Occupation number smearing

See also

Convergence with respect to number of k-point for bulk Cu energy with different smearing methods:

fromase.buildimport bulk
fromgpawimport GPAW, PW
cu = bulk('Cu', 'fcc', a=3.6)
for smearing in [{'name': 'improved-tetrahedron-method'},
 {'name': 'tetrahedron-method'},
 {'name': 'fermi-dirac', 'width': 0.05},
 {'name': 'marzari-vanderbilt', 'width': 0.2}]:
 name = ''.join(word[0].upper() for word in smearing['name'].split('-'))
 width = smearing.get('width')
 if width:
 name += f'-{width}'
 for k in range(8, 21):
 cu.calc = GPAW(
 mode=PW(400),
 kpts=(k, k, k),
 occupations=smearing,
 txt=f'Cu-{name}-{k}.txt')
 e = cu.get_potential_energy()

../_images/cu.png

(made with cu_plot.py). See also figure 3 in Blöchl et. al.

gpaw.occupations.create_occ_calc(dct, *, parallel_layout=None, fixed_magmom_value=None, rcell=None, monkhorst_pack_size=None, bz2ibzmap=None, nspins=None, nelectrons=None, nkpts=None, nbands=None)[source] 

Surprise: Create occupation-number object.

The unit of width is eV and name must be one of:

‘fermi-dirac’
‘marzari-vanderbilt’
‘methfessel-paxton’
‘fixed’
‘tetrahedron-method’
‘improved-tetrahedron-method’
‘orbital-free’

>>> occ = create_occ_calc({'width': 0.0})
>>> occ.calculate(nelectrons=3,
...  eigenvalues=[[0, 1, 2], [0, 2, 3]],
...  weights=[1, 1])
(array([[1., 1., 0.],
 [1., 0., 0.]]), [1.5], 0.0)

gpaw.occupations.fermi_dirac(eig, fermi_level, width)[source] 

Fermi-Dirac distribution function.

>>> f, _, _ = fermi_dirac(np.array([-1.0, 0.0, 1.0]), 0.0, 0.01)
>>> f.round(5)
array([1. , 0.5, 0. ])

gpaw.occupations.marzari_vanderbilt(eig, fermi_level, width)[source] 

Marzari-Vanderbilt distribution (cold smearing).

See: doi: 10.1103/PhysRevLett.82.3296

gpaw.occupations.methfessel_paxton(eig, fermi_level, width, order=0)[source] : Methfessel-Paxton distribution.

classgpaw.occupations.OccupationNumberCalculator(parallel_layout=None)[source] 

Base class for all occupation number calculators.

Object for calculating fermi level(s) and occupation numbers.

If fixmagmom=True then the fixed_magmom_value attribute must be set and two fermi levels will be calculated.

calculate(nelectrons, eigenvalues, weights, spins=None, fermi_levels_guess=None, fix_fermi_level=False)[source] 

Calculate occupation numbers and fermi level(s) from eigenvalues.

nelectrons:: Number of electrons.
eigenvalues: ndarray, shape=(nibzkpts, nbands): Eigenvalues in Hartree.
weights: ndarray, shape=(nibzkpts,): Weights of k-points in IBZ (must sum to 1).
parallel:: Parallel distribution of eigenvalues.
fermi_level_guesses:: Optional guess(es) at fermi level(s).

Returns a tuple containing:

occupation numbers (in the range 0 to 1)
fermi-level in Hartree
entropy as -S*T in Hartree

>>> occ = ZeroWidth()
>>> occ.calculate(1, [[0, 1]], [1])
(array([[1., 0.]]), [0.5], 0.0)

classgpaw.occupations.FixedOccupationNumbers(numbers, parallel_layout=None)[source] 

Fixed occupation numbers.

f_sn: ndarray, shape=(nspins, nbands): Occupation numbers (in the range from 0 to 1)

Example (excited state with 4 electrons):

occ = FixedOccupationNumbers([[1, 0, 1, 0], [1, 1, 0, 0]])

classgpaw.occupations.ParallelLayout(bd, kpt_comm, domain_comm)[source] 

Collection of parallel stuff.

Create new instance of ParallelLayout(bd, kpt_comm, domain_comm)

gpaw.occupations.occupation_numbers(occ, eig_skn, weight_k, nelectrons)[source] 

Calculate occupation numbers from eigenvalues in eV (deprecated).

occ: dict: Example: {‘name’: ‘fermi-dirac’, ‘width’: 0.05} (width in eV).
eps_skn: ndarray, shape=(nspins, nibzkpts, nbands): Eigenvalues.
weight_k: ndarray, shape=(nibzkpts,): Weights of k-points in IBZ (must sum to 1).
nelectrons: int or float: Number of electrons.

Returns a tuple containing:

f_skn (sums to nelectrons)
fermi-level [Hartree]
magnetic moment
entropy as -S*T [Hartree]

Tetrahedron method

classgpaw.tetrahedron.TetrahedronMethod(rcell, size, improved=False, bz2ibzmap=None, parallel_layout=None)[source] 

Tetrahedron method for calculating occupation numbers.

The reciprocal cell, rcell, can be given in arbitrary units (only the shape matters) and size is the size of the Monkhorst-Pack grid. If k-points have been symmetry-reduced the bz2ibzmap parameter mapping BZ k-point indizes to IBZ k-point indices must be given.