# In a plane-wave code, does the Fermi energy depend on the pseudopotentials used?

I am planning to do some work using grand-canonical DFT, I believe that the Fermi energy depends on the pseudo-potentials used (like the total energy).

My understanding is that the Fermi energy has to be referenced to a specific level, for example the vacuum level (workfunction) for it to be comparable in plane-wave DFT context.

With methods like ESM, the Fermi energy is automatically referenced to the vacuum level, and different pseudo-potentials should give the same value (with different levels of accuracy of course). Thus, in this case, fixing the Fermi energy to a specific value makes sense (4.44 V abs value of the SHE typically). But how one would do for a bulk material?

Your understanding is correct; you need to have a well-defined reference level for the Fermi level if you want to make comparisons between different systems (for example, comparing work functions of different surfaces for the same metal or different metals). In addition to subtle variations due to pseudopotential selection, you'll also have different Fermi level / work function predictions depending on how many layers of material you include, how many layers you are freezing, whether you are using a symmetric or asymmetric slab, etc. For example, for metals, it's well-known that the work function converges somewhat slowly in an oscillatory fashion with respect to layer thickness.

For condensed phases with 3D periodicity, you don't have a well-defined reference level. You also cannot have a bulk system that has a finite charge; it's electrostatic potential energy is undefined (in reciprocal space, the electrostatic potential diverges at the gamma point) since you'd have a periodic collection of charged cells. Most planewave DFT codes will insert a uniform background charge (equivalent to setting the electrostatic potential at the gamma point to zero) to cancel the net charge to avoid this and still enable the calculation to converge. For these reasons, you couldn't build an electrochemical model for a bulk (3D periodic) structure or easily use a GC-SCF method.

But essentially, the work function and electrode potential are inherently surface-dependent properties because they require the presence of an interface to even define them in the first place. You'd also need to include some sort of compensating charge distribution (like an electrical double layer) to keep the system at a net neutral charge.

• Thank you for this answer. I also work with nanoparticle (In plane-wave DFT so the nanoparticle is centred in the box). I have been told to perform GC-DFT on this system. When raising this reference issue I have been told that I could just take the electrostatic potential (in a 3D context then, which direction??) far away from the nanoparticle and use it as a reference. Even if possible I doubt this would be comparable to any kind of experimental work? Dec 2, 2022 at 11:02
• Well you can think of an isolated nano particle as just a big molecule. In this case you will have vacuum around the particle (or solvent if you are using something like SCCS, ESM-RISM, VaspSol, etc.), so you can reference the Fermi level of the particle to the vacuum or solvent inner potential. I think you should be able to do GC-SCF in this case. Not sure how ESM will perform in this case since it is designed for slab surfaces, but it could be interesting to test. Dec 2, 2022 at 16:27
• You may also be interested in some of the electrostatic correction schemes implemented in Environ in Quantum ESPRESSO: journals.aps.org/prb/abstract/10.1103/PhysRevB.77.115139 Dec 2, 2022 at 17:24