def get_1p_or_1h(self):
occa, occb, vira, virb = self._get_slices()
self.e_mp2 = 0
hs = []
for a, b in [(0, 1), (1, 0)]:
if a == 0:
occa, occb, vira, virb = self._get_slices()
else:
occb, occa, virb, vira = self._get_slices()
eri_aa_ovov = self.eri[a, a][occa, vira, occa, vira]
eri_ab_ovov = self.eri[a, b][occa, vira, occb, virb]
eo_a = self.hf.e[a][occa]
eo_b = self.hf.e[b][occb]
ev_a = self.hf.e[a][vira]
ev_b = self.hf.e[b][virb]
t2_aa = eri_aa_ovov.copy()
t2_aa /= util.dirsum('i,a,j,b->iajb', eo_a, -ev_a, eo_a, -ev_a)
t2a_aa = t2_aa - t2_aa.swapaxes(0, 2).copy()
t2_ab = eri_ab_ovov.copy()
t2_ab /= util.dirsum('i,a,j,b->iajb', eo_a, -ev_a, eo_b, -ev_b)
self.e_mp2 += util.einsum('iajb,iajb->', t2a_aa, eri_aa_ovov) * 0.5
self.e_mp2 += util.einsum('iajb,iajb->', t2_ab, eri_ab_ovov) * 0.5
h = np.diag(eo_a)
h += util.einsum('a,iakb,jakb->ij', ev_a, t2a_aa, t2a_aa)
h += util.einsum('a,iakb,jakb->ij', ev_a, t2_ab, t2_ab)
h += util.einsum('a,ibka,jbka->ij', ev_b, t2_ab, t2_ab)
h -= util.einsum('k,iakb,jakb->ij', eo_a, t2a_aa, t2a_aa) * 0.5
h -= util.einsum('k,iakb,jakb->ij', eo_b, t2_ab, t2_ab) * 0.5
h -= util.einsum('k,iakb,jakb->ij', eo_b, t2_ab, t2_ab) * 0.5
h -= util.einsum('i,iakb,jakb->ij', eo_a, t2a_aa, t2a_aa) * 0.25
h -= util.einsum('i,iakb,jakb->ij', eo_a, t2_ab, t2_ab) * 0.25
h -= util.einsum('i,iakb,jakb->ij', eo_a, t2_ab, t2_ab) * 0.25
h -= util.einsum('j,iakb,jakb->ij', eo_a, t2a_aa, t2a_aa) * 0.25
h -= util.einsum('j,iakb,jakb->ij', eo_a, t2_ab, t2_ab) * 0.25
h -= util.einsum('j,iakb,jakb->ij', eo_a, t2_ab, t2_ab) * 0.25
h += util.einsum('iakb,jakb->ij', t2a_aa, eri_aa_ovov) * 0.5
h -= util.einsum('iakb,jbka->ij', t2a_aa, eri_aa_ovov) * 0.5
h += util.einsum('iakb,jakb->ij', t2_ab, eri_ab_ovov)
h += util.einsum('jakb,iakb->ij', t2a_aa, eri_aa_ovov) * 0.5
h -= util.einsum('jakb,kaib->ij', t2a_aa, eri_aa_ovov) * 0.5
h += util.einsum('jakb,iakb->ij', t2_ab, eri_ab_ovov)
hs.append(h)
self.h_1p_or_1h = tuple(hs)
def build_se(self):
if self.rpa is None:
self.solve_casida()
e_rpa, v_rpa, xpy = self.rpa
naux_gf = self.gf.naux
chempot = self.chempot
c = self.gf.v[:self.nphys]
co = c[:, self.gf.e < chempot]
cv = c[:, self.gf.e >= chempot]
xyia = util.mo2qo(self.eri, c, co, cv).reshape(self.nphys, naux_gf, -1)
omega = util.einsum('xyk,ks->xys', xyia, xpy)
e_gf = self.gf.e
e_rpa_s = np.outer(np.sign(e_gf - chempot), e_rpa)
e = util.dirsum('i,ij->ij', e_gf, e_rpa_s).flatten()
v = omega.reshape((self.nphys, -1))
self.se = aux.Aux(e, v, chempot=self.chempot)
log.write('naux (se,build) = %d\n' % self.se.naux, self.verbose)
return self.se
def gradient(x, se, fock, nelec, occupancy=2.0, buf=None, return_val=False):
''' Gradient function for the minimization with respect to error.
Parameters
----------
x : float
objective parameter, i.e. chemical potential on the auxiliary
space
buf : ndarray, optional
array to store the extended Fock matrix
occupancy : float, optional
orbital occupancy, i.e. 2 for RHF and 1 for UHF, default 2
Returns
-------
fx : float
objective function value, i.e. error in number of electrons
dx : float
gradient value
'''
w, v, chempot, error = diag_fock_ext(se,
fock,
nelec,
chempot=x,
buf=buf,
occupancy=occupancy)
nocc = np.sum(w < chempot)
phys = slice(None, se.nphys)
aux = slice(se.nphys, None)
occ = slice(None, nocc)
vir = slice(nocc, None)
h_1 = -np.dot(v[aux, vir].conj().T, v[aux, occ])
z_ai = -h_1 / util.dirsum('i,a->ai', w[occ], -w[vir])
c_occ = np.dot(v[phys, vir], z_ai)
rdm1 = np.dot(v[phys, occ], c_occ.conj().T) * 4
ne = np.trace(rdm1).real
d = 2 * error * ne
if return_val:
return error**2, d
else:
return d
def get_1p_or_1h(self):
occ, vir = self._get_slices()
eri_ovov = self.eri[occ, vir, occ, vir]
eo = self.hf.e[occ]
ev = self.hf.e[vir]
t2 = eri_ovov.copy()
t2 /= util.dirsum('i,a,j,b->iajb', eo, -ev, eo, -ev)
t2a = t2 - t2.swapaxes(0, 2).copy()
os = self.options['os_factor']
ss = self.options['ss_factor']
self.e_mp2 = util.einsum('iajb,iajb->', t2, eri_ovov) * (os + ss)
self.e_mp2 -= util.einsum('iajb,ibja->', t2, eri_ovov) * ss
h = np.diag(eo)
h += util.einsum('a,iakb,jakb->ij', ev, t2a, t2a)
h += util.einsum('a,iakb,jakb->ij', ev, t2, t2)
h += util.einsum('a,ibka,jbka->ij', ev, t2, t2)
h -= util.einsum('k,iakb,jakb->ij', eo, t2a, t2a) * 0.5
h -= util.einsum('k,iakb,jakb->ij', eo, t2, t2) * 0.5
h -= util.einsum('k,iakb,jakb->ij', eo, t2, t2) * 0.5
h -= util.einsum('i,iakb,jakb->ij', eo, t2a, t2a) * 0.25
h -= util.einsum('i,iakb,jakb->ij', eo, t2, t2) * 0.25
h -= util.einsum('i,iakb,jakb->ij', eo, t2, t2) * 0.25
h -= util.einsum('j,iakb,jakb->ij', eo, t2a, t2a) * 0.25
h -= util.einsum('j,iakb,jakb->ij', eo, t2, t2) * 0.25
h -= util.einsum('j,iakb,jakb->ij', eo, t2, t2) * 0.25
h += util.einsum('iakb,jakb->ij', t2a, eri_ovov) * 0.5
h -= util.einsum('iakb,jbka->ij', t2a, eri_ovov) * 0.5
h += util.einsum('iakb,jakb->ij', t2, eri_ovov)
h += util.einsum('jakb,iakb->ij', t2a, eri_ovov) * 0.5
h -= util.einsum('jakb,kaib->ij', t2a, eri_ovov) * 0.5
h += util.einsum('jakb,iakb->ij', t2, eri_ovov)
self.h_1p_or_1h = h
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_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