void colvar::rmsd::calc_Jacobian_derivative()
{
// divergence of the rotated coordinates (including only derivatives of the rotation matrix)
cvm::real divergence = 0.0;
if (atoms.b_rotate) {
// gradient of the rotation matrix
cvm::matrix2d<cvm::rvector> grad_rot_mat(3, 3);
// gradients of products of 2 quaternion components
cvm::rvector g11, g22, g33, g01, g02, g03, g12, g13, g23;
for (size_t ia = 0; ia < atoms.size(); ia++) {
// Gradient of optimal quaternion wrt current Cartesian position
cvm::vector1d<cvm::rvector> &dq = atoms.rot.dQ0_1[ia];
g11 = 2.0 * (atoms.rot.q)[1]*dq[1];
g22 = 2.0 * (atoms.rot.q)[2]*dq[2];
g33 = 2.0 * (atoms.rot.q)[3]*dq[3];
g01 = (atoms.rot.q)[0]*dq[1] + (atoms.rot.q)[1]*dq[0];
g02 = (atoms.rot.q)[0]*dq[2] + (atoms.rot.q)[2]*dq[0];
g03 = (atoms.rot.q)[0]*dq[3] + (atoms.rot.q)[3]*dq[0];
g12 = (atoms.rot.q)[1]*dq[2] + (atoms.rot.q)[2]*dq[1];
g13 = (atoms.rot.q)[1]*dq[3] + (atoms.rot.q)[3]*dq[1];
g23 = (atoms.rot.q)[2]*dq[3] + (atoms.rot.q)[3]*dq[2];
// Gradient of the rotation matrix wrt current Cartesian position
grad_rot_mat[0][0] = -2.0 * (g22 + g33);
grad_rot_mat[1][0] = 2.0 * (g12 + g03);
grad_rot_mat[2][0] = 2.0 * (g13 - g02);
grad_rot_mat[0][1] = 2.0 * (g12 - g03);
grad_rot_mat[1][1] = -2.0 * (g11 + g33);
grad_rot_mat[2][1] = 2.0 * (g01 + g23);
grad_rot_mat[0][2] = 2.0 * (g02 + g13);
grad_rot_mat[1][2] = 2.0 * (g23 - g01);
grad_rot_mat[2][2] = -2.0 * (g11 + g22);
cvm::atom_pos &y = ref_pos[ia];
for (size_t alpha = 0; alpha < 3; alpha++) {
for (size_t beta = 0; beta < 3; beta++) {
divergence += grad_rot_mat[beta][alpha][alpha] * y[beta];
// Note: equation was derived for inverse rotation (see colvars paper)
// so here the matrix is transposed
// (eq would give divergence += grad_rot_mat[alpha][beta][alpha] * y[beta];)
}
}
}
}
jd.real_value = x.real_value > 0.0 ? (3.0 * atoms.size() - 4.0 - divergence) / x.real_value : 0.0;
}
void colvar::eigenvector::calc_Jacobian_derivative()
{
// gradient of the rotation matrix
cvm::matrix2d<cvm::rvector> grad_rot_mat(3, 3);
cvm::quaternion &quat0 = atoms->rot.q;
// gradients of products of 2 quaternion components
cvm::rvector g11, g22, g33, g01, g02, g03, g12, g13, g23;
cvm::real sum = 0.0;
for (size_t ia = 0; ia < atoms->size(); ia++) {
// Gradient of optimal quaternion wrt current Cartesian position
// trick: d(R^-1)/dx = d(R^t)/dx = (dR/dx)^t
// we can just transpose the derivatives of the direct matrix
cvm::vector1d<cvm::rvector> &dq_1 = atoms->rot.dQ0_1[ia];
g11 = 2.0 * quat0[1]*dq_1[1];
g22 = 2.0 * quat0[2]*dq_1[2];
g33 = 2.0 * quat0[3]*dq_1[3];
g01 = quat0[0]*dq_1[1] + quat0[1]*dq_1[0];
g02 = quat0[0]*dq_1[2] + quat0[2]*dq_1[0];
g03 = quat0[0]*dq_1[3] + quat0[3]*dq_1[0];
g12 = quat0[1]*dq_1[2] + quat0[2]*dq_1[1];
g13 = quat0[1]*dq_1[3] + quat0[3]*dq_1[1];
g23 = quat0[2]*dq_1[3] + quat0[3]*dq_1[2];
// Gradient of the inverse rotation matrix wrt current Cartesian position
// (transpose of the gradient of the direct rotation)
grad_rot_mat[0][0] = -2.0 * (g22 + g33);
grad_rot_mat[0][1] = 2.0 * (g12 + g03);
grad_rot_mat[0][2] = 2.0 * (g13 - g02);
grad_rot_mat[1][0] = 2.0 * (g12 - g03);
grad_rot_mat[1][1] = -2.0 * (g11 + g33);
grad_rot_mat[1][2] = 2.0 * (g01 + g23);
grad_rot_mat[2][0] = 2.0 * (g02 + g13);
grad_rot_mat[2][1] = 2.0 * (g23 - g01);
grad_rot_mat[2][2] = -2.0 * (g11 + g22);
for (size_t i = 0; i < 3; i++) {
for (size_t j = 0; j < 3; j++) {
sum += grad_rot_mat[i][j][i] * eigenvec[ia][j];
}
}
}
jd.real_value = sum * std::sqrt(eigenvec_invnorm2);
}
