ROTATIONAL DIFFUSION EQUATION FOR
A
POLAR MOLECULE
3277
Solution of the Rotational Diffusion Equation for a Polar Molecule in an Electric Field1 by Donald E. O’Rei& Argonne National Laboratory, Argonne, I ~ ~ ~ n60@9 o k
(Received February 27, 1970)
A formal solution of the rotational diffusion equation for a polar molecule in an electric field is presented. The theory is applied to nuclear magnetic dipolar or quadrupolar relaxation of a polar molecule in the electric
field of an ion. In general, the rotational autocorrelation function decays more rapidly due to the field, the dipolar or quadrupolar interaction is not averaged completely by the rotational diffusion, and translational diffusion or rotation of the molecule about the ion is required to average the nuclear interaction to zero. Estimates of the strength of the effect are made for NH, coordinated with Na+.
1. Introduction A considerable amount of experimental data is currently available on the nuclear relaxation rates of ions and solvent molecules in aqueous solutions, Progress in the interpretation of the data is hampered by the lack of a detailed model for the dynamics of a water molecule in the electric field of an ion. There is evidenceas that HzOmolecules in water obey the rotational diffusion equation since the ratio n / r Zof the dielectric rotational correlation time 71 and the magnetic rotational correlation time r2is approximately equal to 3 as predicted by the rotational diffusion equation. Also TI and rz are approximately given3b with the friction constant 1: equal to the Stokes expre~sion:~1: = 8aaaq where a is an effective molecular radius and q is shear viscosity. The rotational diffusion equation in the presence of an electric field was given by Debye,b who investigated solutions of the equation in weak radiofrequency electric fields of interest in the theory of dielectric absorption. I n the following it will be assumed that the rotational diffusion coefficient is isotropic, but the method can be applied to anisotropic diffusion6 as well.
Let us seek a solution of eq 1 as an infinite series in spherical harmonics Y lm(9) with coefficients which depend on time, i.e.
‘
= Cczrn(t)YP(9) lm
(2)
Placing eq 2 into eq 1 one obtains the following expression
+
2pE cos BcZmY1” pE sin eclm -
be From eq 3 rate equations for the quantities clrn(t)can be obtained by multiplying by Yz?’(st)*and integrating over 9. First, however, note that’ b rn (4) be where L+ = L, iL, and L is the angular momentum operator. Also the following integral’ is needed
+
= -m cot P Y z m eiq - Y
+
11. Theory The rotational diffusion equation in the presence of an electric field isb
V,2 is the surface Laplacian operator, is the friction constant, 1/,(9,t) is the probability of finding the molecule in orientation st a t time t, p is the molecular dipole moment, and E is the strength of the local electric field a t the molecule. Of interest is the explicit time dependence of $ so that the autocorrelation function Gz(t) of Yzm(st)may be computed and hence the rotational correlation time r2 = f,” Gz(t) dt may be evaluated.
(1) Based on work performed under the auspices of the U.S. Atomic Energy Commission. (2) (a) See for example, C. Deverell, P T O ~NMR T. Spectrosc., 4, 301 (1969) for references to the large volume of literature; (b) D. E. O’Reilly and E. M. Peterson, J. Chem. Phys., 51, 4906 (1969); ( 0 ) D. E . O’Reilly and E. M. Peterson, J. Phve. Chem., 74,3280 (1970). (3) (a) K.Krynicki, Physica, 32, 167 (1966); (b) D . E. O’Reilly, J . Chem. Phys., 49, 5416 (1968). (4) H. Lamb, “Hydrodynamics,” Cambridge University Press, London, 1924, p 557. (6) P. Debye, “Polar Molecules,” Dover Publications, New York, N. Y., 1945,p 83. (6) See, for example, W. T. Huntress, J. Phys. Chem., 73,103 (1969). (7) M. E. Rose, “Elementary Theory of Angular Momentum,” Wiley, New York, N. Y., 1957,pp 237, 62. The Journal of Physical Chemi8try, Vol. 74, No. 17, 1970
DONALD E. O'REILLY
3278 where C(111213;mlmzm3)is a Clebsch-Gordon coefficient. The rate equation for ccm is as follows
appearing in eq 14 for each value of p need be considered. At t = 0 consider the molecular dipole moment to be at orientation Qo = (O0,pO) and hence
+(Q,O) = 6(Q
- 00)
= CYl"*(Qo)Ylm(Q)
(15)
Zm
so that clm(0) = Ylm*(Qo). For very long times, i.e., in equilibrium, it may be readily verifi.ed that
C(1 - 111'; mOm)C(l - 111'; 000)
C(l - 111; m
+
+ 1 - lm)C(l - 111; 000)
and thus
(74
and
C(1
C(Z
+ 111; mOm)C(l + 111; 000) +
+ 111; m + 1 - lm)C(Z + 111; 000)
Gzo(t) = f,fJexp(a
lm
+ l)t/c)
(9)
KrK-l =
r,i.e. (10)
3,
where ;i is a diagonal matrix with diagonal elements A, that are the eigenvalues of r. Applying K to eq 9 one obtains (11)
C' = ;ic'
where C'
=
K'c
(12)
Equation 11 may be immediately solved to yield c,
=
c,(O) exp(-A,t)
+ c,(a)
where the quantities c, are the elements of c'. coefficients ctmare then obtained from eq 12 as clm(t>= C(K-'),zm,,[c,(O) exP(--,t) P
+ c,(00)1
(13) The
exp(a cos Oo)Yz"(Qo)czo(t)dQo
(19)
~ ' " ( t )=
1
- Jexp(a cos e o ) Y z o ( n , ) ~ ( ~ - l ) zxo , N P exp( -A,t)dQO
+ 1 s e x p ( a cos eo) X
Y~"(cosOo)cz"( m)d
COS
Oodpo (20)
From eq 17 one readily obtains cz0(a) = (&)'Iz{( 1
+ -$)- _3a coth a}
(21)
Hence Gz"(ta) = ($){(I
2)
3
-CY coth a }'
(22)
As a + 0, Gz"( a)---t a4/180nand as a+ a ,Gzo(a )--+ 5/(4n). It is clear from eq 20 that Gz"(t) will decay with a complicated time dependence (which will be a more rapid decay than Gz"(t)for a! = 0) and the magnetic interaction will not average to zero. Averaging of the dipolar or quadrupolar interaction to zero can only occur by the relative translational diffusion of the molecule or rotation of the molecule-ion complex. More generally, one will be interested
