If the molecular species to be modelled is charged, special consideration must be given to obtaining the correct limit for the total energy, as the energy of a periodically repeated system with a net charge diverges. The usual way around this problem is to introduce a uniform background charge into the supercell in order to make the total charge zero. In the limit of a large cell, the energy calculated in this way is the same as that for an isolated system. Unfortunately, the convergence with cell size will be slow. However, Makov and Payne describe a scheme for calculating a correction to the total energy which leads to more rapid convergence [40].

The energy of the charge-neutral system will converge as a power law
in the size of the supercell *L*. The terms of this series to
may be found and used to correct the calculated total
energy.

The charge density of the system consists of the density of the charged system, , and the background charge, .

where , if *q* is the charge on the molecule and *V*
is the volume of the supercell. This may be split into two components,
adding and subtracting a point charge at giving

The position is chosen so that has no net dipole moment and the origin of coordinates is chosen to be at the centre of the unit cell.

The energy of interaction may be divided into three components:

**:**- on a lattice interacting with itself
This is the Madelung energy of a system of point charges on a cubic lattice immersed in a neutralising background [41],

where is the Madelung constant which depends on the lattice.

**:**- on a lattice interacting with itself
It may be shown that the energy of interaction of a neutral charged system with no net dipole moment on a cubic lattice will converge as [40].

**:**- and on a lattice interacting only
with each other.
Makov and Payne [40] show that this may itself be divided into two components. The interaction of the point charge with has a leading term due to the interaction of the point charge with the second radial moment of , Q. This vanishes due to symmetry for a simple cubic lattice.

The second component is due to the interaction of with the jellium background. This leads to a total contribution of

Therefore the total calculated energy for a simple cubic lattice is given by

where is the desired energy of the isolated charged molecule. An example of the effect of these corrections on the convergence of the energy of an molecule may be seen in Figure .

**Figure:** Graph demonstrating the effect of the
charged molecule corrections on the convergence of the energy of an
molecule with supercell side length.

Wed Sep 24 12:24:18 BST 1997