Beyond the LO EDF.

The 72 second-order isoscalar and isovector coupling constants of the nonlocal EDF, expressed by the 14 second-order finite-range pseudopotential parameters, are collected in the supplemental material according to the following formula using the grouped index defined as $(a,\alpha,Q,a', \alpha',t,\mathcal{L})\equiv \mathcal{A}$,

$$C_{a,\alpha,Q}^{a', \alpha',t,\mathcal{L}} = \sum_{\tilde{t}=0}^1 a_{\mathcal{A},\tilde{t}} C_{00,00}^{20,\til... ...t}} C_{11,00}^{11,\tilde{t}}+e_{\mathcal{A},\tilde{t}} C_{11,11}^{11,\tilde{t}}$$

$$+f_{\mathcal{A},\tilde{t}} C_{11,20}^{11,\tilde{t}}+g_{\mathcal{A},\tilde{t}} C_{11,22}^{11,\tilde{t}}.$$ (17)

In Eq. (18), and in Eqs. (19) and (20) below, numerical coefficients $a_{\mathcal{A},\tilde{t}}$, $b_{\mathcal{A},\tilde{t}}$,..., express the EDF coupling constants as linear combinations of the finite-range pseudopotential parameters.

Similar expressions for the fourth-order (sixth-order) 444 (2020) coupling constants that are related to 30 (52) parameters of the finite-range pseudopotential can also be found in the supplemental material, according to the formulas

$$C_{a,\alpha,Q}^{a', \alpha',t,\mathcal{L}} = \sum_{\tilde{t}=0}^1 a_{\mathcal{A},\tilde{t}} C_{00,00}^{40,\til... ...t}} C_{11,00}^{31,\tilde{t}}+e_{\mathcal{A},\tilde{t}} C_{11,11}^{31,\tilde{t}}$$

$$+ f_{\mathcal{A},\tilde{t}} C_{11,20}^{31,\tilde{t}}+g_{\mathcal{... ...t}} C_{20,00}^{20,\tilde{t}}+j_{\mathcal{A},\tilde{t}} C_{20,20}^{20,\tilde{t}}$$

$$+k_{\mathcal{A},\tilde{t}} C_{20,22}^{22,\tilde{t}} +l_{\mathcal{... ...}} C_{22,20}^{22,\tilde{t}}+o_{\mathcal{A},\tilde{t}} C_{22,22}^{22,\tilde{t}},$$ (18)

and

$$C_{a,\alpha,Q}^{a', \alpha',t,\mathcal{L}} = \sum_{\tilde{t}=0}^1 a_{\mathcal{A},\tilde{t}} C_{00,00}^{60,\til... ...t}} C_{11,00}^{51,\tilde{t}}+e_{\mathcal{A},\tilde{t}} C_{11,11}^{51,\tilde{t}}$$

$$+ f_{\mathcal{A},\tilde{t}} C_{11,20}^{31,\tilde{t}}+g_{\mathcal{... ...t}} C_{20,00}^{40,\tilde{t}}+j_{\mathcal{A},\tilde{t}} C_{20,20}^{40,\tilde{t}}$$

$$+ k_{\mathcal{A},\tilde{t}} C_{20,22}^{42,\tilde{t}}+l_{\mathcal{... ...t}} C_{22,20}^{42,\tilde{t}}+o_{\mathcal{A},\tilde{t}} C_{22,22}^{40,\tilde{t}}$$

$$+p_{\mathcal{A},\tilde{t}} C_{22,22}^{42,\tilde{t}}+ q_{\mathcal{... ...}} C_{31,11}^{31,\tilde{t}}+ u_{\mathcal{A},\tilde{t}} C_{31,20}^{31,\tilde{t}}$$

$$+ v_{\mathcal{A},\tilde{t}} C_{31,22}^{31,\tilde{t}}+w_{\mathcal{... ...t}} C_{33,11}^{33,\tilde{t}}+z_{\mathcal{A},\tilde{t}} C_{33,20}^{33,\tilde{t}}$$

$$+ \beta_{\mathcal{A},\tilde{t}} C_{33,22}^{33,\tilde{t}},$$ (19)

respectively.

Besides the coupling constants expressed through the linear combinations in Eqs. (19) and (20) there are 6 (38) more coupling constants for each isospin channel at fourth-order (sixth-order), corresponding to EDF terms that are allowed by the general symmetries of the functional, but that are forced to be equal to zero once the functional is derived from the pseudopotential. Therefore, the number of terms of the fourth-order (sixth-order) EDF when not derived from the pseudopotential is 228 (1048) in each isospin channel. The complete lists of these fourth-order and sixth-order vanishing coupling constants are included in the supplemental material. Indeed as we verified for the quasilocal EDF obtained from the zero-range pseudopotential, also for the finite-range one, the explicit Galilean invariance of the pseudopotential (3) yields the Galilean invariance of the derived nonlocal functional. The coupling constants of the terms which are not Galilean invariant turn out to be equal to zero when the derivation of the functional from the pseudopotential is performed. Moreover using expressions (18)-(20), one can, in principle, determine constraints between the nonvanishing coupling constants of the EDF that enforce Galilean symmetry. For example, at NLO this task can be realized by inverting two subsets of 14 linear combinations (18), chosen arbitrarily within each local and nonlocal part separately. The only requirement is in selecting linear combinations that lead to a nonsingular matrix. In this way, one can express the remaining coupling constants through the 14 selected as independent ones. The two sets of linear combinations, one for the local component and one for the nonlocal component of the EDF, form together the constraints on the functional obtained by imposing the Galilean invariance.

These linear combinations form subsets of equivalent relations in each isospin space. This is owing to the specularity of the EDF with respect to the isospin. In the local and nonlocal sectors of the functionals, this is owing to the fact that linear combinations between EDF terms can be established only for terms with the same kind of densities, local or nonlocal.

Then, considering at NLO both isospin channels ($t=0, 1$) and both local and nonlocal sectors of the functional ( $\mathcal{L}=L, N$), 44 dependent coupling constants are equal to specific linear combinations of the 28 ones, that is,

$$C_{00,1110,0}^{00,1110,t,\mathcal{L}} = -\frac{C_{00,2011,1}^{00,0011,t,\mathcal{L}}}{3}-\frac{\sqrt{5} C_{00,2211,1}^{00,0011,t,\mathcal{L}}}{3} ,$$ (20)

$$C_{00,1111,1}^{00,1111,t,\mathcal{L}} = -\frac{C_{00,2011,1}^{00,0011,t,\mathcal{L}}}{\sqrt{3}}+\frac{1}{2} \sqrt{\frac{5}{3}} C_{00,2211,1}^{00,0011,t,\mathcal{L}} ,$$ (21)

$$C_{00,1112,2}^{00,1112,t,\mathcal{L}} = -\frac{\sqrt{5} C_{00,2011,1}^{00,0011,t,\mathcal{L}}}{3}-\frac{C_{00,2211,1}^{00,0011,t,\mathcal{L}}}{6} ,$$ (22)

$$C_{00,2000,0}^{00,0000,t,\mathcal{L}} = -C_{00,1101,1}^{00,1101,t,\mathcal{L}} ,$$ (23)

$$C_{11,0000,1}^{00,1111,t,\mathcal{L}} = -C_{11,1101,1}^{00,0011,t,\mathcal{L}} ,$$ (24)

$$C_{11,0011,1}^{00,1101,t,\mathcal{L}} = C_{11,1101,1}^{00,0011,t,\mathcal{L}} ,$$ (25)

$$C_{11,0011,1}^{11,0011,t,\mathcal{L}} = -\frac{2 C_{11,0011,0}^{11,0011,t,\mathcal{L}}}{\sqrt{3}}+\sqrt{\frac{5}{3}} C_{11,0011,2}^{11,0011,t,\mathcal{L}} ,$$ (26)

$$C_{11,1111,0}^{00,0000,t,\mathcal{L}} = C_{11,1101,1}^{00,0011,t,\mathcal{L}} ,$$ (27)

$$C_{20,0000,0}^{00,0000,t,\mathcal{L}} = -C_{11,0000,1}^{11,0000,t,\mathcal{L}} ,$$ (28)

$$C_{20,0011,1}^{00,0011,t,\mathcal{L}} = \frac{C_{11,0011,0}^{11,0011,t,\mathcal{L}}}{3}-\frac{2 \sqrt{5} C_{11,0011,2}^{11,0011,t,\mathcal{L}}}{3} ,$$ (29)

$$C_{22,0011,1}^{00,0011,t,\mathcal{L}} = -\frac{2 \sqrt{5} C_{11,0011,0}^{11,0011,t,\mathcal{L}}}{3}+\frac{2 C_{11,0011,2}^{11,0011,t,\mathcal{L}}}{3}.$$ (30)

Equations (21)-(25) are formally equivalent to the constraints found for the same symmetry in the case of quasilocal EDF [9]. This is so, because the corresponding terms have exactly the same tensor forms as those found in the functional stemming from the zero-range pseudopotential. The remaining linear combinations (26)-(31) have no analogs in the quasilocal EDF, because they involve terms that are absent in the quasilocal version of the functional. We also note that there are no terms corresponding to unrestricted coupling constants; indeed, the Galilean symmetry forces all the nonvanishing coupling constants to enter in specific linear combinations.