Mesh-Cutoff

[Suresh]

Dear All,

Does the Mesh-cutoff depends on the number of atoms in the Unit cell.

Thanks

[Anders Blom]

The mesh cut-off is given as input by the user. It's an energy (default 150 Ry) which corresponds to the fineness of the real-space grid on which the Poisson equation is solved. (Or, grids, there are a couple of different grids, including the real-space electron density and the effective potential.) A higher value of the mesh cut-off gives a finer real-space grid and hence better accuracy.

The fineness or grid point spacing dx is calculated according to the formulas (hmm, isn't there a way to type math prettier...?)

E = hbar² k² / 2m

dx = pi/k (no factor 2 here!)

which if E is in Rydberg and dx is in Bohr becomes

dx = pi/sqrt(E)

k is the equivalent plane-wave cut-off component, but one can not really compare the values used in a plane-wave code and ATK, for several reasons.

This is perhaps obvious to you, but it's best to have a full explanation at hand.

Now, regarding how to choose the mesh cut-off, the only really proper way is to try with one value, then change it and see if the results (say, the current, or band structure) changes. If it does, you keep increasing it until the results converge.

The default (150 Ry) is often sufficient in most cases, in particular for electronic structure and transport calculations. In fact, for carbon nanotubes and other systems you can often decrease it and still get quality results (but not necessarily very fine plots of the density...). For geometry optimizations, a higher value is sometimes needed, in particular for elements with complex d-shells like iron.

So, the number of atoms is not really a determining factor a priori, and in most cases the real-space grids are not a limiting factor in the sense that the calculation uses too much memory. The main impact of increasing the mesh-cut off is rather on calculation time.

[Anders Blom]

Another point, in case somebody is wondering:

Since ATK is a numerical-orbital code, it's easy to think that the real-space mesh determined by the mesh cut-off also controls the basis set functions of the atomic orbitals. This is, however, not the case (which is one of the reasons you cannot compare the value of the cut-off in ATK with plane-wave codes).

The basis set orbitals are expanded on a radial equidistant grid where the point spacing is determined by the keyword radial_sampling_dr (in basisSetParameters()). The default value of this parameter was lowered quite substantially in ATK 2008.02 to improve the default accuracy, which in particular was very helpful in relaxations. The new value should give a sufficiently smooth representation of the basis functions in all normal cases.

For more information on the basis set functions, see the manual section on basisSetParameters().

[Suresh]

Dear Anders Blom,

Thank you for proper guidance.

In case of water molecule, with mesh cutoff 30-200 Ry, the energy is calculated. After 90Ry, the Total energy remains constant. But In my example two probe, I have varied the mesh cut-off from 150-400 Ry, But The Total enrgy is not constant. As compared to water molecule, I should get the Total energy constant for some value of Mesh-cutoff. but I am not getting. should I increase the mesh cut-off above 400Ry.

Thanks

[Anders Blom]

You should not expect the total energy to be a smooth curve that converges to a lot of decimals with increasing the mesh cut-off (or some other parameters like k-point sampling, for that matter). The nature of the problem, and the complexity of the non-linear self-consistent cycle means that there will always be small fluctuations. Therefore, look rather for a general trend of convergence when you plot the energy vs the mesh cut-off.

At the same time, I would not use the total energy as the criterion in a two-probe system, but rather something like the transmission spectrum (even just at the Fermi level).

If you plot this against the mesh cut-off there should be convergence before 400 Ry, hopefully!