contribution. The weighted densities are related to the bare one-body densities, ρ$_{i}$ ( r , ω ), via where ∗ represents the three-dimensional convolution h ( r ) ∗ g ( r ) = ∫ d $^{3}$xh ( x ) g ( x − r ). We have kept the notation of Ref. [52] where the upper index D (disk) is indicative of the number of particle orientations that appear in the weight function (below) or weighted density. The weight functions for species i are given by where δ ( · ) is the Dirac delta distribution, Θ( · ) is the Heaviside step function and × denotes the vector product, such that ω · ( ω ′ × ω ′′ ) is the triple scalar product. Note the modulus in Eqs. (20) and (21). For spatially homogeneous states, Eq. (14) becomes Inserting the definitions of the weighted densities (15)-(17) into the excess free energy (14) we obtain βF$_{exc}$ ([ ρ$_{1}$, ρ$_{2}$ ]) = − 1 2 [ GLYPH<a116> 1 1 GLYPH<a116> + GLYPH<a116> 2 2 GLYPH<a116> + 2 GLYPH<a116> 1 2 $^{GLYPH<a116>}$] + ∫ d x ∫ d ω ∫ d $_{ω}$′ ∫ d $_{ω}$′′ | ω · ( ω ′ × ω ′′ ) | 3 π 2 (23) × n D 2 ( x , ω ) n D $_{2}$( x , ω $^{′}$) n D 2 ( x , ω $^{′′}$) . βF$_{exc}$ ([ ρ$_{1}$, ρ$_{2}$ ]) V = ∫ d ω ∫ d ω ′ [ n DD 1 ( ω ) n D $_{2}$( ω $^{′}$) + 1 24 π n D 2 ( ω ) n DDD 2 ( ω , ω $^{′}$) n D 2 ( ω $^{′}$) ] . (22) w DDD i 2 ( r , ω , ω $^{′}$, ω $^{′′}$) = 8 π | ω · ( ω ′ × ω $^{′′}$) | w D i 2 ( r , ω ) , (21) w DD i 1 ( r , ω , ω $^{′}$) = 2 R$_{i}$ | ω · ( ω ′ × r ) | w D i 1 ( r , ω ) , (20) w D i 2 ( r , ω ) = 2Θ( R$_{i}$ − | r | ) δ ( r · ω ) , (19) w D i 1 ( r , ω ) = δ ( R$_{i}$ − | r | ) δ ( r · ω ) / 8 , (18) n DDD 2 ( r , ω , ω $^{′}$) = 2 ∑ i =1 ∫ d ω $^{′′}$w DDD i 2 ( r , ω , ω $^{′}$, ω $^{′′}$) ∗ ρ$_{i}$ ( r , ω $^{′′}$) , (17) n D 2 ( r , ω ) = 2 ∑ i =1 w D i 2 ( r , ω ) ∗ ρ$_{i}$ ( r , ω ) , (16) n DD 1 ( r , ω ) = 2 ∑ i =1 ∫ d ω $^{′}$w DD i 1 ( r , ω $^{′}$, ω ) ∗ ρ$_{i}$ ( r , ω $^{′}$) , (15) 9