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| /index.md | ||
| /examples/ | ||
| /guide/tutorial.md | ||
| /guide/tutorial.ipynb |
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| # Notation and conventions | ||
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| ## 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](https://github.com/JuliaEditorSupport/) | ||
| to simplify typing them. | ||
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| ## [Symbol conventions](@id symbol-conventions) | ||
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| - **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 | ||
| ```math | ||
| \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**: | ||
| ```math | ||
| 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}``. | ||
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| ## Units | ||
| In DFTK atomic units are used thoughout, most importantly | ||
| lengths are in Bohr and energies in Hartree. | ||
| See [wikipedia](https://en.wikipedia.org/wiki/Hartree_atomic_units) | ||
| for a list of conversion factors. Useful conversion factors | ||
| can also be found in `DFTK.units` and using [`DFTK.unit_to_au`](@ref). | ||
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| ## [Lattices and lattice vectors](@id conventions-lattice) | ||
| 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}``. | ||
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| 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``. | ||
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| ## Lattice coordinate systems | ||
| 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](@ref symbol-conventions)). | ||
| One switches to Cartesian coordinates by | ||
| ```math | ||
| 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 | ||
| ```math | ||
| 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). | ||
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| ## Normalisation convention | ||
| 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 [Tutorial](@ref) for an example where this is demonstrated. |
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