Validity of the CE for the scalar-isovector density, Eq. (30)
for and ,
imposes through Eq. (48) certain specific constraints
on the coupling constants of the functional. At second order we obtain,
Constraints in Eqs. (53)-(56) connect the isoscalar and isovector coupling constants. The numerical coefficients of the corresponding linear combinations are the same as those for the scalar-isoscalar CE, see Eqs. (C1)-(C4) of Ref. [Carlsson et al.(2008)Carlsson, Dobaczewski, and Kortelainen], apart from factors of explained before Eq. (36). However, conditions for the scalar-isoscalar CE keep the coupling constants in the two isospin channels disconnected. Moreover, in the scalar-isovector channel the spin-orbit coupling constants must vanish, Eq. (57), along with the isovector surface coupling constants, Eq. (58). On the other hand, the corresponding isoscalar surface coupling constants, , , and , are left unrestricted.
For the fourth and sixth orders, analogous constraints are presented in Appendix A.