# Advances in chemical physics - Prigogine I.

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ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

221

The computational scheme outlined above was tested with the testing

set

УМ = e

_

COS

n/3

, n/3 .

+ —sm

_ p-ti/2

■>/3

(B.15)

i = 0, ...,499 (B.16)

which behaves approximately like the autocorrelation functions obtained from the dynamics calculations. However, this particular testing function does not satisfy Eq. (B.3). Therefore, in order to get dyldt\Jt and d2y/dt for short times, y(t) was approximately by the interpolating polynomial

2i T

У*т(1) — ao + Z ам*

M

M = 2

where the coefficients aM were determined such that У*М = Ут«д i = 0, ..., 6

(B.17)

(BAS)

KyT(ti) for ti < 10 12 was recovered to within a maximum absolute error

•0.0009.

y(t) from Ky(t)

The problem of developing y(t) numerically from Ky(t) is much simpler than the reverse problem. One reason for this is that Ky(t) is usually a two- or three-parameter, analytic approximation to the true Ky(t) for the system under consideration. Therefore, one need not worry about statistical errors in Ky(t). The following scheme was used in developing y(t) from Ky(t) which depend on properties of y(t) and Ky(t) given in Eqs. (B.l), (B.3), and (B.4).

.K'o) = i

X<i) = 1 - Wo)'.2

y(t2) = 1 - (At)z[KyM + KyOMtJ] X'i+i) = X'i-i) - (biflKyO) + Kjt^yUj]

i-1

-2(a t)2

j = 2

2 < i

(B.19)

(B.20)

(B.21)

(B.22)

Equations (B.21) and (B.22) involve approximating dy/dt|(< by

y(h+i) -y(tj-i)

2A t

222

В. J. BEKNE AND G. D. HARP

and by approximating the integral in Eq. (B.l) by the trapezoidal rule.

The above scheme was tested by using the testing set given in Eq. (B.l5) and (B.16). y(ti) for tt < 1СГ12 s was recovered to within a maximum absolute error of <j;0.0003. All computations for both schemes were done in double precision on an IBM 360.

The error in Ky(t) generated in the first scheme from the experimental autocorrelation function was also examined by taking the generated Ky(t) and using it as input to the 2nd scheme to try to recover the original autocorrelation function. The original autocorrelation functions were all recovered in this manner within a maximum absolute error ^ 0.002 for all times, t < 10"12 s.

APPENDIX C. Properties of the Polynomials He^(x)

This appendix gives some of the properties of the Hermite polynomials, HeN(jt). These polynomials form a basis set for Rahman’s32 expansion of Czs(v)(r, t) and play a fundamental role in the discussion of the non-Gaussian behavior of this latter function. Brief sketches of this expansion and of the calculation of Fs(v) (К, t) are also given.

The polynomials HeN(jt) are defined by92

They are related to the Hermite polynomials HN(x), which are the solutions to the Schrodingen equation for a harmonic oscilator by

Hn(jc) for N = 0, ..., 10 are given in Pauling and Wilson.94 He^x) satisfy the recursion relations

He„(x) = (-1) ~ [e<

(C.l)

H„(x) = 2м'2Нек(хУ2)

(C.3)

He0(x) = 1

He^x) = *

Hejv+1(jt) = л?Недг(л:) — iVHe^.^x) N> 1

(C.4)

(C.5)

(C.6)

They also satisfy the orthogonality relation92

f He*(x)e( *2/2) HeM(x) dx = (2%y/2N\dM>N

“* rf\

(C.7)

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS 223

Finally the first six even polynomials used in the expansion of C?s(v)(r, t) are

He0(jc) = 1 (C.8)

He2(x) = jc2 — 1 (C.9)

He4(x) = jc4 - 6x2 + 3 (C.10)

He6(x) = jc6 - 15x4 + 45x2 - 15 (С. 11)

He8(x) = jc8 - 28x6 + 21 Ox4 - 420x2 + 105 (C.12)

He10(x) = x10 - 45x8 + 630x6 - 3150*4 + 4725x2 - 945 (C.13)

The expansion of Gs{v) (r,t) in the polynomials Не^(лг) proceeds by determining the coefficients b2N(v)(t) in the expression32

GsHr, 0 = E Vv)W He„(x) (C.14)

N = 0

where x2 = 3/*2/<(Ar(v)(/))2> and Ar(v)(/) = r(v)(/) — r(v)(0).

These coefficients are determined such that the following five moment relations on Gs(v)(r, t) are satisfied:

4k f r2 Gs(r, t) dr = 1 (C.15)

•'o

4k Г r2M+2 Gs(r, t) = <(Ar(v)(0)2M> M = 1, 2, 3, 4 (C.16)

Jo

Using the properties of HeN(x) given above and a great deal of algebra, one obtains Rahman’s32 expressions for

f>o(<)0) = 1 (C.17)

62<v>(f) = (,„<“>(<) = 0 (C.18)

f>6<v)(0 = ^«2<v|(0 (C.19)

b8(<l(0 = 3^ C«3<v)(0 - 4c<2(vl(r)] (C.20)

where

bt o{’K>) = 5^5 [«4(v)(0 - Wvl(0 + 10a2(v)(t)] (C.21) , <(Ar<v)(0)2~> * ()_Ся<(Лг'''>(0)2>" ( )

and CN = 1 x 3 x 5 x • • • (IN + l)/3*.

224

В. J. BERNE AND G. D. HARP

The intermediate scattering function for an isotropic system is given by

djr z»00

Fs(v)(k, 0 = t r sin Сkr]Gs(v)(r, t) dr (C.23)

к Jo

Using Eq. (C.14) for G(v)(r, t) and the relation32

f xe{~X2ll) He2tf(*)sin Bx dx = (— l)N+iB[2NB2N~2 — B2iV]e(-B2/2)

0 (C.24)

One obtains Rahman and Nijboer’s95 expression for Fs(v)(k, t):

Fs<v>(/c, 0 = e<-'2> £ a2N(v)(t)y2N (C.25)

N = 0

where

/с2<(Дг<у>(0)2>

У = - ~ -6 (C.26)

ao(v)(0 = 1 (C.27)

a2(v)(0 = О (C.28)

a4M(r) = 2466M(r) = ?^W (C.29)

^6<v)(0 = - [8*>6(v,(<) + 6468<v>(0] = ^ [a3M(0 - За^СО] (C.30) af\t) = 16[fte<v>(0 + 10i>10(,|(0] = ~ - 4a3(v)(0 + ба^СО] (C.31) я,о(v>(<) = -32i>,„(v|(0 = - jy [a4(v)(0 - 5ot3<v)(f) + 10a2<v>(r)] (C.32)

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