deletions | additions       diff --git a/geometry optimization.tex b/geometry optimization.tex  index 2433c86..f7c2511 100644  --- a/geometry optimization.tex  +++ b/geometry optimization.tex 
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This gain of momentum is done through the modification of the time step $\Delta t$ as adaptative parameter, and by introducing the following velocity modification  %  \begin{equation}  \label{eq:velocitymod} \label{eq:velocity_fire}  \vec{v}(t) \rightarrow \vec{V}(t) = (1-\alpha)\vec{v}(t) + \alpha |\vec{v}(t)|\hat{F}(t)\ ,  \end{equation}  % where $\vec{v}$ is the velocity of the atoms, $\alpha$ is an adaptative parameter, and $\hat{F}$ is an unitary vector in the direction of the force $\vec{F}$. By doing this velocity modification, the acceleration of the atoms is given by  %  \begin{equation}  \label{eq:acceleration_fire}  \dot{\vec{v}}{(t)} = &~\dfrac{\vec{F}{(t)}}{m} - \dfrac{\alpha}{\Delta t}|\vec{v}(t)|\left[\hat{v}(t)-\hat{F}(t)\right]\ ,  \end{equation}  %  where the second term is an introduced acceleration in a direcction "steeper" than the usual direction of motion. Oviously, if $\alpha = 0$ then $\vec{V}(t) = \vec{v}(t)$, meaning the velocity modification vanish, and the acceleration $\dot{\vec{v}}{(t)} = \vec{F}{(t)}/m$, as usual.