Zero-order functional

The leading order functional, which depends on four parameters $t_0$, $x_0$, $y_0$, and $z_0$ of the regularized interaction (38), has the form

$$\langle V_0\rangle=\langle V_0^L \rangle + \langle V_0^{N} \rangle ,$$ (69)

with

$$\langle V_0^L \rangle = \int\! \mathrm d\mathbf r_1\,\mathrm d\mathbf r_2\,g_a(\mathbf r_... ...f r_1)\rho_0(\mathbf r_2) + A^{\rho_1}_0 \rho_1(\mathbf r_1)\rho_1(\mathbf r_2)$$

$$+ A^{\mathbf s_0}_0 \mathbf s_0(\mathbf r_1)\cdot\mathbf s_0(\mat... ... A^{\mathbf s_1}_0 \mathbf s_1(\mathbf r_1)\cdot\mathbf s_1(\mathbf r_2) \Bigr]$$ (70)

and

$$\langle V_0^{N} \rangle = \int\! \mathrm d\mathbf r_1\,\mathrm d\mathbf r_2\, g_a(\mathbf r... ...gl[ B^{\rho_0}_0 \rho_0(\mathbf r_2,\mathbf r_1)\rho_0(\mathbf r_1,\mathbf r_2)$$

$$+B^{\rho_1}_0 \rho_1(\mathbf r_2,\mathbf r_1)\rho_1(\mathbf r_1,\mathbf r_2)$$

$$+B^{\mathbf s_0}_0 \mathbf s_0(\mathbf r_2,\mathbf r_1)\cdot\math... ...f s_1(\mathbf r_2,\mathbf r_1)\cdot\mathbf s_1(\mathbf r_1,\mathbf r_2) \Bigr].$$ (71)

Densities $\rho_0(\bm{r})$, $\rho_1(\bm{r})$, $\mathbf s_0(\bm{r})$, and $\mathbf s_1(\bm{r})$, together with their nonlocal counterparts, are the standard scalar and vector densities in spin and isospin spaces, as defined in Ref. [23]. By taking the zero-range limit, which amounts to bringing in Eqs. (71) and (72) the regularized delta function $g_a(\bm{r})$ to its Dirac delta limit, and using relations (62), one recovers the standard local form of the zero-order functional,

$$\langle V_0\rangle= {\textstyle{\frac{1}{2}}}\,t_0\int\! \mathrm d\mathbf r\, \Bigl\{{\t... ...\left(1+z_0\right)+{\textstyle{\frac{1}{2}}}\left(x_0+y_0\right)\right]\rho_1^2$$

$$-\left[{\textstyle{\frac{1}{4}}}\left(1+z_0\right)-{\textstyle{\frac... ...athbf s_0^2 -{\textstyle{\frac{1}{4}}}\left(1+ z_0\right)\mathbf s_1^2 \Bigr\}.$$ (72)

We explicitly verified that the Cartesian zero-order nonlocal EDF (70) is exactly equivalent to that in the spherical-tensor representation. To show this equivalence, we made use of the relations of conversions between spherical and Cartesian local densities [9], which are also valid for nonlocal densities. In the same way, one finds the relations of conversions between the parameters of the zero-order regularized pseudopotential (3) and those of its Cartesian form (38), which read

$$C_{00,00}^{00,0} = t_0+ \frac{1}{2}t_0x_0,$$ (73)

$$C_{00,00}^{00,1} = -\frac{1}{2}t_0z_0-t_0y_0,$$ (74)

$$C_{00,20}^{00,0} = -\frac{\sqrt{3}}{2} t_0x_0 ,$$ (75)

$$C_{00,20}^{00,1} = \frac{\sqrt{3}}{2} t_0z_0.$$ (76)