Recent content by raul_l

So it seems the two definitions are not exactly equivalent. This issue came to my head when reading a text about how to calculate optical properties using ab initio methods. For DFT based methods often the Kohn-Sham orbitals are used to construct the polarizability, which then leads to the...

Yes, I am only trying to show the equivalence for the longitudinal dielectric constant. But if we use the Coulomb gauge and consider only linear response, shouldn't then the two definitions be exactly equal?

Here's another attempt. Generally \mathbf E_{\text{tot}} = \mathbf E_{\text{ext}}\epsilon^{-1} = \mathbf E_{\text{ext}}(1+4\pi\alpha)^{-1} , but in the linear regime we take \mathbf E_{\text{tot}} = \mathbf E_{\text{ext}}(1-4\pi\alpha) . From V_{\text{tot}}(\mathbf{r}, \omega) = \int...

Thanks a lot. The convolution theorem gets me a lot closer to the result, but there are still some problems. First of all, I think both replies were a bit erroneous because the Fourier transform of 1/r should be 4pi/k^2. Using that I arrive at \frac{4\pi}{k^2}...

The two definitions of the dielectric function are equivalent if \alpha(\mathbf{r}, \mathbf{r'},\omega) = \int d\mathbf{r''} \frac{\chi(\mathbf{r''}, \mathbf{r'}, \omega)}{|\mathbf{r} - \mathbf{r''}|}, where \alpha(\mathbf{r}, \mathbf{r'},\omega) is the polarizability (microscopic) or...

Sure it is. See, e.g., Chapter 2. While for the macroscopic response only the same Fourier component get affected as that of the perturbation, \mathbf{E}(\mathbf{q},\omega) = \epsilon_{\text{mac}}^{-1}(\mathbf{q},\omega) \mathbf{E}_{\text{ext}}(\mathbf{q},\omega), at the microscopic level...

I see how Dickfore's observation about \mathbf{r} - \mathbf{r}' somewhat helps, but from what I understand the relations are also supposed to work for the microscopic dielectric function. I think we should instead focus on the fact that we are in the linear regime. The external potential...

Sometimes the dielectric function is defined as the connection between the total electric field in a material and the external field, \mathbf{E}(\mathbf{r},\omega) = \int \epsilon^{-1}(\mathbf{r},\mathbf{r'},\omega) \mathbf{E}_{\text{ext}}(\mathbf{r'},\omega) d \mathbf{r'}, and sometimes...

These give you the most elementary information about the material such as whether it is a metal, semiconductor or insulator (determined by band gap, which is the energy difference between the maximum of the valence band and the minimum of the conduction band) and whether it has a direct or...

I had the same question about electronic correlation and found the answers of cgk and alxm very helpful. If I understand correctly the original paper of Kohn and Hohenberg*, the exchange and correlation energy is given in terms of single and two particle density matrices E_{xc} [n] =...

I see that my question was too complicated. So I'll ask something simpler. Does anyone know of any AD tools that are capable of calculating the derivatives of integrals (like above)? I just want to make sure that I'm not reinventing the wheel by writing my own implementation of this.

I've written a Levenberg-Marquardt nonlinear optimization routine that employs the reverse-mode automatic differentiation algorithm for building the Jacobian. So far it has worked marvelously for me. However, now I have to use functions that contain integrals that cannot be analytically taken...

Homework Statement Hi I'm looking for an algorithm that can perform a fitting procedure on many curves simultaneously. Let's say, for example, that I have 3 exponential decay curves with 3 different decay times but they all share the same initial amplitude. So I have to find a fit for 4...

If you think about the wave function and what it represents I think it's pretty clear that the notion of distance becomes meaningless at such short distances. As was already mentioned, very small movements simply don't exist. If you still want to ask the question of shortest distances you would...

The Fermi level comes from Fermi-Dirac statistics. But it's not the only distribution function. You also have to think about the density of states. And while the Fermi-Dirac function F(E) (E - energy) might be non-zero in the forbidden gap the density of states g(E) is zero. The number of...