Relativistic kinetic-energy correction ($\delta H=-p^4/(8m^3{c}^2)$)

Expanding the relativistic kinetic energy $\sqrt{m^2{c}^4+c^2{p}^2}-m{c}^2$ for $|p|\ll mc$ gives $p^2/(2m)-p^4/(8m^3{c}^2)+\cdots$. The first correction is negative and contributes to atomic fine structure. Its perturbative use requires small momentum compared with mc; the truncated negative quartic term is not a globally stable exact Hamiltonian.