# Alignment of two Density of States obtained from the Kohn-Sham eigenvalues in DFT

## Main Question or Discussion Point

Dear all,

In Density Functional Theory (DFT) the Kohn-Sham eigenvalues are used to construct the band structure and the density of states (DOS). For a 3D extended system the eigenvalues are determined up to a constant since there is no absolute energy reference, while for a 2D extended surface a well-defined energy reference exists (the vacuum level).

I now want to look at

a) Which states that changes (i.e. the energy range) when creating a surface from a bulk system (e.g. to find surface resonances).

b) Which states that changes when creating an interface from two free surfaces.

In order to investigate the change in states I have to have the same energy reference for both systems.

For a) I can for the surface system evaluate the electrostatic (and the exhange-correlation) potential and align my vacuum level with my DOS at the surface layer. Then if my slab is sufficiently thick I can also align the bulk DOS.

How can I proceed for case b) ? For each respective surface calculation I can relate the eigen values to the vacuum level, but for the interface calculation I run into to difficulties since I don't have an absolute reference. As far as I can see it the Fermi levels for the surfaces and the interface can't be aligned w.r.t. each other to due the lack of this absolute reference. Is my conclusion correct or can I still align them at the Fermi level? If the latter is valid, why is it so?

Best regards,

## Answers and Replies

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I think that as long as you don't shift your states so that the Fermi level is at zero then you will be able to properly compare them directly. The eigenvalues are determined up to an additive constant, but that constant will be present in your Hamiltonian. Your potential in either case is $$V(r) = \sum_i -Ze/r_i$$ where i runs over the nuclear positions. Since you don't have an extra constant that is changed from one calculation to another both will give you energies relative to the same reference.