# Why is Energy Cut-off and K-Point Convergence necessary in LCAO-based codes like SIESTA?

I am new to DFT-simulation and I wish to understand how it works behind the scene. As the title suggests I don't understand why LCAO-based codes have to define K-points and cut-off energies. I understand why they are used in Plane-Wave codes. In such codes the cut-off energy defines the max G value of the Fourier series and the k-point defines how fine the Monkhorst-Pack grid is defined. However, in LCAO we work in real space, so both parameters are not inherent to the representation used.

How I understand it now, the K-point is defined to determine how fine the real-space grid is discretized, but I exactly understand how and why this is used instead of using a real-space discretization variable. If this is the case, I still don't see what purpose the Energy cut-off has.

Could someone help me with this or point me in the direction of some resources on this?

The k-point discretization has the same meaning in plane wave codes as LCAO based codes. In fact, it has the same meaning in all DFT related codes. It defines the integration of the Brillouin zone. You want the integration to be good enough to capture the relevant physics (e.g. graphene with k-point sampling touching the Dirac point vs Gamma-only) but also sparse enough to allow high computational throughput.

The mesh cutoff determines the plane wave cutoff of the kinetic energy, so in that sense it relates directly to the energy cutoff from plane wave codes. It defines the density of mesh-points for things such as density, XC functionals etc.

What people typically refer to LCAO in terms of real-space codes is that it is capable of calculating systems so large that only the Gamma-point is necessary (thereby having only real-space components of the wavefunctions). I.e. thousands of atoms.
However, generally LCAO can equally well simulate condensed matter physics where Brillouin zone integrations are important.

Finally let me note that the difficult thing about LCAO codes (Siesta, OpenMX etc) is not the mesh cutoff, nor the k-point sampling (they are easy to converge). It is the basis set which is important. This basis set is generally not transferable between two different chemical environments and tuning of the basis sets are important to correctly describe the physics.

Edit: Perhaps I should clarify that LCAO, generally, does not work in real space. Generally the LCAO model is exactly the same as the PW codes. This is a misconception of what the "real space" refers to in LCAO codes.

• In what part of a LCAO simulation would integration happen in reciprocal space? Say you want to compute the equilibrium structure of bulk Cu. The DFT part can be solved in real-space and the equilibrium calculation wouldn't require any calculations in k-space either. May 31, 2021 at 19:03
• Same as any other code that exploits the translation symmetry due to periodic boundary conditions via k point sampling. A gamma point calculation is just k point sampling with a single k point - you are asserting that a single sampling point is sufficient to accurately perform the reciprocal space integration, and it just so happens that purely real algebra is all that is required due to your choice of k point Jun 1, 2021 at 11:13
• @Brentdb you are missing the point. In LCAO you could do a 250 atom bulk Cu calculation at the Gamma-point and thus find the bulk structure. Or you could do a single atom with a fine Brillouin zone integration using k-points. It turns out the latter is much more efficient. When you can use symmetries, always use them. You really seem to be confusing k-points and real space here... Jun 1, 2021 at 11:17