# Why don't we use the principal quantum number when building the projected density of state?

I am using VASP to calculate the projected DOS, but I am not sure why we aren't using the principal quantum number when doing the projection.

According to the VASP wiki, LORBIT would control VASP to generate PROCAR file, which contains the spd- and site projected wave function character of each orbital. It is called lm-decomposed, where l,m are the angular moment and magnetic quantum numbers.

Since the electronic band structure is composed of atom orbitals, shouldn't we use the principal quantum number for projection as well? Taking Titanium as an example, its electron configuration is 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d². If we only do lm-decomposition, how should we distinguish the contribution from 3s and 4s?

There are three types of pseudopotentials in VASP for Titanium:
Ti_sv: VRHFIN =Ti: 3s3p4s3d ZVAL = 12.000
Ti_pv: VRHFIN =Ti: d3 s1 ZVAL = 10.000
Ti: VRHFIN =Ti: d3 s1 ZVAL = 4.000

• Part of the answer is that electrons of the same atom with different principal quantum number has very different energies, so most of the information coming from the distinction would be trivial.
– Greg
Commented Jan 17, 2022 at 22:28

I think the reason is many-fold. In addition to what Tristan mentioned, there are some other possible reasons:

1. The principle quantum number is a relatively ill-defined concept for an atom in a molecule or material. The valence orbitals of an atom usually shrink somewhat upon bonding (or expand, if the atom becomes noticeably negatively charged during bonding). Do you view a shrunk 4s orbital as a pure 4s orbital, or a linear combination of 3s and 4s (and possibly other s) orbitals? There are some inherent ambiguities here. By contrast, an s orbital is always a pure s orbital, without any other angular momentum contributions, if it simply expands or contracts. Even if it is deformed and acquire p, d, ... components, it is always possible to rigorously separate out these components by harmonic analysis.
2. If one defines a shrunk 4s orbital as still essentially a 4s orbital (and similarly for other principle quantum numbers), then one will find that orbitals with "core" principle quantum numbers are always almost full, and those with "Rydberg" ones always almost empty. Thus for every atom and every angular momentum, there is usually only one principle quantum number for which the orbital has a non-trivial occupation number. Decomposing the DOS using principle quantum numbers thus usually does not lead to any additional insights.
3. With the exception of high-lying Rydberg orbitals (which are generally not interesting in bulk materials anyway, as it is usually more physically useful to describe them using plane waves instead, because any tight binding approximation will fail for these high-lying bands), the PDOSs of all principle quantum numbers of the same angular momentum are usually well-separated in energy. Thus, if you look at the s PDOS of a period 4 element, for example, you will find that the 1s, 2s, 3s and 4s components already stand out as very clearly separated bands, and you can identify their principle quantum numbers by simple counting; from 5s on, the s bands may start to overlap, but you are usually not interested in them anyway. This contrasts with the usual practice where you decompose a mixed band of 3d, 4s and 4p into s, p and d contributions using PDOS, in which case you cannot do the same job simply by looking at the total DOS.

I think this issue is avoided in practical sense. I am not sure any of the psuedopotentials provided with VASP actually include valence electrons of the same angular momentum (so you never have 3s and 4s as valence), since these states are normally part of the core. I don't think there is any reason you cannot do this in theory, but in practice the core is always remaining the same so this doesn't need to be available in VASP.

This is probably not true if you are doing all-electron calculations in a different code.

• There are three types of pseudopotential for Ti element. For example, Ti_sv include 12 valence electrons. So there might be a possibility that 3s and 4s electrons are both valence electrons.
– Jack
Commented Jan 12, 2022 at 20:29
• @Jack I can see looking at the actual potcar the electron config should be "VRHFIN =Ti: 3p4s3d" I think this means there is no overlap, but an expert in reading these files might know. The documentation is very unclear. Commented Jan 12, 2022 at 20:40
• Even with a large core pseudopotential, one can still meaningfully talk about the projected DOS of 5s orbitals. The fact that we do not regularly see e.g. "PDOS of 5s orbitals for Ti metal" requires a different explanation. Commented Jan 13, 2022 at 10:06
• @wzkchem5 If you can elaborate further, please turn that into an answer. This is a good point, although I am not sure how often this is needed in practice (and demand drives features normally for better or worse) Commented Jan 13, 2022 at 15:20
• I would agree that it would be valuable to have a more elaborate version as an answer. Commented Jan 13, 2022 at 16:27