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Advances in chemical physics - Prigogine I.

Prigogine I., Rice S.A. Advances in chemical physics - Advision, 1970. - 167 p.
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as
y(0 = 1 - ; t2 + 77 f4 (B.3)
1 ((d)2) 1 <(a)2>
2 <a2> 4! <a2>
(2) jO0 is also an even function and may be expanded for short times as
K(t)= «*)2> | 1 <a2> + 2
<(a2)>
<a >
<(&)2)
<a2>
Г2 (B.4)
The important point to note here is that the 2nd moment of Ky(t) depends on the 2nd and 4th moments of ^(r). The 2nd moments of each of the three previously mentioned autocorrelation functions may be calculated from ensemble averages of appropriate functions of the positions, velocities, and accelerations created in the dynamics calculations. Likewise, the 4th moment of the dipolar autocorrelation function may also be calculated in this manner. However the 4th moments of the velocity and angular momentum correlation functions depend on the derivative with respect to time of the force and torque acting on a molecule and, hence, cannot be evaluated directly from the primary dynamics information. Therefore, these moments must be calculated in another manner before Eq. (B.3) may be used.
Ky from y(t)
Consider first the problem of developing Ky(t) from .y(0- y(t) from the molecular dynamics calculations is tabulated for the equally spaced times
ti = iAt / = 0,..., 499 (B.5)
where t = 0.05 with time in units of 10“13 s. Therefore, it is necessary to determine Ky(tt) i = 0,499 from y(tt) i = 0,..., 499. It is more advantageous in terms of stability to do this by considering the first derivative of Eq. (B.l) rather than Eq. (B.l) itself:
С'К,(()% (h-ndf i = 0..... 499 (B.6)
at
Approximating the integral on the right-hand side of Eq. (B.6) by a closed quadrature formula89 such as the trapezoidal rule, one obtains.
ON THE CALCULATION OF TIME CORRELATION FUNCTIONS
219
where соj is the weight assigned to the /th point of the integrand, соj will depend on the particular formula used to perform the integration. In this form the right-hand side of Eq. (B.7) depends only on values of Ky(tj) for tj < because
<»,ад^«о) = 0 (B.8)
However, in order to actually use Eq. (B.7) one needs:
(1) dy/dt |f( and d2y/dt2 |fi for i = 1,..., 499
(2) Accurate starting values of Ky{t), i.e.,
Ky(t0), ОД, Ky(t2), Ky(t3)
(3) A convenient and accurate quadrature formula.
The derivatives of y{t) were obtained by two different methods: one used for short times and the other used for long times. For short times, 0 < ti < tM,y(t) was approximated by Eq. (B.3). If the 4th moment of j(0 was not known, then y(t) was first fit via least squares to Eq. (B.3) to obtain <(a2))/<a2). 4th moments of the velocity and angular momentum autocorrelation functions determined this way are tabulated in Table IV. The error quoted for each of these values of <(a)2>/<a2> is the statistical error from the least square fit which amounts to ~10%. The number of points used in the least squares process in general depends on how fast the autocorrelation function changes around t0. For the velocity and angular momentum autocorrelation functions, 8 points were used to determine <(a)2)/<a2>. dyjdt\ti and d2yldt2\ti for t0, ..., r4 were then calculated by evaluating Eq. (B.3) at the points t0,..., t4. For long times, t5 < t% < r499, y(t) was assumed to be represented by the interpolating polynomial for
y(t)--89
y’M = t “x't" (B-9)
N= 0
At each point tif t5 < tt< t491, the coefficients aNl were determined such that
У*М = ) j = i ~ 3, i — 2,..., i + 3 (B.10)
In other words, the exact form of the interpolating polynomials varied from point to point. dyldt\tt and dzyjdt2\tt for ts<ti<! t499 were then calculated by evaluating the first and second derivatives of the appropriate interpolating polynomial, y*t(t).
220
В. J. BERNE AND G. D. HARP
o) or <(&)2>Ka2> was calculated directly from appropriate ensemble averages of the molecular dynamics information (see Tables II, III, and V). Kyfa), Ky(t2), and Ky(t3) were determined by using Day’s90 starting method applied to Eq. (B.3). After applying Day’s method and exploiting the odd property of dy/dt, one obtains three linear equations involving Ky(tt), Ky(t2), and Ky(tz) which can easily be solved:
d2y
At
Ky(,i) dt2 L 24
V'2)=--t4
d2y
~dt‘
d2y
w>--£
At
t2~ т
3 At ti 8
Ky(h)
dy
dt
,, + Krlh)Yt
(B.ll) (B.12) ]
11
(B.13)
Day’s method essentially replaces the integral in Eq. (B.6) by a different quadrature formula at each of the times tu t2, and t3. Each of these quadrature formulas approximates its appropriate integral with an error which is Q(At5). Therefore, this method gives very accurate starting values for Ky{t2), and Ky(t3).
For Ky(ti), t4< tt< tA99, the integral in Eq. (B.6) was approximated by the Gregory formula.90,91
1 ("к, (0 ^ - (') dt = iKM Y,\ + IВД T, I
ANo at dt\tt N=i dt |f(_N
1 Г__ , . dy
dy
i— 1
dy
+
12
1
24
Wi)-h
Ky(t2)
-Ky(t0)if \ +Ky(ti-l)^:
tt-t dt L
+ км
ti-i
dt
dy
dt
tiJ
2 X,(*,-!>
dy
dt
The Gregory formula used here has the advantage that it requires no special considerations as to whether or not the integral involves an odd or even number of points, in contrast to other integration formulas such as the composite Simpson’s rule.
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