How the projected density of states is built

Question

I'm new to DFT. I'm trying to figure out how projected density of states (PDOS) is built in solid state physics software package.

I understand the basics of the density functional theory method: each orbital of each substance is described by basis functions, the concept of electron density $\rho$ is introduced and there is a one-to-one correspondence between the wave functions and the introduced electron density. However, I do not fully understand the principles of computer calculations.

According to the initial position of the atoms, the initial value of $\rho_0$ is specified. The pseudopotential and the exchange-correlation potential are set. Then the Kohn-Sham equations are solved, where $\rho_0$ is specified as the initial approximation, until the next value of $\rho_{i+1}$ differs from the previous $\rho_i$ by the allowable error $\varepsilon$.

How is the PDOS determined further? Indeed, in the compound there are common orbitals, as the algorithm determines from the electron density a specification of the electrons ($s, p ,d, f$) and the energy that they can occupy? (How is the PDOS determined for each individual kind, if the orbitals are common?)

Perhaps I do not fully understand the principles of DFT itself, please explain.

Answer 1

Disclaimer: This answer would have been better placed in a comment but since I have just joined the community I cannot write comments. Ironically, I was tipped about this community after answering a very related question in Physics SE to which I dare giving a link.

The short answer here is that projected DOS is a somewhat ambiguous construction in the context of solid-state systems. What is often done in practice is that one chooses some basis set of functions localized at atomic sites and having well-defined orbital character [that is, the functions are of the form $\chi_l(r) Y_{lm}(\hat{\mathbf{r}})$] and calculates the overlap of Kohn-Sham wavefunctions with these localized functions. Summing these overlaps over bands and $k$-points weighted by occupation numbers (Fermi weights) one gets the occupancy of local states associated with a given atom. For a PDOS as function of $E$ one just multiplies the summation terms by band partial DOS, $\Im [1 / (E + i\delta + \mu - \varepsilon_{\nu\mathbf{k}})]$, with $\mu$ being the Fermi level, $\varepsilon_{\nu\mathbf{k}}$ Kohn-Sham eigenenergies, and $\delta$ a smearing factor.

In many cases, projected DOS does not even have to be properly normalized (because, again, there is no unique procedure to do this) and it can be constructed by choosing very simple local basis functions, such as, e.g., $\Theta(a - r) Y_{lm}(\hat{\mathbf{r}})$ (so-called "theta-projectors"), where $\Theta(a - r)$ simply cuts out a spherical region of radius $a$ around an atom.