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_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 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 R0(expr):
if expr == "IP":
R0 = 0
return R0
elif expr == "DIP":
R0 = 0
return R0
elif expr == "EA":
R0 = 0
return R0
elif expr == "DEA":
R0 = 0
return R0
elif expr == "EE":
R0 = Fraction(1, 1)*AntiSymmetricTensor('R0',(),())
return R0
elif expr == "CCSD":
R0 = 1
return R0
def L2(expr):
j1,j2,j3,j4,j5 = symbols('j1,j2,j3,j4,j5' ,below_fermi=True, cls=Dummy)
b1,b2,b3,b4,b5 = symbols('b1,b2,b3,b4,b5' ,above_fermi=True, cls=Dummy)
if expr == "IP":
L2 = Fraction(1, 2)*AntiSymmetricTensor('l',(j2,j3),(b1,))*Fd(j2)*Fd(j3)*F(b1)
return L2
elif expr == "DIP":
L2 = Fraction(1, 6)*AntiSymmetricTensor('l',(j3,j4,j5),(b1,))*Fd(j3)*Fd(j4)*Fd(j5)*F(b1)
return L2
elif expr == "EA":
L2 = Fraction(1, 2)*AntiSymmetricTensor('l',(j1,),(b2,b3))*Fd(j1)*F(b3)*F(b2)
return L2
elif expr == "DEA":
L2 = Fraction(1, 6)*AntiSymmetricTensor('l',(j1,),(b3,b4,b5))*Fd(j1)*F(b5)*F(b4)*F(b3)
return L2
elif expr == "EE":
L2 = Fraction(1, 4)*AntiSymmetricTensor('l',(j2,j3),(b2,b3))*Fd(j2)*Fd(j3)*F(b3)*F(b2)
return L2
elif expr == "CCSD":
L2 = Fraction(1, 4)*AntiSymmetricTensor('l',(j2,j3),(b2,b3))*Fd(j2)*Fd(j3)*F(b3)*F(b2)
return L2
def 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 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 atv(*args):
return AntiSymmetricTensor("v", args[:2], args[2:])
def test_sympy__physics__secondquant__AntiSymmetricTensor():
from sympy.physics.secondquant import AntiSymmetricTensor
i, j = symbols('i j', below_fermi=True)
a, b = symbols('a b', above_fermi=True)
assert _test_args(AntiSymmetricTensor('v', (a, i), (b, j)))
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 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 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)
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)
p, q, r, s = symbols('p,q,r,s', cls=Dummy)
#Setup Hamiltonian on normal ordered form
E0 = symbols('Eref', real=True, constant=True) #Reference energy
f = AntiSymmetricTensor('f', (p, ), (q, ))
pq = NO((Fd(p) * F(q)))
Fock = f * pq #F is reserved by sympy
V = AntiSymmetricTensor('V', (p, q), (r, s))
pqsr = NO(Fd(p) * Fd(q) * F(s) * F(r))
HI = Rational(1, 4) * V * pqsr
HN = E0 + F + HI
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
print("Compute CCD energy and amplitude equations term by term")
#BCH: HN + [HN,T] + 0.5*[[HN,T],T] + 1/6 * [[[HN,T],T],T] + 1/24 * [[[[HN,T],T],T],T]
from sympy import symbols, latex, WildFunction, collect, Rational, simplify
from sympy.physics.secondquant import F, Fd, wicks, AntiSymmetricTensor, substitute_dummies, NO, evaluate deltas
"""
Define Hamiltonian and the second-quantized representation of a three-body Slater determinant.
"""
# Define Hamiltonian
p, q, r, s = symbols("p q r s", dummy=True)
f = AntiSymmetricTensor("f", (p,), (q,))
pr = Fd(p) * F(q)
v = AntiSymmetricTensor("v", (p, q), (r, s))
pqsr = Fd(p) * Fd(q) * F(s) * F(r)
Hamiltonian = f * pr + Rational(1) / Rational(4) * v * pqsr
a, b, c, d, e, f = symbols("a, b, c, d, e, f", above_fermi=True)
# Create teh representatoin
expression = wicks(F(c) * F(b) * F(a) * Hamiltonian * Fd(d) * Fd(e) * Fd(f), keep_only_fully_contracted=True, simplify_kronecker_deltas=True)
expression = evaluate_deltas(expression)
expression = simplify(expression)
print(latex(expression))
