Hydrogen-like atom

A hydrogen-like ion is any atomic nucleus which has one electron and thus is isoelectronic with hydrogen. These ions carry the positive charge e ( Z − 1 ) {displaystyle e(Z-1)} , where Z {displaystyle Z} is the atomic number of the atom. Examples of hydrogen-like ions are He+, Li2+, Be3+ and B4+. Because hydrogen-like ions are two-particle systems with an interaction depending only on the distance between the two particles, their (non-relativistic) Schrödinger equation can be solved in analytic form, as can the (relativistic) Dirac equation. The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals. Other systems may also be referred to as 'hydrogen-like atoms', such as muonium (an electron orbiting an antimuon), positronium (an electron and a positron), certain exotic atoms (formed with other particles), or Rydberg atoms (in which one electron is in such a high energy state that it sees the rest of the atom practically as a point charge). In the solution to the Schrödinger equation, which is non-relativistic, hydrogen-like atomic orbitals are eigenfunctions of the one-electron angular momentum operator L and its z component Lz. A hydrogen-like atomic orbital is uniquely identified by the values of the principal quantum number n, the angular momentum quantum number l, and the magnetic quantum number m. The energy eigenvalues do not depend on l or m, but solely on n. To these must be added the two-valued spin quantum number ms = ±½, setting the stage for the Aufbau principle. This principle restricts the allowed values of the four quantum numbers in electron configurations of more-electron atoms. In hydrogen-like atoms all degenerate orbitals of fixed n and l, m and s varying between certain values (see below) form an atomic shell. The Schrödinger equation of atoms or atomic ions with more than one electron has not been solved analytically, because of the computational difficulty imposed by the Coulomb interaction between the electrons. Numerical methods must be applied in order to obtain (approximate) wavefunctions or other properties from quantum mechanical calculations. Due to the spherical symmetry (of the Hamiltonian), the total angular momentum J of an atom is a conserved quantity. Many numerical procedures start from products of atomic orbitals that are eigenfunctions of the one-electron operators L and Lz. The radial parts of these atomic orbitals are sometimes numerical tables or are sometimes Slater orbitals. By angular momentum coupling many-electron eigenfunctions of J2 (and possibly S2) are constructed. In quantum chemical calculations hydrogen-like atomic orbitals cannot serve as an expansion basis, because they are not complete. The non-square-integrable continuum (E > 0) states must be included to obtain a complete set, i.e., to span all of one-electron Hilbert space. In the simplest model, the atomic orbitals of hydrogen-like ions are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's law: