# Validity of interpolation for density of states?

A density of states (DoS) curve seems like an example of a curve for which curve fitting doesn't seem to make sense, or does it? How could we interpolate on such a curve, given that we got the curve from a discrete dataset?

I thought of using an integrated DoS curve instead, since it has a nature suitable for fitting. Any suggestions are welcome.

• The curve for the density of state is artificially constructed as a sum of centered Gaussians/Lorentzians. So, if you want to fit it, you can (as it is done in Rietveld x-ray powder analysis), but what you will get is the reconstruction of s sum of Gaussians/Lorentizian... – Camps Oct 21 '20 at 10:49
• And how about using the integrated DoS curve itself for interpolation? Now that I think about it, interpolation on this one seems incorrect as well, since it is built from the DoS curve itself. – Hitanshu Sachania Oct 21 '20 at 10:57

Calculating densities of states is a tricky problem, as you correctly recognized. The density of states is: $$\tag{1} g(E)=\sum_{n}\int\frac{d\mathbf{k}}{(2\pi)^3}\delta(E-E_{n\mathbf{k}}),$$ where $$E_{n\mathbf{k}}$$ are the electronic energies and the integral is over the Brillouin zone. There are various strategies that one can follow, including:

1. Broadening. The delta function is replaced by a smooth function (typically a Gaussian). This method smears out the sharper features, so you may need many $$\mathbf{k}$$-points for convergence.
2. Adaptive broadening. This is a broadening scheme in which the broadening is different at each $$\mathbf{k}$$-point, and related to the gradient of the energy at that $$\mathbf{k}$$-point. This enables the description of sharp features with smaller number of $$\mathbf{k}$$-points. A description of adaptive broadening can be found in this paper.
3. Interpolation. The energy is calculated at a discrete grid covering the Brillouin zone (typically made of tetraheda) and the values in-between the grid points are interpolated, typically using linear or quadratic schemes. This method fails near band crossings because the different points are always joined by their respective $$n$$ index. A description of the tetrahedron method can be found in this paper.
4. Extrapolation. The energy is extrapolated from the calculated $$\mathbf{k}$$-point to allow for band crossings. This is a related paper.

Overall, a very nice overview for calculating densities of states can be found in the paper introducing OPTADOS, a software package that implements the above options.

• Thank you, ProfM. I never tend to pay attention to how equations are implemented in computer algorithms. Your answer was a wake-up call. Stuff like 'band crossing' and 'band kissing' would have never crossed my mind. The paper on OPTADOS was a good briefing. – Hitanshu Sachania Oct 22 '20 at 20:10
• @HitanshuSachania happy to help! – ProfM Oct 22 '20 at 20:19

The density of states (DOS) is the number of different states at a particular energy level that electrons are permitted to occupy, i.e. the number of electron states per unit volume per unit energy.

It is usually summarized by the equation $$DOS = (\frac{dN}{dE})$$

Since the first derivative of N with respect to E exists, you can, in fact, interpolate the integrated DOS via a smoothly varying function. But the choice should be based on the function should be such that it's derivative at certain energies should give the DOS value at that energy.

Hope this helps :)