Sorry, I forgot to mention that the summation on k is restricted to first Brillouin zone only, in that case does ## \sum_{\mathbf{k} \in BZ} \rightarrow \frac{N\Omega}{(2\pi)^3} \int_{BZ} d\mathbf{k} ##
apply? I mean directly restricting the integration region to BZ.
I have some doubts on this...
Since ##
\sum_\mathbf{k}=\sum_\mathbf{n}=\frac{V}{(2\pi)^3} \sum_n \frac{(2\pi)^3}{V}=\frac{V}{(2\pi)^3} \int d\mathbf{n} \frac{(2\pi)^3}{V}=\frac{V}{(2\pi)^3} \int d\mathbf{k}
##
we have made a transformation from discrete k to continuous k, then why can't we use dirac delta function?
Ok so now ## \sum_{\mathbf{k}} \rightarrow \frac{N^2\Omega}{(2\pi)^3} \int_{BZ} d\mathbf{k} ## and
$$
\phi_{n\mathbf{R}}(\mathbf{r}) = \frac{N^{3/2}\Omega}{(2\pi)^3} \int_{BZ} e^{-i\mathbf{k \cdot R}} \psi_{n\mathbf{k}}(\mathbf{r}) d\mathbf{k}
$$
and the final form become
$$
\langle...
It is not? It is in accordance with that on Wikipedia
##\Omega## in this case is the Brillouin zone volume. Most of the references I consult automatically limit the integration region to the first Brillouin zone.
If not then how do I limit the integration region to only the first Brillouin...
Thanks for your reply. I am tempted to use that one because it is closer to the solution I want. But even then the final form is just
$$
\langle \phi_{n\mathbf{R}}(\mathbf{r}) \vert \phi_{m\mathbf{R'}}(\mathbf{r}) \rangle = N \frac{\Omega}{(2\pi)^3} \delta_{mn} \delta_{\mathbf{R,R'}}
$$
which...
I have trouble reconciling orthogonality condition for Wannier functions using both continuous and discrete k-space. I am using the definition of Wannier function and Bloch function as provided by Wikipedia (https://en.wikipedia.org/wiki/Wannier_function).
Wannier function:
Bloch function:
I...
I do not understand what you mean. If you mean the equation at the third part, I have edited it:
$$
\langle w \vert \mathbf{r} \vert w \rangle = \left( \frac{\Omega}{8\pi^3} \right)^2 \int_{BZ} d\mathbf{k} d\mathbf{k}' i e^{i(\mathbf{k-k'}) \cdot r} \langle u_{\mathbf{k}} \vert \nabla...
You are right. I have edited the post and fixed some typos.
Do you mean this?
$$ (2\pi)^3 \delta({\mathbf{k-k'}}) = \int^\infty_{-\infty} e^{i(\mathbf{k-k'}) \cdot \mathbf{r}} d\mathbf{r} $$
But I cannot figure how to factorize the term out.
Homework Statement
I did not manage to get the final form of the equation. My prefactor in the final form always remain quadratic, whereas the solution shows that it is linear,
Homework Equations
w refers to wannier function, which relates to the Bloch function
##\mathbf{R}## is this case...
I am referring to perturbation expansion of density functional Kohn Sham energy expression in
Phys. Rev. A 52, 1096.
In equation (92) the variational form of the second order energy is listed, but I cannot seem to work out the last 3 terms involving XC energy and density. Particularly, the...
Hi. I am currently calculating PDOS of oxygen atoms in cubic phase of BaTiO3 using Abinit package. In cubic phase the 3 oxygen atoms in the primitive cell of BaTiO3 are identical and related by space group symmetry.
Should I expect exactly the same PDOS diagrams for all three oxygen atoms?
I...