Journal of Molacdar Liquids, 62 (1992) 81-96 EXsevier Science Publie.bem B-V_. Amntedam

81

MOLECULAR COORDINATE SYSTEMS FOR RELAXATION PROCESSES+

JAMES McCONNELL School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin 4, Ireland,

(Received October 18,199O)

SUMMARY The choice of sui table coordinate systems for the theoretical investigation of nuclear magnetic relaxation processes is considered for both rigid and nonrigid molecules. Expl4ci t results are obtained for certain correlation functions, spectral densities and relaxation times,

INTRODUCTION

In the discussIon of relaxation processes associated with molecular

liquids 4t is often found convenient for calculational purposes to work

in more than one coordinate system. Thus for dielectric relaxation one

may work in both a laboratory system, which we shalt denote by S, and a

mol ecul ar system. The same i s true for nuclear magnetic rel axation

ari sing from di pal ar 5 nteraction and from spi n-rotat I onal interaction.

When the nuclear magnetic relaxation arises from chemical shift or

electric quadrupolar interact4on, a second coordinate system fixed in

the molecule is also needed, Nuclear magnet4c relaxation by seal ar

interact4on can be dl scussed ent4rely in the system S-

Dielectr4c ret axati on has al ready been treated at some

length (ref, 1). It was then found necessary to take due account of the

inertia of the molecules involved in the relaxat-lon process. This 4s

not required for the present accuracy of nuclear magnetic relaxat4on

experimental results - Thus rotational d4ffusi on theory wi 11 be adequate

for the subsequent sections which are devoted entirely to nuclear magnetic

processes_

*Dedicated to Professor H .G -Hertz

0167-7322/92/$05.00 8 1992 - E%avisr Science Publie.bem B.V. AU righta raewed

82

In the next section a mathematical treatment of the rotational

Brownian motion of a rigid molecule is summarized and a mol ecu1 ar

coordinate system S’ is defined_ A second molecul at- system S’/ is

introduced in the following section for a discussion of quadrupol ar

relaxation. In the succeeding section the general resul ts derived

earl ier are applied to the symmetric and asymmetric rotator molecular

model s. The final section serves as an introduction to the influence of

molecul ar nonri gi di ty on nucl ear magnetic rel axati on processes.

THE INERTIAL COORDINATE SYSTEM

We consider the rotational Brownian motion of a rigid molecule in a

heat bath. Many of the early studies of this subject were based on the

Fokker-Planck equation (ref. 21. A more recent approach is based on a

study of -the stochastic rotation operator R(t). This describes the

rotation of the molecule from its orientation at time zero to its

orientation at time t, and so R(o) is the identity operator. Now R(t)

obeys the equation (ref. 1, chap.71

dR(tl dt = -i (4 . w(t)) R(t) ,

(1)

where J,, J,, J, are the infinitesimal generators of rotation. We can

in principle solve (1) for R(t), provided that we know the angular

velocity g(tl of the rotating molecule. For most rel axation problems

the value of the ensemble average <R(t)> is all that is required.

Indeed the matrlx element <R(t)>,, related to the basis constituted by

first or second rank spherical harmonics with polar

angles fixed in the molecule often plays the part of

function.

In order to find w_(t) we take a set of molecular

with or

axes of

by II,

to make

igin at the centre of mass and axes coinciding

inertia. We shall refer to S’as the inertial

I I, 1 l the principal moments of inertia at the origin. In order

progress with tl-.z sathematical investigation we assume that the

inertial tensor and the rotational friction tensor are simultaneously

diagonalfzable, Then corresponding to I, , I a , I, we shall have

rotational friction coefficients 1,. 1%’ J,, say, and a frictional _-ag

c,~,(t), czwdt). c,.wdtL The heat bath produces a thermal couple with

and azimuthal

a Green’s

coordinate axes S’

with the principal

system. We denote

83

components denoted by N,(t), N,(t), N,(t) and these satisfy the

relations (ref. 3)

<Ni(t)> =o

<Ni(t) NJ(S)> 3 2kTaij Cib(t-S)- (2)

Since S/ is a system of rotating axes, the components of angular

velocity obey the Euler-Langevin equations

I, doAt) - (1 - 1,)~~ (t)w, (t.) = -cLw,(t) + N,(t), etc. - a (3) dt

We emptoy (2) to find stationary solutions of (3). This has bec;l done

for spherical , 1 lnear, symmetric rotator and asyrmnetric rotator mot ecu1 ar

model s _ We can then proceed to solve (1) for R(t) (ref. l)-

For nuclear magnetic relaxation processes a knowledge of the

rotational diffusion limit of <R(t)> i s al 1 that i s requi red for our

purposes. It is given by (ref. 4; ref, 1, p. 194)

<R(t)> =exp 1-i 7 Jft ]_ (4) i=I i

The rotational dl ffuslon tensor i s di agonal and its nonvanl shi ng

elements D,, D,, D, are given by

We see from (4) and (5) that

<R(t), = exp [-(D, J: + D, J; + D, J:)t 1. (6)

If the value of R(t) is required, as it is for the study of nuclear .

magnetic relaxation by spin-rotational interaction, it may be expressed

in series form by

R(t) =(E + cF(‘)(t) +E* F(*)(t) + . . . ) < R(t)> ,

where E is the identfty operator, G is a small dimensionless parameter

and the F(j)’ s are stochastic functfons defined in ref, 4-

84

TWO MOLECULAR COORDINATE FRAMES

We mentioned in the Introduction that in order to investigate nuclear

magnetic relaxation by ani sotropic chemical shift or by electric

quadrupol ar interaction two mol ecu1 ar coordinate systems

Since the mathematical prob’Iem is much the same for these

mechani sms, we shall confine our attention to quadrupol ar

A theory of nuclear magnetic re7 axat-i on appl icabl e

are required.

two relaxation

interaction.

to asymmetric

rigid mol ecu1 es has been based on the stochastl c rotational operator

(ref. 5). We start with a 1 aboratory frame S having cartesi an

coordinates x, y, z. a constant external magnetic field in the

z-direction and a molecular nucleus situated at the point (x, y, z) at

time t. We regard all

denote by V the scalar

charges _ Taking second

other charges in the molecule as fixed and we

electric potential at (x, y, z) of these other

derivatives of V we write

F 3 = $vzz, Fkl = :;i tVLX ’ ‘%y’

F = *1 2% (V,x - vyy z 2ivxy)*

(7)

The five quantities F1 are the components of a spherical tensor of

rank 2 (ref -6, section 9.1). Since V is a scalar, we deduce from (7)

that the five quantities Fi/ defined by

Fg’ = $V,~,N , Fz, = ;$ (V/,/k iV,/y/)

F?: x’x’ - Vy/y’ 52iV)(/yf I

constitute a spherica tensor or rank 2 in the inerti al frame S .

(8)

The array

I

Vx’x’ Vx’y’ Vx’z’

Vy’x’ VY’Y’ Vy’r’

V-’ _ x’ Vz’y’ VZ’Z’

is the field gradient tensor-

(9)

We now take a second body coordinate system S” with origin at the

relaxing nuctelrs and with x” , y”, z” coordinate axes in directions that

make (9) diagonal so that the only nonvanishing elements are Vx~~x~~) Vyfimy”.

Vz”z” ) and we order these in such a manner that IVZ@~z” 1~1 Vxl~x~~ 1~1 Vy~~yl~ 1

We deduce from IS) that we have in SO a spherical tensor with components

F;s, = 3 V =,,= 1, , Fy , =O, F:, = rl ‘7”7: , -

2(6)3

where

n= V,“,” - V,“,”

. Vz”z”

the asymmetry parameter _ It is usual to write VZm~t~~ as eq and

therefore express (10) as

FZ = 1 eq, Fll =O, F;’ = n eq 2(61$

(10)

(11)

It is to be remembered that S’ and S” are both coordinate frames

fixed in the molecule which contains the nucleus, the origins of the

frames being, respectively, the molecular mass centre and the relaxing

nucleus, Since the rotation of .a rigid body about any axi s through the

origin of S’ is equivalent to a similar rotation about the parallel axis

through the origin of S” together with a translation and since

translational motion of the ml ecu1 e does not i nf 1 uence the relaxational

probl em under 5 nvestigati on, we may say that a rotation about any axis

through the origin of S Is equivalent to an equal rotation about the

parallel axis through the origin of S” _ If then we go from S’ to S” by

a rotation specified by the Euler angles Q ) s, y (ref -6, appendix B).

we can relate the functions in (8) and (10) through the equation

(121

where Glrn (a. e,~) is a Wigner function.

In order to derive expressions for the spin-lattice relaxation time T,

and the spin-spin relaxation time T, we first define the spectral

density J(W) by

j( w 1 = 3 {z<F, (01 F, (t)> giW%lt.

where Fo comes from (7). An alternative expression for f(w) is provided

by (ref.3, appendix)

’ J”1 j(W) = l%, mj=_r $,,’ <R+(t) >mm, Fp, dt

, (13)

with Fm’ , F;, defined by (8) and the matrix element of CR+(t)>

calculated for the basis constituted by the spherical harmonics Y,‘,

associated with S’_ We may obtain F,/ for insertion into (13) by

inverting (12) and employing (11). so that we obtajn

F; us.&2 D;;(a, B, Y) F; (14)

= ( 4)L eq YZ-, (0, a) + qeq

ZP i D~z(a,~.y) + Dmy_, (a.e.r)I-

To complete the calculation of j( 0) from (13) it is convenient to

introduce the operator T ( W) by the equat-i on

T( w) =I; <R+(t)> e -i tidt

and the operator Q ( W) by

a( w) = -C (w) + T (w)+ *

so that (13) becomes (ref. 6, appendix D)

Then for a spin one nucleus with electric quadrupole moment eQ the

longitudinal relaxation time T, and the transverse relaxation tSme T,

are gtven by (ref.6, ~~-120, 121)

+ = g t+-12 ij(mo) + 4jt20,) I

+ = $ ($+)2 (3j(o) + 5j( wg) + 2j (20,)I ,

(15)

(16)

where oO is the Larmor angular frequency corresponding to the external

magnetjc field Introduced at the beginning of this section,

APPLICATION TO RIGID MOLECULAR MODELS

We shall now employ results derived in the previous section to

calculate the spectral density j(o) for quadrupolar relaxation when the

relaxing nucleus is sStuated in a rjgid asymmetric molecule.

For simplicity we work in rotational diffusion theory so that CR(t)>

is given by (6). Then (ref, 6. appendix D-3)

a(dmn=

where

A+A* o D+D* o F+F*

0 B+I3* 0 E+E* o

D*D* o C+Cf 0 D+D*

0 E+E* o 8+B* o

F+F* o D+D* o A+A*

A= ac - d’

a(ac-2d’ ) , B=J b , C= a

b’- e’ ac - 2d’

D=- d ac-2d2

,E= -e b2 -e2

,F= *’ a(ac-2d2 )

(17)

(18)

a= DI + DZ + ~DJ + i UI, lp 9 (0, + Dt) + DJ + iw

c= ~(4 + D* ) + i o, d = ($I* e = (9)’ (D, - Dz)

and D, , Dz, DZ are the rotational dlffusjon coefficients associated

with the S’ coordinate axes. From (15) and (17) we deduce that

j(hl) p &t (A+A*) (F’ F 1 Jc + --F’F’* 1 + (B+ If) (FL Fl,f + F,’ F’*)

+(C+C*) F,’ F,1*-: (o:D*) ‘(;I F,‘* + F,’ F’* + F,’ F’* + F” F,‘*) (19) +(E+E*) (FIEF,‘” + F; FlF) +-;F+F*) (F’-;‘* + F”F’* )< _

We note that the molecular shapzenters iitz(l9) through

A, B. C, D. E, F and that the quadrupolar relaxation process enters

through the F’m’s. In order to express the latter 1 n terms of Euler

angles we employ ( 14) and

D&m [a -6,~ =e ) 1

-1 (mb +IIIY )~,m( e)

dm/m ( e I E I(2+m) ! (2-m ) ! (Z+m’ ) ! (2-m’) !I4

X z (-1s (cos ~ft)4m-m’-2s (-sin ~a)m’-m+2s

, s (2-m/-s)! (2+11-s)! (2+m’-m)! si

where the sulmnation is over all values Of S for which the arguments of the

factorials are nonnegative. Then on substitution into (19) we obtaSn after

a lengthy cal cul ati on

(A+A*) fz Sin’B + : Sdn*B COS2Y [Sln‘jEi + COS'iB]

+ 5 12 sin’iB + t sfn’s cos 4 Y + 2 cosmJq )

+ (B+p*) ( 3 ii;

sir&B cc6 2r

+ g [2 SlIP6 (sllr ltl + co!?~eI - sin*@ cos 4~1)

+& tc + P) (3 cafe - 1 + n sin% COSa)*

+ $ (D + D*) (3 COPE (20)

- 1 + n slnB COSa)

x ((314 sina CoSZa + 2n 2 2 [cossjg COS(2a+2Y) +sin+~ecos(2a-2Y)lI

+ (E+E*) (- 3 sirP 20 cos 2a + 2 sin 0 sin 20 16 4

x [ sinzlB cos(b-2~) -cosaiB cos(aO+211 1

+ z sin28 [J sfn’6 COS 2a 12

- co!9 & c0s(&~1-s~n4~Bc0s(2a-4Y 11)

+ (F+F*+) ( 3 x sln’e cos4a

+ g [cos’~Bcos(~+4Y1

Thus by introducing the three coordinate

from S and S” to 5’ we have succeeded in

+ a sin28 [coscae cos(4a+2Y)+si1+ jecos(4a-2r)l

+ s-ln’$fl cos(4a-4Yl +$ sin*BcosqPl) 3

_

frames S.S’,S” and transporting results

deriving an analytSc expression for the

spectral denstty, The values of T, and T, are then obtainable by substStuting

(201 into (161.

It lsobvious from (181 and (201 that the calculation of explicit

results for the asymmetric molecule can be a complicated process. This Is due

basically to the noncorunutatlvity of the operators J:. Ji. J: that appear In (6)_

For this reason one often simplifies the mathematics by taktng the molecular model

to be a symaetrlc rotator. If the axis of rotational symmetry is the third S’ axis,

then D, = D, and (61 yields

<R(t)> = exp E- [ 4 J= + (4 - D, ) J: It,

= exp. (-D,J’ t) exp [(q - 0, ) J: t].

au

(21)

Moreover, from (la),

d=e=O

so that

A z a-1 = (20, + 4D, + iw )-I

B = b-’ = (5D, + D, I- iw )-I

c = c-’ = (6D, + i w ) -’

D=E=F=O.

Hence eq. (20) will be considerably reduced,

In the rep:*esentation with basis consisting of second rank

spherical harmonics associated with S’ the representative of Jz is 6

times the unit matrix and the representatives of J: is the diagonal

matrix with successive elements 4, 1, 0, 1, 4. It follows from (21)

that d R(t)> is represented by -the diagonal matrix with nonvanishing

el ernents e -6D,t ( e4(D,-D,)t ,e(D,-D,)t ) 1. e(D,-D,)t, e4(4 -D,)t ) ?

that is,

e-(2D, +4D, )t , e-(5D, +4 )t , e-6D.t, ,-(5D, + D,)t,e-_(2D, +4D,)t _ (22)

Let us employ (22) to calculate the autocorrelation function

of Yzm (e(t). 9(t)). Or< going from the laboratory frame S” to the

molecular frame S’ we obtain for t + o (ref, 3. appendix)

4’)lZ @ ’ )I *1,

that is,

<YCm ( O(O), ql(o)) Yam ( s(t), a (t)),

= (16*)-l I (3 COSa B m _ 1 I~ e-6Dlt +3 sin* 2 e I e-(5D~+Ih It

+3 sin* 9 ’ -(2[h + 4[h)lz) _ e

(23)

This agrees with a result of Hertz (ref - 7, eq.(l71). when it is noted

that he took the axis of rotatior;al syrmnetry of the molecule to be the

first coordinate axis of S’.

INTERNAL ROTATIONS

Irl the previous sections we have related the coordinate

frame S” to the inertial frame S’ by a rotation specified by constant

Euler angl es. The molecule has so far been regarded as rigid but in

fact very few liquid molecules are rigid, While nonrigid molecules have

been studied both theoretically and experimentally for more than thirty

years, the rrtolecu’l;ar models have been rather simple. Thus Frech and

Hertz (ref.81 distinguish bet&en fast and slow motions of parts of a

nonrigid molecule- It is assumed that the correlation function belonging

to a part with fast internal motion is influenced by the stow motion of

other parts, but that the carrel ation function of a slower part i s never

so influenced by the faster motions that it contains fast decaying

exponenti al s. The part with the slowest motion is referred to as “the

molecule”. It has al so been called “the standard part” (ref. 91 and

“the framework” (ref, 101.

An early theoretical investigatjon of internal rotations was proposed

by Stejskal and Gutowsky who examined nuclear magnetic relaxation by the

intramolecular interaction of the protons in the symmetric top CH,

molecule (ref. 11). A somewhat similar approach to the study of

relaxation in a two-proton system was made by Woessner (ref. 10).

Taking a 1 aboratory coordinate system S with origin at one proton and

initial line in the direction of a constant ekternal field tie he denotes

by r: the position of the second proton and by e,+ the polar and

azimuthal angles in S of an axis cf rotation fixed in the framework.

The vector c makes a constant angle A with the axis of rotation and a’

is the azimuthal angle of c about this axis with reference to the

framework system.

Let n,(t), n,(t), n,(t) be the direction cosines of g(t) referred

to S- Uoessner introduces the functions Fa (t), E, (t), F, (t), not to be

confused with F’s introduced in the previous sections, through the

equations

F, (t) = 1 -3n:. F1 (t) = (n, + ini)n, , F1 (t) = (n, +in, IpI (24)

91

It follows that In terms of spherical harmonics associated with the

poiar ar.d azimuthal angles of the proton-proton axis at time t

F,(t) = -(16/5$, Y-It)

F,(t) = -(8mNj12 Y,(t), h (t)* = @/,5+ y _,(t) (25) F,(t) = (32&5)iY,(t), Fz (t)+ = (32x/15$ Yz;_z (t) .

We recall that, if T is the gyromagnetic ratio of the proton, then (ref.

6. eq. (4.22) and (C.30))

j( w) = 9 I? Y”;,,, (0) Yzm (t)> emiti dt -m

and that the longitudinal 1, and transverse relaxatlon time T,

given by (ref. 6, eq, (4.58) and (4.59))

(26)

are

(27)

where o. is the Larmor angular frequency. From (25) and (26) with m=l

j( w) = e J-I_ <F, to)* F,(tB euiwt dt. (28)

Sfnce we are no longer dealing with a rjgid body, we cannot

employ the rotation operator technique of the two previous sections to

calculate the ensemble average in (28). Woessner empl oyed spherical trigonometry to express nk, n,. n, in terms of e , o , a , +’ by

n, = sin n cos 8 cos * cos 0’ + cos a sin 8 cos I# - sin A sin d sin&l n, = Sin A cos 9 Sin $I’ + Sin A COS 8 sfn * cos 0’ + COS A sdn osjn 0

n, = cos A case -S-inA sin 9 CDS 9’ and on substitution Into (24) found that

F,(t)+3 (3 c&A - 1) sin ecos e+ 1 sin ACOS A (cosze - sin’0 + cos e)e i@’ - t

+ 4 Sin A cos A (cos* 8 - sin* & - cos 9)e-‘*

- f Sfn*A (sin 8 tos 8 + sin 8 ) e2”’ (29) - ) sin’s (sin e cos e - sin 9 )e-2i@‘l ei’ -

92

Wri ti ;rg

1 = g1 , ei*f = g,, e -i*‘, g 1 V e2i V= gN , e-ZiQ/= g 5

we express (29) as

F,(t) = f,g, + fz gz + f, 91 + f* 91. + f, g, I (31)

We remember that a is constant, that the tumbling motion of the

framework is described by o and 9, and that the internal rotation is

described by 0’. Thus the fi’S are related to the tumbling motion and

the gi ‘s are related to the internal rotation.

Woessner now assumes that the tuabl ing rnotjon and the internal

(30)

rotati on are mutually 1 ndependent - Thi s goes beyond the assumptjon of

Frech and Hertz reported at the beginning of this section, but Woessner

claims that hl s assumption 3 s physical ly reasonable. Accepting it

al lows us to deduce from (31) that

<F’: (01 F, (tb = & i.j=l

<f~(ol fj(U> q;(o) gj(t)’ . (32)

The value of <go gj(‘)> for a Brownian motion model is found by

assuming a Gaussi an probabi 1 Ity dl stri butIon for the angle V’; that is

to say, we employ a djstribution function

PI a/. t) = $(*t =c-V exp (- P r&It). (331

where xc is the reciprocal of the rotational diffusion coefficjent

for the one-dimensional mot1 on. From (301 and (33) we find that

+$(o) gkc[t) > vanishes for i#k, and hence that

<FI*(o) Fl(t)r = <fp (0) f-i(t)> <go gi(t)'_ (34)

Moreover (33) yields

93

<exp c-i*‘(o) ] exp [i ia’(t)l> = exp[ -It I kc]

c exp [.2i 6; (9) ] exp [T21 O’(t) ]> = exp [ -41 tl /‘cc I - (35)

To complete the calculation of < FT(o)F,(t) > from (32) we note

that “91 *(o) gf(t)> is obtainable from (30) and (35). In order to

calculate < A?(o) fj (t)> we note that thi S describes the tumbling

motion of the molecule, whtch we assume to be rotationally diffusive_

Woessner now supposes that the molecule is spherIcal. Then one deduces

from (6) that

<R(t)> = exp [ -D J*t ] , (t5o)

where D is the co-n value of D L , D.- , DB . In the representation

having Y,, as basis elements, J* is 6 tjmes the identity E and so

<R(t)> = e-‘6DtE . (36)

We see from (29) that fi(t) may be expressed as a linear combination of

the Y 2m(t) ‘s, Replacing 6D by T,: we deduce from (36) and section 7.4

of ref .l that

<f;(o) fjW> = <If3 (0) I’> e-W)/TCiL > where T c, is independent of f . On collecting the results it is found

that

<F:(o) F,(t) = & (3 cos2d -1)’ exp (-It1 /TCI )

+ 1

!¶I sin22 A exp [- Itl h,! +‘$)I

+ 10 sfnC A exp [- It.1 (7;: + 4$ ) ] I

Let us define xc2 and T,_, by

(37)

(38)

We then deduce from (28). (37) and (38) that

(39)

On emptoylng (25) we imnediately find that (39) is in agreement with

eq- (21) of ref. IO.

If a = 0, the proton-proton axis coincides wf th the axis of rotat-ion

and there I s no internal rotation. For vani shi ng a eq- (37) reduces to

<F, *“l(o) F, (t)> = $- exp (- Itl / Tc,)

and so the presence of internal rotations introduces two terms proportional

to exp (-Id 1~;: + ~~~ I) and exp (-4ltl [T$ + T~‘I )- It has

been shown by Hertz that precisely the same occurs for a symmetric top

molecule (ref.7, eq. (26))_ Thus in place of the right hand side of

(23) we shall have terms with time dependences

exp (-6D,t) , exp (- [6D1 + T;’ 1 ) , exp (- [6ch + 4~Cl3 )

exp (- [5D, +D, ] t) , exp (- [ 5& + DI + r c ‘I t) ,

exp (- [~DI + DS + Will t)

exp (- 12D1 + 4D,1 t) _ exp (- [ 2D, + 4D, + Till t),

exp kC2D, + 4D, + 4~;’ 1 t)-

Hence each term on the right hand side of (23) Is split Into three terms

as a result of the internal rotation.

CONCLUSION

An account has been given of the various coordinate systems employed

95

in the investigation at molecular level of djelectric and nuclear

magnet-it ret axati on processes. Apart from a laboratory coordinate system

the most fundamental system 1 s the inerti al molecular frame. This is

extremely useful for a discussion of the rotational Brownian motion of a

rigid body and it Is appltcable not only to spherical molecules but also

to ljnear, ci rcul ar pl ate, symmetric rotator and asynmtetric rotator

mot ecu1 ar model s.

For nuclear magnetjc ret axation by quadrupol ar interaction or

anisotropIc chemical shift 1t is necessary to employ both the inertial

frame and another coordinate system fSxed in the molecule. Et is shown

for these relaxatjon mechanisms that It is possible to provide an

analytical discussSon wfthout restricting the magnjtude of the

approprl ate asymmetry parameter.

To discuss relaxation for nontmigid molecules we have taken the simple

model in which a part of the molecule is free to rotate about an axis

fjxed in the main body of the molecule. The motion of the molecule then

consists of a slow tumbling motion of the major part combined with a

fast rotation of the smal ler part. In order to obtain a mathematical

solution it has been found necessary to suppose that the two motions

occur independently. Expressions for correlatfon functions and spectral

densities are then derived.

REFERENCES

1.

::

4.

2

:: 9.

10. 11.

J. McConnell, Rotational Browni an Motion and Diet ectric Theory, Academic Press, London, 1980. J. McConnell, J. Molec. Li . in press. G.W. Ford, J.T. Lewis and . McConnell. Phys.Rev.A 19(1979) 3 907-919. J. McConnell, Physica A 128 (1984) 611-630. J. McConnell, Physica A 117 (1983) 251-264. J. McConnell, The Theory of Nuclear Magnetic Ret axatl on in Liquids, Cambrldge Univ. Press. 1987. H.G. Hertz, Progress In NMR Spectroscopy 16 (1983) 115-162. T. Frech and H.G. Hertz, 3. Molec. Liq. 30 (1985) 237-282. M.S. Ansari and H.G. Hertz, Z.Phys.Chem. Neue Folge 137 (1983) 187-220. D.E. Woessner. J.Chem.Phys. 36 (1962) l-4. E-0. Stejskal and H.S. Gutowsky, J.Chem.Phys. 28 (1958) 388-396.