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)