As shown in the picture, E_gcdft = E_dft + nelect_net*(fermi_shift+work_function)
but work_function = -e_fermi - fermi_shift, so combine these two equations we can find E_gcdft is unrelated to the fermi_shift. Is that right?
To be honest, I feel confused about the method to calculate the E_gcdft even though I read the paper "First principles reaction modeling of the electrochemical interface: Consideration and calculation of a tunable surface potential from atomic and electronic structure". Could you please tell me which paper or tutorial I shoud read?

Thanks