diff --git a/doc/src/Eqs/pair_entropy.jpg b/doc/src/Eqs/pair_entropy.jpg new file mode 100644 index 0000000000..44161e6aa0 Binary files /dev/null and b/doc/src/Eqs/pair_entropy.jpg differ diff --git a/doc/src/Eqs/pair_entropy.tex b/doc/src/Eqs/pair_entropy.tex new file mode 100644 index 0000000000..304c9d6138 --- /dev/null +++ b/doc/src/Eqs/pair_entropy.tex @@ -0,0 +1,10 @@ +\documentclass[12pt]{article} + +\begin{document} +\thispagestyle{empty} + +$$ + s_S^i=-2\pi\rho k_B \int\limits_0^{r_m} \left [ g(r) \ln g(r) - g(r) + 1 \right ] r^2 dr , +$$ + +\end{document} diff --git a/doc/src/Eqs/pair_entropy2.jpg b/doc/src/Eqs/pair_entropy2.jpg new file mode 100644 index 0000000000..db3dcb0ca7 Binary files /dev/null and b/doc/src/Eqs/pair_entropy2.jpg differ diff --git a/doc/src/Eqs/pair_entropy2.tex b/doc/src/Eqs/pair_entropy2.tex new file mode 100644 index 0000000000..c91004840f --- /dev/null +++ b/doc/src/Eqs/pair_entropy2.tex @@ -0,0 +1,10 @@ +\documentclass[12pt]{article} + +\begin{document} +\thispagestyle{empty} + +$$ + g_m^i(r) = \frac{1}{4 \pi \rho r^2} \sum\limits_{j} \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(r-r_{ij})^2/(2\sigma^2)} , +$$ + +\end{document} diff --git a/doc/src/Eqs/pair_entropy3.jpg b/doc/src/Eqs/pair_entropy3.jpg new file mode 100644 index 0000000000..0d5e0025b7 Binary files /dev/null and b/doc/src/Eqs/pair_entropy3.jpg differ diff --git a/doc/src/Eqs/pair_entropy3.tex b/doc/src/Eqs/pair_entropy3.tex new file mode 100644 index 0000000000..b0b3c76358 --- /dev/null +++ b/doc/src/Eqs/pair_entropy3.tex @@ -0,0 +1,10 @@ +\documentclass[12pt]{article} + +\begin{document} +\thispagestyle{empty} + +$$ + \bar{s}_S^i = \frac{\sum_j s_S^j + s_S^i}{N + 1} , +$$ + +\end{document} diff --git a/doc/src/Section_commands.txt b/doc/src/Section_commands.txt index 891abc2428..80ebe55345 100644 --- a/doc/src/Section_commands.txt +++ b/doc/src/Section_commands.txt @@ -863,6 +863,7 @@ package"_Section_start.html#start_3. "dpd"_compute_dpd.html, "dpd/atom"_compute_dpd_atom.html, "edpd/temp/atom"_compute_edpd_temp_atom.html, +"entropy/atom"_compute_entropy_atom.html, "fep"_compute_fep.html, "force/tally"_compute_tally.html, "heat/flux/tally"_compute_tally.html, diff --git a/doc/src/compute_entropy_atom.txt b/doc/src/compute_entropy_atom.txt new file mode 100644 index 0000000000..f7e7b8a667 --- /dev/null +++ b/doc/src/compute_entropy_atom.txt @@ -0,0 +1,130 @@ +"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c + +:link(lws,http://lammps.sandia.gov) +:link(ld,Manual.html) +:link(lc,Section_commands.html#comm) + +:line + +compute entropy/atom command :h3 + +[Syntax:] + +compute ID group-ID entropy/atom sigma cutoff keyword value ... :pre + +ID, group-ID are documented in "compute"_compute.html command :l +entropy/atom = style name of this compute command :l +sigma = width of gaussians used in the g(r) smoothening :l +cutoff = cutoff for the g(r) calculation :l +one or more keyword/value pairs may be appended :l +keyword = {avg} or {local} + {avg} values = {yes} or {no} cutoff2 + {yes} = average the pair entropy over neighbors + {no} = do not average the pair entropy over neighbors + cutoff2 = cutoff for the averaging over neighbors + {local} values = {yes} or {no} = use the local density around each atom to normalize the g(r) :pre +:ule + +[Examples:] + +compute 1 all entropy/atom 0.25 5. +compute 1 all entropy/atom 0.25 5. avg yes 5. +compute 1 all entropy/atom 0.125 7.3 avg yes 5.1 local yes :pre + +[Description:] + +Define a computation that calculates the pair entropy fingerprint for +each atom in the group. The fingerprint is useful to distinguish between +ordered and disordered environments, for instance liquid and solid-like +environments, or glassy and crystalline-like environments. Some +applications could be the identification of grain boundaries, a +melt-solid interface, or a solid cluster emerging from the melt. +The advantage of this parameter over others is that no a priori +information about the solid structure is required. + +This parameter for atom i is computed using the following formula from +"(Piaggi)"_#Piaggi and "(Nettleton)"_#Nettleton , + +:c,image(Eqs/pair_entropy.jpg) + +where r is a distance, g(r) is the radial distribution function of atom +i and rho is the density of the system. The g(r) computed for each +atom i can be noisy and therefore it is smoothened using: + +:c,image(Eqs/pair_entropy2.jpg) + +where the sum in j goes through the neighbors of atom i, and sigma is a +parameter to control the smoothening. + +The input parameters are {sigma} the smoothening parameter, and the +{cutoff} for the calculation of g(r). + +If the keyword {avg} has the setting {yes}, then this compute also +averages the parameter over the neighbors of atom i according to: + +:c,image(Eqs/pair_entropy3.jpg) + +where the sum j goes over the neighbors of atom i and N is the number +of neighbors. This procedure provides a sharper distinction between +order and disorder environments. In this case the input parameter +{cutoff2} is the cutoff for the averaging over the neighbors and +must also be specified. + +If the {avg yes} option is used, the effective cutoff of the neighbor +list should be {cutoff}+{cutoff2} and therefore it might be necessary +to increase the skin of the neighbor list with: + +neighbor skin bin :pre + +See "neighbor"_neighbor.html for details. + +If the {local yes} option is used, the g(r) is normalized by the +local density around each atom, that is to say the density around each +atom is the number of neighbors within the neighbor list cutoff divided +by the corresponding volume. This option can be useful when dealing with +inhomogeneus systems such as those that have surfaces. + +Here are typical input parameters for fcc aluminum (lattice +constant 4.05 Angstroms), + +compute 1 all entropy/atom 0.25 5.7 avg yes 3.7 :pre + +and for bcc sodium (lattice constant 4.23 Angstroms), + +compute 1 all entropy/atom 0.25 7.3 avg yes 5.1 :pre + + +[Output info:] + +By default, this compute calculates the pair entropy value for each +atom as a per-atom vector, which can be accessed by any command that +uses per-atom values from a compute as input. See "Section +6.15"_Section_howto.html#howto_15 for an overview of LAMMPS output +options. + +The pair entropy values have units of the Boltzmann constant. They are +always negative, and lower values (lower entropy) correspond to more +ordered environments. + +[Restrictions:] + +This compute is part of the USER-MISC package. It is only enabled if +LAMMPS was built with that package. See the "Making +LAMMPS"_Section_start.html#start_3 section for more info. + +[Related commands:] + +"compute cna/atom"_compute_cna_atom.html +"compute centro/atom"_compute_centro_atom.html + +[Default:] + +The default values for the optional keywords are avg = no and local = no. + +:line + +:link(Piaggi) +[(Piaggi)] Piaggi and Parrinello, J Chem Phys, 147, 114112 (2017). + +:link(Nettleton) +[(Nettleton)] Nettleton and Green, J Chem Phys, 29, 6 (1958). diff --git a/doc/src/computes.txt b/doc/src/computes.txt index 1b64e2e5b4..7848d4db04 100644 --- a/doc/src/computes.txt +++ b/doc/src/computes.txt @@ -31,6 +31,7 @@ Computes :h1 compute_dpd compute_dpd_atom compute_edpd_temp_atom + compute_entropy_atom compute_erotate_asphere compute_erotate_rigid compute_erotate_sphere diff --git a/doc/src/lammps.book b/doc/src/lammps.book index 0764c593f7..716a52b2e3 100644 --- a/doc/src/lammps.book +++ b/doc/src/lammps.book @@ -317,6 +317,7 @@ compute_displace_atom.html compute_dpd.html compute_dpd_atom.html compute_edpd_temp_atom.html +compute_entropy_atom.html compute_erotate_asphere.html compute_erotate_rigid.html compute_erotate_sphere.html diff --git a/examples/USER/misc/entropy/Na_MendelevM_2014.eam.fs b/examples/USER/misc/entropy/Na_MendelevM_2014.eam.fs new file mode 100644 index 0000000000..76ecae7b44 --- /dev/null +++ b/examples/USER/misc/entropy/Na_MendelevM_2014.eam.fs @@ -0,0 +1,6006 @@ +Source: M.I. 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4.0580253230452010e+00 -3.0945361122357586e+00 -8.8323796705297763e-01 +4089 7.2248060554373073e+00 9.0480655644191388e-01 -9.6284911590930766e-01 +2593 3.2503133857492319e+00 -4.1972151630470256e+00 -1.7952974231313504e-01 diff --git a/examples/USER/misc/entropy/in.entropy b/examples/USER/misc/entropy/in.entropy new file mode 100644 index 0000000000..2c25dac320 --- /dev/null +++ b/examples/USER/misc/entropy/in.entropy @@ -0,0 +1,34 @@ +echo both + +units metal +atom_style full + +read_data data.interface +mass 1 22.98977 + +neigh_modify delay 10 every 1 +pair_style eam/fs +pair_coeff * * Na_MendelevM_2014.eam.fs Na +timestep 0.002 +thermo 500 + +neighbor 4. bin + +# Define computes +# Global density, no average +compute 1 all entropy/atom 0.25 7.75 +# Local density, no average +compute 2 all entropy/atom 0.25 7.75 local yes +# Global density, average over neighbors +compute 3 all entropy/atom 0.25 7.75 avg yes 5. +# Local density, average over neighbors +compute 4 all entropy/atom 0.25 7.75 avg yes 5. local yes + +dump myDump all custom 500 dump.interface id type x y z c_1 c_2 c_3 c_4 + + +fix 1 all nph x 1. 1. 10. +fix 2 all temp/csvr 350. 350. 0.1 64582 + +run 1000 + diff --git a/examples/USER/misc/entropy/log.entropy b/examples/USER/misc/entropy/log.entropy new file mode 100644 index 0000000000..c380c62a73 --- /dev/null +++ b/examples/USER/misc/entropy/log.entropy @@ -0,0 +1,124 @@ +LAMMPS (30 Mar 2018) + +units metal +atom_style full + +read_data data.interface +Reading data file ... + triclinic box = (0 0 0) to (138.4 34.57 34.57) with tilt (0 0 0) + 4 by 1 by 1 MPI processor grid + reading atoms ... + 4096 atoms + reading velocities ... + 4096 velocities +Finding 1-2 1-3 1-4 neighbors ... + special bond factors lj: 0 0 0 + special bond factors coul: 0 0 0 + 0 = max # of 1-2 neighbors + 0 = max # of 1-3 neighbors + 0 = max # of 1-4 neighbors + 1 = max # of special neighbors +mass 1 22.98977 + +neigh_modify delay 10 every 1 +pair_style eam/fs +pair_coeff * * Na_MendelevM_2014.eam.fs Na +timestep 0.002 +thermo 500 + +neighbor 4. bin + +# Define computes +# Global density, no average +compute 1 all entropy/atom 0.25 7.75 +# Local density, no average +compute 2 all entropy/atom 0.25 7.75 local yes +# Global density, average over neighbors +compute 3 all entropy/atom 0.25 7.75 avg yes 5. +# Local density, average over neighbors +compute 4 all entropy/atom 0.25 7.75 avg yes 5. local yes + +dump myDump all custom 500 dump.interface id type x y z c_1 c_2 c_3 c_4 + + +fix 1 all nph x 1. 1. 10. +fix 2 all temp/csvr 350. 350. 0.1 64582 + +run 1000 +WARNING: More than one compute entropy/atom (../compute_entropy_atom.cpp:138) +WARNING: More than one compute entropy/atom (../compute_entropy_atom.cpp:138) +WARNING: More than one compute entropy/atom (../compute_entropy_atom.cpp:138) +WARNING: More than one compute entropy/atom (../compute_entropy_atom.cpp:138) +Neighbor list info ... + update every 1 steps, delay 10 steps, check yes + max neighbors/atom: 2000, page size: 100000 + master list distance cutoff = 13.2 + ghost atom cutoff = 13.2 + binsize = 6.6, bins = 21 6 6 + 5 neighbor lists, perpetual/occasional/extra = 5 0 0 + (1) pair eam/fs, perpetual + attributes: half, newton on + pair build: half/bin/newton/tri + stencil: half/bin/3d/newton/tri + bin: standard + (2) compute entropy/atom, perpetual + attributes: full, newton on, ghost + pair build: full/bin/ghost + stencil: full/ghost/bin/3d + bin: standard + (3) compute entropy/atom, perpetual, copy from (2) + attributes: full, newton on, ghost + pair build: copy + stencil: none + bin: none + (4) compute entropy/atom, perpetual, copy from (2) + attributes: full, newton on, ghost + pair build: copy + stencil: none + bin: none + (5) compute entropy/atom, perpetual, copy from (2) + attributes: full, newton on, ghost + pair build: copy + stencil: none + bin: none +Setting up Verlet run ... + Unit style : metal + Current step : 0 + Time step : 0.002 +Per MPI rank memory allocation (min/avg/max) = 25.68 | 25.69 | 25.69 Mbytes +Step Temp E_pair E_mol TotEng Press Volume + 0 346.29871 -4285.222 0 -4101.9191 594.65353 165399.75 + 500 359.33758 -4285.247 0 -4095.0423 471.98587 165847.18 + 1000 348.99659 -4276.2274 0 -4091.4964 149.27188 166966.18 +Loop time of 5.3437 on 4 procs for 1000 steps with 4096 atoms + +Performance: 32.337 ns/day, 0.742 hours/ns, 187.136 timesteps/s +99.8% CPU use with 4 MPI tasks x no OpenMP threads + +MPI task timing breakdown: +Section | min time | avg time | max time |%varavg| %total +--------------------------------------------------------------- +Pair | 4.2832 | 4.3257 | 4.3839 | 1.8 | 80.95 +Bond | 0.00018309 | 0.00019825 | 0.00021418 | 0.0 | 0.00 +Neigh | 0.42195 | 0.42512 | 0.42739 | 0.3 | 7.96 +Comm | 0.051679 | 0.1101 | 0.14916 | 10.8 | 2.06 +Output | 0.40909 | 0.4091 | 0.40911 | 0.0 | 7.66 +Modify | 0.060869 | 0.061921 | 0.06327 | 0.4 | 1.16 +Other | | 0.01161 | | | 0.22 + +Nlocal: 1024 ave 1040 max 1001 min +Histogram: 1 0 0 0 0 0 2 0 0 1 +Nghost: 4614.25 ave 4700 max 4540 min +Histogram: 1 1 0 0 0 0 0 1 0 1 +Neighs: 121747 ave 126398 max 116931 min +Histogram: 1 0 0 1 0 0 1 0 0 1 +FullNghs: 243494 ave 252523 max 233842 min +Histogram: 1 0 0 1 0 0 1 0 0 1 + +Total # of neighbors = 973974 +Ave neighs/atom = 237.787 +Ave special neighs/atom = 0 +Neighbor list builds = 13 +Dangerous builds = 0 + +Total wall time: 0:00:05 diff --git a/src/USER-MISC/README b/src/USER-MISC/README index 222c06b4c4..68a6252d8d 100644 --- a/src/USER-MISC/README +++ b/src/USER-MISC/README @@ -29,6 +29,7 @@ bond_style harmonic/shift/cut, Carsten Svaneborg, science at zqex.dk, 8 Aug 11 compute ackland/atom, Gerolf Ziegenhain, gerolf at ziegenhain.com, 4 Oct 2007 compute basal/atom, Christopher Barrett, cdb333 at cavs.msstate.edu, 3 Mar 2013 compute cnp/atom, Paulo Branicio (USC), branicio at usc.edu, 31 May 2017 +compute entropy/atom, Pablo Piaggi (EPFL), pablo.piaggi at epfl.ch, 15 June 2018 compute temp/rotate, Laurent Joly (U Lyon), ljoly.ulyon at gmail.com, 8 Aug 11 compute PRESSURE/GREM, David Stelter, dstelter@bu.edu, 22 Nov 16 dihedral_style cosine/shift/exp, Carsten Svaneborg, science at zqex.dk, 8 Aug 11 diff --git a/src/USER-MISC/compute_entropy_atom.cpp b/src/USER-MISC/compute_entropy_atom.cpp new file mode 100644 index 0000000000..a2e6765c83 --- /dev/null +++ b/src/USER-MISC/compute_entropy_atom.cpp @@ -0,0 +1,340 @@ +/* ---------------------------------------------------------------------- + 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: Pablo Piaggi (EPFL Lausanne) +------------------------------------------------------------------------- */ + +#include +#include +#include +#include "compute_entropy_atom.h" +#include "atom.h" +#include "update.h" +#include "modify.h" +#include "neighbor.h" +#include "neigh_list.h" +#include "neigh_request.h" +#include "force.h" +#include "pair.h" +#include "comm.h" +#include "math_extra.h" +#include "math_const.h" +#include "memory.h" +#include "error.h" +#include "domain.h" + + +using namespace LAMMPS_NS; +using namespace MathConst; + +/* ---------------------------------------------------------------------- */ + +ComputeEntropyAtom:: +ComputeEntropyAtom(LAMMPS *lmp, int narg, char **arg) : + Compute(lmp, narg, arg), + pair_entropy(NULL), pair_entropy_avg(NULL) +{ + if (narg < 5 || narg > 10) + error->all(FLERR,"Illegal compute entropy/atom command; wrong number" + " of arguments"); + + // Arguments are: sigma cutoff avg yes/no cutoff2 local yes/no + // sigma is the gaussian width + // cutoff is the cutoff for the calculation of g(r) + // avg is optional and allows averaging the pair entropy over neighbors + // the next argument should be yes or no + // cutoff2 is the cutoff for the averaging + // local is optional and allows using the local density to normalize + // the g(r) + + sigma = force->numeric(FLERR,arg[3]); + if (sigma < 0.0) error->all(FLERR,"Illegal compute entropy/atom" + " command; negative sigma"); + cutoff = force->numeric(FLERR,arg[4]); + if (cutoff < 0.0) error->all(FLERR,"Illegal compute entropy/atom" + " command; negative cutoff"); + + avg_flag = 0; + local_flag = 0; + + // optional keywords + int iarg = 5; + while (iarg < narg) { + if (strcmp(arg[iarg],"avg") == 0) { + if (iarg+2 > narg) + error->all(FLERR,"Illegal compute entropy/atom;" + " missing arguments after avg"); + if (strcmp(arg[iarg+1],"yes") == 0) avg_flag = 1; + else if (strcmp(arg[iarg+1],"no") == 0) avg_flag = 0; + else error->all(FLERR,"Illegal compute entropy/atom;" + " argument after avg should be yes or no"); + cutoff2 = force->numeric(FLERR,arg[iarg+2]); + if (cutoff2 < 0.0) error->all(FLERR,"Illegal compute entropy/atom" + " command; negative cutoff2"); + cutsq2 = cutoff2*cutoff2; + iarg += 3; + } else if (strcmp(arg[iarg],"local") == 0) { + if (strcmp(arg[iarg+1],"yes") == 0) local_flag = 1; + else if (strcmp(arg[iarg+1],"no") == 0) local_flag = 0; + else error->all(FLERR,"Illegal compute entropy/atom;" + " argument after local should be yes or no"); + iarg += 2; + } else error->all(FLERR,"Illegal compute entropy/atom; argument after" + " sigma and cutoff should be avg or local"); + } + + + cutsq = cutoff*cutoff; + nbin = static_cast(cutoff / sigma) + 1; + nmax = 0; + maxneigh = 0; + // Number of bins above and below the central one that will be + // considered as affected by the gaussian kernel + // 3 seems a good compromise between speed and good mollification + deltabin = 3; + deltar = sigma; + peratom_flag = 1; + size_peratom_cols = 0; +} + +/* ---------------------------------------------------------------------- */ + +ComputeEntropyAtom::~ComputeEntropyAtom() +{ + memory->destroy(pair_entropy); + if (avg_flag) memory->destroy(pair_entropy_avg); +} + +/* ---------------------------------------------------------------------- */ + +void ComputeEntropyAtom::init() +{ + if (force->pair == NULL) + error->all(FLERR,"Compute centro/atom requires a pair style be" + " defined"); + + if ( (cutoff+cutoff2) > (force->pair->cutforce + neighbor->skin) ) + { + error->all(FLERR,"Compute entropy/atom cutoff is longer than the" + " pairwise cutoff. Increase the neighbor list skin" + " distance."); + } + + int count = 0; + for (int i = 0; i < modify->ncompute; i++) + if (strcmp(modify->compute[i]->style,"entropy/atom") == 0) count++; + if (count > 1 && comm->me == 0) + error->warning(FLERR,"More than one compute entropy/atom"); + + // need a full neighbor list with neighbors of the ghost atoms + + int irequest = neighbor->request(this,instance_me); + neighbor->requests[irequest]->pair = 0; + neighbor->requests[irequest]->compute = 1; + neighbor->requests[irequest]->half = 0; + neighbor->requests[irequest]->full = 1; + neighbor->requests[irequest]->occasional = 0; + neighbor->requests[irequest]->ghost = 1; + +} + +/* ---------------------------------------------------------------------- */ + +void ComputeEntropyAtom::init_list(int id, NeighList *ptr) +{ + list = ptr; +} + +/* ---------------------------------------------------------------------- */ + +void ComputeEntropyAtom::compute_peratom() +{ + int i,j,ii,jj,inum,jnum; + double xtmp,ytmp,ztmp,delx,dely,delz,rsq; + int *ilist,*jlist,*numneigh,**firstneigh; + double rbin[nbin], rbinsq[nbin]; + + invoked_peratom = update->ntimestep; + + // Initialize distance vectors + for (int i = 0; i < nbin; i++) { + rbin[i] = i*deltar; + rbinsq[i] = rbin[i]*rbin[i]; + } + + // grow pair_entropy and pair_entropy_avg array if necessary + + if (atom->nmax > nmax) { + if (!avg_flag) { + memory->destroy(pair_entropy); + nmax = atom->nmax; + memory->create(pair_entropy,nmax,"entropy/atom:pair_entropy"); + vector_atom = pair_entropy; + } else { + memory->destroy(pair_entropy); + memory->destroy(pair_entropy_avg); + nmax = atom->nmax; + memory->create(pair_entropy,nmax,"entropy/atom:pair_entropy"); + memory->create(pair_entropy_avg,nmax, + "entropy/atom:pair_entropy_avg"); + vector_atom = pair_entropy_avg; + } + } + + inum = list->inum + list->gnum; + ilist = list->ilist; + numneigh = list->numneigh; + firstneigh = list->firstneigh; + + // Compute some constants + double sigmasq2=2*sigma*sigma; + double volume = domain->xprd * domain->yprd * domain->zprd; + double density = atom->natoms / volume; + + // compute pair entropy for each atom in group + // use full neighbor list + + double **x = atom->x; + int *mask = atom->mask; + + for (ii = 0; ii < inum; ii++) { + i = ilist[ii]; + if (mask[i] & groupbit) { + xtmp = x[i][0]; + ytmp = x[i][1]; + ztmp = x[i][2]; + jlist = firstneigh[i]; + jnum = numneigh[i]; + + // If local density is used, calculate it + if (local_flag) { + double neigh_cutoff = force->pair->cutforce + neighbor->skin; + double volume = + (4./3.)*MY_PI*neigh_cutoff*neigh_cutoff*neigh_cutoff; + density = jnum / volume; + } + + // calculate kernel normalization + // Normalization of g(r) + double normConstantBase = 4*MY_PI*density; + // Normalization of gaussian + normConstantBase *= sqrt(2.*MY_PI)*sigma; + double invNormConstantBase = 1./normConstantBase; + + // loop over list of all neighbors within force cutoff + + // initialize gofr + double gofr[nbin]; + for(int k=0;k (nbin-1)) minbin=nbin-1; + maxbin=bin + deltabin; + if (maxbin > (nbin-1)) maxbin=nbin-1; + for(int k=minbin;k