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# LREAL

LREAL = .TRUE. | .FALSE. | On (or O) | Auto (or A)
Default: LREAL = .FALSE.

Description: LREAL determines whether the projection operators are evaluated in real-space or in reciprocal space.

 LREAL=.FALSE. projection done in reciprocal space LREAL=.TRUE. projection done in real space, (old, superseded by LREAL=O) LREAL=On or O projection done in real space, projection operators are re-optimized LREAL=Auto or A projection done in real space, fully automatic optimization of projection operators (no user interference required)

The non local part of the pseudopotential requires the evaluation of an expression:

${\displaystyle \sum _{ij}D_{ij}|\beta _{j}\rangle \langle \beta _{i}|{\tilde {\psi }}_{n{\mathbf {k} }}\rangle }$ .

where the "projected wavefunction character" is defined as:

{\displaystyle {\begin{aligned}C_{in{\mathbf {k} }}=\langle \beta _{i}|{\tilde {\psi }}_{n{\mathbf {k} }}\rangle &={\frac {\Omega }{N_{\rm {FFT}}}}\sum _{\mathbf {r} }\langle \beta _{i}|{\mathbf {r} }\rangle \langle {\mathbf {r} }|{\tilde {\psi }}_{n{\mathbf {k} }}\rangle ={\frac {\Omega }{N_{\rm {FFT}}}}\sum _{\mathbf {r} }\beta ({\mathbf {r} }){\tilde {\psi }}_{n{\mathbf {k} }}({\mathbf {r} })\\&=\sum _{\mathbf {G} }\langle \beta _{i}|{\mathbf {k} }+{\mathbf {G} }\rangle \langle {\mathbf {k} }+{\mathbf {G} }|{\tilde {\psi }}_{n{\mathbf {k} }}\rangle =\sum _{\mathbf {G} }{\bar {\beta }}({\mathbf {k} }+{\mathbf {G} })C_{{\mathbf {G} }n{\mathbf {k} }}\end{aligned}}}

This expression can be evaluated in reciprocal or real space: In reciprocal space (second line) the number of operations scales with the size of the basis set (i.e. number of plane-waves). In real space (first line) the projection-operators are confined to spheres around each atom. Therefore the number of operations necessary to evaluate one Cink does not increase with the system size (usually the number of grid points within the cut-off-sphere is between 500 and 2000). One of the major obstacles of the method working in real space is that the projection operators must be optimized, i.e. all high frequency components must be removed from the projection operators. If this is not done "aliasing" can happen (i.e. the high frequency components of the projection operators are downfolded to low frequency components and a random noise is introduced).

Currently VASP supports three different schemes to remove the high frequency components from the projectors. LREAL=.TRUE. is the simplest one. If LREAL=.TRUE. is selected, the real space projectors which have been generated by the pseudopotential generation code are used. This requires no user interference. For LREAL=On the real space projectors are optimized by VASP using an algorithm proposed by King-Smith et al.[1] For LREAL=Auto a new scheme[2] is used which is considerably better (resulting in more localized) projector functions than the King-Smith et al. method. To fine-tune the optimization procedure the flag ROPT can be used if LREAL=Auto or LREAL=On is used.

We recommend to use the real-space projection scheme for systems containing more than 20 atoms. We also recommend to use only LREAL=Auto (for version VASP.4.4 and newer releases) and LREAL=On (for all other versions). Version 4.4 also supports the old mode LREAL=O to allow calculations that are fully compatible to VASP.4.3 (and VASP.3.2). The best performance is generally achieved with LREAL=Auto, but if performance is not that important you can also use LREAL=.TRUE. which generally requires less user interference. You can skip the rest of the paragraph, if you use only LREAL=.TRUE..

For LREAL=O and LREAL=A the projection operators are optimized by VASP on the fly (i.e. on startup). Several flags influence the optimization:

• ENCUT (i.e. the energy cutoff), components beyond the energy cutoff are 'removed' from the projection operators.
• PREC tag specifies how precise the real space projectors should be, and sets the variables ROPT accordingly to the following values:
For LREAL=On
 PREC=Low 700 points in the real space sphere (ROPT=0.67) PREC=Med 1000 points in the real space sphere (ROPT=1.0) PREC=High 1500 points in the real space sphere (ROPT=1.5)
For LREAL=Auto
 PREC=Low accuracy 1 · 10-2 (ROPT=0.01) PREC=Med accuracy 2 · 10-3 (ROPT=0.002) PREC=High accuracy 2 · 10-4 (ROPT=2E-4)
These defaults can be superseded by specifying the ROPT-tag in the INCAR file.

Mind: Real space optimization (LREAL=.TRUE., On, or Auto) always results in a small (not necessarily negligible) error (the error is usually a constant energy shift for each atom). If you are interested in energy differences of a few meV use only calculations with the same setup (i.e. same ENCUT, PREC, LREAL and ROPT setting) for all calculations. For example, if you want to calculate surface energies recalculate the bulk groundstate energy with exactly the same setting you are going to use for the surface. Another possibility is to relax the surface with real space projection, and to do one final total energy calculation with LREAL=.FALSE. to get exact energies. Anyway, for PREC=Med, the errors introduced by the real space projection are usually of the same order magnitude as those introduced by the wrap around errors. For PREC=High errors are usually less than meV. PREC=Low should be used only for high speed MD's, if computer resources are really a problem.

## References

1. R.D. King-Smith, M.C. Payne and J.S. Lin, Phys. Rev B 44, 13063 (1991).
2. G. Kresse, to be published.