as ρ$_{m}$ ( r , t ) = ∑ N$_{m}$ i $_{=1}$M$_{i}$ δ ( r − r$_{i}$ ) in real space, respec- tively. In a similar way we can expand the atomic mo- mentum density about the molecular center of mass: ˜ J $^{A}$( k , t ) = ∑ N$_{m}$ i =1 ∑ N$_{s}$ α $_{=1}$m$_{iα}$ v$_{iα}$ (1 + i k · R$_{iα}$ + . . . ) e i k · r iα . The Fourier transform of the momentum density in the molecular representation can then be defined as i from the center of mass of molecule j . In cases where two sites on two different periodic images of the same molecule may interact, the correct value of r$_{ij}$ = r$_{j}$ − r$_{i}$ corresponding to the particular images of molecule i and j in F$_{iαjβ}$ must be used. Though this is unlikely to hap- pen in diatomic fluids it is particularly important in sim- ulation of long molecules. The momenta appearing in these equations, p$_{iα}$ , p$_{i}$ , are the momenta appearing in the respective atomic and molecular equations of motion Eqs. (21) and (24). 3 A complete procedure for expressing the mass and mo- mentum densities in physical and reciprocal space for atomic and molecular fluids has been discussed in more detail by Todd and Daivis [29]. For a monatomic system the wave-vector dependent pressure tensor is defined as [29] where F$_{i}$ is the force on atom i due to atom j and g ( k ) = ( e i k · r − 1) /i k · r = ∑ ∞ n $_{=0}$( i k · r ) $^{n}$/ ( n + 1)! is the Fourier transform of the Irving-Kirkwood O$_{ij}$ operator [30]. For a molecular system the molecular pressure tensor is the pressure calculated using the intermolecular forces and the molecular center of mass momenta. The atomic pressure on the other hand includes all atomic momenta and all interatomic forces and constraint forces. Thus the wave-vector dependent pressure tensor for constrained di- atomic fluid can be written in atomic representation as where F$_{iαjβ}$ is the LJ force acting on site α of molecule i due to site β of molecule j , and F C iα is the bond constraint force on site α of molecule i . r$_{iαjβ}$ = r$_{jβ}$ − r$_{iα}$ is the minimum image separation of site α of molecule i from site β of molecule j . The pressure tensor for a diatomic molecule in the molecular representation is defined as where, F N ij represents the total intermolecular force on molecule i due to molecule j . r$_{ij}$ = r$_{j}$ − r$_{i}$ is the mini- mum image separation of the center of mass of molecule because the total peculiar kinetic energy and three com- ponents of the momenta are constants of the motion in k$_{B}$ is Boltzmann’s constant. Due to finite time aver- aging and finite system size, the value of C ( k$_{y}$, t = 0) obtained from simulation differs insubstantially from the exact value given by The zero time value of C$_{⊥}$ ( k$_{y}$, t = 0) in the thermody- namic limit is where ρ is the number density of the fluid and ˜ C$_{⊥}$ ( k$_{y}$, ω ) is the Laplace transform of the ensemble averaged transverse momentum density autocorrelation function C$_{⊥}$ ( k$_{y}$, t ), which is defined as The complex wave-vector and frequency dependent vis- cosity can be evaluated by using two different expres- sions in terms of the Fourier-Laplace transform of the transverse momentum density autocorrelation function (ACF), C$_{⊥}$ ( k , t ), and the Fourier-Laplace transform of the stress tensor autocorrelation function, N ( k , t ) [13]. We define the complex Laplace transform (one-sided Fourier transform) as L [ f ( t )] = ˜ f ( w ) = ∫ ∞ 0 f ( t ) e − $^{iωt}$dt . We also note that for the sake of simplicity of notation and consistency with the notation used in previous pub- lications, in what follows, we drop the tilde sign over the Fourier transformed correlation functions and keep the tilde notation over the Fourier-Laplace transformed cor- relation functions only. If we set Γ k = (0 , k$_{y}$, 0) and J$_{x}$ is the component of the momentum density in the x di- rection, the expression for the wave-vector and frequency dependent viscosity in terms of ˜ C$_{⊥}$ ( k$_{y}$, ω ) takes the form [13]: C. Wave-vector and frequency dependent viscosity B. Wave-vector dependent pressure tensor C$_{⊥}$ ( k$_{y}$, t = 0) = ρk$_{B}$T 3 N − 4 3 N (16) C$_{⊥}$ ( k$_{y}$, t = 0) = ρk$_{B}$T. (15) C$_{⊥}$ ( k$_{y}$, t ) = 1 V 〈 J$_{x}$ ( k$_{y}$, t ) J$_{x}$ ( k$_{y}$, t = 0) 〉 . (14) ˜ η ( k$_{y}$, ω ) = ρ k 2 y C$_{⊥}$ ( k$_{y}$, t = 0) − iω ˜ C$_{⊥}$ ( k$_{y}$, ω ) ˜ C$_{⊥}$ ( k$_{y}$, ω ) (13) ˜ P $^{A}$( k , t ) = N$_{m}$ ∑ i =1 2 ∑ α =1 p$_{iα}$p$_{iα}$ m$_{iα}$ e i k · r iα − 1 2 N$_{m}$ ∑ i =1 2 ∑ α =1 N$_{m}$ ∑ j = i 2 ∑ β =1 r$_{iαjβ}$F$_{iαjβ}$ g ( k ) e i k · r iαjβ + N$_{m}$ ∑ i =1 2 ∑ α =1 r$_{iα}$F C $_{iα}$g ( k ) e i k · r iα (11) ˜ P $^{M}$( k , t ) = N$_{m}$ ∑ i =1 p$_{i}$p$_{i}$ M$_{i}$ e i k · r i − 1 2 N$_{m}$ ∑ i =1 N$_{m}$ ∑ j = i r$_{ij}$F N $_{ij}$g ( k ) e i k · r ij (12) ˜ P ( k , t ) = N ∑ i =1 p$_{i}$p$_{i}$ m$_{i}$ e i k · r i − 1 2 N ∑ i =1 N ∑ j = i r$_{ij}$F$_{ij}$ g ( k ) e i k · r i (10) ˜ J $^{M}$( k , t ) = N$_{m}$ ∑ i =1 M$_{i}$ v$_{i}$ ( t ) e i k · r i (9)