Spin-averaged effective Hamiltonian of orders $m{\alpha }^6$ and $m{\alpha }^6(m/M)$ for hydrogen molecular ions

The spin-averaged effective Hamiltonian of orders $m{\alpha }^6$ and $m{\alpha }^6(m/M)$ for a one-electron two-center Coulombic system is derived by using the theory of nonrelativistic quantum electrodynamics (NRQED), without assuming the Born-Oppenheimer approximation. The separated singularities from the first- and second-order perturbations are shown to be canceled out analytically for both order $m{\alpha }^6$ and $m{\alpha }^6(m/M)$ corrections by regularizing the effective Hamiltonian. Our results can be used to perform high-precision spectroscopic calculations of hydrogen molecular ions.

Figure 1

The coordinates for a one-electron two-center system, where $O$ is the origin of the laboratory frame, the electron is denoted as $e$, and the two nuclei are denoted as particle 1 and 2.