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Periodic Boundary Conditions

We now need to extend the above analysis to calculate the potential in the simulation cell when it is just one of an infinite number of periodically repeated, identical simulation cells making up a bulk solid, as in a supercell calculation. The infinite summations of Coulomb potentials from all the charges in the infinite system are only conditionally convergent. It is therefore easier to solve Poisson's equation, again just for the simulation cell, but now imposing periodic boundary conditions to build in the effect of all the periodic images of the simulation cell.

Let us first consider such a crystal with a smoothly varying charge density. The charge density has the periodicity of the primitive lattice and can therefore be written as a Fourier expansion in the reciprocal primitive lattice vectors, tex2html_wrap_inline6775 , as follows


This charge density generates a potential, tex2html_wrap_inline6765 , satisfying Poisson's equation, Eq.(gif). We can also expand the potential in a Fourier Series to obtain a reciprocal space version of Poisson's equation,


Hence one can solve for tex2html_wrap_inline6765


Eq.(gif) is only soluble provided tex2html_wrap_inline6781 , which implies that one can only solve Poisson's equation for an infinite periodic system if each unit cell is neutral. In this case, the value of tex2html_wrap_inline6783 is completely arbitrary, corresponding to a constant shift in potential which does not affect the potential energy of the system as a whole because it applies in equal and opposite amounts to the positive and negative charges in the system. The solution of Poisson's equation subject to periodic boundary conditions is unique up to an arbitrary constant. This can be demonstrated as follows. Suppose that there were two different periodic solutions, tex2html_wrap_inline6785 and tex2html_wrap_inline6787 , to Poisson's equation. The difference between these two solutions,


also satisfies the periodic boundary conditions. tex2html_wrap_inline6765 must also satisfy Laplace's equation


everywhere within the simulation cell and so tex2html_wrap_inline6765 must be a constant throughout the simulation cell as this is the only periodic solution of Laplace's equation. Therefore the two solutions to Poisson's equation are indeed the same to within an arbitrary constant.

It has therefore been shown that the potential due to a smooth charge density can easily be evaluated in reciprocal space. However, in a QMC calculation, the instantaneous positions of the electrons are represented in real space by a set of delta functions. This array of delta functions cannot be represented by a convergent Fourier series in the same way that a smooth charge density can, and so the above method for evaluating the potential breaks down. The method of calculating the potential due to such a charge density of point charges that has been traditionally used in QMC calculations is Ewald summation. This method is described in the following section.

next up previous contents
Next: Ewald Summation Up: Coulomb Interactions in Supercell Previous: Isolated Simulation Cell

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996