-Furthermore special techniques will be outlined, which reduce the complexity of the MD algorithm, though the evaluation of energy and force almost inevitably dictates the overall speed.
-
-\subsection{Verlet integration}
-\label{subsection:integrate_algo}
-
-A numerical method to integrate Newton's equation of motion was presented by Verlet in 1967 \cite{verlet67}.
-The idea of the so-called Verlet and a variant, the velocity Verlet algorithm, which additionaly generates directly the velocities, is explained in the following.
-Starting point is the Taylor series for the particle positions at time $t+\delta t$ and $t-\delta t$
-\begin{equation}
-\vec{r}_i(t+\delta t)=
-\vec{r}_i(t)+\delta t\vec{v}_i(t)+\frac{\delta t^2}{2m_i}\vec{f}_i(t)+
-\frac{\delta t^3}{6}\vec{b}_i(t) + \mathcal{O}(\delta t^4)
-\label{basics:verlet:taylor1}
-\end{equation}
-\begin{equation}
-\vec{r}_i(t-\delta t)=
-\vec{r}_i(t)-\delta t\vec{v}_i(t)+\frac{\delta t^2}{2m_i}\vec{f}_i(t)-
-\frac{\delta t^3}{6}\vec{b}_i(t) + \mathcal{O}(\delta t^4)
-\label{basics:verlet:taylor2}
-\end{equation}
-where $\vec{v}_i=\frac{d}{dt}\vec{r}_i$ are the velocities, $\vec{f}_i=m\frac{d}{dt^2}\vec{r}_i$ are the forces and $\vec{b}_i=\frac{d}{dt^3}\vec{r}_i$ are the third derivatives of the particle positions with respect to time.
-The Verlet algorithm is obtained by summarizing and substracting equations \eqref{basics:verlet:taylor1} and \eqref{basics:verlet:taylor2}
-\begin{equation}
-\vec{r}_i(t+\delta t)=
-2\vec{r}_i(t)-\vec{r}_i(t-\delta t)+\frac{\delta t^2}{m_i}\vec{f}_i(t)+
-\mathcal{O}(\delta t^4)
-\end{equation}
-\begin{equation}
-\vec{v}_i(t)=\frac{1}{2\delta t}[\vec{r}_i(t+\delta t)-\vec{r}_i(t-\delta t)]+
-\mathcal{O}(\delta t^3)
-\end{equation}
-the truncation error of which is of order $\delta t^4$ for the positions and $\delta t^3$ for the velocities.
-The velocities, although not used to update the particle positions, are not synchronously determined with the positions but drag behind one step of discretization.
-The Verlet algorithm can be rewritten into an equivalent form, which updates the velocities and positions in the same step.
-The so-called velocity Verlet algorithm is obtained by combining \eqref{basics:verlet:taylor1} with equation \eqref{basics:verlet:taylor2} displaced in time by $+\delta t$
-\begin{equation}
-\vec{v}_i(t+\delta t)=
-\vec{v}_i(t)+\frac{\delta t}{2m_i}[\vec{f}_i(t)+\vec{f}_i(t+\delta t)]
-\end{equation}
-\begin{equation}
-\vec{r}_i(t+\delta t)=
-\vec{r}_i(t)+\delta t\vec{v}_i(t)+\frac{\delta t^2}{2m_i}\vec{f}_i(t) \text{ .}
-\end{equation}
-Since the forces for the new positions are required to update the velocity the determination of the forces has to be carried out within the integration algorithm.