// ******************************************************************************
   //
   // Title:       Force Field X.
   // Description: Force Field X - Software for Molecular Biophysics.
   // Copyright:   Copyright (c) Michael J. Schnieders 2001-2025.
   //
   // 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
  // and conditions of the license of that module. An independent module is a
  // 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.
  //
  // ******************************************************************************
  package ffx.potential.parameters;
  
  import ffx.utilities.FFXProperty;
  import org.w3c.dom.Document;
  import org.w3c.dom.Element;
  
  import java.io.BufferedReader;
  import java.io.IOException;
  import java.util.Arrays;
  import java.util.Comparator;
  import java.util.HashMap;
  import java.util.Map;
  import java.util.logging.Level;
  import java.util.logging.Logger;
  
  import static ffx.potential.parameters.ForceField.ForceFieldType.TORTORS;
  import static ffx.utilities.Constants.KCAL_TO_KJ;
  import static ffx.utilities.PropertyGroup.EnergyUnitConversion;
  import static ffx.utilities.PropertyGroup.PotentialFunctionParameter;
  import static java.lang.Double.parseDouble;
  import static java.lang.Integer.parseInt;
  import static java.lang.String.format;
  import static java.lang.System.arraycopy;
  import static java.util.Arrays.sort;
  import static org.apache.commons.math3.util.FastMath.abs;
  
  /**
   * The TorsionTorsionType class defines a Torsion-Torsion spline.
   *
   * @author Michael J. Schnieders
   * @since 1.0
   */
  @FFXProperty(name = "tortors", clazz = String[].class, propertyGroup = PotentialFunctionParameter, description = """
      [7 integers, then multiple lines of 2 integers and 1 real]
      Provides the values for a single torsion-torsion parameter.
      The first five integer modifiers give the atom class numbers for the atoms involved in the two adjacent torsional angles to be defined.
      The last two integer modifiers contain the number of data grid points that lie along each axis of the torsion-torsion map.
      For example, this value will be 13 for a 30 degree torsional angle spacing, i.e., 360/30 = 12, but 13
      values are required since data values for -180 and +180 degrees must both be supplied.
      The subsequent lines contain the torsion-torsion map data as the integer values in degrees of each
      torsional angle and the target energy value in kcal/mole.
      """)
  public final class TorsionTorsionType extends BaseType implements Comparator<String> {
  
    /**
     * Default units to convert Torsion-Torsion energy to kcal/mole.
     */
    public static final double DEFAULT_TORTOR_UNIT = 1.0;
    /**
     * Convert Torsion-Torsion energy to kcal/mole.
     */
    @FFXProperty(name = "tortorunit", propertyGroup = EnergyUnitConversion, defaultValue = "1.0", description = """
        Sets the scale factor needed to convert the energy value computed by the torsion-torsion potential into units of kcal/mole.
        The correct value is force field dependent and typically provided in the header of the master force field parameter file.
        """)
    public double torTorUnit = DEFAULT_TORTOR_UNIT;
  
    private static final Logger logger = Logger.getLogger(TorsionTorsionType.class.getName());
    /**
     * Atom classes that form this Torsion-Torsion type.
     */
    public final int[] atomClasses;
   /**
    * Energy values.
    */
   public final double[] energy;
   /**
    * Number of points along x.
    */
   public final int nx;
   /**
    * Number of point along y.
    */
   public final int ny;
   /**
    * Torsion values along x.
    */
   public final double[] tx;
   /**
    * Torsion values along y.
    */
   public final double[] ty;
   /**
    * First derivative along x.
    */
   public final double[] dx;
   /**
    * First derivative along y.
    */
   public final double[] dy;
   /**
    * Second derivatives.
    */
   public final double[] dxy;
   /**
    * Grid points.
    */
   private final int[] gridPoints;
 
   /**
    * Constructor for TorsionTorsionType.
    *
    * @param atomClasses an array of int.
    * @param gridPoints  an array of int.
    * @param torsion1    an array of double.
    * @param torsion2    an array of double.
    * @param energy      an array of double.
    */
   public TorsionTorsionType(int[] atomClasses, int[] gridPoints, double[] torsion1,
                             double[] torsion2, double[] energy) {
     super(TORTORS, sortKey(atomClasses));
     this.atomClasses = atomClasses;
     nx = gridPoints[0];
     ny = gridPoints[1];
     if (nx != ny) {
       logger.severe("Untested TORTOR parameters: nx != ny: " + nx + ", " + ny);
     }
 
     this.energy = energy;
     this.gridPoints = gridPoints;
     tx = new double[nx];
     ty = new double[ny];
     dx = new double[nx * ny];
     dy = new double[nx * ny];
     dxy = new double[nx * ny];
     sort(torsion1);
     sort(torsion2);
     tx[0] = torsion1[0];
     ty[0] = torsion2[0];
     int j1 = 1;
     int j2 = 1;
     for (int i = 1; i < nx; i++) {
       while (torsion1[j1] == tx[i - 1]) {
         j1++;
       }
       while (torsion2[j2] == ty[i - 1]) {
         j2++;
       }
       tx[i] = torsion1[j1];
       ty[i] = torsion2[j2];
     }
 
     // Check for cyclic energy.
     boolean isCyclic = true;
     double eps = 0.0001;
     if (abs(tx[0] - tx[nx - 1]) - 360.0 > eps) {
       isCyclic = false;
       if (logger.isLoggable(Level.FINEST)) {
         logger.finest(" tortor is aperiodic: " + tx[0] + ", " + tx[nx - 1]);
       }
     }
     if (isCyclic) {
       for (int i = 0; i < ny; i++) {
         int k = i * nx;
         if (abs(energy[k] - energy[k + nx - 1]) > eps) {
           isCyclic = false;
           if (logger.isLoggable(Level.FINEST)) {
             logger.finest(" tortor is apreriodic: " + k + ", " + (k + nx - 1) + ": " + abs(
                 energy[k] - energy[k + nx - 1]));
           }
           break;
         }
       }
     }
     if (isCyclic) {
       int k = (ny - 1) * nx;
       for (int i = 0; i < nx; i++) {
         if (abs(energy[i] - energy[i + k]) > eps) {
           if (logger.isLoggable(Level.FINEST)) {
             logger.fine(
                 " tortor is aperiodic: " + i + ", " + i + k + ": " + abs(energy[i] - energy[i + k]));
           }
           isCyclic = false;
           break;
         }
       }
     }
 
     boolean cyclic = isCyclic;
     double[] tmp1 = new double[nx];
     double[] tmp2 = new double[nx];
     double[] tmp3 = new double[nx];
     double[] tmp4 = new double[nx];
     double[] tmp5 = new double[nx];
     double[] tmp6 = new double[nx];
     double[] tmp7 = new double[nx];
     double[] bs = new double[nx];
     double[] cs = new double[nx];
     double[] ds = new double[nx];
 
     // Spline fit the derivatives about the first torsion.
     arraycopy(tx, 0, tmp1, 0, nx);
 
     int m = 0;
     for (int j = 0; j < ny; j++) {
       arraycopy(energy, m, tmp2, 0, nx);
       if (cyclic) {
         cspline(nx - 1, tmp1, tmp2, bs, cs, ds, tmp3, tmp4, tmp5, tmp6, tmp7);
       } else {
         nspline(nx - 1, tmp1, tmp2, 0.0, 0.0, bs, cs, tmp3, tmp4, tmp5, tmp6, tmp7);
       }
       arraycopy(bs, 0, dx, m, nx);
       m = m + nx;
     }
 
     // Spline fit the derivatives about the second torsion.
     arraycopy(ty, 0, tmp1, 0, ny);
 
     m = 0;
     for (int j = 0; j < nx; j++) {
       for (int k = 0; k < ny; k++) {
         tmp2[k] = energy[m + k * nx];
       }
       if (cyclic) {
         cspline(ny - 1, tmp1, tmp2, bs, cs, ds, tmp3, tmp4, tmp5, tmp6, tmp7);
       } else {
         nspline(ny - 1, tmp1, tmp2, 0.0, 0.0, bs, cs, tmp3, tmp4, tmp5, tmp6, tmp7);
       }
       for (int k = 0; k < ny; k++) {
         dy[m + k * nx] = bs[k];
       }
       m = m + 1;
     }
 
     // Spline fit the cross derivatives about both torsions.
     m = 0;
     for (int j = 0; j < nx; j++) {
       for (int k = 0; k < ny; k++) {
         tmp2[k] = dx[m + k * nx];
       }
       if (cyclic) {
         cspline(ny - 1, tmp1, tmp2, bs, cs, ds, tmp3, tmp4, tmp5, tmp6, tmp7);
       } else {
         nspline(ny - 1, tmp1, tmp2, 0.0, 0.0, bs, cs, tmp3, tmp4, tmp5, tmp6, tmp7);
       }
       for (int k = 0; k < ny; k++) {
         dxy[m + k * nx] = bs[k];
       }
       m = m + 1;
     }
   }
 
   /**
    * average.
    *
    * @param torsionTorsionType1 a {@link ffx.potential.parameters.TorsionTorsionType} object.
    * @param torsionTorsionType2 a {@link ffx.potential.parameters.TorsionTorsionType} object.
    * @param atomClasses         an array of {@link int} objects.
    * @return a {@link ffx.potential.parameters.TorsionTorsionType} object.
    */
   public static TorsionTorsionType average(TorsionTorsionType torsionTorsionType1,
                                            TorsionTorsionType torsionTorsionType2, int[] atomClasses) {
     if (torsionTorsionType1 == null || torsionTorsionType2 == null || atomClasses == null) {
       return null;
     }
     return null;
   }
 
   /**
    * Construct a TorsionTorsionType from multiple input lines.
    *
    * @param input  The overall input String.
    * @param tokens The input String tokenized.
    * @param br     a BufferedReader instance.
    * @return a TorsionTorsionType instance.
    */
   public static TorsionTorsionType parse(String input, String[] tokens, BufferedReader br) {
     if (tokens.length < 8) {
       logger.log(Level.WARNING, "Invalid TORTORS type:\n{0}", input);
     } else {
       try {
         int[] atomClasses = new int[5];
         for (int i = 0; i < 5; i++) {
           atomClasses[i] = parseInt(tokens[i + 1]);
         }
         int[] gridPoints = new int[2];
         gridPoints[0] = parseInt(tokens[6]);
         gridPoints[1] = parseInt(tokens[7]);
         int points = gridPoints[0] * gridPoints[1];
         double[] torsion1 = new double[points];
         double[] torsion2 = new double[points];
         double[] energy = new double[points];
         for (int i = 0; i < points; i++) {
           input = br.readLine();
           tokens = input.trim().split(" +");
           if (tokens.length == 3) {
             torsion1[i] = parseDouble(tokens[0]);
             torsion2[i] = parseDouble(tokens[1]);
             energy[i] = parseDouble(tokens[2]);
           } else if (tokens.length == 6) {
             // First entry.
             torsion1[i] = parseDouble(tokens[0]);
             torsion2[i] = parseDouble(tokens[1]);
             energy[i] = parseDouble(tokens[2]);
             // Second entry.
             torsion1[i + 1] = parseDouble(tokens[3]);
             torsion2[i + 1] = parseDouble(tokens[4]);
             energy[i + 1] = parseDouble(tokens[5]);
             // Increment i by 1 since we read 2 values.
             i += 1;
           } else if (tokens.length == 9) {
             // First entry.
             torsion1[i] = parseDouble(tokens[0]);
             torsion2[i] = parseDouble(tokens[1]);
             energy[i] = parseDouble(tokens[2]);
             // Second entry.
             torsion1[i + 1] = parseDouble(tokens[3]);
             torsion2[i + 1] = parseDouble(tokens[4]);
             energy[i + 1] = parseDouble(tokens[5]);
             // Third entry.
             torsion1[i + 2] = parseDouble(tokens[6]);
             torsion2[i + 2] = parseDouble(tokens[7]);
             energy[i + 2] = parseDouble(tokens[8]);
             // Increment i by 2 since we read 3 values.
             i += 2;
           } else {
             logger.log(Level.WARNING, "Invalid TORTORS type:\n{0}", input);
             return null;
           }
         }
         return new TorsionTorsionType(atomClasses, gridPoints, torsion1, torsion2, energy);
       } catch (NumberFormatException | IOException e) {
         String message = "Exception parsing TORTORS type:\n" + input + "\n";
         logger.log(Level.SEVERE, message, e);
       }
     }
     return null;
   }
 
   /**
    * Construct a TorsionTorsionType from a single input line.
    *
    * @param input  The overall input String.
    * @param tokens The input String tokenized.
    * @return a TorsionTorsionType instance.
    */
   public static TorsionTorsionType parse(String input, String[] tokens) {
     if (tokens.length < 8) {
       logger.log(Level.WARNING, "Invalid TORTORS type:\n{0}", input);
     } else {
       try {
         int[] atomClasses = new int[5];
         for (int i = 0; i < 5; i++) {
           atomClasses[i] = parseInt(tokens[i + 1]);
         }
         int[] gridPoints = new int[2];
         gridPoints[0] = parseInt(tokens[6]);
         gridPoints[1] = parseInt(tokens[7]);
         int points = gridPoints[0] * gridPoints[1];
         int numTokens = points * 3 + 8;
         if (tokens.length < numTokens) {
           logger.log(Level.WARNING, "Invalid TORTORS type:\n{0}", input);
           return null;
         }
         double[] torsion1 = new double[points];
         double[] torsion2 = new double[points];
         double[] energy = new double[points];
         int index = 8;
         for (int i = 0; i < points; i++) {
           torsion1[i] = parseDouble(tokens[index++]);
           torsion2[i] = parseDouble(tokens[index++]);
           energy[i] = parseDouble(tokens[index++]);
         }
         return new TorsionTorsionType(atomClasses, gridPoints, torsion1, torsion2, energy);
       } catch (NumberFormatException e) {
         String message = "Exception parsing TORTORS type:\n" + input + "\n";
         logger.log(Level.SEVERE, message, e);
       }
     }
     return null;
   }
 
   /**
    * Reversed key for the Torsion-Torsion lookup.
    *
    * @param c atomClasses
    * @return lookup key
    */
   public static String reverseKey(int[] c) {
     return c[4] + " " + c[3] + " " + c[2] + " " + c[1] + " " + c[0];
   }
 
   /**
    * No sorting is done for the Torsion-Torsion lookup.
    *
    * @param c atomClasses
    * @return lookup key
    */
   public static String sortKey(int[] c) {
     return c[0] + " " + c[1] + " " + c[2] + " " + c[3] + " " + c[4];
   }
 
   /**
    * Solves a system of linear equations for a cyclically tridiagonal, symmetric, positive definite
    * matrix.
    *
    * @param n
    * @param dm
    * @param du
    * @param rs
    * @param c
    * @return positive or negative 1.
    */
   private static int cytsy(int n, double[] dm, double[] du, double[] cr, double[] rs, double[] c) {
     if (n < 3) {
       return -2;
     }
 
     // Factorization of the input matrix.
     if (cytsyp(n, dm, du, cr) == 1) {
       // Update and back-substitute as necessary.
       cytsys(n, dm, du, cr, rs, c);
       return 1;
     }
     return -1;
   }
 
   /**
    * Finds the Cholesky factors of a cyclically tridiagonal symmetric, positive definite matrix given
    * by two vectors.
    *
    * @param n
    * @param dm
    * @param du
    * @param cr
    * @return an integer.
    */
   private static int cytsyp(int n, double[] dm, double[] du, double[] cr) {
     double eps = 0.00000001;
     // Check for n < 3.
     if (n < 3) {
       return -2;
     }
 
     // Check if the matrix is positive definite.
     double row = abs(dm[1]) + abs(du[1]) + abs(du[n]);
     if (row == 0.0) {
       return 0;
     }
     double d = 1.0 / row;
     if (dm[1] < 0.0) {
       return -1;
     } else if (abs(dm[1]) * d < eps) {
       return 0;
     }
 
     // Factoring while checking for a positive definite and strong non-singular matrix.
     double temp1 = du[1];
     double temp2 = 0.0;
     du[1] = du[1] / dm[1];
     cr[1] = du[n] / dm[1];
     for (int i = 2; i < n; i++) {
       row = abs(dm[i]) + abs(du[i]) + abs(temp1);
       if (row == 0.0) {
         return 0;
       }
       d = 1.0 / row;
       dm[i] = dm[i] - temp1 * du[i - 1];
       if (dm[i] < 0.0) {
         return -1;
       } else if (abs(dm[i]) * d < eps) {
         return 0;
       }
       if (i < (n - 1)) {
         cr[i] = -temp1 * cr[i - 1] / dm[i];
         temp1 = du[i];
         du[i] = du[i] / dm[i];
       } else {
         temp2 = du[i];
         du[i] = (du[i] - temp1 * cr[i - 1]) / dm[i];
       }
     }
     row = abs(du[n]) + abs(dm[n]) + abs(temp2);
     if (row == 0.0) {
       return 0;
     }
     d = 1.0 / row;
     dm[n] = dm[n] - dm[n - 1] * du[n - 1] * du[n - 1];
     temp1 = 0.0;
     for (int i = 1; i < n - 1; i++) {
       temp1 = temp1 + dm[i] * cr[i] * cr[i];
     }
     dm[n] = dm[n] - temp1;
     if (dm[n] < 0.0) {
       return -1;
     } else if (abs(dm[n]) * d < eps) {
       return 0;
     }
     return 1;
   }
 
   /**
    * Solves a cyclically tridiagonal linear system given the Cholesky factors.
    *
    * @param n
    * @param dm
    * @param du
    * @param cr
    * @param rs
    * @param c
    */
   private static void cytsys(int n, double[] dm, double[] du, double[] cr, double[] rs, double[] c) {
     // Updating phase.
     double temp = rs[1];
     rs[1] = temp / dm[1];
     double sum = cr[1] * temp;
     for (int i = 2; i < n; i++) {
       temp = rs[i] - du[i - 1] * temp;
       rs[i] = temp / dm[i];
       if (i != (n - 1)) {
         sum = sum + cr[i] * temp;
       }
     }
     temp = rs[n] - du[n - 1] * temp;
     temp = temp - sum;
     rs[n] = temp / dm[n];
 
     // Back-substitution phase.
     c[n] = rs[n];
     c[n - 1] = rs[n - 1] - du[n - 1] * c[n];
     for (int i = n - 2; i >= 1; i--) {
       c[i] = rs[i] - du[i] * c[i + 1] - cr[i] * c[n];
     }
   }
 
   /**
    * {@inheritDoc}
    */
   @Override
   public int compare(String key1, String key2) {
     String[] keys1 = key1.split(" ");
     String[] keys2 = key2.split(" ");
     int c1 = parseInt(keys1[2]);
     int c2 = parseInt(keys2[2]);
     if (c1 < c2) {
       return -1;
     } else if (c1 > c2) {
       return 1;
     }
     return 0;
   }
 
   /**
    * {@inheritDoc}
    */
   @Override
   public boolean equals(Object o) {
     if (this == o) {
       return true;
     }
     if (o == null || getClass() != o.getClass()) {
       return false;
     }
     TorsionTorsionType torsionTorsionType = (TorsionTorsionType) o;
     return Arrays.equals(atomClasses, torsionTorsionType.atomClasses);
   }
 
   /**
    * {@inheritDoc}
    */
   @Override
   public int hashCode() {
     return Arrays.hashCode(atomClasses);
   }
 
   /**
    * Increment the atom classes by a value.
    *
    * @param increment The increment to add to the atom classes.
    */
   public void incrementClasses(int increment) {
     for (int i = 0; i < atomClasses.length; i++) {
       atomClasses[i] += increment;
     }
     setKey(sortKey(atomClasses));
   }
 
   /**
    * Remap new atom classes to known internal ones.
    *
    * @param typeMap a lookup between new atom types and known atom types.
    */
   public void patchClasses(HashMap<AtomType, AtomType> typeMap) {
     int count = 0;
     for (AtomType newType : typeMap.keySet()) {
 
       for (int atomClass : atomClasses) {
         if (atomClass == newType.atomClass) {
           count++;
         }
       }
     }
     if (count > 0 && count < atomClasses.length) {
       for (AtomType newType : typeMap.keySet()) {
         for (int i = 0; i < atomClasses.length; i++) {
           if (atomClasses[i] == newType.atomClass) {
             AtomType knownType = typeMap.get(newType);
             atomClasses[i] = knownType.atomClass;
           }
         }
       }
       setKey(sortKey(atomClasses));
     }
   }
 
   /**
    * {@inheritDoc}
    *
    * <p>Nicely formatted torsion-torsion type.
    */
   @Override
   public String toString() {
     StringBuilder tortorBuffer = new StringBuilder("tortors");
     for (int i : atomClasses) {
       tortorBuffer.append(format("  %5d", i));
     }
     tortorBuffer.append(format("  %2d  %2d", gridPoints[0], gridPoints[1]));
     for (int i = 0; i < energy.length; i++) {
       // Emit one entry.
       int nxi = i % nx;
       int nyi = i / ny;
       tortorBuffer.append(format(" \\\n  % 6.1f  % 6.1f  % 8.5f", tx[nxi], ty[nyi], energy[i]));
       // Emit a second entry if available.
       if (i < energy.length - 1) {
         i++;
         nxi = i % nx;
         nyi = i / ny;
         tortorBuffer.append(format("  % 6.1f  % 6.1f  % 8.5f", tx[nxi], ty[nyi], energy[i]));
       }
       // Emit a third entry if available.
       if (i < energy.length - 1) {
         i++;
         nxi = i % nx;
         nyi = i / ny;
         tortorBuffer.append(format("  % 6.1f  % 6.1f  % 8.5f", tx[nxi], ty[nyi], energy[i]));
       }
     }
     return tortorBuffer.toString();
   }
 
 
   /**
    * Create an AmoebaTorsionTorsionForce Element.
    *
    * @param doc        the Document instance.
    * @param forceField the ForceField instance to grab constants from.
    * @return the AmoebaTorsionTorsionForce Element.
    */
   public static Element getXMLForce(Document doc, ForceField forceField) {
     Map<String, TorsionTorsionType> types = forceField.getTorsionTorsionTypes();
     if (!types.values().isEmpty()) {
       Element node = doc.createElement("AmoebaTorsionTorsionForce");
       int i = 0;
       for (TorsionTorsionType torsionTorsionType : types.values()) {
         torsionTorsionType.toXML(doc, node, i);
         i++;
       }
       return node;
     }
     return null;
   }
 
   /**
    * Write TorsionTorsionType to OpenMM XML format.
    *
    * @param doc        the Document instance.
    * @param torTorNode the Element to attach TorsionTorsion and TorsionTorsionGrid to.
    * @param label      the label used to match TorsionTorsion and TorsionTorsionGrid.
    */
   public void toXML(Document doc, Element torTorNode, int label) {
     Element tortors = doc.createElement("TorsionTorsion");
     Element ttGrid = doc.createElement("TorsionTorsionGrid");
 
     // Set TorsionTorsion node
     tortors.setAttribute("class1", format("%d", atomClasses[0]));
     tortors.setAttribute("class2", format("%d", atomClasses[1]));
     tortors.setAttribute("class3", format("%d", atomClasses[2]));
     tortors.setAttribute("class4", format("%d", atomClasses[3]));
     tortors.setAttribute("class5", format("%d", atomClasses[4]));
     tortors.setAttribute("nx", format("%d", gridPoints[0]));
     tortors.setAttribute("ny", format("%d", gridPoints[1]));
 
     // Set TorsionTorsionGrid
     ttGrid.setAttribute("nx", format("%d", gridPoints[0]));
     ttGrid.setAttribute("ny", format("%d", gridPoints[1]));
     for (int i = 0; i < energy.length; i++) {
       int nxi = i % nx;
       int nyi = i / ny;
       // OpenMM appears to offer some support for storing dE/dX, dE/dY and d2E/dXdY
       // instead of computing these values on the fly.
       Element grid = doc.createElement("Grid");
       grid.setAttribute("angle1", format("%f", tx[nxi]));
       grid.setAttribute("angle2", format("%f", ty[nyi]));
       grid.setAttribute("f", format("%f", energy[i] * KCAL_TO_KJ));
       ttGrid.appendChild(grid);
     }
 
     tortors.setAttribute("grid", String.valueOf(label));
     ttGrid.setAttribute("grid", String.valueOf(label));
     torTorNode.appendChild(tortors);
     torTorNode.appendChild(ttGrid);
   }
 
   /**
    * Computes the coefficients for an aperiodic interpolating cubic spline.
    *
    * @param n
    * @param x0
    * @param y0
    * @param y21
    * @param y2n
    * @param s1
    * @param s2
    * @param h
    * @param g
    * @param dy
    * @param dla
    * @param dmu
    */
   private void nspline(int n, double[] x0, double[] y0, double y21, double y2n, double[] s1,
                        double[] s2, double[] h, double[] g, double[] dy, double[] dla, double[] dmu) {
 
     // Calculate the intervals.
     for (int i = 0; i < n; i++) {
       h[i] = x0[i + 1] - x0[i];
       dy[i] = (y0[i + 1] - y0[i]) / h[i];
     }
 
     // Get the coeffcients.
     for (int i = 1; i < n; i++) {
       dla[i] = h[i] / (h[i] + h[i - 1]);
       dmu[i] = 1.0 - dla[i];
       g[i] = 3.0 * (dla[i] * dy[i - 1] + dmu[i] * dy[i]);
     }
 
     // Set the initial value via natural boundary condition.
     dla[n] = 1.0;
     dla[0] = 0.0;
     dmu[n] = 0.0;
     dmu[0] = 1.0;
     g[0] = 3.0 * dy[0] - 0.5 * h[0] * y21;
     g[n] = 3.0 * dy[n - 1] + 0.5 * h[n - 1] * y2n;
 
     // Solve the triagonal system of linear equations.
     dmu[0] = 0.5 * dmu[0];
     g[0] = 0.5 * g[0];
     for (int i = 1; i <= n; i++) {
       double t = 2.0 - dmu[i - 1] * dla[i];
       dmu[i] = dmu[i] / t;
       g[i] = (g[i] - g[i - 1] * dla[i]) / t;
     }
 
     for (int i = n - 1; i >= 0; i--) {
       g[i] = g[i] - dmu[i] * g[i + 1];
     }
 
     // Get the first derivative at each grid point.
     arraycopy(g, 0, s1, 0, n + 1);
 
     // Get the second derivative at each grid point.
     s2[0] = y21;
     s2[n] = y2n;
     for (int i = 1; i < n; i++) {
       s2[i] =
           6.0 * (y0[i + 1] - y0[i]) / (h[i] * h[i]) - 4.0 * s1[i] / h[i] - 2.0 * s1[i + 1] / h[i];
     }
   }
 
   /**
    * Computes the coefficients for a periodic interpolating cubic spline.
    *
    * @param n
    * @param xn
    * @param fn
    * @param b
    * @param d
    * @param h
    * @param du
    * @param dm
    * @param rc
    * @param rs
    */
   private void cspline(int n, double[] xn, double[] fn, double[] b, double[] c, double[] d,
                        double[] h, double[] du, double[] dm, double[] rc, double[] rs) {
     double eps = 0.000001;
     if (abs(fn[n] - fn[0]) > eps) {
       logger.severe("TORTOR values are not periodic.");
     }
 
     // Make the spline exactly periodic making the ends equal to their mean.
     double mean = 0.5 * (fn[0] + fn[n]);
     fn[0] = mean;
     fn[n] = mean;
 
     // Set up auxiliary variables and matrix elements on first call.
     for (int i = 0; i < n; i++) {
       h[i] = xn[i + 1] - xn[i];
     }
     h[n] = h[0];
 
     if (n - 1 >= 0) {
       arraycopy(h, 1, du, 1, n - 1);
     }
 
     du[n] = h[0];
     for (int i = 1; i <= n; i++) {
       dm[i] = 2.0 * (h[i - 1] + h[i]);
     }
 
     // Compute the RHS.
     double temp1 = (fn[1] - fn[0]) / h[0];
     for (int i = 1; i < n; i++) {
       double temp2 = (fn[i + 1] - fn[i]) / h[i];
       rs[i] = 3.0 * (temp2 - temp1);
       temp1 = temp2;
     }
     rs[n] = 3.0 * ((fn[1] - fn[0]) / h[0] - temp1);
 
     // Solve the linear system with factorization.
     if (cytsy(n, dm, du, rc, rs, c) != 1) {
       return;
     }
 
     // Compute remaining spline coefficients.
     c[0] = c[n];
     for (int i = 0; i < n; i++) {
       b[i] = (fn[i + 1] - fn[i]) / h[i] - h[i] / 3.0 * (c[i + 1] + 2.0 * c[i]);
       d[i] = (c[i + 1] - c[i]) / (3.0 * h[i]);
     }
     b[n] = (fn[1] - fn[n]) / h[n] - h[n] / 3.0 * (c[1] + 2.0 * c[n]);
   }
 }