API reference

DFTK –- The density-functional toolkit. Provides functionality for experimenting with plane-wave density-functional theory algorithms.

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.

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. $\text{Energy} = \sum_a \sum_{ij} \sum_{n} f_n <ψ_n|p_{ai}> D_{ij} <p_{aj}|ψ_n>.$

Blow-up function as used in Abinit.

Blow-up function as proposed in https://arxiv.org/abs/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.

We use a simplification of the Resta model DOI 10.1103/physrevb.16.2717 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 $A=1$).

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 an element name (e.g. "silicon")

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 an element name (e.g. "silicon")

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 an element name (e.g. "silicon")

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.

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.

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) = multiplier(G) ψ(G).

Construct a particular GPU architecture by passing the ArrayType

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 = sum{R} 4π δR, integral of G zero on a unit cell).

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 sumn fn ∫ |∇ψn|^2 * blowup(-i∇Ψ).

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

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 arXiv:2009.01665

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.

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.

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).

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 PlaneWaveBasis using the kinetic energy cutoff Ecut and a Monkhorst-Pack $k$-point grid. The MP grid can either be specified directly with kgrid providing the number of points in each dimension and kshift the shift (0 or 1/2 in each direction). If not specified a grid is generated using kgrid_from_minimal_spacing with a minimal spacing of 2π * 0.022 per Bohr.

Creates a new basis identical to basis, but with a custom set of kpoints

No preconditioning

(simplified version of) Tetter-Payne-Allan preconditioning ↑ M.P. Teter, M.C. Payne and D.C. Allan, Phys. Rev. B 40, 12255 (1989).

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:

• Non-linear core correction
• 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).

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).

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

fft(basis::PlaneWaveBasis, [kpt::Kpoint, ] f_real)

Perform an FFT to obtain the Fourier representation of f_real. If kpt is given, the coefficients are truncated to the k-dependent spherical basis set.

In-place version of ifft.

ifft(basis::PlaneWaveBasis, [kpt::Kpoint, ] f_fourier)

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

Chemical symbol corresponding to an element

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.

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.

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 sgnum keyword argument.

CROP-accelerated root-finding iteration for f, starting from x0 and keeping a history of m steps. Optionally warming specifies the number of non-accelerated steps to perform for warming up the history.

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([architecture=AbstractArchitecture], fft_size::Tuple)

The wave vectors G in reduced (integer) coordinates for a cubic basis set of given sizes.

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)

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

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

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

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

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 arXiv:2009.01665

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

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.

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 arXiv:2009.01665.

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

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.

Flag convergence by using the L2Norm of the change between input density and unpreconditioned output density (ρout)

Flag convergence as soon as total energy change drops below tolerance

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

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

Determine the tolerance used for the next diagonalization. This function takes $|ρnext - ρin|$ 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.

Plot the trace of an SCF, i.e. the absolute error of the total energy at each iteration versus the converged energy in a semilog plot. By default a new plot canvas is generated, but an existing one can be passed and reused along with kwargs for the call to plot!. Requires Plots to be loaded.

Adds simplistic checkpointing to a DFTK self-consistent field calculation. Requires JLD2 to be loaded.

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; kwargs...)

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

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 a symmetry operation to a density.

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.

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

attach_psp(system::AbstractSystem, pspmap::AbstractDict{Symbol,String})
attach_psp(system::AbstractSystem; psps::String...)

Return a new system with the pseudopotential property of all atoms set according to the passed pspmap, which maps from the atomic symbol to a pseudopotential identifier. Alternatively the mapping from atomic symbol to pseudopotential identifier can also be passed as keyword arguments. An empty string can be used to denote elements where the full Coulomb potential should be employed.

Examples

Select pseudopotentials for all silicon and oxygen atoms in the system.

julia> attach_psp(system, Dict(:Si => "hgh/lda/si-q4", :O => "hgh/lda/o-q6")

Same thing but using the kwargs syntax:

julia> attach_psp(system, Si="hgh/lda/si-q4", O="hgh/lda/o-q6")

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 (Fourier transforms of projectors) for an atom centered at 0.

Build projection vectors for a atoms array generated by term_nonlocal

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

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

where $\hat p_i(q) = ∫_{ℝ^3} p_i(r) e^{-iq·r} dr$.

We store $\frac{1}{\sqrt Ω} \hat p_i(k+G)$ in proj_vectors.

 bzmesh_ir_wedge(kgrid_size, symmetries; kshift=[0, 0, 0])

Construct the irreducible wedge of a uniform Brillouin zone mesh for sampling $k$-points, given the crystal symmetries symmetries. Returns the list of irreducible $k$-point (fractional) coordinates, the associated weights and the new symmetries compatible with the grid.

bzmesh_uniform(kgrid_size; kshift=[0, 0, 0])

Construct a (shifted) uniform Brillouin zone mesh for sampling the $k$-points. Returns all $k$-point coordinates, appropriate weights and the identity SymOp.

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.

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.

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.

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

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

Return the total nuclear charge of an atom type

Function to compute exp(2π i x)

Compute the matrix $[A_k]_{m,n} = \langle ψ_m^k | g^{\text{per}}_n \rangle$

$g^{per}_n$ are periodized gaussians whose respective centers are given as an (num_bands,1) array [ [center 1], ... ].

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.

Computes the probability (not charge) current, ∑ fn Im(ψn* ∇ψn)

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.

Total density of states at energy ε

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

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

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

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

Compute occupation given eigenvalues and Fermi level

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 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 the stresses (= 1/Vol dE/d(M*lattice), taken at M=I) of an obtained SCF solution.

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

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

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.

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.

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:

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

Function to compute cos(2π x)

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, l)

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

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, l)

Number of projector radial functions at angular momentum l.

count_n_proj_radial(psp)

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

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

Return the data directory with pseudopotential files

Default selection of a Fermi level determination algorithm

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

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

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

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)

Compute the diameter of the unit cell

Computes the ground state by direct minimization. kwargs... are passed to Optim.Options(). Note that the resulting ψ are not necessarily eigenvectors of the Hamiltonian.

Convenience function to disable all threading in DFTK and assert that Julia threading is off as well.

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.

Compute the electrostatic interaction energy per unit cell between point charges in a uniform background of compensating charge to yield net neutrality.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. If forces is not nothing, minus the derivatives of the energy with respect to positions is computed.

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.

Compute the pairwise interaction energy per unit cell between atomic sites. If forces is not nothing, minus the derivatives of the energy with respect to positions is computed. The potential is expected to decrease quickly at infinity.

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

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 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(psp, q)

Evaluate the atomic core charge density in reciprocal space: ρval(q) = ∫{R^3} ρcore(r) e^{-iqr} dr = 4π ∫{R+} ρcore(r) sin(qr)/qr r^2 dr

eval_psp_density_core_real(psp, r)

Evaluate the atomic core charge density in real space.

eval_psp_density_valence_fourier(psp, q)

Evaluate the atomic valence charge density in reciprocal space: ρval(q) = ∫{R^3} ρval(r) e^{-iqr} dr = 4π ∫{R+} ρval(r) sin(qr)/qr r^2 dr

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 $q \to 0$, using the same correction as in the Fourier-transform of the local potential: math \lim_{q \to 0} 4π N_{\rm elec} ∫_{ℝ_+} (V(r) - C(r)) \frac{\sin(qr)}{qr} r^2 dr + F[C(r)] = 4π N_{\rm elec} ∫_{ℝ_+} (V(r) + Z/r) r^2 dr

eval_psp_local_fourier(psp, q)

Evaluate the local part of the pseudopotential in reciprocal space:

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

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:

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

eval_psp_local_real(psp, r)

Evaluate the local part of the pseudopotential in real space.

eval_psp_projector_fourier(psp, i, l, q)

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

\[\begin{aligned} p(q) &= ∫_{ℝ^3} p_{il}(r) e^{-iq·r} dr \\ &= 4π ∫_{ℝ_+} r^2 p_{il}(r) j_l(qr) dr \end{aligned}\]

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.

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

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 the distributed data of a quantity depending on k-Points on the master process and return it. On the other (non-master) processes nothing is returned.

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 to extract data for post-processing and serialisation to disk.

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

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 $μ_B$.

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

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

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

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

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.

Return the indices of the kpoints shifted by q. That is for each kpoint of the basis: kpoints[ik].coordinate + q = kpoints[indices[ik]].coordinate.

Selects a kgrid 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.

Selects a kgrid size to ensure a minimal spacing (in inverse Bohrs) between kpoints. A reasonable spacing is 0.13 inverse Bohrs (around $2π * 0.04 \AA^{-1}$).

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

Return kpoint coordinates in reduced coordinates

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

list_psp(element; functional, family, core)

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 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)

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

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

Build an LDA model (Perdew & Wang parametrization) from the specified atoms. DOI:10.1103/PhysRevB.45.13244

Build an PBE-GGA model from the specified atoms. DOI:10.1103/PhysRevLett.77.3865

Build a SCAN meta-GGA model from the specified atoms. DOI:10.1103/PhysRevLett.115.036402

Convenience constructor, which builds a standard atomic (kinetic + atomic potential) model. Use extra_terms to add additional terms.

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

Return the Fourier coefficients for ψk · e^{i q·r} in the basis of kpt_out, where ψk is defined on a basis kpt_in.

Return the number of core electrons

Return the number of valence electrons

Number of valence electrons.

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.

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.

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.

Square of the ℓ²-norm.

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.

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

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

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

Compute and plot the band structure. n_bands selects the number of bands to compute. If this value is absent and an scfres is used to start the calculation a default of n_bands_scf + 5sqrt(n_bands_scf) is used. 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). The kline_density is given in number of $k$-points per inverse bohrs (i.e. overall in units of length).

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

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 $t = r_\text{loc} q$ and Q is a polynomial of at most degree 8. This function returns Q.

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

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

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

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)

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

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

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

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 using random Gaussians as guesses. All keyword arguments supported by Wannier90 for the disentanglement may be added as keyword arguments. The function returns the fileprefix.

save_scfres(filename, scfres)

Save an scfres obtained from self_consistent_field to a file. 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. 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 and some metadata (energy, Fermi level, occupation etc.). Supports these keyword arguments:
  • save_ψ: Save the real-space representation of the orbitals as well (may lead to larger files).
  • extra_data: Dict{String,Array} with additional data on the 3D real-space grid to store into the VTK file.

Create a simple anderson-accelerated SCF solver. m specifies the number of steps to keep the history of.

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.

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

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

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 $ρ_\text{next} = α P^{-1} (ρ_\text{out} - ρ_\text{in})$.

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 DFTK.ScfConvergenceDensity(tol) (the default), DFTK.ScfConvergenceEnergy(tol) or DFTK.ScfConvergenceForce(tol).
• maxiter: Maximal number of SCF iterations
• 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.
• callback: Function called at each SCF iteration. Usually takes care of printing the intermediate state.

Function to compute sin(2π x)

solve_ΩplusK(basis::PlaneWaveBasis{T}, ψ, res, occupation;
             tol=1e-10, verbose=false) where {T}

Return δψ where (Ω+K) δψ = rhs

Solve the problem (Ω+K) δψ = rhs 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).

Convert the DFTK atom groups and positions datastructure into a tuple of datastructures for use with spglib. Validity of the input data is assumed. The output positions contains positions per atom, numbers contains the mapping atom to a unique number for each group of indistinguishable atoms, spins contains the $z$-component of the initial magnetic moment on each atom, mapping contains the mapping of the numbers to the element objects in DFTK and collinear whether the atoms mark a case of collinear spin or not. Notice that if collinear is false then spins is garbage.

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::Number)

Returns the spherical Bessel function of the first kind jl(x). Consistent with https://en.wikipedia.org/wiki/Besselfunction#SphericalBesselfunctions and with SpecialFunctions.sphericalbesselj. Specialized for integer 0 <= l <= 5.

Explicit spin components of the KS orbitals and the density

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

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.

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

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

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

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

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

Return the $k$-point symmetry operations associated to a lattice and atoms.

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).

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

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

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

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

Transfer an array ψk_in expanded on kpt_in, and produce $ψ(r) e^{i ΔG·r}$ expanded on kpt_out. It is mostly useful for phonons. Beware: ψk_out can lose information if the shift ΔG is large or if the G_vectors differ between k-points.

Transfer an array ψk defined on basisin $k$-point kptin to basisout $k$-point kptout.

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

" 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).

Sum an array over kpoints, taking weights into account

Write the eigenvalues in a format readable by Wannier90.

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

Returns the (l,m) real spherical harmonic Ylm(r). Consistent with https://en.wikipedia.org/wiki/Tableofsphericalharmonics#Realsphericalharmonics

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.

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