Koopmans' theorem is an approximation in molecular orbital theory, such as density functional theory, or Hartree-Fock theory, in which the first ionization energy of a molecule is equal to the energy of the highest occupied molecular orbital (the HOMO), and the electron affinity is the negative of the energy of the lowest unoccupied, i.e. virtual, orbital (the LUMO). Electron affinities calculated via Koopmans' theorem are usually quite poor. The theorem is named after Tjalling Koopmans.
In Hartree-Fock theory, Koopmans' theorem specifically states that the ionization energies of a molecule are equal to the eigenvalues of the Fock operator associated to the occupied molecular orbitals, and the electron affinity is the negative of the eigenvalues of the Fock operator associated to the lowest unoccupied, i.e. virtual, orbital.
Koopmans' Theorem, like Hartree-Fock theory, operates under the assumption that the electronic wavefunction of a multi-electron atom can be described as the Slater determinant of a set of one-electron wavefunctions, (which are the eigenfunctions of the corresponding Fock operators). In addition, Koopmans' Theorem makes the assumption that upon the addition or subtraction of a single electron to or from the system, the Fock operators for all of the remaining electrons will not change at all. In reality, an added or subtracted electron to or from the initial wavefunction will change the Fock operator of the system, resulting in a re-arranging of the one-electron wavefunctions. This results in an actual final wavefunction which is lower in energy than the Koopmans' Theorem prediction. Koopmans' theorem also assumes that the Hartree-Fock theory is adequate to describe ionization or electron-gain. In reality electron correlation, usually calculated by post Hartree-Fock methods, is important. In many cases, but not all, the corrections arising from rearrangement or relaxation of the orbitals is of opposite sign to the correction arising from the introduction of electron correlation and is of similar magnitude. This explains why Koopmans' theorem is often so successful, but should aways be used with care.