// ******************************************************************************
   //
   // Title:       Force Field X.
   // Description: Force Field X - Software for Molecular Biophysics.
   // Copyright:   Copyright (c) Michael J. Schnieders 2001-2024.
   //
   // This file is part of Force Field X.
   //
   // Force Field X is free software; you can redistribute it and/or modify it
  // under the terms of the GNU General Public License version 3 as published by
  // the Free Software Foundation.
  //
  // Force Field X 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
  // Force Field X; if not, write to the Free Software Foundation, Inc., 59 Temple
  // Place, Suite 330, Boston, MA 02111-1307 USA
  //
  // Linking this library statically or dynamically with other modules is making a
  // combined work based on this library. Thus, the terms and conditions of the
  // GNU General Public License cover the whole combination.
  //
  // As a special exception, the copyright holders of this library give you
  // permission to link this library with independent modules to produce an
  // executable, regardless of the license terms of these independent modules, and
  // to copy and distribute the resulting executable under terms of your choice,
  // provided that you also meet, for each linked independent module, the terms
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  // module which is not derived from or based on this library. If you modify this
  // library, you may extend this exception to your version of the library, but
  // you are not obligated to do so. If you do not wish to do so, delete this
  // exception statement from your version.
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  // ******************************************************************************
  package ffx.potential.utils;
  
  import static ffx.crystal.SymOp.asMatrixString;
  import static ffx.potential.parsers.DistanceMatrixFilter.writeDistanceMatrixRow;
  import static ffx.utilities.StringUtils.writeAtomRanges;
  import static java.lang.String.format;
  import static java.lang.System.arraycopy;
  import static java.util.Arrays.fill;
  import static org.apache.commons.io.FilenameUtils.concat;
  import static org.apache.commons.io.FilenameUtils.getBaseName;
  import static org.apache.commons.io.FilenameUtils.getFullPath;
  import static org.apache.commons.math3.util.FastMath.sqrt;
  
  import edu.rit.mp.DoubleBuf;
  import edu.rit.pj.Comm;
  import ffx.crystal.SymOp;
  import ffx.numerics.math.Double3;
  import ffx.numerics.math.RunningStatistics;
  import ffx.potential.AssemblyState;
  import ffx.potential.ForceFieldEnergy;
  import ffx.potential.MolecularAssembly;
  import ffx.potential.bonded.Atom;
  import ffx.potential.parsers.DistanceMatrixFilter;
  import ffx.potential.parsers.SystemFilter;
  import ffx.potential.parsers.XYZFilter;
  import java.io.File;
  import java.util.ArrayList;
  import java.util.logging.Logger;
  import org.apache.commons.math3.linear.Array2DRowRealMatrix;
  import org.apache.commons.math3.linear.EigenDecomposition;
  
  public class Superpose {
  
    /**
     * Logger for the Superpose Class.
     */
    private static final Logger logger = Logger.getLogger(Superpose.class.getName());
  
    private final SystemFilter baseFilter;
    private final SystemFilter targetFilter;
    private final boolean isSymmetric;
  
    private final int baseSize;
    private final int targetSize;
  
    private int restartRow;
    private int restartColumn;
    private final double[] distRow;
  
    /**
     * Parallel Java world communicator.
     */
    private final Comm world;
    /**
     * Number of processes.
     */
    private final int numProc;
    /**
     * Rank of this process.
     */
    private final int rank;
    /**
    * The distances matrix stores a single RMSD value from each process. The array is of size
    * [numProc][1].
    */
   private final double[][] distances;
   /**
    * Each distance is wrapped inside a DoubleBuf for MPI communication.
    */
   private final DoubleBuf[] buffers;
   /**
    * Convenience reference to the RMSD value for this process.
    */
   private final double[] myDistance;
   /**
    * Convenience reference for the DoubleBuf of this process.
    */
   private final DoubleBuf myBuffer;
 
 
   public Superpose(SystemFilter baseFilter, SystemFilter targetFilter, boolean isSymmetric) {
     this.baseFilter = baseFilter;
     this.targetFilter = targetFilter;
     this.isSymmetric = isSymmetric;
 
     // Number of models to be evaluated.
     baseSize = baseFilter.countNumModels();
     targetSize = targetFilter.countNumModels();
 
     // Initialize the distance matrix row.
     distRow = new double[targetSize];
     fill(distRow, -1.0);
 
     world = Comm.world();
     // Number of processes is equal to world size (often called size).
     numProc = world.size();
     // Each processor gets its own rank (ID of sorts).
     rank = world.rank();
     if (numProc > 1) {
       logger.info(format(" Number of MPI Processes:  %d", numProc));
       logger.info(format(" Rank of this MPI Process: %d", rank));
     }
 
     distances = new double[numProc][1];
 
     // Initialize each distance as -1.0.
     for (int i = 0; i < numProc; i++) {
       fill(distances[i], -1.0);
     }
 
     // DoubleBuf is a wrapper used by Comm to transfer data between processors.
     buffers = new DoubleBuf[numProc];
     for (int i = 0; i < numProc; i++) {
       buffers[i] = DoubleBuf.buffer(distances[i]);
     }
 
     // Reference to each processor's individual task (for convenience).
     myDistance = distances[rank];
     myBuffer = buffers[rank];
   }
 
   /**
    * This method calculates the all versus all RMSD of a multiple model pdb/arc file.
    *
    * @param usedIndices List of atom indices in use.
    * @param dRMSD Compute dRMSDs in addition to RMSD.
    * @param verbose Verbose logging.
    * @param restart Attempt to restart from an RMSD matrix.
    * @param write Write out an RMSD file.
    * @param saveSnapshots Save superposed snapshots.
    */
   public void calculateRMSDs(int[] usedIndices, boolean dRMSD, boolean verbose, boolean restart,
       boolean write, boolean saveSnapshots, boolean printSym) {
 
     String filename = baseFilter.getFile().getAbsolutePath();
     String targetFilename = targetFilter.getFile().getAbsolutePath();
 
     // Prepare to write out superposed snapshots.
     File baseOutputFile = null;
     File targetOutputFile = null;
     SystemFilter baseOutputFilter = null;
     SystemFilter targetOutputFilter = null;
     if (saveSnapshots) {
       String baseOutputName = concat(getFullPath(filename),
           getBaseName(filename) + "_superposed.arc");
       String targetOutputName = concat(getFullPath(filename),
               getBaseName(targetFilename) + "_superposed.arc");
       baseOutputFile = SystemFilter.version(new File(baseOutputName));
       targetOutputFile = SystemFilter.version(new File(targetOutputName));
       MolecularAssembly baseAssembly = baseFilter.getActiveMolecularSystem();
       MolecularAssembly targetAssembly = targetFilter.getActiveMolecularSystem();
       baseOutputFilter = new XYZFilter(baseOutputFile, baseAssembly,
               baseAssembly.getForceField(),
               baseAssembly.getProperties());
       targetOutputFilter = new XYZFilter(targetOutputFile, targetAssembly,
           targetAssembly.getForceField(),
           targetAssembly.getProperties());
     }
 
     String matrixFilename = concat(getFullPath(filename), getBaseName(filename) + ".dst");
     RunningStatistics runningStatistics;
     if (restart) {
       // Define the filename to use for the RMSD values.
       runningStatistics = readMatrix(matrixFilename, isSymmetric, baseSize, targetSize);
       if (runningStatistics == null) {
         runningStatistics = new RunningStatistics();
       }
     } else {
       runningStatistics = new RunningStatistics();
       File file = new File(matrixFilename);
       if (file.exists() && file.delete()) {
         logger.info(format(" RMSD file (%s) was deleted.", matrixFilename));
         logger.info(" To restart from a previous run, use the '-r' flag.");
       }
     }
 
     // Number of atoms used for the superposition
     int nUsed = usedIndices.length;
     int nUsedVars = nUsed * 3;
 
     // Load atomic coordinates for the base system.
     MolecularAssembly baseMolecularAssembly = baseFilter.getActiveMolecularSystem();
     ForceFieldEnergy baseForceFieldEnergy = baseMolecularAssembly.getPotentialEnergy();
     int nVars = baseForceFieldEnergy.getNumberOfVariables();
     double[] baseCoords = new double[nVars];
     baseForceFieldEnergy.getCoordinates(baseCoords);
     double[] baseUsedCoords = new double[nUsedVars];
 
     // Load an array for mass weighting.
     Atom[] atoms = baseMolecularAssembly.getAtomArray();
     double[] mass = new double[nUsed];
     for (int i = 0; i < nUsed; i++) {
       mass[i] = atoms[usedIndices[i]].getMass();
     }
 
     // Allocate space for the superposition
     MolecularAssembly targetMolecularAssembly = targetFilter.getActiveMolecularSystem();
     ForceFieldEnergy targetForceFieldEnergy = targetMolecularAssembly.getPotentialEnergy();
     double[] targetCoords = new double[nVars];
     double[] targetUsedCoords = new double[nUsedVars];
 
     // Read ahead to the base starting conformation.
     for (int row = 0; row < restartRow; row++) {
       baseFilter.readNext(false, false);
     }
 
     // Padding of the target array size (inner loop limit) is for parallelization.
     // Target conformations are parallelized over available nodes.
     // For example, if numProc = 8 and targetSize = 12, then paddedTargetSize = 16.
     int paddedTargetSize = targetSize;
     int extra = targetSize % numProc;
     if (extra != 0) {
       paddedTargetSize = targetSize - extra + numProc;
       logger.fine(format(" Target size %d vs. Padded size %d", targetSize, paddedTargetSize));
     }
 
     // Loop over base structures.
     for (int row = restartRow; row < baseSize; row++) {
 
       // Initialize the distance this rank is responsible for to zero.
       myDistance[0] = -1.0;
 
       if (row == restartRow) {
         if (dRMSD) {
           logger.info(
               "\n Coordinate RMSD\n Snapshots       Original   After Translation   After Rotation     dRMSD");
         } else if (verbose) {
           logger.info(
               "\n Coordinate RMSD\n Snapshots       Original   After Translation   After Rotation");
         }
       }
 
       // Loop over target structures.
       for (int column = restartColumn; column < paddedTargetSize; column++) {
 
         // Make sure this is not a padded value of column.
         if (column < targetSize) {
           int targetRank = column % numProc;
           if (targetRank == rank) {
             if (isSymmetric && row == column) {
               // Fill the diagonal.
               myDistance[0] = 0.0;
               if (verbose) {
                 logger.info(format(" %6d  %6d  %s                             %8.5f",
                     row + 1, column + 1, "Diagonal", myDistance[0]));
               }
             } else if (isSymmetric && row > column) {
               // Skip the calculation of the lower triangle.
               myDistance[0] = -1.0;
             } else {
               // Calculate an upper triangle entry.
 
               // Load base coordinates.
               baseForceFieldEnergy.getCoordinates(baseCoords);
 
               // Save the current coordinates of the target
               AssemblyState origStateB = new AssemblyState(targetMolecularAssembly);
 
               // Load the target coordinates.
               targetForceFieldEnergy.getCoordinates(targetCoords);
 
               extractCoordinates(usedIndices, baseCoords, baseUsedCoords);
               extractCoordinates(usedIndices, targetCoords, targetUsedCoords);
 
               double origRMSD = rmsd(baseUsedCoords, targetUsedCoords, mass);
 
               // Calculate the translation on only the used subset, but apply it to the entire structure.
               double[] baseTranslation = calculateTranslation(baseUsedCoords, mass);
               applyTranslation(baseCoords, baseTranslation);
               double[] targetTranslation = calculateTranslation(targetUsedCoords, mass);
               applyTranslation(targetCoords, targetTranslation);
               // Copy the applied translation to baseUsedCoords and targetUsedCoords.
               extractCoordinates(usedIndices, baseCoords, baseUsedCoords);
               extractCoordinates(usedIndices, targetCoords, targetUsedCoords);
               double translatedRMSD = rmsd(baseUsedCoords, targetUsedCoords, mass);
 
               // Calculate the rotation on only the used subset, but apply it to the entire structure.
               double[][] rotation = calculateRotation(baseUsedCoords, targetUsedCoords, mass);
               applyRotation(targetCoords, rotation);
               // Copy the applied rotation to targetUsedCoords.
               extractCoordinates(usedIndices, targetCoords, targetUsedCoords);
               double rotatedRMSD = rmsd(baseUsedCoords, targetUsedCoords, mass);
 
               if (dRMSD) {
                 double disRMSD = calcDRMSD(baseUsedCoords, targetUsedCoords, nUsed * 3);
                 logger.info(format(" %6d  %6d  %8.5f            %8.5f         %8.5f  %8.5f",
                     row + 1, column + 1, origRMSD, translatedRMSD, rotatedRMSD, disRMSD));
               } else if (verbose) {
                 logger.info(format(" %6d  %6d  %8.5f            %8.5f         %8.5f",
                     row + 1, column + 1, origRMSD, translatedRMSD, rotatedRMSD));
               }
 
               // Store the RMSD result.
               myDistance[0] = rotatedRMSD;
 
               if(printSym){
                 StringBuilder sbSO = new StringBuilder(
                         format("\n Sym Op to move %s onto %s:\nsymop ", targetFilename, filename));
                 StringBuilder sbInv = new StringBuilder(
                         format("\n Inverted Sym Op to move %s onto %s:\nsymop ", filename, targetFilename));
                 SymOp bestBaseSymOp = new SymOp(SymOp.ZERO_ROTATION, SymOp.Tr_0_0_0).append(new SymOp(SymOp.ZERO_ROTATION, baseTranslation).append(SymOp.invertSymOp(new SymOp(rotation, targetTranslation))));
                 double[] inverseBaseTranslation = new double[]{-baseTranslation[0], -baseTranslation[1], -baseTranslation[2]};
                 SymOp bestTargetSymOp = new SymOp(SymOp.ZERO_ROTATION, SymOp.Tr_0_0_0).append(new SymOp(SymOp.ZERO_ROTATION, targetTranslation).append(new SymOp(rotation, inverseBaseTranslation)));
                 ArrayList<Integer> mol1List = new ArrayList<>();
                 ArrayList<Integer> mol2List = new ArrayList<>();
                 Atom[] atomArr1 = baseMolecularAssembly.getAtomArray();
                 Atom[] atomArr2 = targetMolecularAssembly.getAtomArray();
                 int nAtoms = atomArr1.length;
                 for(int i = 0; i < nAtoms; i++){
                   if(atomArr1[i].isActive()){
                     mol1List.add(i);
                   }
                 }
                 nAtoms = atomArr2.length;
                 for(int i = 0; i < nAtoms; i++){
                   if(atomArr2[i].isActive()){
                     mol2List.add(i);
                   }
                 }
                 int[] mol1arr = mol1List.stream().mapToInt(Integer::intValue).toArray();
                 int[] mol2arr = mol2List.stream().mapToInt(Integer::intValue).toArray();
                 sbSO.append(format("    %s     %s", writeAtomRanges(mol2arr), writeAtomRanges(mol1arr)))
                         .append(asMatrixString(bestBaseSymOp));
                 sbInv.append(format("    %s     %s", writeAtomRanges(mol2arr), writeAtomRanges(mol1arr))).append(asMatrixString(bestTargetSymOp));
                 logger.info(format(" %s\n %s", sbSO, sbInv));
               }
 
               // Save a PDB snapshot.
               if (saveSnapshots && numProc == 1) {
                 MolecularAssembly molecularAssembly = targetOutputFilter.getActiveMolecularSystem();
                 molecularAssembly.getPotentialEnergy().setCoordinates(targetCoords);
                 targetOutputFilter.writeFile(targetOutputFile, true);
                 molecularAssembly = baseOutputFilter.getActiveMolecularSystem();
                 molecularAssembly.getPotentialEnergy().setCoordinates(baseCoords);
                 baseOutputFilter.writeFile(baseOutputFile, true);
                 origStateB.revertState();
               }
             }
           }
 
           // Read ahead to the next target structure.
           targetFilter.readNext(false, false);
         }
 
         // Every numProc iterations, send the results of each rank.
         if ((column + 1) % numProc == 0) {
           gatherRMSDs(row, column, runningStatistics);
         }
       }
 
       // Reset the starting column.
       restartColumn = 0;
 
       // Reset the targetFilter and read the first structure.
       targetFilter.readNext(true, false);
 
       // Read the next base structure.
       baseFilter.readNext(false, false);
 
       // Only Rank 0 writes the distance matrix.
       if (rank == 0 && write) {
         int firstColumn = 0;
         if (isSymmetric) {
           firstColumn = row;
         }
         writeDistanceMatrixRow(matrixFilename, distRow, firstColumn);
       }
     }
 
     baseFilter.closeReader();
     targetFilter.closeReader();
 
     logger.info(format(" RMSD Minimum:  %8.6f", runningStatistics.getMin()));
     logger.info(format(" RMSD Maximum:  %8.6f", runningStatistics.getMax()));
     logger.info(format(" RMSD Mean:     %8.6f", runningStatistics.getMean()));
     double variance = runningStatistics.getVariance();
     if (!Double.isNaN(variance)) {
       logger.info(format(" RMSD Variance: %8.6f", variance));
     }
   }
 
   /**
    * This method calls <code>world.AllGather</code> to collect numProc PAC RMSD values.
    *
    * @param row Current row of the PAC RMSD matrix.
    * @param column Current column of the PAC RMSD matrix.
    * @param runningStatistics Stats.
    */
   private void gatherRMSDs(int row, int column, RunningStatistics runningStatistics) {
     try {
       logger.finer(" Receiving results.");
       world.allGather(myBuffer, buffers);
       for (int i = 0; i < numProc; i++) {
         int c = (column + 1) - numProc + i;
         if (c < targetSize) {
           distRow[c] = distances[i][0];
           if (!isSymmetric) {
             runningStatistics.addValue(distRow[c]);
           } else if (c > row) {
             // Only collect stats for the upper triangle.
             runningStatistics.addValue(distRow[c]);
           }
           logger.finer(format(" %d %d %16.8f", row, c, distances[i][0]));
         }
       }
     } catch (Exception e) {
       logger.severe(" Exception collecting distance values." + e);
     }
   }
 
   /**
    * Read in the distance matrix.
    *
    * @param filename The PAC RMSD matrix file to read from.
    * @param isSymmetric Is the distance matrix symmetric.
    * @param expectedRows The expected number of rows.
    * @param expectedColumns The expected number of columns.
    * @return Stats for all read in distance matrix values.
    */
   private RunningStatistics readMatrix(String filename, boolean isSymmetric, int expectedRows,
       int expectedColumns) {
     restartRow = 0;
     restartColumn = 0;
 
     DistanceMatrixFilter distanceMatrixFilter = new DistanceMatrixFilter();
     RunningStatistics runningStatistics = distanceMatrixFilter.readDistanceMatrix(
         filename, expectedRows, expectedColumns);
 
     if (runningStatistics != null && runningStatistics.getCount() > 0) {
       restartRow = distanceMatrixFilter.getRestartRow();
       restartColumn = distanceMatrixFilter.getRestartColumn();
 
       if (isSymmetric) {
         // Only the diagonal entry (0.0) is on the last row for a symmetric matrix.
         if (restartRow == expectedRows && restartColumn == 1) {
           logger.info(format(" Complete symmetric distance matrix found (%d x %d).", restartRow,
               restartRow));
         } else {
           restartColumn = 0;
           logger.info(format(
               " Incomplete symmetric distance matrix found.\n Restarting at row %d, column %d.",
               restartRow + 1, restartColumn + 1));
         }
       } else if (restartRow == expectedRows && restartColumn == expectedColumns) {
         logger.info(format(" Complete distance matrix found (%d x %d).", restartRow, restartColumn));
       } else {
         restartColumn = 0;
         logger.info(format(" Incomplete distance matrix found.\n Restarting at row %d, column %d.",
             restartRow + 1, restartColumn + 1));
       }
     }
 
     return runningStatistics;
   }
 
   /**
    * Extract used coordinate subset from the entire system.
    *
    * @param usedIndices Mapping from the xUsed array to its source in x.
    * @param x All atomic coordinates.
    * @param xUsed The used subset of coordinates.
    */
   public static void extractCoordinates(int[] usedIndices, double[] x, double[] xUsed) {
     int nUsed = usedIndices.length;
     for (int u = 0; u < nUsed; u++) {
       int u3 = 3 * u;
       int i3 = 3 * usedIndices[u];
       arraycopy(x, i3, xUsed, u3, 3);
     }
   }
 
   /**
    * Calculates the dRMSD between to sets of coordinates.
    *
    * @param xUsed A double array containing the xyz coordinates for multiple atoms.
    * @param x2Used A double array containing the xyz coordinates for multiple atoms.
    * @param nUsed The number of atoms that dRMSD is calculated on.
    * @return A double containing the dRMSD value.
    */
   public static double calcDRMSD(double[] xUsed, double[] x2Used, int nUsed) {
     double disRMSD = 0.0;
     int counter = 0;
     for (int i = 0; i < nUsed; i = i + 3) {
       Double3 xi = new Double3(xUsed[i], xUsed[i + 1], xUsed[i + 2]);
       Double3 x2i = new Double3(x2Used[i], x2Used[i + 1], x2Used[i + 2]);
       for (int j = i + 3; j < nUsed; j = j + 3) {
         Double3 xj = new Double3(xUsed[j], xUsed[j + 1], xUsed[j + 2]);
         Double3 x2j = new Double3(x2Used[j], x2Used[j + 1], x2Used[j + 2]);
         double dis1 = xi.sub(xj).length();
         double dis2 = x2i.sub(x2j).length();
         double diff = dis1 - dis2;
         disRMSD += diff * diff;
         counter++;
       }
     }
     disRMSD = disRMSD / counter;
     return sqrt(disRMSD);
   }
 
   /**
    * Apply a rotation matrix to a set of coordinates.
    *
    * @param x2 Cartesian coordinates of the second system. Modified in-place.
    * @param rot A pre-calculated rotation matrix.
    */
   public static void applyRotation(double[] x2, double[][] rot) {
     int n = x2.length / 3;
 
     // Rotate second molecule to best fit with first molecule
     for (int i = 0; i < n; i++) {
       int k = i * 3;
       double xrot = x2[k] * rot[0][0] + x2[k + 1] * rot[0][1] + x2[k + 2] * rot[0][2];
       double yrot = x2[k] * rot[1][0] + x2[k + 1] * rot[1][1] + x2[k + 2] * rot[1][2];
       double zrot = x2[k] * rot[2][0] + x2[k + 1] * rot[2][1] + x2[k + 2] * rot[2][2];
       x2[k] = xrot;
       x2[k + 1] = yrot;
       x2[k + 2] = zrot;
     }
   }
 
   /**
    * Apply a translation matrix [dx,dy,dz] to a molecular system.
    *
    * @param x Coordinates to move; modified in-place.
    * @param translation Translation matrix.
    */
   public static void applyTranslation(double[] x, final double[] translation) {
     int n = x.length / 3;
     for (int i = 0; i < n; i++) {
       int k = i * 3;
       for (int j = 0; j < 3; j++) {
         x[k + j] += translation[j];
       }
     }
   }
 
   /**
    * Calculate a rotation to minimize RMS distance between two sets of atoms using quaternions,
    * overlapping x2 on x1.
    *
    * @param x1 Cartesian coordinates of the first system.
    * @param x2 Cartesian coordinates of the second system.
    * @param mass The mass of each particle in the system.
    * @return A rotation matrix.
    */
   public static double[][] calculateRotation(double[] x1, double[] x2, double[] mass) {
 
     // Build the upper triangle of the quadratic form matrix
     double xxyx = 0.0;
     double xxyy = 0.0;
     double xxyz = 0.0;
     double xyyx = 0.0;
     double xyyy = 0.0;
     double xyyz = 0.0;
     double xzyx = 0.0;
     double xzyy = 0.0;
     double xzyz = 0.0;
 
     int n = x1.length / 3;
     for (int i = 0; i < n; i++) {
       int k = i * 3;
       double weigh = mass[i];
       xxyx = xxyx + weigh * x1[k] * x2[k];
       xxyy = xxyy + weigh * x1[k + 1] * x2[k];
       xxyz = xxyz + weigh * x1[k + 2] * x2[k];
       xyyx = xyyx + weigh * x1[k] * x2[k + 1];
       xyyy = xyyy + weigh * x1[k + 1] * x2[k + 1];
       xyyz = xyyz + weigh * x1[k + 2] * x2[k + 1];
       xzyx = xzyx + weigh * x1[k] * x2[k + 2];
       xzyy = xzyy + weigh * x1[k + 1] * x2[k + 2];
       xzyz = xzyz + weigh * x1[k + 2] * x2[k + 2];
     }
 
     double[][] c = new double[4][4];
     c[0][0] = xxyx + xyyy + xzyz;
     c[0][1] = xzyy - xyyz;
     c[1][0] = c[0][1];
     c[1][1] = xxyx - xyyy - xzyz;
     c[0][2] = xxyz - xzyx;
     c[2][0] = c[0][2];
     c[1][2] = xxyy + xyyx;
     c[2][1] = c[1][2];
     c[2][2] = xyyy - xzyz - xxyx;
     c[0][3] = xyyx - xxyy;
     c[3][0] = c[0][3];
     c[1][3] = xzyx + xxyz;
     c[3][1] = c[1][3];
     c[2][3] = xyyz + xzyy;
     c[3][2] = c[2][3];
     c[3][3] = xzyz - xxyx - xyyy;
 
     // Diagonalize the quadratic form matrix
     Array2DRowRealMatrix cMatrix = new Array2DRowRealMatrix(c, false);
     EigenDecomposition eigenDecomposition = new EigenDecomposition(cMatrix);
 
     // Extract the quaternions.
     double[] q = eigenDecomposition.getEigenvector(0).toArray();
 
     // Assemble rotation matrix that superimposes the molecules
     double q02 = q[0] * q[0];
     double q12 = q[1] * q[1];
     double q22 = q[2] * q[2];
     double q32 = q[3] * q[3];
     double[][] rot = new double[3][3];
     rot[0][0] = q02 + q12 - q22 - q32;
     rot[1][0] = 2.0 * (q[1] * q[2] - q[0] * q[3]);
     rot[2][0] = 2.0 * (q[1] * q[3] + q[0] * q[2]);
     rot[0][1] = 2.0 * (q[1] * q[2] + q[0] * q[3]);
     rot[1][1] = q02 - q12 + q22 - q32;
     rot[2][1] = 2.0 * (q[2] * q[3] - q[0] * q[1]);
     rot[0][2] = 2.0 * (q[1] * q[3] - q[0] * q[2]);
     rot[1][2] = 2.0 * (q[2] * q[3] + q[0] * q[1]);
     rot[2][2] = q02 - q12 - q22 + q32;
 
     return rot;
   }
 
   /**
    * Calculate a translation vector [dx,dy,dz] to move the center of mass to the origin.
    *
    * @param x Coordinates of the system.
    * @param mass Mass of each atom.
    * @return Translation to the origin.
    */
   public static double[] calculateTranslation(double[] x, final double[] mass) {
     double xmid = 0.0;
     double ymid = 0.0;
     double zmid = 0.0;
     double norm = 0.0;
     int n = x.length / 3;
 
     for (int i = 0; i < n; i++) {
       int k = 3 * i;
       double weigh = mass[i];
       xmid += x[k] * weigh;
       ymid += x[k + 1] * weigh;
       zmid += x[k + 2] * weigh;
       norm += weigh;
     }
     xmid /= norm;
     ymid /= norm;
     zmid /= norm;
 
     return new double[] {-xmid, -ymid, -zmid};
   }
 
   /**
    * Compute the RMSD for superimposed atom pairs.
    *
    * @param x1 Cartesian coordinates of the first system.
    * @param x2 Cartesian coordinates of the second system.
    * @param mass The mass of each particle in the system.
    * @return The RMSD.
    */
   public static double rmsd(double[] x1, double[] x2, double[] mass) {
     double rmsfit = 0.0;
     double norm = 0.0;
     int n = x1.length / 3;
     for (int i = 0; i < n; i++) {
       int k = 3 * i;
       double weigh = mass[i];
       double xr = x1[k] - x2[k];
       double yr = x1[k + 1] - x2[k + 1];
       double zr = x1[k + 2] - x2[k + 2];
       double dist2 = xr * xr + yr * yr + zr * zr;
       norm = norm + weigh;
       double rmsterm = dist2 * weigh;
       rmsfit = rmsfit + rmsterm;
     }
     return sqrt(rmsfit / norm);
   }
 
   /**
    * Minimize the RMS distance between two sets of atoms using quaternions.
    *
    * @param x1 Cartesian coordinates of the first system. Unmodified.
    * @param x2 Cartesian coordinates of the second system. Modified in-place.
    * @param mass The mass of each particle in the system.
    */
   public static void rotate(double[] x1, double[] x2, double[] mass) {
     double[][] rotation = calculateRotation(x1, x2, mass);
     applyRotation(x2, rotation);
   }
 
   /**
    * Move the center of mass for both sets of atoms to the origin.
    *
    * @param x1 Cartesian coordinates of the first system.
    * @param mass1 The mass of each particle in the first system.
    * @param x2 Cartesian coordinates of the second system.
    * @param mass2 The mass of each particle in the second system.
    */
   public static void translate(double[] x1, double[] mass1, double[] x2, double[] mass2) {
     // Move the first set of atoms.
     translate(x1, mass1);
     // Move the second set of atoms.
     translate(x2, mass2);
   }
 
   /**
    * Move the center of mass for a set of atoms to the origin.
    *
    * @param x Cartesian coordinates of the system; modified in-place.
    * @param mass The mass of each particle in the system.
    */
   public static void translate(double[] x, final double[] mass) {
     double[] translation = calculateTranslation(x, mass);
     applyTranslation(x, translation);
   }
 
   /**
    * This method completes the following superposition operations and returns an RMSD: 1) translates
    * the x1 coordinates to the origin. 2) translates the x2 coordinates to the origin. 3) rotates x2
    * onto x1. 4) computes and return the RMSD.
    *
    * @param x1 Coordinates of system 1.
    * @param x2 Coordinates of system 2.
    * @param mass Mass of each atom.
    * @return The mass-weighted RMSD following superposition.
    */
   public static double superpose(double[] x1, double[] x2, double[] mass) {
     translate(x1, mass);
     translate(x2, mass);
     rotate(x1, x2, mass);
     return rmsd(x1, x2, mass);
   }
 }