Exact Energy Computation of the One Component Plasma on a Sphere for Even Values of the Coupling Parameter

The two dimensional one component plasma 2dOCP is a classical system consisting of $N$ identical particles with the same charge $q$ confined in a two dimensional surface with a neutralizing background. The Boltzmann factor at temperature $T$ may be expressed as a Vandermonde determinant to the power $\Gamma=q^2/(k_B T)$. Several statistical properties of the 2dOCP have been studied by expanding the Boltzmann factor in the monomial basis for even values of $\Gamma$. In this work, we use this formalism to compute the energy of the 2dOCP on a sphere. Using the same approach the entropy is computed. The entropy as well as the free energy in the thermodynamic limit have a universal finite-size correction term $\frac{\chi}{12}\log N$, where $\chi=2$ is the Euler characteristic of the sphere. A non-recursive formula for coefficients of monomial functions expansion is used for exploring the energy as well as structural properties for sufficiently large values of $\Gamma$ to appreciate the crystallization features for $N=2,3,\ldots,9$ particles. Finally, we make a brief comparison between the exact and numerical energies obtained with the Metropolis method for even values of $\Gamma$.