Merge process for noncollinear systems

[antelmor]

This script merges the contributions of different rotated configurations to obtain the full exchange tensor. It can deal with non-collinear systems.

The algorithm is as follows:

Let $J_{ij}$ be the exchange tensor and $R_i$, $R_j$ the matrices that rotate the local magnetic moments into the $z$-axis. We obtain $J_{ij}' = R_i J_{ji} R_j^T$. Then, we set all the $z$ components to zero and recover the exchange tensor with
$\hat{J_{ij}} = R_i^T J_{ij}' R_j$.

In other to preserve the condition $J_{ij}(\mathbf{d}) = J_{ij}^T(-\mathbf{d})$, we need to repeat the process with the matrices $R_i$ and $R_j$ interchanged. That is, we calculate $\hat{J_{ij}}' = R_j \hat{J_{ij}} R_i^T$ and then discard all of its $z$ components. Finally, we obtain
$\tilde{J_{ij}} = R_j^T \hat{J_{ij}}' R_i$.