API reference

TimerOutput object used to store DFTK timings.

Abstract supertype for architectures supported by DFTK.

Dynamically adapt number of bands to be converged to ensure that the orbitals of lowest occupation are occupied to at most occupation_threshold. To obtain rapid convergence of the eigensolver a gap between the eigenvalues of the last occupied orbital and the last computed (but not converged) orbital of gap_min is ensured.

Algorithm for the tolerance used for the next diagonalization. This function takes $|ρ_{\rm next} - ρ_{\rm in}|$ and multiplies it with ratio_ρdiff to get the next diagtol, ensuring additionally that the returned value is between diagtol_min and diagtol_max and never increases.

Quick and dirty Anderson implementation. Not particularly optimised.

Accelerates the iterative solution of $f(x) = 0$ according to a damped preconditioned scheme

\[ xₙ₊₁ = xₙ + αₙ P⁻¹ f(xₙ)\]

Where $f(x)$ computes the residual (e.g. SCF(x) - x) Further define

• preconditioned residual $Pf(x) = P⁻¹ f(x)$
• fixed-point map $g(x) = x + α Pf(x)$

where the $α$ may vary between steps.

Finds the linear combination $xₙ₊₁ = g(xₙ) + ∑ᵢ βᵢ (g(xᵢ) - g(xₙ))$ such that $|Pf(xₙ) + ∑ᵢ βᵢ (Pf(xᵢ) - Pf(xₙ))|²$ is minimal.

While doing this AndersonAcceleration ensures that the history size (number of $g(xᵢ)$ considered) never exceeds m. This value should ideally be chosen to be the maximal value fitting in memory as we use other measures on top to take care of conditioning issues, namely:

• We follow ^[CDLS21] (adaptive Anderson acceleration) and drop iterates, which do not satisfy

  \[ \|P⁻¹ f(xᵢ)\| < \text{errorfactor} minᵢ \|P⁻¹ f(xᵢ)\|.\]

  This means the best way to save memory is to reduce errorfactor to 1e3 or 100, which reduces the effective window size. Note that in comparison to the adaptive depth reference implementation of ^[CDLS21], we use $\text{errorfactor} = 1/δ$.
• Additionally we monitor the conditioning of the Anderson linear system and if it exceeds maxcond we drop the entries with largest residual norm $\|P⁻¹ f(xᵢ)\|$ (but never the most recent, i.e. $n-1$-st, iterate).

Accelerate the fixed-point scheme

\[ xₙ₊₁ = xₙ + αₙ P⁻¹ f(xₙ)\]

using Anderson acceleration. Requires Pfxₙ is $P⁻¹ f(xₙ)$, $xₙ$ and $αₙ$ and returns $xₙ₊₁$.

Full χ0 application, optionally dropping terms or disabling Sternheimer. All keyword arguments passed to apply_χ0.

Atomic local potential defined by model.atoms.

Nonlocal term coming from norm-conserving pseudopotentials in Kleinmann-Bylander form.

\[\text{Energy} = ∑_a ∑_{ij} ∑_{n} f_n \braket{ψ_n}{{\rm proj}_{ai}} D_{ij} \braket{{\rm proj}_{aj}}{ψ_n}.\]

Blow-up function as used in Abinit.

Blow-up function as proposed in arXiv:2210.00442. The blow-up order of the function is fixed to ensure C^2 regularity of the energies bands away from crossings and Lipschitz continuity at crossings.

Default blow-up corresponding to the standard kinetic energies.

Construct a AtomsCalculators compatible calculator for DFTK. The model_kwargs are passed onto the Model constructor, the basis_kwargs to the PlaneWaveBasis constructor, the scf_kwargs to self_consistent_field. At the very least the DFT functionals and the Ecut needs to be specified.

By default the calculator preserves the symmetries that are stored inside the st (the basis is re-built, but symmetries are fixed and not re-computed).

Calculator-specific keyword arguments are:

• verbose: If true, the SCF iterations are printed.
• enforce_convergence: If false, the calculator does not error out in case of a non-converging SCF.

Example

julia> DFTKCalculator(; model_kwargs=(; functionals=LDA()),
                        basis_kwargs=(; Ecut=10, kgrid=(2, 2, 2)),
                        scf_kwargs=(; tol=1e-4))

We use a simplification of the Resta model and set $χ_0(q) = \frac{C_0 G^2}{4π (1 - C_0 G^2 / k_{TF}^2)}$ where $C_0 = 1 - ε_r$ with $ε_r$ being the macroscopic relative permittivity. We neglect $K_\text{xc}$, such that $J^{-1} ≈ \frac{k_{TF}^2 - C_0 G^2}{ε_r k_{TF}^2 - C_0 G^2}$

By default it assumes a relative permittivity of 10 (similar to Silicon). εr == 1 is equal to SimpleMixing and εr == Inf to KerkerMixing. The mixing is applied to $ρ$ and $ρ_\text{spin}$ in the same way.

A localised dielectric model for $χ_0$:

\[\sqrt{L(x)} \text{IFFT} \frac{C_0 G^2}{4π (1 - C_0 G^2 / k_{TF}^2)} \text{FFT} \sqrt{L(x)}\]

where $C_0 = 1 - ε_r$, L(r) is a real-space localization function and otherwise the same conventions are used as in DielectricMixing.

Nonlocal "divAgrad" operator $-½ ∇ ⋅ (A ∇)$ where $A$ is a scalar field on the real-space grid. The $-½$ is included, such that this operator is a generalisation of the kinetic energy operator (which is obtained for $A=1$).

ElementCohenBergstresser(
    key::Union{AtomsBase.ChemicalSpecies, Integer, Symbol};
    lattice_constant
) -> ElementCohenBergstresser

Element where the interaction with electrons is modelled as in CohenBergstresser1966. Only the homonuclear lattices of the diamond structure are implemented (i.e. Si, Ge, Sn).

key may be an element symbol (like :Si), an atomic number (e.g. 14) or a chemical species (e.g. ChemicalSpecies(:Si)).

ElementCoulomb(
    key::Union{AtomsBase.ChemicalSpecies, Integer, Symbol};
    mass
) -> ElementCoulomb

Element interacting with electrons via a bare Coulomb potential (for all-electron calculations) key may be an element symbol (like :Si), an atomic number (e.g. 14) or a chemical species (e.g. ChemicalSpecies(:He3))

Element interacting with electrons via a Gaussian potential. Symbol is non-mandatory.

Element interacting with electrons via a pseudopotential model. key may be an element symbol (like :Si), an atomic number (e.g. 14) or a chemical species (e.g. ChemicalSpecies(:He3)). psp may be one of:

• a PseudoPotentialData.PseudoFamily to automatically determine the pseudopotential from the specified pseudo family. In this case kwargs are used when calling load_psp to read the pseudopotential from disk.
• a Dict{Symbol,String} mapping an atomic symbol to the pseudopotential to be employed. Again kwargs are used when calling load_psp to read the pseudopotential from disk.
• a pseudopotential object (like PspHgh or PspUpf), usually obtained using load_psp
• nothing (to return a ElementCoulomb)

Examples

Construct an ElementPsp for silicon using a HGH pseudopotential from an identifier

ElementPsp(:Si, load_psp("psp/hgh/Si-q4"))

Construct an ElementPsp again for silicon using the specified pseudpotential family (from the PseudopotentialData package).

using PseudoPotentialData
ElementPsp(:Si, PseudoFamily("dojo.nc.sr.pbe.v0_4_1.oncvpsp3.standard.upf"))

A simple struct to contain a vector of energies, and utilities to print them in a nice format.

Entropy term $-TS$, where $S$ is the electronic entropy. Turns the energy $E$ into the free energy $F=E-TS$. This is in particular useful because the free energy, not the energy, is minimized at self-consistency.

Ewald term: electrostatic energy per unit cell of the array of point charges defined by model.atoms in a uniform background of compensating charge yielding net neutrality.

Explicitly define the k-points along which to perform BZ sampling. (Useful for bandstructure calculations)

External potential from the (unnormalized) Fourier coefficients V(G) G is passed in Cartesian coordinates

External potential from an analytic function V (in Cartesian coordinates). No low-pass filtering is performed.

External potential given as values.

We define the FFTGrid struct, containing all the data required to perform FFTs. Namely:

• fft_size: defines the extent of the real and reciprocal space grids
• opFFT: out-of-place FFT plan
• ipFFT: in-place FFT plan
• opBFFT: out-of-place backward FFT plan
• ipBFFT: in-place backward FFT plan
• fft_normalization: normalization constant for FFTs
• ifft_normalization: normalization constant for backward FFTs
• G_vectors: grid coordinate in reciprocal space
• r_vectors: grid coordinate in real space
• architecture: information about the architecture on which FFT calculations take place

Note that the FFT plans are not normalized. Normalization takes place explicitely when the fft()/ifft() functions are called

Two-stage Fermi level finding algorithm starting from a Gaussian-smearing guess.

In each SCF step converge exactly n_bands_converge, computing along the way exactly n_bands_compute (usually a few more to ease convergence in systems with small gaps).

Fourier space multiplication, like a kinetic energy term:

\[(Hψ)(G) = {\rm multiplier}(G) ψ(G).\]

Construct a particular GPU architecture by passing the ArrayType

A Gaussian-shaped initial guess. Can be used as an approximation of an s- or σ-like orbital.

Hartree term: for a decaying potential V the energy would be

\[1/2 ∫ρ(x)ρ(y)V(x-y) dxdy,\]

with the integral on x in the unit cell and of y in the whole space. For the Coulomb potential with periodic boundary conditions, this is rather

\[1/2 ∫ρ(x)ρ(y) G(x-y) dx dy,\]

where G is the Green's function of the periodic Laplacian with zero mean ($-Δ G = ∑_R 4π δ_R$, integral of G zero on a unit cell).

A hydrogenic initial guess.

α is the diffusivity, $\frac{Z}{a}$ where $Z$ is the atomic number and $a$ is the Bohr radius.

The same as KerkerMixing, but the Thomas-Fermi wavevector is computed from the current density of states at the Fermi level.

Kerker mixing: $J^{-1} ≈ \frac{|G|^2}{k_{TF}^2 + |G|^2}$ where $k_{TF}$ is the Thomas-Fermi wave vector. For spin-polarized calculations by default the spin density is not preconditioned. Unless a non-default value for $ΔDOS_Ω$ is specified. This value should roughly be the expected difference in density of states (per unit volume) between spin-up and spin-down.

Notes:

• Abinit calls $1/k_{TF}$ the dielectric screening length (parameter dielng)

Kinetic energy:

\[1/2 ∑_n f_n ∫ |∇ψ_n|^2 * {\rm blowup}(-i∇Ψ_n).\]

Discretization information for $k$-point-dependent quantities such as orbitals. More generally, a $k$-point is a block of the Hamiltonian; e.g. collinear spin is treated by doubling the number of $k$-points.

Simple wrapper to represent a matrix formed by the concatenation of column blocks: it is mostly equivalent to hcat, but doesn't allocate the full matrix. LazyHcat only supports a few multiplication routines: furthermore, a multiplication involving this structure will always yield a plain array (and not a LazyHcat structure). LazyHcat is a lightweight subset of BlockArrays.jl's functionalities, but has the advantage to be able to store GPU Arrays (BlockArrays is heavily built on Julia's CPU Array).

Represents the LDOS-based $χ_0$ model

\[χ_0(r, r') = (-D_\text{loc}(r) δ(r, r') + D_\text{loc}(r) D_\text{loc}(r') / D)\]

where $D_\text{loc}$ is the local density of states and $D$ the density of states. For details see Herbst, Levitt 2020.

LibxcDensities(
    basis,
    max_derivative::Integer,
    ρ,
    τ
) -> DFTK.LibxcDensities

Compute density in real space and its derivatives starting from ρ

Local nonlinearity, with energy ∫f(ρ) where ρ is the density

Magnetic term $A⋅(-i∇)$. It is assumed (but not checked) that $∇⋅A = 0$.

Magnetic field operator A⋅(-i∇).

Model(system::AbstractSystem; kwargs...)

AtomsBase-compatible Model constructor. Sets structural information (atoms, positions, lattice, n_electrons etc.) from the passed system.

Keyword arguments

• pseudopotentials: Set the pseudopotential information for the atoms of the passed system. Can be (a) a list of pseudopotential objects (one for each atom), where a nothing element indicates that the Coulomb potential should be used for that atom or (b) a PseudoPotentialData.PseudoFamily to automatically determine the pseudopotential from the specified pseudo family or (c) a Dict{Symbol,String} mapping an atomic symbol to the pseudopotential to be employed.

Model(lattice, atoms, positions; n_electrons, magnetic_moments, terms, temperature,
      smearing, spin_polarization, symmetries)

Creates the physical specification of a model (without any discretization information).

n_electrons is taken from atoms if not specified.

spin_polarization is :none by default (paired electrons) unless any of the elements has a non-zero initial magnetic moment. In this case the spin_polarization will be :collinear.

magnetic_moments is only used to determine the symmetry and the spin_polarization; it is not stored inside the datastructure.

smearing is Fermi-Dirac if temperature is non-zero, none otherwise

The symmetries kwarg allows (a) to pass true / false to enable / disable the automatic determination of lattice symmetries or (b) to pass an explicit list of symmetry operations to use for lowering the computational effort. The default behaviour is equal to true, namely that the code checks the specified model in form of the Hamiltonian terms, lattice, atoms and magnetic_moments parameters and from these automatically determines a set of symmetries it can safely use. If you want to pass custom symmetry operations (e.g. a reduced or extended set) use the symmetry_operations function. Notice that this may lead to wrong results if e.g. the external potential breaks some of the passed symmetries. Use false to turn off symmetries completely.

Model(model; [lattice, positions, atoms, kwargs...])
Model{T}(model; [lattice, positions, atoms, kwargs...])

Construct an identical model to model with the option to change some of the contained parameters. This constructor is useful for changing the data type in the model or for changing lattice or positions in geometry/lattice optimisations.

Perform BZ sampling employing a Monkhorst-Pack grid.

NbandsAlgorithm subtypes determine how many bands to compute and converge in each SCF step.

Nonlocal operator in Fourier space in Kleinman-Bylander format, defined by its projectors P matrix and coupling terms D: Hψ = PDP' ψ.

Noop operation: don't do anything. Useful for energy terms that don't depend on the orbitals at all (eg nuclei-nuclei interaction).

PairwisePotential(
    V,
    params;
    max_radius
) -> PairwisePotential

Pairwise terms: Pairwise potential between nuclei, e.g., Van der Waals potentials, such as Lennard—Jones terms. The potential is dependent on the distance between to atomic positions and the pairwise atomic types: For a distance d between to atoms A and B, the potential is V(d, params[(A, B)]). The parameters max_radius is of 100 by default, and gives the maximum distance (in Cartesian coordinates) between nuclei for which we consider interactions.

A plane-wave discretized Model. Normalization conventions:

• Things that are expressed in the G basis are normalized so that if $x$ is the vector, then the actual function is $\sum_G x_G e_G$ with $e_G(x) = e^{iG x} / \sqrt(\Omega)$, where $\Omega$ is the unit cell volume. This is so that, eg $norm(ψ) = 1$ gives the correct normalization. This also holds for the density and the potentials.
• Quantities expressed on the real-space grid are in actual values.

ifft and fft convert between these representations.

Creates a new basis identical to basis, but with a new k-point grid, e.g. an MonkhorstPack or a ExplicitKpoints grid.

Creates a PlaneWaveBasis using the kinetic energy cutoff Ecut and a k-point grid. By default a MonkhorstPack grid is employed, which can be specified as a MonkhorstPack object or by simply passing a vector of three integers as the kgrid. Optionally kshift allows to specify a shift (0 or 1/2 in each direction). If not specified a grid is generated using kgrid_from_maximal_spacing with a maximal spacing of 2π * 0.022 per Bohr.

No preconditioning.

(Simplified version of) Tetter-Payne-Allan preconditioning.

Pseudopotential correction energy. TODO discuss the need for this.

PspHgh(path[, identifier, description])

Construct a Hartwigsen, Goedecker, Teter, Hutter separable dual-space Gaussian pseudopotential (1998) from file.

PspUpf(path[, identifier])

Construct a Unified Pseudopotential Format pseudopotential from file.

Does not support:

• Fully-realtivistic / spin-orbit pseudos
• Bare Coulomb / all-electron potentials
• Semilocal potentials
• Ultrasoft potentials
• Projector-augmented wave potentials
• GIPAW reconstruction data

Linear operators that act on tuples (real, fourier) The main entry point is apply!(out, op, in) which performs the operation out += op*in where out and in are named tuples (; real, fourier). They also implement mul! and Matrix(op) for exploratory use.

Real space multiplication by a potential:

\[(Hψ)(r) = V(r) ψ(r).\]

Result of calling the refine_scfres function.

• basis: Refinement basis, larger than the basis used to run a first self_consistent_field computation.
• ψ, ρ, occupation: Quantities from the scfres, transferred to the refinement basis and with virtual orbitals removed.
• δψ, δρ: First order corrections to the wavefunctions and density. The refined quantities are ψ + δψ and ρ + δρ.
• ΩpK_res: Additional information returned by the inversion of (Ω+K)_11.

Default callback function for solve_ΩplusK, which prints a convergence table.

Flag convergence by using the L2Norm of the density change in one SCF step.

Flag convergence as soon as total energy change drops below a tolerance.

Flag convergence on the change in Cartesian force between two iterations.

Default callback function for self_consistent_field methods, which prints a convergence table.

Adds checkpointing to a DFTK self-consistent field calculation. The checkpointing file is silently overwritten. Requires the package for writing the output file (usually JLD2) to be loaded.

• filename: Name of the checkpointing file.
• compress: Should compression be used on writing (rarely useful)
• save_ψ: Should the bands also be saved (noteworthy additional cost ... use carefully)

Simple mixing: $J^{-1} ≈ 1$

A term with a constant zero energy.

Exchange-correlation term, defined by a list of functionals and usually evaluated through libxc.

Generic mixing function using a model for the susceptibility composed of the sum of the χ0terms. For valid χ0terms See the subtypes of χ0Model. The dielectric model is solved in real space using a GMRES. Either the full kernel (RPA=false) or only the Hartree kernel (RPA=true) are employed. verbose=true lets the GMRES run in verbose mode (useful for debugging).

fft!(
    f_fourier::AbstractArray{T, 3} where T,
    fft_grid::FFTGrid,
    f_real::AbstractArray{T, 3} where T
) -> Any

In-place version of fft!. NOTE: If Gvec_mapping is given, not only f_fourier but also f_real is overwritten.

fft(
    fft_grid::FFTGrid{T, VT} where VT<:Real,
    f_real::AbstractArray{U}
) -> Any

fft(fft_grid::FFTGrid, [Gvec_mapping, ] f_real)

Perform an FFT to obtain the Fourier representation of f_real. If Gvec_mapping is given, the coefficients are truncated to the k-dependent spherical basis set correspoding to a k-point G-vector mapping.

ifft!(
    f_real::AbstractArray{T, 3} where T,
    fft_grid::FFTGrid,
    f_fourier::AbstractArray{T, 3} where T
) -> Any

In-place version of ifft.

ifft!(
    f_real::AbstractArray{T, 3} where T,
    basis::PlaneWaveBasis,
    f_fourier::AbstractArray{T, 3} where T
) -> Any

Forward FFT calls to the PlaneWaveBasis fft_grid field

ifft(fft_grid::FFTGrid, f_fourier::AbstractArray) -> Any

ifft(fft_grid::FFTGrid, [Gvec_mapping, ] f_fourier)

Perform an iFFT to obtain the quantity defined by f_fourier defined on the k-dependent spherical basis set (if Gvec_mapping of a k-point is given) or the k-independent cubic (if it is not) on the real-space grid.

atomic_system(
    lattice::AbstractMatrix{<:Number},
    atoms::Vector{<:DFTK.Element},
    positions::AbstractVector
) -> AtomsBase.FlexibleSystem{_A, _B, AtomsBase.PeriodicCell{_A1, _B1}} where {_A, _B, _A1, _B1}
atomic_system(
    lattice::AbstractMatrix{<:Number},
    atoms::Vector{<:DFTK.Element},
    positions::AbstractVector,
    magnetic_moments::AbstractVector
) -> AtomsBase.FlexibleSystem{_A, _B, AtomsBase.PeriodicCell{_A1, _B1}} where {_A, _B, _A1, _B1}

atomic_system(model::DFTK.Model, magnetic_moments=[])
atomic_system(lattice, atoms, positions, magnetic_moments=[])

Construct an AtomsBase atomic system from a DFTK model and associated magnetic moments or from the usual lattice, atoms and positions list used in DFTK plus magnetic moments.

element_symbol(el::DFTK.Element) -> Symbol

Chemical symbol corresponding to an element

mass(el::DFTK.Element) -> Any

Return the atomic mass of an element

periodic_system(
    model::Model
) -> AtomsBase.FlexibleSystem{_A, _B, AtomsBase.PeriodicCell{_A1, _B1}} where {_A, _B, _A1, _B1}
periodic_system(
    model::Model,
    magnetic_moments
) -> AtomsBase.FlexibleSystem{_A, _B, AtomsBase.PeriodicCell{_A1, _B1}} where {_A, _B, _A1, _B1}

periodic_system(model::DFTK.Model, magnetic_moments=[])
periodic_system(lattice, atoms, positions, magnetic_moments=[])

Construct an AtomsBase atomic system from a DFTK model and associated magnetic moments or from the usual lattice, atoms and positions list used in DFTK plus magnetic moments.

species(_::DFTK.Element) -> AtomsBase.ChemicalSpecies

Return the chemical species corresponding to an element

irrfbz_path(
    model::Model;
    ...
) -> Union{Brillouin.KPaths.KPath{3}, Brillouin.KPaths.KPath{2}, Brillouin.KPaths.KPath{1}}
irrfbz_path(
    model::Model,
    magnetic_moments;
    dim,
    space_group_number
) -> Union{Brillouin.KPaths.KPath{3}, Brillouin.KPaths.KPath{2}, Brillouin.KPaths.KPath{1}}

Extract the high-symmetry $k$-point path corresponding to the passed model using Brillouin. Uses the conventions described in the reference work by Cracknell, Davies, Miller, and Love (CDML). Of note, this has minor differences to the $k$-path reference (Y. Himuma et. al. Comput. Mater. Sci. 128, 140 (2017)) underlying the path-choices of Brillouin.jl, specifically for oA and mC Bravais types.

If the cell is a supercell of a smaller primitive cell, the standard $k$-path of the associated primitive cell is returned. So, the high-symmetry $k$ points are those of the primitive cell Brillouin zone, not those of the supercell Brillouin zone.

The dim argument allows to artificially truncate the dimension of the employed model, e.g. allowing to plot a 2D bandstructure of a 3D model (useful for example for plotting band structures of sheets with dim=2).

Due to lacking support in Spglib.jl for two-dimensional lattices it is (a) assumed that model.lattice is a conventional lattice and (b) required to pass the space group number using the space_group_number keyword argument.

G_vectors(
    basis::PlaneWaveBasis
) -> AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}

G_vectors(basis::PlaneWaveBasis)
G_vectors(basis::PlaneWaveBasis, kpt::Kpoint)

The list of wave vectors $G$ in reduced (integer) coordinates of a basis or a $k$-point kpt.

G_vectors(fft_size::Union{Tuple, AbstractVector}) -> Any

G_vectors(fft_size::Tuple)

of given sizes.

G_vectors_cart(basis::PlaneWaveBasis) -> Any

G_vectors_cart(basis::PlaneWaveBasis)
G_vectors_cart(basis::PlaneWaveBasis, kpt::Kpoint)

The list of $G$ vectors of a given basis or kpt, in Cartesian coordinates.

Gplusk_vectors(basis::PlaneWaveBasis, kpt::Kpoint) -> Any

Gplusk_vectors(basis::PlaneWaveBasis, kpt::Kpoint)

The list of $G + k$ vectors, in reduced coordinates.

Gplusk_vectors_cart(
    basis::PlaneWaveBasis,
    kpt::Kpoint
) -> Any

Gplusk_vectors_cart(basis::PlaneWaveBasis, kpt::Kpoint)

The list of $G + k$ vectors, in Cartesian coordinates.

Gplusk_vectors_in_supercell(
    basis::PlaneWaveBasis,
    basis_supercell::PlaneWaveBasis,
    kpt::Kpoint
) -> Any

Maps all $k+G$ vectors of an given basis as $G$ vectors of the supercell basis, in reduced coordinates.

HybridMixing(
;
    εr,
    kTF,
    localization,
    adjust_temperature,
    kwargs...
) -> χ0Mixing

The model for the susceptibility is

\[\begin{aligned} χ_0(r, r') &= (-D_\text{loc}(r) δ(r, r') + D_\text{loc}(r) D_\text{loc}(r') / D) \\ &+ \sqrt{L(x)} \text{IFFT} \frac{C_0 G^2}{4π (1 - C_0 G^2 / k_{TF}^2)} \text{FFT} \sqrt{L(x)} \end{aligned}\]

where $C_0 = 1 - ε_r$, $D_\text{loc}$ is the local density of states, $D$ is the density of states and the same convention for parameters are used as in DielectricMixing. Additionally there is the real-space localization function L(r). For details see Herbst, Levitt 2020.

Important kwargs passed on to χ0Mixing

• RPA: Is the random-phase approximation used for the kernel (i.e. only Hartree kernel is used and not XC kernel)
• verbose: Run the GMRES in verbose mode.
• reltol: Relative tolerance for GMRES

IncreaseMixingTemperature(
;
    factor,
    above_ρdiff,
    temperature_max
) -> DFTK.var"#callback#775"{DFTK.var"#callback#774#776"{Int64, Float64}}

Increase the temperature used for computing the SCF preconditioners. Initially the temperature is increased by a factor, which is then smoothly lowered towards the temperature used within the model as the SCF converges. Once the density change is below above_ρdiff the mixing temperature is equal to the model temperature.

LDA(; kwargs...) -> Xc

Specify an LDA model (Perdew & Wang parametrization) in conjunction with model_DFT https://doi.org/10.1103/PhysRevB.45.13244

LdosMixing(; adjust_temperature, kwargs...) -> χ0Mixing

The model for the susceptibility is

\[\begin{aligned} χ_0(r, r') &= (-D_\text{loc}(r) δ(r, r') + D_\text{loc}(r) D_\text{loc}(r') / D) \end{aligned}\]

where $D_\text{loc}$ is the local density of states, $D$ is the density of states. For details see Herbst, Levitt 2020.

Important kwargs passed on to χ0Mixing

• RPA: Is the random-phase approximation used for the kernel (i.e. only Hartree kernel is used and not XC kernel)
• verbose: Run the GMRES in verbose mode.
• reltol: Relative tolerance for GMRES

PBE(; kwargs...) -> Xc

Specify an PBE GGA model in conjunction with model_DFT https://doi.org/10.1103/PhysRevLett.77.3865

PBEsol(; kwargs...) -> Xc

Specify an PBEsol GGA model in conjunction with model_DFT https://doi.org/10.1103/physrevlett.100.136406

SCAN(; kwargs...) -> Xc

Specify a SCAN meta-GGA model in conjunction with model_DFT https://doi.org/10.1103/PhysRevLett.115.036402

ScfAcceptImprovingStep(
;
    max_energy_change,
    max_relative_residual
) -> DFTK.var"#accept_step#857"{Float64, Float64}

Accept a step if the energy is at most increasing by max_energy and the residual is at most max_relative_residual times the residual in the previous step.

apply_K(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    δψ,
    ψ,
    ρ,
    occupation
) -> Any

apply_K(basis::PlaneWaveBasis, δψ, ψ, ρ, occupation)

Compute the application of K defined at ψ to δψ. ρ is the density issued from ψ. δψ also generates a δρ, computed with compute_δρ.

apply_kernel(
    basis::PlaneWaveBasis,
    δρ;
    RPA,
    kwargs...
) -> Any

apply_kernel(basis::PlaneWaveBasis, δρ; kwargs...)

Computes the potential response to a perturbation δρ in real space, as a 4D (i,j,k,σ) array.

apply_symop(
    symop::SymOp,
    basis,
    kpoint,
    ψk::AbstractVecOrMat
) -> Tuple{Any, Any}

Apply a symmetry operation to eigenvectors ψk at a given kpoint to obtain an equivalent point in [-0.5, 0.5)^3 and associated eigenvectors (expressed in the basis of the new $k$-point).

apply_symop(symop::SymOp, basis, ρin; kwargs...) -> Any

Apply a symmetry operation to a density.

apply_Ω(δψ, ψ, H::Hamiltonian, Λ) -> Any

apply_Ω(δψ, ψ, H::Hamiltonian, Λ)

Compute the application of Ω defined at ψ to δψ. H is the Hamiltonian computed from ψ and Λ is the set of Rayleigh coefficients ψk' * Hk * ψk at each k-point.

apply_χ0(
    ham,
    ψ,
    occupation,
    εF,
    eigenvalues,
    δV::AbstractArray{TδV};
    occupation_threshold,
    q,
    kwargs_sternheimer...
) -> Any

Get the density variation δρ corresponding to a potential variation δV.

band_data_to_dict(
    band_data::NamedTuple;
    kwargs...
) -> Dict{String, Any}

Convert a band computational result to a dictionary representation. Intended to give a condensed set of results and useful metadata for post processing. See also the todict function for the Model and the PlaneWaveBasis, which are called from this function and the outputs merged. Note, that only the master process returns meaningful data. All other processors still return a dictionary (to simplify code in calling locations), but the data may be dummy.

Some details on the conventions for the returned data:

• εF: Computed Fermi level (if present in band_data)
• labels: A mapping of high-symmetry k-Point labels to the index in the kcoords vector of the corresponding k-point.
• eigenvalues, eigenvalues_error, occupation, residual_norms: (n_bands, n_kpoints, n_spin) arrays of the respective data.
• n_iter: (n_kpoints, n_spin) array of the number of iterations the diagonalization routine required.
• kpt_max_n_G: Maximal number of G-vectors used for any k-point.
• kpt_n_G_vectors: (n_kpoints, n_spin) array, the number of valid G-vectors for each k-point, i.e. the extend along the first axis of ψ where data is valid.
• kpt_G_vectors: (3, max_n_G, n_kpoints, n_spin) array of the integer (reduced) coordinates of the G-points used for each k-point.
• ψ: (max_n_G, n_bands, n_kpoints, n_spin) arrays where max_n_G is the maximal number of G-vectors used for any k-point. The data is zero-padded, i.e. for k-points which have less G-vectors than maxnG, then there are tailing zeros.

build_fft_plans!(
    tmp::Array{ComplexF64}
) -> Tuple{FFTW.cFFTWPlan{ComplexF64, -1, true, N, G} where {N, G<:Tuple{Vararg{Int64}}}, FFTW.cFFTWPlan{ComplexF64, -1, false}, Any, Any}

Plan a FFT of type T and size fft_size, spending some time on finding an optimal algorithm. (Inplace, out-of-place) x (forward, backward) FFT plans are returned.

build_form_factors(
    fun::Function,
    l::Int64,
    G_plus_ks::AbstractArray{<:AbstractArray{StaticArraysCore.SArray{Tuple{3}, TT, 1, 3}, 1}, 1}
) -> Any

Build Fourier transform factors of an atomic function centered at 0 for a given l.

build_projection_vectors(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    kpt::Kpoint,
    psps::AbstractVector{<:DFTK.NormConservingPsp},
    psp_positions
) -> Any

Build projection vectors for a atoms array generated by term_nonlocal

\[\begin{aligned} H_{\rm at} &= \sum_{ij} C_{ij} \ket{{\rm proj}_i} \bra{{\rm proj}_j} \\ H_{\rm per} &= \sum_R \sum_{ij} C_{ij} \ket{{\rm proj}_i(x-R)} \bra{{\rm proj}_j(x-R)} \end{aligned}\]

\[\begin{aligned} \braket{e_k(G') \middle| H_{\rm per}}{e_k(G)} &= \ldots \\ &= \frac{1}{Ω} \sum_{ij} C_{ij} \widehat{\rm proj}_i(k+G') \widehat{\rm proj}_j^*(k+G), \end{aligned}\]

where $\widehat{\rm proj}_i(p) = ∫_{ℝ^3} {\rm proj}_i(r) e^{-ip·r} dr$.

We store $\frac{1}{\sqrt Ω} \widehat{\rm proj}_i(k+G)$ in proj_vectors.

build_projector_form_factors(
    psp::DFTK.NormConservingPsp,
    G_plus_k::AbstractArray{StaticArraysCore.SArray{Tuple{3}, TT, 1, 3}, 1}
) -> Any

Build form factors (Fourier transforms of projectors) for all orbitals of an atom centered at 0.

cell_to_supercell(scfres::NamedTuple) -> NamedTuple

Transpose all data from a given self-consistent-field result from unit cell to supercell conventions. The parameters to adapt are the following:

• basis_supercell and ψ_supercell are computed by the routines above.
• The supercell occupations vector is the concatenation of all input occupations vectors.
• The supercell density is computed with supercell occupations and ψ_supercell.
• Supercell energies are the multiplication of input energies by the number of unit cells in the supercell.

Other parameters stay untouched.

cell_to_supercell(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})}
) -> PlaneWaveBasis{T, _A, Arch, FFTGrid{T, _A, T_G_vectors, T_r_vectors}} where {T<:Real, _A<:Real, Arch<:DFTK.AbstractArchitecture, T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, _A}, 3}}

Construct a plane-wave basis whose unit cell is the supercell associated to an input basis $k$-grid. All other parameters are modified so that the respective physical systems associated to both basis are equivalent.

cell_to_supercell(
    ψ,
    basis::PlaneWaveBasis{T<:Real, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T<:Real, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    basis_supercell::PlaneWaveBasis{T<:Real, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T<:Real, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})}
) -> Any

Re-organize Bloch waves computed in a given basis as Bloch waves of the associated supercell basis. The output ψ_supercell have a single component at $Γ$-point, such that ψ_supercell[Γ][:, k+n] contains ψ[k][:, n], within normalization on the supercell.

cg!(
    x::AbstractArray{T, 1},
    A::LinearMaps.LinearMap{T},
    b::AbstractArray{T, 1};
    precon,
    proj,
    callback,
    tol,
    maxiter,
    miniter,
    comm
) -> NamedTuple{(:x, :converged, :tol, :residual_norm, :n_iter, :maxiter, :stage), <:Tuple{AbstractVector, Bool, Float64, Any, Int64, Int64, Symbol}}

Implementation of the conjugate gradient method which allows for preconditioning and projection operations along iterations.

charge_ionic(el::DFTK.Element) -> Int64

Return the total ionic charge of an element (nuclear charge - core electrons)

charge_nuclear(el::DFTK.Element) -> Int64

Return the total nuclear charge of an element

check_full_occupation(basis::PlaneWaveBasis, occupation)

Check that all orbitals are fully occupied.

cis2pi(x) -> Any

Function to compute exp(2π i x)

compute_amn_kpoint(
    basis::PlaneWaveBasis,
    kpt,
    ψk,
    projections,
    n_bands
) -> Any

Compute the starting matrix for Wannierization.

Wannierization searches for a unitary matrix $U_{m_n}$. As a starting point for the search, we can provide an initial guess function $g$ for the shape of the Wannier functions, based on what we expect from knowledge of the problem or physical intuition. This starting matrix is called $[A_k]_{m,n}$, and is computed as follows: $[A_k]_{m,n} = \langle ψ_m^k | g^{\text{per}}_n \rangle$. The matrix will be orthonormalized by the chosen Wannier program, we don't need to do so ourselves.

Centers are to be given in lattice coordinates and G_vectors in reduced coordinates. The dot product is computed in the Fourier space.

Given an orbital $g_n$, the periodized orbital is defined by : $g^{per}_n = \sum\limits_{R \in {\rm lattice}} g_n( \cdot - R)$. The Fourier coefficient of $g^{per}_n$ at any G is given by the value of the Fourier transform of $g_n$ in G.

Each projection is a callable object that accepts the basis and some p-points as an argument, and returns the Fourier transform of $g_n$ at the p-points.

compute_bands(
    basis_or_scfres,
    kpath::Brillouin.KPaths.KPath;
    kline_density,
    kwargs...
) -> NamedTuple

Compute band data along a specific Brillouin.KPath using a kline_density, the number of $k$-points per inverse bohrs (i.e. overall in units of length).

If not given, the path is determined automatically by inspecting the Model. If you are using spin, you should pass the magnetic_moments as a kwarg to ensure these are taken into account when determining the path.

compute_bands(
    scfres::NamedTuple,
    kgrid::DFTK.AbstractKgrid;
    n_bands,
    kwargs...
) -> NamedTuple{(:basis, :ψ, :eigenvalues, :ρ, :εF, :occupation, :diagonalization), <:Tuple{PlaneWaveBasis{T, _A, Arch, FFTGrid{T, _A, T_G_vectors, T_r_vectors}} where {T<:Real, _A<:Real, Arch<:DFTK.AbstractArchitecture, T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, _A}, 3}}, Any, Any, Any, Any, Any, Vector{T} where T<:NamedTuple}}

Compute band data starting from SCF results. εF and ρ from the scfres are forwarded to the band computation and n_bands is by default selected as n_bands_scf + 5sqrt(n_bands_scf).

compute_bands(
    basis::PlaneWaveBasis,
    kgrid::DFTK.AbstractKgrid;
    n_bands,
    n_extra,
    ρ,
    εF,
    eigensolver,
    tol,
    kwargs...
) -> NamedTuple{(:basis, :ψ, :eigenvalues, :ρ, :εF, :occupation, :diagonalization), <:Tuple{PlaneWaveBasis{T, _A, Arch, FFTGrid{T, _A, T_G_vectors, T_r_vectors}} where {T<:Real, _A<:Real, Arch<:DFTK.AbstractArchitecture, T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, _A}, 3}}, Any, Any, Any, Nothing, Nothing, Vector{T} where T<:NamedTuple}}

Compute n_bands eigenvalues and Bloch waves at the k-Points specified by the kgrid. All kwargs not specified below are passed to diagonalize_all_kblocks:

• kgrid: A custom kgrid to perform the band computation, e.g. a new MonkhorstPack grid.
• tol The default tolerance for the eigensolver is substantially lower than for SCF computations. Increase if higher accuracy desired.
• eigensolver: The diagonalisation method to be employed.

compute_current(
    basis::PlaneWaveBasis,
    ψ,
    occupation
) -> Vector

Computes the probability (not charge) current, $∑_n f_n \Im(ψ_n · ∇ψ_n)$.

compute_density(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SArray{Tuple{3}, VT, 1, 3}, 3}})},
    ψ,
    occupation;
    occupation_threshold
) -> AbstractArray{_A, 4} where _A

compute_density(basis::PlaneWaveBasis, ψ::AbstractVector, occupation::AbstractVector)

Compute the density for a wave function ψ discretized on the plane-wave grid basis, where the individual k-points are occupied according to occupation. ψ should be one coefficient matrix per $k$-point. It is possible to ask only for occupations higher than a certain level to be computed by using an optional occupation_threshold. By default all occupation numbers are considered.

compute_dos(
    ε,
    basis,
    eigenvalues;
    smearing,
    temperature
) -> Any

Total density of states at energy ε

compute_dynmat(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    ψ,
    occupation;
    q,
    ρ,
    ham,
    εF,
    eigenvalues,
    kwargs...
) -> Any

Compute the dynamical matrix in the form of a $3×n_{\rm atoms}×3×n_{\rm atoms}$ tensor in reduced coordinates.

compute_fft_size(
    model::Model{T},
    Ecut;
    ...
) -> NTuple{_A, Any} where _A
compute_fft_size(
    model::Model{T},
    Ecut,
    kgrid;
    ensure_smallprimes,
    algorithm,
    factors,
    kwargs...
) -> NTuple{_A, Any} where _A

Determine the minimal grid size for the cubic basis set to be able to represent product of orbitals (with the default supersampling=2).

Optionally optimize the grid afterwards for the FFT procedure by ensuring factorization into small primes.

The function will determine the smallest parallelepiped containing the wave vectors $|G|^2/2 \leq E_\text{cut} ⋅ \text{supersampling}^2$. For an exact representation of the density resulting from wave functions represented in the spherical basis sets, supersampling should be at least 2.

If factors is not empty, ensure that the resulting fft_size contains all the factors

compute_forces(scfres) -> Any

Compute the forces of an obtained SCF solution. Returns the forces wrt. the fractional lattice vectors. To get Cartesian forces use compute_forces_cart. Returns a list of lists of forces (as SVector{3}) in the same order as the atoms and positions in the underlying Model.

compute_forces_cart(
    basis::PlaneWaveBasis,
    ψ,
    occupation;
    kwargs...
) -> Any

Compute the Cartesian forces of an obtained SCF solution in Hartree / Bohr. Returns a list of lists of forces [[force for atom in positions] for (element, positions) in atoms] which has the same structure as the atoms object passed to the underlying Model.

compute_inverse_lattice(lattice::AbstractArray{T, 2}) -> Any

Compute the inverse of the lattice. Takes special care of 1D or 2D cases.

compute_kernel(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})};
    kwargs...
) -> Any

compute_kernel(basis::PlaneWaveBasis; kwargs...)

Computes a matrix representation of the full response kernel (derivative of potential with respect to density) in real space. For non-spin-polarized calculations the matrix dimension is prod(basis.fft_size) × prod(basis.fft_size) and for collinear spin-polarized cases it is 2prod(basis.fft_size) × 2prod(basis.fft_size). In this case the matrix has effectively 4 blocks

\[\left(\begin{array}{cc} K_{αα} & K_{αβ}\\ K_{βα} & K_{ββ} \end{array}\right)\]

compute_ldos(
    ε,
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    eigenvalues,
    ψ;
    smearing,
    temperature,
    weight_threshold
) -> AbstractArray{_A, 4} where _A

Local density of states, in real space. weight_threshold is a threshold to screen away small contributions to the LDOS.

compute_occupation(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    eigenvalues::AbstractVector,
    εF::Number;
    temperature,
    smearing
) -> NamedTuple{(:occupation, :εF), <:Tuple{Any, Number}}

Compute occupation given eigenvalues and Fermi level

compute_occupation(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    eigenvalues::AbstractVector;
    ...
) -> NamedTuple{(:occupation, :εF), <:Tuple{Any, Number}}
compute_occupation(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    eigenvalues::AbstractVector,
    fermialg::AbstractFermiAlgorithm;
    tol_n_elec,
    temperature,
    smearing
) -> NamedTuple{(:occupation, :εF), <:Tuple{Any, Number}}

Compute occupation and Fermi level given eigenvalues and using fermialg. The tol_n_elec gives the accuracy on the electron count which should be at least achieved.

compute_pdos(
    ε,
    basis,
    eigenvalues,
    ψ,
    i,
    l,
    psp,
    position;
    smearing,
    temperature
) -> Any

Projected density of states at energy ε for an atom with given i and l.

compute_recip_lattice(lattice::AbstractArray{T, 2}) -> Any

Compute the reciprocal lattice. We use the convention that the reciprocal lattice is the set of G vectors such that G ⋅ R ∈ 2π ℤ for all R in the lattice.

compute_stresses_cart(scfres) -> Any

Compute the stresses of an obtained SCF solution. The stress tensor is given by

\[\left( \begin{array}{ccc} σ_{xx} σ_{xy} σ_{xz} \\ σ_{yx} σ_{yy} σ_{yz} \\ σ_{zx} σ_{zy} σ_{zz} \end{array} \right) = \frac{1}{|Ω|} \left. \frac{dE[ (I+ϵ) * L]}{dϵ}\right|_{ϵ=0}\]

where $ϵ$ is the strain tensor. See O. Nielsen, R. Martin Phys. Rev. B. 32, 3792 (1985) for details. In Voigt notation one would use the vector $[σ_{xx} σ_{yy} σ_{zz} σ_{zy} σ_{zx} σ_{yx}]$.

compute_transfer_matrix(
    basis_in::PlaneWaveBasis,
    kpt_in::Kpoint,
    basis_out::PlaneWaveBasis,
    kpt_out::Kpoint
) -> Any

Return a sparse matrix that maps quantities given on basis_in and kpt_in to quantities on basis_out and kpt_out.

compute_transfer_matrix(
    basis_in::PlaneWaveBasis,
    basis_out::PlaneWaveBasis
) -> Vector

Return a list of sparse matrices (one per $k$-point) that map quantities given in the basis_in basis to quantities given in the basis_out basis.

compute_unit_cell_volume(lattice) -> Any

Compute unit cell volume volume. In case of 1D or 2D case, the volume is the length/surface.

compute_δHψ_αs(basis::PlaneWaveBasis, ψ, α, s, q) -> Any

Get $δH·ψ$, with $δH$ the perturbation of the Hamiltonian with respect to a position displacement $e^{iq·r}$ of the $α$ coordinate of atom $s$. δHψ[ik] is $δH·ψ_{k-q}$, expressed in basis.kpoints[ik].

compute_δocc!(
    δocc,
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    ψ,
    εF,
    ε,
    δHψ
) -> NamedTuple{(:δocc, :δεF), <:Tuple{Any, Any}}

Compute the derivatives of the occupations (and of the Fermi level). The derivatives of the occupations are in-place stored in δocc. The tuple (; δocc, δεF) is returned. It is assumed the passed δocc are initialised to zero.

compute_δψ!(
    δψ,
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    H,
    ψ,
    εF,
    ε,
    δHψ;
    ...
)
compute_δψ!(
    δψ,
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    H,
    ψ,
    εF,
    ε,
    δHψ,
    ε_minus_q;
    ψ_extra,
    q,
    kwargs_sternheimer...
)

Perform in-place computations of the derivatives of the wave functions by solving a Sternheimer equation for each k-points. It is assumed the passed δψ are initialised to zero. For phonon, δHψ[ik] is $δH·ψ_{k-q}$, expressed in basis.kpoints[ik].

compute_χ0(ham; temperature) -> Any

Compute the independent-particle susceptibility. Will blow up for large systems. For non-spin-polarized calculations the matrix dimension is prod(basis.fft_size) × prod(basis.fft_size) and for collinear spin-polarized cases it is 2prod(basis.fft_size) × 2prod(basis.fft_size). In this case the matrix has effectively 4 blocks, which are:

\[\left(\begin{array}{cc} (χ_0)_{αα} & (χ_0)_{αβ} \\ (χ_0)_{βα} & (χ_0)_{ββ} \end{array}\right)\]

cos2pi(x) -> Any

Function to compute cos(2π x)

count_n_proj(psps, psp_positions) -> Any

count_n_proj(psps, psp_positions)

Number of projector functions for all angular momenta up to psp.lmax and for all atoms in the system, including angular parts from -m:m.

count_n_proj(psp::DFTK.NormConservingPsp, l::Integer) -> Any

count_n_proj(psp, l)

Number of projector functions for angular momentum l, including angular parts from -m:m.

count_n_proj(psp::DFTK.NormConservingPsp) -> Int64

count_n_proj(psp)

Number of projector functions for all angular momenta up to psp.lmax, including angular parts from -m:m.

count_n_proj_radial(
    psp::DFTK.NormConservingPsp,
    l::Integer
) -> Any

count_n_proj_radial(psp, l)

Number of projector radial functions at angular momentum l.

count_n_proj_radial(psp::DFTK.NormConservingPsp) -> Int64

count_n_proj_radial(psp)

Number of projector radial functions at all angular momenta up to psp.lmax.

create_supercell(
    lattice,
    atoms,
    positions,
    supercell_size
) -> NamedTuple{(:lattice, :atoms, :positions), <:Tuple{Any, Any, Any}}

Construct a supercell of size supercell_size from a unit cell described by its lattice, atoms and their positions.

datadir_psp() -> String

Return the data directory with pseudopotential files

default_fermialg(
    _::DFTK.Smearing.SmearingFunction
) -> FermiBisection

Default selection of a Fermi level determination algorithm

default_symmetries(
    lattice,
    atoms,
    positions,
    magnetic_moments,
    spin_polarization,
    terms;
    tol_symmetry
) -> Union{Vector{SymOp{Bool}}, Vector{SymOp{Float64}}}

Default logic to determine the symmetry operations to be used in the model.

default_wannier_centers(n_wannier) -> Any

Default random Gaussian guess for maximally-localised wannier functions generated in reduced coordinates.

determine_spin_polarization(magnetic_moments) -> Symbol

:none if no element has a magnetic moment, else :collinear or :full.

diagonalize_all_kblocks(
    eigensolver,
    ham::Hamiltonian,
    nev_per_kpoint::Int64;
    ψguess,
    prec_type,
    interpolate_kpoints,
    tol,
    miniter,
    maxiter,
    n_conv_check
) -> NamedTuple{(:λ, :X, :residual_norms, :n_iter, :converged, :n_matvec), <:Tuple{Vector, Vector, Vector, Vector, Union{Missing, Bool}, Any}}

Function for diagonalising each $k$-Point blow of ham one step at a time. Some logic for interpolating between $k$-points is used if interpolate_kpoints is true and if no guesses are given. eigensolver is the iterative eigensolver that really does the work, operating on a single $k$-Block. eigensolver should support the API eigensolver(A, X0; prec, tol, maxiter) prec_type should be a function that returns a preconditioner when called as prec(ham, kpt)

diameter(lattice::AbstractMatrix) -> Any

Compute the diameter of the unit cell

direct_minimization(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})};
    ψ,
    tol,
    is_converged,
    maxiter,
    prec_type,
    callback,
    optim_method,
    alphaguess,
    linesearch,
    kwargs...
) -> Any

Computes the ground state by direct minimization. kwargs... are passed to Optim.Options() and optim_method selects the optim approach which is employed.

disable_threading() -> Union{Nothing, Bool}

Convenience function to disable all threading in DFTK.

divergence_real(operand, basis) -> Any

Compute divergence of an operand function, which returns the Cartesian x,y,z components in real space when called with the arguments 1 to 3. The divergence is also returned as a real-space array.

energy(
    basis::PlaneWaveBasis,
    ψ,
    occupation;
    kwargs...
) -> NamedTuple{(:energies,), <:Tuple{Energies}}

Faster version than energy_hamiltonian for cases where only the energy is needed.

energy_forces_ewald(
    S,
    lattice::AbstractArray{T},
    charges,
    positions,
    q,
    ph_disp;
    η
) -> NamedTuple{(:energy, :forces), <:Tuple{Any, Any}}

Compute the electrostatic energy and forces. The energy is the electrostatic interaction energy per unit cell between point charges in a uniform background of compensating charge to yield net neutrality. The forces is the opposite of the derivative of the energy with respect to positions.

lattice should contain the lattice vectors as columns. charges and positions are the point charges and their positions (as an array of arrays) in fractional coordinates.

For now this function returns zero energy and force on non-3D systems. Use a pairwise potential term if you want to customise this treatment.

For phonon (q ≠ 0), this computes the local energy and forces on the atoms of the reference unit cell 0, for an infinite array of atoms at positions $r_{iR} = {\rm positions}_i + R + {\rm ph_disp}_i e^{-iq·R}$. q is the phonon q-point, and ph_disp a list of displacements to compute the Fourier transform of (only the direct part of) the force constant matrix.

energy_forces_pairwise(
    S,
    lattice::AbstractArray{T},
    symbols,
    positions,
    V,
    params,
    q,
    ph_disp;
    max_radius
) -> NamedTuple{(:energy, :forces), <:Tuple{Any, Any}}

Compute the pairwise energy and forces. The energy is the interaction energy per unit cell between atomic sites. The forces is the opposite of the derivative of the energy with respect to positions.

lattice should contain the lattice vectors as columns. symbols and positions are the atomic elements and their positions (as an array of arrays) in fractional coordinates. V and params are the pairwise potential and its set of parameters (that depends on pairs of symbols).

The potential is expected to decrease quickly at infinity.

For phonons (q ≠ 0), this computes the local energy and forces on the atoms of the reference unit cell 0, for an infinite array of atoms at positions $r_{iR} = {\rm positions}_i + R + {\rm ph_disp}_i e^{-iq·R}$. q is the phonon q-point, and ph_disp a list of displacements to compute the Fourier transform of the force constant matrix.

energy_hamiltonian(
    basis::PlaneWaveBasis,
    ψ,
    occupation;
    kwargs...
) -> NamedTuple{(:energies, :ham), <:Tuple{Energies, Hamiltonian}}

Get energies and Hamiltonian kwargs is additional info that might be useful for the energy terms to precompute (eg the density ρ)

energy_psp_correction(
    lattice::AbstractArray{T, 2},
    atoms,
    atom_groups
) -> Any

Compute the correction term for properly modelling the interaction of the pseudopotential core with the compensating background charge induced by the Ewald term.

enforce_real!(
    fourier_coeffs,
    basis::PlaneWaveBasis
) -> AbstractArray

Ensure its real-space equivalent of passed Fourier-space representation is entirely real by removing wavevectors G that don't have a -G counterpart in the basis.

estimate_integer_lattice_bounds(
    M::AbstractArray{T, 2},
    δ;
    ...
) -> Vector
estimate_integer_lattice_bounds(
    M::AbstractArray{T, 2},
    δ,
    shift;
    tol
) -> Vector

Estimate integer bounds for dense space loops from a given inequality ||Mx|| ≤ δ. For 1D and 2D systems the limit will be zero in the auxiliary dimensions.

eval_psp_density_core_fourier(
    _::DFTK.NormConservingPsp,
    _::Real
) -> Any

eval_psp_density_core_fourier(psp, p)

Evaluate the atomic core charge density in reciprocal space:

\[\begin{aligned} ρ_{\rm core}(p) &= ∫_{ℝ^3} ρ_{\rm core}(r) e^{-ip·r} dr \\ &= 4π ∫_{ℝ_+} ρ_{\rm core}(r) \frac{\sin(p·r)}{ρ·r} r^2 dr. \end{aligned}\]

eval_psp_density_core_real(
    _::DFTK.NormConservingPsp,
    _::Real
) -> Any

eval_psp_density_core_real(psp, r)

Evaluate the atomic core charge density in real space.

eval_psp_density_valence_fourier(
    psp::DFTK.NormConservingPsp,
    p::AbstractVector
) -> Any

eval_psp_density_valence_fourier(psp, p)

Evaluate the atomic valence charge density in reciprocal space:

\[\begin{aligned} ρ_{\rm val}(p) &= ∫_{ℝ^3} ρ_{\rm val}(r) e^{-ip·r} dr \\ &= 4π ∫_{ℝ_+} ρ_{\rm val}(r) \frac{\sin(p·r)}{ρ·r} r^2 dr. \end{aligned}\]

eval_psp_density_valence_real(
    psp::DFTK.NormConservingPsp,
    r::AbstractVector
) -> Any

eval_psp_density_valence_real(psp, r)

Evaluate the atomic valence charge density in real space.

eval_psp_energy_correction([T=Float64,] psp, n_electrons)

Evaluate the energy correction to the Ewald electrostatic interaction energy of one unit cell, which is required compared the Ewald expression for point-like nuclei. n_electrons is the number of electrons per unit cell. This defines the uniform compensating background charge, which is assumed here.

Notice: The returned result is the energy per unit cell and not the energy per volume. To obtain the latter, the caller needs to divide by the unit cell volume.

The energy correction is defined as the limit of the Fourier-transform of the local potential as $p \to 0$, using the same correction as in the Fourier-transform of the local potential:

\[\lim_{p \to 0} 4π N_{\rm elec} ∫_{ℝ_+} (V(r) - C(r)) \frac{\sin(p·r)}{p·r} r^2 dr + F[C(r)] = 4π N_{\rm elec} ∫_{ℝ_+} (V(r) + Z/r) r^2 dr.\]

eval_psp_local_fourier(
    psp::DFTK.NormConservingPsp,
    p::AbstractVector
) -> Any

eval_psp_local_fourier(psp, p)

Evaluate the local part of the pseudopotential in reciprocal space:

\[\begin{aligned} V_{\rm loc}(p) &= ∫_{ℝ^3} V_{\rm loc}(r) e^{-ip·r} dr \\ &= 4π ∫_{ℝ_+} V_{\rm loc}(r) \frac{\sin(p·r)}{p} r dr \end{aligned}\]

In practice, the local potential should be corrected using a Coulomb-like term $C(r) = -Z/r$ to remove the long-range tail of $V_{\rm loc}(r)$ from the integral:

\[\begin{aligned} V_{\rm loc}(p) &= ∫_{ℝ^3} (V_{\rm loc}(r) - C(r)) e^{-ip·r} dr + F[C(r)] \\ &= 4π ∫_{ℝ_+} (V_{\rm loc}(r) + Z/r) \frac{\sin(p·r)}{p·r} r^2 dr - Z/p^2. \end{aligned}\]

eval_psp_local_real(
    psp::DFTK.NormConservingPsp,
    r::AbstractVector
) -> Any

eval_psp_local_real(psp, r)

Evaluate the local part of the pseudopotential in real space.

eval_psp_projector_fourier(
    psp::DFTK.NormConservingPsp,
    p::AbstractVector
)

eval_psp_projector_fourier(psp, i, l, p)

Evaluate the radial part of the i-th projector for angular momentum l at the reciprocal vector with modulus p:

\[\begin{aligned} {\rm proj}(p) &= ∫_{ℝ^3} {\rm proj}_{il}(r) e^{-ip·r} dr \\ &= 4π ∫_{ℝ_+} r^2 {\rm proj}_{il}(r) j_l(p·r) dr. \end{aligned}\]

eval_psp_projector_real(
    psp::DFTK.NormConservingPsp,
    i,
    l,
    r::AbstractVector
) -> Any

eval_psp_projector_real(psp, i, l, r)

Evaluate the radial part of the i-th projector for angular momentum l in real-space at the vector with modulus r.

filled_occupation(model) -> Int64

Maximal occupation of a state (2 for non-spin-polarized electrons, 1 otherwise).

find_equivalent_kpt(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    kcoord,
    spin;
    tol
) -> NamedTuple{(:index, :ΔG), <:Tuple{Int64, Any}}

Find the equivalent index of the coordinate kcoord ∈ ℝ³ in a list kcoords ∈ [-½, ½)³. ΔG is the vector of ℤ³ such that kcoords[index] = kcoord + ΔG.

gather_kpts(
    basis::PlaneWaveBasis
) -> Union{Nothing, PlaneWaveBasis}

Gather the distributed $k$-point data on the master process and return it as a PlaneWaveBasis. On the other (non-master) processes nothing is returned. The returned object should not be used for computations and only for debugging or to extract data for serialisation to disk.

gather_kpts_block!(
    dest,
    basis::PlaneWaveBasis,
    kdata::AbstractArray{A, 1}
) -> Any

Gather the distributed data of a quantity depending on k-Points on the master process and save it in dest as a dense (size(kdata[1])..., n_kpoints) array. On the other (non-master) processes nothing is returned.

gaussian_valence_charge_density_fourier(
    el::DFTK.Element,
    p::Real
) -> Any

Gaussian valence charge density using Abinit's coefficient table, in Fourier space.

guess_density(
    basis::PlaneWaveBasis,
    method::AtomicDensity;
    ...
) -> Any
guess_density(
    basis::PlaneWaveBasis,
    method::AtomicDensity,
    magnetic_moments;
    n_electrons
) -> Any

guess_density(basis::PlaneWaveBasis, method::DensityConstructionMethod,
              magnetic_moments=[]; n_electrons=basis.model.n_electrons)

Build a superposition of atomic densities (SAD) guess density or a rarndom guess density.

The guess atomic densities are taken as one of the following depending on the input method:

-RandomDensity(): A random density, normalized to the number of electrons basis.model.n_electrons. Does not support magnetic moments. -ValenceDensityAuto(): A combination of the ValenceDensityGaussian and ValenceDensityPseudo methods where elements whose pseudopotentials provide numeric valence charge density data use them and elements without use Gaussians. -ValenceDensityGaussian(): Gaussians of length specified by atom_decay_length normalized for the correct number of electrons:

\[\hat{ρ}(G) = Z_{\mathrm{valence}} \exp\left(-(2π \text{length} |G|)^2\right)\]

• ValenceDensityPseudo(): Numerical pseudo-atomic valence charge densities from the

pseudopotentials. Will fail if one or more elements in the system has a pseudopotential that does not have valence charge density data.

When magnetic moments are provided, construct a symmetry-broken density guess. The magnetic moments should be specified in units of $μ_B$.

hamiltonian_with_total_potential(
    ham::Hamiltonian,
    V
) -> Hamiltonian

Returns a new Hamiltonian with local potential replaced by the given one

hankel(
    r::AbstractVector,
    r2_f::AbstractVector,
    l::Integer,
    p::Real
) -> Any

hankel(r, r2_f, l, p)

Compute the Hankel transform

\[ H[f] = 4\pi \int_0^\infty r f(r) j_l(p·r) r dr.\]

The integration is performed by simpson quadrature, and the function takes as input the radial grid r, the precomputed quantity r²f(r) r2_f, angular momentum / spherical bessel order l, and the Hankel coordinate p.

has_core_density(_::DFTK.Element) -> Any

Check presence of model core charge density (non-linear core correction).

index_G_vectors(
    fft_size::Tuple,
    G::AbstractVector{<:Integer}
) -> Any

Return the index tuple I such that G_vectors(basis)[I] == G or the index i such that G_vectors(basis, kpoint)[i] == G. Returns nothing if outside the range of valid wave vectors.

interpolate_density(
    ρ_in::AbstractArray{T, 4},
    basis_in::PlaneWaveBasis,
    basis_out::PlaneWaveBasis
) -> AbstractArray{T, 4} where T

Interpolate a density expressed in a basis basis_in to a basis basis_out. This interpolation uses a very basic real-space algorithm, and makes a DWIM-y attempt to take into account the fact that basis_out can be a supercell of basis_in.

interpolate_density(
    ρ_in::AbstractArray{T, 4},
    grid_in::Tuple{T, T, T} where T,
    grid_out::Tuple{T, T, T} where T,
    lattice_in,
    lattice_out
) -> AbstractArray{T, 4} where T

Interpolate a density in real space from one FFT grid to another, where lattice_in and lattice_out may be supercells of each other.

interpolate_density(
    ρ_in::AbstractArray{T, 4},
    grid_out::Tuple{T, T, T} where T
) -> AbstractArray{T, 4} where T

Interpolate a density in real space from one FFT grid to another. Assumes the lattice is unchanged.

interpolate_kpoint(
    data_in::AbstractVecOrMat,
    basis_in::PlaneWaveBasis,
    kpoint_in::Kpoint,
    basis_out::PlaneWaveBasis,
    kpoint_out::Kpoint
) -> Any

Interpolate some data from one $k$-point to another. The interpolation is fast, but not necessarily exact. Intended only to construct guesses for iterative solvers.

invert_refinement_metric(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    ψ,
    res
) -> Any

Invert the metric operator M.

irfft(
    fft_grid::FFTGrid{T, VT} where VT<:Real,
    f_fourier::AbstractArray
) -> Any

Perform a real valued iFFT; see ifft. Note that this function silently drops the imaginary part.

irreducible_kcoords(
    kgrid::MonkhorstPack,
    symmetries::AbstractVector{<:SymOp};
    check_symmetry
) -> Union{@NamedTuple{kcoords::Vector{StaticArraysCore.SVector{3, Rational{Int64}}}, kweights::Vector{Float64}}, @NamedTuple{kcoords::Vector{StaticArraysCore.SVector{3, Float64}}, kweights::Vector{Float64}}}

Construct the irreducible wedge given the crystal symmetries. Returns the list of k-point coordinates and the associated weights.

irreducible_kcoords_global(
    basis::PlaneWaveBasis
) -> Vector{T} where T<:(StaticArraysCore.SVector{3})

Utilities to get information about the irreducible k-point mesh (in case of duplication) Useful for I/O, where k-point information should not be duplicated

is_metal(eigenvalues, εF; tol) -> Bool

is_metal(eigenvalues, εF; tol)

Determine whether the provided bands indicate the material is a metal, i.e. where bands are cut by the Fermi level.

k_to_kpq_permutation(basis::PlaneWaveBasis, q) -> Vector

Returns a permutation indices of the $k$-points in basis such that kpoints[ik].coordinate + q is equivalent to kpoints[indices[ik]].coordinate.

kgrid_from_maximal_spacing(
    system::AtomsBase.AbstractSystem,
    spacing;
    kshift
) -> MonkhorstPack

Build a MonkhorstPack grid to ensure kpoints are at most this spacing apart (in inverse Bohrs). A reasonable spacing is 0.13 inverse Bohrs (around $2π * 0.04 \, \text{Å}^{-1}$). The kshift keyword argument allows to specify an explicit shift for all $k$-points.

kgrid_from_minimal_n_kpoints(
    system::AtomsBase.AbstractSystem,
    n_kpoints::Integer;
    kshift
) -> MonkhorstPack

Selects a MonkhorstPack grid size which ensures that at least a n_kpoints total number of $k$-points are used. The distribution of $k$-points amongst coordinate directions is as uniformly as possible, trying to achieve an identical minimal spacing in all directions.

krange_spin(basis::PlaneWaveBasis, spin::Integer) -> Any

Return the index range of $k$-points that have a particular spin component.

kwargs_scf_checkpoints(
    basis::DFTK.AbstractBasis;
    filename,
    callback,
    diagtolalg,
    ρ,
    ψ,
    save_ψ,
    kwargs...
) -> NamedTuple{(:callback, :diagtolalg, :ψ, :ρ), <:Tuple{ComposedFunction{ScfDefaultCallback, ScfSaveCheckpoints}, AdaptiveDiagtol, Any, Any}}

Transparently handle checkpointing by either returning kwargs for self_consistent_field, which start checkpointing (if no checkpoint file is present) or that continue a checkpointed run (if a checkpoint file can be loaded). filename is the location where the checkpoint is saved, save_ψ determines whether orbitals are saved in the checkpoint as well. The latter is discouraged, since generally slow.

list_psp(; ...) -> Any
list_psp(element; family, functional, core) -> Any

List the pseudopotential files known to DFTK. Allows various ways to restrict the displayed files.

Examples

julia> list_psp(; family="hgh")

will list all HGH-type pseudopotentials and

julia> list_psp(; family="hgh", functional="lda")

will only list those for LDA (also known as Pade in this context) and

julia> list_psp(:O, core=:semicore)

will list all oxygen semicore pseudopotentials known to DFTK.

load_psp(
    pseudofamily::AbstractDict{Symbol, <:AbstractString},
    system::AtomsBase.AbstractSystem;
    kwargs...
) -> Vector

Load all pseudopotentials from the pseudopotential family pseudofamily corresponding to the atoms of a system. Returns the list of the pseudopotential objects in the same order as the atoms in system. Takes care that each pseudopotential object is only loaded once. Applies the keyword arguments when loading all pseudopotentials. pseudofamily can be a PseudoPotentialData.PseudoFamily or simply a Dict{Symbol,String} which returns a file path when indexed with an element symbol.

load_psp(
    key::AbstractString;
    kwargs...
) -> Union{PspHgh{Float64}, PspUpf{_A, Interpolations.Extrapolation{T, 1, ITPT, IT, Interpolations.Throw{Nothing}}} where {_A, T, ITPT, IT}}

Load a pseudopotential file from the library of pseudopotentials. The file is searched in the directory datadir_psp() and by the key. If the key is a path to a valid file, the extension is used to determine the type of the pseudopotential file format and a respective class is returned.

load_scfres(filename::AbstractString; ...) -> Any
load_scfres(
    filename::AbstractString,
    basis;
    skip_hamiltonian,
    strict
) -> Any

Load back an scfres, which has previously been stored with save_scfres. Note the warning in save_scfres.

If basis is nothing, the basis is also loaded and reconstructed from the file, in which case architecture=CPU(). If a basis is passed, this one is used, which can be used to continue computation on a slightly different model or to avoid the cost of rebuilding the basis. If the stored basis and the passed basis are inconsistent (e.g. different FFT size, Ecut, k-points etc.) the load_scfres will error out.

By default the energies and ham (Hamiltonian object) are recomputed. To avoid this, set skip_hamiltonian=true. On errors the routine exits unless strict=false in which case it tries to recover from the file as much data as it can, but then the resulting scfres might not be fully consistent.

model_DFT(
    system::AtomsBase.AbstractSystem;
    pseudopotentials,
    functionals,
    kwargs...
)

Build a DFT model from the specified atoms with the specified XC functionals.

The functionals keyword argument takes either an Xc object, a list of objects subtyping DftFunctionals.Functional or a list of Symbols. For the latter any functional symbol from libxc can be specified, see examples below. Note, that most DFT models require two symbols in the functionals list (one for the exchange and one for the correlation part).

If functionals=[] (empty list), then a reduced Hartree-Fock model is constructed.

All other keyword arguments but functional are passed to model_atomic and from there to Model.

Note in particular that the pseudopotential keyword argument is mandatory to specify pseudopotential information. This can be easily achieved for example using the PseudoFamily struct from the PseudoPotentialData package as shown below:

Examples

julia> model_DFT(system; functionals=LDA(), temperature=0.01,
                 pseudopotentials=PseudoFamily("dojo.nc.sr.lda.v0_4_1.oncvpsp3.standard.upf"))

builds an LDA model for a passed system with specified smearing temperature.

julia> model_DFT(system; functionals=[:lda_x, :lda_c_pw], temperature=0.01,
                 pseudopotentials=PseudoFamily("dojo.nc.sr.lda.v0_4_1.oncvpsp3.standard.upf"))

Alternative syntax specifying the functionals directly via their libxc codes.

model_atomic(
    system::AtomsBase.AbstractSystem;
    pseudopotentials,
    kwargs...
)

Convenience constructor around Model, which builds a standard atomic (kinetic + atomic potential) model.

Keyword arguments

• pseudopotentials: Set the pseudopotential information for the atoms of the passed system. Can be (a) a list of pseudopotential objects (one for each atom), where a nothing element indicates that the Coulomb potential should be used for that atom or (b) a PseudoPotentialData.PseudoFamily to automatically determine the pseudopotential from the specified pseudo family or (c) a Dict{Symbol,String} mapping an atomic symbol to the pseudopotential to be employed.
• extra_terms: Specify additional terms to be passed to the Model constructor.
• kinetic_blowup: Specify a blowup function for the kinetic energy term, see e.g BlowupCHV.

Examples

julia> model_atomic(system; pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.oncvpsp3.standard.upf"))

Construct an atomic system using the specified pseudo-dojo pseudopotentials for all atoms of the system.

julia> model_atomic(system; pseudopotentials=Dict(:Si => "hgh/lda/si-q4"))

same thing, but specify the pseudopotentials explicitly in a dictionary.

mpi_nprocs() -> Int64
mpi_nprocs(comm) -> Int64

Number of processors used in MPI. Can be called without ensuring initialization.

multiply_ψ_by_blochwave(
    basis::PlaneWaveBasis,
    ψ,
    f_real,
    q
) -> Any

Return the Fourier coefficients for the Bloch waves $f^{\rm real}_{q} ψ_{k-q}$ in an element of basis.kpoints equivalent to $k-q$.

n_elec_core(el::DFTK.Element) -> Int64

Return the number of core electrons

n_elec_valence(el::DFTK.Element) -> Any

Return the number of valence electrons

n_electrons_from_atoms(atoms) -> Any

Number of valence electrons.

newton(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    ψ0;
    tol,
    tol_cg,
    maxiter,
    callback,
    is_converged
) -> NamedTuple{(:ham, :basis, :energies, :converged, :ρ, :eigenvalues, :occupation, :εF, :n_iter, :ψ, :stage, :algorithm, :runtime_ns), <:Tuple{Hamiltonian, PlaneWaveBasis, Energies, Any, AbstractArray{_A, 4} where _A, Vector{Any}, Vector, Nothing, Int64, Any, Symbol, String, UInt64}}

newton(basis::PlaneWaveBasis{T}, ψ0;
       tol=1e-6, tol_cg=tol / 100, maxiter=20, callback=ScfDefaultCallback(),
       is_converged=ScfConvergenceDensity(tol))

Newton algorithm. Be careful that the starting point needs to be not too far from the solution.

next_compatible_fft_size(
    size::Int64;
    smallprimes,
    factors
) -> Int64

Find the next compatible FFT size Sizes must (a) be a product of small primes only and (b) contain the factors. If smallprimes is empty (a) is skipped.

next_density(
    ham::Hamiltonian;
    ...
) -> NamedTuple{(:ψ, :eigenvalues, :occupation, :εF, :ρout, :diagonalization, :n_bands_converge, :occupation_threshold, :n_matvec), <:Tuple{Vector, Vector, Vector, Number, AbstractArray{_A, 4} where _A, NamedTuple{(:λ, :X, :residual_norms, :n_iter, :converged, :n_matvec), <:Tuple{Vector, Vector, Vector, Vector, Union{Missing, Bool}, Any}}, Int64, Float64, Any}}
next_density(
    ham::Hamiltonian,
    nbandsalg::DFTK.NbandsAlgorithm;
    ...
) -> NamedTuple{(:ψ, :eigenvalues, :occupation, :εF, :ρout, :diagonalization, :n_bands_converge, :occupation_threshold, :n_matvec), <:Tuple{Vector, Vector, Vector, Number, AbstractArray{_A, 4} where _A, NamedTuple{(:λ, :X, :residual_norms, :n_iter, :converged, :n_matvec), <:Tuple{Vector, Vector, Vector, Vector, Union{Missing, Bool}, Any}}, Int64, Any, Any}}
next_density(
    ham::Hamiltonian,
    nbandsalg::DFTK.NbandsAlgorithm,
    fermialg::AbstractFermiAlgorithm;
    eigensolver,
    ψ,
    eigenvalues,
    occupation,
    kwargs...
) -> NamedTuple{(:ψ, :eigenvalues, :occupation, :εF, :ρout, :diagonalization, :n_bands_converge, :occupation_threshold, :n_matvec), <:Tuple{Vector, Vector, Vector, Number, AbstractArray{_A, 4} where _A, NamedTuple{(:λ, :X, :residual_norms, :n_iter, :converged, :n_matvec), <:Tuple{Vector, Vector, Vector, Vector, Union{Missing, Bool}, Any}}, Int64, Any, Any}}

Obtain new density ρ by diagonalizing ham. Follows the policy imposed by the bands data structure to determine and adjust the number of bands to be computed.

norm2(G) -> Any

Square of the ℓ²-norm.

norm_cplx(x) -> Any

Complex-analytic extension of LinearAlgebra.norm(x) from real to complex inputs. Needed for phonons as we want to perform a matrix-vector product f'(x)·h, where f is a real-to-real function and h a complex vector. To do this using automatic differentiation, we can extend analytically f to accept complex inputs, then differentiate t -> f(x+t·h). This will fail if non-analytic functions like norm are used for complex inputs, and therefore we have to redefine it.

normalize_kpoint_coordinate(x::Real) -> Any

Bring $k$-point coordinates into the range [-0.5, 0.5)

overlap_Mmn_k_kpb(
    basis::PlaneWaveBasis,
    ψ,
    ik,
    ik_plus_b,
    G_shift,
    n_bands
) -> Any

Computes the matrix $[M^{k,b}]_{m,n} = \langle u_{m,k} | u_{n,k+b} \rangle$ for given k.

G_shift is the "shifting" vector, correction due to the periodicity conditions imposed on $k \to ψ_k$. It is non zero if k_plus_b is taken in another unit cell of the reciprocal lattice. We use here that: $u_{n(k + G_{\rm shift})}(r) = e^{-i*\langle G_{\rm shift},r \rangle} u_{nk}$.

parallel_loop_over_range(fun, range; allocate_local_storage)

Parallelize a loop, calling fun(i) for side effects for all i in range. If allocatelocalstorage is not nothing, fun is called as fun(i, st) where st is a thread-local temporary storage allocated by allocate_local_storage().

pcut_psp_local(psp::PspHgh{T}) -> Any

Estimate an upper bound for the argument p after which abs(eval_psp_local_fourier(psp, p)) is a strictly decreasing function.

pcut_psp_projector(psp::PspHgh{T}, i, l) -> Any

Estimate an upper bound for the argument p after which eval_psp_projector_fourier(psp, p) is a strictly decreasing function.

phonon_modes(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    ψ,
    occupation;
    kwargs...
) -> NamedTuple{(:mass_matrix, :frequencies, :dynmat, :dynmat_cart, :vectors, :vectors_cart), <:NTuple{6, Any}}

Get phonon quantities. We return the frequencies, the mass matrix and reduced and Cartesian eigenvectors and dynamical matrices.

Compute and plot the band structure. Kwargs are like in compute_bands. Requires Plots.jl to be loaded to be defined and working properly. The unit used to plot the bands can be selected using the unit parameter. Like in the rest of DFTK Hartree is used by default. Another standard choices is unit=u"eV" (electron volts).

Plot the density of states over a reasonable range. Requires to load Plots.jl beforehand.

psp_local_polynomial(T, psp::PspHgh) -> Any
psp_local_polynomial(T, psp::PspHgh, t) -> Any

The local potential of a HGH pseudopotentials in reciprocal space can be brought to the form $Q(t) / (t^2 exp(t^2 / 2))$ where $t = r_\text{loc} p$ and Q is a polynomial of at most degree 8. This function returns Q.

psp_projector_polynomial(T, psp::PspHgh, i, l) -> Any
psp_projector_polynomial(T, psp::PspHgh, i, l, t) -> Any

The nonlocal projectors of a HGH pseudopotentials in reciprocal space can be brought to the form $Q(t) exp(-t^2 / 2)$ where $t = r_l p$ and Q is a polynomial. This function returns Q.

r_vectors(
    basis::PlaneWaveBasis
) -> AbstractArray{StaticArraysCore.SVector{3, VT}, 3} where VT<:Real

r_vectors(basis::PlaneWaveBasis)

The list of $r$ vectors, in reduced coordinates. By convention, this is in [0,1)^3.

r_vectors_cart(basis::PlaneWaveBasis) -> Any

r_vectors_cart(basis::PlaneWaveBasis)

The list of $r$ vectors, in Cartesian coordinates.

radial_hydrogenic(
    r::AbstractArray{T<:Real, 1},
    n::Integer
) -> Any
radial_hydrogenic(
    r::AbstractArray{T<:Real, 1},
    n::Integer,
    α::Real
) -> Any

Radial functions from solutions of Hydrogenic Schrödinger equation. Same as Wannier90 user guide Table 3.3.

Arguments

• r: radial grid
• n: principal quantum number
• α: diffusivity, $\frac{Z}{a}$ where $Z$ is the atomic number and $a$ is the Bohr radius.

random_density(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    n_electrons::Integer
) -> Any

Build a random charge density normalized to the provided number of electrons.

read_w90_nnkp(
    fileprefix::String
) -> @NamedTuple{nntot::Int64, nnkpts::Vector{@NamedTuple{ik::Int64, ik_plus_b::Int64, G_shift::Vector{Int64}}}}

Read the .nnkp file provided by the preprocessing routine of Wannier90 (i.e. "wannier90.x -pp prefix") Returns:

1. the array 'nnkpts' of k points, their respective nearest neighbors and associated shifing vectors (non zero if the neighbor is located in another cell).
2. the number 'nntot' of neighbors per k point.

TODO: add the possibility to exclude bands

reducible_kcoords(
    kgrid::MonkhorstPack
) -> @NamedTuple{kcoords::Vector{StaticArraysCore.SVector{3, Rational{Int64}}}}

Construct the coordinates of the k-points in a (shifted) Monkhorst-Pack grid

refine_energies(
    refinement::DFTK.RefinementResult{T}
) -> NamedTuple{(:E, :dE), <:Tuple{Energies, Energies}}

Refine energies using a RefinementResult.

The refined energies can be obtained by E + dE.

refine_forces(
    refinement::DFTK.RefinementResult{T}
) -> NamedTuple{(:F, :dF), <:Tuple{Any, Any}}

Refine forces using a RefinementResult.

The refined forces can be obtained by F + dF.

refine_scfres(
    scfres,
    basis_ref::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})};
    tol,
    occ_threshold,
    kwargs...
) -> DFTK.RefinementResult

Transfer the result of an SCF to a larger basis set, and compute approximate first order corrections ("refinements") to the wavefunctions and density.

Only full occupations are currently supported.

Returns a RefinementResult instance that can be used to refine quantities of interest, through refine_energies and refine_forces.

Wannerize the obtained bands using wannier90. By default all converged bands from the scfres are employed (change with n_bands kwargs) and n_wannier = n_bands wannier functions are computed. Random Gaussians are used as guesses by default, can be changed using the projections kwarg. All keyword arguments supported by Wannier90 for the disentanglement may be added as keyword arguments. The function returns the fileprefix.

save_bands(
    filename::AbstractString,
    band_data::NamedTuple;
    save_ψ
)

Write the computed bands to a file. On all processes, but the master one the filename is ignored. save_ψ determines whether the wavefunction is also saved or not. Note that this function can be both used on the results of compute_bands and self_consistent_field.

save_scfres(
    filename::AbstractString,
    scfres::NamedTuple;
    save_ψ,
    extra_data,
    compress,
    save_ρ
) -> Any

Save an scfres obtained from self_consistent_field to a file. On all processes but the master one the filename is ignored. The format is determined from the file extension. Currently the following file extensions are recognized and supported:

• jld2: A JLD2 file. Stores the complete state and can be used (with load_scfres) to restart an SCF from a checkpoint or post-process an SCF solution. Note that this file is also a valid HDF5 file, which can thus similarly be read by external non-Julia libraries such as h5py or similar. See Saving SCF results on disk and SCF checkpoints for details.
• vts: A VTK file for visualisation e.g. in paraview. Stores the density, spin density, optionally bands and some metadata.
• json: A JSON file with basic information about the SCF run. Stores for example the number of iterations, occupations, some information about the basis, eigenvalues, Fermi level etc.

Keyword arguments:

• save_ψ: Save the orbitals as well (may lead to larger files). This is the default for jld2, but false for all other formats, where this is considerably more expensive.
• save_ρ: Save the density as well (may lead to larger files). This is the default for all but json.
• extra_data: Additional data to place into the file. The data is just copied like fp["key"] = value, where fp is a JLD2.JLDFile, WriteVTK.vtk_grid and so on.
• compress: Apply compression to array data. Requires the CodecZlib package to be available.

scatter_kpts_block(
    basis::PlaneWaveBasis,
    data::Union{Nothing, AbstractArray}
) -> Any

Scatter the data of a quantity depending on k-Points from the master process to the child processes and return it as a Vector{Array}, where the outer vector is a list over all k-points. On non-master processes nothing may be passed.

scf_anderson_solver(
;
    kwargs...
) -> DFTK.var"#anderson#783"{DFTK.var"#anderson#782#784"{Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}}

Create a simple anderson-accelerated SCF solver.

scf_damping_quadratic_model(
    info,
    info_next;
    modeltol
) -> NamedTuple{(:α, :relerror), <:Tuple{Any, Any}}

Use the two iteration states info and info_next to find a damping value from a quadratic model for the SCF energy. Returns nothing if the constructed model is not considered trustworthy, else returns the suggested damping.

scf_damping_solver(
;
    damping
) -> DFTK.var"#fp_solver#779"{DFTK.var"#fp_solver#778#780"{Float64}}

Create a damped SCF solver updating the density as x = damping * x_new + (1 - damping) * x

scfres_to_dict(
    scfres::NamedTuple;
    kwargs...
) -> Dict{String, Any}

Convert an scfres to a dictionary representation. Intended to give a condensed set of results and useful metadata for post processing. See also the todict function for the Model and the PlaneWaveBasis as well as the band_data_to_dict functions, which are called by this function and their outputs merged. Only the master process returns meaningful data.

Some details on the conventions for the returned data:

• ρ: (fftsize[1], fftsize[2], fftsize[3], nspin) array of density on real-space grid.
• energies: Dictionary / subdirectory containing the energy terms
• converged: Has the SCF reached convergence
• norm_Δρ: Most recent change in ρ during an SCF step
• occupation_threshold: Threshold below which orbitals are considered unoccupied
• n_bands_converge: Number of bands that have been fully converged numerically.
• n_iter: Number of iterations.

select_eigenpairs_all_kblocks(eigres, range) -> NamedTuple

Function to select a subset of eigenpairs on each $k$-Point. Works on the Tuple returned by diagonalize_all_kblocks.

self_consistent_field(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})};
    ρ,
    ψ,
    tol,
    is_converged,
    miniter,
    maxiter,
    maxtime,
    mixing,
    damping,
    solver,
    eigensolver,
    diagtolalg,
    nbandsalg,
    fermialg,
    callback,
    compute_consistent_energies,
    response
) -> NamedTuple{(:ham, :basis, :energies, :ρ, :eigenvalues, :occupation, :εF, :ψ, :response, :converged, :occupation_threshold, :α, :n_iter, :n_bands_converge, :diagonalization, :stage, :algorithm, :runtime_ns), <:Tuple{Hamiltonian, PlaneWaveBasis{T, VT, Arch, FFTtype} where {T<:ForwardDiff.Dual, VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})}, Energies, Any, Any, Any, Any, Any, Any, Any, Float64, Float64, Any, Any, Any, Symbol, String, UInt64}}

self_consistent_field(basis; [tol, mixing, damping, ρ, ψ])

Solve the Kohn-Sham equations with a density-based SCF algorithm using damped, preconditioned iterations where $ρ_\text{next} = ρ_\text{in} + α P^{-1} (ρ_\text{out} - ρ_\text{in})$.

Overview of parameters:

• ρ: Initial density
• ψ: Initial orbitals
• tol: Tolerance for the density change ($\|ρ_\text{out} - ρ_\text{in}\|$) to flag convergence. Default is 1e-6.
• is_converged: Convergence control callback. Typical objects passed here are ScfConvergenceDensity(tol) (the default), ScfConvergenceEnergy(tol) or ScfConvergenceForce(tol).
• miniter: Minimal number of SCF iterations
• maxiter: Maximal number of SCF iterations
• maxtime: Maximal time to run the SCF for. If this is reached without convergence, the SCF stops.
• mixing: Mixing method, which determines the preconditioner $P^{-1}$ in the above equation. Typical mixings are LdosMixing, KerkerMixing, SimpleMixing or DielectricMixing. Default is LdosMixing()
• damping: Damping parameter $α$ in the above equation. Default is 0.8.
• nbandsalg: By default DFTK uses nbandsalg=AdaptiveBands(model), which adaptively determines the number of bands to compute. If you want to influence this algorithm or use a predefined number of bands in each SCF step, pass a FixedBands or AdaptiveBands. Beware that with non-zero temperature, the convergence of the SCF algorithm may be limited by the default_occupation_threshold() parameter. For highly accurate calculations we thus recommend increasing the occupation_threshold of the AdaptiveBands.
• callback: Function called at each SCF iteration. Usually takes care of printing the intermediate state.

setup_threading(
;
    n_fft,
    n_blas,
    n_DFTK
) -> Union{Nothing, Bool}

Setup the number of threads used by DFTK's own threading (n_DFTK), by BLAS (n_blas) and by FFTW (n_fft). This is independent from the number of Julia threads (Threads.nthreads()). DFTK and FFTW threading are upper bounded by Threads.nthreads(), but not BLAS, which uses its own threading system. By default, use 1 FFT thread, and Threads.nthreads() BLAS and DFTK threads.

Integrate the integrand function using the nodal points x using Simpson's rule. The function will be called as integrand(i, x[i]) for each integrand point i (not necessarily in order).

simpson(y::AbstractVector, x::AbstractVector) -> Any

Integrate a function represented by the nodal points and function values given by the arrays x, y. Note the order (y comes first).

sin2pi(x) -> Any

Function to compute sin(2π x)

solve_ΩplusK(
    basis::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    ψ,
    rhs,
    occupation;
    callback,
    tol
) -> NamedTuple{(:δψ, :converged, :tol, :residual_norm, :n_iter), <:Tuple{Any, Bool, Float64, Any, Int64}}

Solve density-functional perturbation theory problem, that is return δψ where (Ω+K) δψ = rhs.

solve_ΩplusK_split(
    ham::Hamiltonian,
    ρ::AbstractArray{T},
    ψ,
    occupation,
    εF,
    eigenvalues,
    rhs;
    tol,
    tol_sternheimer,
    verbose,
    occupation_threshold,
    q,
    kwargs...
)

Solve the problem (Ω+K) δψ = rhs (density-functional perturbation theory) using a split algorithm, where rhs is typically -δHextψ (the negative matvec of an external perturbation with the SCF orbitals ψ) and δψ is the corresponding total variation in the orbitals ψ. Additionally returns: - δρ: Total variation in density) - δHψ: Total variation in Hamiltonian applied to orbitals - δeigenvalues: Total variation in eigenvalues - δVind: Change in potential induced by δρ (the term needed on top of δHextψ to get δHψ).

spglib_standardize_cell(
    lattice::AbstractArray{T},
    atom_groups,
    positions;
    ...
) -> NamedTuple{(:lattice, :atom_groups, :positions, :magnetic_moments), <:Tuple{Matrix, Any, Any, Vector{StaticArraysCore.SVector{3, Float64}}}}
spglib_standardize_cell(
    lattice::AbstractArray{T},
    atom_groups,
    positions,
    magnetic_moments;
    correct_symmetry,
    primitive,
    tol_symmetry
) -> NamedTuple{(:lattice, :atom_groups, :positions, :magnetic_moments), <:Tuple{Matrix, Any, Any, Vector{StaticArraysCore.SVector{3, Float64}}}}

Returns crystallographic conventional cell according to the International Table of Crystallography Vol A (ITA) in case primitive=false. If primitive=true the primitive lattice is returned in the convention of the reference work of Cracknell, Davies, Miller, and Love (CDML). Of note this has minor differences to the primitive setting choice made in the ITA.

sphericalbesselj_fast(l::Integer, x) -> Any

sphericalbesselj_fast(l::Integer, x::Number)

Returns the spherical Bessel function of the first kind $j_l(x)$. Consistent with Wikipedia and with SpecialFunctions.sphericalbesselj. Specialized for integer $0 ≤ l ≤ 5$.

spin_components(
    spin_polarization::Symbol
) -> Union{Bool, Tuple{Symbol}, Tuple{Symbol, Symbol}}

Explicit spin components of the KS orbitals and the density

split_evenly(itr, N) -> Any

Split an iterable evenly into N chunks, which will be returned.

standardize_atoms(
    lattice,
    atoms,
    positions;
    ...
) -> NamedTuple{(:lattice, :atoms, :positions, :magnetic_moments), <:Tuple{Matrix, Any, Any, Vector{StaticArraysCore.SVector{3, Float64}}}}
standardize_atoms(
    lattice,
    atoms,
    positions,
    magnetic_moments;
    kwargs...
) -> NamedTuple{(:lattice, :atoms, :positions, :magnetic_moments), <:Tuple{Matrix, Any, Any, Vector{StaticArraysCore.SVector{3, Float64}}}}

Apply various standardisations to a lattice and a list of atoms. It uses spglib to detect symmetries (within tol_symmetry), then cleans up the lattice according to the symmetries (unless correct_symmetry is false) and returns the resulting standard lattice and atoms. If primitive is true (default) the primitive unit cell is returned, else the conventional unit cell is returned.

symmetries_preserving_kgrid(symmetries, kcoords) -> Any

Filter out the symmetry operations that don't respect the symmetries of the discrete BZ grid

symmetries_preserving_rgrid(symmetries, fft_size) -> Any

Filter out the symmetry operations that don't respect the symmetries of the discrete real-space grid

symmetrize_forces(model::Model, forces; symmetries)

Symmetrize the forces in reduced coordinates, forces given as an array forces[iel][α,i].

symmetrize_stresses(model::Model, stresses; symmetries)

Symmetrize the stress tensor, given as a Matrix in Cartesian coordinates

symmetrize_ρ(
    basis,
    ρ::AbstractArray{T};
    symmetries,
    do_lowpass
) -> Any

Symmetrize a density by applying all the basis (by default) symmetries and forming the average.

symmetry_operations(
    lattice,
    atoms,
    positions;
    ...
) -> Union{Vector{SymOp{Bool}}, Vector{SymOp{Float64}}}
symmetry_operations(
    lattice,
    atoms,
    positions,
    magnetic_moments;
    tol_symmetry,
    check_symmetry
) -> Union{Vector{SymOp{Bool}}, Vector{SymOp{Float64}}}

Return the symmetries given an atomic structure with optionally designated magnetic moments on each of the atoms. The symmetries are determined using spglib.

symmetry_operations(
    hall_number::Integer
) -> Vector{SymOp{Float64}}

Return the Symmetry operations given a hall_number.

This function allows to directly access to the space group operations in the spglib database. To specify the space group type with a specific choice, hall_number is used.

The definition of hall_number is found at Space group type.

synchronize_device(_::DFTK.AbstractArchitecture)

Synchronize data and finish all operations on the execution stream of the device. This needs to be called explicitly before a task finishes (e.g. in an @spawn block).

to_cpu(x::AbstractArray) -> Array

Transfer an array from a device (typically a GPU) to the CPU.

to_device(_::DFTK.CPU, x) -> Any

Transfer an array to a particular device (typically a GPU)

todict(energies::Energies) -> Dict

Convert an Energies struct to a dictionary representation

todict(model::Model) -> Dict{String, Any}

Convert a Model struct to a dictionary representation. Intended to give a condensed set of useful metadata to post-processing scripts or for storing computational results (e.g. bands, bloch waves etc.).

Some details on the conventions for the returned data:

• lattice, recip_lattice: Always a zero-padded 3x3 matrix, independent on the actual dimension
• atomic_positions, atomic_positions_cart: Atom positions in fractional or Cartesian coordinates, respectively.
• atomic_symbols: Atomic symbols if known.
• terms: Some rough information on the terms used for the computation.
• n_electrons: Number of electrons, may be missing if εF is fixed instead
• εF: Fixed Fermi level to use, may be missing if n_electrons is is specified instead.

todict(basis::PlaneWaveBasis) -> Dict{String, Any}

Convert a PlaneWaveBasis struct to a dictionary representation. Intended to give a condensed set of useful metadata to post-processing scripts or for storing computational results (e.g. bands, bloch waves etc.). As such the function is lossy and might not keep all data consistently. Returns the same result on all MPI processors. See also the todict function for the Model, which is called from this one to merge the data of both outputs.

Some details on the conventions for the returned data:

• dvol: Volume element for real-space integration
• variational: Is the k-point specific basis (for ψ) variationally consistent with the basis for ρ.
• kweights: Weights for the k-points, summing to 1.0

total_local_potential(ham::Hamiltonian) -> Any

Get the total local potential of the given Hamiltonian, in real space in the spin components.

transfer_blochwave(
    ψ_in,
    basis_in::PlaneWaveBasis,
    basis_out::PlaneWaveBasis
) -> Vector

Transfer Bloch wave between two basis sets. Limited feature set.

transfer_blochwave_equivalent_to_actual(
    basis,
    ψ_plus_q_equivalent,
    q
) -> Any

For Bloch waves $ψ$ such that ψ[ik] is defined in a point in basis.kpoints equivalent to basis.kpoints[ik] + q, return the Bloch waves ψ_plus_q[ik] defined on kpt_plus_q[ik].

transfer_blochwave_kpt(
    ψk_in,
    basis_in::PlaneWaveBasis,
    kpt_in::Kpoint,
    basis_out::PlaneWaveBasis,
    kpt_out::Kpoint
) -> Any

Transfer an array ψk defined on basisin $k$-point kptin to basisout $k$-point `kptout; see [transfer_mapping`](@ref).

transfer_density(
    ρ_in,
    basis_in::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})},
    basis_out::PlaneWaveBasis{T, VT, Arch, FFTtype} where {VT<:Real, Arch<:DFTK.AbstractArchitecture, FFTtype<:(FFTGrid{T, VT, T_G_vectors, T_r_vectors} where {T_G_vectors<:AbstractArray{StaticArraysCore.SVector{3, Int64}, 3}, T_r_vectors<:AbstractArray{StaticArraysCore.SVector{3, VT}, 3}})}
) -> Any

Transfer density (in real space) between two basis sets.

This function is fast by transferring only the Fourier coefficients from the small basis to the big basis.

Note that this implies that for even-sized small FFT grids doing the transfer small -> big -> small is not an identity (as the small basis has an unmatched Fourier component and the identity $c_G = c_{-G}^\ast$ does not fully hold).

Note further that for the direction big -> small employing this function does not give the same answer as using first transfer_blochwave and then compute_density.

transfer_mapping(
    basis_in::PlaneWaveBasis,
    kpt_in::Kpoint,
    basis_out::PlaneWaveBasis,
    kpt_out::Kpoint
) -> Tuple{Any, Any}

Compute the index mapping between two bases. Returns two arrays idcs_in and idcs_out such that ψk_out[idcs_out] = ψk_in[idcs_in] does the transfer from ψk_in (defined on basis_in and kpt_in) to ψk_out (defined on basis_out and kpt_out).

Note that kpt_out does not have to belong to basis_out as long as it is equivalent to some other point in it (kpt_out = kpt_in + ΔG). In that case, the transfer does not change the Bloch wave $ψ$. It does change the periodic part $u$: $e^{i k·x} u_k(x) = e^{i (k+ΔG)·x} (e^{-i ΔG·x} u_k(x))$. Beware: this is a lossy conversion in general.

transfer_mapping(
    basis_in::PlaneWaveBasis,
    basis_out::PlaneWaveBasis
) -> Base.Iterators.Zip{Tuple{Array{CartesianIndices{3, Tuple{UnitRange{Int64}, UnitRange{Int64}, UnitRange{Int64}}}, 3}, Array{CartesianIndices{3, Tuple{UnitRange{Int64}, UnitRange{Int64}, UnitRange{Int64}}}, 3}}}

Compute the index mapping between the global grids of two bases. Returns an iterator of 8 pairs (block_in, block_out). Iterated over these pairs x_out_fourier[block_out, :] = x_in_fourier[block_in, :] does the transfer from the Fourier coefficients x_in_fourier (defined on basis_in) to x_out_fourier (defined on basis_out, equally provided as Fourier coefficients).

Integrate the integrand function using the nodal points x using the trapezoidal rule. The function will be called as integrand(i, x[i]) for each integrand point i (not necessarily in order).

trapezoidal(y::AbstractVector, x::AbstractVector) -> Any

Integrate a function represented by the nodal points and function values given by the arrays x, y. Note the order (y comes first).

unfold_bz(basis::PlaneWaveBasis) -> PlaneWaveBasis

" Convert a basis into one that doesn't use BZ symmetry. This is mainly useful for debug purposes (e.g. in cases we don't want to bother thinking about symmetries).

versioninfo()
versioninfo(io::IO)

DFTK.versioninfo([io::IO=stdout])

Summary of version and configuration of DFTK and its key dependencies.

weighted_ksum(basis::PlaneWaveBasis, array) -> Any

Sum an array over kpoints, taking weights into account

write_w90_eig(fileprefix::String, eigenvalues; n_bands)

Write the eigenvalues in a format readable by Wannier90.

write_w90_win(
    fileprefix::String,
    basis::PlaneWaveBasis;
    bands_plot,
    wannier_plot,
    kwargs...
)

Write a win file at the indicated prefix. Parameters to Wannier90 can be added as kwargs: e.g. num_iter=500.

write_wannier90_files(
    preprocess_call,
    scfres;
    n_bands,
    n_wannier,
    projections,
    fileprefix,
    wannier_plot,
    kwargs...
)

Shared file writing code for Wannier.jl and Wannier90.

ylm_real(
    l::Integer,
    m::Integer,
    rvec::AbstractArray{T, 1}
) -> Any

Returns the $(l,m)$ real spherical harmonic $Y_l^m(r)$. Consistent with Wikipedia.

zeros_like(X::AbstractArray) -> Any
zeros_like(
    X::AbstractArray,
    T::Type,
    dims::Integer...
) -> Any

Create an array of same "array type" as X filled with zeros, minimizing the number of allocations. This unifies CPU and GPU code, as the output will always be on the same device as the input.

Shortened version of the @timeit macro from TimerOutputs, which writes to the DFTK timer.

Fermi-Dirac smearing

Gaussian Smearing

Marzari Vanderbilt cold smearing. NB: The Fermi energy with Marzari-Vanderbilt smearing is not unique.

Methfessel-Paxton smearing of a given order. NB: The Fermi energy with Methfessel-Paxton smearing is not unique.

No smearing

A term in the Hermite delta expansion

Standard Hermite function using physicist's convention.

Entropy. Note that this is a function of the energy x, not of occupation(x). This function satisfies s' = x f' (see https://www.vasp.at/vasp-workshop/k-points.pdf p. 12 and https://arxiv.org/pdf/1805.07144.pdf p. 18.

occupation(S::SmearingFunction, x)

Occupation at x, where in practice x = (ε - εF) / temperature. If temperature is zero, (ε-εF)/temperature = ±∞. The occupation function is required to give 1 and 0 respectively in these cases.

Derivative of the occupation function, approximation to minus the delta function.

(f(x) - f(y))/(x - y), computed stably in the case where x and y are close