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Mathematical formulation
The Schrödinger equation can be expressed like \[\left[ {{\nabla ^2} + E} \right]\psi \left( {\bf{r}} \right) = V\left( {\bf{r}} \right)\psi \left( {\bf{r}} \right)\]
 * $$[\nabla^2 + E]\psi(\mathbf{r}) = V(\mathbf{r})\psi(\mathbf{r})$$

where  $V({\bf{r}})$ $$V(\mathbf{r})$$ is the potential of the solid and   $\psi ({\bf{r}})$$$\psi(\mathbf{r})$$ is the wave function of the electron that has to be calculated.

The unperturbed Green's function is defined as the solution of $\left[ {{\nabla ^2} + E} \right]G\left( {{\bf{r}},{\bf{r'}}} \right) = \delta \left( {{\bf{r}} - {\bf{r'}}} \right)$
 * $$[\nabla^2 + E]G(\mathbf{r},\mathbf{r}') = \delta(\mathbf{r}-\mathbf{r}')$$

A plane wave can be expanded as \[{e^{i{\bf{k}}{\bf{r}}}} = \sum\limits_{} {\left( {2l + 1} \right)} {i^l}{j_l}\left( {kr} \right){P_l}\left( {\cos \theta } \right)\]
 * $$e^{i\mathbf{k}\cdot \mathbf{r}} = \sum_l (2 l + 1) i^l j_l(kr)P_l(\cos\theta)$$

where  ${j_l}\left( {kr} \right)$ $$j_l(kr)$$ are spherical Bessel functions and   ${P_l}\left( {\cos \theta } \right)$ $$P_l(\cos\theta)$$ are Legendre polynomials.

Mathematical formulation
The Schrödinger equation can be expressed like


 * $$\left[ {{\nabla ^2} + E} \right]\psi \left( {\bf{r}} \right) = V\left( {\bf{r}} \right)\psi \left( {\bf{r}} \right)$$
 * $$[ {{\nabla ^2} + E} ]\psi ( {\bf{r}} ) = V( {\bf{r}} )\psi ( {\bf{r}} )$$


 * $$[\nabla^2 + E]\psi(\mathbf{r}) = V(\mathbf{r})\psi(\mathbf{r})$$

where  $$V({\bf{r}})$$  $$V(\mathbf{r})$$ is the potential of the solid and   $$\psi ({\bf{r}})$$ $$\psi(\mathbf{r})$$ is the wave function of the electron that has to be calculated.

The unperturbed Green's function is defined as the solution of
 * $$\left[ {{\nabla ^2} + E} \right]G\left( {{\bf{r}},{\bf{r'}}} \right) = \delta \left( {{\bf{r}} - {\bf{r'}}} \right)$$


 * $$[\nabla^2 + E]G(\mathbf{r},\mathbf{r}') = \delta(\mathbf{r}-\mathbf{r}')$$

A plane wave can be expanded as
 * $${e^{i{\bf{k}}\cdot {\bf{r}}}} = \sum\limits_{} {\left( {2l + 1} \right)} {i^l}{j_l}\left( {kr} \right){P_l}\left( {\cos \theta } \right)$$


 * $$e^{i\mathbf{k}\cdot \mathbf{r}} = \sum_l (2 l + 1) i^l j_l(kr)P_l(\cos\theta)$$

where  $${j_l}\left( {kr} \right)$$  $$j_l(kr)$$ are spherical Bessel functions and   $${P_l}\left( {\cos \theta } \right)$$  $$P_l(\cos\theta)$$ are Legendre polynomials.