Physica I12A (1982) 479-487 North-Holland Publishing Co.

NUCLEAR MAGNETIC SPIN-ROTATIONAL RELAXATION TIMES FOR SYMMETRIC MOLECULES

James McCONNELL

Dublin Institute .for Advanced Studies, Dublin 4, Ireland

Received 12 November 1981

It is shown that the problem of calculating times related to nuclear magnetic spin-rotational interactions may be solved for the symmetric rotator model of a molecule by employing the method already proposed in a general manner for asymmetric molecules that undergo rotational thermal motion. Expressions are derived for the spin-rotational correlation time and for the contributions arising from spin-rotational interactions to the longitudinal and transverse relaxation times.

1. Introduction

A general analytical method of calculating nuclear magnetic relaxation times

resulting f rom spin-rotational interactions has been based on the stochastic rotation opera tor and on the solution of stochastic differential equations1). The

method is applicable in principle to molecules with no special symmet ry which

are subject to rotational thermal motion, but a thorough examinat ion of this problem is hampered by the relatively low order of approximat ion to which

calculations on the rotational Brownian motion of asymmetr ic bodies have so far been taken2). The problem has already been solved for spherical molecules 1,3,4,5).

In the present paper the case of symmetr ic top molecules is examined. The results are true for a molecule which has a principal axis of inertia through the

centre of mass that passes through the nucleus in which we are interested, and such that this principal axis is an axis of symmet ry Cn for the molecule with /1 /> 36).

2. General equations

We consider nuclei in a rotating molecule denoting by Ii the spin operator of the ith nucleus and by hdi the angular momen tum of the molecule3). The spin-rotational Hamil tonian of the ith nucleus

hGi = hi, • C ' • J,, (1)

where C ~ is a dyadic. If C~, with/x, v = 1, 2, 3 are the cartesian components of C ~

0378-4371/82/0000-0000/$02.75 O 1982 North-Hol land

480 J. McCONNELL

in the molecular frame of coordinates, which is taken in the directions of the principal axes of inertia through the centre of mass, we write

i - - " i 1 C E v _ C 1~ + ( 2 ) i

bg=CL, b~,,v=+ V2

Then (1) is expressible as I

O', ~ k k = ViUi, k= 1

where V~ are the spherical components of Ii in the laboratory system and

3 l

E E ' ' = b m~Okm(ai, •i, "Yi)Ji~, v = l m = - - I

D~m being the rotation matrix for the transformation of a spherical tensor and a~,/3~, 7i the Euler angles specifying the molecular system with reference to the laboratory system of coordinates.

To introduce the various relaxation times we define lk Cii(t) by

Ik C,, (t) = (U',(t)uk(o)),

where the angular brackets denote ensemble average for thermal equilibrium. The Laplace transform c~(s) of C~(t) is expressible by 7)

' ' ( I ) 0o c , i ( s )= ~ ~ ~ (-)~b~.b~dJ~ e ~'(R(t)t%(t)to~(O))dt (3) 3h ~,~=l m,~= I ~, m

0

In this I~, Iv are the moments of inertia about the molecular/z - and u - axes, and to~(t), to~(0) the corresponding components of angular velocity of the molecule at times t and zero, respectively. R(t) is the rotation operator that brings the molecular frame at time zero to its orientation at time t. The subscripts n, - m signify the n , - m - m a t r i x element with respect to the basis Y~, 1(/3(0), a(0)), Y10(/3(0), a(0)), Y.(/B(0), a(0)), the a(0),/3(0) being two of the above mentioned Euler angles at time zero.

The contributions (1/T0b (l/T2h from the spin-rotational interactions to the reciprocals I[T~, liT2 of the longitudinal and transverse relaxation times are given by

(~ t ) =2J,(too), (~-~2)=Jt(O)+J,(to0), (4) 1 1

where tOo is the angular velocity of the Larmor precession and

1 O 0 • _[_ 0 0 . J l ( t O ) = 2(Cii ( - - l O t ) ) Ci i ( l t o ) ) . ( 5 )

In the extreme narrowing case we replace too by zero and then denoting the

SPIN-ROTATIONAL RELAXATION TIMES 481

common value of (lIT1)1 and (1/T2)1 by llTsr obtain

-~-~= 2c~(0). (6)

Lastly, the spin-rotational correlation time ¢sr, which is defined as the integral from 0 to o~ of the normalized autocorrelation function of U~(t) is given by s)

3ti 2 c~(0) (7)

q'sr=k'-T ~ ~ ( - ) m b i - m g b i m ~ I g p.=l m=-I

It is seen from the above equations that the central calculational problem is the evaluation of the integral J'o e-~'(R(t)to~(t)to~(O)) dt. This has been done for the asymmetric rotator model of the molecule, and for future reference we quote the resultg):

f f e-~t(R(t)to.(t)to~(O)) dt

= 8~jkTI.[(-G + [ s + B~]I)-'

3 k T 2 ] + ~ r---fi-Tu J ,{ ( -G + [s + B . ] I ) - ' - ( - G + [s + B. + B,]I)-'}

i=1 liJ.J i 2 2 s I ( Ip - I~ '~ (kT)

+ ~ ' \ I~ / Ifl,,(Bp + B~, - B~) 2[(-G + [s + B~]I) -1

- ( - G + [s + Bp + B~]I ) -1 - (Bp + B~ - B~. ) ( -G + [s + B . ] I ) -2]

( kT)2".- e~)[( i_i~[~. ( - Q +__[s +_B~]I) -1 / -~31ta • e. x "L(B~, - Bv)(B. - B~ + Br)

+ ( - G + [s + B t, + Br]I) -~ _ ( - G + [s + B~]I)-~'[ B , (B . - B~ + Br) B , (B. - B~) 1

+ (L - r ~[ ( - G + [s + Ba]I) -l ( - G + [s + B~ + B,]I) -1 "'t(B~ - Bv)(B~ - B~ - B,) Br(B~ - B~ - B~)

( - G + [ s + B,,ll)-"[] _ (kT) 2 + ts + S , l t ) - ' -4 B , ( B ~ - B , ) j j / - i ~ i(J " e,~ x e~') L ~ - ~ , Z ~-~)

( - G + [s + B~]I) -I + ( - G + Is + B~ + B~]I) -I] (kT)2j j~ - B~(B,. - B~) B.B~ _,- I.I,.B,~B~ [ ( - G + sI ) - '

- ( - G + [s + B~]I) - ~ - ( - G + [ s + B~]I) -~

+ ( - G + [ s + B ~ + B ~ ] I ) -1]+ . . - . (8)

In this I is the identity operator. B~, B2, B3 are the frictional constants defined such that I~Brto(t), I2B2to2(t), I3B3to3(t) are the components of the couple resisting the rotation of the molecule, p and tr are the numbers such that txp(r is a cyclic permutation of 123. and r is the number which with /~ and v (# /~)

4 8 2 J . M c C O N N E L L

completes the set 123. Then

3

G : - E D,S~, I=1

where the cartesian components JL, J2, J3 of the rotation operator molecular coordinate system satisfy the commutation relation

(9)

in the

[J~, J,] = - i(J • e, × e,) (1o)

and in the approximation to which we shall be working it is allowable to put

kT kT kT D1 - IIB1' D2- I2B2' D3- I3B/ (11)

3. Calculat ion of c~(O)

In the representation employed in eq. (3) we have

E°'°I I i i°°i J l = 1 0 1 , J z = ~ 2 0 - , J~= 0 0 0 , (12) 0 1 0d 1 0 0 1

which clearly satisfy (10). From (12) we deduce that

1 1

j ~ = 1 , o 1

o -'~ o

I 0 0 0 1 J2J3 = -- i /X/2 0 - i /X/2 ,

0 0 0

I

0 ~ , j~= 0 0 0 , 1 0 0 12

J 3 J 1 = 0 0 ,

0 1/~,/2

I IOo !1 i/2 0 /2 0 i/V2

J1J2 = 0 0 , J~J2 = 0 ,

i/2 0 - i / 2 j i/~/2 _

I0 0 I : 0 : l J1J3 = - - 1/'~/2 0 1/ 2 , J2J1 = 0 ,

0 0 i/2 0 i/2

(13)

and we verify that

J~+J~+J~= 2I.

We take the axis of symmetry of the molecule to be the third coordinate axis and then

I2=I i , B2=BI , D2=DI , (14)

S P I N - R O T A T I O N A L R E L A X A T I O N T I M E S 483

so that from (9) and (13)

- O + aI = I D~ + D3 + a 0

0 2D~ + a 0 0

0 DI + D3 + a

I (D1 + D 3 + a) -1 0

( - G + al) -1 = 0 (2D1 + a) -1 0 0

0 j 0

(DI + D3 + a ) -L

(15)

which is a diagonal matrix but in general not just a multiple of the identity matrix. We now proceed to find c~(0) from (3) and (8), employing the results (12)-(15).

In subsection 4.4 of ref. 1 it was found that the JJ~- terms of (8) gave rise to calculational difficulties, so let us consider these first of all. From (3)

c . ( 0 ) = ~ - ~ ~ I.I~ ~, (-)mb..bm~ (R(t)oJ.(t)oo~(O))dt (16) g,v=l m,n=-I n, -m"

0

We take the 9 different combinations of Ix, v separately and perform the summations over m and n for the JJ~- terms of (8). Let us, for example, take Ix = 2, u = 3. This gives a contribution

(kT) 2 ~ i i -1 z.~ (-)rob nzb m3JzJ3[(-G) - ( - G + B l l ) -1 - ( - G + B3I) -1

3h BIB3 . . . . . 1

+ ( - G + [B~ + B3]I)-1]. (17)

Since the third axis is the axis of symmetry, the components of the dyadic C ~ in the molecular frame satisfy l°)

i i C~3=GI, C ~ . = C 2 2 = C . , C~q=0, ( p ~ q ) .

We then deduce from (2) that

b~l,1 _ C± i iC± i i i _ i _ = + : ~ , b±t,2 ; ~ , b±L3=0, b o l = b 0 2 - 0 , bo3-Cii. (18)

Let us consider

1

( - ) ' b l.2b ~a(JzJ3(-G + al))- ') ._, , . (19) trl,n =--1

Since, by (13) and (15),

__ I ° J2J3(-G + aI) -1 = - ~ (D1 + D3 + a) -~

0

o o 1 0 ( D I + D 3 + a ) -1 , 0 0

484 J. McCONNELL

the sum (19) is equal to

i(b + b %) ~/2(Di + D3 + a ) '

which vanishes by (18). Taking a successively equal to 0, B~, B3, B~ + B3 we deduce that (17) vanishes. It may similarly be shown for all combinations of

and v that

1 (_)%i , nt, b , . . ( J . J~ ( -G + al)-~).. ,. = 0. (20) m , n - - I

Then by putting a successively equal to 0, B~, B~, B. + B~ we conclude from (20) that the JJ~-term of (8) gives no contribution to c~(0) even before the summation over/~, v is carried out.

The contribution to c~(0) from the first term on the right-hand side of (8),

k T ~ f ~ f~_ I. ,.,mE = , (-)mb',,b~,,

I (DI + D3+ B,) * 0

× 0 (2Di + Bu) -1 0 0

o 0

(Dx + D 3 + B . ) -1 .. m

k T + i i i i i i b l~,b l.)(D1 + D3 + B.) t + bo~,bo.(2D1 + B.) ~} = ~ ..... I u{ - (b .-l.b lu +

k T ~ 2I,C 2 k I3C~ (21) ~ t D ~ + D3 + B, . 2DI + B3J'

by (18). The other terms in (8) that give a contribution to c~(0) are proportional to (kT) 2

and they include no term proportional to ( - G + s/) -1. Moreover they are correct only to order (kT)Zl(I2B3), where for the moment we suppress the suffixes of I and B. We may therefore replace ( - G + aI) -~ in them by a II.

Let us examine the contribution to c~(0) from

l B;)' 8uJ ~--~/_-~l Yd 2(1-. B . +

where Yi -- kT/(liB~). This contribution is from (13), (16) and (18)

I . YJ B .

[( ' k T ~_. 3'1 -=W &

, ) l ] i i 2 B. + B~ ~ (- )mb"~'b""(Jt)

m , n = - I n , rn

1 B.)(_2b~ ,~b~ + 2(b~,)2) B . +

SPIN-ROTATIONAL RELAXATION TIMES 485

, ) ] + ~ ~ n~ + ni (-2b~-l~b~)

(kT)2f C 2 + 2I, C 2 213C~ - f ~ r - ] - ~ IaB~B3(BI+ B3) + I tB ,B3(B1 + B3)}" (22)

Similarly it is deduced that the last 8~-term in (8) gives a contribution

2(kT)2C2(I1- I3) 2 - 3h2IlI3B~(B~ + B3)" (23)

Of the two terms proportional to (J • e~ × e~) in (8) the first is approximated for s = 0 by

(kT)2i( j • e~, × e~)[ /~ - L + I~ - I~ ] - - _ _ )

I~I3B~,B,, [B,, + B, B~ + B , ]

and the contribution of this to c~(0) is

1

(kT) 2 ~ /_~_/~[/~_~L + I v - I , ] ~ . (_)mb~ b ~ i ( j . e . × e ~ ) . _ . , "

(24) For the sum of the combinations ~ = 2, v = 3 and/~ = 3, v = 2 this yields

_ 2(kT)2C±CII(I~ - I3) 3h211B1Ba(BI + B3)" (25)

Similarly for the combinations t~ = 3, v = 1 and ~ = 1, v = 3, (24) gives

_ 2(kT)2C±CII(I~ - I3) 3h2IIB1B3(BI + B3)'

and f o r / ~ = l , v = 2 a n d # = 2 , v = l

4 ( k y ) 2 c 2 ( I i - I3) 3h213B~(B1 + B3)"

On summing (25), (26) and (27) we obtain

(kT) 2 ~ I~.I,, [ I ~ - I, I , , - L ] '

4 ( kT )2 ( I i - I3) [ C 2 C±CII] = 3h2B,(B, + B3)Li-~, I,B3 J"

(26)

(27)

m i i • ( - ) b n~.b,.~l(J • e. X e~)n,-,.

(28)

Proceeding in the same way we find that the contribution of the remaining term in (8) to c//°°(0) is given by

(kT) 2 ~-. 1 1 ~,--rr L , 3h ~,,,=1 B,.,B,,(B,, + B,,) . . . . . 1

(kT)21. 4C±C i 2 - f~r-] .B,B3(BI+ B3)+~-~}"

m i i • ( - ) b.~b.,~l(J • e~ x e~).,_.,

(29)

4 8 6 J. M c C O N N E L L

On collecting our results from (21)-(23), (28), (29) we conclude that

k T { 2IjC~ + 13C~ cii(0) 3-~ + D , + D , B ~ + 2 D ~

r i 2 211 213 "~r~2 213(C~+2C±CII)]~ + kT [t-~t + ~ - I,B2(B, + B3)} "-'~ + I,B,B3(B, + B3)JJ" (30)

As a check on this result we examine what it reduces to when the symmetric molecule is spherical. We put

kT [ l = 12 = I, B1 = B 3 = B, D i = D~ = D = IB'

by (l l), and write D/B = kT/(IB 2) = 2/. Then (30) becomes

kr, f 2c: c ~ ' ( 0 ) = ~ t l + 2 v 1 2-t

kTI 2C" " = 3 T ~ - D i : + c ~ - v ( c , - co>} .

We see from (6) that this agrees in the approximation of the present paper with the result of direct calculation~).

4. Spin-rotational relaxation times

Eq. (30) leads immediately to expressions for spin-rotational relaxation times. For the extreme narrowing case we have from (6) that

1 2kT[ 2IIC2~ I3C~ B~ + 2DI

[(2 2,, kT B-~+ I 3 B ~ I,B~(B, + BO I iB1B3(B i - BO j j" (31)

It is possible to generalize this result so as to find (1/T1)1 and (l/T2), from (4) and (5). We see from eq. (3) that in order to do this we would have to retain s when employing (8) for the evaluation of c~(s). This would present no great difficulty but the calculations would be somewhat more complicated.

To find the spin-rotational correlation time from (7) we first deduce from (18) that

3 1

E E ( - ) r o b ' ' " ' ,..b,.,.l~. = II(-2b illb i ,, - 2b'12b'12) + I3(bo3Y ~*=1 m - 1

= 2I~C2~ + I3C~.

SPIN-ROTATIONAL RELAXATION TIMES 487

It then follows that

1 211C 2 + I3C~ "r~r=2I~C~+I3C~{B,+O~+Oi B3 + 2D~

2 211 213(C~ + 2CICII) ]~ - IIBE(B1 + B3)) C2 + IlBiB3(BI + B3)JJ" (32)

5. Conclusion

The general method of finding the contributions to longitudinal and t ransverse relaxation times arising f rom spin-rotational interactions, which had been

proposed for asymmetr ic molecules undergoing rotational Brownian motion, has been successfully applied to molecules with an axis of rotational symmetry .

The calculations are possible because the terms in the Laplace t ransformat ion of (R(t)t%(t)to~(O)), that had caused difficulty for vanishing values of the paramete r s, are found to give zero contributions. The spin-rotational correlation time has

also been calculated.

References

l) J. McConnell, Physica IllA (1982) 85. 2) G.W. Ford, J.T. Lewis and J. McConnell, Phys. Rev. A19 (1979) 907. 3) P.S. Hubbard, Phys. Rev. A9 (1974) 481. 4) P.S. Hubbard, Phys. Rev. A24 (1981) 645(E). 5) R.E.D. McClung, J. chem. Phys. 73 (1980) 2435. 6) C.H. Townes and A.L. Schawlow, Microwave Spectroscopy (Dover, New York, 1975) pp. 48, 49. 7) Ref. l, eq. (2.14) and remark about ensemble average notation after eq. (2.37). 8) Ref. l, eq. (2.27). 9) Ref. l, eq. (4.22).

10) Ref. 6, pp. 219, 220; ref. 3, p. 488. ll) Ref. l, eq. (3.40); ref. 4, eq. (5.15).