dipole_matrix Interface

public interface dipole_matrix

Form either a real or a complex dipole matrix.

The real-typed version is equivalent to $\mathbf q=0$.

$$\boldsymbol{\mathcal A}:=[\mathbf a_1\mathbf a_2\mathbf a_3],\qquad\boldsymbol{\mathcal B}:=[\mathbf b_1\mathbf b_2\mathbf b_3] \\ \boldsymbol{\mathcal B}=2\pi(\boldsymbol{\mathcal A}^{-1})^\mathrm T,\qquad \partial\boldsymbol{\mathcal B}=-\big((\partial\boldsymbol{\mathcal A})\boldsymbol{\mathcal A}^{-1}\big)^\mathrm T\boldsymbol{\mathcal B} \\ \mathbf R_\mathbf n=\boldsymbol{\mathcal A}\mathbf n,\qquad\partial\mathbf R_\mathbf n=(\partial\boldsymbol{\mathcal A})\mathbf n, \\ \mathbf G_\mathbf m=\boldsymbol{\mathcal B}\mathbf m,\qquad \partial\mathbf G_\mathbf m=-\big((\partial\boldsymbol{\mathcal A})\boldsymbol{\mathcal A}^{-1}\big)^\mathrm T\mathbf G_\mathbf m, \\ \frac{\partial G_{\mathbf ma}}{\partial A_{bc}}=-\mathcal A^{-1}_{ca}G_{\mathbf mb}$$

$$ \begin{gathered} \mathbf T_{ij}(\mathbf q)=\sum_{\mathbf n}\mathbf T(\mathbf R_{\mathbf nij})\mathrm e^{-\mathrm i\mathbf q\cdot\mathbf R_{\mathbf nij}},\quad\mathbf R_{\mathbf nij}=\mathbf R_j+\mathbf R_\mathbf n-\mathbf R_i \ \frac{\mathrm d\mathbf R_{\mathbf nij}}{\mathrm d\mathbf R_k}=(\delta_{jk}-\delta_{ik})\mathbf I \ \mathbf{T}{ij}(\mathbf{q})\approx\mathbf{T}^\text{Ew}(\mathbf{q}) =\sum_\mathbf n^{|\mathbf R_{\mathbf nij}|

Module Procedures