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# Monopole Dipole and Quadrupole corrections

For charged cells or for calculations of molecules and surfaces with a large dipole moment, the energy converges very slowly with respect to the size ${\displaystyle L}$ of the supercell. Using methods discussed by Makov et al.[1] and Neugebauer et al.[2], VASP can correct for the leading errors (in many details, we have taken a more general approach, though).

The following flags control the behaviour of VASP:

NELECT determines the total number of electrons in the system. The value may deviate from the default value, which is calculated assuming charge neutrality in the system. If NELECT differs from the default, an additional neutralizing background charge is applied by VASP. In this case, however, the energy converges very slowly with respect to the size L of the super cell. The required first order correction to the energy is given by
${\displaystyle {\frac {e^{2}q^{2}\alpha }{L\epsilon }}}$
where q is the net charge of the system, α the Madelung constant of a point charge q placed in a homogeneous background charge -q, and ε the dielectric constant of the system. For atoms or molecules surrounded by vacuum, ε takes on the vacuum value ε=1. VASP can automatically correct for the leading error, by setting the IDIPOL and EPSILON tags in the INCAR file (see below).
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 EPSILON-tag can be used to supply the dielectric constant of the medium. VASP uses this flag only to scale the calculated monopole and dipole corrections. EPSILON defaults to 1, which is the proper value for isolated atoms and molcules. For solids, the screening properties can and should be determined using the linear response routines of VASP (see LEPSILON and/or LCALCEPS). Ionic contributions to the dielectric tensor should be included, if the ions are allowed to relax. Ionic contributions to the dielectric tensor can be calculated using IBRION=8.
• IDIPOL, type of correction (monopole/dipole and quadrupole):
For systems with a net dipole moment, the energy converges slowly with respect to the size of the super cell as well. The dipole corrections (and quadrupole corrections for charged systems) fall off like 1/L3. Both corrections, dipole and quadrupole for charged systems, will be calculated and added to the total energy if IDIPOL is set.
There are four possible settings for IDIPOL (= 1 | 2 | 3 | 4).
For IDIPOL=1-3, the dipole moment will be calculated only parallel to the direction of the first, second or third lattice vector, respectively. The corrections for the total energy are calculated as the energy difference between a monopole/dipole and quadrupole in the current supercell and the same dipole placed in a super cell with the corresponding lattice vector approaching infinity. This flag should be used for slab calculations.
For IDIPOL=4 the full dipole moment in all directions will be calculated, and the corrections to the total energy are calculated as the energy difference between a monopole/dipole/quadrupole in the current supercell and the same monopole/dipole/quadrupole placed in a vacuum, use this flag for calculations for isolated molecules.
Note: strictly speaking quadrupole corrections is not the proper wording. The relevant quantity is
${\displaystyle \int d^{3}{{\mathbf r}}\rho ({\mathbf r})\Vert {\mathbf r}\Vert ^{2}.}$
• DIPOL, center of the net charge of the cell:
This tag sets the center of the net charge distribution: DIPOL=Rcenter (in direct coordinates). The dipole is then defined as
${\displaystyle \int ({{\mathbf r}}-{{\mathbf R}}_{{\mathrm {center}}})\rho _{{\mathrm {ions+valence}}}({{\mathbf r}})d{{\mathbf r}}}$
If DIPOL is not set, VASP determines, where the charge density averaged over one plane drops to a minimum and calculates the center of the charge distribution by adding half of the lattice vector perpendicular to the plane where the charge density has a minimum (this is a rather reliable approach for orthorhombic cells).
• The tag LDIPOL enables dipole and if required also monopole corrections to the potential:
This tag switch on the potential correction mode. Due to the periodic boundary conditions, not only the total energy converges slowly with respect to the size of the supercell, but also the potential and the forces are affected by finite size errors. This effect can be counterbalanced by setting LDIPOL=.TRUE. (dipole corrections) in the INCAR file. 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. This is in the spirit of Neugebauer et al.[2] (but more general and the total energy is correctly implemented, whereas the Neugebauer paper contains an erroneous factor 2 in the total energy). The biggest advantage of this mode is that leading errors in the forces are corrected, and that the work-function can be evaluated for asymmetric slabs. The disadvantage is that the convergence to the electronic groundstate might slow down considerably (i.e., more electronic iterations might be required to obtain the required precision). It is recommended to use this mode only after pre-converging the orbitals without the LDIPOL flag, and the center of charge should be set in the INCAR file (DIPOL= center of mass). The user must also ensure that the cell is sufficiently large to determine the dipole moment with sufficient accuracy. If the cell is too small, charge might swap through the vacuum, causing very slow convergence (often convergence improves with the size of the supercell).
• The LMONO switches on monopole corrections for charged systems. The correction is calculated only a posteriori for the total energy. No correction to the potential is calculated. If corrections for the potential are desired as well, please use the LDIPOL instead (for LDIPOL=.TRUE., VASP automatically determines whether the system is charged and activates the monopole corrections automatically).

• EFIELD, to apply an electrostatic field:
It is possible to apply an external electrostatic field in slab, or molecular calculations. Presently only a single value can be supplied and the field is applied in the direction selected by IDIPOL=1-3. The field is supplied in units of eV/Å. Dipole corrections to the potential (LDIPOL=.TRUE.) can and should be turned on to avoid interactions between the periodically repeated images.