def d2aa_d1a_mapping(dim, normalization):
"""
Construct dual basis for contracting d2 -> d1
:param dim: linear dimension of the 1-RDM
:param normalization: normalization constant for coeff of D1 elememnts
:return: the dual basis of the contraction
:rtype: DualBasis
"""
db_basis = DualBasis()
dbe_list = []
dim /= 2
dim = int(dim)
for i in range(dim):
for j in range(i, dim):
dbe = DualBasisElement()
for r in range(dim):
# duplicate entries get summed in DualBasisElement
dbe.add_element('cckk', (2 * i, 2 * r, 2 * j, 2 * r), 0.5)
dbe.add_element('cckk', (2 * j, 2 * r, 2 * i, 2 * r), 0.5)
# D1 terms
dbe.add_element('ck', (2 * i, 2 * j), -0.5 * normalization)
dbe.add_element('ck', (2 * j, 2 * i), -0.5 * normalization)
dbe.simplify()
db_basis += dbe
# dbe_list.append(dbe)
# return DualBasis(elements=dbe_list) # db_basis
return db_basis
def d2_to_t1(dim):
"""
Generate the dual basis elements for mapping d2 and d1 to the T1-matrix
The T1 condition is the sum of three-marginals
T1 = <p^ q^ r^ i j k> i j k p^ q^ r^>
in such a fashion that any triples component cancels out. T1 will be
represented over antisymmeterized basis functions to save a significant amount
of space
:param dim: spin-orbital basis dimension
:return:
"""
db = [] # DualBasis()
for p, q, r, i, j, k in product(range(dim), repeat=6):
# if (p * dim**2 + q * dim + r <= i * dim**2 + j * dim + k):
print(p, q, r, i, j, k)
dbe = DualBasisElement()
dbe.dual_scalar -= (
(-1.0) * kdelta(i, p) * kdelta(j, q) * kdelta(k, r) +
(1.0) * kdelta(i, p) * kdelta(j, r) * kdelta(k, q) +
(1.0) * kdelta(i, q) * kdelta(j, p) * kdelta(k, r) +
(-1.0) * kdelta(i, q) * kdelta(j, r) * kdelta(k, p) +
(-1.0) * kdelta(i, r) * kdelta(j, p) * kdelta(k, q) +
(1.0) * kdelta(i, r) * kdelta(j, q) * kdelta(k, p))
dbe.add_element('ck', (r, k), (1.0) * kdelta(i, p) * kdelta(j, q))
dbe.add_element('ck', (q, k), (-1.0) * kdelta(i, p) * kdelta(j, r))
dbe.add_element('ck', (r, j), (-1.0) * kdelta(i, p) * kdelta(k, q))
dbe.add_element('ck', (q, j), (1.0) * kdelta(i, p) * kdelta(k, r))
dbe.add_element('ck', (r, k), (-1.0) * kdelta(i, q) * kdelta(j, p))
dbe.add_element('ck', (p, k), (1.0) * kdelta(i, q) * kdelta(j, r))
dbe.add_element('ck', (r, j), (1.0) * kdelta(i, q) * kdelta(k, p))
dbe.add_element('ck', (p, j), (-1.0) * kdelta(i, q) * kdelta(k, r))
dbe.add_element('ck', (q, k), (1.0) * kdelta(i, r) * kdelta(j, p))
dbe.add_element('ck', (p, k), (-1.0) * kdelta(i, r) * kdelta(j, q))
dbe.add_element('ck', (q, j), (-1.0) * kdelta(i, r) * kdelta(k, p))
dbe.add_element('ck', (p, j), (1.0) * kdelta(i, r) * kdelta(k, q))
dbe.add_element('ck', (r, i), (1.0) * kdelta(j, p) * kdelta(k, q))
dbe.add_element('ck', (q, i), (-1.0) * kdelta(j, p) * kdelta(k, r))
dbe.add_element('ck', (r, i), (-1.0) * kdelta(j, q) * kdelta(k, p))
dbe.add_element('ck', (p, i), (1.0) * kdelta(j, q) * kdelta(k, r))
dbe.add_element('ck', (q, i), (1.0) * kdelta(j, r) * kdelta(k, p))
dbe.add_element('ck', (p, i), (-1.0) * kdelta(j, r) * kdelta(k, q))
dbe.add_element('cckk', (q, r, j, k), (1.0) * kdelta(i, p))
dbe.add_element('cckk', (p, r, j, k), (-1.0) * kdelta(i, q))
dbe.add_element('cckk', (p, q, j, k), (1.0) * kdelta(i, r))
dbe.add_element('cckk', (q, r, i, k), (-1.0) * kdelta(j, p))
dbe.add_element('cckk', (p, r, i, k), (1.0) * kdelta(j, q))
dbe.add_element('cckk', (p, q, i, k), (-1.0) * kdelta(j, r))
dbe.add_element('cckk', (q, r, i, j), (1.0) * kdelta(k, p))
dbe.add_element('cckk', (p, r, i, j), (-1.0) * kdelta(k, q))
dbe.add_element('cckk', (p, q, i, j), (1.0) * kdelta(k, r))
dbe.add_element('t1', (p, q, r, i, j, k), -1.0)
dbe.simplify()
db.append(dbe)
return DualBasis(elements=db)
def d2_e2_mapping(dim, measured_tpdm):
"""
Generate the constraints between the error matrix, d2, and a measured d2.
:param dim: dimension of the spin-orbital basis
:param measured_tpdm: a 4-tensor of the measured 2-p
:return:
"""
db_elements = []
for p, q, r, s in product(range(dim), repeat=4):
if p * dim + q >= r * dim + s:
dbe = DualBasisElement()
# two elements of d2
dbe.add_element('cckk', (p, q, r, s), 0.5)
dbe.add_element('cckk', (r, s, p, q), 0.5)
# add four elements of the error matrix
dbe.add_element('cckk_me', (p * dim + q + dim**2, r * dim + s),
-0.25)
dbe.add_element('cckk_me', (r * dim + s + dim**2, p * dim + q),
-0.25)
dbe.add_element('cckk_me', (p * dim + q, r * dim + s + dim**2),
-0.25)
dbe.add_element('cckk_me', (r * dim + s, p * dim + q + dim**2),
-0.25)
dbe.dual_scalar = measured_tpdm[p, q, r, s].real
dbe.simplify()
# construct the dual basis element for constraining the [0, 0] orthant to be identity matrix
dbe_idenity = DualBasisElement()
if p * dim + q == r * dim + s:
dbe_idenity.add_element('cckk_me', (p * dim + q, r * dim + s),
1.0)
dbe_idenity.dual_scalar = 1.0
else:
# will a symmetric constraint provide variational freedom?
dbe_idenity.add_element('cckk_me', (p * dim + q, r * dim + s),
0.5)
dbe_idenity.add_element('cckk_me', (r * dim + s, p * dim + q),
0.5)
dbe_idenity.dual_scalar = 0.0
db_elements.append(dbe)
db_elements.append(dbe_idenity)
return DualBasis(elements=db_elements)
def _contraction_base(tname_d2, tname_d1, dim, normalization, offset):
db = DualBasis()
for i in range(dim):
for j in range(i + offset, dim):
dbe = DualBasisElement()
for r in range(dim):
dbe.add_element(tname_d2, (i, r, j, r), 0.5)
dbe.add_element(tname_d2, (j, r, i, r), 0.5)
dbe.add_element(tname_d1, (i, j), -0.5 * normalization)
dbe.add_element(tname_d1, (j, i), -0.5 * normalization)
dbe.dual_scalar = 0
dbe.simplify()
db += dbe
return db
def d2bb_d1b_mapping(dim, Nb):
"""
Map the d2_spin-adapted 2-RDM to the D1 rdm
:param Nb: number of beta electrons
:param dim:
:return:
"""
db = DualBasis()
for i in range(dim):
for j in range(i, dim):
dbe = DualBasisElement()
for r in range(dim):
# Not in the basis because always zero
if i == r or j == r:
continue
else:
sir = 1 if i < r else -1
sjr = 1 if j < r else -1
ir_pair = (i, r) if i < r else (r, i)
jr_pair = (j, r) if j < r else (r, j)
if i == j:
dbe.add_element(
'cckk_bb',
(ir_pair[0], ir_pair[1], jr_pair[0], jr_pair[1]),
sir * sjr * 0.5)
else:
# TODO: Remember why I derived a factor of 0.25 (0.5 above) for this equation.
dbe.add_element(
'cckk_bb',
(ir_pair[0], ir_pair[1], jr_pair[0], jr_pair[1]),
sir * sjr * 0.25)
dbe.add_element(
'cckk_bb',
(jr_pair[0], jr_pair[1], ir_pair[0], ir_pair[1]),
sir * sjr * 0.25)
dbe.add_element('ck_b', (i, j), -0.5 * (Nb - 1))
dbe.add_element('ck_b', (j, i), -0.5 * (Nb - 1))
dbe.dual_scalar = 0
dbe.simplify()
db += dbe
return db
def d2ab_d1b_mapping(dim, Na):
"""
Map the d2_spin-adapted 2-RDM to the D1 rdm
:param Nb: number of beta electrons
:param dim:
:return:
"""
db = DualBasis()
for i in range(dim):
for j in range(i, dim):
dbe = DualBasisElement()
for r in range(dim):
dbe.add_element('cckk_ab', (r, i, r, j), 0.5)
dbe.add_element('cckk_ab', (r, j, r, i), 0.5)
dbe.add_element('ck_b', (i, j), -0.5 * Na)
dbe.add_element('ck_b', (j, i), -0.5 * Na)
dbe.dual_scalar = 0
dbe.simplify()
db += dbe
return db
def d1a_d1b_mapping(tname_d1a, tname_d1b, dim):
db = DualBasis()
for i in range(dim):
for j in range(i, dim):
dbe = DualBasisElement()
if i != j:
dbe.add_element(tname_d1a, (i, j), 0.5)
dbe.add_element(tname_d1a, (j, i), 0.5)
dbe.add_element(tname_d1b, (i, j), -0.5)
dbe.add_element(tname_d1b, (j, i), -0.5)
dbe.dual_scalar = 0.0
else:
dbe.add_element(tname_d1a, (i, j), 1.0)
dbe.add_element(tname_d1b, (i, j), -1.0)
dbe.dual_scalar = 0.0
db += dbe.simplify()
return db
def d2_e2_mapping(dim, bas_aa, bas_ab, measured_tpdm_aa, measured_tpdm_bb,
measured_tpdm_ab):
"""
Generate constraints such that the error matrix and the d2 matrices look like the measured matrices
:param dim: spatial basis dimension
:param measured_tpdm_aa: two-marginal of alpha-alpha spins
:param measured_tpdm_bb: two-marginal of beta-beta spins
:param measured_tpdm_ab: two-marginal of alpha-beta spins
:return:
"""
db = DualBasis()
# first constrain the aa-matrix
aa_dim = dim * (dim - 1) / 2
ab_dim = dim**2
# map the aa matrix to the measured_tpdm_aa
for p, q, r, s in product(range(dim), repeat=4):
if p < q and r < s and bas_aa[(p, q)] <= bas_aa[(r, s)]:
dbe = DualBasisElement()
# two elements of D2aa
dbe.add_element('cckk_aa', (p, q, r, s), 0.5)
dbe.add_element('cckk_aa', (r, s, p, q), 0.5)
# four elements of the E2aa
dbe.add_element('cckk_me_aa',
(bas_aa[(p, q)] + aa_dim, bas_aa[(r, s)]), 0.25)
dbe.add_element('cckk_me_aa',
(bas_aa[(r, s)] + aa_dim, bas_aa[(p, q)]), 0.25)
dbe.add_element('cckk_me_aa',
(bas_aa[(p, q)], bas_aa[(r, s)] + aa_dim), 0.25)
dbe.add_element('cckk_me_aa',
(bas_aa[(r, s)], bas_aa[(p, q)] + aa_dim), 0.25)
dbe.dual_scalar = measured_tpdm_aa[bas_aa[(p, q)],
bas_aa[(r, s)]].real
dbe.simplify()
# construct the dbe for constraining the [0, 0] orthant to the idenity matrix
dbe_identity_aa = DualBasisElement()
if bas_aa[(p, q)] == bas_aa[(r, s)]:
dbe_identity_aa.add_element('cckk_me_aa',
(bas_aa[(p, q)], bas_aa[(r, s)]),
1.0)
dbe_identity_aa.dual_scalar = 1.0
else:
dbe_identity_aa.add_element('cckk_me_aa',
(bas_aa[(p, q)], bas_aa[(r, s)]),
0.5)
dbe_identity_aa.add_element('cckk_me_aa',
(bas_aa[(r, s)], bas_aa[(p, q)]),
0.5)
dbe_identity_aa.dual_scalar = 0.0
db += dbe
db += dbe_identity_aa
# map the bb matrix to the measured_tpdm_bb
for p, q, r, s in product(range(dim), repeat=4):
if p < q and r < s and bas_aa[(p, q)] <= bas_aa[(r, s)]:
dbe = DualBasisElement()
# two elements of D2bb
dbe.add_element('cckk_bb', (p, q, r, s), 0.5)
dbe.add_element('cckk_bb', (r, s, p, q), 0.5)
# four elements of the E2bb
dbe.add_element('cckk_me_bb',
(bas_aa[(p, q)] + aa_dim, bas_aa[(r, s)]), 0.25)
dbe.add_element('cckk_me_bb',
(bas_aa[(r, s)] + aa_dim, bas_aa[(p, q)]), 0.25)
dbe.add_element('cckk_me_bb',
(bas_aa[(p, q)], bas_aa[(r, s)] + aa_dim), 0.25)
dbe.add_element('cckk_me_bb',
(bas_aa[(r, s)], bas_aa[(p, q)] + aa_dim), 0.25)
dbe.dual_scalar = measured_tpdm_bb[bas_aa[(p, q)],
bas_aa[(r, s)]].real
dbe.simplify()
# construct the dbe for constraining the [0, 0] orthant to the idenity matrix
dbe_identity_bb = DualBasisElement()
if bas_aa[(p, q)] == bas_aa[(r, s)]:
dbe_identity_bb.add_element('cckk_me_bb',
(bas_aa[(p, q)], bas_aa[(r, s)]),
1.0)
dbe_identity_bb.dual_scalar = 1.0
else:
dbe_identity_bb.add_element('cckk_me_bb',
(bas_aa[(p, q)], bas_aa[(r, s)]),
0.5)
dbe_identity_bb.add_element('cckk_me_bb',
(bas_aa[(r, s)], bas_aa[(p, q)]),
0.5)
dbe_identity_bb.dual_scalar = 0.0
db += dbe
db += dbe_identity_bb
# map the ab matrix to the measured_tpdm_ab
for p, q, r, s in product(range(dim), repeat=4):
if bas_ab[(p, q)] <= bas_ab[(r, s)]:
dbe = DualBasisElement()
# two elements of D2ab
dbe.add_element('cckk_ab', (p, q, r, s), 0.5)
dbe.add_element('cckk_ab', (r, s, p, q), 0.5)
# four elements of the E2ab
dbe.add_element('cckk_me_ab',
(bas_ab[(p, q)] + ab_dim, bas_ab[(r, s)]), 0.25)
dbe.add_element('cckk_me_ab',
(bas_ab[(r, s)] + ab_dim, bas_ab[(p, q)]), 0.25)
dbe.add_element('cckk_me_ab',
(bas_ab[(p, q)], bas_ab[(r, s)] + ab_dim), 0.25)
dbe.add_element('cckk_me_ab',
(bas_ab[(r, s)], bas_ab[(p, q)] + ab_dim), 0.25)
dbe.dual_scalar = measured_tpdm_ab[bas_ab[(p, q)],
bas_ab[(r, s)]].real
dbe.simplify()
# construct the dbe for constraining the [0, 0] orthant to the idenity matrix
dbe_identity_ab = DualBasisElement()
if bas_ab[(p, q)] == bas_ab[(r, s)]:
dbe_identity_ab.add_element('cckk_me_ab',
(bas_ab[(p, q)], bas_ab[(r, s)]),
1.0)
dbe_identity_ab.dual_scalar = 1.0
else:
dbe_identity_ab.add_element('cckk_me_ab',
(bas_ab[(p, q)], bas_ab[(r, s)]),
0.5)
dbe_identity_ab.add_element('cckk_me_ab',
(bas_ab[(r, s)], bas_ab[(p, q)]),
0.5)
dbe_identity_ab.dual_scalar = 0.0
db += dbe
db += dbe_identity_ab
return db
def s_representability_d2ab_to_d2aa(dim):
"""
Constraint the antisymmetric part of the alpha-beta matrix to be equal
to the aa and bb components if a singlet
:param dim:
:return:
"""
sma = dim * (dim - 1) // 2
sms = dim * (dim + 1) // 2
uadapt, d2ab_abas, d2ab_abas_rev, d2ab_sbas, d2ab_sbas_rev = \
gen_trans_2rdm(dim**2, dim)
d2ab_bas = {}
d2aa_bas = {}
cnt_ab = 0
cnt_aa = 0
for p, q in product(range(dim), repeat=2):
d2ab_bas[(p, q)] = cnt_ab
cnt_ab += 1
if p < q:
d2aa_bas[(p, q)] = cnt_aa
cnt_aa += 1
d2ab_rev = dict(zip(d2ab_bas.values(), d2ab_bas.keys()))
d2aa_rev = dict(zip(d2aa_bas.values(), d2aa_bas.keys()))
assert uadapt.shape == (int(dim)**2, int(dim)**2)
dbe_list = []
for r, s in product(range(dim * (dim - 1) // 2), repeat=2):
if r < s:
dbe = DualBasisElement()
# lower triangle
i, j = d2aa_rev[r]
k, l = d2aa_rev[s]
# aa element should equal the triplet block aa
dbe.add_element('cckk_aa', (i, j, k, l), -0.5)
coeff_mat = uadapt[:, [r]] @ uadapt[:, [s]].T
for p, q in product(range(coeff_mat.shape[0]), repeat=2):
if not np.isclose(coeff_mat[p, q], 0):
ii, jj = d2ab_rev[p]
kk, ll = d2ab_rev[q]
dbe.add_element('cckk_ab', (ii, jj, kk, ll),
0.5 * coeff_mat[p, q])
# upper triangle . Hermitian conjugate
dbe.add_element('cckk_aa', (k, l, i, j), -0.5)
coeff_mat = uadapt[:, [s]] @ uadapt[:, [r]].T
for p, q in product(range(coeff_mat.shape[0]), repeat=2):
if not np.isclose(coeff_mat[p, q], 0):
ii, jj = d2ab_rev[p]
kk, ll = d2ab_rev[q]
dbe.add_element('cckk_ab', (ii, jj, kk, ll),
0.5 * coeff_mat[p, q])
dbe.simplify()
dbe_list.append(dbe)
elif r == s:
i, j = d2aa_rev[r]
k, l = d2aa_rev[s]
dbe = DualBasisElement()
# aa element should equal the triplet block aa
dbe.add_element('cckk_aa', (i, j, k, l), -1.0)
coeff_mat = uadapt[:, [r]] @ uadapt[:, [s]].T
for p, q in product(range(coeff_mat.shape[0]), repeat=2):
if not np.isclose(coeff_mat[p, q], 0):
ii, jj = d2ab_rev[p]
kk, ll = d2ab_rev[q]
dbe.add_element('cckk_ab', (ii, jj, kk, ll), coeff_mat[p,
q])
dbe.simplify()
dbe_list.append(dbe)
return DualBasis(elements=dbe_list)
def d2_to_t1_matrix(dim):
"""
Generate the dual basis elements for mapping d2 and d1 to the T1-matrix
The T1 condition is the sum of three-marginals
T1 = <p^ q^ r^ i j k> i j k p^ q^ r^>
in such a fashion that any triples component cancels out. T1 will be
represented over antisymmeterized basis functions to save a significant amount
of space
:param dim: spin-orbital basis dimension
:return:
"""
db = [] # DualBasis()
for p, q, r, i, j, k in product(range(dim), repeat=6):
if (p * dim**2 + q * dim + r <= i * dim**2 + j * dim + k and p != q
and q != r and p != r and i != j and j != k and i != k):
print(p, q, r, i, j, k)
dbe = DualBasisElement()
if np.isclose(p * dim**2 + q * dim + r, i * dim**2 + j * dim + k):
# diagonal term should be treated once
dbe.dual_scalar -= t1_dual_scalar(p, q, r, i, j, k)
for element in t1_opdm_component(p, q, r, i, j, k):
# dbe.join_elements(element)
dbe.add_element(element.primal_tensors_names[0],
element.primal_elements[0],
element.primal_coeffs[0])
for element in t1_tpdm_component(p, q, r, i, j, k):
# dbe.join_elements(element)
dbe.add_element(element.primal_tensors_names[0],
element.primal_elements[0],
element.primal_coeffs[0])
dbe.add_element('t1', (p, q, r, i, j, k), -1.0)
else:
dbe.dual_scalar -= t1_dual_scalar(p, q, r, i, j, k) * 0.5
for element in t1_opdm_component(p, q, r, i, j, k):
# dbe.join_elements(element)
dbe.add_element(element.primal_tensors_names[0],
element.primal_elements[0],
element.primal_coeffs[0] * 0.5)
for element in t1_tpdm_component(p, q, r, i, j, k):
# dbe.join_elements(element)
dbe.add_element(element.primal_tensors_names[0],
element.primal_elements[0],
element.primal_coeffs[0] * 0.5)
dbe.dual_scalar -= t1_dual_scalar(i, j, k, p, q, r) * 0.5
for element in t1_opdm_component(i, j, k, p, q, r):
# dbe.join_elements(element)
dbe.add_element(element.primal_tensors_names[0],
element.primal_elements[0],
element.primal_coeffs[0] * 0.5)
for element in t1_tpdm_component(i, j, k, p, q, r):
# dbe.join_elements(element)
dbe.add_element(element.primal_tensors_names[0],
element.primal_elements[0],
element.primal_coeffs[0] * 0.5)
# This is the weirdest part right here!
# we reference everything else in tensor ordering but then this is
# put into a weird matrix order. The way it works is that it
# just groups them
dbe.add_element('t1', (p, q, r, i, j, k), -0.5)
dbe.add_element('t1', (i, j, k, p, q, r), -0.5)
dbe.simplify()
db.append(dbe)
return DualBasis(elements=db)
def d2_to_t1_matrix_antisym(dim):
"""
Generate the dual basis elements for mapping d2 and d1 to the T1-matrix
The T1 condition is the sum of three-marginals
T1 = <p^ q^ r^ i j k> i j k p^ q^ r^>
in such a fashion that any triples component cancels out. T1 will be
represented over antisymmeterized basis functions to save a significant amount
of space
:param dim: spin-orbital basis dimension
:return:
"""
db = [] # DualBasis()
for p, q, r, i, j, k in product(range(dim), repeat=6):
if (p * dim**2 + q * dim + r <= i * dim**2 + j * dim + k and p < q < r
and i < j < k):
print(p, q, r, i, j, k)
dbe = DualBasisElement()
if np.isclose(p * dim**2 + q * dim + r, i * dim**2 + j * dim + k):
for pp, qq, rr, cparity in _three_parity(p, q, r):
for ii, jj, kk, rparity in _three_parity(i, j, k):
# diagonal term should be treated once
dbe.dual_scalar -= t1_dual_scalar(
pp, qq, rr, ii, jj, kk) * cparity * rparity / 6
for element in t1_opdm_component(
pp, qq, rr, ii, jj, kk):
# dbe.join_elements(element)
dbe.add_element(
element.primal_tensors_names[0],
element.primal_elements[0], cparity * rparity *
element.primal_coeffs[0] / 6)
for element in t1_tpdm_component(
pp, qq, rr, ii, jj, kk):
# dbe.join_elements(element)
dbe.add_element(
element.primal_tensors_names[0],
element.primal_elements[0], cparity * rparity *
element.primal_coeffs[0] / 6)
dbe.add_element('t1', (p, q, r, i, j, k), -1.0)
else:
for pp, qq, rr, cparity in _three_parity(p, q, r):
for ii, jj, kk, rparity in _three_parity(i, j, k):
dbe.dual_scalar -= t1_dual_scalar(
pp, qq, rr, ii, jj, kk) * 0.5 / 6
for element in t1_opdm_component(
pp, qq, rr, ii, jj, kk):
dbe.add_element(
element.primal_tensors_names[0],
element.primal_elements[0], cparity * rparity *
element.primal_coeffs[0] * 0.5 / 6)
for element in t1_tpdm_component(
pp, qq, rr, ii, jj, kk):
dbe.add_element(
element.primal_tensors_names[0],
element.primal_elements[0], cparity * rparity *
element.primal_coeffs[0] * 0.5 / 6)
dbe.add_element('t1', (p, q, r, i, j, k), -0.5)
for pp, qq, rr, cparity in _three_parity(i, j, k):
for ii, jj, kk, rparity in _three_parity(p, q, r):
dbe.dual_scalar -= t1_dual_scalar(
pp, qq, rr, ii, jj, kk) * 0.5 / 6
for element in t1_opdm_component(
pp, qq, rr, ii, jj, kk):
# dbe.join_elements(element)
dbe.add_element(
element.primal_tensors_names[0],
element.primal_elements[0], cparity * rparity *
element.primal_coeffs[0] * 0.5 / 6)
for element in t1_tpdm_component(
pp, qq, rr, ii, jj, kk):
# dbe.join_elements(element)
dbe.add_element(
element.primal_tensors_names[0],
element.primal_elements[0], cparity * rparity *
element.primal_coeffs[0] * 0.5 / 6)
dbe.add_element('t1', (i, j, k, p, q, r), -0.5)
dbe.simplify()
db.append(dbe)
return DualBasis(elements=db)