RealSphericalHarmonic.cpp

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 * SUPPORT OPEN SCIENCE!  If you use OpenMD or its source code in your
 * research, please cite the appropriate papers when you publish your
 * work.  Good starting points are:
 *
 * [1] Meineke, et al., J. Comp. Chem. 26, 252-271 (2005).
 * [2] Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006).
 * [3] Sun, Lin & Gezelter, J. Chem. Phys. 128, 234107 (2008).
 * [4] Vardeman, Stocker & Gezelter, J. Chem. Theory Comput. 7, 834 (2011).
 * [5] Kuang & Gezelter, Mol. Phys., 110, 691-701 (2012).
 * [6] Lamichhane, Gezelter & Newman, J. Chem. Phys. 141, 134109 (2014).
 * [7] Lamichhane, Newman & Gezelter, J. Chem. Phys. 141, 134110 (2014).
 * [8] Bhattarai, Newman & Gezelter, Phys. Rev. B 99, 094106 (2019).
 */
 
#include "math/RealSphericalHarmonic.hpp"
 
#include <cmath>
#include <cstdio>
#include <limits>
 
using namespace OpenMD;
 
RealSphericalHarmonic::RealSphericalHarmonic() {}
 
RealType RealSphericalHarmonic::getValueAt(RealType costheta, RealType phi) {
  RealType p, phase;
 
  // associated Legendre polynomial
  p = LegendreP(L, M, costheta);
 
  if (functionType == RSH_SIN) {
    phase = sin((RealType)M * phi);
  } else {
    phase = cos((RealType)M * phi);
  }
 
  return coefficient * p * phase;
}
 
//---------------------------------------------------------------------------//
//
// RealType LegendreP (int l, int m, RealType x);
//
// Computes the value of the associated Legendre polynomial P_lm (x)
// of order l at a given point.
//
// Input:
//   l  = degree of the polynomial  >= 0
//   m  = parameter satisfying 0 <= m <= l,
//   x  = point in which the computation is performed, range -1 <= x <= 1.
// Returns:
//   value of the polynomial in x
//
//---------------------------------------------------------------------------//
RealType RealSphericalHarmonic::LegendreP(int l, int m, RealType x) {
  // check parameters
  if (m < 0 || m > l || fabs(x) > 1.0) {
    printf("LegendreP got a bad argument: l = %d\tm = %d\tx = %lf\n", l, m, x);
    //    return NAN;
    return std::numeric_limits<RealType>::quiet_NaN();
  }
 
  RealType pmm = 1.0;
  if (m > 0) {
    RealType h = sqrt((1.0 - x) * (1.0 + x)), f = 1.0;
    for (int i = 1; i <= m; i++) {
      pmm *= -f * h;
      f += 2.0;
    }
  }
  if (l == m)
    return pmm;
  else {
    RealType pmmp1 = x * (2 * m + 1) * pmm;
    if (l == (m + 1))
      return pmmp1;
    else {
      RealType pll = 0.0;
      for (int ll = m + 2; ll <= l; ll++) {
        pll   = (x * (2 * ll - 1) * pmmp1 - (ll + m - 1) * pmm) / (ll - m);
        pmm   = pmmp1;
        pmmp1 = pll;
      }
      return pll;
    }
  }
}