Hamiltonian kernel

Conventional Skyrme EDF can be expressed by bilinear forms of six isoscalar ($t=0$) and six isovector ($t=1$) local densities, including the particle $\rho_t$, kinetic $\tau_\tau$, spin $\vec{s}_t$, spin-kinetic $\vec{T}_t$, current $\vec{j}_t$, and spin-current $\stackrel{\leftrightarrow}{J}_t$ densities and their derivatives. Standard definitions of these densities can be found in numerous references, see, e.g., Refs. [9,46] and references quoted therein. It is to be noted that in the standard MF theory the proton and neutron s.p. wave functions are not mixed, i.e., the proton-neutron symmetry is strictly conserved [46,47]. Therefore, the MF isoscalar (isovector) densities are simply sums (differences) of neutron and proton densities since only the third component of the isovector density is nonzero.

The isospin projection essentially preserves the functional form of the Skyrme EDF derived by averaging the isospin-invariant Skyrme interaction over the Slater determinant. All what needs to be done is a replacement of the local density matrix by the corresponding transition density matrix. Moreover, the bilinear terms that depend of the isovector densities must be replaced by the full isoscalar products of the corresponding isovector transition densities. Special care should be taken of the density-dependent term of the Skyrme interaction as the extension of this term to the transition case is undefined in the process of averaging the Skyrme interaction over the Slater determinant. Following the argumentation of Refs. [48,49], we replace the isoscalar density by the transition isoscalar density matrix, that is, $\rho_{n}+\rho_{p} \longrightarrow \tilde \rho_{00}$.

As for the Coulomb-interaction kernel, it depends on the isoscalar and isovector transition densities in the following way:

$$\langle\Phi \vert \hat V^C \hat{R}(\beta)\vert\Phi\rangle =$$

$$\displaystyle = \frac{e^2}{4} \int d \vec{r}_1 \int d \vec{r}_2 ... ...mu = - \lambda}^{\lambda} {\cal V}^{C}_{\lambda \mu }( \vec{r}_1, \vec{r}_2 ),$$ (52)

where

$${\cal V}^{C}_{00} = \frac{1}{2} \left\{ \tilde\rho_{00} (\vec{r}_1 ) \tilde\rho_{00} ... ...ac{1}{3} \tilde\rho_{1} (\vec{r}_1 ) \circ \tilde\rho_{1} (\vec{r}_2 ) \right\}$$

$$- \frac{1}{4} \left\{ \tilde\rho_{00} (\vec{r}_2 , \vec{r}_1 ) \til... ...1} (\vec{r}_2, \vec{r}_1 ) \circ \tilde\rho_{1} (\vec{r}_1, \vec{r}_2 ) \right.$$

$$+ \left. \tilde{\vec s}_{00} (\vec{r}_2 , \vec{r}_1 ) \cdot \tild... ... \vec{r}_1 ) \circ \cdot \tilde{\vec s}_{1} (\vec{r}_1, \vec{r}_2 ) \right\} ,$$ (53)

$${\cal V}^{C}_{10} = - \tilde\rho_{00} (\vec{r}_1 ) \tilde\rho_{10} (\vec{r}_2 ) + \f... ...r}_2, \vec{r}_1 ) \cdot \tilde{\vec s}_{10} (\vec{r}_1, \vec{r}_2 ) \right\} ,$$ (54)

$${\cal V}^{C}_{1\pm 1} = \tilde\rho_{00} (\vec{r}_1 ) \tilde\rho_{1\mp 1} (\vec{r}_2 ) - ... ..., \vec{r}_1 ) \cdot \tilde{\vec s}_{1\mp 1} (\vec{r}_1, \vec{r}_2 ) \right\} ,$$ (55)

$${\cal V}^{C}_{20} = \frac{1}{3} \left\{ \tilde\rho_{10} (\vec{r}_1 ) \tilde\rho_{1 0}... ...c{r}_2 ) + \tilde\rho_{11} (\vec{r}_1 ) \tilde\rho_{1-1} (\vec{r}_2 ) \right\}$$

$$- \frac{1}{6} \left\{ \tilde\rho_{10} (\vec{r}_2, \vec{r}_1 ) \tild... ..._{11} (\vec{r}_2, \vec{r}_1 ) \tilde\rho_{1-1} (\vec{r}_1, \vec{r}_2 ) \right.$$

$$+ \left. \tilde{\vec s}_{10} (\vec{r}_2, \vec{r}_1 ) \cdot \tilde... ...}_2, \vec{r}_1 ) \cdot \tilde{\vec s}_{1-1} (\vec{r}_1, \vec{r}_2 ) \right\} ,$$ (56)

$${\cal V}^{C}_{2\pm 1} = -\frac{1}{\sqrt{3}} \tilde\rho_{10} (\vec{r}_1 ) \tilde\rho_{1\m... ..., \vec{r}_1 ) \cdot \tilde{\vec s}_{1\mp 1} (\vec{r}_1, \vec{r}_2 ) \right\} ,$$ (57)

$${\cal V}^{C}_{2\pm 2} = \frac{1}{\sqrt{6}} \tilde\rho_{1\mp 1} (\vec{r}_1 ) \tilde\rho_{... ..., \vec{r}_1 ) \cdot \tilde{\vec s}_{1\mp 1} (\vec{r}_1, \vec{r}_2 ) \right\} .$$ (58)

Here, the symbols $\cdot$ and $\circ$ stand for the scalar products of vectors and isovectors, respectively.