Basics of pnEDF approach

The pnEDF Kohn-Sham state $\vert\Psi\rangle$ is a single Slater determinant built of the set of $A$ fully occupied single-particle (s.p.) states, that is,

$$\vert\Psi\rangle = \prod_{k=1}^A c^{+}_{k}\vert\rangle,$$ (1)

where $c^{+}_k$ denotes the s.p. state creation operator. This operator can be expressed in terms of the s.p. wave function $V_k(\bbox{r}st)$,

$$c_k^{+} = \int d^3 \bbox{r} \sum_{st} V^\ast_k(\bbox{r}st) a^{+}_{\bbox{r}st} \quad (k \le A),$$ (2)

where $a^{+}_{\bbox{r}st}$ creates the nucleon at point $\bbox{r}$, spin $s$= $\pm\tfrac{1}{2}$, and isospin $t=+\frac{1}{2}$ (neutron) or $-\frac{1}{2}$ (proton). In the p-n mixing framework, the HF s.p. state $c_k^{+}$ contains the neutron and proton components. In the present study, we only consider unpaired systems and particle-hole (p-h) densities, whereupon $V_k(\bbox{r}st)$ are simply the self-consistent HF wave functions. However, expressions given below are also valid within the HFB approach, where $V_k(\bbox{r}st)$ correspond to lower components of the quasiparticle wave functions [30,31].

To fix the notation, we now recall basic expressions introduced and derived in Ref. [17]. The one-body density matrix $\hat{\rho}$ is defined as

$$\hat{\rho}(\bbox{r}st,\bbox{r}'s't') =\langle \Psi \vert a_{\bbox{r}'s't'}^{+}a_{\bbox{r}st}\vert\Psi \rangle$$

$$= \sum_{k=1}^A V_k(\bbox{r}'s't') V_k^\ast(\bbox{r}st),$$ (3)

and the pnEDF can be written as

$$\bar{H}[\hat{\rho}] = \int d^3\bbox{r}\, {\mathcal H}(\bbox{r}) = \int d^3\bbox{r}\, {\mathcal H}_{\rm Sk}(\bbox{r}) + E_{\rm Cou}[\hat{\rho}],$$ (4)

where the Skyrme energy density is

$${\mathcal H}_{\rm Sk}(\bbox{r}) = \frac{\hbar^{2}}{2m} \tau_0(\bbox{r}) +\chi_0(\bbox{r})+\chi_1(\bbox{r})$$ (5)

with $\tau_0(\bbox{r})$ being the isoscalar kinetic-energy density (it is assumed in the following that the neutron and proton masses are equal). The Coulomb energy functional $E_{\rm Cou}$ is the only term that breaks the isospin symmetry. The Slater approximation is used for the Coulomb exchange functional. The p-h Skyrme interaction-energy densities $\chi_0(\bbox{r})$ and $\chi_1(\bbox{r})$ depend quadratically on the isoscalar and isovector densities, respectively. Based on general rules of constructing the energy density [17], one obtains

$$\begin{eqnarray} {\chi}_{0}(\bbox{r}) &=& C_{0}^{\rho} \rho_{... ...^2 + C_{1}^{F} \vec{ \bbox{s}} \cdot \circ~ \vec{ \bbox{F}} , \end{eqnarray}$$

where $\times$ stands for the vector product of vectors in space, $\circ$ stands for the scalar product of isovectors in isospace, and other definitions closely follow those introduced in Ref. [17].

Quasilocal densities $\rho_k$, $\tau_k$, $\bbox{s}_k$, $\bbox{T}_k$, $\bbox{j}_k$, $\bbox{F}_k$, $J_k$, $\bbox{J}_k$, and $\underline{\sf J}_k$, are defined through the particle and spin non-local densities,

$$\rho_k(\bbox{r},\bbox{r}') = \sum_{stt'} \hat{\rho}(\bbox{r}st,\bbox{r}'st') \hat{{\tau}}^k_{t't},$$ (7)

$$\bbox{s}_k(\bbox{r},\bbox{r}') = \sum_{ss'tt'} \hat{\rho}(\bbox{r}st,\bbox{r}'s't') \hat{\bbox{\sigma}}_{s's}\hat{{\tau}}^k_{t't},$$ (8)

where $k$ runs from 0 to 3, $\hat{\bbox{\sigma}}$ and $\hat{\tau}^k (k=1,2,3)$ are the Pauli matrices for spin and isospin, respectively, and $\hat{\tau}^0_{t't} = \delta_{t't}$. The explicit definitions and expressions in the cylindrical basis for the local densities appearing in Eqs. (6) are given in Appendix A. By varying the pnEDF with respect to the density matrices, one obtains the p-h mean-field Hamiltonian:

$$\hat{h}(\bbox{r}'s't',\bbox{r}st) = \frac{\delta \overline{H}[\hat{\rho}]} {\delta \hat{\rho}(\bbox{r}st,\bbox{r}'s't')}$$

$$= - \frac{\hbar^{2}}{2m}\delta(\bbox{r}-\bbox{r}') \bbox{\nabla}\cdot \bbox{\nabla}\,\delta_{s's}\delta_{t't}$$

$$+ \hat{\Gamma} (\bbox{r}'s't',\bbox{r}st) + \hat{\Gamma}_{\text{r}}(\bbox{r}'s't',\bbox{r}st),$$ (9)

where $\hat{\Gamma}$ is the HF potential and $\hat{\Gamma}_{\text{r}}$ is the rearrangement potential.

For the pnEDF depending on quasilocal densities only, such as in Eq. (6), the HF Hamiltonian is a local differential operator,

$$\hat{{h}}(\bbox{r}'s't',\bbox{r}st) = \delta (\bbox{r}-\bbox{r}')\hat{{h}}(\bbox{r};s't',st),$$ (10)

which has a simple isospin structure:

$$\hat{h}(\bbox{r};s't',st) = {h}_0(\bbox{r};s',s)\delta_{t't}+ \vec{h}(\bbox{r};s',s)\circ \hat{\vec{\tau}}_{t't}.$$ (11)

The isoscalar and isovector parts of the HF Skyrme Hamiltonian can be written in the compact form as,

$$h_k(\bbox{r};s',s) =- \frac{\hbar^{2}}{2m}\bbox{\nabla}^2\delta_{s's}\delta_{k0} + U_k \delta_{s's} + \bbox{\Sigma}_k \cdot\hat{\bbox{\sigma}}_{s's}$$

$$+ \frac{1}{2i}\big[\bbox{I}_k \delta_{s's} + ({\mathsf B}_k \cdot\hat{\bbox{\sigma}}_{s's})\big]\cdot\bbox{\nabla}$$

$$+ \frac{1}{2i}\bbox{\nabla}\cdot\big[\bbox{I}_k \delta_{s's} + ({\mathsf B}_k \cdot\hat{\bbox{\sigma}}_{s's})\big]$$

$$- \bbox{\nabla}\cdot\big[M_k \delta_{s's} + \bbox{C}_k \cdot\hat{\bbox{\sigma}}_{s's}\big]\bbox{\nabla}$$

$$- \bbox{\nabla}\cdot \bbox{D}_k \,\hat{\bbox{\sigma}}_{s's}\cdot \bbox{\nabla},$$ (12)

where $k=0,1,2,3$, and

$$({\mathsf B}\cdot\hat{\bbox{\sigma}})_a =\sum_b{\mathsf B}_{ab}\hat{\bbox{\sigma}}^b ,$$ (13)

for $a=x,y,z$, denotes the $a$th component of a space vector. By introducing the unit space tensor ${\delta }$ and the antisymmetric space tensor $(\bbox{\epsilon}\cdot\bbox{J})_{ab} = \sum_{c}\bbox{\epsilon}_{acb}\bbox{J}_c$, the local real potentials can be written as:

All coupling constants $C_t$ in Eqs. (14) are taken with $t=0$ for $k=0$ (isoscalars), and with $t=1$ for $k=1, 2, 3$ (isovectors).¹

The resulting HF equation can be written as a self-consistent eigenvalue problem,

$$\int d^3 \bbox{r} \sum_{st} \hat{h}(\bbox{r}'s't',\bbox{r}st) V_k^\ast(\bbox{r}st) = \varepsilon_k V^\ast_k(\bbox{r}'s't'),$$ (15)

which is solved by filling the lowest $A$ s.p. orbits in the density matrix (3).