Relation between tunneling current and fermi golden rule (Bardeen model)

I am looking into the paper of Tersoff and Hamann - Theory of the scanning tunneling microscope. In this one there is the tunnel current written as $I = \frac{2\pi e}{\hbar} \sum_{\mu ,\nu} f(E_\mu) [1- f(E_\nu +eV)] |M_{\mu\nu}|^2 \delta(E_\mu - E_\nu)$ So in my lectures the tunneling current is written as $I_T = \frac{4\pi e}{\hbar} \sum_{\mu ,\nu} |M_{\mu\nu}|^2\delta(E_\mu - E_\nu) [f(E_\mu - E_F^\mu) - f(E_\nu - E_F^\nu)]$ $\mu$ is the tip and $\nu$ the sample, $E_F$ is the fermi energy. First of all I am wondering if both equation are equal. As far as I see, the one in my lecture already used the limit of $T \to 0$ . But if I try to apply this limit to the formula of Tersoff and Hamann, I still do not get the formula of my lecture. The other question is about how to apply the tunneling current in this model. In my lectures there was said that one can take Fermi golden rule as a start. How is then the tunnel current related to Fermi golden rule?