# Advances in chemical physics - Prigogine I.

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(309)

(311)

(312)

172 В. J. BERNE AND G. D. HARP

The mean square angular deviation <92(0) can be found in the following way. Note that the following integral of the C.M. angular velocity, Q(/)•

I* dty fi(^)

Jo

is a vector whose magnitude is the angular displacement, 6(t). The mean square angular displacement can consequently be written in terms of this integral as

<02w>=72 f '<*i <\dh<m-m>

1 * о Jo

where / is the moment of inertia of the molecule. The correlation function <J(/X) • J(?2)> is a stationary even function of the time—a result which follows from the fact that an equilibrium average is being taken,

<J('i) • J(t2)> = <J(0) • J(t2 - h)>

then

<92(0> = I \'dh Cdt2 <J(0) • J(h - O) (313)

1 •'О •'о

Introduction of the normalized angular momentum correlation functions, Aj(t), into this integral, followed by an integration by parts yields

/*оо

If the integral I dt t Aj(t) exists then the limit above is

КТ л00 KT

Dr = — Jo dt Aj(t) = — Aj(0) (314)

where the equilibrium mean square angular momentum, 21 KT, has been used and Л7(0) is the Laplace transform, Aj(S), of Aj(t) at 5 = 0. The rotational friction coefficient, yj, is so defined that

KT

Dr = t- (315)

iy j

from which it follows that

гГ‘ = Д,(0) (316)

or in terms of the memory function, Kj(t)

ъ = RAO) (317)

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

173

The single relaxation time approximation can be applied to the angular momentum memory function in a Completely analogous way.68 Kj(t) can be interpreted as the time-correlation function of the random torque acting on the molecule. If this random torque has a Lorentzian spectrum or, more restrictively, is a Gaussian-Markov process, Kj(t) is exponential.

The mean square torque is taken from computer experiments. Nevertheless, it could have been found from the infrared bandshapes. Likewise the integral in this expression can be found from the experimental spin rotation relaxation time, or it can be found directly from the computer experiment as it is here. The memory function equation can be solved for this memory. The corresponding angular momentum correlation function has the same form as v|/(/) in Eq. (302) with

There are alternative forms of the Gaussian memories* corresponding to both v|/(/) and Aj(t). From Eq. (169) we see that the formal power series expansions of K^(t) and Kj(t) are

(318)

(319)

where

/м2\ SN2s>

«J = ^ Д,(0) = ^ J/'' MO (320)

The solution is oscillatory if

These solutions are described later.

The Gaussian approximation to Kj(t) is in like manner

(322)

* This form follows from Eq. (188a).

174

В. J. BERNE AND G. D. HARP

If K^{t) is assumed to have a Gaussian form, as suggested by the information theory interpolative model presented in Section II1.F.

Then

K^(t) = В е_“2*2 **(() = Я[1 -Аг + ---]

Comparison of this expansion with Eq. (322) shows that

<*2>

A -

or = -

B <v2y r<d2> <02>

(323)

(324)

(325)

The Laplace transform of K^(t) is

KJS) = В e<s2'4“2> erfc (5/2a) v 2a

from which it follows that the friction coefficient у is

(я)xt2B (я)1/2 <a2>/<d2> <a2>\"1/2

Y = **(0) =

Y<£>_<£2V

\<a2> <v2})

(326)

(327)

<y2>\<^2> <У2>;

Let the factor multiplying я1/2/2 be called ц. Thus we see that if we assume a functional form for the memory function, then it is possible to determine the parameters in the functional form by using the moment theorems of Eq. (162) and to determine, thereby, the transport coefficients, such as the friction coefficient. Moreover, the time correlation function, i|/(/), can also be determined.

Here we see that

'(a2} <а2У

(а2У

*ф(0 = <^>еХР “

t2(<d2> _

.2 \<a2> <v2y]\

(328)

Exactly the same procedure can be carried through for the angular momentum memory function.68 Then the rotational friction coefficient is

(я)1'2

У J = —Z— \Lj

Vj =

2

<N2}

<N2> (N2y

-1/2

<J2> 1<N2} <J2> J 2r<iV2>

<N2s>

Kj(f) = JJiy exP

4Я

<N2y

(329)

(330)

(331)

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

175

Corresponding to the following memory functions are the indicated friction coefficients.

Delta function memory: K(t) = B8(t)

У =

(я)

1/2

(332)

Y j =

(к)

1/2

Hi

Lorentzian memory: K(t) = J5/(l + art2)

n

y=7^

(333)

Yj y/2*'

Gaussian memory: K(t) — Be

Y =

(я)

1/2

Y J =

(я)

1/2

(334)

Exponential memory: K(t) = Be “|f|, (a) adjusted so that the half-life for the exponential memory is identical to the Gaussian memory

У =

Yj =

In 2 2 '

1/2

In 2

1/2

(335)

(Z?) is adjusted so that the half-life for the exponential is identical with the Lorentzian memory:

_ V2

У = ПГ2Ц 72

(336)

This very last procedure is not really justified since the exponential memory starts out with nonvanishing odd time derivatives.

176

В. J. BERNE AND G. D. HARP

Values of <a2> and <a2> are required to compute jx. For this purpose we use the moments determined by Nijboer and Rahman from Rahman computer studies on liquid argon. The results are presented in Table VI.

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