Many-body energy density functional

The Hohenberg-Kohn theorem and the Kohn-Sham equations, which are at the basis of the density-functional theory, are reformulated in terms of a particular many-body density, which is translational and Galilean invariant and therefore is relevant for self-bound systems. In a similar way that there is a unique relation between the one-body density and the external potential that gives rise to it, we demonstrate that there is a unique relation between that particular many-body density and a definite many-body potential. The energy is then a functional of this density, and its minimization leads to the ground-state energy of the system. As a proof of principle, the analogous of the Kohn-Sham equation is solved in the specific case of ${}^4\mathrm{He}$ atomic clusters, to put in evidence the advantages of this formulation in terms of physical insights.

Figure 1

Binding energy per particle obtained solving Eq. (35) with $7.5a_0<\beta <9.0a_0$, the (red) diamonds highlight the best case $(B,\beta )=(7.211\text{K},8.33a_0)$, with the spread denoted by the error bars visible for $N\ge 40$. The GFMC results for the HFD-HE2 potential (black solid line), the DFT results of Ref. [37] (blue squares), and the results of the soft Gaussian potential (SGP) of Refs. [38, 39] (orange triangles) are shown too.

Figure 2

The (reduced) many-body density $\nu \left(\rho \right)$ for selected number of particles (upper panel). The unit radius $r_0$ (black solid points) (lower panel) with error bars corresponding to variations of $\beta$ in the interval $7.5a_0<\beta <9.0a_0$. For the sake of comparison, the GFMC results [36] are shown too (red solid points), together with a fit to these values represented by the (red) dashed line in units of $a_0$.