def test_h2o_rdm(self):
mol = mol_s2
mf = mf_s2
mycc = uccsd.UCCSD(mf)
mycc.frozen = 2
ecc, t1, t2 = mycc.kernel()
l1, l2 = mycc.solve_lambda()
dm1a, dm1b = mycc.make_rdm1(t1, t2, l1, l2)
dm2aa, dm2ab, dm2bb = mycc.make_rdm2(t1, t2, l1, l2)
mo_a = mf.mo_coeff[0]
mo_b = mf.mo_coeff[1]
nmoa = mo_a.shape[1]
nmob = mo_b.shape[1]
eriaa = ao2mo.kernel(mf._eri, mo_a, compact=False).reshape([nmoa] * 4)
eribb = ao2mo.kernel(mf._eri, mo_b, compact=False).reshape([nmob] * 4)
eriab = ao2mo.kernel(mf._eri, (mo_a, mo_a, mo_b, mo_b), compact=False)
eriab = eriab.reshape([nmoa, nmoa, nmob, nmob])
hcore = mf.get_hcore()
h1a = reduce(numpy.dot, (mo_a.T.conj(), hcore, mo_a))
h1b = reduce(numpy.dot, (mo_b.T.conj(), hcore, mo_b))
e1 = numpy.einsum('ij,ji', h1a, dm1a)
e1 += numpy.einsum('ij,ji', h1b, dm1b)
e1 += numpy.einsum('ijkl,ijkl', eriaa, dm2aa) * .5
e1 += numpy.einsum('ijkl,ijkl', eriab, dm2ab)
e1 += numpy.einsum('ijkl,ijkl', eribb, dm2bb) * .5
e1 += mol.energy_nuc()
self.assertAlmostEqual(e1, mycc.e_tot, 7)
d1 = uccsd_rdm._gamma1_intermediates(mycc, mycc.t1, mycc.t2, mycc.l1,
mycc.l2)
mycc.max_memory = 0
d2 = uccsd_rdm._gamma2_intermediates(mycc, mycc.t1, mycc.t2, mycc.l1,
mycc.l2, True)
dm2 = uccsd_rdm._make_rdm2(mycc,
d1,
d2,
with_dm1=True,
with_frozen=True)
e1 = numpy.einsum('ij,ji', h1a, dm1a)
e1 += numpy.einsum('ij,ji', h1b, dm1b)
e1 += numpy.einsum('ijkl,ijkl', eriaa, dm2[0]) * .5
e1 += numpy.einsum('ijkl,ijkl', eriab, dm2[1])
e1 += numpy.einsum('ijkl,ijkl', eribb, dm2[2]) * .5
e1 += mol.energy_nuc()
self.assertAlmostEqual(e1, mycc.e_tot, 7)
def test_h2o_rdm(self):
mol = mol_s2
mf = mf_s2
mycc = uccsd.UCCSD(mf)
mycc.frozen = 2
ecc, t1, t2 = mycc.kernel()
l1, l2 = mycc.solve_lambda()
dm1a,dm1b = mycc.make_rdm1(t1, t2, l1, l2)
dm2aa,dm2ab,dm2bb = mycc.make_rdm2(t1, t2, l1, l2)
mo_a = mf.mo_coeff[0]
mo_b = mf.mo_coeff[1]
nmoa = mo_a.shape[1]
nmob = mo_b.shape[1]
eriaa = ao2mo.kernel(mf._eri, mo_a, compact=False).reshape([nmoa]*4)
eribb = ao2mo.kernel(mf._eri, mo_b, compact=False).reshape([nmob]*4)
eriab = ao2mo.kernel(mf._eri, (mo_a,mo_a,mo_b,mo_b), compact=False)
eriab = eriab.reshape([nmoa,nmoa,nmob,nmob])
hcore = mf.get_hcore()
h1a = reduce(numpy.dot, (mo_a.T.conj(), hcore, mo_a))
h1b = reduce(numpy.dot, (mo_b.T.conj(), hcore, mo_b))
e1 = numpy.einsum('ij,ji', h1a, dm1a)
e1+= numpy.einsum('ij,ji', h1b, dm1b)
e1+= numpy.einsum('ijkl,ijkl', eriaa, dm2aa) * .5
e1+= numpy.einsum('ijkl,ijkl', eriab, dm2ab)
e1+= numpy.einsum('ijkl,ijkl', eribb, dm2bb) * .5
e1+= mol.energy_nuc()
self.assertAlmostEqual(e1, mycc.e_tot, 7)
d1 = uccsd_rdm._gamma1_intermediates(mycc, mycc.t1, mycc.t2, mycc.l1, mycc.l2)
mycc.max_memory = 0
d2 = uccsd_rdm._gamma2_intermediates(mycc, mycc.t1, mycc.t2, mycc.l1, mycc.l2, True)
dm2 = uccsd_rdm._make_rdm2(mycc, d1, d2, with_dm1=True, with_frozen=True)
e1 = numpy.einsum('ij,ji', h1a, dm1a)
e1+= numpy.einsum('ij,ji', h1b, dm1b)
e1+= numpy.einsum('ijkl,ijkl', eriaa, dm2[0]) * .5
e1+= numpy.einsum('ijkl,ijkl', eriab, dm2[1])
e1+= numpy.einsum('ijkl,ijkl', eribb, dm2[2]) * .5
e1+= mol.energy_nuc()
self.assertAlmostEqual(e1, mycc.e_tot, 7)
def _gamma2_intermediates(mycc,
t1,
t2,
l1,
l2,
eris=None,
compress_vvvv=False):
d2 = uccsd_rdm._gamma2_intermediates(mycc, t1, t2, l1, l2)
if eris is None: eris = mycc.ao2mo()
dovov, dovOV, dOVov, dOVOV = d2[0]
dvvvv, dvvVV, dVVvv, dVVVV = d2[1]
doooo, dooOO, dOOoo, dOOOO = d2[2]
doovv, dooVV, dOOvv, dOOVV = d2[3]
dovvo, dovVO, dOVvo, dOVVO = d2[4]
dvvov, dvvOV, dVVov, dVVOV = d2[5]
dovvv, dovVV, dOVvv, dOVVV = d2[6]
dooov, dooOV, dOOov, dOOOV = d2[7]
t1a, t1b = t1
t2aa, t2ab, t2bb = t2
nocca, noccb, nvira, nvirb = t2ab.shape
nmoa = eris.focka.shape[0]
nmob = eris.fockb.shape[0]
mo_ea = eris.focka.diagonal().real
mo_eb = eris.fockb.diagonal().real
eia = mo_ea[:nocca, None] - mo_ea[nocca:]
eIA = mo_eb[:noccb, None] - mo_eb[noccb:]
fvo = eris.focka[nocca:, :nocca]
fVO = eris.fockb[noccb:, :noccb]
# aaa
d3 = lib.direct_sum('ia+jb+kc->ijkabc', eia, eia, eia)
w = numpy.einsum('ijae,kceb->ijkabc', t2aa,
numpy.asarray(eris.get_ovvv()).conj())
w -= numpy.einsum('mkbc,iajm->ijkabc', t2aa,
numpy.asarray(eris.ovoo.conj()))
v = numpy.einsum('jbkc,ia->ijkabc', numpy.asarray(eris.ovov).conj(), t1a)
v += numpy.einsum('jkbc,ai->ijkabc', t2aa, fvo) * .5
rw = r6(p6(w)) / d3
wvd = r6(p6(w * 2 + v)) / d3
dovov += numpy.einsum('ia,ijkabc->jbkc', t1a, rw.conj()) * 0.25
# *(1/8) instead of (1/4) because ooov appears 4 times in the 2pdm tensor due
# to symmetrization, and its contribution is scaled by 1/2 in Tr(H,2pdm)
dooov -= numpy.einsum('mkbc,ijkabc->jmia', t2aa, wvd.conj()) * .125
dovvv += numpy.einsum('kjcf,ijkabc->iafb', t2aa, wvd.conj()) * .125
# bbb
d3 = lib.direct_sum('ia+jb+kc->ijkabc', eIA, eIA, eIA)
w = numpy.einsum('ijae,kceb->ijkabc', t2bb,
numpy.asarray(eris.get_OVVV()).conj())
w -= numpy.einsum('imab,kcjm->ijkabc', t2bb,
numpy.asarray(eris.OVOO.conj()))
v = numpy.einsum('jbkc,ia->ijkabc', numpy.asarray(eris.OVOV).conj(), t1b)
v += numpy.einsum('jkbc,ai->ijkabc', t2bb, fVO) * .5
rw = r6(p6(w)) / d3
wvd = r6(p6(w * 2 + v)) / d3
dOVOV += numpy.einsum('ia,ijkabc->jbkc', t1b, rw.conj()) * .25
dOOOV -= numpy.einsum('mkbc,ijkabc->jmia', t2bb, wvd.conj()) * .125
dOVVV += numpy.einsum('kjcf,ijkabc->iafb', t2bb, wvd.conj()) * .125
# baa
d3 = lib.direct_sum('ia+jb+kc->ijkabc', eIA, eia, eia)
w = numpy.einsum('jIeA,kceb->IjkAbc', t2ab,
numpy.asarray(eris.get_ovvv()).conj()) * 2
w += numpy.einsum('jIbE,kcEA->IjkAbc', t2ab,
numpy.asarray(eris.get_ovVV()).conj()) * 2
w += numpy.einsum('jkbe,IAec->IjkAbc', t2aa,
numpy.asarray(eris.get_OVvv()).conj())
w -= numpy.einsum('mIbA,kcjm->IjkAbc', t2ab,
numpy.asarray(eris.ovoo).conj()) * 2
w -= numpy.einsum('jMbA,kcIM->IjkAbc', t2ab,
numpy.asarray(eris.ovOO).conj()) * 2
w -= numpy.einsum('jmbc,IAkm->IjkAbc', t2aa,
numpy.asarray(eris.OVoo).conj())
v = numpy.einsum('jbkc,IA->IjkAbc', numpy.asarray(eris.ovov).conj(), t1b)
v += numpy.einsum('kcIA,jb->IjkAbc', numpy.asarray(eris.ovOV).conj(), t1a)
v += numpy.einsum('kcIA,jb->IjkAbc', numpy.asarray(eris.ovOV).conj(), t1a)
v += numpy.einsum('jkbc,AI->IjkAbc', t2aa, fVO) * .5
v += numpy.einsum('kIcA,bj->IjkAbc', t2ab, fvo) * 2
rw = r4(w) / d3
wvd = r4(w * 2 + v) / d3
dovvv += numpy.einsum('jiea,ijkabc->kceb', t2ab, wvd.conj()) * .25
dovVV += numpy.einsum('jibe,ijkabc->kcea', t2ab, wvd.conj()) * .25
dOVvv += numpy.einsum('jkbe,ijkabc->iaec', t2aa, wvd.conj()) * .125
dooov -= numpy.einsum('miba,ijkabc->jmkc', t2ab, wvd.conj()) * .25
dOOov -= numpy.einsum('jmba,ijkabc->imkc', t2ab, wvd.conj()) * .25
dooOV -= numpy.einsum('jmbc,ijkabc->kmia', t2aa, wvd.conj()) * .125
dovov += numpy.einsum('ia,ijkabc->jbkc', t1b, rw.conj()) * .25
#dOVov += numpy.einsum('jb,ijkabc->iakc', t1a, rw.conj()) * .25
dovOV += numpy.einsum('jb,ijkabc->kcia', t1a, rw.conj()) * .25
# bba
d3 = lib.direct_sum('ia+jb+kc->ijkabc', eia, eIA, eIA)
w = numpy.einsum('ijae,kceb->ijkabc', t2ab,
numpy.asarray(eris.get_OVVV()).conj()) * 2
w += numpy.einsum('ijeb,kcea->ijkabc', t2ab,
numpy.asarray(eris.get_OVvv()).conj()) * 2
w += numpy.einsum('jkbe,iaec->ijkabc', t2bb,
numpy.asarray(eris.get_ovVV()).conj())
w -= numpy.einsum('imab,kcjm->ijkabc', t2ab,
numpy.asarray(eris.OVOO).conj()) * 2
w -= numpy.einsum('mjab,kcim->ijkabc', t2ab,
numpy.asarray(eris.OVoo).conj()) * 2
w -= numpy.einsum('jmbc,iakm->ijkabc', t2bb,
numpy.asarray(eris.ovOO).conj())
v = numpy.einsum('jbkc,ia->ijkabc', numpy.asarray(eris.OVOV).conj(), t1a)
v += numpy.einsum('iakc,jb->ijkabc', numpy.asarray(eris.ovOV).conj(), t1b)
v += numpy.einsum('iakc,jb->ijkabc', numpy.asarray(eris.ovOV).conj(), t1b)
v += numpy.einsum('JKBC,ai->iJKaBC', t2bb, fvo) * .5
v += numpy.einsum('iKaC,BJ->iJKaBC', t2ab, fVO) * 2
rw = r4(w) / d3
wvd = r4(w * 2 + v) / d3
dOVVV += numpy.einsum('ijae,ijkabc->kceb', t2ab, wvd.conj()) * .25
dOVvv += numpy.einsum('ijeb,ijkabc->kcea', t2ab, wvd.conj()) * .25
dovVV += numpy.einsum('jkbe,ijkabc->iaec', t2bb, wvd.conj()) * .125
dOOOV -= numpy.einsum('imab,ijkabc->jmkc', t2ab, wvd.conj()) * .25
dooOV -= numpy.einsum('mjab,ijkabc->imkc', t2ab, wvd.conj()) * .25
dOOov -= numpy.einsum('jmbc,ijkabc->kmia', t2bb, wvd.conj()) * .125
dOVOV += numpy.einsum('ia,ijkabc->jbkc', t1a, rw.conj()) * .25
dovOV += numpy.einsum('jb,ijkabc->iakc', t1b, rw.conj()) * .25
#dOVov += numpy.einsum('jb,ijkabc->kcia', t1b, rw.conj()) * .25
if compress_vvvv:
nmoa, nmob = mycc.nmo
nocca, noccb, nvira, nvirb = t2ab.shape
idxa = numpy.tril_indices(nvira)
idxa = idxa[0] * nvira + idxa[1]
idxb = numpy.tril_indices(nvirb)
idxb = idxb[0] * nvirb + idxb[1]
dvvvv = dvvvv + dvvvv.transpose(1, 0, 2, 3)
dvvvv = lib.take_2d(dvvvv.reshape(nvira**2, nvira**2), idxa, idxa)
dvvvv *= .5
dvvVV = dvvVV + dvvVV.transpose(1, 0, 2, 3)
dvvVV = lib.take_2d(dvvVV.reshape(nvira**2, nvirb**2), idxa, idxb)
dVVVV = dVVVV + dVVVV.transpose(1, 0, 2, 3)
dVVVV = lib.take_2d(dVVVV.reshape(nvirb**2, nvirb**2), idxb, idxb)
dVVVV *= .5
d2 = ((dovov, dovOV, dOVov, dOVOV), (dvvvv, dvvVV, dVVvv, dVVVV),
(doooo, dooOO, dOOoo, dOOOO), (doovv, dooVV, dOOvv, dOOVV),
(dovvo, dovVO, dOVvo, dOVVO), (dvvov, dvvOV, dVVov, dVVOV),
(dovvv, dovVV, dOVvv, dOVVV), (dooov, dooOV, dOOov, dOOOV))
return d2
def _gamma2_intermediates(mycc, t1, t2, l1, l2, eris=None,
compress_vvvv=False):
d2 = uccsd_rdm._gamma2_intermediates(mycc, t1, t2, l1, l2)
if eris is None: eris = mycc.ao2mo()
dovov, dovOV, dOVov, dOVOV = d2[0]
dvvvv, dvvVV, dVVvv, dVVVV = d2[1]
doooo, dooOO, dOOoo, dOOOO = d2[2]
doovv, dooVV, dOOvv, dOOVV = d2[3]
dovvo, dovVO, dOVvo, dOVVO = d2[4]
dvvov, dvvOV, dVVov, dVVOV = d2[5]
dovvv, dovVV, dOVvv, dOVVV = d2[6]
dooov, dooOV, dOOov, dOOOV = d2[7]
t1a, t1b = t1
t2aa, t2ab, t2bb = t2
nocca, noccb, nvira, nvirb = t2ab.shape
nmoa = eris.focka.shape[0]
nmob = eris.fockb.shape[0]
mo_ea, mo_eb = eris.mo_energy
eia = mo_ea[:nocca,None] - mo_ea[nocca:]
eIA = mo_eb[:noccb,None] - mo_eb[noccb:]
fvo = eris.focka[nocca:,:nocca]
fVO = eris.fockb[noccb:,:noccb]
# aaa
d3 = lib.direct_sum('ia+jb+kc->ijkabc', eia, eia, eia)
w = numpy.einsum('ijae,kceb->ijkabc', t2aa, numpy.asarray(eris.get_ovvv()).conj())
w-= numpy.einsum('mkbc,iajm->ijkabc', t2aa, numpy.asarray(eris.ovoo.conj()))
v = numpy.einsum('jbkc,ia->ijkabc', numpy.asarray(eris.ovov).conj(), t1a)
v+= numpy.einsum('jkbc,ai->ijkabc', t2aa, fvo) * .5
rw = r6(p6(w)) / d3
wvd = r6(p6(w * 2 + v)) / d3
dovov += numpy.einsum('ia,ijkabc->jbkc', t1a, rw.conj()) * 0.25
# *(1/8) instead of (1/4) because ooov appears 4 times in the 2pdm tensor due
# to symmetrization, and its contribution is scaled by 1/2 in Tr(H,2pdm)
dooov -= numpy.einsum('mkbc,ijkabc->jmia', t2aa, wvd.conj()) * .125
dovvv += numpy.einsum('kjcf,ijkabc->iafb', t2aa, wvd.conj()) * .125
# bbb
d3 = lib.direct_sum('ia+jb+kc->ijkabc', eIA, eIA, eIA)
w = numpy.einsum('ijae,kceb->ijkabc', t2bb, numpy.asarray(eris.get_OVVV()).conj())
w-= numpy.einsum('imab,kcjm->ijkabc', t2bb, numpy.asarray(eris.OVOO.conj()))
v = numpy.einsum('jbkc,ia->ijkabc', numpy.asarray(eris.OVOV).conj(), t1b)
v+= numpy.einsum('jkbc,ai->ijkabc', t2bb, fVO) * .5
rw = r6(p6(w)) / d3
wvd = r6(p6(w * 2 + v)) / d3
dOVOV += numpy.einsum('ia,ijkabc->jbkc', t1b, rw.conj()) * .25
dOOOV -= numpy.einsum('mkbc,ijkabc->jmia', t2bb, wvd.conj()) * .125
dOVVV += numpy.einsum('kjcf,ijkabc->iafb', t2bb, wvd.conj()) * .125
# baa
d3 = lib.direct_sum('ia+jb+kc->ijkabc', eIA, eia, eia)
w = numpy.einsum('jIeA,kceb->IjkAbc', t2ab, numpy.asarray(eris.get_ovvv()).conj()) * 2
w += numpy.einsum('jIbE,kcEA->IjkAbc', t2ab, numpy.asarray(eris.get_ovVV()).conj()) * 2
w += numpy.einsum('jkbe,IAec->IjkAbc', t2aa, numpy.asarray(eris.get_OVvv()).conj())
w -= numpy.einsum('mIbA,kcjm->IjkAbc', t2ab, numpy.asarray(eris.ovoo).conj()) * 2
w -= numpy.einsum('jMbA,kcIM->IjkAbc', t2ab, numpy.asarray(eris.ovOO).conj()) * 2
w -= numpy.einsum('jmbc,IAkm->IjkAbc', t2aa, numpy.asarray(eris.OVoo).conj())
v = numpy.einsum('jbkc,IA->IjkAbc', numpy.asarray(eris.ovov).conj(), t1b)
v += numpy.einsum('kcIA,jb->IjkAbc', numpy.asarray(eris.ovOV).conj(), t1a)
v += numpy.einsum('kcIA,jb->IjkAbc', numpy.asarray(eris.ovOV).conj(), t1a)
v += numpy.einsum('jkbc,AI->IjkAbc', t2aa, fVO) * .5
v += numpy.einsum('kIcA,bj->IjkAbc', t2ab, fvo) * 2
rw = r4(w) / d3
wvd = r4(w * 2 + v) / d3
dovvv += numpy.einsum('jiea,ijkabc->kceb', t2ab, wvd.conj()) * .25
dovVV += numpy.einsum('jibe,ijkabc->kcea', t2ab, wvd.conj()) * .25
dOVvv += numpy.einsum('jkbe,ijkabc->iaec', t2aa, wvd.conj()) * .125
dooov -= numpy.einsum('miba,ijkabc->jmkc', t2ab, wvd.conj()) * .25
dOOov -= numpy.einsum('jmba,ijkabc->imkc', t2ab, wvd.conj()) * .25
dooOV -= numpy.einsum('jmbc,ijkabc->kmia', t2aa, wvd.conj()) * .125
dovov += numpy.einsum('ia,ijkabc->jbkc', t1b, rw.conj()) * .25
#dOVov += numpy.einsum('jb,ijkabc->iakc', t1a, rw.conj()) * .25
dovOV += numpy.einsum('jb,ijkabc->kcia', t1a, rw.conj()) * .25
# bba
d3 = lib.direct_sum('ia+jb+kc->ijkabc', eia, eIA, eIA)
w = numpy.einsum('ijae,kceb->ijkabc', t2ab, numpy.asarray(eris.get_OVVV()).conj()) * 2
w += numpy.einsum('ijeb,kcea->ijkabc', t2ab, numpy.asarray(eris.get_OVvv()).conj()) * 2
w += numpy.einsum('jkbe,iaec->ijkabc', t2bb, numpy.asarray(eris.get_ovVV()).conj())
w -= numpy.einsum('imab,kcjm->ijkabc', t2ab, numpy.asarray(eris.OVOO).conj()) * 2
w -= numpy.einsum('mjab,kcim->ijkabc', t2ab, numpy.asarray(eris.OVoo).conj()) * 2
w -= numpy.einsum('jmbc,iakm->ijkabc', t2bb, numpy.asarray(eris.ovOO).conj())
v = numpy.einsum('jbkc,ia->ijkabc', numpy.asarray(eris.OVOV).conj(), t1a)
v += numpy.einsum('iakc,jb->ijkabc', numpy.asarray(eris.ovOV).conj(), t1b)
v += numpy.einsum('iakc,jb->ijkabc', numpy.asarray(eris.ovOV).conj(), t1b)
v += numpy.einsum('JKBC,ai->iJKaBC', t2bb, fvo) * .5
v += numpy.einsum('iKaC,BJ->iJKaBC', t2ab, fVO) * 2
rw = r4(w) / d3
wvd = r4(w * 2 + v) / d3
dOVVV += numpy.einsum('ijae,ijkabc->kceb', t2ab, wvd.conj()) * .25
dOVvv += numpy.einsum('ijeb,ijkabc->kcea', t2ab, wvd.conj()) * .25
dovVV += numpy.einsum('jkbe,ijkabc->iaec', t2bb, wvd.conj()) * .125
dOOOV -= numpy.einsum('imab,ijkabc->jmkc', t2ab, wvd.conj()) * .25
dooOV -= numpy.einsum('mjab,ijkabc->imkc', t2ab, wvd.conj()) * .25
dOOov -= numpy.einsum('jmbc,ijkabc->kmia', t2bb, wvd.conj()) * .125
dOVOV += numpy.einsum('ia,ijkabc->jbkc', t1a, rw.conj()) * .25
dovOV += numpy.einsum('jb,ijkabc->iakc', t1b, rw.conj()) * .25
#dOVov += numpy.einsum('jb,ijkabc->kcia', t1b, rw.conj()) * .25
if compress_vvvv:
nmoa, nmob = mycc.nmo
nocca, noccb, nvira, nvirb = t2ab.shape
idxa = numpy.tril_indices(nvira)
idxa = idxa[0] * nvira + idxa[1]
idxb = numpy.tril_indices(nvirb)
idxb = idxb[0] * nvirb + idxb[1]
dvvvv = dvvvv + dvvvv.transpose(1,0,2,3)
dvvvv = lib.take_2d(dvvvv.reshape(nvira**2,nvira**2), idxa, idxa)
dvvvv *= .5
dvvVV = dvvVV + dvvVV.transpose(1,0,2,3)
dvvVV = lib.take_2d(dvvVV.reshape(nvira**2,nvirb**2), idxa, idxb)
dVVVV = dVVVV + dVVVV.transpose(1,0,2,3)
dVVVV = lib.take_2d(dVVVV.reshape(nvirb**2,nvirb**2), idxb, idxb)
dVVVV *= .5
d2 = ((dovov, dovOV, dOVov, dOVOV),
(dvvvv, dvvVV, dVVvv, dVVVV),
(doooo, dooOO, dOOoo, dOOOO),
(doovv, dooVV, dOOvv, dOOVV),
(dovvo, dovVO, dOVvo, dOVVO),
(dvvov, dvvOV, dVVov, dVVOV),
(dovvv, dovVV, dOVvv, dOVVV),
(dooov, dooOV, dOOov, dOOOV))
return d2
