Force calculation

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

$$\rho_{i\alpha,j\beta} = \sum_{n (occ.)}C_{i\alpha}^{(n)}C_{j\beta}^{(n)}$$ (6.17)

The cohesive energy thus becomes

$$E_{\rm tot} = 2\sum_{i\alpha,j\beta} \rho_{j\beta,i\alpha} H_{i\alpha,j\beta} + U_{\rm rep}$$ (6.18)

The forces acting on the atoms are then obtained by differentiating the cohesive energy with respect to atomic positions, that is

$${\bf F}_k = - \frac{\partial E_{\rm tot}}{\partial {\bf r}_k}$$ (6.19)

$$= - \left \{ 2\sum_{i\alpha,j\beta} \rho_{j\beta,i\alpha} \frac{\pa... ...partial {\bf r}_k} + \frac{\partial U_{\rm rep}}{\partial {\bf r}_k} \right \}.$$ (6.20)