Notation and conventions

Usage of unicode characters

DFTK liberally uses unicode characters to represent Greek characters (e.g. ψ, ρ, ε...). Make sure you use the proper Julia plugins to simplify typing them.

Symbol conventions

  • Reciprocal-space vectors: $k$ for vectors in the Brillouin zone, $G$ for vectors of the reciprocal lattice, $q$ for general vectors
  • Real-space vectors: $R$ for lattice vectors, $r$ and $x$ are usually used for unit for vectors in the unit cell or general real-space vectors, respectively. This convention is, however, less consistently applied.
  • $\Omega$ is the unit cell, and $|\Omega|$ (or sometimes just $\Omega$) is its volume.
  • $A$ are the real-space lattice vectors (model.lattice) and $B$ the Brillouin zone lattice vectors (model.recip_lattice).
  • The Bloch waves are

    \[\psi_{nk}(x) = e^{ik\cdot x} u_{nk}(x),\]

    where $n$ is the band index and $k$ the $k$-point. In the code we sometimes use $\psi$ and $u$ interchangeably.
  • $\varepsilon$ are the eigenvalues, $\varepsilon_F$ is the Fermi level.
  • $\rho$ is the density.
  • In the code we use normalized plane waves:

    \[e_G(r) = \frac 1 {\sqrt{\Omega}} e^{i G \cdot r}.\]

  • $Y^l_m$ are the complex spherical harmonics, and $Y_{lm}$ the real ones.
  • $j_l$ are the Bessel functions. In particular, $j_{0}(x) = \frac{\sin x}{x}$.

Units

In DFTK, atomic units are used throughout, most importantly lengths are in Bohr and energies in Hartree. See wikipedia for a list of conversion factors. Useful conversion factors can also be found in DFTK.units and using DFTK.unit_to_au:

import DFTK.units: eV
10eV      # 10eV in Hartree
0.3674932248
import DFTK.units: Å
1.2 / Å  # 1.2 Bohr in Ångström
0.635012652804
Differing unit conventions

Different electronic-structure codes use different unit conventions. For example for lattice vectors the common length units are Bohr (used by DFTK) and Ångström (used e.g. by ASE, 1Å ≈ 1.80 Bohr). When setting up a calculation for DFTK one needs to ensure to convert to Bohr and atomic units. For some Python libraries (currently ASE, pymatgen and abipy) DFTK directly ships conversion tools in form of the load_lattice and load_atoms functions, which take care of such conversions. Examples which demonstrate this are Creating slabs with ASE and Creating supercells with pymatgen.

Lattices and lattice vectors

Both the real-space lattice (i.e. model.lattice) and reciprocal-space lattice (model.recip_lattice) contain the lattice vectors in columns. If 1D or 2D problems are to be treated these arrays are still $3 \times 3$ matrices, but contain two or one zero-columns, respectively. The real-space lattice vectors are sometimes referred to by $A$ and the reciprocal-space lattice vectors by $B = 2\pi A^{-T}$.

Row-major versus column-major storage order

Julia stores matrices as column-major, but other languages (notably Python and C) use row-major ordering. Care therefore needs to be taken to properly transpose the unit cell matrices $A$ before using it with DFTK. For the supported third-party packages load_lattice and load_atoms again handle such conversion automatically.

We use the convention that the unit cell in real space is $[0, 1)^3$ in reduced coordinates and the unit cell in reciprocal space (the reducible Brillouin zone) is $[-1/2, 1/2)^3$.

Reduced and cartesian coordinates

Unless denoted otherwise the code uses reduced coordinates for reciprocal-space vectors such as $k$, $G$, $q$ or real-space vectors like $r$ and $R$ (see Symbol conventions). One switches to Cartesian coordinates by

\[x_\text{cart} = M x_\text{red}\]

where $M$ is either $A$ / model.lattice (for real-space vectors) or $B$ / model.recip_lattice (for reciprocal-space vectors). A useful relationship is

\[b_\text{cart} \cdot a_\text{cart}=2\pi b_\text{red} \cdot a_\text{red}\]

if $a$ and $b$ are real-space and reciprocal-space vectors respectively. Other names for reduced coordinates are integer coordinates (usually for $G$-vectors) or fractional coordinates (usually for $k$-points).

Normalization conventions

The normalization conventions used in the code is that quantities stored in reciprocal space are coefficients in the $e_{G}$ basis, and quantities stored in real space use real physical values. This means for instance that wavefunctions in the real space grid are normalized as $\frac{|\Omega|}{N} \sum_{r} |\psi(r)|^{2} = 1$ where $N$ is the number of grid points and in reciprocal space its coefficients are $\ell^{2}$-normalized, see the discussion in section PlaneWaveBasis and plane-wave discretisations where this is demonstrated.