def mb_ph_ops(hlat, perturb, omega, nimp, nocc, pemb, pcore, pvirt):
"""
external ops for part-hole Green's function
"""
nimp_sp = nimp * 2
nsites_sp = hlat.shape[0] * 2
nocc_sp = nocc * 2
# transform h into imp+bath/ext(all) basis
pall = N.hstack([pemb, pcore, pvirt])
hall = N.dot(pall.T, N.dot(hlat, pall))
hall_sp = _spinify(hall)
# t1amp in (all) basis
t1amp_ao = make_t1amp_ao(hlat, omega, perturb, nocc)
t1amp_all = N.dot(pall.T, N.dot(t1amp_ao, pall))
t1amp_all_sp = _spinify(t1amp_all)
# mb_ext_operators
sites_imp = range(0, nimp_sp * 2)
sites_ext = range(nimp_sp * 2, nsites_sp)
cops = []
dops = []
project = False
if project:
li_t1amp_ext_imp = _linear_orthonormal_basis(
t1amp_all_sp[sites_ext, :][:, sites_imp], orient='c')
print "retained %i out of %i vectors" % (li_t1amp_ext_imp.shape[1],
len(sites_imp))
li_t1amp_imp_ext = _linear_orthonormal_basis(
t1amp_all_sp[sites_imp, :][:, sites_ext], orient='r')
print "retained %i out of %i vectors" % (li_t1amp_imp_ext.shape[0],
len(sites_imp))
for i in xrange(li_t1amp_ext_imp.shape[1]):
cops.append(models.ContractedC(sites_ext, li_t1amp_ext_imp[:, i]))
for i in xrange(li_t1amp_imp_ext.shape[0]):
dops.append(models.ContractedD(sites_ext, li_t1amp_imp_ext[i, :]))
else:
for imp in sites_imp:
coeff_c = t1amp_all_sp[sites_ext, imp]
cops.append(models.ContractedC(sites_ext, coeff_c))
coeff_d = t1amp_all_sp[imp, sites_ext]
dops.append(models.ContractedD(sites_ext, coeff_d))
cd_coeffs = {}
for i in sites_ext:
for j in sites_ext:
if abs(t1amp_all_sp[i, j]) > 1.e-8:
cd_coeffs[i, j] = t1amp_all_sp[i, j]
cdop = models.ContractedCD(cd_coeffs)
ops = {"1": models.Unit(), "c": cops, "d": dops, "cd": cdop}
return ops
def hubbard_imp_ext_h(h0, u, nocc, nimp, nsites):
nsites_sp = nsites * 2
sites_imp = range(0, 4 * nimp)
sites_ext = range(4 * nimp, 2 * nsites)
t_dict = {}
for i in xrange(nsites_sp):
for j in xrange(nsites_sp):
if abs(h0[i, j]) > 1.e-12:
t_dict[i, j] = h0[i, j]
t_dict[0, 0] = 0
t_dict[1, 1] = 0
u_dict = {}
u_dict[0, 1] = u
# h in imp+ext space
hop = models.GeneralHubbard(t_dict, u_dict)
t_imp = {}
for i in sites_imp:
for j in sites_imp:
if abs(h0[i, j]) > 1.e-12:
t_imp[i, j] = h0[i, j]
t_imp[0, 0] = 0
t_imp[1, 1] = 0
t_ext = {}
for i in sites_ext:
for j in sites_ext:
if abs(h0[i, j]) > 1.e-12:
t_ext[i, j] = h0[i, j]
hext = models.ContractedCD(t_ext)
himp = models.GeneralHubbard(t_imp, u_dict)
hcs_imp = []
hds_imp = []
hcs_ext = []
hds_ext = []
for i in sites_imp:
imp_coeffs = utils.zeros(len(sites_imp))
imp_coeffs[i] = 1.
hcs_imp.append(models.ContractedC(sites_imp, imp_coeffs))
hds_imp.append(models.ContractedD(sites_imp, imp_coeffs))
hcs_ext.append(models.ContractedC(sites_ext, h0[sites_ext, i]))
hds_ext.append(models.ContractedD(sites_ext, h0[i, sites_ext]))
return hop, himp, hcs_imp, hds_imp, hcs_ext, hds_ext, hext
def mb_sp_ops(hlat, perturb, omega, nimp, nocc, pemb, pcore, pvirt):
"""
single particle external ops
(for single particle Green's function)
"""
nimp_sp = nimp * 2
nsites_sp = hlat.shape[0] * 2
nocc_sp = nocc * 2
# transform h into imp+bath/ext(all) basis
pall = N.hstack([pemb, pcore, pvirt])
hall = N.dot(pall.T, N.dot(hlat, pall))
hall_sp = _spinify(hall)
# green's function amps in (all) basis
damp_ao, camp_ao = make_greens_ao(hlat, omega, perturb, nocc)
damp_all = N.dot(pall.T, damp_ao)
camp_all = N.dot(pall.T, camp_ao)
#print N.dot(damp_ao,pall)
damp_all_sp = _spinify_vec(damp_all)
camp_all_sp = _spinify_vec(camp_all)
# mb_ext operators
sites_imp = range(0, nimp_sp * 2)
sites_ext = range(nimp_sp * 2, nsites_sp)
coeff_c = camp_all_sp[sites_ext]
cops = [models.ContractedC(sites_ext, coeff_c)]
coeff_d = damp_all_sp[sites_ext]
dops = [models.ContractedD(sites_ext, coeff_d)]
ops = {"1": models.Unit(), "c": cops, "d": dops}
return ops
def sp_greens():
# main loop for single-particle GF
utils.dtype = N.complex128
nimp = 1
nimp_sp = 2 * nimp
nocc = 20
nsites = 40
nsites_sp = nsites * 2
ndim = 2
sz = 0
# sites numbered in order as imp+bath ... ext
sites_imp = range(0, 2 * nimp_sp)
sites_ext = range(2 * nimp_sp, nsites_sp)
ncore = 2 * nocc - 2 * nimp
cocc = range(2 * nimp_sp, 2 * nimp_sp + ncore)
vocc = range(2 * nimp_sp + ncore, nsites_sp)
N.set_printoptions(precision=3)
t = -1. # hopping
delta = 0.2 # broadening
# for u in [0.0,4.0,10.0]:
for u in [1.0]:
# Single impurity Anderson model
mu = 0.
hlat, hall, hlat_sp, hall_sp, e0 = models_embed.si_anderson_imp_ext_ni_h(
t, nocc, nimp, nsites)
hop, himp, hcs_imp, hds_imp, hcs_ext, hds_ext, hext = models_embed.si_anderson_imp_ext_h(
hall_sp, u, nocc, nimp, nsites)
# single site Hubbard in DMET basis
# mu=u/2
# hlat,hall,hlat_sp,hall_sp,e0=models_embed.hubbard_imp_ext_ni_h(t,u,nocc,nimp,nsites)
# hop,himp,hcs_imp,hds_imp,hcs_ext,hds_ext,hext=models_embed.hubbard_imp_ext_h(hall_sp,u,nocc,nimp,nsites)
# g.s embedding basis
pemb, pcore, pvirt = embed.embed(hlat, nimp, nocc)
# perturbation operator
perturb = utils.zeros([nsites])
perturb[0] = 1.
p_coeffs = N.array([1])
perturb_dop = models.ContractedD(N.array([0]), p_coeffs)
perturb_cop = models.ContractedC(N.array([0]), p_coeffs)
fd = file("ph_siam.out." + str(u), "w")
for omega in N.arange(-2, 2, 0.1):
ops_dict = response_embed.mb_sp_ops(hlat, perturb,
omega + 1j * delta, nimp, nocc,
pemb, pcore, pvirt)
configs_dict = response_embed.mb_configs(nsites, nimp, nimp_sp,
2 * nocc - nimp_sp, 0)
neutral_ops_configs, plus_ops_configs, minus_ops_configs = response_embed.mb_sp_ops_configs(
ops_dict, configs_dict)
# basis is setup, now build matrix representations
perturb_dop_mat = opbasis_ni.oimp_matrix_bra_ket_form(
perturb_dop, minus_ops_configs, neutral_ops_configs, cocc,
vocc)
perturb_cop_mat = opbasis_ni.oimp_matrix_bra_ket_form(
perturb_cop, plus_ops_configs, neutral_ops_configs, cocc, vocc)
# h, N-1 configs
himp_mat_minus = opbasis_ni.oimp_matrix_form(
himp, minus_ops_configs, cocc, vocc)
himp_ext_mat_minus = opbasis_ni.himp_ext_matrix_form(
hcs_imp, hds_ext, hds_imp, hcs_ext, minus_ops_configs, cocc,
vocc)
hext_mat_minus = opbasis_ni.hext_matrix_form(
hext, minus_ops_configs, cocc, vocc, e0)
hmat_minus = himp_mat_minus + himp_ext_mat_minus + hext_mat_minus
unit_mat_minus = opbasis_ni.oimp_matrix_form(
models.Unit(), minus_ops_configs, cocc, vocc)
# h, N+1 configs
himp_mat_plus = opbasis_ni.oimp_matrix_form(
himp, plus_ops_configs, cocc, vocc)
himp_ext_mat_plus = opbasis_ni.himp_ext_matrix_form(
hcs_imp, hds_ext, hds_imp, hcs_ext, plus_ops_configs, cocc,
vocc)
hext_mat_plus = opbasis_ni.hext_matrix_form(
hext, plus_ops_configs, cocc, vocc, e0)
hmat_plus = himp_mat_plus + himp_ext_mat_plus + hext_mat_plus
unit_mat_plus = opbasis_ni.oimp_matrix_form(
models.Unit(), plus_ops_configs, cocc, vocc)
# h, neutral configs
himp_mat = opbasis_ni.oimp_matrix_form(himp, neutral_ops_configs,
cocc, vocc)
himp_ext_mat = opbasis_ni.himp_ext_matrix_form(
hcs_imp, hds_ext, hds_imp, hcs_ext, neutral_ops_configs, cocc,
vocc)
hext_mat = opbasis_ni.hext_matrix_form(hext, neutral_ops_configs,
cocc, vocc, e0)
hmat = himp_mat + himp_ext_mat + hext_mat
unit_mat = opbasis_ni.oimp_matrix_form(models.Unit(),
neutral_ops_configs, cocc,
vocc)
# get neutral ground-state energy
es, cs = la.peigh(hmat, unit_mat)
e0 = es[0]
psi0 = cs[:, 0]
print e0
print psi0
# all matrices setup, solve linear response (complex)
psi1_minus = la.solve_perturb(omega + 1j * delta,
unit_mat_minus,
hmat_minus,
e0,
psi0,
perturb_dop_mat,
project=False,
sign_ham=-1.)
gfminus = N.dot(psi1_minus, N.dot(perturb_dop_mat, psi0))
psi1_plus = la.solve_perturb(omega + 1j * delta,
unit_mat_plus,
hmat_plus,
e0,
psi0,
perturb_cop_mat,
project=False)
gfplus = N.dot(psi1_plus, N.dot(perturb_cop_mat, psi0))
d_gf, c_gf = _sp_gf(hlat, omega + 1j * delta, perturb, nocc)
print omega, gfplus.imag, c_gf.imag, gfminus.imag, d_gf.imag
print >> fd, omega - mu, gfplus.imag + gfminus.imag, c_gf.imag + d_gf.imag
def main():
# main loop for single-particle GF
utils.dtype = N.complex128
nimp = 4
nimp_sp = 2 * nimp
nsites = 8
nocc = nsites / 2
nsites_sp = nsites * 2
ndim = 2
sz = 0
# sites numbered in order as imp+bath ... ext
sites_imp = range(0, 2 * nimp_sp)
sites_ext = range(2 * nimp_sp, nsites_sp)
ncore = 2 * nocc - 2 * nimp
cocc = range(2 * nimp_sp, 2 * nimp_sp + ncore)
vocc = range(2 * nimp_sp + ncore, nsites_sp)
N.set_printoptions(precision=3)
#t=1. # hopping
gamma = 0.005
rho0 = (nsites - 2) / 2.
t = 0.1 #N.sqrt(gamma/(rho0*N.pi))
#gamma = (t**2)*pi*0.1
delta = 0.005 # broadening
#a=12.5
# for u in [0.0,4.0,10.0]:
for u in [0., 2.5, 5., 7.5]:
u *= t
# Single impurity Anderson model
mu = 0.
hlat, hall, hlat_sp, hall_sp, e0 = models_embed.si_anderson_imp_ext_ni_h(
t, u, nocc, nimp, nsites)
hop, himp, hcs_imp, hds_imp, hcs_ext, hds_ext, hext = models_embed.si_anderson_imp_ext_h(
hall_sp, u, nocc, nimp, nsites)
#print hext
# single site Hubbard in DMET basis
#mu=u/2
#hlat,hall,hlat_sp,hall_sp,e0=models_embed.hubbard_imp_ext_ni_h(t,u,nocc,nimp,nsites)
#hop,himp,hcs_imp,hds_imp,hcs_ext,hds_ext,hext=models_embed.hubbard_imp_ext_h(hall_sp,u,nocc,nimp,nsites)
# g.s embedding basis
pemb, pcore, pvirt = embed.embed(hlat, nimp, nocc)
# perturbation operator
perturb = utils.zeros([nsites])
perturb[0] = 1.
p_coeffs = N.array([1])
perturb_dop = models.ContractedD(N.array([0]), p_coeffs)
perturb_cop = models.ContractedC(N.array([0]), p_coeffs)
fd = file(
"siam_emb_dos_nsites" + str(nsites) + '_U' + str(u) + '_nimp' +
str(nimp) + '_eta' + str(delta), "w")
for omega in N.arange(-1., 1., 0.01):
ops_dict = response_embed.mb_sp_ops(hlat, perturb,
omega + 1j * delta, nimp, nocc,
pemb, pcore, pvirt)
configs_dict = response_embed.mb_configs(nsites, nimp, nimp_sp,
2 * nocc - nimp_sp, 0)
neutral_ops_configs, plus_ops_configs, minus_ops_configs = response_embed.mb_sp_ops_configs(
ops_dict, configs_dict)
#print minus_ops_configs
# basis is setup, now build matrix representations
perturb_dop_mat = opbasis_ni.oimp_matrix_bra_ket_form(
perturb_dop, minus_ops_configs, neutral_ops_configs, cocc,
vocc)
perturb_cop_mat = opbasis_ni.oimp_matrix_bra_ket_form(
perturb_cop, plus_ops_configs, neutral_ops_configs, cocc, vocc)
#print plus_ops_configs
# h, N-1 configs
#print minus_ops_configs
himp_mat_minus = opbasis_ni.oimp_matrix_form(
himp, minus_ops_configs, cocc, vocc)
himp_ext_mat_minus = opbasis_ni.himp_ext_matrix_form(
hcs_imp, hds_ext, hds_imp, hcs_ext, minus_ops_configs, cocc,
vocc)
#print himp_ext_mat_minus
hext_mat_minus = opbasis_ni.hext_matrix_form(
hext, minus_ops_configs, cocc, vocc, e0)
hmat_minus = himp_mat_minus + himp_ext_mat_minus + hext_mat_minus
unit_mat_minus = opbasis_ni.oimp_matrix_form(
models.Unit(), minus_ops_configs, cocc, vocc)
#print hext_mat_minus
# h, N+1 configs
himp_mat_plus = opbasis_ni.oimp_matrix_form(
himp, plus_ops_configs, cocc, vocc)
himp_ext_mat_plus = opbasis_ni.himp_ext_matrix_form(
hcs_imp, hds_ext, hds_imp, hcs_ext, plus_ops_configs, cocc,
vocc)
hext_mat_plus = opbasis_ni.hext_matrix_form(
hext, plus_ops_configs, cocc, vocc, e0)
hmat_plus = himp_mat_plus + himp_ext_mat_plus + hext_mat_plus
unit_mat_plus = opbasis_ni.oimp_matrix_form(
models.Unit(), plus_ops_configs, cocc, vocc)
# h, neutral configs
himp_mat = opbasis_ni.oimp_matrix_form(himp, neutral_ops_configs,
cocc, vocc)
himp_ext_mat = opbasis_ni.himp_ext_matrix_form(
hcs_imp, hds_ext, hds_imp, hcs_ext, neutral_ops_configs, cocc,
vocc)
hext_mat = opbasis_ni.hext_matrix_form(hext, neutral_ops_configs,
cocc, vocc, e0)
hmat = himp_mat + himp_ext_mat + hext_mat
unit_mat = opbasis_ni.oimp_matrix_form(models.Unit(),
neutral_ops_configs, cocc,
vocc)
# get neutral ground-state energy
es, cs = la.peigh(hmat, unit_mat)
e_gs = es[0]
psi0 = cs[:, 0]
#print e0
#print psi0
#print N.dot(perturb_dop_mat,psi0)
# all matrices setup, solve linear response (complex)
psi1_minus = la.solve_perturb(omega + 1j * delta,
unit_mat_minus,
hmat_minus,
e_gs,
psi0,
perturb_dop_mat,
project=False,
sign_ham=-1.)
gfminus = N.dot(psi1_minus, N.dot(perturb_dop_mat, psi0))
psi1_plus = la.solve_perturb(omega + 1j * delta,
unit_mat_plus,
hmat_plus,
e_gs,
psi0,
perturb_cop_mat,
project=False)
#print N.dot(perturb_cop_mat,psi0)
#print hmat
gfplus = N.dot(psi1_plus, N.dot(perturb_cop_mat, psi0))
d_gf, c_gf = _sp_gf(hlat, omega + 1j * delta, perturb, nocc)
#print omega, gfplus.imag,c_gf.imag, gfminus.imag,d_gf.imag
#print >>fd, omega-mu, -1.*(gfplus.imag),-1.*(c_gf.imag+d_gf.imag)
print >> fd, omega - mu, -(1 / N.pi) * (gfplus.imag + gfminus.imag)
def main():
# main loop for single-particle GF
utils.dtype = N.complex128
nimp = 1
nimp_sp = 2 * nimp
nsites = 300
nocc = nsites / 2
nsites_sp = nsites * 2
ndim = 2
sz = 0
# sites numbered in order as imp+bath ... ext
sites_imp = range(0, 2 * nimp_sp)
sites_ext = range(2 * nimp_sp, nsites_sp)
ncore = 2 * nocc - 2 * nimp
cocc = range(2 * nimp_sp, 2 * nimp_sp + ncore)
vocc = range(2 * nimp_sp + ncore, nsites_sp)
N.set_printoptions(precision=3)
#t=1. # hopping
gamma = 0.05
rho0 = (nsites - 2) / 2.
t = N.sqrt(gamma / (rho0 * N.pi))
#t = 0.1
#gamma = (t**2)*pi*0.1
delta = 1e-12 # broadening
#a=12.5
# for u in [0.0,4.0,10.0]:
for u in [10.]:
u *= gamma
# Single impurity Anderson model
mu = 0.
hlat, hall, hlat_sp, hall_sp, e0 = models_embed.si_anderson_imp_ext_ni_h(
t, u, nocc, nimp, nsites)
hop, himp, hcs_imp, hds_imp, hcs_ext, hds_ext, hext = models_embed.si_anderson_imp_ext_h(
hall_sp, u, nocc, nimp, nsites)
#print hext
# single site Hubbard in DMET basis
#mu=u/2
#hlat,hall,hlat_sp,hall_sp,e0=models_embed.hubbard_imp_ext_ni_h(t,u,nocc,nimp,nsites)
#hop,himp,hcs_imp,hds_imp,hcs_ext,hds_ext,hext=models_embed.hubbard_imp_ext_h(hall_sp,u,nocc,nimp,nsites)
# g.s embedding basis
pemb, pcore, pvirt = embed.embed(hlat, nimp, nocc)
# perturbation operator
perturb = utils.zeros([nsites])
perturb[0] = 1.
p_coeffs = N.array([1])
perturb_dop = models.ContractedD(N.array([0]), p_coeffs)
perturb_cop = models.ContractedC(N.array([0]), p_coeffs)
#fd=file("siam_emb_dos_nsites"+str(nsites)+'_U'+str(u)+'_nimp'+str(nimp)+'_eta'+str(delta),"w")
fdEnminus = file("En_minus", "w")
fdCnminus = file("Cn_minus", "w")
fdEnplus = file("En_plus", "w")
fdCnplus = file("Cn_plus", "w")
for omega in [1e-08]: #N.arange(-1.,1.,0.1):
#print omega
ops_dict = response_embed.mb_sp_ops(hlat, perturb,
omega + 1j * delta, nimp, nocc,
pemb, pcore, pvirt)
configs_dict = response_embed.mb_configs(nsites, nimp, nimp_sp,
2 * nocc - nimp_sp, 0)
neutral_ops_configs, plus_ops_configs, minus_ops_configs = response_embed.mb_sp_ops_configs(
ops_dict, configs_dict)
#print _ops_configs
# basis is setup, now build matrix representations
perturb_dop_mat = opbasis_ni.oimp_matrix_bra_ket_form(
perturb_dop, minus_ops_configs, neutral_ops_configs, cocc,
vocc)
perturb_cop_mat = opbasis_ni.oimp_matrix_bra_ket_form(
perturb_cop, plus_ops_configs, neutral_ops_configs, cocc, vocc)
# h, N-1 configs
#print ops_dict['d'][0]
himp_mat_minus = opbasis_ni.oimp_matrix_form(
himp, minus_ops_configs, cocc, vocc)
himp_ext_mat_minus = opbasis_ni.himp_ext_matrix_form(
hcs_imp, hds_ext, hds_imp, hcs_ext, minus_ops_configs, cocc,
vocc)
#print himp_ext_mat_minus
hext_mat_minus = opbasis_ni.hext_matrix_form(
hext, minus_ops_configs, cocc, vocc, e0)
hmat_minus = himp_mat_minus + himp_ext_mat_minus + hext_mat_minus
unit_mat_minus = opbasis_ni.oimp_matrix_form(
models.Unit(), minus_ops_configs, cocc, vocc)
#print hext_mat_minus
# h, N+1 configs
himp_mat_plus = opbasis_ni.oimp_matrix_form(
himp, plus_ops_configs, cocc, vocc)
himp_ext_mat_plus = opbasis_ni.himp_ext_matrix_form(
hcs_imp, hds_ext, hds_imp, hcs_ext, plus_ops_configs, cocc,
vocc)
hext_mat_plus = opbasis_ni.hext_matrix_form(
hext, plus_ops_configs, cocc, vocc, e0)
hmat_plus = himp_mat_plus + himp_ext_mat_plus + hext_mat_plus
unit_mat_plus = opbasis_ni.oimp_matrix_form(
models.Unit(), plus_ops_configs, cocc, vocc)
# h, neutral configs
himp_mat = opbasis_ni.oimp_matrix_form(himp, neutral_ops_configs,
cocc, vocc)
himp_ext_mat = opbasis_ni.himp_ext_matrix_form(
hcs_imp, hds_ext, hds_imp, hcs_ext, neutral_ops_configs, cocc,
vocc)
hext_mat = opbasis_ni.hext_matrix_form(hext, neutral_ops_configs,
cocc, vocc, e0)
hmat = himp_mat + himp_ext_mat + hext_mat
unit_mat = opbasis_ni.oimp_matrix_form(models.Unit(),
neutral_ops_configs, cocc,
vocc)
#print hext_mat_minus
# get neutral ground-state energy
es, cs = la.peigh(hmat, unit_mat)
e_gs = es[0]
psi0 = cs[:, 0]
es_minus, cs_minus = la.peigh(hmat_minus, unit_mat_minus)
Enarray = es_minus - e_gs * N.ones(len(es_minus))
print >> fdEnminus, omega, str(list(Enarray))[1:-1].translate(
None, ',')
es_plus, cs_plus = la.peigh(hmat_plus, unit_mat_plus)
Enarray = es_plus - e_gs * N.ones(len(es_minus))
print >> fdEnplus, omega, str(list(Enarray))[1:-1].translate(
None, ',')
gfminus = 0.
mat_el_list_minus = []
for i in xrange(len(es_minus)):
ket = N.dot(perturb_dop_mat, psi0)
bra = N.transpose(cs_minus[:, i].conjugate())
mat_el = N.dot(bra, ket)
mat_el *= mat_el.conjugate()
mat_el_list_minus.append(mat_el.real)
den = omega + 1j * delta - e_gs + es_minus[i]
gfminus += (mat_el / den)
print >> fdCnminus, omega, str(mat_el_list_minus)[1:-1].translate(
None, ',')
gfplus = 0.
mat_el_list_plus = []
for i in xrange(len(es_plus)):
ket = N.dot(perturb_cop_mat, psi0)
bra = N.transpose(cs_plus[:, i].conjugate())
mat_el = N.dot(bra, ket)
mat_el *= mat_el.conjugate()
mat_el_list_plus.append(mat_el.real)
den = omega + 1j * delta + e_gs - es_plus[i]
gfplus += (mat_el / den)
print >> fdCnplus, omega, str(mat_el_list_plus)[1:-1].translate(
None, ',')