Quantities of interest

Once at self-consistency the bandstructure energy can be calculated by occupying the energy levels corresponding to $E^b = \sum_nf_n\varepsilon_n$ or expanded in the implicit basis:

$$E^b = \rho S^{-1} H = Zf\Lambda Z^*$$ (22)

where the quantities without indices are matrices to be multiplied as such.

A more smoothly varying quantity is the first order energy:

$$E_1 = Tr(\rho H^0) = \sum_{RLR'L'} \rho_{RLR'L'} H^0_{RLR'L'} = \rho : H^0$$ (23)

Together with the second order energy,

$$E^U_2 = \frac{1}{2}\sum_R \left( (U_R -\frac{1}{2}I_R)\delta q_R^2 - \frac{1}{2}I_Rm_R^2 \right)$$ (24)

the classical term $E_{class} = \frac{1}{2}\sum_{RR'} V^{class}_{RR'}(R'-R)$ and the electrostatics contribution combine to form the total energy.

The corresponding forces are given by:

$$F_R = -2 \sum_{LR'L'} H'_{RLR'L'} \rho_{RLR'L'} + 2\sum_{LR'L'} S'_{RLR'L'} E^b_{RLR'L'} - \sum_{R'} V'^{class}_{RR'}(\vert R'-R\vert)$$ (25)

Where $F_R$, $H'_{RLR'L'} = \partial H_{RLR'L'} /\partial R$ and $S'_{RLR'L'} = \partial S_{RLR'L'} /\partial R$ are 3D vectors. Additionally the pressure may be obtained through the radial derivatives:

$$P = -\sum_{RLR'L'} H^{'r}_{RLR'L'} \rho_{RLR'L'} + \sum_{RLR'L'} S^{'r}_{RLR'L'} E^b_{RLR'L'} = - H^{'r} : \rho + S^{'r}:E^b$$ (26)

NB There is no magnetic contribution to the forces.