symmetries_finite.c

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/*
 *  Brute force symmetry analyzer.
 *  This is actually C++ program, masquerading as a C one!
 *
 *  (C) 2011 X. Andrade Converted to a Fortran library.
 *  (C) 1996, 2003 S. Patchkovskii, Serguei.Patchkovskii@sympatico.ca
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, write to the Free Software
 *  Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
 *
 * Revision 1.16  2003/04/04  13:05:03  patchkov
 * Revision 1.15  2000/01/25  16:47:17  patchkov
 * Revision 1.14  2000/01/25  16:39:08  patchkov
 * Revision 1.13  1996/05/24  12:32:08  ps
 * Revision 1.12  1996/05/23  16:10:47  ps
 * First reasonably stable version.
 *
 */
 
#include <config.h>
 
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
 
#define DIMENSION 3
#define MAXPARAM 7
 
typedef struct {
  int type;
  double x[DIMENSION];
} ATOM;
 
/*
 *  All specific structures should have corresponding elements in the
 *  same position generic structure does.
 *
 *  Planes are characterized by the surface normal direction
 *  (taken in the direction *from* the coordinate origin)
 *  and distance from the coordinate origin to the plane
 *  in the direction of the surface normal.
 *
 *  Inversion is characterized by location of the inversion center.
 *
 *  Rotation is characterized by a vector (distance+direction) from the origin
 *  to the rotation axis, axis direction and rotation order. Rotations
 *  are in the clockwise direction looking opposite to the direction
 *  of the axis. Note that this definition of the rotation axis
 *  is *not* unique, since an arbitrary multiple of the axis direction
 *  can be added to the position vector without changing actual operation.
 *
 *  Mirror rotation is defined by the same parameters as normal rotation,
 *  but the origin is now unambiguous since it defines the position of the
 *  plane associated with the axis.
 *
 */
 
typedef struct _SYMMETRY_ELEMENT_ {
  void (*transform_atom)(struct _SYMMETRY_ELEMENT_ *el, ATOM *from, ATOM *to);
  int *transform; /*   Correspondence table for the transformation         */
  int order;      /*   Applying transformation this many times is identity */
  int nparam;     /*   4 for inversion and planes, 7 for axes              */
  double maxdev;  /*   Larges error associated with the element            */
  double distance;
  double normal[DIMENSION];
  double direction[DIMENSION];
} SYMMETRY_ELEMENT;
 
typedef struct {
  char *group_name;    /* Canonical group name                              */
  char *symmetry_code; /* Group symmetry code                               */
  int (*check)(void);  /* Additional verification routine, not used         */
} POINT_GROUP;
 
static double ToleranceSame = 1e-3;
static double TolerancePrimary = 5e-2;
static double ToleranceFinal = 1e-4;
static double MaxOptStep = 5e-1;
static double MinOptStep = 1e-7;
static double GradientStep = 1e-7;
static double OptChangeThreshold = 1e-10;
static double CenterOfSomething[DIMENSION];
static double *DistanceFromCenter = NULL;
static int verbose = 0;
static int MaxOptCycles = 200;
static int OptChangeHits = 5;
static int MaxAxisOrder = 20;
static int AtomsCount = 0;
static ATOM *Atoms = NULL;
static int PlanesCount = 0;
static SYMMETRY_ELEMENT **Planes = NULL;
static SYMMETRY_ELEMENT *MolecularPlane = NULL;
static int InversionCentersCount = 0;
static SYMMETRY_ELEMENT **InversionCenters = NULL;
static int NormalAxesCount = 0;
static SYMMETRY_ELEMENT **NormalAxes = NULL;
static int ImproperAxesCount = 0;
static SYMMETRY_ELEMENT **ImproperAxes = NULL;
static int *NormalAxesCounts = NULL;
static int *ImproperAxesCounts = NULL;
static int BadOptimization = 0;
static char *SymmetryCode = NULL; /*"" ;*/
/*
 *    Statistics
 */
static long StatTotal = 0;
static long StatEarly = 0;
static long StatPairs = 0;
static long StatDups = 0;
static long StatOrder = 0;
static long StatOpt = 0;
static long StatAccept = 0;
 
/*
 *    Point groups I know about
 */
int true(void) { return 1; }
POINT_GROUP PointGroups[] = {
    {"C1", "", true},
    {"Cs", "(sigma) ", true},
    {"Ci", "(i) ", true},
    {"C2", "(C2) ", true},
    {"C3", "(C3) ", true},
    {"C4", "(C4) (C2) ", true},
    {"C5", "(C5) ", true},
    {"C6", "(C6) (C3) (C2) ", true},
    {"C7", "(C7) ", true},
    {"C8", "(C8) (C4) (C2) ", true},
    {"D2", "3*(C2) ", true},
    {"D3", "(C3) 3*(C2) ", true},
    {"D4", "(C4) 5*(C2) ", true},
    {"D5", "(C5) 5*(C2) ", true},
    {"D6", "(C6) (C3) 7*(C2) ", true},
    {"D7", "(C7) 7*(C2) ", true},
    {"D8", "(C8) (C4) 9*(C2) ", true},
    {"C2v", "(C2) 2*(sigma) ", true},
    {"C3v", "(C3) 3*(sigma) ", true},
    {"C4v", "(C4) (C2) 4*(sigma) ", true},
    {"C5v", "(C5) 5*(sigma) ", true},
    {"C6v", "(C6) (C3) (C2) 6*(sigma) ", true},
    {"C7v", "(C7) 7*(sigma) ", true},
    {"C8v", "(C8) (C4) (C2) 8*(sigma) ", true},
    {"C2h", "(i) (C2) (sigma) ", true},
    {"C3h", "(C3) (S3) (sigma) ", true},
    {"C4h", "(i) (C4) (C2) (S4) (sigma) ", true},
    {"C5h", "(C5) (S5) (sigma) ", true},
    {"C6h", "(i) (C6) (C3) (C2) (S6) (S3) (sigma) ", true},
    {"C7h", "(C7) (S7) (sigma) ", true},
    {"C8h", "(i) (C8) (C4) (C2) (S8) (S4) (sigma) ", true},
    {"D2h", "(i) 3*(C2) 3*(sigma) ", true},
    {"D3h", "(C3) 3*(C2) (S3) 4*(sigma) ", true},
    {"D4h", "(i) (C4) 5*(C2) (S4) 5*(sigma) ", true},
    {"D5h", "(C5) 5*(C2) (S5) 6*(sigma) ", true},
    {"D6h", "(i) (C6) (C3) 7*(C2) (S6) (S3) 7*(sigma) ", true},
    {"D7h", "(C7) 7*(C2) (S7) 8*(sigma) ", true},
    {"D8h", "(i) (C8) (C4) 9*(C2) (S8) (S4) 9*(sigma) ", true},
    {"D2d", "3*(C2) (S4) 2*(sigma) ", true},
    {"D3d", "(i) (C3) 3*(C2) (S6) 3*(sigma) ", true},
    {"D4d", "(C4) 5*(C2) (S8) 4*(sigma) ", true},
    {"D5d", "(i) (C5) 5*(C2) (S10) 5*(sigma) ", true},
    {"D6d", "(C6) (C3) 7*(C2) (S12) (S4) 6*(sigma) ", true},
    {"D7d", "(i) (C7) 7*(C2) (S14) 7*(sigma) ", true},
    {"D8d", "(C8) (C4) 9*(C2) (S16) 8*(sigma) ", true},
    {"S4", "(C2) (S4) ", true},
    {"S6", "(i) (C3) (S6) ", true},
    {"S8", "(C4) (C2) (S8) ", true},
    {"T", "4*(C3) 3*(C2) ", true},
    {"Th", "(i) 4*(C3) 3*(C2) 4*(S6) 3*(sigma) ", true},
    {"Td", "4*(C3) 3*(C2) 3*(S4) 6*(sigma) ", true},
    {"O", "3*(C4) 4*(C3) 9*(C2) ", true},
    {"Oh", "(i) 3*(C4) 4*(C3) 9*(C2) 4*(S6) 3*(S4) 9*(sigma) ", true},
    {"Cinfv", "(Cinf) (sigma) ", true},
    {"Dinfh", "(i) (Cinf) (C2) 2*(sigma) ", true},
    {"I", "6*(C5) 10*(C3) 15*(C2) ", true},
    {"Ih", "(i) 6*(C5) 10*(C3) 15*(C2) 6*(S10) 10*(S6) 15*(sigma) ", true},
    {"Kh", "(i) (Cinf) (sigma) ", true},
};
#define PointGroupsCount (sizeof(PointGroups) / sizeof(POINT_GROUP))
char *PointGroupRejectionReason = NULL;
 
/*
 *   Generic functions
 */
 
double pow2(double x) { return x * x; }
 
int establish_pairs(SYMMETRY_ELEMENT *elem) {
  int i, j, k, best_j;
  char *atom_used = calloc(AtomsCount, 1);
  double distance, best_distance;
  ATOM symmetric;
 
  if (atom_used == NULL) {
    fprintf(stderr, "Out of memory for tagging array in establish_pairs()\n");
    exit(EXIT_FAILURE);
  }
  for (i = 0; i < AtomsCount; i++) {
    if (elem->transform[i] >= AtomsCount) { /* No symmetric atom yet          */
      if (verbose > 2)
        printf("        looking for a pair for %d\n", i);
      elem->transform_atom(elem, Atoms + i, &symmetric);
      if (verbose > 2)
        printf("        new coordinates are: (%g,%g,%g)\n", symmetric.x[0],
               symmetric.x[1], symmetric.x[2]);
      best_j = i;
      best_distance = 2 * TolerancePrimary; /* Performance value we'll reject */
      for (j = 0; j < AtomsCount; j++) {
        if (Atoms[j].type != symmetric.type || atom_used[j])
          continue;
        for (k = 0, distance = 0; k < DIMENSION; k++) {
          distance += pow2(symmetric.x[k] - Atoms[j].x[k]);
        }
        distance = sqrt(distance);
        if (verbose > 2)
          printf("        distance to %d is %g\n", j, distance);
        if (distance < best_distance) {
          best_j = j;
          best_distance = distance;
        }
      }
      if (best_distance >
          TolerancePrimary) { /* Too bad, there is no symmetric atom */
        if (verbose > 0)
          printf("        no pair for atom %d - best was %d with err = %g\n", i,
                 best_j, best_distance);
        free(atom_used);
        return -1;
      }
      elem->transform[i] = best_j;
      atom_used[best_j] = 1;
      if (verbose > 1)
        printf("        atom %d transforms to the atom %d, err = %g\n", i,
               best_j, best_distance);
    }
  }
  free(atom_used);
  return 0;
}
 
int check_transform_order(SYMMETRY_ELEMENT *elem) {
  int i, j, k;
  void rotate_reflect_atom(SYMMETRY_ELEMENT *, ATOM *, ATOM *);
 
  for (i = 0; i < AtomsCount; i++) {
    if (elem->transform[i] == i) /* Identity transform is Ok for any order */
      continue;
    if (elem->transform_atom == rotate_reflect_atom) {
      j = elem->transform[i];
      if (elem->transform[j] == i)
        continue; /* Second-order transform is Ok for improper axis */
    }
    for (j = elem->order - 1, k = elem->transform[i]; j > 0;
         j--, k = elem->transform[k]) {
      if (k == i) {
        if (verbose > 0)
          printf("        transform looped %d steps too early from atom %d\n",
                 j, i);
        return -1;
      }
    }
    if (k != i && elem->transform_atom == rotate_reflect_atom) {
      /* For improper axes, the complete loop may also take twice the order */
      for (j = elem->order; j > 0; j--, k = elem->transform[k]) {
        if (k == i) {
          if (verbose > 0)
            printf("        (improper) transform looped %d steps too early "
                   "from atom %d\n",
                   j, i);
          return -1;
        }
      }
    }
    if (k != i) {
      if (verbose > 0)
        printf("        transform failed to loop after %d steps from atom %d\n",
               elem->order, i);
      return -1;
    }
  }
  return 0;
}
 
int same_transform(SYMMETRY_ELEMENT *a, SYMMETRY_ELEMENT *b) {
  int i, j;
  int code;
 
  if ((a->order != b->order) || (a->nparam != b->nparam) ||
      (a->transform_atom != b->transform_atom))
    return 0;
  for (i = 0, code = 1; i < AtomsCount; i++) {
    if (a->transform[i] != b->transform[i]) {
      code = 0;
      break;
    }
  }
  if (code == 0 &&
      a->order > 2) { /* b can also be a reverse transformation for a */
    for (i = 0; i < AtomsCount; i++) {
      j = a->transform[i];
      if (b->transform[j] != i)
        return 0;
    }
    return 1;
  }
  return code;
}
 
SYMMETRY_ELEMENT *alloc_symmetry_element(void) {
  SYMMETRY_ELEMENT *elem = calloc(1, sizeof(SYMMETRY_ELEMENT));
  int i;
 
  if (elem == NULL) {
    fprintf(stderr, "Out of memory allocating symmetry element\n");
    exit(EXIT_FAILURE);
  }
  elem->transform = calloc(AtomsCount, sizeof(int));
  if (elem->transform == NULL) {
    fprintf(stderr,
            "Out of memory allocating transform table for symmetry element\n");
    exit(EXIT_FAILURE);
  }
  for (i = 0; i < AtomsCount; i++) {
    elem->transform[i] = AtomsCount + 1; /* An impossible value */
  }
  return elem;
}
 
void destroy_symmetry_element(SYMMETRY_ELEMENT *elem) {
  if (elem != NULL) {
    if (elem->transform != NULL)
      free(elem->transform);
    free(elem);
  }
}
 
int check_transform_quality(SYMMETRY_ELEMENT *elem) {
  int i, j, k;
  ATOM symmetric;
  double r, max_r;
 
  for (i = 0, max_r = 0; i < AtomsCount; i++) {
    j = elem->transform[i];
    elem->transform_atom(elem, Atoms + i, &symmetric);
    for (k = 0, r = 0; k < DIMENSION; k++) {
      r += pow2(symmetric.x[k] - Atoms[j].x[k]);
    }
    r = sqrt(r);
    if (r > ToleranceFinal) {
      if (verbose > 0)
        printf("        distance to symmetric atom (%g) is too big for %d\n", r,
               i);
      return -1;
    }
    if (r > max_r)
      max_r = r;
  }
  elem->maxdev = max_r;
  return 0;
}
 
double eval_optimization_target_function(SYMMETRY_ELEMENT *elem, int *finish) {
  int i, j, k;
  ATOM symmetric;
  double target, r, maxr;
 
  if (elem->nparam >= 4) {
    for (k = 0, r = 0; k < DIMENSION; k++) {
      r += elem->normal[k] * elem->normal[k];
    }
    r = sqrt(r);
    if (r < ToleranceSame) {
      fprintf(stderr, "Normal collapsed!\n");
      exit(EXIT_FAILURE);
    }
    for (k = 0; k < DIMENSION; k++) {
      elem->normal[k] /= r;
    }
    if (elem->distance < 0) {
      elem->distance = -elem->distance;
      for (k = 0; k < DIMENSION; k++) {
        elem->normal[k] = -elem->normal[k];
      }
    }
  }
  if (elem->nparam >= 7) {
    for (k = 0, r = 0; k < DIMENSION; k++) {
      r += elem->direction[k] * elem->direction[k];
    }
    r = sqrt(r);
    if (r < ToleranceSame) {
      fprintf(stderr, "Direction collapsed!\n");
      exit(EXIT_FAILURE);
    }
    for (k = 0; k < DIMENSION; k++) {
      elem->direction[k] /= r;
    }
  }
  for (i = 0, target = maxr = 0; i < AtomsCount; i++) {
    elem->transform_atom(elem, Atoms + i, &symmetric);
    j = elem->transform[i];
    for (k = 0, r = 0; k < DIMENSION; k++) {
      r += pow2(Atoms[j].x[k] - symmetric.x[k]);
    }
    if (r > maxr)
      maxr = r;
    target += r;
  }
  if (finish != NULL) {
    *finish = 0;
    if (sqrt(maxr) < ToleranceFinal)
      *finish = 1;
  }
  return target;
}
 
void get_params(SYMMETRY_ELEMENT *elem, double values[]) {
  memcpy(values, &elem->distance, elem->nparam * sizeof(double));
}
 
void set_params(SYMMETRY_ELEMENT *elem, double values[]) {
  memcpy(&elem->distance, values, elem->nparam * sizeof(double));
}
 
void optimize_transformation_params(SYMMETRY_ELEMENT *elem) {
  double values[MAXPARAM];
  double grad[MAXPARAM];
  double force[MAXPARAM];
  double step[MAXPARAM];
  double f, fold, fnew, fnew2, fdn, fup, snorm;
  double a, b, x;
  int vars = elem->nparam;
  int cycle = 0;
  int i, finish;
  int hits = 0;
 
  if (vars > MAXPARAM) {
    fprintf(stderr, "Catastrophe in optimize_transformation_params()!\n");
    exit(EXIT_FAILURE);
  }
  f = 0;
  do {
    fold = f;
    f = eval_optimization_target_function(elem, &finish);
    /* Evaluate function, gradient and diagonal force constants */
    if (verbose > 1)
      printf("            function value = %g\n", f);
    if (finish) {
      if (verbose > 1)
        printf("        function value is small enough\n");
      break;
    }
    if (cycle > 0) {
      if (fabs(f - fold) > OptChangeThreshold)
        hits = 0;
      else
        hits++;
      if (hits >= OptChangeHits) {
        if (verbose > 1)
          printf("        no progress is made, stop optimization\n");
        break;
      }
    }
    get_params(elem, values);
    for (i = 0; i < vars; i++) {
      values[i] -= GradientStep;
      set_params(elem, values);
      fdn = eval_optimization_target_function(elem, NULL);
      values[i] += 2 * GradientStep;
      set_params(elem, values);
      fup = eval_optimization_target_function(elem, NULL);
      values[i] -= GradientStep;
      grad[i] = (fup - fdn) / (2 * GradientStep);
      force[i] = (fup + fdn - 2 * f) / (GradientStep * GradientStep);
      if (verbose > 1)
        printf("        i = %d, grad = %12.6e, force = %12.6e\n", i, grad[i],
               force[i]);
    }
    /* Do a quasy-Newton step */
    for (i = 0, snorm = 0; i < vars; i++) {
      if (force[i] < 0)
        force[i] = -force[i];
      if (force[i] < 1e-3)
        force[i] = 1e-3;
      if (force[i] > 1e3)
        force[i] = 1e3;
      step[i] = -grad[i] / force[i];
      snorm += step[i] * step[i];
    }
    snorm = sqrt(snorm);
    if (snorm > MaxOptStep) { /* Renormalize step */
      for (i = 0; i < vars; i++)
        step[i] *= MaxOptStep / snorm;
      snorm = MaxOptStep;
    }
    do {
      for (i = 0; i < vars; i++) {
        values[i] += step[i];
      }
      set_params(elem, values);
      fnew = eval_optimization_target_function(elem, NULL);
      if (fnew < f)
        break;
      for (i = 0; i < vars; i++) {
        values[i] -= step[i];
        step[i] /= 2;
      }
      set_params(elem, values);
      snorm /= 2;
    } while (snorm > MinOptStep);
    if ((snorm > MinOptStep) &&
        (snorm < MaxOptStep / 2)) { /* try to do quadratic interpolation */
      for (i = 0; i < vars; i++)
        values[i] += step[i];
      set_params(elem, values);
      fnew2 = eval_optimization_target_function(elem, NULL);
      if (verbose > 1)
        printf("        interpolation base points: %g, %g, %g\n", f, fnew,
               fnew2);
      for (i = 0; i < vars; i++)
        values[i] -= 2 * step[i];
      a = (4 * f - fnew2 - 3 * fnew) / 2;
      b = (f + fnew2 - 2 * fnew) / 2;
      if (verbose > 1)
        printf("        linear interpolation coefficients %g, %g\n", a, b);
      if (b > 0) {
        x = -a / (2 * b);
        if (x > 0.2 && x < 1.8) {
          if (verbose > 1)
            printf("        interpolated: %g\n", x);
          for (i = 0; i < vars; i++)
            values[i] += x * step[i];
        } else
          b = 0;
      }
      if (b <= 0) {
        if (fnew2 < fnew) {
          for (i = 0; i < vars; i++)
            values[i] += 2 * step[i];
        } else {
          for (i = 0; i < vars; i++)
            values[i] += step[i];
        }
      }
      set_params(elem, values);
    }
  } while (snorm > MinOptStep && ++cycle < MaxOptCycles);
  f = eval_optimization_target_function(elem, NULL);
  if (cycle >= MaxOptCycles)
    BadOptimization = 1;
  if (verbose > 0) {
    if (cycle >= MaxOptCycles)
      printf("        maximum number of optimization cycles made\n");
    printf("        optimization completed after %d cycles with f = %g\n",
           cycle, f);
  }
}
 
int refine_symmetry_element(SYMMETRY_ELEMENT *elem, int build_table) {
  int i;
 
  if (build_table && (establish_pairs(elem) < 0)) {
    StatPairs++;
    if (verbose > 0)
      printf("        no transformation correspondence table can be "
             "constructed\n");
    return -1;
  }
  for (i = 0; i < PlanesCount; i++) {
    if (same_transform(Planes[i], elem)) {
      StatDups++;
      if (verbose > 0)
        printf("        transformation is identical to plane %d\n", i);
      return -1;
    }
  }
  for (i = 0; i < InversionCentersCount; i++) {
    if (same_transform(InversionCenters[i], elem)) {
      StatDups++;
      if (verbose > 0)
        printf("        transformation is identical to inversion center %d\n",
               i);
      return -1;
    }
  }
  for (i = 0; i < NormalAxesCount; i++) {
    if (same_transform(NormalAxes[i], elem)) {
      StatDups++;
      if (verbose > 0)
        printf("        transformation is identical to normal axis %d\n", i);
      return -1;
    }
  }
  for (i = 0; i < ImproperAxesCount; i++) {
    if (same_transform(ImproperAxes[i], elem)) {
      StatDups++;
      if (verbose > 0)
        printf("        transformation is identical to improper axis %d\n", i);
      return -1;
    }
  }
  if (check_transform_order(elem) < 0) {
    StatOrder++;
    if (verbose > 0)
      printf("        incorrect transformation order\n");
    return -1;
  }
  optimize_transformation_params(elem);
  if (check_transform_quality(elem) < 0) {
    StatOpt++;
    if (verbose > 0)
      printf("        refined transformation does not pass the numeric "
             "threshold\n");
    return -1;
  }
  StatAccept++;
  return 0;
}
 
/*
 *   Plane-specific functions
 */
 
void mirror_atom(SYMMETRY_ELEMENT *plane, ATOM *from, ATOM *to) {
  int i;
  double r;
 
  for (i = 0, r = plane->distance; i < DIMENSION; i++) {
    r -= from->x[i] * plane->normal[i];
  }
  to->type = from->type;
  for (i = 0; i < DIMENSION; i++) {
    to->x[i] = from->x[i] + 2 * r * plane->normal[i];
  }
}
 
SYMMETRY_ELEMENT *init_mirror_plane(int i, int j) {
  SYMMETRY_ELEMENT *plane = alloc_symmetry_element();
  double dx[DIMENSION], midpoint[DIMENSION], rab, r;
  int k;
 
  if (verbose > 0)
    printf("Trying mirror plane for atoms %d,%d\n", i, j);
  StatTotal++;
  plane->transform_atom = mirror_atom;
  plane->order = 2;
  plane->nparam = 4;
  for (k = 0, rab = 0; k < DIMENSION; k++) {
    dx[k] = Atoms[i].x[k] - Atoms[j].x[k];
    midpoint[k] = (Atoms[i].x[k] + Atoms[j].x[k]) / 2.0;
    rab += dx[k] * dx[k];
  }
  rab = sqrt(rab);
  if (rab < ToleranceSame) {
    fprintf(stderr, "Atoms %d and %d coincide (r = %g)\n", i, j, rab);
    exit(EXIT_FAILURE);
  }
  for (k = 0, r = 0; k < DIMENSION; k++) {
    plane->normal[k] = dx[k] / rab;
    r += midpoint[k] * plane->normal[k];
  }
  if (r < 0) { /* Reverce normal direction, distance is always positive! */
    r = -r;
    for (k = 0; k < DIMENSION; k++) {
      plane->normal[k] = -plane->normal[k];
    }
  }
  plane->distance = r;
  if (verbose > 0)
    printf("    initial plane is at %g from the origin\n", r);
  if (refine_symmetry_element(plane, 1) < 0) {
    if (verbose > 0)
      printf("    refinement failed for the plane\n");
    destroy_symmetry_element(plane);
    return NULL;
  }
  return plane;
}
 
SYMMETRY_ELEMENT *init_ultimate_plane(void) {
  SYMMETRY_ELEMENT *plane = alloc_symmetry_element();
  double d0[DIMENSION], d1[DIMENSION], d2[DIMENSION];
  double p[DIMENSION];
  double r, s0, s1, s2;
  double *d;
  int i, j, k;
 
  if (verbose > 0)
    printf("Trying whole-molecule mirror plane\n");
  StatTotal++;
  plane->transform_atom = mirror_atom;
  plane->order = 1;
  plane->nparam = 4;
  for (k = 0; k < DIMENSION; k++)
    d0[k] = d1[k] = d2[k] = 0;
  d0[0] = 1;
  d1[1] = 1;
  d2[2] = 1;
  for (i = 1; i < AtomsCount; i++) {
    for (j = 0; j < i; j++) {
      for (k = 0, r = 0; k < DIMENSION; k++) {
        p[k] = Atoms[i].x[k] - Atoms[j].x[k];
        r += p[k] * p[k];
      }
      r = sqrt(r);
      for (k = 0, s0 = s1 = s2 = 0; k < DIMENSION; k++) {
        p[k] /= r;
        s0 += p[k] * d0[k];
        s1 += p[k] * d1[k];
        s2 += p[k] * d2[k];
      }
      for (k = 0; k < DIMENSION; k++) {
        d0[k] -= s0 * p[k];
        d1[k] -= s1 * p[k];
        d2[k] -= s2 * p[k];
      }
    }
  }
  for (k = 0, s0 = s1 = s2 = 0; k < DIMENSION; k++) {
    s0 += d0[k];
    s1 += d1[k];
    s2 += d2[k];
  }
  d = NULL;
  if (s0 >= s1 && s0 >= s2)
    d = d0;
  if (s1 >= s0 && s1 >= s2)
    d = d1;
  if (s2 >= s0 && s2 >= s1)
    d = d2;
  if (d == NULL) {
    fprintf(stderr,
            "Catastrophe in init_ultimate_plane(): %g, %g and %g have no "
            "ordering!\n",
            s0, s1, s2);
    exit(EXIT_FAILURE);
  }
  for (k = 0, r = 0; k < DIMENSION; k++)
    r += d[k] * d[k];
  r = sqrt(r);
  if (r > 0) {
    for (k = 0; k < DIMENSION; k++)
      plane->normal[k] = d[k] / r;
  } else {
    for (k = 1; k < DIMENSION; k++)
      plane->normal[k] = 0;
    plane->normal[0] = 1;
  }
  for (k = 0, r = 0; k < DIMENSION; k++)
    r += CenterOfSomething[k] * plane->normal[k];
  plane->distance = r;
  for (k = 0; k < AtomsCount; k++)
    plane->transform[k] = k;
  if (refine_symmetry_element(plane, 0) < 0) {
    if (verbose > 0)
      printf("    refinement failed for the plane\n");
    destroy_symmetry_element(plane);
    return NULL;
  }
  return plane;
}
/*
 *   Inversion-center specific functions
 */
void invert_atom(SYMMETRY_ELEMENT *center, ATOM *from, ATOM *to) {
  int i;
 
  to->type = from->type;
  for (i = 0; i < DIMENSION; i++) {
    to->x[i] = 2 * center->distance * center->normal[i] - from->x[i];
  }
}
 
SYMMETRY_ELEMENT *init_inversion_center(void) {
  SYMMETRY_ELEMENT *center = alloc_symmetry_element();
  int k;
  double r;
 
  if (verbose > 0)
    printf("Trying inversion center at the center of something\n");
  StatTotal++;
  center->transform_atom = invert_atom;
  center->order = 2;
  center->nparam = 4;
  for (k = 0, r = 0; k < DIMENSION; k++)
    r += CenterOfSomething[k] * CenterOfSomething[k];
  r = sqrt(r);
  if (r > 0) {
    for (k = 0; k < DIMENSION; k++)
      center->normal[k] = CenterOfSomething[k] / r;
  } else {
    center->normal[0] = 1;
    for (k = 1; k < DIMENSION; k++)
      center->normal[k] = 0;
  }
  center->distance = r;
  if (verbose > 0)
    printf("    initial inversion center is at %g from the origin\n", r);
  if (refine_symmetry_element(center, 1) < 0) {
    if (verbose > 0)
      printf("    refinement failed for the inversion center\n");
    destroy_symmetry_element(center);
    return NULL;
  }
  return center;
}
 
/*
 *   Normal rotation axis-specific routines.
 */
void rotate_atom(SYMMETRY_ELEMENT *axis, ATOM *from, ATOM *to) {
  double x[3], y[3], a[3], b[3], c[3];
  double angle = axis->order ? 2 * M_PI / axis->order : 1.0;
  double a_sin = sin(angle);
  double a_cos = cos(angle);
  double dot;
  int i;
 
  if (DIMENSION != 3) {
    fprintf(stderr, "Catastrophe in rotate_atom!\n");
    exit(EXIT_FAILURE);
  }
  for (i = 0; i < 3; i++)
    x[i] = from->x[i] - axis->distance * axis->normal[i];
  for (i = 0, dot = 0; i < 3; i++)
    dot += x[i] * axis->direction[i];
  for (i = 0; i < 3; i++)
    a[i] = axis->direction[i] * dot;
  for (i = 0; i < 3; i++)
    b[i] = x[i] - a[i];
  c[0] = b[1] * axis->direction[2] - b[2] * axis->direction[1];
  c[1] = b[2] * axis->direction[0] - b[0] * axis->direction[2];
  c[2] = b[0] * axis->direction[1] - b[1] * axis->direction[0];
  for (i = 0; i < 3; i++)
    y[i] = a[i] + b[i] * a_cos + c[i] * a_sin;
  for (i = 0; i < 3; i++)
    to->x[i] = y[i] + axis->distance * axis->normal[i];
  to->type = from->type;
}
 
SYMMETRY_ELEMENT *init_ultimate_axis(void) {
  SYMMETRY_ELEMENT *axis = alloc_symmetry_element();
  double dir[DIMENSION], rel[DIMENSION];
  double s;
  int i, k;
 
  if (verbose > 0)
    printf("Trying infinity axis\n");
  StatTotal++;
  axis->transform_atom = rotate_atom;
  axis->order = 0;
  axis->nparam = 7;
  for (k = 0; k < DIMENSION; k++)
    dir[k] = 0;
  for (i = 0; i < AtomsCount; i++) {
    for (k = 0, s = 0; k < DIMENSION; k++) {
      rel[k] = Atoms[i].x[k] - CenterOfSomething[k];
      s += rel[k] * dir[k];
    }
    if (s >= 0)
      for (k = 0; k < DIMENSION; k++)
        dir[k] += rel[k];
    else
      for (k = 0; k < DIMENSION; k++)
        dir[k] -= rel[k];
  }
  for (k = 0, s = 0; k < DIMENSION; k++)
    s += pow2(dir[k]);
  s = sqrt(s);
  if (s > 0)
    for (k = 0; k < DIMENSION; k++)
      dir[k] /= s;
  else
    dir[0] = 1;
  for (k = 0; k < DIMENSION; k++)
    axis->direction[k] = dir[k];
  for (k = 0, s = 0; k < DIMENSION; k++)
    s += pow2(CenterOfSomething[k]);
  s = sqrt(s);
  if (s > 0)
    for (k = 0; k < DIMENSION; k++)
      axis->normal[k] = CenterOfSomething[k] / s;
  else {
    for (k = 1; k < DIMENSION; k++)
      axis->normal[k] = 0;
    axis->normal[0] = 1;
  }
  axis->distance = s;
  for (k = 0; k < AtomsCount; k++)
    axis->transform[k] = k;
  if (refine_symmetry_element(axis, 0) < 0) {
    if (verbose > 0)
      printf("    refinement failed for the infinity axis\n");
    destroy_symmetry_element(axis);
    return NULL;
  }
  return axis;
}
 
SYMMETRY_ELEMENT *init_c2_axis(int i, int j, double support[DIMENSION]) {
  SYMMETRY_ELEMENT *axis;
  int k;
  double ris, rjs;
  double r, center[DIMENSION];
 
  if (verbose > 0)
    printf("Trying c2 axis for the pair (%d,%d) with the support (%g,%g,%g)\n",
           i, j, support[0], support[1], support[2]);
  StatTotal++;
  /* First, do a quick sanity check */
  for (k = 0, ris = rjs = 0; k < DIMENSION; k++) {
    ris += pow2(Atoms[i].x[k] - support[k]);
    rjs += pow2(Atoms[j].x[k] - support[k]);
  }
  ris = sqrt(ris);
  rjs = sqrt(rjs);
  if (fabs(ris - rjs) > TolerancePrimary) {
    StatEarly++;
    if (verbose > 0)
      printf("    Support can't actually define a rotation axis\n");
    return NULL;
  }
  axis = alloc_symmetry_element();
  axis->transform_atom = rotate_atom;
  axis->order = 2;
  axis->nparam = 7;
  for (k = 0, r = 0; k < DIMENSION; k++)
    r += CenterOfSomething[k] * CenterOfSomething[k];
  r = sqrt(r);
  if (r > 0) {
    for (k = 0; k < DIMENSION; k++)
      axis->normal[k] = CenterOfSomething[k] / r;
  } else {
    axis->normal[0] = 1;
    for (k = 1; k < DIMENSION; k++)
      axis->normal[k] = 0;
  }
  axis->distance = r;
  for (k = 0, r = 0; k < DIMENSION; k++) {
    center[k] = (Atoms[i].x[k] + Atoms[j].x[k]) / 2 - support[k];
    r += center[k] * center[k];
  }
  r = sqrt(r);
  if (r <=
      TolerancePrimary) { /* c2 is underdefined, let's do something special */
    if (MolecularPlane != NULL) {
      if (verbose > 0)
        printf("    c2 is underdefined, but there is a molecular plane\n");
      for (k = 0; k < DIMENSION; k++)
        axis->direction[k] = MolecularPlane->normal[k];
    } else {
      if (verbose > 0)
        printf("    c2 is underdefined, trying random direction\n");
      for (k = 0; k < DIMENSION; k++)
        center[k] = Atoms[i].x[k] - Atoms[j].x[k];
      if (fabs(center[2]) + fabs(center[1]) > ToleranceSame) {
        axis->direction[0] = 0;
        axis->direction[1] = center[2];
        axis->direction[2] = -center[1];
      } else {
        axis->direction[0] = -center[2];
        axis->direction[1] = 0;
        axis->direction[2] = center[0];
      }
      for (k = 0, r = 0; k < DIMENSION; k++)
        r += axis->direction[k] * axis->direction[k];
      r = sqrt(r);
      for (k = 0; k < DIMENSION; k++)
        axis->direction[k] /= r;
    }
  } else { /* direction is Ok, renormalize it */
    for (k = 0; k < DIMENSION; k++)
      axis->direction[k] = center[k] / r;
  }
  if (refine_symmetry_element(axis, 1) < 0) {
    if (verbose > 0)
      printf("    refinement failed for the c2 axis\n");
    destroy_symmetry_element(axis);
    return NULL;
  }
  return axis;
}
 
SYMMETRY_ELEMENT *init_axis_parameters(double a[3], double b[3], double c[3]) {
  SYMMETRY_ELEMENT *axis;
  int i, order, sign;
  double ra, rb, rc, rab, rbc, rac, r;
  double angle;
 
  ra = rb = rc = rab = rbc = rac = 0;
  for (i = 0; i < DIMENSION; i++) {
    ra += a[i] * a[i];
    rb += b[i] * b[i];
    rc += c[i] * c[i];
  }
  ra = sqrt(ra);
  rb = sqrt(rb);
  rc = sqrt(rc);
  if (fabs(ra - rb) > TolerancePrimary || fabs(ra - rc) > TolerancePrimary ||
      fabs(rb - rc) > TolerancePrimary) {
    StatEarly++;
    if (verbose > 0)
      printf("    points are not on a sphere\n");
    return NULL;
  }
  for (i = 0; i < DIMENSION; i++) {
    rab += (a[i] - b[i]) * (a[i] - b[i]);
    rac += (a[i] - c[i]) * (a[i] - c[i]);
    rbc += (c[i] - b[i]) * (c[i] - b[i]);
  }
  rab = sqrt(rab);
  rac = sqrt(rac);
  rbc = sqrt(rbc);
  if (fabs(rab - rbc) > TolerancePrimary) {
    StatEarly++;
    if (verbose > 0)
      printf("    points can't be rotation-equivalent\n");
    return NULL;
  }
  if (rab <= ToleranceSame || rbc <= ToleranceSame || rac <= ToleranceSame) {
    StatEarly++;
    if (verbose > 0)
      printf("    rotation is underdefined by these points\n");
    return NULL;
  }
  rab = (rab + rbc) / 2;
  angle = M_PI - 2 * asin(rac / (2 * rab));
  if (verbose > 1)
    printf("    rotation angle is %f\n", angle);
  if (fabs(angle) <= M_PI / (MaxAxisOrder + 1)) {
    StatEarly++;
    if (verbose > 0)
      printf("    atoms are too close to a straight line\n");
    return NULL;
  }
  order = floor((2 * M_PI) / angle + 0.5);
  if (order <= 2 || order > MaxAxisOrder) {
    StatEarly++;
    if (verbose > 0)
      printf("    rotation axis order (%d) is not from 3 to %d\n", order,
             MaxAxisOrder);
    return NULL;
  }
  axis = alloc_symmetry_element();
  axis->order = order;
  axis->nparam = 7;
  for (i = 0, r = 0; i < DIMENSION; i++)
    r += CenterOfSomething[i] * CenterOfSomething[i];
  r = sqrt(r);
  if (r > 0) {
    for (i = 0; i < DIMENSION; i++)
      axis->normal[i] = CenterOfSomething[i] / r;
  } else {
    axis->normal[0] = 1;
    for (i = 1; i < DIMENSION; i++)
      axis->normal[i] = 0;
  }
  axis->distance = r;
  axis->direction[0] =
      (b[1] - a[1]) * (c[2] - b[2]) - (b[2] - a[2]) * (c[1] - b[1]);
  axis->direction[1] =
      (b[2] - a[2]) * (c[0] - b[0]) - (b[0] - a[0]) * (c[2] - b[2]);
  axis->direction[2] =
      (b[0] - a[0]) * (c[1] - b[1]) - (b[1] - a[1]) * (c[0] - b[0]);
  /*
   *  Arbitrarily select axis direction so that first non-zero component
   *  of the direction is positive.
   */
  sign = 0;
  if (axis->direction[0] <= 0) {
    if (axis->direction[0] < 0) {
      sign = 1;
    } else if (axis->direction[1] <= 0) {
      if (axis->direction[1] < 0) {
        sign = 1;
      } else if (axis->direction[2] < 0) {
        sign = 1;
      }
    }
  }
  if (sign)
    for (i = 0; i < DIMENSION; i++)
      axis->direction[i] = -axis->direction[i];
  for (i = 0, r = 0; i < DIMENSION; i++)
    r += axis->direction[i] * axis->direction[i];
  r = sqrt(r);
  for (i = 0; i < DIMENSION; i++)
    axis->direction[i] /= r;
  if (verbose > 1) {
    printf("    axis origin is at (%g,%g,%g)\n",
           axis->normal[0] * axis->distance, axis->normal[1] * axis->distance,
           axis->normal[2] * axis->distance);
    printf("    axis is in the direction (%g,%g,%g)\n", axis->direction[0],
           axis->direction[1], axis->direction[2]);
  }
  return axis;
}
 
SYMMETRY_ELEMENT *init_higher_axis(int ia, int ib, int ic) {
  SYMMETRY_ELEMENT *axis;
  double a[DIMENSION], b[DIMENSION], c[DIMENSION];
  int i;
 
  if (verbose > 0)
    printf("Trying cn axis for the triplet (%d,%d,%d)\n", ia, ib, ic);
  StatTotal++;
  /* Do a quick check of geometry validity */
  for (i = 0; i < DIMENSION; i++) {
    a[i] = Atoms[ia].x[i] - CenterOfSomething[i];
    b[i] = Atoms[ib].x[i] - CenterOfSomething[i];
    c[i] = Atoms[ic].x[i] - CenterOfSomething[i];
  }
  if ((axis = init_axis_parameters(a, b, c)) == NULL) {
    if (verbose > 0)
      printf("    no coherent axis is defined by the points\n");
    return NULL;
  }
  axis->transform_atom = rotate_atom;
  if (refine_symmetry_element(axis, 1) < 0) {
    if (verbose > 0)
      printf("    refinement failed for the c%d axis\n", axis->order);
    destroy_symmetry_element(axis);
    return NULL;
  }
  return axis;
}
 
/*
 *   Improper axes-specific routines.
 *   These are obtained by slight modifications of normal rotation
 *       routines.
 */
void rotate_reflect_atom(SYMMETRY_ELEMENT *axis, ATOM *from, ATOM *to) {
  double x[3], y[3], a[3], b[3], c[3];
  double angle = 2 * M_PI / axis->order;
  double a_sin = sin(angle);
  double a_cos = cos(angle);
  double dot;
  int i;
 
  if (DIMENSION != 3) {
    fprintf(stderr, "Catastrophe in rotate_reflect_atom!\n");
    exit(EXIT_FAILURE);
  }
  for (i = 0; i < 3; i++)
    x[i] = from->x[i] - axis->distance * axis->normal[i];
  for (i = 0, dot = 0; i < 3; i++)
    dot += x[i] * axis->direction[i];
  for (i = 0; i < 3; i++)
    a[i] = axis->direction[i] * dot;
  for (i = 0; i < 3; i++)
    b[i] = x[i] - a[i];
  c[0] = b[1] * axis->direction[2] - b[2] * axis->direction[1];
  c[1] = b[2] * axis->direction[0] - b[0] * axis->direction[2];
  c[2] = b[0] * axis->direction[1] - b[1] * axis->direction[0];
  for (i = 0; i < 3; i++)
    y[i] = -a[i] + b[i] * a_cos + c[i] * a_sin;
  for (i = 0; i < 3; i++)
    to->x[i] = y[i] + axis->distance * axis->normal[i];
  to->type = from->type;
}
 
SYMMETRY_ELEMENT *init_improper_axis(int ia, int ib, int ic) {
  SYMMETRY_ELEMENT *axis;
  double a[DIMENSION], b[DIMENSION], c[DIMENSION];
  double centerpoint[DIMENSION];
  double r;
  int i;
 
  if (verbose > 0)
    printf("Trying sn axis for the triplet (%d,%d,%d)\n", ia, ib, ic);
  StatTotal++;
  /* First, reduce the problem to Cn case */
  for (i = 0; i < DIMENSION; i++) {
    a[i] = Atoms[ia].x[i] - CenterOfSomething[i];
    b[i] = Atoms[ib].x[i] - CenterOfSomething[i];
    c[i] = Atoms[ic].x[i] - CenterOfSomething[i];
  }
  for (i = 0, r = 0; i < DIMENSION; i++) {
    centerpoint[i] = a[i] + c[i] + 2 * b[i];
    r += centerpoint[i] * centerpoint[i];
  }
  r = sqrt(r);
  if (r <= ToleranceSame) {
    StatEarly++;
    if (verbose > 0)
      printf(
          "    atoms can not define improper axis of the order more than 2\n");
    return NULL;
  }
  for (i = 0; i < DIMENSION; i++)
    centerpoint[i] /= r;
  for (i = 0, r = 0; i < DIMENSION; i++)
    r += centerpoint[i] * b[i];
  for (i = 0; i < DIMENSION; i++)
    b[i] = 2 * r * centerpoint[i] - b[i];
  /* Do a quick check of geometry validity */
  if ((axis = init_axis_parameters(a, b, c)) == NULL) {
    if (verbose > 0)
      printf("    no coherent improper axis is defined by the points\n");
    return NULL;
  }
  axis->transform_atom = rotate_reflect_atom;
  if (refine_symmetry_element(axis, 1) < 0) {
    if (verbose > 0)
      printf("    refinement failed for the s%d axis\n", axis->order);
    destroy_symmetry_element(axis);
    return NULL;
  }
  return axis;
}
 
/*
 *   Control routines
 */
 
void find_center_of_something(void) {
  int i, j;
  double coord_sum[DIMENSION];
  double r;
 
  for (j = 0; j < DIMENSION; j++)
    coord_sum[j] = 0;
  for (i = 0; i < AtomsCount; i++) {
    for (j = 0; j < DIMENSION; j++)
      coord_sum[j] += Atoms[i].x[j];
  }
  for (j = 0; j < DIMENSION; j++)
    CenterOfSomething[j] = coord_sum[j] / AtomsCount;
  if (verbose > 0)
    printf("Center of something is at %15.10f, %15.10f, %15.10f\n",
           CenterOfSomething[0], CenterOfSomething[1], CenterOfSomething[2]);
  DistanceFromCenter = (double *)calloc(AtomsCount, sizeof(double));
  if (DistanceFromCenter == NULL) {
    fprintf(stderr, "Unable to allocate array for the distances\n");
    exit(EXIT_FAILURE);
  }
  for (i = 0; i < AtomsCount; i++) {
    for (j = 0, r = 0; j < DIMENSION; j++)
      r += pow2(Atoms[i].x[j] - CenterOfSomething[j]);
    DistanceFromCenter[i] = r;
  }
}
 
void find_planes(void) {
  int i, j;
  SYMMETRY_ELEMENT *plane = NULL;
 
  plane = init_ultimate_plane();
  if (plane != NULL) {
    MolecularPlane = plane;
    PlanesCount++;
    Planes = (SYMMETRY_ELEMENT **)realloc(Planes, sizeof(SYMMETRY_ELEMENT *) *
                                                      PlanesCount);
    if (Planes == NULL) {
      perror("Out of memory in find_planes");
      exit(EXIT_FAILURE);
    }
    Planes[PlanesCount - 1] = plane;
  }
  for (i = 1; i < AtomsCount; i++) {
    for (j = 0; j < i; j++) {
      if (Atoms[i].type != Atoms[j].type)
        continue;
      if ((plane = init_mirror_plane(i, j)) != NULL) {
        PlanesCount++;
        Planes = (SYMMETRY_ELEMENT **)realloc(
            Planes, sizeof(SYMMETRY_ELEMENT *) * PlanesCount);
        if (Planes == NULL) {
          perror("Out of memory in find_planes");
          exit(EXIT_FAILURE);
        }
        Planes[PlanesCount - 1] = plane;
      }
    }
  }
}
 
void destroy_planes(void) {
  int i;
  for (i = 0; i < PlanesCount; i++)
    destroy_symmetry_element(Planes[i]);
  PlanesCount = 0;
  free(Planes);
  MolecularPlane = NULL;
  Planes = NULL;
}
 
void find_inversion_centers(void) {
  SYMMETRY_ELEMENT *center;
 
  if ((center = init_inversion_center()) != NULL) {
    InversionCenters =
        (SYMMETRY_ELEMENT **)calloc(1, sizeof(SYMMETRY_ELEMENT *));
    InversionCenters[0] = center;
    InversionCentersCount = 1;
  }
}
 
void destroy_inversion_centers(void) {
  int i;
  for (i = 0; i < InversionCentersCount; i++)
    destroy_symmetry_element(InversionCenters[i]);
  InversionCentersCount = 0;
  free(InversionCenters);
  InversionCenters = NULL;
}
 
void find_infinity_axis(void) {
  SYMMETRY_ELEMENT *axis;
 
  if ((axis = init_ultimate_axis()) != NULL) {
    NormalAxesCount++;
    NormalAxes = (SYMMETRY_ELEMENT **)realloc(
        NormalAxes, sizeof(SYMMETRY_ELEMENT *) * NormalAxesCount);
    if (NormalAxes == NULL) {
      perror("Out of memory in find_infinity_axes()");
      exit(EXIT_FAILURE);
    }
    NormalAxes[NormalAxesCount - 1] = axis;
  }
}
 
void find_c2_axes(void) {
  int i, j, k, l, m;
  double center[DIMENSION];
  double *distances = calloc(AtomsCount, sizeof(double));
  double r;
  SYMMETRY_ELEMENT *axis;
 
  if (distances == NULL) {
    fprintf(stderr, "Out of memory in find_c2_axes()\n");
    exit(EXIT_FAILURE);
  }
  for (i = 1; i < AtomsCount; i++) {
    for (j = 0; j < i; j++) {
      if (Atoms[i].type != Atoms[j].type)
        continue;
      if (fabs(DistanceFromCenter[i] - DistanceFromCenter[j]) >
          TolerancePrimary)
        continue; /* A very cheap, but quite effective check */
      /*
       *   First, let's try to get it cheap and use CenterOfSomething
       */
      for (k = 0, r = 0; k < DIMENSION; k++) {
        center[k] = (Atoms[i].x[k] + Atoms[j].x[k]) / 2;
        r += pow2(center[k] - CenterOfSomething[k]);
      }
      r = sqrt(r);
      if (r > 5 * TolerancePrimary) { /* It's Ok to use CenterOfSomething */
        if ((axis = init_c2_axis(i, j, CenterOfSomething)) != NULL) {
          NormalAxesCount++;
          NormalAxes = (SYMMETRY_ELEMENT **)realloc(
              NormalAxes, sizeof(SYMMETRY_ELEMENT *) * NormalAxesCount);
          if (NormalAxes == NULL) {
            perror("Out of memory in find_c2_axes");
            exit(EXIT_FAILURE);
          }
          NormalAxes[NormalAxesCount - 1] = axis;
        }
        continue;
      }
      /*
       *  Now, C2 axis can either pass through an atom, or through the
       *  middle of the other pair.
       */
      for (k = 0; k < AtomsCount; k++) {
        if ((axis = init_c2_axis(i, j, Atoms[k].x)) != NULL) {
          NormalAxesCount++;
          NormalAxes = (SYMMETRY_ELEMENT **)realloc(
              NormalAxes, sizeof(SYMMETRY_ELEMENT *) * NormalAxesCount);
          if (NormalAxes == NULL) {
            perror("Out of memory in find_c2_axes");
            exit(EXIT_FAILURE);
          }
          NormalAxes[NormalAxesCount - 1] = axis;
        }
      }
      /*
       *  Prepare data for an additional pre-screening check
       */
      for (k = 0; k < AtomsCount; k++) {
        for (l = 0, r = 0; l < DIMENSION; l++)
          r += pow2(Atoms[k].x[l] - center[l]);
        distances[k] = sqrt(r);
      }
      for (k = 0; k < AtomsCount; k++) {
        for (l = 0; l < AtomsCount; l++) {
          if (Atoms[k].type != Atoms[l].type)
            continue;
          if (fabs(DistanceFromCenter[k] - DistanceFromCenter[l]) >
                  TolerancePrimary ||
              fabs(distances[k] - distances[l]) > TolerancePrimary)
            continue; /* We really need this one to run reasonably fast! */
          for (m = 0; m < DIMENSION; m++)
            center[m] = (Atoms[k].x[m] + Atoms[l].x[m]) / 2;
          if ((axis = init_c2_axis(i, j, center)) != NULL) {
            NormalAxesCount++;
            NormalAxes = (SYMMETRY_ELEMENT **)realloc(
                NormalAxes, sizeof(SYMMETRY_ELEMENT *) * NormalAxesCount);
            if (NormalAxes == NULL) {
              perror("Out of memory in find_c2_axes");
              exit(EXIT_FAILURE);
            }
            NormalAxes[NormalAxesCount - 1] = axis;
          }
        }
      }
    }
  }
  free(distances);
}
 
void destroy_normal_axes() {
  int i = 0;
  for (i = 0; i < NormalAxesCount; i++)
    destroy_symmetry_element(NormalAxes[i]);
  NormalAxesCount = 0;
  free(NormalAxes);
  NormalAxes = NULL;
}
 
void destroy_improper_axes() {
  int i = 0;
  for (i = 0; i < ImproperAxesCount; i++)
    destroy_symmetry_element(ImproperAxes[i]);
  ImproperAxesCount = 0;
  free(ImproperAxes);
  ImproperAxes = NULL;
}
 
void find_higher_axes(void) {
  int i, j, k;
  SYMMETRY_ELEMENT *axis;
 
  for (i = 0; i < AtomsCount; i++) {
    for (j = i + 1; j < AtomsCount; j++) {
      if (Atoms[i].type != Atoms[j].type)
        continue;
      if (fabs(DistanceFromCenter[i] - DistanceFromCenter[j]) >
          TolerancePrimary)
        continue; /* A very cheap, but quite effective check */
      for (k = 0; k < AtomsCount; k++) {
        if (Atoms[i].type != Atoms[k].type)
          continue;
        if ((fabs(DistanceFromCenter[i] - DistanceFromCenter[k]) >
             TolerancePrimary) ||
            (fabs(DistanceFromCenter[j] - DistanceFromCenter[k]) >
             TolerancePrimary))
          continue;
        if ((axis = init_higher_axis(i, j, k)) != NULL) {
          NormalAxesCount++;
          NormalAxes = (SYMMETRY_ELEMENT **)realloc(
              NormalAxes, sizeof(SYMMETRY_ELEMENT *) * NormalAxesCount);
          if (NormalAxes == NULL) {
            perror("Out of memory in find_higher_axes");
            exit(EXIT_FAILURE);
          }
          NormalAxes[NormalAxesCount - 1] = axis;
        }
      }
    }
  }
}
 
void find_improper_axes(void) {
  int i, j, k;
  SYMMETRY_ELEMENT *axis;
 
  for (i = 0; i < AtomsCount; i++) {
    for (j = i + 1; j < AtomsCount; j++) {
      for (k = 0; k < AtomsCount; k++) {
        if ((axis = init_improper_axis(i, j, k)) != NULL) {
          ImproperAxesCount++;
          ImproperAxes = (SYMMETRY_ELEMENT **)realloc(
              ImproperAxes, sizeof(SYMMETRY_ELEMENT *) * ImproperAxesCount);
          if (ImproperAxes == NULL) {
            perror("Out of memory in find_higher_axes");
            exit(EXIT_FAILURE);
          }
          ImproperAxes[ImproperAxesCount - 1] = axis;
        }
      }
    }
  }
}
 
void report_planes(void) {
  int i;
 
  if (PlanesCount == 0)
    printf("There are no planes of symmetry in the molecule\n");
  else {
    if (PlanesCount == 1)
      printf("There is a plane of symmetry in the molecule\n");
    else
      printf("There are %d planes of symmetry in the molecule\n", PlanesCount);
    printf(
        "     Residual          Direction of the normal           Distance\n");
    for (i = 0; i < PlanesCount; i++) {
      printf("%3d %8.4e ", i, Planes[i]->maxdev);
      printf("(%11.8f,%11.8f,%11.8f) ", Planes[i]->normal[0],
             Planes[i]->normal[1], Planes[i]->normal[2]);
      printf("%14.8f\n", Planes[i]->distance);
    }
  }
}
 
void report_inversion_centers(void) {
  if (InversionCentersCount == 0)
    printf("There is no inversion center in the molecule\n");
  else {
    printf("There in an inversion center in the molecule\n");
    printf("     Residual                      Position\n");
    printf("   %8.4e ", InversionCenters[0]->maxdev);
    printf("(%14.8f,%14.8f,%14.8f)\n",
           InversionCenters[0]->distance * InversionCenters[0]->normal[0],
           InversionCenters[0]->distance * InversionCenters[0]->normal[1],
           InversionCenters[0]->distance * InversionCenters[0]->normal[2]);
  }
}
 
void report_axes(void) {
  int i;
 
  if (NormalAxesCount == 0)
    printf("There are no normal axes in the molecule\n");
  else {
    if (NormalAxesCount == 1)
      printf("There is a normal axis in the molecule\n");
    else
      printf("There are %d normal axes in the molecule\n", NormalAxesCount);
    printf("     Residual  Order         Direction of the axis                 "
           "        Supporting point\n");
    for (i = 0; i < NormalAxesCount; i++) {
      printf("%3d %8.4e ", i, NormalAxes[i]->maxdev);
      if (NormalAxes[i]->order == 0)
        printf("Inf ");
      else
        printf("%3d ", NormalAxes[i]->order);
      printf("(%11.8f,%11.8f,%11.8f) ", NormalAxes[i]->direction[0],
             NormalAxes[i]->direction[1], NormalAxes[i]->direction[2]);
      printf("(%14.8f,%14.8f,%14.8f)\n",
             NormalAxes[0]->distance * NormalAxes[0]->normal[0],
             NormalAxes[0]->distance * NormalAxes[0]->normal[1],
             NormalAxes[0]->distance * NormalAxes[0]->normal[2]);
    }
  }
}
 
void report_improper_axes(void) {
  int i;
 
  if (ImproperAxesCount == 0)
    printf("There are no improper axes in the molecule\n");
  else {
    if (ImproperAxesCount == 1)
      printf("There is an improper axis in the molecule\n");
    else
      printf("There are %d improper axes in the molecule\n", ImproperAxesCount);
    printf("     Residual  Order         Direction of the axis                 "
           "        Supporting point\n");
    for (i = 0; i < ImproperAxesCount; i++) {
      printf("%3d %8.4e ", i, ImproperAxes[i]->maxdev);
      if (ImproperAxes[i]->order == 0)
        printf("Inf ");
      else
        printf("%3d ", ImproperAxes[i]->order);
      printf("(%11.8f,%11.8f,%11.8f) ", ImproperAxes[i]->direction[0],
             ImproperAxes[i]->direction[1], ImproperAxes[i]->direction[2]);
      printf("(%14.8f,%14.8f,%14.8f)\n",
             ImproperAxes[0]->distance * ImproperAxes[0]->normal[0],
             ImproperAxes[0]->distance * ImproperAxes[0]->normal[1],
             ImproperAxes[0]->distance * ImproperAxes[0]->normal[2]);
    }
  }
}
 
/*
 *  General symmetry handling
 */
void report_and_reset_counters(void) {
  printf("  %10ld candidates examined\n"
         "  %10ld removed early\n"
         "  %10ld removed during initial mating stage\n"
         "  %10ld removed as duplicates\n"
         "  %10ld removed because of the wrong transformation order\n"
         "  %10ld removed after unsuccessful optimization\n"
         "  %10ld accepted\n",
         StatTotal, StatEarly, StatPairs, StatDups, StatOrder, StatOpt,
         StatAccept);
  StatTotal = StatEarly = StatPairs = StatDups = StatOrder = StatOpt =
      StatAccept = 0;
}
 
void find_symmetry_elements(void) {
  find_center_of_something();
  if (verbose > -1) {
    printf("Looking for the inversion center\n");
  }
  find_inversion_centers();
  if (verbose > -1) {
    report_and_reset_counters();
    printf("Looking for the planes of symmetry\n");
  }
  find_planes();
  if (verbose > -1) {
    report_and_reset_counters();
    printf("Looking for infinity axis\n");
  }
  find_infinity_axis();
  if (verbose > -1) {
    report_and_reset_counters();
    printf("Looking for C2 axes\n");
  }
  find_c2_axes();
  if (verbose > -1) {
    report_and_reset_counters();
    printf("Looking for higher axes\n");
  }
  find_higher_axes();
  if (verbose > -1) {
    report_and_reset_counters();
    printf("Looking for the improper axes\n");
  }
  find_improper_axes();
  if (verbose > -1) {
    report_and_reset_counters();
  }
}
 
int compare_axes(const void *a, const void *b) {
  SYMMETRY_ELEMENT *axis_a = *(SYMMETRY_ELEMENT **)a;
  SYMMETRY_ELEMENT *axis_b = *(SYMMETRY_ELEMENT **)b;
  int i, order_a, order_b;
 
  order_a = axis_a->order;
  if (order_a == 0)
    order_a = 10000;
  order_b = axis_b->order;
  if (order_b == 0)
    order_b = 10000;
  if ((i = order_b - order_a) != 0)
    return i;
  if (axis_a->maxdev > axis_b->maxdev)
    return -1;
  if (axis_a->maxdev < axis_b->maxdev)
    return 1;
  return 0;
}
 
void sort_symmetry_elements(void) {
  if (PlanesCount > 1) {
    qsort(Planes, PlanesCount, sizeof(SYMMETRY_ELEMENT *), compare_axes);
  }
  if (NormalAxesCount > 1) {
    qsort(NormalAxes, NormalAxesCount, sizeof(SYMMETRY_ELEMENT *),
          compare_axes);
  }
  if (ImproperAxesCount > 1) {
    qsort(ImproperAxes, ImproperAxesCount, sizeof(SYMMETRY_ELEMENT *),
          compare_axes);
  }
}
 
void report_symmetry_elements_verbose(void) {
  report_inversion_centers();
  report_axes();
  report_improper_axes();
  report_planes();
}
 
void summarize_symmetry_elements(void) {
  int i;
 
  NormalAxesCounts = (int *)calloc(MaxAxisOrder + 1, sizeof(int));
  ImproperAxesCounts = (int *)calloc(MaxAxisOrder + 1, sizeof(int));
  for (i = 0; i < NormalAxesCount; i++)
    NormalAxesCounts[NormalAxes[i]->order]++;
  for (i = 0; i < ImproperAxesCount; i++)
    ImproperAxesCounts[ImproperAxes[i]->order]++;
}
 
void report_symmetry_elements_brief() {
  int i;
  char *symmetry_code =
      calloc(1, 10 * (PlanesCount + NormalAxesCount + ImproperAxesCount +
                      InversionCentersCount + 2));
  char buf[100];
 
  if (symmetry_code == NULL) {
    fprintf(stderr, "Unable to allocate memory for symmetry ID code in "
                    "report_symmetry_elements_brief()\n");
    exit(EXIT_FAILURE);
  }
  if (PlanesCount + NormalAxesCount + ImproperAxesCount +
          InversionCentersCount ==
      0) {
    if (verbose > -1)
      printf("Molecule has no symmetry elements\n");
  } else {
    if (verbose > -1)
      printf("Molecule has the following symmetry elements: ");
    if (InversionCentersCount > 0)
      strcat(symmetry_code, "(i) ");
    if (NormalAxesCounts[0] == 1)
      strcat(symmetry_code, "(Cinf) ");
    if (NormalAxesCounts[0] > 1) {
      sprintf(buf, "%d*(Cinf) ", NormalAxesCounts[0]);
      strcat(symmetry_code, buf);
    }
    for (i = MaxAxisOrder; i >= 2; i--) {
      if (NormalAxesCounts[i] == 1) {
        sprintf(buf, "(C%d) ", i);
        strcat(symmetry_code, buf);
      }
      if (NormalAxesCounts[i] > 1) {
        sprintf(buf, "%d*(C%d) ", NormalAxesCounts[i], i);
        strcat(symmetry_code, buf);
      }
    }
    for (i = MaxAxisOrder; i >= 2; i--) {
      if (ImproperAxesCounts[i] == 1) {
        sprintf(buf, "(S%d) ", i);
        strcat(symmetry_code, buf);
      }
      if (ImproperAxesCounts[i] > 1) {
        sprintf(buf, "%d*(S%d) ", ImproperAxesCounts[i], i);
        strcat(symmetry_code, buf);
      }
    }
    if (PlanesCount == 1)
      strcat(symmetry_code, "(sigma) ");
    if (PlanesCount > 1) {
      sprintf(buf, "%d*(sigma) ", PlanesCount);
      strcat(symmetry_code, buf);
    }
    if (verbose > -1)
      printf("%s\n", symmetry_code);
  }
  SymmetryCode = symmetry_code;
}
 
void identify_point_group(int *point_group) {
  int i;
  int last_matching = -1;
  int matching_count = 0;
 
  for (i = 0; i < PointGroupsCount; i++) {
    if (strcmp(SymmetryCode, PointGroups[i].symmetry_code) == 0) {
      if (PointGroups[i].check() == 1) {
        last_matching = i;
        matching_count++;
      } else {
        if (verbose > -2) {
          printf("It looks very much like %s, but it is not since %s\n",
                 PointGroups[i].group_name, PointGroupRejectionReason);
        }
      }
    }
  }
  if (matching_count == 0) {
    printf("These symmetry elements match no point group I know of. Sorry.\n");
  }
  if (matching_count > 1) {
    printf("These symmetry elements match more than one group I know of.\n"
           "SOMETHING IS VERY WRONG\n");
    printf("Matching groups are:\n");
    for (i = 0; i < PointGroupsCount; i++) {
      if ((strcmp(SymmetryCode, PointGroups[i].symmetry_code) == 0) &&
          (PointGroups[i].check() == 1)) {
        printf("    %s\n", PointGroups[i].group_name);
      }
    }
  }
  if (verbose > 0 && matching_count == 1) {
    printf("It seems to be the %s point group\n",
           PointGroups[last_matching].group_name);
  }
 
  *point_group = last_matching;
}
 
/*
 *  Input/Output
 */
 
void FC_FUNC_(symmetries_finite_init,
              SYMMETRIES_FINITE_INIT)(const int *natoms, const int *types,
                                      const double *positions,
                                      const int *verbosity, int *point_group) {
  int i;
 
  verbose = *verbosity;
  AtomsCount = *natoms;
  Atoms = calloc(AtomsCount, sizeof(ATOM));
  for (i = 0; i < AtomsCount; i++) {
    Atoms[i].type = types[i];
    Atoms[i].x[0] = positions[3 * i + 0];
    Atoms[i].x[1] = positions[3 * i + 1];
    Atoms[i].x[2] = positions[3 * i + 2];
    if (verbose > 5)
      printf("%d %f %f %f\n", Atoms[i].type, Atoms[i].x[0], Atoms[i].x[1],
             Atoms[i].x[2]);
  }
 
  if (AtomsCount > 0)
    find_symmetry_elements();
  sort_symmetry_elements();
 
  summarize_symmetry_elements();
  if (BadOptimization)
    printf("Refinement of some symmetry elements was terminated before "
           "convergence was reached.\n"
           "Some symmetry elements may remain unidentified.\n");
  if (verbose >= 0)
    report_symmetry_elements_verbose();
  report_symmetry_elements_brief();
  identify_point_group(point_group);
}
 
void symmetries_finite_get_group_name(const int *point_group,
                                                char * name) {
 
  strcpy(name, PointGroups[*point_group].group_name);
}
 
void symmetries_finite_get_group_elements(
    const int *point_group, char * elements) {
 
  strcpy(elements, PointGroups[*point_group].symmetry_code);
}
 
void FC_FUNC_(symmetries_finite_end, SYMMETRIES_FINITE_END)() {
  destroy_inversion_centers();
  destroy_planes();
  destroy_normal_axes();
  destroy_improper_axes();
 
  free(SymmetryCode);
  SymmetryCode = NULL;
 
  free(DistanceFromCenter);
  DistanceFromCenter = NULL;
 
  free(NormalAxesCounts);
  NormalAxesCounts = NULL;
 
  free(ImproperAxesCounts);
  ImproperAxesCounts = NULL;
 
  free(Atoms);
  Atoms = NULL;
}