High precision variational calculations of non-relativistic energy levels of the hydrogen molecular ion

We have performed a benchmark calculation of non-relativistic energy level of H$_{2}^{+}$, using Hylleraas coordinates [1] containing three non-linear parameters so that three inter-particle radial coordinates $r_1$, $r_2$, and $r_{12}$ can be described independently. Rayleigh-Ritz variational principle is used to find out minimum of the expectation value of Hamiltonian of this system. Configuration of base varies according to the total angular momenta $J$ of the system, the bigger $J$ is, the more blocks are needed. To solve the matrix equation, power method is used to identify the ground state as well as some other excited states. The non-relativistic ground state energy of H$_{2}^{+}$ has been calculated to a few parts in 10$^{33}$, which represents the best energy level reported so far. \\[4pt] [1] M. M. Cassar and G. W. F. Drake, \textit{J. Phys. B.} \textbf{37} 2485 (2004)