The Five-Electron Case of Thomson’s Problem

Abstract

We give a rigorous computer-assisted proof that the triangular bipyramid is the unique configuration of five points on the sphere that globally minimizes the Coulomb (1/r) potential. We also prove the same result for the (1/r ²) potential. The main mathematical contribution of the paper is a fairly efficient energy estimate that works for any number of points and any power-law potential.

2000 AMS Subject Classification:

• 51-04
• 51F99

Keywords:

• 5-electron problem
• power-law potential
• triangular bi-pyramid
• electron configurations

Acknowledgments

I would like to thank Henry Cohn for helpful conversations about placing electrons on a sphere. Henry’s great colloquium talk at Brown university in the fall of 2009 inspired me to work on this problem. I would also like to thank Jeff Hoffstein and Jill Pipher for their interest and encouragement while I worked on this problem. I would like to thank John Hughes for a very interesting discussion about interval arithmetic. Finally, I would like to say that I learned how to do interval arithmetic in Java by reading the source files from Tim Hickey’s implementation http://interval.sourceforge.net/interval/index.html. This work was supported by N.S.F. Research Grant DMS-0072607

Notes

¹The website http://www-wales.ch.cam.ac.uk/wales/CCD/Thomson/table.html has a list of experimentally determined (candidate) minimizers for the Coulomb potential for n=2, … , 972.

²Lacking a handy reference for these experiments, we performed our own.

³See http://www.math.brown.edu/res/Electron/index.html .