There are many different basis sets used to represent the wavefunctions in DFT, and it is important to understand their advantages and limitations. In this tutorial, I will discuss two of the most commonly used basis sets: plane waves; and atomic orbitals[1]. Unlike the basis sets generally used in undergraduate quantum mechanics courses, these basis sets cannot be complete, and the effects of using a finite basis set must be both understood and tested carefully.

Plane waves are the foundation of almost all of the computational solid state DFT modelling that has been done for the last thirty years. They have two key advantages: the completeness of the basis set can be increased systematically; and they offer great computational efficiency. The completeness is improved simply by increasing the largest wavevector used in the plane wave expansion. This is often expressed as an energy, \(E_{\mathrm{cutoff}} = \hbar^2 k_{max}^2/2m_e\), which represents the largest kinetic energy that can be sustained by the basis (and is also related to the finest spatial features that can be resolved).

The mathematical and computational simplicity comes partly from the orthgonality of the basis functions, and also from the exceptional efficiency of the fast Fourier transform (FFT) as implemented on modern computers. Any transformation from real to reciprocal space, or vice versa, relies on FFTs, and these are generally very fast[2].

However, plane waves are not a very good representation of the electronic wavefunctions in molecules and solids, particularly in areas where the potential varies rapidly (for instance near the nuclear core). Generally they are used either with pseudopotentials (which I will write about soon) or with an auxiliary basis set that describes the rapid variation well. They also extend over all space, which can affect performance with very large system sizes.

Atomic orbitals (which I will refer to as AOs - often we have either numerical AOs, NAOs, or pseudo-AOs, PAOs) consist of a radial function multiplied by a spherical harmonic. The radial function can be a gaussian (as used extensively in quantum chemistry) or the solution to an atom confined in a potential, to ensure that its wavefunctions go to zero at a finite distance. The question of how to choose the confinement is complex, and there is no good, single answer. Typically, AOs use several radial functions in each angular momentum channel (each value of \(l\)) with different radii, to allow for flexibility. The angular momenta for the valence electrons must be included, but it is common to add higher empty angular momentum shells (called polarisation orbitals).

As well as the problem of confinement, there is no systematic way to improve the completeness of an AO basis set. It is not clear whether the number of radial functions (often referred to as the number of zetas) or the maximum angular momentum would offer better convergence, or if both should be increased. In solid state calculations, the double zeta plus polarisation (DZP) basis set is often used as a standard, with two radial functions for the valence electrons and a single function for the first empty shell[3].

PAOs are not an orthogonal basis set, which changes the matrix representation of the Schrodinger equation. If we write \(\vert \phi_{i\alpha}\rangle\) for a PAO on atom \(i\) with \(\alpha\) a combined index for zeta and angular momentum, and expand the eigenstates in this basis, we find:

$$\vert \psi_{n}\rangle = \sum_{i\alpha} c^{n}_{i\alpha}\vert \phi_{i\alpha}\rangle\\ \hat{H}\vert \psi_{n}\rangle = \epsilon_n \vert \psi_{n}\rangle \\ \sum_{j\beta} H_{i\alpha j\beta} c^{n}_{j\beta} = \epsilon_{n} \sum_{j\beta} S_{i\alpha j\beta} c^{n}_{j\beta}\\ S_{i\alpha j\beta} = \langle \phi_{i\alpha} \vert \phi_{j\beta}\rangle\\ H_{i\alpha j\beta} = \langle \phi_{i\alpha} \vert \hat{H} \vert \phi_{j\beta}\rangle$$

This is a generalised eigenvalue equation, which can be solved in a variety of ways. It’s important to note that the number of PAOs is generally much smaller than the number of PWs (13 per atom for a DZP basis for C or Si compared to several hundred or several thousand plane waves), so that the ground state can often be found by direct diagonalisation, rather an a minimisation. The locality of the PAOs is also important: the Hamiltonian and overlap matrices (H and S above) are sparse, and the construction of them can be made to scale linearly with system size, even if the solution does not.

There is no ideal basis set for electronic structure calculations, but it is important to know about the basis set you use, and to characterise it. The more you learn and think about the calculations you are doing, the better the science that you are performing will be.

[1] Note that in many cases we actually use the orbitals that come from using a pseudopotential with the atom, giving pseudo-atomic orbitals.

[2] FFTs are very good for small and medium sized systems; however, they require a certain level of all-to-all communication when implemented in parallel, which ultimately limits their scaling to very large systems or numbers of processes.

[3] The correlation consistent basis sets of quantum chemistry form a group of basis sets where, for each new level of accuracy, an extra radial function is added to every existing angular momentum, and a new angular momentum channel is added. This leads to a series: single zeta (SZ); double zeta plus polarisation (DZP); triple zeta, double polarisation plus f (TZDPF); etc.