## 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 file.

The sections of this page are as follows:

- Nonbond Coulomb
- Nonbond Lennard-Jones
- Mixing Rules for Lennard-Jones
- Bonds
- Angles
- Dihedrals
- Impropers
- Class II 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, and PPPM 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

### 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

### 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. The one exception to this is 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

#### (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

### Dihedrals

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

#### (1) harmonic

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

      phi = radians (computed by LAMMPS)

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

      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
      in the LAMMPS force field, the trans position = 180 degrees, while
        in some force fields trans = 0 degrees

      3 coeffs are listed in data file

#### (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

### 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)

      in data file, listing of 4 atoms requires atom-1 as central atom
      some force fields (AMBER,Discover) have atom-2 as central atom - it is really
        an out-of-plane torsion, may need to treat as dihedral in LAMMPS

      2 coeffs are listed in data file

#### (2) class2

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

      2 coeffs are listed in data file

### Class II Force Field

If class II force fields are selected in the input command file, additional cross terms are computed as part of the force field.

#### Bond-Bond (computed within class II 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 II 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 II 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 II 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 II 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 II 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 II dihedral)

      (undocumented)

#### Angle-Angle (computed within class II 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