# Advances in chemical physics - Prigogine I.

**Download**(direct link)

**:**

**24**> 25 26 27 28 29 30 .. 51 >> Next

4. The Lennard-Jones plus Quadrupole-Quadrupole Simulation of Nitrogen

The intermolecular potential consists of the sum of Eqs. (176) and (179) and was done for 216 molecules.

130

В. J. BERNE AND G. D. HARP

Б. Summary and Discussion of Errors

Our original goal was to study liquids of carbon monoxide and nitrogen. Ideally this would involve solving the equations of motion for a macroscopic number of molecules (~1023). However, in practice we only considered ~512 molecules in an infinite periodic environment. Even this number strained both the storage capacity and computing ability of our IBM 7094. For example, the 6144 first-order differential equations for the modified Stockmayer simulation for 512 carbon monoxide molecules took 5.1 min of 7094 time/step or a total of 76.5 hr of 7094 time for the 900 steps of the equilibration and production phase of this calculation. The data reduction for this calculation took approximately another 75 hr of 7094 time.

We have tried to assess the effects of the finite number of molecules, or, equivalently, the periodic boundary effects by comparing the results of simulations done with 216 and 512 molecules. For equilibrium properties such <iV2> and <F2>, the primary effect of increasing the number of molecules is to reduce the measured variances of these quantities (see Tables II 'and III). We therefore feel that these quantities are within a few percent of

TABLE II

Equilibrium Properties from Modified Stockmayer Simulation of CO

216

(69.45 ± .43)°K (68.87 ± 3.38)°K (70.31 ± 3.91)°K (10.07 ± 1.05) x 10-11 (dyne)2 (35.32 ± 3.00) x 10"28 (dyne-cm)2 (28.14 ± 1.60) x 10-54 (g cm2/s)2 1.82 x 10-5 cm2/s

~ —8.2 x 10“14 erg ~ —11 x 10"14 erg

N 512

TK (67.43 ± 1.26)°K

Гг (66.35 ± 1.72)°K

TR (69.06 ± 2.47)°K

<F2> (9.460 ± .657) x 10"11 (dyne)2

<AT2> (35.74 ± 1.46) x 10“28 (dyne-cm)2

</2> (27.63 ± .991) x 10~54 (g cm2/s)2

D# 1.88 x 10"5 cm2/s

<VC>

(-8.21 ± .05) x 10-14 erg

(—11.34 ± .04) X 10-14 erg

thosre measured for an infinite number of molecules. For the correlation functions discussed here, the primary effect of increasing the number of molecules is to reduce fluctuations in these functions that occur for t% A x 10~13 s. This effect on the velocity autocorrelation function, i[/(0> for the Stockmayer simulation is illustrated in Figure 4. v|/(/) is defined by

= M (.80)

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

131

TABLE III

Equilibrium Properties from Stockmayer Simulation of CO

N 512 216

TK (69.18 ± .97)°K (67.86 ± 1.55)°K

TT (70.03 ± 1.60)°K (67.89 ± 2.54)°K

TR (67.91 ± .12)°K (67.80 ± .29)°K

<F2> (8.722 ± .541) x 10-“ (dyne)2 (8.696 ± .878) ± 10“11 (dyne)2

<./V2> (6.716 ± .144) x 10-31 (dyne-cm)2 (6.635 ± .434) x 10-31 (dyne-cm)2

<У2> (27.17 ± .04) x 10~54 (g cm2/s)2 (27.13 ± .12) x 10“54 (g cm2/s)2

£># 2.39 x 10"5 cm2/s 2.48 x 10“5 cm2/s

(-8.49 ± .03) x 10-14erg ~ -8.4 x 10"14erg

N < VT)

—— (-8.49 ± .03) x 10“14erg ~ -8.4 x 10"14erg N

Time t in 10 s

Fig. 4. The velocity autocorrelation functions from the Stockmayer simulation of CO using 216 and 512 molecules.

where V is the center of mass velocity of a molecule. This function will be discussed in greater detail shortly. Because of the boundary effect, we feel that the fine details of the correlation functions from simulations involving

132

В. J. BERNfc AND G. D. HARP

216 molecules and for times ^4 x 10“13 s should not be taken too seriously.

We have also tried to assess the effects of integrating Hamilton’s equations numerically. This is a rather difficult task since the exact solutions to these equations are not known. However, we can use the observed conservation of total energy and linear momentum as an indication that the equations are being integrated properly. For the Stockmayer and modified Stockmayer simulations the total energy and linear momentum were conserved to ~0.05 and ~ 0.0006%, respectively, over the 600 integration steps of the production phase of these calculations.

In comparing our systems to real liquids of carbon monoxide and nitrogen, we are assuming implicitly that these real liquids behave like classical systems of rigid rotors. That is, quantum effects are relatively small. The usual criteria that have to be satisfied for this to be true are:

(1) The De Broglie wavelength of a molecule must be small compared to the average distance between molecules, i.e., (h2/3MKT)ll2J(plM)113 < 1.

(2) Many rotational states must be occupied or, equivalently, the rotational energy spacing must be small with respect to KT, i.e., h2j2IKT < 1.

(3) The molecules must be predominantly in their ground state vibrational level, i.e., h(aecjKT < 1, where с is the velocity of light and (oe is the energy separation of successive levels in wave numbers. For carbon monoxide at68°Kwith p = 0.8558 g/cc and (oe = 2.170 x 103 cm-1,45 the above factors are ~2 x 10-1, ~5 x 10-2, and ~5 x 101, respectively. Therefore, to a first approximation real liquid carbon monoxide at this temperature and density behaves classically, and our comparisons will be justified.

**24**> 25 26 27 28 29 30 .. 51 >> Next