\(\renewcommand{\AA}{\text{Å}}\)

# pair_style lj/cut/dipole/cut command

Accelerator Variants: lj/cut/dipole/cut/gpu, lj/cut/dipole/cut/kk, lj/cut/dipole/cut/omp

# pair_style lj/sf/dipole/sf command

Accelerator Variants: lj/sf/dipole/sf/gpu, lj/sf/dipole/sf/omp

# pair_style lj/cut/dipole/long command

Accelerator Variants: lj/cut/dipole/long/gpu

# pair_style lj/long/dipole/long command

## Syntax

    pair_style lj/cut/dipole/cut cutoff (cutoff2)
    pair_style lj/sf/dipole/sf cutoff (cutoff2)
    pair_style lj/cut/dipole/long cutoff (cutoff2)
    pair_style lj/long/dipole/long flag_lj flag_coul cutoff (cutoff2)

- cutoff = global cutoff LJ (and Coulombic if only 1 arg) (distance units)

- cutoff2 = global cutoff for Coulombic and dipole (optional) (distance units)

- flag_lj = long or cut or off

      long = use long-range damping on dispersion 1/r^6 term
      cut = use a cutoff on dispersion 1/r^6 term
      off = omit disperion 1/r^6 term entirely

- flag_coul = long or off

      long = use long-range damping on Coulombic 1/r and point-dipole terms
      off = omit Coulombic and point-dipole terms entirely

## Examples

    pair_style lj/cut/dipole/cut 2.5 5.0
    pair_coeff * * 1.0 1.0
    pair_coeff 2 3 0.8 1.0 2.5 4.0

    pair_style lj/sf/dipole/sf 9.0
    pair_coeff * * 1.0 1.0
    pair_coeff 2 3 1.0 1.0 2.5 4.0 scale 0.5
    pair_coeff 2 3 0.8 1.0 2.5 4.0

    pair_style lj/cut/dipole/long 2.5 3.5
    pair_coeff * * 1.0 1.0
    pair_coeff 2 3 0.8 1.0 3.0

    pair_style lj/long/dipole/long long long 3.5
    pair_coeff * * 1.0 1.0
    pair_coeff 2 3 0.8 1.0

    pair_style lj/long/dipole/long cut long 2.5 3.5
    pair_coeff * * 1.0 1.0
    pair_coeff 2 3 0.8 1.0 3.0

## Description

Style lj/cut/dipole/cut computes interactions between pairs of particles that each have a charge and/or a point dipole moment. In addition to the usual Lennard-Jones interaction between the particles (Elj) the charge-charge (Eqq), charge-dipole (Eqp), and dipole-dipole (Epp) interactions are computed by these formulas for the energy (E), force (F), and torque (T) between particles I and J.

\[\begin{split}E_{LJ} = & 4 \epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right] \\ E_{qq} = & \frac{q_i q_j}{r} \\ E_{qp} = & \frac{q}{r^3} (p \bullet \vec{r}) \\ E_{pp} = & \frac{1}{r^3} (\vec{p}_i \bullet \vec{p}_j) - \frac{3}{r^5} (\vec{p}_i \bullet \vec{r}) (\vec{p}_j \bullet \vec{r}) \\ & \\ F_{qq} = & \frac{q_i q_j}{r^3} \vec{r} \\ F_{qp} = & -\frac{q}{r^3} \vec{p} + \frac{3q}{r^5} (\vec{p} \bullet \vec{r}) \vec{r} \\ F_{pp} = & \frac{3}{r^5} (\vec{p}_i \bullet \vec{p}_j) \vec{r} - \frac{15}{r^7} (\vec{p}_i \bullet \vec{r}) (\vec{p}_j \bullet \vec{r}) \vec{r} + \frac{3}{r^5} \left[ (\vec{p}_j \bullet \vec{r}) \vec{p}_i + (\vec{p}_i \bullet \vec{r}) \vec{p}_j \right] \\ & \\ T_{pq} = T_{ij} = & \frac{q_j}{r^3} (\vec{p}_i \times \vec{r}) \\ T_{qp} = T_{ji} = & - \frac{q_i}{r^3} (\vec{p}_j \times \vec{r}) \\ T_{pp} = T_{ij} = & -\frac{1}{r^3} (\vec{p}_i \times \vec{p}_j) + \frac{3}{r^5} (\vec{p}_j \bullet \vec{r}) (\vec{p}_i \times \vec{r}) \\ T_{pp} = T_{ji} = & -\frac{1}{r^3} (\vec{p}_j \times \vec{p}_i) + \frac{3}{r^5} (\vec{p}_i \bullet \vec{r}) (\vec{p}_j \times \vec{r})\end{split}\]

where \(q_i\) and \(q_j\) are the charges on the two particles, \(\vec{p}_i\) and \(\vec{p}_j\) are the dipole moment vectors of the two particles, r is their separation distance, and the vector r = Ri - Rj is the separation vector between the two particles. Note that Eqq and Fqq are simply Coulombic energy and force, Fij = -Fji as symmetric forces, and Tij != -Tji since the torques do not act symmetrically. These formulas are discussed in (AllenTildesley) and in (Toukmaji).

Also note, that in the code, all of these terms (except Elj) have a \(C/\epsilon\) prefactor, the same as the Coulombic term in the LJ + Coulombic pair styles discussed here. C is an energy-conversion constant and \(\epsilon\) is the dielectric constant which can be set by the dielectric command. The same is true of the equations that follow for other dipole pair styles.

Style lj/sf/dipole/sf computes “shifted-force” interactions between pairs of particles that each have a charge and/or a point dipole moment. In general, a shifted-force potential is a (slightly) modified potential containing extra terms that make both the energy and its derivative go to zero at the cutoff distance; this removes (cutoff-related) problems in energy conservation and any numerical instability in the equations of motion (AllenTildesley). Shifted-force interactions for the Lennard-Jones (E_LJ), charge-charge (Eqq), charge-dipole (Eqp), dipole-charge (Epq) and dipole-dipole (Epp) potentials are computed by these formulas for the energy (E), force (F), and torque (T) between particles I and J:

\[ \begin{align}\begin{aligned}\begin{split} E_{LJ} = & 4\epsilon \left\{ \left[ \left( \frac{\sigma}{r} \right)^{\!12} - \left( \frac{\sigma}{r} \right)^{\!6} \right] + \left[ 6\left( \frac{\sigma}{r_c} \right)^{\!12} - 3\left(\frac{\sigma}{r_c}\right)^{\!6}\right]\left(\frac{r}{r_c}\right)^{\!2} - 7\left( \frac{\sigma}{r_c} \right)^{\!12} + 4\left( \frac{\sigma}{r_c} \right)^{\!6}\right\} \\ E_{qq} = & \frac{q_i q_j}{r}\left(1-\frac{r}{r_c}\right)^{\!2} \\ E_{pq} = & E_{ji} = -\frac{q}{r^3} \left[ 1 - 3\left(\frac{r}{r_c}\right)^{\!2} + 2\left(\frac{r}{r_c}\right)^{\!3}\right] (\vec{p}\bullet\vec{r}) \\ E_{qp} = & E_{ij} = \frac{q}{r^3} \left[ 1 - 3\left(\frac{r}{r_c}\right)^{\!2} + 2\left(\frac{r}{r_c}\right)^{\!3}\right] (\vec{p}\bullet\vec{r}) \\ E_{pp} = & \left[1-4\left(\frac{r}{r_c}\right)^{\!3} + 3\left(\frac{r}{r_c}\right)^{\!4}\right]\left[\frac{1}{r^3} (\vec{p}_i \bullet \vec{p}_j) - \frac{3}{r^5} (\vec{p}_i \bullet \vec{r}) (\vec{p}_j \bullet \vec{r})\right] \\ & \\\end{split}\\\begin{split}F_{LJ} = & \left\{\left[48\epsilon \left(\frac{\sigma}{r}\right)^{\!12} - 24\epsilon \left(\frac{\sigma}{r}\right)^{\!6} \right]\frac{1}{r^2} - \left[48\epsilon \left(\frac{\sigma}{r_c}\right)^{\!12} - 24\epsilon \left(\frac{\sigma}{r_c}\right)^{\!6} \right]\frac{1}{r_c^2}\right\}\vec{r}\\ F_{qq} = & \frac{q_i q_j}{r}\left(\frac{1}{r^2} - \frac{1}{r_c^2}\right)\vec{r} \\ F_{pq} = & F_{ij } = -\frac{3q}{r^5} \left[ 1 - \left(\frac{r}{r_c}\right)^{\!2}\right](\vec{p}\bullet\vec{r})\vec{r} + \frac{q}{r^3}\left[1-3\left(\frac{r}{r_c}\right)^{\!2} + 2\left(\frac{r}{r_c}\right)^{\!3}\right] \vec{p} \\ F_{qp} = & F_{ij} = \frac{3q}{r^5} \left[ 1 - \left(\frac{r}{r_c}\right)^{\!2}\right] (\vec{p}\bullet\vec{r})\vec{r} - \frac{q}{r^3}\left[1-3\left(\frac{r}{r_c}\right)^{\!2} + 2\left(\frac{r}{r_c}\right)^{\!3}\right] \vec{p} \\ F_{pp} = &\frac{3}{r^5}\Bigg\{\left[1-\left(\frac{r}{r_c}\right)^{\!4}\right] \left[(\vec{p}_i\bullet\vec{p}_j) - \frac{3}{r^2} (\vec{p}_i\bullet\vec{r}) (\vec{p}_j \bullet \vec{r})\right] \vec{r} + \\ & \left[1 - 4\left(\frac{r}{r_c}\right)^{\!3}+3\left(\frac{r}{r_c}\right)^{\!4}\right] \left[ (\vec{p}_j \bullet \vec{r}) \vec{p}_i + (\vec{p}_i \bullet \vec{r}) \vec{p}_j -\frac{2}{r^2} (\vec{p}_i \bullet \vec{r}) (\vec{p}_j \bullet \vec{r})\vec{r}\right] \Bigg\}\end{split}\end{aligned}\end{align} \]
\[\begin{split} T_{pq} = T_{ij} = & \frac{q_j}{r^3} \left[ 1 - 3\left(\frac{r}{r_c}\right)^{\!2} + 2\left(\frac{r}{r_c}\right)^{\!3}\right] (\vec{p}_i\times\vec{r}) \\ T_{qp} = T_{ji} = & - \frac{q_i}{r^3} \left[ 1 - 3\left(\frac{r}{r_c}\right)^{\!2} + 2\left(\frac{r}{r_c}\right)^{\!3} \right] (\vec{p}_j\times\vec{r}) \\ T_{pp} = T_{ij} = & -\frac{1}{r^3}\left[1-4\left(\frac{r}{r_c}\right)^{\!3} + e3\left(\frac{r}{r_c}\right)^{\!4}\right] (\vec{p}_i \times \vec{p}_j) + \\ & \frac{3}{r^5}\left[1-4\left(\frac{r}{r_c}\right)^{\!3} + 3\left(\frac{r}{r_c}\right)^{\!4}\right] (\vec{p}_j\bullet\vec{r}) (\vec{p}_i \times \vec{r}) \\ T_{pp} = T_{ji} = & -\frac{1}{r^3}\left[1-4\left(\frac{r}{r_c}\right)^{\!3} + 3\left(\frac{r}{r_c}\right)^{\!4}\right](\vec{p}_j \times \vec{p}_i) + \\ & \frac{3}{r^5}\left[1-4\left(\frac{r}{r_c}\right)^{\!3} + 3\left(\frac{r}{r_c}\right)^{\!4}\right] (\vec{p}_i \bullet \vec{r}) (\vec{p}_j \times \vec{r})\end{split}\]

where \(\epsilon\) and \(\sigma\) are the standard LJ parameters, \(r_c\) is the cutoff, \(q_i\) and \(q_j\) are the charges on the two particles, \(\vec{p}_i\) and \(\vec{p}_j\) are the dipole moment vectors of the two particles, r is their separation distance, and the vector r = Ri - Rj is the separation vector between the two particles. Note that Eqq and Fqq are simply Coulombic energy and force, Fij = -Fji as symmetric forces, and Tij != -Tji since the torques do not act symmetrically. The shifted-force formula for the Lennard-Jones potential is reported in (Stoddard). The original (non-shifted) formulas for the electrostatic potentials, forces and torques can be found in (Price). The shifted-force electrostatic potentials have been obtained by applying equation 5.13 of (AllenTildesley). The formulas for the corresponding forces and torques have been obtained by applying the ‘chain rule’ as in appendix C.3 of (AllenTildesley).

If one cutoff is specified in the pair_style command, it is used for both the LJ and Coulombic (q,p) terms. If two cutoffs are specified, they are used as cutoffs for the LJ and Coulombic (q,p) terms respectively. This pair style also supports an optional scale keyword as part of a pair_coeff statement, where the interactions can be scaled according to this factor. This scale factor is also made available for use with fix adapt.

Style lj/cut/dipole/long computes the short-range portion of point-dipole interactions as discussed in (Toukmaji). Dipole-dipole, dipole-charge, and charge-charge interactions are all supported, along with the standard 12/6 Lennard-Jones interactions, which are computed with a cutoff. A kspace_style must be defined to use this pair style. If only dipoles (not point charges) are included in the model, the kspace style can be one of these 3 options, all of which compute the long-range portion of dipole-dipole interactions. If the model includes point charges (in addition to dipoles), then only the first of these kspace styles can be used:

- kspace_style ewald/disp

- kspace_style ewald/dipole

- kspace_style pppm/dipole

Style lj/long/dipole/long has the same functionality as style lj/cut/dipole/long, except it also has an option to compute 12/6 Lennard-Jones interactions for use with a long-range dispersion kspace style. This is done by setting its flag_lj argument to long. For long-range LJ interactions, the kspace_style ewald/disp command must be used.

The following coefficients must be defined for each pair of atoms types via the pair_coeff command as in the examples above, or in the data file or restart files read by the read_data or read_restart commands, or by mixing as described below:

- \(\epsilon\) (energy units)

- \(\sigma\) (distance units)

- cutoff1 (distance units)

- cutoff2 (distance units)

The latter 2 coefficients are optional. If not specified, the global LJ and Coulombic cutoffs specified in the pair_style command are used. If only one cutoff is specified, it is used as the cutoff for both LJ and Coulombic interactions for this type pair. If both coefficients are specified, they are used as the LJ and Coulombic cutoffs for this type pair. When using a long-rang Coulomb solver, only a global Coulomb cutoff may be used and only the LJ cutoff may be changed with the pair_coeff command. When using the lj/long/dipole/long pair style with long long setting, only a single global cutoff may be provided and no cutoff for the pair_coeff command.

Note that for systems using these pair styles, typically particles should be able to exert torque on each other via their dipole moments so that the particle and its dipole moment can rotate. This requires they not be point particles, but finite-size spheres. Thus you should use a command like atom_style hybrid sphere dipole to use particles with both attributes.

The magnitude and orientation of the dipole moment for each particle can be defined by the set command or in the “Atoms” section of the data file read in by the read_data command.

Rotating finite-size particles have 6 degrees of freedom (DOFs), translation and rotational. You can use the compute temp/sphere command to monitor a temperature which includes all these DOFs.

Finite-size particles with dipole moments should be integrated using one of these options:

- fix nve/sphere update dipole

- fix nve/sphere update dipole plus fix langevin omega yes

- fix nvt/sphere update dipole

- fix npt/sphere update dipole

In all cases the “update dipole” setting ensures the dipole moments are also rotated when the finite-size spheres rotate. The 2nd and 3rd bullets perform thermostatting; in the case of a Langevin thermostat the “omega yes” option also thermostats the rotational degrees of freedom (if desired). The 4th bullet performs thermostatting and barostatting.

Styles with a gpu, intel, kk, omp, or opt suffix are functionally the same as the corresponding style without the suffix. They have been optimized to run faster, depending on your available hardware, as discussed on the Accelerator packages page. The accelerated styles take the same arguments and should produce the same results, except for round-off and precision issues.

These accelerated styles are part of the GPU, INTEL, KOKKOS, OPENMP, and OPT packages, respectively. They are only enabled if LAMMPS was built with those packages. See the Build package page for more info.

You can specify the accelerated styles explicitly in your input script by including their suffix, or you can use the -suffix command-line switch when you invoke LAMMPS, or you can use the suffix command in your input script.

See the Accelerator packages page for more instructions on how to use the accelerated styles effectively.

## Mixing, shift, table, tail correction, restart, rRESPA info

For atom type pairs I,J and I != J, the epsilon and sigma coefficients and cutoff distances for this pair style can be mixed. The default mix value is geometric. See the “pair_modify” command for details.

For atom type pairs I,J and I != J, the A, sigma, d1, and d2 coefficients and cutoff distance for this pair style can be mixed. A is an energy value mixed like a LJ epsilon. D1 and d2 are distance values and are mixed like sigma. The default mix value is geometric. See the “pair_modify” command for details.

This pair style does not support the pair_modify shift option for the energy of the Lennard-Jones portion of the pair interaction; such energy goes to zero at the cutoff by construction.

The pair_modify table option is not relevant for this pair style.

This pair style does not support the pair_modify tail option for adding long-range tail corrections to energy and pressure.

This pair style writes its information to binary restart files, so pair_style and pair_coeff commands do not need to be specified in an input script that reads a restart file.

This pair style can only be used via the pair keyword of the run_style respa command. It does not support the inner, middle, outer keywords.

## Restrictions

The lj/cut/dipole/cut, lj/cut/dipole/long, lj/long/dipole/long, and lj/sf/dipole/sf* styles are part of the DIPOLE package. They are only enabled if LAMMPS was built with that package. See the Build package page for more info.

Using dipole pair styles with electron units is not currently supported.

## Related commands

pair_coeff, set, read_data, fix nve/sphere, fix nvt/sphere

## Default

none

(AllenTildesley) Allen and Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987.

(Toukmaji) Toukmaji, Sagui, Board, and Darden, J Chem Phys, 113, 10913 (2000).

(Stoddard) Stoddard and Ford, Phys Rev A, 8, 1504 (1973).

(Price) Price, Stone and Alderton, Mol Phys, 52, 987 (1984).