series expansion of the stochastic rotation operator

Series Expansion of the Stochastic Rotation OperatorAuthor(s): James McConnellSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 84A, No. 1 (1984), pp. 9-26Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20489189 .

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SERIES EXPANSION OF THE STOCHASTIC ROTATION OPERATOR

By James McConnell, M.R.I.A.

School of Theoretical Physics, Dublin Institute for Advanced Studies

[Received 31 January 198J. Read 24 March 1983. Published 31 August 1984.1

ABSTRACT

A series expansion of the ensemble average of the stochastic rotation operator is considered from the viewpoint of its accuracy when employed to calculate physical quantities related to dielectric and nuclear magnetic relaxation processes caused by random thermal motion. The problem is approached by associating the expansion with an exponential function for simpler molecular models. In making comparison with experiment it is found reasonable to approximate the ensemble average by a few terms of the series.

1. Introduction

Within the past ten years a number of studies were made of rotational Brownian

motion and of its implications for dielectric relaxation. Lewis et al. (1976) studied the

orientational polarization due to a molecule modelled as a disc or a sphere. which is

subject to random thermal motion. They incltuded inertial effects and emploved an

iterative method based on an equation giving the rate of change of a unit vector in the

direction of the dipole moment of the molecule. The results were in agreement with

those of Fokker-Planck equation studies by Sack (1957a. 1957b).

A fresh approach to the problem of a polar spherical molecule made by Ford ei al.

(1976) was based on the averaging method of solution of a stochastic differential

equation and a graphical technique. It yielded an exponential expression for correlation

functions, in which the exponent consists of a series of terms of descending order of

magnitude. The exponential expression is not one that can be integrated immediately so

as to yield correlation times. A similar situation arises for a linear polar molecule

(McConnell 1978). This paper presents a simple derivation of series expressions for correlation

functions that do not suffer the above disadvantage. The results are applicable to the

disc, spherical and linear rotators. Moreover in order to cater for relaxation problems

other than those occurring in dielectric theory, e.g. for nuclear magnetic relaxation

problems, we present our investigations in terms of the stochastic rotation operator. We

also discuss the questions of the reliability of the series expansions and of their term-by

term integration, and we consider briefly rotators that do not fall into the above three

categories.

The subvention granted by the Dublin Institute for Advanced Studies towards the cost of publication

of papers by members of its staff is gratefullv acknowledged by the Royal Irish Academy.

Proc. R. Ir. Acad. Vol. 84A, No. 1, 9 - 26 (1984)



10 Proceedings of the Royal Irish Academy

2. The stochastic rotation operator

The theory of thermal relaxation priocesses which we shall discuss is based on the

rotational Brownian motion of a rigidl molecule. The principal axes of inertia through

the centre of mass are taken as a rotating frame of coordinate axes and R(t) is the

rotation operator that specifies the orientation of the molecule at time t with respect to

its orientation at the earlier time zero, so that R(o) is the identitv 1. R(t) obeys the

eqluation

dRQ() i- . a4t))R(t),(1 dt

where J denotes the angular momentum operator divided by h. The angular velocity of

the molecule w(t) is a random variable satisfying a set of Etiler-Langevin equations.

Consequently RQ) is a stochastic operator. From the steady state solLutions of the Euler

Langevin equations the correlation function < wi(tk)w/tl) > of two components of

angular velocity may be found. The solution of (1) consists of a slowly varying ensemble

average < R( >, about which there are random fluctuations, and the solution is expressed by (Ford et al. 1976; McConnell 1980a, section 5.4)

R(t) (I + eF(1(t) + IF(2)(t) + 3F(3)() + . . ) < R() > (2)

where e is a small constant parameter and F(Pl(W, F(2)(t) . ...... are stochastic functions

with zero means. The non-stochastic < R) > obeys an equation

(I <

R() >

= (.Q(l)(t) + e2Q(2)(I) + E3c(3)(t) + ...) <R(t) >.

dt

It is found (McConnell 1980a, section 11.2) that Q(2r+1)(t) vanishes, so the last relation mav be written

d < R) > = (2f2(2)(t) + E4Q(4)(t) + 6Q6(6)(Q) + ,) < R(t) >. (3) di

Using the value of the above correlation function we deduce from (1) - (3) a set of

equations which provides eF')Q(). c2F(2)(t), . . . e2Q(2)(t), 4(4)(t) . . . for substitution

into (2) and (3).

Though it may be possible to solve (3) for < R(t) >, the form of the solution may be

unsuitable for calculating such quantities as complex permittivity, spectral densities and

correlation times. Ford et al. (1979) applied a procedure due originally to Krylov and

Bogoliubov (1947) in order to write

< R(t) > = V(t)eGt (4)

V(t) I + e2V2)(t) + e4V(4)(t) + 66V(6)(t) + (5)

G = c2G(2 + e4G(4) + c6G(6) + (6)

G being a time independent operator and V(t) being bounded for all values of t. In the

Debye approximation (5) and (6) become (McConnell 1980a, pp 194, 195)

V(t) = . G =- G(2).



MCCONNELL-Series expansion of the stochastic rotation operator 11

so that

< R(t) > = exp ( e2G(2)t). (7)

The above considerations apply to a rigid rotator of any shape. We now specialize to the cases where it is a disc or sphere or linear rotator. In the first case the only operator that appears is the identity; in the second case the only other operator is J2.

which is a multiple of the identity; in the third case there appears also J32. which

commutes with the identity (McConnell 1980a, sections 8.3, 9.2. 10.2).

Since we are dealing only with commuting operators, we may integrate (3) with the

initial condition < R(o) > = I to obtain

t1 K R(t) > = exp { f di1[e2Q(2)(t1) + e4A2(4)(t1) + e6A(6)(tl) + . . . (8)

In order to put this in the form of (4) we express (8) as

t < R(t) > = exp {J dtI[O2(Q(2)(t1)

- G(2)) + -4(D(4)(tj) G(4)) + c6(Q(6t1) - G(6)) + . (

x exp [(e2G(2) + 64G(4) + e6G(6) + . .

Since V(t) in (4) is bounded for t = oo. we must require that

G(2r) = Q(2r)(c). (1 0)

Writing

Q(2r)(tl) - G(2r) = 02r(I) (tl)

we see that (9) is equivalent to (4) with

V(t) - exp { dt, (6202(tl) + e404(I )+ g6 06(t1) + 6808(1) + , )4. (1?)

On expanding the exponential in powers of' 2 wve deduce that

V(t)=-I + g2 dtl02(tl )+ e4 { 040tf

+ + [I d 02

}

+ e6 {j (tt f dt102(t1) dt 04 (tI) +6 [J fi d )

(13)

?8g {fXdtl#8(t)+f d1#2(t1)f dtl #6(tl) +2- Ffdt4ho1I

1 g 12 t I r' 4 + 2[r dtIO2(tl] X0)4(tl)dtl + iI dt1b2(t1)i (+ 2 [ Js 24 [j0j

which has the form of (5).




In order to compare (13) with the results of the application of the Krylov

Bogoliubov method it is useful to employ the theorem (Ford et al. 1976, appendix):

ti tJI t tn - I

f0dt1 ... dtj dtf0dtd&f db 1ldtIh(tI . . . t1n t1 *.t")

(14)

dt Xi dtj Xdtj . . I dtn h(ta t 9 ta2' * an)

O O O O

s~~~~~~l,a2--an where the summation is over all the permutations of 1,2,...n under the conditions that

a, a2 . . ., aj are in ascending order and that a + I aj+2, .-. an are also in ascending order. The theorem is readily extended to more than two sets of ti variables.

From (5) and (13)

t V(2)(t) fd102201). (15)

Then, by (14),

42 [11dtl 2(tl di I dt212(tl)02(t2)

1 dt?J dt2O2(tl)#02(t2)- &102(t IV(2)(t I

by (15), so that

V(4) (t) f 4(tl)dtI + 0 dt#I 0(t 1 ) V(2)@). (16)

When putting down the coefficient of g6 in (13) we note that (14) and its extension to

three sets of variables give

dt, dt2f2(t0')04(t2) dt, 01dt2 {,2G)04#I2) + 0400020A

4 f dt I dt2 dt 302 0)#02, 203) f dt1 df 2 f dt302(tO0#2@3),

and so deduce that

t t t1

V =6)(t) f dt 06Qtl f'dt1#4(L f) dt2j2(t2)

+ f dtl 02) {f dt204@(t) 2 sf dt2#2(t2) fd302(t3)} (17)

=fd?#601 --fd#(4)t t = dtl06 (t) I+ dtl 04(tl) V(2) (t I) + dtIO2(tI)V(4)(t1).




Similarly it is found that

V(8)(t)=5 dtl 08Q1)l dt1#6(tl) VZ(t1) dt1#4(tI)V(4)(t)I dt#2Q 1)06)(tI). (18)

Equations (15) - (17) when combined with (11) agree with results obtained by the

Krylov-Bogoliubov method (McConnell 1980a, sections 11.5 and 12.1).

Let us write

5 #2r(ti )dtl = --2r(t) (19)

and deduce from (12) that

V(t) = exp {e2X2(t) + e4X4 (t) + e6 x 6t) + 8X8 ... }. (20)

On expanding this exponential in series of powers of 52 and comparing with (5) we

obtain

VA2)(t) = X2(t), V4) ()X4(t) + - Ix2(t)]2 2

V(6)(t) = X6(t) + X2(t)X4(t) + I

[X2(t)] (21) 6

J48)(t) - x8(t) + x2(t)x6(t) + - [x4(t)12 + [ [x2Q)P2x4(t) + [2Q [2(1)]4.

Hence in order to obtain V8)(t) we need only calculate the four integrals x2(t), X4(t),

x6 (t), x8(t). This is less laborious than performing the repeated integrations required in

(18). Moreover the method of this section shows how the terms V(2)(), 14)(t),... in (5)

originate. When the molecule has a shape other than that of a disc or sphere or linear rotator.

the values of Q(2)(t), Q(4)(t),... involve the non-commuting Jl, J2, A and eq. (3) cannot

be integrated as in (8). Equation (10) no longer holds except for r = 1. Nonetheless one

may obtain a solution of the form (4). The calculations are diffictult and have been taken

only as far as E2 p(2)(t) in (5) and c4GN4) in (6) (Ford ut al. 1979). No pattern is

discernible for A2)(f), 4)(t),. Once we have a reliable expression for < R(t) > we can immediatelv deduce the

correlation functions for spherical harmonics from the relation (McConnell 1980a,

sections 7.4 and 9.4)

< Y;m(f(o), a(o))Yjm@ (1(t), a(t)) > = (<RQ) >?im)

that is, 1/(47r) - times the complex conjugate of the m'm -element of the matrix

representative of < R(t) > in the representation with basis Yj3_j, Y;j, ... Yfi. The

above correlation functions are adequate for the study of dielectric relaxation and of

nuclear magnetic relaxation by spin-lattice, dipole-dipole and quadrupole interactions or

by anisotropic chemical shift (Abragam 1961, ch. VIII).

In the next three sections we shall examine in turn the values of V(t) for the disc,

sphere and linear rotator.




3. The disc model

By a disc we shall understand a body which rotates about an axis through its centre

of mass that points in a fixed direction in space. It is not necessarily a flat body. We

denote by I the moment of inertia about the axis of rotation and we define the frictional

constant B by saying that the frictional couple resisting the rotational motion is IB times

the angular velocity. For this model (McConnell 1980a, pp 135, 136)

e2Q(2)(t) - yB( - e-Bt) I, 0 = E4Q(4)(t) = e62(6)(t) = ... , (22)

where kT (23)

with k the Boltzmann constant and T the absolute temperature. The value of the

dimensionless constant y deduced from dielectric experiments seems to be of the order

of one per cent (Herzfeld 1964, Leroy et al. 1967, McConnell 1980a, p.245).

We obtain from (10) and (22)

=2G(2) = 212(2)(co) =-yBI, 0 = e4G(4) e6G(6)- . (24)

so that, from (11) and (19),

e2x2(t) =

62 { ((2)(t) G(2))d1 =Al -e-Bt)I

0 = E4X4(t) = c6X6t)

Hence, from (20),

V(t) =1+ y( - e-Bt) + ' y2 (1 e-Bt)2 + ? y3(1- e-Be)3 + ...lL (25) VQ)= ~~~

~~ ~~2 - 61

Since 0 < 1 - e-Bt < 1, this is a rapidly converging series. From (4). (5), (6), (24), (25)

<R(t)>= { [ I +Al -e-Bt) + 2 ) (1 - e-Bt)2 + I

y3(1-e-B9)3 + ... e -Bt Ji. (26)

This may also be expressed as

< R(t) > = exp [-ABt - I + e-Bt)1I.

4. The spherical model

In this and in the next section we shall often express results in terms of multiple

integrals that are convolutions. These integrals are of the general type Ji(2r)(t), where

r' rt It2r- I

I (2r)(t) dt I dt2 v w dt2re-BQ1 ? t2 ?+3

? t2r-1 -t2r), (27)

there being equal numbers of positive and negative signs in the exponent. It is clear that

IJ(2r)(t) vanishes for t = 0, and that it is a positive and increasing function of t. It follows

that dt(r)

is positive for t > 0. The values of the functions Ii(2r)(t)

that we shall encounter are given explicitly in the Appendix.



MCCoNNELL--Series expansion of the stochastic rotation operator IS

When the molecule is spherical, it is known that (McConnell 1980a, p. 156)

IS2D(2)QI)dtI_= - JIL(2)(t), f e4D(4)(Q1)dt1= J2 (?I )212(4)(t)

;6 {D(6) (tI)dI = -J (JJ I ) (16)(t) + 4I4(6)(O) (28)

68D(8)I)dtI = - P ({ j ) (14(8)(t) + 85(8)(t) + [_4J2 + 16I]I7(8)(t)

+ [- 1I02 + 38 I1I8(8)(t)).

This is as far as the present analytical investigation of the rotating sphere allows us to go. From the values of If(2r(t) in eq. (A.9) of the Appendix and (23), (28) we find that

2 G(2) =e2.I222)(oo) = J2 P T dI2(t) Yj2B I dt t =o - y

=4G(4) - y2 J2B, 66G(6) = _ 7 y3 J2B (29) 2 ~~~~~~~1 2

e8G(8)=( 17- J2 y4j2B

so that, by (6),

G~ =-yB{ 1 7

2 + 7 J2 \3 8 J2

Let us consider the form of the functions I,(2r)(t) that appear in our calculations. Since V(t) is bounded for all values of t, so are V2)(t), V(4)(t), ... and hence, by (21), so also are x20t). Thus, from (1 1) and (19), Q(2rt) - G(2r) vanishes for t = oo. Hence, from

d8,i (2r)tis a constant at infinity, so that I(2r)(t) is a linear function of t for t = oo.

Such functions are discussed in the Appendix and it is shown as in (A.8) that

lim (B2rIi(2r)(t)) = aBt -b, t _ ao

where a and b are positive dimensionless constants. This is confirmed by the explicit results of (A.9).

We shall now calculate e2rX2r(t). From (11), (28) and (29)

ge 0(t J k

-j(2(t ?yP2B, (30) I dt where. by (27),

J(2)(t) = f dt I dt2e-B(tl -t2) = B-2(Bt - 1 + e-Bt).




Hence, from (19),

COX2Q) = J c2(t l)dt - yJ2(1- e-). (31)

Similarly we deduce from (28), (29) and (A.9)

E4 X4(t)= _J2 ( I2 )1(4)(t) + -

y2 J2Bt (2 c4x_0 -- P ~ ) +I y2fii (32)

=y2J2 {+-[Bt + leVB +e2Bt }

E6X6t) = y3J2 - B2t2 + 2Bt + I ]eR5- 14 Bt+ 1 Jew2-- e- 3& } (33)

98x,(I)= 74 17 1 J2 )Bt

- (kT ) J2 {I 4(8)(t) + 8I5(8)(t) + [-4J2 + 1611I78(t) + L-IIOJ2 + 381]I8(8)(t) } (34)

We shall not write out the lengthy explicit expression for c8xj(:) in which, as we know

from above, the coefficient of Bt is zero.

An inspection of (31), (32), (33) will show that e2X2(t), 64X4(t), 6X6(t) are products

of J2 and positive functions of Bt, their values increasing steadily from zero at t= 0 to

y2J2 y3J2, '4 '9

respectively, for t = oo. In the consideration of e8x8(t) the coefficients of I78)(t) and

I8(8)(t) create difficulties, since for j > 2 these coefficients are negative. On referring to

the explicit expressions for I4(8)(t), 15(8)(t), I7(8)(t), I8(8)(t) in (A.9) we find that

e8x8(0)=0, 68x8(mc) (4 223 8643 864 864 0

Forj = 1,2, 68x8(co) is positive but we cannot say that e8x8(t) is positive for all values

of t.

We see from (6) that c8G(8) gives in G a correction of order e6 relative to the first

term of the series. To obtain an e6-correction to the first term in (5) we need go only as

far as the 6 V(6)(t)-term and this does not involve e8x8(t). Hence to obtain an e6

correction to the Debye result (7) we may simplify (20) to

V(t) = exp {02X2(t) + E4X4() + 66X6(t) }.

We deduce from (21), (31), (32), (33) that

2 2)(t) yP(l - e-BI)

E4 V4(t) y2 {J2 - (Bt + l)eBt - + e-2Bt + (J2)2 -eBt + + e-tl I




e6V(6)(t)=y3 {J2 [ - (4 B2t2+2Bt+1)e (--(Bt+1 ) e2&-- e4B] (35)

)[+ (Bt

+ + )e +Bt

+ ( +

)e-2Bt+

4 -3BtI

+

(J2)3 [

- eBt + -

e - - e3Bt] 6 2

~ 2 6 and that E2 V(2)(), E4 V(4)(t), g6 V(6)(t) are each a positive and increasing function of t.

5. The linear rotator model

For a linear molecule (McConnell 1982)

g2Q(2)(tl) kT

dI2 -J2) 02)(t 1) 3 dt1

e4Q(4)(t1)= k( 2

(2J2 - 5J 2) dI 4)(t1) I / dt1

e6A2(6)(t)=-( ) {[9(J2 J 32)J32 +4J2 -.16J32] d3(6)(t1)

\I fdt1

+ [21(J _PJ3 2J3 2 + 20J2- 68h3 2] d' (6)(t1) 1

dt J'

where I is the moment of inertia about a line through the centre of mass and

perpendicular to the molecule. It follows that

e2G(2) =y2 J3 2)B, e4G(4)=-y2 (J2- 3 2 )B

(36)

&6G(6)=_ y3 [42 J_3 2 3 2+ - J2 - 3 2 B

I.3 Hence, from (11), (19) and (A.9),

62X2(t) -YJ2 - J3 2)[1

- e(Bt)

e4X 4(t) y2 (2 - 5J32) {+ -[Bt + Ie-Bt- e-2Bt} 4 4 ~~~~~~~~~~~~(37)

c6 (t)=y3 + 4J2-{16J3 29]JB2t2P [6 2 e)

+ ( Bt + 1 )e-2B1




+421 P2-J32 132+20J2-68J321 [ y- (+- + )e

- (+ Bt+ )+ e-2Bt-Ae3Bt] }

An inspection of the time functions in the brackets of the right hand sides of (37) shows

5 5 that they increase steadily from zero at t = 0 to 1, 4, 1, 18 , respectively, for t = x.

Thus the signs of the elements of the matrix representatives of e 2x2(t), e4x4(0,

6x6(t) will depend entirely on the operators multiplying the above time functions. We

shall consider the representation of these operators with respect to the basis consisting of the spherical harmonics Yjm, the rows and columns being labelled by m which goes progressively from -j to j. Then J2 is represented by j(j + 1)1 and J3 2 is represented by the diagonal matrix with diagonal elementsj2, (j - 1)2,... 1, 0, . . j2. It follows from

(4), (5), (6), (21), (36), (37) that <R(t) >,m is a diagodial matrix, that <R(t) >jmm =< R(t) >,m , and that the matrix elements 62X2(t)mm C4X4(t)mm E66X6(t)mm are found

from (37) when J2 is replaced by j(] + 1) and J32 by M2. We shall now examine the values of these matrix elements for] = 1,2, which are the cases of chief physical interest.

In the j = 1 representation we deduce that

2X2 (000 = 2y(1 - eB)

4 ~ ~5 1B 1 2B 6 X4tWO=4O=2 [ -(Bt + 1)e - e-> J

c6x6(t). = y { 8 [1(- B2t2 + 2 )eB + ( Bt + I )e2BtI

+ 40 5[-A (- jBt ) e-Bt Bt + )e-2Bt--e-3Bt] }

and so the exponent of the 00-element of

exp { OXc2(t)

+ e4X4(t)

+ e6X6(t)

} (38)

increases steadily from zero at t = 0 to 2y (1 + 2 Y + 6

y2 ) for t = cc. We similarly 2 9 /

deduce that

62x2(t)11 - - Bt

.64 4(l =-Y2 - - (Bt + I)e-Bt-I e-2B




6x6 ( =y3{ [I ( Bt( + 2 e-Bt + IBt+ 1 )e-2BtI

7 +8 (4) Bt - + )eB +Bt+) e-2Bt+- e-3Bt] B

which show that e2x2(t)11 > 0, e4x4(t)I < 0 and that the sign of e6x6(t)1I is doubtful.

For t = oa the 11 -element of the exponent of (38) is

(l Sy_5 172\

Before leaving the case ofj = 1 we recall that for a gas dielectric or for a very dilute

solution of a liquid dielectric in a nonpolar solvent the complex relative permittivity e(co) is given by (McConnell 1981)

--- - X ico 1w

< R(t) > IPe-wtdt. (39)

When this is expressed in terms of Ji2, J22, J32 (McConnell 1980a, section 11.5), the

operator J32 gives no contribution to the integral in (39). It is thus permissible to put

J3 2 = 0 in the expression for < R(t) >, when we are dealing with the dielectric relaxation

problem. In the j = 2 representation

e2x2(0 = 61 - e-RB)

64X 4(00o 12y2 [ - (Bt + 1)eB B- I

er2It]

e6x 6(t00=y { 24 [1 ( B2t2 + 2 )e Bt + ( Bt + 1 e-2Bt

[ 18 2 4 \4 4 2 J

and so the exponent of the 00-element of (38) increases steadily with z from 0 to

6 y I + 2 y + 8

y2 p Similarly we have

e2x 2(t)1= 5Y( 1 - eBt)

e4x 4(t)1= 7y2 - - (Bt + 1) e-BJ- Ie- ]

e6x6(t) = y3 53 [1 ( B2t2 + 2 )eB + ( Bt + ) e-2




+ 157 [- --- ( Bt+ )e-Bt (+-Bt ++ e-2Bt_Ae3 j [ 8 (2 4 ) (4 Bt+2 )-S-Ie3Bt6

which show that the exponential of the 11-element of (38) increases steadily from 0

to 5y (I + y + 19 29 y2 ).Finally we obtain

E2 X2 (22 = 2A1 - erBt)

64X4(t)22- 8y2 [4-(Bt + I)e11'- e--2Bt1

?6X6()22 = y3 {32 [ - ( B2t2 + 2 e-Bt + (+Bt + 1) e-2B]

+ 16 [-A- (+Bt- - t (-Bt + + )e-2t&e-3Bt] }

On account of the - 8y2 in g4x4(t)22 the sign of the 22-element of the exponential in (38)

is not necessarily positive. The value of the element for t == o is 2y - 5y +18 + y2)

The quantity c6G(6) in (36) gives a correction of order y2 to the Debye result (7). To

the same order we need take the series in (5) only as far as 64 V(4)(t). We deduce from

(21) and (37) that

c2 V2)(t) = y(2 - J3 2) (1 - e-Bt)

4 V(4)(t) = y2 {(2J2 - 5J3 2) [4 (Bt + I)e- + J (40)

+ (J2 - J32)2 [+-e Bt+2e21 }I.

6 The asymmetric rotator model

In establishing an expression for < RQ) > when a molecule is asymmetric, rotating

coordinate axes are taken through the centre of mass and in the directions of the

principal axes of inertia, the moments of inertia being 1p, 12, 13. It is then assumed that

the friction tensor is diagonal with respect to the coordinate axes, the B employed for

the spherical molecule being generalized to B , B2, B3. This would be so, for example, if

the coordinate planes were planes of symmetry for the molecule and in particular if the

molecule were a symmetric rotator. Under the above assumption regarding the friction

tensor the diffusion tensor is diagonal with nonvanishing components Di given by (Ford

et al. 1979)




kT Di= + Di T2)h (41)

where the first term on the right hand side is the diffusion coefficient that appears in the

Perrin theory (Perrin 1934) and D/2) is a correction of relative order y. It is seen from

(41) that the motion is anisotropic. For a general asymmetric body it would be

necessary to consider both the shape of the molecular surface and the location of the

nuclei before postulating that the inertial and frictional tensors be simultaneouslv

diagonalizable. When the molecule is asymmetric, the three non-commuting operators J1, J2, J3

appear in the calculations and < R(t) > cannot be found by the method of section 2. It

has, however, been calculated by the Krylov-Bogoliubov method to a correction of first

order in y to the Debye approximation so that (McConnell 1980a. eq. (11, 5.24))

( 3 kT(1e 'J2 lG <R(t) > ={I + E IB2 (I -e-it)ji2 + ***}e GI+ (42)

where =l i iB

3 ~ I(kfl2 3 B-B G=- Z D1J12-7 (J22J32-J32J22) I I (43)

1 2 3 i = I k

i, j, k is a cyclic permutation of 1, 2, 3 and D, is given by (41). Since in (42) we have succeeded in calculating only the 62 V2)(t)-term of the series,

we have no way of discussing the properties of VQ). We may nevertheless argue that the

value of < RQ) > given bv (42) and (43) is reliable by observing that when applied to the

sphere by putting I, = I2 = I3 = I, B I =B2 = B3 = B the equations yield (McConnell 1980a, section 1 1.5)

<R(t)>={I+yJ2(I-e-B)+**exp [-( TI + D(2) J2t +

= 1 + yJ2(1 -e-Bt) + ... exp [(Y + I

y2 ) 2Bt + ...]

which in the notation of section 4 is expressed as

R(t) >- { I + e2V(2)(t) + . . . } exp I(e2G(2) ? e4Gt4) + )tl

This approximation with the small experimental value of y is usually quite adequate for

the sphere and we may expect the same to be true for (42) when applied to the

asymmetric rotator.

The above considerations are applicable to a molecule with an axis of symmetry. If

this is the third coordinate axis, the value of < R(t) > may be found by ptutting 12 = 1' B2 =-B in (42) and (43).

7. Application to relaxation problems

In applying the theory of rotational Brownian motion to dielectric and nuclear

magnetic relaxation problems one often employs expressions that may be derived from




I(co) defined by

rr I(@) - < R(t) > r1a di (44)

i(McConnell 1980a, ch. 13, 1980b). We deduce from (4) and (5) that

1(c) = f (I + 2 V(2)(t) + g4 V4)(1) + .. .) exp [(G - iw)tldt. (45)

If an infinite series is uniformly convergent, it is allowable to integrate it term-by-term. ror the disc model eq. (26) shows that the integrand in (44) is the identity operator multiplied by

[ + y(1-e-B') + I y2(1 - E-B9)2 + y3(1- e-Bt)3 + ... e-Ye-iw1. (46) I~1+ ;41 ~ +2! 3!

Since for t positive 0 < I - e-" < 1, the absolute value of the remainder after n terms of

(46) does not exceed the remainde'r after n terms of the fixed convergent series of

positive terms

+ y + 2 + I y3+ 2! 3!

Hence (46) is a uniformly convergent series, and so the right hand side of (45) for the

disc may be integrated term-by-term.

The investigations of the other molecular models in sections 4, 5, 6 gave values of

O2rrV2r)(t) in (35), (40) and (42) which show no clear pattern for the structure of the

series V(t). For the spherical and linear rotators we found that the exponent in (20)

could contain negative terms. When this is so, we would obtain both positive and

negative terms in the expansion of the exponential and this could make the truncation of

the series V(t) a procedure of questionable validity. However on account of the small

experimental value of y the truncation seems to be reasonable. Indeed the negative signs of greatest possible consequence arose for the linear rotator with j= 1, m ? 1 and

i = 2, m = + 2 and there the negative signs appeared in corrections of order y2 to the

Debye approximation, which would be obtained on replacing (38) by 1. When the series is truncated, it is finite and then the integration in (45) may be performed term-by-term.

It should be pointed out that a correction of order y to the Debye result may have

important implications for experiment. Thus, if a liquid dielectric is composed of polar

molecules having an axis of symmetry, the curve for the rate of absorption of

electromagnetic radiation as a function of wave number has a plateau in the Debye

theory but falls down towards zero in the submillimetre region when corrections of order y are included (McConnell 1980a, section 14.2). For dielectric absorption we are

concerned only with j = 1, m = 0, as we saw in (39), and then the elements of the exponent in (38) are positive up to corrections of order y3, which is more than we need

for the linear rotator equations (36) and (40). The elements of the exponent are positive to the same order also for the spherical model. This makes it reasonable to suppose that

calculations based on the expansion of < RQ) > are reliable up to order y3 for the study of dielectric relaxation associated with polar spherical molecules.




A correction of order y to the Debye theory has no great significance for nuclear

magnetic relaxation experiments in the present state of their accuracy (McConnell 1982,

1983). It thus seems allowable to employ the expansion of < RQ) > in the theoretical

investigation of nuclear magnetic relaxation processes. An abstract study of the way in which the exact value of < R(t) > might be obtained

from (4) by taking successive approximations to V(t) and G in (5) and (6) is included in

a general investigation of the averaging method by Frigerio et al. (1981). For the case of

the spherical rotator they find upper limits for the norms of

< R(t) > - exp [e2G(2)t]

and

< R(t) > - [I + e2V(2)(t)] exp [(e2G(2) + e4G(4))t].

In our notation the respective upper limits are

y(a + byBt), y2(c + dyBt),

where a,b,c,d are positive constants. This is as far as their calculations have been taken.

We may then seek further guidance from the investigations of the previous sections

when relating theoretical with experimental-fresults.

8. Conclusion

The expansion of the ensemble average < R(t) > of the stochastic rotation operator

in a series suitable for applications to thermal relaxation problems has been studied for

the disc, spherical, linear, symmetric and asymmetric rotator models of a molecule. In

the case of the disc model the expansion gives exact results for < R(t), and for the

integral I(co) defined in (44). For the disc, linear and spherical rotators the series is

constructed by relating it to the exponential of an integral as in (12). On account of the

small experimental value of the dimensionless parameter y defined in (23) it is likely that

a few terms of the series expansion will yield a value of < R(t) > sufficiently accurate for

the calculation of such quantities as orientational correlation functions, complex permittivities, spectral densities and correlation times encountered in dielectric and nuclear magnetic relaxation theories.

Appendix

The application of Laplace transforms to the evaluation of integrals occurring in the

calculation of < R(t) > has been discussed already in Lewis et al. 1976, Ford et al. 1976

and McConnell 1980a. We shall supplement these discussions by deriving results of

special relevance to the calculations of the present paper.

The Laplace transform L,{f(t) I of a function f(t) -is defined by

00

Ls{f(t) I = e-sf(t)dt. (A. 1)

It follows that for m zero or a positive integer




L,{tme-bt } =I + (A.2)

The convolutionfi * f2 * ... * f, given by

(f *f2 *... *f,Xt)

i A ^ tI tn2

= ft J dt2 . .fl dtn.1f1(t-tl)f2(tl- t2) . . fn_l(tn-2 -

tn-Of(A.3)

has the property

Lj{f *f2 * .1 . *.f} = L,{f1 } Ls1f2} . . . Ls{fn }. (A.4)

As an illustration let us consider I6(8)(t) defined by

J6(8)(t) J Jt,i dt2 . . J'7dt8 e-B(1-t2 +3 +14 +t5-6-17-1--8) (A.5)

To express this in the form of a convolution we write

tl - t2 + t3 + t4 + ts - t6 - t7 -t8 (A.6)

= 0(e-tl)+(tl-t2)+(Xt2-t3)+(t3-t4)+2(t4-t5)+ 3(t-t6)+ 2(t6-t7)+(t7-t8)+Ot8

and we note that the coefficients of the brackets are non-negative. As may be seen from

(27), in our integrals the coefficient of t1 will always be + 1, the coefficient of t2r will

always be - 1 and the other coefficients will have equal numbers of + l's and - l's.

Hence in writing out the multiplier of -B in the exponent of the integrand in (27) we

shall always begin with 0(t - t,) and end with Ot2r, and the coefficients of the tf -t+ will be non-negative. On substituting from (A.6) into (A.5) and (A.3) we obtain

6(8)(t)1 * eBI * 1 * e-Bt * e-2Bt *e-30 * e-M * e * 1

Then from (A.2) and (A.4)

LS14l)( = I(A.7) s3(s + B)3(s + 2B)2(s + 3B)

By inverting we may deduce the value of I6(8)(t). The s-3-factor comes from the three

zeros in (A.6). If the minimum number of two zeros had been present, we would have

had s2 in the denominator of the right hand side of (A.7). We see from (A.1) that for t very large we obtain the greatest contribution to the

integral for s very satll. Hence in order to study the behaviour of I6(8)(t) as t tends to

infinity we take s very small in the right hand side of (A.7) and expand in powers of s/B,

thus obtaining

}s I -m 16(8)()}=13 _ _ 13t

Ls { iM i6 (8)(t) ==72S t -0 0012B6s3 36B7S2 24B6 -36B7




by (A.2). We deduce that

lim [B8I6(8)(t)i = - B2t2 - 36 Bt + I - COf

6 24 3

From the method of constructing (A.7) it is clear for any i and r that in the expression

for

lim [B2rI,(2r)(t)

the second order term will have a negative sign. Moreover the above reasoning shows that

lim IB2rli(2r)(t)'-aBt-b, (A.8) f oo

where a and b are positive dimensionless constants, if and only if there are not more

than two zeros in the equation corresponding to (A.6).

An investigation similar to the above with an expansion in terms of large s will show

that

lim [B2rIj 2r)()i = (1 - fBt + 1-0 i

~~~(2r)!

where f is positive and dimensionless.

We quote from the previous references the values of B2r I.(2r)(t) required in the

present paper:

B21(2)(t) Bt - 1 + e-B'

1 51 B4I2(4)(t)- = Bt - + [Bt + lie-BI + e-2B

B6R (6)(1) I

Bt- 1 + 2 e-B - R Bi + I e-2Bt 3 4 1

12 I6 [ 2 1 i ] eB e-2B' + 864( (t B-t----+ I Bt-Ri~e-Bt + B+e3B

12 18 [2 4J [4 2t JU 36

1 11 I1 33 B814(8)(t)= Bt-

- B2t2 + 3Bt--3 e-Bi'

8 1 6 + 6- 2 (A.9)

+ + Bt + 6 e

185(8)(t) 2 Bt-29 i 3 1 5 B8IS (8)(t) Bt

- ~~~~~~~~~

+ -

-B2/2 --Ri+

-B

24 144 4 4




[8 B2t2 + 3

Bt +- | e-2Bt 1

-3Bt - -Bt ? I e211 + [8 4 161 72

72 432 4 Bt12 e-B'+8 [2t28 +If+ l16w2B

11 7 13B

[36 t +54

1 47 [1 1 112B fi8I (8))~ fiBt- 7+ | fit--I e-B+l -Bt + ]e 144 1728 [12 9J [8 161

? 3

fi +27

] 3t+576

REFERENCES

[1] Abragam, A. 1961 The principles of nuclear magnetism. Oxford. Clarendon Press.

[2] Ford, G. W., Lewis, J. T. and McConnell, J. 1976 Graphical study of rotational Brownian

motion. Proc. R. Ir. Acad. 76A, 117-43.

[3] Ford, G. W., Lewis, J. T. and McConnell, J. 1979 Rotational Brownian motion of an

asymmetric top. Phys. Rev. A19, 907-19.

[4] Frigerio, A., Lewis, J. T. and Pul?, J. V. 1981 The averaging method for asymptotic evolutions.

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[9] McConnell, J. 1978 Correlation functions for spherical harmonics resulting from rotational

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[12] McConnell, J. 1981 Modified Rocard relation for complex permittivity. Physica 105A, 593-600.

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[14] McConnell, J. 1983 Nuclear magnetic relaxation by quadrupole interactions in non-spherical molecules. Physica 117A, 251-64.

[15] Perrin, F. 1934 Mouvement brownien d'un ellipsoide (I). Dispersion di?lectrique pour des

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[16] Sack, R. A. 1975a Relaxation processes and inertial effects - I: Free rotation about a fixed axis.

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series expansion of the stochastic rotation operator

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