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.

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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	projection done in real space, (old, superseded by LREAL=O)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:

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

where the "projected wavefunction character" is defined as:

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

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 $C_{in\mathbf {k} }$ 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).

Related Tags and Sections

ROPT, PREC

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