content_image/366199.jpg content_image/366198.jpg content_image/366197.jpg content_image/366194.jpg content_image/366193.jpg content_image/366196.jpg content_image/366195.jpg content_image/366192.jpg content_image/366191.jpg
contribution. The weighted densities are related to the bare one-body densities, ρ$_{i}$ ( r , ω ), via 

<img src='content_image/366199.jpg'>

<img src='content_image/366198.jpg'>

<img src='content_image/366197.jpg'>

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 

<img src='content_image/366194.jpg'>

<img src='content_image/366193.jpg'>

<img src='content_image/366196.jpg'>

<img src='content_image/366195.jpg'>

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 

<img src='content_image/366192.jpg'>

Inserting the definitions of the weighted densities (15)-(17) into the excess free energy (14) we obtain 

<img src='content_image/366191.jpg'>

