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Derivatives of the Local Energy

To evaluate the derivatives of the local energy we need to calculate the following terms:

$\displaystyle \frac{\partial_a \vec{\nabla}_i \left\vert A \right\vert}{\left\v...
...\frac{\partial_a \nabla^2_i \left\vert A \right\vert}{\left\vert A \right\vert}$ (B.23)

we obtain:
$\displaystyle \frac{\partial_a \vec{\nabla}_i \left\vert A \right\vert}{\left\vert A \right\vert}$ $\displaystyle =$ $\displaystyle \frac{1}{\left\vert A \right\vert} \sum_{n,l,m} \frac{\partial \l...
...\vert A \right\vert}{\partial{a_{in}}} \partial_a \vec{\nabla}_i \Phi (r_i,r_n)$  
  $\displaystyle =$ $\displaystyle \sum_{n,l,m}\left( A^{-1}_{in}A_{ml}^{-1} - A^{-1}_{il}A^{-1}_{mn...
...al_a \Phi(r_l,r_m) + \sum _n A_{ni}^-1 \partial_a \vec{\nabla}_i \Phi (r_i,r_n)$  

and a similar formula for the derivative of the laplacian. If only the orbital $ k$ depends by $ a$ we have:
$\displaystyle \partial_a \vec{\nabla}_i \Phi(r_i,r_j) = \partial_a \vec{\nabla}...
...phi_m(r_j) + \partial_a \phi_k(r_j)\sum_m\lambda_{mk}\vec{\nabla}_i \phi_m(r_i)$      

Claudio Attaccalite 2005-11-07