Particle-Number-Projected DFT

According to the DFT, the energy density of the system, $E_{\mbox{\rm\scriptsize {DFT}}}(\rho,\chi,\chi^*)$, can be written as a function of the local particle $\rho({\bf r})$ and pairing $\chi({\bf r})$ densities obtained as the diagonal elements of the corresponding density matrices:

$$\begin{array}{rcl} \rho({\bf r}) &\equiv& \sum_\sigma \rh... ...sigma}u_nv_n \vert\varphi_n({\bf r}\sigma)\vert^2 . \end{array}$$ (30)

The nuclear density functionals for time-even systems also depend on kinetic $\tau$ and spin-orbit ${\mathsf J}$ densities. An even larger set of densities enters the energy density for time-odd systems [38,13]. For simplicity, we discuss here the dependence on the particle density only, because extension to other densities is straightforward.

We note in passing that the densities corresponding to the shifted HFB state $\vert\Phi(z)\rangle$ (6) can be written as:

$$\begin{array}{lll} \rho^z({\bf r}) &=& \displaystyle \su... ...~ \sum_\sigma\vert\varphi_n({\bf r}\sigma)\vert^2. \end{array}$$ (31)