Cluster model of 12C in density functional theory framework

We employ the constrained density functional theory to investigate cluster phenomena for the $^{12}$C nucleus. The proton and neutron densities are generated from the placement of three $^{4}$He nuclei (alpha particles) geometrically. These densities are then used in a density constrained Hartree-Fock calculation that produces an antisymmetrized state with the same densities through energy minimization. In the calculations no \textit{a priori} analytic form for the single-particle states is assumed and the full energy density functional is utilized. The geometrical scan of the energy landscape provides the ground state of $^{12}$C as an equilateral triangular configuration of three alphas with molecular bond like structures. The use of the nucleon localization function provides further insight to these configurations. One can conclude that these configurations are a hybrid between a pure mean-field and a pure alpha particle condensate. This development could facilitate DFT based fusion calculations with a more realistic $^{12}$C ground state.