and there is no stable means to identify the corresponding active orbital in the bath, b. In other words, without state averaging, b is chosen as that orbital that most helps to lower the energy of the ground state; however, in this regime, the ground state really does not mix with the bath and so the choice of b will be numerically unstable. As a result, as we change the fermi level, we will likely see instabilities of the excited states. Indeed, these instabilities will be shown below. Note that this unstable structure can be easily removed, however, if we simply give some weight to the excited state so that the latter is chosen robustly. For this reason, we will perform state-averaging below.