SphericalHarmonic.cpp

/*
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 *    this software without specific prior written permission.
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 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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 * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 * POSSIBILITY OF SUCH DAMAGE.
 *
 * 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/SphericalHarmonic.hpp"
 
#include <config.h>
 
#include <cmath>
#include <cstdio>
#include <limits>
 
#include "utils/Constants.hpp"
#include "utils/simError.h"
 
using namespace OpenMD;
 
SphericalHarmonic::SphericalHarmonic() {}
 
ComplexType SphericalHarmonic::getValueAt(RealType costheta, RealType phi) {
  RealType p;
 
  // associated Legendre polynomial
  p = Ptilde(L, M, costheta);
  ComplexType phase(0.0, (RealType)M * phi);
 
  return exp(phase) * (ComplexType)p;
}
//
// Routine to calculate the associated Legendre polynomials for m>=0
//
RealType SphericalHarmonic::LegendreP(int l, int m, RealType x) {
  RealType result;
 
  if (fabs(x) > 1.0) {
    printf("LegendreP: x out of range: l = %d\tm = %d\tx = %lf\n", l, m, x);
    return std::numeric_limits<RealType>::quiet_NaN();
  }
 
  if (m > l) {
    printf("LegendreP: m > l: l = %d\tm = %d\tx = %lf\n", l, m, x);
    return std::numeric_limits<RealType>::quiet_NaN();
  }
 
  if (m < 0) {
    printf("LegendreP: m < 0: l = %d\tm = %d\tx = %lf\n", l, m, x);
    return std::numeric_limits<RealType>::quiet_NaN();
  } else {
    RealType temp1, temp2(0.0), temp3, temp4, temp5;
    temp3 = 1.0;
 
    if (m > 0) {
      temp1 = sqrt(1.0 - pow(x, 2));
      temp5 = 1.0;
      for (int i = 1; i <= m; ++i) {
        temp3 *= -temp5 * temp1;
        temp5 += 2.0;
      }
    }
    if (l == m) {
      result = temp3;
    } else {
      temp4 = x * (2. * m + 1.) * temp3;
      if (l == (m + 1)) {
        result = temp4;
      } else {
        for (int ll = (m + 2); ll <= l; ++ll) {
          temp2 = (x * (2. * ll - 1.) * temp4 - (ll + m - 1.) * temp3) /
                  (RealType)(ll - m);
          temp3 = temp4;
          temp4 = temp2;
        }
        result = temp2;
      }
    }
  }
  return result;
}
 
//
// Routine to calculate the associated Legendre polynomials for all m...
//
RealType SphericalHarmonic::Legendre(int l, int m, RealType x) {
  RealType result;
  if (m > l || m < -l) {
    printf("Legendre got a bad argument: l = %d\tm = %d\tx = %lf\n", l, m, x);
    return std::numeric_limits<RealType>::quiet_NaN();
  } else if (m >= 0) {
    result = LegendreP(l, m, x);
  } else {
    // result = mpow(-m)*LegendreP(l,-m,x);
    result = mpow(-m) * Fact(l + m) / Fact(l - m) * LegendreP(l, -m, x);
  }
  result *= mpow(m);
  return result;
}
//
// Routine to calculate the normalized associated Legendre polynomials...
//
RealType SphericalHarmonic::Ptilde(int l, int m, RealType x) {
  RealType result;
  if (m > l || m < -l) {
    result = 0.;
  } else {
    RealType y = (RealType)(2. * l + 1.) * Fact(l - m) / Fact(l + m);
    result = mpow(m) * sqrt(y) * Legendre(l, m, x) / sqrt(4.0 * Constants::PI);
  }
  return result;
}
//
// mpow returns (-1)**m
//
RealType SphericalHarmonic::mpow(int m) {
  int result;
  if (m < 0) m = -m;
  if (m & 0x1)
    result = -1;
  else
    result = 1;
  return result;
}
//
// factorial_list is a lookup table for n!
//
static RealType factorial_list[171] = {1.,
                                       1.,
                                       2.,
                                       6.,
                                       24.,
                                       120.,
                                       720.,
                                       5040.,
                                       40320.,
                                       362880.,
                                       3.6288e6,
                                       3.99168e7,
                                       4.790016e8,
                                       6.2270208e9,
                                       8.71782912e10,
                                       1.307674368e12,
                                       2.0922789888e13,
                                       3.55687428096e14,
                                       6.402373705728e15,
                                       1.21645100408832e17,
                                       2.43290200817664e18,
                                       5.109094217170944e19,
                                       1.1240007277776077e21,
                                       2.585201673888498e22,
                                       6.204484017332394e23,
                                       1.5511210043330986e25,
                                       4.0329146112660565e26,
                                       1.0888869450418352e28,
                                       3.0488834461171387e29,
                                       8.841761993739702e30,
                                       2.6525285981219107e32,
                                       8.222838654177922e33,
                                       2.631308369336935e35,
                                       8.683317618811886e36,
                                       2.9523279903960416e38,
                                       1.0333147966386145e40,
                                       3.7199332678990125e41,
                                       1.3763753091226346e43,
                                       5.230226174666011e44,
                                       2.0397882081197444e46,
                                       8.159152832478977e47,
                                       3.345252661316381e49,
                                       1.40500611775288e51,
                                       6.041526306337383e52,
                                       2.658271574788449e54,
                                       1.1962222086548019e56,
                                       5.502622159812089e57,
                                       2.5862324151116818e59,
                                       1.2413915592536073e61,
                                       6.082818640342675e62,
                                       3.0414093201713376e64,
                                       1.5511187532873822e66,
                                       8.065817517094388e67,
                                       4.2748832840600255e69,
                                       2.308436973392414e71,
                                       1.2696403353658276e73,
                                       7.109985878048635e74,
                                       4.0526919504877214e76,
                                       2.3505613312828785e78,
                                       1.3868311854568984e80,
                                       8.32098711274139e81,
                                       5.075802138772248e83,
                                       3.146997326038794e85,
                                       1.98260831540444e87,
                                       1.2688693218588417e89,
                                       8.247650592082472e90,
                                       5.443449390774431e92,
                                       3.647111091818868e94,
                                       2.4800355424368305e96,
                                       1.711224524281413e98,
                                       1.1978571669969892e100,
                                       8.504785885678623e101,
                                       6.1234458376886085e103,
                                       4.4701154615126844e105,
                                       3.307885441519386e107,
                                       2.48091408113954e109,
                                       1.8854947016660504e111,
                                       1.4518309202828587e113,
                                       1.1324281178206297e115,
                                       8.946182130782976e116,
                                       7.156945704626381e118,
                                       5.797126020747368e120,
                                       4.753643337012842e122,
                                       3.945523969720659e124,
                                       3.314240134565353e126,
                                       2.81710411438055e128,
                                       2.4227095383672734e130,
                                       2.107757298379528e132,
                                       1.8548264225739844e134,
                                       1.650795516090846e136,
                                       1.4857159644817615e138,
                                       1.352001527678403e140,
                                       1.2438414054641308e142,
                                       1.1567725070816416e144,
                                       1.087366156656743e146,
                                       1.032997848823906e148,
                                       9.916779348709496e149,
                                       9.619275968248212e151,
                                       9.426890448883248e153,
                                       9.332621544394415e155,
                                       9.332621544394415e157,
                                       9.42594775983836e159,
                                       9.614466715035127e161,
                                       9.90290071648618e163,
                                       1.0299016745145628e166,
                                       1.081396758240291e168,
                                       1.1462805637347084e170,
                                       1.226520203196138e172,
                                       1.324641819451829e174,
                                       1.4438595832024937e176,
                                       1.588245541522743e178,
                                       1.7629525510902446e180,
                                       1.974506857221074e182,
                                       2.2311927486598138e184,
                                       2.5435597334721877e186,
                                       2.925093693493016e188,
                                       3.393108684451898e190,
                                       3.969937160808721e192,
                                       4.684525849754291e194,
                                       5.574585761207606e196,
                                       6.689502913449127e198,
                                       8.094298525273444e200,
                                       9.875044200833601e202,
                                       1.214630436702533e205,
                                       1.506141741511141e207,
                                       1.882677176888926e209,
                                       2.372173242880047e211,
                                       3.0126600184576594e213,
                                       3.856204823625804e215,
                                       4.974504222477287e217,
                                       6.466855489220474e219,
                                       8.47158069087882e221,
                                       1.1182486511960043e224,
                                       1.4872707060906857e226,
                                       1.9929427461615188e228,
                                       2.6904727073180504e230,
                                       3.659042881952549e232,
                                       5.012888748274992e234,
                                       6.917786472619489e236,
                                       9.615723196941089e238,
                                       1.3462012475717526e241,
                                       1.898143759076171e243,
                                       2.695364137888163e245,
                                       3.854370717180073e247,
                                       5.5502938327393044e249,
                                       8.047926057471992e251,
                                       1.1749972043909107e254,
                                       1.727245890454639e256,
                                       2.5563239178728654e258,
                                       3.80892263763057e260,
                                       5.713383956445855e262,
                                       8.62720977423324e264,
                                       1.3113358856834524e267,
                                       2.0063439050956823e269,
                                       3.0897696138473508e271,
                                       4.789142901463394e273,
                                       7.471062926282894e275,
                                       1.1729568794264145e278,
                                       1.853271869493735e280,
                                       2.9467022724950384e282,
                                       4.7147236359920616e284,
                                       7.590705053947219e286,
                                       1.2296942187394494e289,
                                       2.0044015765453026e291,
                                       3.287218585534296e293,
                                       5.423910666131589e295,
                                       9.003691705778438e297,
                                       1.503616514864999e300,
                                       2.5260757449731984e302,
                                       4.269068009004705e304,
                                       7.257415615307999e306};
 
//
// Routine to return the factorial of j
//
RealType SphericalHarmonic::Fact(int j) {
  if (j <= 170 && j >= 0) return factorial_list[j];
 
  snprintf(painCave.errMsg, MAX_SIM_ERROR_MSG_LENGTH, "Fact(j) for j >= 171\n");
  painCave.isFatal = 0;
  simError();
  return 0.;
}