$L=-5\,\mathrm{MeV}$), the radius $R_{1.4}$ and dimensionless tidal
deformability $\Lambda_{1.4}$ of $1.4M_{\odot}$ neutron star and the maximum
mass $M_{\mathrm{TOV}}$. The parameter sets with $5\,\mathrm{MeV}\leq L\leq
85\,\mathrm{MeV}$ satisfy the constraint of $\Lambda_{1.4}\leqslant 580$.

##  VII SUMMARY AND OUTLOOK

Based on the N3LO Skyrme pseudopotential, we have obtained new extended Skyrme
interactions by modifying the density-dependent term in the spirit of the
Fermi momentum expansion. The Haimiltonian density and single-nucleon
potential of the new extended Skyrme effective interactions are derived in the
Hartree-Fock approximation. The momentum dependence of the single-nucleon
potential is regulated to fit the empirical nucleon optical potential (and its
extrapolation), and so the new Skyrme interactions can be used in transport
model simulations for HICs up to $1.5\,\mathrm{GeV}$/nucleon. At the same
time, the extension of the density-dependent term can provide very flexible
freedom to adjust the density dependence of the symmetry energy as well as the
high density behavior of symmetric nuclear matter, and thus can essentially
satisfy all the constraints on the neutron stars from the multimessenger
astrophysical observations. We have constructed a series of interactions,
denoted as SP6X, with the density slope parameter $L$ of the symmetry energy
ranging from $-5\,\mathrm{MeV}$ to $125\,\mathrm{MeV}$, and we find the
following results: (i) To simultaneously satisfy constraints from microscopic
calculations on the EOS of pure neutron matter $E_{\mathrm{PNM}}(\rho)$ and
astrophysical observations of neutron star mass-radius relations of both PSR
J0030+0451 and PSR J0740+6620, $L$ should fall within the range of
$5\,\mathrm{MeV}\leq L\leq 75\,\mathrm{MeV}$; (ii) To further additionally
describe the neutron star with small mass and radius in CCO of HESS J1731-347,
a range of $-5\,\mathrm{MeV}\leq L\leq 35\,\mathrm{MeV}$ is necessary; (iii) A
distinct peak structure in the squared sound speed for neutron star matter
emerges for interactions with a soft symmetry energy around saturation
density, especially when $-5\,\mathrm{MeV}\leq L\leq 25\,\mathrm{MeV}$.

In addition, we have constructed a versatile parameter set family consisting
of parameter sets with different density dependencies of the symmetry energy
as well as various momentum dependencies of the single-nucleon potential and
the symmetry potential (i.e., various isoscalar and isovector nucleon
effective masses). This parameter set family will be useful for investigating
in a more transparent way the effects of the symmetry energy and nucleon
effective masses, and serves as a useful theoretical tool to extract the
density dependence of the symmetry energy and the nucleon effective masses by
analyzing the data from different nuclear experiments and astrophysical
observations.

The unified non-relativistic nuclear energy density functional constructed in
the present work can be simultaneously applied in the study of finite nuclei,
HICs and neutron stars. Next we will also extend our model to finite
temperature to explore the thermal effects on the nuclear EOS and construct
the corresponding EOS tables. These EOS tables can be used to study the
liquid-gas phase transition as well as to simulate core-collapse supernovae
and binary neutron star mergers. Our work will be useful for the determination
of the high density behavior of the symmetry energy and in-medium nucleon
effective masses by analyzing various data from nuclear experiments and
multimessenger observations of neutron stars in a single unified nuclear
energy density functional.

###### Acknowledgements.

This work was supported in part by the National Natural Science Foundation of
China under Grant Nos. 12235010 and 11625521, and the National SKA Program of
China No. 2020SKA0120300.

##  Appendix A Expressions of Macroscopic quantities

The values of $E_{0}(\rho)$, $P_{\mathrm{SNM}}(\rho)$, $K_{0}(\rho)$,
$J_{0}(\rho)$, $a_{2}(\rho)$, $a_{4}(\rho)$, $a_{6}(\rho)$, $b_{2}(\rho)$,
$b_{4}(\rho)$, $b_{6}(\rho)$, $E_{\mathrm{sym}}(\rho)$, $L(\rho)$,
$K_{\mathrm{sym}}(\rho)$ and $J_{\mathrm{sym}}(\rho)$ at arbitrary density,
can be expressed as a sum of powers of $k_{F}$ ($k_{F}=a\rho^{1/3}$ with
$a=(3\pi^{2}/2)^{1/3}$), where the coefficients can be expressed in terms of
linear combinations of 14 parameters: $t_{0}$, $t_{3}^{[1]}$, $t_{3}^{[3]}$,
$t_{3}^{[5]}$, $C^{[2]}$, $D^{[2]}$, $C^{[4]}$, $D^{[4]}$, $C^{[6]}$,
$D^{[6]}$, $t_{0}x_{0}$, $t_{3}^{[1]}x_{3}^{[1]}$, $t_{3}^{[3]}x_{3}^{[3]}$
and $t_{3}^{[5]}x_{3}^{[5]}$. Note that $x_{0}$ ($x_{3}^{[1]}$, $x_{3}^{[3]}$
and $x_{3}^{[5]}$) are coupled with $t_{0}$ ($t_{3}^{[1]}$, $t_{3}^{[3]}$ and
$t_{3}^{[5]}$).

At arbitrary density $\rho$, the 14 macroscopic quantities can be expressed as
linear combinations of the 14 parameters, i.e.,

$\mathbf{Q}(\rho)=\mathbf{M}(\rho)\cdot\mathbf{P}+\mathbf{C}(\rho),$ (53)

with

$\mathbf{Q}(\rho)=\left[E_{0}(\rho),P_{\mathrm{SNM}}(\rho),K_{0}(\rho),J_{0}(\rho),a_{2}(\rho),a_{4}(\rho),a_{6}(\rho),b_{2}(\rho),b_{4}(\rho),b_{6}(\rho),E_{\mathrm{sym}}(\rho),L(\rho),K_{\mathrm{sym}}(\rho),J_{\mathrm{sym}}(\rho)\right]^{\mathrm{T}}$
(54)

being the vector of the 14 quantities at $\rho$,

$\mathbf{P}=\left[t_{0},t_{3}^{[1]},t_{3}^{[3]},t_{3}^{[5]},C^{[2]},D^{[2]},C^{[4]},D^{[4]},C^{[6]},D^{[6]},t_{0}x_{0},t_{3}^{[1]}x_{3}^{[1]},t_{3}^{[3]}x_{3}^{[3]},t_{3}^{[5]}x_{3}^{[5]}\right]^{\mathrm{T}}$
(55)

being the vector of the 14 parameters,

$\mathbf{C}(\rho)=\left[\frac{3\hbar^{2}k_{F}^{2}}{10m},\frac{2\hbar^{2}k_{F}^{5}}{15m\pi^{2}},-\frac{3\hbar^{2}k_{F}^{2}}{5m},\frac{12\hbar^{2}k_{F}^{2}}{5m},0,0,0,0,0,0,\frac{\hbar^{2}k_{F}^{2}}{6m},\frac{\hbar^{2}k_{F}^{2}}{3m},-\frac{\hbar^{2}k_{F}^{2}}{3m},\frac{4\hbar^{2}k_{F}^{2}}{3m}\right]^{\mathrm{T}}$
(56)

being the parameter-irrelevant term, and

$\mathbf{M}(\rho)=\left(\begin{array}[]{cccccccccccccc}\frac{3\rho}{8}&\frac{\rho^{4/3}}{16}&\frac{\rho^{2}}{16}&\frac{\rho^{8/3}}{16}&\frac{k_{F}^{5}}{20\pi^{2}}&\frac{k_{F}^{5}}{40\pi^{2}}&\frac{3k_{F}^{7}}{70\pi^{2}}&\frac{3k_{F}^{7}}{140\pi^{2}}&\frac{8k_{F}^{9}}{45\pi^{2}}&\frac{4k_{F}^{9}}{45\pi^{2}}&0&0&0&0\\\
\frac{3\rho^{2}}{8}&\frac{\rho^{7/3}}{12}&\frac{\rho^{3}}{8}&\frac{\rho^{11/3}}{6}&\frac{k_{F}^{8}}{18\pi^{4}}&\frac{k_{F}^{8}}{36\pi^{4}}&\frac{k_{F}^{10}}{15\pi^{4}}&\frac{k_{F}^{10}}{30\pi^{4}}&\frac{16k_{F}^{12}}{45\pi^{4}}&\frac{8k_{F}^{12}}{45\pi^{4}}&0&0&0&0\\\
0&\frac{\rho^{4/3}}{4}&\frac{9\rho^{2}}{8}&\frac{5\rho^{8/3}}{2}&\frac{k_{F}^{5}}{2\pi^{2}}&\frac{k_{F}^{5}}{4\pi^{2}}&\frac{6k_{F}^{7}}{5\pi^{2}}&\frac{3k_{F}^{7}}{5\pi^{2}}&\frac{48k_{F}^{9}}{5\pi^{2}}&\frac{24k_{F}^{9}}{5\pi^{2}}&0&0&0&0\\\
0&-\frac{\rho^{4/3}}{2}&0&5\rho^{8/3}&-\frac{k_{F}^{5}}{2\pi^{2}}&-\frac{k_{F}^{5}}{4\pi^{2}}&\frac{6k_{F}^{7}}{5\pi^{2}}&\frac{3k_{F}^{7}}{5\pi^{2}}&\frac{144k_{F}^{9}}{5\pi^{2}}&\frac{72k_{F}^{9}}{5\pi^{2}}&0&0&0&0\\\
0&0&0&0&\frac{k_{F}^{3}}{12\pi^{2}}&\frac{k_{F}^{3}}{24\pi^{2}}&\frac{k_{F}^{5}}{12\pi^{2}}&\frac{k_{F}^{5}}{24\pi^{2}}&\frac{k_{F}^{7}}{4\pi^{2}}&\frac{k_{F}^{7}}{8\pi^{2}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{k_{F}^{3}}{24\pi^{2}}&\frac{k_{F}^{3}}{48\pi^{2}}&\frac{7k_{F}^{5}}{20\pi^{2}}&\frac{7k_{F}^{5}}{40\pi^{2}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&\frac{k_{F}^{3}}{12\pi^{2}}&\frac{k_{F}^{3}}{24\pi^{2}}&0&0&0&0\\\
0&0&0&0&0&\frac{k_{F}^{3}}{24\pi^{2}}&0&\frac{5k_{F}^{5}}{72\pi^{2}}&0&\frac{7k_{F}^{7}}{24\pi^{2}}&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{k_{F}^{3}}{48\pi^{2}}&0&\frac{7k_{F}^{5}}{24\pi^{2}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&\frac{k_{F}^{3}}{24\pi^{2}}&0&0&0&0\\\
-\frac{\rho}{8}&-\frac{\rho^{4/3}}{48}&-\frac{\rho^{2}}{48}&-\frac{\rho^{8/3}}{48}&\frac{k_{F}^{5}}{36\pi^{2}}&\frac{k_{F}^{5}}{18\pi^{2}}&\frac{k_{F}^{7}}{18\pi^{2}}&\frac{k_{F}^{7}}{12\pi^{2}}&\frac{2k_{F}^{9}}{5\pi^{2}}&\frac{8k_{F}^{9}}{15\pi^{2}}&-\frac{\rho}{4}&-\frac{\rho^{4/3}}{24}&-\frac{\rho^{2}}{24}&-\frac{\rho^{8/3}}{24}\\\
-\frac{3\rho}{8}&-\frac{\rho^{4/3}}{12}&-\frac{\rho^{2}}{8}&-\frac{\rho^{8/3}}{6}&\frac{5k_{F}^{5}}{36\pi^{2}}&\frac{5k_{F}^{5}}{18\pi^{2}}&\frac{7k_{F}^{7}}{18\pi^{2}}&\frac{7k_{F}^{7}}{12\pi^{2}}&\frac{18k_{F}^{9}}{5\pi^{2}}&\frac{24k_{F}^{9}}{5\pi^{2}}&-\frac{3\rho}{4}&-\frac{\rho^{4/3}}{6}&-\frac{\rho^{2}}{4}&-\frac{\rho^{8/3}}{3}\\\
0&-\frac{\rho^{4/3}}{12}&-\frac{3\rho^{2}}{8}&-\frac{5\rho^{8/3}}{6}&\frac{5k_{F}^{5}}{18\pi^{2}}&\frac{5k_{F}^{5}}{9\pi^{2}}&\frac{14k_{F}^{7}}{9\pi^{2}}&\frac{7k_{F}^{7}}{3\pi^{2}}&\frac{108k_{F}^{9}}{5\pi^{2}}&\frac{144k_{F}^{9}}{5\pi^{2}}&0&-\frac{\rho^{4/3}}{6}&-\frac{3\rho^{2}}{4}&-\frac{5\rho^{8/3}}{3}\\\
0&\frac{\rho^{4/3}}{6}&0&-\frac{5\rho^{8/3}}{3}&-\frac{5k_{F}^{5}}{18\pi^{2}}&-\frac{5k_{F}^{5}}{9\pi^{2}}&\frac{14k_{F}^{7}}{9\pi^{2}}&\frac{7k_{F}^{7}}{3\pi^{2}}&\frac{324k_{F}^{9}}{5\pi^{2}}&\frac{432k_{F}^{9}}{5\pi^{2}}&0&\frac{\rho^{4/3}}{3}&0&-\frac{10\rho^{8/3}}{3}\\\
\end{array}\right)$ (57)

being the representation matrix.

In practice, we usually encounter the macroscopic quantities, and subsequently
employ them to inversely infer the values of parameters. From Eq. (53), we
obtain

$\mathbf{P}=\mathbf{M}^{-1}(\rho)\cdot\mathbf{Q}(\rho)-\mathbf{C}^{\prime}(\rho)$
(58)

where

$\displaystyle\mathbf{C}^{\prime}(\rho)$
$\displaystyle=\mathbf{M}^{-1}(\rho)\cdot\mathbf{C}(\rho)$ (59)
$\displaystyle=\left[\frac{64a^{2}\hbar^{2}}{25m\rho^{1/3}},-\frac{72a^{2}\hbar^{2}}{5m\rho^{2/3}},\frac{24a^{2}\hbar^{2}}{5m\rho^{4/3}},-\frac{24a^{2}\hbar^{2}}{25m\rho^{2}},0,0,0,0,0,0,-\frac{256a^{2}\hbar^{2}}{75m\rho^{1/3}},\frac{96a^{2}\hbar^{2}}{5m\rho^{2/3}},-\frac{32a^{2}\hbar^{2}}{5m\rho^{4/3}},\frac{32a^{2}\hbar^{2}}{25m\rho^{2}}\right]^{\mathrm{T}},$

and

$\displaystyle\mathbf{M}^{-1}(\rho)=$
$\displaystyle\left(\begin{array}[]{cccccccccccccc}\frac{512}{15\rho}&-\frac{472}{15\rho^{2}}&\frac{8}{5\rho}&-\frac{8}{45\rho}&\frac{8a^{2}}{25\rho^{1/3}}&-\frac{208a^{3}k_{F}}{175}&\frac{10936a^{6}\rho}{1125}&0&0&0&0&0&0&0\\\
-\frac{288}{\rho^{4/3}}&\frac{288}{\rho^{7/3}}&-\frac{16}{\rho^{4/3}}&\frac{2}{\rho^{4/3}}&-\frac{36a^{2}}{5\rho^{2/3}}&\frac{792a^{4}}{35}&-\frac{3756a^{4}k_{F}^{2}}{25}&0&0&0&0&0&0&0\\\
\frac{128}{\rho^{2}}&-\frac{128}{\rho^{3}}&\frac{8}{\rho^{2}}&-\frac{4}{3\rho^{2}}&-\frac{24a^{2}}{5\rho^{4/3}}&-\frac{48a^{4}}{7\rho^{2/3}}&\frac{1928a^{6}}{15}&0&0&0&0&0&0&0\\\
-\frac{144}{5\rho^{8/3}}&\frac{144}{5\rho^{11/3}}&-\frac{8}{5\rho^{8/3}}&\frac{2}{5\rho^{8/3}}&\frac{12a^{2}}{25\rho^{2}}&-\frac{1032a^{4}}{175\rho^{4/3}}&-\frac{6684a^{6}}{125\rho^{2/3}}&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{8}{\rho}&-\frac{16a^{2}}{\rho^{1/3}}&\frac{216a^{3}k_{F}}{5}&-\frac{8}{\rho}&\frac{80a^{2}}{3\rho^{1/3}}&-\frac{392a^{3}k_{F}}{3}&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{16}{\rho}&-\frac{160a^{2}}{3\rho^{1/3}}&\frac{784a^{3}k_{F}}{3}&0&0&0&0\\\
0&0&0&0&0&\frac{16}{\rho}&-\frac{336a^{2}}{5\rho^{1/3}}&0&-\frac{16}{\rho}&\frac{112a^{2}}{\rho^{1/3}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&\frac{32}{\rho}&-\frac{224a^{2}}{\rho^{1/3}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{8}{\rho}&0&0&-\frac{8}{\rho}&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&\frac{16}{\rho}&0&0&0&0\\\
-\frac{256}{15\rho}&\frac{236}{15\rho^{2}}&-\frac{4}{5\rho}&\frac{4}{45\rho}&-\frac{32a^{2}}{75\rho^{1/3}}&\frac{384a^{3}k_{F}}{175}&-\frac{33728a^{6}\rho}{1125}&-\frac{4a^{2}}{5\rho^{1/3}}&\frac{24a^{3}k_{F}}{5}&-60a^{6}\rho&-\frac{256}{5\rho}&\frac{236}{15\rho}&-\frac{12}{5\rho}&\frac{4}{15\rho}\\\
\frac{144}{\rho^{4/3}}&-\frac{144}{\rho^{7/3}}&\frac{8}{\rho^{4/3}}&-\frac{1}{\rho^{4/3}}&\frac{48a^{2}}{5\rho^{2/3}}&-\frac{1376a^{4}}{35}&\frac{10848a^{4}k_{F}^{2}}{25}&\frac{18a^{2}}{\rho^{2/3}}&-92a^{4}&950a^{6}\rho^{2/3}&\frac{432}{\rho^{4/3}}&-\frac{144}{\rho^{4/3}}&\frac{24}{\rho^{4/3}}&-\frac{3}{\rho^{4/3}}\\\
-\frac{64}{\rho^{2}}&\frac{64}{\rho^{3}}&-\frac{4}{\rho^{2}}&\frac{2}{3\rho^{2}}&\frac{32a^{2}}{5\rho^{4/3}}&\frac{192a^{4}}{7\rho^{2/3}}&-\frac{6976a^{6}}{15}&\frac{12a^{2}}{\rho^{4/3}}&\frac{24a^{4}}{\rho^{2/3}}&-732a^{6}&-\frac{192}{\rho^{2}}&\frac{64}{\rho^{2}}&-\frac{12}{\rho^{2}}&\frac{2}{\rho^{2}}\\\
\frac{72}{5\rho^{8/3}}&-\frac{72}{5\rho^{11/3}}&\frac{4}{5\rho^{8/3}}&-\frac{1}{5\rho^{8/3}}&-\frac{16a^{2}}{25\rho^{2}}&\frac{2336a^{4}}{175\rho^{4/3}}&\frac{30432a^{6}}{125\rho^{2/3}}&-\frac{6a^{2}}{5\rho^{2}}&\frac{116a^{4}}{5\rho^{4/3}}&\frac{278a^{6}}{\rho^{2/3}}&\frac{216}{5\rho^{8/3}}&-\frac{72}{5\rho^{8/3}}&\frac{12}{5\rho^{8/3}}&-\frac{3}{5\rho^{8/3}}\\\
\end{array}\right).$ (74)

##  Appendix B Macroscopic quantities at saturation density

The values of the macroscopic quantities at saturation density
$\rho_{0}=0.16\,\mathrm{fm}^{-3}$ are used in the fitting procedure. Here we
provide the values of the elements in $\mathbf{M}(\rho_{0})$,
$\mathbf{C}(\rho_{0})$, $\mathbf{M}^{-1}(\rho_{0})$ and
$\mathbf{C}^{\prime}(\rho_{0})$.

$\mathbf{M}(\rho_{0})$ and $\mathbf{C}(\rho_{0})$ can be calculated as

$\displaystyle\mathbf{M}(\rho_{0})=$
$\displaystyle\left(\begin{array}[]{cccccccccccccc}6.00\mathrm{E}\text{-}2&5.43\mathrm{E}\text{-}3&1.60\mathrm{E}\text{-}3&4.72\mathrm{E}\text{-}4&2.13\mathrm{E}\text{-}2&1.07\mathrm{E}\text{-}2&3.25\mathrm{E}\text{-}2&1.62\mathrm{E}\text{-}2&2.39\mathrm{E}\text{-}1&1.20\mathrm{E}\text{-}1&0&0&0&0\\\
9.60\mathrm{E}\text{-}3&1.16\mathrm{E}\text{-}3&5.12\mathrm{E}\text{-}4&2.01\mathrm{E}\text{-}4&5.69\mathrm{E}\text{-}3&2.84\mathrm{E}\text{-}3&1.21\mathrm{E}\text{-}2&6.06\mathrm{E}\text{-}3&1.15\mathrm{E}\text{-}1&5.75\mathrm{E}\text{-}2&0&0&0&0\\\
0&2.17\mathrm{E}\text{-}2&2.88\mathrm{E}\text{-}2&1.89\mathrm{E}\text{-}2&2.13\mathrm{E}\text{-}1&1.07\mathrm{E}\text{-}1&9.09\mathrm{E}\text{-}1&4.55\mathrm{E}\text{-}1&1.29\mathrm{E}1&6.46&0&0&0&0\\\
0&-4.34\mathrm{E}\text{-}2&0&3.77\mathrm{E}\text{-}2&-2.13\mathrm{E}\text{-}1&-1.07\mathrm{E}\text{-}1&9.09\mathrm{E}\text{-}1&4.55\mathrm{E}\text{-}1&3.88\mathrm{E}1&1.94\mathrm{E}1&0&0&0&0\\\
0&0&0&0&2.00\mathrm{E}\text{-}2&1.00\mathrm{E}\text{-}2&3.55\mathrm{E}\text{-}2&1.78\mathrm{E}\text{-}2&1.89\mathrm{E}\text{-}1&9.47\mathrm{E}\text{-}2&0&0&0&0\\\
0&0&0&0&0&0&1.00\mathrm{E}\text{-}2&5.00\mathrm{E}\text{-}3&1.49\mathrm{E}\text{-}1&7.46\mathrm{E}\text{-}2&0&0&0&0\\\
0&0&0&0&0&0&0&0&2.00\mathrm{E}\text{-}2&1.00\mathrm{E}\text{-}2&0&0&0&0\\\
0&0&0&0&0&1.00\mathrm{E}\text{-}2&0&2.96\mathrm{E}\text{-}2&0&2.21\mathrm{E}\text{-}1&0&0&0&0\\\
0&0&0&0&0&0&0&5.00\mathrm{E}\text{-}3&0&1.24\mathrm{E}\text{-}1&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&1.00\mathrm{E}\text{-}2&0&0&0&0\\\
-2.00\mathrm{E}\text{-}2&-1.81\mathrm{E}\text{-}3&-5.33\mathrm{E}\text{-}4&-1.57\mathrm{E}\text{-}4&1.18\mathrm{E}\text{-}2&2.37\mathrm{E}\text{-}2&4.21\mathrm{E}\text{-}2&6.32\mathrm{E}\text{-}2&5.39\mathrm{E}\text{-}1&7.18\mathrm{E}\text{-}1&-4.00\mathrm{E}\text{-}2&-3.62\mathrm{E}\text{-}3&-1.07\mathrm{E}\text{-}3&-3.14\mathrm{E}\text{-}4\\\
-6.00\mathrm{E}\text{-}2&-7.24\mathrm{E}\text{-}3&-3.20\mathrm{E}\text{-}3&-1.26\mathrm{E}\text{-}3&5.92\mathrm{E}\text{-}2&1.18\mathrm{E}\text{-}1&2.95\mathrm{E}\text{-}1&4.42\mathrm{E}\text{-}1&4.85&6.46&-1.20\mathrm{E}\text{-}1&-1.45\mathrm{E}\text{-}2&-6.40\mathrm{E}\text{-}3&-2.51\mathrm{E}\text{-}3\\\
0&-7.24\mathrm{E}\text{-}3&-9.60\mathrm{E}\text{-}3&-6.29\mathrm{E}\text{-}3&1.18\mathrm{E}\text{-}1&2.37\mathrm{E}\text{-}1&1.18&1.77&2.91\mathrm{E}1&3.88\mathrm{E}1&0&-1.45\mathrm{E}\text{-}2&-1.92\mathrm{E}\text{-}2&-1.26\mathrm{E}\text{-}2\\\
0&1.45\mathrm{E}\text{-}2&0&-1.26\mathrm{E}\text{-}2&-1.18\mathrm{E}\text{-}1&-2.37\mathrm{E}\text{-}1&1.18&1.77&8.73\mathrm{E}1&1.16\mathrm{E}2&0&2.90\mathrm{E}\text{-}2&0&-2.51\mathrm{E}\text{-}2\\\
\end{array}\right),$ (89)

and

$\mathbf{C}(\rho_{0})=\left[22.1,~{}2.36,~{}-44.2,~{}177,~{}0,~{}0,~{}0,~{}0,~{}0,~{}0,~{}12.3,~{}24.6,~{}-24.6,~{}98.3\right]^{\mathrm{T}}.$
(90)

$\mathbf{M}^{-1}(\rho_{0})$ and $\mathbf{C}^{\prime}(\rho_{0})$ can be
obtained as

$\displaystyle\mathbf{M}^{-1}(\rho_{0})=$
$\displaystyle\left(\begin{array}[]{cccccccccccccc}2.13\mathrm{E}2&-1.23\mathrm{E}3&1.00\mathrm{E}1&-1.11&3.55&-2.35\mathrm{E}1&3.41\mathrm{E}2&0&0&0&0&0&0&0\\\
-3.32\mathrm{E}3&2.07\mathrm{E}4&-1.84\mathrm{E}2&2.30\mathrm{E}1&-1.47\mathrm{E}2&8.23\mathrm{E}2&-9.70\mathrm{E}3&0&0&0&0&0&0&0\\\
5.00\mathrm{E}3&-3.12\mathrm{E}4&3.13\mathrm{E}2&-5.21\mathrm{E}1&-3.33\mathrm{E}2&-8.46\mathrm{E}2&2.82\mathrm{E}4&0&0&0&0&0&0&0\\\
-3.82\mathrm{E}3&2.39\mathrm{E}4&-2.12\mathrm{E}2&5.30\mathrm{E}1&1.13\mathrm{E}2&-2.47\mathrm{E}3&-3.98\mathrm{E}4&0&0&0&0&0&0&0\\\
0&0&0&0&5.00\mathrm{E}1&-1.78\mathrm{E}2&8.53\mathrm{E}2&-5.00\mathrm{E}1&2.96\mathrm{E}2&-2.58\mathrm{E}3&0&0&0&0\\\
0&0&0&0&0&0&0&1.00\mathrm{E}2&-5.92\mathrm{E}2&5.16\mathrm{E}3&0&0&0&0\\\
0&0&0&0&0&1.00\mathrm{E}2&-7.46\mathrm{E}2&0&-1.00\mathrm{E}2&1.24\mathrm{E}3&0&0&0&0\\\
0&0&0&0&0&0&0&0&2.00\mathrm{E}2&-2.49\mathrm{E}3&0&0&0&0\\\
0&0&0&0&0&0&5.00\mathrm{E}1&0&0&-5.00\mathrm{E}1&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&1.00\mathrm{E}2&0&0&0&0\\\
-1.07\mathrm{E}2&6.15\mathrm{E}2&-5.00&5.56\mathrm{E}\text{-}1&-4.74&4.33\mathrm{E}1&-1.05\mathrm{E}3&-8.88&9.47\mathrm{E}1&-2.10\mathrm{E}3&-3.20\mathrm{E}2&9.83\mathrm{E}1&-1.50\mathrm{E}1&1.67\\\
1.66\mathrm{E}3&-1.04\mathrm{E}4&9.21\mathrm{E}1&-1.15\mathrm{E}1&1.96\mathrm{E}2&-1.43\mathrm{E}3&2.80\mathrm{E}4&3.68\mathrm{E}2&-3.34\mathrm{E}3&6.14\mathrm{E}4&4.97\mathrm{E}3&-1.66\mathrm{E}3&2.76\mathrm{E}2&-3.45\mathrm{E}1\\\
-2.50\mathrm{E}3&1.56\mathrm{E}4&-1.56\mathrm{E}2&2.60\mathrm{E}1&4.44\mathrm{E}2&3.38\mathrm{E}3&-1.02\mathrm{E}5&8.33\mathrm{E}2&2.96\mathrm{E}3&-1.60\mathrm{E}5&-7.50\mathrm{E}3&2.50\mathrm{E}3&-4.69\mathrm{E}2&7.81\mathrm{E}1\\\
1.91\mathrm{E}3&-1.19\mathrm{E}4&1.06\mathrm{E}2&-2.65\mathrm{E}1&-1.51\mathrm{E}2&5.59\mathrm{E}3&1.81\mathrm{E}5&-2.83\mathrm{E}2&9.71\mathrm{E}3&2.07\mathrm{E}5&5.73\mathrm{E}3&-1.91\mathrm{E}3&3.18\mathrm{E}2&-7.95\mathrm{E}1\\\
\end{array}\right),$ (105)

and

$\mathbf{C}^{\prime}(\rho_{0})=\left[1180,~{}12200,~{}-13800,~{}9380,0~{},0~{},0~{},0~{},0~{},0~{},1570~{},16300~{},18400~{},-12500\right]^{\mathrm{T}}.$
(106)

Readers can solve for the corresponding set of parameter values based on a
given set of macroscopic quantities through Eq. (58), and vice versa through
Eq. (53). The units of the parameters and macroscopic quantities are presented
in Table 1 and Table 2.

##  Appendix C The fourth-order symmetry energy, kurtosis coefficients, linear
isospin splitting coefficient and the nucleon effective masses

In addition to the quantities used in the fitting procedure, for completeness,
we hereby present the expressions of $E_{\mathrm{sym},4}(\rho)$, $I_{0}$,
$I_{\mathrm{sym}}$, $\Delta m_{1}^{\ast}(\rho)$, $m_{s}$, and $m_{v}$.
Furthermore, we discuss the relationship between $E_{\mathrm{sym},4}(\rho)$,
$\Delta m_{1}^{\ast}(\rho)$ and $m_{s}$, $m_{v}$.

The fourth-order symmetry energy defined in Eq. (23) can be expressed as

$\displaystyle E_{\mathrm{sym},4}(\rho)$
$\displaystyle\equiv\left.\frac{1}{4!}\frac{\partial^{4}E(\rho,\delta)}{\partial\delta^{4}}\right|_{\delta=0}$
(107)
$\displaystyle=\frac{\hbar^{2}}{162m}a^{2}\rho^{2/3}+\frac{1}{648}a^{2}\rho^{5/3}\left(C^{[2]}-D^{[2]}\right)+\frac{1}{648}a^{4}\rho^{7/3}\left(8C^{[4]}+3D^{[4]}\right)+\frac{2}{135}a^{6}\rho^{3}\left(13C^{[6]}+9D^{[6]}\right).$

The kurtosis coefficients of $E_{0}(\rho)$ and $E_{\mathrm{sym}}(\rho)$ are
expressed as

$\displaystyle I_{0}(\rho)\equiv$ $\displaystyle
81\rho^{4}\frac{d^{4}E_{0}(\rho)}{d\rho^{4}}$ (108) $\displaystyle=$
$\displaystyle-\frac{84\hbar^{2}}{5m}a^{2}\rho^{2/3}+\sum_{n=1,3,5}\left[\frac{t_{3}^{[n]}}{16}(n-6)(n-3)n(n+3)\rho^{\frac{n}{3}+1}\right]$
$\displaystyle+\frac{3}{2}a^{2}\rho^{5/3}\left(2C^{[2]}+D^{[2]}\right)+\frac{9}{10}a^{4}\rho^{7/3}\left(2C^{[4]}+D^{[4]}\right),$

and

$\displaystyle I_{\mathrm{sym}}(\rho)\equiv$ $\displaystyle
81\rho^{4}\frac{d^{4}E_{\mathrm{sym}}(\rho)}{d\rho^{4}}$ (109)
$\displaystyle=$
$\displaystyle-\frac{28\hbar^{2}}{3m}a^{2}\rho^{2/3}-\sum_{n=1,3,5}\left[\frac{t_{3}^{[n]}}{16}\left(2x_{3}^{[n]}+1\right)(n-6)(n-3)n(n+3)\rho^{\frac{n}{3}+1}\right]$
$\displaystyle+\frac{5}{3}a^{2}\rho^{5/3}\left(C^{[2]}+2D^{[2]}\right)-\frac{7}{3}a^{4}\rho^{7/3}\left(2C^{[4]}+3D^{[4]}\right),$

respectively. The isospin splitting coefficients $\Delta
m_{2n-1}^{\ast}(\rho)$ are defined in Eq. (35), while the linear coefficient
can be expressed as

$\displaystyle\Delta m_{1}^{\ast}(\rho)$
$\displaystyle\equiv\left.\frac{\partial
m_{\mathrm{n}\text{-}\mathrm{p}}^{\ast}(\rho,\delta)}{\partial\delta}\right|_{\delta=0}$
(110) $\displaystyle=-\frac{80m\hbar^{2}\left[15\rho
D^{[2]}+10a^{2}\rho^{5/3}\left(2C^{[4]}+5D^{[4]}\right)+72a^{4}\rho^{7/3}\left(4C^{[6]}+7D^{[6]}\right)\right]}{3\left[40\hbar^{2}+5m\rho\left(2C^{[2]}+D^{[2]}\right)+10ma^{2}\rho^{5/3}\left(2C^{[4]}+D^{[4]}\right)+72ma^{4}\rho^{7/3}\left(2C^{[6]}+D^{[6]}\right)\right]^{2}}.$

The isoscalar nucleon effective mass $m_{s}^{\ast}$ and isovector nucleon
effective mass $m_{v}^{\ast}$ are momentum dependent in the N3LO Skyrme
pseudopotential interactions. We define $\tilde{M}_{s}\equiv m/m_{s}$ and
$\tilde{M}_{v}\equiv m/m_{v}$. Thus we have

$\displaystyle\tilde{M}_{s}(\rho,p)=1+\frac{m}{p}\frac{dU_{0}(\rho,p)}{dp}=$
$\displaystyle
1+\frac{m}{8\hbar^{2}}\rho\left(2C^{[2]}+D^{[2]}\right)+\frac{m}{8\hbar^{2}}a^{2}\rho^{5/3}\left(2C^{[4]}+D^{[4]}\right)+\frac{3m}{8\hbar^{2}}a^{4}\rho^{7/3}\left(2C^{[6]}+D^{[6]}\right)$
(111)
$\displaystyle+\frac{p^{2}}{\hbar^{2}}\left[\frac{m}{8\hbar^{2}}\rho\left(2C^{[4]}+D^{[4]}\right)+\frac{21m}{20\hbar^{2}}a^{2}\rho^{5/3}\left(2C^{[6]}+D^{[6]}\right)\right]$
$\displaystyle+\frac{p^{4}}{\hbar^{4}}\left[\frac{3m}{8\hbar^{2}}\rho\left(2C^{[6]}+D^{[6]}\right)\right],$

and

$\displaystyle\tilde{M}_{v}(\rho,p)=$ $\displaystyle
1+\frac{m}{p}\frac{dU_{\tau}(\rho,-\tau,p)}{dp}=1+\frac{m}{4\hbar^{2}}\rho
C^{[2]}+\frac{m}{4\hbar^{2}}2^{2/3}a^{2}\rho^{5/3}C^{[4]}+\frac{m}{4\hbar^{2}}2^{4/3}a^{4}\rho^{7/3}C^{[6]}$
(112) $\displaystyle+\frac{p^{2}}{\hbar^{2}}\left[\frac{m}{4\hbar^{2}}\rho
C^{[4]}+\frac{21m}{20\hbar^{2}}2^{2/3}a^{2}\rho^{5/3}C^{[6]}\right]+\frac{p^{4}}{\hbar^{4}}\left[\frac{3m}{4\hbar^{2}}\rho
C^{[6]}\right],$

with $\tau=1\,[-1]$ for neutron [proton]. The derivatives of $\tilde{M}_{s}$
and $\tilde{M}_{v}$ with respect to momentum are expressed as

$\displaystyle\frac{d^{2}\tilde{M}_{s}(\rho,p)}{dp^{2}}=\frac{2!}{\hbar^{2}}\left[\frac{m}{8\hbar^{2}}\rho\left(2C^{[4]}+D^{[4]}\right)+\frac{21m}{20\hbar^{2}}a^{2}\rho^{5/3}\left(2C^{[6]}+D^{[6]}\right)\right]+\frac{12p^{2}}{\hbar^{4}}\left[\frac{3m}{8\hbar^{2}}\rho\left(2C^{[6]}+D^{[6]}\right)\right],$
(113)
$\displaystyle\frac{d^{4}\tilde{M}_{s}(\rho,p)}{dp^{4}}=\frac{4!}{\hbar^{4}}\left[\frac{3m}{8\hbar^{2}}\rho\left(2C^{[6]}+D^{[6]}\right)\right],$
(114)
$\displaystyle\frac{d^{2}\tilde{M}_{v}(\rho,p)}{dp^{2}}=\frac{2!}{\hbar^{2}}\left[\frac{m}{4\hbar^{2}}\rho
C^{[4]}+\frac{21m}{20\hbar^{2}}2^{2/3}a^{2}\rho^{5/3}C^{[6]}\right]+\frac{12p^{2}}{\hbar^{4}}\left[\frac{3m}{4\hbar^{2}}\rho
C^{[6]}\right],$ (115)

and

$\displaystyle\frac{d^{4}\tilde{M}_{v}(\rho,p)}{dp^{4}}=\frac{4!}{\hbar^{4}}\left[\frac{3m}{4\hbar^{2}}\rho
C^{[6]}\right].$ (116)

Compare Eqs. (111)-(116) with Eq. (107) and Eq. (110), we can obtain

$\displaystyle E_{\mathrm{sym},4}(\rho)=$
$\displaystyle\frac{\hbar^{2}}{162m}a^{2}\rho^{2/3}\left[3\tilde{M}_{v}(\rho,p=0)-2\tilde{M}_{s}(\rho,p=p_{F})\right]$
(117) $\displaystyle+\frac{\hbar^{4}}{324m}a^{4}\rho^{4/3}\left[(2-3\cdot
2^{2/3})\left.\frac{d^{2}\tilde{M}_{v}(\rho,p)}{dp^{2}}\right|_{p=0}+10\left.\frac{d^{2}\tilde{M}_{s}(\rho,p)}{dp^{2}}\right|_{p=p_{F}}\right]$
$\displaystyle+\frac{\hbar^{6}}{2430m}a^{6}\rho^{2}\left[\frac{27\cdot
2^{1/3}-14\cdot
2^{2/3}-40}{4}\frac{d^{4}\tilde{M}_{v}(\rho,p)}{dp^{4}}-13\frac{d^{4}\tilde{M}_{s}(\rho,p)}{dp^{4}}\right],$

and

$\displaystyle\Delta m_{1}^{\ast}(\rho)=$
$\displaystyle\frac{1}{3\left[\tilde{M}_{s}(\rho,p=p_{F})\right]^{2}}\Bigg{\\{}6\left[\tilde{M}_{v}(\rho,p=0)-\tilde{M}_{s}(\rho,p=p_{F})\right]$
(118) $\displaystyle+\hbar^{2}a^{2}\rho^{2/3}\left[(8-3\cdot
2^{2/3})\left.\frac{d^{2}\tilde{M}_{v}(\rho,p)}{dp^{2}}\right|_{p=0}-4\left.\frac{d^{2}\tilde{M}_{s}(\rho,p)}{dp^{2}}\right|_{p=p_{F}}\right]$
$\displaystyle+\left.\hbar^{4}a^{4}\rho^{4/3}\left[\frac{27\cdot
2^{1/3}-56\cdot
2^{2/3}+60}{30}\frac{d^{4}\tilde{M}_{v}(\rho,p)}{dp^{4}}+\frac{4}{3}\frac{d^{4}\tilde{M}_{s}(\rho,p)}{dp^{4}}\right]\right\\},$

where $p_{F}=\hbar(3\pi^{2}\rho/2)^{1/3}$.

In standard SHF and eSHF models, where $C^{[4]}$, $C^{[6]}$, $D^{[4]}$ and
$D^{[6]}$ all equal zero, $m_{s}^{\ast}$ and $m_{v}^{\ast}$ are independent of
momentum. Consequently, Eq. (117) and Eq. (118) reduce to the very
straightforward forms (i.e., Eq. (33) in Ref. [96] and Eq. (8) in Ref. [105],
respectively):

$E_{\mathrm{sym},4}(\rho)=\frac{\hbar^{2}}{162m}a^{2}\rho^{2/3}\left[\frac{3m}{m_{v}(\rho)}-\frac{2m}{m_{s}(\rho)}\right],$
(119)

and

$\displaystyle\Delta
m_{1}^{\ast}(\rho)=2\frac{m_{s}(\rho)}{m}\left[\frac{m_{s}(\rho)}{m_{v}(\rho)}-1\right].$
(120)

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