LMAXFOCKAE

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NMAXFOCKAE and LMAXFOCKAE

Default: LMAXFOCKAE	= -1	for Hartree-Fock and hybrid functionals
	= 4	for post DFT methods

Description: NMAXFOCKAE and LMAXFOCKAE determine whether the overlap densities in the Fock exchange and correlated wave function methods are accurately reconstructed on the plane wave grid. This flag generally only applies to the Fock-exchange part as well as many-body post DFT methods (GW, RPA, MP2, etc.).

In the PAW method, the difference between the charge density of the all-electron partial waves $\phi_\beta$ and the pseudo partial waves $\tilde \phi_\beta$ ${\displaystyle Q_{\alpha\beta}(r)= \phi^*_\alpha(r)\phi_\beta(r) - \tilde \phi^*_\alpha(r)\tilde \phi_\beta(r) }$ is usually restored on spherical grids centered at each atom (one-center terms inside the PAW spheres). To describe long range electrostatic effect, the moments of the differences of the all-electron and pseudo charge density also need to added to the pseudo density on the plane wave grid. This is done up to a certain l quantum number. These augmentation charges exactly restore the moments of the all-electron density on the plane wave grid. For the charge densities used in the Hartree and DFT term, this augmentation is done exactly up to the maximum l quantum number required by the POTCAR files, whereas for the Fock exchange, the augmentation on the plane wave grid is controlled by LMAXFOCK.

For the RPA, GW, and most post DFT methods, the one-center terms are, however, presently not implemented. Depending on the material, this can cause sizable errors in particular for 3d and (to a lesser extent) 2p, 4d and 5d elements. To correct for this error, an alternative treatment is implemented on the plane wave grid. This allows to restore the all-electron densities accurately on the plane wave grid instead of the one-center grids by specifying the flags LMAXFOCKAE and NMAXFOCKAE.

To achieve this improved treatment on the plane wave grid, $Q_{\alpha\beta}(r)$ is Fourier transformed to reciprocal space $Q_{\alpha\beta}(q)$ and then expanded in a set of orthogonal functions localized at each atomic site. These augmentation charges are then added to the pseudo charge densities on the plane wave grid.

For LMAXFOCKAE=-1 (the default for DFT and Hartree-Fock calculations), only the moments of the all-electron charge densities are restored on the plane wave grid. This setting is exact for Hartree-Fock since the one-center terms are implemented.

If LMAXFOCKAE is set to values larger than -1 (and NMAXFOCKAE=1), not only the moments of the all-electron charge density are restored, but also the all-electron charge density is restored up to a typical plane wave energy of 140 eV. This setting yields very accurate results for post DFT methods (MP2, RPA, GW, etc.) for most sp bonded materials. LMAXFOCKAE is used to specify the maximum spherical (l) quantum number up to which this more accurate treatment is used. The default is LMAXFOCKAE=4 for post DFT methods. If no accurate augmentation is desired by the user, simply set LMAXFOCKAE=-1 in the INCAR file.

If LMAXFOCKAE is set to values larger than -1 and NMAXFOCKAE=2, the charge density is restored accurately on the plane wave grid up to a typical plane wave energy of 380 eV. As before, LMAXFOCKAE can be used to specify the maximum spherical (l) quantum number up to which this more accurate treatment is used. NMAXFOCKAE=2 yields very accurate results for post DFT methods (MP2, RPA, GW) even for difficult 3d elements. For RPA and MP2 total energy calculations, differences between NMAXFOCKAE=1 and NMAXFOCKAE=2 are usually tiny for total energy differences. Since the absolute correlation energies might change, it is vital to use the same setting for NMAXFOCKAE and LMAXFOCKAE, if energy differences are calculated. For GW calculations, increasing NMAXFOCKAE from 1 to 2 might change QP energies by 100-200 meV for 3d and late 4d and 5d elements.

The setting for LMAXFOCKAE should be also considered carefully. Generally, it suffices to set LMAXFOCKAE to twice the maximum l quantum number found in the POTCAR file. For instance for sp elements, LMAXFOCKAE = 2 suffices. For d elements, LMAXFOCKAE = 4 suffices (a d electron can create charge densities with l-quantum numbers up to 4), whereas for f elements, users should test whether LMAXFOCKAE = 6 is required.

In summary, useful manual settings of NMAXFOCKAE and LMAXFOCKAE are:

LMAXFOCKAE=-1, to switch off the accurate augmentation altogether
LMAXFOCKAE=4 (or larger) to force an accurate treatment for the charge augmentation on the plane wave grid (can be selected even in Hartree-Fock type calculations).
NMAXFOCKAE=2, to select the very accurate augmentation on the plane wave grid. Please check whether the VASP default setting for LMAXFOCKAE suffices (OUTCAR file).

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