Electron-phonon interaction and the ground state of metallic hydrogen

Time-independent perturbation theory has been developed for a many-body system of electrons and nuclei, where the unperturbed states are formed from the product of oscillator functions and Slater determinants of plane waves. The theory is applied to metallic hydrogen. The expansion is simplified by isolating certain subseries which correspond to the energy levels of known simpler systems, and the remaining terms are computed through second order. The perturbed energy levels are given in terms of two parameters: the Wigner-Seitz parameter $r_s$ and the ratio m/M of the electron to the nuclear mass. Anharmonicity relative to harmonic terms depends upon (m/$r_s$M${)}^{1/2}$. An expansion for the energy difference between two states is obtained by subtracting the leading terms of a pair of perturbation expansions. A condition is proposed which indicates when the normal state is the ground state of the system. The normal state has a particularly simple parameter expansion. A numerical series for this state has been computed, and it is found that the lattice dynamics has a large influence on the binding energy.