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.. index:: compute born/matrix
compute born/matrix command
===========================
Syntax
""""""
.. parsed-literal::
compute ID group-ID born/matrix
* ID, group-ID are documented in :doc:`compute <compute>` command
* born/matrix = style name of this compute command
Examples
""""""""
.. code-block:: LAMMPS
compute 1 all born/matrix
Description
"""""""""""
Define a compute that calculates
:math:`\frac{\partial{}^2U}{\partial\varepsilon_{i}\partial\varepsilon_{j}}` the
second derivatives of the potential energy :math:`U` with regard to strain
tensor :math:`\varepsilon` elements. These values are related to:
.. math::
C^{B}_{i,j}=\frac{1}{V}\frac{\partial{}^2U}{\partial{}\varepsilon_{i}\partial\varepsilon_{j}}
also called the Born term of elastic constants in the stress-stress fluctuation
formalism. This quantity can be used to compute the elastic constant tensor.
Using the symmetric Voigt notation, the elastic constant tensor can be written
as a 6x6 symmetric matrix:
.. math::
C_{i,j} = \langle{}C^{B}_{i,j}\rangle
+ \frac{V}{k_{B}T}\left(\langle\sigma_{i}\sigma_{j}\rangle\right.
\left.- \langle\sigma_{i}\rangle\langle\sigma_{j}\rangle\right)
+ \frac{Nk_{B}T}{V}
\left(\delta_{i,j}+(\delta_{1,i}+\delta_{2,i}+\delta_{3,i})\right.
\left.*(\delta_{1,j}+\delta_{2,j}+\delta_{3,j})\right)
In the above expression, :math:`\sigma` stands for the virial stress
tensor, :math:`\delta` is the Kronecker delta and the usual notation apply for
the number of particle, the temperature and volume respectively :math:`N`,
:math:`T` and :math:`V`. :math:`k_{B}` is the Boltzmann constant.
The Born term is a symmetric 6x6 matrix by construction and as such can be
expressed as 21 independent terms. The terms are ordered corresponding to the
following matrix element:
.. math::
\begin{matrix}
C_{1} & C_{7} & C_{8} & C_{9} & C_{10} & C_{11} \\
C_{7} & C_{2} & C_{12} & C_{13} & C_{14} & C_{15} \\
\vdots & C_{12} & C_{3} & C_{16} & C_{17} & C_{18} \\
\vdots & C_{13} & C_{16} & C_{4} & C_{19} & C_{20} \\
\vdots & \vdots & \vdots & C_{19} & C_{5} & C_{21} \\
\vdots & \vdots & \vdots & \vdots & C_{21} & C_{6}
\end{matrix}
in this matrix the indices of :math:`C_{k}` value are the corresponding index
:math:`k` in the compute output. Each term comes from the sum of every
interactions derivatives in the system as explained in :ref:`(VanWorkum)
<VanWorkum>` or :ref:`(Voyiatzis) <Voyiatzis>`.
The output can be accessed using usual Lammps routines:
.. code-block:: LAMMPS
compute 1 all born/matrix
compute 2 all pressure NULL virial
variable S1 equal -c_2[1]
variable S2 equal -c_2[2]
variable S3 equal -c_2[3]
variable S4 equal -c_2[4]
variable S5 equal -c_2[5]
variable S6 equal -c_2[6]
fix 1 all ave/time 1 1 1 v_S1 v_S2 v_S3 v_S4 v_S5 v_S6 c_1[*] file born.out
In this example, the file *born.out* will contain the information needed to
compute the first and second terms of the elastic constant matrix in a post
processing procedure. The other required quantities can be accessed using any
other *LAMMPS* usual method.
NOTE: In the above :math:`C_{i,j}` computation, the term involving the virial
stress tensor :math:`\sigma` is the covariance between each elements. In a
solid the virial stress can have large variations between timesteps and average
values can be slow to converge. This term is better computed using
instantaneous values.
**Output info:**
This compute calculates a global array with the number of rows=21.
The values are ordered as explained above. These values can be used
by any command that uses a global values from a compute as input. See
the :doc:`Howto output <Howto_output>` doc page for an overview of
LAMMPS output options.
The array values calculated by this compute are all "extensive".
Restrictions
""""""""""""
This compute is part of the EXTRA-COMPUTE package. It is only enabled if
LAMMPS was built with that package. See the :doc:`Build package <Build_package>` page for more info.
The Born term can be decomposed as a product of two terms. The first one
is a general term which depends on the configuration. The second one is
specific to every interaction composing your force field (non-bonded,
bonds, angle...). Currently not all interaction implement the *born_matrix*
method giving first and second order derivatives and a warning will
be raised if you try to use this compute with such interactions. The returned
values of this force field component is currently zero.
Default
"""""""
none
----------
.. _VanWorkum:
**(Van Workum)** K. Van Workum et al., J. Chem. Phys. 125 144506 (2006)
.. _Voyiatzis:
**(Voyiatzis)** E. Voyiatzis, Computer Physics Communications 184(2013)27-33

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# Numerical difference calculation
# of Born matrix
# adjustable parameters
variable nsteps index 50 # length of run
variable nthermo index 10 # thermo output interval
variable nlat index 3 # size of box
variable delta index 1.0e-6 # strain size
variable temp index 0.3 # temperature
units lj
atom_style atomic
lattice fcc 0.8442
region box block 0 ${nlat} 0 ${nlat} 0 ${nlat}
create_box 1 box
create_atoms 1 box
mass 1 1.0
velocity all create ${temp} 87287 loop geom
pair_style lj/cut 2.5
pair_coeff 1 1 1.0 1.0
neighbor 0.0 bin
neigh_modify every 1 delay 0 check no
fix 1 all nve
compute born all born/matrix
thermo ${nthermo}
thermo_style custom step temp pe press c_born[*]
run ${nsteps}

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# Numerical difference calculation
# of Born matrix
# adjustable parameters
variable nsteps index 50 # length of run
variable nthermo index 10 # thermo output interval
variable nlat index 3 # size of box
variable delta index 1.0e-6 # strain size
variable temp index 0.3 # temperature
units lj
atom_style atomic
lattice fcc 0.8442
region box block 0 ${nlat} 0 ${nlat} 0 ${nlat}
create_box 1 box
create_atoms 1 box
mass 1 1.0
velocity all create ${temp} 87287 loop geom
pair_style lj/cut 2.5
pair_coeff 1 1 1.0 1.0
neighbor 0.0 bin
neigh_modify every 1 delay 0 check no
fix 1 all nve
compute myvirial all pressure NULL virial
compute born all born/matrix numdiff ${delta} myvirial
thermo ${nthermo}
thermo_style custom step temp pe press c_born[*]
run ${nsteps}

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LAMMPS (7 Jan 2022)
# Numerical difference calculation
# of Born matrix
# adjustable parameters
variable nsteps index 50 # length of run
variable nthermo index 10 # thermo output interval
variable nlat index 3 # size of box
variable delta index 1.0e-6 # strain size
variable temp index 0.3 # temperature
units lj
atom_style atomic
lattice fcc 0.8442
Lattice spacing in x,y,z = 1.6795962 1.6795962 1.6795962
region box block 0 ${nlat} 0 ${nlat} 0 ${nlat}
region box block 0 3 0 ${nlat} 0 ${nlat}
region box block 0 3 0 3 0 ${nlat}
region box block 0 3 0 3 0 3
create_box 1 box
Created orthogonal box = (0 0 0) to (5.0387886 5.0387886 5.0387886)
1 by 1 by 1 MPI processor grid
create_atoms 1 box
Created 108 atoms
using lattice units in orthogonal box = (0 0 0) to (5.0387886 5.0387886 5.0387886)
create_atoms CPU = 0.000 seconds
mass 1 1.0
velocity all create ${temp} 87287 loop geom
velocity all create 0.3 87287 loop geom
pair_style lj/cut 2.5
pair_coeff 1 1 1.0 1.0
neighbor 0.0 bin
neigh_modify every 1 delay 0 check no
fix 1 all nve
compute born all born/matrix
thermo ${nthermo}
thermo 10
thermo_style custom step temp pe press c_born[*]
run ${nsteps}
run 50
generated 0 of 0 mixed pair_coeff terms from geometric mixing rule
Neighbor list info ...
update every 1 steps, delay 0 steps, check no
max neighbors/atom: 2000, page size: 100000
master list distance cutoff = 2.5
ghost atom cutoff = 2.5
binsize = 1.25, bins = 5 5 5
2 neighbor lists, perpetual/occasional/extra = 1 1 0
(1) pair lj/cut, perpetual
attributes: half, newton on
pair build: half/bin/atomonly/newton
stencil: half/bin/3d
bin: standard
(2) compute born/matrix, occasional, copy from (1)
attributes: half, newton on
pair build: copy
stencil: none
bin: none
Per MPI rank memory allocation (min/avg/max) = 2.574 | 2.574 | 2.574 Mbytes
Step Temp PotEng Press c_born[1] c_born[2] c_born[3] c_born[4] c_born[5] c_born[6] c_born[7] c_born[8] c_born[9] c_born[10] c_born[11] c_born[12] c_born[13] c_born[14] c_born[15] c_born[16] c_born[17] c_born[18] c_born[19] c_born[20] c_born[21]
0 0.3 -6.7733681 -5.9844023 411.5161 411.5161 411.5161 595.57522 595.57522 595.57522 595.57522 595.57522 -2.7200464e-15 -6.1062266e-15 7.7715612e-16 595.57522 -1.1518564e-14 -3.3306691e-15 -5.3013149e-15 -1.0630385e-14 -6.800116e-15 -2.6367797e-15 -3.4139358e-15 -3.7747583e-15 -2.7200464e-15
10 0.25283594 -6.703292 -5.5569889 891.93989 928.15437 944.55768 872.69866 852.24088 816.0675 816.0675 852.24088 10.111334 3.6921122 -6.3839362 872.69866 -4.6709643 4.4335317 -9.7018557 -4.5201548 3.7624526 11.200776 11.200776 4.4335317 10.111334
20 0.13298345 -6.5218396 -4.5683635 1958.2215 2130.5784 2172.9636 1516.3032 1426.6019 1323.0834 1323.0834 1426.6019 39.867158 11.406477 -25.845853 1516.3032 3.3483658 27.179119 -36.809191 10.147948 7.0018777 41.995438 41.995438 27.179119 39.867158
30 0.089302796 -6.4604227 -4.1807379 2401.2346 2625.1676 2638.9312 1724.973 1625.4639 1602.8817 1602.8817 1625.4639 55.594925 -101.03316 -41.689385 1724.973 35.280816 26.589804 -40.046683 63.973353 -78.086653 41.559746 41.559746 26.589804 55.594925
40 0.12149354 -6.5076059 -4.4000835 2234.612 2345.8821 2337.9552 1513.845 1486.9674 1564.2263 1564.2263 1486.9674 56.775913 -199.0226 -45.302207 1513.845 27.333643 -9.5839331 -16.768077 120.5015 -141.06684 17.802047 17.802047 -9.5839331 56.775913
50 0.13053032 -6.5204024 -4.5064456 2097.1985 2270.3252 2180.7348 1458.5923 1396.6336 1507.4402 1507.4402 1396.6336 37.603097 -213.85355 22.82129 1458.5923 47.47204 -20.546302 54.328025 195.36071 -169.33495 -24.624815 -24.624815 -20.546302 37.603097
Loop time of 0.00790344 on 1 procs for 50 steps with 108 atoms
Performance: 2732988.192 tau/day, 6326.362 timesteps/s
98.9% CPU use with 1 MPI tasks x no OpenMP threads
MPI task timing breakdown:
Section | min time | avg time | max time |%varavg| %total
---------------------------------------------------------------
Pair | 0.0014593 | 0.0014593 | 0.0014593 | 0.0 | 18.46
Neigh | 0.0047863 | 0.0047863 | 0.0047863 | 0.0 | 60.56
Comm | 0.00057975 | 0.00057975 | 0.00057975 | 0.0 | 7.34
Output | 0.00096954 | 0.00096954 | 0.00096954 | 0.0 | 12.27
Modify | 5.3432e-05 | 5.3432e-05 | 5.3432e-05 | 0.0 | 0.68
Other | | 5.508e-05 | | | 0.70
Nlocal: 108 ave 108 max 108 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Nghost: 700 ave 700 max 700 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Neighs: 2909 ave 2909 max 2909 min
Histogram: 1 0 0 0 0 0 0 0 0 0
Total # of neighbors = 2909
Ave neighs/atom = 26.935185
Neighbor list builds = 50
Dangerous builds not checked
Total wall time: 0:00:00

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LAMMPS (7 Jan 2022)
# Numerical difference calculation
# of Born matrix
# adjustable parameters
variable nsteps index 50 # length of run
variable nthermo index 10 # thermo output interval
variable nlat index 3 # size of box
variable delta index 1.0e-6 # strain size
variable temp index 0.3 # temperature
units lj
atom_style atomic
lattice fcc 0.8442
Lattice spacing in x,y,z = 1.6795962 1.6795962 1.6795962
region box block 0 ${nlat} 0 ${nlat} 0 ${nlat}
region box block 0 3 0 ${nlat} 0 ${nlat}
region box block 0 3 0 3 0 ${nlat}
region box block 0 3 0 3 0 3
create_box 1 box
Created orthogonal box = (0 0 0) to (5.0387886 5.0387886 5.0387886)
1 by 2 by 2 MPI processor grid
create_atoms 1 box
Created 108 atoms
using lattice units in orthogonal box = (0 0 0) to (5.0387886 5.0387886 5.0387886)
create_atoms CPU = 0.000 seconds
mass 1 1.0
velocity all create ${temp} 87287 loop geom
velocity all create 0.3 87287 loop geom
pair_style lj/cut 2.5
pair_coeff 1 1 1.0 1.0
neighbor 0.0 bin
neigh_modify every 1 delay 0 check no
fix 1 all nve
compute born all born/matrix
thermo ${nthermo}
thermo 10
thermo_style custom step temp pe press c_born[*]
run ${nsteps}
run 50
generated 0 of 0 mixed pair_coeff terms from geometric mixing rule
Neighbor list info ...
update every 1 steps, delay 0 steps, check no
max neighbors/atom: 2000, page size: 100000
master list distance cutoff = 2.5
ghost atom cutoff = 2.5
binsize = 1.25, bins = 5 5 5
2 neighbor lists, perpetual/occasional/extra = 1 1 0
(1) pair lj/cut, perpetual
attributes: half, newton on
pair build: half/bin/atomonly/newton
stencil: half/bin/3d
bin: standard
(2) compute born/matrix, occasional, copy from (1)
attributes: half, newton on
pair build: copy
stencil: none
bin: none
Per MPI rank memory allocation (min/avg/max) = 2.561 | 2.561 | 2.561 Mbytes
Step Temp PotEng Press c_born[1] c_born[2] c_born[3] c_born[4] c_born[5] c_born[6] c_born[7] c_born[8] c_born[9] c_born[10] c_born[11] c_born[12] c_born[13] c_born[14] c_born[15] c_born[16] c_born[17] c_born[18] c_born[19] c_born[20] c_born[21]
0 0.3 -6.7733681 -5.9844023 411.5161 411.5161 411.5161 595.57522 595.57522 595.57522 595.57522 595.57522 -2.9143354e-15 -1.9984014e-15 1.8873791e-15 595.57522 -9.7560848e-15 -3.3029135e-15 -9.6311847e-15 -1.2281842e-14 -5.3568261e-15 -2.5257574e-15 -3.2196468e-15 -3.5804693e-15 -2.553513e-15
10 0.25283594 -6.703292 -5.5569889 891.93989 928.15437 944.55768 872.69866 852.24088 816.0675 816.0675 852.24088 10.111334 3.6921122 -6.3839362 872.69866 -4.6709643 4.4335317 -9.7018557 -4.5201548 3.7624526 11.200776 11.200776 4.4335317 10.111334
20 0.13298345 -6.5218396 -4.5683635 1958.2215 2130.5784 2172.9636 1516.3032 1426.6019 1323.0834 1323.0834 1426.6019 39.867158 11.406477 -25.845853 1516.3032 3.3483658 27.179119 -36.809191 10.147948 7.0018777 41.995438 41.995438 27.179119 39.867158
30 0.089302796 -6.4604227 -4.1807379 2401.2346 2625.1676 2638.9312 1724.973 1625.4639 1602.8817 1602.8817 1625.4639 55.594925 -101.03316 -41.689385 1724.973 35.280816 26.589804 -40.046683 63.973353 -78.086653 41.559746 41.559746 26.589804 55.594925
40 0.12149354 -6.5076059 -4.4000835 2234.612 2345.8821 2337.9552 1513.845 1486.9674 1564.2263 1564.2263 1486.9674 56.775913 -199.0226 -45.302207 1513.845 27.333643 -9.5839331 -16.768077 120.5015 -141.06684 17.802047 17.802047 -9.5839331 56.775913
50 0.13053032 -6.5204024 -4.5064456 2097.1985 2270.3252 2180.7348 1458.5923 1396.6336 1507.4402 1507.4402 1396.6336 37.603097 -213.85355 22.82129 1458.5923 47.47204 -20.546302 54.328025 195.36071 -169.33495 -24.624815 -24.624815 -20.546302 37.603097
Loop time of 0.00293746 on 4 procs for 50 steps with 108 atoms
Performance: 7353280.980 tau/day, 17021.484 timesteps/s
99.2% CPU use with 4 MPI tasks x no OpenMP threads
MPI task timing breakdown:
Section | min time | avg time | max time |%varavg| %total
---------------------------------------------------------------
Pair | 0.00037884 | 0.00040147 | 0.00041828 | 0.0 | 13.67
Neigh | 0.0010222 | 0.0011142 | 0.0011623 | 0.2 | 37.93
Comm | 0.00098867 | 0.0010309 | 0.0011341 | 0.2 | 35.09
Output | 0.00030288 | 0.00031473 | 0.00034979 | 0.0 | 10.71
Modify | 2.0147e-05 | 2.218e-05 | 2.3691e-05 | 0.0 | 0.76
Other | | 5.406e-05 | | | 1.84
Nlocal: 27 ave 31 max 25 min
Histogram: 2 0 0 1 0 0 0 0 0 1
Nghost: 429.75 ave 450 max 409 min
Histogram: 1 0 0 0 0 2 0 0 0 1
Neighs: 727.25 ave 845 max 651 min
Histogram: 1 0 1 1 0 0 0 0 0 1
Total # of neighbors = 2909
Ave neighs/atom = 26.935185
Neighbor list builds = 50
Dangerous builds not checked
Total wall time: 0:00:00

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/* ----------------------------------------------------------------------
LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
http://www.lammps.org/, Sandia National Laboratories
Steve Plimpton, sjplimp@sandia.gov
Copyright (2003) Sandia Corporation. Under the terms of Contract
DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
certain rights in this software. This software is distributed under
the GNU General Public License.
See the README file in the top-level LAMMPS directory.
------------------------------------------------------------------------- */
/*------------------------------------------------------------------------
Contributing Authors : Germain Clavier (TUe)
--------------------------------------------------------------------------*/
#ifdef COMPUTE_CLASS
// clang-format off
ComputeStyle(born/matrix,ComputeBornMatrix);
// clang-format on
#else
#ifndef LMP_COMPUTE_BORN_MATRIX_H
#define LMP_COMPUTE_BORN_MATRIX_H
#include "compute.h"
namespace LAMMPS_NS {
class ComputeBornMatrix : public Compute {
public:
ComputeBornMatrix(class LAMMPS *, int, char **);
virtual ~ComputeBornMatrix() override;
void init() override;
void init_list(int, class NeighList *) override;
void compute_vector() override;
double memory_usage() override;
private:
// Born matrix contributions
void compute_pairs(); // pair and manybody
void compute_bonds(); // bonds
void compute_angles(); // angles
void compute_dihedrals(); // dihedrals
void compute_numdiff(); // stress virial finite differences
void displace_atoms(int, int, double); // displace atoms
void force_clear(int); // zero out force array
void update_virial(); // recalculate the virial
void restore_atoms(int, int); // restore atom positions
void virial_addon(); // restore atom positions
void reallocate(); // grow the atom arrays
int me; // process rank
int nvalues; // length of elastic tensor
int numflag; // 1 if using finite differences
double numdelta; // size of finite strain
int maxatom; // allocated size of atom arrays
int pairflag, bondflag, angleflag;
int dihedflag, impflag, kspaceflag;
double *values_local,*values_global;
double pos,pos1,dt,nktv2p,ftm2v;
class NeighList *list;
char *id_virial; // name of virial compute
class Compute *compute_virial; // pointer to virial compute
static constexpr int NDIR_VIRIAL = 6; // dimension of virial and strain vectors
static constexpr int NXYZ_VIRIAL = 3; // number of Cartesian coordinates
int revalbe[NDIR_VIRIAL][NDIR_VIRIAL];
int voigt3VtoM[NDIR_VIRIAL][2];
int voigt3MtoV[NXYZ_VIRIAL][NXYZ_VIRIAL];
int virialMtoV[NXYZ_VIRIAL][NXYZ_VIRIAL];
int kronecker[NXYZ_VIRIAL][NXYZ_VIRIAL];
double **temp_x; // original coords
double **temp_f; // original forces
double fixedpoint[NXYZ_VIRIAL]; // displacement field origin
int dirlist[NDIR_VIRIAL][2]; // strain cartesian indices
};
}
#endif
#endif
/* ERROR/WARNING messages:
E: Illegal ... command
Self-explanatory. Check the input script syntax and compare to the
documentation for the command. You can use -echo screen as a
command-line option when running LAMMPS to see the offending line.
E: ... style does not support compute born/matrix
Some component of the force field (pair, bond, angle...) does not provide
a function to return the Born term contribution.
*/

1
src/pair.cpp
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@ -58,6 +58,7 @@ Pair::Pair(LAMMPS *lmp) : Pointers(lmp)
comm_forward = comm_reverse = comm_reverse_off = 0;
single_enable = 1;
born_matrix_enable = 0;
single_hessian_enable = 0;
restartinfo = 1;
respa_enable = 0;

8
src/pair.h
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@ -52,6 +52,7 @@ class Pair : protected Pointers {
int single_enable; // 1 if single() routine exists
int single_hessian_enable; // 1 if single_hessian() routine exists
int born_matrix_enable; // 1 if born_matrix() routine exists
int restartinfo; // 1 if pair style writes restart info
int respa_enable; // 1 if inner/middle/outer rRESPA routines
int one_coeff; // 1 if allows only one coeff * * call
@ -168,6 +169,13 @@ class Pair : protected Pointers {
return 0.0;
}
virtual void born_matrix(int /*i*/, int /*j*/, int /*itype*/, int /*jtype*/, double /*rsq*/,
double /*factor_coul*/, double /*factor_lj*/, double& du, double& du2)
{
du = 0.0;
du2 = 0.0;
}
virtual void settings(int, char **) = 0;
virtual void coeff(int, char **) = 0;

23
src/pair_lj_cut.cpp
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@ -40,6 +40,7 @@ using namespace MathConst;
PairLJCut::PairLJCut(LAMMPS *lmp) : Pair(lmp)
{
respa_enable = 1;
born_matrix_enable = 1;
writedata = 1;
}
@ -680,6 +681,28 @@ double PairLJCut::single(int /*i*/, int /*j*/, int itype, int jtype, double rsq,
/* ---------------------------------------------------------------------- */
void PairLJCut::born_matrix(int /*i*/, int /*j*/, int itype, int jtype, double rsq,
double /*factor_coul*/, double factor_lj,
double &dupair, double &du2pair)
{
double rinv,r2inv,r6inv,du,du2;
r2inv = 1.0/rsq;
rinv = sqrt(r2inv);
r6inv = r2inv*r2inv*r2inv;
// Reminder: lj1 = 48*e*s^12, lj2 = 24*e*s^6
// so dupair = -forcelj/r = -fforce*r (forcelj from single method)
du = r6inv * rinv * (lj2[itype][jtype] - lj1[itype][jtype]*r6inv);
du2 = r6inv * r2inv * (13*lj1[itype][jtype]*r6inv - 7*lj2[itype][jtype]);
dupair = factor_lj*du;
du2pair = factor_lj*du2;
}
/* ---------------------------------------------------------------------- */
void *PairLJCut::extract(const char *str, int &dim)
{
dim = 2;

1
src/pair_lj_cut.h
View File

@ -40,6 +40,7 @@ class PairLJCut : public Pair {
void write_data(FILE *) override;
void write_data_all(FILE *) override;
double single(int, int, int, int, double, double, double, double &) override;
void born_matrix(int, int, int, int, double, double, double, double &, double &) override;
void *extract(const char *, int &) override;
void compute_inner() override;