# How to discretize a finite crystal?

I am trying to find a general method to discretize a finite crystal system.

## How I have been discretizing systems so far (using Wannier functions):

When you have an infinite crystal, you may apply Bloch theorem to conclude your energy eigenfunctions are also eigenfunctions of crystal-translation operators, in which case they can be written as $\psi_{\lambda,\vec{k}}(\vec{r})=\exp\left(i\vec{k}\cdot\vec{r}\right)u_{\lambda,\vec{k}}\left(\vec{r}\right)$ where $u_{\lambda,\vec{k}}\left(\vec{r}\right)$ have the periodicity of the Hamiltonian: $u_{\lambda,\vec{k}}\left(\vec{r}\right)=u_{\lambda,\vec{k}}\left(\vec{r}+\vec{R}\right)$ where $\vec{R}$ are the lattice vectors, and $\psi_{\lambda,\vec{k}}(\vec{r})$ is periodic in $\vec{k}$: $\psi_{\lambda,\vec{k}}(\vec{r}) = \psi_{\lambda,\vec{k}+\vec{K}}(\vec{r})$ where $\vec{K}$ are the reciprocal lattice vectors to $\vec{R}$.

Due to the periodicity in $\vec{k}$, we may expand $\psi_{\lambda,\vec{k}}(\vec{r})$ in a Fourier series as $\psi_{\lambda,\vec{k}}(\vec{r}) = \sum_{\vec{R}} \exp\left(i\vec{k}\cdot\vec{R}\right)\phi_{\lambda,\vec{R}}\left(\vec{r}\right)$. The $\phi_{\lambda,\vec{R}}\left(\vec{r}\right)$ functions are called "Wannier" functions and they are orthogonal (between different bands $\lambda$ and different values of $\vec{R}$). Using this decomposition we are able to write a general wave-function $\Psi\left(x\right)$ as \begin{align} \Psi\left(x\right)&=\sum_{\lambda,\vec{k}}\alpha_{\lambda,\vec{k}}\psi_{\lambda,\vec{k}}(\vec{r})\\&=\sum_{\lambda,\vec{k}}\alpha_{\lambda,\vec{k}}\sum_{\vec{R}}\exp\left(i\vec{k}\cdot\vec{R}\right)\phi_{\lambda,\,\vec{R}}\left(\vec{r}\right)\\&=\sum_{\vec{R}}\sum_{\lambda}\underbrace{\left(\sum_{\vec{k}}\alpha_{\lambda,\vec{k}}\exp\left(i\vec{k}\cdot\vec{R}\right)\right)}_{\beta_{\lambda,\,\vec{R}}}\phi_{\lambda,\,\vec{R}}\left(\vec{r}\right)\\&=\sum_{\vec{R}}\sum_{\lambda}\beta_{\lambda,\,\vec{R}}\phi_{\lambda,\,\vec{R}}\left(\vec{r}\right) \end{align}

As a result, we can work with $\beta_{\lambda,\,\vec{R}}$ as wave-functions that depend on a discrete parameter ($\lambda, \vec{R}$) instead of $\Psi\left(x\right)$ and write everywhere (infinite) matrix equations instead of differential equations for the Schroedinger equation (by writing matrix-elements with the Wannier functions).

For example, instead of writing $$H\Psi\left(x\right)=E\Psi\left(x\right)$$ (a differential equation) we can write $$\sum_{\lambda',\vec{R}'}H_{\lambda,\vec{R},\lambda',\vec{R}'}\beta_{\lambda',\,\vec{R}'} = E \beta_{\lambda,\,\vec{R}} \tag{2}$$ (an (infinite) matrix equation), where $H_{\lambda,\vec{R},\lambda',\vec{R}'} \equiv \int_{\mathbb{R}^3}\phi_{\lambda,\vec{R}}\left(\vec{r}\right)^\ast H\phi_{\lambda,\vec{R}}\left(\vec{r}\right)d^3\vec{r}$.

## My question Is

How do I convert the Schroedinger equation to an equation of the form $\left(2\right)$ in situations when I cannot apply the Bloch theorem (and so may not build Wannier functions)? Such instances could be: (a) the existence of a perturbing potential which does not have the periodicity of the potential or (b) the finite nature of the crystal where we do not take to have periodic boundary conditions but actual edge boundary conditions (such as here).

I am looking for a general way to do this process.

I have read some articles that lead me to pessimistic conclusions that this might be complicated, for example Wilkinson (1998).

• Not sure about Wannier functions, but it seems possible to at least use Bloch theorem to partially solve the problem with homogeneous Dirichlet boundary conditions. Namely, if you take linear combinations of Bloch states with opposite quasimomenta and equal energy, you may be able to satisfy the Dirichlet conditions (similarly to solving equation for particle in infinite potential box). – Ruslan Nov 14 '14 at 15:01

You do not necessarily need Bloch states with the full translational symmetry to construct Wannier states. You can also do this in the presence of inhomogeneities and there is actually a well-defined approach as long as your band gap is well defined (and possibly beyond). The Wannier states at different sites may then have a different shape, but they remain orthogonal. Are you interested in an explicit construction procedure?

On the other hand, you could also work in the basis of the bare Wannier functions of the system with full translational symmetry. Also in the presence of other potentials, they remain a perfectly valid basis. Different bands may however be coupled by the additional potential.

• Thanks for your answer. I would be interested in an explicit construction of the first alternative you propose. However, how should I know what the band-structure is a-priori? About your second suggestion, when you say coupling, you mean that the expansion of an arbitrary $\Psi\left(x\right)$ would change, how from what I've written? – PPR Nov 14 '14 at 19:44
• One method that can directly be extended to inhomogeneous systems is described in this paper in the (supplemental material) journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.185307, let me know if you need more details. Instead of initial Bloch states, you need to calculate the single particle eigenstates in your inhomogeneous system. There are various ways to do this. – ulf Nov 15 '14 at 11:53
• journals.aps.org/prb/pdf/10.1103/PhysRevB.8.2485 – PPR Nov 16 '14 at 22:57