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 quantum chemistry, molecular orbitals are the statistical states electrons can have within molecules. ... Density functional theory (DFT) is one of the most popular approaches to quantum mechanical many-body electronic structure calculations of molecular and condensed matter systems. ... In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. ... The ionization energy (IE) of an atom or of a molecule is the energy required to strip it of an electron. ... Look up homo in Wiktionary, the free dictionary. ... In chemistry, electron affinity is the amount of energy absorbed when an electron is added to a neutral isolated gaseous atom to form a gaseous ion with a 1- charge. ... HOMO and LUMO are acronyms for Highest Occupied Molecular Orbitals and Lowest Unoccupied Molecular Orbitals respectively. ... Tjalling Charles Koopmans (August 28, 1910 – February 26, 1985) was the joint winner, with Leonid Kantorovich, of the 1975 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel. ...

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. In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors. ...

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 ionisation 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 rearrangment 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. A Slater determinant (named after the physicist John C. Slater) is an expression in quantum mechanics for the wavefunction of a many-fermion system, which by construction satisfies the Pauli principle. ... In quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors. ... In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. ... Electronic correlation refers to the interaction between electrons in a quantum system whose electronic structure is being considered. ... In computational chemistry, Post-Hartree-Fock methods are the set of methods developed to improve on the Hartree-Fock (HF), or self-consistent field (SCF) method. ...

External links

• http://www.chemistry.emory.edu/faculty/bowman/old_classes/chem531/lectures/koopman's_theorem.pdf
• http://nobelprize.org/economics/laureates/1975/koopmans-autobio.html