Volume 39, number 3 CHEMICAL PHySICS LETTERS ! M2y 1976

A GENERALISED FRKXfON MODEL FOR-THE EVALUATION

OF ANGULAR MOMENTUM AUTO CORRELATION FUNCTIONS

Myron EVANS Physical Chemistry Laboratory, Oxford OXi 3QZ, UK

Received 4 February 1976

A theory of brownian mo:ion where the friction coefficient is a kite correlation function of molecular torque is used to derive an expression for the ar@a~ velocity sutocorrelation function, @(r). This is related to the orientational a.c.f.,&, using an m&Gs of bandshapes due to Kubo 2nd extended by Shiiizu. Both functions G(t) and J,(t) contain inherent weak- necks since q(r) has a MacLaurin expvlsion even up to t4 only, and’@(t) does not reduce to the Kummer function for an ensemble of free rotors in the so-called limit of vanishing friction. Reasons for this behaviour are discussed.

1. Introduction and theoretical considerations

To describe the process of general random rotation of a linear molecule, Shimizu [l] has used an analysis by Kubo [2 ] of the shape of spectral bands. This has been discussed further by Wyllie [3] whose comments are recalled briefly here as an introduction.

Let the angular velocity of the rotor at time t be w(t), then since t = Cl it has been displaced through an angle:

t 6 =sw(t’) dt’ ,

0

c i (1)

and the normal&d autocorrelation function of a unit vector in the axis of the tinear molecule is:

(cos 0(t)> s &OS [_fko(t’) dr’]) E Re {(exp [if; o(t’) dt’])} , (2)

where the angular brackets ( ) denote an average over the different w(O), and the different w(t’). Regarding w as a random variable, its equilibrium distribution (i.e. that ofw(0)) will be gaussian, and we can

expand (cos e(t)>, the general term being:

The r.h.s. now involves multiple self-correlations of w, and can be regrouped in terms of cumulatt averages (( ),),

where, for example, (w)~= W; ++w$~= <WIW$ - <wt) (w$. We ,can see immkdiately from eq. (3) that if all the odd correlations of o vanish, then the autocorrelation function I$() = &OS O(t)) will be-positive throughout its domain of definition, and therefore will not reproduce the well known form of I$@) for free rotation, the hyper-

601

,--Volul& 39; &ll&r~ 3. -_ ; -. :. j -’ . CHEMICAL PHYSICS LkTTERS_ 1 May 1976 ,

’ geometric Kumtier 5m&& 131, which for large t approaches zero from below the time ax&. The sanre is true if the Reties in eq. (3) is trt&ated m any way such as to delete the imaginary terms. F&r exampie, if it is assumed *at the muhivarkte &n&y function for o(fr) ___&n) is a gaussian, then all cumulants beyond (w~c+)~ vanish &id the process LJ isStatistically determined by its mean (zero) and first self-correlation alone, i.e.

and +(t) is inevitably badly behaved in the free rotor limit at long times, in that it cannot become negative. This is a point which does not seem to have been emphasized sufficiently. For example, Kluk and Powles [4]

recently used the exponential form:

&(O)w(t)l= (~~(0)) exp(-Et/I) , (9

which is consequential upon Langevin’s treatment [3] of rotational brownian motion with .$ as the constant friction coefficient and I the molecular moment of inertia. Using kq. (5) in (4), then:

G(i) = exp{-(2kTI@) [exp(-@I) + ,$‘I1 - l]) , (6)

.a form previously given by Wyllie and others [3] for a rotator with one degree of freedom. Here we give it another (laboratory) frame so that:

(w2(oj)=2kT/~.

Eq. (6) is identical with that of Kluk and Powles for the fust harmonic of the orientational correlation function (j = 1 in their notation). It has the correct initial short time expansion [S] for the classical autocorrelation function:

G(t)= l- (kT/I)t2+ S(t3), (7)

and reduces to. a pure exponential for high 4, but not surprisingly, does not reduce [4] to the Kummer function as -E+O (the limit of vanishing friction).

Eq. (5) is an unsatisfactory form for at least two reasons:

(i) Gordon [5], Berne [6], and others have shown that the angular velocity correlation function is classically an even function of time, so that [6], for linear molecules:

~(t)=<o(O)w(t)>/<w*(O)>= 1-to(Y)2)t2/41kT+(~(V)2)t4148kTI- . . . ,

where (0(Tq2> is the mean square torque on a molecule due to the others, and (&V)2) its derivative. G(t) in turn ought to be even in this sense.

(ii) It truncates the series in eq. (3) so as to leave no imaginary terms, and therefore from the start has no hope of producing a e,(t) which is well behaved as the free rotor limit is approached. It seems that cumulants beyond (~1 w$;, both even and odd, must be used before G(t) can be induced to go negative and remain an even function of time (in the sense that its MacLaurin expansion contains no odd powers oft).

An improvement over eq. (S), but one which still has the second weakness built in, is to use generalised Langevin theory-in deriving (w(O) o(t)1 instead of the simple equation used by Kluk and Powles. This would at least involve the eqr&m’hrm intermolecular properties (0(V)2) and <& V)2) to be used in G(t), properties that can be simulated by computerised molecular dynamics of large molecular ensembles.

-_ Kubo [7] and others [3,6] have presented a general form of the classical Lange- equation. This is now written as:

t G(f)=- s K(t - 7)0(r) dr + F(t) ,

--0

602

03)

V&me 39, nuinber 3 CHEMICAL PHYSICS LETTERS’ 1 May.1976

where K is a timedependei;t friction coefficient known as the memory function. m F(i) is the fluctuating part of. ~ the torque on a molecule of mass m. It follows from eq. (S) that:

The last term is zero since ~(0) an&I’(t) are assumed to be kncorrelated statistically. Mori [S, 61 has shown that the set of memory functions K,(f) . ..K.(t) obey the set of ccupled Volterra equa-

tions such that;

X,_,(r),&= - jK,,(r - r)Kn_l(T) dr . 0

Solving by Laplace transformation we have:

E:(p) = c(o) = cm = p + Ko(p) p f ho --- ’

P+&(P)

(10)

(11)

with e(p) as the Laplace transform of C(t) = (w(O) w(r),. Several authors [S, 6,9] have shown how the method of truncating this continued fraction using simple forms for K&).can be used to derive complicated but more realistic forms of C(t) and its Fourier transform, the complex spectral function C(-iw).

The equilibrium averages K@(O), K1(0), etc. are related [6] to the terms in the MacLaurin expansion of C(t), so that if we define:

$(t) = C(r)/<w2(0)> = Fan @r/(24! > (12)

with nn alternatively positive and negative, we have:

K,(O) = -al = (O(V)2)/2ZkT , 03)

K&O) = al - a2/al = (b(V)2)/(O(V)2) - W(V)2)/2kTI, (141

and so on. Here, we truncate the series (11) with:

Kl(t) =Kl(O) exp (-_yf) , (19

where 7-l is an empirical correlation tune, and find that Q(r) is even up to the t4 term in its series expansion, but thereafter contains all the odd powers. Higher order truncations than that of eq. (15) merely introduce higher derivatives of (O(V)2}, which in general is unknown analytically. Therefore, by using eq. (15) we introduce the

“empirical” terms (O(V)2) and (b(V)2). The advantage of using this form for KI(t) and Jl(t) is that 4(t) will have the odd powers only after &, and will be very well behaved at short times away from the free rotor limit.

From eqs. (11) and (15) we have:

where

(16)

V&me 39, number 3 CHEMICAL PHYSICS LETTERS 1 May.1976 _

The terms s1 and s2 are defmid-by:

‘cl= i-+5+(&A3 +$52)1/2]1/3, &= [--35-(~A3+~5z)f/2]li3 ; : .

with

A=Ko(0)+#,(0)-~~2~, B =+r[+g+2Ko(o) -Iq(o)J .

The mgular velocity correlatioti tfrne, ru , is now;

(17)

Using eq . (4), we have :

Q(~)=exp~-(2kTII)[A~(e-Q1’cos~~-l)~-5le-”l’sinpr+~(e-a2f-l)/(i+~)~+C~~]) , (18)

where

A,= 3+ 2,p2i’--p2 2&9+ (cq f rtY2) @2 - &4,

(1 +r)(C$tp2)2 ; B,= *_

4(1 +r) (a; f p2)2 ; q= 2&l

(1+ IxqP2)

2. Diiirssion

Some idea of the forms of+(t) and 9(t) is given in fig. 1 for reasonable values of CU(V)2) and its derivative_ It is seen that as t.ke torque is increased #(z) osciifates abcut the time axis and 4(f) develops damped oscillations at short times, becoming exponential (logarithmic) thereafter. As (U(V)2) + 0, G,(r) should, by eq. (S), become con- stant at unity for all positive t, and Q(P) should show a negative region. The former is true for ally, but the latter is not observed in the Emit of vtishing torque, since 4/(t) contains all the odd powers after ,&.

”

t

V t Cps,

Fii. 1. (a) @(t) (cur& 1) and-@(t) fcurve 2) for&(O) = 0.65 1/2kT;Kl(O) = 2.25 1/2kT;r = 1.85 (T/2kT)“.whereT= moment of inertia= 6.31XIO-3ggqn2. T= temperatiue = 298K. (5) Q(f) (curve l).&nd e(f) (curve2) for&,(O) = 7991/2ktiK1(0) = 2?.7.61!2kR~;r=32.7 UiZkT)“;IattdTasforf~,La.

Volume 39, m&k 3 CHEMICALPHYSICS LETTERS ., 1 -May 1976 -

it would be fruitfh for further work in this field to apply geneklised Langevin theory to approximate odd cum&n& such a~ (w(~&J(~&&&, and to abandon the idea that the mtiltiva&e den& function for 00~) . . . w(tJ is a gaussian.

Acknowledgement

SW is thanked for a fellowship.

Appendix

The overall memory function K,,,(t) corresponding to G(t), related by the equation:

i(t) = -jiQ(t-d$(r)dr , 0

is given [using eq. (15)] by:

J$(t)=exp(-7t)(cosat++7asinat)Ko(0), for K1(0) >,+r2 ;

= exp (- 7t) Cl+ 3 7 t)rC,(O) , fOi Kl(0) =&72 e 9

=exp(-rt)(coshbt++7b sinhbt)KO(0), for K1(0)<ir2 .

Berne and Harp have shown that the ideal K,(t) has a MacLaurin series starting:

so that in the absence of molecular interaction, K,,,(t) should reduce to zero, i.e. it can be looked upon as repre- senting the “molecular memory” of past interaction. From eqs. (13) and (14) we have Kg(O) and ICI(O) of our approximate 4(-t) disappearing in this limit of collision free angular motion.

References

[l] H. Shimizu. J. Chem. Phys. 48 (1968) 2494. [2] R. Kubo. in;_Fluctuation, relaxaticn and resonance in magnetic systems, ed. D. ter Haar (Ol&er and Floyd, Edinburgh, 1962)

p. 23. 131 G. Wyllie, in: Dielectric and related molecular processes, Vol. 1 (The Chemical Society, London,-1972) p. 41. [4] E. IUuk and J.G. Powles. Mol. Phys. 30 (1975) 1109. [S] R.G. Gordon, Advan. Mag. Reson. 3 (1968) 1. [6] B.J. Beme and G.D. Harp, Advan. Chem. Phys. 17 (1970) 63.. [7] R. Kubo, Lectures in theoretical physics, Vol. 1 (Interscience, New York, 1959); Statistical me&a&x of equilibrium and non-

equilibrium (North-Holland, Amsterdam, 1965). [8] H.‘Mori, Pro@. Theoret. Phys. 33 (1965) 423. [9] C. E+ot. ia: Dielectric and related molecular processes. Vol. 2 me <ShemU Society, London, 1975) p. 1; -

B. Qugn@ec..wd P. Bezot, Mol. Phys. 27 (1?74) 879; F. Mot. C. Abbar and E. Co+ant, MoL Phys. 24 (1972) 241; F. Eliot and E. Constant, Chem. Phys.,Letters 18 (1973) 253; 29 (1974) 618.

‘605