deletions | additions       diff --git a/Forces and geometry optimization.tex b/Forces and geometry optimization.tex  index f1485a9..5ea9e8a 100644  --- a/Forces and geometry optimization.tex  +++ b/Forces and geometry optimization.tex 
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\begin{equation}  \label{eq:forcespot}  \vec{F}_{\alpha} = \vec{F}_{\alpha}^{\mathrm{ion-ion}}  -\sum_{n}\int -\sum_{n} \left< \varphi_{n} \left| \frac{\partial \hat{v}_{\alpha}{\partial \vec{R}_{\alpha}} \right| \varphi_{n} \right>\ .  %\int  \mathrm{d}\vec{r} \varphi^*_{n}(\vec{r})\frac{\partial %\varphi^*_{n}(\vec{r})\frac{\partial  v_{\alpha}(\vec{r}- \vec{R}_\alpha)}{\partial %\vec{R}_\alpha)}{\partial  \vec{R}_{\alpha}}\varphi_{n}(\vec{r})\ . \end{equation}  %  (For simplicity, we consider only local potentials here, but the results are valid for non-local potentials as well.) 
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%  \begin{equation}  \label{eq:forcesgrad}  \vec{F}_\alpha = \vec{F}_\alpha^{\mathrm{ion-ion}}+\sum_{n}\int \mathrm{d}\vec{r} \vec{F}_\alpha^{\mathrm{ion-ion}}+\sum_{n} \left[ \left<  \frac{\partial \varphi_{n}}{\partial r_{\vec{r}} \left| \partial \hat{v}_{\alpha} \right| \varphi_{n} \right> + \mathrm{cc.} \right]\,.  %\int \mathrm{d}\vec{r}  %\frac{\partial  \varphi^*_{n}(\vec{r})}{\partial \vec{r}}v_{\alpha}(\vec{r}- \vec{R}_\alpha)\varphi_{n}(\vec{r}) %\vec{R}_\alpha)\varphi_{n}(\vec{r})  + \mathrm{cc.}\,. \end{equation}  %  The first advantage of this formulation is that it is easier to implement than eq.~\eqref{eq:forcespot}, as it does not require the derivatives of the potential, which can be quite complex and difficult to code, especially when relativistic corrections are included.