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LJ.f90
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! This performs the calculations necessary to run a Lennard-Jones (LJ)
! simulation.
!
! Copyright (C) 2013, Joshua More and Michele Ceriotti
!
! Permission is hereby granted, free of charge, to any person obtaining
! a copy of this software and associated documentation files (the
! "Software"), to deal in the Software without restriction, including
! without limitation the rights to use, copy, modify, merge, publish,
! distribute, sublicense, and/or sell copies of the Software, and to
! permit persons to whom the Software is furnished to do so, subject to
! the following conditions:
!
! The above copyright notice and this permission notice shall be included
! in all copies or substantial portions of the Software.
!
! THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
! EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
! MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
! IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
! CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
! TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
! SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
!
!
! This contains the functions that calculate the potential, forces and
! virial tensor of a single-component LJ system.
! Includes functions which calculate the long-range correction terms for a
! simulation with a sharp nearest-neighbour cut-off.
!
! Functions:
! LJ_functions: Calculates the LJ pair potential and the magnitude of the
! forces acting on a pair of atoms.
! LJ_fij: Calculates the LJ pair potential and force vector for the
! interaction of a pair of atoms.
! LJ_longrange: Calculates the long range correction to the potential
! and virial.
! LJ_getall: Calculates the potential and virial of the system and the
! forces acting on all the atoms.
MODULE LJ
USE DISTANCE
IMPLICIT NONE
DOUBLE PRECISION, PARAMETER :: four_tau_by_3 = 8.3775804095727811d0
CONTAINS
SUBROUTINE LJ_functions(sigma, eps, r, pot, force)
! Calculates the magnitude of the LJ force and potential between
! a pair of atoms at a given distance from each other.
!
! Args:
! sigma: The LJ distance parameter.
! eps: The LJ energy parameter.
! r: The separation of the atoms.
! pot: The LJ interaction potential.
! force: The magnitude of the LJ force.
DOUBLE PRECISION, INTENT(IN) :: sigma
DOUBLE PRECISION, INTENT(IN) :: eps
DOUBLE PRECISION, INTENT(IN) :: r
DOUBLE PRECISION, INTENT(OUT) :: pot
DOUBLE PRECISION, INTENT(OUT) :: force
DOUBLE PRECISION sigma_by_r6
sigma_by_r6 = sigma/r
sigma_by_r6 = sigma_by_r6*sigma_by_r6*sigma_by_r6
sigma_by_r6 = sigma_by_r6*sigma_by_r6
pot = 4*eps*(sigma_by_r6*(sigma_by_r6 - 1))
force = 24*eps*(sigma_by_r6*(2*sigma_by_r6 - 1)/r)
END SUBROUTINE
SUBROUTINE LJ_fij(sigma, eps, rij, r, pot, fij)
! This calculates the LJ potential energy and the magnitude and
! direction of the force acting on a pair of atoms.
!
! Args:
! sigma: The LJ distance parameter.
! eps: The LJ energy parameter.
! rij: The vector joining the two atoms.
! r: The separation of the two atoms.
! pot: The LJ interaction potential.
! fij: The LJ force vector.
DOUBLE PRECISION, INTENT(IN) :: sigma
DOUBLE PRECISION, INTENT(IN) :: eps
DOUBLE PRECISION, DIMENSION(3), INTENT(IN) :: rij
DOUBLE PRECISION, INTENT(IN) :: r
DOUBLE PRECISION, INTENT(OUT) :: pot
DOUBLE PRECISION, DIMENSION(3), INTENT(OUT) :: fij
DOUBLE PRECISION f_tot
CALL LJ_functions(sigma, eps, r, pot, f_tot)
fij = f_tot*rij/r
END SUBROUTINE
SUBROUTINE LJ_longrange(rc, sigma, eps, natoms, volume, pot_lr, vir_lr)
! Calculates the long range correction to the total potential and
! virial pressure.
!
! Uses the tail correction for a sharp cut-off, with no smoothing
! function, as derived in Martyna and Hughes, Journal of Chemical
! Physics, 110, 3275, (1999).
!
! Args:
! rc: The cut-off radius.
! sigma: The LJ distance parameter.
! eps: The LJ energy parameter.
! natoms: The number of atoms in the system.
! volume: The volume of the system box.
! pot_lr: The tail correction to the LJ interaction potential.
! vir_lr: The tail correction to the LJ virial pressure.
DOUBLE PRECISION, INTENT(IN) :: rc
DOUBLE PRECISION, INTENT(IN) :: sigma
DOUBLE PRECISION, INTENT(IN) :: eps
INTEGER, INTENT(IN) :: natoms
DOUBLE PRECISION, INTENT(IN) :: volume
DOUBLE PRECISION, INTENT(OUT) :: pot_lr
DOUBLE PRECISION, INTENT(OUT) :: vir_lr
DOUBLE PRECISION sbyr, s3byr3, s6byr3, s6byr6, prefactor
sbyr = sigma/rc
s3byr3 = sbyr*sbyr*sbyr
s6byr6 = s3byr3*s3byr3
prefactor = four_tau_by_3*natoms*natoms*eps/volume
prefactor = prefactor*s3byr3*sigma*sigma*sigma
pot_lr = prefactor*(s6byr6/3-1)
vir_lr = prefactor*(s6byr6-1) + pot_lr
END SUBROUTINE
SUBROUTINE LJ_getall(rc, sigma, eps, natoms, atoms, cell_h, cell_ih, index_list, n_list, pot, forces, virial)
! Calculates the LJ potential energy and virial and the forces
! acting on all the atoms.
!
! Args:
! rc: The cut-off radius.
! sigma: The LJ distance parameter.
! eps: The LJ energy parameter.
! natoms: The number of atoms in the system.
! atoms: A vector holding all the atom positions.
! cell_h: The simulation box cell vector matrix.
! cell_ih: The inverse of the simulation box cell vector matrix.
! index_list: A array giving the last index of n_list that
! gives the neighbours of a given atom.
! n_list: An array giving the indices of the atoms that neighbour
! the atom determined by index_list.
! pot: The total potential energy of the system.
! forces: An array giving the forces acting on all the atoms.
! virial: The virial tensor, not divided by the volume.
DOUBLE PRECISION, INTENT(IN) :: rc
DOUBLE PRECISION, INTENT(IN) :: sigma
DOUBLE PRECISION, INTENT(IN) :: eps
INTEGER, INTENT(IN) :: natoms
DOUBLE PRECISION, DIMENSION(natoms,3), INTENT(IN) :: atoms
DOUBLE PRECISION, DIMENSION(3,3), INTENT(IN) :: cell_h
DOUBLE PRECISION, DIMENSION(3,3), INTENT(IN) :: cell_ih
INTEGER, DIMENSION(natoms), INTENT(IN) :: index_list
INTEGER, DIMENSION(natoms*(natoms-1)/2), INTENT(IN) :: n_list
DOUBLE PRECISION, INTENT(OUT) :: pot
DOUBLE PRECISION, DIMENSION(natoms,3), INTENT(OUT) :: forces
DOUBLE PRECISION, DIMENSION(3,3), INTENT(OUT) :: virial
INTEGER i, j, k, l, start
DOUBLE PRECISION, DIMENSION(3) :: fij, rij
DOUBLE PRECISION r2, pot_ij, pot_lr, vir_lr, volume
forces = 0.0d0
pot = 0.0d0
virial = 0.0d0
start = 1
DO i = 1, natoms - 1
! Only loops over the neigbour list, not all the atoms.
DO j = start, index_list(i)
CALL vector_separation(cell_h, cell_ih, atoms(i,:), atoms(n_list(j),:), rij, r2)
IF (r2 < rc*rc) THEN ! Only calculates contributions between neighbouring particles.
CALL LJ_fij(sigma, eps, rij, sqrt(r2), pot_ij, fij)
forces(i,:) = forces(i,:) + fij
forces(n_list(j),:) = forces(n_list(j),:) - fij
pot = pot + pot_ij
DO k = 1, 3
DO l = k, 3
! Only the upper triangular elements calculated.
virial(k,l) = virial(k,l) + fij(k)*rij(l)
ENDDO
ENDDO
ENDIF
ENDDO
start = index_list(i) + 1
ENDDO
! Assuming an upper-triangular vector matrix for the simulation box.
volume = cell_h(1,1)*cell_h(2,2)*cell_h(3,3)
CALL LJ_longrange(rc, sigma, eps, natoms, volume, pot_lr, vir_lr)
pot = pot + pot_lr
DO k = 1, 3
virial(k,k) = virial(k,k) + vir_lr
ENDDO
END SUBROUTINE
END MODULE
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