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I was calculating a charged monolayer using VASP5.4.4 and the slab model. Since my supercell is not cubic, I can not obtain dipole corrected potential by setting LDIPOL = .TRUE. because "Quadrupole corrections are only correct for cubic supercells".
On the wiki page "Monopole_Dipole_and_Quadrupole_corrections", it said "For LDIPOL=.TRUE., a linear correction and for charged cells a quadratic electrostatic potential is added to the local potential in order to correct the errors introduced by the periodic boundary conditions." My question is why the correction is linear and quadratic for charged cells. And what is the formula for the linear or quadratic correction?

By setting IDIPOL = 3, I can obtain corrected energy for charged system. Since "Quadrupole corrections are only correct for cubic supercells", can I use this energy? I also followed the wiki page, by setting IDIPOL= 3 and LMONO = .TRUE. I obtain monopole corrected energies at different L, and extrapolate to L = infinity. This is suggested for negatively charged system.

I quote from the relevant section of the manual

Vasp manual wrote:It is important to emphasize that the total energy can not be corrected for charged slabs, since a charged slab results in an electrostatic potential that grows linearly with the distance from the slab (corresponding to a fixed electrostatic field). It is fairly simple to show that as a result of the interaction between the charged slab and the compensating background, the total energy depends linearly on the width of the vacuum. Presently, no simple a posteriori correction scheme is known or implemented in VASP. Total energies from charged slab calculations are hence useless, and can not be used to determine relative energies.
Note: If you are not convinced, simply vary the vacuum width and draw the energy versus the vacuum width.

The corrections that are implemented in Vasp are intended for charged bulk materials, atoms, and molecules. The dipol corrections work for more classes of systems.