def project_dm_nr2nr(cell1, dm1, cell2, kpts=None):
r''' Project density matrix representation from basis set 1 (cell1) to basis
set 2 (cell2).
.. math::
|AO2\rangle DM_AO2 \langle AO2|
= |AO2\rangle P DM_AO1 P \langle AO2|
DM_AO2 = P DM_AO1 P
P = S_{AO2}^{-1}\langle AO2|AO1\rangle
'''
s22 = cell2.pbc_intor('int1e_ovlp', hermi=1, kpts=kpts)
s21 = pbcgto.intor_cross('int1e_ovlp', cell2, cell1, kpts=kpts)
if kpts is None or numpy.shape(kpts) == (3, ): # A single k-point
p21 = scipy.linalg.solve(s22, s21, sym_pos=True)
return numpy.einsum(
"pi,...ij,qj->...pq", p21, dm1, p21.conj(),
optimize=True) # Ellipsis for potential spin dimension in UHF
else:
p21 = [
scipy.linalg.solve(s22[k], s21[k], sym_pos=True)
for k, kpt in enumerate(kpts)
]
return numpy.einsum("kpi,...kij,kqj->...kpq",
p21,
dm1,
p21.conj(),
optimize=True)
def get_nuc_less_accurate(mydf, kpts=None):
log = logger.Logger(mydf.stdout, mydf.verbose)
t1 = t0 = (time.clock(), time.time())
if kpts is None:
kpts_lst = numpy.zeros((1, 3))
else:
kpts_lst = numpy.reshape(kpts, (-1, 3))
nkpts = len(kpts_lst)
if mydf._cderi is None:
mydf.build()
cell = mydf.cell
fused_cell, fuse = fuse_auxcell_(mydf, mydf.auxcell)
nao = cell.nao_nr()
charge = -cell.atom_charges()
j2c = pgto.intor_cross("cint2c2e_sph", fused_cell, _fake_nuc(cell))
jaux = j2c.dot(charge)
jaux -= charge.sum() * mydf.auxbar(fused_cell)
Gv = cell.get_Gv(mydf.gs)
SI = cell.get_SI(Gv)
# The normal nuclues have been considered in function get_gth_vlocG_part1
# The result vG is the potential in G-space for erf part of the pp nuclues and
# "numpy.dot(charge, SI) * coulG" for normal nuclues.
vpplocG = pgto.pseudo.pp_int.get_gth_vlocG_part1(cell, Gv)
vG = -1.0 / cell.vol * numpy.einsum("ij,ij->j", SI, vpplocG)
kpt_allow = numpy.zeros(3)
if is_zero(kpts_lst):
vj = numpy.zeros((nkpts, nao ** 2))
else:
vj = numpy.zeros((nkpts, nao ** 2), dtype=numpy.complex128)
max_memory = max(2000, mydf.max_memory - lib.current_memory()[0])
for k, pqkR, pqkI, p0, p1 in mydf.ft_loop(cell, mydf.gs, kpt_allow, kpts_lst, max_memory=max_memory):
if not gamma_point(kpts_lst[k]):
vj[k] += numpy.einsum("k,xk->x", vG.real, pqkI) * 1j
vj[k] += numpy.einsum("k,xk->x", vG.imag, pqkR) * -1j
vj[k] += numpy.einsum("k,xk->x", vG.real, pqkR)
vj[k] += numpy.einsum("k,xk->x", vG.imag, pqkI)
pqkR = pqkI = None
Gv = cell.get_Gv(mydf.gs)
aoaux = ft_ao.ft_ao(fused_cell, Gv)
jaux -= numpy.einsum("x,xj->j", vG.real, aoaux.real)
jaux -= numpy.einsum("x,xj->j", vG.imag, aoaux.imag)
jaux = fuse(jaux)
vj = vj.reshape(-1, nao, nao)
for k, kpt in enumerate(kpts_lst):
with mydf.load_Lpq((kpt, kpt)) as Lpq:
v = 0
for p0, p1 in lib.prange(0, jaux.size, mydf.blockdim):
v += numpy.dot(jaux[p0:p1], numpy.asarray(Lpq[p0:p1]))
if gamma_point(kpt):
vj[k] += lib.unpack_tril(numpy.asarray(v.real, order="C"))
else:
vj[k] += lib.unpack_tril(v)
if kpts is None or numpy.shape(kpts) == (3,):
vj = vj[0]
return vj
def get_int1e_cross(intor, cell1, cell2, kpt=None, comp=1):
r'''1-electron integrals from two molecules like
.. math::
\langle \mu | intor | \nu \rangle, \mu \in cell1, \nu \in cell2
'''
return pgto.intor_cross(intor, cell1, cell2, comp, 0, kpt)
def get_int1e_cross(intor, cell1, cell2, kpt=None, comp=1):
r'''1-electron integrals from two molecules like
.. math::
\langle \mu | intor | \nu \rangle, \mu \in cell1, \nu \in cell2
'''
return pgto.intor_cross(intor, cell1, cell2, comp, 0, kpt)
def get_hcore(self, cell=None, kpts=None):
if cell is None: cell = self.cell
if kpts is None:
kpts_lst = numpy.zeros((1, 3))
else:
kpts_lst = numpy.reshape(kpts, (-1, 3))
xcell, contr_coeff = self.get_xmol(cell)
with_df = aft.AFTDF(xcell)
c = lib.param.LIGHT_SPEED
assert ('1E' in self.approx.upper())
if 'ATOM' in self.approx.upper():
atom_slices = xcell.offset_nr_by_atom()
nao = xcell.nao_nr()
x = numpy.zeros((nao, nao))
vloc = numpy.zeros((nao, nao))
wloc = numpy.zeros((nao, nao))
for ia in range(xcell.natm):
ish0, ish1, p0, p1 = atom_slices[ia]
shls_slice = (ish0, ish1, ish0, ish1)
t1 = xcell.intor('int1e_kin', shls_slice=shls_slice)
s1 = xcell.intor('int1e_ovlp', shls_slice=shls_slice)
with xcell.with_rinv_at_nucleus(ia):
z = -xcell.atom_charge(ia)
v1 = z * xcell.intor('int1e_rinv', shls_slice=shls_slice)
w1 = z * xcell.intor('int1e_prinvp', shls_slice=shls_slice)
vloc[p0:p1, p0:p1] = v1
wloc[p0:p1, p0:p1] = w1
x[p0:p1, p0:p1] = x2c._x2c1e_xmatrix(t1, v1, w1, s1, c)
else:
raise NotImplementedError
t = xcell.pbc_intor('int1e_kin', 1, lib.HERMITIAN, kpts_lst)
s = xcell.pbc_intor('int1e_ovlp', 1, lib.HERMITIAN, kpts_lst)
v = with_df.get_nuc(kpts_lst)
#w = get_pnucp(with_df, kpts_lst)
if self.basis is not None:
s22 = s
s21 = pbcgto.intor_cross('int1e_ovlp', xcell, cell, kpts=kpts_lst)
h1_kpts = []
for k in range(len(kpts_lst)):
# The treatment of pnucp local part has huge effects to hcore
#h1 = x2c._get_hcore_fw(t[k], vloc, wloc, s[k], x, c) - vloc + v[k]
#h1 = x2c._get_hcore_fw(t[k], v[k], w[k], s[k], x, c)
h1 = x2c._get_hcore_fw(t[k], v[k], wloc, s[k], x, c)
if self.basis is not None:
c = lib.cho_solve(s22[k], s21[k])
h1 = reduce(numpy.dot, (c.T, h1, c))
if self.xuncontract and contr_coeff is not None:
h1 = reduce(numpy.dot, (contr_coeff.T, h1, contr_coeff))
h1_kpts.append(h1)
if kpts is None or numpy.shape(kpts) == (3, ):
h1_kpts = h1_kpts[0]
return lib.asarray(h1_kpts)
def get_hcore(self, cell=None, kpts=None):
if cell is None: cell = self.cell
if kpts is None:
kpts_lst = numpy.zeros((1,3))
else:
kpts_lst = numpy.reshape(kpts, (-1,3))
xcell, contr_coeff = self.get_xmol(cell)
with_df = aft.AFTDF(xcell)
c = lib.param.LIGHT_SPEED
assert('1E' in self.approx.upper())
if 'ATOM' in self.approx.upper():
atom_slices = xcell.offset_nr_by_atom()
nao = xcell.nao_nr()
x = numpy.zeros((nao,nao))
vloc = numpy.zeros((nao,nao))
wloc = numpy.zeros((nao,nao))
for ia in range(xcell.natm):
ish0, ish1, p0, p1 = atom_slices[ia]
shls_slice = (ish0, ish1, ish0, ish1)
t1 = xcell.intor('int1e_kin', shls_slice=shls_slice)
s1 = xcell.intor('int1e_ovlp', shls_slice=shls_slice)
with xcell.with_rinv_as_nucleus(ia):
z = -xcell.atom_charge(ia)
v1 = z * xcell.intor('int1e_rinv', shls_slice=shls_slice)
w1 = z * xcell.intor('int1e_prinvp', shls_slice=shls_slice)
vloc[p0:p1,p0:p1] = v1
wloc[p0:p1,p0:p1] = w1
x[p0:p1,p0:p1] = x2c._x2c1e_xmatrix(t1, v1, w1, s1, c)
else:
raise NotImplementedError
t = xcell.pbc_intor('int1e_kin', 1, lib.HERMITIAN, kpts_lst)
s = xcell.pbc_intor('int1e_ovlp', 1, lib.HERMITIAN, kpts_lst)
v = with_df.get_nuc(kpts_lst)
#w = get_pnucp(with_df, kpts_lst)
if self.basis is not None:
s22 = s
s21 = pbcgto.intor_cross('int1e_ovlp', xcell, cell, kpts=kpts_lst)
h1_kpts = []
for k in range(len(kpts_lst)):
# The treatment of pnucp local part has huge effects to hcore
#h1 = x2c._get_hcore_fw(t[k], vloc, wloc, s[k], x, c) - vloc + v[k]
#h1 = x2c._get_hcore_fw(t[k], v[k], w[k], s[k], x, c)
h1 = x2c._get_hcore_fw(t[k], v[k], wloc, s[k], x, c)
if self.basis is not None:
c = lib.cho_solve(s22[k], s21[k])
h1 = reduce(numpy.dot, (c.T, h1, c))
if self.xuncontract and contr_coeff is not None:
h1 = reduce(numpy.dot, (contr_coeff.T, h1, contr_coeff))
h1_kpts.append(h1)
if kpts is None or numpy.shape(kpts) == (3,):
h1_kpts = h1_kpts[0]
return lib.asarray(h1_kpts)
def project_mo_nr2nr(cell1, mo1, cell2, kpt=None):
r''' Project orbital coefficients
.. math::
|\psi1> = |AO1> C1
|\psi2> = P |\psi1> = |AO2>S^{-1}<AO2| AO1> C1 = |AO2> C2
C2 = S^{-1}<AO2|AO1> C1
'''
s22 = cell2.pbc_intor('cint1e_ovlp_sph', hermi=1, kpts=kpt)
s21 = pbcgto.intor_cross('cint1e_ovlp_sph', cell2, cell1, kpts=kpt)
mo2 = numpy.dot(s21, mo1)
return scipy.linalg.cho_solve(scipy.linalg.cho_factor(s22), mo2)
def project_mo_nr2nr(cell1, mo1, cell2, kpt=None):
r''' Project orbital coefficients
.. math::
|\psi1> = |AO1> C1
|\psi2> = P |\psi1> = |AO2>S^{-1}<AO2| AO1> C1 = |AO2> C2
C2 = S^{-1}<AO2|AO1> C1
'''
s22 = cell2.pbc_intor('cint1e_ovlp_sph', hermi=1, kpts=kpt)
s21 = pbcgto.intor_cross('cint1e_ovlp_sph', cell2, cell1, kpts=kpt)
mo2 = numpy.dot(s21, mo1)
return scipy.linalg.cho_solve(scipy.linalg.cho_factor(s22), mo2)
def project_mo_nr2nr(cell1, mo1, cell2, kpts=None):
r''' Project orbital coefficients
.. math::
|\psi1> = |AO1> C1
|\psi2> = P |\psi1> = |AO2>S^{-1}<AO2| AO1> C1 = |AO2> C2
C2 = S^{-1}<AO2|AO1> C1
'''
s22 = cell2.pbc_intor('int1e_ovlp', hermi=1, kpts=kpts)
s21 = pbcgto.intor_cross('int1e_ovlp', cell2, cell1, kpts=kpts)
if kpts is None or numpy.shape(kpts) == (3,): # A single k-point
return scipy.linalg.solve(s22, s21.dot(mo1), sym_pos=True)
else:
assert(len(kpts) == len(mo1))
return [scipy.linalg.solve(s22[k], s21[k].dot(mo1[k]), sym_pos=True)
for k, kpt in enumerate(kpts)]
def project_mo_nr2nr(cell1, mo1, cell2, kpts=None):
r''' Project orbital coefficients
.. math::
|\psi1> = |AO1> C1
|\psi2> = P |\psi1> = |AO2>S^{-1}<AO2| AO1> C1 = |AO2> C2
C2 = S^{-1}<AO2|AO1> C1
'''
s22 = cell2.pbc_intor('int1e_ovlp', hermi=1, kpts=kpts)
s21 = pbcgto.intor_cross('int1e_ovlp', cell2, cell1, kpts=kpts)
if kpts is None or numpy.shape(kpts) == (3,): # A single k-point
return scipy.linalg.solve(s22, s21.dot(mo1), sym_pos=True)
else:
assert(len(kpts) == len(mo1))
return [scipy.linalg.solve(s22[k], s21[k].dot(mo1[k]), sym_pos=True)
for k, kpt in enumerate(kpts)]
def get_nuc(mydf, kpts=None):
cell = mydf.cell
if kpts is None:
kpts_lst = numpy.zeros((1, 3))
else:
kpts_lst = numpy.reshape(kpts, (-1, 3))
if mydf._cderi is None:
mydf.build()
log = logger.Logger(mydf.stdout, mydf.verbose)
t1 = t0 = (time.clock(), time.time())
fused_cell, fuse = fuse_auxcell_(mydf, mydf.auxcell)
nuccell = make_modchg_basis(cell, mydf.eta, 0)
nuccell._bas = numpy.asarray(nuccell._bas[nuccell._bas[:,
gto.ANG_OF] == 0],
dtype=numpy.int32,
order='C')
charge = -cell.atom_charges()
nucbar = sum([z / nuccell.bas_exp(i)[0] for i, z in enumerate(charge)])
nucbar *= numpy.pi / cell.vol
vj = [v.ravel() for v in _int_nuc_vloc(cell, nuccell, kpts_lst)]
t1 = log.timer_debug1('vnuc pass1: analytic int', *t1)
j2c = pgto.intor_cross('cint2c2e_sph', fused_cell, nuccell)
jaux = j2c.dot(charge)
kpt_allow = numpy.zeros(3)
coulG = tools.get_coulG(cell, kpt_allow, gs=mydf.gs) / cell.vol
Gv = cell.get_Gv(mydf.gs)
aoaux = ft_ao.ft_ao(nuccell, Gv)
vGR = numpy.einsum('i,xi->x', charge, aoaux.real) * coulG
vGI = numpy.einsum('i,xi->x', charge, aoaux.imag) * coulG
max_memory = mydf.max_memory - lib.current_memory()[0]
for k, pqkR, pqkI, p0, p1 \
in mydf.ft_loop(cell, mydf.gs, kpt_allow, kpts_lst, max_memory=max_memory):
# rho_ij(G) nuc(-G) / G^2
# = [Re(rho_ij(G)) + Im(rho_ij(G))*1j] [Re(nuc(G)) - Im(nuc(G))*1j] / G^2
if not gamma_point(kpts_lst[k]):
vj[k] += numpy.einsum('k,xk->x', vGR[p0:p1], pqkI) * 1j
vj[k] += numpy.einsum('k,xk->x', vGI[p0:p1], pqkR) * -1j
vj[k] += numpy.einsum('k,xk->x', vGR[p0:p1], pqkR)
vj[k] += numpy.einsum('k,xk->x', vGI[p0:p1], pqkI)
pqkR = pqkI = None
t1 = log.timer_debug1('contracting Vnuc', *t1)
vG = numpy.einsum('i,xi,x->x', charge, ft_ao.ft_ao(nuccell, Gv), coulG)
aoaux = ft_ao.ft_ao(fused_cell, Gv)
jaux -= numpy.einsum('x,xj->j', vG.real, aoaux.real)
jaux -= numpy.einsum('x,xj->j', vG.imag, aoaux.imag)
jaux -= charge.sum() * mydf.auxbar(fused_cell)
jaux = fuse(jaux)
aoaux = None
ovlp = cell.pbc_intor('cint1e_ovlp_sph', 1, lib.HERMITIAN, kpts_lst)
nao = cell.nao_nr()
nao_pair = nao * (nao + 1) // 2
max_memory = max(2000, mydf.max_memory - lib.current_memory()[0])
blksize = max(16, min(int(max_memory * 1e6 / 16 / nao_pair),
mydf.blockdim))
for k, kpt in enumerate(kpts_lst):
with mydf.load_Lpq((kpt, kpt)) as Lpq:
v = 0
for p0, p1 in lib.prange(0, jaux.size, blksize):
v += numpy.dot(jaux[p0:p1], numpy.asarray(Lpq[p0:p1]))
vj[k] = vj[k].reshape(nao, nao) - nucbar * ovlp[k]
if gamma_point(kpt):
vj[k] += lib.unpack_tril(numpy.asarray(v.real, order='C'))
else:
vj[k] += lib.unpack_tril(v)
if kpts is None or numpy.shape(kpts) == (3, ):
vj = vj[0]
return vj
def get_nuc_less_accurate(mydf, kpts=None):
log = logger.Logger(mydf.stdout, mydf.verbose)
t1 = t0 = (time.clock(), time.time())
if kpts is None:
kpts_lst = numpy.zeros((1, 3))
else:
kpts_lst = numpy.reshape(kpts, (-1, 3))
nkpts = len(kpts_lst)
if mydf._cderi is None:
mydf.build()
cell = mydf.cell
fused_cell, fuse = fuse_auxcell_(mydf, mydf.auxcell)
nao = cell.nao_nr()
charge = -cell.atom_charges()
j2c = pgto.intor_cross('cint2c2e_sph', fused_cell, _fake_nuc(cell))
jaux = j2c.dot(charge)
jaux -= charge.sum() * mydf.auxbar(fused_cell)
Gv = cell.get_Gv(mydf.gs)
SI = cell.get_SI(Gv)
# The normal nuclues have been considered in function get_gth_vlocG_part1
# The result vG is the potential in G-space for erf part of the pp nuclues and
# "numpy.dot(charge, SI) * coulG" for normal nuclues.
vpplocG = pgto.pseudo.pp_int.get_gth_vlocG_part1(cell, Gv)
vG = -1. / cell.vol * numpy.einsum('ij,ij->j', SI, vpplocG)
kpt_allow = numpy.zeros(3)
if is_zero(kpts_lst):
vj = numpy.zeros((nkpts, nao**2))
else:
vj = numpy.zeros((nkpts, nao**2), dtype=numpy.complex128)
max_memory = max(2000, mydf.max_memory - lib.current_memory()[0])
for k, pqkR, pqkI, p0, p1 \
in mydf.ft_loop(cell, mydf.gs, kpt_allow, kpts_lst, max_memory=max_memory):
if not gamma_point(kpts_lst[k]):
vj[k] += numpy.einsum('k,xk->x', vG.real, pqkI) * 1j
vj[k] += numpy.einsum('k,xk->x', vG.imag, pqkR) * -1j
vj[k] += numpy.einsum('k,xk->x', vG.real, pqkR)
vj[k] += numpy.einsum('k,xk->x', vG.imag, pqkI)
pqkR = pqkI = None
Gv = cell.get_Gv(mydf.gs)
aoaux = ft_ao.ft_ao(fused_cell, Gv)
jaux -= numpy.einsum('x,xj->j', vG.real, aoaux.real)
jaux -= numpy.einsum('x,xj->j', vG.imag, aoaux.imag)
jaux = fuse(jaux)
vj = vj.reshape(-1, nao, nao)
for k, kpt in enumerate(kpts_lst):
with mydf.load_Lpq((kpt, kpt)) as Lpq:
v = 0
for p0, p1 in lib.prange(0, jaux.size, mydf.blockdim):
v += numpy.dot(jaux[p0:p1], numpy.asarray(Lpq[p0:p1]))
if gamma_point(kpt):
vj[k] += lib.unpack_tril(numpy.asarray(v.real, order='C'))
else:
vj[k] += lib.unpack_tril(v)
if kpts is None or numpy.shape(kpts) == (3, ):
vj = vj[0]
return vj
def get_nuc(mydf, kpts=None):
cell = mydf.cell
if kpts is None:
kpts_lst = numpy.zeros((1,3))
else:
kpts_lst = numpy.reshape(kpts, (-1,3))
if mydf._cderi is None:
mydf.build()
log = logger.Logger(mydf.stdout, mydf.verbose)
t1 = t0 = (time.clock(), time.time())
auxcell = mydf.auxcell
nuccell = make_modchg_basis(cell, mydf.eta, 0)
nuccell._bas = numpy.asarray(nuccell._bas[nuccell._bas[:,gto.ANG_OF]==0],
dtype=numpy.int32, order='C')
charge = -cell.atom_charges()
nucbar = sum([z/nuccell.bas_exp(i)[0] for i,z in enumerate(charge)])
nucbar *= numpy.pi/cell.vol
vj = [v.ravel() for v in _int_nuc_vloc(cell, nuccell, kpts_lst)]
t1 = log.timer_debug1('vnuc pass1: analytic int', *t1)
j2c = pgto.intor_cross('cint2c2e_sph', auxcell, nuccell)
jaux = j2c.dot(charge)
kpt_allow = numpy.zeros(3)
coulG = tools.get_coulG(cell, kpt_allow, gs=mydf.gs) / cell.vol
Gv = cell.get_Gv(mydf.gs)
aoaux = ft_ao.ft_ao(nuccell, Gv)
vGR = numpy.einsum('i,xi->x', charge, aoaux.real) * coulG
vGI = numpy.einsum('i,xi->x', charge, aoaux.imag) * coulG
max_memory = mydf.max_memory - lib.current_memory()[0]
for k, pqkR, pqkI, p0, p1 \
in mydf.ft_loop(cell, mydf.gs, kpt_allow, kpts_lst, max_memory=max_memory):
# rho_ij(G) nuc(-G) / G^2
# = [Re(rho_ij(G)) + Im(rho_ij(G))*1j] [Re(nuc(G)) - Im(nuc(G))*1j] / G^2
if not gamma_point(kpts_lst[k]):
vj[k] += numpy.einsum('k,xk->x', vGR[p0:p1], pqkI) * 1j
vj[k] += numpy.einsum('k,xk->x', vGI[p0:p1], pqkR) *-1j
vj[k] += numpy.einsum('k,xk->x', vGR[p0:p1], pqkR)
vj[k] += numpy.einsum('k,xk->x', vGI[p0:p1], pqkI)
t1 = log.timer_debug1('contracting Vnuc', *t1)
# Append nuccell to auxcell, so that they can be FT together in pw_loop
# the first [:naux] of ft_ao are aux fitting functions.
nuccell._atm, nuccell._bas, nuccell._env = \
gto.conc_env(auxcell._atm, auxcell._bas, auxcell._env,
nuccell._atm, nuccell._bas, nuccell._env)
naux = auxcell.nao_nr()
aoaux = ft_ao.ft_ao(nuccell, Gv)
vG = numpy.einsum('i,xi,x->x', charge, aoaux[:,naux:], coulG)
jaux -= numpy.einsum('x,xj->j', vG.real, aoaux[:,:naux].real)
jaux -= numpy.einsum('x,xj->j', vG.imag, aoaux[:,:naux].imag)
jaux -= charge.sum() * mydf.auxbar(auxcell)
ovlp = cell.pbc_intor('cint1e_ovlp_sph', 1, lib.HERMITIAN, kpts_lst)
nao = cell.nao_nr()
nao_pair = nao * (nao+1) // 2
max_memory = max(2000, mydf.max_memory-lib.current_memory()[0])
blksize = max(16, min(int(max_memory*1e6/16/nao_pair), mydf.blockdim))
for k, kpt in enumerate(kpts_lst):
with mydf.load_Lpq((kpt,kpt)) as Lpq:
v = 0
for p0, p1 in lib.prange(0, jaux.size, blksize):
v += numpy.dot(jaux[p0:p1], numpy.asarray(Lpq[p0:p1]))
vj[k] = vj[k].reshape(nao,nao) - nucbar * ovlp[k]
if gamma_point(kpt):
vj[k] += lib.unpack_tril(numpy.asarray(v.real,order='C'))
else:
vj[k] += lib.unpack_tril(v)
if kpts is None or numpy.shape(kpts) == (3,):
vj = vj[0]
return vj
