pymatgen.electronic_structure.dos module

This module defines classes to represent the density of states, etc.

class CompleteDos(structure: Structure, total_dos: Dos, pdoss: Mapping[PeriodicSite, Mapping[Orbital, Mapping[Spin, _SupportsArray[dtype] | _NestedSequence[_SupportsArray[dtype]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes]]]], normalize: bool = False)

Bases: Dos

This wrapper class defines a total dos, and also provides a list of PDos. Mainly used by pymatgen.io.vasp.Vasprun to create a complete Dos from a vasprun.xml file. You are unlikely to try to generate this object manually.

structure

Structure associated with the CompleteDos.

pdos

Dict of partial densities of the form {Site:{Orbital:{Spin:Densities}}}

Parameters:

• structure – Structure associated with this particular DOS.

• total_dos – total Dos for structure

• pdoss – The pdoss are supplied as an {Site: {Orbital: {Spin:Densities}}}

• normalize – Whether to normalize the densities by the volume of the structure. If True, the units of the densities are states/eV/Angstrom^3. Otherwise, the units are states/eV.

as_dict() → dict

JSON-serializable dict representation of CompleteDos.

static fp_to_dict(fp: NamedTuple) → dict

Converts a fingerprint into a dictionary

Parameters:

fp – The DOS fingerprint to be converted into a dictionary

Returns:

A dict of the fingerprint Keys=type, Values=np.ndarray(energies, densities)

Return type:

dict

classmethod from_dict(d) → CompleteDos

Returns CompleteDos object from dict representation.

get_band_center(band: OrbitalType = OrbitalType.d, elements: list[SpeciesLike] | None = None, sites: list[PeriodicSite] | None = None, spin: Spin | None = None, erange: list[float] | None = None) → float

Computes the orbital-projected band center, defined as the first moment relative to the Fermi level

int_{-inf}^{+inf} rho(E)*E dE/int_{-inf}^{+inf} rho(E) dE

based on the work of Hammer and Norskov, Surf. Sci., 343 (1995) where the limits of the integration can be modified by erange and E is the set of energies taken with respect to the Fermi level. Note that the band center is often highly sensitive to the selected erange.

Parameters:

• band – Orbital type to get the band center of (default is d-band)

• elements – Elements to get the band center of (cannot be used in conjunction with site)

• sites – Sites to get the band center of (cannot be used in conjunction with el)

• spin – Spin channel to use. By default, the spin channels will be combined.

• erange – [min, max] energy range to consider, with respect to the Fermi level. Default is None, which means all energies are considered.

Returns:

band center in eV, often denoted epsilon_d for the d-band center

get_band_filling(band: OrbitalType = OrbitalType.d, elements: list[SpeciesLike] | None = None, sites: list[PeriodicSite] | None = None, spin: Spin | None = None) → float

Computes the orbital-projected band filling, defined as the zeroth moment up to the Fermi level

Parameters:

• band – Orbital type to get the band center of (default is d-band)

• elements – Elements to get the band center of (cannot be used in conjunction with site)

• sites – Sites to get the band center of (cannot be used in conjunction with el)

• spin – Spin channel to use. By default, the spin channels will be combined.

Returns:

band filling in eV, often denoted f_d for the d-band

get_band_kurtosis(band: OrbitalType = OrbitalType.d, elements: list[SpeciesLike] | None = None, sites: list[PeriodicSite] | None = None, spin: Spin | None = None, erange: list[float] | None = None) → float

Get the orbital-projected kurtosis, defined as the fourth standardized moment

int_{-inf}^{+inf} rho(E)*(E-E_center)^4 dE/int_{-inf}^{+inf} rho(E) dE) / (int_{-inf}^{+inf} rho(E)*(E-E_center)^2 dE/int_{-inf}^{+inf} rho(E) dE))^2

where E_center is the orbital-projected band center, the limits of the integration can be modified by erange, and E is the set of energies taken with respect to the Fermi level. Note that the skewness is often highly sensitive to the selected erange.

Parameters:

• band – Orbital type to get the band center of (default is d-band)

• elements – Elements to get the band center of (cannot be used in conjunction with site)

• sites – Sites to get the band center of (cannot be used in conjunction with el)

• spin – Spin channel to use. By default, the spin channels will be combined.

• erange – [min, max] energy range to consider, with respect to the Fermi level. Default is None, which means all energies are considered.

Returns:

Orbital-projected kurtosis in eV

get_band_skewness(band: OrbitalType = OrbitalType.d, elements: list[SpeciesLike] | None = None, sites: list[PeriodicSite] | None = None, spin: Spin | None = None, erange: list[float] | None = None) → float

Get the orbital-projected skewness, defined as the third standardized moment

int_{-inf}^{+inf} rho(E)*(E-E_center)^3 dE/int_{-inf}^{+inf} rho(E) dE) / (int_{-inf}^{+inf} rho(E)*(E-E_center)^2 dE/int_{-inf}^{+inf} rho(E) dE))^(3/2)

where E_center is the orbital-projected band center, the limits of the integration can be modified by erange, and E is the set of energies taken with respect to the Fermi level. Note that the skewness is often highly sensitive to the selected erange.

Parameters:

• band – Orbitals to get the band center of (default is d-band)

• elements – Elements to get the band center of (cannot be used in conjunction with site)

• sites – Sites to get the band center of (cannot be used in conjunction with el)

• spin – Spin channel to use. By default, the spin channels will be combined.

• erange – [min, max] energy range to consider, with respect to the Fermi level. Default is None, which means all energies are considered.

Returns:

Orbital-projected skewness in eV

get_band_width(band: OrbitalType = OrbitalType.d, elements: list[SpeciesLike] | None = None, sites: list[PeriodicSite] | None = None, spin: Spin | None = None, erange: list[float] | None = None) → float

Get the orbital-projected band width, defined as the square root of the second moment

sqrt(int_{-inf}^{+inf} rho(E)*(E-E_center)^2 dE/int_{-inf}^{+inf} rho(E) dE)

where E_center is the orbital-projected band center, the limits of the integration can be modified by erange, and E is the set of energies taken with respect to the Fermi level. Note that the band width is often highly sensitive to the selected erange.

Parameters:

• band – Orbital type to get the band center of (default is d-band)

• elements – Elements to get the band center of (cannot be used in conjunction with site)

• sites – Sites to get the band center of (cannot be used in conjunction with el)

• spin – Spin channel to use. By default, the spin channels will be combined.

• erange – [min, max] energy range to consider, with respect to the Fermi level. Default is None, which means all energies are considered.

Returns:

Orbital-projected band width in eV

get_dos_fp(type: str = 'summed_pdos', binning: bool = True, min_e: float | None = None, max_e: float | None = None, n_bins: int = 256, normalize: bool = True) → NamedTuple

Generates the DOS fingerprint based on work of F. Knoop, T. A. r Purcell, M. Scheffler, C. Carbogno, J. Open Source Softw. 2020, 5, 2671. Source - https://gitlab.com/vibes-developers/vibes/-/tree/master/vibes/materials_fp Copyright (c) 2020 Florian Knoop, Thomas A.R.Purcell, Matthias Scheffler, Christian Carbogno

Parameters:

• type (str) – Specify fingerprint type needed can accept ‘{s/p/d/f/}summed_{pdos/tdos}’

• summed_pdos) ((default is) –

• min_e (float) – The minimum mode energy to include in the fingerprint (default is None)

• max_e (float) – The maximum mode energy to include in the fingerprint (default is None)

• n_bins (int) – Number of bins to be used in the fingerprint (default is 256)

• normalize (bool) – If true, normalizes the area under fp to equal to 1 (default is True)

Raises:

ValueError – If type is not one of the accepted values {s/p/d/f/}summed_{pdos/tdos}.

Returns:

The electronic density of states fingerprint of format (energies, densities, type, n_bins)

Return type:

Fingerprint(namedtuple)

static get_dos_fp_similarity(fp1: NamedTuple, fp2: NamedTuple, col: int = 1, pt: int | str = 'All', normalize: bool = False, tanimoto: bool = False) → float

Calculates the similarity index (dot product) of two fingerprints

Parameters:

• fp1 (NamedTuple) – The 1st dos fingerprint object

• fp2 (NamedTuple) – The 2nd dos fingerprint object

• col (int) – The item in the fingerprints (0:energies,1: densities) to take the dot product of (default is 1)

• pt (int or str) – The index of the point that the dot product is to be taken (default is All)

• normalize (bool) – If True normalize the scalar product to 1 (default is False)

• tanimoto (bool) – If True will compute Tanimoto index (default is False)

Raises:

ValueError – If both tanimoto and normalize are set to True.

Returns: Similarity index (float): The value of dot product

get_element_dos() → dict[Union[str, pymatgen.core.periodic_table.Element, pymatgen.core.periodic_table.Species, pymatgen.core.periodic_table.DummySpecies], pymatgen.electronic_structure.dos.Dos]

Get element projected Dos.

Returns:

Dos}

Return type:

dict of {Element

get_element_spd_dos(el: str | Element | Species | DummySpecies) → dict[pymatgen.electronic_structure.core.OrbitalType, pymatgen.electronic_structure.dos.Dos]

Get element and spd projected Dos

Parameters:

el – Element in Structure.composition associated with CompleteDos

Returns:

Dos}, e.g. {OrbitalType.s: Dos object, …}

Return type:

dict of {OrbitalType

get_hilbert_transform(band: OrbitalType = OrbitalType.d, elements: list[SpeciesLike] | None = None, sites: list[PeriodicSite] | None = None) → Dos

Returns the Hilbert transform of the orbital-projected density of states, often plotted for a Newns-Anderson analysis.

Parameters:

• elements – Elements to get the band center of (cannot be used in conjunction with site)

• sites – Sites to get the band center of (cannot be used in conjunction with el)

• band – Orbitals to get the band center of (default is d-band)

Returns:

Hilbert transformation of the projected DOS.

get_n_moment(n: int, band: OrbitalType = OrbitalType.d, elements: list[SpeciesLike] | None = None, sites: list[PeriodicSite] | None = None, spin: Spin | None = None, erange: list[float] | None = None, center: bool = True) → float

Get the nth moment of the DOS centered around the orbital-projected band center, defined as

int_{-inf}^{+inf} rho(E)*(E-E_center)^n dE/int_{-inf}^{+inf} rho(E) dE

where n is the order, E_center is the orbital-projected band center, the limits of the integration can be modified by erange, and E is the set of energies taken with respect to the Fermi level. If center is False, then the E_center reference is not used.

Parameters:

• n – The order for the moment

• band – Orbital type to get the band center of (default is d-band)

• elements – Elements to get the band center of (cannot be used in conjunction with site)

• sites – Sites to get the band center of (cannot be used in conjunction with el)

• spin – Spin channel to use. By default, the spin channels will be combined.

• erange – [min, max] energy range to consider, with respect to the Fermi level. Default is None, which means all energies are considered.

• center – Take moments with respect to the band center

Returns:

Orbital-projected nth moment in eV

get_site_dos(site: PeriodicSite) → Dos

Get the total Dos for a site (all orbitals).

Parameters:

site – Site in Structure associated with CompleteDos.

Returns:

Dos containing summed orbital densities for site.

get_site_orbital_dos(site: PeriodicSite, orbital: Orbital) → Dos

Get the Dos for a particular orbital of a particular site.

Parameters:

• site – Site in Structure associated with CompleteDos.

• orbital – Orbital in the site.

Returns:

Dos containing densities for orbital of site.

get_site_spd_dos(site: PeriodicSite) → dict[pymatgen.electronic_structure.core.OrbitalType, pymatgen.electronic_structure.dos.Dos]

Get orbital projected Dos of a particular site

Parameters:

site – Site in Structure associated with CompleteDos.

Returns:

Dos}, e.g. {OrbitalType.s: Dos object, …}

Return type:

dict of {OrbitalType

get_site_t2g_eg_resolved_dos(site: PeriodicSite) → dict[str, pymatgen.electronic_structure.dos.Dos]

Get the t2g, eg projected DOS for a particular site.

Parameters:

site – Site in Structure associated with CompleteDos.

Returns:

A dict {“e_g”: Dos, “t2g”: Dos} containing summed e_g and t2g DOS for the site.

Return type:

dict[str, Dos]

get_spd_dos() → dict[pymatgen.electronic_structure.core.OrbitalType, pymatgen.electronic_structure.dos.Dos]

Get orbital projected Dos.

Returns:

Dos}, e.g. {OrbitalType.s: Dos object, …}

Return type:

dict of {OrbitalType

get_upper_band_edge(band: OrbitalType = OrbitalType.d, elements: list[SpeciesLike] | None = None, sites: list[PeriodicSite] | None = None, spin: Spin | None = None, erange: list[float] | None = None) → float

Get the orbital-projected upper band edge. The definition by Xin et al. Phys. Rev. B, 89, 115114 (2014) is used, which is the highest peak position of the Hilbert transform of the orbital-projected DOS.

Parameters:

• band – Orbital type to get the band center of (default is d-band)

• elements – Elements to get the band center of (cannot be used in conjunction with site)

• sites – Sites to get the band center of (cannot be used in conjunction with el)

• spin – Spin channel to use. By default, the spin channels will be combined.

• erange – [min, max] energy range to consider, with respect to the Fermi level. Default is None, which means all energies are considered.

Returns:

Upper band edge in eV, often denoted epsilon_u

property spin_polarization: float | None

Calculates spin polarization at Fermi level. If the calculation is not spin-polarized, None will be returned.

See Sanvito et al., doi: 10.1126/sciadv.1602241 for an example usage.

Return (float):

spin polarization in range [0, 1],

will also return NaN if spin polarization ill-defined (e.g. for insulator)

class DOS(energies: _SupportsArray[dtype] | _NestedSequence[_SupportsArray[dtype]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes], densities: _SupportsArray[dtype] | _NestedSequence[_SupportsArray[dtype]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes], efermi: float)

Bases: Spectrum

Replacement basic DOS object. All other DOS objects are extended versions of this object. Work in progress.

Parameters:

• energies – A sequence of energies

• densities (ndarray) – Either a Nx1 or a Nx2 array. If former, it is interpreted as a Spin.up only density. Otherwise, the first column is interpreted as Spin.up and the other is Spin.down.

• efermi – Fermi level energy.

XLABEL = 'Energy'

YLABEL = 'Density'

get_cbm_vbm(tol: float = 0.001, abs_tol: bool = False, spin=None)

Expects a DOS object and finds the cbm and vbm.

Parameters:

• tol – tolerance in occupations for determining the gap

• abs_tol – An absolute tolerance (True) and a relative one (False)

• spin – Possible values are None - finds the gap in the summed densities, Up - finds the gap in the up spin channel, Down - finds the gap in the down spin channel.

Returns:

float in eV corresponding to the gap

Return type:

(cbm, vbm)

get_gap(tol: float = 0.001, abs_tol: bool = False, spin: Spin | None = None)

Expects a DOS object and finds the gap.

Parameters:

• tol – tolerance in occupations for determining the gap

• abs_tol – An absolute tolerance (True) and a relative one (False)

• spin – Possible values are None - finds the gap in the summed densities, Up - finds the gap in the up spin channel, Down - finds the gap in the down spin channel.

Returns:

gap in eV

get_interpolated_gap(tol: float = 0.001, abs_tol: bool = False, spin: Spin | None = None)

Expects a DOS object and finds the gap

Parameters:

• tol – tolerance in occupations for determining the gap

• abs_tol – Set to True for an absolute tolerance and False for a relative one.

• spin – Possible values are None - finds the gap in the summed densities, Up - finds the gap in the up spin channel, Down - finds the gap in the down spin channel.

Returns:

Tuple of floats in eV corresponding to the gap, cbm and vbm.

Return type:

(gap, cbm, vbm)

class Dos(efermi: float, energies: ArrayLike, densities: Mapping[Spin, ArrayLike], norm_vol: float | None = None)

Bases: MSONable

Basic DOS object. All other DOS objects are extended versions of this object.

Parameters:

• efermi – Fermi level energy

• energies – A sequences of energies

• (dict[Spin (densities) – np.array]): representing the density of states for each Spin.

• norm_vol – The volume used to normalize the densities. Defaults to 1 if None which will not perform any normalization. If not None, the resulting density will have units of states/eV/Angstrom^3, otherwise the density will be in states/eV.

as_dict() → dict

JSON-serializable dict representation of Dos.

classmethod from_dict(d) → Dos

Returns Dos object from dict representation of Dos.

get_cbm_vbm(tol: float = 0.001, abs_tol: bool = False, spin: Spin | None = None)

Expects a DOS object and finds the cbm and vbm.

Parameters:

• tol – tolerance in occupations for determining the gap

• abs_tol – An absolute tolerance (True) and a relative one (False)

• spin – Possible values are None - finds the gap in the summed densities, Up - finds the gap in the up spin channel, Down - finds the gap in the down spin channel.

Returns:

float in eV corresponding to the gap

Return type:

(cbm, vbm)

get_densities(spin: Spin | None = None)

Returns the density of states for a particular spin.

Parameters:

spin – Spin

Returns:

Returns the density of states for a particular spin. If Spin is None, the sum of all spins is returned.

get_gap(tol: float = 0.001, abs_tol: bool = False, spin: Spin | None = None)

Expects a DOS object and finds the gap.

Parameters:

• tol – tolerance in occupations for determining the gap

• abs_tol – An absolute tolerance (True) and a relative one (False)

• spin – Possible values are None - finds the gap in the summed densities, Up - finds the gap in the up spin channel, Down - finds the gap in the down spin channel.

Returns:

gap in eV

get_interpolated_gap(tol: float = 0.001, abs_tol: bool = False, spin: Spin | None = None)

Expects a DOS object and finds the gap

Parameters:

• tol – tolerance in occupations for determining the gap

• abs_tol – Set to True for an absolute tolerance and False for a relative one.

• spin – Possible values are None - finds the gap in the summed densities, Up - finds the gap in the up spin channel, Down - finds the gap in the down spin channel.

Returns:

Tuple of floats in eV corresponding to the gap, cbm and vbm.

Return type:

(gap, cbm, vbm)

get_interpolated_value(energy: float)

Returns interpolated density for a particular energy.

Parameters:

energy – Energy to return the density for.

get_smeared_densities(sigma: float)

Returns the Dict representation of the densities, {Spin: densities}, but with a Gaussian smearing of std dev sigma

Parameters:

sigma – Std dev of Gaussian smearing function.

Returns:

Dict of Gaussian-smeared densities.

class FermiDos(dos: Dos, structure: Structure | None = None, nelecs: float | None = None, bandgap: float | None = None)

Bases: Dos, MSONable

This wrapper class helps relate the density of states, doping levels (i.e. carrier concentrations) and corresponding fermi levels. A negative doping concentration indicates the majority carriers are electrons (n-type doping); a positive doping concentration indicates holes are the majority carriers (p-type doping).

Parameters:

• dos – Pymatgen Dos object.

• structure – A structure. If not provided, the structure of the dos object will be used. If the dos does not have an associated structure object, an error will be thrown.

• nelecs – The number of electrons included in the energy range of dos. It is used for normalizing the densities. Default is the total number of electrons in the structure.

• bandgap – If set, the energy values are scissored so that the electronic band gap matches this value.

as_dict() → dict

JSON-serializable dict representation of Dos.

classmethod from_dict(d) → FermiDos

Returns Dos object from dict representation of Dos.

get_doping(fermi_level: float, temperature: float) → float

Calculate the doping (majority carrier concentration) at a given fermi level and temperature. A simple Left Riemann sum is used for integrating the density of states over energy & equilibrium Fermi-Dirac distribution.

Parameters:

• fermi_level – The fermi_level level in eV.

• temperature – The temperature in Kelvin.

Returns:

The doping concentration in units of 1/cm^3. Negative values indicate that the majority carriers are electrons (n-type doping) whereas positive values indicates the majority carriers are holes (p-type doping).

get_fermi(concentration: float, temperature: float, rtol: float = 0.01, nstep: int = 50, step: float = 0.1, precision: int = 8) → float

Finds the fermi level at which the doping concentration at the given temperature (T) is equal to concentration. A greedy algorithm is used where the relative error is minimized by calculating the doping at a grid which continually becomes finer.

Parameters:

• concentration – The doping concentration in 1/cm^3. Negative values represent n-type doping and positive values represent p-type doping.

• temperature – The temperature in Kelvin.

• rtol – The maximum acceptable relative error.

• nstep – THe number of steps checked around a given fermi level.

• step – Initial step in energy when searching for the Fermi level.

• precision – Essentially the decimal places of calculated Fermi level.

Raises:

ValueError – If the Fermi level cannot be found.

Returns:

The fermi level in eV. Note that this is different from the default dos.efermi.

get_fermi_interextrapolated(concentration: float, temperature: float, warn: bool = True, c_ref: float = 10000000000.0, **kwargs) → float

Similar to get_fermi except that when get_fermi fails to converge, an interpolated or extrapolated fermi is returned with the assumption that the fermi level changes linearly with log(abs(concentration)).

Parameters:

• concentration – The doping concentration in 1/cm^3. Negative values represent n-type doping and positive values represent p-type doping.

• temperature – The temperature in Kelvin.

• warn – Whether to give a warning the first time the fermi cannot be found.

• c_ref – A doping concentration where get_fermi returns a value without error for both c_ref and -c_ref.

• **kwargs – Keyword arguments passed to the get_fermi function.

Returns:

The Fermi level. Note, the value is possibly interpolated or extrapolated and must be used with caution.

class LobsterCompleteDos(structure: Structure, total_dos: Dos, pdoss: Mapping[PeriodicSite, Mapping[Orbital, Mapping[Spin, _SupportsArray[dtype] | _NestedSequence[_SupportsArray[dtype]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes]]]], normalize: bool = False)

Bases: CompleteDos

Extended CompleteDOS for Lobster

Parameters:

• structure – Structure associated with this particular DOS.

• total_dos – total Dos for structure

• pdoss – The pdoss are supplied as an {Site: {Orbital: {Spin:Densities}}}

• normalize – Whether to normalize the densities by the volume of the structure. If True, the units of the densities are states/eV/Angstrom^3. Otherwise, the units are states/eV.

classmethod from_dict(d) → LobsterCompleteDos

Returns: CompleteDos object from dict representation.

get_element_spd_dos(el: str | Element | Species | DummySpecies) → dict[str, pymatgen.electronic_structure.dos.Dos]

Get element and spd projected Dos

Parameters:

el – Element in Structure.composition associated with LobsterCompleteDos

Returns:

densities, OrbitalType.p: densities, OrbitalType.d: densities}

Return type:

dict of {OrbitalType.s

get_site_orbital_dos(site: PeriodicSite, orbital: str) → Dos

Get the Dos for a particular orbital of a particular site.

Parameters:

• site – Site in Structure associated with CompleteDos.

• orbital – principal quantum number and orbital in string format, e.g. “4s”. possible orbitals are: “s”, “p_y”, “p_z”, “p_x”, “d_xy”, “d_yz”, “d_z^2”, “d_xz”, “d_x^2-y^2”, “f_y(3x^2-y^2)”, “f_xyz”, “f_yz^2”, “f_z^3”, “f_xz^2”, “f_z(x^2-y^2)”, “f_x(x^2-3y^2)” In contrast to the Cohpcar and the Cohplist objects, the strings from the Lobster files are used

Returns:

Dos containing densities of an orbital of a specific site.

get_site_t2g_eg_resolved_dos(site: PeriodicSite) → dict[str, pymatgen.electronic_structure.dos.Dos]

Get the t2g, eg projected DOS for a particular site. :param site: Site in Structure associated with CompleteDos.

Returns:

Dos, “t2g”: Dos} containing summed e_g and t2g DOS for the site.

Return type:

A dict {“e_g”

get_spd_dos() → dict[str, pymatgen.electronic_structure.dos.Dos]

Get orbital projected Dos. For example, if 3s and 4s are included in the basis of some element, they will be both summed in the orbital projected DOS

Returns:

Dos}, e.g. {“s”: Dos object, …}

Return type:

dict of {orbital

add_densities(density1: Mapping[Spin, _SupportsArray[dtype] | _NestedSequence[_SupportsArray[dtype]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes]], density2: Mapping[Spin, _SupportsArray[dtype] | _NestedSequence[_SupportsArray[dtype]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes]]) → dict[pymatgen.electronic_structure.core.Spin, numpy.ndarray]

Method to sum two densities.

Parameters:

• density1 – First density.

• density2 – Second density.

Returns:

dict[Spin, np.ndarray]

f0(E, fermi, T) → float

Returns the equilibrium fermi-dirac.

Parameters:

• E (float) – energy in eV

• fermi (float) – the fermi level in eV

• T (float) – the temperature in kelvin

Returns:

float