6 TABLE II: Zero frequency, zero wave-vector shear viscosity and fitted parameter values for monatomic and diatomic systems FIG. 1: ˜ η ( k$_{y}$ ) versus k$_{y}$ for chlorine calculated using atomic and molecular formalisms ( ρ$_{a}$ = 1 . 088, T = 0 . 97, N$_{a}$ = 1728). statistical uncertainty in both atomic and molecular for- malisms. It has been shown previously that numerous one pa- rameter functions failed to capture the behaviour of the reciprocal space kernel data [20]. We therefore present the best fits with two or more fitting parameters. We have identified two functional forms that fit the data well: We present the best fits of the data to (i) a two-term Gaussian function with freely estimated amplitudes (i.e. unconstrained fitting) termed as ˜ η$_{G}$$_{2}$ , (ii) to a two-term Gaussian function with interdependent amplitudes (i.e. constrained fitting ∑ N$_{G}$ j A$_{j}$ = 1) given by Hansen et al. [20] and termed as ˜ η$_{G}$$_{2}$$_{H}$ , (iii) to a four-term Gaus- sian function with freely estimated amplitudes, termed as ˜ η$_{G}$$_{4}$ and (iv) to the Lorentzian type function, Eq. (28). In order to measure the magnitude of the residuals we use the residual standard deviation defined as s$_{r}$ = √∑ n$_{s}$ n $_{=1}$r$^{2}$/ ( n$_{s}$ − n$_{p}$ ) where n$_{s}$ is the number of data points, n$_{p}$ is the number of fitting parameters, and r is the residual [48]. After an iterative curve fitting proce- dure the accurate estimation of η$_{0}$ was kept fixed allowing all other parameters in Eqs. (27) and (28) to be used as fitting parameters. In Table II we have listed the fitting parameters for monatomic and diatomic molecular fluids and compared to the previous results where possible. and a Lorentzian type function an N$_{G}$ term Gaussian function ˜ η$_{G}$ ( k$_{y}$ ) = η$_{0}$ N$_{G}$ ∑ j A$_{j}$ exp( − k 2 $_{y}$/ 2 σ 2 $_{j}$) A$_{j}$, σ$_{j}$ ∈ R$_{+}$ (27) ˜ η$_{L}$ ( k$_{y}$ ) = η$_{0}$ 1 + α | k$_{y}$ | β α, β ∈ R$_{+}$ , (28) WCA WCA WCA LJ 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 System size N$_{a}$ 2048 1728 η 0 0 . 265(0 . 273[20]) 0 . 392 2 . 290 2 . 810 2 . 650 6 . 889 2-term Gaussian , A 2 = 1 − A 1 Eq. (27) A 0 . 189(0 . 440[20]) 0 . 309 0 . 155 0 . 093 0 . 107 0 . 407 σ 1 12 . 48(4 . 750[20]) 6 . 916 8 . 122 10 . 04 9 . 088 5 . 377 σ 2 2 . 116(1 . 376[20]) 1 . 835 2 . 592 2 . 778 2 . 759 1 . 236 s$_{r}$ 0 . 007(0 . 005[20]) 0 . 013 0 . 044 0 . 021 0 . 027 0 . 082 2-term Gaussian Eq. (27) A 1 0 . 792 0 . 687 0 . 874 0 . 907 0 . 892 0 . 592 A 2 0 . 174 0 . 254 0 . 155 0 . 094 0 . 106 0 . 407 σ 1 2 . 245 2 . 113 2 . 592 2 . 776 2 . 765 1 . 237 σ 2 13 . 36 7 . 745 8 . 124 10 . 02 9 . 127 5 . 377 s$_{r}$ 0 . 007 0 . 011 0 . 035 0 . 022 0 . 031 0 . 081 4-term Gaussian Eq. (27) A 1 0 . 432 0 . 566 0 . 778 0 . 689 0 . 868 0 . 398 A 2 0 . 394 0 . 248 0 . 118 0 . 190 0 . 047 0 . 538 A 3 0 . 120 0 . 138 0 . 088 0 . 114 0 . 089 0 . 055 A 4 0 . 056 0 . 047 0 . 017 0 . 017 0 . 020 0 . 008 σ 1 3 . 228 2 . 826 2 . 950 2 . 709 2 . 814 4 . 355 σ 2 1 . 261 0 . 821 0 . 651 2 . 709 0 . 145 1 . 155 σ 3 8 . 165 6 . 973 8 . 496 7 . 037 7 . 628 10 . 46 σ 4 15 . 19 14 . 74 23 . 66 24 . 94 19 . 99 37 . 56 s$_{r}$ 0 . 008 0 . 002 0 . 012 0 . 014 0 . 024 0 . 018 Lorentzian-type Eq. (28) α 0 . 198(0 . 180[20]) 0 . 170 0 . 062 0 . 041 0 . 043 0 . 239 β 1 . 562(1 . 662[20]) 1 . 715 2 . 326 2 . 602 2 . 572 1 . 667 s$_{r}$ 0 . 002(0 . 005[20]) 0 . 005 0 . 042 0 . 018 0 . 042 0 . 016  0  1  2  3  4  5  6  7  8  0  5  10  15  20  25  30  35  40 ~ (k y ) k$_{y}$ Chlorine Atomic formalism Molecular formalism