def t1_tpdm_component(p, q, r, i, j, k):
"""
Iterate through two-RDM mapping components of T1 map
:param Int p: index for T1 term
:param Int q: index for T1 term
:param Int r: index for T1 term
:param Int i: index for T1 term
:param Int j: index for T1 term
:param Int k: index for T1 term
:return: Generator yielding DualbasisElements
"""
terms = [
DualElementStructGenerator(1.0, [(i, p)], 'cckk', (q, r, j, k)),
DualElementStructGenerator(-1.0, [(i, q)], 'cckk', (p, r, j, k)),
DualElementStructGenerator(1.0, [(i, r)], 'cckk', (p, q, j, k)),
DualElementStructGenerator(-1.0, [(j, p)], 'cckk', (q, r, i, k)),
DualElementStructGenerator(1.0, [(j, q)], 'cckk', (p, r, i, k)),
DualElementStructGenerator(-1.0, [(j, r)], 'cckk', (p, q, i, k)),
DualElementStructGenerator(1.0, [(k, p)], 'cckk', (q, r, i, j)),
DualElementStructGenerator(-1.0, [(k, q)], 'cckk', (p, r, i, j)),
DualElementStructGenerator(1.0, [(k, r)], 'cckk', (p, q, i, j))
]
# create the generator that yeilds the next non-zero 2-RDM component
for desg_term in terms:
dbe = DualBasisElement()
delta_val = 1.0
for krond_pair in desg_term.deltas:
delta_val *= kdelta(*krond_pair)
if np.isclose(delta_val, 1.0):
dbe.add_element(desg_term.tensor_name, desg_term.tensor_element,
desg_term.coeff)
yield dbe
def d2bb_d1b_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 + 1, 2 * r + 1, 2 * j + 1, 2 * r + 1),
0.5)
dbe.add_element('cckk',
(2 * j + 1, 2 * r + 1, 2 * i + 1, 2 * r + 1),
0.5)
# D1 terms
dbe.add_element('ck', (2 * i + 1, 2 * j + 1), -0.5 * normalization)
dbe.add_element('ck', (2 * j + 1, 2 * i + 1), -0.5 * normalization)
dbe.simplify()
db_basis += dbe
# dbe_list.append(dbe)
# return DualBasis(elements=dbe_list) # db_basis
return db_basis
def d1_q1_mapping(dim):
"""
Map the ck to kc
D1 + Q1 = I
:param dim: linear dimension of the 1-RDM
:return: the dual basis of the mapping
:rtype: DualBasis
"""
db = DualBasis()
dbe_list = []
for i in range(dim):
for j in range(i, dim):
dbe = DualBasisElement()
if i != j:
dbe.add_element('ck', (i, j), 0.5)
dbe.add_element('ck', (j, i), 0.5)
dbe.add_element('kc', (j, i), 0.5)
dbe.add_element('kc', (i, j), 0.5)
dbe.dual_scalar = 0.0
else:
dbe.add_element('ck', (i, j), 1.0)
dbe.add_element('kc', (i, j), 1.0)
dbe.dual_scalar = 1.0
# db += dbe
dbe_list.append(dbe)
return DualBasis(elements=dbe_list) # db
def _trace_map(tname, dim, normalization):
dbe = DualBasisElement()
for i, j in product(range(dim), repeat=2):
if i < j:
dbe.add_element(tname, (i, j, i, j), 1.0)
dbe.dual_scalar = normalization
return dbe
def trace_d2_bb(dim, Nb):
dbe = DualBasisElement()
for i, j in product(range(dim), repeat=2):
if i < j:
dbe.add_element('cckk_bb', (i, j, i, j), 2.0)
dbe.dual_scalar = Nb * (Nb - 1)
return dbe
def antisymmetry_constraints(dim):
"""
The dual basis elements representing the antisymmetry constraints
:param dim: spinless Fermion basis rank
:return: the dual basis of antisymmetry_constraints
:rtype: DualBasis
"""
# dual_basis = DualBasis()
dbe_list = []
for p, q, r, s in product(range(dim), repeat=4):
if p * dim + q <= r * dim + s:
if p < q and r < s:
tensor_elements = [
tuple(indices) for indices in _coord_generator(p, q, r, s)
]
tensor_names = ['cckk'] * len(tensor_elements)
tensor_coeffs = [0.5] * len(tensor_elements)
dbe = DualBasisElement()
for n, e, c in zip(tensor_names, tensor_elements,
tensor_coeffs):
dbe.add_element(n, e, c)
# dual_basis += dbe
dbe_list.append(dbe)
return DualBasis(elements=dbe_list)
def nb_constraint(dim, nb):
"""
:param dim:
:param sz:
:return:
"""
dbe = DualBasisElement()
for i in range(dim // 2):
dbe.add_element('ck', (2 * i + 1, 2 * i + 1), 1.0)
dbe.dual_scalar = nb
return DualBasis(elements=[dbe])
def d2_to_t1_from_iterator(dim):
"""
Generate T1 from the iteratively generated dbe elements
:param dim:
:return:
"""
db = []
# NOTE: Figure out why join_elements is not working
for p, q, r, i, j, k in product(range(dim), repeat=6):
if 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()
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)
# print(element.primal_tensors_names, element.primal_elements, element.primal_coeffs)
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)
# print(element.primal_tensors_names, element.primal_elements, element.primal_coeffs)
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)
db.append(dbe)
return DualBasis(elements=db)
def d2q2element_ab(p, q, r, s, factor, tname_d1_1, tname_d1_2, tname_d2,
tname_q2):
if tname_d1_1 != 'ck_a':
raise TypeError("For some reason I am expecting a ck_a. Ask Nick")
dbe = DualBasisElement()
dbe.add_element(tname_d1_1, (p, r), krond[q, s] * factor)
dbe.add_element(tname_d1_2, (q, s), krond[p, r] * factor)
dbe.add_element(tname_q2, (r, s, p, q), 1.0 * factor)
dbe.add_element(tname_d2, (p, q, r, s), -1.0 * factor)
dbe.dual_scalar = krond[q, s] * krond[p, r] * factor
return dbe
def s_representability_d2ab(dim, N, M, S):
"""
Constraint for S-representability
PHYSICAL REVIEW A 72, 052505 2005
:param dim: number of spatial basis functions
:param N: Total number of electrons
:param M: Sz expected value
:param S: S(S + 1) is eigenvalue of S^{2}
:return:
"""
dbe = DualBasisElement()
for i, j in product(range(dim), repeat=2):
dbe.add_element('cckk_ab', (i, j, j, i), 1.0)
dbe.dual_scalar = N / 2.0 + M**2 - S * (S + 1)
return dbe
def _d1_q1_mapping(tname_d1, tname_q1, dim):
db = DualBasis()
for i in range(dim):
for j in range(i, dim):
dbe = DualBasisElement()
if i != j:
dbe.add_element(tname_d1, (i, j), 0.5)
dbe.add_element(tname_d1, (j, i), 0.5)
dbe.add_element(tname_q1, (i, j), 0.5)
dbe.add_element(tname_q1, (j, i), 0.5)
dbe.dual_scalar = 0.0
else:
dbe.add_element(tname_d1, (i, j), 1.0)
dbe.add_element(tname_q1, (i, j), 1.0)
dbe.dual_scalar = 1.0
db += dbe # .simplify()
return db
def t1_opdm_component(p, q, r, i, j, k):
"""
Iterate through one-RDM mapping components of T1 map
:param Int p: Index for T1 matrix
:param Int q: Index for T1 matrix
:param Int r: Index for T1 matrix
:param Int i: Index for T1 matrix
:param Int j: Index for T1 matrix
:param Int k: Index for T1 matrix
:return: yield non-zero elements
"""
terms = [
DualElementStructGenerator(1.0, [(i, p), (j, q)], 'ck', (r, k)),
DualElementStructGenerator(-1.0, [(i, p), (j, r)], 'ck', (q, k)),
DualElementStructGenerator(-1.0, [(i, p), (k, q)], 'ck', (r, j)),
DualElementStructGenerator(1.0, [(i, p), (k, r)], 'ck', (q, j)),
DualElementStructGenerator(-1.0, [(i, q), (j, p)], 'ck', (r, k)),
DualElementStructGenerator(1.0, [(i, q), (j, r)], 'ck', (p, k)),
DualElementStructGenerator(1.0, [(i, q), (k, p)], 'ck', (r, j)),
DualElementStructGenerator(-1.0, [(i, q), (k, r)], 'ck', (p, j)),
DualElementStructGenerator(1.0, [(i, r), (j, p)], 'ck', (q, k)),
DualElementStructGenerator(-1.0, [(i, r), (j, q)], 'ck', (p, k)),
DualElementStructGenerator(-1.0, [(i, r), (k, p)], 'ck', (q, j)),
DualElementStructGenerator(1.0, [(i, r), (k, q)], 'ck', (p, j)),
DualElementStructGenerator(1.0, [(j, p), (k, q)], 'ck', (r, i)),
DualElementStructGenerator(-1.0, [(j, p), (k, r)], 'ck', (q, i)),
DualElementStructGenerator(-1.0, [(j, q), (k, p)], 'ck', (r, i)),
DualElementStructGenerator(1.0, [(j, q), (k, r)], 'ck', (p, i)),
DualElementStructGenerator(1.0, [(j, r), (k, p)], 'ck', (q, i)),
DualElementStructGenerator(-1.0, [(j, r), (k, q)], 'ck', (p, i))
]
# Create the generator that yeilds the next non-zero 1-RDM component
for desg_term in terms:
dbe = DualBasisElement()
delta_val = 1.0
for krond_pair in desg_term.deltas:
delta_val *= kdelta(*krond_pair)
if np.isclose(delta_val, 1.0):
dbe.add_element(desg_term.tensor_name, desg_term.tensor_element,
desg_term.coeff)
yield dbe
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 g2d2map_aabb(p, q, r, s, dim, key, factor=1.0):
"""
Accept pqrs of G2 and map to D2
"""
dbe = DualBasisElement()
# this is ugly. :(
quad = {'aabb': [0, 1], 'bbaa': [1, 0]}
dbe.add_element('ckck_aabb', (p * dim + q + quad[key][0] * dim**2,
r * dim + s + quad[key][1] * dim**2),
1.0 * factor)
dbe.add_element('ckck_aabb',
(r * dim + s + quad[key[::-1]][0] * dim**2,
p * dim + q + quad[key[::-1]][1] * dim**2),
1.0 * factor)
dbe.add_element('cckk_ab', (p, s, q, r), -1.0 * factor)
dbe.add_element('cckk_ab', (q, r, p, s), -1.0 * factor)
dbe.dual_scalar = 0.0
return dbe
def g2d2map_aa_or_bb(p, q, r, s, dim, key, factor=1.0):
"""
Accept pqrs of G2 and map to D2
"""
dbe = DualBasisElement()
quad = {'aa': [0, 0], 'bb': [1, 1]}
dbe.add_element('ckck_aabb', (p * dim + q + quad[key][0] * dim**2,
r * dim + s + quad[key][1] * dim**2),
-1.0 * factor)
dbe.add_element('ck_' + key[0], (p, r), krond[q, s] * factor)
if p != s and r != q:
gem1 = tuple(sorted([p, s]))
gem2 = tuple(sorted([r, q]))
parity = (-1)**(p < s) * (-1)**(r < q)
dbe.add_element('cckk_' + key,
(gem1[0], gem1[1], gem2[0], gem2[1]),
parity * -0.5 * factor)
dbe.dual_scalar = 0
return dbe
def sz_constraint(dim, sz):
"""
Sz constraint is on the 1-RDM
:param dim:
:param sz:
:return:
"""
dbe = DualBasisElement()
for i in range(dim // 2):
dbe.add_element('ck', (2 * i, 2 * i), 0.5)
dbe.add_element('ck', (2 * i + 1, 2 * i + 1), -0.5)
dbe.dual_scalar = sz
return DualBasis(elements=[dbe])
def d2q2element(p, q, r, s, factor):
"""
Build the dual basis element for symmetric form of 2-marginal
:param p: tensor index
:param q: tensor index
:param r: tensor index
:param s: tensor index
:param factor: scaling coeff for a symmetric constraint
:return: the dual basis of the mapping
"""
dbe = DualBasisElement()
dbe.add_element('cckk', (p, q, r, s), -1.0 * factor)
dbe.add_element('kkcc', (r, s, p, q), +1.0 * factor)
dbe.add_element('ck', (p, r), krond[q, s] * factor)
dbe.add_element('ck', (q, s), krond[p, r] * factor)
dbe.add_element('ck', (p, s), -1. * krond[q, r] * factor)
dbe.add_element('ck', (q, r), -1. * krond[p, s] * factor)
dbe.dual_scalar = (krond[q, s] * krond[p, r] -
krond[q, r] * krond[p, s]) * factor
return dbe
def trace_constraint(dim, normalization):
"""
Generate the trace constraint on the 2-RDM
:param dim: spinless Fermion basis rank
:return: the dual basis element
:rtype: DualBasisElement
"""
tensor_elements = [(i, j, i, j) for i, j in product(range(dim), repeat=2)]
tensor_names = ['cckk'] * (dim**2)
tensor_coeffs = [1.0] * (dim**2)
bias = 0
return DualBasisElement(tensor_names=tensor_names,
tensor_elements=tensor_elements,
tensor_coeffs=tensor_coeffs,
bias=bias,
scalar=normalization)
def sz_representability(dim, M):
"""
Constraint for S_z-representability
Helgaker, Jorgensen, Olsen. Sz is one-body RDM constraint
:param dim: number of spatial basis functions
:param M: Sz expected value
:return:
"""
dbe = DualBasisElement()
for i in range(dim):
dbe.add_element('ck_a', (i, i), 0.5)
dbe.add_element('ck_b', (i, i), -0.5)
dbe.dual_scalar = M
return dbe
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 g2d2map(p, q, r, s, factor=1):
"""
Build the dual basis element for a symmetric 2-marginal
:param p: tensor index
:param q: tensor index
:param r: tensor index
:param s: tensor index
:param factor: weighting of the element
:return: the dual basis element
"""
dbe = DualBasisElement()
dbe.add_element('ck', (p, r), -1. * krond[q, s] * factor)
dbe.add_element('ckck', (p, s, r, q), 1.0 * factor)
dbe.add_element('cckk', (p, q, r, s), 1.0 * factor)
dbe.dual_scalar = 0
return dbe
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 g2d2map_ab(p, q, r, s, key, factor=1.0):
dbe = DualBasisElement()
if key == 'ab':
if q == s:
dbe.add_element('ck_' + key[0], (p, r), krond[q, s] * factor)
dbe.add_element('cckk_' + key, (p, s, r, q), -1.0 * factor)
elif key == 'ba':
if q == s:
dbe.add_element('ck_' + key[0], (p, r), krond[q, s] * factor)
dbe.add_element('cckk_ab', (s, p, q, r), -1.0 * factor)
else:
raise TypeError("I only accept ab or ba blocks")
dbe.add_element('ckck_' + key, (p, q, r, s), -1.0 * factor)
dbe.dual_scalar = 0.0
return dbe
def d2q2element_ab(p, q, r, s, factor):
dbe = DualBasisElement()
if q == s:
dbe.add_element('ck_a', (p, r), factor)
if p == r:
dbe.add_element('kc_b', (s, q), -factor)
dbe.add_element('kkcc_ab', (r, s, p, q), factor)
dbe.add_element('cckk_ab', (p, q, r, s), -factor)
# dbe.dual_scalar = -krond[q, s]*krond[p, r] * factor
dbe.dual_scalar = 0
return dbe
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)
else:
dbe.add_element(
'cckk_bb',
(ir_pair[0], ir_pair[1], jr_pair[0], jr_pair[1]),
sir * sjr * 0.5)
dbe.add_element(
'cckk_bb',
(jr_pair[0], jr_pair[1], ir_pair[0], ir_pair[1]),
sir * sjr * 0.5)
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 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(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 trace_d2_ab(dim, Na, Nb):
dbe = DualBasisElement()
for i, j in product(range(dim), repeat=2):
dbe.add_element('cckk_ab', (i, j, i, j), 1.0)
dbe.dual_scalar = Na * Nb
return dbe
def d2q2element(p, q, r, s, factor, tname_d1_1, tname_d2,
tname_q2): # , spin_string):
"""
# ( 1.00000) cre(r) cre(s) des(q) des(p)
# ( 1.00000) kdelta(p,s) cre(r) des(q)
# ( -1.00000) kdelta(p,r) cre(s) des(q)
# ( -1.00000) kdelta(q,s) cre(r) des(p)
# ( 1.00000) kdelta(q,r) cre(s) des(p)
# ( -1.00000) kdelta(p,s) kdelta(q,r)
# ( 1.00000) kdelta(p,r) kdelta(q,s)
"""
dbe = DualBasisElement()
dbe.add_element(tname_q2, (p, q, r, s), -factor)
dbe.add_element(tname_d2, (r, s, p, q), factor)
if p == s:
dbe.add_element(tname_d1_1, (r, q), factor)
if p == r:
dbe.add_element(tname_d1_1, (s, q), -factor)
if q == s:
dbe.add_element(tname_d1_1, (r, p), -factor)
if q == r:
dbe.add_element(tname_d1_1, (s, p), factor)
# remember the negative sign because AX = b
dbe.dual_scalar = -factor * (krond[p, r] * krond[q, s] -
krond[p, s] * krond[q, r])
# dbe.add_element('kkcc_' + spin_string + spin_string, (r, s, p, q), factor)
# if q == s:
# dbe.add_element('ck_' + spin_string, (p, r), factor)
# if p == s:
# dbe.add_element('ck_' + spin_string, (q, r), -factor)
# if q == r:
# dbe.add_element('kc_' + spin_string, (s, p), factor)
# if p == r:
# dbe.add_element('kc_' + spin_string, (s, q), -factor)
# dbe.add_element('cckk_' + spin_string + spin_string, (p, q, r, s), -factor)
# dbe.dual_scalar = 0
return dbe
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)