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_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 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 get_CC_operators():
"""
Returns a tuple (T1,T2) of unique operators.
"""
i = symbols('i', below_fermi=True, cls=Dummy)
a = symbols('a', above_fermi=True, cls=Dummy)
t_ai = AntiSymmetricTensor('t', (a,), (i,))
ai = NO(Fd(a)*F(i))
i, j = symbols('i,j', below_fermi=True, cls=Dummy)
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)
def test_fully_contracted():
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)
Fock = (AntiSymmetricTensor('f', (p, ), (q, )) * NO(Fd(p) * F(q)))
V = (AntiSymmetricTensor('v', (p, q),
(r, s)) * NO(Fd(p) * Fd(q) * F(s) * F(r))) / 4
Fai = wicks(NO(Fd(i) * F(a)) * Fock,
keep_only_fully_contracted=True,
simplify_kronecker_deltas=True)
assert Fai == AntiSymmetricTensor('f', (a, ), (i, ))
Vabij = wicks(NO(Fd(i) * Fd(j) * F(b) * F(a)) * V,
keep_only_fully_contracted=True,
simplify_kronecker_deltas=True)
assert Vabij == AntiSymmetricTensor('v', (a, b), (i, j))
def get_ccsd_lambda_operators(ast_symb="l"):
i = symbols("i", below_fermi=True, cls=Dummy)
a = symbols("a", above_fermi=True, cls=Dummy)
l_ia = AntiSymmetricTensor(ast_symb, (i, ), (a, ))
c_ia = NO(Fd(i) * F(a))
i, j = symbols("i, j", below_fermi=True, cls=Dummy)
a, b = symbols("a, b", above_fermi=True, cls=Dummy)
l_ijab = AntiSymmetricTensor(ast_symb, (i, j), (a, b))
c_ijab = NO(Fd(i) * Fd(j) * F(b) * F(a))
L_1 = l_ia * c_ia
L_2 = Rational(1, 4) * l_ijab * c_ijab
return (L_1, L_2)
def get_ccsd_t_operators(ast_symb="t"):
i = symbols("i", below_fermi=True, cls=Dummy)
a = symbols("a", above_fermi=True, cls=Dummy)
t_ai = AntiSymmetricTensor(ast_symb, (a, ), (i, ))
c_ai = NO(Fd(a) * F(i))
i, j = symbols("i, j", below_fermi=True, cls=Dummy)
a, b = symbols("a, b", above_fermi=True, cls=Dummy)
t_abij = AntiSymmetricTensor(ast_symb, (a, b), (i, j))
c_abij = NO(Fd(a) * Fd(b) * F(j) * F(i))
T_1 = t_ai * c_ai
T_2 = Rational(1, 4) * t_abij * c_abij
return (T_1, T_2)
def test_is_commutative():
from sympy.physics.secondquant import NO, F, Fd
m = Symbol('m', commutative=False)
for f in (Sum, Product, Integral):
assert f(z, (z, 1, 1)).is_commutative is True
assert f(z * y, (z, 1, 6)).is_commutative is True
assert f(m * x, (x, 1, 2)).is_commutative is False
assert f(NO(Fd(x) * F(y)) * z, (z, 1, 2)).is_commutative is False
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 R2(expr):
i1,i2,i3,i4,i5 = symbols('i1,i2,i3,i4,i5' ,below_fermi=True, cls=Dummy)
a1,a2,a3,a4,a5 = symbols('a1,a2,a3,a4,a5' ,above_fermi=True, cls=Dummy)
if expr == "IP":
R2 = Fraction(1, 2)*AntiSymmetricTensor('r',(a1,),(i2,i3))*Fd(a1)*F(i3)*F(i2)
return R2
elif expr == "DIP":
R2 = Fraction(1, 6)*AntiSymmetricTensor('r',(a1,),(i3,i4,i5))*Fd(a1)*F(i5)*F(i4)*F(i3)
return R2
elif expr == "EA":
R2 = Fraction(1, 2)*AntiSymmetricTensor('r',(a2,a3),(i1,))*Fd(a2)*Fd(a3)*F(i1)
return R2
elif expr == "DEA":
R2 = Fraction(1, 6)*AntiSymmetricTensor('r',(a3,a4,a5),(i1,))*NO(Fd(a3)*Fd(a4)*Fd(a5)*F(i1))
return R2
elif expr == "EE":
R2 = Fraction(1, 4)*AntiSymmetricTensor('r',(a2,a3),(i2,i3))*Fd(a2)*Fd(a3)*F(i3)*F(i2)
return R2
elif expr == "CCSD":
R2 = 0
return R2
def R1(expr):
i1,i2,i3 = symbols('i1,i2,i3' ,below_fermi=True, cls=Dummy)
a1,a2,a3 = symbols('a1,a2,i3' ,above_fermi=True, cls=Dummy)
if expr == "IP":
R1 = Fraction(1, 1)*AntiSymmetricTensor('r',(),(i1,))*(F(i1))
return R1
elif expr == "DIP":
R1 = Fraction(1, 2)*AntiSymmetricTensor('r',(),(i1,i2))*F(i2)*F(i1)
return R1
elif expr == "EA":
R1 = Fraction(1, 1)*AntiSymmetricTensor('r',(a1,),())*Fd(a1)
return R1
elif expr == "DEA":
R1 = Fraction(1, 2)*AntiSymmetricTensor('r',(a1,a2),())*NO(Fd(a1)*Fd(a2))
return R1
elif expr == "EE":
R1 = Fraction(1, 1)*AntiSymmetricTensor('r',(a1,),(i1,))*Fd(a1)*F(i1)
return R1
elif expr == "CCSD":
R1 = 0
return R1
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)))
def test_sympy__physics__secondquant__NO():
from sympy.physics.secondquant import NO, F, Fd
assert _test_args(NO(Fd(x) * F(y)))
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_substitute_dummies_NO_operator():
i, j = symbols("i j", cls=Dummy)
assert (
substitute_dummies(att(i, j) * NO(Fd(i) * F(j)) - att(j, i) * NO(Fd(j) * F(i)))
== 0
)
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)
)
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]
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)
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_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_get_subNO():
p, q, r = symbols("p,q,r")
assert NO(F(p) * F(q) * F(r)).get_subNO(1) == NO(F(p) * F(r))
assert NO(F(p) * F(q) * F(r)).get_subNO(0) == NO(F(q) * F(r))
assert NO(F(p) * F(q) * F(r)).get_subNO(2) == NO(F(p) * F(q))
Eref = evaluate_deltas(wicks(H, keep_only_fully_contracted=True))
Eref = substitute_dummies(Eref,
new_indices=True,
pretty_indices=pretty_dummies_dict)
print("Eref: ", latex(Eref))
print()
print("Normal ordered Hamiltonian")
print(latex(H_N))
print()
#Setup Hamiltonian on normal ordered form
E0 = symbols('Eref', real=True, constant=True) #Reference energy
f = AntiSymmetricTensor('f', (p, ), (q, ))
pq = NO(ap_dagger * aq)
V = AntiSymmetricTensor('V', (p, q), (r, s))
pqsr = NO(Fd(p) * Fd(q) * F(s) * F(r))
HI = Rational(1, 4) * V * pqsr
Fock = f * pq #F is reserved by sympy
HN = E0 + Fock + HI
#Compute <c|F|Phi_i^a>
#Define indices above and below Fermi level
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
