with the scale factor c$_{0}$ which represents the effect of incompressibility of nuclear matter in the nucleus and is determined by the so-called constant volume condition [19]. Y$_{lm}$ ( θ, φ ) are the spherical harmonics. R = r$_{0}$A 1 / 3 and a denote the radius and surface diffuseness of the single particle potential, respectively. Here, we assume and set the radius and diffuseness of the single particle potential of protons equal to those of neutrons for simplicity. For protons the Coulomb potential is additionally involved (see [19] for details). The relation between the isospin-asymmetric part V$_{s}$ of the single particle potential depth in the microscopic part and the symmetry energy coefficient in the macroscopic part is investigated based on the Skyrme energy density functional together with the ETF approach. In this approach, the central one-body potential is described by V$_{q}$ = δε ( r ) δρ$_{q}$ ( r ) with the energy density functional ε ( r ) (see Eq.(9) in Ref. [7] for details). The difference between the neutron ( q = n ) and proton ( q = p ) potentials of nuclear matter is written as with the kinetic energy density τ$_{q}$ which can be expressed as τ$_{q}$ = 3 $_{5}$(3 π $^{2}$)2 / $^{3}$ρ 5 / 3 q in the Thomas-Fermi approximation, the isospin asymmetry δ = ( ρ$_{n}$ − ρ$_{p}$ ) /ρ , and the coefficient c$_{k}$ = (3 π $^{2}$/ 2) 2 / $^{3}$. B$_{2}$ , B$_{8}$ and B$_{4}$ (notations in [7]) are some combinations of Skyrme param- eters, given by B$_{2}$ = − 1 $_{2}$t$_{0}$ ( 1 2 + x$_{0}$ ), B$_{8}$ = − 1 $_{12}$t$_{3}$ ( 1 2 + x$_{3}$ ) and B$_{4}$ = − 1 $_{4}$[ t$_{1}$ ( 1 2 + x$_{1}$ ) − t$_{2}$ ( 1 2 + x$_{2}$ )]. The symmetry potential V$_{sym}$ may be written as The symmetry energy coefficient of nuclear matter J is written as [20] with Θ$_{s}$ = 3 t$_{1}$x$_{1}$ − t$_{2}$ (4+5 x$_{2}$ ). The Θ$_{s}$ term and the last term of Eq.(16) give the contributions of the effective-mass [7] and the kinetic energy to the J , respectively. From the above equations for V$_{sym}$ and J , one can get the relation between them, 9 J = 1 2 V$_{sym}$ − 1 24 [Θ$_{s}$ + 6 B$_{4}$ ] c$_{k}$ρ 5 / 3 + 1 3 ( ℏ 2 2 M ) c$_{k}$ρ 2 / $^{3}$. (17) J = 1 2 B$_{2}$ρ + 1 2 B$_{8}$ρ α +1 − 1 24 Θ$_{s}$ c$_{k}$ρ 5 / 3 + 1 3 ( ℏ 2 2 M ) c$_{k}$ρ 2 / 3 (16) V$_{sym}$ = V$_{n}$ − V$_{p}$ 2 δ = B$_{2}$ρ + B$_{8}$ρ α +1 + 1 2 B$_{4}$c$_{k}$ρ 5 / 3 + O ( δ $^{2}$) . (15) V$_{n}$ − V$_{p}$ = 2 B$_{2}$ρδ + 2 B$_{8}$ρ α $^{+1}$δ + B$_{4}$ ( τ$_{n}$ − τ$_{p}$ ) = 2 B$_{2}$ρδ + 2 B$_{8}$ρ α $^{+1}$δ + B$_{4}$c$_{k}$ρ 5 / $^{3}$δ + O ( δ $^{3}$) (14) D. Symmetry Potential and Symmetry Energy Coefficient