inline double imagePotential(double eps, double epsSolv, double radius, const Eigen::Vector3d & origin,
const Eigen::Vector3d & sp, const Eigen::Vector3d & pp) {
Eigen::Vector3d sp_origin = sp - origin;
double sp_origin_norm = sp_origin.norm();
Eigen::Vector3d pp_origin = pp - origin;
double pp_origin_norm = pp_origin.norm();
double cos_gamma = sp_origin.dot(pp_origin) / (sp_origin.norm() * pp_origin.norm());
// Clean-up cos_gamma, Legendre polynomials are only defined for -1 <= x <= 1
if (numericalZero(cos_gamma - 1)) cos_gamma = 1.0;
if (numericalZero(cos_gamma + 1)) cos_gamma = -1.0;
// Image charge position
Eigen::Vector3d r_img = origin + std::pow(radius / pp_origin_norm, 2) * pp_origin;
double sp_image_norm = (sp - r_img).norm();
// Image charge
double q_img = radius / pp_origin_norm;
// Permittivity factor
double factor = (eps - epsSolv) / (eps + epsSolv);
// Image Green's function
double G_img = factor * (q_img / sp_image_norm - q_img / sp_origin_norm);
// Image Green's function
// Accumulate Legendre polynomial expansion of image potential
double f_0 = radius / (sp_origin_norm * pp_origin_norm);
double f_l = f_0;
for (int l = 1; l <= 200; ++l) {
f_l = f_l * radius * f_0;
double C_0_l = (eps - epsSolv) * l / ((eps + epsSolv) * l + epsSolv);
double pl_x = boost::math::legendre_p(l, cos_gamma);
G_img += f_l * (C_0_l - factor) * pl_x;
}
return G_img / epsSolv;
}
inline double derivativeImagePotential(double eps, double epsSolv, double radius, const Eigen::Vector3d & origin,
const Eigen::Vector3d & sp,
const Eigen::Vector3d & ppNormal, const Eigen::Vector3d & pp) {
Eigen::Vector3d sp_origin = sp - origin;
double sp_origin_norm = sp_origin.norm();
Eigen::Vector3d pp_origin = pp - origin;
double pp_origin_norm = pp_origin.norm();
double cos_gamma = sp_origin.dot(pp_origin) / (sp_origin_norm * pp_origin_norm);
// Clean-up cos_gamma, Legendre polynomials are only defined for -1 <= x <= 1
if (numericalZero(cos_gamma - 1)) cos_gamma = 1.0;
if (numericalZero(cos_gamma + 1)) cos_gamma = -1.0;
double pp_origin_norm_3 = std::pow(pp_origin_norm, 3);
Eigen::Vector3d r_img = origin + std::pow(radius / pp_origin_norm, 2) * pp_origin;
Eigen::Vector3d pp_image = pp - r_img;
double pp_image_norm_3 = std::pow(pp_image.norm(), 3);
double factor = (eps - epsSolv) / (eps + epsSolv);
double der_G_img = factor * (radius / sp_origin_norm) * (pp_origin.dot(ppNormal) / pp_origin_norm_3 - pp_image.dot(ppNormal) / pp_image_norm_3);
/*
// Accumulate Legendre polynomial expansion of image potential
double f_0 = radius / (sp_origin_norm * pp_origin_norm);
double f_l = f_0;
double pp_origin_norm_l_3 = pp_origin_norm_3; // To accumulate (pp-origin).norm()^(l+3)
double pp_origin_norm_l_1 = pp_origin_norm; // To accumulate (pp-origin).norm()^(l+1)
for (int l = 1; l <= 200; ++l) {
f_l = f_l * radius * f_0;
pp_origin_norm_l_3 *= pp_origin_norm;
pp_origin_norm_l_1 *= pp_origin_norm;
double pl_x = boost::math::legendre_p(l, cos_gamma); // P_l(cos_gamma)
double pl_1_x = boost::math::legendre_p(l+1, cos_gamma); // P_(l+1)(cos_gamma)
double cos_denom = std::pow(cos_gamma, 2) - 1;
double tmp_a = ((l+1) * pl_x * pp_origin.dot(ppNormal)) / pp_origin_norm_l_3;
double tmp_b = ((l+1) * (pl_1_x - cos_gamma * pl_x) ) / (pp_origin_norm_l_1 * cos_denom);
double tmp_c = sp_origin.dot(ppNormal) / (sp_origin_norm * pp_origin_norm);
double tmp_d = (pp_origin.dot(sp_origin) * pp_origin.dot(ppNormal)) / (sp_origin_norm * pp_origin_norm_3);
double tmp_e = tmp_b * (tmp_c - tmp_d);
double C_0_l = (eps - epsSolv) * l / ((eps + epsSolv) * l + epsSolv);
double C_l = C_0_l - factor;
der_G_img += f_l * C_l * (-tmp_a + tmp_e);
}
*/
return der_G_img / epsSolv;
}
/*! Provides a functor for the evaluation of the system
* of first-order ODEs needed by Boost.Odeint
* The second-order ODE and the system of first-order ODEs
* are reported in the manuscript.
* \param[in] rho state vector holding the function and its first derivative
* \param[out] drhodr state vector holding the first and second derivative
* \param[in] r position on the integration grid
*/
void operator()(const StateType & rho, StateType & drhodr, const double r)
{
// Evaluate the dielectric profile
double eps = 0.0, epsPrime = 0.0;
pcm::tie(eps, epsPrime) = eval_(r);
if (numericalZero(eps)) throw std::domain_error("Division by zero!");
double gamma_epsilon = epsPrime / eps;
// System of equations is defined here
drhodr[0] = rho[1];
drhodr[1] = -rho[1] * (rho[1] + 2.0/r + gamma_epsilon) + l_ * (l_ + 1) / std::pow(r, 2);
}
/*! \fn inline void symmetryBlocking(Eigen::MatrixXd & matrix, int cavitySize, int ntsirr, int nr_irrep)
* \param[out] matrix the matrix to be block-diagonalized
* \param[in] cavitySize the size of the cavity (size of the matrix)
* \param[in] ntsirr the size of the irreducible portion of the cavity (size of the blocks)
* \param[in] nr_irrep the number of irreducible representations (number of blocks)
*/
inline void symmetryBlocking(Eigen::MatrixXd & matrix, size_t cavitySize, int ntsirr,
int nr_irrep)
{
// This function implements the simmetry-blocking of the PCM
// matrix due to point group symmetry as reported in:
// L. Frediani, R. Cammi, C. S. Pomelli, J. Tomasi and K. Ruud, J. Comput.Chem. 25, 375 (2003)
// u is the character table for the group (t in the paper)
Eigen::MatrixXd u = Eigen::MatrixXd::Zero(nr_irrep, nr_irrep);
for (int i = 0; i < nr_irrep; ++i) {
for (int j = 0; j < nr_irrep; ++j) {
u(i, j) = parity(i&j);
}
}
// Naming of indices:
// a, b, c, d run over the total size of the cavity (N)
// i, j, k, l run over the number of irreps (n)
// p, q, r, s run over the irreducible size of the cavity (N/n)
// Instead of forming U (T in the paper) and then perform the dense
// multiplication, we multiply block-by-block using just the u matrix.
// matrix = U * matrix * Ut; U * Ut = Ut * U = id
// First half-transformation, i.e. first_half = matrix * Ut
Eigen::MatrixXd first_half = Eigen::MatrixXd::Zero(cavitySize, cavitySize);
for (int i = 0; i < nr_irrep; ++i) {
int ioff = i * ntsirr;
for (int k = 0; k < nr_irrep; ++k) {
int koff = k * ntsirr;
for (int j = 0; j < nr_irrep; ++j) {
int joff = j * ntsirr;
double ujk = u(j, k) / nr_irrep;
for (int p = 0; p < ntsirr; ++p) {
int a = ioff + p;
for (int q = 0; q < ntsirr; ++q) {
int b = joff + q;
int c = koff + q;
first_half(a, c) += matrix(a, b) * ujk;
}
}
}
}
}
// Second half-transformation, i.e. matrix = U * first_half
matrix.setZero(cavitySize, cavitySize);
for (int i = 0; i < nr_irrep; ++i) {
int ioff = i * ntsirr;
for (int k = 0; k < nr_irrep; ++k) {
int koff = k * ntsirr;
for (int j = 0; j < nr_irrep; ++j) {
int joff = j * ntsirr;
double uij = u(i, j);
for (int p = 0; p < ntsirr; ++p) {
int a = ioff + p;
int b = joff + p;
for (int q = 0; q < ntsirr; ++q) {
int c = koff + q;
matrix(a, c) += uij * first_half(b, c);
}
}
}
}
}
// Traverse the matrix and discard numerical zeros
for (size_t a = 0; a < cavitySize; ++a) {
for (size_t b = 0; b < cavitySize; ++b) {
if (numericalZero(matrix(a, b))) {
matrix(a, b) = 0.0;
}
}
}
}
