What should happen to the fermi energy as the lattice is relaxed?

If you would not mind helping me think about what the fermi energy means in terms of structural optimization and scf calculations, please share your thoughts on what I am observing here.

I am looking at scf calculations on two simillar but distinct complexes. In one case, as the atomic positions are relaxed, and the fermi energy is calculated with each set of atomic positions, the fermi energy increases throught this process - each set of new corrdinates results in a larger fermi energy than the previous.

In the other case (a calculation which I am less confident about whether I set it up correctly), I am seeing a decrease in the fermi energy with each set of positions.

Is one or the other certainly wrong or a bad sign? What if anything can be inferred from a rising or falling fermi energy as the lattice is relaxed?

I am using quantum espresso, btw. But my question is more of a theoretical nature. If you have any comments at all, please and thank you in advance.