manual/defunits.tex

   1 % This file is part of the GROMACS molecular simulation package.
   2 %
   3 % Copyright (c) 2013, by the GROMACS development team, led by
   4 % David van der Spoel, Berk Hess, Erik Lindahl, and including many
   5 % others, as listed in the AUTHORS file in the top-level source
   6 % directory and at http://www.gromacs.org.
   7 %
   8 % GROMACS is free software; you can redistribute it and/or
   9 % modify it under the terms of the GNU Lesser General Public License
  10 % as published by the Free Software Foundation; either version 2.1
  11 % of the License, or (at your option) any later version.
  12 %
  13 % GROMACS is distributed in the hope that it will be useful,
  14 % but WITHOUT ANY WARRANTY; without even the implied warranty of
  15 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
  16 % Lesser General Public License for more details.
  17 %
  18 % You should have received a copy of the GNU Lesser General Public
  19 % License along with GROMACS; if not, see
  20 % http://www.gnu.org/licenses, or write to the Free Software Foundation,
  21 % Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA.
  22 %
  23 % If you want to redistribute modifications to GROMACS, please
  24 % consider that scientific software is very special. Version
  25 % control is crucial - bugs must be traceable. We will be happy to
  26 % consider code for inclusion in the official distribution, but
  27 % derived work must not be called official GROMACS. Details are found
  28 % in the README & COPYING files - if they are missing, get the
  29 % official version at http://www.gromacs.org.
  30 %
  31 % To help us fund GROMACS development, we humbly ask that you cite
  32 % the research papers on the package. Check out http://www.gromacs.org
  33
  34 \chapter{Definitions and Units}
  35 \label{ch:defunits}
  36 \section{Notation}
  37 The following conventions for mathematical typesetting
  38 are used throughout this document:
  39
  40 \centerline{
  41 \begin{tabular}{l|l|c}
  42 Item            &       Notation        & Example       \\
  43 \hline
  44 Vector          &       Bold italic     & $\rvi$        \\
  45 Vector Length   &       Italic          & $r_i$         \\
  46 \end{tabular}
  47 }
  48
  49 We define the {\em lowercase} subscripts
  50 $i$, $j$, $k$ and $l$ to denote particles:
  51 $\rvi$ is the {\em position vector} of particle $i$, and using this
  52 notation:
  53 \bea
  54 \rvij   =       \rvj-\rvi       \\
  55 \rij    =       | \rvij |
  56 \eea
  57 The force on particle $i$ is denoted by $\ve{F}_i$ and
  58 \beq
  59 \ve{F}_{ij} = \mbox{force on $i$ exerted by $j$}
  60 \eeq
  61 Please note that we changed notation as of version 2.0 to $\rvij=\rvj-\rvi$ since this
  62 is the notation commonly used. If you encounter an error, let us know.
  63
  64 \section{\normindex{MD units}\index{units}}
  65 {\gromacs} uses a consistent set of units that produce values in the
  66 vicinity of unity for most relevant molecular quantities. Let us call
  67 them {\em MD units}. The basic units in this system are nm, ps, K,
  68 electron charge (e) and atomic mass unit (u), see
  69 \tabref{basicunits}.
  70 \begin{table}
  71 \centerline{
  72 \begin{tabular}{|l|c|l|}
  73 \dline
  74 Quantity        & Symbol&  Unit                                         \\
  75 \hline
  76 length          &  r    &  nm $= 10^{-9}$ m                             \\
  77 mass            &  m    &  u (atomic mass unit) $=$
  78                                 1.6605402(10)$\times 10^{-27}$ kg       \\
  79                 &       &  ($1/12$ the mass of a $^{12}$C atom)         \\
  80                 &       &  $1.6605402(10)\times 10^{-27}$ kg            \\
  81 time            &  t    &  ps $= 10^{-12}$ s                            \\
  82 charge          &  q    &  {\it e} $=$ electronic charge $=
  83                                 1.60217733(49)\times 10^{-19}$ C        \\
  84 temperature     &  T    &  K                                            \\
  85 \dline
  86 \end{tabular}
  87 }
  88 \caption[Basic units used in {\gromacs}.]{Basic units used in
  89 {\gromacs}. Numbers in parentheses give accuracy.}
  90 \label{tab:basicunits}
  91 \end{table}
  92
  93 Consistent with these units are a set of derived units, given in
  94 \tabref{derivedunits}.
  95 \begin{table}
  96 \centerline{
  97 \begin{tabular}{|l|c|l|}
  98 \dline
  99 Quantity        & Symbol   & Unit                               \\
 100 \hline
 101 energy          & $E,V$    & kJ~mol$^{-1}$                      \\
 102 Force           & $\ve{F}$ & kJ~mol$^{-1}$~nm$^{-1}$            \\
 103 pressure        & $p$      & kJ~mol$^{-1}$~nm$^{-3} =
 104                                 10^{30}/N_{AV}$~Pa              \\
 105                 &          & $1.660\,54\times 10^6$~Pa $=
 106                                 16.6054$~bar                    \\
 107 velocity        & $v$      & nm~ps$^{-1} = 1000$ m s$^{-1}$             \\
 108 dipole moment   & $\mu$    & \emph{e}~nm                                \\
 109 electric potential& $\Phi$ & kJ~mol$^{-1}$~\emph{e}$^{-1} =
 110                                 0.010\,364\,272(3)$ Volt        \\
 111 electric field  & $E$      & kJ~mol$^{-1}$~nm$^{-1}$~\emph{e}$^{-1} =
 112                                 1.036\,427\,2(3) \times 10^7$~V m$^{-1}$        \\
 113 \dline
 114 \end{tabular}
 115 }
 116 \caption{Derived units}
 117 \label{tab:derivedunits}
 118 \end{table}
 119
 120 The {\bf electric conversion factor} $f=\frac{1}{4 \pi
 121 \varepsilon_o}=138.935\,485(9)$ kJ~mol$^{-1}$~nm~e$^{-2}$. It relates
 122 the mechanical quantities to the electrical quantities as in
 123 \beq
 124  V = f \frac{q^2}{r} \mbox{\ \ or\ \ } F = f \frac{q^2}{r^2}
 125 \eeq
 126
 127 Electric potentials $\Phi$ and electric fields $\ve{E}$ are
 128 intermediate quantities in the calculation of energies and
 129 forces. They do not occur inside {\gromacs}. If they are used in
 130 evaluations, there is a choice of equations and related units. We
 131 strongly recommend following the usual practice of including the factor
 132 $f$ in expressions that evaluate $\Phi$ and $\ve{E}$:
 133 \bea
 134 \Phi(\ve{r}) = f \sum_j \frac{q_j}{|\ve{r}-\ve{r}_j|}   \\
 135 \ve{E}(\ve{r}) = f \sum_j q_j \frac{(\ve{r}-\ve{r}_j)}{|\ve{r}-\ve{r}_j|^3}
 136 \eea
 137 With these definitions, $q\Phi$ is an energy and $q\ve{E}$ is a
 138 force. The units are those given in \tabref{derivedunits}:
 139 about 10 mV for potential. Thus, the potential of an electronic charge
 140 at a distance of 1 nm equals $f \approx 140$ units $\approx
 141 1.4$~V. (exact value: 1.439965 V)
 142
 143 {\bf Note} that these units are mutually consistent; changing any of the
 144 units is likely to produce inconsistencies and is therefore {\em
 145 strongly discouraged\/}! In particular: if \AA \ are used instead of
 146 nm, the unit of time changes to 0.1 ps. If kcal mol$^{-1}$ (= 4.184
 147 kJ mol$^{-1}$) is used instead of kJ mol$^{-1}$ for energy, the unit of time becomes
 148 0.488882 ps and the unit of temperature changes to 4.184 K. But in
 149 both cases all electrical energies go wrong, because they will still
 150 be computed in kJ mol$^{-1}$, expecting nm as the unit of length. Although
 151 careful rescaling of charges may still yield consistency, it is clear
 152 that such confusions must be rigidly avoided.
 153
 154 In terms of the MD units, the usual physical constants take on
 155 different values (see \tabref{consts}). All quantities are per mol rather than per
 156 molecule. There is no distinction between Boltzmann's constant $k$ and
 157 the gas constant $R$: their value is
 158 $0.008\,314\,51$~kJ~mol$^{-1}$~K$^{-1}$.
 159 \begin{table}
 160 \centerline{
 161 \begin{tabular}{|c|l|l|}
 162 \dline
 163 Symbol  & Name                  & Value                                 \\
 164 \hline
 165 $N_{AV}$& Avogadro's number     & $6.022\,136\,7(36)\times 10^{23}$  mol$^{-1}$ \\
 166 $R$     & gas constant  & $8.314\,510(70)\times 10^{-3}$~kJ~mol$^{-1}$~K$^{-1}$ \\
 167 $k_B$   & Boltzmann's constant  & \emph{idem}   \\
 168 $h$     & Planck's constant     & $0.399\,031\,32(24)$~kJ~mol$^{-1}$~ps \\
 169 $\hbar$ & Dirac's constant      & $0.063\,507\,807(38)$~kJ~mol$^{-1}$~ps \\
 170 $c$     & velocity of light     & $299\,792.458$~nm~ps$^{-1}$ \\
 171 \dline
 172 \end{tabular}
 173 }
 174 \caption{Some Physical Constants}
 175 \label{tab:consts}
 176 \end{table}
 177
 178 \section{Reduced units\index{reduced units}}
 179 When simulating Lennard-Jones (LJ) systems, it might be advantageous to
 180 use reduced units ({\ie}, setting
 181 $\epsilon_{ii}=\sigma_{ii}=m_i=k_B=1$ for one type of atoms). This is
 182 possible. When specifying the input in reduced units, the output will
 183 also be in reduced units. The one exception is the {\em
 184 temperature}, which is expressed in $0.008\,314\,51$ reduced
 185 units. This is a consequence of using Boltzmann's constant in the
 186 evaluation of temperature in the code. Thus not $T$, but $k_BT$, is the
 187 reduced temperature. A {\gromacs} temperature $T=1$ means a reduced
 188 temperature of $0.008\ldots$ units; if a reduced temperature of 1 is
 189 required, the {\gromacs} temperature should be 120.2717.
 190
 191 In \tabref{reduced} quantities are given for LJ potentials:
 192 \beq
 193 V_{LJ} = 4\epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]
 194 \eeq
 195
 196 \begin{table}
 197 \centerline{
 198 \begin{tabular}{|l|c|l|}
 199 \dline
 200 Quantity        & Symbol        & Relation to SI                        \\
 201 \hline
 202 Length          & r$^*$         & r $\sigma^{-1}$                       \\
 203 Mass            & m$^*$         & m M$^{-1}$                            \\
 204 Time            & t$^*$         & t $\sigma^{-1}$ $\sqrt{\epsilon/M}$ \\
 205 Temperature     & T$^*$         & k$_B$T $\epsilon^{-1}$                \\
 206 Energy          & E$^*$         & E $\epsilon^{-1}$             \\
 207 Force           & F$^*$         & F $\sigma~\epsilon^{-1}$              \\
 208 Pressure        & P$^*$         & P $\sigma ^3 \epsilon^{-1}$           \\
 209 Velocity        & v$^*$         & v $\sqrt{M/\epsilon}$         \\
 210 Density         & $\rho^*$      & N $\sigma ^3~V^{-1}$          \\
 211 \dline
 212 \end{tabular}
 213 }
 214 \caption{Reduced Lennard-Jones quantities}
 215 \label{tab:reduced}
 216 \end{table}
 217
 218
 219 % LocalWords:  ij basicunits derivedunits kJ mol mV kcal consts LJ BT
 220 % LocalWords:  nm ps