def build_part(gf_occ, gf_vir, eri, sym_in='s2'):
''' Builds the truncated occupied (or virtual) self-energy.
Parameters
----------
gf_occ : Aux
Occupied (or virtual) Green's function
gf_vir : Aux
Virtual (or occupied) Green's function
eri : ndarray
Cholesky-decomposed DF ERI tensor
sym_in : str, optional
Symmetry of `eri`, default 's2'
Returns
-------
se : Aux
Occupied (or virtual) truncated self-energy
'''
syms = dict(sym_in=sym_in, sym_out='s1')
nphys = gf_occ.nphys
nocc = gf_occ.naux
nvir = gf_vir.naux
ixq = OptRAGF2.ao2mo(eri, gf_occ.v, np.eye(nphys), **syms).T
qja = OptRAGF2.ao2mo(eri, gf_occ.v, gf_vir.v, **syms)
vv = np.zeros((nphys, nphys), dtype=types.float64)
vev = np.zeros((nphys, nphys), dtype=types.float64)
if libagf2._libagf2 is not None:
vv, vev = libagf2.build_part_loop_rhf(ixq,
qja,
gf_occ,
gf_vir,
0,
nocc,
vv=vv,
vev=vev)
else:
buf1 = np.zeros((nphys, nocc * nvir), dtype=types.float64)
buf2 = np.zeros((nocc * nphys, nvir), dtype=types.float64)
for i in range(nocc):
xja = np.dot(ixq[i * nphys:(i + 1) * nphys], qja, out=buf1)
xia = np.dot(ixq, qja[:, i * nvir:(i + 1) * nvir], out=buf2)
xia = util.reshape_internal(xia, (nocc, nphys, nvir), (0, 1),
(nphys, nocc * nvir))
eja = util.outer_sum([gf_occ.e[i] + gf_occ.e, -gf_vir.e])
eja = eja.ravel()
vv = util.dgemm(xja, xja.T, alpha=2, beta=1, c=vv)
vv = util.dgemm(xja, xia.T, alpha=-1, beta=1, c=vv)
vev = util.dgemm(xja * eja[None],
xja.T,
alpha=2,
beta=1,
c=vev)
vev = util.dgemm(xja * eja[None],
xia.T,
alpha=-1,
beta=1,
c=vev)
b = np.linalg.cholesky(vv).T
b_inv = np.linalg.inv(b)
m = np.dot(np.dot(b_inv.T, vev), b_inv)
e, c = util.eigh(m)
c = np.dot(b.T, c[:nphys])
se = gf_occ.new(e, c)
return se
def build_dfump2_part(eo, ev, ixq, qja, wtol=1e-12, ss_factor=1.0, os_factor=1.0):
''' Builds a set of auxiliaries representing all (i,j,a) or (a,b,i)
diagrams for an unrestricted reference.
Parameters
----------
eo : 2-tuple of (o) ndarray
occupied (virtual) energies for alpha, beta (beta, alpha) spin
ev : 2-tuple of (v) ndarray
virtual (occupied) energies for alpha, beta (beta, alpha) spin
ixq : 1-tuple or 2-tuple of (n,o,o,v)
density-fitted two-electron integrals index as occupied,
physical, auxiliary (virtual, physical, auxiliary) for alpha,
beta (beta, alpha) spin. Only alpha (beta) is required.
qja : 2-tuple of (n,o,o,v)
density-fitted two-electron integrals index as auxiliary,
occupied, virtual (auxiliary, virtual, occupied) for alpha,
beta (beta, alpha) spin.
wtol : float, optional
threshold for an eigenvalue to be considered zero
ss_factor : float, optional
same spin factor, default 1.0
os_factor : float, optional
opposite spin factor, default 1.0
Returns
-------
e : (m) ndarray
auxiliary energies
v : (n,m) ndarray
auxiliary couplings
'''
nphys = ixq[0].shape[1]
ndf, nocca, nvira = qja[0].shape
_, noccb, nvirb = qja[1].shape
npoles = nvira * nocca * (nocca-1) // 2
npoles += nvirb * nocca * noccb
e = np.zeros((npoles), dtype=types.float64)
v = np.zeros((nphys, npoles), dtype=ixq[0].dtype)
a_factor = np.sqrt(ss_factor)
b_factor = np.sqrt(os_factor)
ixq = (ixq[0].reshape((nocca*nphys, ndf)),)
qja = (qja[0].reshape((ndf, nocca*nvira)),
qja[1].reshape((ndf, noccb*nvirb)))
n0 = 0
for i in range(nocca):
nja_a = i * nvira
nja_b = noccb * nvirb
jm = slice(None, i)
am = slice(n0, n0+nja_a)
bm = slice(n0+nja_a, n0+nja_a+nja_b)
xq_a = ixq[0][i*nphys:(i+1)*nphys]
qa_a = qja[0][:,i*nvira:(i+1)*nvira]
xja_aa = np.dot(ixq[0][:i*nphys].conj(), qa_a)
xja_aa = util.reshape_internal(xja_aa, (i, nphys, nvira), (0,1), (nphys, i*nvira))
xia_aa = np.dot(xq_a.conj(), qja[0][:,:i*nvira]).reshape((nphys,-1))
xja_ab = np.dot(xq_a.conj(), qja[1]).reshape((nphys,-1))
xija_aa = np.dot(ixq[0].conj(), qja[0]).reshape((nocca, nphys, nocca, nvira)).swapaxes(0,1)
xija_ab = np.dot(ixq[0].conj(), qja[1]).reshape((nocca, nphys, noccb, nvirb)).swapaxes(0,1)
assert np.allclose(xia_aa, xija_aa[:,i,:i].reshape((nphys, -1)))
assert np.allclose(xja_aa, xija_aa[:,:i,i].reshape((nphys, -1)))
assert np.allclose(xja_ab, xija_ab[:,i].reshape((nphys, -1)))
e[am] = eo[0][i] + np.subtract.outer(eo[0][jm], ev[0]).flatten()
e[bm] = eo[0][i] + np.subtract.outer(eo[1], ev[1]).flatten()
v[:,am] = a_factor * (xia_aa - xja_aa)
v[:,bm] = b_factor * xja_ab
n0 += nja_a + nja_b
mask = np.absolute(util.complex_sum(v*v, axis=0)) >= wtol
e = e[mask]
v = v[:,mask]
e = e[:n0]
v = v[:,:n0]
assert e.shape[0] == v.shape[1]
return e, v
def _build_part(s=slice(None)):
vv = np.zeros((nphys, nphys), dtype=types.float64)
vev = np.zeros((nphys, nphys), dtype=types.float64)
if libagf2._libagf2 is not None:
vv, vev = libagf2.build_part_loop_uhf(ixq[s],
qja[s],
gf_occ[s],
gf_vir[s],
0,
nocc[s][0],
vv=vv,
vev=vev)
else:
nocca, noccb = nocc[s]
nvira, nvirb = nvir[s]
ixq_a, ixq_b = ixq[s]
qja_a, qja_b = qja[s]
gf_occ_a, gf_occ_b = gf_occ[s]
gf_vir_a, gf_vir_b = gf_vir[s]
buf1 = np.zeros((nphys, nocca * nvira), dtype=types.float64)
buf2 = np.zeros((nocca * nphys, nvira), dtype=types.float64)
buf3 = np.zeros((nphys, noccb * nvirb), dtype=types.float64)
for i in range(nocca):
xja_aa = np.dot(ixq_a[i * nphys:(i + 1) * nphys],
qja_a,
out=buf1)
xia_aa = np.dot(ixq_a,
qja_a[:, i * nvira:(i + 1) * nvira],
out=buf2)
xia_aa = util.reshape_internal(xia_aa,
(nocca, nphys, nvira),
(0, 1),
(nphys, nocca * nvira))
xja_ab = np.dot(ixq_a[i * nphys:(i + 1) * nphys],
qja_b,
out=buf3)
eja_aa = util.outer_sum(
[gf_occ_a.e[i] + gf_occ_a.e, -gf_vir_a.e]).flatten()
eja_ab = util.outer_sum(
[gf_occ_a.e[i] + gf_occ_b.e, -gf_vir_b.e]).flatten()
vv = util.dgemm(xja_aa, xja_aa.T, alpha=1, beta=1, c=vv)
vv = util.dgemm(xja_aa, xia_aa.T, alpha=-1, beta=1, c=vv)
vv = util.dgemm(xja_ab, xja_ab.T, alpha=1, beta=1, c=vv)
vev = util.dgemm(xja_aa * eja_aa[None],
xja_aa.T,
alpha=1,
beta=1,
c=vev)
vev = util.dgemm(xja_aa * eja_aa[None],
xia_aa.T,
alpha=-1,
beta=1,
c=vev)
vev = util.dgemm(xja_ab * eja_ab[None],
xja_ab.T,
alpha=1,
beta=1,
c=vev)
b = np.linalg.cholesky(vv).T
b_inv = np.linalg.inv(b)
m = np.dot(np.dot(b_inv.T, vev), b_inv)
e, c = util.eigh(m)
c = np.dot(b.T, c[:nphys])
se = gf_occ[s][0].new(e, c)
return se
def build_dfump2_part_direct(eo, ev, ixq, qja, wtol=1e-12, ss_factor=1.0, os_factor=1.0):
''' Builds a set of auxiliaries representing all (i,j,a) or (a,b,i)
diagrams for an unrestricted reference. Uses a generator which
iterates over blocks.
Parameters
----------
eo : 2-tuple of (o) ndarray
occupied (virtual) energies for alpha, beta (beta, alpha) spin
ev : 2-tuple of (v) ndarray
virtual (occupied) energies for alpha, beta (beta, alpha) spin
ixq : 1-tuple or 2-tuple of (n,o,o,v)
density-fitted two-electron integrals index as occupied,
physical, auxiliary (virtual, physical, auxiliary) for alpha,
beta (beta, alpha) spin. Only alpha (beta) is required.
qja : 2-tuple of (n,o,o,v)
density-fitted two-electron integrals index as auxiliary,
occupied, virtual (auxiliary, virtual, occupied) for alpha,
beta (beta, alpha) spin.
wtol : float, optional
threshold for an eigenvalue to be considered zero
ss_factor : float, optional
same spin factor, default 1.0
os_factor : float, optional
opposite spin factor, default 1.0
Yields
------
e : (m) ndarray
auxiliary energies
v : (n,m) ndarray
auxiliary couplings
'''
nphys = ixq[0].shape[1]
ndf, nocca, nvira = qja[0].shape
_, noccb, nvirb = qja[1].shape
npoles = nvira * nocca * (nocca-1) // 2
npoles += nvirb * nocca * noccb
a_factor = np.sqrt(ss_factor)
b_factor = np.sqrt(os_factor)
ixq = (ixq[0].reshape((nocca*nphys, ndf)),)
qja = (qja[0].reshape((ndf, nocca*nvira)), qja[1].reshape((ndf, noccb*nvirb)))
for i in range(nocca):
nja_a = i * nvira
nja_b = noccb * nvirb
jm = slice(None, i)
xq_a = ixq[0][i*nphys:(i+1)*nphys]
qa_a = qja[0][:,i*nvira:(i+1)*nvira]
xja_aa = np.dot(ixq[0][:i*nphys].conj(), qa_a)
xja_aa = util.reshape_internal(xja_aa, (i, nphys, nvira), (0,1), (nphys, i*nvira))
xia_aa = np.dot(xq_a.conj(), qja[0][:,:i*nvira]).reshape((nphys,-1))
xja_ab = np.dot(ixq[0][i*nphys:(i+1)*nphys].conj(), qja[1]).reshape((nphys,-1))
ea = eo[0][i] + np.subtract.outer(eo[0][jm], ev[0]).flatten()
eb = eo[0][i] + np.subtract.outer(eo[1], ev[1]).flatten()
va = a_factor * (xja_aa - xia_aa)
vb = b_factor * xja_ab
if len(ea):
yield ea, va
if len(eb):
yield eb, vb
def build_dfump2_part_se_direct(eo, ev, ixq, qja, grid, chempot=0.0, ordering='feynman'):
''' Builds a set of auxiliaries representing all (i,j,a) or (a,b,i)
diagrams for an unrestricted reference. Poles are summed straight
into the self-energy and returns an `ndarray` instead of `Aux`.
Parameters
----------
eo : (o) ndarray
occupied (virtual) energies
ev : (v) ndarray
virtual (occupied) energies
ixq : 1-tuple or 2-tuple of (n,o,o,v)
density-fitted two-electron integrals index as occupied,
physical, auxiliary (virtual, physical, auxiliary) for alpha,
beta (beta, alpha) spin. Only alpha (beta) is required.
qja : 2-tuple of (n,o,o,v)
density-fitted two-electron integrals index as auxiliary,
occupied, virtual (auxiliary, virtual, occupied) for alpha,
beta (beta, alpha) spin.
grid : (k) ImFqGrid, ImFqQuad or ReFqGrid
grid
chempot : tuple of float, optional
chemical potential for alpha, beta (beta, alpha) spin
ordering : str
ordering of the poles {'feynman', 'advanced', 'retarded'}
(default 'feynman')
Returns
-------
se : (k,n,n) ndarray
frequency-dependent self-energy
'''
if not _is_tuple(chempot):
chempot = (chempot, chempot)
#TODO write in C
if grid.axis == 'imag':
if ordering == 'feynman':
get_s = lambda x : np.sign(x)
elif ordering == 'advanced':
get_s = lambda x : np.ones(x.shape, dtype=types.int64)
elif ordering == 'retarded':
get_s = lambda x : -np.ones(x.shape, dtype=types.int64)
else:
get_s = lambda x : 0.0
w = grid.prefac * grid.values
nphys = ixq[0].shape[1]
ndf, nocca, nvira = qja[0].shape
_, noccb, nvirb = qja[1].shape
se = np.zeros((grid.shape[0], nphys, nphys), dtype=types.complex128)
ixq = (ixq[0].reshape((nocca*nphys, ndf)),)
qja = (qja[0].reshape((ndf, nocca*nvira)),
qja[1].reshape((ndf, noccb*nvirb)))
eova = util.outer_sum([eo[0], -ev[0]]).flatten() - chempot[0]
eovb = util.outer_sum([eo[1], -ev[1]]).flatten() - chempot[0]
for i in range(nocca):
ei_a = eo[0][i] + eova
ei_b = eo[0][i] + eovb
xq_a = ixq[0][i*nphys:(i+1)*nphys]
vi_a = np.dot(xq_a.conj(), qja[0]).reshape((nphys, -1))
vi_b = np.dot(xq_a.conj(), qja[1]).reshape((nphys, -1))
vip_a = np.dot(ixq[0].conj(), qja[0][:,i*nvira:(i+1)*nvira])
vip_a = util.reshape_internal(vip_a, (nocca, nphys, nvira), (0,1), (nphys, nocca*nvira))
di_a = 1.0 / util.outer_sum([w, -ei_a + get_s(ei_a) * grid.eta * 1.0j])
di_b = 1.0 / util.outer_sum([w, -ei_b + get_s(ei_b) * grid.eta * 1.0j])
se += util.einsum('wk,xk,yk->wxy', di_a, vi_a, (vi_a - vip_a).conj())
se += util.einsum('wk,xk,yk->wxy', di_b, vi_b, vi_b.conj())
return se
def build_dfump2_iter(se, h_phys, eri_mo, **kwargs):
''' Builds a set of auxiliaries representing all (i,j,a) and (a,b,i)
diagrams by iterating the current set of auxiliaries according
to the eigenvalue form of the Dyson equation.
Unlike the other routines in this module, this builds for both
spins.
Parameters
----------
aux : tuple of Aux
auxiliaries of previous iteration as (alpha, beta)
h_phys : (2,n,n) ndarray
physical space Hamiltonian for alpha and beta spin, if `ndim==2`
then spin-symmetry in the Hamiltonian is assumed
eri_mo : (2,q,n,n) ndarray
density-fitted two-electron repulsion integrals in MO basis,
where the first index is spin
wtol : float, optional
threshold for an eigenvalue to be considered zero
ss_factor : float, optional
same spin factor, default 1.0
os_factor : float, optional
opposite spin factor, default 1.0
Returns
-------
poles_a : Aux
auxiliaries for alpha spin
poles_b : Aux
auxiliaries for beta spin
'''
h_phys = np.asarray(h_phys, dtype=types.float64)
if h_phys.ndim == 2:
h_phys = np.stack((h_phys, h_phys))
ea, ca = se[0].eig(h_phys[0])
eb, cb = se[1].eig(h_phys[1])
oa = ea < se[0].chempot
ob = eb < se[1].chempot
va = ea >= se[0].chempot
vb = eb >= se[1].chempot
nphys = se[0].nphys
nocca = np.sum(oa)
noccb = np.sum(ob)
nvira = np.sum(va)
nvirb = np.sum(vb)
eo = (ea[oa], eb[ob])
ev = (ea[va], eb[vb])
co = (ca[:se[0].nphys,oa], cb[:se[1].nphys,ob])
cv = (ca[:se[0].nphys,va], cb[:se[1].nphys,vb])
eye = np.eye(nphys)
ixq_a = util.ao2mo_df(eri_mo[0], co[0], eye)
ixq_a = util.reshape_internal(ixq_a, (-1, nocca*nphys), (0,1), (nocca, nphys, -1))
qja_a = util.ao2mo_df(eri_mo[0], co[0], cv[0])
qja_b = util.ao2mo_df(eri_mo[1], co[1], cv[1])
eija_a, vija_a = build_dfump2_part(eo, ev, (ixq_a,), (qja_a, qja_b), **kwargs)
del ixq_a
ixq_b = util.ao2mo_df(eri_mo[1], co[1], eye)
ixq_b = util.reshape_internal(ixq_b, (-1, noccb*nphys), (0,1), (noccb, nphys, -1))
eija_b, vija_b = build_dfump2_part(eo[::-1], ev[::-1], (ixq_b,), (qja_b, qja_a), **kwargs)
del ixq_b, qja_a, qja_b
axq_a = util.ao2mo_df(eri_mo[0], cv[0], eye)
axq_a = util.reshape_internal(axq_a, (-1, nvira*nphys), (0,1), (nvira, nphys, -1))
qbi_a = util.ao2mo_df(eri_mo[0], cv[0], co[0])
qbi_b = util.ao2mo_df(eri_mo[1], cv[1], co[1])
eabi_a, vabi_a = build_dfump2_part(ev, eo, (axq_a,), (qbi_a, qbi_b), **kwargs)
del axq_a
axq_b = util.ao2mo_df(eri_mo[1], cv[1], eye)
axq_b = util.reshape_internal(axq_b, (-1, nvirb*nphys), (0,1), (nvirb, nphys, -1))
eabi_b, vabi_b = build_dfump2_part(ev[::-1], eo[::-1], (axq_b,), (qbi_b, qbi_a), **kwargs)
del axq_b, qbi_a, qbi_b
ea = np.concatenate((eija_a, eabi_a), axis=0)
eb = np.concatenate((eija_b, eabi_b), axis=0)
va = np.concatenate((vija_a, vabi_a), axis=1)
vb = np.concatenate((vija_b, vabi_b), axis=1)
poles_a = se[0].new(ea, va)
poles_b = se[1].new(eb, vb)
return poles_a, poles_b
def build_dfrmp2_part(eo,
ev,
ixq,
qja,
wtol=1e-12,
ss_factor=1.0,
os_factor=1.0):
''' Builds a set of auxiliaries representing all (i,j,a) or (a,b,i)
diagrams for a restricted reference.
Parameters
----------
eo : (o) ndarray
occupied (virtual) energies
ev : (v) ndarray
virtual (occupied) energies
ixq : (o,n,q) ndarray
density-fitted two-electron integrals indexed as occupied,
physical, auxiliary (physical, virtual, auxiliary)
qja : (q,o,v) ndarray
density-fitted two-electron integrals indexed as auxiliary,
occupied, virtual (auxiliary, virtual, occupied)
wtol : float, optional
threshold for an eigenvalue to be considered zero
ss_factor : float, optional
same spin factor, default 1.0
os_factor : float, optional
opposite spin factor, default 1.0
Returns
-------
e : (m) ndarray
auxiliary energies
v : (n,m) ndarray
auxiliary couplings
'''
nphys = ixq.shape[1]
ndf, nocc, nvir = qja.shape
npoles = nocc * nocc * nvir
e = np.zeros((npoles), dtype=types.float64)
v = np.zeros((nphys, npoles), dtype=ixq.dtype)
pos_factor = np.sqrt(0.5 * os_factor)
neg_factor = np.sqrt(0.5 * os_factor + ss_factor)
dia_factor = np.sqrt(os_factor)
ixq = ixq.reshape((nocc * nphys, ndf))
qja = qja.reshape((ndf, nocc * nvir))
n0 = 0
for i in range(nocc):
nja = i * nvir
am = slice(n0, n0 + nja)
bm = slice(n0 + nja, n0 + nja * 2)
cm = slice(n0 + nja * 2, n0 + nja * 2 + nvir)
xq = ixq[i * nphys:(i + 1) * nphys]
qa = qja[:, i * nvir:(i + 1) * nvir]
xja = np.dot(ixq[:i * nphys].conj(), qa)
xja = util.reshape_internal(xja, (i, nphys, nvir), (0, 1),
(nphys, i * nvir))
xia = np.dot(xq.conj(), qja[:, :i * nvir]).reshape((nphys, -1))
xa = np.dot(xq.conj(), qa)
e[am] = e[bm] = eo[i] + util.dirsum('i,a->ia', eo[:i], -ev).ravel()
e[cm] = 2 * eo[i] - ev
v[:, am] = neg_factor * (xja - xia)
v[:, bm] = pos_factor * (xja + xia)
v[:, cm] = dia_factor * xa
n0 += nja * 2 + nvir
mask = np.absolute(util.complex_sum(v * v, axis=0)) >= wtol
e = e[mask]
v = v[:, mask]
assert e.shape[0] == v.shape[1]
return e, v
def build_dfrmp2_part_se_direct(eo,
ev,
ixq,
qja,
grid,
chempot=0.0,
ordering='feynman'):
''' Builds a set of auxiliaries representing all (i,j,a) or (a,b,i)
diagrams for a restricted reference. Poles are summed straight
into the self-energy and returns an `ndarray` instead of `Aux`.
Parameters
----------
eo : (o) ndarray
occupied (virtual) energies
ev : (v) ndarray
virtual (occupied) energies
ixq : (o,n,q) ndarray
density-fitted two-electron integrals indexed as occupied,
physical, auxiliary (physical, virtual, auxiliary)
qja : (q,o,v) ndarray
density-fitted two-electron integrals indexed as auxiliary,
occupied, virtual (auxiliary, virtual, occupied)
grid : (k) ImFqGrid, ImFqQuad or ReFqGrid
grid
chempot : float, optional
chemical potential
ordering : str
ordering of the poles {'feynman', 'advanced', 'retarded'}
(default 'feynman')
Returns
-------
se : (k,n,n) ndarray
frequency-dependent self-energy
'''
if grid.axis == 'imag':
if ordering == 'feynman':
get_s = lambda x: np.sign(x)
elif ordering == 'advanced':
get_s = lambda x: np.ones(x.shape, dtype=types.int64)
elif ordering == 'retarded':
get_s = lambda x: -np.ones(x.shape, dtype=types.int64)
else:
get_s = lambda x: 0.0
w = grid.prefac * grid.values
nphys = ixq.shape[1]
ndf, nocc, nvir = qja.shape
npoles = nocc * nocc * nvir
se = np.zeros((grid.shape[0], nphys, nphys), dtype=types.complex128)
ixq = ixq.reshape((nocc * nphys, ndf))
qja = qja.reshape((ndf, nocc * nvir))
eov = util.outer_sum([eo, -ev]).flatten()
for i in range(nocc):
ei = eo[i] + eov - chempot
vi = np.dot(ixq[i * nphys:(i + 1) * nphys].conj(), qja).reshape(
(nphys, -1))
vip = np.dot(ixq.conj(), qja[:, i * nvir:(i + 1) * nvir])
vip = util.reshape_internal(vip, (nocc, nphys, -1), (0, 1),
(nphys, -1))
di = 1.0 / util.outer_sum([w, -ei + get_s(ei) * grid.eta * 1.0j])
se += util.einsum('wk,xk,yk->wxy', di, vi, (2 * vi - vip).conj())
return se
def build_dfrmp2_part_direct(eo,
ev,
ixq,
qja,
wtol=1e-12,
ss_factor=1.0,
os_factor=1.0):
''' Builds a set of auxiliaries representing all (i,j,a) or (a,b,i)
diagrams for a restricted reference. Uses a generator which
iterates over blocks.
Parameters
----------
eo : (o) ndarray
occupied (virtual) energies
ev : (v) ndarray
virtual (occupied) energies
ixq : (o,n,q) ndarray
density-fitted two-electron integrals indexed as occupied,
physical, auxiliary (physical, virtual, auxiliary)
qja : (q,o,v) ndarray
density-fitted two-electron integrals indexed as auxiliary,
occupied, virtual (auxiliary, virtual, occupied)
wtol : float, optional
threshold for an eigenvalue to be considered zero
ss_factor : float, optional
same spin factor, default 1.0
os_factor : float, optional
opposite spin factor, deafult 1.0
Yields
------
e : (m) ndarray
auxiliary energies
v : (n,m) ndarray
auxiliary couplings
'''
nphys = ixq.shape[1]
ndf, nocc, nvir = qja.shape
npoles = nocc * nocc * nvir
pos_factor = np.sqrt(0.5 * os_factor)
neg_factor = np.sqrt(0.5 * os_factor + ss_factor)
dia_factor = np.sqrt(os_factor)
ixq = ixq.reshape((nocc * nphys, ndf))
qja = qja.reshape((ndf, nocc * nvir))
for i in range(nocc):
nja = i * nvir
xq = ixq[i * nphys:(i + 1) * nphys]
qa = qja[:, i * nvir:(i + 1) * nvir]
xja = np.dot(ixq[:i * nphys].conj(), qa)
xja = util.reshape_internal(xja, (i, nphys, nvir), (0, 1),
(nphys, i * nvir))
xia = np.dot(xq.conj(), qja[:, :i * nvir]).reshape((nphys, -1))
xa = np.dot(xq.conj(), qa)
ea = eb = eo[i] + util.dirsum('i,a->ia', eo[:i], -ev).ravel()
ec = 2 * eo[i] - ev
va = neg_factor * (xja - xia)
vb = pos_factor * (xja + xia)
vc = dia_factor * xa
if len(ea):
yield ea, va
if len(eb):
yield eb, vb
if len(ec):
yield ec, vc
def build_dfrmp2_iter(se, h_phys, eri_mo, **kwargs):
''' Builds a set of auxiliaries representing all (i,j,a) and (a,b,i)
diagrams by iterating the current set of auxiliaries according
to the eigenvalue form of the Dyson equation.
Parameters
----------
se : Aux
auxiliaries of previous iteration
h_phys : (n,n) ndarray
physical space hamiltonian
eri_mo : (q,n,n) ndarray
density-fitted two-electron repulsion integrals in MO basis
wtol : float, optional
threshold for an eigenvalue to be considered zero
ss_factor : float, optional
same spin factor, default 1.0
os_factor : float, optional
opposite spin factor, deafult 1.0
Returns
-------
poles : Aux
auxiliaries
'''
e, c = se.eig(h_phys)
o = e < se.chempot
v = e >= se.chempot
nphys = se.nphys
nocc = np.sum(o)
nvir = np.sum(v)
eo = e[o]
ev = e[v]
co = c[:se.nphys, o]
cv = c[:se.nphys, v]
eye = np.eye(se.nphys)
ixq = util.ao2mo_df(eri_mo, co, eye)
ixq = util.reshape_internal(ixq, (-1, nocc * nphys), (0, 1),
(nocc, nphys, -1))
qja = util.ao2mo_df(eri_mo, co, cv)
eija, vija = build_dfrmp2_part(eo, ev, ixq, qja, **kwargs)
del ixq, qja
axq = util.ao2mo_df(eri_mo, cv, eye)
axq = util.reshape_internal(axq, (-1, nvir * nphys), (0, 1),
(nvir, nphys, -1))
qbi = util.ao2mo_df(eri_mo, cv, co)
eabi, vabi = build_dfrmp2_part(ev, eo, axq, qbi, **kwargs)
del axq, qbi
e = np.concatenate((eija, eabi), axis=0)
v = np.concatenate((vija, vabi), axis=1)
poles = se.new(e, v)
return poles