def test_first_hyperpolarizability_or_rhf_wigner_explicit():
mol = molecule_water_sto3g_angstrom()
mol.build()
mf = pyscf.scf.RHF(mol)
mf.kernel()
C = utils.fix_mocoeffs_shape(mf.mo_coeff)
E = utils.fix_moenergies_shape(mf.mo_energy)
occupations = utils.occupations_from_pyscf_mol(mol, C)
nocc_alph, nvirt_alph, _, _ = occupations
nov_alph = nocc_alph * nvirt_alph
norb = nocc_alph + nvirt_alph
# calculate linear response vectors for electric dipole operator
f1 = 0.0
f2 = 0.0773178
frequencies = [f1, f2]
calculator = electric.Polarizability(Program.PySCF,
mol,
C,
E,
occupations,
frequencies=frequencies)
calculator.form_operators()
calculator.run()
calculator.form_results()
polarizability_1 = calculator.polarizabilities[0]
polarizability_2 = calculator.polarizabilities[1]
print("polarizability (static)")
print(polarizability_1)
print("polarizability: {} a.u.".format(f2))
print(polarizability_2)
# each operator contains multiple sets of response vectors, one
# set of components for each frequency
assert isinstance(calculator.driver.solver.operators, list)
assert len(calculator.driver.solver.operators) == 1
operator = calculator.driver.solver.operators[0]
rhsvecs = operator.mo_integrals_ai_supervector_alph
assert isinstance(operator.rspvecs_alph, list)
assert len(operator.rspvecs_alph) == 2
rspvecs_1 = operator.rspvecs_alph[0]
rspvecs_2 = operator.rspvecs_alph[1]
## Form the full [norb, norb] representation of everything.
# Response vectors: transform X_{ia} and Y_{ia} -> U_{p,q}
# 0. 'a' is fast index, 'i' slow
# 1. rspvec == [X Y]
# 2. U_{p, q} -> zero
# 3. place X_{ia} into U_{i, a}
# 4. place Y_{ia} into U_{a, i}
ncomp = rhsvecs.shape[0]
rspmats_1 = np.zeros(shape=(ncomp, norb, norb))
rspmats_2 = np.zeros(shape=(ncomp, norb, norb))
for i in range(ncomp):
rspvec_1 = rspvecs_1[i, :, 0]
rspvec_2 = rspvecs_2[i, :, 0]
x_1 = rspvec_1[:nov_alph]
y_1 = rspvec_1[nov_alph:]
x_2 = rspvec_2[:nov_alph]
y_2 = rspvec_2[nov_alph:]
x_full_1 = utils.repack_vector_to_matrix(x_1, (nvirt_alph, nocc_alph))
y_full_1 = utils.repack_vector_to_matrix(y_1, (nvirt_alph, nocc_alph))
x_full_2 = utils.repack_vector_to_matrix(x_2, (nvirt_alph, nocc_alph))
y_full_2 = utils.repack_vector_to_matrix(y_2, (nvirt_alph, nocc_alph))
rspmats_1[i, :nocc_alph, nocc_alph:] = y_full_1.T
rspmats_1[i, nocc_alph:, :nocc_alph] = x_full_1
rspmats_2[i, :nocc_alph, nocc_alph:] = y_full_2.T
rspmats_2[i, nocc_alph:, :nocc_alph] = x_full_2
rhsmats = np.zeros(shape=(ncomp, norb, norb))
for i in range(ncomp):
rhsvec = rhsvecs[i, :, 0]
rhsvec_top = rhsvec[:nov_alph]
rhsvec_bot = rhsvec[nov_alph:]
rhsvec_top_mat = utils.repack_vector_to_matrix(rhsvec_top,
(nvirt_alph, nocc_alph))
rhsvec_bot_mat = utils.repack_vector_to_matrix(rhsvec_bot,
(nvirt_alph, nocc_alph))
rhsmats[i, :nocc_alph, nocc_alph:] = rhsvec_top_mat.T
rhsmats[i, nocc_alph:, :nocc_alph] = rhsvec_bot_mat
polarizability_full_1 = np.empty_like(polarizability_1)
polarizability_full_2 = np.empty_like(polarizability_2)
for a in (0, 1, 2):
for b in (0, 1, 2):
polarizability_full_1[a, b] = 2 * np.trace(rhsmats[a, ...].T.dot(
rspmats_1[b, ...]))
polarizability_full_2[a, b] = 2 * np.trace(rhsmats[a, ...].T.dot(
rspmats_2[b, ...]))
np.testing.assert_almost_equal(polarizability_1, -polarizability_full_1)
np.testing.assert_almost_equal(polarizability_2, -polarizability_full_2)
# V_{p,q} <- full MO transformation of right hand side
integrals_ao = operator.ao_integrals
integrals_mo = np.empty_like(integrals_ao)
for i in range(ncomp):
integrals_mo[i, ...] = (C[0, ...].T).dot(integrals_ao[i,
...]).dot(C[0,
...])
# from pyresponse.ao2mo import AO2MOpyscf
# ao2mo = AO2MOpyscf(C, pyscfmol=mol)
# ao2mo.perform_rhf_full()
# tei_mo = ao2mo.tei_mo[0]
G_1 = np.empty_like(rspmats_1)
G_2 = np.empty_like(rspmats_2)
C = mf.mo_coeff
# TODO I feel as though if I have all the MO-basis two-electron
# integrals, I shouldn't need another JK build.
for i in range(ncomp):
V = integrals_mo[i, ...]
Dl_1 = (C[:, :nocc_alph]).dot(rspmats_1[i, :nocc_alph, :]).dot(C.T)
Dr_1 = (-C).dot(rspmats_1[i, :, :nocc_alph]).dot(C[:, :nocc_alph].T)
D_1 = Dl_1 + Dr_1
Dl_2 = (C[:, :nocc_alph]).dot(rspmats_2[i, :nocc_alph, :]).dot(C.T)
Dr_2 = (-C).dot(rspmats_2[i, :, :nocc_alph]).dot(C[:, :nocc_alph].T)
D_2 = Dl_2 + Dr_2
J_1, K_1 = mf.get_jk(mol, D_1, hermi=0)
J_2, K_2 = mf.get_jk(mol, D_2, hermi=0)
F_AO_1 = 2 * J_1 - K_1
F_AO_2 = 2 * J_2 - K_2
F_MO_1 = (C.T).dot(F_AO_1).dot(C)
F_MO_2 = (C.T).dot(F_AO_2).dot(C)
G_1[i, ...] = V + F_MO_1
G_2[i, ...] = V + F_MO_2
E_diag = np.diag(E[0, ...])
epsilon_1 = G_1.copy()
epsilon_2 = G_2.copy()
for i in range(ncomp):
eoU_1 = (E_diag[..., np.newaxis] + f1) * rspmats_1[i, ...]
Ue_1 = rspmats_1[i, ...] * E_diag[np.newaxis, ...]
epsilon_1[i, ...] += eoU_1 - Ue_1
eoU_2 = (E_diag[..., np.newaxis] + f2) * rspmats_2[i, ...]
Ue_2 = rspmats_2[i, ...] * E_diag[np.newaxis, ...]
epsilon_2[i, ...] += eoU_2 - Ue_2
# Assume some symmetry and calculate only part of the tensor.
hyperpolarizability = np.zeros(shape=(6, 3))
off1 = [0, 1, 2, 0, 0, 1]
off2 = [0, 1, 2, 1, 2, 2]
for r in range(6):
b = off1[r]
c = off2[r]
for a in range(3):
# _1 -> 0
# _2 -> +w
# a is _1, b is _2, c is _2 transposed/negated
tl1 = np.trace(rspmats_1[a, ...].dot(G_2[b, ...]).dot(
rspmats_2[c, ...].T)[:nocc_alph, :nocc_alph])
tl2 = np.trace(rspmats_2[c, ...].T.dot(G_2[b, ...]).dot(
rspmats_1[a, ...])[:nocc_alph, :nocc_alph])
tl3 = np.trace(rspmats_1[a, ...].dot(-G_2[c, ...].T).dot(
rspmats_2[b, ...])[:nocc_alph, :nocc_alph])
tl4 = np.trace(rspmats_2[b, ...].dot(-G_2[c, ...].T).dot(
rspmats_1[a, ...])[:nocc_alph, :nocc_alph])
tl5 = np.trace(rspmats_2[c, ...].T.dot(G_1[a, ...]).dot(
rspmats_2[b, ...])[:nocc_alph, :nocc_alph])
tl6 = np.trace(rspmats_2[b, ...].dot(G_1[a, ...]).dot(
rspmats_2[c, ...].T)[:nocc_alph, :nocc_alph])
tr1 = np.trace(rspmats_2[c, ...].T.dot(rspmats_2[b, ...]).dot(
epsilon_1[a, ...])[:nocc_alph, :nocc_alph])
tr2 = np.trace(rspmats_2[b, ...].dot(rspmats_2[c, ...].T).dot(
epsilon_1[a, ...])[:nocc_alph, :nocc_alph])
tr3 = np.trace(rspmats_2[c, ...].T.dot(rspmats_1[a, ...]).dot(
epsilon_2[b, ...])[:nocc_alph, :nocc_alph])
tr4 = np.trace(rspmats_1[a, ...].dot(rspmats_2[c, ...].T).dot(
epsilon_2[b, ...])[:nocc_alph, :nocc_alph])
tr5 = np.trace(rspmats_2[b, ...].dot(
rspmats_1[a,
...]).dot(-epsilon_2[c,
...].T)[:nocc_alph, :nocc_alph])
tr6 = np.trace(rspmats_1[a, ...].dot(
rspmats_2[b,
...]).dot(-epsilon_2[c,
...].T)[:nocc_alph, :nocc_alph])
tl = tl1 + tl2 + tl3 + tl4 + tl5 + tl6
tr = tr1 + tr2 + tr3 + tr4 + tr5 + tr6
hyperpolarizability[r, a] = -2 * (tl - tr)
# pylint: disable=C0326
ref = np.array([
[-9.02854579, 0.92998934, -0.52377445],
[2.01080066, 5.23470702, -3.01208409],
[0.66669794, 1.66112712, -0.87205853],
[0.92021130, 2.01769267, -1.11067223],
[-0.51824440, -1.11067586, 0.67140102],
[-1.10887175, -3.00950655, 1.65659586],
])
ref_avgs = np.array([6.34331713, -7.81628395, 4.40251201])
ref_avg = 10.98699590
thresh = 4.0e-5
assert np.all(np.abs(ref - hyperpolarizability) < thresh)
print("hyperpolarizability: OR, (0; {}, -{}), symmetry-unique components".
format(f2, f2))
print(hyperpolarizability)
return
def test_first_hyperpolarizability_shg_rhf_wigner_explicit_psi4numpy_pyscf_large():
mol = molecule_physicists_water_augccpvdz()
mol.build()
mf = pyscf.scf.RHF(mol)
mf.kernel()
C = utils.fix_mocoeffs_shape(mf.mo_coeff)
E = utils.fix_moenergies_shape(mf.mo_energy)
occupations = occupations_from_pyscf_mol(mol, C)
nocc_alph, nvirt_alph, _, _ = occupations
nov_alph = nocc_alph * nvirt_alph
norb = nocc_alph + nvirt_alph
# calculate linear response vectors for electric dipole operator
f1 = 0.0773178
f2 = 2 * f1
frequencies = [f1, f2]
calculator = electric.Polarizability(
Program.PySCF,
mol,
cphf.CPHF(solvers.ExactInv(C, E, occupations)),
C,
E,
occupations,
frequencies=frequencies,
)
calculator.form_operators()
calculator.run(hamiltonian=Hamiltonian.RPA, spin=Spin.singlet)
calculator.form_results()
polarizability_1 = calculator.polarizabilities[0]
polarizability_2 = calculator.polarizabilities[1]
print("polarizability: {} a.u.".format(f1))
print(polarizability_1)
print("polarizability: {} a.u. (frequency doubled)".format(f2))
print(polarizability_2)
# each operator contains multiple sets of response vectors, one
# set of components for each frequency
assert isinstance(calculator.driver.solver.operators, list)
assert len(calculator.driver.solver.operators) == 1
operator = calculator.driver.solver.operators[0]
rhsvecs = operator.mo_integrals_ai_supervector_alph
assert isinstance(operator.rspvecs_alph, list)
assert len(operator.rspvecs_alph) == 2
rspvecs_1 = operator.rspvecs_alph[0]
rspvecs_2 = operator.rspvecs_alph[1]
## Form the full [norb, norb] representation of everything.
# Response vectors: transform X_{ia} and Y_{ia} -> U_{p,q}
# 0. 'a' is fast index, 'i' slow
# 1. rspvec == [X Y]
# 2. U_{p, q} -> zero
# 3. place X_{ia} into U_{i, a}
# 4. place Y_{ia} into U_{a, i}
ncomp = rhsvecs.shape[0]
rspmats_1 = np.zeros(shape=(ncomp, norb, norb))
rspmats_2 = np.zeros(shape=(ncomp, norb, norb))
for i in range(ncomp):
rspvec_1 = rspvecs_1[i, :, 0]
rspvec_2 = rspvecs_2[i, :, 0]
x_1 = rspvec_1[:nov_alph]
y_1 = rspvec_1[nov_alph:]
x_2 = rspvec_2[:nov_alph]
y_2 = rspvec_2[nov_alph:]
x_full_1 = utils.repack_vector_to_matrix(x_1, (nvirt_alph, nocc_alph))
y_full_1 = utils.repack_vector_to_matrix(y_1, (nvirt_alph, nocc_alph))
x_full_2 = utils.repack_vector_to_matrix(x_2, (nvirt_alph, nocc_alph))
y_full_2 = utils.repack_vector_to_matrix(y_2, (nvirt_alph, nocc_alph))
rspmats_1[i, :nocc_alph, nocc_alph:] = y_full_1.T
rspmats_1[i, nocc_alph:, :nocc_alph] = x_full_1
rspmats_2[i, :nocc_alph, nocc_alph:] = y_full_2.T
rspmats_2[i, nocc_alph:, :nocc_alph] = x_full_2
rhsmats = np.zeros(shape=(ncomp, norb, norb))
for i in range(ncomp):
rhsvec = rhsvecs[i, :, 0]
rhsvec_top = rhsvec[:nov_alph]
rhsvec_bot = rhsvec[nov_alph:]
rhsvec_top_mat = utils.repack_vector_to_matrix(rhsvec_top, (nvirt_alph, nocc_alph))
rhsvec_bot_mat = utils.repack_vector_to_matrix(rhsvec_bot, (nvirt_alph, nocc_alph))
rhsmats[i, :nocc_alph, nocc_alph:] = rhsvec_top_mat.T
rhsmats[i, nocc_alph:, :nocc_alph] = rhsvec_bot_mat
polarizability_full_1 = np.empty_like(polarizability_1)
polarizability_full_2 = np.empty_like(polarizability_2)
for a in (0, 1, 2):
for b in (0, 1, 2):
polarizability_full_1[a, b] = 2 * np.trace(np.dot(rhsmats[a].T, rspmats_1[b]))
polarizability_full_2[a, b] = 2 * np.trace(np.dot(rhsmats[a].T, rspmats_2[b]))
np.testing.assert_almost_equal(polarizability_1, -polarizability_full_1)
np.testing.assert_almost_equal(polarizability_2, -polarizability_full_2)
# V_{p,q} <- full MO transformation of right hand side
integrals_ao = operator.ao_integrals
integrals_mo = np.empty_like(integrals_ao)
for i in range(ncomp):
integrals_mo[i] = (C[0].T).dot(integrals_ao[i]).dot(C[0])
G_1 = np.empty_like(rspmats_1)
G_2 = np.empty_like(rspmats_2)
C = mf.mo_coeff
# TODO I feel as though if I have all the MO-basis two-electron
# integrals, I shouldn't need another JK build.
for i in range(ncomp):
V = integrals_mo[i]
Dl_1 = (C[:, :nocc_alph]).dot(rspmats_1[i, :nocc_alph, :]).dot(C.T)
Dr_1 = (-C).dot(rspmats_1[i, :, :nocc_alph]).dot(C[:, :nocc_alph].T)
D_1 = Dl_1 + Dr_1
Dl_2 = (C[:, :nocc_alph]).dot(rspmats_2[i, :nocc_alph, :]).dot(C.T)
Dr_2 = (-C).dot(rspmats_2[i, :, :nocc_alph]).dot(C[:, :nocc_alph].T)
D_2 = Dl_2 + Dr_2
J_1, K_1 = mf.get_jk(mol, D_1, hermi=0)
J_2, K_2 = mf.get_jk(mol, D_2, hermi=0)
F_AO_1 = 2 * J_1 - K_1
F_AO_2 = 2 * J_2 - K_2
F_MO_1 = (C.T).dot(F_AO_1).dot(C)
F_MO_2 = (C.T).dot(F_AO_2).dot(C)
G_1[i] = V + F_MO_1
G_2[i] = V + F_MO_2
E_diag = np.diag(E[0, ...])
epsilon_1 = G_1.copy()
epsilon_2 = G_2.copy()
for i in range(ncomp):
eoU_1 = (E_diag[..., np.newaxis] + f1) * rspmats_1[i]
Ue_1 = rspmats_1[i] * E_diag[np.newaxis, ...]
epsilon_1[i] += eoU_1 - Ue_1
eoU_2 = (E_diag[..., np.newaxis] + f2) * rspmats_2[i]
Ue_2 = rspmats_2[i] * E_diag[np.newaxis, ...]
epsilon_2[i] += eoU_2 - Ue_2
# Assume some symmetry and calculate only part of the tensor.
hyperpolarizability = np.zeros(shape=(6, 3))
off1 = [0, 1, 2, 0, 0, 1]
off2 = [0, 1, 2, 1, 2, 2]
for r in range(6):
b = off1[r]
c = off2[r]
for a in range(3):
tl1 = np.trace(rspmats_2[a].T.dot(G_1[b]).dot(rspmats_1[c])[:nocc_alph, :nocc_alph])
tl2 = np.trace(rspmats_1[c].dot(G_1[b]).dot(rspmats_2[a].T)[:nocc_alph, :nocc_alph])
tl3 = np.trace(rspmats_2[a].T.dot(G_1[c]).dot(rspmats_1[b])[:nocc_alph, :nocc_alph])
tl4 = np.trace(rspmats_1[b].dot(G_1[c]).dot(rspmats_2[a].T)[:nocc_alph, :nocc_alph])
tl5 = np.trace(rspmats_1[c].dot(-G_2[a].T).dot(rspmats_1[b])[:nocc_alph, :nocc_alph])
tl6 = np.trace(rspmats_1[b].dot(-G_2[a].T).dot(rspmats_1[c])[:nocc_alph, :nocc_alph])
tr1 = np.trace(
rspmats_1[c].dot(rspmats_1[b]).dot(-epsilon_2[a].T)[:nocc_alph, :nocc_alph]
)
tr2 = np.trace(
rspmats_1[b].dot(rspmats_1[c]).dot(-epsilon_2[a].T)[:nocc_alph, :nocc_alph]
)
tr3 = np.trace(
rspmats_1[c].dot(rspmats_2[a].T).dot(epsilon_1[b])[:nocc_alph, :nocc_alph]
)
tr4 = np.trace(
rspmats_2[a].T.dot(rspmats_1[c]).dot(epsilon_1[b])[:nocc_alph, :nocc_alph]
)
tr5 = np.trace(
rspmats_1[b].dot(rspmats_2[a].T).dot(epsilon_1[c])[:nocc_alph, :nocc_alph]
)
tr6 = np.trace(
rspmats_2[a].T.dot(rspmats_1[b]).dot(epsilon_1[c])[:nocc_alph, :nocc_alph]
)
tl = tl1 + tl2 + tl3 + tl4 + tl5 + tl6
tr = tr1 + tr2 + tr3 + tr4 + tr5 + tr6
hyperpolarizability[r, a] = -2 * (tl - tr)
# pylint: disable=C0326
ref = np.array(
[
[0.00000000, 0.00000000, 1.92505358],
[0.00000000, 0.00000000, -31.33652886],
[0.00000000, 0.00000000, -13.92830863],
[0.00000000, 0.00000000, 0.00000000],
[-1.80626084, 0.00000000, 0.00000000],
[0.00000000, -31.13504192, 0.00000000],
]
)
ref_avgs = np.array([0.00000000, 0.00000000, 45.69300223])
ref_avg = 45.69300223
diff = np.abs(ref - hyperpolarizability)
print("abs diff")
print(diff)
thresh = 2.5e-04
assert np.all(diff < thresh)
print("hyperpolarizability: SHG, (-{}; {}, {}), symmetry-unique components".format(f2, f1, f1))
print(hyperpolarizability)
print("ref")
print(ref)
# Transpose all frequency-doubled quantities (+2w) to get -2w.
for i in range(ncomp):
rspmats_2[i] = rspmats_2[i].T
G_2[i] = -G_2[i].T
epsilon_2[i] = -epsilon_2[i].T
# Assume some symmetry and calculate only part of the tensor. This
# time, work with the in-place manipulated quantities (this tests
# their correctness).
mU = (rspmats_2, rspmats_1)
mG = (G_2, G_1)
me = (epsilon_2, epsilon_1)
hyperpolarizability = np.zeros(shape=(6, 3))
off1 = [0, 1, 2, 0, 0, 1]
off2 = [0, 1, 2, 1, 2, 2]
for r in range(6):
b = off1[r]
c = off2[r]
for a in range(3):
tl1 = np.trace(mU[0][a].dot(mG[1][b]).dot(mU[1][c])[:nocc_alph, :nocc_alph])
tl2 = np.trace(mU[1][c].dot(mG[1][b]).dot(mU[0][a])[:nocc_alph, :nocc_alph])
tl3 = np.trace(mU[0][a].dot(mG[1][c]).dot(mU[1][b])[:nocc_alph, :nocc_alph])
tl4 = np.trace(mU[1][b].dot(mG[1][c]).dot(mU[0][a])[:nocc_alph, :nocc_alph])
tl5 = np.trace(mU[1][c].dot(mG[0][a]).dot(mU[1][b])[:nocc_alph, :nocc_alph])
tl6 = np.trace(mU[1][b].dot(mG[0][a]).dot(mU[1][c])[:nocc_alph, :nocc_alph])
tr1 = np.trace(mU[1][c].dot(mU[1][b]).dot(me[0][a])[:nocc_alph, :nocc_alph])
tr2 = np.trace(mU[1][b].dot(mU[1][c]).dot(me[0][a])[:nocc_alph, :nocc_alph])
tr3 = np.trace(mU[1][c].dot(mU[0][a]).dot(me[1][b])[:nocc_alph, :nocc_alph])
tr4 = np.trace(mU[0][a].dot(mU[1][c]).dot(me[1][b])[:nocc_alph, :nocc_alph])
tr5 = np.trace(mU[1][b].dot(mU[0][a]).dot(me[1][c])[:nocc_alph, :nocc_alph])
tr6 = np.trace(mU[0][a].dot(mU[1][b]).dot(me[1][c])[:nocc_alph, :nocc_alph])
tl = [tl1, tl2, tl3, tl4, tl5, tl6]
tr = [tr1, tr2, tr3, tr4, tr5, tr6]
hyperpolarizability[r, a] = -2 * (sum(tl) - sum(tr))
assert np.all(np.abs(ref - hyperpolarizability) < thresh)
# Assume no symmetry and calculate the full tensor.
hyperpolarizability_full = np.zeros(shape=(3, 3, 3))
# components x, y, z
for ip, p in enumerate(list(product(range(3), range(3), range(3)))):
a, b, c = p
tl, tr = [], []
# 1st tuple -> index a, b, c (*not* x, y, z!)
# 2nd tuple -> index frequency (0 -> -2w, 1 -> +w)
for iq, q in enumerate(list(permutations(zip(p, (0, 1, 1)), 3))):
d, e, f = q
tlp = (mU[d[1]][d[0]]).dot(mG[e[1]][e[0]]).dot(mU[f[1]][f[0]])
tle = np.trace(tlp[:nocc_alph, :nocc_alph])
tl.append(tle)
trp = (mU[d[1]][d[0]]).dot(mU[e[1]][e[0]]).dot(me[f[1]][f[0]])
tre = np.trace(trp[:nocc_alph, :nocc_alph])
tr.append(tre)
hyperpolarizability_full[a, b, c] = -2 * (sum(tl) - sum(tr))
print("hyperpolarizability: SHG, (-{}; {}, {}), full tensor".format(f2, f1, f1))
print(hyperpolarizability_full)
# Check that the elements of the reduced and full tensors are
# equivalent.
for r in range(6):
b = off1[r]
c = off2[r]
for a in range(3):
diff = hyperpolarizability[r, a] - hyperpolarizability_full[a, b, c]
# TODO why not 14?
assert abs(diff) < 1.0e-13
# Compute averages and compare to reference.
avgs, avg = utils.form_first_hyperpolarizability_averages(hyperpolarizability_full)
assert np.allclose(ref_avgs, avgs, rtol=0, atol=1.0e-3)
assert np.allclose([ref_avg], [avg], rtol=0, atol=1.0e-3)
print(avgs)
print(avg)
return
def test_first_hyperpolarizability_static_rhf_wigner_explicit():
mol = molecule_water_sto3g_angstrom()
mol.build()
mf = pyscf.scf.RHF(mol)
mf.kernel()
C = utils.fix_mocoeffs_shape(mf.mo_coeff)
E = utils.fix_moenergies_shape(mf.mo_energy)
occupations = utils.occupations_from_pyscf_mol(mol, C)
nocc_alph, nvirt_alph, _, _ = occupations
nov_alph = nocc_alph * nvirt_alph
norb = nocc_alph + nvirt_alph
# calculate linear response vectors for electric dipole operator
calculator = electric.Polarizability(Program.PySCF,
mol,
C,
E,
occupations,
frequencies=[0.0])
calculator.form_operators()
calculator.run()
calculator.form_results()
polarizability = calculator.polarizabilities[0]
print("polarizability (static)")
print(polarizability)
operator = calculator.driver.solver.operators[0]
rhsvecs = operator.mo_integrals_ai_supervector_alph
rspvecs = operator.rspvecs_alph[0]
## Form the full [norb, norb] representation of everything.
# Response vectors: transform X_{ia} and Y_{ia} -> U_{p,q}
# 0. 'a' is fast index, 'i' slow
# 1. rspvec == [X Y]
# 2. U_{p, q} -> zero
# 3. place X_{ia} into U_{i, a}
# 4. place Y_{ia} into U_{a, i}
ncomp = rhsvecs.shape[0]
rspmats = np.zeros(shape=(ncomp, norb, norb))
for i in range(ncomp):
rspvec = rspvecs[i, :, 0]
x = rspvec[:nov_alph]
y = rspvec[nov_alph:]
x_full = utils.repack_vector_to_matrix(x, (nvirt_alph, nocc_alph))
y_full = utils.repack_vector_to_matrix(y, (nvirt_alph, nocc_alph))
rspmats[i, :nocc_alph, nocc_alph:] = x_full.T
rspmats[i, nocc_alph:, :nocc_alph] = y_full
rhsmats = np.zeros(shape=(ncomp, norb, norb))
for i in range(ncomp):
rhsvec = rhsvecs[i, :, 0]
rhsvec_top = rhsvec[:nov_alph]
rhsvec_bot = rhsvec[nov_alph:]
rhsvec_top_mat = utils.repack_vector_to_matrix(rhsvec_top,
(nvirt_alph, nocc_alph))
rhsvec_bot_mat = utils.repack_vector_to_matrix(rhsvec_bot,
(nvirt_alph, nocc_alph))
rhsmats[i, :nocc_alph, nocc_alph:] = rhsvec_top_mat.T
rhsmats[i, nocc_alph:, :nocc_alph] = rhsvec_bot_mat
polarizability_full = np.empty_like(polarizability)
for a in (0, 1, 2):
for b in (0, 1, 2):
polarizability_full[a, b] = 2 * np.trace(rhsmats[a, ...].T.dot(
rspmats[b, ...]))
np.testing.assert_almost_equal(polarizability, polarizability_full)
# V_{p,q} <- full MO transformation of right hand side
integrals_ao = operator.ao_integrals
integrals_mo = np.empty_like(integrals_ao)
for i in range(ncomp):
integrals_mo[i, ...] = (C[0, ...].T).dot(integrals_ao[i,
...]).dot(C[0,
...])
G = np.empty_like(rspmats)
C = mf.mo_coeff
# TODO I feel as though if I have all the MO-basis two-electron
# integrals, I shouldn't need another JK build.
for i in range(ncomp):
V = integrals_mo[i, ...]
Dl = (C[:, nocc_alph:].dot(
utils.repack_vector_to_matrix(rspvecs[i, :nov_alph, 0],
(nvirt_alph, nocc_alph))).dot(
C[:, :nocc_alph].T))
J, K = mf.get_jk(mol, Dl, hermi=0)
F_AO = -(4 * J - K - K.T)
F_MO = (C.T).dot(F_AO).dot(C)
G[i, ...] = V + F_MO
E_diag = np.diag(E[0, ...])
epsilon = G.copy()
omega = 0
for i in range(ncomp):
eoU = (E_diag[..., np.newaxis] + omega) * rspmats[i, ...]
Ue = rspmats[i, ...] * E_diag[np.newaxis, ...]
epsilon[i, ...] += eoU - Ue
# Assume some symmetry and calculate only part of the tensor.
hyperpolarizability = np.zeros(shape=(6, 3))
off1 = [0, 1, 2, 0, 0, 1]
off2 = [0, 1, 2, 1, 2, 2]
for r in range(6):
b = off1[r]
c = off2[r]
for a in range(3):
tl1 = 2 * np.trace(rspmats[a, ...].dot(G[b, ...]).dot(
rspmats[c, ...])[:nocc_alph, :nocc_alph])
tl2 = 2 * np.trace(rspmats[a, ...].dot(G[c, ...]).dot(
rspmats[b, ...])[:nocc_alph, :nocc_alph])
tl3 = 2 * np.trace(rspmats[c, ...].dot(G[a, ...]).dot(
rspmats[b, ...])[:nocc_alph, :nocc_alph])
tr1 = np.trace(rspmats[c, ...].dot(rspmats[b, ...]).dot(
epsilon[a, ...])[:nocc_alph, :nocc_alph])
tr2 = np.trace(rspmats[b, ...].dot(rspmats[c, ...]).dot(
epsilon[a, ...])[:nocc_alph, :nocc_alph])
tr3 = np.trace(rspmats[c, ...].dot(rspmats[a, ...]).dot(
epsilon[b, ...])[:nocc_alph, :nocc_alph])
tr4 = np.trace(rspmats[a, ...].dot(rspmats[c, ...]).dot(
epsilon[b, ...])[:nocc_alph, :nocc_alph])
tr5 = np.trace(rspmats[b, ...].dot(rspmats[a, ...]).dot(
epsilon[c, ...])[:nocc_alph, :nocc_alph])
tr6 = np.trace(rspmats[a, ...].dot(rspmats[b, ...]).dot(
epsilon[c, ...])[:nocc_alph, :nocc_alph])
tl = tl1 + tl2 + tl3
tr = tr1 + tr2 + tr3 + tr4 + tr5 + tr6
hyperpolarizability[r, a] = 2 * (tl - tr)
# pylint: disable=C0326
ref = np.array([
[-8.86822254, 0.90192130, -0.50796586],
[1.98744058, 5.13635628, -2.95319400],
[0.66008119, 1.62699646, -0.85632412],
[0.90192130, 1.98744058, -1.09505123],
[-0.50796586, -1.09505123, 0.66008119],
[-1.09505123, -2.95319400, 1.62699646],
])
ref_avgs = np.array([6.22070078, -7.66527404, 4.31748398])
ref_avg = 10.77470242
thresh = 1.5e-4
assert np.all(np.abs(ref - hyperpolarizability) < thresh)
print("hyperpolarizability (static), symmetry-unique components")
print(hyperpolarizability)
# Assume no symmetry and calculate the full tensor.
hyperpolarizability_full = np.zeros(shape=(3, 3, 3))
for p in product(range(3), range(3), range(3)):
a, b, c = p
tl, tr = 0, 0
for q in permutations(p, 3):
d, e, f = q
tl += np.trace(rspmats[d, ...].dot(G[e, ...]).dot(
rspmats[f, ...])[:nocc_alph, :nocc_alph])
tr += np.trace(rspmats[d, ...].dot(rspmats[e, ...]).dot(
epsilon[f, ...])[:nocc_alph, :nocc_alph])
hyperpolarizability_full[a, b, c] = 2 * (tl - tr)
print("hyperpolarizability (static), full tensor")
print(hyperpolarizability_full)
# Check that the elements of the reduced and full tensors are
# equivalent.
thresh = 1.0e-14
for r in range(6):
b = off1[r]
c = off2[r]
for a in range(3):
diff = hyperpolarizability[r, a] - hyperpolarizability_full[a, b,
c]
assert abs(diff) < thresh
# Compute averages and compare to reference.
# This is the slow way.
# avgs = []
# for i in range(3):
# avg_c = 0
# for j in range(3):
# avg_c += hyperpolarizability_full[i, j, j] + hyperpolarizability_full[j, i, j] + hyperpolarizability_full[j, j, i]
# avgs.append((-1/3) * avg_c)
# print(np.asarray(avgs))
x = hyperpolarizability_full
# This is the simplest non-einsum way.
# avgs = (-1 / 3) * np.asarray([np.trace(x[i, :, :] + x[:, i, :] + x[:, :, i]) for i in range(3)])
# This is the best way.
avgs = (-1 / 3) * (np.einsum("ijj->i", x) + np.einsum("jij->i", x) +
np.einsum("jji->i", x))
# print(list(set([''.join(p) for p in list(permutations('ijj', 3))])))
assert np.allclose(ref_avgs, avgs, rtol=0, atol=1.0e-3)
avg = np.sum(avgs**2)**(1 / 2)
assert np.allclose([ref_avg], [avg], rtol=0, atol=1.0e-3)
print(avgs)
print(avg)
utils_avgs, utils_avg = utils.form_first_hyperpolarizability_averages(x)
assert np.allclose(avgs, utils_avgs, rtol=0, atol=1.0e-13)
assert np.allclose([avg], [utils_avg], rtol=0, atol=1.0e-13)
return