venv/lib/python3.11/site-packages/sympy/physics/tests/test_secondquant.py

from sympy.functions.elementary.complexes import conjugate
from sympy.functions.elementary.exponential import exp
from sympy.physics.secondquant import (
    Dagger, Bd, VarBosonicBasis, BBra, B, BKet, FixedBosonicBasis,
    matrix_rep, apply_operators, InnerProduct, Commutator, KroneckerDelta,
    AnnihilateBoson, CreateBoson, BosonicOperator,
    F, Fd, FKet, BosonState, CreateFermion, AnnihilateFermion,
    evaluate_deltas, AntiSymmetricTensor, contraction, NO, wicks,
    PermutationOperator, simplify_index_permutations,
    _sort_anticommuting_fermions, _get_ordered_dummies,
    substitute_dummies, FockStateBosonKet,
    ContractionAppliesOnlyToFermions
)

from sympy.concrete.summations import Sum
from sympy.core.function import (Function, expand)
from sympy.core.numbers import (I, Rational)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.printing.repr import srepr
from sympy.simplify.simplify import simplify

from sympy.testing.pytest import slow, raises
from sympy.printing.latex import latex


def test_PermutationOperator():
    p, q, r, s = symbols('p,q,r,s')
    f, g, h, i = map(Function, 'fghi')
    P = PermutationOperator
    assert P(p, q).get_permuted(f(p)*g(q)) == -f(q)*g(p)
    assert P(p, q).get_permuted(f(p, q)) == -f(q, p)
    assert P(p, q).get_permuted(f(p)) == f(p)
    expr = (f(p)*g(q)*h(r)*i(s)
        - f(q)*g(p)*h(r)*i(s)
        - f(p)*g(q)*h(s)*i(r)
        + f(q)*g(p)*h(s)*i(r))
    perms = [P(p, q), P(r, s)]
    assert (simplify_index_permutations(expr, perms) ==
        P(p, q)*P(r, s)*f(p)*g(q)*h(r)*i(s))
    assert latex(P(p, q)) == 'P(pq)'

    p1, p2 = symbols('p1,p2')
    assert latex(P(p1,p2) == 'P(p_{1}p_{2})')

def test_index_permutations_with_dummies():
    a, b, c, d = symbols('a b c d')
    p, q, r, s = symbols('p q r s', cls=Dummy)
    f, g = map(Function, 'fg')
    P = PermutationOperator

    # No dummy substitution necessary
    expr = f(a, b, p, q) - f(b, a, p, q)
    assert simplify_index_permutations(
        expr, [P(a, b)]) == P(a, b)*f(a, b, p, q)

    # Cases where dummy substitution is needed
    expected = P(a, b)*substitute_dummies(f(a, b, p, q))

    expr = f(a, b, p, q) - f(b, a, q, p)
    result = simplify_index_permutations(expr, [P(a, b)])
    assert expected == substitute_dummies(result)

    expr = f(a, b, q, p) - f(b, a, p, q)
    result = simplify_index_permutations(expr, [P(a, b)])
    assert expected == substitute_dummies(result)

    # A case where nothing can be done
    expr = f(a, b, q, p) - g(b, a, p, q)
    result = simplify_index_permutations(expr, [P(a, b)])
    assert expr == result


def test_dagger():
    i, j, n, m = symbols('i,j,n,m')
    assert Dagger(1) == 1
    assert Dagger(1.0) == 1.0
    assert Dagger(2*I) == -2*I
    assert Dagger(S.Half*I/3.0) == I*Rational(-1, 2)/3.0
    assert Dagger(BKet([n])) == BBra([n])
    assert Dagger(B(0)) == Bd(0)
    assert Dagger(Bd(0)) == B(0)
    assert Dagger(B(n)) == Bd(n)
    assert Dagger(Bd(n)) == B(n)
    assert Dagger(B(0) + B(1)) == Bd(0) + Bd(1)
    assert Dagger(n*m) == Dagger(n)*Dagger(m)  # n, m commute
    assert Dagger(B(n)*B(m)) == Bd(m)*Bd(n)
    assert Dagger(B(n)**10) == Dagger(B(n))**10
    assert Dagger('a') == Dagger(Symbol('a'))
    assert Dagger(Dagger('a')) == Symbol('a')
    assert Dagger(exp(2 * I)) == exp(-2 * I)
    assert Dagger(i) == conjugate(i)


def test_operator():
    i, j = symbols('i,j')
    o = BosonicOperator(i)
    assert o.state == i
    assert o.is_symbolic
    o = BosonicOperator(1)
    assert o.state == 1
    assert not o.is_symbolic


def test_create():
    i, j, n, m, p1 = symbols('i,j,n,m,p1')
    o = Bd(i)
    assert latex(o) == "{b^\\dagger_{i}}"
    assert latex(Bd(p1)) == "{b^\\dagger_{p_{1}}}"
    assert isinstance(o, CreateBoson)
    o = o.subs(i, j)
    assert o.atoms(Symbol) == {j}
    o = Bd(0)
    assert o.apply_operator(BKet([n])) == sqrt(n + 1)*BKet([n + 1])
    o = Bd(n)
    assert o.apply_operator(BKet([n])) == o*BKet([n])


def test_annihilate():
    i, j, n, m, p1 = symbols('i,j,n,m,p1')
    o = B(i)
    assert latex(o) == "b_{i}"
    assert latex(B(p1)) == "b_{p_{1}}"
    assert isinstance(o, AnnihilateBoson)
    o = o.subs(i, j)
    assert o.atoms(Symbol) == {j}
    o = B(0)
    assert o.apply_operator(BKet([n])) == sqrt(n)*BKet([n - 1])
    o = B(n)
    assert o.apply_operator(BKet([n])) == o*BKet([n])


def test_basic_state():
    i, j, n, m = symbols('i,j,n,m')
    s = BosonState([0, 1, 2, 3, 4])
    assert len(s) == 5
    assert s.args[0] == tuple(range(5))
    assert s.up(0) == BosonState([1, 1, 2, 3, 4])
    assert s.down(4) == BosonState([0, 1, 2, 3, 3])
    for i in range(5):
        assert s.up(i).down(i) == s
    assert s.down(0) == 0
    for i in range(5):
        assert s[i] == i
    s = BosonState([n, m])
    assert s.down(0) == BosonState([n - 1, m])
    assert s.up(0) == BosonState([n + 1, m])


def test_basic_apply():
    n = symbols("n")
    e = B(0)*BKet([n])
    assert apply_operators(e) == sqrt(n)*BKet([n - 1])
    e = Bd(0)*BKet([n])
    assert apply_operators(e) == sqrt(n + 1)*BKet([n + 1])


def test_complex_apply():
    n, m = symbols("n,m")
    o = Bd(0)*B(0)*Bd(1)*B(0)
    e = apply_operators(o*BKet([n, m]))
    answer = sqrt(n)*sqrt(m + 1)*(-1 + n)*BKet([-1 + n, 1 + m])
    assert expand(e) == expand(answer)


def test_number_operator():
    n = symbols("n")
    o = Bd(0)*B(0)
    e = apply_operators(o*BKet([n]))
    assert e == n*BKet([n])


def test_inner_product():
    i, j, k, l = symbols('i,j,k,l')
    s1 = BBra([0])
    s2 = BKet([1])
    assert InnerProduct(s1, Dagger(s1)) == 1
    assert InnerProduct(s1, s2) == 0
    s1 = BBra([i, j])
    s2 = BKet([k, l])
    r = InnerProduct(s1, s2)
    assert r == KroneckerDelta(i, k)*KroneckerDelta(j, l)


def test_symbolic_matrix_elements():
    n, m = symbols('n,m')
    s1 = BBra([n])
    s2 = BKet([m])
    o = B(0)
    e = apply_operators(s1*o*s2)
    assert e == sqrt(m)*KroneckerDelta(n, m - 1)


def test_matrix_elements():
    b = VarBosonicBasis(5)
    o = B(0)
    m = matrix_rep(o, b)
    for i in range(4):
        assert m[i, i + 1] == sqrt(i + 1)
    o = Bd(0)
    m = matrix_rep(o, b)
    for i in range(4):
        assert m[i + 1, i] == sqrt(i + 1)


def test_fixed_bosonic_basis():
    b = FixedBosonicBasis(2, 2)
    # assert b == [FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]
    state = b.state(1)
    assert state == FockStateBosonKet((1, 1))
    assert b.index(state) == 1
    assert b.state(1) == b[1]
    assert len(b) == 3
    assert str(b) == '[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]'
    assert repr(b) == '[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]'
    assert srepr(b) == '[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]'


@slow
def test_sho():
    n, m = symbols('n,m')
    h_n = Bd(n)*B(n)*(n + S.Half)
    H = Sum(h_n, (n, 0, 5))
    o = H.doit(deep=False)
    b = FixedBosonicBasis(2, 6)
    m = matrix_rep(o, b)
    # We need to double check these energy values to make sure that they
    # are correct and have the proper degeneracies!
    diag = [1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11]
    for i in range(len(diag)):
        assert diag[i] == m[i, i]


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_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')
    p1 = symbols("p1")

    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}}'
    assert latex(Fd(p1)) == r'{a^\dagger_{p_{1}}}'
    assert latex(FKet([a,i], 1)) == r"\left|\left( a, \  i\right)\right\rangle"
    assert latex(FKet([j,i,b,a], 2)) == r"\left|\left( a, \  b, \  i, \  j\right)\right\rangle"


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')
    p1 = symbols('p1')

    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}'
    assert latex(F(p1)) == 'a_{p_{1}}'


def test_create_b():
    i, j, n, m = symbols('i,j,n,m')
    o = Bd(i)
    assert isinstance(o, CreateBoson)
    o = o.subs(i, j)
    assert o.atoms(Symbol) == {j}
    o = Bd(0)
    assert o.apply_operator(BKet([n])) == sqrt(n + 1)*BKet([n + 1])
    o = Bd(n)
    assert o.apply_operator(BKet([n])) == o*BKet([n])


def test_annihilate_b():
    i, j, n, m = symbols('i,j,n,m')
    o = B(i)
    assert isinstance(o, AnnihilateBoson)
    o = o.subs(i, j)
    assert o.atoms(Symbol) == {j}
    o = B(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)))


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 = list(no.iter_q_creators())
    assert l1 == [0, 1]
    l2 = list(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_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_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 test_evaluate_deltas():
    i, j, k = symbols('i,j,k')

    r = KroneckerDelta(i, j) * KroneckerDelta(j, k)
    assert evaluate_deltas(r) == KroneckerDelta(i, k)

    r = KroneckerDelta(i, 0) * KroneckerDelta(j, k)
    assert evaluate_deltas(r) == KroneckerDelta(i, 0) * KroneckerDelta(j, k)

    r = KroneckerDelta(1, j) * KroneckerDelta(j, k)
    assert evaluate_deltas(r) == KroneckerDelta(1, k)

    r = KroneckerDelta(j, 2) * KroneckerDelta(k, j)
    assert evaluate_deltas(r) == KroneckerDelta(2, k)

    r = KroneckerDelta(i, 0) * KroneckerDelta(i, j) * KroneckerDelta(j, 1)
    assert evaluate_deltas(r) == 0

    r = (KroneckerDelta(0, i) * KroneckerDelta(0, j)
         * KroneckerDelta(1, j) * KroneckerDelta(1, j))
    assert evaluate_deltas(r) == 0


def test_Tensors():
    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')

    AT = AntiSymmetricTensor
    assert AT('t', (a, b), (i, j)) == -AT('t', (b, a), (i, j))
    assert AT('t', (a, b), (i, j)) == AT('t', (b, a), (j, i))
    assert AT('t', (a, b), (i, j)) == -AT('t', (a, b), (j, i))
    assert AT('t', (a, a), (i, j)) == 0
    assert AT('t', (a, b), (i, i)) == 0
    assert AT('t', (a, b, c), (i, j)) == -AT('t', (b, a, c), (i, j))
    assert AT('t', (a, b, c), (i, j, k)) == AT('t', (b, a, c), (i, k, j))

    tabij = AT('t', (a, b), (i, j))
    assert tabij.has(a)
    assert tabij.has(b)
    assert tabij.has(i)
    assert tabij.has(j)
    assert tabij.subs(b, c) == AT('t', (a, c), (i, j))
    assert (2*tabij).subs(i, c) == 2*AT('t', (a, b), (c, j))
    assert tabij.symbol == Symbol('t')
    assert latex(tabij) == '{t^{ab}_{ij}}'
    assert str(tabij) == 't((_a, _b),(_i, _j))'

    assert AT('t', (a, a), (i, j)).subs(a, b) == AT('t', (b, b), (i, j))
    assert AT('t', (a, i), (a, j)).subs(a, b) == AT('t', (b, i), (b, j))

    a1, a2, a3, a4 = symbols('alpha1:5')
    u_alpha1234 = AntiSymmetricTensor("u", (a1, a2), (a3, a4))

    assert latex(u_alpha1234) == r'{u^{\alpha_{1}\alpha_{2}}_{\alpha_{3}\alpha_{4}}}'
    assert str(u_alpha1234) == 'u((alpha1, alpha2),(alpha3, alpha4))'


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_substitute_dummies_without_dummies():
    i, j = symbols('i,j')
    assert substitute_dummies(att(i, j) + 2) == att(i, j) + 2
    assert substitute_dummies(att(i, j) + 1) == att(i, j) + 1


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_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_substitute_dummies_new_indices():
    i, j = symbols('i j', below_fermi=True, cls=Dummy)
    a, b = symbols('a b', above_fermi=True, cls=Dummy)
    p, q = symbols('p q', cls=Dummy)
    f = Function('f')
    assert substitute_dummies(f(i, a, p) - f(j, b, q), new_indices=True) == 0


def test_substitute_dummies_substitution_order():
    i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy)
    f = Function('f')
    from sympy.utilities.iterables import variations
    for permut in variations([i, j, k, l], 4):
        assert substitute_dummies(f(*permut) - f(i, j, k, l)) == 0


def test_dummy_order_inner_outer_lines_VT1T1T1():
    ii = symbols('i', below_fermi=True)
    aa = symbols('a', above_fermi=True)
    k, l = symbols('k l', below_fermi=True, cls=Dummy)
    c, d = symbols('c d', above_fermi=True, cls=Dummy)

    v = Function('v')
    t = Function('t')
    dums = _get_ordered_dummies

    # Coupled-Cluster T1 terms with V*T1*T1*T1
    # t^{a}_{k} t^{c}_{i} t^{d}_{l} v^{lk}_{dc}
    exprs = [
        # permut v and t <=> swapping internal lines, equivalent
        # irrespective of symmetries in v
        v(k, l, c, d)*t(c, ii)*t(d, l)*t(aa, k),
        v(l, k, c, d)*t(c, ii)*t(d, k)*t(aa, l),
        v(k, l, d, c)*t(d, ii)*t(c, l)*t(aa, k),
        v(l, k, d, c)*t(d, ii)*t(c, k)*t(aa, l),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_dummy_order_inner_outer_lines_VT1T1T1T1():
    ii, jj = symbols('i j', below_fermi=True)
    aa, bb = symbols('a b', above_fermi=True)
    k, l = symbols('k l', below_fermi=True, cls=Dummy)
    c, d = symbols('c d', above_fermi=True, cls=Dummy)

    v = Function('v')
    t = Function('t')
    dums = _get_ordered_dummies

    # Coupled-Cluster T2 terms with V*T1*T1*T1*T1
    exprs = [
        # permut t <=> swapping external lines, not equivalent
        # except if v has certain symmetries.
        v(k, l, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
        v(k, l, c, d)*t(c, jj)*t(d, ii)*t(aa, k)*t(bb, l),
        v(k, l, c, d)*t(c, ii)*t(d, jj)*t(bb, k)*t(aa, l),
        v(k, l, c, d)*t(c, jj)*t(d, ii)*t(bb, k)*t(aa, l),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
    exprs = [
        # permut v <=> swapping external lines, not equivalent
        # except if v has certain symmetries.
        #
        # Note that in contrast to above, these permutations have identical
        # dummy order.  That is because the proximity to external indices
        # has higher influence on the canonical dummy ordering than the
        # position of a dummy on the factors.  In fact, the terms here are
        # similar in structure as the result of the dummy substitutions above.
        v(k, l, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
        v(l, k, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
        v(k, l, d, c)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
        v(l, k, d, c)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) == dums(permut)
        assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
    exprs = [
        # permut t and v <=> swapping internal lines, equivalent.
        # Canonical dummy order is different, and a consistent
        # substitution reveals the equivalence.
        v(k, l, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
        v(k, l, d, c)*t(c, jj)*t(d, ii)*t(aa, k)*t(bb, l),
        v(l, k, c, d)*t(c, ii)*t(d, jj)*t(bb, k)*t(aa, l),
        v(l, k, d, c)*t(c, jj)*t(d, ii)*t(bb, k)*t(aa, l),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


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_equivalent_internal_lines_VT1T1():
    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)

    v = Function('v')
    t = Function('t')
    dums = _get_ordered_dummies

    exprs = [  # permute v.  Different dummy order. Not equivalent.
        v(i, j, a, b)*t(a, i)*t(b, j),
        v(j, i, a, b)*t(a, i)*t(b, j),
        v(i, j, b, a)*t(a, i)*t(b, j),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) != substitute_dummies(permut)

    exprs = [  # permute v.  Different dummy order. Equivalent
        v(i, j, a, b)*t(a, i)*t(b, j),
        v(j, i, b, a)*t(a, i)*t(b, j),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)

    exprs = [  # permute t.  Same dummy order, not equivalent.
        v(i, j, a, b)*t(a, i)*t(b, j),
        v(i, j, a, b)*t(b, i)*t(a, j),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) == dums(permut)
        assert substitute_dummies(exprs[0]) != substitute_dummies(permut)

    exprs = [  # permute v and t.  Different dummy order, equivalent
        v(i, j, a, b)*t(a, i)*t(b, j),
        v(j, i, a, b)*t(a, j)*t(b, i),
        v(i, j, b, a)*t(b, i)*t(a, j),
        v(j, i, b, a)*t(b, j)*t(a, i),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_equivalent_internal_lines_VT2conjT2():
    # this diagram requires special handling in TCE
    i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy)
    a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy)
    p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy)
    h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy)

    from sympy.utilities.iterables import variations

    v = Function('v')
    t = Function('t')
    dums = _get_ordered_dummies

    # v(abcd)t(abij)t(ijcd)
    template = v(p1, p2, p3, p4)*t(p1, p2, i, j)*t(i, j, p3, p4)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert dums(base) != dums(expr)
        assert substitute_dummies(expr) == substitute_dummies(base)
    template = v(p1, p2, p3, p4)*t(p1, p2, j, i)*t(j, i, p3, p4)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert dums(base) != dums(expr)
        assert substitute_dummies(expr) == substitute_dummies(base)

    # v(abcd)t(abij)t(jicd)
    template = v(p1, p2, p3, p4)*t(p1, p2, i, j)*t(j, i, p3, p4)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert dums(base) != dums(expr)
        assert substitute_dummies(expr) == substitute_dummies(base)
    template = v(p1, p2, p3, p4)*t(p1, p2, j, i)*t(i, j, p3, p4)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert dums(base) != dums(expr)
        assert substitute_dummies(expr) == substitute_dummies(base)


def test_equivalent_internal_lines_VT2conjT2_ambiguous_order():
    # These diagrams invokes _determine_ambiguous() because the
    # dummies can not be ordered unambiguously by the key alone
    i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy)
    a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy)
    p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy)
    h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy)

    from sympy.utilities.iterables import variations

    v = Function('v')
    t = Function('t')
    dums = _get_ordered_dummies

    # v(abcd)t(abij)t(cdij)
    template = v(p1, p2, p3, p4)*t(p1, p2, i, j)*t(p3, p4, i, j)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert dums(base) != dums(expr)
        assert substitute_dummies(expr) == substitute_dummies(base)
    template = v(p1, p2, p3, p4)*t(p1, p2, j, i)*t(p3, p4, i, j)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert dums(base) != dums(expr)
        assert substitute_dummies(expr) == substitute_dummies(base)


def test_equivalent_internal_lines_VT2():
    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)

    v = Function('v')
    t = Function('t')
    dums = _get_ordered_dummies
    exprs = [
        # permute v. Same dummy order, not equivalent.
        #
        # This test show that the dummy order may not be sensitive to all
        # index permutations.  The following expressions have identical
        # structure as the resulting terms from of the dummy substitutions
        # in the test above.  Here, all expressions have the same dummy
        # order, so they cannot be simplified by means of dummy
        # substitution.  In order to simplify further, it is necessary to
        # exploit symmetries in the objects, for instance if t or v is
        # antisymmetric.
        v(i, j, a, b)*t(a, b, i, j),
        v(j, i, a, b)*t(a, b, i, j),
        v(i, j, b, a)*t(a, b, i, j),
        v(j, i, b, a)*t(a, b, i, j),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) == dums(permut)
        assert substitute_dummies(exprs[0]) != substitute_dummies(permut)

    exprs = [
        # permute t.
        v(i, j, a, b)*t(a, b, i, j),
        v(i, j, a, b)*t(b, a, i, j),
        v(i, j, a, b)*t(a, b, j, i),
        v(i, j, a, b)*t(b, a, j, i),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) != substitute_dummies(permut)

    exprs = [  # permute v and t.  Relabelling of dummies should be equivalent.
        v(i, j, a, b)*t(a, b, i, j),
        v(j, i, a, b)*t(a, b, j, i),
        v(i, j, b, a)*t(b, a, i, j),
        v(j, i, b, a)*t(b, a, j, i),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_internal_external_VT2T2():
    ii, jj = symbols('i j', below_fermi=True)
    aa, bb = symbols('a b', above_fermi=True)
    k, l = symbols('k l', below_fermi=True, cls=Dummy)
    c, d = symbols('c d', above_fermi=True, cls=Dummy)

    v = Function('v')
    t = Function('t')
    dums = _get_ordered_dummies

    exprs = [
        v(k, l, c, d)*t(aa, c, ii, k)*t(bb, d, jj, l),
        v(l, k, c, d)*t(aa, c, ii, l)*t(bb, d, jj, k),
        v(k, l, d, c)*t(aa, d, ii, k)*t(bb, c, jj, l),
        v(l, k, d, c)*t(aa, d, ii, l)*t(bb, c, jj, k),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
    exprs = [
        v(k, l, c, d)*t(aa, c, ii, k)*t(d, bb, jj, l),
        v(l, k, c, d)*t(aa, c, ii, l)*t(d, bb, jj, k),
        v(k, l, d, c)*t(aa, d, ii, k)*t(c, bb, jj, l),
        v(l, k, d, c)*t(aa, d, ii, l)*t(c, bb, jj, k),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
    exprs = [
        v(k, l, c, d)*t(c, aa, ii, k)*t(bb, d, jj, l),
        v(l, k, c, d)*t(c, aa, ii, l)*t(bb, d, jj, k),
        v(k, l, d, c)*t(d, aa, ii, k)*t(bb, c, jj, l),
        v(l, k, d, c)*t(d, aa, ii, l)*t(bb, c, jj, k),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_internal_external_pqrs():
    ii, jj = symbols('i j')
    aa, bb = symbols('a b')
    k, l = symbols('k l', cls=Dummy)
    c, d = symbols('c d', cls=Dummy)

    v = Function('v')
    t = Function('t')
    dums = _get_ordered_dummies

    exprs = [
        v(k, l, c, d)*t(aa, c, ii, k)*t(bb, d, jj, l),
        v(l, k, c, d)*t(aa, c, ii, l)*t(bb, d, jj, k),
        v(k, l, d, c)*t(aa, d, ii, k)*t(bb, c, jj, l),
        v(l, k, d, c)*t(aa, d, ii, l)*t(bb, c, jj, k),
    ]
    for permut in exprs[1:]:
        assert dums(exprs[0]) != dums(permut)
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_dummy_order_well_defined():
    aa, bb = symbols('a b', above_fermi=True)
    k, l, m = symbols('k l m', below_fermi=True, cls=Dummy)
    c, d = symbols('c d', above_fermi=True, cls=Dummy)
    p, q = symbols('p q', cls=Dummy)

    A = Function('A')
    B = Function('B')
    C = Function('C')
    dums = _get_ordered_dummies

    # We go through all key components in the order of increasing priority,
    # and consider only fully orderable expressions.  Non-orderable expressions
    # are tested elsewhere.

    # pos in first factor determines sort order
    assert dums(A(k, l)*B(l, k)) == [k, l]
    assert dums(A(l, k)*B(l, k)) == [l, k]
    assert dums(A(k, l)*B(k, l)) == [k, l]
    assert dums(A(l, k)*B(k, l)) == [l, k]

    # factors involving the index
    assert dums(A(k, l)*B(l, m)*C(k, m)) == [l, k, m]
    assert dums(A(k, l)*B(l, m)*C(m, k)) == [l, k, m]
    assert dums(A(l, k)*B(l, m)*C(k, m)) == [l, k, m]
    assert dums(A(l, k)*B(l, m)*C(m, k)) == [l, k, m]
    assert dums(A(k, l)*B(m, l)*C(k, m)) == [l, k, m]
    assert dums(A(k, l)*B(m, l)*C(m, k)) == [l, k, m]
    assert dums(A(l, k)*B(m, l)*C(k, m)) == [l, k, m]
    assert dums(A(l, k)*B(m, l)*C(m, k)) == [l, k, m]

    # same, but with factor order determined by non-dummies
    assert dums(A(k, aa, l)*A(l, bb, m)*A(bb, k, m)) == [l, k, m]
    assert dums(A(k, aa, l)*A(l, bb, m)*A(bb, m, k)) == [l, k, m]
    assert dums(A(k, aa, l)*A(m, bb, l)*A(bb, k, m)) == [l, k, m]
    assert dums(A(k, aa, l)*A(m, bb, l)*A(bb, m, k)) == [l, k, m]
    assert dums(A(l, aa, k)*A(l, bb, m)*A(bb, k, m)) == [l, k, m]
    assert dums(A(l, aa, k)*A(l, bb, m)*A(bb, m, k)) == [l, k, m]
    assert dums(A(l, aa, k)*A(m, bb, l)*A(bb, k, m)) == [l, k, m]
    assert dums(A(l, aa, k)*A(m, bb, l)*A(bb, m, k)) == [l, k, m]

    # index range
    assert dums(A(p, c, k)*B(p, c, k)) == [k, c, p]
    assert dums(A(p, k, c)*B(p, c, k)) == [k, c, p]
    assert dums(A(c, k, p)*B(p, c, k)) == [k, c, p]
    assert dums(A(c, p, k)*B(p, c, k)) == [k, c, p]
    assert dums(A(k, c, p)*B(p, c, k)) == [k, c, p]
    assert dums(A(k, p, c)*B(p, c, k)) == [k, c, p]
    assert dums(B(p, c, k)*A(p, c, k)) == [k, c, p]
    assert dums(B(p, k, c)*A(p, c, k)) == [k, c, p]
    assert dums(B(c, k, p)*A(p, c, k)) == [k, c, p]
    assert dums(B(c, p, k)*A(p, c, k)) == [k, c, p]
    assert dums(B(k, c, p)*A(p, c, k)) == [k, c, p]
    assert dums(B(k, p, c)*A(p, c, k)) == [k, c, p]


def test_dummy_order_ambiguous():
    aa, bb = symbols('a b', above_fermi=True)
    i, j, k, l, m = symbols('i j k l m', below_fermi=True, cls=Dummy)
    a, b, c, d, e = symbols('a b c d e', above_fermi=True, cls=Dummy)
    p, q = symbols('p q', cls=Dummy)
    p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy)
    p5, p6, p7, p8 = symbols('p5 p6 p7 p8', above_fermi=True, cls=Dummy)
    h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy)
    h5, h6, h7, h8 = symbols('h5 h6 h7 h8', below_fermi=True, cls=Dummy)

    A = Function('A')
    B = Function('B')

    from sympy.utilities.iterables import variations

    # A*A*A*A*B  --  ordering of p5 and p4 is used to figure out the rest
    template = A(p1, p2)*A(p4, p1)*A(p2, p3)*A(p3, p5)*B(p5, p4)
    permutator = variations([a, b, c, d, e], 5)
    base = template.subs(zip([p1, p2, p3, p4, p5], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4, p5], permut)
        expr = template.subs(subslist)
        assert substitute_dummies(expr) == substitute_dummies(base)

    # A*A*A*A*A  --  an arbitrary index is assigned and the rest are figured out
    template = A(p1, p2)*A(p4, p1)*A(p2, p3)*A(p3, p5)*A(p5, p4)
    permutator = variations([a, b, c, d, e], 5)
    base = template.subs(zip([p1, p2, p3, p4, p5], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4, p5], permut)
        expr = template.subs(subslist)
        assert substitute_dummies(expr) == substitute_dummies(base)

    # A*A*A  --  ordering of p5 and p4 is used to figure out the rest
    template = A(p1, p2, p4, p1)*A(p2, p3, p3, p5)*A(p5, p4)
    permutator = variations([a, b, c, d, e], 5)
    base = template.subs(zip([p1, p2, p3, p4, p5], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4, p5], permut)
        expr = template.subs(subslist)
        assert substitute_dummies(expr) == substitute_dummies(base)


def atv(*args):
    return AntiSymmetricTensor('v', args[:2], args[2:] )


def att(*args):
    if len(args) == 4:
        return AntiSymmetricTensor('t', args[:2], args[2:] )
    elif len(args) == 2:
        return AntiSymmetricTensor('t', (args[0],), (args[1],))


def test_dummy_order_inner_outer_lines_VT1T1T1_AT():
    ii = symbols('i', below_fermi=True)
    aa = symbols('a', above_fermi=True)
    k, l = symbols('k l', below_fermi=True, cls=Dummy)
    c, d = symbols('c d', above_fermi=True, cls=Dummy)

    # Coupled-Cluster T1 terms with V*T1*T1*T1
    # t^{a}_{k} t^{c}_{i} t^{d}_{l} v^{lk}_{dc}
    exprs = [
        # permut v and t <=> swapping internal lines, equivalent
        # irrespective of symmetries in v
        atv(k, l, c, d)*att(c, ii)*att(d, l)*att(aa, k),
        atv(l, k, c, d)*att(c, ii)*att(d, k)*att(aa, l),
        atv(k, l, d, c)*att(d, ii)*att(c, l)*att(aa, k),
        atv(l, k, d, c)*att(d, ii)*att(c, k)*att(aa, l),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_dummy_order_inner_outer_lines_VT1T1T1T1_AT():
    ii, jj = symbols('i j', below_fermi=True)
    aa, bb = symbols('a b', above_fermi=True)
    k, l = symbols('k l', below_fermi=True, cls=Dummy)
    c, d = symbols('c d', above_fermi=True, cls=Dummy)

    # Coupled-Cluster T2 terms with V*T1*T1*T1*T1
    # non-equivalent substitutions (change of sign)
    exprs = [
        # permut t <=> swapping external lines
        atv(k, l, c, d)*att(c, ii)*att(d, jj)*att(aa, k)*att(bb, l),
        atv(k, l, c, d)*att(c, jj)*att(d, ii)*att(aa, k)*att(bb, l),
        atv(k, l, c, d)*att(c, ii)*att(d, jj)*att(bb, k)*att(aa, l),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) == -substitute_dummies(permut)

    # equivalent substitutions
    exprs = [
        atv(k, l, c, d)*att(c, ii)*att(d, jj)*att(aa, k)*att(bb, l),
        # permut t <=> swapping external lines
        atv(k, l, c, d)*att(c, jj)*att(d, ii)*att(bb, k)*att(aa, l),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_equivalent_internal_lines_VT1T1_AT():
    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)

    exprs = [  # permute v.  Different dummy order. Not equivalent.
        atv(i, j, a, b)*att(a, i)*att(b, j),
        atv(j, i, a, b)*att(a, i)*att(b, j),
        atv(i, j, b, a)*att(a, i)*att(b, j),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) != substitute_dummies(permut)

    exprs = [  # permute v.  Different dummy order. Equivalent
        atv(i, j, a, b)*att(a, i)*att(b, j),
        atv(j, i, b, a)*att(a, i)*att(b, j),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)

    exprs = [  # permute t.  Same dummy order, not equivalent.
        atv(i, j, a, b)*att(a, i)*att(b, j),
        atv(i, j, a, b)*att(b, i)*att(a, j),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) != substitute_dummies(permut)

    exprs = [  # permute v and t.  Different dummy order, equivalent
        atv(i, j, a, b)*att(a, i)*att(b, j),
        atv(j, i, a, b)*att(a, j)*att(b, i),
        atv(i, j, b, a)*att(b, i)*att(a, j),
        atv(j, i, b, a)*att(b, j)*att(a, i),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_equivalent_internal_lines_VT2conjT2_AT():
    # this diagram requires special handling in TCE
    i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy)
    a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy)
    p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy)
    h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy)

    from sympy.utilities.iterables import variations

    # atv(abcd)att(abij)att(ijcd)
    template = atv(p1, p2, p3, p4)*att(p1, p2, i, j)*att(i, j, p3, p4)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert substitute_dummies(expr) == substitute_dummies(base)
    template = atv(p1, p2, p3, p4)*att(p1, p2, j, i)*att(j, i, p3, p4)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert substitute_dummies(expr) == substitute_dummies(base)

    # atv(abcd)att(abij)att(jicd)
    template = atv(p1, p2, p3, p4)*att(p1, p2, i, j)*att(j, i, p3, p4)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert substitute_dummies(expr) == substitute_dummies(base)
    template = atv(p1, p2, p3, p4)*att(p1, p2, j, i)*att(i, j, p3, p4)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert substitute_dummies(expr) == substitute_dummies(base)


def test_equivalent_internal_lines_VT2conjT2_ambiguous_order_AT():
    # These diagrams invokes _determine_ambiguous() because the
    # dummies can not be ordered unambiguously by the key alone
    i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy)
    a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy)
    p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy)
    h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy)

    from sympy.utilities.iterables import variations

    # atv(abcd)att(abij)att(cdij)
    template = atv(p1, p2, p3, p4)*att(p1, p2, i, j)*att(p3, p4, i, j)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert substitute_dummies(expr) == substitute_dummies(base)
    template = atv(p1, p2, p3, p4)*att(p1, p2, j, i)*att(p3, p4, i, j)
    permutator = variations([a, b, c, d], 4)
    base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
    for permut in permutator:
        subslist = zip([p1, p2, p3, p4], permut)
        expr = template.subs(subslist)
        assert substitute_dummies(expr) == substitute_dummies(base)


def test_equivalent_internal_lines_VT2_AT():
    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)

    exprs = [
        # permute v. Same dummy order, not equivalent.
        atv(i, j, a, b)*att(a, b, i, j),
        atv(j, i, a, b)*att(a, b, i, j),
        atv(i, j, b, a)*att(a, b, i, j),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) != substitute_dummies(permut)

    exprs = [
        # permute t.
        atv(i, j, a, b)*att(a, b, i, j),
        atv(i, j, a, b)*att(b, a, i, j),
        atv(i, j, a, b)*att(a, b, j, i),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) != substitute_dummies(permut)

    exprs = [  # permute v and t.  Relabelling of dummies should be equivalent.
        atv(i, j, a, b)*att(a, b, i, j),
        atv(j, i, a, b)*att(a, b, j, i),
        atv(i, j, b, a)*att(b, a, i, j),
        atv(j, i, b, a)*att(b, a, j, i),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_internal_external_VT2T2_AT():
    ii, jj = symbols('i j', below_fermi=True)
    aa, bb = symbols('a b', above_fermi=True)
    k, l = symbols('k l', below_fermi=True, cls=Dummy)
    c, d = symbols('c d', above_fermi=True, cls=Dummy)

    exprs = [
        atv(k, l, c, d)*att(aa, c, ii, k)*att(bb, d, jj, l),
        atv(l, k, c, d)*att(aa, c, ii, l)*att(bb, d, jj, k),
        atv(k, l, d, c)*att(aa, d, ii, k)*att(bb, c, jj, l),
        atv(l, k, d, c)*att(aa, d, ii, l)*att(bb, c, jj, k),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
    exprs = [
        atv(k, l, c, d)*att(aa, c, ii, k)*att(d, bb, jj, l),
        atv(l, k, c, d)*att(aa, c, ii, l)*att(d, bb, jj, k),
        atv(k, l, d, c)*att(aa, d, ii, k)*att(c, bb, jj, l),
        atv(l, k, d, c)*att(aa, d, ii, l)*att(c, bb, jj, k),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
    exprs = [
        atv(k, l, c, d)*att(c, aa, ii, k)*att(bb, d, jj, l),
        atv(l, k, c, d)*att(c, aa, ii, l)*att(bb, d, jj, k),
        atv(k, l, d, c)*att(d, aa, ii, k)*att(bb, c, jj, l),
        atv(l, k, d, c)*att(d, aa, ii, l)*att(bb, c, jj, k),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_internal_external_pqrs_AT():
    ii, jj = symbols('i j')
    aa, bb = symbols('a b')
    k, l = symbols('k l', cls=Dummy)
    c, d = symbols('c d', cls=Dummy)

    exprs = [
        atv(k, l, c, d)*att(aa, c, ii, k)*att(bb, d, jj, l),
        atv(l, k, c, d)*att(aa, c, ii, l)*att(bb, d, jj, k),
        atv(k, l, d, c)*att(aa, d, ii, k)*att(bb, c, jj, l),
        atv(l, k, d, c)*att(aa, d, ii, l)*att(bb, c, jj, k),
    ]
    for permut in exprs[1:]:
        assert substitute_dummies(exprs[0]) == substitute_dummies(permut)


def test_issue_19661():
    a = Symbol('0')
    assert latex(Commutator(Bd(a)**2, B(a))
                 ) == '- \\left[b_{0},{b^\\dagger_{0}}^{2}\\right]'


def test_canonical_ordering_AntiSymmetricTensor():
    v = symbols("v")

    c, d = symbols(('c','d'), above_fermi=True,
                                   cls=Dummy)
    k, l = symbols(('k','l'), below_fermi=True,
                                   cls=Dummy)

    # formerly, the left gave either the left or the right
    assert AntiSymmetricTensor(v, (k, l), (d, c)
        ) == -AntiSymmetricTensor(v, (l, k), (d, c))