Lennard-Jones potential with Coulomb interactions accounted through Wolf summations

doc/src/pair_lj.txt

@@ -32,6 +32,7 @@ pair_style lj/cut/coul/long/omp command :h3
 pair_style lj/cut/coul/msm command :h3
 pair_style lj/cut/coul/msm/gpu command :h3
 pair_style lj/cut/coul/msm/omp command :h3
+pair_style lj/cut/coul/wolf command :h3
 pair_style lj/cut/tip4p/cut command :h3
 pair_style lj/cut/tip4p/cut/omp command :h3
 pair_style lj/cut/tip4p/long command :h3
@@ -63,6 +64,10 @@ args = list of arguments for a particular style :ul
   {lj/cut/coul/msm} args = cutoff (cutoff2)
     cutoff = global cutoff for LJ (and Coulombic if only 1 arg) (distance units)
     cutoff2 = global cutoff for Coulombic (optional) (distance units)
+  {lj/cut/coul/wolf} args = alpha cutoff (cutoff2)
+    alpha = damping parameter (inverse distance units)
+    cutoff = global cutoff for LJ (and Coulombic if only 2 arg) (distance units)
+    cutoff2 = global cutoff for Coulombic (optional) (distance units)
   {lj/cut/tip4p/cut} args = otype htype btype atype qdist cutoff (cutoff2)
     otype,htype = atom types for TIP4P O and H
     btype,atype = bond and angle types for TIP4P waters
@@ -115,6 +120,10 @@ pair_style lj/cut/tip4p/cut 1 2 7 8 0.15 12.0 10.0
 pair_coeff * * 100.0 3.0
 pair_coeff 1 1 100.0 3.5 9.0 :pre
 
+pair_style lj/cut/coul/wolf 0.2 5. 10.0
+pair_coeff * * 1.0 1.0
+pair_coeff 1 1 1.0 1.0 2.5 :pre
+
 pair_style lj/cut/tip4p/long 1 2 7 8 0.15 12.0
 pair_style lj/cut/tip4p/long 1 2 7 8 0.15 12.0 10.0
 pair_coeff * * 100.0 3.0
@@ -179,6 +188,24 @@ that a term is added for the "core/shell
 model"_Section_howto.html#howto_25 to allow charges on core and shell
 particles to be separated by r = 0.0.
 
+Style {coul/wolf} adds a Coulombic pairwise interaction via the Wolf
+summation method, described in "Wolf"_#Wolf1, given by:
+
+:c,image(Eqs/pair_coul_wolf.jpg)
+
+where {alpha} is the damping parameter, and erfc() is the
+complementary error-function terms.  This potential
+is essentially a short-range, spherically-truncated,
+charge-neutralized, shifted, pairwise {1/r} summation.  With a
+manipulation of adding and subtracting a self term (for i = j) to the
+first and second term on the right-hand-side, respectively, and a
+small enough {alpha} damping parameter, the second term shrinks and
+the potential becomes a rapidly-converging real-space summation.  With
+a long enough cutoff and small enough alpha parameter, the energy and
+forces calculated by the Wolf summation method approach those of the
+Ewald sum.  So it is a means of getting effective long-range
+interactions with a short-range potential.
+
 Styles {lj/cut/tip4p/cut} and {lj/cut/tip4p/long} implement the TIP4P
 water model of "(Jorgensen)"_#Jorgensen2, which introduces a massless
 site located a short distance away from the oxygen atom along the

src/pair_lj_cut_coul_wolf.cpp

+/* ----------------------------------------------------------------------
+   LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
+   http://lammps.sandia.gov, 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 author: Vishal Boddu (FAU)
+------------------------------------------------------------------------- */
+
+#include <math.h>
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include "pair_lj_cut_coul_wolf.h"
+#include "atom.h"
+#include "comm.h"
+#include "force.h"
+#include "neighbor.h"
+#include "neigh_list.h"
+#include "neigh_request.h"
+#include "update.h"
+#include "integrate.h"
+#include "respa.h"
+#include "math_const.h"
+#include "memory.h"
+#include "error.h"
+
+using namespace LAMMPS_NS;
+using namespace MathConst;
+
+/* ---------------------------------------------------------------------- */
+
+PairLJCutCoulWolf::PairLJCutCoulWolf(LAMMPS *lmp) : Pair(lmp)
+{
+  single_enable = 0;        // NOTE: single() method is not implemented
+  respa_enable = 1;
+  writedata = 1;
+}
+
+/* ---------------------------------------------------------------------- */
+
+PairLJCutCoulWolf::~PairLJCutCoulWolf()
+{
+  if (allocated) {
+    memory->destroy(setflag);
+    memory->destroy(cutsq);
+
+    memory->destroy(cut_lj);
+    memory->destroy(cut_ljsq);
+    memory->destroy(epsilon);
+    memory->destroy(sigma);
+    memory->destroy(lj1);
+    memory->destroy(lj2);
+    memory->destroy(lj3);
+    memory->destroy(lj4);
+    memory->destroy(offset);
+  }
+}
+
+/* ---------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::compute(int eflag, int vflag)
+{
+  int i,j,ii,jj,inum,jnum,itype,jtype;
+  double qtmp,xtmp,ytmp,ztmp,delx,dely,delz,evdwl,ecoul,fpair;
+  double rsq,r2inv,r6inv,forcelj,factor_lj,forcecoul,factor_coul;
+  double prefactor;
+  double r;
+  int *ilist,*jlist,*numneigh,**firstneigh;
+  double erfcc,erfcd,v_sh,dvdrr,e_self,e_shift,f_shift,qisq;
+
+  evdwl = 0.0;
+  ecoul = 0.0;
+  if (eflag || vflag) ev_setup(eflag,vflag);
+  else evflag = vflag_fdotr = 0;
+
+  double **x = atom->x;
+  double **f = atom->f;
+  double *q = atom->q;
+  int *type = atom->type;
+  int nlocal = atom->nlocal;
+  double *special_lj = force->special_lj;
+  double *special_coul = force->special_coul;
+  int newton_pair = force->newton_pair;
+  double qqrd2e = force->qqrd2e;
+
+  // self and shifted Coulombic energy
+
+  e_self = v_sh = 0.0;
+  e_shift = erfc(alf*cut_coul)/cut_coul;
+  f_shift = -(e_shift+ 2.0*alf/MY_PIS * exp(-alf*alf*cut_coul*cut_coul)) /
+    cut_coul;
+
+  inum = list->inum;
+  ilist = list->ilist;
+  numneigh = list->numneigh;
+  firstneigh = list->firstneigh;
+
+  // loop over neighbors of my atoms
+
+  for (ii = 0; ii < inum; ii++) {
+    i = ilist[ii];
+    qtmp = q[i];
+    xtmp = x[i][0];
+    ytmp = x[i][1];
+    ztmp = x[i][2];
+    itype = type[i];
+    jlist = firstneigh[i];
+    jnum = numneigh[i];
+
+    qisq = qtmp*qtmp;
+    e_self = -(e_shift/2.0 + alf/MY_PIS) * qisq*qqrd2e;
+    if (eflag) ev_tally(i,i,nlocal,0,0.0,e_self,0.0,0.0,0.0,0.0);
+
+    for (jj = 0; jj < jnum; jj++) {
+      j = jlist[jj];
+      factor_lj = special_lj[sbmask(j)];
+      factor_coul = special_coul[sbmask(j)];
+      j &= NEIGHMASK;
+
+      delx = xtmp - x[j][0];
+      dely = ytmp - x[j][1];
+      delz = ztmp - x[j][2];
+      rsq = delx*delx + dely*dely + delz*delz;
+      jtype = type[j];
+
+      if (rsq < cutsq[itype][jtype]) {
+        r2inv = 1.0/rsq;
+
+        if (rsq < cut_coulsq) {
+          r = sqrt(rsq);
+          prefactor = qqrd2e*qtmp*q[j]/r;
+          erfcc = erfc(alf*r);
+          erfcd = exp(-alf*alf*r*r);
+          v_sh = (erfcc - e_shift*r) * prefactor;
+          dvdrr = (erfcc/rsq + 2.0*alf/MY_PIS * erfcd/r) + f_shift;
+          forcecoul = dvdrr*rsq*prefactor;
+          if (factor_coul < 1.0) forcecoul -= (1.0-factor_coul)*prefactor;
+        } else forcecoul = 0.0;
+
+        if (rsq < cut_ljsq[itype][jtype]) {
+          r6inv = r2inv*r2inv*r2inv;
+          forcelj = r6inv * (lj1[itype][jtype]*r6inv - lj2[itype][jtype]);
+        } else forcelj = 0.0;
+
+        fpair = (forcecoul + factor_lj*forcelj) * r2inv;
+
+        f[i][0] += delx*fpair;
+        f[i][1] += dely*fpair;
+        f[i][2] += delz*fpair;
+        if (newton_pair || j < nlocal) {
+          f[j][0] -= delx*fpair;
+          f[j][1] -= dely*fpair;
+          f[j][2] -= delz*fpair;
+        }
+
+        if (eflag) {
+          if (rsq < cut_ljsq[itype][jtype]) {
+            evdwl = r6inv*(lj3[itype][jtype]*r6inv-lj4[itype][jtype]) -
+                    offset[itype][jtype];
+            evdwl *= factor_lj;
+          } else evdwl = 0.0;
+          if (rsq < cut_coulsq) {
+            ecoul = v_sh;
+          if (factor_coul < 1.0) ecoul -= (1.0-factor_coul)*prefactor;
+          } else ecoul = 0.0;
+        }
+
+        if (evflag) ev_tally(i,j,nlocal,newton_pair,
+                             evdwl,ecoul,fpair,delx,dely,delz);
+      }
+    }
+  }
+
+  if (vflag_fdotr) virial_fdotr_compute();
+}
+
+/* ----------------------------------------------------------------------
+   allocate all arrays
+------------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::allocate()
+{
+  allocated = 1;
+  int n = atom->ntypes;
+
+  memory->create(setflag,n+1,n+1,"pair:setflag");
+  for (int i = 1; i <= n; i++)
+    for (int j = i; j <= n; j++)
+      setflag[i][j] = 0;
+
+  memory->create(cutsq,n+1,n+1,"pair:cutsq");
+  memory->create(cut_lj,n+1,n+1,"pair:cut");
+  memory->create(cut_ljsq,n+1,n+1,"pair:cut_ljsq");
+  memory->create(epsilon,n+1,n+1,"pair:epsilon");
+  memory->create(sigma,n+1,n+1,"pair:sigma");
+  memory->create(lj1,n+1,n+1,"pair:lj1");
+  memory->create(lj2,n+1,n+1,"pair:lj2");
+  memory->create(lj3,n+1,n+1,"pair:lj3");
+  memory->create(lj4,n+1,n+1,"pair:lj4");
+  memory->create(offset,n+1,n+1,"pair:offset");
+}
+
+/* ----------------------------------------------------------------------
+   global settings
+------------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::settings(int narg, char **arg)
+{
+  if (narg < 2 || narg > 3) error->all(FLERR,"Illegal pair_style command");
+
+  alf = force->numeric(FLERR,arg[0]);
+  cut_lj_global = force->numeric(FLERR,arg[1]);
+  if (narg == 2) cut_coul = cut_lj_global;
+
+  if (allocated) {
+    int i,j;
+    for (i = 1; i <= atom->ntypes; i++)
+      for (j = i; j <= atom->ntypes; j++)
+        if (setflag[i][j]) cut_lj[i][j] = cut_lj_global;
+  }
+}
+
+/* ----------------------------------------------------------------------
+   set coeffs for one or more type pairs
+------------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::coeff(int narg, char **arg)
+{
+  if (narg < 4 || narg > 5)
+    error->all(FLERR,"Incorrect args for pair coefficients");
+  if (!allocated) allocate();
+
+  int ilo,ihi,jlo,jhi;
+  force->bounds(FLERR,arg[0],atom->ntypes,ilo,ihi);
+  force->bounds(FLERR,arg[1],atom->ntypes,jlo,jhi);
+
+  double epsilon_one = force->numeric(FLERR,arg[2]);
+  double sigma_one = force->numeric(FLERR,arg[3]);
+
+  double cut_lj_one = cut_lj_global;
+  if (narg == 5) cut_lj_one = force->numeric(FLERR,arg[4]);
+
+  int count = 0;
+  for (int i = ilo; i <= ihi; i++) {
+    for (int j = MAX(jlo,i); j <= jhi; j++) {
+      epsilon[i][j] = epsilon_one;
+      sigma[i][j] = sigma_one;
+      cut_lj[i][j] = cut_lj_one;
+      setflag[i][j] = 1;
+      count++;
+    }
+  }
+
+  if (count == 0) error->all(FLERR,"Incorrect args for pair coefficients");
+}
+
+/* ----------------------------------------------------------------------
+   init specific to this pair style
+------------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::init_style()
+{
+  if (!atom->q_flag)
+    error->all(FLERR,"Pair style lj/cut/coul/wolf requires atom attribute q");
+
+  cut_coulsq = cut_coul * cut_coul;
+
+  // request regular or rRESPA neighbor list
+
+  int irequest;
+  int respa = 0;
+
+  if (update->whichflag == 1 && strstr(update->integrate_style,"respa")) {
+    if (((Respa *) update->integrate)->level_inner >= 0) respa = 1;
+    if (((Respa *) update->integrate)->level_middle >= 0) respa = 2;
+  }
+
+  irequest = neighbor->request(this,instance_me);
+
+  if (respa >= 1) {
+    neighbor->requests[irequest]->respaouter = 1;
+    neighbor->requests[irequest]->respainner = 1;
+  }
+  if (respa == 2) neighbor->requests[irequest]->respamiddle = 1;
+
+  // set rRESPA cutoffs
+
+  if (strstr(update->integrate_style,"respa") &&
+      ((Respa *) update->integrate)->level_inner >= 0)
+    cut_respa = ((Respa *) update->integrate)->cutoff;
+  else cut_respa = NULL;
+}
+
+/* ----------------------------------------------------------------------
+   init for one type pair i,j and corresponding j,i
+------------------------------------------------------------------------- */
+
+double PairLJCutCoulWolf::init_one(int i, int j)
+{
+  if (setflag[i][j] == 0) {
+    epsilon[i][j] = mix_energy(epsilon[i][i],epsilon[j][j],
+                               sigma[i][i],sigma[j][j]);
+    sigma[i][j] = mix_distance(sigma[i][i],sigma[j][j]);
+    cut_lj[i][j] = mix_distance(cut_lj[i][i],cut_lj[j][j]);
+  }
+
+  double cut = MAX(cut_lj[i][j],cut_coul);
+  cut_ljsq[i][j] = cut_lj[i][j] * cut_lj[i][j];
+
+  lj1[i][j] = 48.0 * epsilon[i][j] * pow(sigma[i][j],12.0);
+  lj2[i][j] = 24.0 * epsilon[i][j] * pow(sigma[i][j],6.0);
+  lj3[i][j] = 4.0 * epsilon[i][j] * pow(sigma[i][j],12.0);
+  lj4[i][j] = 4.0 * epsilon[i][j] * pow(sigma[i][j],6.0);
+
+  if (offset_flag && (cut_lj[i][j] > 0.0)) {
+    double ratio = sigma[i][j] / cut_lj[i][j];
+    offset[i][j] = 4.0 * epsilon[i][j] * (pow(ratio,12.0) - pow(ratio,6.0));
+  } else offset[i][j] = 0.0;
+
+  lj1[j][i] = lj1[i][j];
+  lj2[j][i] = lj2[i][j];
+  lj3[j][i] = lj3[i][j];
+  lj4[j][i] = lj4[i][j];
+  offset[j][i] = offset[i][j];
+
+  // check interior rRESPA cutoff
+
+  if (cut_respa && cut_lj[i][j] < cut_respa[3])
+    error->all(FLERR,"Pair cutoff < Respa interior cutoff");
+
+  // compute I,J contribution to long-range tail correction
+  // count total # of atoms of type I and J via Allreduce
+
+  if (tail_flag) {
+    int *type = atom->type;
+    int nlocal = atom->nlocal;
+
+    double count[2],all[2];
+    count[0] = count[1] = 0.0;
+    for (int k = 0; k < nlocal; k++) {
+      if (type[k] == i) count[0] += 1.0;
+      if (type[k] == j) count[1] += 1.0;
+    }
+    MPI_Allreduce(count,all,2,MPI_DOUBLE,MPI_SUM,world);
+
+    double sig2 = sigma[i][j]*sigma[i][j];
+    double sig6 = sig2*sig2*sig2;
+    double rc3 = cut_lj[i][j]*cut_lj[i][j]*cut_lj[i][j];
+    double rc6 = rc3*rc3;
+    double rc9 = rc3*rc6;
+    etail_ij = 8.0*MY_PI*all[0]*all[1]*epsilon[i][j] *
+      sig6 * (sig6 - 3.0*rc6) / (9.0*rc9);
+    ptail_ij = 16.0*MY_PI*all[0]*all[1]*epsilon[i][j] *
+      sig6 * (2.0*sig6 - 3.0*rc6) / (9.0*rc9);
+  }
+
+  return cut;
+}
+
+/* ----------------------------------------------------------------------
+   proc 0 writes to restart file
+------------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::write_restart(FILE *fp)
+{
+  write_restart_settings(fp);
+
+  int i,j;
+  for (i = 1; i <= atom->ntypes; i++)
+    for (j = i; j <= atom->ntypes; j++) {
+      fwrite(&setflag[i][j],sizeof(int),1,fp);
+      if (setflag[i][j]) {
+        fwrite(&epsilon[i][j],sizeof(double),1,fp);
+        fwrite(&sigma[i][j],sizeof(double),1,fp);
+        fwrite(&cut_lj[i][j],sizeof(double),1,fp);
+      }
+    }
+}
+
+/* ----------------------------------------------------------------------
+   proc 0 reads from restart file, bcasts
+------------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::read_restart(FILE *fp)
+{
+  read_restart_settings(fp);
+  allocate();
+
+  int i,j;
+  int me = comm->me;
+  for (i = 1; i <= atom->ntypes; i++)
+    for (j = i; j <= atom->ntypes; j++) {
+      if (me == 0) fread(&setflag[i][j],sizeof(int),1,fp);
+      MPI_Bcast(&setflag[i][j],1,MPI_INT,0,world);
+      if (setflag[i][j]) {
+        if (me == 0) {
+          fread(&epsilon[i][j],sizeof(double),1,fp);
+          fread(&sigma[i][j],sizeof(double),1,fp);
+          fread(&cut_lj[i][j],sizeof(double),1,fp);
+        }
+        MPI_Bcast(&epsilon[i][j],1,MPI_DOUBLE,0,world);
+        MPI_Bcast(&sigma[i][j],1,MPI_DOUBLE,0,world);
+        MPI_Bcast(&cut_lj[i][j],1,MPI_DOUBLE,0,world);
+      }
+    }
+}
+
+/* ----------------------------------------------------------------------
+   proc 0 writes to restart file
+------------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::write_restart_settings(FILE *fp)
+{
+  fwrite(&alf,sizeof(double),1,fp);
+  fwrite(&cut_lj_global,sizeof(double),1,fp);
+  fwrite(&cut_coul,sizeof(double),1,fp);
+  fwrite(&offset_flag,sizeof(int),1,fp);
+  fwrite(&mix_flag,sizeof(int),1,fp);
+  fwrite(&tail_flag,sizeof(int),1,fp);
+}
+
+/* ----------------------------------------------------------------------
+   proc 0 reads from restart file, bcasts
+------------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::read_restart_settings(FILE *fp)
+{
+  int me = comm->me;
+  if (me == 0) {
+    fread(&alf,sizeof(double),1,fp);
+    fread(&cut_lj_global,sizeof(double),1,fp);
+    fread(&cut_coul,sizeof(double),1,fp);
+    fread(&offset_flag,sizeof(int),1,fp);
+    fread(&mix_flag,sizeof(int),1,fp);
+    fread(&tail_flag,sizeof(int),1,fp);
+  }
+  MPI_Bcast(&alf,1,MPI_DOUBLE,0,world);
+  MPI_Bcast(&cut_lj_global,1,MPI_DOUBLE,0,world);
+  MPI_Bcast(&cut_coul,1,MPI_DOUBLE,0,world);
+  MPI_Bcast(&offset_flag,1,MPI_INT,0,world);
+  MPI_Bcast(&mix_flag,1,MPI_INT,0,world);
+  MPI_Bcast(&tail_flag,1,MPI_INT,0,world);
+}
+
+/* ----------------------------------------------------------------------
+   proc 0 writes to data file
+------------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::write_data(FILE *fp)
+{
+  for (int i = 1; i <= atom->ntypes; i++)
+    fprintf(fp,"%d %g %g\n",i,epsilon[i][i],sigma[i][i]);
+}
+
+/* ----------------------------------------------------------------------
+   proc 0 writes all pairs to data file
+------------------------------------------------------------------------- */
+
+void PairLJCutCoulWolf::write_data_all(FILE *fp)
+{
+  for (int i = 1; i <= atom->ntypes; i++)
+    for (int j = i; j <= atom->ntypes; j++)
+      fprintf(fp,"%d %d %g %g %g\n",i,j,epsilon[i][j],sigma[i][j],cut_lj[i][j]);
+}

src/pair_lj_cut_coul_wolf.h

+/* -*- c++ -*- ----------------------------------------------------------
+   LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
+   http://lammps.sandia.gov, 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.
+------------------------------------------------------------------------- */
+
+#ifdef PAIR_CLASS
+
+PairStyle(lj/cut/coul/wolf,PairLJCutCoulWolf)
+
+#else
+
+#ifndef LMP_PAIR_LJ_CUT_COUL_WOLF_H
+#define LMP_PAIR_LJ_CUT_COUL_WOLF_H
+
+#include "pair.h"
+
+namespace LAMMPS_NS {
+
+class PairLJCutCoulWolf : public Pair {
+ public:
+  PairLJCutCoulWolf(class LAMMPS *);
+  virtual ~PairLJCutCoulWolf();
+  virtual void compute(int, int);
+  void settings(int, char **);
+  void coeff(int, char **);
+  void init_style();
+  double init_one(int, int);
+  void write_restart(FILE *);
+  void read_restart(FILE *);
+  void write_restart_settings(FILE *);
+  void read_restart_settings(FILE *);
+  void write_data(FILE *);
+  void write_data_all(FILE *);
+
+ protected:
+  double cut_lj_global;
+  double **cut_lj,**cut_ljsq;
+  double **epsilon,**sigma;
+  double **lj1,**lj2,**lj3,**lj4,**offset;
+  double *cut_respa;
+  double cut_coul,cut_coulsq,alf;
+
+  virtual void allocate();
+};
+
+}
+
+#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: Incorrect args for pair coefficients
+
+Self-explanatory.  Check the input script or data file.
+
+E: Pair cutoff < Respa interior cutoff
+
+One or more pairwise cutoffs are too short to use with the specified
+rRESPA cutoffs.
+
+*/