We have examined within a fully self-consistent theoretical framework the energy difference between the anti-analog giant dipole resonance (AGDR) and the isobaric analog state (IAS), E-AGDR - E-IAS, as an indicator of the neutron skin and of the density behavior of the symmetry energy. We have improved two specific points in our HF+RPA calculations: (1) the exchange term of the two-body Coulomb interaction is treated exactly without Slater approximation; and (2) the two-parameters spin-orbit interaction is treated in a consistent way within the energy density functional theory. The estimated values for the neutron skin in Pb-208 and the slope parameter of symmetry energy are compared with previous analysis available in the literature.
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