Relations between the pseudopotential and Energy Density Functional with conserved spherical symmetry

In this Section, we assume the spherical, space-inversion, and time-reversal symmetries of the EDF, see Sec. IV of Ref. [2]. In this way we make our results applicable to the simplest case of spherical even-even nuclei. Below we fully show explicit results for the case of gauge symmetry conserved, whereas the full results pertaining to the case of Galilean symmetry are given in the supplemental material [24].

When the gauge symmetry is imposed on the EDF and the isospin degree of freedom is taken into account, we have 8 independent spherical EDF terms at second order, 6 at fourth order, and 6 at sixth order. The 8 corresponding second-order coupling constants can then be expressed by the 7 second-order pseudopotential parameters. Similarly, both at forth and sixth orders, 6 coupling constants can then be expressed by 6 pseudopotential parameters.

As is well known, at second order the isoscalar and isovector spin-orbit coupling constants depend both on one spin-orbit pseudopotential parameter, namely,

$$C_{11,1111}^{0000,0} = -\frac{3}{4}C_{11,11}^{11} ,$$ (34)

$$C_{11,1111}^{0000,1} = -\frac{\sqrt{3}}{4}C_{11,11}^{11} ,$$ (35)

which gives one constraint on the spin-orbit coupling constants,

$$C_{11,1111}^{0000,1}=\frac{1}{\sqrt{3}}C_{11,1111}^{0000,0} .$$ (36)

The remaining 6 spherical EDF coupling constants expressed through 6 pseudopotential parameters are given in Table 21. Similar expressions at fourth and sixth orders are given in Tables 22 and 23. As in Sec. 3.1, from these results we can obtain the inverse expressions relating the parameters of the pseudopotential to the coupling constants of the spherical EDF; these are given in Tables 24-26.

Table 21: Second-order coupling constants of the EDF as functions of parameters of the pseudopotential when the gauge and the spherical symmetries are simultaneously imposed, according to the formula $C_{mI,n L v J}^{n' L' v' J', t}= A(a C_{00,00}^{20}+b C_{00,20}^{20}+c C_{00,22}^{22}+d C_{11,00}^{11}+e C_{11,20}^{11}+f C_{11,22}^{11})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$
$C_{20,0000}^{0000,0}$	$\frac{1}{32}$	$-$3	$-\sqrt{3}$	0	5	$-\sqrt{3}$	0
$C_{20,0000}^{0000,1}$	$\frac{1}{32}$	$\sqrt{3}$	$-3$	0	$\sqrt{3}$	$-3$	0
$C_{00,2000}^{0000,0}$	$\frac{1}{16}$	3	$\sqrt{3}$	0	5	$-\sqrt{3}$	0
$C_{00,2000}^{0000,1}$	$\frac{1}{16}$	$-\sqrt{3}$	$3$	0	$\sqrt{3}$	$-3$	0
$C_{00,1111}^{1111,0}$	$\frac{1}{48}$	3	$5\sqrt{3}$	$-\sqrt{15}$	$-$3	$3\sqrt{3}$	$-3\sqrt{15}$
$C_{00,1111}^{1111,1}$	$\frac{1}{16}$	$\sqrt{3}$	1	$\sqrt{5}$	$-\sqrt{3}$	$-$1	$-\sqrt{5}$

Table 22: Same as in Table 21 but for the fourth-order coupling constants of the EDF, according to the formula $C_{mI,n L v J}^{n' L' v' J', t}= A(a C_{11,00}^{31}+b C_{11,20}^{31}+c C_{11,22}^{33}+d C_{22,00}^{22}+e C_{22,20}^{22}+f C_{22,22}^{22})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$
$C_{40,0000}^{0000,0}$	$\frac{1}{640}$	$-$25	$5\sqrt{3}$	0	$18\sqrt{5}$	$6\sqrt{15}$	0
$C_{40,0000}^{0000,1}$	$\frac{1}{640}$	$-5\sqrt{3}$	15	0	$-6\sqrt{15}$	$18\sqrt{5}$	0
$C_{00,2202}^{2202,0}$	$\frac{1}{96}$	$5\sqrt{5}$	$-\sqrt{15}$	0	18	$6\sqrt{3}$	0
$C_{00,2202}^{2202,1}$	$\frac{1}{96}$	$\sqrt{15}$	$-3\sqrt{5}$	0	$-6\sqrt{3}$	18	0
$C_{00,3111}^{1111,0}$	$\frac{1}{80}$	$-$5	$5\sqrt{3}$	$-15\sqrt{7}$	$6\sqrt{5}$	$10\sqrt{15}$	$-2\sqrt{105}$
$C_{00,3111}^{1111,1}$	$\frac{1}{80}$	$-5\sqrt{3}$	$-$5	$-5\sqrt{21}$	$6\sqrt{15}$	$6\sqrt{5}$	$6\sqrt{35}$

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Table 23: Same as in Table 21 but for the sixth-order coupling constants of the EDF, according to the formula $C_{mI,n L v J}^{n' L' v' J', t}= A(a C_{11,22}^{53}+b C_{22,00}^{42}+c C_{22,20}^{42}+d C_{22,22}^{44}+e C_{33,00}^{33}+f C_{33,20}^{33})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$
$C_{60,0000}^{0000,0}$	$\frac{1}{4480}$	0	$-21\sqrt{5}$	$-7\sqrt{15}$	0	$75\sqrt{21}$	$-45\sqrt{7}$
$C_{60,0000}^{0000,1}$	$\frac{1}{4480}$	0	$7\sqrt{15}$	$-21\sqrt{5}$	0	$45\sqrt{7}$	$-45\sqrt{21}$
$C_{00,6000}^{0000,0}$	$\frac{1}{2240}$	0	$21\sqrt{5}$	$7\sqrt{15}$	0	$75\sqrt{21}$	$-45\sqrt{7}$
$C_{00,6000}^{0000,1}$	$\frac{1}{2240}$	0	$-7\sqrt{15}$	$21\sqrt{5}$	0	$45\sqrt{7}$	$-45\sqrt{21}$
$C_{00,3111}^{3111,0}$	$\frac{1}{800}$	$-135\sqrt{7}$	$21\sqrt{5}$	$35\sqrt{15}$	$-105\sqrt{3}$	$-45\sqrt{21}$	$135\sqrt{7}$
$C_{00,3111}^{3111,1}$	$-\frac{3}{800}$	$15\sqrt{21}$	$-7\sqrt{15}$	$-7\sqrt{5}$	$-$105	$45\sqrt{7}$	$15\sqrt{21}$

Table: Second-order parameters of the pseudopotential (spin-orbit term not included) as functions of the coupling constants of the EDF when the gauge and the spherical symmetries are simultaneously imposed, according to the formula $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}= a C_{20,0000}^{0000,0... ...0000,0}+ d C_{00,2000}^{0000,1}+ e C_{00,1111}^{1111,0}+ f C_{00,1111}^{1111,1}$.

	$a$	$b$	$c$	$d$	$e$	$f$
$C_{00,00}^{20}$	$-$4	$\frac{4}{\sqrt{3}}$	2	$-\frac{2}{\sqrt{3}}$	0	0
$C_{00,20}^{20}$	$-\frac{4}{\sqrt{3}}$	$-$4	$\frac{2}{\sqrt{3}}$	2	0	0
$C_{00,22}^{22}$	$\frac{16}{\sqrt{15}}$	$-\frac{16}{\sqrt{5}}$	$\frac{4}{\sqrt{15}}$	$-\frac{4}{\sqrt{5}}$	$-4\sqrt{\frac{3}{5}}$	$\frac{12}{\sqrt{5}}$
$C_{11,00}^{11}$	4	$-\frac{4}{\sqrt{3}}$	2	$-\frac{2}{\sqrt{3}}$	0	0
$C_{11,20}^{11}$	$\frac{4}{\sqrt{3}}$	$-\frac{20}{3}$	$\frac{2}{\sqrt{3}}$	$-\frac{10}{3}$	0	0
$C_{11,22}^{11}$	$-\frac{16}{\sqrt{15}}$	$-\frac{16}{3\sqrt{5}}$	$\frac{4}{\sqrt{15}}$	$\frac{4}{3\sqrt{5}}$	$-4\sqrt{\frac{3}{5}}$	$-\frac{4}{\sqrt{5}}$

Table: Fourth-order parameters of the pseudopotential as functions of the coupling constants of the EDF when the gauge and the spherical symmetries are simultaneously imposed, according to the formula $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}= a C_{40,0000}^{0000,0... ...2202,0}+ d C_{00,2202}^{2202,1}+ e C_{00,3111}^{1111,0}+ f C_{00,3111}^{1111,1}$.

	$a$	$b$	$c$	$d$	$e$	$f$
$C_{11,00}^{31}$	$-16$	$\frac{16}{\sqrt{3}}$	$\frac{12}{\sqrt{5}}$	$-4\sqrt{\frac{3}{5}}$	0	0
$C_{11,20}^{31}$	$-\frac{16}{\sqrt{3}}$	$\frac{80}{3}$	$4\sqrt{\frac{3}{5}}$	$-4\sqrt{5}$	0	0
$C_{11,22}^{33}$	$\frac{64}{3\sqrt{7}}$	$\frac{64}{3\sqrt{21}}$	$\frac{8}{\sqrt{35}}$	$\frac{8}{\sqrt{105}}$	$-\frac{4}{\sqrt{7}}$	$-\frac{4}{\sqrt{21}}$
$C_{22,00}^{22}$	$\frac{8\sqrt{5}}{3}$	$-\frac{8}{3}\sqrt{\frac{5}{3}}$	2	$-\frac{2}{\sqrt{3}}$	0	0
$C_{22,20}^{22}$	$\frac{8}{3}\sqrt{\frac{5}{3}}$	$\frac{8\sqrt{5}}{3}$	$\frac{2}{\sqrt{3}}$	2	0	0
$C_{22,22}^{22}$	$-\frac{32}{3}\sqrt{\frac{5}{21}}$	$\frac{32}{3}\sqrt{\frac{5}{7}}$	$\frac{4}{\sqrt{21}}$	$-\frac{4}{\sqrt{7}}$	$-2\sqrt{\frac{5}{21}}$	$2\sqrt{\frac{5}{7}}$

Table: Same as in Table 25 but for the sixth-order parameters of the pseudopotential, according to the formula $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}= a C_{60,0000}^{0000,0... ...0000,0}+ d C_{00,6000}^{0000,1}+ e C_{00,3111}^{3111,0}+ f C_{00,3111}^{3111,1}$.

	$a$	$b$	$c$	$d$	$e$	$f$
$C_{11,22}^{53}$	$-\frac{64\sqrt{7}}{9}$	$-\frac{64}{9}\sqrt{\frac{7}{3}}$	$\frac{16\sqrt{7}}{9}$	$\frac{16}{9}\sqrt{\frac{7}{3}}$	$-\frac{40}{9\sqrt{7}}$	$-\frac{40}{9\sqrt{21}}$
$C_{22,00}^{42}$	$-16\sqrt{5}$	$16\sqrt{\frac{5}{3}}$	$8\sqrt{5}$	$-8\sqrt{\frac{5}{3}}$	0	0
$C_{22,20}^{42}$	$-16\sqrt{\frac{5}{3}}$	$-16\sqrt{5}$	$8\sqrt{\frac{5}{3}}$	$8\sqrt{5}$	0	0
$C_{22,22}^{44}$	$\frac{64}{3\sqrt{3}}$	$-\frac{64}{3}$	$\frac{16}{3\sqrt{3}}$	$-\frac{16}{3}$	$-\frac{40}{21\sqrt{3}}$	$\frac{40}{21}$
$C_{33,00}^{33}$	$\frac{16}{3}\sqrt{\frac{7}{3}}$	$-\frac{16\sqrt{7}}{9}$	$\frac{8}{3}\sqrt{\frac{7}{3}}$	$-\frac{8}{9}\sqrt{7}$	0	0
$C_{33,20}^{33}$	$\frac{16\sqrt{7}}{9}$	$-\frac{80}{9}\sqrt{\frac{7}{3}}$	$\frac{8\sqrt{7}}{9}$	$-\frac{40}{9}\sqrt{\frac{7}{3}}$	0	0

At second order, the gauge and Galilean symmetries are equivalent to one another [2]. When at higher orders the Galilean invariance is imposed on the spherical EDF, we have at fourth (sixth) order 18 (32) independent terms, of which 4 (8) are of the spin-orbit character. It turns out that, in the same way as for the second order, the higher-order spin-orbit coupling constants are related only to the spin-orbit pseudopotential parameters. Namely, at fourth order we have

$$C_{31,1111}^{0000,0} = \frac{3}{16}C_{11,11}^{31}-\frac{1}{8}\sqrt{\frac{3}{5}}C_{22,11}^{22} ,$$ (37)

$$C_{31,1111}^{0000,1} = \frac{1}{16}\sqrt{3}C_{11,11}^{31}+\frac{3}{8}\sqrt{\frac{1}{5}} C_{22,11}^{22} ,$$ (38)

$$C_{11,3111}^{0000,0} = -\frac{3}{16}C_{11,11}^{31}-\frac{1}{8}\sqrt{\frac{3}{5}}C_{22,11}^{22} ,$$ (39)

$$C_{11,3111}^{0000,1} = -\frac{1}{16}\sqrt{3}C_{11,11}^{31}+\frac{3}{8}\sqrt{\frac{1}{5}} C_{22,11}^{22} ,$$ (40)

which gives the following constraints on the spin-orbit coupling constants:

$$C_{31,1111}^{0000,1} = -\frac{1}{\sqrt{3}}C_{31,1111}^{0000,0} - \frac{2}{\sqrt{3}}C_{11,3111}^{0000,0} ,$$ (41)

$$C_{11,3111}^{0000,1} = -\frac{2}{\sqrt{3}}C_{31,1111}^{0000,0}-\frac{1}{\sqrt{3}}C_{11,3111}^{0000,0} ,$$ (42)

and at sixth order we have

$$C_{51,1111}^{0000,0}= -\frac{3}{64}C_{11,11}^{51}+\frac{1}{32}\sqrt{\frac{3}{5}} C_{22,11}^{42}-\frac{3}{64}C_{31,11}^{31}$$

$$-\frac{27}{80}\sqrt{\frac{1}{14}}C_{33,11}^{33},$$ (43)

$$C_{51,1111}^{0000,1}= -\frac{\sqrt{3}}{64}C_{11,11}^{51}- \frac{3}{32}\sqrt{\frac{1}{5}}C_{22,11}^{42}-\frac{\sqrt{3}}{64}C_{31,11}^{31}$$

$$-\frac{9}{80}\sqrt{\frac{3}{14}}C_{33,11}^{33},$$ (44)

$$C_{11,5111}^{0000,0}= -\frac{3}{64}C_{11,11}^{51}-\frac{1}{32}\sqrt{\frac{3}{5}}C_{22,11}^{42} -\frac{3}{64}C_{31,11}^{31}$$

$$-\frac{27}{80}\sqrt{\frac{1}{14}}C_{33,11}^{33},$$ (45)

$$C_{11,5111}^{0000,1}= -\frac{\sqrt{3}}{64}C_{11,11}^{51}+\frac{3}{32}\sqrt{\frac{1}{5}}C_{22,11}^{42}-\frac{\sqrt{3}}{64}C_{31,11}^{31}$$

$$-\frac{9}{80}\sqrt{\frac{3}{14}}C_{33,11}^{33},$$ (46)

$$C_{31,3111}^{0000,0}= \frac{21}{160}C_{11,11}^{51}+\frac{9}{160}C_{31,11}^{31}$$

$$-\frac{27}{200}\sqrt{\frac{7}{2}}C_{33,11}^{33},$$ (47)

$$C_{31,3111}^{0000,1}= \frac{7}{160}\sqrt{3}C_{11,11}^{51} +\frac{3}{160}\sqrt{3}C_{31,11}^{31}$$

$$-\frac{9}{200}\sqrt{\frac{21}{2}}C_{33,11}^{33},$$ (48)

$$C_{33,3313}^{0000,0}= \frac{1}{24}\sqrt{\frac{7}{2}}C_{11,11}^{51}-\frac{1}{24}\sqrt{\frac{7}{2}}C_{31,11}^{31}$$

$$+\frac{3}{80}C_{33,11}^{33},$$ (49)

$$C_{33,3313}^{0000,1}= \frac{1}{24}\sqrt{\frac{7}{6}}C_{11,11}^{51}-\frac{1}{24}\sqrt{\frac{7}{6}}C_{31,11}^{31}$$

$$+\frac{\sqrt{3}}{80}C_{33,11}^{33} ,$$ (50)

which gives the constraints:

$$C_{51,1111}^{0000,1} = -\frac{1}{\sqrt{3}}C_{51,1111}^{0000,0} + \frac{2}{\sqrt{3}}C_{11,5111}^{0000,0} ,$$ (51)

$$C_{11,5111}^{0000,1} = \frac{2}{\sqrt{3}}C_{51,1111}^{0000,0}-\frac{1}{\sqrt{3}}C_{11,5111}^{0000,0} ,$$ (52)

$$C_{31,3111}^{0000,1} = \frac{1}{\sqrt{3}}C_{31,3111}^{0000,0} ,$$ (53)

$$C_{33,3313}^{0000,1} = \frac{1}{\sqrt{3}}C_{33,3313}^{0000,0} .$$ (54)

If now we consider the Galilean-invariant and spherical EDF without spin-orbit terms, we obtain at fourth (sixth) order 1 (2) possible constraints among the remaining 14 (24) coupling constants related to the remaining 13 (22) parameters of the pseudopotential. These results are available in the supplemental material [24]. Of course, such constraints can be imposed in very many different ways. We have checked that, in fact, not any of the 1 (2) coupling constants of the fourth (sixth) order spherical EDF can be considered as being dependent on all the other coupling constants. In the supplemental material we present one example of a possible choice, whereby at fourth (sixth) order the coupling constants $C_{22,1111}^{1111,1}$ ( $C_{00,3111}^{3111,0}$ and $C_{00,3111}^{3111,1}$) are selected to be dependent. A comparison between the numbers of terms of the Galilean-invariant and gauge-invariant spherical EDF with and without constraints coming from the reference to the pseudopotential is plotted in Fig. 2.

Figure 2: (Color online) Number of terms of the spherical EDF that is related to a pseudopotential (solid lines). Full squares and circles show results for the Galilean and gauge invariance, respectively. For reference, dashed lines with open squares and circles show the corresponding results for the general spherical EDF studied in Ref. [2].