## LAMMPS Force Fields

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This file outlines the force-field formulas used in LAMMPS. Read this file in conjunction with the data_format and units files.

The sections of this page are as follows:

- Nonbond Coulomb
- Nonbond Lennard-Jones
- Mixing Rules for Lennard-Jones
- Bonds
- Angles
- Dihedrals
- Impropers
- Class 2 Force Field

### Nonbond Coulomb

Whatever Coulomb style is specified in the input command file, the short-range Coulombic interactions are computed by this formula, modified by an appropriate smoother for the smooth, Ewald, PPPM, charmm, and debye styles.

      E = C q1 q2 / (epsilon * r)

      r = distance (computed by LAMMPS)
      C = hardwired constant to convert to energy units
      q1,q2 = charge of each atom in electron units (proton = +1),
        specified in "Atoms" entry in data file
      epsilon = dielectric constant (vacuum = 1.0),
        set by user in input command file

For the debye style, the smoother is exp(-kappa*r) where kappa is an input parameter.

### Nonbond Lennard-Jones

The style of nonbond potential is specified in the input command file.

#### (1) lj/cutoff

      E = 4 epsilon [ (sigma/r)^12 - (sigma/r)^6 ]

      standard Lennard Jones potential

      r = distance (computed by LAMMPS)

      coeff1 = epsilon (energy)
      coeff2 = sigma (distance)

      2 coeffs are listed in data file or set in input script
      1 cutoff is set in input script

#### (2) lj/switch

      E = 4 epsilon [ (sigma/r)^12 - (sigma/r)^6 ]  for  r < r_inner
        = spline fit    for  r_inner < r < cutoff
        = 0             for r > cutoff

      switching function (spline fit) is applied to standard LJ
        within a switching region (from r_inner to cutoff) so that
        energy and force go smoothly to zero
      spline coefficients are computed by LAMMPS
        so that at inner cutoff (r_inner) the potential, force,
        and 1st-derivative of force are all continuous,
        and at outer cutoff (cutoff) the potential and force
        both go to zero

      r = distance (computed by LAMMPS)

      coeff1 = epsilon (energy)
      coeff2 = sigma (distance)

      2 coeffs are listed in data file or set in input script
      2 cutoffs (r_inner and cutoff) are set in input script

#### (3) lj/shift

      E = 4 epsilon [ (sigma/(r - delta))^12 - (sigma/(r - delta))^6 ]

      same as lj/cutoff except that r is shifted by delta

      r = distance (computed by LAMMPS)

      coeff1 = epsilon (energy)
      coeff2 = sigma (distance)
      coeff3 = delta (distance)

      3 coeffs are listed in data file or set in input script
      1 cutoff is set in input script

#### (4) soft

      E = A * [ 1 + cos( pi * r / cutoff ) ]

      useful for pushing apart overlapping atoms by ramping A over time

      r = distance (computed by LAMMPS)

      coeff1 = prefactor A at start of run (energy)
      coeff2 = prefactor A at end of run (energy)

      2 coeffs are listed in data file or set in input script
      1 cutoff is set in input script

#### (5) class2/cutoff

      E = epsilon [ 2 (sigma/r)^9 - 3 (sigma/r)^6 ]

      used with class2 bonded force field

      r = distance (computed by LAMMPS)

      coeff1 = epsilon (energy)
      coeff2 = sigma (distance)

      2 coeffs are listed in data file or set in input script
      1 cutoff is set in input script

#### 6) lj/charmm

      E = 4 epsilon [ (sigma/r)^12 - (sigma/r)^6 ]  for  r < r_inner
        = switch * E    for  r_inner < r < cutoff
        = 0             for r > cutoff

      where

      switch = [(cutoff^2 - r^2)^2 * (cutoff^2 + 2*r^2 - 3*r_inner)] /
               [(cutoff^2 - r_inner^2)^3]

      switching function is applied to standard LJ
        within a switching region (from r_inner to cutoff) so that
        energy and force go smoothly to zero
      switching function causes that at inner cutoff (r_inner)
        the potential and force are continuous,
        and at outer cutoff (cutoff) the potential and force
        both go to zero

      r = distance (computed by LAMMPS)

      coeff1 = epsilon (energy)
      coeff2 = sigma (distance)
      coeff3 = epsilon for 1-4 interactions (energy)
      coeff4 = sigma for 1-4 interactions (distance)

      4 coeffs are listed in data file or set in input script
      2 cutoffs (r_inner and cutoff) are set in input script

### Mixing Rules for Lennard-Jones

The coefficients for each nonbond style are input in either the data file by the "read data" command or in the input script using the "nonbond coeff" command. In the former case, only one set of coefficients is input for each atom type. The cross-type coeffs are computed using one of three possible mixing rules:

     geometric:  epsilon_ij = sqrt(epsilon_i * epsilon_j)
                 sigma_ij = sqrt(sigma_i * sigma_j)

     arithmetic: epsilon_ij = sqrt(epsilon_i * epsilon_j)
                 sigma_ij = (sigma_i + sigma_j) / 2

     sixthpower: epsilon_ij =
                   (2 * sqrt(epsilon_i*epsilon_j) * sigma_i^3 * sigma_j^3) /
                   (sigma_i^6 + sigma_j^6)
                 sigma_ij=  ((sigma_i**6 + sigma_j**6) / 2) ^ (1/6)

The default mixing rule for nonbond styles lj/cutoff, lj/switch, lj/shift, and soft is "geometric". The default for nonbond style class2/cutoff is "sixthpower".

The default can be overridden using the "mixing style" command. Two exceptions to this are for the nonbond style soft, for which only an epsilon prefactor is input. This is always mixed geometrically. Also, for nonbond style lj/shift, the delta coefficient is always mixed using the rule

- delta_ij = (delta_i + delta_j) / 2

### Bonds

The style of bond potential is specified in the input command file.

#### (1) harmonic

      E = K (r - r0)^2

      standard harmonic spring

      r = distance (computed by LAMMPS)

      coeff1 = K (energy/distance^2)  (the usual 1/2 is included in the K)
      coeff2 = r0 (distance)

      2 coeffs are listed in data file or set in input script

#### (2) FENE/standard

      E = -0.5 K R0^2 * ln[1 - (r/R0)^2] +
        4 epsilon [(sigma/r)^12 - (sigma/r)^6] + epsilon

      finite extensible nonlinear elastic (FENE) potential for
        polymer bead-spring models
      see Kremer, Grest, J Chem Phys, 92, p 5057 (1990)

      r = distance (computed by LAMMPS)

      coeff1 = K (energy/distance^2)
      coeff2 = R0 (distance)
      coeff3 = epsilon (energy)
      coeff4 = sigma (distance)

      1st term is attraction, 2nd term is repulsion (shifted LJ)
      1st term extends to R0
      2nd term only extends to the minimum of the LJ potential,
        a cutoff distance computed by LAMMPS (2^(1/6) * sigma)

      4 coeffs are listed in data file or set in input script

#### (3) FENE/shift

      E = -0.5 K R0^2 * ln[1 - ((r - delta)/R0)^2] +
        4 epsilon [(sigma/(r - delta))^12 - (sigma/(r - delta))^6] + epsilon

      same as FENE/standard expect that r is shifted by delta

      r = distance (computed by LAMMPS)

      coeff1 = K (energy/distance^2)
      coeff2 = R0 (distance)
      coeff3 = epsilon (energy)
      coeff4 = sigma (distance)
      coeff5 = delta (distance)

      1st term is attraction, 2nd term is repulsion (shifted LJ)
      1st term extends to R0
      2nd term only extends to the minimum of the LJ potential,
        a cutoff distance computed by LAMMPS (2^(1/6) * sigma + delta)

      5 coeffs are listed in data file or set in input script

#### (4) nonlinear

      E = epsilon (r - r0)^2 / [ lamda^2 - (r - r0)^2 ]

      non-harmonic spring of equilibrium length r0
        with finite extension of lamda
      see Rector, Van Swol, Henderson, Molecular Physics, 82, p 1009 (1994)

      r = distance (computed by LAMMPS)

      coeff1 = epsilon (energy)
      coeff2 = r0 (distance)
      coeff3 = lamda (distance)

      3 coeffs are listed in data file or set in input script

#### (5) class2

      E = K2 (r - r0)^2  +  K3 (r - r0)^3  +  K4 (r - r0)^4

      r = distance (computed by LAMMPS)

      coeff1 = r0 (distance)
      coeff2 = K2 (energy/distance^2)
      coeff3 = K3 (energy/distance^3)
      coeff4 = K4 (energy/distance^4)

      4 coeffs are listed in data file - cannot be set in input script

### Angles

The style of angle potential is specified in the input command file.

#### (1) harmonic

      E = K (theta - theta0)^2

      theta = radians (computed by LAMMPS)

      coeff1 = K (energy/radian^2) (the usual 1/2 is included in the K)
      coeff2 = theta0 (degrees) (converted to radians within LAMMPS)

      2 coeffs are listed in data file or set in input script

#### (2) class2

      E = K2 (theta - theta0)^2 +  K3 (theta - theta0)^3 +
           K4 (theta - theta0)^4

      theta = radians (computed by LAMMPS)

      coeff1 = theta0 (degrees) (converted to radians within LAMMPS)
      coeff2 = K2 (energy/radian^2)
      coeff3 = K3 (energy/radian^3)
      coeff4 = K4 (energy/radian^4)

      4 coeffs are listed in data file - cannot be set in input script

#### (3) charmm

      (harmonic + Urey-Bradley)

      E = K (theta - theta0)^2 + K_UB (r_13 - r_UB)^2

      theta = radians (computed by LAMMPS)
      r_13 = distance (computed by LAMMPS)

      coeff1 = K (energy/radian^2) (the usual 1/2 is included in the K)
      coeff2 = theta0 (degrees) (converted to radians within LAMMPS)
      coeff3 = K_UB (energy/distance^2)
      coeff4 = r_UB (distance)

      4 coeffs are listed in data file or set in input script

#### (4) cosine

      E = K (1 + cos(theta))

      theta = radians (computed by LAMMPS)

      coeff1 = K (energy)

      1 coeff is listed in data file or set in input script

### Dihedrals

The style of dihedral potential is specified in the input command file. IMPORTANT NOTE for all these dihedral styles: in the LAMMPS force field the trans position = 180 degrees, while in some force fields trans = 0 degrees.

#### (1) harmonic

      E = K [1 + d * cos (n*phi) ]

      phi = radians (computed by LAMMPS)

      coeff1 = K (energy)
      coeff2 = d (+1 or -1)
      coeff3 = n (1,2,3,4,6)

      Additional cautions when comparing to other force fields:

      some force fields reverse the sign convention on d so that
        E = K [1 - d * cos(n*phi)]
      some force fields divide/multiply K by the number of multiple
        torsions that contain the j-k bond in an i-j-k-l torsion
      some force fields let n be positive or negative which
        corresponds to d = 1,-1

      3 coeffs are listed in data file or set in input script

#### (2) class2

      E = SUM(n=1,3) { K_n [ 1 - cos( n*Phi - Phi0_n ) ] }

      phi = radians (computed by LAMMPS)

      coeff1 = K_1 (energy)
      coeff2 = Phi0_1 (degrees) (converted to radians within LAMMPS)
      coeff3 = K_2 (energy)
      coeff4 = Phi0_2 (degrees) (converted to radians within LAMMPS)
      coeff5 = K_3 (energy)
      coeff6 = Phi0_3 (degrees) (converted to radians within LAMMPS)

      6 coeffs are listed in data file - cannot be set in input script

#### (3) multiharmonic

      E = SUM(n=1,5) { A_n * cos(Phi)^(n-1) }

      phi = radians (computed by LAMMPS)

      coeff1 = A_1
      coeff2 = A_2
      coeff3 = A_3
      coeff4 = A_4
      coeff5 = A_5

      5 coeffs are listed in data file or set in input script

#### (4) charmm

    (harmonic + 1-4 interactions)

      E = K [1 + cos (n*phi + d) ]

      phi = radians (computed by LAMMPS)

      coeff1 = K (energy)
      coeff2 = n (1,2,3,4,6)
      coeff3 = d (0 or 180 degrees) (converted to radians within LAMMPS)
      coeff4 = weighting factor to turn on/off 1-4 neighbor nonbond interactions

      coeff4 weight values are from 0.0 to 1.0 and are used to multiply the
      energy and force interaction (both Coulombic and LJ) between the 2 atoms
      weight of 0.0 means no interaction
      weight of 1.0 means full interaction

      must be used with the special bonds charmm command
      "special bonds 0 0 0") which shuts off the uniform special bonds and
      allows pair-specific special bonds for the 1-4 interactions to be
      defined in the data file

      LAMMPS assumes that all 1-4 interaction distances, which are
      generally less than 6 Angstroms, are less than the smallest of the
      inner LJ and Coulombic cutoffs, which are generally at least 8
      Angstroms.

      4 coeffs are listed in data file or set in input script

### Impropers

The style of improper potential is specified in the input command file.

#### (1) harmonic

      E = K (chi - chi0)^2

      chi = radians (computed by LAMMPS)

      coeff1 = K (energy/radian^2) (the usual 1/2 is included in the K)
      coeff2 = chi0 (degrees) (converted to radians within LAMMPS)

      2 coeffs are listed in data file or set in input script

#### (2) cvff

      E = K [1 + d * cos (n*chi) ]

      chi = radians (computed by LAMMPS)

      coeff1 = K (energy)
      coeff2 = d (+1 or -1)
      coeff3 = n (0,1,2,3,4,6)

      3 coeffs are listed in data file or set in input script

#### (3) class2

      same formula, coeffs, and meaning as "harmonic" except that LAMMPS
        averages all 3 angle-contributions to chi
      in class 2 this is called a Wilson out-of-plane interaction

      2 coeffs are listed in data file - cannot be set in input script

### Class 2 Force Field

If class 2 force fields are selected in the input command file, additional cross terms are computed as part of the force field. All class 2 coefficients must be set in the data file; they cannot be set in the input script.

#### Bond-Bond (computed within class 2 angles)

      E = K (r - r0) * (r' - r0')

      r,r' = distance (computed by LAMMPS)

      coeff1 = K (energy/distance^2)
      coeff2 = r0 (distance)
      coeff3 = r0' (distance)

      3 coeffs are input in data file

#### Bond-Angle (computed within class 2 angles for each of 2 bonds)

      E = K_n (r - r0_n) * (theta - theta0)

      r = distance (computed by LAMMPS)
      theta = radians (computed by LAMMPS)

      coeff1 = K_1 (energy/distance-radians)
      coeff2 = K_2 (energy/distance-radians)
      coeff3 = r0_1 (distance)
      coeff4 = r0_2 (distance)

      Note: theta0 is known from angle coeffs so don't need it specified here

      4 coeffs are listed in data file

#### Middle-Bond-Torsion (computed within class 2 dihedral)

      E = (r - r0) * [ F1*cos(phi) + F2*cos(2*phi) + F3*cos(3*phi) ]

      r = distance (computed by LAMMPS)
      phi = radians (computed by LAMMPS)

      coeff1 = F1 (energy/distance)
      coeff2 = F2 (energy/distance)
      coeff3 = F3 (energy/distance)
      coeff4 = r0 (distance)

      4 coeffs are listed in data file

#### End-Bond-Torsion (computed within class 2 dihedral for each of 2 bonds)

      E = (r - r0_n) * [ F1_n*cos(phi) + F2_n*cos(2*phi) + F3_n*cos(3*phi) ]

      r = distance (computed by LAMMPS)
      phi = radians (computed by LAMMPS)

      coeff1 = F1_1 (energy/distance)
      coeff2 = F2_1 (energy/distance)
      coeff3 = F3_1 (energy/distance)
      coeff4 = F1_2 (energy/distance)
      coeff5 = F2_3 (energy/distance)
      coeff6 = F3_3 (energy/distance)
      coeff7 = r0_1 (distance)
      coeff8 = r0_2 (distance)

      8 coeffs are listed in data file

#### Angle-Torsion (computed within class 2 dihedral for each of 2 angles)

      E = (theta - theta0) * [ F1_n*cos(phi) + F2_n*cos(2*phi) + F3_n*cos(3*phi) ]

      theta = radians (computed by LAMMPS)
      phi = radians (computed by LAMMPS)

      coeff1 = F1_1 (energy/radians)
      coeff2 = F2_1 (energy/radians)
      coeff3 = F3_1 (energy/radians)
      coeff4 = F1_2 (energy/radians)
      coeff5 = F2_3 (energy/radians)
      coeff6 = F3_3 (energy/radians)
      coeff7 = theta0_1 (degrees) (converted to radians within LAMMPS)
      coeff8 = theta0_2 (degrees) (converted to radians within LAMMPS)

      8 coeffs are listed in data file

#### Angle-Angle-Torsion (computed within class 2 dihedral)

      E = K (theta - theta0) * (theta' - theta0') * (phi - phi0)

      theta,theta' = radians (computed by LAMMPS)
      phi = radians (computed by LAMMPS)

      coeff1 = K (energy/radians^3)
      coeff2 = theta0 (degrees) (converted to radians within LAMMPS)
      coeff3 = theta0' (degrees) (converted to radians within LAMMPS)

      Note: phi0 is known from dihedral coeffs so don't need it specified here

      3 coeffs are listed in data file

#### Bond-Bond-13-Torsion (computed within class 2 dihedral)

      E = K * (r1 - r10)*(r3 - r30)

      r1,r3 = bond lengths of bonds 1 and 3 (computed by LAMMPS)

      coeff1 = K (energy/distance^2)
      coeff2 = r10 (distance) = equilibrium bond length for bond 1
      coeff3 = r30 (distance) = equilibrium bond length for bond 3

      K is only non-zero for aromatic rings

      3 coeffs are listed in data file

#### Angle-Angle (computed within class 2 improper for each of 3 pairs of angles)

      E = K_n (theta - theta0_n) * (theta' - theta0_n')

      theta,theta' = radians (computed by LAMMPS)

      coeff1 = K_1 (energy/radians^2)
      coeff2 = K_2 (energy/radians^2)
      coeff3 = K_3 (energy/radians^2)
      coeff4 = theta0_1 (degrees) (converted to radians within LAMMPS)
      coeff5 = theta0_2 (degrees) (converted to radians within LAMMPS)
      coeff6 = theta0_3 (degrees) (converted to radians within LAMMPS)

      6 coeffs are listed in data file