\displaystyle \frac{2}{3t}\overbrace{\sin\Big(\frac{\pi}{3}\Big) \sin\Big(\frac{2\pi}{3}\Big)}^{\displaystyle \frac{3}{4}} \Big[ \frac{E+2t\cos(\frac{\pi}{3})}{2t})-i\sqrt{1-\Big(\frac{E+2t\cos(\frac{\pi}{3})}{2t}\Big)^2}\Big]

\displaystyle \frac{2}{3t}\underbrace{\sin\Big(\frac{2\pi}{3}\Big) \sin\Big(\frac{4\pi}{3}\Big)}_{\displaystyle -\frac{3}{4}} \Big[ \frac{E+2t\cos(\frac{2\pi}{3})}{2t})-i\sqrt{1-\Big(\frac{E+2t\cos(\frac{2\pi}{3})}{2t}\Big)^2}\Big]

+

+\frac{1}{2}

-\frac{1}{2}

\displaystyle g_{12} =\frac{1}{2t} \Big[ 1-\frac{i t}{2t_c^2} (\Gamma_1-\Gamma_2)\Big]