LMAXFOCKAE

LMAXFOCKAE = [integer]

Default: LMAXFOCKAE	= -1	DFT, Hartree-Fock
	= 4	post-DFT methods

Description: Specifies the maximum spherical angular momentum quantum number ([math]\displaystyle{ L }[/math]) up to which augmentation charges are accurately restored on the plane-wave grid. This parameter controls the quality of shape restoration for the PAW method.

In the PAW method, the difference between the charge density of the all-electron partial waves [math]\displaystyle{ \phi_n }[/math] and the pseudo partial waves [math]\displaystyle{ \tilde \phi_n }[/math]

[math]\displaystyle{ Q_{nm}({\mathbf r})= \phi^*_n({\mathbf r})\phi_m({\mathbf r}) - \tilde \phi^*_n({\mathbf r})\tilde \phi_m({\mathbf r}) }[/math]

is usually treated on spherical grids centered at each atom (one-center terms inside the PAW spheres, see PAW method method and more detailed description for LMAXFOCK). To describe long-range electrostatic effects, the moments of the differences of the all-electron and pseudo charge density also need to be added on the plane wave grid (compensation density, see PAW method). These compensation charges exactly restore the moments of the all-electron density on the plane wave grid. For the Fock exchange the maximum L quantum number up to which the augmentation is done is controlled by LMAXFOCK.

For GW, RPA, and post-DFT methods, the one-center terms are, however, presently not implemented, which can cause sizable errors, particularly 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 for the exchange and correlation contributions, allowing for accurate restoration of the all-electron densities.

This improved treatment — termed shape restoration — is available for exchange as well as many-body correlation contributions. More details on shape restoration are explained in Ref. ^[1] and ^[2].

Shape restoration allows restoration of the all-electron densities accurately already on the plane-wave grid by specifying the tags LMAXFOCKAE and NMAXFOCKAE. This accurate augmentation charge [math]\displaystyle{ \hat n }[/math] is calculated for the total orbital and angular momentum quantum numbers [math]\displaystyle{ L }[/math] (up to LMAXFOCKAE) and [math]\displaystyle{ M }[/math] using shape-restoring radial functions [math]\displaystyle{ \Delta g_L(r) }[/math] with coefficients [math]\displaystyle{ c^L_{nm} }[/math] to be determined:

[math]\displaystyle{ \hat n=\sum_{LM\atop n,m}\rho_{LM}\!\left[q^{LM}_{nm}Y_{LM}(\Omega)+c^L_{nm}\,\Delta g_L(r)Y_{LM}(\Omega)\right] }[/math]

where [math]\displaystyle{ Y_{LM}(\Omega) }[/math] are the spherical harmonics and [math]\displaystyle{ q^{LM}_{nm} }[/math] are the moments of the charge difference.

NMAXFOCKAE controls how accurately this compensation is performed by specifying the maximum expansion order of spherical Bessel functions [math]\displaystyle{ j_L }[/math] with carefully chosen coefficients [math]\displaystyle{ \alpha_{\beta L} }[/math] and [math]\displaystyle{ q_{\beta L} }[/math] (cf. Appendix of Ref. ^[2] for details).

[math]\displaystyle{ \;\Delta g_L(r)=\sum_{\beta=1}^{NMAXFOCKAE}\alpha_{\beta L}\, j_L\!\left(q_{\beta L}\, r\right) }[/math]

Mind: This tag generally only applies to the Fock-exchange and many-body perturbation theory (MBPT) (GW, RPA, MP2, etc). The default for DFT and Hartree-Fock, LMAXFOCKAE = -1, restores only the moments of the all-electron charge densities on the plane-wave grid. For Hartree-Fock, the setting is very precise since the one-center terms are implemented in radial grids as for DFT.

Recommendations

Element-Dependent

The setting for LMAXFOCKAE should be chosen based on the elements in the calculation:

s and p elements: LMAXFOCKAE = 2
d elements: LMAXFOCKAE = 4 (a d electron can create charge densities with L-quantum numbers up to 4)
f elements: LMAXFOCKAE = 6 may be required - test for each case

The general rule is to set LMAXFOCKAE to approximately twice the maximum l quantum number found in the POTCAR file.

General

Setting LMAXFOCKAE = 4 (or larger) forces an accurate treatment for charge augmentation on the plane-wave grid. This can be selected even in Hartree-Fock-type calculations, but causes some additional noise.
If no accurate augmentation is desired set LMAXFOCKAE = -1, which will reduce the noise in the correlation energies.
For DFT calculations, the exact one-center terms are implemented. This means shape restoration is not required and should not change the results. For post-DFT calculations, shape restoration significantly improves accuracy.

Relationship with NMAXFOCKAE

LMAXFOCKAE and NMAXFOCKAE are strongly coupled. It is vital to use consistent settings for both when comparing energy differences.

NMAXFOCKAE = 1 and LMAXFOCKAE ≥ 0: Charge density is restored up to approximately 150 eV (controlled by QMAXFOCKAE).
NMAXFOCKAE = 2 and LMAXFOCKAE ≥ 0: Charge density is restored up to approximately 400 eV.

Important: NMAXFOCKAE = 2 is for very accurate augmentation on the plane-wave grid. Use NMAXFOCKAE = 2 only with care, as it can result in very noisy data for coarse FFT grids.

References

[shishkin:prb:2006-1] M. Shishkin and G. Kresse, Phys. Rev. B 74, 035101 (2006).

[unzog:prb:2022-2] M. Unzog, A. Tal, G. Kresse, X-ray absorption using the projector augmented-wave method and the Bethe-Salpeter equation, Phys. Rev. B 106, 155133 (2022).

[1]

[2]