Zero-order (LO) EDF.

In order to illustrate how the nonlocal EDF derives from a finite-range pseudopotential, we show explicitly the functional obtained from the averaging of the zero-order finite-range pseudopotential $\hat{V}^{(0)}$ over the nuclear Slater determinant. Below, symbols $\hat{V}^{(n)}$ and ${\cal E}^{(n)}$ denote, respectively, the order-$n$ regularized pseudopotential and functional.

The four terms of the pseudopotential at zero order are,

$$\hat{V}^{(0)}=C_{00,00}^{00,0} \hat{V}_{00,00}^{00,0}+C_{... ...V}_{00,20}^{00,0}+C_{00,20}^{00,1} \hat{V}_{00,20}^{00,1} .$$ (15)

The complete expression for the zero-order nonlocal EDF obtained from (16) reads

$${\cal E}^{(0)} = \left(\frac{C_{00,00}^{00,0}}{2}+\frac{C_{00,00}^{00,1}}{4} \right) T^{00,0000, 0,L}_{00,000,0}$$

$$+\left( -\frac{C_{00,20}^{00,0}}{2} -\frac{C_{00,20}^{00,1}}{4}\right) T^{00,0011, 0,L}_{00,0011,1}$$

$$+\left(\frac{\sqrt{3}}{4}C_{00,00}^{00,1} \right) T^{00,0000, 1,L}_{00,0000,0}$$

$$+\left( - \frac{\sqrt{3}}{4}C_{00,20}^{00,1}\right) T^{00,0011, 1,L}_{00,0011,1}$$

$$+ \left(-\frac{C_{00,00}^{00,0}}{8} - \frac{C_{00,00}^{00,1}}{4} ... ...0}^{00,0}+\frac{\sqrt{3}}{4}C_{00,20}^{00,1}\right) T^{00,0000, 0,N}_{00,000,0}$$

$$+\left(-\frac{\sqrt{3}}{8}C_{00,00}^{00,0}-\frac{\sqrt{3}}{4}C_{0... ...0,20}^{00,0}}{8} -\frac{C_{00,20}^{00,1}}{4}\right)T^{00,0011, 0,N}_{00,0011,1}$$

$$+\left(-\frac{\sqrt{3}}{8}C_{00,00}^{00,0}+\frac{3}{8}C_{00,20}^{00,0} \right) T^{00,0000, 1,N}_{00,0000,0}$$

$$+\left(-\frac{3}{8}C_{00,00}^{00,0}-\frac{\sqrt{3}}{8}C_{00,20}^{00,0} \right) T^{00,0011, 1,N}_{00,0011,1}.$$ (16)

In Eq. (17), we have eight zero-order, 4 local and 4 nonlocal, or 4 isoscalar and 4 isovector. coupling constants expressed explicitly as linear combinations of the four zero-order pseudopotential parameters given in Eq. (16).