# Energy cut-off test meaning in atomic-like basis functions' DFT

I have been using SIESTA code to perform some practicals in my college and I have run energy cut-off tests before calculating properties of semi-conductors or insulator. As far as I am concerned, SIESTA is a code using atomic-like basis sets, instead of plane-wave based ones. In the case of the plane-wave representation, the energy cut-off is the kinetic energy of the plane wave with highest plane-wave vector. The longer this cut-off is, the more waves will be used to represent the orbitals, the more accurate the description of the system will be...

However, what is really the meaning or the physics behind this test for atomic-like basis sets?

There is no energy cut off test for calculations that employ atomic basis sets, in general: the calculation is well-defined with just the atomic basis set.

For comparison, the Gaussian-basis PySCF program implements four ways to compute the Coulomb interactions in crystalline systems:

1. Gaussian-basis density fitting
2. mixed Gaussian-plane wave density fitting
3. plane-wave density fitting
4. exact Gaussian integrals through a range separated approach (https://arxiv.org/abs/2012.07929)

As to your question, SIESTA employs pseudopotentials, and option 3 is what they use. If you look at the newest paper on the code, J. Chem. Phys. 152, 204108 (2020)

The auxiliary real-space grid is an essential ingredient of the method as it allows the efficient representation of charge densities and potentials as well as the computation of the matrix elements of the Hamiltonian that cannot be handled as two-center integrals. This grid can be seen as the reciprocal space of a set of plane waves, and its fineness is most conveniently parameterized by an energy cutoff (the “density” cutoff of plane-wave methods). There are limits to the softness of the functions that can be described with such a grid, so core electrons are not considered (although semi-core electrons usually are), and their effect is incorporated into pseudopotentials. The real-space grid is also used to solve the Poisson equation involved in the computation of the electrostatic potential from the charge density through the use of a fast-Fourier-transform method. This means that SIESTA uses periodic boundary conditions (PBC). Non-periodic systems, such as molecules, tubes, or slabs, are treated using appropriate supercells.

This means that the plane-wave cutoff in SIESTA affects the evaluation of the Coulomb interaction. The higher the cutoff, the better SIESTA is able to describe the interaction of the atomic basis functions. With $$E_\text{cut} \to \infty$$ you recover the exact solution in the used atomic basis set.