation (22) ^to equation (26) ^becomes question-

able. According ^to the smaller value of AE (b ) ^{in the}

3D model, the above approximation will remain valid for smaller values of f30 b than in the 1D case.

In the typical examples shown in table I, f30 b

ranges from 3 to 8 and we are rather in the short range coupling ^case. Numerical values of k were

obtained by taking ^{a number} density ^of proton

NB 21 3

acceptors N B - ^{1021 cm- 3. It can be seen that a large}

variety of values of k can be expected. They range between 106 S-1 ¹ for a significant ^barrier height ^and 1011 S-1 ¹ for a low barrier height. ^In the first case

proton tunnelling is very difficult to observe while the second case corresponds ^{to a} very special ^situ-

ation in which ^we have a shallow ground ^state energy level for each spherical ^{well. The most relevant}

situation for observing diffusion induced proton tunnelling corresponds ^{to the case} (ii) ^where

k - 109 s-1. This transfer rate is larger ^{or at least of}

the same order of magnitude ^than typical ^diffusion

controlled processes ^{at room} temperature [1]. ^As pointed ^out by Kloffler and Brickman, ^{a normal} activated proton jump ^{mechanism behaves like} exp (- EA/kT ) ^where EA ^{is an} activation energy, and increases with temperature. ^The proton ^transfer

rate studied in this work and given by equation (34)

is inversely proportional ^to D and behaves like

’YJ I kT, ^{with the} viscosity increasing exponentially

with 1 / T. Consequently ^{k is} increasing ^{with decreas-} ing temperature. Qualitatively ^{when T} decreases, tc, the lifetime of the tunnelling process increases, favouring the transfer rate. As the temperature ^is lowered, it is clear that the diffusion induced proton transfer may dominate the normal activation _process.

In section 2.1, ^we considered the situation where each spherical ^well only ^{has one} bound level corre-

sponding ^{to an s state. For C}

7r, ^we also have a p excited level of energy Ei. ^{A resonant} proton transfer mechanism is possible between the two wells with a new resonance frequency AEI (R )/h.

The corresponding ^transfer probability WI is, ^accord- ing ^to equation (32), essentially ^driven by ^{the factor}

ð-E1 (b )2. According ^to the lower height ^{of the} potential barrier for the proton ^{in the} excited state,

Wl > WAB. Despite of the fact that the wave func- tions are slightly different, ð-E1 (b) ^{behaves essen-} tially ^like exp (- B1 b ) ^where {31

(- ² MEllh2)112

^C (yl)112 (see Eq. (6b)). Taking

the relative populations ^{of the} ground and excited levels into account, the ratio of the resonant proton transfer rates kl/k ^{for the two levels is}

We checked on several examples ^{that at}

T

273 K, kllk.-5 ^{10- 2. This} simply ^{means that in} equation (37) the reduction Boltzman factor is much

more efficient than the increase of the barrier transparency expressed by ^{the first} exponential ^fac-

tor. Furthermore as shown below, ^{the absolute value}

of k is very weak for values of C involving ^excited

levels. Thus, ^the proton ^transfer through ^excited

levels does not seem to be important ^{in normal}

situations.

We have also estimated the isotope ^{effect on the}

low temperature ^{transfer rate. The} effect of deuteron substitution can be easily ^evaluated by replacing ^C given by equation (5c) by ^C N/2. Considering ^an

intermediate barrier height ^{for the} proton, ^case (ii)

of table I, ^the corresponding ^value of Eo I ^{for the}

deuteron is 0.286 eV and k - 102 s- 1. Thus, ^the isotope ^{effect is} considerable and k can be reduced

by ^{a factor -} 107. With the typical values of the

spherical potential ^well parameters ^taken above, ^the diffusion induced deuteron transfer becomes inob- servable.

We have been able to give ^an analytical ^treatment

of the low temperature proton ^transfer in solutions

by using ^a simple 3D model. We have underlined the main differences with a 1D model and we have shown that the 3D spatial extension of the proton

wave functions has to be taken into account in any

realistic treatment of the direct tunnelling.

Our model still completely neglects ^{the internal}

molecular structure of the reactive units, ^{and this} represents only ^{a first} approximation. ^The theory

will be improved by considering ^{that the} spherical potential ^{wells are not at the center} of the donor and acceptor molecules A and B. Since the distance between these wells depend ^on both the relative

position ^{of the reactants and of their} orientations the random rotational motion of the molecules should be included in the theory. This will be done through

a procedure ^{similar to} that used in the theory ^{of the}

intermolecular dipolar relaxation [18].

A further improvement will be the consideration of a more accurate shape ^{of the} individual proton potential ^wells beyond ^the simple ^{form of a box with}

a flat floor. But this can only ^be performed ^numeri- cally. These effects will be treated in subsequent

papers.

Acknowledgments.

We are grateful ^{to Dr. Y.} Marechal, ^{Pr. D.} Gag- naire, and Pr. C. B6guin ^for fruitful discussions and to Ph. Michallon for technical assistance.

Appendix ^A.

MATRIX ELEMENTS OF THE DOUBLE WELL ENERGY SPLITTING.

Calculation of f/I A! VA !f/lB) = f/I A! VB !f/lB):

The various distances are represented ⁱⁿ figure ^2.

We have

and similar expressions ^for I/1B ^obtained by replacing rA by rB. Then,

sin À ’A COS À ’A

Using the addition theorems [12b] giving À ’A and À ’A ^{in terms} of Rand y-g, ^we easily obtain, setting A = i f3 0

where jt ^and hi ^are spherical ^{Bessel functions} (hi = jQ ⁺ iyf) ^{and 0 is the} angle ^between OBOA

^{R and} OBM

rB.

Replacing ^the expression (A.3) ⁱⁿ (A.2), only ^{the f}

^{0 term in} the series will contribute to the integral

because of its spherical symmetry. Thus, ^{we obtain}

with

Then,

The integral appearing ^{in the second member of} (A.5) ^is very easily calculated :

- w u ’ v

and finally ^we find the value of the matrix element given by equation (11) :

Calcultation of ^S

We have

the factor 2 in the first integral arising ^{from the} symmetry ^with respect ^to vA ^and vB. Denoting by ^{I the second} integral, ^from equation (A.2), ^{we have}

with, according ^to equations (A.1),

We introduce the elliptical coordinates

and the integral appearing ⁱⁿ equation (A.10) ^{can be rewritten} using the volume element [lOb]

In order to integrate ^over the volume v (rA

a, rB

a ), ^{the limits are} readily ^{seen to be as follows :}

from 0 to 2

for 0, from ^{1 to 00 for A} and from - A + 2a/R to A - 2a/R ^{R R} for u when ^{1 A 1} + 2 R a ^{R and from}

-1 to 1 for , when k--l+ 2 a ^The expression (A.12) ^becomes

R

and is integrated ^{in a} straightforward way. Using equation (A. 10), ^we finally obtain the value of I given by equation (12b) :

Calculation of 41 A I VB I t/1 A) :

According ^to equations (A.1) ^{we have}

Denoting by ^{J the} integral appearing in the second member of equation (A.15), ^{it is seen that}

Using ^the expansion (A.3), ^with 2 j6o, replacing j6o, again ^the only contribution to the integral ^{is the} f

0 term, ^{so that}

r

From equations (A.4) ^we get

or

We have then

or

Introducing ^the exponential integrals ^defined by [12a]

we obtain

From equations (A. 15) ^and (A.23) ^we immediately obtain the expression (13) of (/1 A 1 VB 1«/1 A) .

Finally, the matrix elements («/1AIVBI/1B)’ Sand («/1AIVBI/1A) ^{given by} equations (11), (12), (13) respectively, ^are expressed ^{in terms} of the reduced quantities ^defined by equation (4), ^{which are more}

suitable for computing purposes : Defining

We find :

and

Appendix ^B.

CALCULATION OF THE RESONANT PROTON TRANS- FER RATE.

We start from the expression (27) ^of WAB

Setting

and introducing F (a), ^the Laplace transform of the function f ( T ),

we have

Denoting by P2 (Ro, R, u) ^the Laplace transform of

P2(RO, R, t ), ^{we obtain}

Ayant ^{et al.} [17] have shown that with a hard

sphere ^{condition at R}

b, ^{we have an} expansion

where y is the angle between the directions R and

Ro and where the C f (Ro, R, o-- ) ^are functions of

Ro, R, a ^{and D.}

According ^{to the} spherical symmetry ^of åE (R)

and AE (RO), only the f

^{0 term} contributes in

expansion (B.6), ^and

Introducing

we have [17] ^{for R} Ro

and a similar expression ^obtained by permutting ^R

and Ro ^{when R}

Ro. ^In equation (B.9) ko ^and io ^{are modified} spherical ^{Bessel functions}

and ko, i o their derivatives. As we only ^{need the}

value of F(cr) ^{for cr}

^{0, we} have for R Ro

and a similar expression ^{with R} replacing Ro ^when

R

Ro. Using equation (B.l1) ⁱⁿ equation (B.7) ^for

o- ⁼

0, ^we obtain from equation (B.4)

which is identical to expression (29).

Note that in the free diffusion case the coefficient

Co(Ro, R, cr ), corresponding ^{to the Gaussian} prob- ability (28), ^{is still} given by equation (B.9) ^with

b = 0 and has the same limiting ^value (B.11) ^for

0" ⁼

0. This explains why ^the expression (27) ^of WAB ^{leads to the same result} (B.12) ^or (29) indepen- dently of the existence of a reflecting boundary

condition at R

b in the diffusion equation.

References

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