def test_create_f():
i, j, n, m = symbols('i,j,n,m')
o = Fd(i)
assert isinstance(o, CreateFermion)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = Fd(1)
assert o.apply_operator(FKet([n])) == FKet([1, n])
assert o.apply_operator(FKet([n])) == -FKet([n, 1])
o = Fd(n)
assert o.apply_operator(FKet([])) == FKet([n])
vacuum = FKet([], fermi_level=4)
assert vacuum == FKet([], fermi_level=4)
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
assert Fd(i).apply_operator(FKet([i, j, k], 4)) == FKet([j, k], 4)
assert Fd(a).apply_operator(FKet([i, b, k], 4)) == FKet([a, i, b, k], 4)
assert Dagger(B(p)).apply_operator(q) == q*CreateBoson(p)
assert repr(Fd(p)) == 'CreateFermion(p)'
assert srepr(Fd(p)) == "CreateFermion(Symbol('p'))"
assert latex(Fd(p)) == r'a^\dagger_{p}'
def test_create_f():
i, j, n, m = symbols('i j n m')
o = Fd(i)
assert isinstance(o, CreateFermion)
o = o.subs(i, j)
assert o.atoms(Symbol) == set([j])
o = Fd(1)
assert o.apply_operator(FKet([n])) == FKet([1,n])
assert o.apply_operator(FKet([n])) ==-FKet([n,1])
o = Fd(n)
assert o.apply_operator(FKet([])) == FKet([n])
vacuum = FKet([],fermi_level=4)
assert vacuum == FKet([],fermi_level=4)
i,j,k,l = symbols('ijkl',below_fermi=True)
a,b,c,d = symbols('abcd',above_fermi=True)
p,q,r,s = symbols('pqrs')
assert Fd(i).apply_operator(FKet([i,j,k],4)) == FKet([j,k],4)
assert Fd(a).apply_operator(FKet([i,b,k],4)) == FKet([a,i,b,k],4)
def test_create_f():
i, j, n, m = symbols('i,j,n,m')
o = Fd(i)
assert isinstance(o, CreateFermion)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = Fd(1)
assert o.apply_operator(FKet([n])) == FKet([1, n])
assert o.apply_operator(FKet([n])) == -FKet([n, 1])
o = Fd(n)
assert o.apply_operator(FKet([])) == FKet([n])
vacuum = FKet([], fermi_level=4)
assert vacuum == FKet([], fermi_level=4)
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
assert Fd(i).apply_operator(FKet([i, j, k], 4)) == FKet([j, k], 4)
assert Fd(a).apply_operator(FKet([i, b, k], 4)) == FKet([a, i, b, k], 4)
def test_commutation():
n, m = symbols("n,m", above_fermi=True)
c = Commutator(B(0), Bd(0))
assert c == 1
c = Commutator(Bd(0), B(0))
assert c == -1
c = Commutator(B(n), Bd(0))
assert c == KroneckerDelta(n, 0)
c = Commutator(B(0), Bd(0))
e = simplify(apply_operators(c * BKet([n])))
assert e == BKet([n])
c = Commutator(B(0), B(1))
e = simplify(apply_operators(c * BKet([n, m])))
assert e == 0
c = Commutator(F(m), Fd(m))
assert c == +1 - 2 * NO(Fd(m) * F(m))
c = Commutator(Fd(m), F(m))
assert c.expand() == -1 + 2 * NO(Fd(m) * F(m))
C = Commutator
X, Y, Z = symbols('X,Y,Z', commutative=False)
assert C(C(X, Y), Z) != 0
assert C(C(X, Z), Y) != 0
assert C(Y, C(X, Z)) != 0
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
D = KroneckerDelta
assert C(Fd(a), F(i)) == -2 * NO(F(i) * Fd(a))
assert C(Fd(j), NO(Fd(a) * F(i))).doit(wicks=True) == -D(j, i) * Fd(a)
assert C(Fd(a) * F(i), Fd(b) * F(j)).doit(wicks=True) == 0
def main():
print()
print("Calculates the Coupled-Cluster energy- and amplitude equations")
print("See 'An Introduction to Coupled Cluster Theory' by")
print("T. Daniel Crawford and Henry F. Schaefer III")
print(
"Reference to a Lecture Series: http://vergil.chemistry.gatech.edu/notes/sahan-cc-2010.pdf"
)
print()
# setup hamiltonian
p, q, r, s = symbols('p,q,r,s', cls=Dummy)
f = AntiSymmetricTensor('f', (p, ), (q, ))
pr = NO(Fd(p) * F(q))
v = AntiSymmetricTensor('v', (p, q), (r, s))
pqsr = NO(Fd(p) * Fd(q) * F(s) * F(r))
H = f * pr + Rational(1, 4) * v * pqsr
print("Using the hamiltonian:", latex(H))
print("Calculating 4 nested commutators")
C = Commutator
T1, T2 = get_CC_operators()
T = T1 + T2
print("commutator 1...")
comm1 = wicks(C(H, T))
comm1 = evaluate_deltas(comm1)
comm1 = substitute_dummies(comm1)
T1, T2 = get_CC_operators()
T = T1 + T2
print("commutator 2...")
comm2 = wicks(C(comm1, T))
comm2 = evaluate_deltas(comm2)
comm2 = substitute_dummies(comm2)
T1, T2 = get_CC_operators()
T = T1 + T2
print("commutator 3...")
comm3 = wicks(C(comm2, T))
comm3 = evaluate_deltas(comm3)
comm3 = substitute_dummies(comm3)
T1, T2 = get_CC_operators()
T = T1 + T2
print("commutator 4...")
comm4 = wicks(C(comm3, T))
comm4 = evaluate_deltas(comm4)
comm4 = substitute_dummies(comm4)
print("construct Hausdorff expansion...")
eq = H + comm1 + comm2 / 2 + comm3 / 6 + comm4 / 24
eq = eq.expand()
eq = evaluate_deltas(eq)
eq = substitute_dummies(eq,
new_indices=True,
pretty_indices=pretty_dummies_dict)
print("*********************")
print()
print("extracting CC equations from full Hbar")
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
print()
print("CC Energy:")
print(
latex(wicks(eq, simplify_dummies=True,
keep_only_fully_contracted=True)))
# print("HERE")
# print("HERE")
# print("HERE")
# print(pycode(wicks(eq, simplify_dummies=True,
# keep_only_fully_contracted=True)))
# with open("cc_energy.py", "w") as f:
# f.
with open("ccsd.jl", "w") as f:
eq_energy = wicks(eq,
simplify_dummies=True,
keep_only_fully_contracted=True)
f.write(julia_code(eq_energy))
print()
print("CC T1:")
eqT1 = wicks(NO(Fd(i) * F(a)) * eq,
simplify_kronecker_deltas=True,
keep_only_fully_contracted=True)
eqT1 = substitute_dummies(eqT1)
print(latex(eqT1))
print()
print("CC T2:")
eqT2 = wicks(NO(Fd(i) * Fd(j) * F(b) * F(a)) * eq,
simplify_dummies=True,
keep_only_fully_contracted=True,
simplify_kronecker_deltas=True)
# P = PermutationOperator
# eqT2 = simplify_index_permutations(eqT2, [P(a, b), P(i, j)])
print(latex(eqT2))
print(latex(simplify(eqT2)))
from sympy import symbols, latex, WildFunction, collect, Rational, simplify
from sympy.physics.secondquant import F, Fd, wicks, AntiSymmetricTensor, substitute_dummies, NO, evaluate deltas
"""
Define Hamiltonian and the second-quantized representation of a three-body Slater determinant.
"""
# Define Hamiltonian
p, q, r, s = symbols("p q r s", dummy=True)
f = AntiSymmetricTensor("f", (p,), (q,))
pr = Fd(p) * F(q)
v = AntiSymmetricTensor("v", (p, q), (r, s))
pqsr = Fd(p) * Fd(q) * F(s) * F(r)
Hamiltonian = f * pr + Rational(1) / Rational(4) * v * pqsr
a, b, c, d, e, f = symbols("a, b, c, d, e, f", above_fermi=True)
# Create teh representatoin
expression = wicks(F(c) * F(b) * F(a) * Hamiltonian * Fd(d) * Fd(e) * Fd(f), keep_only_fully_contracted=True, simplify_kronecker_deltas=True)
expression = evaluate_deltas(expression)
expression = simplify(expression)
print(latex(expression))
def main():
print()
print("Calculates the Coupled-Cluster energy- and amplitude equations")
print("See 'An Introduction to Coupled Cluster Theory' by")
print("T. Daniel Crawford and Henry F. Schaefer III")
print("http://www.ccc.uga.edu/lec_top/cc/html/review.html")
print()
# setup hamiltonian
p, q, r, s = symbols('p,q,r,s', cls=Dummy)
f = AntiSymmetricTensor('f', (p,), (q,))
pr = NO((Fd(p)*F(q)))
v = AntiSymmetricTensor('v', (p, q), (r, s))
pqsr = NO(Fd(p)*Fd(q)*F(s)*F(r))
H = f*pr + Rational(1, 4)*v*pqsr
print("Using the hamiltonian:", latex(H))
print("Calculating 4 nested commutators")
C = Commutator
T1, T2 = get_CC_operators()
T = T1 + T2
print("commutator 1...")
comm1 = wicks(C(H, T))
comm1 = evaluate_deltas(comm1)
comm1 = substitute_dummies(comm1)
T1, T2 = get_CC_operators()
T = T1 + T2
print("commutator 2...")
comm2 = wicks(C(comm1, T))
comm2 = evaluate_deltas(comm2)
comm2 = substitute_dummies(comm2)
T1, T2 = get_CC_operators()
T = T1 + T2
print("commutator 3...")
comm3 = wicks(C(comm2, T))
comm3 = evaluate_deltas(comm3)
comm3 = substitute_dummies(comm3)
T1, T2 = get_CC_operators()
T = T1 + T2
print("commutator 4...")
comm4 = wicks(C(comm3, T))
comm4 = evaluate_deltas(comm4)
comm4 = substitute_dummies(comm4)
print("construct Hausdoff expansion...")
eq = H + comm1 + comm2/2 + comm3/6 + comm4/24
eq = eq.expand()
eq = evaluate_deltas(eq)
eq = substitute_dummies(eq, new_indices=True,
pretty_indices=pretty_dummies_dict)
print("*********************")
print()
print("extracting CC equations from full Hbar")
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
print()
print("CC Energy:")
print(latex(wicks(eq, simplify_dummies=True,
keep_only_fully_contracted=True)))
print()
print("CC T1:")
eqT1 = wicks(NO(Fd(i)*F(a))*eq, simplify_kronecker_deltas=True, keep_only_fully_contracted=True)
eqT1 = substitute_dummies(eqT1)
print(latex(eqT1))
print()
print("CC T2:")
eqT2 = wicks(NO(Fd(i)*Fd(j)*F(b)*F(a))*eq, simplify_dummies=True, keep_only_fully_contracted=True, simplify_kronecker_deltas=True)
P = PermutationOperator
eqT2 = simplify_index_permutations(eqT2, [P(a, b), P(i, j)])
print(latex(eqT2))
def test_sorting():
i, j = symbols('i,j', below_fermi=True)
a, b = symbols('a,b', above_fermi=True)
p, q = symbols('p,q')
# p, q
assert _sort_anticommuting_fermions([Fd(p), F(q)]) == ([Fd(p), F(q)], 0)
assert _sort_anticommuting_fermions([F(p), Fd(q)]) == ([Fd(q), F(p)], 1)
# i, p
assert _sort_anticommuting_fermions([F(p), Fd(i)]) == ([F(p), Fd(i)], 0)
assert _sort_anticommuting_fermions([Fd(i), F(p)]) == ([F(p), Fd(i)], 1)
assert _sort_anticommuting_fermions([Fd(p), Fd(i)]) == ([Fd(p), Fd(i)], 0)
assert _sort_anticommuting_fermions([Fd(i), Fd(p)]) == ([Fd(p), Fd(i)], 1)
assert _sort_anticommuting_fermions([F(p), F(i)]) == ([F(i), F(p)], 1)
assert _sort_anticommuting_fermions([F(i), F(p)]) == ([F(i), F(p)], 0)
assert _sort_anticommuting_fermions([Fd(p), F(i)]) == ([F(i), Fd(p)], 1)
assert _sort_anticommuting_fermions([F(i), Fd(p)]) == ([F(i), Fd(p)], 0)
# a, p
assert _sort_anticommuting_fermions([F(p), Fd(a)]) == ([Fd(a), F(p)], 1)
assert _sort_anticommuting_fermions([Fd(a), F(p)]) == ([Fd(a), F(p)], 0)
assert _sort_anticommuting_fermions([Fd(p), Fd(a)]) == ([Fd(a), Fd(p)], 1)
assert _sort_anticommuting_fermions([Fd(a), Fd(p)]) == ([Fd(a), Fd(p)], 0)
assert _sort_anticommuting_fermions([F(p), F(a)]) == ([F(p), F(a)], 0)
assert _sort_anticommuting_fermions([F(a), F(p)]) == ([F(p), F(a)], 1)
assert _sort_anticommuting_fermions([Fd(p), F(a)]) == ([Fd(p), F(a)], 0)
assert _sort_anticommuting_fermions([F(a), Fd(p)]) == ([Fd(p), F(a)], 1)
# i, a
assert _sort_anticommuting_fermions([F(i), Fd(j)]) == ([F(i), Fd(j)], 0)
assert _sort_anticommuting_fermions([Fd(j), F(i)]) == ([F(i), Fd(j)], 1)
assert _sort_anticommuting_fermions([Fd(a), Fd(i)]) == ([Fd(a), Fd(i)], 0)
assert _sort_anticommuting_fermions([Fd(i), Fd(a)]) == ([Fd(a), Fd(i)], 1)
assert _sort_anticommuting_fermions([F(a), F(i)]) == ([F(i), F(a)], 1)
assert _sort_anticommuting_fermions([F(i), F(a)]) == ([F(i), F(a)], 0)
def L2(expr):
j1,j2,j3,j4,j5 = symbols('j1,j2,j3,j4,j5' ,below_fermi=True, cls=Dummy)
b1,b2,b3,b4,b5 = symbols('b1,b2,b3,b4,b5' ,above_fermi=True, cls=Dummy)
if expr == "IP":
L2 = Fraction(1, 2)*AntiSymmetricTensor('l',(j2,j3),(b1,))*Fd(j2)*Fd(j3)*F(b1)
return L2
elif expr == "DIP":
L2 = Fraction(1, 6)*AntiSymmetricTensor('l',(j3,j4,j5),(b1,))*Fd(j3)*Fd(j4)*Fd(j5)*F(b1)
return L2
elif expr == "EA":
L2 = Fraction(1, 2)*AntiSymmetricTensor('l',(j1,),(b2,b3))*Fd(j1)*F(b3)*F(b2)
return L2
elif expr == "DEA":
L2 = Fraction(1, 6)*AntiSymmetricTensor('l',(j1,),(b3,b4,b5))*Fd(j1)*F(b5)*F(b4)*F(b3)
return L2
elif expr == "EE":
L2 = Fraction(1, 4)*AntiSymmetricTensor('l',(j2,j3),(b2,b3))*Fd(j2)*Fd(j3)*F(b3)*F(b2)
return L2
elif expr == "CCSD":
L2 = Fraction(1, 4)*AntiSymmetricTensor('l',(j2,j3),(b2,b3))*Fd(j2)*Fd(j3)*F(b3)*F(b2)
return L2
def TPDM(L, R, flavor):
display(
Markdown(
rf""" Computing TPDM for {flavor} (skipping summation for dummy variables)"""
))
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
oooo = Fd(i) * Fd(j) * F(l) * F(k)
cc = BCH.level(oooo, "SD")
g_oooo = evaluate_deltas(wicks(L * cc * R,
keep_only_fully_contracted=True))
PermutList = [PermutationOperator(i,j),PermutationOperator(i,k), \
PermutationOperator(i,l),PermutationOperator(j,k), \
PermutationOperator(j,l),PermutationOperator(k,l)]
g_oooo = simplify_index_permutations(g_oooo, PermutList)
index_rule = {'below': 'mnop', 'above': 'abcde'}
g_oooo = substitute_dummies(g_oooo,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_{ijkl}')
final_eq = Eq(gamma, g_oooo)
display(final_eq)
ooov = Fd(i) * Fd(j) * F(a) * F(k)
cc = BCH.level(ooov, "SD")
g_ooov = evaluate_deltas(wicks(L * cc * R,
keep_only_fully_contracted=True))
PermutList = [PermutationOperator(i,j),PermutationOperator(i,k), \
PermutationOperator(j,k)]
g_ooov = simplify_index_permutations(g_ooov, PermutList)
index_rule = {'below': 'lmnop', 'above': 'bcdef'}
g_ooov = substitute_dummies(g_ooov,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_{ijka}')
final_eq = Eq(gamma, g_oo)
display(final_eq)
ooov = Fd(i) * Fd(a) * F(k) * F(j)
cc = BCH.level(ooov, "SD")
g_ovoo = evaluate_deltas(wicks(L * cc * R,
keep_only_fully_contracted=True))
PermutList = [PermutationOperator(i,j),PermutationOperator(i,k), \
PermutationOperator(j,k)]
g_ovoo = simplify_index_permutations(g_ovoo, PermutList)
index_rule = {'below': 'lmnop', 'above': 'bcdef'}
g_ovoo = substitute_dummies(g_ovoo,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_{iajk}')
final_eq = Eq(gamma, g_ovoo)
display(final_eq)
ovov = Fd(i) * Fd(a) * F(b) * F(j)
cc = BCH.level(ovov, "SD")
g_ovov = evaluate_deltas(wicks(L * cc * R,
keep_only_fully_contracted=True))
PermutList = [PermutationOperator(i, j), PermutationOperator(a, b)]
g_ovov = simplify_index_permutations(g_ovov, PermutList)
index_rule = {'below': 'klmno', 'above': 'cdef'}
g_ovov = substitute_dummies(g_ovov,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_{iajb}')
final_eq = Eq(gamma, g_ovov)
display(final_eq)
ovvv = Fd(i) * Fd(a) * F(c) * F(b)
cc = BCH.level(ovvv, "SD")
g_ovvv = evaluate_deltas(wicks(L * cc * R,
keep_only_fully_contracted=True))
PermutList = [PermutationOperator(a,b),PermutationOperator(a,c), \
PermutationOperator(b,c)]
g_ovvv = simplify_index_permutations(g_ovvv, PermutList)
index_rule = {'below': 'jklmn', 'above': 'defg'}
g_ovvv = substitute_dummies(g_ovvv,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_{iabc}')
final_eq = Eq(gamma, g_ovvv)
display(final_eq)
oovv = Fd(i) * Fd(j) * F(b) * F(a)
cc = BCH.level(oovv, "SD")
g_oovv = evaluate_deltas(wicks(L * cc * R,
keep_only_fully_contracted=True))
PermutList = [PermutationOperator(i, j), PermutationOperator(a, b)]
g_oovv = simplify_index_permutations(g_oovv, PermutList)
index_rule = {'below': 'klmn', 'above': 'cdefg'}
g_oovv = substitute_dummies(g_oovv,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_{ijab}')
final_eq = Eq(gamma, g_oovv)
display(final_eq)
symbols("p", above_fermi=True, cls=Dummy),
symbols("q", below_fermi=True, cls=Dummy),
),
),
(
"rho^{a}_{i} = ",
(
symbols("p", below_fermi=True, cls=Dummy),
symbols("q", above_fermi=True, cls=Dummy),
),
),
("rho^{j}_{i} = ", symbols("p, q", below_fermi=True, cls=Dummy)),
]
for label, (p, q) in symbol_list:
c_pq = Fd(p) * F(q)
T = sum(get_ccsd_t_operators())
L = sum(get_ccsd_lambda_operators())
# Only keep non-zero terms
rho_eq = eval_equation(c_pq)
rho_eq += eval_equation(Commutator(c_pq, T))
rho_eq += eval_equation(L * c_pq)
comm = Commutator(c_pq, T)
rho_eq += eval_equation(L * comm)
comm = Commutator(comm, sum(get_ccsd_t_operators()))
rho_eq += Rational(1, 2) * eval_equation(L * comm)
rho = rho_eq.expand()
rho = evaluate_deltas(rho)
def LVECTORS(L0, L1, L2, flavor):
display(
Markdown(
rf""" Computing left sigma amplitudes for {flavor} (skipping summation for dummy variables)"""
))
p, q, r, s = symbols('p,q,r,s', cls=Dummy)
f = AntiSymmetricTensor('f', (p, ), (q, ))
pr = NO((Fd(p) * F(q)))
v = AntiSymmetricTensor('v', (p, q), (r, s))
pqsr = NO(Fd(p) * Fd(q) * F(s) * F(r))
ham = f * pr + Rational(1, 4) * v * pqsr
cc = BCH.level(ham, "SD")
E_cc = evaluate_deltas(wicks(cc, keep_only_fully_contracted=True))
i, j, k = symbols('i,j,k', below_fermi=True)
a, b, c = symbols('a,b,c', above_fermi=True)
if flavor == "IP":
sig11 = evaluate_deltas(
wicks(L1 * (cc - E_cc) * F(i), keep_only_fully_contracted=True))
index_rule = {'below': 'jklmno', 'above': 'abcdefg'}
sig11 = substitute_dummies(sig11,
new_indices=True,
pretty_indices=index_rule)
if flavor == "EA":
sig11 = evaluate_deltas(
wicks(L1 * (cc - E_cc) * Fd(a), keep_only_fully_contracted=True))
index_rule = {'below': 'ijklmno', 'above': 'bcdefg'}
sig11 = substitute_dummies(sig11,
new_indices=True,
pretty_indices=index_rule)
if flavor == "DIP":
PermutList = [PermutationOperator(i, j)]
sig11 = evaluate_deltas(
wicks(L1 * (cc - E_cc) * F(j) * F(i),
keep_only_fully_contracted=True))
index_rule = {'below': 'klmno', 'above': 'abcdefg'}
sig11 = substitute_dummies(sig11,
new_indices=True,
pretty_indices=index_rule)
sig11 = simplify_index_permutations(sig11, PermutList)
if flavor == "DEA":
PermutList = [PermutationOperator(a, b)]
sig11 = evaluate_deltas(
wicks(L1 * (cc - E_cc) * Fd(a) * Fd(b),
keep_only_fully_contracted=True))
index_rule = {'below': 'ijklmno', 'above': 'cdefg'}
sig11 = substitute_dummies(sig11,
new_indices=True,
pretty_indices=index_rule)
sig11 = simplify_index_permutations(sig11, PermutList)
sigma_11 = Symbol('(L_{1}(\overline{H}_{SS}-E_{cc}))')
final_eq = Eq(sigma_11, sig11)
display(final_eq)
if flavor == "IP":
PermutList = [PermutationOperator(i, j)]
sig12 = evaluate_deltas(
wicks((L2 * cc) * F(i), keep_only_fully_contracted=True))
index_rule = {'below': 'jklmno', 'above': 'abcdefg'}
sig12 = substitute_dummies(sig12,
new_indices=True,
pretty_indices=index_rule)
sig12 = simplify_index_permutations(sig12, PermutList)
if flavor == "EA":
PermutList = [PermutationOperator(a, b)]
sig12 = evaluate_deltas(
wicks((L2 * cc) * Fd(a), keep_only_fully_contracted=True))
index_rule = {'below': 'ijklmno', 'above': 'bcdefg'}
sig12 = substitute_dummies(sig12,
new_indices=True,
pretty_indices=index_rule)
sig12 = simplify_index_permutations(sig12, PermutList)
if flavor == "DIP":
PermutList = [
PermutationOperator(i, j),
PermutationOperator(j, k),
PermutationOperator(i, k)
]
sig12 = evaluate_deltas(
wicks(L2 * cc * F(j) * F(i), keep_only_fully_contracted=True))
index_rule = {'below': 'klmno', 'above': 'abcdefg'}
sig12 = substitute_dummies(sig12,
new_indices=True,
pretty_indices=index_rule)
sig12 = simplify_index_permutations(sig12, PermutList)
if flavor == "DEA":
PermutList = [
PermutationOperator(a, b),
PermutationOperator(b, c),
PermutationOperator(a, c)
]
sig12 = evaluate_deltas(
wicks((L2 * cc) * Fd(a) * Fd(b), keep_only_fully_contracted=True))
index_rule = {'below': 'ijklmno', 'above': 'cdefg'}
sig12 = substitute_dummies(sig12,
new_indices=True,
pretty_indices=index_rule)
sig12 = simplify_index_permutations(sig12, PermutList)
sigma_12 = Symbol('(L_{2}\overline{H}_{DS})')
final_eq = Eq(sigma_12, sig12)
display(final_eq)
if flavor == "IP":
sig21 = evaluate_deltas(
wicks((L1 * cc) * Fd(a) * F(j) * F(i),
keep_only_fully_contracted=True))
index_rule = {'below': 'klmno', 'above': 'bcdefgh'}
sig21 = substitute_dummies(sig21,
new_indices=True,
pretty_indices=index_rule)
if flavor == "EA":
sig21 = evaluate_deltas(
wicks((L1 * cc) * Fd(a) * Fd(b) * F(i),
keep_only_fully_contracted=True))
index_rule = {'below': 'jklmno', 'above': 'cdefgh'}
sig21 = substitute_dummies(sig21,
new_indices=True,
pretty_indices=index_rule)
if flavor == "DIP":
PermutList = [PermutationOperator(i, j)]
sig21 = evaluate_deltas(
wicks(L1 * cc * Fd(a) * F(k) * F(j) * F(i),
keep_only_fully_contracted=True))
index_rule = {'below': 'lmno', 'above': 'bcdefgh'}
sig21 = substitute_dummies(sig21,
new_indices=True,
pretty_indices=index_rule)
sig21 = simplify_index_permutations(sig21, PermutList)
if flavor == "DEA":
PermutList = [PermutationOperator(a, b)]
sig21 = evaluate_deltas(
wicks((L1 * cc) * Fd(a) * Fd(b) * Fd(c) * F(i),
keep_only_fully_contracted=True))
index_rule = {'below': 'jklmno', 'above': 'defgh'}
sig21 = substitute_dummies(sig21,
new_indices=True,
pretty_indices=index_rule)
sig21 = simplify_index_permutations(sig21, PermutList)
sigma_21 = Symbol('(L_{1}\overline{H}_{SD})')
final_eq = Eq(sigma_21, sig21)
display(final_eq)
if flavor == "IP":
PermutList = [PermutationOperator(i, j)]
sig22 = evaluate_deltas(
wicks(L2 * (cc - E_cc) * Fd(a) * F(j) * F(i),
keep_only_fully_contracted=True))
index_rule = {'below': 'klmno', 'above': 'bcdefgh'}
sig22 = substitute_dummies(sig22,
new_indices=True,
pretty_indices=index_rule)
sig22 = simplify_index_permutations(sig22, PermutList)
if flavor == "EA":
PermutList = [PermutationOperator(a, b)]
sig22 = evaluate_deltas(
wicks(L2 * (cc - E_cc) * Fd(a) * Fd(b) * F(i),
keep_only_fully_contracted=True))
index_rule = {'below': 'jklmno', 'above': 'cdefgh'}
sig22 = substitute_dummies(sig22,
new_indices=True,
pretty_indices=index_rule)
sig22 = simplify_index_permutations(sig22, PermutList)
if flavor == "DIP":
PermutList = [
PermutationOperator(i, j),
PermutationOperator(j, k),
PermutationOperator(i, k)
]
sig22 = evaluate_deltas(
wicks(L2 * (cc - E_cc) * Fd(a) * F(k) * F(j) * F(i),
keep_only_fully_contracted=True))
index_rule = {'below': 'lmno', 'above': 'bcdefgh'}
sig22 = substitute_dummies(sig22,
new_indices=True,
pretty_indices=index_rule)
sig22 = simplify_index_permutations(sig22, PermutList)
if flavor == "DEA":
PermutList = [
PermutationOperator(a, b),
PermutationOperator(b, c),
PermutationOperator(a, c)
]
sig22 = evaluate_deltas(
wicks(L2 * (cc - E_cc) * Fd(a) * Fd(b) * Fd(c) * F(i),
keep_only_fully_contracted=True))
index_rule = {'below': 'jklmno', 'above': 'defgh'}
sig22 = substitute_dummies(sig22,
new_indices=True,
pretty_indices=index_rule)
sig22 = simplify_index_permutations(sig22, PermutList)
sigma_22 = Symbol('(L_{2}(\overline{H}_{DD}-E_{cc}))')
final_eq = Eq(sigma_22, sig22)
display(final_eq)
a, b = symbols('a,b', above_fermi=True, cls=Dummy)
t_abij = AntiSymmetricTensor('t', (a, b), (i, j))
abji = NO(Fd(a) * Fd(b) * F(j) * F(i))
T1 = t_ai * ai
T2 = Rational(1, 4) * t_abij * abji
return (T1, T2)
p, q, r, s = symbols('p,q,r,s', cls=Dummy)
#Setup Hamiltonian on normal ordered form
E0 = symbols('Eref', real=True, constant=True) #Reference energy
f = AntiSymmetricTensor('f', (p, ), (q, ))
pq = NO((Fd(p) * F(q)))
Fock = f * pq #F is reserved by sympy
V = AntiSymmetricTensor('V', (p, q), (r, s))
pqsr = NO(Fd(p) * Fd(q) * F(s) * F(r))
HI = Rational(1, 4) * V * pqsr
HN = E0 + F + HI
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
print("Compute CCD energy and amplitude equations term by term")
#BCH: HN + [HN,T] + 0.5*[[HN,T],T] + 1/6 * [[[HN,T],T],T] + 1/24 * [[[[HN,T],T],T],T]
from sympy.physics.secondquant import Commutator, F, Fd, wicks, NO
from sympy import symbols, simplify
a = symbols("b0:8")
exp = F(a[1])*F(a[0])*Fd(a[3])*Fd(a[2]) * F(a[4])*F(a[5])*Fd(a[6])*Fd(a[7])
exp = wicks(exp)
print(exp)
def test_wicks():
p, q, r, s = symbols('p,q,r,s', above_fermi=True)
# Testing for particles only
str = F(p) * Fd(q)
assert wicks(str) == NO(F(p) * Fd(q)) + KroneckerDelta(p, q)
str = Fd(p) * F(q)
assert wicks(str) == NO(Fd(p) * F(q))
str = F(p) * Fd(q) * F(r) * Fd(s)
nstr = wicks(str)
fasit = NO(
KroneckerDelta(p, q) * KroneckerDelta(r, s) +
KroneckerDelta(p, q) * AnnihilateFermion(r) * CreateFermion(s) +
KroneckerDelta(r, s) * AnnihilateFermion(p) * CreateFermion(q) -
KroneckerDelta(p, s) * AnnihilateFermion(r) * CreateFermion(q) -
AnnihilateFermion(p) * AnnihilateFermion(r) * CreateFermion(q) *
CreateFermion(s))
assert nstr == fasit
assert (p * q * nstr).expand() == wicks(p * q * str)
assert (nstr * p * q * 2).expand() == wicks(str * p * q * 2)
# Testing CC equations particles and holes
i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy)
a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy)
p, q, r, s = symbols('p q r s', cls=Dummy)
assert (wicks(F(a) * NO(F(i) * F(j)) *
Fd(b)) == NO(F(a) * F(i) * F(j) * Fd(b)) +
KroneckerDelta(a, b) * NO(F(i) * F(j)))
assert (wicks(F(a) * NO(F(i) * F(j) * F(k)) *
Fd(b)) == NO(F(a) * F(i) * F(j) * F(k) * Fd(b)) -
KroneckerDelta(a, b) * NO(F(i) * F(j) * F(k)))
expr = wicks(Fd(i) * NO(Fd(j) * F(k)) * F(l))
assert (expr == -KroneckerDelta(i, k) * NO(Fd(j) * F(l)) -
KroneckerDelta(j, l) * NO(Fd(i) * F(k)) -
KroneckerDelta(i, k) * KroneckerDelta(j, l) +
KroneckerDelta(i, l) * NO(Fd(j) * F(k)) +
NO(Fd(i) * Fd(j) * F(k) * F(l)))
expr = wicks(F(a) * NO(F(b) * Fd(c)) * Fd(d))
assert (expr == -KroneckerDelta(a, c) * NO(F(b) * Fd(d)) -
KroneckerDelta(b, d) * NO(F(a) * Fd(c)) -
KroneckerDelta(a, c) * KroneckerDelta(b, d) +
KroneckerDelta(a, d) * NO(F(b) * Fd(c)) +
NO(F(a) * F(b) * Fd(c) * Fd(d)))
def test_commutation():
n, m = symbols("n,m", above_fermi=True)
c = Commutator(B(0), Bd(0))
assert c == 1
c = Commutator(Bd(0), B(0))
assert c == -1
c = Commutator(B(n), Bd(0))
assert c == KroneckerDelta(n, 0)
c = Commutator(B(0), B(0))
assert c == 0
c = Commutator(B(0), Bd(0))
e = simplify(apply_operators(c * BKet([n])))
assert e == BKet([n])
c = Commutator(B(0), B(1))
e = simplify(apply_operators(c * BKet([n, m])))
assert e == 0
c = Commutator(F(m), Fd(m))
assert c == +1 - 2 * NO(Fd(m) * F(m))
c = Commutator(Fd(m), F(m))
assert c.expand() == -1 + 2 * NO(Fd(m) * F(m))
C = Commutator
X, Y, Z = symbols('X,Y,Z', commutative=False)
assert C(C(X, Y), Z) != 0
assert C(C(X, Z), Y) != 0
assert C(Y, C(X, Z)) != 0
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
D = KroneckerDelta
assert C(Fd(a), F(i)) == -2 * NO(F(i) * Fd(a))
assert C(Fd(j), NO(Fd(a) * F(i))).doit(wicks=True) == -D(j, i) * Fd(a)
assert C(Fd(a) * F(i), Fd(b) * F(j)).doit(wicks=True) == 0
c1 = Commutator(F(a), Fd(a))
assert Commutator.eval(c1, c1) == 0
c = Commutator(Fd(a) * F(i), Fd(b) * F(j))
assert latex(c) == r'\left[a^\dagger_{a} a_{i},a^\dagger_{b} a_{j}\right]'
assert repr(
c
) == 'Commutator(CreateFermion(a)*AnnihilateFermion(i),CreateFermion(b)*AnnihilateFermion(j))'
assert str(
c
) == '[CreateFermion(a)*AnnihilateFermion(i),CreateFermion(b)*AnnihilateFermion(j)]'
def test_NO():
i, j, k, l = symbols('i j k l', below_fermi=True)
a, b, c, d = symbols('a b c d', above_fermi=True)
p, q, r, s = symbols('p q r s', cls=Dummy)
assert (NO(Fd(p) * F(q) + Fd(a) * F(b)) == NO(Fd(p) * F(q)) +
NO(Fd(a) * F(b)))
assert (NO(Fd(i) * NO(F(j) * Fd(a))) == NO(Fd(i) * F(j) * Fd(a)))
assert NO(1) == 1
assert NO(i) == i
assert (NO(Fd(a) * Fd(b) * (F(c) + F(d))) == NO(Fd(a) * Fd(b) * F(c)) +
NO(Fd(a) * Fd(b) * F(d)))
assert NO(Fd(a) * F(b))._remove_brackets() == Fd(a) * F(b)
assert NO(F(j) * Fd(i))._remove_brackets() == F(j) * Fd(i)
assert (NO(Fd(p) * F(q)).subs(Fd(p),
Fd(a) + Fd(i)) == NO(Fd(a) * F(q)) +
NO(Fd(i) * F(q)))
assert (NO(Fd(p) * F(q)).subs(F(q),
F(a) + F(i)) == NO(Fd(p) * F(a)) +
NO(Fd(p) * F(i)))
expr = NO(Fd(p) * F(q))._remove_brackets()
assert wicks(expr) == NO(expr)
assert NO(Fd(a) * F(b)) == -NO(F(b) * Fd(a))
no = NO(Fd(a) * F(i) * F(b) * Fd(j))
l1 = [ind for ind in no.iter_q_creators()]
assert l1 == [0, 1]
l2 = [ind for ind in no.iter_q_annihilators()]
assert l2 == [3, 2]
def test_create_f():
i, j, n, m = symbols('i,j,n,m')
o = Fd(i)
assert isinstance(o, CreateFermion)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = Fd(1)
assert o.apply_operator(FKet([n])) == FKet([1, n])
assert o.apply_operator(FKet([n])) == -FKet([n, 1])
o = Fd(n)
assert o.apply_operator(FKet([])) == FKet([n])
vacuum = FKet([], fermi_level=4)
assert vacuum == FKet([], fermi_level=4)
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
assert Fd(i).apply_operator(FKet([i, j, k], 4)) == FKet([j, k], 4)
assert Fd(a).apply_operator(FKet([i, b, k], 4)) == FKet([a, i, b, k], 4)
assert Dagger(B(p)).apply_operator(q) == q * CreateBoson(p)
assert repr(Fd(p)) == 'CreateFermion(p)'
assert srepr(Fd(p)) == "CreateFermion(Symbol('p'))"
assert latex(Fd(p)) == r'a^\dagger_{p}'
def test_substitute_dummies_SQ_operator():
i, j = symbols('i j', cls=Dummy)
assert substitute_dummies(
att(i, j) * Fd(i) * F(j) - att(j, i) * Fd(j) * F(i)) == 0
def test_NO():
i, j, k, l = symbols('i j k l', below_fermi=True)
a, b, c, d = symbols('a b c d', above_fermi=True)
p, q, r, s = symbols('p q r s', cls=Dummy)
assert (NO(Fd(p) * F(q) + Fd(a) * F(b)) == NO(Fd(p) * F(q)) +
NO(Fd(a) * F(b)))
assert (NO(Fd(i) * NO(F(j) * Fd(a))) == NO(Fd(i) * F(j) * Fd(a)))
assert NO(1) == 1
assert NO(i) == i
assert (NO(Fd(a) * Fd(b) * (F(c) + F(d))) == NO(Fd(a) * Fd(b) * F(c)) +
NO(Fd(a) * Fd(b) * F(d)))
assert NO(Fd(a) * F(b))._remove_brackets() == Fd(a) * F(b)
assert NO(F(j) * Fd(i))._remove_brackets() == F(j) * Fd(i)
assert (NO(Fd(p) * F(q)).subs(Fd(p),
Fd(a) + Fd(i)) == NO(Fd(a) * F(q)) +
NO(Fd(i) * F(q)))
assert (NO(Fd(p) * F(q)).subs(F(q),
F(a) + F(i)) == NO(Fd(p) * F(a)) +
NO(Fd(p) * F(i)))
expr = NO(Fd(p) * F(q))._remove_brackets()
assert wicks(expr) == NO(expr)
assert NO(Fd(a) * F(b)) == -NO(F(b) * Fd(a))
no = NO(Fd(a) * F(i) * F(b) * Fd(j))
l1 = [ind for ind in no.iter_q_creators()]
assert l1 == [0, 1]
l2 = [ind for ind in no.iter_q_annihilators()]
assert l2 == [3, 2]
no = NO(Fd(a) * Fd(i))
assert no.has_q_creators == 1
assert no.has_q_annihilators == -1
assert str(no) == ':CreateFermion(a)*CreateFermion(i):'
assert repr(no) == 'NO(CreateFermion(a)*CreateFermion(i))'
assert latex(no) == r'\left\{a^\dagger_{a} a^\dagger_{i}\right\}'
raises(NotImplementedError, lambda: NO(Bd(p) * F(q)))
def test_sympy__physics__secondquant__NO():
from sympy.physics.secondquant import NO, F, Fd
assert _test_args(NO(Fd(x) * F(y)))
Commutator, simplify_index_permutations,
PermutationOperator, contraction)
from sympy import (symbols, expand, pprint, Rational, latex, Dummy)
#Link to Second quant documentation: https://docs.sympy.org/latest/modules/physics/secondquant.html
pretty_dummies_dict = {
'above': 'abcdefgh',
'below': 'ijklmno',
'general': 'pqrstu'
}
p, q, r, s = symbols('p,q,r,s', cls=Dummy)
#Setup creation and annihilation operators
ap_dagger = Fd(p)
aq = F(q)
#Perform a contraction
contr = evaluate_deltas(contraction(ap_dagger, aq))
print("Example outputs")
print()
print("contraction(a_p^\dagger a_q): ", latex(contr))
print()
#Setup Hamiltonian, not on normal order form
h = AntiSymmetricTensor('h', (p, ), (q, ))
pq = ap_dagger * aq
V = AntiSymmetricTensor('V', (p, q), (r, s))
pqsr = Fd(p) * Fd(q) * F(s) * F(r)
