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(b) Molecular interactions enter in the tA term which is positive. Therefore, interactions will delay the decay of the gas phase function.
These points are all illustrated in Figures 6, 7, and 8. That is, the dipolar correlation functions all have the same initial curvature, and the functions from simulations using strongly angular dependent potentials decay more slowly than the gas phase function. The memory functions for the Stockmayer and modified Stockmayer simulations are presented in Figure 19; the angular momentum autocorrelation function from this latter simulation is also shown. The memory for the gas phase or Stockmayer dipolar function decays monotonically and is positive for 0 < t < 10-12 s. On the other hand, the modified Stockmayer memory decays in an entirely different fashion. It goes negative in ~2 x 10“13 s and is approximately equal to the angular momentum autocorrelation function for this simulation. This is a very important observation because it presents the possibility of obtaining approximate angular momentum correlation functions from infrared bandshape studies. Looking closer at Eq. (211) we see that
KMIKD(0) = l-j + • • • (224)
This function’s decay will be dominated initially at least by molecular interactions provided ((NZ}I2/</2)2) < 1. This is actually not a difficult condition to satisfy. In the modified Stockmayer simulation this ratio is ~9.8 and experimentally this ratio is ~4.5 for liquid carbon monoxide at 78°K.59 There are probably other physical systems for which this ratio is much larger. In the event that this criteria is satisfied KD(t)/KD(0) /V (0 to terms in t2 at least. In the case of the modified Stockmayer simulation we have just seen that this approximation is actually valid throughout the interesting negative region of Aj(t). Hopefully, this approximation will also be valid in real systems, and the interesting negative region of Aj(t) can be verified experimentally from infrared bandshape studies.
For completeness consider also the correlation function <Р2(ц(0) * ц(0))
< P, ( u (o) • u (t )) >
t (in I0_l3s)
Fig. 7. The autocorrelation function <Л(ц(0) • |л(/>» and <P2(|*(0) • f*(0)> for CO from the modified Stockmayer simulation.
t (in I0",3s)
Fig. 8. The autocorrelation function <Pi(pi(0) • [л(0)> from (a) the Stockmayer simulation of CO with a dipole moment of 1.172 Debye, and (6) the Lennard-Jones plus quadrupole-quadrupole simulation of N2.
ON THE CALCULATION OF TIME CORRELATION FUNCTIONS
which can also be obtained from the Fourier inversion of rotation-vibra-tion Raman bandshapes.15 The short-time expansion of this function is60
3t2<J2> , /<У4> , <N2}'
/<У4> <iV2>\ \ 214 + 8/2 /'
From Eq. (225) it is seen that this function will: (a) have a time dependence in the absence of interactions, (b) initially decay faster than <ц(0) • ц(0>, and (с) decay slower in the presence of interactions than in their absence. The behavior of this function in the gas phase is given by
<Р2(ц(0) • |1(0)>g = I /“ COS (“)P(J) dJ+ i (226)
In the limit t -*■ oo this equation goes to 1/4 whereas in a system with interactions <Р2(ц(0) • ц(0)> Soes to zero in this limit. These characteristics are all illustrated in Figures 6 and 7 where the results from the Stockmayer and modified Stockmayer simulations and from a system of gas phase molecules (Eq. 226) are presented.
Before discussing other results it is informative to first consider some correlation and memory functions obtained from a few simple models of rotational and translational motion in liquids. One might expect a fluid molecule to behave in some respects like a Brownian particle. That is, its actual motion is very erratic due to the rapidly varying forces and torques that other molecules exert on it. To a first approximation its motion might then be governed by the Langevin equations for a Brownian particle:61
M^+Crv = F(0 (227)
^+C*J = N(0 (228)
where F(f) and N(0 are small stochastic forces and torques whose time averages are zero, and £r and C,R are translational and rotational friction coefficients. This is an oversimplification of the actual motion of a molecule surrounded by other molecules of similar mass but nevertheless is an interesting situation to consider. The linear and angular momentum autocorrelation and memory functions obtained from the solutions to the Langevin are simply:
v|,(t) = (229)
X*(<) = ^5(0 (230)
Aj(t) = (231)
KAt) = Um (232)
В. J. BERNE AND G. D. HARP
Debye62 showed that for a Brownian particle whose molecular orientation changes through small erratic angular displacements, <ц(0) • ц(0> and <Р2(ц(0) ‘ Ц(0)) are also exponentials. In particular under these conditions these functions are given by
where DR is the rotational diffusion coefficient. There are two points of interest here:
(1) All of the autocorrelation functions are exponentials and, as such, are always >0