Some suggestions:

1. You can try to do a pre-optimization with a semiclassical MD scheme to get the ions in a better position for a full relaxation

2. You can then start with a coarse k-mesh (even Gamma-only) and low cut-off energy and a non-magnetic setting (in case your system is magnetic).

3. In case you system is metallic, consider a relatively large smearing value to avoid unexpected oscillations in charge distribution near the Fermi level for the first runs.

4. Also, it's better to start with standard conjugate-gradiant method and broyden mixing.

5. After completing each step, keep the charge density and use it as the initial guess for the next step.

5. When all these done, you can do a final optimization with a more accurate setting.

In priciple if you have any energy dispersion in your supercell (i.e. if the energy levels have significant difference for different k-points) you have interacting supercells and your defects are interacting strongly. If you use 5 k-points in one direction for unit cell and you are satisfied with energy convergence, you have the same wave functions (apart from numerical errors) with supercell having 5 units in given directions. That is the meaning of Bloch theorem. In math language Bloch waves are irreducible representation of translation group. Advantage of using just unit cell and k points instead of big supercell is the knowledge that wave functions belonging to different k points belong to different irreducible representation, therefore integrals of the kind:

The perfect answer by Mohammad Saeed Bahramy . And also use VASPKIT to get the initial guess of Kpoints.

Use a crude k mesh, save the results, and continue with a fine mesh, compare your results, and continue with an even finer mesh until convergence is reached.

Defects often change the electronic structure, and so you need to re-check your k-point convergence in any case. Since it's the force you care most about, that's what you need to base your convergence criteria on. I suggest you:

1. Fully optimise the unit cell (1x1x1) with your converged cut-off energy and k-point sampling. Use slightly stricter geometry optimisation tolerances for your optimisation here -- it's cheap for the unit cell, and it'll save you time for the supercell.

2. Construct a pristine 2x2x2 supercell, and introduce the defect you wish to study.

3. Perform a k-point convergence test on the new supercell, examining how the force on the perturbed atom(s) converges. Note that these are simple energy-force calculations, not geometry optimisations, so they are relatively quick.

You are now ready to optimise the supercell, knowing the k-point spacing you need to get the forces right. This may or may not be the same as the k-point spacing for the (1x1x1) unit cell -- defects can change the electronic structure quite substantially, depending on the nature of the defect and the material. Assuming you're actually interested in the dilute limit of defect concentration, you will want to perform a fixed-cell geometry optimisation rather than a variable-cell one, i.e. only optimise the atomic positions not the lattice vectors.

You don't say which geometry optimisation method you're using, but it's always worth using a good preconditioner. If you have a good guess for the mean optical phonon frequency then you can use that to initialise the Hessian, or if you have a reasonable forcefield you may be able to use that. There are a lot of possibilities here, so I'll just refer you to some papers I've found helpful:

Article Relaxation of Crystals With the Quasi-Newton Method

Article A universal preconditioner for simulating condensed phase materials

I find L-BFGS is as good an algorithm as any, unless you have a lot of constraints in which case two-point steepest descent (TPSD/2PSD) is sometimes quicker.

If you have a good delocalised internal coordinate method implemented, then this is probably worth using for large defect supercells. However, creating good delocalised internal coordinates for condensed phases is quite difficult, and many programs don't have one.

If you're using absolute coordinates instead (Cartesian or fractional cell) and are not using a sophisticated preconditioner, then as your geometry optimisation proceeds you will see a characteristic displacement pattern whereby the 1st nearest neighbour atoms (relative to the defect) have the largest forces and move most in the first few optimisation steps, then the 2nd nearest neighbours get large forces and start to move, then the 3rd nearest neighbours etc. This is typical, but very unhelpful as it slows your geometry optimisation enormously.

The basic problem is that the material "wants" to keep the atoms separated by the bulk inter-atomic spacing. Your initial geometry has all the right spacings, except that the defect introduces an additional force on the 1st nearest neighbours (and other atoms too, of course, but it's largest on the nearest atoms). The forces on the other atoms are still small, so your optimisation algorithm will mostly move the 1st nearest neighbours. Unfortunately, moving these atoms changes the distance between the 1st and 2nd nearest neighbours, so now there is a force on the 2nd nearest neighbours so they will move in the next iteration -- causing a force on the 3rd nearest neighbours, and so on. The forces do get smaller each time, but not always by very much.

This "displacement wave" will pass from the defect outwards, until it meets either a fixed atom or the periodic image of the displacement wave travelling in the opposite direction; in either case, the wave will then be reflected and start travelling back towards the defect. In the best case, when the wave reaches the defect again the displacements are all so small that you have essentially got your optimised geometry -- if your supercell is N atoms in the longest direction, this will take N geometry optimistion steps. In the worst case, the forces and displacements are still too large and the wave reflects again and travels outwards, and so it continues. In any case, the number of optimisation steps grows linearly with the supercell size, which is definitely not what you want! For more details, see:

Article A faster way to relax interfaces in supercells

You can also break this cycle yourself, and dramatically improve convergence. If you examine the displacement of the atoms from their original positions, you will see that for the most part the closest atoms to the defect moved the furthest, and the further away an atom is from the defect, the less it moved. If you knew what the desired displacement direction was (e.g. radially towards a point defect), you could approximate the atomic displacements by moving all the atoms in this direction, but moving them less if they were further away. A reasonable approximation is to distribute the strain uniformly across the material, so that each of the inter-atomic distances is changed by the same amount.

This method is only useful if you know the strain pattern already. How could you know the displacement direction without doing the full optimisation? There are several approaches, but I think the simplest are either:

a) Fix all the atoms in the supercell except the ones close to the defect (definitely the 1st nearest neighbours, but it's often best to include more, e.g. everything within 4 A of the defect). Fully optimise these atomic positions, keeping the other atoms fixed. You now know the displacement directions, although be aware that the magnitude of the displacements will be less than the optimum, because you fixed the surrounding atoms;

or

b) Fix the fractional coordinates of all the atoms, and do a variable-cell (lattice) optimisation. The optimised lattice will approximate the zero-strain geometry, so the change in the lattice can be used to infer what the strain pattern is for the fixed-cell defect supercell.

Once you know the strain pattern, you can displace all the atoms so as to uniformly distribute the strain. This is then the starting point for your full geometry optimisation, but by taking care of the long-range strain field yourself you should find it converges in far fewer iterations, and the number of iterations no longer increases for larger supercells.

I do not know what materials you work on. However, try to review k-point in literature I am sure that you will get an idea on how to choose a proper/correct k-point for your work/materials (elelemnts, Binary, ternary, or quarternary)

For Fermi surface calculation in CASTEP, the k-point spacing need to be less than 0.01. But when I going to select this spacing, the number of k-points exceeding the CASTEP limit. So in that case, how can I solve the problem??

M N H Liton There shouldn't be a limit to the number of k-points in CASTEP. What error message do you get?

Dear Philip James Hasnip

When the k-point spacing less than 0.01, an error message shown which indicates that if the k-spacing is not less than 0.01 then the fermi surface and electron density difference might be incorrect.

M N H Liton are you using Materials Studio? This doesn't look like a CASTEP error. If it is an error from CASTEP itself, I'd be happy to investigate it - just send me the CASTEP input files (either via email or a direct message on ResearchGate).