def angle_hessian(xyz, iangles):
n_atoms, three = xyz.shape
if three != 3:
raise TypeError('xyz must be of length 3 in the last dimension.')
n_angles, three = iangles.shape
if three != 3:
raise TypeError('angles must have 3 columns.')
qa = internal.angles(xyz.reshape(1, n_atoms, 3), iangles).flatten()
jacobian = internal_derivs.angle_derivs(xyz, iangles)
hessian = np.zeros((n_angles, n_atoms, 3, n_atoms, 3))
for i, (m, o, n) in enumerate(iangles):
u_prime = xyz[m] - xyz[o]
v_prime = xyz[n] - xyz[o]
lambda_u = np.linalg.norm(u_prime)
lambda_v = np.linalg.norm(v_prime)
u = u_prime / lambda_u
v = v_prime / lambda_v
jac = jacobian[i]
cos = np.cos(qa[i])
sin = np.sin(qa[i])
uv = np.outer(u, v)
uu = np.outer(u, u)
vv = np.outer(v, v)
eye = np.eye(3)
term1 = (uv + uv.T + (-3 * uu + eye) * cos) / (lambda_u**2 * sin)
term2 = (uv + uv.T + (-3 * vv + eye) * cos) / (lambda_v**2 * sin)
term3 = (uu + vv - uv * cos - eye) / (lambda_u * lambda_v * sin)
term4 = (uu + vv - uv.T * cos - eye) / (lambda_u * lambda_v * sin)
hessian[i] = -(cos / sin) * np.outer(
jac.flatten(), jac.flatten()).reshape(n_atoms, 3, n_atoms, 3)
for a in [m, n, o]:
for b in [m, n, o]:
sign6(a, m, o, b, m, o)
hessian[i, a, :, b, :] += sign6(a, m, o, b, m, o) * term1
hessian[i, a, :, b, :] += sign6(a, n, o, b, n, o) * term2
hessian[i, a, :, b, :] += sign6(a, m, o, b, n, o) * term3
hessian[i, a, :, b, :] += sign6(a, n, o, b, m, o) * term4
return hessian
def angle_hessian(xyz, iangles):
n_atoms, three = xyz.shape
if three != 3:
raise TypeError('xyz must be of length 3 in the last dimension.')
n_angles, three = iangles.shape
if three != 3:
raise TypeError('angles must have 3 columns.')
qa = internal.angles(xyz.reshape(1, n_atoms, 3), iangles).flatten()
jacobian = internal_derivs.angle_derivs(xyz, iangles)
hessian = np.zeros((n_angles, n_atoms, 3, n_atoms, 3))
for i, (m, o, n) in enumerate(iangles):
u_prime = xyz[m] - xyz[o]
v_prime = xyz[n] - xyz[o]
lambda_u = np.linalg.norm(u_prime)
lambda_v = np.linalg.norm(v_prime)
u = u_prime / lambda_u
v = v_prime / lambda_v
jac = jacobian[i]
cos = np.cos(qa[i])
sin = np.sin(qa[i])
uv = np.outer(u, v)
uu = np.outer(u, u)
vv = np.outer(v, v)
eye = np.eye(3)
term1 = (uv + uv.T + (-3 * uu + eye) * cos) / (lambda_u**2 * sin)
term2 = (uv + uv.T + (-3 * vv + eye) * cos) / (lambda_v**2 * sin)
term3 = (uu + vv - uv * cos - eye) / (lambda_u * lambda_v * sin)
term4 = (uu + vv - uv.T * cos - eye) / (lambda_u * lambda_v * sin)
hessian[i] = -(cos / sin) * np.outer(jac.flatten(), jac.flatten()).reshape(n_atoms, 3, n_atoms, 3)
for a in [m, n, o]:
for b in [m, n, o]:
sign6(a,m,o, b,m,o)
hessian[i, a, :, b, :] += sign6(a,m,o, b,m,o) * term1
hessian[i, a, :, b, :] += sign6(a,n,o, b,n,o) * term2
hessian[i, a, :, b, :] += sign6(a,m,o, b,n,o) * term3
hessian[i, a, :, b, :] += sign6(a,n,o, b,m,o) * term4
return hessian
xyz = np.random.randn(4, 3)
xyz2 = xyz.copy()
xyz2[1, 1] += h
ibonds = np.array([[0, 1], [0, 2]])
iangles = np.array([[0, 1, 2], [1, 2, 3]])
idihedrals = np.array([[0, 1, 2, 3]])
# print 'TESTING BOND HESSIAN'
# jac1 = internal_derivs.bond_derivs(xyz, ibonds)
# jac2 = internal_derivs.bond_derivs(xyz2, ibonds)
# hessian = bond_hessian(xyz, ibonds)
# print ((jac2-jac1)/h)[0]
# print hessian[0, 1, 1]
print '\nTESTING ANGLE HESSIAN'
jac1 = internal_derivs.angle_derivs(xyz, iangles)
jac2 = internal_derivs.angle_derivs(xyz2, iangles)
hessian = angle_hessian(xyz, iangles)
print((jac2 - jac1) / h)[0]
print
print hessian[0, 1, 1]
print '\nTESTING DIHEDRAL HESSIAN'
jac1, hessian = dihedral_hessian(xyz, idihedrals)
jac2, hessian = dihedral_hessian(xyz2, idihedrals)
print 'These matricies should match'
print((jac2 - jac1) / h)[0]
print
print hessian[0][1, 1]
xyz2 = xyz.copy()
xyz2[1,1] += h
ibonds = np.array([[0,1], [0,2]])
iangles = np.array([[0,1,2], [1,2,3]])
idihedrals = np.array([[0,1,2,3]])
# print 'TESTING BOND HESSIAN'
# jac1 = internal_derivs.bond_derivs(xyz, ibonds)
# jac2 = internal_derivs.bond_derivs(xyz2, ibonds)
# hessian = bond_hessian(xyz, ibonds)
# print ((jac2-jac1)/h)[0]
# print hessian[0, 1, 1]
print '\nTESTING ANGLE HESSIAN'
jac1 = internal_derivs.angle_derivs(xyz, iangles)
jac2 = internal_derivs.angle_derivs(xyz2, iangles)
hessian = angle_hessian(xyz, iangles)
print ((jac2-jac1)/h)[0]
print
print hessian[0, 1, 1]
print '\nTESTING DIHEDRAL HESSIAN'
jac1, hessian = dihedral_hessian(xyz, idihedrals)
jac2, hessian = dihedral_hessian(xyz2, idihedrals)
print 'These matricies should match'
print ((jac2-jac1)/h)[0]
print
print hessian[0][1,1]