• The Poisson equation tells us how the electrostatic potential due to a distribution of charged atoms varies throughout space.
• So how do we solve it?
• Ideally, we'd like to find an analytical solution, i.e. we'd like to be able to write that

  

• Unfortunately this can only be done for very simple geometries and charge distributions, e.g. spheres. Macromolecules are of arbitrary shape, so in general we have to solve the equation numerically.

Finite Difference Methods

• In the finite difference method we discretize space, i.e. we chop it up into discrete chunks.
• Our aim is to calculate the electrostatic potential only at these grid points. If later we want to know the electrostatic potential at some point between grid points, we simply interpolate from the values at the nearest grid points.
• We re-cast the equations into forms consistent with space no longer being continuous. Here is one form of the finite difference solution of the Poisson equation:
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