7 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 ~$^{ }$ (k y ) /  ~  (0) k$_{y}$ Chlorine data G$_{2H}$ G$_{4}$ L (a) 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 ~$^{ }$ (k y ) /  ~  (0) k$_{y}$ LJ data G$_{2H}$ G$_{4}$ L (b) -0.06 -0.04 -0.02 0 0.02 0.04 0 5 10 15 20 25 30 35 40 ~ (k y ) /  ~$^{ }$ (0) k$_{y}$ Chlorine G$_{2H}$ - data G$_{4}$ - data L - data (c) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0 5 10 15 20 25 30 35 40 ~ (k y ) /  ~$^{ }$ (0) k$_{y}$ Chlorine G$_{2H}$ -  G$_{4}$ L -  G$_{4}$ L -  G$_{2H}$ (d) FIG. 2: Normalized kernel data, best fit of Eq. (27) with N$_{G}$ = 2 and 4, and Eq. (28) and difference between the fits: (a) Best fits to normalized kernel for chlorine fluid ( ρ$_{a}$ = 1 . 088, T = 0 . 97, N$_{a}$ = 1728); (b) Best fits to normalized kernel for LJ fluid ( ρ$_{a}$ = 0 . 840, T = 1 . 0, N$_{a}$ = 2048); (c) Differences between the kernels and simulation data (a); (d) Differences between the fitted kernels (b). A useful check of the fitting can be performed by calcu- lating the total Gaussian amplitudes which should con- verge to the value of 1, Table III. The reciprocal space results presented in figure 2 for LJ and chlorine systems demonstrate that the four-term Gaussian function fits the data much better than the other two forms with a difference between the data and the fit of less than 0.5%, see figure 2(c). The two- term Gaussian ˜ η$_{G}$$_{2}$$_{H}$ fits the kernel data better than the Lorentzian-type function in the low- k$_{y}$ region, figure 2(a), which suggests a more Gaussian-like behaviour in the low- k$_{y}$ region, a fact previously observed by Hansen et. al [20] for atomic fluids modeled with WCA potentials. Nevertheless the difference between the two-term Gaus- sian fit and data is less than 2% which still makes the ˜ η$_{G}$$_{2}$$_{H}$ a good analytical three parameter approximation of the reciprocal space viscosity kernel. The maximum difference between the Lorentzian-type fit and Gaussian fits are around 4% while the maximum difference between the Gaussian fits is about 2%, see figure 2(d). Essen- Figure 3(a) shows the kernel data for a WCA fluid at two different densities along with two sets of data pub- lished previously by Hansen et al. [20]. EMD is the set obtained from an equilibrium MD simulation at the same state point ( ρ$_{a}$ = 0 . 375, T = 0 . 765) and NEMD is the set obtained from a nonequilibrium MD simulation based on the sinusoidal transverse force (STF) method. An excellent agreement between both sets of data was found. Figure 3(b) shows the normalized fit to Eq. (28). The normalized kernels, figure 3(b), show a similar be- havior for k$_{n}$ ≤ 4. Though the higher density kernel is slightly lower for k$_{n}$ ≥ 4 they show a similar limiting be- havior. This effect was not seen by Hansen et al. due to lack of data for high wave vectors. Figure 3(d) indi- cates that despite the difference between the interaction potentials, the results for LJ and WCA fluids are very close. This confirms that transport is dominated by re- pulsive interactions, rather than attractive. The sharper kernel for the diatomic system, figure 3(d), suggests a more Lorentzian-type behavior in the low wave-vector region. It is also important to mention that even though tially, this suggests that, when computing the real space kernels, the four-term Gaussian functional form is to be trusted. It is obvious that eight parameters in the four- term Gaussian make its use less convenient, but on the other hand the Gaussian function can analytically be in- verse Fourier transformed while the inverse Fourier trans- form of the Lorentzian-type function can only be evalu- ated numerically for general values of β . 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 . 0 0 . 97 2-term Gaussian ∑ 2 j $_{=1}$A 0 . 966 0 . 941 1 . 029 1 . 001 0 . 998 0 . 999 4-term Gaussian 4 j $_{=1}$A 1 . 002 0 . 999 1 . 001 1 . 010 1 . 024 0 . 999 TABLE III: Total amplitude for Gaussian functional form