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<H2>
LAMMPS Force Fields</H2>
<P>
<A HREF="README.html">Return</A> to top-level of LAMMPS documentation</P>
<P>
This file outlines the force-field formulas used in LAMMPS. Read this
file in conjunction with the <A HREF="data_format.html">data_format</A>
and <A HREF="units.html">units</A> files.</P>
<P>
The sections of this page are as follows:</P>
<UL>
<LI>
<A HREF="#_cch3_930957465">Nonbond Coulomb</A>
<LI>
<A HREF="#_cch3_930957471">Nonbond Lennard-Jones</A>
<LI>
<A HREF="#_cch3_930957478">Mixing Rules for Lennard-Jones</A>
<LI>
<A HREF="#_cch3_930957482">Bonds</A>
<LI>
<A HREF="#_cch3_930957488">Angles</A>
<LI>
<A HREF="#_cch3_930957509">Dihedrals</A>
<LI>
<A HREF="#_cch3_930957513">Impropers</A>
<LI>
<A HREF="#_cch3_930957527">Class 2 Force Field</A>
</UL>
<HR>
<H3>
<A NAME="_cch3_930957465">Nonbond Coulomb</A></H3>
<P>
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.</P>
<PRE>
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
</PRE>
For the debye style, the smoother is exp(-kappa*r) where kappa is an
input parameter.
<HR>
<H3>
<A NAME="_cch3_930957471">Nonbond Lennard-Jones </A></H3>
<P>
The style of nonbond potential is specified in the input command file. </P>
<H4>
(1) lj/cutoff </H4>
<PRE>
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
</PRE>
<H4>
(2) lj/switch </H4>
<PRE>
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
</PRE>
<H4>
(3) lj/shift </H4>
<PRE>
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
</PRE>
<H4>
(4) soft </H4>
<PRE>
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
</PRE>
<H4>
(5) class2/cutoff </H4>
<PRE>
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
</PRE>
<H4>
6) lj/charmm </H4>
<PRE>
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
</PRE>
<HR>
<H3>
<A NAME="_cch3_930957478">Mixing Rules for Lennard-Jones</A></H3>
<P>
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: </P>
<PRE>
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)
</PRE>
<P>
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". </P>
<P>
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 </P>
<UL>
<LI>
delta_ij = (delta_i + delta_j) / 2
</UL>
<HR>
<H3>
<A NAME="_cch3_930957482">Bonds</A></H3>
<P>
The style of bond potential is specified in the input command file.</P>
<H4>
(1) harmonic </H4>
<PRE>
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
</PRE>
<H4>
(2) FENE/standard </H4>
<PRE>
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
</PRE>
<H4>
(3) FENE/shift </H4>
<PRE>
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
</PRE>
<H4>
(4) nonlinear </H4>
<PRE>
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
</PRE>
<H4>
(5) class2 </H4>
<PRE>
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
</PRE>
<HR>
<H3>
<A NAME="_cch3_930957488">Angles </A></H3>
<P>
The style of angle potential is specified in the input command file. </P>
<H4>
(1) harmonic </H4>
<PRE>
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
</PRE>
<H4>
(2) class2 </H4>
<PRE>
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
</PRE>
<H4>
(3) charmm </H4>
<PRE>
(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
</PRE>
<H4>
(4) cosine </H4>
<PRE>
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
</PRE>
<H3>
<A NAME="_cch3_930957509">Dihedrals </A></H3>
<P>
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. </P>
<H4>
(1) harmonic </H4>
<PRE>
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
</PRE>
<H4>
(2) class2 </H4>
<PRE>
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
</PRE>
<H4>
(3) multiharmonic </H4>
<PRE>
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
</PRE>
<H4>
(4) charmm </H4>
<PRE>
(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
</PRE>
<HR>
<H3>
<A NAME="_cch3_930957513">Impropers</A></H3>
<P>
The style of improper potential is specified in the input command file. </P>
<H4>
(1) harmonic </H4>
<PRE>
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
</PRE>
<H4>
(2) cvff </H4>
<PRE>
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
</PRE>
<H4>
(3) class2 </H4>
<PRE>
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
</PRE>
<HR>
<H3>
<A NAME="_cch3_930957527">Class 2 Force Field</A></H3>
<P>
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.</P>
<H4>
Bond-Bond (computed within class 2 angles) </H4>
<PRE>
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
</PRE>
<H4>
Bond-Angle (computed within class 2 angles for each of 2 bonds) </H4>
<PRE>
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
</PRE>
<H4>
Middle-Bond-Torsion (computed within class 2 dihedral) </H4>
<PRE>
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
</PRE>
<H4>
End-Bond-Torsion (computed within class 2 dihedral for each of 2 bonds) </H4>
<PRE>
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
</PRE>
<H4>
Angle-Torsion (computed within class 2 dihedral for each of 2 angles) </H4>
<PRE>
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
</PRE>
<H4>
Angle-Angle-Torsion (computed within class 2 dihedral) </H4>
<PRE>
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
</PRE>
<H4>
Bond-Bond-13-Torsion (computed within class 2 dihedral) </H4>
<PRE>
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
</PRE>
<H4>
Angle-Angle (computed within class 2 improper for each of 3 pairs of
angles) </H4>
<PRE>
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
</PRE>
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