Time correlation functions of isotropic intermolecular site-site interactions in liquids : effects of the site eccentricity and of the molecular distribution

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Time correlation functions of isotropic intermolecular site-site interactions in liquids : effects of the site

eccentricity and of the molecular distribution

P.H. Fries, E. Belorizky

To cite this version:

P.H. Fries, E. Belorizky. Time correlation functions of isotropic intermolecular site-site interactions in

liquids : effects of the site eccentricity and of the molecular distribution. Journal de Physique, 1989,

50 (22), pp.3347-3363. �10.1051/jphys:0198900500220334700�. �jpa-00211148�

Time correlation functions of isotropic intermolecular site-site interactions in liquids : effects of the site eccentricity ^{and of the}

molecular distribution

P. H. Fries (1) ^{and E.} Belorizky (2)

(1) ^{Centre d’Etudes} Nucléaires de Grenoble, D.R.F., Laboratoires de Chimie (*), 85X, ³⁸⁰⁴¹

Grenoble Cedex, ^France

(2) Laboratoire de Spectrométrie Physique, ^{associé au} C.N.R.S., Université Joseph-

Fourier/Grenoble 1, ^B.P. 87, 38402 Saint-Martin d’Hères Cedex, ^France (Reçu le ^{29 mai} 1989, accepté le ²⁴ juillet 1989)

Résumé. 2014 Dans les théories de l’échange ^entre spins électroniques, de la relaxation nucléaire scalaire par des impuretés paramagnétiques, ^{et du} mécanisme, contrôlé par la diffusion, ^de

transfert direct des protons par effet tunnel dans les solutions, ^on doit calculer la fonction de corrélation ou la densité spectrale ^d’un couplage intermoléculaire purement radial, ^{dont les} fluctuations résultent de la diffusion des molécules. Pour les molécules polyatomiques, ^le couplage interspin ^{ou intersite} dépend à la fois des mouvements aléatoires de translation et de rotation. En utilisant la solution exacte de l’équation ^de diffusion, ^nous calculons la densité

spectrale ^d’un couplage intersite de la forme F(r)

(C/r) exp (- 03BBs r) que l’on développe ^sous

forme d’invariants rotationnels. L’importance des effets d’excentricité des spins ^{et de la}

distribution non uniforme des molécules à l’équilibre ^{est discutée et} illustrée par des exemples numériques. ^{La théorie est ensuite} appliquée ^au mécanisme de transfert intermoléculaire des protons par effet tunnel dans les solutions.

Abstract.

In the theories of electron spin exchange, of scalar nuclear relaxation by paramagnetic impurities, ^{and of} diffusion induced direct proton tunnelling ⁱⁿ solution, ^{one has to} calculate the correlation function or the spectral density ^{of a} purely ^radial intermolecular

coupling, the fluctuations of which result from the diffusion of the molecules. For polyatomic

molecules the interspin ^{or intersite} coupling depends ^on both the translational and rotational random motion. By using ^{the exact} solutions of the diffusion equation, ^we calculate the spectral density ^{for an intersite} coupling of the form F(r)

(C/r)exp(- 03BBs r) ^{which is} expanded ⁱⁿ

rotational invariants. The importance of the effects of eccentricity ^{of the} spins and of the non

uniform equilibrium distribution of the molecules is discussed and illustrated by ^numerical examples. ^The theory ^{is then} applied ^to the intermolecular tunnelling proton transfer mechanism in solutions.

Classification

Physics ^Abstracts

05.40 66.10 2013 82.20

(*) Equipe ^« Chimie de Coordination ».

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500220334700

1. Introduction.

In liquids the relative translational motion of interacting molecules is of fundamental

importance ^for many processes such ^as nuclear magnetic relaxation [la], dynamic ^nuclear polarization by free radicals [2-4] ^{or electron} spin exchange ^between free radicals [5, 6].

Besides the dipolar intermolecular interactions which are involved in relaxation processes and have been extensively ^studied [la, ^{2-4, 7,} 8], ^the isotropic scalar interactions must also be considered for electron nuclear Overhauser effects in liquids containing ^{free radicals} [la, 2-4]

and for electron spin exchange mechanisms [5, 6, 9]. This scalar coupling ^{between two} spins ^is

of the general ^forms J, (r) SI . S2, ^{where r is the} intermolecular distance between the diffusing spins. ^{Several different} expressions ^for J,(r) ^{have been used like the} exponential decay

e- s r[6, 9] ^or (Alr) e- k, r12], ^{or still a constant value Jo for r -- ro with J(r) = 0 for}

r :> ro [6]. Pedersen and Freed [10] have also considered several alternate shapes ^of JS (r).

Note that the index s which reminds of the scalar nature of the interaction is introduced in order to ensure the uniqueness ^{of the notations in this} paper and the forthcoming ^ones.

Recently [11, 12], ^{it was} shown that the direct intermolecular proton transfer between

equivalent ^{molecules A and B in} solution, ^AH+ + B ;:± A + HB + , by ^{diffusion induced} tunnelling, ^{involves an} isotropic coupling ^term AE(r) = (Air) e - /30 r between ^{the acceptor}

and donor molecular sites. For evaluating the relevant physical quantities ^{like the relaxation}

times or the proton transfer rate, ^it is _necessary to calculate the correlation functions

g (t ) ^or their Fourier transforms j (oi ) ^{of the random functions} Js (r) ^or AE (r), further denoted

by F (r). ^This question ^was investigated ^{with some details} [13] ^{when the} sites located at the molecular centres undergo ^a purely translational relative motion. By using ^{the exact solutions}

of the usual diffusion equation ^{it was shown that at low} frequencies, ^the spectral density j (ùj ), ^denoted by J (£o ) for centred sites, ^{is :}

where /(0) ^{and A are} independent ^{of the} procedure ^{used to} incorporate ^{the molecular} impenetrability. ^{On the other} hand, ^at high frequencies, J(£O) strongly depends ^{on the} boundary conditions, decreasing like w - 2in the presence of a reflecting ^{wall at the molecular}

contact. The general expression ^for J(w ) ^{will be recalled in the} following ^section.

However, ^for polyatomic ^{molecules in} liquids, ^{the relative} position ^{of the} interacting spins

of two molecules (or the relative position ^{of the donor and} acceptor sites for the proton intermolecular transfer in chemical reactions [12]) depends ^not only ^on the translational motion of these molecules, ^{but also on} their rotational motion, except ^{if the} spins ^{are at the}

centres of their own molecules. Thus, the correlation functions of the intermolecular coupling

in polyatomic ^molecules depends ^{on both their} translational and rotational diffusive motions.

This effect has been considerably investigated ^{in the case of the} magnetic dipolar

intermolecular coupling ^{which is relevant} for nuclear magnetic relaxation, ^{both on} theoretical

[14, ^15, 16] ^and experimental [17, 18] respects. ^{It was} shown that the eccentricity ^{effects can}

be of fundamental importance, leading ^{to a} measurable increase of the relaxation rates,

mainly ^{in the} high frequency range. These effects are considerably enhanced when a non

uniform relative equilibrium distribution geq(R) ^{of the pairs of} molecules, originating ^from

their correlations of positions, ^is taken into ^account [19, 20, 21]. The combined effects of the site eccentricity and of the pair correlations can increase the dipolar spectral ^densities by ^more

than one order of magnitude [20, 21].

It can be expected that for the isotropic intermolecular couplings ^{which are} usually ^short

range interactions, ^the eccentricity ^{effects will} play ^{an even more} important role than for the

slowly decaying dipolar coupling, ^{and that} these effects will also be more efficiently ^enhanced by pair correlations. The purpose of this article is to study ^these phenomena.

In section 2 the general formalism will be developed ^{for an} isotropic coupling of the form.

F (r) = (C Ir) exp (-A, r).

Numerical illustrations will be provided in section 3 and our theory ^{will be} applied ^{to the}

intermolecular tunnelling proton ^{transfer mechanism} in section 4.

2. Theory.

2.1 CENTRED SITES.

First, ^we briefly ^{recall the main} results obtained for centred sites [13].

Assuming ^that the random relative translational motion of the centres of two interacting

molecules in liquids ^can be described by ^{a diffusion} equation, ^and denoting by b the minimum distance of approach ^{of these centres} (b

^{2 a for identical} spherical particles ^{of radius} a), ^we

want to evaluate the time correlation function :

or, equivalently, its Fourier transform or spectral density

where N is the number of interacting ^molecules per unit volume. Here the equilibrium

statistical density ^N of the molecules is assumed to be uniform and the conditional probability P (Ro, R, t) ^{is the solution of the diffusion} equation

with the initial condition

and the hard sphere boundary ^condition

It was shown [13] ^{that the} Laplace ^transform Õ (u) ^of G (t ) ^is given by :

with k

(u / D )1/2, ^D being the relative diffusion constant (D

² Da ^{for identical} molecules).

The spectral density J(w ) ^{is then} simply proportional ^{to the real} part ^of Ù (o- ) :

At low frequencies (£or .-. 1, ^{with T}

b2lD), J(w) ^{has the} limiting ^behaviour (1) ^{with :}

and

In the specific ^{case where :}

we obtain from equations (1), (9) ^and (10) :

When F(7?) ^is given by equation (11), ^{for wr »} 1,

Now, assuming ^{a non uniform} equilibrium ^relative density Ngeq(Ro) ^{of the} molecules which interact with one of the studied particles, the time correlation function (2) ^is generalized by :

where the conditional probability P (IZO, R, t) is the solution of the generalized ^diffusion equation (Smoluchowski equation) [7, ^8, 19] :

The effective potential Veq (R ) is related to geq (R ) ^by

The probability P (Ro, R, t ) ^{can be} expanded ^as

where y is the angle ^between Ro ^{and R.}

The function ét (Ro, ^R, u) ^{which is the} Laplace ^{transform of} Ce (RO, R, t ), obeys ^a partial

differential equation [19] ^{which is} directly derived from the Smoluchowski equation (15) :

with

The hard sphere boundary condition is expressed by [19] :

or, equivalently, by

From equations (14) ^and (17), expressing Pl (cos y ) ^{in terms of} spherical harmonics, ^{it is} straightforward ^{to obtain :}

The function Co(Ro, R, o- ) obeys equations (18) ^and (20) ^{with f}

^{0. A} detailed numerical

procedure ^for solving ^these equations ^is provided in reference [19]. ^{It is} possible ^{to calculate}

Ô (o- ) ^and consequently J(w ) ^for various realistic pair correlation functions geq (R ) ^or

potentials V eq ^(R).

2.2 ANGULAR EXPANSION OF THE INTERMOLECULAR COUPLING BETWEEN OFF-CENTRE SITES.

We consider two molecules A and B with off-centred spins (or sites) ^{as shown in} figure 1. We denote by ^R

0 AOB ^{the vector} joining ^{the centres} of the molecules and by

pA, pB the positions ^{of the} spins ^with respect ^to the molecular centres. Introducing ^{the vector r}

between the two spins ^{we have}

The intermolecular coupling ^is

with r = Irl.

We have to evaluate the time correlation function of F (r). ^For this purpose it is necessary ^to

expand F(r) ^{in terms} of radial and angular ^variables.

Fig. ^1.

Characteristic parameters of the relative position ^{r of the} interacting ^sites SA, SB.

It is clear that F (r ) ^{is invariant through} any rotation which simultaneously ^{acts on} R, PA, ^, PB, i.e. through ^an arbitrary rotation a of the whole system

Denoting by il, (dA and COB the orientations of R, pA, pB with respect ^to any fixed reference frame we introduce the following rotational invariants :

which are invariant through any rotation a of the system.

In equation (24) ^the symbols ÎA (XA ÎB e ^B À/ ^{) are the 3-j} coefficients [22]. ^This expression ^is simply ^{the linear} combination of products ^{of three} spherical ^harmonics transforming according ^to the irreducible representation 2)() of the rotation group. The site-site coupling F(r) ^can be then rewritten in terms of the rotational invariants (24)

where F pA pB t(PA, pB, R) ^are purely radial functions to be determined.

Introducing

we have

with R --::» p . Using ^the expansion ofc ^e À sr in ^terms of spherical Bessel functions [23a] ^we

r

have

where y is the angle between R and _p and where i p and kl ^{are defined} by

where If + 1/2 ^and KI , 1/2 ^are usual modified Bessel functions of half integer ^order [23b]. Using

the addition theorem of the spherical harmonics, ^{we obtain :}

where w is the direction of the vector p.

Furthermore, the addition theorem [23c]

can be rewritten

Replacing ^p by its definition (26) ^and using equation (31) ^{twice for} developing ek.PA and

e - k . PB, ^{we obtain :}

Using the formula [22]

in order to evaluate the integral ^over d,f2k ^of equation (33), ^{we obtain from} equations (30) ^and (33)

According ^to the definition (24) of the rotational invariants, F(r) takes the form (25) ^with

Note that in equation (25) ^{the sum} over l ranges between 0 and 00, and that the values of

f A and f B ^{are those} compatible ^{with f} using ^{the usual} triangular inequality of addition of

angular ^{momenta. The sum} + QA + f B ^{must be even.} Obviously, A A, À B, ^{À must be summed}

between - fA ^and fA, - f Band f B, - f ^{and f} respectively ^{and we must have} ÀA + À B ^{+ À = 0.}

Having separated F(r) into radial and angular parts, it is then easy ^to write the correlation

function g (t) ^{of the} random function F [r(t)] ^{if we assume that there is no} correlation

between the translational and rotational motions of the molecules. We have, PA and

P B being ^fixed parameters,

In this equation Ngeq(Ro) ^{is the non uniform} equilibrium ^relative density ^{of the} interacting

molecules as in equation (14) for the centred case and the translational conditional probability

is a solution of equations (15), (5), ^and (19). Using ^the expansion (17) and the addition theorem of the spherical ^{harmonics we have}

In equation (37) ^we have also assumed a uniform equilibrium distribution 1/4 ’TT’ of the directions wA and (a) B of pA and pB. The rotational conditional probability P A «(a) Ao’ ^{COA, t )}

obeys ^{the usual} rotational diffusion equation

with

DÂ being the rotational diffusion constant of the molecule A.

Consequently, ^{we have} [1b]

with

We have a similar expression ^{for P} «(J)Bo’ ^{COB, t) with constants DÉ,} Tt2B.

We replace the three conditional probabilities by ^their expression (38), (40a) ^{in the}

correlation

function g ^(t) fiven ^{by equation (37) and we use the explicit} forms of the rotational

invariants q,tAtBt and q, ÁtBt’ ^defined by equation (24). ^After integration ^{over the} angular

variables ^we obtain a considerable simplification ^{because the} only ^non vanishing elements in the various summations are such that the couples l, ^À obey ^{to :}

Then using the relation [22]

we easily ^{find the desired} expression ^{of the} correlation function

where the « radial » functions F pA pB e are _{given by} equation (36).

Introducing ^the auxiliary correlation functions

we simply ^have

Using ^the Laplace ^transform 9 (u) ^of g (t ) ^we obtain the final expression

with

The functions è f (Ro, R, u) ^are solutions of the equations (18) ^and (20).

The relevant spectral densities j «(ù ) ^{are then} simply derived from the expression (44a) ^of

§(o-) using the relation (8).

Note that if _PA

PB

0 (centred case), ^the only ^non vanishing ^{term in} equation (36) ^is F°°°(R )

⁸ 7T5/2(C IR) exp (- À, R ) ^and equation (44a) ^{reduces to}

with F (R) = (C /R ) exp (- Às R), ^{i. e. to the} expression (21) ^of G (u) previously obtained for the centred case.

3. Numerical results.

In this section we shall illustrate the above general theory by ^{some numerical} examples ⁱⁿ

order to evaluate the relative importance ^{of both} eccentricity ^{effects and} pair correlation functions for an isotropic ^intersite coupling of the form (23).

We consider two spherical molécules of radius a (b = 2 a ) ^{with pA}

^{p B. The} translational correlation time is ^T

b21D ^{with D}

2Da. ^{We assume that} DÂ

DÉ

kiT. ^{Using the}

Stokes formulae [1] ^{we have k}

^{3/2, but for} nearly spherical molecules, k ^is significantly larger [24] and for the purpose of numerical illustration we shall assume a typical ^value

k = 5.

Introducing the dimensionless spectral density, / ( ) ^{defined by}

where j(CJJ) ^is directly ^{obtained from} equations (8) ^and (44), ^{we have} calculated the values of

j (co ) ^{for the cases pA}

^{0 and} (p AI b)2 = 0.1 and for various ranges of the interacting ^{force :}

As b

1, 3, ^5. Increasing ^{values of} As correspond ^{to a} decreasing range of the interaction.

The calculations were performed : (i) ^{for a uniform} distribution of the molecules

geq(R)

^{1 ;} (ii) ^{for a} pure liquid ^{made of hard} spheres ^{with a} typical ^reduced density ^0.8 (number of molecules in the volume b3) ^{and a} pair correlation function given by the very accurate Verlet and Weis approximation [25] yielding ^{a contact value} geq (b )

4.03 of the radial distribution function.

In figure ^{2 we} represent ^the frequency dependence of the reduced spectral ^densities il(ù),r) for various values of the interaction range 1/À, ^{when the spins} (or sites) ^{are at the}

molecular centres (PA = p B 0 ) ^{and when} the radial distribution geq (R)

¹ for R ==== b is uniform. The function î 1 (£ù -r considerably increases with the interaction range 1/ As ^{at every} frequency ^since F(7?) = (C /R ) exp (- As R) ^also increases with 1 /,k, for all distances R. The value of i (,wr ) ^decreases by 4, ³ and 2 orders of magnitude ^for As b

1, ^{3 and 5} respectively

when ^wT varies between 0 and 100. This simply results from the shortening of the correlation time of F (R ), ^when k, increases, leading ^{to a} smoothing ^{of the} spectral density. ^As expected (see ^Section 2), ^{at low} frequencies, j (w T ) ^decreases linearly ^with (WT)1/2.

Fig. ^2. - Frequency dependence of the reduced spectral density j , (tor ) ^{in the case} of centred spins

located on uniformely distributed molecules for three values of the range parameter À S.

The eccentricity ^{effects vs. cvT are shown in} figure 3 for the uniform distribution

9 eq (R) ^{= 1} by representing the ratio of the spectral ^densities for off-centre and centred sites

Te(úJT /Tc(úJT ». For co = 0 this ratio is 1.07, 1.94 and 6.03 when Às b = 1, ^{3 and 5}

respectively, while for w -r

100 it is 2.8, 9.2 and 46 for the same k, ^values. Thus, ^the

eccentricity effects increase both with À sand ^úJT. When the range of interaction decreases the

Fig. ^3.

Frequency dependence ^{of the} ratio 7. ) /j ,(£or ) of ^{the spectral} densities for off-centre and centred sites in the case of a uniform molecular distribution for three values of k,. ^The eccentricity parameters ^are (p,lb)’

(p,lb )2 = 0. 1.

random function F (r) ^{becomes more sensitive to} the relative orientation of the molecules.

This effect is particularly important ^{when the two} interacting ^{molecules are close} together,

i.e. for short times corresponding ^to large ^wr.

The influence of the ^non uniform molecular distribution is first illustrated in figure ^{4 wherè}

we represent ^the frequency dependence ^{of the} ratio jlc c(-,r )/i , (cor ) ^{of the} spectral ^densities

Fig. ^4.

Frequency dependence ^{of the} ratio PIC (cor )/] c (wr ) for centred sites where Pcc ^{is the} spectral density ^{relative to} the Verlet and Weis radial distribution of model ii and where Tc corresponds ^to

geq(R)

¹ (model i)..

associated to the non uniform (model ii) ^{and to} the uniform (model i) ^molecular distribution for centred sites. As expected ^the higher ^{contact value} geq (b ) for model ii results in ^a

significant increase of the spectral densities. Again ^{this effect} is all the more pronounced ^as Às increases and can reach a value - 4 for k, b ^{= 5.} Indeed, the Verlet and Weis function of

noticeably greater ^than unity ^{in the} vicinity of R b and the increase ofy will be all the larger

as this integration range includes all the significant ^values of F (R ). Obviously, ^this completely

occurs for large ^{values of} k,.

For the achievement of the present theory the combined effects of the eccentricity ^{of the} interacting ^sites together ^{with the non} uniform molecular distribution are simultaneously

taken into account. For this purpose ^we report ^the ratio j Pc (, 7- )lj , (w -r ) ⁱⁿ figure ^{5 where}

jPc ^{is the} spectral density associated to model ii for off-centre sites and where Tc corresponds

to model i for centred sites. Again ^we observe that this ratio increases with the value of

À s ; it also increases with WT because the increasing ^{behaviour of the} eccentricity effects with

frequency dominates the non monotonous variation of the pair correlation effects (see Figs. ^{3 and} 4). ^{In the case} of nuclear relaxation induced by ^{a scalar} coupling ^{with an electronic} spin, where k, b ^{is -} 5, ^these combined effects can enhance the relaxation rate by one or even two orders of magnitude.

Fig. 5. - Frequency dependence ^{of the} ratio P"C(wr)lic(cùr) where pc ^{is the} spectral density

associated to model ii for off-centre sites (PAlb )2 =(p Blb )2 =0. 1 ^and where Tc corresponds ^{to model i}

for centred sites.

4. Intermolecular proton ^{transfer mechanism.}

In a recent study [12] ^a theoretical model for the intermolecular proton ^transfer AH+ + B :;:± A + HB + between identical molecules A and B in solution was proposed. ^In

this model the proton ^is initially ^{assumed to} be in the ground ^{bound state of a three}

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

molecules A and B collide, ^the proton ^{can tunnel} through ^{a 3D} potential barrier and can be

trapped ^{in an} equivalent ^3D spherical ^well belonging ^to molecule B. This corresponds ^{to a}

resonant process in ^a 3D double potential ^{well. An} analytic expression of the transfer rate can

be obtained by evaluating ^{the résonance} frequency of the double well within the LCAO

approximation, ^and by using ^{a classical} description of the relative translational diffusion of the wells. Typical values of the tunnelling ^{transfer rate were} provided ^{and it was} shown that this mechanism may dominate the normal activated process ^as the temperature is lowered. More

precisely, assuming ^that the attractive potential ^{of the} proton with the molécule A is an

isotropic 3D well of finite constant depth

it was shown that, ^{to a} good approximation, the energy splitting ^{of the double} spherical ^{well is} given by

where ^r is the distance between the two centres of the wells located on the molecules A and B.

In equation (47a) /3 o ^is given by

where Eo ^{is the} ground ^state energy of the proton ^{with mass M} in the individual potential (46)

and where C is a complicated expression ^of Vo, Eo, and ai, given by equation (14b) ^of

reference [12b].

Denoting by NB the number of molécules B (acceptors) ^the proton ^{transfer rate is} simply given by [12]

where the average probability WAB ^{of a proton} transition per unit time taken ^{over a} statistical ensemble of pairs ^of interacting molécules is

According ^to equations (37) ^and (45) ^the proton transfer rate k can be rewritten

where N

NB V’ ^{V being} the volume of the sample.

First, assume as in reference [12] that the relative translational motion of the two well centres is represented by ^the solutions of the usual diffusion equation ^{such as} b is the minimal distance of approach ^{of these centres} and D is their relative translational diffusion constant

approximated ^to that of the two molecules. The transfer rate, kl, ^is according ^to equation (12)

Expressing ^{C as a} function of the maximal energy splitting DE(bl) through equation (47a),

this rate becomes

which is the formula (34) of reference [12b].

Now, ^{we have to} consider the important fact that the proton acceptor and donor sites are not usually ^{at the centres} of the molecules A and B and, consequently, that the energy

splitting AE(r) is also affected by the rotational random diffusion of the molecules. The geometry ^{of the} problem is described in figure ⁶ and it is possible ^to apply ^the general

formalism presented in the above sections. When the eccentricity ^{effects are} taken into account the proton ^{transfer rate,} k, ^is given by equation (49) ^{where the} Laplace ^transform g(0) and the associated spectral density j (0) ^are expressed by equations (44) ^and (45).

Fig. 6. - Various distances involved in the proton tunnelling between the two sites SA, SB. ^The

minimum distance of approach of the molecular centers is b

2 a ; the minimal intersite distance is

bl ^{= b - 2 p and} corresponds ^{to the situation} where SA and SB ^{are on} OA OB when the molecules collide.

According ^{to the} rapid convergence of the series (44), specially for low values of WT (here

w

0), it is reasonable to restrict the series to the term f = fA = f B

^{0 as a first}

approximation. ^The proton ^{transfer rate,} k, simplifies ^{then to}

where Gooo ^is given by equation (44b). ^From equation (36), if PA

Pa,

or, equivalently,

where dE(R) is defined by equation (47). ^From equations (53) ^and (44b), ^the proton ^transfer rate, k, ^is approximated by

where G(0) ^{is the} Laplace ^transform (21) ^which corresponds ^to centred sites.

For the uniform radial distribution geq (R) ⁼ 1 of model i, /(0) = 1. G (0) ^is given by equation

w

(12) ^{and the} approximated expression (53) ^{of k has the} analytical ^form

Note that in the above formulae b is the minimal distance of approach ^{of the centres of the} diffusing molecules and not the minimal distance of the well centres as in equations (50-52).

If the effects of eccentricity ^were neglected (pA = 0) ^{the factor K defined} by

would be unity. Thus this factor gives ^a rough estimate of these effects.

.

In référence [12] ^we considered individual spherical potential ^{wells of} radius al

0.1 À,

with a minimal distance of approach bl

^0.85 À. Taking ^a reasonable diameter of the molecules A and B, ^b

³ À, ^{we have}

so that the eccentricity parameter ^is (p Alb)2

0.13. For the three values of 1/130

^{0. 30} À,

0.16 À and 0.1 À previously chosen, ^{we obtain K}

634, ^{1.4 x} 107 and 2.2 x 1013.

This result was not « a priori » ^{obvious : on one} side with off-centred sites, the minimum distance of approach ^{of the two sites is} considerably lowered, ^thus increasing the transfer rate, but on the other side, because of the rotation of the molecules, ^{the sites} spend less time in front of each other. We have shown that the first effect is the dominant _one, thus increasing

the proton ^transfer.

Within the framework of our description ^the proton transfer rate k

NB WAB is, according .

to equations (49) ^and (45),

In table 1 we report the values of J(O), in the centred case /c(0) (model i), ^{in the case of off-}

centred wells Je(O) (model i), and in the case of off-centred wells JC(O) ^{with a non uniform}

radial distribution of the molecules (model ii) for the three values of l/j3o ^{discussed above.}

We also give the value of the transfer rate k obtained with /(0) together ^{with our} previous

estimate ki ^{of k} [12] given by equation (52). ^{We see that the} ratio Je(O)/Jc(O) varies from

103 to 1015 which shows the importance ^{of the effects of} eccentricity ^{which was} already

predicted by ^the large values of the factor K given by equation (57). On the other hand the

effect of the pair correlation function characterized by ^the factor j p‘ (0 ) j e (0 ) ^is only ^{of the}

order of 5. The values of k obtained in table 1 are lower than the estimate kl calculated in our

previous paper [12], mainly for short range interactions. The ^reason is that our previous

model was equivalent ^to the centred case described here and that we took an irrealistic small value b1 ^{1 = b}

0.85 Â for the minimal distance of approach ^of the molecular centres. This

simplification amounts to excessively enhance the statistical weight of the molecular collisions where the sites are at the minimal distance bl, ^{to the} prejudice of those much more frequent

where the molecular orientations give ^much larger intersite distances which are considerably

less favourable to the proton ^transfer.

However, ^{with the new values of k} obtained with the more accurate description ^{of this}

paper, the general conclusions previously drawn in reference [12] ^remain valid. The proton transfer rate k is inversely proportional ^{to D} and behaves as TI / kT, ^{with the} viscosity increasing exponentially ^with 1 / T. Consequently ^k increases with decreasing temperature ^and in the case of a low barrier height, ^{the diffusion induced} proton ^transfer (k - ¹⁰⁹ s-1 ) ^may

dominate the usual activation _process.

Table 1. - Reduced spectral densities 1 (0) ^and transfer ^{rates k and} k, of the proton between

typical spherical potential ^wells o f constant depth V o and radius al

0.1 Â. The rate k is given by equation (58) with j-(O) = iec(O) ^{and the rate} kl of reference [12] ^is calculated by equation (52). 1/,So ^and AE(bl) ^are defined ^{in the text and in} reference [12]. ^{The various}

parameters ^{are :} relative diffusion constant, D

10- 5 cm2 S- 1; minimal distances o f approach o f the molecular centres b

3 Â and o f ^{the well centres} bl = 0.85 Â ; eccentricity parameter

(p Alb)2 = ^{0 .13 ;} rotational diffusion ^constant Dr

^{5.5 x 10 10} S-l ; ^number density ofproton

acceptors N BIV

^{1021 cm- 3.}

5. Conclusion.

We have developed ^a general ^{formalism for} calculating ^the spectral densities of random time

functions of the distance between two sites (or spins) belonging ^to translating ^and rotating

spherical molecules. The only assumption ^{is the} stochastic independence of the rotational and translational molecular motions. Our model allows the introduction of pair correlation effects in the translational dynamics. The relevant spectral ^{densities are obtained as series} expansions

of the eccentricity parameters. For the random functions C exp (- k, r)lr, ^{these series}

converge rapidly, mainly ^{for low wr} values, and the number of significant ^{terms seldom}

exceeds about 10 for reasonable values of wr -- 100. The first application ^{of this} theory ^was

the improvement ^{of our} previous ^{model for} intermolecular direct proton transfer between identical molecules in solution. This formalism will be further applied ^to the calculation of the scalar hyperfine contribution to the relaxation rate of nuclear spins in presence of

paramagnetic nitroxide radicals.

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