## LAMMPS Force Fields

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

### 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, and charmm 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 van der Waals

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

#### (1) lj/cutoff

```  E = Do [ (Ro/r)^12 - 2 (Ro/r)^6 ]

standard Lennard Jones potential

r = distance (computed by LAMMPS)

coeff1 = Do (energy)
coeff2 = Ro (distance)

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

```

#### (2) lj/switch

```  E = Do [ (Ro/r)^12 - 2(Ro/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 = Do (energy)
coeff2 = Ro (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 = Do [ (Ro/(r - delta))^12 - 2(Ro/(r - delta))^6 ]

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

r = distance (computed by LAMMPS)

coeff1 = Do (energy)
coeff2 = Ro (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 = Do [ (Ro/r)^12 - 2(Ro/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 = Do (energy)
coeff2 = Ro (distance)
coeff3 = Do for 1-4 interactions (energy)
coeff4 = Ro 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```

#### 7) expo_6/cutoff

```  E = Do [(6/6-y) exp (y(1-r/Ro) -  (y/y-6)(Ro/r)^6 ]  for  r < cutoff
= 0             for r > cutoff

Input parameters are Do, Ro, and y. The internal representation of expo_6 is```
`  E = A*exp(-CR) - B/r^6`
`with combination rules calculated with A,B,C as follows:`
`   Aij = sqrt(Aii Ajj)  Bij = sqrt(Bii Bjj) Cij = 1/2(Cii + Cjj)`

#### 8) expo_6/spline

```  E = Do [(6/6-y) exp (y(1-r/Ro) -  (y/y-6)(Ro/r)^6 ]  for  r < r_inner
= switch * E    for  r_inner < r < cutoff
= 0             for r > cutoff

where

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

switching function ensures continuity of energy and forces at r_inner.
Forces are not monotonically vanishing to zero. A ratio
r_outer/r_inner >= 5/4 is recommended.```

#### 9) expo_6/smooth

```  E = Do [(6/6-y) exp (y(1-r/Ro) -  (y/y-6)(Ro/r)^6 ]  for  r < cutoff
= switch   for  cutoff < r < r_outer
= 0             for r > r_outer

where

switch = a*(r-cutoff)^3 + d1*(r-cutoff) + e1```
`  where`
`  a = -4 d1^3/27 e1^2`
`  d1 = dE/dr at r=cutoff`
`  e1 = E(r=cutoff)`
```  r_outer = cutoff - ( 3e1/2d1 )

switching function ensures continuity of energy and forces at r=cutoff.
Forces and energy monotonically vanish to zero at r_outer. The ratio
r_outer/r_inner is approximately equal to 5/4.  The info flag 1 prints out
the values of the coefficients and cutoffs.```

### Mixing Rules for Exponential-6

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 the mixing rule:

Aij = sqrt(Aii * Ajj)
Bij = sqrt(Bii * Bjj)
Cij = 1/2(Cii * Cjj)

where the exponential-6   E = Do [(6/6-y) exp (y(1-r/Ro) -  (y/y-6)(Ro/r)^6 ]  is internally handled as

E(r)=A*exp(-Cr) - B/r^6

• The default cannot be overridden using the "mixing style" command.

• ### 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:  Do_ij = sqrt(Do_i * Do_j)
Ro_ij = sqrt(Ro_i * Ro_j)

arithmetic: Do_ij = sqrt(Do_i * Do_j)
Ro_ij = (Ro_i + Ro_j) / 2

sixthpower: Do_ij =
(2 * sqrt(Do_i*Do_j) * Ro_i^3 * Ro_j^3) /
(Ro_i^6 + Ro_j^6)
Ro_ij=  ((Ro_i**6 + Ro_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. Three exceptions to this are (1) the exponential-6 form, (2) for the nonbond style soft, for which only an Do 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
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 = 1/2 K (theta - theta0)^2

theta = radians (computed by LAMMPS)

coeff1 = K (energy/radian^2) (the usual 1/2 is NOT 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)

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

```

### 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 (+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 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 }

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```

#### (5) dreiding

```  E = SUM 1/2 Kn [1 - d * cos (n*phi) ]
n
phi = radians (computed by LAMMPS), 0 for the cis conformation, 180 trans
phi is positive angle clockwise when looking form J toward K.

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

Cautions when comparing to other force fields:

notice the 1/2 in front of the input force constant (data file)
as well as the negative sign in front of d.  The value of
Kn in dreiding is divided by the number of multiple
torsions that contain the j-k bond in an i-j-k-l torsion.
Kn represents 1/2 the rotational barrier, n the periodicity.
d=+1, the cis conformation is the minimum, while for d=-1 the
cis conformation is a maximum.  In addition the value of K
is rescaled for exocyclic "aromatic rings" according to the
force field used in the translation of the file that creates
the data file.  All fourier terms (up to six) can be specified
although one is sufficient often.  Use the same type for all terms
in consecutive lines in the data or input files.
3 coeffs are listed in data file or set in input script for each
term in each dihedral type dreiding.```

### 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)

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)

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)

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)