def energy_mp2_aux(mo, se, both_sides=False):
''' Calculates the two-body contribution to the electronic energy
using the MOs and the auxiliary representation of the
self-energy according the the MP2 form of the Galitskii-Migdal
formula.
Parameters
----------
mo : (n) ndarray
MO energies
se : Aux
auxiliary representation of self-energy
both_sides : bool, optional
if True, calculate both halves of the functional and return
the mean, default False
Returns
-------
e2b : float
two-body contribution to electronic energy
'''
if isinstance(se, (tuple, list)):
n = len(se)
if util.iter_depth(mo) == 2:
return sum([
energy_mp2_aux(mo[i], se[i], both_sides=both_sides)
for i in range(n)
]) / n
else:
return sum([
energy_mp2_aux(mo, se[i], both_sides=both_sides)
for i in range(n)
]) / n
nphys = se.nphys
occ = mo < se.chempot
vir = mo >= se.chempot
vxk = se.v_vir[occ]
dxk = 1.0 / util.outer_sum([mo[occ], -se.e_vir])
e2b = util.einsum('xk,xk,xk->', vxk, vxk.conj(), dxk)
if both_sides:
vxk = se.v_occ[vir]
dxk = -1.0 / util.outer_sum([mo[vir], -se.e_occ])
e2b += util.einsum('xk,xk,xk->', vxk, vxk.conj(), dxk)
e2b *= 0.5
return np.ravel(e2b.real)[0]
def iterate_qp(self):
# Should e_qp become e_gf or is that only for evGW?
e_qp = self.e_qp
e_qp_0 = e_qp.copy()
e = self.gf.e
v = self.gf.v
for niter in range(1, self.options['qp_maxiter'] + 1):
denom = 1.0 / util.outer_sum(e_qp, -e)
se = v * v * denom
z = se * denom
se = np.sum(se, axis=1)
z = np.sum(z, axis=1)
z = 1.0 / (1.0 + z)
e_qp_prev = e_qp.copy()
e_qp = e_qp_0 + z.real * se
error = np.max(np.absolute(e_qp.real - e_qp_prev.real))
if error < self.options['qp_etol']:
break
self.e_qp = e_qp
return e_qp
def solve_casida(self):
#TODO: this step is n^6 and inefficient in memory, rethink
e_ia = util.outer_sum([-self.gf.e_occ, self.gf.e_vir])
co = self.gf.v[:self.nphys, self.gf.e < self.chempot]
cv = self.gf.v[:self.nphys, self.gf.e >= self.chempot]
iajb = util.ao2mo(self.eri, co, cv, co, cv).reshape((e_ia.size, ) * 2)
apb = np.diag(e_ia.flatten()) + 4.0 * iajb
amb = np.diag(np.sqrt(e_ia.flatten()))
h_rpa = util.dots((amb, apb, amb))
e_rpa, v_rpa = util.eigh(h_rpa)
e_rpa = np.sqrt(e_rpa)
xpy = util.einsum('ij,jk,k->ik', amb, v_rpa, 1.0 / np.sqrt(e_rpa))
xpy *= np.sqrt(2.0)
self.rpa = (e_rpa, v_rpa, xpy)
return self.rpa
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_ump2_part_se_direct(eo,
ev,
xija,
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
xija : (n,o,o,v)
two-electron integrals indexed as physical, occupied, occupied,
virtual (physical, virtual, virtual, occupied)
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, nocca, _, nvira = xija[0].shape
se = np.zeros((grid.shape[0], nphys, nphys), dtype=types.complex128)
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
vi_a = xija[0][:, i].reshape((nphys, -1))
vip_a = xija[0][:, :, i].reshape((nphys, -1))
vi_b = xija[1][:, i].reshape((nphys, -1))
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 run(self):
t_amp = np.zeros((self.nvir, self.nvir, self.nocc, self.nocc), dtype=types.float64)
o = slice(None, self.nocc)
v = slice(self.nocc, None)
opdm_corr = np.zeros((self.nso,)*2, dtype=types.float64)
opdm_ref = opdm_corr.copy()
opdm_ref[o,o] = np.eye(self.nocc)
tpdm_corr = np.zeros((self.nso,)*4, dtype=types.float64)
x = opdm_corr.copy()
eija = util.outer_sum([-self.e[v], -self.e[v], self.e[o], self.e[o]])
eija = 1.0 / eija
if self.diis:
diis = util.DIIS(self.diis_space)
for niter in range(1, self.maxiter+1):
f = self.h1e_mo + util.einsum('piqi->pq', self.eri_mo[:,o,:,o])
fp = f.copy()
np.fill_diagonal(fp, 0.0)
e = f.diagonal()
t1 = self.eri_mo[v,v,o,o]
t2 = util.einsum('ac,cbij->abij', fp[v,v], t_amp)
t3 = util.einsum('ki,abkj->abij', fp[o,o], t_amp)
t_amp = t1.copy()
t_amp += t2 - t2.transpose(1,0,2,3)
t_amp -= t3 - t3.transpose(0,1,3,2)
t_amp *= eija
if niter > 1:
if self.diis:
t_amp = diis.update(t_amp)
if self.damping > 0.0:
damping = self.damping
t_amp = (1.0 - damping) * t_amp + damping * self._t_prev
if self.damping > 0.0:
self._t_prev = t_amp.copy()
opdm_corr[v,v] = util.einsum('ijac,bcij->ba', t_amp.T, t_amp) * 0.5
opdm_corr[o,o] = util.einsum('jkab,abik->ji', t_amp.T, t_amp) * -0.5
opdm = opdm_corr + opdm_ref
tpdm_corr[v,v,o,o] = t_amp
tpdm_corr[o,o,v,v] = t_amp.T
tpdm2 = util.einsum('rp,sq->rspq', opdm_corr, opdm_ref)
tpdm3 = util.einsum('rp,sq->rspq', opdm_ref, opdm_ref)
tpdm = tpdm_corr.copy()
tpdm += tpdm2 - tpdm2.transpose(1,0,2,3)
tpdm -= tpdm2.transpose(0,1,3,2) - tpdm2.transpose(1,0,3,2)
tpdm += tpdm3 - tpdm3.transpose(1,0,2,3)
fnr = util.einsum('pr,rq->pq', self.h1e_mo, opdm)
fnr += util.einsum('prst,stqr->pq', self.eri_mo, tpdm) * 0.5
x[v,o] = ((fnr - fnr.T)[v,o]) / util.outer_sum([-self.e[v], self.e[o]])
u = expm(x - x.T)
c = np.dot(self.c, u)
self.c = c
self.h1e_mo = util.ao2mo(self.h1e_ao, c, c)
self.eri_mo = util.ao2mo(self.eri_ao, c, c, c, c)
e_prev = self.e_tot
self.e_1body = util.einsum('pq,qp->', self.h1e_mo, opdm)
self.e_1body += self.hf.e_nuc
self.e_2body = 0.25 * util.einsum('pqrs,rspq->', self.eri_mo, tpdm)
if abs(self.e_tot - e_prev) < self.etol:
break
return self
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_rmp2_part_se_direct(eo,
ev,
xija,
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
xija : (n,o,o,v)
two-electron integrals indexed as physical, occupied, occupied,
virtual (physical, virtual, 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
'''
#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, nocc, _, nvir = xija.shape
se = np.zeros((grid.shape[0], nphys, nphys), dtype=types.complex128)
eov = util.outer_sum([eo, -ev]).flatten()
for i in range(nocc):
ei = eo[i] + eov - chempot
vi = xija[:, i].reshape((nphys, -1))
vip = xija[:, :, i].reshape((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_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_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