Hydrogenoid

Hydrogenoid is a term used in physics to describe an atom or ion that contains a single electron orbiting a nucleus with charge Z e, so its electronic structure is governed predominantly by a Coulomb potential. The hydrogen atom (Z = 1) is the simplest example, but many one-electron ions, such as He+, Li2+, and Be3+, are also hydrogenoids. In these systems, the Schrödinger equation for the electron resembles that of hydrogen, with the nuclear charge Z scaling the potential.

In the nonrelativistic, single-electron approximation, the energy levels follow E_n = - μ e^4 Z^2 /(2 h^2 n^2), where μ is the reduced mass of the electron–nucleus system. In atomic units, this becomes E_n = - (Z^2 μ)/(2 n^2). The wavefunctions are hydrogenic, with radial parts that scale roughly as 1/Z and characteristic sizes that shrink with increasing Z. The nonrelativistic Coulomb problem yields degenerate energy levels depending only on the principal quantum number n, with a degeneracy of n^2.

Corrections beyond the basic model include relativistic fine structure (from the Dirac equation) and quantum electrodynamics effects such as the Lamb shift, as well as finite-nuclear-mass (reduced mass) effects and hyperfine interactions. Hydrogenoid systems are important as solvable models in quantum mechanics, benchmarks for spectroscopic techniques, and in astrophysical and plasma contexts where single-electron ions are common. The term is often used interchangeably with hydrogen-like or hydrogenic to describe any one-electron system with a Coulombic core.