The Densities

The main ingredients of the nuclear DFT are the local nucleonic densities. Following the standard definitions [14,10], one considers local particle-hole (p-h) densities: particle $\rho (\mbox{\boldmath${r}$\unboldmath })$, kinetic $\tau (\mbox{\boldmath${r}$\unboldmath })$, spin ${\mbox{\boldmath${s}$\unboldmath }}_k (\mbox{\boldmath${r}$\unboldmath })$, spin-kinetic ${\mbox{\boldmath${T}$\unboldmath }}_k (\mbox{\boldmath${r}$\unboldmath })$, current ${\mbox{\boldmath${j}$\unboldmath }}_k (\mbox{\boldmath${r}$\unboldmath })$, tensor-kinetic ${\mbox{\boldmath${F}$\unboldmath }}_{k} (\mbox{\boldmath${r}$\unboldmath })$, spin-current ${\mathsf{J}}_{kl}(\mbox{\boldmath${r}$\unboldmath })$, as well as the corresponding local particle-particle (p-p; or pairing) densities: $\tilde{\rho}(\mbox{\boldmath${r}$\unboldmath })$, $\tilde{\tau} (\mbox{\boldmath${r}$\unboldmath })$, ${\tilde{\mbox{\boldmath${s}$\unboldmath }}}_k (\mbox{\boldmath${r}$\unboldmath })$, ${\tilde{\mbox{\boldmath${T}$\unboldmath }}}_k (\mbox{\boldmath${r}$\unboldmath })$, ${\tilde{\mbox{\boldmath${j}$\unboldmath }}}_k (\mbox{\boldmath${r}$\unboldmath })$, ${\tilde{\mbox{\boldmath${F}$\unboldmath }}}_{k} (\mbox{\boldmath${r}$\unboldmath })$, and ${\tilde{\mathsf{J}}}_{kl}(\mbox{\boldmath${r}$\unboldmath })$.

The local p-h and p-p densities are defined by the spin-dependent one-body density matrices:

$$\begin{array}{rcl} \rho (\mbox{\boldmath${r}$\unboldmath }\s... ...th },\mbox{\boldmath${r}$\unboldmath }^{\prime }) . \end{array}$$ (1)

For instance,

$$\begin{array}{rcl} \rho (\mbox{\boldmath${r}$\unboldmath }) ... ...h }^{\prime }\mbox{\boldmath${=r}$\unboldmath }}\;. \end{array}$$ (2)

Since the nuclear DFT deals with two kinds of nucleons, the isospin degree of freedom has to be introduced and the isoscalar and isovector densities have to be considered[10].