Twoelectron

Two-electron systems are atomic or molecular systems that contain exactly two electrons. The most familiar example is the helium atom, along with its ions that retain two electrons, such as He+ (and other helium-like ions like Li2+). These systems occupy an essential place in quantum chemistry and atomic physics because they are the simplest nontrivial many-body problems involving electron–electron interactions.

In nonrelativistic quantum mechanics, the two-electron Hamiltonian in atomic units is often written as H = -1/2 ∇1^2 - 1/2 ∇2^2 - Z/r1 - Z/r2 + 1/r12, where r1 and r2 are the distances of the electrons from the nucleus, Z is the nuclear charge, and r12 is the distance between the two electrons. The term 1/r12 represents the repulsive electron–electron interaction and couples the two electrons, preventing a closed-form solution in general. Exact solutions exist only for one-electron (hydrogenic) systems; two-electron systems must be solved numerically or approximately.

Common approaches include variational methods with carefully chosen trial wavefunctions, including Hylleraas coordinates that use r1, r2, and r12 to capture electron correlation. Mean-field methods such as Hartree–Fock approximate the problem by independent particles, while post-Hartree–Fock methods (configuration interaction, coupled-cluster, quantum Monte Carlo) account for electron correlation more accurately.

Spin states in two-electron systems comprise singlet (total spin S=0) and triplet (S=1) configurations, influencing the symmetry of the spatial part of the wavefunction due to the Pauli principle. The ground state of helium is the singlet 1s^2 configuration, with higher excited states arising from various 1s-nl configurations.

Two-electron systems are central to testing theories of electron correlation and underpin much of modern quantum chemistry and spectroscopy.