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Force calculations

We can express the forces acting on the atoms in a compact form, by first defining the density matrix


\begin{displaymath}\rho_{i\alpha,j\beta} = \sum_{n (occ.)}C_{i\alpha}^{(n)}C_{j\beta}^{(n)}
\end{displaymath} (7.23)

The cohesive energy thus becomes


$\displaystyle U_{\rm coh}$ = $\displaystyle 2\sum_{i\alpha,j\beta} \rho_{j\beta,i\alpha}
H_{i\alpha,j\beta} + U_{\rm rep}
- \sum_{i\alpha} N_{i\alpha}^{\rm atom} \varepsilon_{i\alpha}$  
  = $\displaystyle 2\sum_{i\alpha \neq j\beta} \rho_{j\beta,i\alpha} H_{i\alpha,j\be...
...{i\alpha,i\alpha} - N_{i\alpha}^{\rm atom}]
\varepsilon_{i\alpha}
+ U_{\rm rep}$ (7.24)

Note that the first term contains only off-diagonal elements of the Hamiltonian matrix while the second term contains on-site elements. The forces acting on the atoms are then obtained by differentiating the cohesive energy with respect to atomic positions, that is


$\displaystyle {\bf F}_k$ = $\displaystyle - \frac{\partial U_{\rm coh}}{\partial {\bf r}_k}$ (7.25)
  = $\displaystyle - \left \{ 2\sum_{i\alpha,j\beta} \rho_{j\beta,i\alpha}
\frac{\pa...
...partial {\bf r}_k}
+ \frac{\partial U_{\rm rep}}{\partial {\bf r}_k} \right \}.$ (7.26)

When self-consistency is required (see above), or when the on-site elements of the Hamiltonian are constant, the second term appearing in the forces vanishes, and we obtain


\begin{displaymath}{\bf F}_k = - \left \{ 2\sum_{i\alpha,j\beta} \rho_{j\beta,i\...
...
+ \frac{\partial U_{\rm rep}}{\partial {\bf r}_k} \right \}.
\end{displaymath} (7.27)


next up previous contents
Next: k sampling Up: The tight binding model Previous: Self consistency
David Saada
2000-06-22