9 FIG. 4: Real space kernel of monatomic and diatomic fluids as predicted by Gaussian and Lorentzian fits of reciprocal space kernel data: (a) chlorine ( ρ$_{a}$ = 1 . 088, T = 0 . 97); (b) LJ ( ρ$_{a}$ = 0 . 840, T = 1 . 0); (c) Differences between the kernels shown in (a); (d) Differences between the kernels shown in (b). 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 (y) y Chlorine G$_{2H}$ G$_{4}$ L (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 (y) y LJ G$_{2H}$ G$_{4}$ L (b) -2 -1.5 -1 -0.5 0 0 1 2 3 4 5 (y) y Chlorine $_{G$_{2H}$}$ -  G$_{4}$ $_{L}$ -  G$_{4}$ $_{L}$ -  G$_{2H}$ (c) -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0 1 2 3 4 5 (y) y LJ $_{G$_{2H}$}$ -  G$_{4}$ $_{L}$ -  G$_{4}$ $_{L}$ -  G$_{2H}$ (d) fluids, figure 4(b). For monatomic systems at relatively low densities ( ρ$_{a}$ = 0 . 375 − 0 . 480), the real space kernels are affected considerably by the functional form chosen to fit the re- ciprocal space kernel, figure 5. For instance, the equally weighted two-term Gaussian function, figure 5(a), dis- torts the real space kernels and predicts a noticeably higher η ( y = 0) value. As we increase the density, the dis- crepancy between Gaussian functions, as well as between Gaussian and Lorentzian-type functions, only partially reduces, figure 5(b,d). The width of the kernel for WCA fluids at low density is roughly 2-4 atomic diameters. In figures 6(a) and 6(b) we compare the unnormalized kernel data in y space extracted from four-term Gaussian and Lorentzian-type functional forms for all the simu- lated systems. Despite the fact that the difference be- tween the reciprocal kernels is less than 4%, figure 2(d) (e.g. chlorine - dashed line), the kernels for the corre- sponding systems in real space look noticeably different (figure 6(a) and figure 6(b)), for all the systems (e.g. chlo- rine - dashed dotted line). However, the zero wave-vector viscosities obtained from both functional forms are very close, with less than 2% error. We can determine the zero wave-vector viscosities η$_{0}$ = η ( k = 0 , ω = 0) by integrating the real space kernel over y , and thus test our numerical analysis. The zero wave- vector viscosity η$_{0}$ obtained by a Gaussian function η$_{G}$ ( y ) It is of interest to discuss the real space viscosity ker- nels for monatomic and diatomic systems from a struc- tural point of view. For this purpose we define a struc- Since the general analytical expression for η$_{L}$ ( y ) does not exist [20] we evaluate the integral numerically and present the results from all functional forms in Table IV. A comparison of the viscosities in Table IV with the sim- ulated zero frequency zero wave-vector shear viscosities given in Table II shows an integration error of less than 3%. This confirms the accuracy of our numerical analysis techniques. WCA LJ Chlorine State Point ρ$_{a}$ 0 . 375 0 . 480 0 . 840 0 . 840 0 . 840 1 . 088 T 0 . 765 0 . 765 1 . 000 0 . 765 1 . 000 0 . 97 2-term Gaussian η 0 0 . 265 0 . 392 2 . 290 2 . 723 2 . 614 6 . 881 4-term Gaussian η 0 0 . 265 0 . 390 2 . 288 2 . 807 2 . 653 6 . 897 Lorentzian η 0 0 . 269 0 . 428 2 . 320 2 . 913 2 . 711 7 . 049 TABLE IV: Effective viscosities evaluated from η$_{G}$$_{4}$ ( y ), η$_{G}$$_{2}$$_{H}$ ( y ) and numerically from ˜ η$_{L}$ ( k ). The values can be compared to the zero frequency, zero wave-vector viscosities shown in Table II. is η$_{0}$ = ∞ ∫ −∞ η$_{G}$ ( y ) dy = η$_{0}$ √$_{2}$$_{π}$ ∞ ∫ −∞ { ∑ j A$_{j}$ σ$_{j}$ exp[ − ( σ$_{j}$y ) $^{2}$/ 2] } dy. (31)