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Charged Molecules

 

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 tex2html_wrap_inline2631 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, tex2html_wrap_inline2633 , and the background charge, tex2html_wrap_inline2635 .

equation432

where tex2html_wrap_inline2637 , 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 tex2html_wrap_inline2643 giving

equation436

The position tex2html_wrap_inline2643 is chosen so that tex2html_wrap_inline2647 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:

tex2html_wrap_inline2649 :
tex2html_wrap_inline2651 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],

equation445

where tex2html_wrap_inline2653 is the Madelung constant which depends on the lattice.

tex2html_wrap_inline2655 :
tex2html_wrap_inline2657 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 tex2html_wrap_inline2631 [40].

tex2html_wrap_inline2661 :
tex2html_wrap_inline2651 and tex2html_wrap_inline2657 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 tex2html_wrap_inline2657 has a leading term due to the interaction of the point charge with the second radial moment of tex2html_wrap_inline2657 , Q. This vanishes due to symmetry for a simple cubic lattice.

The second component is due to the interaction of tex2html_wrap_inline2657 with the jellium background. This leads to a total contribution of

equation455

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

equation463

where tex2html_wrap_inline2467 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 tex2html_wrap_inline2675 molecule may be seen in Figure gif.

   figure471
Figure: Graph demonstrating the effect of the charged molecule corrections on the convergence of the energy of an tex2html_wrap_inline2679 molecule with supercell side length.


next up previous contents
Next: Summary Up: The Total Energy Pseudopotential Previous: Nuclear Degrees of Freedom

Matthew Segall
Wed Sep 24 12:24:18 BST 1997