Relations between the pseudopotential and Energy Density Functional

The EDF related to the pseudopotential is obtained by averaging the pseudopotential $\hat{V}$ over the uncorrelated wavefunction (a Slater determinant), that is,

$${\cal E} = \frac{1}{4} \int {\rm d}\, \bm{r}_{1}\bm{r}_{2}\bm{r}'_{1}\bm{r}'... ... s_1 t_1,\bm{r}'_{1} s'_1 t'_1) \rho(\bm{r}_{2} s_2 t_2,\bm{r}'_{2} s'_2 t'_2),$$ (31)

where the two-body spin-isospin matrix element of the pseudopotential is defined as

$$\hat{V}(\bm{r}'_{1} s'_1 t'_1 \bm{r}'_{2} s'_2 t'_2, \bm{r}_{1} s... ...t_2) =\langle s'_1 t'_1, s'_2 t'_2 \vert\hat{V}\vert s_1 t_1 , s_2 t_2\rangle ,$$ (32)

and $\rho(\bm{r}_{1} s_1 t_1,\bm{r}'_{1} s'_1 t'_1)$ and $\rho(\bm{r}_{2} s_2 t_2,\bm{r}'_{2} s'_2 t'_2)$, are the one-body densities in spin-isospin channels. (For definitions, see, e.g., Ref. [19].) In this lengthy calculation, one must consider as intermediate step the recoupling of the relative-momentum operators, so as to recast the gradients in such a way that each tensor affects only one particle at a time [19]. Such recoupling was performed with the aid of symbolic programming, and is not, for the sake of brevity, reported in this paper.

For each term of the pseudopotential (1), we can write the result of the averaging in the following way,

$$\langle C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{... ...,n L v J}^{n' L' v' J',t} T_{m I, n L v J}^{n' L' v' J', t} ,$$ (33)

where $C_{m I,n L v J}^{n' L' v' J', t}$ and $T_{m I, n L v J}^{n' L' v' J', t}$ denote, respectively, the coupling constants and terms of the EDF according to the formalism developed in Ref. [2]. Since here we treat the isospin degree of freedom explicitly, to the notation of Ref. [2] we have added superscripts $t$, which denote the isoscalar ($t=0$) and isovector ($t=1$) channels.

Once relations (33) are evaluated for each term of the pseudopotential, all terms of the N$^3$LO EDF are generated, with the EDF coupling constants $C_{m I,n L v J}^{n' L' v' J', t}$ becoming linear combinations of the pseudopotential strength parameters $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}$. Since the pseudopotentials are Galilean-invariant, the obtained EDF coupling constants obey the Galilean-invariance constraints [2]. Similarly, when parameters of the pseudopotential are restricted to obey the gauge-invariance conditions defined in Sec. 2.3, the resulting coupling constants correspond to a gauge-invariant EDF.

The 12 second-order isoscalar (isovector) coupling constants expressed by the 7 second-order pseudopotential parameters are given in Table 10 (Table 11). Similar expressions relating at fourth (sixth) order 45 (129) isoscalar and isovector coupling constants to 15 (26) pseudopotential parameters, are available in the supplemental material [24].

Table 10: Second-order coupling constants of the isoscalar EDF $(t=0)$ as functions of parameters of the pseudopotential, expressed by the formula $C_{mI,n L v J}^{n' L' v' J', 0}= A(a C_{00,00}^{20}+b C_{00,20}^{20}+c C_{00,22}^{22}+d C_{11,00}^{11}+e C_{11,20}^{11}+f C_{11,11}^{11}+g C_{11,22}^{11})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$	$g$
$C_{20,0000}^{0000,0}$	$\frac{1}{32}$	$-$3	$-\sqrt{3}$	0	5	$-\sqrt{3}$	0	0
$C_{00,2000}^{0000,0}$	$\frac{1}{16}$	3	$\sqrt{3}$	0	5	$-\sqrt{3}$	0	0
$C_{00,1110}^{1110,0}$	$\frac{1}{48}$	$\sqrt{3}$	$5$	$2\sqrt{5}$	$-\sqrt{3}$	3	0	$6\sqrt{5}$
$C_{00,1111}^{1111,0}$	$\frac{1}{48}$	3	$5\sqrt{3}$	$-\sqrt{15}$	$-$3	$3\sqrt{3}$	0	$-3\sqrt{15}$
$C_{00,1112}^{1112,0}$	$\frac{1}{48}$	$\sqrt{15}$	$5\sqrt{5}$	1	$-\sqrt{15}$	$3\sqrt{5}$	0	3
$C_{11,1111}^{0000,0}$	$-\frac{3}{4}$	0	0	0	0	0	1	0
$C_{00,1101}^{1101,0}$	$\frac{1}{16}$	$-$3	$-\sqrt{3}$	0	$-$5	$\sqrt{3}$	0	0
$C_{20,0011}^{0011,0}$	$\frac{1}{32}$	$\sqrt{3}$	5	0	$\sqrt{3}$	$-$3	0	0
$C_{22,0011}^{0011,0}$	$\frac{1}{16}$	0	0	1	0	0	0	$-$3
$C_{00,2011}^{0011,0}$	$\frac{1}{16}$	$-\sqrt{3}$	$-$5	0	$\sqrt{3}$	$-$3	0	0
$C_{00,2211}^{0011,0}$	$\frac{1}{8}$	0	0	$-$1	0	0	0	$-$3
$C_{11,0011}^{1101,0}$	$-\frac{3}{4}$	0	0	0	0	0	1	0

Table 11: Same as in Table 10 but for isovector EDF $(t=1)$, according to the formula $C_{mI,n L v J}^{n' L' v' J', 1}= A(a C_{00,00}^{20}+b C_{00,20}^{20}+c C_{00,22}^{22}+d C_{11,00}^{11}+e C_{11,20}^{11}+f C_{11,11}^{11}+g C_{11,22}^{11})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$	$g$
$C_{20,0000}^{0000,1}$	$\frac{1}{32}$	$\sqrt{3}$	$-$3	0	$\sqrt{3}$	$-$3	0	0
$C_{00,2000}^{0000,1}$	$\frac{1}{16}$	$-\sqrt{3}$	3	0	$\sqrt{3}$	$-$3	0	0
$C_{00,1110}^{1110,1}$	$\frac{1}{48}$	3	$\sqrt{3}$	$-2\sqrt{15}$	$-$3	$-\sqrt{3}$	0	$2\sqrt{15}$
$C_{00,1111}^{1111,1}$	$\frac{1}{16}$	$\sqrt{3}$	$1$	$\sqrt{5}$	$-\sqrt{3}$	$-1$	0	$-\sqrt{5}$
$C_{00,1112}^{1112,1}$	$\frac{1}{48}$	$3\sqrt{5}$	$\sqrt{15}$	$-\sqrt{3}$	$-3\sqrt{5}$	$-\sqrt{15}$	0	$\sqrt{3}$
$C_{11,1111}^{0000,1}$	$-\frac{1}{4}\sqrt{3}$	0	0	0	0	0	1	0
$C_{00,1101}^{1101,1}$	$\frac{1}{16}$	$\sqrt{3}$	$-$3	0	$-\sqrt{3}$	3	0	0
$C_{20,0011}^{0011,1}$	$\frac{1}{32}$	3	$\sqrt{3}$	0	3	$\sqrt{3}$	0	0
$C_{22,0011}^{0011,1}$	$-\frac{1}{16}\sqrt{3}$	0	0	1	0	0	0	1
$C_{00,2011}^{0011,1}$	$\frac{1}{16}$	$-$3	$-\sqrt{3}$	0	3	$\sqrt{3}$	0	0
$C_{00,2211}^{0011,1}$	$\frac{1}{8}\sqrt{3}$	0	0	1	0	0	0	$-$1
$C_{11,0011}^{1101,1}$	$-\frac{1}{4}\sqrt{3}$	0	0	0	0	0	1	0