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# Energy cut off and FFT mesh

In general, the energy-cut-off must be chosen according to the pseudopotential. All POTCAR files contain a default energy cut off. Use this energy cut-off, but please also perform some bulk calculations with different energy cut-off to find out whether the recommended setting is correct. The cut-off which is specified in the POTCAR file will usually result in an error in the cohesive energy which is less than 10 meV.

You should be aware of the difference between absolute and relative convergence. The absolute convergence with respect to the energy cut off ENCUT is the convergence speed of the total energy, whereas relative convergence is the convergence speed of energy differences between different phases (e.g. energy of fcc minus energy of bcc structure). Energy differences converge much faster than the total energy. This is especially true if both situations are rather similar (e.g. hcp - fcc). In this case the error due to the finite cut-off is 'transferable' from one situation to the other situation. If two configurations differ strongly from each other (different distribution of s p and d electrons, different hybridization) absolute convergence gets more and more critical.

There are some rules of thumb, which you should check whenever making a calculation: For bulk materials the number of plane waves per atom should be between 50-100. A smaller basis set might result in serious errors. A larger basis set is rarely necessary, and is a hint for a badly optimized pseudopotential. If a large vacuum is included the number of plane waves will be larger (i.e. $50\%$ of your super cell vacuum $\to$ number of plane waves increases by a factor of 2).

More problematic than ENCUT is the choice of the FFT mesh, because this error is not easily transferable from one situation to the next. For an exact calculation the FFT mesh must contain all wave vectors up to $2G_{{{\rm {cut}}}}$ if $E_{{{\rm {cut}}}}={\frac {\hbar ^{{2}}}{2m}}G_{{{\rm {cut}}}}^{2}$ , $E_{{{\rm {cut}}}}$ being the used energy-cut-off. Increasing the FFT-mesh from this value does not change the results, except for a possibly very small change due to the changed exchange-correlation potential. The reasons for this behavior are explained in Wrap-around errors.

Nevertheless it is not always possible and necessary to use such a large FFT mesh. In general only 'high quality' calculations (as defined in Technical errors) require a mesh which avoids all wrap around errors. For most calculations, and in particular for the supplied pseudopotentials with the default cut off, it is sufficient to set NGX, NGY and NGZ to $3/4$ of the required values (set PREC=Medium or PREC=Low in the INCAR file before running the makeparam utility or VASP.4.X). The values which strictly avoid any wrap-around errors are also written to the OUTCAR file:

 WARNING: wrap around error must be expected
set NGX to  22

WARNING: wrap around error must be expected
set NGY to  22

WARNING: wrap around error must be expected
set NGZ to  22


Just search for the string 'wrap'. As a rule of thumb the $3/4$ will result in FFT mesh, which contain approximately 8x8x8=256 FFT points per atom (assuming that there is no vacuum).

One hint, that the FFT mesh is sufficient, is given by the lines

 soft charge-density along one line
0      1      2      3      4      5      6      7      8
x    32.0000 -.7711 1.9743  .0141  .3397 -.0569 -.0162 -.0006  .0000
y    32.0000 6.7863  .0205  .2353  .1237 -.1729 -.0269 -.0006  .0000
z    32.0000 -.7057 -.7680 -.0557  .1610 -.2262 -.0042 -.0069  .0000


also written to the file OUTCAR (search for the string 'along'). These lines contain the charge density in reciprocal space at the positions

${\mathbf {G}}=2\pi m_{x}{\mathbf {g}}^{{(x)}},\quad {\mathbf {G}}=2\pi m_{y}{\mathbf {g}}^{{(y)}},\quad {\mathbf {G}}=2\pi m_{z}{\mathbf {g}}^{{(z)}}.$ The last number will always be 0 (it is set explicitly by VASP), but as a rule of thumb the previous value divided by the total number of electrons should be smaller than $10^{{-4}}$ . To be more precise: Because of the wrap-around errors, certain parts of the charge density are wrapped to the other side of the grid, and the size of the 'wrapped' charge density divided by the number of electrons should be less than $10^{{-3}}-10^{{-4}}$ .

Another important hint that the wrap around errors are too large is given by the forces. If there is a considerable drift in the forces, increase the FFT mesh. Search for the string 'total drift' in the OUTCAR file, it is located beneath the line TOTAL-FORCE:

    total drift:                -.00273      -.01048       .03856


The drift should definitely not exceed the magnitude of the forces, in general it should be smaller than the size of the forces you are interested in (usually 0.1 eV/$\AA$ ).

For the representation of the augmentation charges a second more accurate FFT mesh is used. Generally the time spent for the calculation on this mesh is relatively small, therefore there is no need to worry too much about the size of the mesh, and relying on the defaults of the makeparam utility is in most cases safe. In some rare cases like Cu, Fe$\_$ pv with extremely 'hard' augmentation charges, it might be necessary to increase NGXF in comparison to the default setting. This can be done either by hand (setting NGXF in the param.inc file) or by giving a value for ENAUG in the {{TAG|INCAR} file.

As for the soft part of the charge density the total charge density (which is the sum of augmentation charges and soft part) is also written to the file OUTCAR:

 total charge-density along one line
0      1      2      3      4      5      6      7      8
x    32.0000 -.7711 1.9743  .0141  .3397 -.0569 -.0162 -.0006  .0000
y    32.0000 6.7863  .0205  .2353  .1237 -.1729 -.0269 -.0006  .0000
z    32.0000 -.7057 -.7680 -.0557  .1610 -.2262 -.0042 -.0069  .0000


The same criterion which holds for the soft part should hold for the total charge density. If the second mesh is too small the forces might also be wrong (leading to a 'total drift' in the forces).

Mind: The second mesh is only used in conjunction with US-pseudopotentials. For normconserving pseudopotentials neither the charge density nor the local potentials are set on the fine mesh. In this case set NG(X,Y,Z)F to NGX, Y, Z or simply to 1. Both settings result in the same storage allocation.

Mind: If very hard non-linear/partial core corrections are included the convergence of the exchange-correlation potential with respect to the FFT grid might cause problems. All supplied pseudopotentials have been tested in this respect and are safe.

## When to set ENCUT (and ENAUG) by hand

In most cases once can safely use the default values for ENCUT and ENAUG, which are read from the POTCAR file. But there are some cases where this can results in small, easily avoidable inaccuracies.

For instance, if you are interested in the energy difference between bulk phases with different compositions (i.e. Co - CoSi - Si). In this case the default ENCUT will be different for the calculations of pure Co and pure Si, but it is preferable to use the same cut off for all calculations. In this case determine the maximal ENCUT and ENAUG from the POTCAR files and use this value for all calculations.

Another example is the calculation of adsorption energies of molecules on surfaces. To minimize (for instance) non-transferable wrap errors one should calculate the energy of an isolated molecule, of the surface only, and of the adsorbate/surface complex in the same super cell, using the same cut off. This usually requires to fix ENCUT and ENAUG by hand in the INCAR file. If one also wants to use real space optimization (LREAL=On), it is recommended to use LREAL=On for all three calculations as well (the ROPT flag should also be similar for all calculations).