# Meaning of density of states

1. Oct 24, 2012

While studying about k-points, etc. I came across the terms density of states. What is it's physical meaning. research papers often have DOS graphs in which they segregate s, p, d contributions and talk about fermi level etc. Is this DOS the same as the kohn-sham orbitals that are solved for in standard DFT?.. because actually for many body systems there should be no orbitals and stuff

Also, what is the difference between total DOS and projected DOS ?

2. Oct 24, 2012

### Spinnor

3. Oct 24, 2012

well of course i googled it and as usual the answers were either completely drenched with maths or were incomprehensible. here at PF, I am hoping to people who can give me the physical insight without me having to search the entire net.

4. Oct 25, 2012

### daveyrocket

The density of states tells you how many states exist at a certain energy level. This can be calculated, and often is, from the Kohn-Sham orbitals in DFT. This can be compared to photoemission experiments, for occupied states. Since the DOS is calculated from the energy levels of each individual state, you can decompose the states into s,p,d,f and only factor in the (say) d contribution of states to get a partial DOS for d orbitals.

There is a many-body generalization of the density of states called the spectral function. This can be obtained from models which take interactions into account and often agrees better with experimental spectra than the DOS.

5. Oct 25, 2012

### DrDu

I am quite sure there is a model independent definition e.g. in terms of Greens functions.

6. Oct 25, 2012

### daveyrocket

Yes, that's the spectral function A(w) = -1/pi * Im G(w)
In the non-interacting case you can show that it's exactly equal to the density of states.

7. Oct 25, 2012

### cgk

Additionally to the spectral function/density of states: As the Green's function comes with orbital labels, one can also define a local density of states (e.g., for certain k or certain orbitals or even certain points in space). For example, the (hole) Green's function for a wave function $|\Psi\rangle$ is essentially
$$g_{rs}(\Delta t) = \langle\Psi| c^\dagger_r \exp(-i H\cdot \Delta t/\hbar)\,c_s |\Psi\rangle,$$
(take or give some factors of i/-1/pi/2) where the operator in the middle is time propagation operator ($\exp(\Delta t \cdot \partial_t)$), and the creation/annihilation operators a refer to some arbitrary one-particle basis set ($g_{rs}(t)$ is thus the same as the corresponding density matrix at t=0 [not frequency = 0]). The frequency-dependent Green's function is obtained by Fourier-transforming Δt.

Now, you can, if you want, just form the Green's function, say, "g_{rs}(w)" with r and s both restricted to s or p or d orbitals (or bloch waves formed from them). Then you get a density of states for those states only. Or you can put in different operators than the creation/destruction operators (say, density at a certain orbital, or dipole moment operators) to get different effects.

If you are dealing with one-particle wave functions (like Kohn-Sham or Hartree-Fock), then all such transformations can actually be done in practice, at the one-particle level. This is where all those colorful pictures of DOS from DFT programs come from. However, in principle one *can* define analogs of those pictures for correlated theories, too. Evaluating them from first principles, of course, is a different question.

8. Oct 25, 2012

### Useful nucleus

By projected DOS one means the contribution of a certain element in a compound to the total density of states. One can do this by defining a radius for each atom and the states that fall within this radius are assigned to that particular atom.

In addition to what is posted above I'd like to add that simply the density of states for an "aggregate" of atoms is equivalent to the discrete shells of a single atom [recall the simple picture of H atom in elementary chemistry]. when atoms "aggregate" together these discrete shells merge and form continuous energy bands.

9. Oct 25, 2012

### Useful nucleus

One more thing. In DFT parlance , when DOS is mentioned it implies Kohn-Sham DOS. But the problem is people tend to forget this and even more tend to forget that even with the exact density functional, the Kohn-Sham DOS will remain different from the "true" DOS.

10. Oct 26, 2012

### DrDu

cgk defined a greens function matrix g_rs. There should be a set of orbitals which diagonalizes this matrix (and I think I even once knew their names). Especially for these orbitals, the interpretation as a DOS should be especially meaningful.