# Advances in chemical physics - Prigogine I.

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18ЯГ

<(AR(0) Уd —

M(oD2

1 -

sinco^

(On t

(184)

136

В. J. BERNE AND G. D. HARP

where co^ = KQDlh. This function goes as (3KTjM)t2 for short times and approaches the asymptotic value of \SKT/MmD2 in a damped oscillatory fashion. It reaches its maximum value of 22KTlMaD2 at t~4A/G)D. Therefore, <(AR(0)2>d behaves at least qualitatively like Rahman’s function for solid argon. The characteristic temperatures for solid argon and nitrogen are 85 and 68°K,55 respectively. It follows that <(AR(f))2>D in solid argon at 84°K should reach its maximum value of /v 0.3 A2 in ~4x 10"13 s. These values are approximately a factor of 1.5 smaller than what Rahman actually observed. For solid nitrogen at 68°K, <(AR(f))2>B should reach its maximum value of ~0.6 A2 in ~5 x 10“13 s. Since we have not seen the mean square center of mass displacements approach any asymptotic values over a time range five times larger than this value, we conclude again that we are dealing with liquids.

G. Equilibrium Properties

Tables II, III, and VI contain a few equilibrium properties for the 4

TABLE VIе Equilibrium Properties

Numerical value of friction coefficients (x 10-12 s-1)

Corresponding

Memory function friction coefficient у Y'

Gaussian (I)"- 5.98 14.34

Delta function 0”' 5.98 14.34

Lorentzian Ш V- 10.60 25.41

(a) 2 " In 2 8.10 19.43

Exponential

(b) 4/2' In 2 1/2 V- 9.73 16.38

Experimental® 9.38 17.54

“ [A is calculated from moments given in Nijboer and Rahman to be 4.77 x 1012 s-1, [Xj from moments given in Table V.

b The experimental friction coefficient comes from Nijboer and Rahman.95

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

137

systems we studied. The data is presented in the form of the mean value of a given property averaged over 600 blocks plus or minus the variance of that property within the 600 blocks. For example, the mean square force is given by (see Eq. 5):

1 600

<^> = бооД<^> (185)

use)

and the variance of the mean square force, oF, is then given by:

2\i2n

[<*) > - <**>]

1/2

(187)

The kinetic temperatures, TK,TR, and T T, are mainly given for completeness. The fact that these three temperatures are not equal indicates that our heuristic method for forcing each system to equilibrate at a preselected temperature was not 100% effective. The largest differences between these temperatures occur in the Stockmayer simulation of CO with ц = 1.172 Debyes and in the Lennard-Jones etc., simulation of N2-

The mean square force is one test of the validity of the pair potentials used in the dynamics calculations. The calculated mean square forces for the four simulations are not very different: they only vary between ~9 x 10“11 and ~10 x 10"11 (dynes)2. The experimental values51 of the mean square force in solid CO at 68°K and in liquid CO at 77.5°K are ~15 x 10-11 and ~ 14 x 10“11 (dynes)2, respectively. Therefore, our mean square forces are ~30% too small.

The mean square torque is another test of the pair potentials used. The calculated mean square torques are very potential dependent: they range from ~7 x 10~31 to ~36 x 10-28 (dyne-cm)2. The experimental values51 of the mean square torque in solid CO at 68°K and in liquid CO at 77.5°K are ~19 x 10-28 and ~21 x 10-28 (dyne-cm)2, respectively. Therefore, the Stockmayer potential clearly does not represent the noncentral forces in liquid CO, i.e., this potential is much too weak. On the other hand, the noncentral part of the modified Stockmayer potential is too strong. However, as pointed out previously, this problem can easily be solved by using a smaller quadrupole moment. The mean square torques from the other two potentials agree quite favorably with the experimental values. We conclude from the above that the quadrupole-quadrupole interaction can easily account for observed mean square torques in liquid CO.

138

В. J. BERNE AND G. D. HARP

A further test of the pair potentials used is the translational diffusion coefficients, D. The coefficients in Tables II, III, and IV were all evaluated using the Einstein-Kubo relation:

<V2> г00

D = (188)

5 J о

The experimental value52 of D in liquid CO at 69°K is (2.25 + .1) x 10“5 cm2/s. The calculated values of D for the Stockmayer and modified Stockmayer simulations of CO are 2.39 x 10“5 and 1.88 x 10"5 cm2/s, respectively. Therefore, the diffusion coefficients from these two simulations agree fairly well with the experiment.

The average central or Lennard-Jones potential energy per molecule, <Fc>/iV, and the average total potential energy per molecule, <FT)/7V, are also given in Tables I, II, and III. The noncentral part of the Stockmayer potential contributes nothing to this system’s total potential energy. On the other hand, the noncentral parts of the other three potentials contribute from 22 to 38 % to their system’s total potential energies.

H. The Classical Limit

In the previous section it was shown how classical many-body systems can be studied by computer experiments. Actual laboratory experiments probe real systems which are, strictly speaking, entirely quantum-mechanical in nature. What, then, is the relationship between the classical and quantum-mechanical time-correlation function of the dynamical variable To expedite this discussion consider the one sided function <{/+,(0)L/j(f)>. This correlation function is in general complex with real part ф',(0 and imaginary part ф"К0 so that

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