One-electron relativistic molecules with Coulomb interaction

Abstract

As an approximation to a relativistic one-electron molecule, we study the operator$H = ( - \Delta + m^2 )^{1/2} - e^2 \sum\limits_{j = 1}^K {Z_j } |x - R_j |^{ - 1}$ withZ _j ≧0,e ⁻²=137.04.H is bounded below if and only ife ² Z _j ≦2/π allj. Assuming this condition, the system is unstable whene ²∑Z _j >2/π in the sense thatE ₀=inf spec(H)→−∞ as the R _j →0, allj. We prove that the nuclear Coulomb repulsion more than restores stability; namely$E_0 + 0.069e^2 \sum\limits_{i< j} {Z_i Z_j } |R_i - R_j |^{ - 1} \geqq 0$. We also show thatE ₀ is an increasing function of the internuclear distances |R _i −R _j |.