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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> file.</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 II 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, and PPPM
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 &quot;Atoms&quot; entry in data file
epsilon = dielectric constant (vacuum = 1.0),
set by user in input command file
</PRE>
<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 &lt; r_inner
= spline fit for r_inner &lt; r &lt; cutoff
= 0 for r &gt; 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>
<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 &quot;read data&quot; command or in the input script using
the &quot;nonbond coeff&quot; 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 &quot;geometric&quot;. The default for nonbond
style class2/cutoff is &quot;sixthpower&quot;. </P>
<P>
The default can be overridden using the &quot;mixing style&quot;
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. </P>
<P>
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
</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
</PRE>
<HR>
<H3>
<A NAME="_cch3_930957509">Dihedrals </A></H3>
<P>
The style of dihedral potential is specified in the input command file. </P>
<H4>
(1) harmonic </H4>
<PRE>
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
</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
</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)
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
</PRE>
<H4>
(2) class2 </H4>
<PRE>
same formula, coeffs, and meaning as &quot;harmonic&quot; 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
</PRE>
<HR>
<H3>
<A NAME="_cch3_930957527">Class II Force Field</A></H3>
<P>
If class II force fields are selected in the input command file,
additional cross terms are computed as part of the force field.</P>
<H4>
Bond-Bond (computed within class II 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 II 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 II 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 II 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 II 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 II 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 II dihedral) </H4>
<PRE>
(undocumented)
</PRE>
<H4>
Angle-Angle (computed within class II 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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