Background
Below, a condensed overview is provided on the background of plane wave density functional theory, mainly scoping these programs with respect to localized-orbital density functional theory. A detailed treatise on the mathematical background behind PyPWDFT can be found via this link. At the bottom of this page, useful references are provided to more reading material and other code sources.
Contemporary electronic structure calculations are predominantly performed using density functional theory. Two important flavors of density functional theory (DFT) exist, being localized orbital DFT and plane-wave DFT. In both methods, the electronic structure problem is solved by means of the introduction of a basis set. The nature of this basis set however differs.
In localized orbital DFT, the basis functions correspond to quasi atomic orbitals, such as sets of Slater orbitals or combinations of Gaussian type orbitals. Furthermore, the atomic system is placed inside an infinite vacuum. In contrast, in plane wave DFT, it is not the atoms that “spawn” the basis functions, but the unit cell itself. On top of that, the system is assumed to be inherently periodic, although “vacuum” or “gas-phase” systems can be imitated by using relatively large unit cells.
Irrespective of whether one uses localized orbital DFT or plane wave DFT, in the end the Roothaan equation is being solved. The way this equation is solved however differs. In localized orbital DFT, the Hamiltonian matrix is fairly small and one can perform its diagonalization to compute the eigenvalue and -vector pairs. In contrast, in plane wave DFT, due to the sheer number of plane waves being used, the Hamiltonian matrix is very large and is in fact not even stored in the calculation. To find the eigenvector and -value pairs, a partial matrix solver procedure is used. This matrix solver does not even require the full Hamiltonian matrix to be known, but only its effect on an arbitrary vector \(\vec{c}\) in Hilbert space.
Similar to localized-orbital theory, electron-electron interactions can be more efficiently evaluated by solving the Poisson equation. Since the plane wave basis set can also be used to describe the electron density, solving for the Hartree potential effectively utilizes highly-efficient Fast Fourier transforms. In a similar fashion, also the nuclear attraction energy is solved via (analytical) Fourier transforms.
Within PyPWDFT, all these aspects which make a plane wave DFT code stand out from a “regular” localized orbital code are shown. Despite the ambition to show the most salient details, certain aspects are considered out of scope, though these aspects are important in commercial codes. For example, \(\vec{k}\)-point sampling is not shown, nor are pseudopotentials being discussed. Furthermore, although the code would support non-cubic unit cells, none of the examples use these. When using the code, please take these limitations in mind.
Other software
Besides PyPWDFT, there are a number of other educational electronic structure codes written by this author. Below, a list is provided.
Further reading
A detailed background on electronic structure calculations, including a derivation of the Hartree-Fock and localized-orbital density functional theory calculations can be found in this open-access textbook.