 Two-spin exchange spectrum with unequal signed magnetic moments

For $H=(\Delta /4){\mathbf{\sigma }}_1\cdot {\mathbf{\sigma }}_2-({\mu }_{0}B/2)(g_1{\sigma }_{1z}+g_2{\sigma }_{2z})$, the parallel-spin energies are $\Delta /4\mp {\mu }_{0}B(g_1+g_2)/2$. The zero-projection block has energies $-\Delta /4\pm \sqrt{{\Delta }^2+[{\mu }_{0}B(g_1-g_2){]}^2}/2$. Signed moment conventions must be fixed before assigning a physical opposite-charge case.