Reference

AbstractElectronSystem <: AbstractMatterModel

Abstract base type for electron matter models.

Interface requirements

Concrete subtypes must implement the following functions:

1. Matter model interface:

• electron_density(sys::AbstractElectronSystem)::Real
• temperature(sys::AbstractElectronSystem)::Real

1. Electronic system interface:

• imag_dynamic_response(sys::AbstractElectronSystem, om_q::NTuple{2, Real})::Real
• real_dynamic_response(sys::AbstractElectronSystem, om_q::NTuple{2, Real})::Real

AbstractInteractingElectronSystem

Abstract base type for electronic system with interacting electrons by introducing screening potentials. Interface functions to be implemented are:

1. Matter model interface:

   • electron_density(::AbstractElectronSystem)::Real
   • temperature(::AbstractElectronSystem)::Real

2. Electronic system interface:

   • imag_dynamic_response(::AbstractElectronSystem, om_q::NTuple{2,Real})::Real
   • `realdynamicresponse(::AbstractElectronSystem, om_q::

3. Interacting electronic system interface:

   • proper_electron_system(::AbstractInteractingElectronSystem)::AbstractProperElectronSystem
   • screening(::AbstractInteractingElectronSystem)::AbstractScreening

AbstractMatterModel

Abstract base type representing a generic matter model.

Concrete subtypes should implement interface functions such as temperature and electron_density to provide physical properties of the matter system.

AbstractScreening

Abstract base type for screening models. Interface functions, which must be implemented:

• pseudo_potential(::AbstractInteractingElectronSystem)::AbstractPseudoPotential
• local_field_correction(::AbstractInteractingElectronSystem)::AbstractLocalFieldCorrection

CoulombPseudoPotential <: AbstractPseudoPotential

Represents the bare Coulomb pseudo potential: V(q) = e² / q².

GridLookupTable(
    cols::T1,
    rows::T2,
    values::T3
) where {T1<:AbstractVector{T},T2<:AbstractVector{T},T3<:AbstractMatrix{T}} where {T<:Real}

TBW

IdealElectronSystem{T, A} <: AbstractIdealElectronSystem

Represents a non-interacting (ideal) electron system, parameterized by a numerical type (T) and a response approximation model (A). The system may be used to model either a finite-temperature or an effective zero-temperature electron gas. The latter is obtained by choosing approx = ZeroTemperatureApprox().

Type Parameters

• T <: Real: Internal numeric type (e.g. Float64).
• A <: AbstractResponseApproximation: Approximation model used for the response function (e.g. NoApprox() or ZeroTemperatureApprox()).

Fields

• electron_density::T: Electron number density (stored in atomic units).
• temperature::T: Temperature (stored in atomic units).
• approx::A: Approximation model for the response function.

Constructors

IdealElectronSystem(electron_density::T1, temperature::T2, approx::A)

Construct an ideal electron system with a given electron density, temperature, and response approximation model.

IdealElectronSystem(electron_density::T1, approx::ZeroTemperatureApprox)

Construct an effectively zero-temperature system.

IdealElectronSystem(electron_density::T1, temperature::T2)

Construct an ideal electron system using the default response approximation (NoApprox()).

Examples

```julia using Unitful

Zero-temperature system

sys1 = IdealElectronSystem(1e23u"cm^(-3)", ZeroTemperatureApprox())

Finite-temperature system with explicit approximation

sys2 = IdealElectronSystem(1e23u"cm^(-3)", 300.0u"K", NonDegenerated())

Finite-temperature system with default approximation

sys3 = IdealElectronSystem(1e23u"cm^(-3)", 300.0u"K")

InteractingElectronSystem{PES, S} <: AbstractInteractingElectronSystem

Represents an interacting electron system consisting of a proper electron system (e.g., a free or ideal Fermi gas) and a screening model that modifies its response.

Type Parameters

• PES <: AbstractProperElectronSystem: Type of the proper (non-interacting) electron system.
• S <: AbstractScreening: Type of the screening model.

Fields

• proper_elec_system::PES: The underlying proper electron system.
• screening::S: The screening model used to compute interactions.

Constructors

InteractingElectronSystem(pelsys::PES, scr::S)

Construct an interacting electron system directly from a proper electron system and a screening model.

InteractingElectronSystem{PES}(electron_density::T1, temp::T2, scr::S)

Construct an interacting system by providing physical parameters, where PES is the desired proper electron system type.

Examples

iesys = InteractingElectronSystem{IdealFermiGas}(1e23, 300.0, Screening())

NoLocalFieldCorrection <: AbstractLocalFieldCorrection

Represents a model with no local field correction (i.e., the correction is zero).

NoScreening <: AbstractScreening

A screening model with no dielectric screening; dielectric function is unity, and corrections vanish.

Screening{PP, LFC} <: AbstractScreening

A general screening model that combines a pseudo potential and a local field correction.

Fields

• pseudo_potential: An instance of AbstractPseudoPotential.
• lfc: An instance of AbstractLocalFieldCorrection.

dielectric_function(::AbstractIdealElectronSystem,::AbstractScreening,om_q)

Return the value of the dielectric function for the given proper response function and given local effective potential.

_imag_ideal_dynamic_response(
    sys::AbstractIdealElectronSystem,
    approx::AbstractResponseApproximation,
    ombar::T,
    qbar::T
) where {T<:Real}

Compute the imaginary part of the dynamic response function of an ideal electron system using the given response approximation. The input frequencies ombar and wavevectors qbar are given in dimensionless units (normalized to Fermi energy and Fermi wavevector, respectively).

This is a low-level method, typically implemented by concrete approximation types.

real_ideal_dynamic_response(
    sys::AbstractIdealElectronSystem,
    approx::AbstractResponseApproximation,
    ombar::T,
    qbar::T
) where {T<:Real}

Compute the real part of the dynamic response function of an ideal electron system using the given response approximation. The input frequencies ombar and wavevectors qbar are given in dimensionless units (normalized to Fermi energy and Fermi wavevector, respectively).

This is a low-level interface meant for implementation by specific approximation models.

_stable_log_term(x::Real)

Stable version of

\[\log\left(\left\vert\frac{1+x}{1-x}\right\vert\right)\]

beta(esys::AbstractMatterModel) -> Real

Compute the inverse temperature β = 1 / T of the system esys.

Arguments

• esys: An instance of AbstractMatterModel representing the electronic system.

Returns

• Inverse temperature β as a real number.

betabar(esys::AbstractMatterModel) -> Real

Compute the dimensionless inverse temperature $\bar\beta= E_F / T$ of the system esys.

Arguments

• esys: An instance of AbstractMatterModel representing the electronic system.

Returns

• Dimensionless inverse temperature β̄ as a real number.

dielectric_function(iesys::AbstractInteractingElectronSystem, om_q) -> Real

Delegates to the dielectric_function using the proper electron system and screening model associated with iesys.

Arguments

• iesys: An interacting electron system.
• om_q: Tuple (ω, q) representing frequency and wavevector magnitude.

Returns

• Dielectric function ϵ(q, ω) as a real number.

dielectric_function(::AbstractIdealElectronSystem,::AbstractScreening,om_q)

Return the value of the dielectric function for the given electron system and given screening.

dielectric_function(esys, scr::NoScreening, om_q) -> Real

Return the dielectric function for a NoScreening model, which is always 1.

dynamic_response(iesys::AbstractInteractingElectronSystem, om_q) -> Complex

Compute the dynamically screened response function of the interacting electron system.

The response is computed as

\[χ(q, ω) = \frac{χ_0(q, ω)}{ϵ(q, ω)}\]

where χ₀ is the proper response and ϵ is the dielectric function derived from the screening model.

Arguments

• iesys: An interacting electron system.
• om_q: Tuple (ω, q) representing frequency and wavevector magnitude.

Returns

Complex-valued dynamic response χ(q, ω).

electron_density(m::AbstractMatterModel)

Return the electron density of the given matter model m.

This is an interface function that must be implemented by all concrete matter models.

fermi_energy(esys::AbstractMatterModel) -> Real

Compute the Fermi energy of a given electronic system esys, using its Fermi wave vector.

Arguments

• esys: An instance of AbstractMatterModel representing the electronic system.

Returns

• Fermi energy EF as a real number.

fermi_wave_vector(esys::AbstractMatterModel) -> Real

Compute the Fermi wave vector for a given electronic system esys, based on its electron density.

Arguments

• esys: An instance of AbstractMatterModel representing the electronic system.

Returns

• Fermi wave vector kF as a real number.

imag_dynamic_response(sys::AbstractElectronSystem, om_q::NTuple{2, T}) where {T<:Real}

Return the imaginary part of the dynamic response function for the given electron system sys at frequency and wavevector specified by om_q = (ω, q).

imag_dynamic_response(iesys::AbstractInteractingElectronSystem, om_q) -> Real

Return the imaginary part of the dynamic response function of the interacting electron system.

Returns

• Imaginary part of χ(q, ω).

local_effective_potential(iesys::AbstractInteractingElectronSystem, om_q) -> Real

Delegates to the local_effective_potential method of the screening model associated with iesys.

Arguments

• iesys: An interacting electron system.
• om_q: Tuple (ω, q) representing frequency and wavevector magnitude.

Returns

• Local effective potential V_eff(q, ω) as a real number.

local_effective_potential(scr::NoScreening, om_q) -> Real

Return the local effective potential for the NoScreening model, which is zero.

local_field_correction(::AbstractScreening, om_q::Tuple)

Interface function: return value of the local field correction for interacting electron systems

local_field_correction(scr::NoScreening, om_q) -> Real

Return the local field correction for the NoScreening model, which is zero.

lookup(
method::AboveInput,
column::Real,
row::Real,
table::GridLookupTable)

TBW

lookup(
method::BelowInput,
column::Real,
row::Real,
table::GridLookupTable)

TBW

lookup(
method::InterpolEndValue,
column::Real,
row::Real,
table::GridLookupTable)

TBW

lookup(
method::InterpolExtrapol,
column::Real,
row::Real,
table::GridLookupTable)

TBW

lookup(
method::NearestInput,
column::Real,
row::Real,
table::GridLookupTable)

TBW

proper_electron_system(::AbstractInteractingElectronSystem)

Interface function: return the proper electron system associated with the interacting system.

pseudo_potential(::AbstractScreening, om_q::Tuple)

Interface function: return value of the pseudo potential for interacting electron systems

pseudo_potential(iesys::AbstractInteractingElectronSystem, om_q) -> Real

Delegates to the pseudo_potential method of the screening model associated with iesys.

Arguments

• iesys: An interacting electron system.
• om_q: Tuple (ω, q) representing frequency and wavevector magnitude.

Returns

• Pseudo potential V(q, ω) as a real number.

pseudo_potential(scr::NoScreening, om_q) -> Real

Return the pseudo potential for the NoScreening model, which is zero.

real_dynamic_response(sys::AbstractElectronSystem, om_q::NTuple{2, T}) where {T<:Real}

Return the real part of the dynamic response function for the given electron system sys at frequency and wavevector specified by om_q = (ω, q).

real_dynamic_response(iesys::AbstractInteractingElectronSystem, om_q) -> Real

Return the real part of the dynamic response function of the interacting electron system.

Returns

• Real part of χ(q, ω).

response_approximation(sys::AbstractIdealElectronSystem)

Return the response approximation model associated with the given ideal electron system.

screening(::AbstractInteractingElectronSystem)

Interface function: return the screening associated with the interacting system.

temperature(m::AbstractMatterModel)

Return the temperature of the given matter model m in internal units.

This is an interface function that must be implemented by all concrete matter models.