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Evaluation of Forces

Now the electronic system is self-consistent it is possible to evaluate the forces on the atoms in order to optimise the structure.

The forces are determined from the gradient of the free energy, F determined above, with respect to displacement,

f_{la} = -\frac{\partial F}{\partial R_{la}}\end{displaymath} (42)

These forces can be determined analytically by considering the change in each term in the total energy for a displacement of $\Delta
R_{la}$. Thus

\Delta E = \sum_{ij} b_{i,j} \Delta \left\{ T_{ij} + V_{ij}^...
+ \Delta \tilde{E}_H + \Delta \tilde{E}_{xc} + \Delta E_{i-i},\end{displaymath}


\Delta \tilde{E}_H = \sum_{kl}c_kG_{kl}\Delta c_l + \frac{1}{2}
\sum_{kl} c_k c_l \Delta G_{kl},\end{displaymath}

\Delta \tilde{E}_{xc} = \sum_{k,s} \varepsilon_{k,s}\Delta d_{k,s} +
\sum_{k,s} d_{k,s}\Delta \varepsilon_{k,s}.\end{displaymath}

These are rearranged and substitutions made to eliminate $\Delta c_l$and $\Delta d_{l,s}$, leaving terms in only $\Delta S_{ij}, \Delta
G_{kl}, \Delta H_{kl}$ and $\Delta \varepsilon_{k,s}$. $\Delta \varepsilon_{k,s}$contains terms in $\Delta \langle \tilde{n}_s \rangle_k$ and $\Delta
\langle \tilde{n}^2_s \rangle_k$ which can be determined from Equations 3.2.23 and 3.2.27. Although Tij and Sij only depend on Rla through the basis functions, $\phi_i({\bf r} -
{\bf R}_a)$, the pseudopotential term is also dependent on Rla due to $V_a^{ps}({\bf r} - {\bf R}_a)$; this is determined by integrating by parts to replace with terms in $\Delta \phi$:

The forces are relatively quick to evaluate compared to determination of the self-consistent energy.

Chris Ewels