库普曼斯定理

库普曼斯定理（Koopmans' theorem）表明，在闭壳层的哈特里－福克近似下，一个体系的第一电离能等于其最高占据轨道（HOMO）的能量的负值，这个定理是1934年佳林·库普曼斯提出的。库普曼斯于1975年获得了诺贝尔奖，但不是在物理或化学领域，而是在诺贝尔经济学奖领域。
在哈特里－福克理论的框架之中，并引入电离前后轨道不发生改变这一假设后，库普曼斯定理精确成立。通过这一方法计算得到的电离能与实验结果基本一致，对于小分子来说，误差一般在 2 电子伏特以内。因此，Koopmans定理的有效性与HF波函数的准确性密切相关。两个主要的误差来源为：
将实验结果与高级从头算方法的计算结果比较的经验表明，在大部分情况下，上述两部分误差相互抵消，但并不总是这样。
在密度泛函理论中有着将第一电离能与电子亲和能分别与HOMO/LUMO的单电子轨道能关联起来的类似结论，尽管其推导过程与确切表述都与原始的库普曼斯定理有差异。通常，从密度泛函理论轨道能计算得到的单电子轨道能误差较大，可远大于2电子伏特，并与交换相关势的形式相关。在通常的近似下，DFT方法得到的LUMO轨道能与电子亲和能的相关性并不好。在密度泛函理论的库普曼斯定理中，误差的主要来源是交换相关能量泛函的不准确性，这点与哈特里－福克近似不同，因此在改进计算结果上有较大的提高空间。
2O molecule is (1a1)2 (2a1)2 (1b2)2 (3a1)2 (1b1)2, where the symbols a1, b2 and b1 are orbital labels based on molecular symmetry. From Koopmans’ theorem the energy of the 1b1 HOMO corresponds to the ionization energy to form the H2O+ ion in its ground state (1a1)2 (2a1)2 (1b2)2 (3a1)2 (1b1)1. The energy of the second-highest MO 3a1 refers to the ion in the excited state (1a1)2 (2a1)2 (1b2)2 (3a1)1 (1b1)2, and so on. In this case the order of the ion electronic states corresponds to the order of the orbital energies. Excited-state ionization energies can be measured by photoelectron spectroscopy.
For H2O, minus the near-Hartree-Fock orbital energies of these orbitals in eV, are 1a1 559.5, 2a1 36.7 1b2 19.5, 3a1 15.9 and 1b1 13.8. The corresponding ionization energies are 539.7, 32.2, 18.5, 14.7 and 12.6 eV. As explained above, the deviations are due to the effects of orbital relaxation as well as differences in electron correlation energy between the molecular and the various ionized states.
Koopmans' theorem for electron affinities.
It is sometimes claimed 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, Koopmans' 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' to calculate the electron affinity in this sense.
Calculations of electron affinities using this statement of Koopmans' theorem have been criticized 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. As the basis set becomes 
more complete; more and more "molecular" orbitals that are not really "on" the molecule of interest will appear, and care must be taken not to use 
these orbitals for estimating electron affinities.
Comparisons with experiment and higher-quality calculations show that electron affinities predicted in this manner are generally quite poor.
Koopmans' theorem for open-shell systems.
Koopmans' theorem is also applicable to open-shell systems. It was previously believed that this was only in the case for removing the unpaired electron, but
the validity of Koopmans’ theorem for ROHF in general has been proven provided that the correct orbital energies are used.
The spin up (alpha) and spin down (beta) orbital energies do not necessarily have to be the same.
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" − δ"N" for 0 < δ"N" < 1, and is equal to the electron affinity when the total number of electrons is in the range "N" + δ"N" for 0 < δ"N" < 1 (with the δ"N" occupying what is the LUMO of the "N"-electron state). 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, "ni" is equal to the corresponding orbital energy, "εi":
A tuning procedure of density functional theory approximation imposes the Koopmans' theorem on DFT approximations.