4 our simulations. We also note that the number of degrees of freedom in the simulated system has not been taken into account in Eq. (15), therefore we use the simulated value in our calculations to ensure numerical consistency of the computed properties. The expression for the wave-vector and frequency de- pendent viscosity in terms of the autocorrelation function of the shear stress ˜ N ( k$_{y}$, ω ) takes the form: where In the zero wave-vector limit, a generalization of the Green-Kubo expression for the shear viscosity allows the transverse momentum flux to be in an arbitrary direction rather than along a coordinate axis and can be written in terms of the stress tensor as [31, 32]: where the os superscript denotes the traceless symmet- ric part of the stress tensor P $^{os}$( t ) = 1 $_{2}$[ P ( t ) + P $^{T}$( t )] − 1 $_{3}$tr [ P ( t )] 1 and V is the simulation volume. In an isotropic fluid, because the tensor, P $^{os}$, appearing in Eq. (19) is traceless and symmetric, the shear viscosity is also traceless and symmetric. Consequently, the shear viscosity can be expressed in terms of the invariant I of the shear viscosity tensor as η = I/ 10. In the case ω → 0 and k$_{y}$ → 0, relation (17) reduces to the Green-Kubo formula [31]. All the non-zero wave- vector integrals, Eq. (18), converge to zero [13] there- fore the relation in Eq. (13) must be used when non- zero wave-vector viscosities are calculated. By comput- ing the integrals one can computationally verify the zero values of the zero-frequency limit of the function ˜ N ( k , ω ) and thus demonstrate why neither substitution of ω = 0 into Eq.(17) nor evaluation of Eq. (18) at non-zero wave- vector yields the zero-frequency wave-vector dependent viscosity. We use the Edberg, Evans, and Morriss algorithm [33–35] with an improved cell neighbour list construc- tion algorithm [36] to perform equilibrium simulations at constant N , V , T$_{M}$ . N is either the number of atoms or molecules and T$_{M}$ is the molecular temperature as defined by Eq. (26). For the atomic fluids studied in this work, the atoms interact via a Lennard-Jones (LJ) or Weeks-Chandler-Andersen (WCA) potential energy function [37]. The LJ atoms have an interaction poten- tial truncated at r$_{c}$ = 2 . 5 σ and WCA atoms have an where i denotes atom i . r$_{i}$ is the position, p$_{i}$ is the momentum and m$_{i}$ is the mass of the designated atom. F$_{i}$ is the force on atom i due to other atoms and ζ$_{A}$ is the atomic thermostat multiplier. The thermostat multiplier is chosen so as to fix the kinetic temperature. We use the value of ζ$_{A}$ that results from the application of Gauss’ principle of least constraint to the imposition of constant kinetic temperature: carried out in a cubic box with periodic boundary condi- tions. The fifth-order Gear predictor corrector algorithm [44, 45] with time step δt = 0 . 001 was employed to solve the equations of motion. The equations of motion can be written for a monatomic fluid in the isokinetic ensemble (at equilibrium) as [13]: Our diatomic model of liquid chlorine is similar the the one used by Edberg et al. [38], Hounkonnou et al. [39], Travis et al. [40, 41] and more recently by Matin et al. [42, 43] to allow a direct comparison of our results with previous work. This model represents chlorine as a diatomic LJ molecule with r$_{c}$ = 2 . 5 σ and a fixed bond length of 0 . 63 σ . For an adequate representation of the properties of chlorine the LJ parameters are: σ = 3 . 332 ˚ A and ϵ/k$_{B}$ = 178 . 3K. Liquid chlorine systems of 108 and 864 molecules are studied at a reduced site number den- sity of ρ$_{a}$ = 1 . 088 and a reduced molecular temperature T$_{M}$ = 0 . 970. We summarize the most important simu- lation parameters in Table I. All our simulations were where r$_{ij}$ is the interatomic separation, ϵ is the poten- tial well depth, and σ is the value of r$_{ij}$ at which the unshifted potential is zero. The shift Φ$_{c}$ is the value of the unshifted potential at the cutoff r$_{ij}$ = r$_{c}$ , and is in- troduced to eliminate the discontinuity in the potential energy. At distances greater than the cutoff distance r$_{c}$ , the potential is zero. interaction potential truncated at r$_{c}$ = 2 1 / $^{6}$σ [37]. In general: ˜ η ( k$_{y}$, ω ) = ˜ N ( k$_{y}$, ω ) C ( k$_{y}$, t = 0) /ρk$_{B}$T − k 2 ˜ N ( k$_{y}$, ω ) /iωρ (17) ˜ N ( k$_{y}$, ω ) = 1 V k$_{B}$T L [〈 P$_{xy}$ ( k$_{y}$, t ) P$_{xy}$ ( k$_{y}$, 0) 〉] . (18) η = V 10 k$_{B}$T ∫ ∞ 0 〈 P $^{os}$( t ) : P $^{os}$(0) 〉 dt (19) ˙r$_{i}$ = p$_{i}$ m$_{i}$ , ˙p$_{i}$ = F$_{i}$ − ζ$_{A}$ p$_{i}$ (21) ζ$_{A}$ = ∑ N i $_{=1}$F$_{i}$ · p$_{i}$ N i $_{=1}$p 2 i . (22) III. SIMULATION DETAILS Φ$_{ij}$( r$_{ij}$ ) =    $^{4}$$^{ϵ}$ [ ( σ r$_{ij}$ ) 12 − ( σ r$_{ij}$ ) 6 ] − Φ$_{c}$ , r$_{ij}$ < r$_{c}$ 0 , r$_{ij}$ ≥ r$_{c}$ (20) WCA LJ Chlorine Site number density, ρ$_{a}$ 0 . 375 , 0 . 480 , 0 . 840 0 . 840 1 . 088 Temperature, T 0 . 765 , 1 . 0 0 . 765 , 1 . 0 0 . 97 Number of atoms, N$_{a}$ 108 , 2048 , 6912 108 , 2048 , 6912 216 , 1728 Number of sites, N$_{s}$ 1 1 2 Bond length, l - - 0 . 63 LJ cutoff, r$_{c}$ 2 1 / 6 2 . 5 2 . 5 TABLE I: Simulation details