# Wannier Functions as Discrete Basis

In solid state physics, using Bloch's theorem we know that the one-electron energy eigen-function can be written as $\psi_{\lambda,\vec{k}}(\vec{r})$ where $\lambda$ indexes eigenvalues of $\hat{H}$ and $\vec{k}$ indexes eigenvalues of $\hat{T}_\vec{R}$, the translation by lattice-vector-$\vec{R}$ operator.

Because of Bloch's theorem, we know that $\psi_{\lambda,\vec{k}}(\vec{r})$ is periodic in $\vec{k}$ by reciprocal-lattice-vectors $\vec{K}$ and so we may write a Fourier series of it as: $$\psi_{\lambda,\vec{k}}(\vec{r})=\sum_{\vec{R}}\tilde{\psi}_{\lambda,\vec{R}}(\vec{r})\exp\left({i\vec{R}\cdot\vec{k}}\right)$$

$\tilde{\psi}_{\lambda,\vec{R}}(\vec{r})$ are called Wannier functions. One can prove that $\tilde{\psi}_{\lambda,\vec{R}}(\vec{r})$ depends only on the difference $\vec{R}-\vec{r}$ as well as show that the Wannier functions are orthogonal between different values of $\lambda$ or $\vec{R}$.

As a result we may write an arbitrary wave-function $\psi(\vec{r})$ as $$\psi(\vec{r}) = \sum_{\lambda,\,\vec{R}} \alpha_{\lambda,\,\vec{R}}\tilde{\psi}_{\lambda,\vec{R}}(\vec{r})$$

where $\alpha_{\lambda,\,\vec{R}}=\int_{\mathbb{R}^3}\psi(\vec{r})\tilde{\psi}_{\lambda,\vec{R}}(\vec{r})d^3\vec{r}$ are the expansion coefficients in the Wannier basis.

My question is:

1) Expressed in the Wannier basis $\left\{\tilde{\psi}_{\lambda,\vec{R}}(\vec{r})\right\}_{\lambda,\,\vec{R}}$, why is it true that the potential energy of the electron is proportional to a delta-function?

$$V_{\lambda,\lambda',\vec{R},\vec{R'}} \propto \delta_{\vec{R},\vec{R'}}$$

It is clear intuitively because $$V_{\lambda,\lambda',\vec{R},\vec{R'}} \equiv \int_{\mathbb{R}^3}\tilde{\psi}_{\lambda,\vec{R}}(\vec{r})^{\ast}V(\vec{r})\tilde{\psi}_{\lambda',\vec{R'}}(\vec{r})d^3\vec{r}$$

and coming from the tight-binding model (or alternatively using some dubious claim that the Wannier functions can always be chosen to be maximally-localized using the gauge-freedom in the definition of the wave functions) we know that $\tilde{\psi}_{\lambda,\vec{R}}(\vec{r})$ is "sort of" around one particular lattice site $\vec{R}$ and so the integrand is non-zero only if the two lattice sites match.

But I would like a more rigorous (or hopefully much more direct) derivation of this. I am assuming that there is a simple easier reason I don't see.

2) My second question is how to express the kinetic energy, $\frac{-\hbar^2}{2m}\vec{\nabla}^2$, in the Wannier basis? I can't think of how to proceed from $$T_{\lambda,\lambda',\vec{R},\vec{R'}} = \frac{-\hbar^2}{2m} \int_{\mathbb{R}^3}\tilde{\psi}_{\lambda,\vec{R}}(\vec{r})^{\ast}\vec{\nabla}^2\tilde{\psi}_{\lambda',\vec{R'}}(\vec{r})d^3\vec{r}$$

• 1- I'm almost 100% confident that this is an approximation. 2- you're done. use that and the second equation. – Adam Oct 20 '14 at 18:10
• It seems that you're involved with Wannier functions. I recently started to learn about it and I would be happy if you can suggest me a reference for a comprehensive derivation of Wannier functions from Bloch functions? – Sina Dec 22 '16 at 14:27
• @Sina, have a look at the end of the tight-binding chapter in Ashcroft and Mermin. – PPR Dec 22 '16 at 14:30