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 $L$ of the supercell. Using methods discussed in Ref. [55,56] VASP is able to correct for the leading errors, but one should stress, that in many details, we have taken a more general approach than that one outlined in Ref. [55].

The following flags control the behavior of VASP.

• NELECT, charged systems

  NELECT determines the total number of electrons in the system (see Sec. 6.34). For charged systems this value has to be supplied by hand and a neutralizing background charge is assumed by VASP. For these systems the energy converges very slowly with respect to the size of the super cell. The required first order energy correction is given by
  

  $$e^2 q^2 \alpha / L / \epsilon$$
  
  
  where $q$ is the net charge of the system, $\alpha$ the Madelung constant of a point charge $q$ placed in a homogeneous background charge $-q$, and $\epsilon$ the dielectric constant of the system. For atoms or molecules surrounded by vacuum, $\epsilon$ takes the vacuum value $\epsilon=1$. In that case VASP.4.X can correct for the leading error if the IDIPOL tag is set (see below).

• Dipol and quadrupole corrections

  For systems with a net dipole moment the energy also converges slowly with respect to the size of the super cell. The dipole corrections (and quadrupole corrections for charged systems) fall of like $1/L^3$. Both corrections (quadrupole only for charged systems) will be calculated and added to the total energy if the IDIPOL flag is set.

• IDIPOL tag

  If set in the INCAR file monopole/dipole and quadrupole corrections will be calculated. There are four possible settings for IDIPOL

  
   IDIPOL = 1-4
  

  For 1 to 3, the dipole moment will be calculated only into the direction of the first, second or third lattice vector. 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.

• DIPOL tag

  
    DIPOL = center of cell (in direct, fractional coordinates)
  

  This tag determines as in VASP.3.2 the center of the net charge distribution. The dipol is defined as
  

  $$\int ({\bf r}-{\bf R}_{\rm center}) \rho_{\rm ions+valence}{\bf r} d^3 {\bf r},$$ (15)

  
  
  where ${\bf R}_{\rm center}$ is position as defined by the DIPOL tag. If the flag is not set VASP, determines the points where the charge density averaged over one plane drops to a minimum and deduces 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).

• LDIPOL tag

  This tag switches 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 ``wrong''. This effect can be counterbalanced by setting LDIPOL=.TRUE. in the INCAR file. In that case a linear and (in the case of a charged system) a quadratic electrostatic potential is added to the local potential correcting the errors introduced by the periodic boundary conditions. This is in the spirit of Ref. [56] (but more general and the total energy has been correctly implemented). The biggest advantage of this mode is that the leading errors in the forces are corrected, and that the workfunction can be evaluated for asymetric 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 wavefunctions without the LDIPOL flag, and the center of charge should be set by hand (DIPOL = center of mass). The user must also make sure that the cell is sufficiently large to determine the dipol moment with good accuracy. If the cell is too small, it is usually very difficult to tell whether charge is located on the ``left'' or ``right'' side of the slab, causing very slow convergence (often convergence improves with the size of the supercell).

For the current implementation, there are several restriction; please be careful:

• Charged systems:

  Quadrupole corrections are only correct for cubic supercells (this means that the calculated $1/L^3$ corrections are wrong for charged supercells if the supercell is not cubic). In addition we have found empirically that for charged systems with excess electrons (NELECT$>$NELECT $_{\rm neutral}$) more reliable results can be obtained if the energy after correction of the linear error ($1/L$) is plotted against $1/L^3$ to extrapolate results manually for $L\to \infty$. This is due to the uncertainties in extracting the quadrupole moment of systems with excess electrons.

• Potential corrections are only possible for orthorhombic cells (at least the direction in which the potential is corrected must be orthogonal to the other two directions).