def MakeVlocForRdm(NewRdm, nOcc, EmbFock, h0_for_vloc):
# okay... we have the target eigensystem. We now still
# need an operator EmbFock + vloc which generates it as
# a ground state density. The condition here is that
#
# dot(h0_for_vloc, vloc1) =: K v
#
# has the same eigensystem as the one we're calculating,
# or, equivalently, that it commutes with the rdm we have.
# We assert that the resulting Fock matrix does not have
# components mixing occupied and virtual orbitals, which is
# equivalent.
# Note that we cannot assume that F is diagonal in the target rdm
# eigenvectors, because the eigensystem is degenerate
# (has only 0 and 1 evs)
#
# Also note that in general vloc has not enough variational
# freedom to represent arbitrary eigensystems, so technically we
# are returning the best approximation which is possible.
ew, ev = la.eigh(NewRdm)
O, V = ev[:, nOcc:], ev[:, :nOcc]
lhs = mdot(O.T, EmbFock, V).flatten()
VlocSize = h0_for_vloc.shape[-1]
h0_for_vloc_reshape = h0_for_vloc.reshape((nEmb, nEmb, VlocSize))
K = einsum("ro,rsp,sv->ovp", O, h0_for_vloc_reshape,
V).reshape(O.shape[1] * V.shape[1], VlocSize)
rhs = -la.lstsq(K, lhs)[0]
return dot(h0_for_vloc_reshape, rhs)
def MakeFockOp(Rdm, CoreH, Int2e_Frs, ORB_OCC, Int2e_4ix=None, Orb=None, ExchFactor=1.):
"""make closed-shell/spin ensemble Fock matrix:
f[rs] = h[rs] + j[rs] - .5 k[rs].
where j[rs] = (rs|tu) rdm[tu]
and k[rs] = (rs|tu) rdm[su]
Return (f, j, k)."""
assert(Int2e_Frs is None or Int2e_4ix is None)
if Int2e_Frs is not None:
if Orb is None:
OccNum,Orb = eigh(-Rdm); OccNum *= -1.
else:
OccNum = diag(mdot(Orb.T, Rdm, Orb))
# filter out occupation numbers slightly broken due to rounding.
OccNum[OccNum < 0.] = 0.
OccNum[OccNum > ORB_OCC] = ORB_OCC
nFit, nOrb = Int2e_Frs.shape[0], Int2e_Frs.shape[1]
# make coulomb.
gamma = einsum("Frs,rs",Int2e_Frs, Rdm)
j = einsum("Frs,F",Int2e_Frs,gamma)
# make exchange -- first need to find the occupied orbitals.
nOcc = len([o for o in OccNum if abs(o-ORB_OCC)<1e-8])
OccOrb = Orb[:,:nOcc] * OccNum[:nOcc]**.5
k1 = einsum("Frs,si->Fir", Int2e_Frs, OccOrb).reshape((nFit * nOcc, nOrb))
k = dot(k1.T, k1)
else:
assert(Int2e_4ix is not None)
j = einsum("rstu,tu",Int2e_4ix, Rdm)
k = einsum("rtsu,tu",Int2e_4ix, Rdm)
Fock = CoreH + j - (ExchFactor/ORB_OCC) * k
return Fock, j, k
def MakeVlocForRdm(NewRdm, nOcc, EmbFock, h0_for_vloc):
# okay... we have the target eigensystem. We now still
# need an operator EmbFock + vloc which generates it as
# a ground state density. The condition here is that
#
# dot(h0_for_vloc, vloc1) =: K v
#
# has the same eigensystem as the one we're calculating,
# or, equivalently, that it commutes with the rdm we have.
# We assert that the resulting Fock matrix does not have
# components mixing occupied and virtual orbitals, which is
# equivalent.
# Note that we cannot assume that F is diagonal in the target rdm
# eigenvectors, because the eigensystem is degenerate
# (has only 0 and 1 evs)
#
# Also note that in general vloc has not enough variational
# freedom to represent arbitrary eigensystems, so technically we
# are returning the best approximation which is possible.
ew,ev = la.eigh(NewRdm)
O,V = ev[:,nOcc:], ev[:,:nOcc]
lhs = mdot(O.T,EmbFock,V).flatten()
VlocSize = h0_for_vloc.shape[-1]
h0_for_vloc_reshape = h0_for_vloc.reshape((nEmb,nEmb,VlocSize))
K = einsum("ro,rsp,sv->ovp",O,h0_for_vloc_reshape,V).reshape(O.shape[1]*V.shape[1],VlocSize)
rhs = -la.lstsq(K,lhs)[0]
return dot(h0_for_vloc_reshape,rhs)
def RunVcorUpdate(VcorLarge):
#assert(np.allclose(VcorLarge[1::2, ::2],0.))
#assert(np.allclose(VcorLarge[ ::2,1::2],0.))
with Log.Section("lmf","FULL SYSTEM -- MEAN FIELD CALCULATION", 1):
#vFock = self.FullSystem.FockT[0] - self.FullSystem.CoreH[0]
self.FullSystem.CoreH[0] = FullCoreH_Unpatched[0] + VcorLarge
#self.FullSystem.FockT[0] = self.FullSystem.CoreH[0] + vFock
self.FullSystem.FockT[0] = FullCoreH_Unpatched[0] + VcorLarge
# ^- this adds the changed vloc also to the FockT initial guess.
# required if iterations are disabled.
FullSystemHfResult = self.FullSystem._RunHf(Log=Log)
FullSystemHfResult = self.FullSystem
jgr = self.RunFragments(FullSystemHfResult, Log)
jgr.MeanFieldEnergy = FullSystemHfResult.MeanFieldEnergy
jgr.TotalEnergy = jgr.TotalElecEnergy + self.FullSystem.CoreEnergy
self.FitCorrelationPotentials(jgr, FullSystemHfResult, PrintEmb)
# now: add changed potentials to large system, and run again.
dVcorLarge = np.zeros(VcorLarge.shape)
assert(g.VcorType == "Local" or g.VcorType == "Diagonal")
for (Fragment, VcorFull) in zip(self.Fragments, jgr.FragmentPotentials):
nImp = len(Fragment.Sites)
Vcor = Fragment.Factor * VcorFull[:nImp,:nImp]
for Replica in Fragment.Replicas:
VcorR = mdot(Replica.Trafo, Vcor, Replica.Trafo.T)
E = list(enumerate(Replica.Sites))
for (i,iSite),(j,jSite) in it.product(E,E):
dVcorLarge[iSite,jSite] += VcorR[i,j]
return dVcorLarge, jgr, FullSystemHfResult
def RunVcorUpdate(VcorLarge):
#assert(np.allclose(VcorLarge[1::2, ::2],0.))
#assert(np.allclose(VcorLarge[ ::2,1::2],0.))
with Log.Section("lmf","FULL SYSTEM -- MEAN FIELD CALCULATION", 1):
#vFock = self.FullSystem.FockT[0] - self.FullSystem.CoreH[0]
self.FullSystem.CoreH[0] = FullCoreH_Unpatched[0] + VcorLarge
#self.FullSystem.FockT[0] = self.FullSystem.CoreH[0] + vFock
self.FullSystem.FockT[0] = FullCoreH_Unpatched[0] + VcorLarge
# ^- this adds the changed vloc also to the FockT initial guess.
# required if iterations are disabled.
FullSystemHfResult = self.FullSystem._RunHf(Log=Log)
FullSystemHfResult = self.FullSystem
jgr = self.RunFragments(FullSystemHfResult, Log)
jgr.MeanFieldEnergy = FullSystemHfResult.MeanFieldEnergy
jgr.TotalEnergy = jgr.TotalElecEnergy + self.FullSystem.CoreEnergy
self.FitCorrelationPotentials(jgr, FullSystemHfResult, PrintEmb)
# now: add changed potentials to large system, and run again.
dVcorLarge = np.zeros(VcorLarge.shape)
assert(g.VcorType == "Local" or g.VcorType == "Diagonal")
for (Fragment, VcorFull) in zip(self.Fragments, jgr.FragmentPotentials):
nImp = len(Fragment.Sites)
Vcor = Fragment.Factor * VcorFull[:nImp,:nImp]
for Replica in Fragment.Replicas:
VcorR = mdot(Replica.Trafo, Vcor, Replica.Trafo.T)
E = list(enumerate(Replica.Sites))
for (i,iSite),(j,jSite) in it.product(E,E):
dVcorLarge[iSite,jSite] += VcorR[i,j]
return dVcorLarge, jgr, FullSystemHfResult
def TransformToEmb(self, OrbL, InpT, OrbR):
"""return Op[k,l] = OrbL[r,k]^* Op[r,s] OrbR[s,l], where
Op: T x U x U is a periodically symmetric operator expressed
in the super-cell x unit-cell basis, and OrbL/OrbR are one-particle
transformations with the super-cell basis as row dimension."""
nSitesU, nSitesS = self.nSitesU, self.nSites
assert (InpT.shape == (self.nUnitCells, nSitesU, nSitesU)
or InpT.shape == (nSitesS, nSitesU))
assert (OrbL.shape[0] == nSitesS and OrbR.shape[0] == nSitesS)
Op = InpT.reshape((self.nUnitCells, nSitesU, nSitesU))
def FindRequiredTs(Op):
# make a list of Ts we need to consider in the transformation.
# if Op[iT,:,:] is zero, then we can skip the T.
iTs = []
for iT in range(self.nUnitCells):
if not np.allclose(Op[iT, :, :], 0.):
iTs.append(iT)
return np.array(iTs)
iOpTs = FindRequiredTs(Op)
#print "iTs with support: %i of %i: %s" % (len(iOpTs), self.nUnitCells, iOpTs)
Out = np.zeros((OrbL.shape[1], OrbR.shape[1]), Op.dtype)
if len(iOpTs) >= .3 * self.nUnitCells:
Op = InpT.reshape((nSitesS, nSitesU))
for (i, ijk) in enumerate(self.iTs):
RowI = self._ConstructTranslatedRows(Op, ijk).T
# ^- that's the Op[nSitesU*i:nSitesU*(i+1),:] subset of the
# operator expressed in the full nSitesS x nSitesS basis.
# (see FSuperCell._ExpandFull)
Out += mdot(np.conj(OrbL[nSitesU * i:nSitesU * (i + 1), :].T),
RowI, OrbR)
elif 1:
Op = Op[iOpTs, :, :]
OrbR = OrbR.reshape((self.nUnitCells, nSitesU, OrbR.shape[1]))
for (i, ijk) in enumerate(self.iTs):
PF, I = self._MakeUnitCellPhasesForT_Restricted(ijk, iOpTs)
RowIR = np.einsum("t,trs,trl->sl", PF, Op, OrbR[I, :, :])
Out += np.einsum(
"rk,rl->kl",
np.conj(OrbL[nSitesU * i:nSitesU * (i + 1), :]), RowIR)
else:
nT = len(iOpTs)
Op = Op[iOpTs, :, :].reshape((nT * nSitesU, nSitesU))
nOrbR = OrbR.shape[1]
OrbR = OrbR.reshape((self.nUnitCells, nSitesU, nOrbR))
PermShape = (nT * nSitesU, nOrbR)
PfShape = (nT, 1, 1)
for (i, ijk) in enumerate(self.iTs):
PF, I = self._MakeUnitCellPhasesForT_Restricted(ijk, iOpTs)
RowIR = np.dot(Op.T, (PF.reshape(PfShape) *
OrbR[I, :, :]).reshape(PermShape))
Out += np.dot(
np.conj(OrbL[nSitesU * i:nSitesU * (i + 1), :].T), RowIR)
return Out
def TransformToEmb(self, OrbL, InpT, OrbR):
"""return Op[k,l] = OrbL[r,k]^* Op[r,s] OrbR[s,l], where
Op: T x U x U is a periodically symmetric operator expressed
in the super-cell x unit-cell basis, and OrbL/OrbR are one-particle
transformations with the super-cell basis as row dimension."""
nSitesU, nSitesS = self.nSitesU, self.nSites
assert(InpT.shape == (self.nUnitCells, nSitesU, nSitesU) or
InpT.shape == (nSitesS, nSitesU))
assert(OrbL.shape[0] == nSitesS and OrbR.shape[0] == nSitesS)
Op = InpT.reshape((self.nUnitCells, nSitesU, nSitesU))
def FindRequiredTs(Op):
# make a list of Ts we need to consider in the transformation.
# if Op[iT,:,:] is zero, then we can skip the T.
iTs = []
for iT in range(self.nUnitCells):
if not np.allclose(Op[iT,:,:], 0.):
iTs.append(iT)
return np.array(iTs)
iOpTs = FindRequiredTs(Op)
#print "iTs with support: %i of %i: %s" % (len(iOpTs), self.nUnitCells, iOpTs)
Out = np.zeros((OrbL.shape[1],OrbR.shape[1]),Op.dtype)
if len(iOpTs) >= .3 * self.nUnitCells:
Op = InpT.reshape((nSitesS, nSitesU))
for (i,ijk) in enumerate(self.iTs):
RowI = self._ConstructTranslatedRows(Op, ijk).T
# ^- that's the Op[nSitesU*i:nSitesU*(i+1),:] subset of the
# operator expressed in the full nSitesS x nSitesS basis.
# (see FSuperCell._ExpandFull)
Out += mdot(np.conj(OrbL[nSitesU*i:nSitesU*(i+1),:].T),
RowI, OrbR)
elif 0:
Op = Op[iOpTs,:,:]
OrbR = OrbR.reshape((self.nUnitCells, nSitesU, OrbR.shape[1]))
for (i,ijk) in enumerate(self.iTs):
PF,I = self._MakeUnitCellPhasesForT_Restricted(ijk,iOpTs)
RowIR = np.einsum("t,trs,trl->sl",PF,Op,OrbR[I,:,:])
Out += np.einsum("rk,rl->kl",np.conj(OrbL[nSitesU*i:nSitesU*(i+1),:]), RowIR)
else:
nT = len(iOpTs)
Op = Op[iOpTs,:,:].reshape((nT*nSitesU,nSitesU))
nOrbR = OrbR.shape[1]
OrbR = OrbR.reshape((self.nUnitCells, nSitesU, nOrbR))
PermShape = (nT*nSitesU,nOrbR)
PfShape = (nT,1,1)
for (i,ijk) in enumerate(self.iTs):
PF,I = self._MakeUnitCellPhasesForT_Restricted(ijk,iOpTs)
RowIR = np.dot(Op.T, (PF.reshape(PfShape)*OrbR[I,:,:]).reshape(PermShape))
Out += np.dot(np.conj(OrbL[nSitesU*i:nSitesU*(i+1),:].T), RowIR)
return Out
def MakeIdempotentApproxRdm(TargetRdm):
# make a idempotent density matrix which exactly
# coincides with TargetRdm on the imp x imp subset.
A = TargetRdm[:nImp, :nImp]
C = TargetRdm[nImp:, nImp:]
assert (A.shape == C.shape)
C = np.eye(A.shape[0]) - A
# ^-
# WARNING: at this moment this does assume that the embedding
# basis is constructed as full Smh basis, i.e., that the hole
# state corresponding to imp site i is at nImp + i. For other
# cases we need additional rotations on C (not simply overwrite
# it) and fix the eigenvalues in some other way.
def MakeRho(A, B, C):
return np.vstack(
[np.hstack([A, B]),
np.hstack([np.conj(B.T), C])])
L = nImp
ewA, evA = la.eigh(A)
ewC, evC = la.eigh(C)
basis = MakeRho(evA, zeros_like(A), evC)
R = mdot(basis.T, TargetRdm, basis)
M = 0. * basis
M[:L, :L] = diag(ewA)
M[L:, L:] = diag(ewC)
ewB = (ewA - ewA**2)**.5
# ^- we can still put in random phases here and retain an
# idempotent target matrix. we get them from the
# transformation of the target density matrix, hopefully
# in such a way that also the off-diagonals are not way off,
# even if we are only fitting on ImpRdm instead of FullRdm.
# (see idempotency_gen.py)
#
# This freedom may be related to the freedom in choosing orbital
# occupation.
ewB_ref = np.diag(R[nImp:, :nImp][::-1, :])
ewB *= np.sign(ewB_ref)
M[L:, :L] = np.diag(ewB)[::-1, :]
M[:L, L:] = np.conj(M[L:, :L]).T
rho = dot(basis, np.dot(M, basis.T))
assert (np.allclose(np.dot(rho, rho) - rho, 0.0))
# ^- will work for HF in HF, but not for FCI in HF
return rho
def MakeIdempotentApproxRdm(TargetRdm):
# make a idempotent density matrix which exactly
# coincides with TargetRdm on the imp x imp subset.
A = TargetRdm[:nImp,:nImp]
C = TargetRdm[nImp:,nImp:]
assert(A.shape == C.shape)
C = np.eye(A.shape[0]) - A
# ^-
# WARNING: at this moment this does assume that the embedding
# basis is constructed as full Smh basis, i.e., that the hole
# state corresponding to imp site i is at nImp + i. For other
# cases we need additional rotations on C (not simply overwrite
# it) and fix the eigenvalues in some other way.
def MakeRho(A,B,C):
return np.vstack([np.hstack([A,B]),np.hstack([np.conj(B.T),C])])
L = nImp
ewA,evA = la.eigh(A)
ewC,evC = la.eigh(C)
basis = MakeRho(evA,zeros_like(A),evC)
R = mdot(basis.T, TargetRdm, basis)
M = 0.*basis
M[:L,:L] = diag(ewA)
M[L:,L:] = diag(ewC)
ewB = (ewA - ewA**2)**.5
# ^- we can still put in random phases here and retain an
# idempotent target matrix. we get them from the
# transformation of the target density matrix, hopefully
# in such a way that also the off-diagonals are not way off,
# even if we are only fitting on ImpRdm instead of FullRdm.
# (see idempotency_gen.py)
#
# This freedom may be related to the freedom in choosing orbital
# occupation.
ewB_ref = np.diag(R[nImp:,:nImp][::-1,:])
ewB *= np.sign(ewB_ref)
M[L:,:L] = np.diag(ewB)[::-1,:]
M[:L,L:] = np.conj(M[L:,:L]).T
rho = dot(basis,np.dot(M,basis.T))
assert(np.allclose(np.dot(rho,rho)-rho,0.0))
# ^- will work for HF in HF, but not for FCI in HF
return rho
def GetImpRdmFitAnalytical(RdmHl, nImp, VLOC_TYPE, VLOC_FIT_TYPE):
# this does the same as resmin, but
# solves the target fitting equation analyically.
# note that we can setup an equivalent routine for fullrdm fit.
assert(VLOC_TYPE == "Local")
# ^- we could do Diagonal/ImpRdm by making an idempotent
# approximation to a RDM obtained by taking the previous HF
# RDM and just fixing the diagonal entries to the CI RDM.
def MakeIdempotentApproxRdm(TargetRdm):
# make a idempotent density matrix which exactly
# coincides with TargetRdm on the imp x imp subset.
A = TargetRdm[:nImp,:nImp]
C = TargetRdm[nImp:,nImp:]
assert(A.shape == C.shape)
C = np.eye(A.shape[0]) - A
# ^-
# WARNING: at this moment this does assume that the embedding
# basis is constructed as full Smh basis, i.e., that the hole
# state corresponding to imp site i is at nImp + i. For other
# cases we need additional rotations on C (not simply overwrite
# it) and fix the eigenvalues in some other way.
def MakeRho(A,B,C):
return np.vstack([np.hstack([A,B]),np.hstack([np.conj(B.T),C])])
L = nImp
ewA,evA = la.eigh(A)
ewC,evC = la.eigh(C)
basis = MakeRho(evA,zeros_like(A),evC)
R = mdot(basis.T, TargetRdm, basis)
M = 0.*basis
M[:L,:L] = diag(ewA)
M[L:,L:] = diag(ewC)
ewB = (ewA - ewA**2)**.5
# ^- we can still put in random phases here and retain an
# idempotent target matrix. we get them from the
# transformation of the target density matrix, hopefully
# in such a way that also the off-diagonals are not way off,
# even if we are only fitting on ImpRdm instead of FullRdm.
# (see idempotency_gen.py)
#
# This freedom may be related to the freedom in choosing orbital
# occupation.
ewB_ref = np.diag(R[nImp:,:nImp][::-1,:])
ewB *= np.sign(ewB_ref)
M[L:,:L] = np.diag(ewB)[::-1,:]
M[:L,L:] = np.conj(M[L:,:L]).T
rho = dot(basis,np.dot(M,basis.T))
assert(np.allclose(np.dot(rho,rho)-rho,0.0))
# ^- will work for HF in HF, but not for FCI in HF
return rho
def MakeVlocForRdm(NewRdm, nOcc, EmbFock, h0_for_vloc):
# okay... we have the target eigensystem. We now still
# need an operator EmbFock + vloc which generates it as
# a ground state density. The condition here is that
#
# dot(h0_for_vloc, vloc1) =: K v
#
# has the same eigensystem as the one we're calculating,
# or, equivalently, that it commutes with the rdm we have.
# We assert that the resulting Fock matrix does not have
# components mixing occupied and virtual orbitals, which is
# equivalent.
# Note that we cannot assume that F is diagonal in the target rdm
# eigenvectors, because the eigensystem is degenerate
# (has only 0 and 1 evs)
#
# Also note that in general vloc has not enough variational
# freedom to represent arbitrary eigensystems, so technically we
# are returning the best approximation which is possible.
ew,ev = la.eigh(NewRdm)
O,V = ev[:,nOcc:], ev[:,:nOcc]
lhs = mdot(O.T,EmbFock,V).flatten()
VlocSize = h0_for_vloc.shape[-1]
h0_for_vloc_reshape = h0_for_vloc.reshape((nEmb,nEmb,VlocSize))
K = einsum("ro,rsp,sv->ovp",O,h0_for_vloc_reshape,V).reshape(O.shape[1]*V.shape[1],VlocSize)
rhs = -la.lstsq(K,lhs)[0]
return dot(h0_for_vloc_reshape,rhs)
# get number of orbitals we need to occupy in embedded HF.
fOcc = np.trace(RdmHl)
nOcc = int(fOcc+.5)
assert(abs(fOcc - nOcc) < 1e-8)
if VLOC_FIT_TYPE == "ImpRdm":
# make an idempotent density matrix which exactly
# coincides with RdmHl on the Imp x Imp part.
FitRdmHf = MakeIdempotentApproxRdm(RdmHl)
elif VLOC_FIT_TYPE == "FullRdm":
# make the best idempotent approximation to the FCI
# RDM. That's simply the matrix with the same natural
# orbitals, but all occupation numbers rounded to 0
# (if < .5) or 1 (if > .5).
if 0:
OccNum,NatOrb = la.eigh(-RdmHl); OccNum *= -1.
OccNumHf = zeros_like(OccNum)
OccNumHf[:nOcc] = 1.
FitRdmHf = mdot(NatOrb, diag(OccNumHf), NatOrb.T)
# ^- okay.. that is not *entirely* right:
# here we first approximate the target RDM, and then
# approximate the potential changes required to most
# closely obtain the target RDM as a HF solution.
FitRdmHf = RdmHl
else:
# not supported.
assert(0)
def TestMakeVlocForRdm():
vloc = MakeVlocForRdm(FitRdmHf, nOcc, EmbFock, h0_for_vloc)
NewF = EmbFock + vloc
ew,ev = la.eigh(NewF)
TestRdm = mdot(ev[:,:nOcc], ev[:,:nOcc].T)
print TestRdm-FitRdmHf
return MakeVlocForRdm(FitRdmHf, nOcc, EmbFock, h0_for_vloc)
def TestMakeVlocForRdm():
vloc = MakeVlocForRdm(FitRdmHf, nOcc, EmbFock, h0_for_vloc)
NewF = EmbFock + vloc
ew,ev = la.eigh(NewF)
TestRdm = mdot(ev[:,:nOcc], ev[:,:nOcc].T)
print TestRdm-FitRdmHf
def GetImpRdmFitAnalytical(RdmHl, nImp, VLOC_TYPE, VLOC_FIT_TYPE):
# this does the same as resmin, but
# solves the target fitting equation analyically.
# note that we can setup an equivalent routine for fullrdm fit.
assert (VLOC_TYPE == "Local")
# ^- we could do Diagonal/ImpRdm by making an idempotent
# approximation to a RDM obtained by taking the previous HF
# RDM and just fixing the diagonal entries to the CI RDM.
def MakeIdempotentApproxRdm(TargetRdm):
# make a idempotent density matrix which exactly
# coincides with TargetRdm on the imp x imp subset.
A = TargetRdm[:nImp, :nImp]
C = TargetRdm[nImp:, nImp:]
assert (A.shape == C.shape)
C = np.eye(A.shape[0]) - A
# ^-
# WARNING: at this moment this does assume that the embedding
# basis is constructed as full Smh basis, i.e., that the hole
# state corresponding to imp site i is at nImp + i. For other
# cases we need additional rotations on C (not simply overwrite
# it) and fix the eigenvalues in some other way.
def MakeRho(A, B, C):
return np.vstack(
[np.hstack([A, B]),
np.hstack([np.conj(B.T), C])])
L = nImp
ewA, evA = la.eigh(A)
ewC, evC = la.eigh(C)
basis = MakeRho(evA, zeros_like(A), evC)
R = mdot(basis.T, TargetRdm, basis)
M = 0. * basis
M[:L, :L] = diag(ewA)
M[L:, L:] = diag(ewC)
ewB = (ewA - ewA**2)**.5
# ^- we can still put in random phases here and retain an
# idempotent target matrix. we get them from the
# transformation of the target density matrix, hopefully
# in such a way that also the off-diagonals are not way off,
# even if we are only fitting on ImpRdm instead of FullRdm.
# (see idempotency_gen.py)
#
# This freedom may be related to the freedom in choosing orbital
# occupation.
ewB_ref = np.diag(R[nImp:, :nImp][::-1, :])
ewB *= np.sign(ewB_ref)
M[L:, :L] = np.diag(ewB)[::-1, :]
M[:L, L:] = np.conj(M[L:, :L]).T
rho = dot(basis, np.dot(M, basis.T))
assert (np.allclose(np.dot(rho, rho) - rho, 0.0))
# ^- will work for HF in HF, but not for FCI in HF
return rho
def MakeVlocForRdm(NewRdm, nOcc, EmbFock, h0_for_vloc):
# okay... we have the target eigensystem. We now still
# need an operator EmbFock + vloc which generates it as
# a ground state density. The condition here is that
#
# dot(h0_for_vloc, vloc1) =: K v
#
# has the same eigensystem as the one we're calculating,
# or, equivalently, that it commutes with the rdm we have.
# We assert that the resulting Fock matrix does not have
# components mixing occupied and virtual orbitals, which is
# equivalent.
# Note that we cannot assume that F is diagonal in the target rdm
# eigenvectors, because the eigensystem is degenerate
# (has only 0 and 1 evs)
#
# Also note that in general vloc has not enough variational
# freedom to represent arbitrary eigensystems, so technically we
# are returning the best approximation which is possible.
ew, ev = la.eigh(NewRdm)
O, V = ev[:, nOcc:], ev[:, :nOcc]
lhs = mdot(O.T, EmbFock, V).flatten()
VlocSize = h0_for_vloc.shape[-1]
h0_for_vloc_reshape = h0_for_vloc.reshape((nEmb, nEmb, VlocSize))
K = einsum("ro,rsp,sv->ovp", O, h0_for_vloc_reshape,
V).reshape(O.shape[1] * V.shape[1], VlocSize)
rhs = -la.lstsq(K, lhs)[0]
return dot(h0_for_vloc_reshape, rhs)
# get number of orbitals we need to occupy in embedded HF.
fOcc = np.trace(RdmHl)
nOcc = int(fOcc + .5)
assert (abs(fOcc - nOcc) < 1e-8)
if VLOC_FIT_TYPE == "ImpRdm":
# make an idempotent density matrix which exactly
# coincides with RdmHl on the Imp x Imp part.
FitRdmHf = MakeIdempotentApproxRdm(RdmHl)
elif VLOC_FIT_TYPE == "FullRdm":
# make the best idempotent approximation to the FCI
# RDM. That's simply the matrix with the same natural
# orbitals, but all occupation numbers rounded to 0
# (if < .5) or 1 (if > .5).
if 0:
OccNum, NatOrb = la.eigh(-RdmHl)
OccNum *= -1.
OccNumHf = zeros_like(OccNum)
OccNumHf[:nOcc] = 1.
FitRdmHf = mdot(NatOrb, diag(OccNumHf), NatOrb.T)
# ^- okay.. that is not *entirely* right:
# here we first approximate the target RDM, and then
# approximate the potential changes required to most
# closely obtain the target RDM as a HF solution.
FitRdmHf = RdmHl
else:
# not supported.
assert (0)
def TestMakeVlocForRdm():
vloc = MakeVlocForRdm(FitRdmHf, nOcc, EmbFock, h0_for_vloc)
NewF = EmbFock + vloc
ew, ev = la.eigh(NewF)
TestRdm = mdot(ev[:, :nOcc], ev[:, :nOcc].T)
print TestRdm - FitRdmHf
return MakeVlocForRdm(FitRdmHf, nOcc, EmbFock, h0_for_vloc)
def TestMakeVlocForRdm():
vloc = MakeVlocForRdm(FitRdmHf, nOcc, EmbFock, h0_for_vloc)
NewF = EmbFock + vloc
ew, ev = la.eigh(NewF)
TestRdm = mdot(ev[:, :nOcc], ev[:, :nOcc].T)
print TestRdm - FitRdmHf
def GetRdm(self):
#return 2.0 * dot(self.Orbs[:,:nOcc], self.Orbs[:,:nOcc].T)
return mdot(self.Orbs, diag(self.OccNumbers), self.Orbs.T)
def _TestSuperCells():
from numpy import set_printoptions, nan
set_printoptions(precision=5,linewidth=10060,suppress=True,threshold=nan)
if 1:
L = 12
nImp = 1
Hub1d = FHubbardModel1d(nImp, {'t': 1., 'U': 4.})
sc = FSuperCell(Hub1d, [L/nImp], [-1])
sc = FSuperCell(Hub1d, [L/nImp], [np.exp(.221j)])
#sc = FSuperCell(Hub1d, [L/nImp], [1])
else:
#Lx,Ly = 320,120
# ^- FIXME: putting lots of sites there crashes the program. (out of memory)
#nImpX,nImpY = 2,1
PhaseShift = None
FCls = FTestModel2d
#FCls = FHubbardModel2dSquare
Lx,Ly=4,3
#PhaseShift=[-1.,-1.] # anti-periodic boundary conditions
PhaseShift=[1.,1.] # periodic boundary conditions
PhaseShift=[np.exp(.2j),np.exp(-.72j)] # wtf-like boundary conditions
nImpX,nImpY = 2,1
#nImpX,nImpY = 1,1
nImp = nImpX * nImpY
Hub2dSq = FCls((nImpX,nImpY), {'t': 1., 'U': 4.})
sc = FSuperCell(Hub2dSq, [Lx/nImpX,Ly/nImpY], PhaseShift)
#sc = FSuperCell(Hub1d, [L/nImp,L/nImp], None)
#sc = FSuperCell(Hub1d, [L/nImp,L/nImp], None)
CoreH = sc.MakeTijMatrix()
#print "CoreH via super cells:\n%s" % repr(CoreH)
print "CoreH spectrum:\n%s" % la.eigh(CoreH)[0]
#Orb = sc._MakeRealspaceOrb(np.eye(len(sc.Model.UnitCell)), (0,0))
#print "RealSpaceOrb k == (0,0):\n%s" % Orb
#print "norm(orb) == ", la.norm(Orb)
SymBasis = np.hstack([sc._MakeRealspaceOrb(np.eye(len(sc.Model.UnitCell)), ijk) for ijk in sc.iKs])
print "symmetry adapted basis:\n%s" % SymBasis
#print "sc.nUnitCells = ", sc.nUnitCells
#print "sc.nSites = ", sc.nSites
#print "SymBasis.shape = ", SymBasis.shape
print "norm( dot(conj(SymBasis.T),SymBasis) - eye(nSites) ): %s" % (la.norm( np.dot(np.conj(SymBasis.T),SymBasis) - np.eye(sc.nSites) ))
print "CoreH via super cells:\n%s" % CoreH
print "CoreH in symmetry basis:\n%s" % (mdot(np.conj(SymBasis.T), CoreH, SymBasis))
#print "CoreH -- 1site transformed:\n%s" % (mdot(CoreH, SymBasis))
nSitesS = SymBasis.shape[1]
nSitesU = nImp
nUnitCells = nSitesS/nSitesU
CoreH_K = sc.TransformToK(CoreH[:nSitesS,:nSitesU])
print "And via .TransformToK:\n%s" % CoreH_K
CoreH_SymUqBt = sc.TransformToT(CoreH_K)
print "...back to t-space:\n%s" % CoreH_SymUqBt
print "difference norm to orig: %.2e" % la.norm(CoreH_SymUqBt - CoreH[:nSitesS,:nSitesU])
raise SystemExit
#raise Exception("^- that does not look very diagonal (note: works with pbc and apbc, in both directions)")
nSitesU = len(sc.Model.UnitCell)
nSitesS = len(sc.Sites)
#for ijk in sc.iKs.T:
#Orb = sc._MakeRealspaceOrb(np.eye(len(sc.Model.UnitCell)), ijk)
#print "ev x evT for ijk = %s:\n%s" % (ijk, np.dot(Orb,np.conj(Orb.T)))
CoreH_SymUq = CoreH[:nSitesU,:nSitesS]
print "symmetry unique part of CoreH:\n%s" % CoreH_SymUq
CoreH_R = np.zeros((nSitesS,nSitesS),complex)
for (i,ijk) in enumerate(sc.iTs.T):
CoreH_R[nSitesU*i:nSitesU*(i+1),:] = sc._ConstructTranslatedRows(CoreH_SymUq, ijk)
print "CoreH orig:\n%s" % CoreH
print "CoreH reconstructed:\n%s" % CoreH_R
print "difference norm to orig: %.2e" % la.norm(CoreH_R - CoreH)
def _TestSuperCells():
from numpy import set_printoptions, nan
set_printoptions(precision=5,
linewidth=10060,
suppress=True,
threshold=nan)
if 1:
L = 12
nImp = 1
Hub1d = FHubbardModel1d(nImp, {'t': 1., 'U': 4.})
sc = FSuperCell(Hub1d, [L / nImp], [-1])
sc = FSuperCell(Hub1d, [L / nImp], [np.exp(.221j)])
#sc = FSuperCell(Hub1d, [L/nImp], [1])
else:
#Lx,Ly = 320,120
# ^- FIXME: putting lots of sites there crashes the program. (out of memory)
#nImpX,nImpY = 2,1
PhaseShift = None
FCls = FTestModel2d
#FCls = FHubbardModel2dSquare
Lx, Ly = 4, 3
#PhaseShift=[-1.,-1.] # anti-periodic boundary conditions
PhaseShift = [1., 1.] # periodic boundary conditions
PhaseShift = [np.exp(.2j),
np.exp(-.72j)] # wtf-like boundary conditions
nImpX, nImpY = 2, 1
#nImpX,nImpY = 1,1
nImp = nImpX * nImpY
Hub2dSq = FCls((nImpX, nImpY), {'t': 1., 'U': 4.})
sc = FSuperCell(Hub2dSq, [Lx / nImpX, Ly / nImpY], PhaseShift)
#sc = FSuperCell(Hub1d, [L/nImp,L/nImp], None)
#sc = FSuperCell(Hub1d, [L/nImp,L/nImp], None)
CoreH = sc.MakeTijMatrix()
#print "CoreH via super cells:\n%s" % repr(CoreH)
print "CoreH spectrum:\n%s" % la.eigh(CoreH)[0]
#Orb = sc._MakeRealspaceOrb(np.eye(len(sc.Model.UnitCell)), (0,0))
#print "RealSpaceOrb k == (0,0):\n%s" % Orb
#print "norm(orb) == ", la.norm(Orb)
SymBasis = np.hstack([
sc._MakeRealspaceOrb(np.eye(len(sc.Model.UnitCell)), ijk)
for ijk in sc.iKs
])
print "symmetry adapted basis:\n%s" % SymBasis
#print "sc.nUnitCells = ", sc.nUnitCells
#print "sc.nSites = ", sc.nSites
#print "SymBasis.shape = ", SymBasis.shape
print "norm( dot(conj(SymBasis.T),SymBasis) - eye(nSites) ): %s" % (
la.norm(np.dot(np.conj(SymBasis.T), SymBasis) - np.eye(sc.nSites)))
print "CoreH via super cells:\n%s" % CoreH
print "CoreH in symmetry basis:\n%s" % (mdot(np.conj(SymBasis.T), CoreH,
SymBasis))
#print "CoreH -- 1site transformed:\n%s" % (mdot(CoreH, SymBasis))
nSitesS = SymBasis.shape[1]
nSitesU = nImp
nUnitCells = nSitesS / nSitesU
CoreH_K = sc.TransformToK(CoreH[:nSitesS, :nSitesU])
print "And via .TransformToK:\n%s" % CoreH_K
CoreH_SymUqBt = sc.TransformToT(CoreH_K)
print "...back to t-space:\n%s" % CoreH_SymUqBt
print "difference norm to orig: %.2e" % la.norm(CoreH_SymUqBt -
CoreH[:nSitesS, :nSitesU])
raise SystemExit
#raise Exception("^- that does not look very diagonal (note: works with pbc and apbc, in both directions)")
nSitesU = len(sc.Model.UnitCell)
nSitesS = len(sc.Sites)
#for ijk in sc.iKs.T:
#Orb = sc._MakeRealspaceOrb(np.eye(len(sc.Model.UnitCell)), ijk)
#print "ev x evT for ijk = %s:\n%s" % (ijk, np.dot(Orb,np.conj(Orb.T)))
CoreH_SymUq = CoreH[:nSitesU, :nSitesS]
print "symmetry unique part of CoreH:\n%s" % CoreH_SymUq
CoreH_R = np.zeros((nSitesS, nSitesS), complex)
for (i, ijk) in enumerate(sc.iTs.T):
CoreH_R[nSitesU * i:nSitesU *
(i + 1), :] = sc._ConstructTranslatedRows(CoreH_SymUq, ijk)
print "CoreH orig:\n%s" % CoreH
print "CoreH reconstructed:\n%s" % CoreH_R
print "difference norm to orig: %.2e" % la.norm(CoreH_R - CoreH)
def GetRdm(self):
#return 2.0 * dot(self.Orbs[:,:nOcc], self.Orbs[:,:nOcc].T)
return mdot(self.Orbs, diag(self.OccNumbers), self.Orbs.T)
