%
% This file is part of the GROMACS molecular simulation package.
%
-% Copyright (c) 2013,2014,2015,2016,2017, by the GROMACS development team, led by
+% Copyright (c) 2013,2014,2015,2016,2017,2018, by the GROMACS development team, led by
% Mark Abraham, David van der Spoel, Berk Hess, and Erik Lindahl,
% and including many others, as listed in the AUTHORS file in the
% top-level source directory and at http://www.gromacs.org.
\bea \frac {\de^2\ve{r}_i}{\de t^2} & = & \frac{\ve{F}_i}{m_i} -
\ve{M} \frac{\de \ve{r}_i}{\de t} , \\ \ve{M} & = & \ve{b}^{-1} \left[
\ve{b} \frac{\de \ve{b}'}{\de t} + \frac{\de \ve{b}}{\de t} \ve{b}'
- \right] \ve{b}'^{-1}. \eea The (inverse) mass parameter matrix
+ \right] \ve{b}'^{-1}.
+ \eea
+This extra term has the appearance of a friction, but it should be
+noted that it is ficticious, and rather an effect of the
+Parrinello-Rahman equations of motion being defined with all
+particle coordinates represented relative to the box vectors, while
+{\gromacs] uses normal Cartesian coordinates for positions,
+velocities and forces. It is worth noting that the kinetic energy too
+should formally be calculated based on velocities relative to the
+box vectors. This can have an effect e.g. for external constant stress,
+but for now we only support coupling to constant external
+pressures, and for any normal simulation the velocities of box
+vectors should be extremely small compared to particle velocities.
+Gang Liu has done some work on deriving this for Cartesian
+coordinates\cite{Liu2015} that we will try to implement at
+some point in the future together with support for external stress.
+
+The (inverse) mass parameter matrix
$\ve{W}^{-1}$ determines the strength of the coupling, and how the box
can be deformed. The box restriction (\ref{eqn:box_rot}) will be
fulfilled automatically if the corresponding elements of $\ve{W}^{-1}$
%
% This file is part of the GROMACS molecular simulation package.
%
-% Copyright (c) 2013,2014,2015,2016,2017, by the GROMACS development team, led by
+% Copyright (c) 2013,2014,2015,2016,2017,2018, by the GROMACS development team, led by
% Mark Abraham, David van der Spoel, Berk Hess, and Erik Lindahl,
% and including many others, as listed in the AUTHORS file in the
% top-level source directory and at http://www.gromacs.org.
\end{enumerate}
\section{Non-bonded interactions}
-Non-bonded interactions in {\gromacs} are pair-additive and centro-symmetric:
+Non-bonded interactions in {\gromacs} are pair-additive:
\beq
V(\ve{r}_1,\ldots \ve{r}_N) = \sum_{i<j}V_{ij}(\rvij);
\eeq
\beq
-\ve{F}_i = -\sum_j \frac{dV_{ij}(r_{ij})}{dr_{ij}} \frac{\rvij}{r_{ij}} = -\ve{F}_j
+\ve{F}_i = -\sum_j \frac{dV_{ij}(r_{ij})}{dr_{ij}} \frac{\rvij}{r_{ij}}
\eeq
-The non-bonded interactions contain a \normindex{repulsion} term,
+Since the potential only depends on the scalar distance, interactions
+will be centro-symmetric, i.e.\ the vectorial partial force on particle $i$ from
+the pairwise interaction $V_{ij}(r_{ij})$ has the opposite direction of the partial force on
+particle $j$. For efficiency reasons, interactions are calculated by loops over interactions and
+updating both partial forces rather than summing one complete nonbonded force at
+a time. The non-bonded interactions contain a \normindex{repulsion} term,
a \normindex{dispersion}
term, and a Coulomb term. The repulsion and dispersion term are
combined in either the Lennard-Jones (or 6-12 interaction), or the
%
% This file is part of the GROMACS molecular simulation package.
%
-% Copyright (c) 2013,2014,2015, by the GROMACS development team, led by
+% Copyright (c) 2013,2014,2015,2017,2018, by the GROMACS development team, led by
% Mark Abraham, David van der Spoel, Berk Hess, and Erik Lindahl,
% and including many others, as listed in the AUTHORS file in the
% top-level source directory and at http://www.gromacs.org.
\subsection{Virial from non-bonded forces}
Here the derivation for the single sum virial in the {\em non-bonded force}
-routine is given. $i \neq j$ in all formulae below.
+routine is given. There are a couple of considerations that are special
+to {\gromacs} that we take into account:
+\begin{itemize}
+\item When calculating short-range interactions, we apply the
+{\em minimum image convention} and only consider the closest
+image of each neighbor - and in particular we never allow interactions
+between a particle and any of its periodic images. For all the
+equations below, this means $i \neq j$.
+\item In general, either the $i$ or $j$ particle might be shifted to a neighbor
+cell to get the closest interaction (shift $\delta_{ij}$). However, with minimum image
+convention there can be at most 27 different shifts for particles in the central cell,
+and for typical (very short-ranged) biomolecular interactions there are typically only a few
+different shifts involved for each particle, not to mention that each interaction can
+only be present for one shift.
+\item For the {\gromacs} nonbonded interactions
+we use this to split the neighborlist of each $i$ particle into multiple
+separate lists, where each list has a constant shift $\delta_i$ for the $i$ partlcle. We
+can represent this as a sum over shifts (for which we use index $s$), with the constraint that
+each particle interaction can only contribute to one of the terms in this sum, and the
+shift is no longer dependent on the $j$ particles. For any sum that does not contain
+complex dependence on $s$, this means the sum trivially reduces to just the sum
+over $i$ and/or $j$.
+\item To simplify some of the sums, we replace sums over $j<i$ with double sums over
+all particles (remember, $i \neq j$) and divide by 2.
+\end{itemize}
+
+Starting from the above definition of the virial, we then get
\newcommand{\di}{\delta_{i}}
\newcommand{\qrt}{\frac{1}{4}}
\bea
-\Xi
-&~=~&-\half~\sum_{i < j}^{N}~\rnij\otimes\Fvij \\
-&~=~&-\qrt\sum_{i=1}^N~\sum_{j=1}^N ~(\rvi+\di-\rvj)\otimes\Fvij \\
-&~=~&-\qrt\sum_{i=1}^N~\sum_{j=1}^N ~(\rvi+\di)\otimes\Fvij-\rvj\otimes\Fvij \\
-&~=~&-\qrt\left(\sum_{i=1}^N~\sum_{j=1}^N ~(\rvi+\di)\otimes\Fvij~-~\sum_{i=1}^N~\sum_{j=1}^N ~\rvj\otimes\Fvij\right) \\
-&~=~&-\qrt\left(\sum_{i=1}^N~(\rvi+\di)\otimes\sum_{j=1}^N~\Fvij~-~\sum_{j=1}^N ~\rvj\otimes\sum_{i=1}^N~\Fvij\right) \\
-&~=~&-\qrt\left(\sum_{i=1}^N~(\rvi+\di)\otimes\Fvi~+~\sum_{j=1}^N ~\rvj\otimes\Fvj\right) \\
-&~=~&-\qrt\left(2~\sum_{i=1}^N~\rvi\otimes\Fvi+\sum_{i=1}^N~\di\otimes\Fvi\right)
+\Xi
+&~=~&-\half~\sum_{i < j}^{N}~{\mathbf r}^n_{ij} \otimes {\mathbf F}_{ij} \nonumber \\
+&~=~&-\half~\sum_{i < j}^{N}~\left( {\mathbf r}_i + \delta_{ij} - {\mathbf r}_j \right) \otimes {\mathbf F}_{ij} \nonumber \\
+&~=~&-\qrt~\sum_{i=1}^{N}~\sum_{j=1}^{N}~\left( {\mathbf r}_i + \delta_{ij} - {\mathbf r}_j \right) \otimes {\mathbf F}_{ij} \nonumber \\
+&~=~&-\qrt~\sum_{i=1}^{N}~\sum_{s}~\sum_{j=1}^{N}~\left( {\mathbf r}_i + \delta_{i,s} - {\mathbf r}_j \right) \otimes {\mathbf F}_{ij,s} \nonumber \\
+&~=~&-\qrt~\sum_{i=}^{N}~\sum_{s}~\sum_{j=1}^{N}~\left( \left( {\mathbf r}_i + \delta_{i,s} \right) \otimes {\mathbf F}_{ij,s} -{\mathbf r}_j \otimes {\mathbf F}_{ij,s} \right) \nonumber \\
+&~=~&-\qrt~\sum_{i=1}^{N}~\sum_{s}~\sum_{j=1}^N ~\left( {\mathbf r}_i + \delta_{i,s} \right) \otimes {\mathbf F}_{ij,s} + \qrt \sum_{i=1}^{N}~\sum_{s}~\sum_{j=1}^{N} {\mathbf r}_j \otimes {\mathbf F}_{ij,s} \nonumber \\
+&~=~&-\qrt~\sum_{i=1}^{N}~\sum_{s}~\sum_{j=1}^N ~\left( {\mathbf r}_i + \delta_{i,s} \right) \otimes {\mathbf F}_{ij,s} + \qrt \sum_{i=1}^{N}~\sum_{j=1}^{N} {\mathbf r}_j \otimes {\mathbf F}_{ij} \nonumber \\
+&~=~&-\qrt~\sum_{s}~\sum_{i=1}^{N}~\left( {\mathbf r}_i + \delta_{i,s} \right) \otimes ~\sum_{j=1}^N {\mathbf F}_{ij,s} + \qrt \sum_{j=1}^N {\mathbf r}_j \otimes \sum_{i=1}^{N} {\mathbf F}_{ij} \nonumber \\
+&~=~&-\qrt~\sum_{s}~\sum_{i=1}^{N}~\left( {\mathbf r}_i + \delta_{i,s} \right) \otimes ~\sum_{j=1}^N {\mathbf F}_{ij,s} - \qrt \sum_{j=1}^N {\mathbf r}_j \otimes \sum_{i=1}^{N} {\mathbf F}_{ji} \nonumber \\
+&~=~&-\qrt~\sum_{s}~\sum_{i=1}^{N}~\left( {\mathbf r}_i + \delta_{i,s} \right) \otimes {\mathbf F}_{i,s} - \qrt \sum_{j=1}^N~{\mathbf r}_j \otimes {\mathbf F}_{j} \nonumber \\
+&~=~&-\qrt~\left(\sum_{i=1}^{N}~{\mathbf r}_i \otimes {\mathbf F}_{i} + \sum_{j=1}^N~{\mathbf r}_j \otimes {\mathbf F}_{j} \right) - \qrt \sum_{s}~\sum_{i=1}^{N} \delta_{i,s} \otimes {\mathbf F}_{i,s} \nonumber \\
+&~=~&-\half \sum_{i=1}^{N}~{\mathbf r}_i \otimes {\mathbf F}_{i} -\qrt \sum_{s}~\sum_{i=1}^{N}~\delta_{i,s} \otimes {\mathbf F}_{i,s} \nonumber \\
+&~=~&-\half \sum_{i=1}^{N}~{\mathbf r}_i \otimes {\mathbf F}_{i} -\qrt \sum_{s}~\delta_{s} \otimes {\mathbf F}_{s} \nonumber \\
+&~=~&\Xi_0 + \Xi_1
\eea
-In these formulae we introduced:
+In the second-last stage, we have used the property that each shift vector itself does not depend on the coordinates of particle $i$, so it is possible to sum
+up all forces corresponding to each shift vector (in the nonbonded kernels), and then just use a sum over the different shift vectors outside the kernels.
+We have also used
\bea
\Fvi&~=~&\sum_{j=1}^N~\Fvij \\
\Fvj&~=~&\sum_{i=1}^N~\Fvji
\beq
\Fvij~=~-\Fvji
\eeq
-we must, in the implementation, double the term containing the shift $\delta_i$.
+we must, in the implementation, double the term containing the shift $\delta_i$. Similarly, in a few places we have summed the shift-dependent force
+over all shifts to come up with the total force per interaction or particle.
+
+This separates the total virial $\Xi$ into a component $\Xi_0$ that is a single sum over particles, and a second component $\Xi_1$ that describes the influence of
+the particle shifts, and that is only a sum over the different shift vectors.
\subsection{The intra-molecular shift (mol-shift)}
For the bonded forces and SHAKE it is possible to make a {\em mol-shift}
biod.pnpi.spb.ru / alexxy / gromacs.git / commitdiff