  A fluid of hard right isosceles triangles was studied using an extension of
Scaled-Particle Density-Functional Theory which includes the exact third virial
coefficient. We show that the only orientationally ordered stable
liquid-crystal phase predicted by the theory is the uniaxial nematic phase, in
agreement with the second-order virial theory. By contrast, Monte Carlo
simulations predict exotic liquid-crystal phases exhibiting tetratic and
octatic correlations, with orientational distribution functions having four and
eight equivalent peaks, respectively. This demonstrates the failure of the
standard Density Functional Theory based on two and three-body correlations to
describe high-symmetry orientational phases in two-dimensional hard
right-triangle fluids, and points to the necessity to reformulate the theory to
take into account high-order body correlations and ultimately particle
self-assembling and clustering effects. This avenue may represent a great
challenge for future research, and we discuss some fundamental ideas to
construct a modified version of Density-Functional Theory to account for these
clustering effects.
