# API reference

This page provides a plain list of all documented functions, structs, modules and macros in DFTK. Note that this list is neither structured, complete nor particularly clean, so it only provides rough orientation at the moment. The best reference is the code itself.

`DFTK.DFTK`

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

`DFTK.DFTK_DATADIR`

— ConstantThe default search location for Pseudopotential data files

`DFTK.timer`

— ConstantTimerOutput object used to store DFTK timings.

`DFTK.Applyχ0Model`

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

.

`DFTK.AtomicLocal`

— TypeAtomic local potential defined by `model.atoms`

.

`DFTK.AtomicNonlocal`

— TypeNonlocal term coming from norm-conserving pseudopotentials in Kleinmann-Bylander form. $\text{Energy} = \sum_a \sum_{ij} \sum_{n} f_n <ψ_n|p_{ai}> D_{ij} <p_{aj}|ψ_n>.$

`DFTK.DielectricMixing`

— TypeWe 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.

`DFTK.DielectricModel`

— TypeA 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`

.

`DFTK.ElementCohenBergstresser`

— MethodElement 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"`

)

`DFTK.ElementCoulomb`

— MethodElement 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"`

)

`DFTK.ElementPsp`

— MethodElement 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"`

)

`DFTK.Energies`

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

`DFTK.Entropy`

— TypeEntropy 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.

`DFTK.Ewald`

— TypeEwald 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.

`DFTK.ExternalFromFourier`

— TypeExternal potential from the (unnormalized) Fourier coefficients `V(G)`

G is passed in cartesian coordinates

`DFTK.ExternalFromReal`

— TypeExternal potential from an analytic function `V`

(in cartesian coordinates). No low-pass filtering is performed.

`DFTK.FourierMultiplication`

— TypeFourier space multiplication, like a kinetic energy term: (Hψ)(G) = multiplier(G) ψ(G)

`DFTK.Hartree`

— TypeHartree 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 δ*R, integral of G zero on a unit cell).

`DFTK.KerkerDosMixing`

— TypeThe same as `KerkerMixing`

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

`DFTK.KerkerMixing`

— TypeKerker 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*)

`DFTK.Kinetic`

— TypeKinetic energy: 1/2 sum*n f*n ∫ |∇ψn|^2.

`DFTK.Kpoint`

— TypeDiscretization 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.

`DFTK.LdosModel`

— TypeRepresents 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

`DFTK.LibxcDensity`

— MethodCompute density in real space and its derivatives starting from ρ

`DFTK.LocalNonlinearity`

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

`DFTK.Magnetic`

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

`DFTK.MagneticFieldOperator`

— TypeMagnetic field operator A⋅(-i∇).

`DFTK.Model`

— Method```
Model(lattice; n_electrons, atoms, magnetic_moments, terms, temperature,
smearing, spin_polarization, symmetry)
```

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.

`DFTK.NonlocalOperator`

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

`DFTK.NoopOperator`

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

`DFTK.PlaneWaveBasis`

— TypeCreates a new basis identical to `basis`

, but with a custom set of kpoints

`DFTK.PlaneWaveBasis`

— TypeA 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.

`G_to_r`

and `r_to_G`

convert between these representations.

`DFTK.PlaneWaveBasis`

— MethodCreates 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.

If `use_symmetry`

is `true`

(default) the symmetries of the crystal are used to reduce the number of $k$-points which are treated explicitly. In this case all guess densities and potential functions must agree with the crystal symmetries or the result is undefined.

`DFTK.PreconditionerNone`

— TypeNo preconditioning

`DFTK.PreconditionerTPA`

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

`DFTK.PspCorrection`

— TypePseudopotential correction energy. TODO discuss the need for this.

`DFTK.PspHgh`

— Method```
PspHgh(Zion::Number, rloc::Number, cloc::Vector, rp::Vector, h::Vector;
identifier="", description="")
```

Construct a Hartwigsen, Goedecker, Teter, Hutter separable dual-space Gaussian pseudopotential (1998). The required parameters are the ionic charge `Zion`

(total charge - valence electrons), the range for the local Gaussian charge distribution `rloc`

, the coefficients for the local part `cloc`

, the projector radius `rp`

(one per AM channel) and the non-local coupling coefficients between the projectors `h`

(one matrix per AM channel).

`DFTK.RealFourierOperator`

— TypeLinear 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.

`DFTK.RealSpaceMultiplication`

— TypeReal space multiplication by a potential: (Hψ)(r) V(r) ψ(r)

`DFTK.SimpleMixing`

— TypeSimple mixing: $J^{-1} ≈ 1$

`DFTK.TermNoop`

— TypeA term with a constant zero energy.

`DFTK.Xc`

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

`DFTK.χ0Mixing`

— TypeGeneric 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).

`DFTK.CROP`

— FunctionCROP-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.

`DFTK.G_to_r!`

— MethodIn-place version of `G_to_r`

.

`DFTK.G_to_r`

— Method`G_to_r(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.

`DFTK.G_vectors`

— Method```
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`

.

`DFTK.G_vectors`

— Method`G_vectors(fft_size::Tuple)`

The wave vectors `G`

in reduced (integer) coordinates for a cubic basis set of given sizes.

`DFTK.G_vectors_cart`

— Method```
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.

`DFTK.Gplusk_vectors`

— Method`Gplusk_vectors(basis::PlaneWaveBasis, kpt::Kpoint)`

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

`DFTK.Gplusk_vectors_cart`

— Method`Gplusk_vectors_cart(basis::PlaneWaveBasis, kpt::Kpoint)`

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

`DFTK.HybridMixing`

— MethodThe 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.

`DFTK.IncreaseMixingTemperature`

— MethodIncrease 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.

`DFTK.ScfAcceptImprovingStep`

— MethodAccept 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.

`DFTK.ScfConvergenceDensity`

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

`DFTK.ScfConvergenceEnergy`

— MethodFlag convergence as soon as total energy change drops below tolerance

`DFTK.ScfDefaultCallback`

— MethodDefault callback function for `self_consistent_field`

and `newton`

, which prints a convergence table.

`DFTK.ScfDiagtol`

— MethodDetermine 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.

`DFTK.ScfPlotTrace`

— FunctionPlot 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.

`DFTK.ScfSaveCheckpoints`

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

`DFTK.apply_K`

— Method`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`

.

`DFTK.apply_kernel`

— Method`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.

`DFTK.apply_ksymop`

— MethodApply 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).

`DFTK.apply_ksymop`

— MethodApply a `k`

-point symmetry operation (the tuple (S, τ)) to a partial density.

`DFTK.apply_Ω`

— Method`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.

`DFTK.apply_χ0`

— MethodGet the density variation δρ corresponding to a total potential variation δV.

Note: This function assumes that all bands contained in `ψ`

and `eigenvalues`

are sufficiently converged. By default the `self_consistent_field`

routine of `DFTK`

returns `3`

extra bands, which are not converged by the eigensolver (see `n_ep_extra`

parameter). These should be discarded before using this function.

`DFTK.ase_atoms_translation_map`

— MethodMap translating between the `atoms`

datastructure used in DFTK and the indices of the respective atoms in the ASE.Atoms object in `pyobj`

.

`DFTK.atom_decay_length`

— MethodGet the lengthscale of the valence density for an atom with `n_elec_core`

core and `n_elec_valence`

valence electrons.

`DFTK.atomic_symbol`

— MethodChemical symbol corresponding to an element

`DFTK.build_fft_plans`

— MethodPlan 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.

`DFTK.build_form_factors`

— MethodBuild form factors (Fourier transforms of projectors) for an atom centered at 0.

`DFTK.build_projection_vectors_`

— MethodBuild projection vectors for a atoms array generated by term_nonlocal

H*at = sum*ij Cij |pi> <pj| H*per = sum*R sum*ij Cij |pi(x-R)> <pj(x-R)| = sum*R sum_ij Cij |pi(x-R)> <pj(x-R)|

<e*kG'|H*per|e*kG> = ... = 1/Ω sum*ij Cij pihat(k+G') pjhat(k+G)^*

where pihat(q) = ∫_R^3 pi(r) e^{-iqr} dr

We store 1/√Ω pihat(k+G) in proj_vectors.

`DFTK.bzmesh_ir_wedge`

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

Construct the irreducible wedge of a uniform Brillouin zone mesh for sampling $k$-points. The function returns a tuple `(kcoords, ksymops)`

, where `kcoords`

are the list of irreducible $k$-points and `ksymops`

are a list of symmetry operations for regenerating the full mesh. `symmetries`

is the tuple returned from `symmetry_operations(lattice, atoms, magnetic_moments)`

. `tol_symmetry`

is the tolerance used for searching for symmetry operations.

`DFTK.bzmesh_uniform`

— Method`bzmesh_uniform(kgrid_size; kshift=[0, 0, 0])`

Construct a (shifted) uniform Brillouin zone mesh for sampling the $k$-points. The function returns a tuple `(kcoords, ksymops)`

, where `kcoords`

are the list of $k$-points and `ksymops`

are a list of symmetry operations (for interface compatibility with `PlaneWaveBasis`

and `bzmesh_irreducible`

. No symmetry reduction is attempted, such that there will be `prod(kgrid_size)`

$k$-points returned and all symmetry operations are the identity.

`DFTK.charge_ionic`

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

`DFTK.charge_nuclear`

— MethodReturn the total nuclear charge of an atom type

`DFTK.chemical_formula`

— MethodReturns the sum formula of the atoms list as a string.

`DFTK.compute_Ak_gaussian_guess`

— MethodCompute 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.

`DFTK.compute_current`

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

`DFTK.compute_density`

— Method`compute_density(basis::PlaneWaveBasis, ψ::AbstractVector, occupation::AbstractVector)`

Compute the density and spin 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. If the `Model`

underlying the basis is not collinear the spin density is `nothing`

.

`DFTK.compute_dos`

— MethodTotal density of states at energy ε

`DFTK.compute_fft_size`

— MethodDetermine 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`

.

`DFTK.compute_forces`

— MethodCompute 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 `[[force for atom in positions] for (element, positions) in atoms]`

which has the same structure as the `atoms`

object passed to the underlying `Model`

.

`DFTK.compute_forces_cart`

— MethodCompute 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`

.

`DFTK.compute_kernel`

— Method`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

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

`DFTK.compute_ldos`

— MethodLocal density of states, in real space

`DFTK.compute_nos`

— Method```
compute_nos(ε, basis, eigenvalues; smearing=basis.model.smearing,
temperature=basis.model.temperature)
```

The number of Kohn-Sham states in a temperature window of width `temperature`

around the energy `ε`

contributing to the DOS at temperature `T`

.

This quantity is not a physical quantity, but rather a dimensionless approximate measure for how well properties near the Fermi surface are sampled with the passed `smearing`

and temperature `T`

. It increases with both `T`

and better sampling of the BZ with $k$-points. A value $\gg 1$ indicates a good sampling of properties near the Fermi surface.

`DFTK.compute_occupation`

— MethodFind the occupation and Fermi level.

`DFTK.compute_occupation_bandgap`

— MethodFind Fermi level and occupation for the given parameters, assuming a band gap and zero temperature. This function is for DEBUG purposes only, and the finite-temperature version with 0 temperature should be preferred.

`DFTK.compute_partial_density!`

— MethodCompute the partial density at the indicated $k$-Point and return it (in Fourier space).

`DFTK.compute_recip_lattice`

— MethodCompute the reciprocal lattice. Takes special care of 1D or 2D cases. 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.

`DFTK.compute_stresses`

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

`DFTK.compute_transfer_matrix`

— MethodReturn a sparse matrix that maps quantities given on `basis_in`

and `kpt_in`

to quantities on `basis_out`

and `kpt_out`

.

`DFTK.compute_transfer_matrix`

— MethodReturn 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.

`DFTK.compute_unit_cell_volume`

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

`DFTK.compute_χ0`

— MethodCompute 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:

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

`DFTK.datadir_psp`

— MethodReturn the data directory with pseudopotential files

`DFTK.default_spin_polarization`

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

`DFTK.default_symmetries`

— MethodDefault logic to determine the symmetry operations to be used in the model.

`DFTK.default_wannier_centres`

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

`DFTK.diagonalize_all_kblocks`

— MethodFunction 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)`

`DFTK.diameter`

— MethodCompute the diameter of the unit cell

`DFTK.direct_minimization`

— MethodComputes the ground state by direct minimization. `kwargs...`

are passed to `Optim.Options()`

. Note that the resulting ψ are not necessarily eigenvectors of the Hamiltonian.

`DFTK.disable_threading`

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

`DFTK.divergence_real`

— MethodCompute 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.

`DFTK.energy_ewald`

— MethodCompute the electrostatic interaction energy per unit cell between point charges in a uniform background of compensating charge to yield net neutrality. the `lattice`

and `recip_lattice`

should contain the lattice and reciprocal 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.

`DFTK.energy_per_particle`

— MethodVWN5 LDA correlation according to Vosko, Wilk, and Nusair, (DOI 10.1139/p80-159).

`DFTK.energy_per_particle`

— MethodLDA Slater exchange (DOI: 10.1017/S0305004100016108 and 10.1007/BF01340281)

`DFTK.energy_psp_correction`

— MethodCompute the correction term for properly modelling the interaction of the pseudopotential core with the compensating background charge induced by the `Ewald`

term.

`DFTK.eval_psp_energy_correction`

— Method`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.

`DFTK.eval_psp_local_fourier`

— Method`eval_psp_local_fourier(psp, q)`

Evaluate the local part of the pseudopotential in reciprocal space: V(q) = ∫*R^3 Vloc(r) e^{-iqr} dr = 4π ∫*{R+} sin(qr)/q r e^{-iqr} dr

`DFTK.eval_psp_local_real`

— Method`eval_psp_local_real(psp, r)`

Evaluate the local part of the pseudopotential in real space.

`DFTK.eval_psp_projector_fourier`

— Method`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`

: p(q) = ∫*R^3 p*{il}(r) e^{-iqr} dr = 4π ∫*{R+} r^2 p*{il}(r) j_l(q r) dr

`DFTK.eval_psp_projector_real`

— Method`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`

.

`DFTK.fermi_level`

— MethodFind the Fermi level.

`DFTK.filled_occupation`

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

`DFTK.find_irreducible_kpoints`

— MethodImplements a primitive search to find an irreducible subset of kpoints amongst the provided kpoints.

`DFTK.gather_kpts`

— MethodGather 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.

`DFTK.gather_kpts`

— MethodGather 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.

`DFTK.gaussian_superposition`

— MethodBuild a superposition of Gaussians as a guess for the density and magnetisation. Expects a list of tuples `(coefficient, length, position)`

for each of the Gaussian, which follow the functional form

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

and are placed at `position`

(in fractional coordinates).

`DFTK.get_spglib_lattice`

— MethodReturns crystallographic conventional cell according to the International Table of Crystallography Vol A (ITA) in case `to_primitive=false`

. If `to_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.

`DFTK.guess_density`

— Function`guess_density(basis, magnetic_moments)`

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

We take for the guess density a gaussian centered around the atom, of length specified by `atom_decay_length`

, normalized to get the right number of electrons

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

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

`DFTK.hamiltonian_with_total_potential`

— MethodReturns a new Hamiltonian with local potential replaced by the given one

`DFTK.high_symmetry_kpath`

— MethodExtract the high-symmetry $k$-point path corresponding to the passed model using `Brillouin.jl`

. 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. The `kline_density`

is given in number of $k$-points per inverse bohrs (i.e. overall in units of length).

Issues a warning in case the passed lattice does not match the expected primitive.

`DFTK.index_G_vectors`

— MethodReturn 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.

`DFTK.interpolate_density`

— MethodInterpolate 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

`DFTK.interpolate_kpoint`

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

`DFTK.is_metal`

— Function`is_metal(band_data, εF, tol)`

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

`DFTK.kgrid_from_minimal_n_kpoints`

— MethodSelects 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.

`DFTK.kgrid_from_minimal_spacing`

— MethodSelects a kgrid size to ensure a minimal spacing (in inverse Bohrs) between kpoints. A reasonable spacing is `0.13`

inverse Bohrs (around $2π * 0.04 \AA^{-1}$).

`DFTK.kgrid_monkhorst_pack`

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

`DFTK.krange_spin`

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

`DFTK.list_psp`

— Function`list_psp(element; functional, family, core, datadir_psp)`

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.

`DFTK.load_atoms`

— MethodReturn a DFTK-compatible `atoms`

object loaded from an ASE `Atoms`

, a pymatgen `Structure`

or a compatible file (e.g. xyz file, cif file etc.)

`DFTK.load_atoms_pymatgen`

— MethodLoad a DFTK-compatible atoms representation from a supported pymatgen object. All atoms are using a Coulomb model.

`DFTK.load_lattice`

— MethodReturn a DFTK-compatible `lattice`

object loaded from an ASE `Atoms`

, a pymatgen `Structure`

or a compatible file (e.g. xyz file, cif file etc.)

`DFTK.load_lattice`

— MethodLoad a DFTK-compatible lattice object from a supported python object (e.g. pymatgen or ASE)

`DFTK.load_psp`

— MethodLoad 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.

`DFTK.load_scfres`

— Function`load_scfres(filename)`

Load back an `scfres`

, which has previously been stored with `save_scfres`

. Note the warning in `save_scfres`

.

`DFTK.local_potential_fourier`

— MethodRadial local potential, in Fourier space: V(q) = int_{R^3} V(x) e^{-iqx} dx.

`DFTK.local_potential_real`

— MethodRadial local potential, in real space.

`DFTK.model_DFT`

— MethodBuild a DFT model from the specified atoms, with the specified functionals.

`DFTK.model_LDA`

— MethodBuild an LDA model (Teter93 parametrization) from the specified atoms.

`DFTK.model_PBE`

— MethodBuild an PBE-GGA model from the specified atoms.

`DFTK.model_atomic`

— MethodConvenience constructor, which builds a standard atomic (kinetic + atomic potential) model. Use `extra_terms`

to add additional terms.

`DFTK.mpi_nprocs`

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

`DFTK.n_elec_core`

— MethodReturn the number of core electrons

`DFTK.n_elec_valence`

— MethodReturn the number of valence electrons

`DFTK.newton`

— Method```
newton(basis::PlaneWaveBasis{T}; ψ0=nothing,
tol=1e-6, tol_cg=1e-10, maxiter=20, verbose=false,
callback=NewtonDefaultCallback(),
is_converged=NewtonConvergenceDensity(tol))
```

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

`DFTK.next_density`

— MethodObtain new density ρ by diagonalizing `ham`

.

`DFTK.normalize_kpoint_coordinate`

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

`DFTK.overlap_Mmn_k_kpb`

— MethodComputes 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 -> 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*shift)}(r) = e^{-i*\langle G*shift,r \rangle} u_{nk}``

`DFTK.plot_bandstructure`

— MethodCompute 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).

`DFTK.plot_dos`

— FunctionPlot the density of states over a reasonable range

`DFTK.psp_local_polynomial`

— FunctionThe 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`

.

`DFTK.psp_projector_polynomial`

— FunctionThe 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`

.

`DFTK.qcut_psp_local`

— MethodEstimate an upper bound for the argument `q`

after which `abs(eval_psp_local_fourier(psp, q))`

is a strictly decreasing function.

`DFTK.qcut_psp_projector`

— MethodEstimate an upper bound for the argument `q`

after which `eval_psp_projector_fourier(psp, q)`

is a strictly decreasing function.

`DFTK.r_to_G!`

— MethodIn-place version of `r_to_G!`

. NOTE: If `kpt`

is given, not only `f_fourier`

but also `f_real`

is overwritten.

`DFTK.r_to_G`

— Method`r_to_G(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.

`DFTK.r_vectors`

— Method`r_vectors(basis::PlaneWaveBasis)`

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

`DFTK.r_vectors_cart`

— Method`r_vectors_cart(basis::PlaneWaveBasis)`

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

`DFTK.random_density`

— MethodGenerate a physically valid random density integrating to the given number of electrons.

`DFTK.read_w90_nnkp`

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

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

TODO: add the possibility to exclude bands

`DFTK.run_abinit_scf`

— MethodRun an SCF in ABINIT starting from a DFTK `Model`

and some extra parameters. Write the result to the `output`

directory in ETSF Nanoquanta format and return the `EtsfFolder`

object.

`DFTK.run_abinit_scf`

— MethodRun an SCF in ABINIT starting from the input file `infile`

represented as a `abipy.abilab.AbinitInput`

python object. Write the result to the `output`

directory in ETSF Nanoquanta format and return the result as an `EtsfFolder`

object.

`DFTK.run_wannier90`

— FunctionWannerize 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`

.

Currently this is an experimental feature, which has not yet been tested to full depth. The interface is considered unstable and may change incompatibly in the future. Use at your own risk and please report bugs in case you encounter any.

`DFTK.save_scfres`

— Method`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.

No guarantees are made with respect to this function at this point. It may change incompatibly between DFTK versions or stop working / be removed in the future.

`DFTK.scf_anderson_solver`

— FunctionCreate a simple anderson-accelerated SCF solver. `m`

specifies the number of steps to keep the history of.

`DFTK.scf_damping_quadratic_model`

— MethodUse 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.

`DFTK.scf_damping_solver`

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

`DFTK.scf_nlsolve_solver`

— FunctionCreate a NLSolve-based SCF solver, by default using an Anderson-accelerated fixed-point scheme, keeping `m`

steps for Anderson acceleration. See the NLSolve documentation for details about the other parameters and methods.

`DFTK.select_eigenpairs_all_kblocks`

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

.

`DFTK.self_consistent_field`

— MethodSolve the Kohn-Sham equations with a SCF algorithm, starting at ρ.

`DFTK.solve_ΩplusK`

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

Return δψ where (Ω+K) δψ = rhs

`DFTK.spglib_atoms`

— FunctionConvert the DFTK atoms datastructure into a tuple of datastructures for use with spglib. `positions`

contains positions per atom, `numbers`

contains the mapping atom to a unique number for each indistinguishable element, `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.

`DFTK.spin_components`

— MethodExplicit spin components of the KS orbitals and the density

`DFTK.split_evenly`

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

`DFTK.standardize_atoms`

— FunctionApply 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.

`DFTK.symmetries_preserving_kgrid`

— MethodFilter out the symmetry operations that respect the symmetries of the discrete BZ grid

`DFTK.symmetrize_ρ`

— MethodSymmetrize a density by applying all the model symmetries (by default) and forming the average.

`DFTK.symmetry_operations`

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

`DFTK.total_local_potential`

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

`DFTK.transfer_blochwave`

— MethodTransfer Bloch wave between two basis sets. Limited feature set.

`DFTK.transfer_blochwave_kpt`

— MethodTransfer an array ψk defined on basis*in $k$-point kpt*in to basis*out $k$-point kpt*out.

`DFTK.transfer_mapping`

— MethodCompute 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`

).

`DFTK.unfold_bz`

— Method" 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).

`DFTK.weighted_ksum`

— MethodSum an array over kpoints, taking weights into account

`DFTK.write_w90_eig`

— MethodWrite the eigenvalues in a format readable by Wannier90.

`DFTK.write_w90_win`

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

.

`DFTK.ylm_real`

— MethodReturns the (l,m) real spherical harmonic Y*lm(r). Consistent with https://en.wikipedia.org/wiki/Table*of*spherical*harmonics#Real*spherical*harmonics

`DFTK.@timing`

— MacroShortened version of the `@timeit`

macro from `TimerOutputs`

, which writes to the DFTK timer.

`DFTK.@timing_seq`

— MacroSimilar to `@timing`

, but disabled in parallel runs. Should be used to time threaded regions, since TimerOutputs is not thread-safe and breaks otherwise.

`DFTK.Smearing.A`

— Method`A`

term in the Hermite delta expansion

`DFTK.Smearing.H`

— MethodStandard Hermite function using physicist's convention.

`DFTK.Smearing.entropy`

— MethodEntropy. 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.

`DFTK.Smearing.occupation`

— MethodOccupation at `x`

, where in practice x = (ε - εF) / T.

`DFTK.Smearing.occupation_derivative`

— MethodDerivative of the occupation function, approximation to minus the delta function.

`DFTK.Smearing.occupation_divided_difference`

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