Translational diffusion in liquids: Specific features of the spectral densities for isotropic intermolecular interactions

Volume 145, number 1

CHEMICAL PHYSICS LETTERS

18 March 1988

TRANSLATIONAL DIFFUSION IN LIQUIDS: SPECIFIC FEATURES OF THE SPECTRAL DENSITIES FOR ISOTROPIC INTERMOLECULAR INTERACTIONS E. BELORIZKY Laboratoire de SpectromCtrie Physique (associk au CNRS). Universitt!Scientifique, Technologique et Medicale de Grenoble, Boik postale No. 87, 38402 Saint-Martin-d’H&es Cedex, France

and P.H. FRIES CEN Grenoble, DRF, Laboratoires de Chimie ‘, Boilepostale No. 85, 38041 Grenoble Cedex, France

Received 26 November 1987

In theories of electron spin exchange and diffusion-induced direct proton tunnelling in solution, one has to calculate the spectral densities J(o) of a purely radial intermolecular coupling resulting from a translational diffusion. By using the exact solutions of the diffusion equation, it is shown that at low frequencies, J(w) =J( 0) -Aw “2, where J(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 as o -* in the presence of a reflecting wall at the molecular contact.

1. Introduction In liquids the relative translational motion of interacting molecules is of fundamental importance for many processes such as nuclear magnetic relaxation [ 11, dynamic nuclear polarization by free radicals [ 2-41 or electron spin exchange between free radicals [ 5,6]. Besides the dipolar intermolecular interactions involved in relaxation processes which have been extensively studied [ l-4,7,8] the isotropic scalar interactions must also be considered for electron-nucleus Overhauser effects in liquids containing free radicals [ l-41 and for electron spin exchange mechanisms [ 5,6,9]. This scalar coupling between two spins is of the general form J(R)& l S,, where R is the intermolecular distance between the diffusing spins. Several different expressions for J(R) have been used such as the exponential decay A exp( -pR) [6,9] or (AIR) exp( -j?R) [21, or a constant value Jo for R R. [ 61. Pedersen and Freed [ lo] have also considered several alternate shapes for J(R). Recently [ 11,121 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 Al?(R) = (AIR) exp( -j?R) between the acceptor and donor molecules. In order to evaluate relevant physical quantities like relaxation times or the proton transfer rate, it is necessary to calculate the correlation functions G(t) or their Fourier transforms J(o) of the functions J(R) or Al?(R) to be denoted by F(R). Assuming that the random translational motion of interacting molecules in liquids can be described by a diffusion equation, the minimum distance of approach of these molecules must be taken into account. It has been shown [ 7,8] that for spin-carrying molecules with dipolar intermolecular interactions the ususal calculation of the associated spectral densities, J(o), from the clasical solution of the diffusion equation [ 1,2] does ’ Equipe Chimie de Coordination.

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not correctly take into account the boundary conditions in a hard sphere model. A more rigorous treatment of the problem was subsequently developed [ 7,8]. Rather different results were then obtained, especially in the high frequency range, WT> 1, T being the translational correlation time. In this case J(o) decreases as m -’ instead of o-“~. Furthermore, it was shown that in the low frequency range (COT<< l), the spectral density can be written as ~(CB)=.f(0)-~(201T)1'2,

(1)

with

where N is the molecule number density, D is the relative diffusion constant and T= b21D, b denoting the minimal distance of approach of the molecules. Although J( 0) is model dependent, the constant A is independent [ 13,141 of the radial distribution of the molecules and is not modified when the effects of eccentricity of the spins are also considered.

2. Correlation function and spectral density Consider an intermolecular coupling of the form F(R), where R is the relative distance between the centers of the interacting diffusing molecules approximated as hard spheres. The function F(R) is assumed to be real and to decrease faster than Rw3 as R-m. We want to evaluate the correlation function G(t) =N j$Vo,

R, t) F(R,) F(R) d3R d3& ,

(3)

or, equivalently, its Fourier transform or spectral density J(w)= & T G(1) exp( -iwt) --to

dt,

(4)

where N is the number of interacting molecules per unit volume. The equilibrium relative statistical density of the molecules is assumed to be uniform. The conditional probability p(R,, R, t) is the solution of the diffusion equation

Wh R 0 at

=DV2~(Ro,

R, t) ,

(5)

with the initial condition P&R,

0) =&R-&J

(6)

and the hard sphere boundary condition

ap (aR >n=b=o

(7)

In eq. (5), D is the relative diffusion constant of the interacting molecules. Introducing the Laplace transform p( Ro, R, a) of the conditional probability satisfying eqs. ( 5)-( 7), 34

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I

PUG,R, 4 = exd -ofI (Ro,R 0 dt,

(8)

0

it was established by Ayant et al. [ 71 that

HRo,R, 0) =

z

C/CR,, R, u) f’dcos Y) ,

(9)

I=0

where y is the angle between R. and R, and C’,is given by C/(R,, R, u) = w

(k!(kR)

i,(kR,)

- $f$ I

k,(kR)

k,(kR,))

(10)

for R> Ro, and a similar expression obtained by permuting R and R. for R-c Ro. In eq. (10) i, and k, are modified spherical Bessel functions i,(x)=x-“2Z,+~,~(x).

k,(x)=x-“2k,+,,z(x)

(11)

and k is defined by k= (a/D) “2 .

(12)

In what follows it is important to note that the hard sphere condition (7) only yields the second term in the

large parentheses on the right-hand side of eq. (10). This is easily seen by considering the usual solution of eq. (5) without the boundary condition (7), p(R,,R, t)=(4nDt)-3’2

exp( - (Ri2)2).

(13)

Its Laplace transform [ 151, ~(Ro,R,o)=(4~D~R-Ro~)-‘exp[-~R-R,~(alD)1’2]

(14)

can be expanded [ 151 as in eq. (9)) with i,(kR,) ,

C,(R,, R, r~)= wk,(kR)

(15)

which is simply the first term in eq. (10). Introducing G(o), the Laplace transform of the correlation function G(t), we have from eq. (3), G(o)=Nj-jp(Ro,R,~)F(Ro)F(R)d3Rod3R.

(16)

Replacing p( R,, R, a) by its expansion ( 9)) it is easily seen that because of the spherical symmetry of the function F(R), only the I=0 term contributes, so that G(a)=

(4n)Wjj

Co ( R,, R, a) R;F(R,)

From our definitions l/2

R*F(R)

dRo dR.

(17)

’

(18)

(11)) we have

. +,

ko(x)=

; 0

l+exp(2x)X

x-l

)/ ,

“2 exp( -x) x

(19) 35

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and from eq. (10) k

Co(RoJA=~

exp( -kR) sinh kRo + exp[ -k(R+R,)] kR 7 2k2RRo 0

1+ev(2kb)

kb-1

kb+l

(20)

for R > Ro. with a similar expression obtained by permuting R and R. for R -cRo. In eq. (17), replacing C,( Ro, R, a) by the above expression we obtain G’(a)=~~exp(-kR)f(R)RdRj(exp(kR,)+exp(-kR,)exp(2kb)~)F(R,)R,dR,. b

(21)

b

The spectral density J(w) defined by eq. (4) is then immediately deduced through the relation J(w)= iRe[&o=iw)]

(22)

,

where Re denotes the real part of the function G( io).

3. Low frequencies We consider the case wr K 1, with t = b*/D. This long-time limit is, according to eq. (12), equivalent to the condition kb c 1. We have simply, to first order in k,

C,(Ro,Ro)-&j x - (%)“2] [’ (R>Ro),

(23)

and from eqs. (17) and (22)) we obtain the general result J(w)=J(O)-AU”2

)

(24)

with R

J(~)=~~F(R)R~RJF(R,)R;~R, b

b

A=(;)“‘+(R)

R2

(25)

and

dR); .

(26)

Note that, in this case, the form (23) of Co(Ro R, a) can be straightforwardly deduced from eq. (15). This simply means that in eq. (20) the second term is negligible with respect to the first. This is physically understandable because for long times the reflecting boundary condition at R = b is not important. In fact it is easy to check that the first corrective term resulting from this condition in eq. (23) is only of the order of 0 and does not contribute to J( 0). The situation is somewhat different from that of the anisotropic intermolecular dipolar coupling case (1 ), where J(0) deduced from the simplified Gaussian conditional probability (13) is 10% lower than the exact result obtained from expansion (lo), while the value of A is model independent. For any isotropic radial coupling considered in this work, both J( 0) and A can be correctly obtained from the purely Gaussian distribution and are given by eqs. (25) and (26). We illustrate our general results with two specific examples. For intermolecular exchange coupling [ IO] or diffusion-induced proton tunnelling [ 11,121,

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F(R)=

CHEMICAL PHYSICS LETTERS

18 March 1988

;exp(-PR)

(27)

)

where l/p is the range of the interaction. From eqs. (24)-( 26), one has for kb e 1, J(w)=

4Nexp( -2j.Ib)r 8’b

1

(l+ 1/,9b)2

1+2pb-

2112

(w7y2)

(024:

1) .

(28)

For an intermolecular coupling [ 101 F(R)=R-”

(n>3)

(29)

we find J(@)=4Nrb’-2”

2 (n_2)(2n_5)

1 - 21/2(~_3)*

(30)

4. High frequencies We consider the case WTB 1. This short-time limit is equivalent to the condition kb~ 1. For hard sphere boundary conditions, it has been established [ 71 on general grounds that J( co) behaves as CO-~.This can be physically understood because at the limit where R. = b, and for short times t << r, the conditional probability p(Ro, R, t) is close to half a Gaussian distribution. The average value of R- b is of the order of (Dt) “* and there is a critical shell around R= b, with a thickness of the order of (Dt) “2, in which the volume of integration gives a significant contribution to G(t) in eq. (3). Thus, the correlation function behaves like G(t) = G( 0) -at, which implies the CC* law for J(o). If we consider the intermolecular interaction F(R) given by eq. (27), we obtain from eq. (2 1) C?(a)= ~4xN exp(-2gb) Dk

1

2&k+)

--

1 -kb-1 1 k2-,T2 + kb+ 1 2(j+k)2

(31)

and for k/3FP 1,

(32) and J(o)=

4Nexp(-Wb) bW+Vb)

(wr)*

(wrx= 1).

(33)

Note that the incorrect Gaussian conditional probability would have led to the wrong spectral density J(o)=

2”*Nexp( -2/3b) bz ? (coq3’*

which decreases as oP3’*.

5. Conclusion

We have calculated the behaviour of the spectral densities J( CO)resulting from a random translational motion in liquids for a purely radial intermolecular coupling. This has been done by correctly taking into account the 37

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hard sphere reflecting boundary condition at the molecular contact. When 07 ec 1, J(o) is given by the general equations (24)-(26) and both J(0) and A can be obtained from the simple Gaussian law (13) without approximation. In particular, the behaviour J(o) =J( 0) -A ( 202)“~ is characteristic of a long-time dependence of the correlation function according to a t-3’2 law. On the other hand, when tit s 1, J(o) decreases as m-2 and this results from the short-time behaviour G(t) = G( 0) -a?, of the correlation function. The Gaussian law (13) would have led to the wrong behaviour for a radial coupling F(R) =exp( -BR)/R.The situation is very similar to the case of dipolar intermolecular coupling [ 71.

References [ 1 ] A. Abragam, The principles of nuclear magnetism (Clarendon Press, Oxford, 1962). [ 21 P.S. Hubbard, Proc. Roy. Sot. A29 1 (I 966) 537. [ 31 K.H. Hausser and D. Stehlik, Advan. Magn. Reson. 3 (1968) 79. [4] W. Muller-Warmuth and K. Meise-Gresch, Advan. Magn. Reson. 11 (1983) 1. [5] P.W. Atkins and G.T. Evans, Advan. Chem. Phys. 35 (1976) 1. [6] Y.Ayant, J. Phys. (Paris) 37 (1976) 219. [7] Y. Ayant, E. Belorizky, J. Alizon and J. Gallice,.J. Phys. (Paris) 36 (1975) 991. [ 81 L.P. Huang and J.H. Freed, J. Chem. Phys. 63 (1975) 4017. [9] J.H. Freed and J. Boyden Pedersen, Advan. Magn. Reson. 8 (1976) 1. [ IO] J. Boyden Pedersen and J.H. Freed, J. Chem. Phys. 59 (1973) 2869. [ 1I] M. Klijfller and J. Brickmann, Ber. Bunsenges. Physik. Chem. 86 (1982) 203. [ 121 P.H. Fries and E. Belorizky, Proceedings of the International Meeting on Chemical Reactivity in Liquids, Fundamental aspects, Paris (1987); E. Belorizky and P.H. Fries, J. Phys. (Paris), submitted for publication. [13] C.A. Sholl, J. Phys. Cl4 (1981) 447. [ 141 E. Belorizky and P.H. Fries,J. Phys. Cl4 (1981) L521; P.H. Fries, J.P. Albrand, M.C. Taieb, E. Belorizky and M. Minier, J. Magn. Reson. 54 (1983) 177. [ 151 M. Abramowitz and LA. Stegun, Handbook of mathematical functions (Dover, New York, 1970) pp. 445, 1026.

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