or due to improper structural factor arises. A four-term Gaussian only shows a slightly narrower kernels which suggests a need for a more comprehensive structural nor- malization. 11 The wave-vector dependent viscosity of monatomic Lennard-Jones, monatomic Weeks-Chandler-Andersen and diatomic (liquid chlorine) fluids over a large wave- vector range and for a variety of state points has been computed. The equilibrium molecular dynamics calcula- tion involved the evaluation of the transverse momentum density and shear stress autocorrelation functions in both atomic and molecular hydrodynamic representations for molecular fluids. The main results can be summarized as follows: (i) For monatomic fluids the shape of the normalized viscosity kernel in reciprocal space in the low wave-vector region is the same for a whole range of densities consid- ered here. Though the normalized reciprocal kernels in- significantly decreases with the density they show a sim- ilar limiting behaviour at high k$_{y}$ values. For the LJ potential compared to a WCA potential we find higher viscosities in the low wave-vector region but the normal- ized shape of the kernels are almost identical. (ii) For liquid chlorine, the wave-vector dependent vis- cosity shows a similar behaviour in both atomic and molecular formalisms within statistical uncertainty. (iii) While a relatively simple Lorentzian-type function fits the atomic and diatomic data well over the entire range of k$_{y}$ at all the state points it is not possible to analytically inverse Fourier transform it to the real space domain. Therefore one may consider an expansion up to a four-term Gaussian which gives better accuracy in reciprocal space compared to the Lorentzian-type func- tion. Our analysis of the high k$_{y}$ regime reveals that the two-term equally weighted Gaussian functional form is inaccurate in predicting the real space kernels whilst the unequally weighted Gaussian only slightly improves the fit. (iv) The overall conclusion is that the real space viscos- ity kernel for chlorine has a width of roughly 4-6 atomic diameters while for monatomic systems at high densities the width is about 3-5 atomic diameters and 2-4 atomic diameters at low densities. This means that generalized hydrodynamics must be used in predicting the flow prop- erties of molecular fluids on length scales where the gra- dient in the strain rate varies significantly on these scales. Consequently a nonlocal constitutive equations should be invoked for a complete description of flows at atomic and molecular scales under such conditions. Finally, our results for molecular fluids should also pro- vide a good test for more complex molecular systems and the methodology can easily be used for instance in chain- like molecules. The normalized kernels with respect to η ( y = 0) and normalization factor ξ$_{g}$ are shown in figures 8(a) and 8(b). While generally the width of unnormalized ker- nels increases as we increase the density (figures 6(a) and 6(b)), the width of the normalized kernels of WCA flu- ids decreases marginally as we increase the density from 0 . 376 (continuous line) to 0 . 840 (short-dashed line). The LJ system shows a slightly narrower kernel (dotted line) compared to the WCA system at the same state point (short-dashed line). Though the kernels obtained from both functional forms are quite close to each other (al- most identical for values of y of about half of the atomic or molecular diameters, i.e. y = 0 . 5 σ ), we can see in figure 8(a) and 8(b) that the structural normalization did not completely remove the discrepancy between the normalized kernels of the WCA system at different densi- ties, and the normalized kernels of WCA, LJ and chlorine systems, for values higher than y = σ . If we recall that figure 8(b) is based on a Lorentzian-type fit, a further question as to whether the kernel differences are due to numerical analysis, i.e. the choice of the fitting function point ρ$_{a}$ = 0 . 375, T = 0 . 756 to 0 . 730 at ρ$_{a}$ = 0 . 480, T = 1 . 0 and only slightly increases as we increase the cutoff distance, i.e. switch from WCA system to a LJ system at the same state point. ξ$_{g}$ for chlorine at state point ρ$_{a}$ = 1 . 088, T = 0 . 97 was found to be 0 . 585. FIG. 7: Pair-distance correlation function g ( r ) and normaliza- tion factors ξ$_{g}$ , Eq. (32): WCA [(a) ρ$_{a}$ = 0 . 375, (b) ρ$_{a}$ = 0 . 480 both at T = 0 . 765], (c) WCA ( ρ$_{a}$ = 0 . 840, T = 1 . 0); LJ (d) ( ρ$_{a}$ = 0 . 840, T = 1 . 0); chlorine (e) ( ρ$_{a}$ = 1 . 088, T = 0 . 97). For clarity, the RDFs are shifted upwards by 3 units. V. CONCLUSION  0  2  4  6  8  10  12  14  0  1  2  3  4  5 g(r) r (a) WCA  = 0.375,  $_{g}$=0.605 (b) WCA  = 0.480,  $_{g}$=0.628 (c) WCA  = 0.840,  $_{g}$=0.730 (d) LJ  = 0.840,  $_{g}$=0.739 (e) Chlorine   = 1.088,  $_{g}$=0.585