Optimized the bonded thread force reduction

[gromacs/AngularHB.git] / docs / manual / defunits.tex

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\chapter{Definitions and Units}
\label{ch:defunits}
\section{Notation}
The following conventions for mathematical typesetting 
are used throughout this document:

\centerline{
\begin{tabular}{l|l|c}
Item            &       Notation        & Example       \\
\hline  
Vector          &       Bold italic     & $\rvi$        \\
Vector Length   &       Italic          & $r_i$         \\
\end{tabular}
}

We define the {\em lowercase} subscripts 
$i$, $j$, $k$ and $l$ to denote particles:
$\rvi$ is the {\em position vector} of particle $i$, and using this 
notation:
\bea
\rvij   =       \rvj-\rvi       \\
\rij    =       | \rvij |
\eea
The force on particle $i$ is denoted by $\ve{F}_i$ and 
\beq
\ve{F}_{ij} = \mbox{force on $i$ exerted by $j$}
\eeq
Please note that we changed notation as of version 2.0 to $\rvij=\rvj-\rvi$ since this
is the notation commonly used. If you encounter an error, let us know.

\section{\normindex{MD units}\index{units}}
{\gromacs} uses a consistent set of units that produce values in the
vicinity of unity for most relevant molecular quantities. Let us call
them {\em MD units}. The basic units in this system are nm, ps, K,
electron charge (e) and atomic mass unit (u), see
\tabref{basicunits}. The values used in {\gromacs}  are taken from the
CODATA Internationally recommended 2010 values of 
fundamental physical constants (see \verb+http://nist.gov+).
\begin{table}
\centerline{
\begin{tabular}{|l|c|l|}
\dline
Quantity        & Symbol&  Unit                                         \\
\hline                  
length          &  r    &  nm $= 10^{-9}$ m                             \\
mass            &  m    &  u (unified atomic mass unit) $=$ 
                                $1.660\,538\,921(73) \times 10^{-27}$ kg        \\
time            &  t    &  ps $= 10^{-12}$ s                            \\
charge          &  q    &  {\it e} $=$ elementary charge $=
                                1.602\,176\,565(35)\times 10^{-19}$ C   \\
temperature     &  T    &  K                                            \\
\dline
\end{tabular}
}
\caption[Basic units used in {\gromacs}.]{Basic units used in
{\gromacs}. Numbers in parentheses give accuracy.}
\label{tab:basicunits}
\end{table}

Consistent with these units are a set of derived units, given in
\tabref{derivedunits}.
\begin{table}
\centerline{
\begin{tabular}{|l|c|l|}
\dline
Quantity        & Symbol   & Unit                               \\
\hline
energy          & $E,V$    & kJ~mol$^{-1}$                      \\
Force           & $\ve{F}$ & kJ~mol$^{-1}$~nm$^{-1}$            \\
pressure        & $p$      & kJ~mol$^{-1}$~nm$^{-3} =
                                10^{30}/N_{AV}$~Pa              \\
                &          & $1.660\,538\,921\times 10^6$~Pa $= 
                                16.605\,389\,21$~bar                    \\
velocity        & $v$      & nm~ps$^{-1} = 1000$ m s$^{-1}$             \\
dipole moment   & $\mu$    & \emph{e}~nm                                \\ 
electric potential& $\Phi$ & kJ~mol$^{-1}$~\emph{e}$^{-1} = 
                                0.010\,364\,269\,19(32)$ Volt           \\
electric field  & $E$      & kJ~mol$^{-1}$~nm$^{-1}$~\emph{e}$^{-1} =
                                1.036\,426\,919(32) \times 10^7$~V m$^{-1}$     \\
\dline
\end{tabular}
}
\caption{Derived units}
\label{tab:derivedunits}
\end{table}

The {\bf electric conversion factor} $f=\frac{1}{4 \pi
\varepsilon_o}=138.935\,457\,8(39)$ kJ~mol$^{-1}$~nm~e$^{-2}$. It relates
the mechanical quantities to the electrical quantities as in
\beq
 V = f \frac{q^2}{r} \mbox{\ \ or\ \ } F = f \frac{q^2}{r^2}
\eeq

Electric potentials $\Phi$ and electric fields $\ve{E}$ are
intermediate quantities in the calculation of energies and
forces. They do not occur inside {\gromacs}. If they are used in
evaluations, there is a choice of equations and related units. We
strongly recommend following the usual practice of including the factor
$f$ in expressions that evaluate $\Phi$ and $\ve{E}$:
\bea
\Phi(\ve{r}) = f \sum_j \frac{q_j}{|\ve{r}-\ve{r}_j|}   \\
\ve{E}(\ve{r}) = f \sum_j q_j \frac{(\ve{r}-\ve{r}_j)}{|\ve{r}-\ve{r}_j|^3}
\eea
With these definitions, $q\Phi$ is an energy and $q\ve{E}$ is a
force. The units are those given in \tabref{derivedunits}:
about 10 mV for potential. Thus, the potential of an electronic charge
at a distance of 1 nm equals $f \approx 140$ units $\approx
1.4$~V. (exact value: $1.439\,964\,5$ V)

{\bf Note} that these units are mutually consistent; changing any of the
units is likely to produce inconsistencies and is therefore {\em
strongly discouraged\/}! In particular: if \AA \ are used instead of
nm, the unit of time changes to 0.1 ps. If kcal mol$^{-1}$ (= 4.184
kJ mol$^{-1}$) is used instead of kJ mol$^{-1}$ for energy, the unit of time becomes
0.488882 ps and the unit of temperature changes to 4.184 K. But in
both cases all electrical energies go wrong, because they will still
be computed in kJ mol$^{-1}$, expecting nm as the unit of length. Although
careful rescaling of charges may still yield consistency, it is clear
that such confusions must be rigidly avoided.
  
In terms of the MD units, the usual physical constants take on
different values (see \tabref{consts}). All quantities are per mol rather than per
molecule. There is no distinction between Boltzmann's constant $k$ and
the gas constant $R$: their value is
$0.008\,314\,462\,1$~kJ~mol$^{-1}$~K$^{-1}$.
\begin{table}
\centerline{
\begin{tabular}{|c|l|l|}
\dline
Symbol  & Name                  & Value                                 \\
\hline
$N_{AV}$& Avogadro's number     & $6.022\,141\,29(27)\times 10^{23}$  mol$^{-1}$        \\
$R$     & gas constant  & $8.314\,462\,1(75)\times 10^{-3}$~kJ~mol$^{-1}$~K$^{-1}$      \\
$k_B$   & Boltzmann's constant  & \emph{idem}   \\
$h$     & Planck's constant     & $0.399\,031\,271(17)$~kJ~mol$^{-1}$~ps \\
$\hbar$ & Dirac's constant      & $0.063\,507\,799\,3(28)$~kJ~mol$^{-1}$~ps \\
$c$     & velocity of light     & $299\,792.458$~nm~ps$^{-1}$ \\
\dline
\end{tabular}
}
\caption{Some Physical Constants}
\label{tab:consts}
\end{table}

\section{Reduced units\index{reduced units}}
When simulating Lennard-Jones (LJ) systems, it might be advantageous to
use reduced units ({\ie}, setting
$\epsilon_{ii}=\sigma_{ii}=m_i=k_B=1$ for one type of atoms). This is
possible. When specifying the input in reduced units, the output will
also be in reduced units. The one exception is the {\em
temperature}, which is expressed in $0.008\,314\,462\,1$ reduced
units. This is a consequence of using Boltzmann's constant in the
evaluation of temperature in the code. Thus not $T$, but $k_BT$, is the
reduced temperature. A {\gromacs} temperature $T=1$ means a reduced
temperature of $0.008\ldots$ units; if a reduced temperature of 1 is
required, the {\gromacs} temperature should be $120.272\,36$.

In \tabref{reduced} quantities are given for LJ potentials:
\beq
V_{LJ} = 4\epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]
\eeq

\begin{table}
\centerline{
\begin{tabular}{|l|c|l|}
\dline
Quantity        & Symbol        & Relation to SI                        \\
\hline
Length          & r$^*$         & r $\sigma^{-1}$                       \\
Mass            & m$^*$         & m M$^{-1}$                            \\
Time            & t$^*$         & t $\sigma^{-1}$ $\sqrt{\epsilon/M}$ \\
Temperature     & T$^*$         & k$_B$T $\epsilon^{-1}$                \\
Energy          & E$^*$         & E $\epsilon^{-1}$             \\
Force           & F$^*$         & F $\sigma~\epsilon^{-1}$              \\
Pressure        & P$^*$         & P $\sigma ^3 \epsilon^{-1}$           \\
Velocity        & v$^*$         & v $\sqrt{M/\epsilon}$         \\
Density         & $\rho^*$      & N $\sigma ^3~V^{-1}$          \\
\dline
\end{tabular}
}
\caption{Reduced Lennard-Jones quantities}
\label{tab:reduced}
\end{table}


\section{Mixed or Double precision}
{\gromacs} can be compiled in either mixed\index{mixed
precision|see{precision, mixed}}\index{precision, mixed} or
\pawsindex{double}{precision}. Documentation of previous {\gromacs}
versions referred to ``single precision'', but the implementation
has made selective use of double precision for many years.
Using single precision
for all variables would lead to a significant reduction in accuracy.
Although in ``mixed precision'' all state vectors, i.e. particle coordinates,
velocities and forces, are stored in single precision, critical variables
are double precision. A typical example of the latter is the virial,
which is a sum over all forces in the system, which have varying signs.
In addition, in many parts of the code we managed to avoid double precision
for arithmetic, by paying attention to summation order or reorganization
of mathematical expressions. The default configuration uses mixed precision,
but it is easy to turn on double precision by adding the option
{\tt -DGMX_DOUBLE=on} to {\tt cmake}. Double precision
will be 20 to 100\% slower than mixed precision depending on the
architecture you are running on. Double precision will use somewhat
more memory and run input, energy and full-precision trajectory files
will be almost twice as large.

The energies in mixed precision are accurate up to the last decimal,
the last one or two decimals of the forces are non-significant.
The virial is less accurate than the forces, since the virial is only one
order of magnitude larger than the size of each element in the sum over
all atoms (\secref{virial}).
In most cases this is not really a problem, since the fluctuations in the
virial can be two orders of magnitude larger than the average.
Using cut-offs for the Coulomb interactions cause large errors
in the energies, forces, and virial.
Even when using a reaction-field or lattice sum method, the errors
are larger than, or comparable to, the errors due to the partial use of
single precision.
Since MD is chaotic, trajectories with very similar starting conditions will
diverge rapidly, the divergence is faster in mixed precision than in double
precision.

For most simulations, mixed precision is accurate enough.
In some cases double precision is required to get reasonable results:
\begin{itemize}
\item normal mode analysis,
for the conjugate gradient or l-bfgs minimization and the calculation and
diagonalization of the Hessian
\item long-term energy conservation, especially for large systems
\end{itemize}


% LocalWords:  ij basicunits derivedunits kJ mol mV kcal consts LJ BT
% LocalWords:  nm ps