Exploring the energy landscape of the Thomson problem: local minima and stationary states

We conducted a comprehensive numerical investigation of the energy landscape of the Thomson problem for systems up to $N=150$. Our results show the number of distinct configurations grows exponentially with $N$, but significantly faster than previously reported. Furthermore, we find that the average energy gap between independent configurations at a given $N$ decays exponentially with $N$, dramatically increasing the computational complexity for larger systems. Finally, we developed a novel approach that reformulates the search for stationary points in the Thomson problem (or similar systems) as an equivalent minimization problem using a specifically designed potential. Leveraging this method, we performed a detailed exploration of the solution landscape for $N\leq24$ and estimated the growth of the number of stationary states to be exponential in $N$.