# Definitions and Units¶

## Notation¶

The following conventions for mathematical typesetting are used throughout this document:

Item Notation Example
Vector Bold italic $${\mathbf{r}_i}$$
Vector Length Italic $$r_i$$

We define the lowercase subscripts $$i$$, $$j$$, $$k$$ and $$l$$ to denote particles: $$\mathbf{r}_i$$ is the position vector of particle $$i$$, and using this notation:

(1)\begin{split}\begin{aligned} \mathbf{r}_{ij} = \mathbf{r}_j-\mathbf{r}_i\\ r_{ij}= | \mathbf{r}_{ij} | \end{aligned}\end{split}

The force on particle $$i$$ is denoted by $$\mathbf{F}_i$$ and

(2)$\mathbf{F}_{ij} = \mbox{force on i exerted by j}$

## MD 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 MD units. The basic units in this system are nm, ps, K, electron charge (e) and atomic mass unit (u), see Table 2 The values used in GROMACS are taken from the CODATA Internationally recommended 2010 values of fundamental physical constants (see NIST homepage).

Table 2 Basic units used in GROMACS
Quantity Symbol Unit
length r $$\mathrm{nm = }10^{-9}\ m$$
mass m u (unified atomic mass unit) = $$1.660\,538\,921 \times 10^{-27}\ kg$$
time t $$\mathrm{ps = }10^{-12}\ s$$
charge q e = elementary charge = $$1.602\,176\,565 \times 10^{-19}\ C$$
temperature T K

Consistent with these units are a set of derived units, given in Table 3

Table 3 Derived units. Note that an additional conversion factor of 10$$^{28}$$ a.m.u ($$\approx$$ 16.6) is applied to get bar instead of internal MD units in the energy and log files
Quantity Symbol Unit
energy $$E,V$$ $$\mathrm{kJ~mol}^{-1}$$
Force $$\mathbf{F}$$ $$\mathrm{kJ~mol}^{-1}~\mathrm{nm}^{-1}$$
pressure $$p$$ bar
velocity $$v$$ $$\mathrm{nm~ps}^{-1} = 1000\mathrm{~m~s}^{-1}$$
dipole moment $$\mu$$ $$\mathrm{e\ nm}$$
electric potential $$\Phi$$ $$\mathrm{kJ~mol}^{-1}\mathrm{~e}^{-1} =$$ $$0.010\,364\,269\,19$$ Volt
electric field $$E$$ $$\mathrm{kJ~mol}^{-1}\mathrm{~nm}^{-1}\ \mathrm{e}^{-1} =$$ $$1.036\,426\,919 \times 10^7\mathrm{~V m}^{-1}$$

The electric conversion factor $$f=\frac{1}{4 \pi \varepsilon_o}={138.935\,458}$$ $$\mathrm{kJ}~\mathrm{mol}^{-1}\mathrm{nm}~\mathrm{ e}^{-2}$$. It relates the mechanical quantities to the electrical quantities as in

(3)$V = f \frac{q^2}{r} \mbox{\ \ or\ \ } F = f \frac{q^2}{r^2}$

Electric potentials $$\Phi$$ and electric fields $$\mathbf{E}$$ 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 $$\mathbf{E}$$:

(4)\begin{split}\begin{aligned} \Phi(\mathbf{r}) = f \sum_j \frac{q_j}{| \mathbf{r}-\mathbf{r}_j | } \\ \mathbf{E}(\mathbf{r}) = f \sum_j q_j \frac{(\mathbf{r}-\mathbf{r}_j)}{| \mathbf{r}-\mathbf{r}_j| ^3}\end{aligned}\end{split}

With these definitions, $$q\Phi$$ is an energy and $$q\mathbf{E}$$ is a force. The units are those given in Table 3 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)

Note that these units are mutually consistent; changing any of the units is likely to produce inconsistencies and is therefore strongly discouraged! In particular: if Å are used instead of nm, the unit of time changes to 0.1 ps. If $$\mathrm{kcal}~\mathrm{mol}^{-1}$$ (= 4.184 $$\mathrm{kJ~mol}^{-1}$$) is used instead of $$\mathrm{kJ~mol}^{-1}$$ 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 $$\mathrm{kJ~mol}^{-1}$$, 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 Table 4). 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 $$0.008\,314\,462\,1\mathrm{kJ~mol}^{-1} \mathrm{K}^{-1}$$.

Table 4 Some Physical Constants
Symbol Name Value
$$N_{AV}$$ Avogadro’s number $$6.022\,141\,29\times 10^{23}~\mathrm{mol}^{-1}$$
$$R$$ gas constant $$8.314\,462\,1\times 10^{-3}~\mathrm{kJ~mol}^{-1}~\mathrm{K}^{-1}$$
$$k_B$$ Boltzmann’s constant idem
$$h$$ Planck’s constant $$0.399\,031\,271~\mathrm{kJ~mol}^{-1}~\mathrm{ps}$$
$$\hbar$$ Dirac’s constant $$0.063\,507\,799\,3~\mathrm{kJ~mol}^{-1}~\mathrm{ps}$$
$$c$$ velocity of light $$299\,792.458~\mathrm{nm~ps}^{-1}$$

## Reduced units¶

When simulating Lennard-Jones (LJ) systems, it might be advantageous to use reduced units (i.e., 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 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 Table 5 quantities are given for LJ potentials:

(5)$V_{LJ} = 4\epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]$
Table 5 Reduced Lennard-Jones quantities
Quantity Symbol Relation to SI
Length r$$^*$$ r$$\sigma^{-1}$$
Mass m$$^*$$ m M$$^{-1}$$
Time t$$^*$$ t$$\sigma^{-1}~\sqrt{\epsilon/M}$$
Temperature T$$^*$$ k$$_B\mathrm{T}~\epsilon^{-1}$$
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$$\sigma ^3~V^{-1}$$

## Mixed or Double precision¶

GROMACS can be compiled in either mixed or 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 -DGMX\_DOUBLE=on to 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 (sec. Virial and pressure). 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:

• normal mode analysis, for the conjugate gradient or l-bfgs minimization and the calculation and diagonalization of the Hessian
• long-term energy conservation, especially for large systems