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Koopmans' theorem

Koopmans' theorem states that in closed-shell Hartree–Fock theory (HF), 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 if it is assumed that the orbitals of the ion are identical to those of the neutral molecule (the frozen orbital approximation[2]). 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.[3][4][5] 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.[6][7]

A similar theorem (Janak's 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.[8] 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.[3][4] The LUMO energy shows little correlation with the electron affinity with typical approximations.[9] 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.

  1. ^ Koopmans, Tjalling (1934). "Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den einzelnen Elektronen eines Atoms". Physica. 1 (1–6): 104–113. Bibcode:1934Phy.....1..104K. doi:10.1016/S0031-8914(34)90011-2.
  2. ^ Szabo, Attila; Ostlund, Neil S. (1996). Modern quantum chemistry. Dover Publications. p. 128. ISBN 0-486-69186-1. OCLC 34357385.
  3. ^ a b Politzer, Peter; Abu-Awwad, Fakher (1998). "A comparative analysis of Hartree–Fock and Kohn–Sham orbital energies". Theoretical Chemistry Accounts: Theory, Computation, and Modeling. 99 (2): 83–87. doi:10.1007/s002140050307. S2CID 96583645.
  4. ^ a b Hamel, Sebastien; Duffy, Patrick; Casida, Mark E.; Salahub, Dennis R. (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.
  5. ^ See, for example, Szabo, A.; Ostlund, N. S. (1982). "Chapter 3". Modern Quantum Chemistry. ISBN 978-0-02-949710-4.
  6. ^ Michl, Josef; Bonačić-Koutecký, Vlasta (1990). Electronic Aspects of Organic Photochemistry. Wiley. p. 35. ISBN 978-0-471-89626-5.
  7. ^ Hehre, Warren J.; Radom, Leo; Schleyer, Paul v.R.; Pople, John A. (1986). Ab initio molecular orbital theory. Wiley. p. 24. ISBN 978-0-471-81241-8.
  8. ^ Cite error: The named reference :0 was invoked but never defined (see the help page).
  9. ^ Zhang, Gang; Musgrave, Charles B. (2007). "Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations". The Journal of Physical Chemistry A. 111 (8): 1554–1561. Bibcode:2007JPCA..111.1554Z. doi:10.1021/jp061633o. PMID 17279730. S2CID 1516019.

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