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The atomic temperature T$_{A}$ for a system of N$_{a}$ atoms with no internal degrees of freedom is defined as 

Here angled brackets denote an ensemble average, d is the dimensionality of the atomic system, N$_{c}$ is the number of constraints on the system (including constraints for conserved quantities). 

The equations of motion (EOM) for a molecular fluid can be formulated in either atomic or molecular ver- sions. In fact the molecular versions of the homogeneous isothermal EOM with a molecular thermostat at equilib- rium are similar to atomic EOM with a molecular ther- mostat, provided that all of the relevant forces are in- cluded [29]. The thermostatted EOM for molecular sys- tems are given by [29] 

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where iα denotes site α on molecule i . r$_{iα}$ is the posi- tion, p$_{iα}$ is the momentum and m$_{iα}$ is the mass of the site. The force on a site is separated into two terms: F$_{iα}$ is the contribution due to the Lennard-Jones type interactions on site α of molecule i and F C iα is either the constraint force or the bonding force. ζ$_{M}$ is the molecular thermostat multiplier, given by 

where F$_{i}$ is the total force acting on molecule i and M$_{i}$ is the mass of molecule i . ζ$_{M}$ is derived from Gauss’ prin- ciple of least constraint and acts to keep the molecular center of mass kinetic temperature T$_{M}$ constant. Sev- eral algorithms are available for this purpose [47]. The molecular temperature T$_{M}$ is defined by 

where N$_{c}$ is the number of translational center-of-mass degrees of freedom and depends on the total number of sites and the number of constraints on the system. The details of the constraint algorithm used to calcu- late F C iα have been discussed previously [33, 38, 46]. All the systems were equilibrated for at least 10 6 time steps. The results from production runs were ensemble aver- aged over 14 runs, each of length 10 6 steps (i.e. a total of 1 . 4 × 10 7 time steps). The transverse momentum den- sity ACFs were computed over at least 20 reduced time units and the stress ACFs were computed over at least 40 reduced time units. Both the transverse momentum density and stress ACFs were computed at wave-vectors k$_{yn}$ = 2 πn/L$_{y}$ where mode number n is from 0 to 40 with increment 2 and L$_{y}$ = [ N$_{a}$/ρ ] 1 / $^{3}$. For the remainder 

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of this paper we drop the n index in k$_{yn}$ for simplic- ity. The ACFs were Laplace transformed with respect to time using Filon’s rule [47]. We do not report the frequency dependent viscosities in this work. The wave- vector and frequency dependent viscosities were calcu- lated using Eqs. (13) and (17). Eq. (13) was used to obtain the non-zero wave-vector and frequency depen- dent viscosities and Eq. (17) was used to obtain the zero wave-vector viscosity. 

In this work, all quantities are expressed in reduced Lennard-Jones units. Thus, our reference units are the: reduced length r ∗ = r/σ , reduced number density ρ ∗ = ρ/σ $^{3}$, reduced temperature T ∗ = k$_{B}$T/ϵ , reduced time t ∗ = t/ ( σ ( m/ϵ ) 1 / $^{2}$, reduced pressure P ∗ = P ( σ $^{3}$/ϵ ), reduced energy E ∗ = E/ϵ and reduced viscosity η ∗ = ησ $^{2}$/ √ ( mϵ ). For the remainder of this paper we apply these units and omit writing the asterisk. We will not distinguish between T$_{M}$ and T$_{A}$ , but simply use T to indicate the temperature. 

The autocorrelation functions were evaluated for both 108 and 864 molecule system in order to determine whether the results were system size dependent. There were no observed differences within the statistical errors in both monatomic and diatomic systems. We also note that in order to limit the number of figures, we do not display the results for the transverse momentum density and stress autocorrelation functions. However, we must mention that for monatomic systems both quantities (i.e. the transverse momentum density and stress ACFs) were in good agreement with those previously observed for Lennard-Jones monatomic liquids and their running in- tegrals have fully converged which suggests that the cor- relation functions have decayed to zero. 

## A. Viscosity kernels in reciprocal space 

The reciprocal space kernels for monatomic and di- atomic fluids are plotted in figures (1-3). The error bars are smaller than the symbol sizes and therefore omit- ted in figures 2 and 3. Generally the statistical relia- bility of reciprocal space kernel data increases as k$_{y}$ in- creases. Our zero wave-vector, zero frequency viscosities for monatomic fluids agree well with those available in the literature. For the WCA system at the state point ( ρ$_{a}$ = 0 . 375, T = 0 . 765) we found η$_{0}$ = 0 . 27 ± 0 . 01 which agrees with the results of Hansen et al. 0 . 273 [12, 20], while at the state point ( ρ$_{a}$ = 0 . 840, T = 1 . 0) we found η$_{0}$ = 2 . 29 ± 0 . 07, in agreement with the results of Matin et al. (2 . 1 ± 0 . 2) [42]. For chlorine we found η$_{0}$ = 6 . 89 ± 0 . 32, which agrees with the limiting values (6.7 ± 0.4) of the shear or elongational viscosities at zero strain rate [42]. 

The wave-vector dependent viscosity for diatomic sys- tems, figure 1, depicts a similar behaviour within the 

## IV. RESULTS AND DISCUSSION 

