# 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. Appropriate unit conversion can can be performed using the `Unitful`

and `UnitfulAtomic`

packages:

```
using Unitful
using UnitfulAtomic
austrip(10u"eV") # 10eV in Hartree
```

`0.36749322175518595`

```
using Unitful: Å
using UnitfulAtomic
auconvert(Å, 1.2) # 1.2 Bohr in Ångström
```

`0.6350126530835999 Å`

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. When structures are provided as AtomsBase.jl-compatible objects, this unit conversion is automatically performed behind the scenes. See AtomsBase integration for details.

## 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. For example, `model.lattice[:, 1]`

is the first real-space lattice vector. 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 $B = 2\pi A^{-T}$.

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`

, `load_positions`

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.