def get_local_density_of_states(self,projected=False,width=0.05,window=None,npts=501):
"""
Return state density for all atoms as a function of energy.
parameters:
===========
projected: return local density of states projected for
angular momenta 0,1 and 2 (s,p and d)
( sum of pldos over angular momenta = ldos )
width: energy broadening (in eV)
window: energy window around Fermi-energy; 2-tuple (eV)
npts: number of grid points for energy
return: projected==False:
energy grid, ldos[atom,grid]
projected==True:
energy grid,
ldos[atom, grid],
pldos[atom, angmom, grid]
"""
mn, mx = self.emin, self.emax
if window is not None:
mn, mx = window
# only states within window
kl,al,el = [],[],[]
for k in range(self.nk):
for a in range(self.norb):
if mn<=self.e[k,a]<=mx:
kl.append(k)
al.append(a)
el.append(self.e[k,a])
ldos = np.zeros((self.N,npts))
if projected:
pldos = np.zeros((self.N,3,npts))
for i in range(self.N):
q = [ self.get_atom_wf_mulliken(i,k,a,wk=True) for k,a in zip(kl,al) ]
egrid, ldos[i,:] = mix.broaden( el,q,width=width,N=npts,a=mn,b=mx )
if projected:
q = np.array( [self.get_atom_wf_all_angmom_mulliken(i,k,a,wk=True) for k,a in zip(kl,al)] )
for l in range(3):
egrid, pldos[i,l] = mix.broaden( el,q[:,l],width=width,N=npts,a=mn,b=mx )
if projected:
assert np.all( abs(ldos-pldos.sum(axis=1))<1E-6 )
return egrid, ldos, pldos
else:
return egrid, ldos
def plot_spectrum(self, filename, width=0.2, xlim=None):
""" Make pretty plot of the linear response.
Parameters:
===========
filename: output file name (&format, supported by matplotlib)
width: width of Lorenzian broadening
xlim: energy range for plotting tuple (emin,emax)
"""
import pylab as pl
if not self.done:
self.run()
e, f = mix.broaden(self.omega * Hartree,
self.F,
width=width,
N=1000,
extend=True)
f = f / max(abs(f))
pl.plot(e, f, lw=2)
xs, ys = pl.poly_between(e, 0, f)
pl.fill(xs, ys, fc='b', ec='b', alpha=0.5)
pl.ylim(0, 1.2)
if xlim == None:
pl.xlim(0, self.emax * Hartree * 1.2)
else:
pl.xlim(xlim)
pl.xlabel('energy (eV)')
pl.ylabel('linear optical response')
pl.title('Optical response')
pl.savefig(filename)
#pl.show()
pl.close()
def plot_spectrum(self, filename, width=0.2, xlim=None):
""" Make pretty plot of the linear response.
Parameters:
===========
filename: output file name (&format, supported by matplotlib)
width: width of Lorenzian broadening
xlim: energy range for plotting tuple (emin,emax)
"""
import pylab as pl
if not self.done:
self.run()
e, f = mix.broaden(self.omega * Hartree, self.F, width=width, N=1000, extend=True)
f = f / max(abs(f))
pl.plot(e, f, lw=2)
xs, ys = pl.poly_between(e, 0, f)
pl.fill(xs, ys, fc="b", ec="b", alpha=0.5)
pl.ylim(0, 1.2)
if xlim == None:
pl.xlim(0, self.emax * Hartree * 1.2)
else:
pl.xlim(xlim)
pl.xlabel("energy (eV)")
pl.ylabel("linear optical response")
pl.title("Optical response")
pl.savefig(filename)
# pl.show()
pl.close()
def get_density_of_states(self,
broaden=False,
projected=False,
occu=False,
width=0.05,
window=None,
npts=501):
"""
Return the full density of states.
Sum of states over k-points. Zero is the Fermi-level.
Spin-degeneracy is NOT counted.
parameters:
===========
broaden: * If True, return broadened DOS in regular grid
in given energy window.
* If False, return energies of all states, followed
by their k-point weights.
projected: project DOS for angular momenta
occu: for not broadened case, return also state occupations
width: Gaussian broadening (eV)
window: energy window around Fermi-energy; 2-tuple (eV)
npts: number of data points in output
return: * if projected: e[:],dos[:],pdos[l,:] (angmom l=0,1,2)
* if not projected: e[:],dos[:]
* if broaden: e[:] is on regular grid, otherwise e[:] are
eigenvalues and dos[...] corresponding weights
* if occu: e[:],dos[:],occu[:]
"""
if broaden and occu or projected and occu:
raise AssertionError(
'Occupation numbers cannot be returned with broadened DOS.')
if projected:
e, dos, pdos = self.get_local_density_of_states(
True, width, window, npts)
return e, dos.sum(axis=0), pdos.sum(axis=0)
mn, mx = self.emin, self.emax
if window is not None:
mn, mx = window
x, y, f = [], [], []
for a in range(self.calc.el.norb):
x = np.concatenate((x, self.e[:, a]))
y = np.concatenate((y, self.calc.st.wk))
f = np.concatenate((f, self.calc.st.f[:, a]))
x = np.array(x)
y = np.array(y)
f = np.array(f)
if broaden:
e, dos = mix.broaden(x, y, width=width, N=npts, a=mn, b=mx)
else:
e, dos = x, y
if occu:
return e, dos, f
else:
return e, dos
def get_density_of_states(self,broaden=False,projected=False,occu=False,width=0.05,window=None,npts=501):
"""
Return the full density of states.
Sum of states over k-points. Zero is the Fermi-level.
Spin-degeneracy is NOT counted.
parameters:
===========
broaden: * If True, return broadened DOS in regular grid
in given energy window.
* If False, return energies of all states, followed
by their k-point weights.
projected: project DOS for angular momenta
occu: for not broadened case, return also state occupations
width: Gaussian broadening (eV)
window: energy window around Fermi-energy; 2-tuple (eV)
npts: number of data points in output
return: * if projected: e[:],dos[:],pdos[l,:] (angmom l=0,1,2)
* if not projected: e[:],dos[:]
* if broaden: e[:] is on regular grid, otherwise e[:] are
eigenvalues and dos[...] corresponding weights
* if occu: e[:],dos[:],occu[:]
"""
if broaden and occu or projected and occu:
raise AssertionError('Occupation numbers cannot be returned with broadened DOS.')
if projected:
e,dos,pdos = self.get_local_density_of_states(True,width,window,npts)
return e,dos.sum(axis=0),pdos.sum(axis=0)
mn, mx = self.emin, self.emax
if window is not None:
mn, mx = window
x, y, f = [],[],[]
for a in range(self.calc.el.norb):
x = np.concatenate( (x,self.e[:,a]) )
y = np.concatenate( (y,self.calc.st.wk) )
f = np.concatenate( (f,self.calc.st.f[:,a]) )
x=np.array(x)
y=np.array(y)
f=np.array(f)
if broaden:
e,dos = mix.broaden(x, y, width=width, N=npts, a=mn, b=mx)
else:
e,dos = x,y
if occu:
return e,dos,f
else:
return e,dos
def get_dielectric_function(self,width=0.05,cutoff=None,N=400):
"""
Return the imaginary part of the dielectric function for non-SCC.
Note: Uses approximation that requires that the orientation of
neighboring unit cells does not change much.
(Exact for Bravais lattice.)
See, e.g., Marder, Condensed Matter Physics, or
Popov New J. Phys 6, 17 (2004)
parameters:
-----------
width: energy broadening in eV
cutoff: cutoff energy in eV
N: number of points in energy grid
return:
-------
e[:], d[:,0:2]
"""
self.start_timing('dielectric function')
width = width/Hartree
otol = 0.05 # tolerance for occupations
if cutoff==None:
cutoff = 1E10
else:
cutoff = cutoff/Hartree
st = self.st
nk, e, f, wk = st.nk, st.e, st.f, st.wk
ex, wt = [], []
for k in range(nk):
wf = st.wf[k]
wfc = wf.conjugate()
dS = st.dS[k].transpose((0,2,1))
ek = e[k]
fk = f[k]
kweight = wk[k]
# electron excitation ka-->kb; restrict the search:
bmin = list(fk<2-otol).index(True)
amin = list(ek>ek[bmin]-cutoff).index(True)
amax = list(fk<otol).index(True)
for a in range(amin,amax+1):
bmax = list(ek>ek[a]+cutoff).index(True)
for b in range(max(a+1,bmin),bmax+1):
de = ek[b]-ek[a]
df = fk[a]-fk[b]
if df<otol:
continue
# P = < ka | P | kb >
P = 1j*hbar*np.dot(wfc[a],np.dot(dS,wf[b]))
ex.append( de )
wt.append( kweight*df*np.abs(P)**2 )
ex, wt = np.array(ex), np.array(wt)
cutoff = min( ex.max(),cutoff )
y = np.zeros((N,3))
for d in range(3):
# Lorenzian should be used, but long tail would bring divergence at zero energy
x,y[:,d] = broaden( ex,wt[:,d],width,'gaussian',N=N,a=width,b=cutoff )
y[:,d] = y[:,d]/x**2
const = (4*np.pi**2/hbar)
self.stop_timing('dielectric function')
return x*Hartree, y*const #y also in eV, Ang
from ase import *
from pylab import *
from hotbit import *
from box import mix
from hotbit.analysis import LinearResponse
# equilibrium length
Na2 = Atoms('Na2', [(0, 0, 0), (3.0, 0, 0)], cell=(6, 4, 4), pbc=False)
Na2.center()
calc = Hotbit(parameters=testpar, SCC=True, txt='optical.cal')
Na2.set_calculator(calc)
calc.solve_ground_state(Na2)
lr = LinearResponse(calc, energy_cut=10, txt='lr.out')
lr.run()
lr.info()
#get excitation energies and oscillator strengths
omega, F = lr.get_linear_response()
e, f = mix.broaden(omega, F, width=0.1, function='lorentzian')
plot(e, f)
savefig('lr.pdf')
from ase.units import Hartree
Na3 = Atoms('Na3',
positions=[(1.69649997, 0, 0), (-1.69649997, 0, 0),
(0, 2.9384241, 0)],
cell=(50, 50, 50),
pbc=False)
tm = mix.Timer()
calc = Hotbit(charge=1, SCC=True, txt='test_lr.cal', **default_param)
Na3.set_calculator(calc)
calc.solve_ground_state(Na3)
lr = LinearResponse(calc)
omega, F = lr.get_linear_response()
e, f = mix.broaden(omega, F, width=0.1)
if plot:
pl.scatter(e2, f2)
pl.plot(e, f, label='python')
pl.legend()
pl.show()
calc = Hotbit(SCC=True, txt='test_lr.cal', **default_param)
C60 = Atoms(read('C60.xyz'))
C60.set_calculator(calc)
calc.solve_ground_state(C60)
lr = LinearResponse(calc, energy_cut=10 / Hartree)
omega, F = lr.get_linear_response()
e, f = mix.broaden(omega, F, width=0.1)
import numpy as np
from box import mix
from ase.units import Hartree
Na3=Atoms('Na3',positions=[(1.69649997,0,0),(-1.69649997,0,0),(0,2.9384241,0)],cell=(50,50,50),pbc=False)
tm=mix.Timer()
calc=Hotbit(charge=1,SCC=True,txt='test_lr.cal',**default_param)
Na3.set_calculator(calc)
calc.solve_ground_state(Na3)
lr=LinearResponse(calc)
omega,F=lr.get_linear_response()
e,f=mix.broaden(omega,F,width=0.1)
if plot:
pl.scatter(e2,f2)
pl.plot(e,f,label='python')
pl.legend()
pl.show()
calc=Hotbit(SCC=True,txt='test_lr.cal',**default_param)
C60=Atoms(read('C60.xyz'))
C60.set_calculator(calc)
calc.solve_ground_state(C60)
lr=LinearResponse(calc,energy_cut=10/Hartree)
omega,F=lr.get_linear_response()
e,f=mix.broaden(omega,F,width=0.1)
def get_covalent_energy(self,mode='default',i=None,j=None,width=None,window=None,npts=501):
"""
Return covalent bond energies in different modes. (eV)
ecov is described in
Bornsen, Meyer, Grotheer, Fahnle, J. Phys.:Condens. Matter 11, L287 (1999) and
Koskinen, Makinen Comput. Mat. Sci. 47, 237 (2009)
parameters:
===========
mode: 'default' total covalent energy
'orbitals' covalent energy for orbital pairs
'atoms' covalent energy for atom pairs
'angmom' covalent energy for angular momentum components
i,j: atom or orbital indices, or angular momentum pairs
width: * energy broadening (in eV) for ecov
* if None, return energy eigenvalues and corresponding
covalent energies in arrays, directly
window: energy window (in eV wrt Fermi-level) for broadened ecov
npts: number of points in energy grid (only with broadening)
return:
=======
x,y: * if width==None, x is list of energy eigenvalues (including k-points)
and y covalent energies of those eigenstates
* if width!=None, x is energy grid for ecov.
* energies (both energy grid and ecov) are in eV.
Note: energies are always shifted so that Fermi-level is at zero.
Occupations are not otherwise take into account (while k-point weights are)
"""
eps = 1E-6
x, y = [], []
wf = self.st.wf
energy = self.st.e - self.st.occu.get_mu()
if window==None:
mn,mx = energy.flatten().min()-eps, energy.flatten().max()+eps
else:
mn,mx = window[0]/Hartree, window[1]/Hartree
if mode=='angmom':
lorbs = [[],[],[]]
for m,orb in enumerate(self.calc.el.orbitals()):
lorbs[orb['angmom']].append(m)
oi, oj = np.array(lorbs[i]), np.array(lorbs[j])
elif mode=='atoms':
o1i, noi = self.calc.el.get_property_lists(['o1','no'])[i]
o1j, noj = self.calc.el.get_property_lists(['o1','no'])[j]
for k,wk in enumerate(self.wk):
for a in range(self.norb):
if not mn<=energy[k,a]<=mx:
continue
x.append( energy[k,a] )
if mode == 'default':
e = 0.0
for m in range(self.norb):
e += wk*np.sum( (wf[k,a,m]*wf[k,a,:].conj()*self.HS[k,:,m]) )
y.append(e)
elif mode == 'orbitals':
if i!=j:
y.append( wk*2*(wf[k,a,i]*wf[k,a,j].conj()*self.HS[k,j,i]).real )
else:
y.append( wk*wf[k,a,i]*wf[k,a,j].conj()*self.HS[k,j,i])
elif mode == 'atoms':
e = 0.0
if i!=j:
for m in range(o1i,o1i+noi):
for n in range(o1j,o1j+noj):
e += wk*2*(wf[k,a,m]*wf[k,a,n].conj()*self.HS[k,n,m]).real
else:
for m in range(o1i,o1i+noi):
for n in range(o1j,o1j+noj):
e += wk*(wf[k,a,m]*wf[k,a,n].conj()*self.HS[k,n,m])
y.append(e)
elif mode == 'angmom':
e = 0.0
for m in lorbs[i]:
e += wk*np.sum( wf[k,a,m]*wf[k,a,oj].conj()*self.HS[k,oj,m] )
if i!=j:
e += e.conj()
y.append(e)
else:
raise NotImplementedError('Unknown covalent energy mode "%s".' %mode)
x,y = np.array(x), np.array(y)
assert np.all( abs(y.imag)<1E-12 )
y=y.real
if width==None:
return x * Hartree, y * Hartree
else:
e,ecov = mix.broaden(x, y, width=width/Hartree, N=npts, a=mn, b=mx)
return e * Hartree, ecov * Hartree
from ase import *
from pylab import *
from hotbit import *
from box import mix
from hotbit.analysis import LinearResponse
# equilibrium length
Na2=Atoms('Na2',[(0,0,0),(3.0,0,0)],cell=(6,4,4),pbc=False)
Na2.center()
calc=Hotbit(parameters=testpar,SCC=True,txt='optical.cal')
Na2.set_calculator(calc)
calc.solve_ground_state(Na2)
lr=LinearResponse(calc,energy_cut=10,txt='lr.out')
lr.run()
lr.info()
#get excitation energies and oscillator strengths
omega, F=lr.get_linear_response()
e,f=mix.broaden(omega,F,width=0.1,function='lorentzian')
plot(e,f)
savefig('lr.pdf')
def get_dielectric_function(self, width=0.05, cutoff=None, N=400):
"""
Return the imaginary part of the dielectric function for non-SCC.
Note: Uses approximation that requires that the orientation of
neighboring unit cells does not change much.
(Exact for Bravais lattice.)
See, e.g., Marder, Condensed Matter Physics, or
Popov New J. Phys 6, 17 (2004)
parameters:
-----------
width: energy broadening in eV
cutoff: cutoff energy in eV
N: number of points in energy grid
return:
-------
e[:], d[:,0:2]
"""
self.start_timing("dielectric function")
width = width / Hartree
otol = 0.05 # tolerance for occupations
if cutoff == None:
cutoff = 1e10
else:
cutoff = cutoff / Hartree
st = self.st
nk, e, f, wk = st.nk, st.e, st.f, st.wk
ex, wt = [], []
for k in range(nk):
wf = st.wf[k]
wfc = wf.conjugate()
dS = st.dS[k].transpose((0, 2, 1))
ek = e[k]
fk = f[k]
kweight = wk[k]
# electron excitation ka-->kb; restrict the search:
bmin = list(fk < 2 - otol).index(True)
amin = list(ek > ek[bmin] - cutoff).index(True)
amax = list(fk < otol).index(True)
for a in xrange(amin, amax + 1):
bmax = list(ek > ek[a] + cutoff).index(True)
for b in range(max(a + 1, bmin), bmax + 1):
de = ek[b] - ek[a]
df = fk[a] - fk[b]
if df < otol:
continue
# P = < ka | P | kb >
P = 1j * hbar * np.dot(wfc[a], np.dot(dS, wf[b]))
ex.append(de)
wt.append(kweight * df * np.abs(P) ** 2)
ex, wt = np.array(ex), np.array(wt)
cutoff = min(ex.max(), cutoff)
y = np.zeros((N, 3))
for d in range(3):
# Lorenzian should be used, but long tail would bring divergence at zero energy
x, y[:, d] = broaden(ex, wt[:, d], width, "gaussian", N=N, a=width, b=cutoff)
y[:, d] = y[:, d] / x ** 2
const = 4 * np.pi ** 2 / hbar
self.stop_timing("dielectric function")
return x * Hartree, y * const # y also in eV, Ang
def get_covalent_energy(self,
mode='default',
i=None,
j=None,
width=None,
window=None,
npts=501):
"""
Return covalent bond energies in different modes. (eV)
ecov is described in
Bornsen, Meyer, Grotheer, Fahnle, J. Phys.:Condens. Matter 11, L287 (1999) and
Koskinen, Makinen Comput. Mat. Sci. 47, 237 (2009)
parameters:
===========
mode: 'default' total covalent energy
'orbitals' covalent energy for orbital pairs
'atoms' covalent energy for atom pairs
'angmom' covalent energy for angular momentum components
i,j: atom or orbital indices, or angular momentum pairs
width: * energy broadening (in eV) for ecov
* if None, return energy eigenvalues and corresponding
covalent energies in arrays, directly
window: energy window (in eV wrt Fermi-level) for broadened ecov
npts: number of points in energy grid (only with broadening)
return:
=======
x,y: * if width==None, x is list of energy eigenvalues (including k-points)
and y covalent energies of those eigenstates
* if width!=None, x is energy grid for ecov.
* energies (both energy grid and ecov) are in eV.
Note: energies are always shifted so that Fermi-level is at zero.
Occupations are not otherwise take into account (while k-point weights are)
"""
eps = 1E-6
x, y = [], []
wf = self.st.wf
energy = self.st.e - self.st.occu.get_mu()
if window == None:
mn, mx = energy.flatten().min() - eps, energy.flatten().max() + eps
else:
mn, mx = window[0] / Hartree, window[1] / Hartree
if mode == 'angmom':
lorbs = [[], [], []]
for m, orb in enumerate(self.calc.el.orbitals()):
lorbs[orb['angmom']].append(m)
oi, oj = np.array(lorbs[i]), np.array(lorbs[j])
elif mode == 'atoms':
o1i, noi = self.calc.el.get_property_lists(['o1', 'no'])[i]
o1j, noj = self.calc.el.get_property_lists(['o1', 'no'])[j]
for k, wk in enumerate(self.wk):
for a in range(self.norb):
if not mn <= energy[k, a] <= mx:
continue
x.append(energy[k, a])
if mode == 'default':
e = 0.0
for m in range(self.norb):
e += wk * np.sum((wf[k, a, m] * wf[k, a, :].conj() *
self.HS[k, :, m]))
y.append(e)
elif mode == 'orbitals':
if i != j:
y.append(wk * 2 * (wf[k, a, i] * wf[k, a, j].conj() *
self.HS[k, j, i]).real)
else:
y.append(wk * wf[k, a, i] * wf[k, a, j].conj() *
self.HS[k, j, i])
elif mode == 'atoms':
e = 0.0
if i != j:
for m in range(o1i, o1i + noi):
for n in range(o1j, o1j + noj):
e += wk * 2 * (wf[k, a, m] *
wf[k, a, n].conj() *
self.HS[k, n, m]).real
else:
for m in range(o1i, o1i + noi):
for n in range(o1j, o1j + noj):
e += wk * (wf[k, a, m] * wf[k, a, n].conj() *
self.HS[k, n, m])
y.append(e)
elif mode == 'angmom':
e = 0.0
for m in lorbs[i]:
e += wk * np.sum(wf[k, a, m] * wf[k, a, oj].conj() *
self.HS[k, oj, m])
if i != j:
e += e.conj()
y.append(e)
else:
raise NotImplementedError(
'Unknown covalent energy mode "%s".' % mode)
x, y = np.array(x), np.array(y)
assert np.all(abs(y.imag) < 1E-12)
y = y.real
if width == None:
return x * Hartree, y * Hartree
else:
e, ecov = mix.broaden(x,
y,
width=width / Hartree,
N=npts,
a=mn,
b=mx)
return e * Hartree, ecov * Hartree
def get_local_density_of_states(self,
projected=False,
width=0.05,
window=None,
npts=501):
"""
Return state density for all atoms as a function of energy.
parameters:
===========
projected: return local density of states projected for
angular momenta 0,1 and 2 (s,p and d)
( sum of pldos over angular momenta = ldos )
width: energy broadening (in eV)
window: energy window around Fermi-energy; 2-tuple (eV)
npts: number of grid points for energy
return: projected==False:
energy grid, ldos[atom,grid]
projected==True:
energy grid,
ldos[atom, grid],
pldos[atom, angmom, grid]
"""
mn, mx = self.emin, self.emax
if window is not None:
mn, mx = window
# only states within window
kl, al, el = [], [], []
for k in range(self.nk):
for a in range(self.norb):
if mn <= self.e[k, a] <= mx:
kl.append(k)
al.append(a)
el.append(self.e[k, a])
ldos = np.zeros((self.N, npts))
if projected:
pldos = np.zeros((self.N, 3, npts))
for i in range(self.N):
q = [
self.get_atom_wf_mulliken(i, k, a, wk=True)
for k, a in zip(kl, al)
]
egrid, ldos[i, :] = mix.broaden(el,
q,
width=width,
N=npts,
a=mn,
b=mx)
if projected:
q = np.array([
self.get_atom_wf_all_angmom_mulliken(i, k, a, wk=True)
for k, a in zip(kl, al)
])
for l in range(3):
egrid, pldos[i, l] = mix.broaden(el,
q[:, l],
width=width,
N=npts,
a=mn,
b=mx)
if projected:
assert np.all(abs(ldos - pldos.sum(axis=1)) < 1E-6)
return egrid, ldos, pldos
else:
return egrid, ldos
