Koopmans' theorem states that in closed-shell Hartree-Fock theory, the first ionization energy of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO). This theorem is named after Tjalling Koopmans, who published this result in 1934.^[1]

Koopmans' theorem is exact in the context of restricted Hartree-Fock theory with the frozen orbital approximation. Ionization energies calculated this way are in qualitative agreement with experiment – the first ionization energy of small molecules is often calculated with an error of less than two electron volts.^{[2][3][4][citation needed]} Therefore, the validity of Koopmans' theorem is intimately tied to the accuracy of the underlying Hartree-Fock wavefunction.^{[citation needed]} The two main sources of error are:

• orbital relaxation, which refers to the changes in the Fock operator and Hartree-Fock orbitals when changing the number of electrons in the system, and

• electron correlation, referring to the validity of representing the entire many-body wavefunction using the Hartree-Fock wavefunction, i.e. a single Slater determinant composed of orbitals that are the eigenfunctions of the corresponding self-consistent Fock operator.

Empirical comparisons with experimental values and higher-quality ab initio calculations suggest that in many cases, but not all, the energetic corrections due to relaxation effects nearly cancel the corrections due to electron correlation.^{[citation needed]}

A similar theorem exists in density functional theory (DFT) for relating the exact first vertical ionization energy and electron affinity to the HOMO and LUMO energies, although both the derivation and the precise statement differ from that of Koopmans' theorem. Ionization energies calculated from DFT orbital energies are usually poorer than those of Koopmans' theorem, with errors much larger than two electron volts possible depending on the exchange-correlation approximation employed.^[2][3] The LUMO energy shows little correlation with the electron affinity with typical approximations.^[5] The error in the DFT counterpart of Koopmans' theorem is a result of the approximation employed for the exchange correlation energy functional so that, unlike in HF theory, there is the possibility of improved results with the development of better approximations.

Contents 1 Generalizations of Koopmans' theorem 1.1 Koopmans' theorem for electron affinities 1.2 Koopmans' theorem for open-shell systems 2 Counterpart in density functional theory 3 References 4 External links

[edit] Generalizations of Koopmans' theorem

While Koopmans' theorem was originally stated for calculating ionization energies from restricted (closed-shell) Hartree-Fock wavefunctions, the moniker has since taken on a more generalized meaning as a way of using orbital energies to calculate energy changes due to changes in the number of electrons in a system.

[edit] Koopmans' theorem for electron affinities

It is sometimes claimed^[6] that Koopmans' theorem also allows the calculation of electron affinities as the energy of the lowest unoccupied molecular orbitals (LUMO) of the respective systems. However, Koopman's original paper makes no claim with regard to the significance of eigenvalues of the Fock operator other than that corresponding to the HOMO. Nevertheless, it is straightforward to generalize the original statement of Koopmans's to calculate the electron affinity in this sense.

Calculations of electron affinities using this statement of Koopman's theorem have been criticized^{[7][citation needed]} on the grounds that virtual (unoccupied) orbitals do not have well-founded physical interpretations, and that their orbital energies are very sensitive to the choice of basis set used in the calculation.

Comparisons with experiment and higher-quality calculations show that electron affinities predicted in this manner are generally quite poor.

[edit] Koopmans' theorem for open-shell systems

Koopmans' theorem is also applicable to open-shell systems, but only when one is interested in removing the unpaired electron.^{[8][citation needed]}

[edit] Counterpart in density functional theory

In density functional theory (DFT) a similar theorem exists that relates the first ionization energy and electron affinity to the HOMO and LUMO energies. This is sometimes called the DFT-Koopmans' theorem. More generally, for a fixed geometry defined by the N-electron system, the HOMO energy is equal to the ionization energy of the N-electron system when the total number of electrons is in the range N − δN for 0 < δN < 1, and is equal to the electron affinity when the total number of electrons is in the range N + δN for 0 < δN < 1 (with the δN occupying what is the LUMO of the N-electron state).^[9][10] While these are exact statements in the formalism of DFT, the use of approximate exchange-correlation potentials makes the calculated energies approximate. As in HF theory, the electron affinity calculated in this way is less accurate than the ionization energy.

A proof of the DFT counterpart of Koopmans' theorem usually employs Janak's theorem: that the derivative of the total DFT energy, E, with respect to the occupation of a given orbital, n_i is equal to the corresponding orbital energy, ε_i:^[11]

[edit] References

1. ^ Koopmans, Tjalling (1934). "Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen Elektronen Eines Atoms". Physica (Elsevier) 1 (1–6): 104–113. doi:10.1016/S0031-8914(34)90011-2. 
2. ^ ^{a b} Politzer, Peter; Fakher Abu-Awwad (1998). "A comparative analysis of Hartree–Fock and Kohn–Sham orbital energies". Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta) 99 (2): 83–87. doi:10.1007/s002140050307. 
3. ^ ^{a b} Hamel, Sebastien; Patrick Duffyc, Mark E. Casidad and Dennis R. Salahub (2002). "Kohn–Sham orbitals and orbital energies: fictitious constructs but good approximations all the same". Journal of Electron Spectroscopy and Related Phenomena 123 (2–3): 345–363. doi:10.1016/S0368-2048(02)00032-4. 
4. ^ See, for example, A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, Chapter 3.
5. ^ Zhang, Gang; Charles B. Musgrave (2007). "Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations". J. Phys. Chem. A 111 (8): 1554–1561. doi:10.1021/jp061633o. 
6. ^ See, for example, A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, p. 127.
7. ^ Jensen, Frank (1990). Introduction to Computational Chemistry. Wiley. pp. 64–65. 
8. ^ Fulde, Peter (1995). Electron correlations in molecules and solids. Springer. pp. 25–26. ISBN 3-540-59364-0. 
9. ^ Perdew, John P.; Robert G. Parr, Mel Levy, and Jose L. Balduz, Jr. (1982). "Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy". Phys. Rev. Lett. 49 (23): 1691–1694. doi:10.1103/PhysRevLett.49.1691. 
10. ^ Perdew, John P.; Mel Levy (1997). "Comment on "Significance of the highest occupied Kohn–Sham eigenvalue"". Phys. Rev. B 56 (24): 16021–16028. doi:10.1103/PhysRevB.56.16021. 
11. ^ J. F. Janak (1978). "Proof that ∂E / ∂n_i = ε_i in density-functional theory". Phys. Rev. B 18 (12): 7165–7168. doi:10.1103/PhysRevB.18.7165. 

[edit] External links

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