def OPDM(L, R, flavor):
display(
Markdown(
rf""" Computing OPDM for {flavor} (skipping summation for dummy variables)"""
))
i, j = symbols('i,j', below_fermi=True)
a, b = symbols('a,b', above_fermi=True)
PermutList = [PermutationOperator(i, j), PermutationOperator(a, b)]
oo = Fd(i) * F(j)
cc = BCH.level(oo, "SD")
g_oo = evaluate_deltas(wicks(L * cc * R, keep_only_fully_contracted=True))
g_oo = simplify_index_permutations(g_oo, PermutList)
index_rule = {'below': 'klmno', 'above': 'abcde'}
g_oo = substitute_dummies(g_oo,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_{ij}')
final_eq = Eq(gamma, g_oo)
display(final_eq)
ov = Fd(i) * F(a)
cc = BCH.level(ov, "SD")
g_ov = evaluate_deltas(wicks(L * cc * R, keep_only_fully_contracted=True))
g_ov = simplify_index_permutations(g_ov, PermutList)
index_rule = {'below': 'jklmn', 'above': 'bcdef'}
g_ov = substitute_dummies(g_ov,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_{ia}')
final_eq = Eq(gamma, g_ov)
display(final_eq)
vo = Fd(a) * F(i)
cc = BCH.level(vo, "SD")
g_vo = evaluate_deltas(wicks(L * cc * R, keep_only_fully_contracted=True))
g_vo = simplify_index_permutations(g_vo, PermutList)
index_rule = {'below': 'jklmn', 'above': 'bcdef'}
g_vo = substitute_dummies(g_vo,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_{ai}')
final_eq = Eq(gamma, g_vo)
display(final_eq)
vv = Fd(a) * F(b)
cc = BCH.level(vv, "SD")
g_vv = evaluate_deltas(wicks(L * cc * R, keep_only_fully_contracted=True))
g_vv = simplify_index_permutations(g_vv, PermutList)
index_rule = {'below': 'ijklm', 'above': 'cdefg'}
g_vv = substitute_dummies(g_vv,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_{ab}')
final_eq = Eq(gamma, g_vv)
display(final_eq)
def main():
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
H1, H2 = getHamiltonian()
eq_H1 = computeHausdorff(H1) #e^(-T)H1e^T
eq_H2 = computeHausdorff(H2)
print("CC energy:")
Energy = wicks(eq_H1 + eq_H2,
simplify_dummies=True,
keep_only_fully_contracted=True,
simplify_kronecker_deltas=True)
Energy = substitute_dummies(Energy,
new_indices=True,
pretty_indices=pretty_dummies_dict)
print(latex(Energy))
print()
"""
print("CC dE_H1/(dLia):")
eqH1T1 = wicks(Fd(i)*F(a)*eq_H1, simplify_dummies=True, keep_only_fully_contracted=True, simplify_kronecker_deltas=True)
P = PermutationOperator
eqH1T1 = simplify_index_permutations(eqH1T1, [P(a, b), P(i, j)])
eqH1T1 = substitute_dummies(eqH1T1,new_indices=True, pretty_indices=pretty_dummies_dict)
print(latex(eqH1T1))
print()
print("CC dE_H2/(dLia):")
eqH2T1 = wicks(Fd(i)*F(a)*eq_H2, simplify_dummies=True, keep_only_fully_contracted=True, simplify_kronecker_deltas=True)
P = PermutationOperator
eqH2T1 = simplify_index_permutations(eqH2T1, [P(a, b), P(i, j)])
eqH2T1 = substitute_dummies(eqH2T1,new_indices=True, pretty_indices=pretty_dummies_dict)
print(latex(eqH2T1))
print()
"""
print("CC dE_H1/(dLijab):")
eqH1T2 = wicks(Fd(j) * F(b) * Fd(i) * F(a) * eq_H1,
simplify_dummies=True,
keep_only_fully_contracted=True,
simplify_kronecker_deltas=True)
P = PermutationOperator
eqH1T2 = simplify_index_permutations(eqH1T2, [P(a, b), P(i, j)])
eqH1T2 = substitute_dummies(eqH1T2,
new_indices=True,
pretty_indices=pretty_dummies_dict)
print(latex(eqH1T2))
print()
print("CC dE_H2/(dLijab):")
eqH2T2 = wicks(Fd(j) * F(b) * Fd(i) * F(a) * eq_H2,
simplify_dummies=True,
keep_only_fully_contracted=True,
simplify_kronecker_deltas=True)
P = PermutationOperator
eqH2T2 = simplify_index_permutations(eqH2T2, [P(i, j), P(a, b)])
eqH2T2 = substitute_dummies(eqH2T2,
new_indices=True,
pretty_indices=pretty_dummies_dict)
print(latex(eqH2T2))
print()
def get_doubles_cluster_operator():
i, j = symbols("i, j", below_fermi=True, cls=Dummy)
a, b = symbols("a, b", above_fermi=True, cls=Dummy)
t = AntiSymmetricTensor("t", (a, b), (i, j))
t = t * Fd(a) * F(i) * Fd(b) * F(j)
return [Rational(1, 4) * t]
def OPTDM(Lf1, Rf1, Lf2, Rf2, flavor1, flavor2):
display(
Markdown(
rf""" Computing Dyson OPTDM between {flavor1} $\rightarrow$ {flavor2} (skipping summation for dummy variables)"""
))
i = symbols('i', below_fermi=True)
a = symbols('a', above_fermi=True)
index_rule = {'below': 'jklmn', 'above': 'bcde'}
oo = Fd(i)
cc = BCH.level(oo, "SD")
g_oo = evaluate_deltas(
wicks(Lf2 * cc * Rf1, keep_only_fully_contracted=True))
g_oo = substitute_dummies(g_oo,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_i^{R}')
final_eq = Eq(gamma, g_oo)
display(final_eq)
ov = Fd(a)
cc = BCH.level(ov, "SD")
g_ov = evaluate_deltas(
wicks(Lf2 * cc * Rf1, keep_only_fully_contracted=True))
index_rule = {'below': 'jklmn', 'above': 'bcdef'}
g_ov = substitute_dummies(g_ov,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_a^{R}')
final_eq = Eq(gamma, g_ov)
display(final_eq)
vo = F(i)
cc = BCH.level(vo, "SD")
g_vo = evaluate_deltas(
wicks(Lf1 * cc * Rf2, keep_only_fully_contracted=True))
index_rule = {'below': 'jklmn', 'above': 'bcdef'}
g_vo = substitute_dummies(g_vo,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_i^{L}')
final_eq = Eq(gamma, g_vo)
display(final_eq)
vv = F(a)
cc = BCH.level(vv, "SD")
g_vv = evaluate_deltas(
wicks(Lf1 * cc * Rf2, keep_only_fully_contracted=True))
index_rule = {'below': 'ijklm', 'above': 'cdefg'}
g_vv = substitute_dummies(g_vv,
new_indices=True,
pretty_indices=index_rule)
gamma = Symbol('\gamma_a^{L}')
final_eq = Eq(gamma, g_vv)
display(final_eq)
def getHamiltonian():
p, q, r, s = symbols('p,q,r,s', cls=Dummy)
h = AntiSymmetricTensor('h', (p, ), (q, ))
pq = Fd(p) * F(q)
u = AntiSymmetricTensor('u', (p, r), (q, s))
pqsr = Fd(p) * Fd(r) * F(s) * F(q)
H1 = h * pq
H2 = Rational(1, 4) * u * pqsr
return (H1, H2)
def get_two_body_equation(equation_h, equation_u):
two_body_eq = wicks(
Fd(j) * F(b) * Fd(i) * F(a) * equation_u, **wicks_kwargs)
p = PermutationOperator
two_body_eq = simplify_index_permutations(two_body_eq, [p(i, j), p(a, b)])
two_body_eq = substitute_dummies(two_body_eq, **sub_kwargs)
return two_body_eq
def get_one_body_equation(equation_h, equation_u):
one_body_eq = wicks(
Fd(j) * F(b) * Fd(i) * F(a) * equation_h, **wicks_kwargs)
p = PermutationOperator
one_body_eq = simplify_index_permutations(one_body_eq, [p(a, b), p(i, j)])
one_body_eq = substitute_dummies(one_body_eq, **sub_kwargs)
return one_body_eq
def get_hamiltonian():
p, q, r, s = symbols("p, q, r, s", cls=Dummy)
h = AntiSymmetricTensor("h", (p, ), (q, ))
u = AntiSymmetricTensor("u", (p, r), (q, s))
h = h * Fd(p) * F(q)
u = u * Fd(p) * Fd(r) * F(s) * F(q)
return h, Rational(1, 4) * u
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():
base = sys.argv[0]
filename = os.path.splitext(base)[0]
fo = open(filename + '.tex', "w")
fo.write("\documentclass{article}\n")
fo.write("\usepackage[fleqn]{amsmath}\n")
fo.write("\usepackage{latexsym, amsfonts}\n")
fo.write("\usepackage{pdflscape}\n")
fo.write(r'\parindent=0pt\relax')
fo.write("\n")
fo.write(r'\begin{document}')
fo.write("\n")
fo.write(r'\begin{landscape}')
fo.write("\n")
# Hamiltonian
p, q, r, s = symbols('p,q,r,s', cls=Dummy)
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)
H = f * pr + Rational(1, 4) * v * pqsr
# Print Hamiltonian
fo.write("The spin orbital hamiltonian:\n")
Equation2Tex(latex(H), fo)
# Exciation and de-excitation operators
i, j = symbols('i,j', below_fermi=True)
a, b = symbols('a,b', above_fermi=True)
ia = Fd(a) * F(i)
jb = Fd(j) * F(b)
C = Commutator
#First com
comm1 = C(H, ia)
comm2 = C(jb, comm1)
fo.write("Evaluating operators:\n")
Equation2Tex(latex(comm2), fo)
comm2 = comm2.doit()
comm2 = comm2.expand()
#comm2= Permute_Str(comm2)
comm2 = Permute_Op(comm2)
Equation2Tex(latex(comm2), fo)
#finish writing
fo.write("\end{landscape}\n")
fo.write("\end{document}\n")
fo.close()
os.system("pdflatex " + filename + '.tex')
def test_limit_subs():
for F in (Sum, Product, Integral):
assert F(a * exp(a), (a, -2, 2)) == F(a * exp(a),
(a, -b, b)).subs(b, 2)
assert F(a, (a, F(b, (b, 1, 2)), 4)).subs(F(b, (b, 1, 2)), c) == \
F(a, (a, c, 4))
assert F(x, (x, 1, x + y)).subs(x, 1) == F(x, (x, 1, y + 1))
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 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 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 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_contraction():
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 contraction(Fd(i), F(j)) == KroneckerDelta(i, j)
assert contraction(F(a), Fd(b)) == KroneckerDelta(a, b)
assert contraction(F(a), Fd(i)) == 0
assert contraction(Fd(a), F(i)) == 0
assert contraction(F(i), Fd(a)) == 0
assert contraction(Fd(i), F(a)) == 0
assert contraction(Fd(i), F(p)) == KroneckerDelta(i, p)
restr = evaluate_deltas(contraction(Fd(p), F(q)))
assert restr.is_only_below_fermi
restr = evaluate_deltas(contraction(F(p), Fd(q)))
assert restr.is_only_above_fermi
def test_contraction():
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 contraction(Fd(i), F(j)) == KroneckerDelta(i, j)
assert contraction(F(a), Fd(b)) == KroneckerDelta(a, b)
assert contraction(F(a), Fd(i)) == 0
assert contraction(Fd(a), F(i)) == 0
assert contraction(F(i), Fd(a)) == 0
assert contraction(Fd(i), F(a)) == 0
assert contraction(Fd(i), F(p)) == KroneckerDelta(i, p)
restr = evaluate_deltas(contraction(Fd(p), F(q)))
assert restr.is_only_below_fermi
restr = evaluate_deltas(contraction(F(p), Fd(q)))
assert restr.is_only_above_fermi
raises(ContractionAppliesOnlyToFermions, lambda: contraction(B(a), Fd(b)))
def L1(expr):
j1,j2,j3 = symbols('j1,j2,j3' ,below_fermi=True, cls=Dummy)
b1,b2,b3 = symbols('b1,b2,b3' ,above_fermi=True, cls=Dummy)
if expr == "IP":
L1 = Fraction(1, 1)*AntiSymmetricTensor('l',(j1,),())*Fd(j1)
return L1
elif expr == "DIP":
L1 = Fraction(1, 2)*AntiSymmetricTensor('l',(j1,j2),())*Fd(j1)*Fd(j2)
return L1
elif expr == "EA":
L1 = Fraction(1, 1)*AntiSymmetricTensor('l',(),(b1,))*F(b1)
return L1
elif expr == "DEA":
L1 = Fraction(1, 2)*AntiSymmetricTensor('l',(),(b1,b2))*F(b2)*F(b1)
return L1
elif expr == "EE":
L1 = Fraction(1, 1)*AntiSymmetricTensor('l',(j1,),(b1,))*Fd(j1)*F(b1)
return L1
elif expr == "CCSD":
L1 = Fraction(1, 1)*AntiSymmetricTensor('l',(j1,),(b1,))*Fd(j1)*F(b1)
return L1
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 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 test_annihilate_f():
i, j, n, m = symbols('i,j,n,m')
o = F(i)
assert isinstance(o, AnnihilateFermion)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = F(1)
assert o.apply_operator(FKet([1, n])) == FKet([n])
assert o.apply_operator(FKet([n, 1])) == -FKet([n])
o = F(n)
assert o.apply_operator(FKet([n])) == FKet([])
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 F(i).apply_operator(FKet([i, j, k], 4)) == 0
assert F(a).apply_operator(FKet([i, b, k], 4)) == 0
assert F(l).apply_operator(FKet([i, j, k], 3)) == 0
assert F(l).apply_operator(FKet([i, j, k], 4)) == FKet([l, i, j, k], 4)
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 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))
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_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 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_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_annihilate_f():
i, j, n, m = symbols("i,j,n,m")
o = F(i)
assert isinstance(o, AnnihilateFermion)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = F(1)
assert o.apply_operator(FKet([1, n])) == FKet([n])
assert o.apply_operator(FKet([n, 1])) == -FKet([n])
o = F(n)
assert o.apply_operator(FKet([n])) == FKet([])
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 F(i).apply_operator(FKet([i, j, k], 4)) == 0
assert F(a).apply_operator(FKet([i, b, k], 4)) == 0
assert F(l).apply_operator(FKet([i, j, k], 3)) == 0
assert F(l).apply_operator(FKet([i, j, k], 4)) == FKet([l, i, j, k], 4)
assert str(F(p)) == "f(p)"
assert repr(F(p)) == "AnnihilateFermion(p)"
assert srepr(F(p)) == "AnnihilateFermion(Symbol('p'))"
assert latex(F(p)) == "a_{p}"
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_equality():
# if this fails remove special handling below
raises(ValueError, lambda: Sum(x, x))
r = symbols('x', real=True)
for F in (Sum, Product, Integral):
try:
assert F(x, x) != F(y, y)
assert F(x, (x, 1, 2)) != F(x, x)
assert F(x, (x, x)) != F(x, x) # or else they print the same
assert F(1, x) != F(1, y)
except ValueError:
pass
assert F(a, (x, 1, 2)) != F(a, (x, 1, 3))
assert F(a, (x, 1, 2)) != F(b, (x, 1, 2))
assert F(x, (x, 1, 2)) != F(r, (r, 1, 2))
assert F(1, (x, 1, x)) != F(1, (y, 1, x))
assert F(1, (x, 1, x)) != F(1, (y, 1, y))
# issue 5265
assert Sum(x, (x, 1, x)).subs(x, a) == Sum(x, (x, 1, a))
